Advanced Optical Communication Systems and Networks [1 ed.] 9781608075560, 9781608075553

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Advanced Optical Communication Systems and Networks

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For a complete listing of titles in the Artech House Applied Photonics Series turn to the back of this book.

Cvijetic_CIP.indd ii

11/15/2012 3:26:18 PM

Advanced Optical Communication Systems and Networks Milorad Cvijetic Ivan B. Djordjevic

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11/15/2012 3:26:18 PM

Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the U.S. Library of Congress. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library. Cover design by Vicki Kane

ISBN 13: 978-1-60807-555-3

© 2013 ARTECH HOUSE 685 Canton Street Norwood, MA 02062

All rights reserved. Printed and bound in the United States of America. No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the publisher. All terms mentioned in this book that are known to be trademarks or service marks have been appropriately capitalized. Artech House cannot attest to the accuracy of this information. Use of a term in this book should not be regarded as affecting the validity of any trademark or service mark.

10 9 8 7 6 5 4 3 2 1

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To Rada —M. C. To Milena —I. Dj.

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Contents PREFACE….…………………………………………………………………. xix

Chapter 1: Introduction to Optical Communications 1 1.1 THE ROLE OF THE OPTICAL NETWORKING............................................2 1.1.1 The Need for Connectivity and Capacity…………………………………2 1.1.2 Optical Networking and Lightpaths………………………………………3 1.2 HISTORICAL PERSPECTIVE……………………………………………….7 1.2.1 The Early Beginning and the First Generation of Optical Communications…………………………………………………………..8 1.2.2 The Second and Third Generations …………………………………......10 1.2.3 The Fourth and Fifth Generations of Optical Systems and Networks….. 14 1.3 ETHERNET AS A FOUNDATION OF PACKET-BASED NETWORKING…………………………………………………………….. 18 1.3.1 Ethernet as a Layer-2 Networking Technology… ………………………18 1.3.2 100-Gigabit Ethernet as a High-Speed Networking Tool……………… 20 1.3.3 OTN Technology as a High-Speed Transmission Tool……………….... 22 1.4 CLASIFICATION AND BASIC CONCEPT OF PHOTONIC TRANSMISSION SYSTEMS AND NETWORKS………………………….23 1.4.1 Optical Fiber as a Foundation for Transmission and Networking……… 23 1.4.2 Optical Transmission Systems…………………………………………. 24 1.4.3 Optical Networking Parameters……………………………………….... 26 1.4.4 Optical Channel Capacity and Basic Signal References………………...28 1.5 FUTURE PERSPECTIVE…………………………………………………... 30 1.6 ORGANIZATION OF THE BOOK………………………………………… 31 PROBLEMS………………………………………………………………….......34 References……………………………………………………………………….. 35

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Chapter 2: Optical Components and Modules

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2.1 KEY OPTICAL COMPONENTS……………………………………………39 2.2 OPTICAL FIBER…………………………………………………………….42 2.2.1 Optical Fibers Manufacturing and Cabling ……………………………..43 2.2.2 Special Optical Fibers…………………………………………………... 46 2.2.3 Optical Fiber Types with Respect to Transmission Properties…………. 49 2.2.4 Multicore and Few-Mode Optical Fibers………………………………. 54 2.3 THE LIGHT SOURCES ................................................................................. 56 2.3.1 Semiconductor Lasers…………………………………………………... 56 2.3.2 Light Emitting Diodes………………………………………………….. .69 2.4 OPTICAL FILTERS AND MULTIPLEXERS ............................................... 72 2.4.1 The Fabry-Perot Filters…………………………………………………. 72 2.4.2 Mach-Zehnder Filter……………………………………………………..75 2.4.3 Optical Grating Filters…………………………………………………...76 2.4.4 Tunable Optical Filters…………………………………………………..78 2.4.5 Optical Multiplexers and Demultiplexers………………………………. 80 2.5 OPTICAL MODULATORS ........................................................................... 83 2.5.1 Direct Optical Modulation……………………………………………… 84 2.5.2 External Modulation of Optical Signals………………………………... 86 2.6 OPTICAL AMPLIFIERS ................................................................................ 94 2.6.1 Semiconductor Optical Amplifiers………………………………….….. 95 2.6.2 Fiber Doped Amplifiers………………………………………………… 95 2.7 PHOTODIODES ........................................................................................... 110 2.8 PROCESSING OPTICAL COMPONENTS ................................................. 116 2.8.1 Components for Coupling, Isolation, and Adjustments of Optical Power…………………………………………………………..117 2.8.2 Optical Switches………………………………………………………..121 2.8.3 Wavelength Converters………………………………………………...127 2.9 SUMMARY .................................................................................................. 132 PROBLEMS........................................................................................................ 132 References……………………………………………………………………… 136

Contents

Chapter 3: Signal Propagation in Optical Fibers

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141

3.1 OPTICAL FIBER LOSSES .......................................................................... 141 3.2 WAVEGUIDE THEORY OF OPTICAL FIBERS ....................................... 145 3.2.1 Electromagnetic Field and Wave Equations…………………………... 145 3.2.2 Optical Modes in Step-Index Optical Fibers………………………...... 148 3.2.3 Definition of a Single-Mode Regime…………………………………. 154 3.2.4 Modes in Grade-Index Optical Fibers………………………………… 156 3.3 SIGNAL DISPERSION IN SINGLE-MODE OPTICAL FIBERS ............... 160 3.3.1 Modal Dispersion……………………………………………………… 160 3.3.2 Chromatic Dispersion…………………………………………………..160 3.3.3 Polarization Mode Dispersion………………………………………….165 3.3.4 Self-Phase Modulation in Optical Fibers…………………………........ 171 3.4 PULSE PROPAGATION IN SINGLE-MODE OPTICAL FIBERS ............ 179 3.4.1 Single-Channel Propagation……………………………………………180 3.5 MULTICHANNEL PROPAGATION IN OPTICAL FIBERS ..................... 211 3.5.1 Cross-phase Modulation……………………………………………… 213 3.5.2 Four-Wave Mixing (FWM)………………………………………….... 214 3.5.3 Nonlinear Schrodinger Equation for Multichannel Transmission…….. 217 3.6 SIGNAL PROPAGATION IN MULTIMODE OPTICAL FIBERS ............ 218 3.6.1 Mode Coupling in Multimode Fibers…………………………………. 219 3.6.2 Mode Coupling in Curved Multimode Optical Fibers………………… 223 3.6.3 Mode Coupling in Dual-Mode Optical Fibers………………………… 225 3.7 SUMMARY .................................................................................................. 225 PROBLEMS........................................................................................................ 226 References……………………………………………………………………… 229

Chapter 4: Noise Sources and Channel Impairment 233 4.1 OPTICAL CHANNEL NOISE ..................................................................... 233 4.1.1 Mode Partition Noise………………………………………………….. 236 4.1.2 Modal Noise…………………………………………………………… 237 4.1.3 Laser Phase and Intensity Noise………………………………………. 239

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4.1.4 Quantum Shot Noise……………………………………………........... 243 4.1.5 Dark Current Noise……………………………………………………. 245 4.1.6 The Thermal Noise……………………………………………………..247 4.1.7 Spontaneous Emission Noise………………………………………….. 248 4.1.8 Beating Components of Noise in the Optical Receiver……………….. 249 4.1.9 The Cross-talk Noise…………………………………………………...252 4.2 DEFINITION OF BER, SNR, AND RECEIVER SENSITIVITY ............... 254 4.2.1 Bit Error Rate and Signal-to-Noise Ratio for IM/DD Scheme………... 255 4.2.2 Optical Receiver Sensitivity……………………………………………259 4.2.3 Optical Signal-to-Noise Ratio…………………………………………. 269 4.3 SIGNAL IMPAIRMENTS ............................................................................ 270 4.3.1 The Impact of Mode Dispersion in Multimode Fibers…………………272 4.3.2 The Impact of Chromatic Dispersion…………………………..............273 4.3.3 Polarization Mode Dispersion Impact………………………………….277 4.3.4 The Impact of Nonlinear Effects on System Performance……………..279 4.3.5 The Impact of the Extinction Ratio…………………………………….287 4.3.6 The Impact of the Intensity Noise and the Mode Partition Noise……...289 4.3.7 The Impact of the Timing Jitter………………………………………...292 4.3.8 The Impact of Signal Cross-talk……………………………………… 293 4.3.9 Impact of Raman Amplification to Signal Distortion…………………. 295 4.3.10 Impact of the Accumulated Noise…………………………………… 301 4.4 OPTICAL TRANSMISSION LINK LIMITS ............................................... 306 4.4.1 Power-Budget Limited Point-to Point Lightwave Systems…………… 306 4.4.2 Bandwidth-Limited Point-to Point Lightwave Systems………………. 309 4.4.3 OSNR Evaluation in High-Speed Optical Transmission Systems……. 313 4.5 SUMMARY .................................................................................................. 314 PROBLEMS........................................................................................................ 315 References……………………………………………………………………… 319

Chapter 5: Advanced Modulation Schemes

323

5.1 SIGNAL-SPACE THEORY AND PASSBAND DIGITAL OPTICAL TRANSMISSION ......................................................................................... 323

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5.1.1 Generic Optical Digital Communication System……………………... 323 5.1.2 Geometric Representation of Signals in Modulators and Demodulators………………………………………………………….. 326 5.1.3 M-ary Baseband Pulse Amplitude Modulation (PAM)………………...331 5.1.4 Passband Digital Transmission………………………………………... 334 5.1.5 Quadrature Amplitude Modulation (QAM)…………………………... 335 5.1.6 Frequency Shift Keying (FSK)……………………………………….. 342 5.2 MULTILEVEL MODULATION SCHEMES .............................................. 343 5.2.1 I/Q and Polar Modulators……………………………………………... 343 5.2.2 M-ary PSK Transmitters………………………………………………. 347 5.2.3 Star-QAM Transmitters……………………………………………….. 350 5.2.4 Square/Cross QAM Transmitters……………………………………... 351 5.3 POLARIZATION-DIVISION MULTIPLEXING AND FOURDIMENSIONAL SIGNALING…………………………………….. ……...356 5.4 SPACE-DIVISION MULTIPLEXING AND MULTIDIMENSIONAL HYBRID MODULATION SCHEMES…………………………………….360 5.5 OPTIMUM SIGNAL CONSTELLATION DESIGN ................................... 365 5.6 ORTHOGONAL FREQUENCY DIVISION MULTIPLEXING (OFDM) FOR OPTICAL COMMUNICATIONS…………………………………… 369 5.6.1 Generation of OFDM Signals by Inverse Fast Fourier Transform......... 370 5.6.2 Cyclic Extension and Windowing……………………………………...373 5.6.3 Bandwidth Efficiency of CO-OFDM…………………………………..378 5.6.4 OFDM Signal Processing and Parallel Optical Channel Decomposition………………………………………………………… 379 5.6.5 Discrete Multitone (DMT) in Multimode Fiber Links………………... 384 5.7 MIMO OPTICAL COMMUNICATIONS .................................................... 389 5.7.1 Parallel Decomposition of MIMO Optical Channels…………………. 393 5.7.2 Space-Time Coding for MIMO Optical Channels…………………….. 395 5.7.3 Polarization-Time Coding and MIMO-OFDM……………………….. 398 5.8 SUMMARY .................................................................................................. 401 PROBLEMS........................................................................................................ 401 References……………………………………………………………………… 407

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Chapter 6: Advanced Detection Schemes

413

6.1 DETECTION THEORY FUNDAMENTALS .............................................. 413 6.1.1 Geometric Representation of Received Signal, and Theorem of Irrelevance…………………………………...........................................413 6.1.2 Equivalence of Euclidean, Correlation, and Matched Filter Receivers.. 418 6.1.3 Detection of Known Signal and Optimum Receiver Design………….. 422 6.1.4 Error Probability in the Receiver……………………………………… 426 6.1.5 Estimation Theory, ML Estimation, and Cramér-Rao Bound………… 435 6.2 COHERENT DETECTION OF OPTICAL SIGNALS ................................. 437 6.2.1 Coherent Optical Detection Basics……………………………………. 438 6.2.2 Optical Hybrids and Balanced Coherent Receivers…………………… 443 6.2.3 Phase, Polarization, and Intensity Noise Sources in a Coherent Optical Detector……………………………………………………….. 446 6.2.4 Homodyne Coherent Detection………………………………………...449 6.2.5 Phase Diversity Homodyne Receivers………………………………… 449 6.2.6 Polarization Control and Polarization Diversity in Coherent Receivers …………………………………………………….451 6.2.7 Polarization-Division Multiplexing (PDM) and Coded Modulation….. 452 6.3 OPTICAL CHANNEL EQUALIZATION ................................................... 454 6.3.1 ISI-Free Optical Transmission and Partial-Response Signaling……….454 6.3.2 Zero-Forcing Equalizers……………………………………………… 458 6.3.3 Optimum Linear Equalizer……………………………………………. 463 6.3.4 Wiener Filtering………………………………………………………. 465 6.3.5 Adaptive Equalization………………………………………………… 466 6.3.6 Decision Feedback Equalizer…………………………………………. 468 6.3.7 MLSD or Viterbi Equalizer…………………………………………… 469 6.3.8 Blind Equalization…………………………………………………….. 471 6.3.9 Volterra Series-Based Equalization…………………………………….476 6.4 DIGITAL BACKPROPAGATION .............................................................. 478 6.5 SYNCHRONIZATION ................................................................................. 481 6.6 COHERENT OPTICAL OFDM DETECTION ............................................ 486 6.6.1 DFT Window Synchronization………………………………………... 488

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6.6.2 Frequency Synchronization in Optical OFDM Systems…………….... 489 6.6.3 Phase Estimation in Optical OFDM Systems…………………………. 490 6.6.4 Channel Estimation in OFDM Systems……………………………….. 491 6.7 OPTICAL MIMO DETECTION .................................................................. 494 6.7.1 MIMO Model of Few-Mode Fibers…………………………………… 495 6.7.2 Linear and Decision-Feedback MIMO Receivers…………………….. 496 6.7.3 Space-Time Coding (STC)-Based MIMO Detection Schemes………...499 6.8 SUMMARY .................................................................................................. 503 PROBLEMS........................................................................................................ 503 References……………………………………………………………………… 509

Chapter 7: Advanced Coding Schemes

515

7.1 CHANNEL CODING PRELIMINARIES .................................................... 515 7.1.1 Channel Coding Principles………………………………………......... 516 7.1.2 Mutual Information and Channel Capacity…………………………… 519 7.1.3 Channel Coding and Information Capacity Theorems………………....521 7.2 LINEAR BLOCK CODES............................................................................ 522 7.2.1 Generator Matrix……………………………………………………… 523 7.2.2 Parity-Check Matrix…………………………………………………... 524 7.2.3 Code Distance Properties…………………………………………….... 526 7.2.4 Coding Gain…………………………………………………………….527 7.2.5 Syndrome Decoding and Standard Array………………………………528 7.2.6 Important Coding Bounds……………………………………………... 531 7.3 CYCLIC CODES .......................................................................................... 532 7.4 BOSE-CHAUDHURI-HOCQUENGHEM (BCH) CODES ......................... 538 7.5 REED-SOLOMON CODES, CONCATENATED CODES, AND PRODUCT CODES………………………………………………………... 545 7.6 TURBO CODES ........................................................................................... 548 7.7 TURBO-PRODUCT CODES ....................................................................... 551 7.8 LOW-DENSITY PARITY-CHECK (LDPC) CODES ................................. 553 7.8.1 Quasi-Cyclic (QC) Binary LDPC Codes……………………………… 555 7.8.2 Decoding of Binary LDPC Codes and BER Performance Evaluation... 557 7.8.3 Nonbinary LDPC Codes………………………………………………..561

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7.8.4 FPGA Implementation of Decoders for Large-Girth QC-LDPC Codes.562 7.9 CODED MODULATION ............................................................................. 563 7.9.1 Multilevel Coding and Block-Interleaved Coded Modulation……….. 563 7.9.2 Polarization-Division Multiplexed Coded OFDM……………………. 569 7.9.3 Nonbinary LDPC-Coded Modulation…………………………………. 571 7.9.4 Multidimensional Coded Modulation…………………………………. 575 7.9.5 Adaptive Nonbinary LDPC-Coded Modulation………………………. 579 7.10 LDPC-CODED TURBO EQUALIZATION ............................................. 581 7.10.1 MAP Detection……………………………………………………… 581 7.10.2 Multilevel Turbo Equalization……………………………………….. 582 7.10.3 Multilevel Turbo Equalizer for I/Q Imbalance and Polarization Offset…………………………………………………….588 7.10.4 Multilevel Turbo Equalization with Digital Back-Propagation…….. 592 7.11 SUMMARY ................................................................................................ 593 PROBLEMS........................................................................................................ 594 References……………………………………………………………………… 604

Chapter 8: Advanced Optical Networking

609

8.1 OPTICAL NETWORKING AS PART OF THE ISO MODEL ................... 609 8.1.1 ISO Model of Networking…………………………………………….. 609 8.1.2 Optical Network Definition and Role…………………………………. 612 8.1.3 Cross-Layer Interworking with Upper Layers………………………… 613 8.1.4 Electrical Client Layers………………………………………………...614 8.2 OPTICAL NETWORKING ELEMENTS .................................................... 625 8.2.1 Optical Line Terminals………………………………………………... 626 8.2.2 Optical Add-Drop Multiplexer (OADM)……………………………... 627 8.2.3 Optical Interconnect Devices…………………………………………..629 8.2.4 Reconfigurable Optical Add-Drop Multiplexers (ROADM)………… 633 8.2.5 Optical Cross-Connect (OXC)………………………………………. 636 8.3 LIGHTPATH ROUTING IN OPTICAL NETWORKS................................ 638 8.3.1 The Lightpath Topologies and Their Impact on Wavelength Routing ...639 8.3.2 Modeling of Lightpath Topology……………………………………... 642

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8.3.3 Optimizing Multihop Network Topologies…………………………….644 8.4 IMPAIRMENT AWARE ROUTING ........................................................... 652 8.4.1 Optical Performance Monitoring……………………………………… 654 8.4.2 Impairment Aware Constraints………………………………………... 657 8.5 CONTROL AND MANAGEMENT OF OPTICAL NETWORKS .............. 661 8.5.1 Signaling and Resource Reservation………………………………….. 661 8.5.2 Routing and Wavelength Assignment………………………………… 663 8.5.3 Fault Management and Network Restoration…………………………. 664 8.6 CONTROL PLANE FOR AN OPTICAL NETWORK ................................ 671 8.7 OPTICAL PACKET AND BURST SWITCHING ....................................... 674 8.7.1 Optical Packet Switching……………………………………………… 675 8.7.2 Optical Burst Switching……………………………………………….. 679 8.8 OPTICAL NETWORK APPLICATION SEGMENTS ................................ 681 8.8.1 Optical Access Networks……………………………………………… 681 8.8.2 Optical Metro Networks………………………………………………..689 8.8.3 Optical Core Network…………………………………………………. 690 8.8.4 Data Center Networks (DCN)………………………………………... 691 8.9 ADVANCED MUTIDIMENSIONAL AND DYNAMIC OPTICAL NETWORKING…………………………………………………………….693 8.10 SUMMARY ................................................................................................ 698 PROBLEMS........................................................................................................ 698 References……………………………………………………………………… 703

Chapter 9: Optical Channel Capacity and Energy Efficiency

709

9.1 CAPACITY OF CONTINUOUS CHANNELS ............................................ 709 9.2 CAPACITY OF CHANNELS WITH MEMORY………………………… 714 9.2.1 Markov Sources and Their Entropy…………………………………... 714 9.2.2 McMillan Sources and Their Entropy………………………………… 718 9.2.3 McMillan-Khinchin Model for Channel Capacity Evaluation………....718 9.3 MODELING OF SIGNAL PROPAGATION ............................................... 721 9.3.1 The Nonlinear Schrödinger Equation (NSE)………………………….. 721 9.3.2 Step-Size Selection in Split-Step Fourier Algorithms………………… 723

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9.3.3 Multichannel Propagation……………………………………………... 724 9.3.4 Propagation Equation for Polarization-Division Multiplexed Systems. 724 9.4 CALCULATION OF INFORMATION CAPACITY BY FORWARD RECURSION OF THE BCJR ALGORITHM……………………………...725 9.5 INFORMATION CAPACITY OF SYSTEMS WITH COHERENT DETECTION………………………………………………………………..728 9.6 CALCULATION OF CAPACITY OF OPTICAL MIMO-OFDM SYSTEMS ………………………………………………………………….731 9.6.1 Modeling of Few-Mode Fibers in a Strong-Coupling Regime with Mode-Dependent Loss………………………………………………… 731 9.6.2 Optical MIMO-OFDM Channel Capacity…………………………….. 732 9.7 ENERGY-EFFICIENT OPTICAL TRANSMISSION ................................. 734 9.7.1 Energy-Efficient Signal Constellation Design Based on Concepts of Statistical Physics………………………………………………………734 9.7.2 Energy-Efficient Multidimensional Coded Modulation………………. 737 9.7.3 Energy-Efficient Photonic Devices…………………………………… 739 9.8 SUMMARY .................................................................................................. 740 PROBLEMS........................................................................................................ 740 References……………………………………………………………………… 744

Chapter 10: Engineering Tool Box

749

10.1 PHYSICAL QUANTITIES AND UNITS USED IN THIS BOOK ............ 749 10.2 FREQUENCY AND WAVELENGTH OF THE OPTICAL SIGNAL ...... 750 10.3 STIMULATED EMISSION OF THE LIGHT ............................................ 751 10.4 BASIC PHYSICS OF SEMICONDUCTOR JUNCTIONS ........................ 753 10.5 BASIC VECTOR ANALYSIS ................................................................... 758 10.6 BESSEL FUNCTIONS ............................................................................... 760 10.7 MODULATION OF AN OPTICAL SIGNAL ............................................ 761 10.8 DIGITAL-TO-ANALOG AND ANALOG-TO-DIGITAL CONVERSION ............................................................................................................................. 762 10.9 OPTICAL RECEIVER TRANSFER FUNCTION ..................................... 764 10.10 THE Z-TRANFORM AND ITS APPLICATIONS .................................. 766 10.10.1 Bilateral z-Transform……………………………………………….. 766 10.10.2 Properties of z-Transform and Common z-Transform Pairs………...768 10.10.3 The Inversion of the z-Transform…………………………………... 769

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10.10.4 The System Function………………………………………………...770 10.11 ABSTRACT ALGEBRA BASICS ........................................................... 772 10.11.1 Concept of Groups…………………………………………………...772 10.11.2 Concept of Fields…………………………………………………… 774 10.11.3 Concept of Finite Fields…………………………………………….. 775 10.12 PULSE-POSITION MODULATION ....................................................... 777 10.13 STOKES VECTOR AND POINCARE SPHERE..................................... 779 References……………………………………………………………………… 780 ACRONYMS ..................................................................................................... 783 ABOUT THE AUTHORS ................................................................................ 791 INDEX ................................................................................................................ 793

Preface This book aims to introduce and treat a series of advanced and emerging topics in the field of optical communications and networking. The material is largely based on the fifth generation of optical transmission systems and networks, characterized by exploitation of all optical signal parameters (amplitude, phase, frequency, and polarization) via processing in time, frequency, and space domains. Both optical signal transmission and networking topics are treated with equal depth, and feature a gradual introduction to more sophisticated topics. The reader will find fundamentals about optical components and signal generation, propagation, and detection discussed in detail in Chapters 1 to 4. A number of advanced topics such as MIMO, OFDM, coded-modulation, LDPCcoded turbo equalization, polarization-time coding, spatial-domain-based modulation and coding, multidimensional signaling, optimum signal constellation design, and digital compensation of linear and nonlinear impairments are discussed in Chapters 5 to 7. These chapters also include description of the most relevant post-detection techniques such as linear and adaptive equalization, maximum-likelihood sequence detection, digital back-propagation, and Wiener filtering. In Chapter 8, the principles of advanced optical networking are described, including definitions of primary optical network models and network design parameters in common topologies. In Chapter 9, optical channel capacity computation is presented based on modeling an optical fiber as a channel with memory, while in Chapter 10, useful tools to aid understanding of topics associated with optical transmission systems and networks are provided. This book is self-contained and structured to provide straightforward guidance to readers looking to capture fundamentals and gain both theoretical and practical knowledge that can be readily applied in research and practical applications. Every chapter of the book has an extensive list of references, as well as a number of exercises, all with the goal to help the reader build up the knowledge and skills necessary for research and engineering tasks. The book is structured in such a way that the reader can easily understand the general topic without looking to outside literature. The intended audience for the book are senior-year undergraduate students and graduate students of electrical engineering, optical sciences, and physics, research engineers and scientists, development and planning engineers, and attendees of leading industry conferences (e.g. OFC, ECOC, CLEO, OECC). The background knowledge necessary to study this book and fully understand the

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topics is that of a typical senior-year undergraduate engineering/science student. This book is well-suited for mixed undergraduate/graduate-level courses, containing a large range of problems with varying degrees of difficulty. The subject of the book is not limited to any particular geographical region or any specific transmission and networking scenario. We are very grateful to numerous colleagues from both industry and academia for the useful discussions we have had in the past and for their helpful suggestions and comments. We would like also to extend personal thanks and deep gratitude to our families for their unconditional support and understanding.

Chapter 1 Introduction to Optical Communications This chapter describes the role of optical networking in our information society and explains the key technologies that serve as foundations of packet-based optical networking. Historical perspective and significance of the fifth generation of optical systems and networks, as well as the outline for future direction, are also presented. Finally, a classification and basic concept of advanced optical systems and networks will be introduced. We defined the fifth generation of optical transmission systems and networks by several distinct features listed later. Some of them, such as multilevel modulation format or coherent detection schemes, were already in place several decades ago, but with no real traction since they could not be implemented in an efficient and practical manner. The other ones, such as OFDM, MIMO, and LDPC coding, have been effectively used in wireless communications for some time now, and have found a way to serve as enablers of advanced optical transmission and networking. We also consider high-speed digital signal processing to be an essential part and enabler of both optical transmission and optical networking. There are multiple degrees of freedom with respect to spectral arrangement, spatial multiplexing in an optical fiber, and advanced coded modulation that can be utilized in design of high-capacity optical transmission systems, as well as in realization of the multidimensional and elastic optical network architecture. We assume that the fifth generation of optical networking is a part of an integral networking scheme in which Ethernet and IP/MPLS technologies serve as client layers that interwork with an optical layer in the process of delivery of data packets through network cloud to a variety of end users. With that respect, 100-Gb Ethernet serves as one of the identifiers of the fifth generation. Our definition and classification of optical transmission systems and networks with respect to historical perspective may be different than those from some other authors since we consider that both transmission and networking aspects and enabling technologies are mutually interrelated and will be treated as such throughout of this book. The purpose of the material discussed in this book is to provide both the fundamental information and the advanced topics that characterize the fifth generation and beyond.

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Advanced Optical Communication Systems and Networks

1.1 THE ROLE OF OPTICAL NETWORKING The beginning of the second decade of the twenty-first century is characterized by paradigm shift in the overall meaning of information society. We are now in the era where communications are possible whenever and wherever they are needed, with a number of entities exchanging a different kind of information among them. 1.1.1 The Need for Connectivity and Capacity Any information exchange in the network is characterized by a data flow, so a number of flows can occur simultaneously. Each of the information flows in the network is characterized by speed or the bandwidth and the quality of transmission. The growing need for the bandwidth and connectivity is driven by demand from both residential and business users, as well as from the scientific community asking for infrastructure that would support large-scale data transport and information exchange. The bandwidth requirements are mainly driven by IP traffic, which now includes video services, such as IPTV, videoconference, and streaming applications. In addition, telemedicine, social networking, and transaction-intensive Web 2.0+ applications all require a massive deployment of various network elements, such as IP routers, carrier Ethernet switches, or dense wavelength division multiplexing (DWDM) terminals and switches. An information network is increasingly becoming a cloud of information flows connecting different participants, as illustrated in Figure 1.1.

Figure 1.1 Information Infrastructure

Video Conference

Super/cloud Computing

(business ,government)

Video Conference

Mobile business office

Metro optical network

Metro optical network

3G/4G/4G+ Base Station

Access optical network

Core optical network

Web Server Data center

Access optical network

IPTV IP Phone Residential Access (Home Networks)

Remote medical Mobile connection

Hosted Business Appl. (Storage, VoIP, Security)

(Image processing)

IPTV/VoD Center

Figure 1.1 High-speed networking.

Since the Internet became the synonym of the information era, there have been a numerous efforts to make it comprehensive and affordable. Carrier-grade Ethernet became both the transport and networking engine of the IP traffic loaded by various applications mentioned above. We can expect that the bandwidth

Introduction to Optical Communications

3

requirements in the second decade of this century will be ranging from 100 Mb/s to 1 Gb/s for residential access users, 10 Gb/s to 40 Gb/s for the majority of business users, and 100 Gb/s to 1 Tb/s for some institutions, such as government agencies or major research labs. The bandwidth requirements are the cause for the rapid growth of Internet traffic over past decade, and that growth has been exponential. The annual IP traffic is now measured by exabytes. The prediction is that by 2015 the total IP traffic in the United States alone will be around 1,000 exabytes per year, which is equal to 1 zettabyte, and then a count will begin towards the yottabyte (1024 bytes). The network architecture that supports IP traffic is structured to accommodate packet transport over optical bandwidth pipes [1–7]. The question is how to provide enough bandwidth for all users, while dealing with different granularities, quality of services, and energy constraints. There is wide consensus today that Ethernet technology will remain to be the best option for high-speed statistical bandwidth sharing. Different Ethernet networking speeds (10M/100M/1G/10G) were successfully introduced over the past two decades. That hierarchy was extended by the introduction of 40 GbE (stands for Ethernet at 40 Gb/s speed) and 100 GbE in 2010. The expectation is that Ethernet speed will reach 1 Tb/s by 2015 or so. The next stop after 1 Tb/s would likely be 4 Tb/s followed by 10 Tb/s. 1.1.2 Optical Networking and Lightpaths In terms of the ownership, networks and transmission systems symbolized by clouds in Figure 1.1 can either belong to private enterprises or be owned by telecommunication carriers. The ownership can be related either to networking equipment and infrastructure associated with a specified network topology or to a logical entity known as virtual private network that resides within the physical network topology. We can recognize several segments within the structure from Figure 1.1, all with respect to the size of the area that they cover. The central part of this structure is a long-haul core network interconnecting big cities or major communication hubs by high-capacity optical fiber links. At the same time, the connections between major hubs on different continents have been done through submarine optical transmission links. The core network is a generic name, but very often it is referred to as either a wide area network (WAN) if belongs to an enterprise, or as the interchange carrier (IXC) public network if operated by telecommunication carriers. The nodes in a core network are known as central offices, although the terms POP (point of presence) or hubs are also used. The distance between nodes in a core network can range anywhere from several hundred to several thousand kilometers. As an example, the distance between nodes in core networks of major North American telecom carriers or major cable companies is much longer than the distance between core network nodes of major carriers in Europe or network nodes of major carriers in Asia.

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On the other hand, WAN networks are for private use, owned and operated by big corporations. Typical examples of such networks are networks of major ASP/ISP (application/Internet service providers), where network nodes coincide with the data centers, while connections are established between these data centers. A majority of other corporations still use the services provided by telecom carriers by implementing their private networking within the carriers’ network infrastructure. Since there is the emulation of one network within the other, logical connections that serve corporations are known as virtual private networks (VPNs). The second part of the overall optical network structure is known as the edge network, which is deployed within a smaller geographical area, such as a metropolitan area, or a smaller geographic region. The distance between the nodes connected by optical fiber links in edge networks ranges from tens to a few hundred kilometers. The edge network is often recognized as a metropolitan area network (MAN) if owned by an enterprise or as a local exchange carrier (LEC) if operated by telecommunication carriers. Finally, the access network is a peripheral part of the overall network related to the last-mile access and bandwidth distribution to individual end users, which could be enterprises, government agencies, medical institutions, scientific labs, or residential customers. Two examples of the access networks are an enterprise local area network (LAN), and a distribution network that connects the carrier’s central office location with individual users. The distance between two nodes in an access network usually ranges from several hundred meters to several kilometers. As we can see from Figure 1.1, application providers and big processing centers may have access to any of the segments mentioned above. We can expect that in the future the network will be a unified information cloud with various participants, each having specified access to it. Accordingly, there is constant transformation of logical network structure to comply with business models of various service providers, as well as with the service requirements of various endusers. Some of these business models include ownership either of the entire network (infrastructure, equipment, network management) or of some portion needed for delivering IP-based services to customers. The other models are based on leasing network infrastructure from a third party or on performing a bandwidth brokerage in carrier hotels. The ownership of the network includes responsibility for network planning and traffic engineering, while leasing and brokering are more related to service delivery and billing arrangements, often related to a service layer agreement (SLA). The possible way that end users will communicate with a specified amount of bandwidth and with specified services is illustrated in Figure 1.2, which presents a high-level scheme of cloud computing. The cloud computing concept relates to delivery of both computing and storage capacity as services to a number of end users. Individual end users can access cloud-based applications through a Web browser or mobile applications. In a cloud computing scheme we can have the following: (1) infrastructure as a service, (2) platform as a service, and (3)

Introduction to Optical Communications

5

software (application) as a service. Cloud computing as concept is designed to allow enterprises to obtain and run specific applications faster an in more manageable manner, while information technology (IT) resources can be adjusted to satisfy dynamic business demands.

Servers

Application (monitoring, collaboration, content, finance) Smartphones

Platform

Laptops

(object storage, database, run time, identity)

Infrastructure (compute, network, file storage)

Tablets Desktops

Figure 1.2 Cloud computing networking concept.

The importance of network intelligence in future networks can be further outlined by software-defined networking (SDN), which is concept in which control is decoupled from hardware and transferred to a specific software application. The SDN has an essential impact on the way that a packet travels through the network since its trajectory and attributes will not be determined by embedded software (firmware) in the switching node. Instead, separate software can be used to shape traffic in a centralized manner. Network administrator can determine switching rules by putting and removing priorities on specific traffic (bandwidth flows). This concept goes together with cloud computing architecture since traffic loads can be managed in a flexible and more efficient manner. The physical network topology that best supports traffic demand is generally different in different segments of the optical networking structure shown in Figure 1.1. It could vary from the mesh (deployed mostly in the core networks), to ring (deployed mainly in in metro areas), to a star topology deployed in an access networks to enable efficient statistical multiplexing and bandwidth distribution. At the same time, submarine optical links, whose length could be up to tens of thousands of kilometers and more, have a point-to-point character. From the optical transmission perspective, the optical network configuration enables an end-to-end connection through the optical signal flow, which can be referred as the lightpath. Each lightpath is related to the physical layer of the optical network and takes into account the optical signal properties, characteristics of deployed

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Advanced Optical Communication Systems and Networks

optical elements (fibers, lasers, amplifiers) along the path, the impact of the networking topology, and the type of service that is requested. Each service has the source and destination node and can be either point-to point or broadcast nature. Any connection is associated with quality of services (QoS) requirements, which is then related to each individual lightpath.

Submarine System

Optical Core Network

Optical Metro Network

Optical Access Network

Figure 1.3 Lightpaths in optical network.

The term lightpath means that an optical signal propagates between the source and destination without experiencing any opto-electrical-opto (O-E-O) conversion. Several examples of the lightpaths across topologies discussed above are presented in the Figure 1.3. In general, the lightpaths differ in lengths and information capacity that is carried along. As an example, the lightwave path in a submarine transmission system can be several thousand kilometers long while carrying information capacity measured by tens of terabits. On the other hand, a typical lightpath within the metro area is measured by tens of kilometers while carrying the information capacity measured by gigabits. It is also important to mention that lightpath is usually associated with a single optical channel (single optical carrier wavelength), which means that several lightpaths can be established in parallel between the source and destination The lightpath length is one of the most important variables from a transmission perspective, since most of impairments have an accumulating effect proportional that length. The lightpath, as an optical connection between two distinct locations, is realized by assigning a dedicated optical channel between them. The lightpaths are propagating through an optical fiber within a single (in single-mode optical fibers) or multiple spatial modes (in multimode optical

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fibers). A lightpath can have a permanent character if it is set up during the network deployment and does not experience any change. On the other hand, it can have a temporary character if it is set up and torn down on request. There are several key optical components that can be placed along the lightpath, such as optical amplifiers, optical switches, and optical filters. Several optical signals from different lightpaths can be also combined in a composite signal at some point and continue to travel together throughout the network. In such a case, an equalization of the optical signal parameters from different lightpaths might be necessary in order to provide an equal transmission quality for all channels. Such equalization can include signal level adjustment, adaptive dispersion compensation, and optical filtering. From a signal transmission perspective, the lightwave path can be considered as a bandwidth pipe or a wrapper for lower-speed information flows. There are a number of publications and books dealing with optical transmission and networking aspects, and some of them are listed at the end of this chapter [1–13]. In next section, we will present a brief historical perspective that reflects the evolution of optical systems and networks. We will also include a brief historical note with respect to the networking content that is wrapped up by a lightpath.

1.2 HISTORICAL PERSPECTIVE The idea to use light for communication purposes is very old. There is evidence that that some of oldest civilizations (Chinese, Middle Eastern, Native Americans) used visual signals to send messages over some distance. The fire and smoke signals, or mirror reflected light, were used for that purpose. The first optomechanical system that could relay digital messages (in a telegraph-like manner) was deployed in France in 1794, and was used in different countries in the first half of eighteenth century. The position and the angle of the bar visible from some distance were used to code and transmit messages. These telegraphlike systems were soon replaced by electrical telegraphs that provided much faster operation over longer distances. Even transatlantic transmission with telegraphs was achieved in 1866. Since then, electrical communications became the only means to effectively communicate over longer distances. (Interestingly enough, the telegraphic transmission as the first step in electrical communications was fully digital.) The twisted cooper pair was widely used until the mid-twentieth century, when transmission through coax cables and free space was proposed. Both of these transmission methods offered much wider transmission capacity, which at that time meant that more analog voice signals (each with the bandwidth around 4 kHz) could be transmitted. The transmission bandwidth that was utilized was initially around 10 MHz, gradually increasing towards 100 MHz by the midtwentieth century. When digital communications became dominant in the second half of twentieth century, transmission capacity was measured by maximum

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aggregate bit rate that could be transmitted over a certain distance. That bit rate has reached several hundreds of megabits per second over distance of 1 km by the mid-1970s, while more advanced schemes for signal modulation and detection have been applied, and the maximum channel capacity has been studied [13–16]. 1.2.1 The Early Beginning and the First Generation of Optical Communications The idea to use light signals for communication purpose was reintroduced soon after laser (light emitting by stimulated radiation) was invented in 1960. Free space communication by using lasers as light sources looked very promising for shorter distances and when atmospheric conditions were good (no fog, rain, snow, moderate atmospheric turbulence). However, the real beginning of optical communications began in 1966, when it was suggested that optical fibers can be used for guided propagation of the light signals [17]. The biggest obstacle at the very beginning was an extremely high attenuation of the optical signal in the glass optical fibers (at that time more than 1,000 dB/km). Fortunately, soon after that, the process was invented by Corning to reduce the attenuation from 1,000 dB/km down to less than 20 dB/km at optical carrier wavelengths around 1 m. That was possible by using silica as the main material and ensuring that the concentration of different impurities (ions different than SiO2) is kept below a certain threshold. By improving optical fiber manufacturing technology, the attenuation of the optical signal rapidly decreased to the typical values presented by curve in Figure 1.4. As we can see, the curve has its hills and valleys, and several distinct wavelength regions can be recognized. The most notable regions are around 1,300 nm and 1,550 nm, where attenuation has a relative and absolute minimum, respectively. The region between 800 nm and 900 nm has some historical significance since early light sources operated at the wavelengths from that region.

Figure 1.4 Typical attenuation curve of silica-based optical fibers.

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The invention of semiconductor GaAs lasers operating in continuous wave (CW) regime provided another component essential for the massive deployment of optical transmission systems over optical fibers. What followed from 1970 to the first decade of 21st century was an extraordinary race that led to enormous increase in both the maximum transmission capacities and the transmission distances. Previously we described a brief history of optical communications. In parallel with this, as digital communications became dominant, communication and information theory explained the way to increase the total transmission capacity of twisted pairs, coax cables, and microwave channels by applying more sophisticated modulation formats and signal detection schemes. Optical communications, however, did not follow such a trend, since a basic digital on-off modulation of output optical power was still predominant. On the other hand, digital networking on an electrical level followed the progress in digital transmission. Networking protocols were invented to facilitate communications between processors associated with individual networking nodes. As processors became faster, it was possible to have more efficient routing of the digital information packets from the origin to the destination. The reader can look at the references [18–33] to find more information about evolution of networking methods and protocols. 1987

1977

First Generation

Channel rate Distance

Number of Channels

System Enablers

Optical Networking

Subchannel Networking

100 Mb/s - 500 Mb/s Up to 10 km

Single 800 nm-900 nm wavelength region

Multimode fibers Fabry-Perot lasers

No, just Point to point Systems Asynchronous PCM Systems

Second Generation

650 Mb/s - 2.5 Gb/s Up to 50 km

1997

Third Generation

2.5 Gb/s - 10 Gb/s Up to 1000 km

Single 1310 nm wavelength region

1 – 128 1530 nm -1560 nm wavelength region

Single-mode fibers DFB lasers

Optical EFDA Amps WDM MuX Dispersion compensation

No, just Point to point Systems Asynchronous PCM Systems ATM Ethernet

WDM MUX Static OADM

SONET/SDH IP ATM Ethernet ESCON/FICON

2007

Fourth Generation

10 Gb/s - 40 Gb/s Up to 10000

40 – 160 1520 nm -1610 nm wavelength region

Fifth Generation

100 Gb/s – 1 Tb/s Up to 10000 km

96+ 1520 nm -1610 nm wavelength region

Optical EFDA Amps Raman Amplifiers FEC Tunable lasers

Multilevel modulation Coherent Detection Advanced FEC Intense DSP OFDM MIMO

DWDM MUX ROADM/OXC Layered control plane

CDC ROADM ROADM/OXC OFDMA MIMO

SONET/SDH/OTN IP MPLS Ethernet

OTN IP/MPLS Ethernet

Figure 1.5 The evolution path of optical transmission systems and networks.

The evolution path of the optical transmission systems and networks is illustrated in Figure 1.5. The first generation of optical transmission systems operated in the wavelength optical window from 800 nm to 900 nm. Transmission

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Advanced Optical Communication Systems and Networks

was done over multimode optical fibers, while multimode semiconductor lasers of the Fabry-Perot type were used as the light sources. At the same time, siliconbased photodiodes have been used as photodetectors. In some cases, for very short distances, even light emitting diodes (LED) as the optical signal sources were deployed. The total transmission distance was measured by tens of kilometers and was limited by either optical fiber attenuation or spatial mode dispersion. Optical light sources were modulated by digital signals belonging to the asynchronous pulse code modulation (PCM) hierarchy with the direct intensity modulation (IM) of the light source, accompanied a with direct detection (DD) scheme. These optical systems are recognized as IM/DD ones. Since no more than one optical channel (lightpath) was transmitted, optical networking was not enabled. All networking functions were performed by utilizing an electrical time division multiplex as a networking tool. 1.2.2 The Second and Third Generations In the second generation, there was a shift from multimode to single-mode optical fibers, since technology was available to have more efficient source-fiber coupling and interconnection/splicing of individual optical fiber lengths. By using singlemode optical fibers, spatial mode dispersion was effectively eliminated and attention has shifted to transmission in the wavelength window around 1,310 nm, where attenuation in optical fibers were considerably lower (typically below 0.5 dB/km) than the attenuation associated with the window between 800 nm and 900 nm. However, the major advantage of using transmission in 1,310 nm wavelength region came from the fact that material dispersion, closely related to refractive properties of silica-based optical fibers, has its minimum. Single-mode semiconductor lasers, known as distributed feedback (DFB) ones, based on the InGaAsP semiconductor structure have been introduced as operating in the region close to 1,310 nm. The IM/DD transmission of bit rates up to 2 Gb/s, still belonging to asynchronous hierarchy of PCM voice-related signals, was performed on a single optical channel over distances of up to 50 km before regeneration was required. In order to achieve distances longer than 50 km, a number of optoelectronic regenerators, where signal was photodetected and digitally reshaped, were put in place along the transmission line. Novel protocols, such as Ethernet and asynchronous transfer mode (ATM), were also introduced to handle data transmission, but with no real connection with the optical channel. The third generation of optical systems is characterized by the shift of the carrier wavelength to the wavelength region around 1,550 nm. The main reason for that was an absolute minimum of fiber attenuation around that specific wavelength. The major concern for system designers became not fiber attenuation, but chromatic dispersion in standard single-mode fibers (SSMF). More complex multilayer design of the optical fiber core in single-mode fibers helped to mitigate chromatic dispersion with waveguide dispersion component (which is mainly associated with the optical power distribution over a multilayer cross-sectional

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area of the fiber core). A new category of single-mode fibers, called dispersion shifted fibers (DSF), were introduced and deployed by some major carriers with the goal to maximize the transmission distance before optoelectronic regeneration takes place. Initially, a single optical channel was deployed, with optical regenerators spaced 80 km apart along the transmission line. The optical signal was modulated by a format that now belonged to a new synchronous hierarchy aimed to transmit a number of multiplexed voice channels (each of them at a 64 kb/s bit rate). The synchronous hierarchy is known as SONET (Synchronous Optical Network) in North America or SDH (Synchronous Digital Hierarchy) in the rest of the world. Even a minor difference between SONET and SDH was not good from transoceanic transmission perspective. For data transmission, proprietary protocols such as ESCON and FICON were still widely used. An optical carrier was modulated directly by ESCON/FICON signals for data transfer over shorter distances. The SONET/SDH was also enabled to accept pure data signals by incorporating networking protocols, such ATM or IP, into the SONET/SDH payload. For a moment, it looked as if some good technologies were in place to handle both digital transmission and networking. In that period of time, the research community began working on optical transmission systems that would use phase/frequency digital modulation formats, but this time they were coupled with coherent optical detection. The goal was to increase the transmission distance by increasing receiver sensitivity, while enabling chromatic dispersion compensation on the electrical level. Transmission capacity was eventually increased by several times and the 10 Gb/s SONET bit rate (known as OC-192) was introduced. However, from an optical transmission perspective, IM/DD scheme was still in place. It was clear, however, that an increase in SONET bit rates would not be enough to satisfy growing bandwidth requirements due to increase of the data related traffic, while voice-based traffic showed just a modest growth. Technology maturity and introduction of several technologies enabled unprecedented growth with respect to both total transmission capacity and transmission link length before regeneration was needed. First, it was possible to multiplex a number of individual optical channels in an aggregated signal by using wavelength division multiplex (WDM). Second, it was possible to amplify optical signal by using the stimulated emission principle, therefore without optoelectronic conversion. Finally, it was possible to compensate chromatic dispersion by periodical deployment of special optical fibers, called dispersion compensating fibers (DCF), that would reverse the chromatic dispersion impact. Both optical amplifiers and DCF were collocated and periodically deployed along the transmission line, thus enabling the distances of up to 1,000 km. That distance was gradually increased and it was possible to have transoceanic transmission without any OEO regeneration. The best part of that was that both optical amplifiers and DCF could handle multiple wavelengths within the WDM optical signal.

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The introduction of WDM signaled a real start to optical networking, since each wavelength within the WDM spectrum could eventually follow its distinct path or the lightpath. Also, each WDM wavelength could be modulated with a different kind of the digital signal, such as SONET, ATM, or ESCON/FICON, which was often case for transmission within the metro regions. The new class of WDM systems called the Metro WDM was recognized. The functionalities known as add-drop signal multiplexing or signal cross-connect have become applicable to optical signals as well, and the first generation of optical add/drop multiplexers (OADM) and optical crossconnects (OXC) was introduced. These functions were initially done in a simplified static manner. The lightpaths have been transmission tools for lower bandwidth services required between source and destination. Historically, the lightpath has acted as a bandwidth wrapper for lower-speed transmission channels (often referred to as virtual circuits). The time division multiplexing technique in the electrical domain has been applied to aggregate the bandwidth of lower-speed transmission channels before they are wrapped in the lightwave paths. There have been two forms of multiplexing known as fixed multiplexing and statistical multiplexing. In the fixed multiplexing case, the aggregate bandwidth is divided among individual virtual circuits in such a way that each circuit receives a guaranteed amount of the bandwidth, often referred to as a bandwidth pipe. If there is an unused portion of the bandwidth within any specific bandwidth pipe, it is wasted since a dynamic exchange of contents between any pair of bandwidth pipes is not possible. Therefore, it can happen that some bandwidth pipes are almost empty at any given moment, while others can be quite full. Statistical multiplexing can minimize the waste of bandwidth, since it can be used more efficiently. It is done by breaking up the data content in each bandwidth pipe into data packets, which can be handled independently. By doing this, packet switching is enabled. If there is an overload in any specific bandwidth pipe, some packets can be redirected to other unloaded ones. In addition, statistical multiplexing permits distribution of the aggregated bandwidth between individual virtual circuits with a finer granularity. The fixed multiplexing of virtual circuits has been defined by the SONET standard applied in North America and SDH applied worldwide [26]. Both standards define a synchronous frame structure, bandwidth partitioning, multiplexing patterns, and supervisory functions for transport of different kind of digital signals. Although just digital voice signals with time division multiplexing (TDM) were originally packed into a SONET/SDH frame, it gradually became a mean for framing and transport of different kind of data channels. These data channels are usually arranged according to specified standards [18–25]. A review of digital bandwidth channels mostly used in the past two decades is presented in Table 1.1. There are two separate columns in this table related to the TDM/synchronous channels and to data/asynchronous bandwidth channels. The bandwidth channels are lined up based on the bit rate. Table 1.2 shows high-speed channels that are unified in terms of the content inside.

Introduction to Optical Communications

TDM/synchronous bandwidth channels DS-1 E-1 OC-1 OC-3=STM-1 OC-12=STM-4

OC-48=STM-16 OC-192=STM-64 OC-768=STM-256

Table 1.1 The bandwidth channels Bit rate Data/asynchronous bandwidth channels 1.544 Mb/s 10-BaseT Ethernet 2.048 Mb/s 100-BaseT Ethernet 51.84 Mb/s FDDI 155.52 Mb/s ESCON Fiber Channel-I 602.08 Mb/s Fiber Channel-II Fiber Channel-III Fiber Channel IV 2.488 Gb/s 9.953 Gb/s 39.813 Gb/s

Gb Ethernet 10 Gb Ethernet 40 Gb Ethernet

13

Bit rate 10 Mb/s 100 Mb/s 100 Mb/s 200 Mb/s 200 Mb/s 400 Mb/s 800 Mb/s 4 Gb/s 1 Gb/s 10 Gb/s 40 Gb/s

Table 1.2 Unified data channels (introduced and envisioned) 100 Gb Ethernet 107-123 Gb/s 400 Gb Ethernet 400 Gb/s 1 Tb Ethernet 1000 Gb/s 4 Tb/10 Tb Ethernet 4,000/10,000 Gb/s

The basic building block of the TDM hierarchy has been 64 kb/s, corresponding to one digitalized voice channel. The next level is obtained by multiplexing either 24 channels (DS-1 format in North America) or 30 channels (E-1 format outside North America). The higher-levels in asynchronous TDM hierarchy were built up by multiplexing several lower level signals. It has been common to take four lower-level streams and combine then into a higher-level aggregate signal. In such a way, a third level, corresponding to DS-3 (44.736 Mb/s) in North America, E-3 (34.368 Mb/s) in Europe, or DS-3J (32.064 Mb/s) in Japan, has been obtained. The lack of a unified approach becomes an obstacle for a high-speed connection around the globe, due to interoperability issues. The introduction of SONET/SDH standards was the first step towards a global networking. Although the corresponding building blocks, which are OC-1 (optical carrier - 1) in SONET and STM-1 (synchronous transport module - 1) in SDH, are different in bit rates, they belong to the same synchronous hierarchy. Optical transmission systems widely deployed today operate at bit rates equal to 2.488 Gb/s and 9.953 Gb/s. These bit rates are commonly referred to as 2.5 and 10 Gb/s. The high-speed transmission systems operating at 40 Gb/s, which corresponds to the OC-768/STM-256, have also been deployed. Another ITU standard, known as Optical Transport Network (OTN) [3], based on fixed multiplexing has been specially designed for optical transport networks and high bit rates. It defines an optical channel (OCh) in a flexible way that provides more efficient data accommodation, and better OA&M (operation, administration, and maintenance) capabilities. Since the OTN technology is one of the key

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Advanced Optical Communication Systems and Networks

transmission tools for fifth generation of optical systems and networks, it will be described separately later on in this chapter. Data channels presented on the right side of Table 1.1 do not follow any strict hierarchy. Some of them, such as Enterprise Serial Connection (ESCON) or Fiber Channel, have been designed for interconnecting computers with other computers and peripherals (such as large memories and data storages). They have been widely used in storage area networks (SAN), for backup operations, financial transactions, and so forth. The bit rates presented in Tables 1.1 and 1.2 are related just to data payload or useful data. However, these bit rates are increased when some line coding is applied before the data is sent. For example, ESCON data channel uses (8B, 10B) line code, which increases the line bit rate to 250 Mb/s, while FDDI utilizes (4B, 5B) line code that increases the line bit rate to 125 Mb/s. It is important to notice that the Fiber Channel standard defines three data-payload bit rates that eventually become equal to 256.6, 531.2, and 1,062.6 Mb/s, respectively, after (8B/10B) line code is applied. Both ESCON and Fiber Channel, if deployed individually, use low-speed and low-cost optical components. However, these channels can be also multiplexed in order to improve the transport efficiency. Multiplexing could be done by using the WDM technique with optical channels loaded by data bit rates presented in Table 1.1. It could be also done by placing them into the SONET/SDH frame in accordance with flexible bit rate standard specifications [23, 24]. The FDDI and the Ethernet data channels have been originally designed for data bandwidth sharing in the LAN environment. The FDDI is based on the token ring access method, while the Ethernet standard is based on the well-known CSMA/CD medium access scheme. The size of shared bandwidth channel, as defined by the Ethernet standards, went from 10 Mb/s at the very beginning while ago to 100 Gb/s today, while it is widely anticipated that 400 Gb/s and 1 Tb/s bits rate will be captured by Ethernet standards in near future. The enhancement of both bandwidth channel size and the bandwidth sharing capability has been accompanied by an effort to define Ethernet as a transport standard, which would be suitable not just for LAN and MAN environment, but for WAN applications as well. It is also worth to mention that the gigabit Ethernet bit rate goes up to 1.25 Gb/s after line coding is applied. Therefore, from the transmission perspective, the line bit rate is a parameter that should be taken into account. Since the Ethernet technology is foundation of high speed statistical networking that belong to the fifth generation of optical transmission systems and networks, it will be discussed in more detail later in this chapter. 1.2.3 The Fourth and Fifth Generations of Optical Systems and Networks The fourth generation of optical transmission systems and networks have brought progress on different fronts. First, EDFA amplifiers have been enhanced by adding some additional dopants ions (such as aluminum or thulium) to erbium doped fiber so they can cover almost entire wavelength region where attenuation

Introduction to Optical Communications

15

has a minimum. In addition, Raman amplifiers based on stimulated emission in regular optical fibers have been introduced to provide an additional gain [34–35]. Since Raman amplification is distributed in nature, it has a beneficial effect to the overall signal-to-noise ratio during signal transmission. A higher number of WDM channels could be now transmitted in the wavelength region covered by EDFA gain. The wavelength spacing of the neighboring WDM channel was standardized under ITU-T (International Telecommunications Union) and has scaled from 50 GHz to 200 GHz. Since WDM channels are densely spaced in the wavelength region around 1,550 nm, the acronym DWDM (Dense WDM) was adopted to reflect such a reality. Second, the forward error correction (FEC) coding technique, originally applied just for submarine optical links, was introduced and became indispensable part of the overall optical transmission system design. The FEC effectively relaxed the requirements for the signal-to-noise ratio (SNR) needed to achieve the required transmission quality since it has been capable to detect and correct some errors that appeared during signal propagation. In addition, tunable optical elements (lasers, filters, dispersion compensators) have been introduced, providing much-needed flexibility and dynamic arrangement of DWDM channels. The technology enablers mentioned above made transmission at bit rates of 40 Gb/s possible, and an initial deployment of systems loaded by OC-768 SONET has started. The first generation of 40 Gb/s-based optical transmission systems utilized traditional IM/DD process of signal modulation and detection. However, some other modulation formats, such as differential phase shift keying (DPSK), were considered, but in combination with the direct detection scheme. It has been clear that moving from an intensity modulation to more sophisticated formats is just a matter of time. However, also, designers began thinking about how to effectively reintroduce a coherent detection technique. The fourth generation of optical transmission systems and networks has been characterized by considerable progress in the networking domain. Optical adddrop multiplexers became reconfigurable, and the generic name ROADM (reconfigurable optical add drop mux) was adopted. Reconfigurable design has enabled the dynamic assignment of lightpaths and has been utilized for provisioning or restoration purposes. The control plane that facilities interoperability between different logical network layers became more effective. As for subwavelength signal networking, and as the data-originated traffic outpaced voice-originated traffic, ATM technology became outdated. A new approach known as multiple protocol label switching (MPLS) became an effective tool to perform a connection-oriented operation on IP data streams. The MPLS function is very similar in nature to that performed by ATM, but this time it was more efficient in terms of speed and routing processing. Massive employment of MPLS enabled dedicated data packet transport, while keeping quality of service (QoS) parameters usually attributed to circuit switching. At the same time, the Ethernet has kept its function as the most efficient networking toll for statistical multiplexing and distribution of data. SONET technology has been still utilized as

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Advanced Optical Communication Systems and Networks

an efficient wrapper of different protocols (IP/MPLS, FICON, and Ethernet), which could be transferred to arbitrary distances while providing the most reliable operation and maintenance (OAM) function. However, the newly adopted Optical Transport Network (OTN) standard as a successor of SONET/SDH became the most attractive transport engine to be directly carried by an optical channel. The transition from circuit-based traffic in the electrical domain to a fully packet-based one has an overall impact to optical networking. The optical channel is associated with a specific optical wavelength. As such, it is data content agnostic and does not belong to any digital hierarchy. However, an optical channel can be considered as a building block of the overall optical bandwidth that can be carried along the lightpath [36, 37]. Several optical channels can be grouped together to form an optical subband. The amount of bandwidth carried by an optical subband depends on the number of wavelengths inside the subband, and the bit rates of signals carried by each individual wavelength. It is important from a routing and provisioning perspective that wavelength subband can be treated as a lightpath, the same way a single wavelength channel is treated. The next level above the optical subband is the entire optical band, such as C, L, and S. Each individual wavelength band can be considered as an aggregation of optical channels and wavelength subbands. In some cases, an optical band can be considered as a single lightpath, if all wavelength components follow the same path from the source to the destination. The fifth generation of optical transmission systems and networks has started before the beginning of the second decade of this century and has been characterized by a number of technologies that have been either introduced or became more effective [36–60]. We can start with the fact that the IM/DD scheme, although still applicable and cost-effective for lower bit rates, has been replaced by sophisticated multilevel modulation formats and coherent detection scheme. The phase and polarization of the optical signal, in addition to its amplitude, have been effectively utilized to increase the overall transmission capacity by enhancing the signal spectral efficiency. Modulation formats that have been widely considered include schemes such as: M-QAM (M-ary quaternary amplitude modulation), and N-PSK (N-ary phase shift keying). The number of modulation states readily recognized in a constellation diagram has been increasing rapidly, and in 2012 we had cases where a successful transmission with the overall capacity over 100 Tb/s were performed by using these modulation formats [40, 41]. Multilevel modulation at the transmission side is accompanied by polarization multiplexing, which means that the total transmission bit rate is doubled as compared with the case when no polarization multiplex is applied. The combination of multilevel modulation formats and polarization multiplexing enabled a spectral efficiency of 14 b/s/Hz and more [41]. The frequency of the optical carrier has been effectively utilized as well. Namely, by applying orthogonal frequency division multiplexing (OFDM), a number of frequency subcarriers can be generated along the frequency spectrum [43, 46]. Although

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they overlap with each other, there is no crosstalk between neighboring carriers since they are orthogonally positioned in accordance with the Nyquist criterion. Each optical subcarrier can be modulated independently by employing multilevel modulation formats mentioned above. Therefore, multilevel modulation formats, in combination with polarization multiplexing and OFDM, are the most effective tools for efficient usage of the optical spectrum. Coherent detection of optical signals became a mainstream detection scheme for the fifth generation of optical transmission systems. An optical hybrid coupler is used for the design of a balanced optical receiver where in-phase and quadrature signal components, each of them as a combination of input signal and local oscillator input, are detected separately, and this scheme is applied two times since there are two input polarization states. The frequency and phase recovery of the detected electrical signal in balanced optical receiver is done by using intense digital signal processing (DSP). In addition, DSP includes digital filtering for chromatic and polarization mode dispersion mitigation. The FEC decoding process in a coherent receiver is a function heavily dependent on DSP. It is important to outline that DSP is enabled by digital-to-analog conversion (DAC) and analog-to-digital conversion (ADC) of the electrical signal that are applied in the optical transmitter and optical receiver, respectively. The total bandwidth and the speed of ADC and DAC components are of the essential importance to enable the most effective DSP during modulation and detection processes. The efficient FEC methods became one of the most important tools to enhance the capacity of the optical channel. Advanced FEC methods that include soft-decision deciding are an essential part of the fifth generation of optical transmission systems. There are different flavors of soft-decision FEC such as turbo codes or low-density parity check (LDPC) codes that can bring the coding gain of 10 dB and more, which is far superior to traditional Reed-Solomon codes widely used in the fourth generation of optical transmission systems. The multiple-input multiple-output (MIMO) technique, well known in wireless telecommunications, became the important enabling technique not just for increasing the total transmission capacity, but also for enhancing the optical networking capabilities. The MIMO is enabled by the fact that a number of independent physical paths could exist even in the same optical fiber. Such paths are associated with either polarization states or spatial modes. We can expect that both OFDM and MIMO will play a vital role in any future advancement of the fifth generation of optical systems and networks and beyond. From an optical networking perspective, the fifth generation is characterized by employment of CDC (colorless, directionless, and contentionless) ROADM that finally provides a full optical switching of lightpaths in any specified direction and without any signal contention. It appears that wavelength selective switches (WSS) will continue to play a key role in the future design of both ROADM and OXC (optical cross-connects) for optical networking. Also, networking became more dynamic in terms of wavelength spacing that can be

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Advanced Optical Communication Systems and Networks

processed by ROADM and OXC, which effectively means that ITU-T wavelength grid may not be the only determining factor anymore. Subwavelength networking at the electrical level, as a part of the fifth generation, is done by utilizing IP/MPLS, Ethernet, and OTN technologies for distribution of fully packetized optical bandwidth. Each of them has a specific function: IP/MPLS to provide virtual circuits for connection-oriented bandwidth delivery, Ethernet to provide the most efficient bandwidth grooming and sharing, and OTN to provide reliable transport of the packet traffic over arbitrary distances, while keeping a robust OAM functionality. In the next section, we will briefly describe the basic features of high-speed Ethernet (100 Gb Ethernet) and OTN networking standards since they are essential parts of the fifth generation of optical networks. What is important is that SONET and packet (Ethernet)-based technology finally converged at bit rates of 100 Gb/s and above. The assumption is that, after 100-Gb Ethernet, future standards will define the following: 400-Gb Ethernet, 1,000-Gb/s Ethernet (1-Tb Ethernet), 4-Tb Ethernet, and eventually 10-Tb Ethernet, as indicated in Table 1.1. These bit rates will be accommodated in corresponding OTU formats within the OTN standard, which will account for the bit rate increase due to the FEC overhead (expected to be in the range of 7% to around 20%).

1.3 ETHERNET AS A FOUNDATION OF PACKET-BASED NETWORKING The Ethernet is one of the oldest networking technologies, but it has been continuously evolving, so we can say that it is one of the newest networking technologies as well. The Ethernet became the most commonly used networking technology not just in local area networks (LAN), as it was its original purpose, but also for delivering networking solutions over wide area networks (WAN). What distinguished the Ethernet among competing networking technologies was proven performance, flexibility, and interoperability, all accompanied with its highly affordable cost. 1.3.1 Ethernet as a Layer-2 Networking Technology The Ethernet technology that we know today [2] is derived from the specification invented by Robert Metcalfe and codeveloped in 1980 by Xerox, Intel, and Digital. Over the past three decades, Ethernet standards have continuously advanced to meet the network requirements related to the speed, flexibility, and quality of services. The main standard body that has been dealing with Ethernet specifications is the IEEE (Institute of Electrical and Electronic Engineers), or specifically its group, known by the name 802.3. The Ethernet has been originally defined for the speed of 10 Mb/s, and evolved to 100 Mb/s in the early 1980s. The

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bit rate of 100 Mb/s is well known as the fast Ethernet (FE). The next evolution steps involved bit rates of 1 Gb/s (gigabit Ethernet or GbE) and 10 Gb/s (10 GbE). Finally, in 2010, new classes of Ethernet technology were defined at bit rates of 40 Gb/s (40 GbE) and 100Gb/s (100 GbE). However, some activities already started in the IEEE and ITU-T standard forums with the goal to define the specifications for next generation Ethernet, which will likely be at 400 Gb/s (400 GbE), and soon after that at 1,000 Gb/s (1-Terabit Ethernet). There is also a prediction that, by 2020, 4-TbE and 10-TbE standards will be in place. The Ethernet is often a synonym for Layer 2 networking technology, based on the OSI (Open Systems Interconnection) reference model recommended by the International Standard Organization (ISO) in the mid-1970s. At that time, communication equipment complexity, from both a hardware and software perspective, as well as the implementation of unstructured programs, created a difficult situation for testing, modification, and interoperability. The ISO helped to simplify the problem by introducing a layered network structure with each layer performing well defined functions. The layers of the communication system perform either network-dependent functions (Layers 1–4) or application/serviceoriented functions (Layers 5–7). The OSI structure will be described in Chapter 8. Ethernet network technology belongs to Layer 2, also recognized as MAC (medium access layer) of OSI reference model. The Ethernet defines access to network topology by CSMA/CD (carrier sense multiple access/collision detection), which is a congestion resolution-based scheme. In fact, the Ethernet is the only widely used representative of multiuser CDMA/CD scheme. If some node (user) wants to use network resources to transmit data frame, it can be done only if the medium connecting all nodes is not utilized by another user at that specific moment. The transmitted Ethernet frame, per original definition, has had a structure shown in Figure 1.6. The MAC protocol works as follows: Step 1, if the medium is idle, transmission will start, otherwise go to Step 2; Step 2, if the medium is busy, monitor it until the “idle” state is detected, then move back to Step 1; Step 3, if collision is detected during the transmission process, send the jamming signal that notifies all nodes of the frame collision, and discontinue transmission; and Step 4, reattempt transmission sometime after the jamming signal has been transmitted. Header (HD)

Preamble

SDF

DA

SA

LC

Payload (PD)

Frame Checking (FC)

Data (Packets)

Pad

SDF – Start Frame Delimiter DA – Destination Address SA – Source Address LC – Length Count FCS – Frame Check Sequence

Figure 1.6 Ethernet MAC frame format in accordance with IEEE 802.3.

FSC

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Advanced Optical Communication Systems and Networks

The minimum MAC frame size is determined in accordance with the maximum propagation delay of the medium. In our case the medium is the optical fiber, which presents the physical layer in accordance with the OSI model. The IEEE 803.2 standard defines Ethernet technology as a two-part structure that consists of MAC portion (which belongs to Layer 2) and the physical layer portion. The MAC portion frame structure of any subsequent standard release covering higher bit rates does not differ from the structure associated with lowerspeed Ethernet standards. Accordingly, the 100-GbE frame structure is similar to the structure of 1 GbE and 10 GbE, just adjusted to the 100 Gb/s speed. 1.3.2 100 Gigabit Ethernet as a High-Speed Networking Tool From now on, we will concentrate on 100-GbE features, since they are in the latest standard release at this moment and will represent a networking foundation for a number of years to come. The 100-Gb/s Ethernet structure, with an illustration of the individual portions (sublayers), is shown in Figure 1.7. The MAC sublayer in 100 GbE converts the packets received from higher layers into Ethernet frames in accordance with the scheme from Figure 1.6 and converts received Ethernet frames back to packets. The transmitting side of the MAC sublayer from Figure 1.7 assembles the data coming from upper layers (commonly the IP/MPLS layer) into Ethernet frames, while adding the header to the leading edge of the packet, and the frame checking sequence (FCS) at the trailing edge of the packet. The FCS is usually generated by multiplication of the payload content with a specified polynomial. The receiving side of the MAC sublayer proceeds with FSC verification and discard the Ethernet frames containing errors. The reconciliation sublayer (RS) from Figure 1.7 serves as an adjustment between MAC sublayer and physical layer (PHY) by changing an abstract interface into a logical 100-Gb/s media independent interface (CGMII). In this sublayer, the MAC serial data stream is converted to parallel data path that consists of 64-bit-wide data signal accompanied with 8-bit-wide control signal. Also, RS performs a detection of link faults and notification relevant to layers above and below. The physical layer consists of three sublayers: PCS, PMA, and PMD. The PCS performs encoding and decoding function, while PMA functionality is related to bit-level multiplexing and demultiplexing, accompanied with clock and data recovery. Finally, PMD sublayer serves as a connection of PHY layer to a physical medium, which is either optical fiber for distances from several tens of meters to more than 40 km in accordance to current IEEE specification, or copper cable for shorter distances measured by meters and tens of meters. IEEE standard 802.3ba specifies several cases if fibers are used. First, multimode fibers can be used over distances of at least 100m through 100GBASE-SR10 interface, which generates 10 parallel optical streams, each with a 10 Gb/s bit rate. Single-mode

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optical fibers are used for interfaces known as 100GBASE-LR4 and 100GBASEER4 for distances of at least 10 km and 40 km, respectively. In this case, PMD generates four streams of optical signals each with the speed of 25 Gb/s. IP/MPLS Layer

MAC Sublayer

IP#1

HD

IP#2

IP Packets

IP#3

Payload (IP/MPLS packets)

FCS

HD

Payload (IP/MPLS packets)

IP#N

FCS

Ethernet

Reconciliation Sublayer (RS) Ethernet Frame

CGMII

Ethernet Frame

Physical Coding Sublayer (PCS) 64/66 Line Code Physical Medium Attachment Sublayer (PMA) Physical Medium Dependent Sublayer (PMD) MDI

1 1 0 0 1 1 0 1 1 1 0 0 0 1 1 1 0 1 0 1

Physical Medium

Optical or electrical signal

Figure 1.7 100-Gb Ethernet architecture.

The copper interface for 100 GbE is known as 100GBASE-CR10 and generates 10 streams, but without electro-optical conversion. For this interface, there could be an optional sublayer for FEC to improve the overall transmission quality over copper cables. The FEC applied here should bring more than 2-dB coding gain and should deal with the burst errors. As we mentioned, multilane parallel transmission is adopted in the physical layer structure (PCS, PMA, PMD sublayers). The exact bit rate that is coming to PMA and PMD sublayers is 103.125 Gb/s, since 64B/66B line coding is adopted in PCS. The exit from PCS block is then split to either 10 or 4 physical lane configurations, while taking care of skew issues (skew is difference in data arrival times between individual lanes). As we noticed above, 100-Gb Ethernet interfaces are defined by the IEEE and include distances from several meters to over 40 km. These interfaces are in essence LAN-designed ones to support transmission over relatively small geographical area. These interfaces are tailored for a number of applications such as video, latency-sensitive stock market financial services, and distributed network computing, most of them directly related to the data center service ports. However, some portions of LAN client signals need to be transferred over longer distances between different states, big cities, or universities. The longer distances are likely to include several domains, where multidomain interworking is needed. A domain is often network belonging to a specific telecom carrier, which means

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Advanced Optical Communication Systems and Networks

that domain interworking has to deal with OAMP (operation and maintenance) functions. The OTN (optical transport network) technology, which was mentioned earlier, serves as a transport engine for Ethernet services. 1.3.3 OTN Technology as a High-Speed Transmission Tool The OTN standard has been envisioned to unify infrastructure for different services by adjusting the data container size in accordance with the speed of the data payload, while introducing a reach set of OAMP functions. The original OTN standard was approved in 2001 by ITU-T 709 recommendations, but a number of additions were made from that time including those aimed to accommodate 100Gb Ethernet. The client signal, in this case 100-Gb Ethernet, is wrapped into OTN containers by following the generic scheme illustrated in Figure 1.8. The client signal is combined with the overhead bytes to form an optical-channel data unit (ODU). Several ODUs are hierarchically multiplexed to form an optical-channel transport unit (OTU). In addition, FEC bytes are added before the OTU is converted to the optical form and eventually combined with other wavelengths within the same optical fiber. The total OTU frame size is 4 rows x 4080 columns, and it is filled with overhead, payload, and FEC bytes, respectively, as illustrated in Figure 1.9. While payload bytes carry an information signal, overhead bytes provide very rich OAM capability. That OAM is often referred as “carrier class OAM,” just as a reminder that it complies with stringent requirements in terms of monitoring signal quality and enabling interoperability among different domains.

OTU on l1

ODU ODU

Client #1

Client #2

DWDM MUX on optical fiber/spatial modes

OTU on lN

ODU

Client #M 100 GbE

Figure 1.8 Combining 100 GbE and other client signals into OTN.

The ITU-T has defined different sizes of the OTU frame to better capture the content belonging to different clients. Accordingly, we have the following units and corresponding bit rates: OTU1 ~ 2.67 Gb/s, OTU2 ~ 10.71 Gb/s, OTU3 ~ 43.02 Gb/s, and OTU4 ~ 111.81 Gb/s. As for 100 Ethernet, it is obvious that OTU4 was tailed to wrap up the 100 GbE with the addition of FEC bytes. Since

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the size of the OTU frame is limited, only standard hard-decision FEC codes, such as the well-known Reed-Solomon (255,239) code, which introduces 7% additional bytes, can be accommodated. COLUMNS 3824 1

3825

4080

16 17

ROWS

1 2

Overhead

Payload

FEC

3 4

Figure 1.9 OTN frame structure.

As for more advanced FEC schemes, such as LDPC [42], which utilize softdecision algorithms and can introduce anywhere from 7% to 35%, new frame sizes have to be adopted. It is agreed within the ITU-T to introduce the OTU4v size to capture soft-decision FEC overhead ranging around 20%. We will witness some other proposals for OTU sizes to capture not only 100 GbE, but also 400 GbE and 1 TbE together with the most efficient coding schemes.

1.4 CLASIFICATION AND BASIC CONCEPT OF PHOTONIC TRANSMISSION SYSTEMS AND NETWORKS The basic difference between optical and any other transmission systems is the fact that optical carrier frequencies have much higher values as compared to any other carrier frequencies used in wireline or wireless transmissions. The optical carrier frequencies are somewhere around 200 THz, which is at least 10 5 times higher than carrier frequencies of wireless mobile signals, or signals propagating through a coax cable. The higher carrier frequency enables a wider signal modulation bandwidth around that frequency, which means that the information capacity of the optical channels is far larger than that of any other known medium used today. 1.4.1 Optical Fiber as a Foundation for Transmission and Networking Optical fiber is the foundation of optical signal transmission. It offers wider available bandwidth, lower signal attenuation, and smaller signal distortion than other wired physical media. The total available optical bandwidth is illustrated in Figure 1.10, which shows a typical attenuation curve of a single-mode optical fiber that is used today. The total optical bandwidth is approximately 400 nm, or

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Advanced Optical Communication Systems and Networks

around 50 THz, if it is related to the wavelength region with fiber attenuation lower than 0.5 dB/km. The resultant information capacity can be calculated by multiplying the available optical bandwidth with the spectral efficiency, which is expressed in bit/s per 1 Hz of bandwidth. The total information capacity can be more than 500 Tb/s by using advanced modulation techniques that improve spectral efficiency far over 1 bit/s/Hz.

0.6

0.4

S

C

L

E

O

U

0.2

1100

1200

1300

1400

1500

1600

1700

Wavelength in nm

Figure 1.10 Bandwidth of optical fibers.

The usable optical bandwidth is usually split to several wavelength bands. The bands around the minimum attenuation region, usually referred to as C and L bands, have been considered the most suitable ones for high channel count DWDM transmission, and have been already widely used for transmission purposes. The C (conventional) band occupies wavelengths approximately from 1,530 nm to 1,565 nm, while the L (long) band includes wavelengths between 1,565 nm and 1,625 nm. The S (short) band covers shorter wavelengths between 1,450 nm and 1,530 nm, where optical fiber attenuation is slightly higher than in the C and L bands. The wavelength region around 1,300 nm is less favorable for optical signal transmission since signal attenuation is higher than attenuation associated with wavelengths from S, C, and L bands. On the other side, the bandwidth around 1300 nm is quite usable for some specific purposes, such as transmission of CATV signals. In addition, the CWDM (course wavelength division multiplexing) technique can be easily employed in this band. Recently, although not standardized, two bands around a wavelength of 1,310 nm have been recognized: O (original) band from 1,260 nm to 1,360 nm, and E (extended) band from 1,360–1,460 nm. In addition, one more band at the wavelengths longer than 1,625 nm has been considered. It is known as the U (ultra-long) band, which ranges from 1,625 nm to 1,675 nm. 1.4.2 Optical Transmission Systems The simplest optical transmission system is a point-to-point connection by utilizing a single optical wavelength as a lightpath that propagates through an

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optical fiber. The upgrade to this topology is deployment of the WDM technology, where multiple optical wavelengths are combined to travel over the same physical route (again as a single lightpath just with increased capacity). The WDM has proven to be a cost-effective mean of increasing the bandwidth of an installed fiber plant. While the technology originally only served to increase the aggregate bit rate, it gradually became the foundation for optical networks.

Client #1

Tx

100 GbE

Tx

100 GbE

Tx

Client #N

Tx

Optical Multiplexers

Optical Fiber Link

Reconfigurable Optical Add/Drop or Optical Crossconnect

Optical Fiber Link

Optical Multiplexers

Optical Fibers

Optical Multiplexer

Optical transmitters

Optical receivers

Rx

Rx Rx AMP

AMP

AMP

AMP

Rx

Optical Amplifiers

Optical Amplifiers

Figure 1.11 Optical transmission and networking.

A generic scheme of an optical transmission system with networking elements inside is shown in Figure 1.11. We should assume that the signal transmission is generally bidirectional, while the unidirectional character from Figure 1.11 is just for an illustration purpose. Several optical channels, associated with specified information bandwidth, have been combined by WDM technology, and sent to the optical fiber line. The aggregated signal is then transported over some distance before it is demultiplexed and converted back to an electrical domain by photodetection process. The optical signal transmission path can include a number of optical amplifiers, optical cross-connects, and optical adddrop multiplexers. The optical signal on its way from the source to destination can be processed by various optical components, such as optical filters, optical attenuators, and optical isolators. Generally speaking, each wavelength in a scheme from Figure 1.11 can be considered as distinct lightpath since it can be routed independently by reconfigurable optical add-drop multiplexed or optical cross-connect. The routing process of the optical signal is done automatically by using either network management system or a specialized control plane software. The routing and add/drop processes are associated with one of the following networking functions: service initiation, service teardown, network protection and restoration, grooming of services, and bandwidth on demand.

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Advanced Optical Communication Systems and Networks

1.4.3 Optical Networking Parameters The set of parameters that are related either to the enabling technologies and components or to the transmission and networking issues can be attached to the system from Figure 1.11. The optical signal transmission quality is usually defined by the bit error rate (BER) at the end of lightpath or between two separate intermediate points. The lightpath connection needs to be properly engineered to provide stable and reliable operation. The arrangement of optical transmission and networking parameters is shown in Figure 1.12. As we can see, most of them are either optical power related or optical wavelength related. They can also be time-invariant or time-variant. The optical power-related system parameters are optical power per channel, optical fiber attenuation, the extinction ratio, and optical component losses. The optical wavelength-related parameters are optical wavelength itself (through the wavelength stability) and optical channel bandwidth. The third group of parameters, time-variant parameters, includes the optical channel bit rate, the firstorder polarization mode dispersion (PMD), and signal timing jitter. Power level Extinction ratio Fiber loss Insertion loss

Optical Power

Modulation format Quantum noise BER PDL Optical noise OAMP gain Crosstalk FWM, SRS SBS SPM XPM

Wavelength

Wavelength stability Optical bandwidth Optical multiplexing Laser chirp Chromatic dispersion PMD (second order)

Time

Jitter Bit rate PMD (first order)

Electrical noise (Thermal noise)

Figure 1.12 Optical signal parameters.

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The next group of parameters consists of optical power-related but also time variant, and it includes: optical modulation format, quantum noise, polarizationdependent losses (PDL), and the bit error rate. There is yet another group that contains parameters dependent on both optical power and optical wavelength, and includes optical amplifier gain, optical noise, cross-talk between optical channels, and some nonlinear effects such as four-wave mixing (FWM) and stimulated Raman scattering (SRS). Parameters that are dependent on wavelength, but are also time variant are: chromatic dispersion, laser frequency chirp, and the secondorder PMD. Finally, there are some parameters that are dependent on both optical power and optical wavelength, and are also time variant, and they are: stimulated Brillouin scattering (SBS), self-phase modulation (SPM), and cross-phase modulation (XPM). It is important to mention that in addition to the parameters listed above, there is an electrical noise caused by the thermal effect, which can be considered as a background parameter, which is present in any transmission scenario. This parameter does not have any specific connection to either optical power or signal wavelength. The design of optical transmission systems involves accounting for all effects that can alter an optical signal on its way from the source (laser) to the destination (photodiode) and then to the threshold decision point. Different impairments will degrade and compromise the integrity of the signal before it arrives to the decision point to be recovered from corruptive additives (noise, cross-talk, and interference). The transmission quality is measured by the received signal-to-noise ratio (SNR), which is defined as the ratio of the signal level to the noise level at the threshold point. The bit-error-rate (BER) parameter is also used to signal quality evaluation since it defines the probability that a digital signal space (or a logical “0”) will be mistaken for a digital signal mark (a logical “1”), and vice versa. Evaluating the BER requires determination of the received signal level at the threshold point, calculation of the noise power, and quantification of the impacts of various impairments, all with respect to modulation format and detection scheme applied. However, any designers of optical transmission systems and networks have at their disposal of set of advanced technologies for amplification, routing, and optical processing of the signals along the lightwave path. Also, advanced modulation and detection schemes can be utilized to improve signal quality, to increase the transmission capacity and to enhance network flexibility and dynamics. Multilevel modulation schemes, such as M-QAM, have been combined with coherent detection schemes to increase both the optical channel capacity and quality of the transmission. Advanced multiplexing schemes, such as OFDM, polarization multiplexing, and optical spatial MIMO, can be used to increase the capacity, but also serve as the optical networking sublayer that can work together with Layers 2 and 3.

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Advanced Optical Communication Systems and Networks

1.4.4 Optical Channel Capacity and Basic Signal References The major objective when designing optical transmission system is to maximize optical channel capacity and optical network traffic throughput. The capacity can be increased by applying the advanced modulation/detection schemes to increase the spectral efficiency. Advanced multiplexing schemes, as well as the advanced FEC, are also applied to serve the same purpose. The ultimate channel capacity as defined by well-known Shannon principle [14] that the channel capacity is equal [

]

[

]

(1.1)

where C is the maximum rate of the information within a specified bandwidth W, while the signal and the noise level levels are defined by S and N, respectively. We also used SNR=S/N as a conventional designation for the signal-to-noise ratio. Generally, the signal and nose levels are defined in different ways in electrical and optical domains. The electrical signals are represented either by current or voltage amplitudes, while noise, as a statistical process, is represented by the root mean square (RMS) of its magnitude. Therefore, we can assume that S ~ I(t), or S ~ U(t), where I(t) and U(t) represent the magnitudes of the current and voltage attributed to the signal. At the same time, we have that N ~ 1/2, where ne(t) represent a stochastic electrical parameter associated with the noise. The ratio S/N is also often known just as SNR and has been expressed in decibel (dB) units as (

), in dB

(1.2)

where we put a subscript e to reflect the fact that SNR relates to electrical domain. At the same time, signals in the optical domain have been represented by their optical power. Optical power is also relevant for different types of optical noise (such as amplified spontaneous noise in optical amplifiers, or intensity noise in semiconductor lasers). Therefore, now we have that S ~ P(t), where P(t) represent optical signal power, and N ~ 1/2 where np(t) describes stochastic variations of the optical noise power. The SNR in optical domain can be expressed as (

), in [dB]

(1.3)

where subscript o stands for optical. We still have the signal-to-noise ratio expressed in decibel units, but this time the logarithmic ratio is multiplied by factor 10 instead of 20. That simply reflects the fact that that we are now dealing with the power, not with a magnitude (power is proportional to the square of the magnitude.)

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The decibel, as a relative unit, has been also used as a measure of the ratio between two optical powers P1 and P2, so we have that (

), in [dB]

(1.4)

If that ratio is 3 dB, it means that power acquired gain and doubled in value, while –3 dB means that power lost a half of the original value between two points in question. Another illustrative case is -20 dB, which means that optical signal had lost 99% of its original value when arrived to the second control point. The special case appears if P2 = 1 mW, since then the power ratio reflected the absolute value of the optical signal. Instead of dB notation, we now have dB m, where m stands for milliwatt. As a reference, it is useful to remember that zero dBm represent the optical power of 1 mW. Although the SNR parameter for both optical and electrical signals is used to evaluate the signal health and make transmission quality assessments, it is BER that is commonly used to evaluate transmission quality of digital signals. The topics in Chapters 4, 5, 6, and 7 will have a detailed analysis of both SNR and BER parameters in different transmission scenarios. Talking about channel capacity, both the performance and the ultimate channel capacity, in accordance with Equation (1.1), are limited by the noise level in the receiving end. Equation (1.1) will give us the maximum possible bit rate in a noisy communication channel, which is the case when the noise with a Gaussian distribution is present. However, although the Gaussian noise is present in optical communication channel, Equation (1.1) cannot be exactly applied to determine the channel capacity in all cases. It is because the impact of nonlinear effects in optical fibers becomes dominant when optical power crosses a certain level [51]. In such a case, any additional increase in the signal level to enhance the SNR will not bring the increase of the parameter C in Equation (1.1) since transmission quality will fall below the acceptable value. If we assume a generic case of optical transmission system, illustrated in Figure 1.10, and assuming that there are M optical channels (lightpaths) loaded with bit rates B1, B2, B2… and BM, respectively, the total transmission capacity B will be equal to sum of the individual bit rates, that is ∑

(1.5)

Very often, instead of parameter B, we use the product of total transmission capacity and the distance L to characterize the transmission capability of an optical link. The parameter BL is defined as BL=B·L

(1.6)

It is clear that the total transmission capacity of an optical transmission system and the throughput of optical networks will depend on the density of the

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optical channels within the specific wavelength band. The ITU-T originally defined a wavelength grid for neighboring DWDM channels, which specifies that neighboring channels are placed 100 GHz apart in both C and L bands. That specification was first updated with 50-GHz spacing, and consequently 25-GHz spacing will also introduced. However, there is a new reality that no rigid, but rather a flexible wavelength positioning will be preferred as we move to most advanced optical networking architectures, so ITU-T has introduced a variable channel spacing that can be changed with 12.5-GHz granularity [47]. Another parameter mentioned earlier, commonly known as spectral efficiency (SE), is used to characterize the transmission capabilities in optical transmission systems and networks. Spectral efficiency in multichannel optical system is defined as (1.7) where  is the optical frequency band occupied by multichannel signal. The parameter SE is expressed in b/s/Hz, and serves as a good reference for comparison of various modulation, detection, and coding schemes. The employment of the most advanced modulation, coding, and detection schemes in the fifth generation of optical transmission systems and networks led to more than a tenfold increase in spectral efficiency as compared to the spectral efficiency in previous generations. As an example, the transmission capacity of B = 100 Tb/s in C and L bands with spectral efficiency of 11 b/s/Hz was achieved in 2011 [40] with employment of polarization diversity 128QAM-OFDM modulation scheme followed by coherent detection. We can expect that this trend will continue in years to come, as there is much more room for spectral efficiency improvement.

1.5 FUTURE PERSPECTIVE A steady progress has been made in optical transmission and networking during past several decades and that trend will continue. We assume that the future trend will be defined by further increase in the optical channel capacity (both on the optical channel level and the optical link level) and further enhancement of optical networking with respect to flexibility, throughput, and dynamics. We expect that the following trend will dominate throughout the next two decades. Multicarier transport will be gradually introduced to optical access, with a new generation of passive optical networks, such as WDM PON or OFDM PON, deployed. These networks should deliver Ethernet services to individual and corporate users, as well as connect multiple antenna sites in the next generation of wireless systems. We can expect that 1-Gb Ethernet will become a standard bandwidth pipe for residential users, while 10-Gb Ethernet will become a commodity bandwidth pipe serving mid-size corporate users. At the same time,

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the 100-Gb Ethernet will be the unit for data packet delivery over nationwide networks. The high demand for bandwidth will speed up the introduction of higher Ethernet rates, and we can expect that 400-Gb Ethernet and 1-Tb Ethernet will be introduced in this decade, while 4-Tb Ethernet and 10-Tb Ethernet should be in place by 2025. Overall, there is a list of topics that will be addressed in the future, such as: (1) design of high capacity optical channels by utilizing all available degrees of freedom, (2) design of multidimensional and elastic optical networking schemes, (3) employment of novel technologies and materials for optical components (including optical fibers), (4) large scale optoelectronic integration with increase in the maximum data throughput and decrease of the relative power consumption, and (5) efficient schemes for distribution of data packets within data center networks, just to name few. The purpose of this book is to provide both foundation and guidance that will help to address the topics listed above.

1.6 ORGANIZATION OF THE BOOK The chapters that follow are structured to provide a comprehensive material for understanding the physical phenomena behind the topic in question and understanding and implementing both basic and the most advanced principles, algorithms, and schemes that have been discussed. The material is organized in a textbook format and mostly deals with advanced topics of optical communications and networking. It is organized to provide straightforward guidance to readers looking to capture fundamentals and gain theoretical and practical knowledge to better understand the fifth generation of optical communication systems and networks. Every chapter of the book has an extensive list of references. There is also a list of problems at the end of each chapter. In addition, this book provides guidance for simulation and modeling with the goal of helping the readers build up the knowledge and skills necessary for their research and engineering tasks. The sequence of topics is arranged to take a reader from description of optical components, through detailed analysis of the signal propagation in an optical fibers and analysis of the optical channel impairments and noise sources. All this is followed by description of the advanced methods for signal modulation, detection, and coding. Optical networking principles and advanced networking architectures are described in a separate chapter, although the optical networking theme will be present in most of the chapters. A brief summary of the book chapters is provided next. This chapter outlines the role of optical networking in our information society, and explains the Ethernet technology as a foundation of packet-based networking. Historical perspective and significance of the fifth generation of optical systems and networks are also provided, with more specifics of the role of

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different functional blocks used for communications and networking. Finally, classification and basic concepts of optical systems and networks are introduced. Chapter 2 describes characteristics of the optical components that are commonly deployed along a lightpath, thus having an impact on light propagation, optical networking, or advanced signal processing. The topics in this chapter are related to optical fibers, light sources, optical modulators, photodetectors, optical amplifiers, optical filters, optical switches, and optical multiplexers. The structure of optical fibers and cables, as well as the role of the other components, such as optical isolators, couplers, or circulators, is explained. The special attention will be paid to the features that are important from both a transmission and a networking perspective. In addition, system aspects of key elements, such as advanced optical modulators and optical filters/switches, will be outlined. In Chapter 3, propagation properties of different types of optical fibers will be described in detail. The nonlinear propagation equation will be used to analyze the impact of both linear and nonlinear effects to the pulse shape in single-mode optical fibers. We will use the Gaussian pulse shape to calculate the impact of chromatic dispersion and self-phase modulation. In addition, transmission properties of special optical fibers (few-mode and few-core optical fibers) that support multiple spatial modes and serve as a foundation for employment of the spatial MIMO (multiple-input multiple-outputs) technique will also be described. Finally, propagation characteristics of optical fibers for special applications, such as those for dispersion compensation, will be analyzed. Chapter 4 is related to the assessment of the optical channel impairments and their impact to the signal transmission along the lightpath. Special attention will be paid to the chromatic dispersion, polarization mode dispersion, self-phase and cross-phase modulation, four-wave mixing, and Raman scattering. These phenomena will be explained in details and expressions for the evaluations of their impact will be obtained. All noise components relevant to design of optical systems and networks will be analyzed in detail. In addition to detailed analysis of the photodetection-related noise components, the impact of the beat noise components caused by optical amplifiers, as well as the laser amplitude and phase noise components will be examined. In Chapter 5, we will describe advanced formats including the following: multilevel modulations such as N-ary phase-shift keying (N-PSK) and M-ary quadrature amplitude modulation (M-QAM); multidimensional constellations, such as four-dimensional signal constellations suitable for communication over single-mode fiber links, multidimensional orbital angular momentum (OAM) modulation suitable for communication over few-mode fibers, and hybrid multidimensional signal constellation employing all available degrees of freedom; and orthogonal frequency division multiplexing (OFDM). In addition, the concepts of polarization-division multiplexing and spatial-division-multiplexing will be introduced. The chapter will also provide the concepts of signal-space theory and optimum signal constellation design.

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Chapter 6 is devoted to advanced detection concepts that include: coherent detection, coherent-optical OFDM, channel equalization, MIMO detection, and compensation of various impairments. The following post-detection compensation techniques will be described: feed-forward equalizer, decision-feedback equalizer, Viterbi equalizer, turbo equalization method, and digital back-propagation method. Also, compensation of chromatic dispersion and polarization mode dispersion by OFDM will be described. Finally, various MIMO detection techniques will be described in detail. Chapter 7 represents an overview of advanced FEC techniques for optical communication. Topics include: codes on graphs, coded modulation, rate-adaptive coded modulation, and turbo equalization. The main objectives of this chapter include description of different classes of codes on graphs that are of interest for optical communications, description methods used to combine multilevel modulation and channel coding, explanation of algorithms used to perform equalization and soft-decoding jointly, and, finally, explanation of methods that are efficiently used for joint demodulation, decoding, and equalization process. The codes on graphs that will be explained include turbo codes, turbo-product codes, and LDPC codes. We will also discuss an FPGA implementation of decoders for binary LDPC codes. In Chapter 8, we will describe different network topologies and characteristics of lightpath routing with respect to initial routing and signaling constraints. Key optical routing elements (ROADM and OXC) and the role of advanced wavelength selective switches (WSS) will be described in detail. Multilayer aspects of optical networking will be outlined, with a description of the role of optical networking client layers (IP/MPLS, Ethernet, OTN) in the interworking process. Multipath optical network design for both packet-based and circuit-based switching will be discussed, and its implementation in the key networking segments (access networks, metro networks, core networks, and data center networks) will be outlined. Finally, the spatial-spectral concept of advanced photonic networking that involves MIMO and OFDM technologies will be explained. Chapter 9 is dedicated to modeling and simulation of optical-communication systems and evaluation of optical channel capacity. We will describe models suitable for the study of various linear channel impairments, models for the study of OFDM and MIMO techniques, and propagation models suitable to the study of intrachannel and interchannel nonlinearities. We will also describe the evaluation of optical fiber information capacity by considering fiber as a channel having memory. In addition, we will also describe how to determine the uniform information capacity as a lower bound of the channel capacity. In Chapter 10, we will provide the explanation and fundamental description with respect to the physics of semiconductors and signal information theory. The definition of special mathematical functions and their relations will be also provided to help the reader to easily follow material covered in earlier chapters.

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The reader is advised to consult the topics inside this chapter even before starting to study the topics covered in earlier chapters.

PROBLEMS 1.1 What are the advantages of statistical versus time division multiplexing (TDM) of digital signals? Explain both the major advantages and disadvantages of circuit-based and packet-based networking. 1.2 What is difference among IP/MPLS, Ethernet, and OTN networking technologies? Which one is the most closely associated with the lightpaths and why? 1.3 What was the turning point in considering optical fiber as a signal transmission medium? Explain why. 1.4 Explain why transmission at optical frequencies can enable a higher information capacity as compared with any kind of copper cables and microwave links. 1.5 Calculate the optical fiber link length over which the optical power will be losing 99% of its original value if the fibers with losses of 0.2 dB/km, 10 dB/km, and 1,000 dB/km are used. 1.6 The optical laser source is capable of producing output power of 0.05W. Half of that power is injected into optical fiber link with the goal to achieve 200 km transmission. In order to achieve that, an in-line optical amplifier is needed. Assume that the transmission is done in C band around 1,550 nm and calculate the gain of an in-line optical amplifier needed to achieve the total transmission distance of 200 km. Assume that the minimum required optical power needed before both the amplification and detection processes is –21 dBm. 1.7 In previous problem, you assumed that received optical power equals –21 dBm. How many photons are coming within a time slot at the receiving side if transmission bit rate is 100 Mb/s, 1 Gb/s, and 100 Gb/s? 1.8 Calculate the actual bandwidth in Hertz of the O, E, S, C, L, and U wavelength regions from Figure 1.9. What is the total transmission capacity for each of them if spectral efficiency of the transmission systems is 0.1 b/s/Hz? Compare the available transmission capacity of optical fibers to the capacity of high-quality coaxial cables. 1.9 Explain the ways to enhance the total transmission capacity in a given spectral range. What compromise should you take into account in an attempt to increase the overall capacity? 1.10 What signal-to-noise ratio can guarantee that spectral efficiency is higher than 4 b/s/Hz? What would be the SNR increment to double the spectral efficiency? 1.11 What are the major enablers of the fifth generation of optical transmission systems and networks? What would be the ultimate goal in optical

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communications and networking? What key elements are necessary to achieve that goal? 1.12 Optical transmission systems of the first, second, and third generations were operating at 900 nm, 1,310 nm, and 1,550 nm, respectively. Calculate the photon energy (in eV) of the signal at the carrier frequency in each case.

References [1]

Cvijetic, M., “Advanced technologies for next-generation fiber networks,” Proc. of Optical Fiber Communication Conference, San Diego, CA, 2010, paper OWY1.

[2]

IEEE 802.3ba 40 Gb/s and 100 Gb/s Ethernet Standard, June 2010, www.ieee802.org/3/ba.

[3]

ITU-T Rec. G. 709, “Interfaces for the Optical Transport Network – OTN,” 2009.

[4]

Cvijetic, M., and Magill, P., (coeditors): “100 Gigabit Ethernet, Seed Delivering on the Promise,” IEEE Communication Magazine, Vol. 45, 2007, Supplements 3 and 4.

[5]

Baldine, I., et al, “A unified architecture for cross-layer design in future optical Internet,” Proc. European Conference on Optical communications (ECOC), Vienna, 2009, paper 2.5.1.

[6]

McGuire, A., et al., (editors) IEEE Communications Magazine: Series on Carrier Scale Ethernet, Vol. 46, Sept 2008.

[7]

Internet Engineering Task Force (IETF), MPLS (Multiple Protocol Label Switching) Protocol www.ietf.org/rfc/rfc3031.txt, www.ietf.org/rfc/rfc3032.txt, www.ietf.org/rfc/rfc5462.txt, www.ietf.org/rfc/rfc5921.txt.

[8]

Gower, J., Optical Communication Systems, 2nd edition, Upper Saddle River, NJ: Prentice Hall, 1993.

[9]

Agrawal, G. P., Fiber Optic Communication Systems, 4th edition, New York: Wiley, 2010.

[10] Ramaswami, R., Sivarajan, K. N., and Sasaki, G., Optical Networks, a Practical Perspective, 3rd edition, San Francisco, CA: Morgan Kaufmann Publishers, 2010. [11] Kazovski, L., Benedetto S., and Willner A., Optical Fiber Communication Systems, Norwood, MA: Artech House, 1996. [12] Mukherjee, B., Optical WDM Networks, New York: Springer, 2006. [13] Cvijetic, M., Coherent and Nonlinear Lightwave Communications, Norwood, MA: Artech House, 1996. [14] Shannon, C. E., “A Mathematical Theory of Communications,” Bell Syst. Techn. J., Vol. 27, Jul. and Oct. 1948, pp. 379–423 and 623–656. [15] Schwartz, M., Information Transmission, Modulation and Noise, 4th edition, New York: McGraw-Hill, 1990. [16] Proakis, J. G., Digital Communications, 5th edition, New York: McGraw Hill, 2007. [17] Kao, K. C., and Hockman, G. A., “Dielectric fiber surface waveguides for optical frequencies,” Proc. IEE, 133(1966), pp. 191–198. [18] Bellcore, G.R., SONET Transport Systems: Common Generic Criteria, GR-253-CORE, Bellcore 1995.

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[19] ITU-T Rec. G. 681: “Functional characteristics of interoffice and long-haul line systems using optical amplifiers, including optical multiplexing,” ITU-T (10/96), 1996. [20] ITU-T, Rec. G.798 “Characteristics of Optical Transport Network Hierarchy Functional Blocks,” ITU-T (02/01), 2009. [21] ITU-T, Rec G.694.2, “Spectral grids for WDM applications: CWDM wavelength grid,” ITU-T (06/02), 2002. [22] ITU-T, Rec G.704, “Synchronous frame structures used at 1544, 6312, 2048, 8448 and 44,736 kbit/s hierarchical levels,” ITU-T (10/98), 1998. [23] ITU-T, Rec G.7041/Y.1303, “Generic framing procedure (GFP),” ITU-T (12/01), 2001. [24] ITU-T, Rec G.7042/Y.1305, “Link capacity adjustment scheme (LCAS) for virtual concatenated signals,” ITU-T (11/01), 2001. [25] ITU-T, Rec G.8110/Y.1370.1, “Architecture of transport MPLS layer network,” ITU-T (11/06), 2006. [26] Kartalopoulos, S. V., Understanding SONET/SDH and ATM, Piscataway, NJ: IEEE Press, 1999. [27] Loshin, P., and Kastenholz F., Essential Ethernet Standards: RFCs and Protocols Made Practical, New York: John Wiley & Sons, 1999. [28] Huitema, C., Routing in the Internet, 2nd edition, Upper Saddle River, NJ: Prentice Hall, 1999. [29] Calta, S. A., et al., “Enterprise system connection (ESCON) architecture – system overview,” IBM Journal of Research and Development, 36(1992), pp. 535–551. [30] Sachs, M. W., and Varma, A., “Fiber Channel and related standards,” IEEE Commun. Magazine, 34(1996), pp. 40–49. [31] Ross, F. E., “FDDI – a tutorial,” IEEE Commun. Magazine, 24(1986), pp. 10–17. [32] Yuan, P., et al., “The IEEE 802.17 media access protocol for high-speed metropoliten area resilient ring packets,” IEEE Network, 18(2004), pp 8–15. [33] Farrel, A., and Bryskin, I., GMPLS: Architecture and Applications, San Francisco, CA: Morgan Kaufmann Publishers, 2006. [34] Cvijetic, M., Optical Transmission Systems Engineering, Norwood, MA: Artech House, 2003. [35] Lightwave Optical Engineering Sourcebook, Worldwide Directory, Lightwave 2003 Edition, Nashua, NH: PennWell, 2003 [36] Kovacevic, M., and Acampora, A. S., “On the benefits of wavelength translation in all optical clear-channel networks,” IEEE JSAC/JLT Special Issue on Optical Networks, Vol. 14, 1996, pp. 868–880. [37] Lee, M., et al., “Design of hierarchical crossconnect WDM networks employing two-stage multiplexing scheme at waveband and wavelength,” IEEE J. Sel. Areas in Comm., Vol. 20, pp. 116–171. [38] Fukuchi, K., et al, “10.92 Tb/s (273x40 Gb/s) triple band ultra-dense WDM optical-repeated transmission experiment,” Optical Fiber Conference-OFC, Anaheim, CA, 2001, PD 26. [39] Zhou, X., et al., “32Tb/s (320x114Gb/s) PDM-RZ-8QAM Transmission over 580km of SMF-28 Ultra-Low-Loss Fiber,” Optical Fiber Conference-OFC, San Diego, CA, 2009, PDPB4. [40] Quin, D., et al., “101.7-Tb/s (370×294-Gb/s) PDM-128QAM-OFDM Transmission over 3×55km SSMF using Pilot-based Phase Noise Mitigation,” Optical Fiber Conference-OFC, Los Angeles, CA, 2011, PDPB5.

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[41] Sakaguchi, J., et al., “109 Tb/s (9 x 79 x 172 Gb/s SDM/WDM/PDM) QPSK transmission through 16.8 Homogenous Multicore Fiber,” Optical Fiber Conference-OFC, Los Angeles, CA, 2011, PDPB6. [42] Djordjevic, I. B., et al., “Next generation FEC for High-capacity Communication in Optical Transport network,” IEEE J. Lightwave Techn., Vol. 27, Aug. 2009, pp. 3518–3530. [43] Yu, J., et al., “Ultra-High capacity DWDM Transmission System for 100G and Beyond,” IEEE Commun. Magazine, Vol. 48, 2010, pp. 56–64. [44] Chandrasekhar, S., et al., “Transmission of a 1.2 Tb/s 24-carrier No-Guard-interval Coherent OFDM Super Channel over 7200 km of Ultra-large-Area- Fiber,” Optical Fiber ConferenceOFC, San Diego, CA, 2009, PD 2.6. [45] Lam, C. F., (editor), Passive Optical Networks: Principles and Practice, San Diego CA: Academic Press, 2007. [46] Cvijetic, N., et al., “100 Gb/s optical access based on optical orthogonal frequency-division multiplexing,” IEEE Commun. Magazine, Vol. 48, No 7, 2010, pp. 70–77. [47] Gringeri, S., et al., “Flexible Architectures for Optical Transport Nodes and Networks,” IEEE Commun. Magazine, Vol. 48, No 7, 2010, pp. 40–50. [48] Qiao, C., et al., “Extending Generalized Multiprotocol Label Switching (GMPLS) for Polymorphous Agile and Transparent Optical Network,” IEEE Commun. Magazine, Vol. 44, No 12, 2006, pp. 104–114. [49] Tomkos, I., et al., “Performance engineering of metropolitan area optical networks through impairment constraint routing,” IEEE Commun. Magazine, Vol. 42, No 7, 2004, pp. 40–47. [50] Alferness, R., et al., (editors), IEEE/OSA Journ. of Lightwave Techn., Special Issue, Vol. 26, May 2008. [51] Essiambre, R. J., et al., “Capacity Limits of Optical Fiber Networks,” IEEE J. of Lightwave Techn., Vol. LT-28, 2010, pp. 662–701. [52] Sheih, W., and Djordjevic, I. B., OFDM for Optical Communications, Burlington, MA: Elsevier, 2010 [53] Djordjevic, I. B., et al., “Optical LDPC decoders for beyond 100 Gbit/s optical transmission,” Opt. Letters, Vol. 34, May 2009, pp. 1420–1422. [54] Xia, T. J., et al., “Field experiment with mixed line-rate transmission (112-Gb/s, 450-Gb/s, and 1.15-Tb/s) over 3,560 km of installed fiber using filterless coherent receiver and EDFAs only,” Optical Fiber Conference, Los Angeles, CA, 2011, PDPA3. [55] Ryf, R., et al., “Space-division multiplexing over 10 km of three-mode fiber using coherent 6 × 6 MIMO processing, ” Optical Fiber Conference, Los Angeles, CA, 2011, PDPA3. [56] Sitch, J., “High-speed digital signal processing for optical communications,” Proc. of European Conference on Optical Communications, Brussels, 2008, paper Th.1.A.1. [57] Morioka, T., et al., “Enhancing Optical Communications with Brand New Fibers,” IEEE Commun. Mag., Vol. 50, no. 2, pp. 40–50. [58] Zhu, B., et al., “Seven-Core Multicore Fiber Transmission for Passive Optical Network.” Optics Express, Vol. 18, May 2010, pp. 117–122. [59] Gerstel, O., et al., “Elastic Optical Networking: a New Down for the Optical Layer?,” IEEE Commun. Mag., Vol. 50, pp. 512–520.

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[60] Myslivets, E., and Radic, S., “Advanced fiber optic parametric synthesis and characterization,” Optical Fiber Conference-OFC, Los Angeles, CA, Mar. 2011, paper OWL5.

Chapter 2 Optical Components and Modules There are a number of different optical components that can be deployed along the lightpath either in a point-to-point transmission system or in an optical networking scenario. Some of them, such as light sources, optical fibers, and photodetectors, are deployed at any lightpath, while the others, such as optical amplifiers, optical filters, and optical switches, are employed based on the overall system design and network functions. In this chapter we will describe the fundamentals of optical components used in optical communication systems and networks and identify the key parameters that are important from a systems and network perspective. A detailed analysis of signal propagation through optical fibers will be presented in Chapter 3. The well-established references [1–22] deal with mathematical, physical, and transmission aspects of optical devices, and were consulted in writing this chapter.

2.1 KEY OPTICAL COMPONENTS Optical components that could be deployed along the lightpath are shown in Figure 2.1. The first row shows components that are essential for optical signal generation, modulation, multiplexing, amplification, and detection, while the second row shows components that play an important role with respect to specific functions related to optical signal processing. The optical signal would follow the path from the right to left in the upper row of Figure 2.1 experiencing the impact of the individual elements. Some of the components from Figure 2.1 may not be commonly present. However, the elements that contribute to the creation, propagation, and detection of the optical signal are part of any lightpath and we will briefly describe their role. Semiconductor light sources (lasers and light emitting diodes) convert an electrical signal to optical radiation, thus producing an optical signal. These semiconductor devices are directly polarized, where the bias current flows through a semiconductor p-n junction, initiating a recombination of electrons and holes. The recombination process produces the flow of photons that present the output

39

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optical signal. If the current flowing through the p-n junction is higher than a certain threshold, a stimulated emission of radiation can be achieved in special semiconductor structures. In these structures, well known as semiconductor lasers, the recombination occurs in an organized way, with a strong correlation in the phase, frequency, and direction of radiated photons that form the output optical signal. Semiconductor lasers could either be directly modulated by an electrical signal, or just biased by a DC voltage and operate in combination with an external optical modulator. Each laser generates a specified optical wavelength or optical carrier, but a spectral linewidth around the carrier is associated with the generated optical signal as well. These lasers are known as single-mode ones (SML), characterized by a distinguished single longitudinal mode in the optical spectrum. If a set of separated longitudinal modes can be recognized under the optical spectrum envelope, we are talking about multimode lasers (MML).

Light Source (LED, LD)

Modulator Optical (MZ, EA) Multiplexer

Optical Isolator

Optical Filter

Optical Fiber

Optical Switch

Optical Photodiode Optical Amplifier Demultiplexer (PIN, APD)

Optical Coupler

Wavelength Converter

Figure 2.1 Lineup of optical components for systems and networks.

Another component, an external optical modulator, accepts the signal from semiconductor lasers and imprints the information content into it. The external modulation is applied in high-speed transmission systems since it helps to suppress the impacts of laser frequency chirp and chromatic dispersion in optical fibers. In addition, it enables the implementation of complex modulation schemes with amplitude and phase shift keying. Optical modulators accept a continuous wave (CW) optical signal from the laser and change its parameters (amplitude and phase) by an applied modulation voltage. There are two types of external optical modulators commonly used today, Mach-Zehnder (MZ) and electro-absorption (EA) ones. Optical fibers are the central part of any system or network configuration. They transport an optical signal from the source to its destination. The combination of low signal loss and extremely wide transmission bandwidth allows high-speed optical signals to be transmitted over long distances before regeneration becomes necessary. There are two groups of optical fibers. The first one, multimode optical fibers (MMF), transfers light through multiple spatial or transversal modes. Each mode, defined through a specified combination of electrical and magnetic field components, occupies a different cross-sectional area

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41

of the optical fiber core, and takes a slightly distinguished path along the optical fiber. The difference in mode path lengths causes a difference in arrival times at the receiving point. This phenomenon is known as multimode dispersion and causes signal distortion and limitations in fiber transmission bandwidth. A special group of multimode fibers, called few-mode fibers (FMF), became very attractive recently as a good candidate to increase the total transmission capacity and to enhance the total networking capabilities by applying the optical MIMO (multiple-input multiple-output) technique for signal transmission. This technique will be discussed in more detail in Chapter 4. The second group of optical fibers effectively eliminates multimode dispersion by limiting the number of modes to just one through a much smaller core diameter and appropriate cross-sectional refractive index profile. These fibers, called single-mode optical fibers (SMF), do introduce another signal impairment known as chromatic dispersion. Chromatic dispersion is caused by a difference in velocities among different spectral components within the same pulse. There are several methods to minimize chromatic dispersion at a specified wavelength, involving a special design of single-mode optical fibers, or the utilization of different dispersion compensation methods. Another dispersion phenomenon, known as polarization mode dispersion, is potentially a serious impairment in single-mode optical fibers, especially at very high optical signal bit rates. If the optical power propagating through the optical fiber is high enough, the impacts of nonlinear effects should be taken into account. There are several flavors of nonlinear effects (self-phase modulation, cross-phase modulation, fourwave mixing, Brillouin scattering, Raman scattering), and they can contribute to signal attenuation, signal dispersion, or signal crosstalk. Each of these three outcomes is undesired one. However, in some cases, nonlinear effects can be utilized favorably (such as for Raman amplification or chromatic dispersion suppression). Optical amplifiers serve to amplify weak incoming optical signals through the process of stimulated emission. Optical amplifiers can be considered as a part of any longer optical fiber transmission line since they are periodically inserted along the lightpath. Also, amplifiers can be positioned after the optical source to boost the output signal level, before the photodetector to increase the receiver sensitivity, or at any other specific place to compensate for the signal loss occurred before that point. Optical amplifiers should provide enough gain to amplify a specified number of optical channels. There are several types of optical amplifiers currently in use, such as semiconductor optical amplifiers (SOA), erbium doped fiber amplifiers (EDFA), or Raman amplifiers. Amplifier parameters important from a system perspective are the total gain, the gain flatness over amplification bandwidth, output optical power, the total signal bandwidth, and noise power. The noise generated in an optical amplifier occurs due to a spontaneous emission process that is not correlated with the signal. All amplifiers degrade the signal-to-noise

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ratio (SNR) of the output signal because of amplified spontaneous emission (ASE) that adds itself to the signal during the amplification process. The SNR degradation is measured by the noise figure. Photodiodes are key elements of any transmission line since they convert an incoming optical signal back to the electrical level through a process just opposite to one that takes place in lasers. There are two types of photodiodes commonly used: PIN and avalanche photodiodes (APD). The process within the PIN photodiodes is characterized by quantum efficiency, which is the probability that each photon will generate an electron-hole pair. As for the avalanche photodiode, each primary electron-hole pair is accelerated in a strong electric field, which can cause the generation of several secondary electron-hole pairs through the effect of impact ionization. This process is random in nature and avalanche-like. During the photodetection process, several kinds of noise (thermal, quantum, shot) are generated and mixed with the signal, and additional measures are necessary to suppress their impact. Optical filters, optical switches, optical multiplexers, and wavelength convertors are essential components in any optical networking scenario and their characteristics will be also described in more detail in this chapter. There are also optical components used to either enhance the signal quality by preventing detrimental effects, or to perform a specific function related to the optical signal processing. Such optical components are optical couplers, optical isolators, optical circulators, and variable optical attenuators and they will be described as well. Although there is a specific set of parameters for each of optical components listed above that should be taken into consideration from system and networking perspective, some parameters, such as insertion loss and sensitivity to optical signal polarization state, are common for all of them.

2.2 OPTICAL FIBERS Optical fibers serve as a foundation of an optical transmission system since they transport optical signals from the source to destination. The combination of low signal loss and extremely wide transmission bandwidth allows high-speed optical signals to be transmitted over long distances before regeneration becomes necessary. In a generic representation, optical fiber is a cylindrical structure with two distinct waveguide regions, known a fiber core and fiber cladding, surrounded by a protective buffer cladding as illustrated in Figure 2.2. The waveguide property of an optical fiber is enabled by keeping the difference in the refractive indexes nco and ncl between core and cladding. However, the waveguide structure of optical fibers can be for more complex that the one illustrated in Figure 2.2, since fiber core and cladding by themselves can contain several discrete waveguide layers along the radius r. In some cases, the refractive index in the core can have a graded profile nco(r) with the maximum in the core center (r = 0) and a gradual descent to the value of ncl with r = a (a is the radius of the fiber core).

Optical Components and Modules

Fiber core

Fiber cladding

Protective cladding

43

nco(r)

r

ncl

a

b

Figure 2.2 Optical fiber structure: (a) physical structure, and (b) refractive index distribution.

A low-loss optical fiber is manufactured from several different materials. The base row material is ultrapure silica, which is mixed with different additives, or dopants, to adjust the refractive index in the optical fiber core and to influence the propagation characteristics of the fiber. The optical fiber waveguide structure is protected by buffer coating before any cabling process. 2.2.1 Optical Fibers Manufacturing and Cabling Optical fiber is produced from a cylindrical preform, as illustrated in Figure 2.3. The preform has the same structure as the fiber, with fiber core and fiber cladding distinguished from each other [23, 24]. The base of the preform is heated in a specially shaped furnace at temperature usually higher than 2000K. The fiber drawing can begin when preform starts melting. The fiber diameter is continuously monitored and controlled by an automatic servo process during the drawing process. The control is done through adjusting the winding rate of the drum, in such a way that it permits just small diameter variations (usually as small as 0.1%). The protective coating is also inserted during the drawing process to provide mechanical protection and isolation of the fiber waveguide structure. Finally, the protected fiber is drawn to the receiving rotation spool. The overall diameter of the manufactured fiber can vary from several hundreds of microns to close to 1 mm, which depends on the thickness of the protective coating. It usually takes several hours to transfer preform rod, which is usually 1 meter long and 2 cm in diameter, into an optical fiber with the length of about 5 km. The shape of the optical fiber preform should be cylindrical to prevent the occurrence of polarization mode dispersion (see Chapter 3). There are several established methods for perform preparation, such as vapor axial deposition (VAD), outer vapor deposition (OVD), modified chemical vapor deposition (MCVD), the sol-gel method, and the plasma process [24, 26]. The three of them (VAD, OVD, and MCVD) are widely used for massive production of optical fibers. The main purpose of the preform manufacturing is to provide a homogenous radial distribution of dopants that change the refractive index profile.

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All dopants, such as germanium (Ge), boron (B), phosphorus (P), and fluoride (F), are added in vapor phase together with silicon, which serves as a base material. The chemicals added are not pure chemical elements, but rather a mix of chlorines and oxides that are eventually merged with the oxygen. Preform feed mechanism

Oxygen O2

POCl3 Oxygen O2

Flow meters

Preform

Furnace Gas

Silica tube

Bare fiber

Cl2 , O2

Diameter monitor

GeCl4

Coating applicator

Oxygen O2 Multiburner torch

Torch movement SiF4, BCl3

Automatic control

SiCl4

Coated fiber

Strength monitoring Takeup drum

(a)

(b)

Figure 2.3 Optical fiber manufacturing: (a) preform preparation, and (b) fiber drawing.

The chemical interaction takes place under the high temperature torch and leads to creation of numerous layers. The chemical interaction and deposition of the layers take place in an organized and controlled manner. The refractive index of a specific layer is controlled by changing the content of dopants that undergo the chemical interaction. An even and radial character of deposition is achieved by the rotation of the tube and by the movement of the torch. The modified chemical vapor deposition process (MCVD) is characterized by a deposition that is done on the inner surface of the rotating silica tube, while a sliding torch is used to maintain a very high temperature, as illustrated in Figure 2.3(a). The deposited material will eventually form the core of an optical preform that has a prescribed refractive index distribution. The temperature of the multiburner torch is raised after the deposition of the all layers, and that causes tube to collapse around deposited structure and form a solid rode, also known as the fiber preform. The tube used for deposition serves as a preform cladding. A slightly different approach is taken during fiber preform production by using the VAD process. The VAD process is characterized by a frontal deposition of chemicals and a vertical growth of the perform rod. However, the end result of any manufacturing process is the fiber preform as a macroscopic version of the optical fiber structure.

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The manufactured optical fibers should be incorporated in some type of cable structures before any practical application takes place. The cabling is necessary to protected fibers from possible damages during transportation and the installation process. In addition, cabling structure provides more stable environment for optical fibers during their lifetime. Optical cable design is generally different for different applications, and it may vary from a simple light-duty structure including just a plastic jacket around the fiber, to very robust mechanical structures that contains some strengthening elements. The light duty optical cables can be designed with either a tight jacket or a loose polyethylene tube, as shown in Figure 2.4(a). A tight jacket, with thickness up to 1 mm, is applied directly to the optical fiber primary coating. The tight jacket puts some internal pressure to the optical fiber structure, which generally leads to an increase in microbending losses. The loose tube design prevents tight mechanical contacts between fiber and protective jacket, which nearly eliminates microbending losses. Several elementary cable structures can be composed together and wrapped inside an additional tube, which effectively produces a multifiber cable. Polyethylene Optical fiber

Steel wire

Polyethylene Tight tube

Optical fibers

Polyurethane

Fiberglass Kevlar

Polyurethane

Loose tube

Heavy duty tube

(a)

(b)

(c)

Figure 2.4 Optical cables: (a) light-duty cables, (b) heavy-duty cables, and (c) multifiber cable.

A more robust optical fiber structure is usually needed for different outdoor applications. Such cables can be buried directly in the ground, pulled into underground or ducts between two buildings, buried under water, or installed at the outdoor poles. Although each application requires a specific cable design, there are requirements that they should be installed by the same type of equipment and installation technique that is used for installation of other conventional cables. One of the most important mechanical properties of heavy-duty optical cables is the maximum allowable axial load. Optical fibers are not strong enough to be load-bearing elements, as opposed to cooper wires in conventional cables. In addition, optical fibers cannot sustain any serious elongation without an irreparable damage. The endurable value before the fiber breaks is usually about 0.5%. However, the elongation that might happen during the cable manufacturing and installation should be limited to up to about 0.1% to prevent any damage.

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To reinforce the strength of optical cables, either steel or fiberglass wires can be used, as shown in Figure 2.4(b). The fiberglass rods embedded in polyurethane are preferred for some applications to reduce the overall optical cable weight and to avoid the effect of electromagnetic induction that might occur if steel wires were used. Nonmetallic protection structure might also include a high-tensile strength Kevlar jacket and the polyethylene jacket. Any construction of heavyduty optical cables should provide a room for fiber to move when the cable is bended or stretched. Fiber ribbons with fibers placed between two polyester tapes are sometimes used to pack a large number of optical fibers within a single optical cable, as shown in Figure 2.4(c). Although the number of fibers per ribbon can vary, usually 12 fibers are placed between the tapes. The number of ribbons stacked together can also vary, but the best mechanical stability is achieved by a rectangular array shape. Optical cables should have an outer sheath to protect fibers inside from impact loads. It is because an optical fiber has a low tolerance to absorb the energy due to impacts. The outer sheath should also provide protection from transversal forces and should be resistant to corrosion. The outer sheath is usually made from polyethylene, although a metal sleeve can be sometimes used. In addition, the construction of optical cables should prevent eventual water intrusion into cable structure, which means that special filler should be put in any empty space all along the cable. Both optical fibers and cables are manufactured with some nominal length (usually up to 10 km long), which means that longer optical fiber link is formed by joining several nominal lengths together [27]. In addition, there is a need to link an optical fiber with optical transmitters, receivers, and amplifiers to establish a lightpath. Optical fusion splices are commonly used to joint optical fiber nominal lengths, while optical connectors are used to link the fiber with transmitters, receivers, and optical amplifiers. Optical fiber ends must be properly prepared, cleaned, and aligned in order to minimize the insertion losses. 2.2.2 Special Optical Fibers Although the silica-based fiber structure from Figure 2.2 is the most commonly used one, there are some fiber types that could be called special ones, since they use either unconventional waveguide structure or some material other than silica to build on the core/cladding structure. Herewith we will pay attention to photonic crystal fibers that have more complex waveguide structure, and to plastic optical fibers that use plastic materials during manufacturing process. 2.2.2.1 Photonic Crystal Fibers Photonic crystal fibers (PCF) are not created from a bulk material, but rather from a periodic structure that involves some hollows inside [28]. These fibers were introduced in 1996 and the purpose was to demonstrate some new effects in terms

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of fiber dispersion, nonlinearities, and so forth. As such, they can be utilized for a variety of functions, such as optical signal amplification and wavelength conversion by nonlinear effects and compensation of chromatic dispersion in regular optical fibers. The best example of PCF is holey optical fibers, in which the glass material is manufactured to contain a specific pattern of holes, as illustrated in Figure 2.5. As we can see, the cross-sectional area has a symmetric pattern containing a number of the holes that may have the same or different sizes. They can be produced if a number of glass tubes is bundled together to form a preform before fiber drawing by a standard process described earlier. It is clear, though, that this process needs to be very precise.

(a)

(b)

Figure 2.5 Design of holey optical fibers: (a) index guiding structure, and (b) photonic bandgap structure.

The operation of photonic crystal fibers can be based on index guiding or on photonic bandgap physical principles, although these two principles can be mixed in a hybrid version. The index guiding structure has a periodical two-dimensional cross-sectional pattern (x-y coordinates). The central part does not have any hole. This structure is similar to the one presented in Figure 2.2(b), with a difference that there is a number of air holes in the cladding region. The core region is part of the overall structure, and we put a dashed circle in Figure 2.2(a) to indicate the comparison with a conventional design. Although the exact analysis of the waveguide properties is rather complex, we can assume a simplistic picture just to illustrate guiding principles. Namely, the index in the cladding can be calculated as an average value between the refractive index in the glass and the refractive index in the air (which is unity). Therefore, by increasing the number of holes, the refractive index in the cladding will be decreasing, thus permitting a strong guidance of the optical signal through the overall structure. It is important to mention that holes can be filled with a material other than the air. It could be different gases or liquid crystals, with a distinct impact to guiding properties. There is freedom for designers to create various patterns that will affect the guiding structure of the index guided photonic crystal fibers. The well-known

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fiber from this category is the Corning ClearCurve fiber, where holes are positioned as a ring in the cladding area. Since these holes have diameter of several hundred nanometers, the waveguide design is sometimes referred as nanostructure. The major advantage of nanostructures is that they can be bent much more tightly than ordinary optical fibers and can be effectively used for indoor applications. If fiber core is defined by the mini glass tube positioned at the very center, as in Figure 2.5(b), than the inside of that tube acts as a core that support light. Since the air inside has a lower refractive index than the surrounding glass area, the waveguide principle is not based on total internal reflection, as in conventional representation of the waveguide process. However, the periodic structure of the holes creates a photonic bandgap that identifies the spectral band or range of the wavelengths for which the propagation will be suppressed. This is the same principle as one used in Bragg gratings, which will be described later on in this chapter. The point is that periodicity in the gaps presents a periodic variation in the refractive index profile that enables destructive interference of certain wavelengths that try to spread around in the radial direction. That periodical variation should be designed to have a bandgap just for wavelength band that would be guided through the fibers (since it cannot escape the fiber structure). Therefore, that wavelength band is confined in the fiber core, or the central tube, and very small area around it. The term photonic bandgap is invoked in analogy with the electronic bandgap in semiconductor structure [3]. 2.2.2.2 Plastic Optical Fibers In these fibers, a plastic material is used instead of silica glass as a base for waveguide structure. Plastic optical fibers are used as an alternative to different copper cables for short distances (mainly for in-house or in-office connections), since they have wider transmission bandwidth than copper cables. Also they are less sensitive to mechanical degradation over the time as compared to glass optical fibers, while maintenance is much simpler. The total diameter (core plus cladding) of the plastic fibers is approximately 10 times larger than the total diameter of conventional optical fiber structure presented in Figure 2.2. The overall diameter is in the 1mm range, but the fiber core diameter usually accounts for more than 95% of that value. The plastic material often used is polymethyl methacrylate (PPMA), having a step-index profile with the refractive index in the core nco=1.49, and the refractive index in the cladding ncl=1.42. The B·L product of the fiber bandwidth and distance, as defined by Equation (1.7), for plastic optical fibers is around 10 MHz·km. Transmission over polymethyl methacrylate-based plastic optical fibers is done by using inexpensive light sources from the visible spectrum (around 650 nm), since it falls to the attenuation minimum of their attenuation curve. There is another type of plastic optical fiber, known as the perflourinated graded index fiber (POF) that can support transmission at the wavelengths around

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850 nm, so affordable laser sources, such as VCSELs, can be used for high-speed in-home or in-office networking. To support higher bit rates, the fiber core is reduced. There is always a trade-off between the transmission bandwidth of a specific design of plastic optical fibers and the practicality of their application, since a lower diameter means more difficult maintenance and less resistance to mechanical failures. 2.2.3 Optical Fiber Types with Respect to Transmission Properties As we already mentioned, optical fiber is a cylindrical waveguide structure that operates at optical frequencies. It confines the electromagnetic energy within the waveguide structure and guides it along its axis. The transmission properties are dictated by the waveguide structure and by the properties of the material used for fiber manufacturing. Generally, the propagation of the light along the axis can be described through set of guided electromagnetic waves, which are commonly known as guided modes. Only a certain number of discrete modes are allowed to propagate along the optical fiber axis. If just a fundamental mode exists and propagates in an optical fiber, these fibers can be identified as single-mode ones. If waveguide structure of the fiber allows the existence of a number of guided modes, that fiber is recognized as a multimode one. The properties of the guided modes in the optical fibers will be discussed in details in Chapter 3 by using waveguide theory description, but in this section we will use the geometric optics approach just to explain the overall picture related to guided modes. Readers can consult references [1–21] for more detailed information or an explanation of the physics, optical principles, and mathematical methods that have been used. The geometric structure with a refractive index distribution for common types of optical fibers is illustrated in Figure 2.6. In the first case, shown in in Figure 2.6(a), the refractive index is uniform throughout the core region and has a value nco, but undergoes the abrupt change and decreases to a value ncl in the cladding region. This type is known as step-index multimode fiber (SI-MMF), because of its step-index profile and capability to support a multiple of guided modes. The core diameter of SI-MMF silica-based optical fibers is 50 m, while the diameter of the fiber cladding is 125 m. (We should remember that the cladding is surrounded by protective plastic coat with a diameter usually more than 200 m.) In the second type of multimode optical fibers, known as graded index multimode optical fibers (GI-MMF), the refractive index is arranged to vary as a function of the core radius r. This was done with the goal of enhancing the transmission properties of multimode optical fibers, which will be discussed below. Finally, the third fiber structure, presented in Figure 2.6(c), is known as a single-mode optical fiber (SMF). These fibers have a much smaller core diameter, and it could vary anywhere from 5 m to 15 m [18, 19]. The refractive index distribution is considered to have a step profile, although it became more complex for various types of SMF. A smaller core diameter and a smaller difference between nco and ncl values are the main reasons why just fundamental mode is

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supported. From an optical signal transmission perspective, in a general case, single-mode optical fibers can support a much higher transmission capacity than multimode ones. But it is also important to outline that novel spatial multiplexing techniques based on MIMO (multiple-input multiple-output) principle can greatly increase the overall transmission capacity of multimode fibers [29]. Step-index Multimode fiber

Graded-index Multimode fiber

Step-index Singlemode fiber

n(r)

n(r)

n(r)

nco

nco

ncl

ncl

nco ncl r

r

r

2a

2a

2a Cladding diameter

(a)

(b)

(c)

Figure 2.6 Types of optical fibers: (a) multimode with step-index profile, (b) multimode with gradedindex profile, and (c) single mode optical fiber.

2.2.3.1 Step-Index Optical Fibers The relative difference  between refractive indexes in core and cladding in stepindex fibers is approximately 1%, so we have that  = (nco - ncl )/ nco ≈ 0.01. The waveguide principle for step-index fibers is illustrated in Figure 2.7(a) by using geometrical representation for the incident light rays. A number of incident rays will find the way to continue its propagation through the optical fiber. The group of propagating rays that are confined to meridional planes (planes containing the z-axis) of the fiber are known as congregation of meridional rays. Meridional ray Refracted ray (unguided mode)

f

n0 = 1

q0

ncl

nco

q z-axis

(a)

Figure 2.7 Geometrical representations of: (a) meridional and (b) skew rays.

..

..

(b)

.

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The other propagating rays, skew rays, are not confined to a single plane, but instead follow the spatial zigzag path around the fiber axis, as shown in Figure 2.7(b). The skew rays are very sensitive to any bending of the fiber and can go out of the core region more easily than meridional rays. Although the exact analysis of the modes associated with meridional and skew rays will be performed in Chapter 3 by applying the waveguide theory, it is useful here to use geometric optics [1] on meridional rays just to make some estimation about the impact of modal propagation properties. If we apply the geometrical optics principles to the step-index multimode fiber structure from Figure 2.7, the following relation can be written for the rays entering the optical fiber through the air-glass interface:

q

q

(2.1)

where q0 and q are the angles of the incident ray with respect to z-axis in the air and in the fiber core, respectively, while n0 = 1 and nco are the corresponding refractive indexes. Relation (2.1) is just the expression of the Snell law applied to air-glass interference [1]. When the ray hits the core-cladding interference, it could be refracted again and continue propagation through the fiber cladding (the dash ray from Figure 2.7) or be reflected from the core-cladding boundary. The refraction at the core-cladding boundary can be described by applying the Snell law to that interference. The transition from refraction to reflection comes for an angle f = fc from Figure 2.7 that satisfies the relation n (2.2) sin fc  cl nco where the value fc is known as a critical angle. The total internal reflection at the core-cladding interference occurs if condition f ≥ fc is satisfied. By using the basic trigonometric relations, we can transform Equations (2.1) and (2.2) to the following equation

q

f

√

q

√

(2.3)

The product n0·sinq0 is known as numerical aperture, or simply NA, of the optical fiber, and defines the maximum acceptance angle for the rays that will be guided through the optical fiber core. Since the angle q0 that defines the numerical aperture is relatively small, we can assume that it is sinq0 ≈ qNA ≈ nco (2)1/2, where qNA now defines the maximum acceptance angle. It would be very beneficial to have an optical fiber that would have a large acceptance angle, which could be done by increasing the value of the relative index difference . By increasing the acceptance angle, more rays will be guided through the optical fiber. However, these rays will follow the paths with different lengths while traveling along the z-axis. The end result is that arrival times at the end of the optical fiber will also be different and dispersed around some mean

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value. The ray entering under the angle q0 = 0 will follow the z-axis and have the shortest path, while the ray entering under the angle q0 ≈ qNA will experience the longest path. The phenomenon just described is known as the modal dispersion in multimode fibers, and it is caused by behavior of different rays (or modes) that can propagate through the fiber core. The difference in the arrival times between the fastest and slowest rays, or the time delay of the slowest ray as compared to the fastest one, can be calculated as  SI 

nco c

2  L  L nco  L   c ncl  sin fc 

(2.4)

We can roughly assume that delay SI should be lower than the duration of the half of the bit period T of a digital signal that modulates the optical carrier. The reader should recall that it is T=1/B for digital signals that are transmitted through the fiber, where B is signal bit rate defined by Equation (1.5). Therefore, by making replacement SI < 1/(2B) in Equation (2.4), we can find the parameter BL=B·L, introduced in Equation (1.6) as a measure of total transmission capability, and it is c ncl (2.5) BL  2 2 nco The parameter BL for step-index multimode optical fibers is relatively small, and just good enough to permit transmission of signals with B = 100 Mb/s (fast Ethernet speeds) over distances of 1 km or so. 2.2.3.2 Graded-Index Optical Fibers In multimode optical fibers with a graded index profile in the core, the refractive index decreases continuously with the radial coordinate r from the center of the fiber until it reaches the value ncl at the core-cladding boundary, as shown in Figure 2.6(b). Most commonly, the function n(r) is represented in the form ( )

{

[ (

( ) ] )

(2.6)

where r is the core radius and  is the exponent that takes positive values. The propagation of the meridional rays in graded-index optical fiber is illustrated in Figure 2.8. Due to the gradual decrease in the refractive index, the trajectory of a ray is not a zigzag line anymore, but has a shape that resembles either the sine function form for a meridional plane trajectory, or the helical curve for skew rays. It is important to notice that trajectory curves gradually and reverses the direction even before reaching core-cladding boundary. Skew rays are basically concentrated within the tube with some inner radius r1 and some outer

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radius r2. Since the rays in graded index optical fibers follow complex spatial trajectories, we can use vector r shown in Figure 2.8 for the position characterization. Here, r is the local unity ray vector placed as a tangent to the trajectory.

n0 = 1

skew ray trajectory

ncl

#3

nco(r)

#1

#2

z axis

Figure 2.8 Ray trajectories in a graded-index optical fiber.

In graded-index multimode fibers, there is periodic convergence of the rays and equalization of their paths. It is because the rays travelling along the longer geometrical trajectory (such as ray #1 from Figure 2.8) experience lower refractive index value, thus traveling faster than the rays with shorter paths that experience a larger refractive index value (such as ray #2 from Figure 2.8). The shortest path is attributed to the ray following the direction of the z-axis. However, this axis ray is also the slowest one since the refractive index value it faces is the highest one. It was found in [30] that the refractive index profile that minimizes the difference in arrival times of individual rays (modes) has a near inverse parabolic shape with parameter  = 2. The direction of the trajectories in graded-index optical fiber is determined by the direction of the refractive-index gradient [1]. The trajectories can be determined by solving the following differential equation d 2r 1 dnr   dr 2 nr  dr

(2.7)

where r and z are radial and axial coordinates, respectively. The difference in lengths between the longest and shortest trajectories can be found by solving Equation (2.7) as in [16]. It is clear that the solution of Equation (2.7) will depend on the refractive index profile or on the value of the parameter . The minimum difference among arrival times occurs for a nearly parabolic profile shape when = 2(1-) ≈ 2. Similar to Equation (2.4), valid for step-index multimode fibers, we can have the expression for the time difference in the arrival time applied to the graded-index case, and it is

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Lnco 2 (2.8) ncl Assuming again that GI < 1/2B, where B is signal bit rate, the parameter BL=B·L for this case becomes  GI 

4c (2.9) nco 2 By comparing Equations (2.5) and (2.9), we can conclude that, for typical values of nco ≈ 1.45 and ≈ 0.01, the total transmission capacity of graded-index multimode optical fibers is about 1,000 times larger than that of the step-index ones. Because of that, as we recall from Chapter 1, graded-index multimode optical fibers were widely employed in the first generation of optical transmission systems. It is important to outline that the graded-index profile is common for plastic optical fibers discussed in Section 2.2.2.2. BL 

2.2.4 Multicore and Few-Mode Optical Fibers Although the total transmission capacity of the optical fibers is huge, as illustrated in Figure 1.8, it has been realized that there are fundamental and practical limits that will ultimately determine the maximum transmission capacity. The fundamental limit is related to the Shannon formula (1.1) and optical fiber nonlinearities, while the practical limit is related to the bandwidth of optical amplifiers and the availability of the optical components operating in a specific wavelength range. The total transmission capacity can be ultimately increased by applying the space-domain multiplexing for signals propagating through optical fiber. The space-domain multiplexing is implemented by establishing independent diverse lightpaths in a single optical fiber, and it comes in two flavors: as space-division multiplexing and mode-division multiplexing [29, 31–33]. The space-division multiplexing method is done by using multicore optical fibers (MCF), where a number of cylindrical waveguides are positioned across the fiber cross-section. That picture is similar in nature to the picture of holey optical fibers from Figure 2.5(b), just assuming that the air holes are now filled with the glass having higher refractive index than the surrounding area, thus acting as fiber cores. The cylindrical glass cores can act as independent transmission channels if the signal leak among them is above some level that presents a critical impairment. The second way of doing space-division multiplexing is by using spatial modes in multimode optical fibers to achieve the diverse lightpaths. In fact, since the number of the modes that are used is rather limited, they are recognized as few-mode fibers (FMF). The major problem in both multicore and few-mode fibers is to prevent an excessive coupling between lightpaths that would diminish the transmission quality. In addition, a proper modulation (multiplexing),

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amplification, and detection (demultiplexing) of the spatial components should be achieved. Signal transmission over multiple diverse lightpaths in these optical fibers is achieved by applying the multiple-input multiple-output (MIMO) technique, which is well known and widely used in different wireless links [12]. There are two options when utilizing multicore fiber for signal transmission. The first one is to provide as low cross-talk among independent lightpaths as possible. The number of cores in this MCF structure is typically up to 10, with separation between cores of about 30–40 microns. The aggregating lens could provide a good beam separation at the output of the optical fiber [31]. It is possible to have cross-talk as low as –60 dB among the signals from different cores. Therefore, low cross-talk transmission over multicore fibers can be considered as a case of transmission over aggregated single mode fibers. This case is also very attractive from a networking perspective, which will be discussed in Chapter 8. Another case of transmission over multicore optical fibers is where the cores are used as spatial elements in order to excite spatial modes over entire crosssectional area. A different kind of spatial of modes can be excited at the entrance of the optical fiber. However, an effective usage of Laguerre-Gaussian (LG) modes, associated with skew rays from Figure 2.8(b) can be achieved in this case. An LG mode, with a helical wavefront, shows a phase singularity at the very center of the mode resulting in zero beam intensity, which is quite opposite to the modes associated with the rays traveling along meridional planes. The LG modes can carry the orbital angular momentum (OAM) due to rotating character of the wavefront vector. The superposition of LG modes can illuminate the overall crosssection of the multicore fiber. It was shown in [31] that the orbital angular momentum is preserved when several modes are overlapped and coupled with a number of cores in multicore fibers. The beam of the incoming signal should be adjusted by lenses to have the diameter be approximately equal to the diameter of the overall multicore structure, which could be between 80 and 120 microns. In general, multicore fibers operate similarly to imaging fiber, but here each core is single mode in nature and can carry high-speed signals. The few-mode multimode fibers are designed to support just several basic modes. It was done by adjusting the value of optical fiber parameters (core and the refractive index difference ). It is also possible to utilize few-mode fiber with specially designed refractive-index profile that will minimize group delay among basic modes. The refractive-index profile that practically eliminates the group delay among two basic modes was proposed in [34]. With the inclusion of several fundamental modes, each with two polarization states, four to eight independent lightpaths can be created. We can expect that the proper multiplexing technique will eventually suppress the cross-talk between individual modes to values less than –25 dB. Signal propagation in both multicore and few-mode optical fibers will be discussed in more detail in Chapter 3.

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2.3 THE LIGHT SOURCES The optical signal that propagates through an optical fiber was created in semiconductor light sources. The light generation process occurs in certain semiconductors when they are under direct bias voltage due to the recombination of electrons with holes at the n-p junction [35–42]. The reader is advised to refer to Chapter 10, Section 10.2 to learn more about physics of semiconductor junctions. The recombination process can be either spontaneous or stimulated. In spontaneous emission, photons are generated by the electron-hole recombination in a process that that has a random character, and there is no phase relationship between generated photons. Recombination process from energy perspective can be understood as a transition of the electron from an upper energy level that belongs to the conduction band to a lower level residing within the valence band. This process is accompanied by radiation of the photons. In a stimulated emission, the recombination process is initiated by another photon. A newly generated photon matches the original one in direction and phase. The nature of the recombination process determines the type of light sources. There are two major types of the semiconductor optical sources: semiconductor lasers and the light emitting diodes, and they will be described next. 2.3.1 Semiconductor Lasers The operation principle of semiconductor lasers can be understood by following the general scheme explained in Chapter 10. For this purpose, we will redraw the diagram from Figure 10.3 and show it in Figure 2.9, together with the basic structure of a semiconductor laser. The semiconductor laser is composed from ntype and p-type basic materials that form a p-n juncture as shown in Figure 2.9(a). Therefore, a laser is a monolithically integrated semiconductor device with sandwich-like p-n junction and terminal contacts. The light is generated by stimulated emission in the junction region, but propagates outside the junction region and out of the semiconductor’s structure. The sandwich-like structure is also a waveguide, since the relationship n1 > n2, and n1 > n3 is valid for the refractive indexes belonging to the junction and surrounding layers. In this case, the radiation is lateral through the edges of the semiconductor structure. The precondition for stimulated emission is that the inverse population between two energy levels in a composite sandwich-like structure is achieved. That means that higher energy level E2 is more populated than the lower energy level E1, which is quite opposite to regular situation in semiconductors. The difference E = E2 – E1 between the energy levels in conduction and valence bands, or the energy gap value, will determine the wavelength and the frequency of the optical radiation. It is expressed through the well-known equation h  E  E2  E1

(2.10)

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where h is Plank’s constant and  is the frequency of optical radiation. The output frequency is determined by the semiconductor energy gap value that is specific for each semiconductor structure. Both energy levels should also fall in between Fermi’s levels related to conduction and valence bands, respectively. E

Bias current

p-type

n3

n-type

n1 n2

E2

Conduction band - - - - Electrons - - - - -

E1 Holes

Fermi levels

+

The light beam to optical fiber

+ + + + +++ + + +++

Valence band Facet mirrors

(a)

k (b)

Figure 2.9 (a) Basic structure of a semiconductor laser, and (b) energy diagram and population inversion in semiconductors.

Semiconductor lasers that have been used for the second generation of optical communication systems and after that were developed by using a quaternary compound of semiconductors belonging to groups III and V. The widely used compound has been In1-xGaxAsyP1-y, which was obtained as multilayer by applying the epitaxial growth on InP substrate. The fractions x and y are not freely chosen since they should satisfy the ratio x/y = 0.45 in order to ensure the matching of the crystal lattice constant. The energy gap value E from Equation (2.1), expressed in electron-volts (eV) units, can be found as [42] ( )

(2.11)

where 0 ≤ y ≤ 1. The energy gap value and, consequently, the longest wavelength value of the radiated photons, are related to parameter y = 1, and they are: E = 1 eV, and  = 1,650 nm, respectively. The wavelength of the radiated photons, expressed in micrometers, can be calculated from the energy gap value, expressed in eV, by the following simple relation

 ≈ 1.24/E

(2.12)

There are also reflection coatings in laser structure that cover the semiconductor facets. These coatings act as mirrors to capture generated light within resonant cavity, so the light will make multiple paths between the mirrors. When the reflection coefficient of one of the facets is lower than 100%, a portion

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Output optical power

of the light will come out, while the rest will continue the back and forth oscillation. This device is well known as the Fabry-Perot (FP) type laser. The bias current flow through the laser p-n junction stimulates the recombination of electrons and holes that leads to the generation of photons. The stimulated emission of radiation occurs if the current is higher than a certain threshold value, since the inverse population takes place and the recombination occurs in an organized way, with a strong correlation in phase, frequency, and direction of radiated photons that form the output optical signal. The bias current threshold where the stimulated radiation process really starts, or where the amplification of the light within cavity picks up, can be easily recognized by a sharp slope increase at the functional curve presenting output light power versus bias current as illustrated in Figure 2.10a. This curve is otherwise known either as the “P–I”, or “L–I” curve (L stands for light), since it presents the functional dependence of the emitted light power P that leaves the facet from the current I. The optical radiation spectrum of FP lasers contains several distinct wavelength peaks placed under the spectral envelope, as shown in Figure 2.10(b). These peaks are known as longitudinal or spectral modes, and they are typical for any structure based on the Fabry-Perot resonator. The existence of several longitudinal modes is the reason why these lasers are well recognized as the multimode lasers (MML).

Spectral (longitudinal) modes

Spectral envelope

Temperature increase; Aging Bias current Threshold current (a)

Wavelength  (b)

Figure 2.10 (a) The P-I curve and (b) the output optical spectrum of Fabry-Perot semiconductor lasers.

The light radiation from semiconductor lasers can be are characterized through the rate equations describing the change in number of photons and electrons with time. The rate equations can be expressed as [14, 16] dP P  GP  Gnsp  dt p

(2.13)

dN N I  GP   dt e q

(2.14)

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where P is the radiated optical power measured by the number of photons involved in the process, N is the number of carriers (electrons) involved in the process, G is the net gain related to the stimulated emission, I is the bias current, p and e are the lifetimes of electrons and photons, q is the electron charge. The factor nsp is the spontaneous emission factor expressed through the electron populations N2 and N1 at the upper and lower energy levels, respectively, nsp = N2/(N2 – N1)

(2.15)

The electron lifetime e represents the time that an electron can spend at the upper metastable level before being recombined, while the photon lifetime p is related to the time the photon can spend in the laser resonant cavity. Equation (2.14) shows that the number of generated photons is directly proportional to the gain and inversely proportional to the photon lifetime. The gain and photon lifetime are dependent on the material structure of the resonant cavity, and can be expressed as G  vg g m

p 

1 vg cavity

(2.16)



1 vg ( int   mirror )

(2.17)

where vg is the group velocity of the light,  is the cavity confinement factor, gm is the material gain at the spectral mode frequency, and cavity is the resonant cavity loss, which consists of internal losses in the material int and the loss due to leaking through mirrors mirror. It was shown in [42] that the net rate of stimulated emission can also be expressed as a linear function of the number of electrons (

)

(2.18)

where GN = vgg/V and N0=NTV. Parameters g and V represent gain crosssection and active cavity volume, respectively, while NT is the transparency value of the carrier density. Typical values of these parameters for InGaAsP compound lasers are NT ~ [1.0 to 1.5]·1018 cm-3 and g ~ [2 to 3] ·10-16 cm2, respectively. The number of electrons is enhanced by direct bias current, while it is being depreciated by the recombination process. The deprecation is expressed by the first term on the right side of Equation (2.14). The rate of depreciation is faster if the lifetime is smaller. Equations (2.13) and (2.1) can be solved either for the continuous wave (CW) regime in order to evaluate the P-I curve, or for a dynamic regime related to the optical signal modulation if it happens. A solution of the rate

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equations is commonly done numerically, often as a part of the modeling and simulation software package. There is one more rate equation that can be established to express the phase f of the output radiation. It is [42] df  chirp  1  G  dt 2  p

   

(2.19)

where chirp is an amplitude-phase coupling parameter which determines the ratio between the refractive index change and the gain change. It is defined as

 chirp 

dn / dN dG / dN

(2.20)

where n, N, and G are the refractive index in the laser cavity, the number of carriers, and the laser gain, respectively. The parameter chirp is also associated with the frequency chirp of the output optical signal. That is the reason why this parameter is often called simply the laser chirp factor. It takes the values ranging from 2 to 8 for different types of semiconductor lasers [43]. The chirp factor introduced by Equation (2.20) reflects the fact that any change in the carrier population N causes the change in the refractive index within the laser resonant cavity. The change in the refractive index means that some amount of phase modulation always accompanies intensity modulation of an optical signal. Equations (2.13), (2.14), and (2.19) can be generalized by including terms that represent the noise impact, so they become dP P  GP  Gnsp   FP ( t ) dt p dN N I  GP    FN (t ) dt e q

df  chirp  1  G  dt 2  p

   Ff (t )  

(2.21) (2.22) (2.23)

where FP(t), FN(t), and Ff(t) are Langevin forces related to fluctuations of the intensity, the number of carriers, and the phase of output optical radiation, respectively [43]. It is often assumed that the Langevin forces are Gaussian random processes. The noise characteristics in semiconductor lasers by using Langevin forces will be studied in Chapter 4. The P-I curve, presented in Figure 2.10(a) is essential for understanding of the radiation characteristics of the semiconductor lasers. It is difficult to find two laser chips that would have identical P-I curves, even if they belong to the same

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group and have exactly the same structure. This curve is temperature-dependent since both the slope and the value of the threshold current change with the temperature, as indicated in Figure 2.10. The slope of the curve gradually decreases, while the value of the threshold current gradually increases with the temperature increase. The threshold current Ith dependence from temperature Θ can be expressed as ( )

(

)

(2.24)

where I0 is a current constant, while is the temperature constant that is material specific, and varies in the range from 50K to around 70K for InGaAsP-based semiconductor lasers. A similar effect can be observed with the chip aging, with slope going down, and with the threshold current going up. The effects of the temperature and the aging need to compensated by establishing the temperature control and output power monitoring loops. The threshold current can be calculated from Equations (2.13) and (2.14) by assuming that there is a continuous wave operation and that spontaneous emission can be neglected (nsp=0). By eliminating all time derivatives from these equations, we can assume that the threshold is reached for the current value where Gp = 1, which leads to [14] q 1  (2.25) I th   N 0    e  GN p  The number of photons P, when the current is above the threshold value, can be expressed as

P

e q

I  Ith 

(2.26)

The total optical power Plas emitted by the laser can be calculated by multiplying the number of photons by photon energy and the rate at which they escape the resonant cavity through two mirrors. So, we have that Plas= P·h·vgmirror)/2

(2.27)

The factor (1/2) appears since there are two mirrors and escape through just one is counted. Now, by using the Equations (2.17), (2.25), and (2.26), the following equation that will describe the P-I curve from Figure 2.10(a) can be established

Plas 

h int mirror 2q  cavity

(2.28)

Please note that the factor has been introduced in Equation (2.28) to indicate the internal quantum efficiency, or the ratio of the total number of the

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photons and the total number of the injected electrons. We can assume that this ratio is very close to 1 if the current is higher than the threshold value. The slope of the P-I curve when the current is higher than the threshold value is therefore dPlas h int mirror  (2.29)   slope    dI 2q   cavity  where efficiency (or P-I curve slope) that determines the slope of the output power increase with the injection current increase. In addition, one more quantum efficiency parameter, this time the external quantum efficiency can be introduced as

ext 

2 Plas / h 2q   slope I/q h

 I th  1  I 

(2.30)

In a general case, as we see from Equation (2.30), the external quantum efficiency is smaller than the internal quantum efficiency due to internal cavity loss. 2.3.1.1 Spectral (Longitudinal Modes) in Semiconductor Lasers We mentioned that the laser cavity is a Fabry-Perot resonator with two cleaved facets that act as mirrors. The facet reflectivity rfac can be calculated as 2

rfac

 n  n   n 1    1 0    1   n1  n0   n1  1 

2

(2.31)

where n1 and n0=1 are refractive indices in the resonant cavity and in the outside air, respectively. For III-V semiconductor compound, it is n1 ≈ 3.5, which produces reflectivity 0.31. The loss of the radiation due to internal material absorption and outflow through the facets is compensated by stimulated emission and amplification process during the round trip through Fabry-Perot resonator. The constructive interference of the generated electromagnetic waves will produce laser modes. A laser mode is characterized by its propagation constant  [1]. When a stimulated emission is achieved, one round trip of the mode through the resonant cavity is characterized by the following balance equation (

)

(

)

(

)

(2.32)

where E0 is the mode field amplitude, Lcav is the resonator length, r1 and r2 are reflectivity of two facets, g and int are the power gain at the spectral mode frequency and the loss in material. Factor 2 in the imaginary part of the Equation

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(2.32) accounts for a round trip of the wave. The Equation (2.32) can be converted to two equalities with respect to amplitude and phase, which leads to { [

]}

(2.33)

The phase portion of Equation (2.33) points to the fact that the resonant condition and constructive interference is reached at the set of frequencies νm determined by an integer m. These frequencies define the spectral modes in a Fabry-Perot laser. These modes are also known as longitudinal modes since their frequency is directly related to the length of the resonator. The spacing FP between spectral modes is constant and equal to the free spectral range of any other Fabry-Perot resonator—please refer to Equation (2.48). A typical value for it is around 150 GHz for a cavity length of 250 microns. We should notice that from Equations (2.32) and (2.33) is not the same as material gain gm from Equation (2.16). Since the optical mode exists beyond the active layer, while the gain exists only inside the active layer, it is =gm

(2.34)

where  (typically  < 0.4) is confinement factor defined by Equation (2.16). Several longitudinal modes can be emitted simultaneously under the spectral envelope, as illustrated in Figure 2.10(b). The point is that the envelope is determined by the gain spectrum g = g() from Equation (2.33). That spectrum is approximately 10 THz, so it can cover almost 100 longitudinal modes. The dominant mode is aligned in frequency with the gain peak. The majority of the signal power is carried out by the dominant mode and its two neighbors from the right and the left. From a transmission perspective, the multimode nature of the Fabry-Perot laser is not favorable since modes have different propagation speeds through optical fiber and that limits the total transmission capacity, which will be analyzed in detail in Chapter 3. Some additional elements in the semiconductor laser structure are needed to effectively select just one mode from the Fabry-Perot spectrum, and suppress the rest of them. This also contributes to the improvement of the laser modulation speed, thus making them more suitable for high-speed applications. The improvement is usually done by insertion of the optical filtering element in the Fabry-Perot structure. That filtering element is usually grating-type element that is placed either in the resonant cavity or outside of the cavity, as shown in Figure 2.11. If the selective grating is placed within the cavity, lasers are referred as the distributed feedback (DFB) ones and have a structure shown in Figure 2.11(a). The lasers with an outside Bragg grating, or gratings placed instead of the facet mirrors, are referred as distributed Bragg reflector (DBR) lasers, as shown in Figure 2.11(b). The distributed feedback in DFB lasers occurs due to Bragg diffraction phenomenon [1] that couples electromagnetic waves propagating in

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forward and in backward directions. The coupling happens only if the Bragg condition is met, which means only for wavelength DFB that satisfies condition

DFB = 2nmo/m

(2.35)

where  is grating period, m is the order of Bragg grating, and nmo is the mode index (nmo=  ) which is related to the mode propagation constant . The coupling between forward and backward waves, which enable positive feedback and stimulated emission around wavelength DFB, is the strongest for m = 1. As an example, the grating that supports wavelength at  ≈ 1,550 nm has period equal 235 nm. As for the Bragg grating in the laser structure, the grating period relates to the periodic variations in the refractive index that are commonly imprinted in semiconductor material. Bias current Bragg grating +

Bragg gratings +

The light beam to optical fiber

The light beam to optical fiber

p-type

p-type

n-type

n-type

Bias current

Active region Facet mirrors (a)

(b)

Waveguide ridge Silica layer

p-type

Silica layer

p-type

Cavity

n+ type

(c)

Figure 2.11 (a) Structure of DFB lasers, (b) structure of DBR lasers, and (c) lateral view of ridgewaveguide laser.

Phase-shifted DFB lasers have the grating shifted by DFB/4 in the middle of the cavity to produce /2 phase shift of the coupled mode. This is very beneficial since it results in much larger mode gain. There is also a class of DFB lasers known as gain-coupled ones [37], where both the mode gain and mode index vary periodically along the resonator length. As for DBR lasers, gratings serve as mirrors that provide a reflectivity maximum at the wavelength DFB.

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The manufacturing process for DFB and DBR lasers is more complicated than that for Fabry-Perot lasers, since the grating is etched onto one of the layers surrounding the active layer (it is in p-type on Figure 2.11). Holographic methods are used to form the grating with a pitch of approximately 0.2 micron. The waveguide properties of the sandwich-like structures shown in Figure 2.11 is enhanced by confinement of the mode not just in the transversal direction, but in the lateral one as well. It can be done by either ridge-waveguide structure, as shown in Figure 2.11(c), or buried heterostructure [42]. By doing so, the crosssectional symmetry of the dominant mode is greatly improved from its original highly elliptical shape to more circular shape.

+

The light beam to optical fiber

p-type multilayer

Insulation Active region Insulation

n-type multilayer

Mode suppression ratio

Bias current

Spectral width 

contacts

(a)

Wavelength (b)

Figure 2.12 (a) VCSEL structure, and (b) single mode laser spectrum.

There is another type of semiconductor lasers known as the Vertical Cavity Surface Emitting Laser (VCSEL) [44], which is a monolithic multi-layer semiconductor device, as shown in Figure 2.12(a). This type is usually based on In-GaAs-P layers that act as Bragg reflectors, thus enabling the positive feedback and stimulated emission. The layers are very thin, with a layer width of about onefourth of a wavelength. The VCSEL devices emit a regular circular light beam, as opposed to the other semiconductor lasers mentioned above. This feature is much more convenient in terms of launching the light into an optical fiber. It is also important to outline that VCSEL exhibits extremely low threshold current, which could be even less than 1 mA. The VCSEL lasers have been effectively used for different applications over shorter distances. All three laser types mentioned above (DFB, DBR, and VCSEL) are known as single-mode lasers (SML) that are characterized by a distinguished single longitudinal mode in the optical spectrum, as shown in Figure 2.12(b). The point is that the grating optical filter suppressed all longitudinal modes except the dominant one. The side modes still exist, but their power is much lower than the power of the dominant one. The ratio between these powers is known as modesuppression ratio (MSR). The MSR should be 30dB or more to have a quality single-mode laser. The spectral curve of the single-mode laser can be represented by a Lorentzian line given as [43]

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g ( ) 

 2 (  0 ) 2  ( / 2) 2



(2.36)



where 0 is central optical frequency, and  represents the spectral linewidth. The function g() is normalized so that the total area under the curve equals 1. The spectral linewidth, defined as the full width at half maximum (FWHM) of the Lorentzian spectral line, can be expressed  



2 nspG 1   chirp



(2.37)

4 P0

where nsp is the spontaneous emission factor, G is the net rate of the stimulated emission, and chirp is the chirp-factor, all introduced by Equations (2.13) and (2.20), respectively. Parameter P0 presents the stationary value of the output optical power (measured by number of photons). It is important to notice that the linewidth  decreases with the optical power increase. In addition, the linewidth can be reduced by decreasing values of chirp factor and the rate of spontaneous emission. Parameter chirp can be reduced by using so-called MQW (multiquantumwell) laser design [45–52], while the rate of spontaneous emission can be reduced by increasing the resonant cavity length. Although with some special design the linewidth of DFB single-mode lasers can go down to several hundred kilohertz, the linewidth of most DFB lasers is in the range from 5 to 10 MHz for an output power level of approximately 10 mW. The frequency response of the semiconductor laser is largely defined by relaxation frequency, since the modulation response starts to decrease rapidly when modulation frequency exceeds the relaxation frequency value. The 3-dB modulation bandwidth, defined as the frequency at which modulation output is reduced by two times (or 3 dB) with respect to output corresponding continuous wave case, is given as [46] M 

R 3 2

1/ 2

 3G P    N2 b  4   p  

1/ 2

 3G ( I  I )    N b2 th  4 q  

(2.38)

where R is the relaxation frequency, GN is the gain coefficient of stimulated emission from Equation (2.18), Pb is the output power for bias current, p is the photon lifetime on excited energy level, and Ib and Ith define bias current and threshold current, respectively. It is worth to notice the square root dependence of the frequency response on the output power. The modulation bandwidth of the semiconductor lasers is very important when laser is directly modulated, which means that the total current is a sum of Ib and Im(t), where Im(t) is the time-changing modulation component. Although modulation bandwidth of DFB lasers can exceed 30 GHz, due to the laser chirp,

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the application is limited to bit rates up to 40 Gb/s and shorter distances. In addition, direct modulation is not suitable for sophisticated modulation formats. These formats are effectively produced by an external optical modulator fed by the output from the laser that is biased just by a DC voltage to produce a continuous wave radiation. 2.3.1.2 Tunable Lasers It is highly desirable from the application perspective to have lasers that can be frequency manageable in an organized manner. In such a way, the same physical device can be used for different optical channels within the DWDM system, just by changing the frequency (the wavelength) of the output optical signal. This feature is quite welcomed from operation and maintenance perspective. In addition, some other applications, such as optical packet switching, are feasible if the change in the output wavelength is very fast. The simplest way to change the output wavelength of a semiconductor laser is by changing its injection current. If the injection current changes, there is a change in carrier density that causes change in both the optical gain and the refractive index. Consequently, changes in the optical gain lead to the intensity modulation of the output optical signal, while changes in the refractive index lead to optical phase variations and a shift in instantaneous frequency from the steady-state value 0. The wavelength changes with a current with the pace which is typically around 0.02 nm/mA. The output wavelength can be tuned by changing the laser operating temperature that causes the refractive index change and optical phase variations. The temperature change has a rate of about 0.08 nm/K, but it is not advisable to tune the wavelength by a making temperature changes larger than 20 degrees since this will cause drop in the output signal level and will have a negative impact to the reliability and lifetime expectations. In a general case, the frequency of the emitted radiation can be changed by changing the optical length of the resonant cavity and by altering filtering conditions related to the facet reflections. The relative change  in the output frequency is proportional to the change in the optical length, which can be expressed by proportion 



~

K1Lcav K 2 nmo  Lcav nmo

(2.39)

where K1 and K2 are arbitrary constants that characterize this empiric approach, and Lcav and nmo are cavity length and mode refractive index introduced in Equations (2.32) and (2.35), respectively. There are several practical methods, which are used today to design a wavelength tunable semiconductor lasers. The first of them is related to deployment an external tunable filter between the mirrors. An external mirror is

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used to extend the resonant cavity and to find a room in which to put the tunable filter. Then, by changing the optical filter spectrum, either through the refractive index tuning or by temperature tuning, the output wavelength of the laser is changed. The wavelength tuning can also be done if a diffraction grating, such as that shown in Figure 2.13(a), is placed at the end of an external resonator. The Bragg reflection condition is changed by mechanical rotation of the grating, which changes the effective grating period and selects the lasing wavelengths. Mechanical tuning can enable a wide tuning range, but these lasers may be bulky, which limits their applications.

The light beam to optical fiber

Bias current

Active section

Bias current

Focusing lens

Phase-shift current

Tuning current

Phase-shift section

-selective section

Facet mirror

a)

b)

Figure 2.13 Tunable lasers: (a) external cavity laser, and (b) three-section tunable laser.

The same principle, based on adjustable Bragg grating, can be used in different monolithic DBR constructions. In this case there is no mechanical movement involved to change position of the Bragg mirror since the Bragg condition is changed by a designated current injection. To make this principle workable, a DBR laser is made as a multisection device, such as one shown in Figure 2.13(b). A number of sections can vary from two to four and more, [53– 59]. In the two-section device there is a pair of electrodes to carry injection currents, one pair for the regular active region, and the other for controlling Bragg reflection conditions through the refractive index change. The two-section structure prevents the output optical power from falling down, but it does not provide a wider continuous tunable range. In the three-section structure, presented in Figure 2.13(b), there is a third section inserted between the active region and Bragg grating section. This section is phase-shift section and serves to change the phase of the light entering into the Bragg grating section. The phase shift, achieved through independent current bias, helps to spread the continuous tunable range, which has been up to 10 nm. Any structure involving four or more sections can be considered as a cascade of two or more Fabry-Perot filters, each of them with different free spectral range (FSR)— please refer to Section 2.4. In a cascaded multisectional structure, Bragg mirror function occurs not just once but a number of times, which effectively provides a continuous tunable range of over 40 nm on a single monolithic chip [58]. The Bragg gratings from Figure 2.13(b) can be replaced by ring resonators [59]. The number of rings can be two or more and they effectively play the role of sections in a multisectional laser. The rings are waveguide structures that are

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monolithically integrated and do not cause any bulky structure. A thermo-optic heater can be built around ring resonators to have temperature tuning of the refractive index and the effective change of the output wavelength. Broadly tunable light sources can be also produced by using optical fiber loops with integrated optical filters in the loop. The fiber loops serve as an active medium that amplifies the light, while the filter helps to select a specified wavelength at the output side.

1 Tb/s

1 Tb/s

1 Tb/s

1 Tb/s

Output power

Tuning range

1 Tb/s

1 Tb/s

Wavelength

Figure 2.14 Typical spectrum of a tunable laser.

The current versions of wavelength selectable lasers used in DWDM systems are mainly based on multisectional/multiring schemes described above. The wavelength range covered by a single laser device varies from several nanometers to close to 200 nm, while the output power can be up to 20 mW [37]. The wavelength-selectable lasers operate in combination with external modulators, very often integrated with them and equipped with a wavelength locker that helps to preserve the values of the output wavelength after the tuning has been done. A typical spectrum of a tunable laser is shown in Figure 2.14. 2.3.2 Light Emitting Diodes Light emitting diode (LED) is a monolithically integrated semiconductor structure with p and n layers forming p-n junction that is directly polarized, as illustrated in Figure 2.15. The light is generated by spontaneous emission in the junction region, but propagates outside the junction region and out of the semiconductor’s structure. This simplest structure is known as surface radiation LED, shown in Figure 2.15(a). If there is at least one more semiconductor layer, as in Figure 2.15(b), the active region can be structured as a waveguide, since the relationship n1>n2, and n1>n3 is valid for the refractive indexes belonging to the junction and surrounding layers. In such a case, the radiation is lateral through the edges of the semiconductor structure. The output power of the LED is relatively low, usually up to 0.1 mW, for bias current exceeding 100 mA. Semiconductor material that was used from the very beginning to produce LED operating in 800–900 nm wavelength region was ternary alloy Ga1-xAlxAs (if x=0.08, the radiation peak is at 810 nm). The quaternary alloy In 1-xGaxAsyP1-y has

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been used for LED manufacturing if they operate in the region between 1,000 nm and 1,600 nm by varying the mole fractions x and y. Both of these compounds have been chosen because it has been possible to match crystal lattice parameters that results in higher quantum efficiency. The light beam to optical fiber

Bias current

Bias current The light beam to optical fiber

n3 n1 n2

p-type

p-type

n-type n-type contacts a)

b)

Figure 2.15 Light Emitting Diodes: (a) surface emission, and (b) edge emission.

Quantum efficiency of LED can be estimated by assuming that only spontaneous emission is present in Equations (2.13) and (2.14) applied earlier to the laser case. We can assume that in a steady state there is a balance between the injected carriers and those lost either through radiative or nonradiative recombination process. The rate of generated photons is equal to ( ) where is internal quantum efficiency introduced in Equation (2.28), while I/q is carrier injection rate as in Equation (2.14). If the external quantum efficiency is also introduced to express the fraction of photons escaping semiconductor structure and going to air, the total power radiated by LED can be expressed as [

(

)]

(2.40)

The external quantum efficiency is related to the reflection conditions at the semiconductor-air interface, and can be expressed as [14] 1 (2.41) ext  2 nn  1 where n is the refractive index in the semiconductor material. With n ≈ 3.5, Equation (2.41) produces the value of 1.4%, which means that only small amount of the optical energy finds its way out. Even that smaller value has a wide radiation angle so that only few percent of the radiated power can be coupled to an optical fiber. Therefore, even the total internal power produced by the electronhole recombination in the p-n junction can exceed 10 mW, only about 100 W can continue to propagate through an optical fiber, which is a 20-dB loss just at the very beginning of the transmission process. The slope efficiency of a LED can be expressed as

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dPLED h (2.42)  intiext dI q and it is typically around 0.01 mW/mA, which is up to 10 times lower than the quantum slope efficiency in laser diodes expressed by Equation (2.29). The slope expressed by Equation (2.42) is not constant, but depends both on the temperature and the current value, which means that curve PLED(I) is not linear, but bends down with the current and the temperature. The output LEC spectrum is dependent of the rate of spontaneous emission and can be expressed in a normalized form as [39]

 slope 

( )

[

√

]

(2.43)

where kB is the Boltzman constant, is the absolute temperature, and Eg is the energy bandgap between conduction and valence bands. The LED spectral curve has a Gaussian shape with a maximum at 0 and the spectral full-width at half maximum LED, given as

0 

E

g

 k B  / 2 h

;  

 2  2k  2k B  ;     B h  c  h

(2.44)

where the relationship =c/ is used in last portion of Equation (2.44). It can be easily calculated that the full width half maximum (FWHM) for light emitting diodes is ranging from 30 nm at wavelengths around 850 nm, to more than 60 nm if operated in the wavelength region above 1,300 nm. If LED is intensity modulated by some electrical signal using the PLED(I) functional curve, the frequency response can be easily found by using Equation (2.14) and dropping the last term related to stimulated emission. The frequency bandwidth of the LED is limited by large diffusion capacitance in the active region. The following relation connects the output optical power PLED with the frequency f of the modulation current



PLED ( f )  P0 1  (2 f e ) 2



1 / 2

(2.45)

where P0 is the output optical power for DC current, and e is the effective carrier (electron) lifetime ranging from 2 ns to 10 ns. The futures mentioned above limit the application of LED to lower speeds (up to 200 Mb/s bit rates) and lower distances (up to several tens of kilometers).

Parameter Output power

Table 2.1 Typical values of semiconductor sources parameters LED FP lasers DFB lasers Up to 150 W

Up to 10 mW

Up to 20 mW

VSCEL Up to 7 mW

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Advanced Optical Communication Systems and Networks Spectral width (FWHM) Modulation bandwidth

30-60 nm

1-3 nm

Up to 200 MHz

Up to 3 GHz

0.000001-0.0004 nm (0.12-50 MHz) Up to 35 GHz

0.1-1 nm Up to 20 GHz

In a summary, typical values of optical source parameters that we discussed in this section are summarized in Table 2.1. The numbers shown are typical values extracted from product-related literature, such as [54], or data sheets of different manufacturers. It is important to mention that the output powers for both FabryPerot and DFB lasers can be much higher if they are designed to serve as power pumps in optical amplifier schemes. In such a case, the power can be as high as 400–500 mW for FP lasers, and up to 300 mW for DFB lasers.

2.4 OPTICAL FILTERS AND MULTIPLEXERS The role of the optical filters and multiplexers has become very important in multichannel WDM transmission systems and the optical networking environment [60]. While the operation of an optical filter is based either on optical spectral interference or on absorption of a light signal, the operation of optical multiplexers is mainly based on interferometric principle [1]. From that perspective, optical multiplexers can be considered as a special class of optical filters. The spectral interference results in a vector sum of two or more electromagnetic waves that originate from the same source and travel along the paths with slightly different lengths. The difference in lengths will produce the difference in the phases, which will have an impact to the summation result after the waves are rejoined again. The interference effect implicates that there are at least two versions of the same optical signal that are combined together after passing different paths. There are two main interferometer schemes that are used for the design of optical filters. They are Fabry-Perot (FP) and Mach-Zehnder (MZ) interferometers. These names have been mentioned a number of times in this book in association with different optical elements (laser diodes, optical modulators), which means that the same physical phenomenon is a base for different optical components. 2.4.1 The Fabry-Perot Filters A Fabry-Perot filter, often known as the “etalon,” is a basic interferometer structure. It consists of resonant cavity established by two parallel mirrors, as shown in Figure 2.16. For the sake of illustration, we assumed that light is entering from incoming fiber and is collated to an outgoing fiber. Light enters the cavity through the outer side of Mirror 1, which is transparent to the incoming signal. After one pass through the cavity length L, part of the light signal leaves the cavity through Mirror 2, while a part is reflected from it and goes

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back to Mirror 1. Now Mirror 1 reflects that part, and so on. The light signal that entered the cavity is therefore bounced back and forth between mirrors. There will be the forward and backward propagating waves that will either contribute to each other or cancel each other, which depends on the characteristics of the resonator. Fabry-Perot resonator

(a)

Mirror 1

Mirror 2

Width L

(b)

FP filter transfer function

FSR 1

3

2

N

1

2

r ~ 0.80

…

r ~ 0.90

r ~ 0.99

1

FP

2

3

N

Optical frequency

Figure 2.16 (a) the Fabry-Perot filter interferometric structure and (b) its transfer function.

The resonator is tuned to its resonant position if the cavity length is adjusted to take the value L = i/2n (2.46) where n is the refractive index within the cavity, while i is an integer known as the filter order. A number of reflections contribute constructively to the filter response before the light signal decays and eventually vanishes following a zigzag path. Signal decay is caused by losses due to the exit through Mirror 2 or due to internal absorption within the cavity. A full constructive interference will no longer take a place if the resonator length is detuned from the value L that is given by Equation (2.4). Accordingly, the light output through Mirror 2 will be suppressed. The powers of optical signals that come to Mirror 1 and that exit through Mirror 2 will be related by each other by the transfer function [61] Pout ( )  Pin ( )

(1    r ) 2 (1  r ) 2  4r sin 2 2  

(2.47)

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where  is the optical frequency,  is a single-pass propagation time across the resonant cavity, r is the power reflectivity at each of two mirrors, and  is the optical power absorption coefficient. (It is r =1 t, where t is the transmission coefficient of the mirror [1].) The right side of Equation (2.47) defines a periodic function known as the Airy function [9]. The frequency at which function transmission peaks are periodically repeated is known as the free spectral rage (FSR), defined as FSR 

1 c  2 2nL

(2.48)

The Airy’s shape of transfer function of FP filters is illustrated in Figure 2.16 for several values of the mirror power reflectivity. The Fabry-Perot interferometric structure is also characterized by the filter finesse F, which is a parameter, defined as

F

FSR r   FP 1  r

(2.49)

where FP is related to the width of the transmission peaks, as shown in Figure 2.16. This width, which determines the filter bandwidth, is defined by the points at which the transfer function decreases to the half of the transmission peak value. The relation between the filter finesse and the mirror reflectivity in Equation (2.49) is obtained by assuming that the optical power absorption coefficient  can be neglected. If the FP filter is used for channel selection in an optical multichannel system, the free spectral range of the filter should be larger than the combined bandwidth of the multichannel signal, that is FSR S =N·B/SE (S is channel spacing, B is the bit rate and SE is the spectral efficiency defined by Equation (1.7)). The number of optical channels that can be effectively resolved in such multichannel environment is measured by the filter finesse. The simplest way to produce a high finesse FP optical filter is by cascading several stages. There are two approaches how to achieve the cascade of FP resonators. The first one is to employ several resonators as a simple chain, where the output signal from preceding resonator becomes the input signal to the following stage. In the second scheme, the optical signal from the output mirror (Mirror 2 from Figure 2.16) is reflected back to enter the cavity again. In such a scheme, light passes twice through the same cavity. The total number of signal passes through the cavity can be additionally increased if the output signal (this time from Mirror 1) is directed back to enter the cavity one more time. The effective FSR of the cascaded FP structure increases proportionally to the number of the light passes through the resonator, while the filter bandwidth decreases in proportion to the numbers of cavities that are cascaded. The cascaded structure of FP filters that are used in practical applications consists of several

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dielectric layers. Such a design does not need classical mirrors since the difference in refractive indexes between neighbor layers is used to reflect the light signals back and forth. 2.4.2 Mach-Zehnder Filter

Input 1

l

Optical coupler 1

Delay ~l

Input 2

Output 1 Optical coupler 2

Output 2 a

MZ filter transfer function

A Mach-Zehnder (MZ) filter operation is based on a Mach-Zehnder interferometer that consists of two directional optical couplers connected by optical waveguides that have different lengths, as shown in Figure 2.17(a). The waveguides can be either optical fibers or planar optical waveguides that are produced on the structure such as silica on silicon. In general, there is a difference l between twowaveguide arms, which causes the signal delay .

Optical frequency b

Figure 2.17 Mach-Zehnder interferometer: (a) interferometer scheme, and (b) transfer function shape.

Optical power is equally split between two arms at Coupler 1 in Figure 2.17(a), but the signal in one arm has a phase shift of /2 with respect to the other (we can assume that signal in lower arm is shifted in phase by /2). There is the following relationship between the power of the optical signals at the Outputs 1 and 2 and the optical signal entering the Input 1 in Figure 2.17(a) [39] Pout,1 ( )  Pin,1 ( ) sin 2   

Pout,2 ( )  Pin,1 ( ) cos    2

(2.50) (2.51)

where  is the frequency of the optical signal. Therefore, the transfer functions that correlate the powers at input and output ports are raised sine functions, which are out of phase with one another. The optical filter transfer function presented by Equation (2.51) has a shape that contains the main lobe and periodical arcades, as illustrated in Figure 2.17(b). That is the reason why the Mach-Zehnder filter belongs to class of periodic optical filters. Several Mach-Zehnder elementary structures can be cascaded to form a multistage Mach-Zehnder interferometer. The total path difference between the two portions of the signal that travel through the chain is calculated as a sum of differences associated to individual stages. If there are M interferometers in the chain, the signal from Output 2 can be expressed by the signal at the Input 1 as

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M

Pout,2 ( )  Pin,1 ( ) cos 2  i  

(2.52)

i 1

An optical filter that is based on the chain of MZ interferometers can be designed by adjusting the optical path parameters from Equation (2.52). Such a filter will be able to isolate one of N=2M – 1 optical channels, which are mutually separated by frequency spacing if the delay of the i-th interferometer in the chain is 1 (2.53)  i  i 2  As an example, the chain that consists of six stages is capable to distinguish one out of 64 channels, while the chain that consists of seven stages it will provide selection among 128 channels. 2.4.3 Optical Grating Filters Optical gratings have been used throughout the centuries to separate a composite light into its constituent wavelengths. In such a way, just a selected wavelength is allowed to come to the specific place, which produces a filtering effect. Optical grating filters found a widespread application in WDM systems to either separate or to combine individual wavelengths, thus acting as optical multiplexers or demultiplexers. All gratings can be divided into two groups: transmission and reflection ones. The operation principle of optical gratings is illustrated in Figure 2.18. Grating Period 

……………… qi

Grating plane

Imaging plane

1 2 3 (a)

…

N

1 2 3

Imaging plane

Si

qi

qd

N

…

Si

……………… Grating plane

(b)

Figure 2.18 Operational principle of filter gratings: (a) transmission grating, and (b) reflection grating.

The input composite signal is either refracted or reflected from the grating plane. The grating plane is produced from an optical material in such a way that periodic changes in geometrical structure are imprinted. The changes follow the specified pattern with multiple narrow slits separated by pitch, which is known as the grating period. The grating causes diffraction of the input signal, a well-known phenomenon in optics [1]. Diffracted light is spread in different directions and eventually produces an interferometric picture at a plane placed in parallel to the grating plane. Constractive interference of any particular wavelength is dependent on the position at the plane, which is a function of the diffraction angle of that

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specific wavelength. Any other wavelength will experience distractive interference at that specific place, which means that there is a spatial filtering of the input composite signal, as shown in Figure 2.18. The constructive interference at the wavelength i occurs at the imaging plane among the rays diffracted under the angle qd if the following condition is satisfied

[sin(qi) – sin(qd )]=mi

(2.54)

where m is an integer known as an order of the grating. The energy of a single wavelength is distributed over all discrete angles that satisfy condition (2.54). However, the output is collected just from one of the angles, while the remaining energy is lost. The most energy will be concentrated at the point where qi =qd and m=0, but wavelengths cannot be separated at this point and all energy is wasted. In order to move interference maximum to some other point defined by Equation (2.54), the reflection slits are inclined at a certain angle to the grating plane. The reflected energy will now have a maximum at the grating orders m associated with that angle. The majority of the gratings used today for optical networking applications are reflection gratings. The main reason for that is that they are much easier to fabricate. In addition to plane design as shown in Figure 2.18, these gratings are fabricated by using concave geometry since the concave design leads to more convenient placement of the other elements that come together with the grating, such as mirrors, lenses, fiber inputs, and so forth. Bragg gratings are special class of gratings in which periodic perturbation in the propagating medium serve as a grating. We already introduced the example of Bragg gratings in Section 2.3.1 when we discussed DFB laser structures. Bragg gratings can be imprinted in semiconductor structures, planar waveguides, or in the optical fiber core, thus serving as filters that select specified wavelengths. The grating condition in Bragg gratings is given by Equation (2.35), which can be rewritten as 2nmo] = mBragg

(2.55)

Where  is the grating period, m is the order of Bragg grating, and nmo is mode index (nmo=  ) associated with the propagation constant of the mode propagating through the medium. The coupling between forward and backward waves that enables constructive interference at wavelength Bragg is the strongest for m=1. It is clear that Equation (2.55) can be obtained from Equation (2.54) if we put that qi = /2 and qd = – /2, while assuming that the medium is not the air, but a specified medium with mode index higher than 1. Optical fiber Bragg gratings (FBG) are special optical filters that are imprinted in optical fibers [63]. Due to such a design, they are relatively cheap and easy for packaging and coupling with other optical fibers. These gratings are

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usually written in conventional germanium-doped silica fibers that are more photosensitive than other fiber types. The gratings are formed by changing the refractive index in the fiber core, which occurs under the impact of ultraviolet radiation. A permanent grating can be written if the fiber core is exposed to two interfering ultraviolet signals since the light intensity of resulting wave varies periodically along the fiber length. The grating is formed by an increase in the refractive index at the places where the resultant wave has maximums since the index stays unchanged at the other places. The increase in the refractive index is around 0.005% to 0.01%. There are two types of fiber Bragg gratings: short-period and long-period ones. In short-period gratings, the Bragg period is comparable to wavelengths of the propagating optical signal. The optical signal filtering due to the Bragg reflection of a specific wavelength occurs if the distance between the grating lines equals the wavelength value. The Bragg structure can be imprinted with parameter  gradually increasing/decreasing with the length. In such a case, not just a single wavelength but a waveband will be reflected from the imprinted structure, thus performing a passband optical filtering. The short-period fiber Bragg gratings have low insertion loss (usually lower than 0.1 dB), and relatively sharp transition between the passband and the rest of the spectrum. In addition, these fiber Bragg grating (FBG) filters have a flat top of the passband and low sensitivity to the polarization states of the incoming optical signal. The FBG filters are commonly used in chromatic dispersion compensation schemes, and for optical signal filtering in optical add-drop multiplexers. The other group of optical fiber gratings has the grating period that is much longer than the signal wavelength. The optical signal energy is not reflected back and forth as in short period fiber Bragg gratings, but it is rather coupled to the vanishing fiber cladding modes. These long-period fiber gratings are widely used to flatten spectral gain profile in a different kind of multichannel optical amplifiers, or as band-rejection filters, since their spectral profile can be precisely shaped. 2.4.4 Tunable Optical Filters Tunable optical filters are very useful optical components since they can dynamically select a specified range of optical wavelengths [62, 64]. It is desirable that tunable optical filters have wide tuning range, high filter finesse, and flat top of the transfer function peaks, and that they can be tuned as fast as possible. However, some of these general requirements may not be relevant for some applications. Both Mach-Zehnder (MZ) and Fabry-Perot (FP) optical filters structures are potentially tunable. Tunable MZ structure is achieved through an active control of the optical signal delay , which is done by changing the refractive index in the waveguide arms—please refer to Figure 2.16. The simplest way to control the refractive index is through thermo-optic effect by using the temperature change

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imposed by a thin-film heater around the arms. The refractive index also can be altered by an external electrical field applied to the waveguide, which is referred to as the electro-optical effect [1]. As a result, the resonant wavelengths that are associated with the transmission passband peaks can be dynamically shifted by changing the refractive index. The passband tuning of the FP filter can be achieved by either changing the cavity length, or by varying the refractive index within the cavity. Both methods lead to change in propagation delay  and to shift of the resonant wavelength away from its initial position. The simplest way to change the cavity length is by mechanical movement of one of the mirrors. It can be done, for example, by an all-fiber design that changes the air gap between two optical fibers. The two fibers face each other in an enclosed piezoelectric chamber [64]. The cavity length, which is the air space between two polished fiber ends that act as mirrors, is changed electronically through piezoelectric contraction. The finesse of the allfiber FP filters is higher than 100 and can be additionally increased by putting two filters in a cascade. The tuning range of such filters is up to 20 nm, while the tuning speed is relatively slow and can be more than 1 second. The tunable FP filters that use the refractive index change, rather than mechanical movement of the mirrors, are based on special materials, such as liquid crystals or semiconductors. The resonant cavity is formed by placing the some of these materials between two mirrors, while the tuning is done electronically by changing the refractive index of the material in the resonant cavity. The finesse of these filters can be over 300, while the tuning range can be over 50 nm, with a tuning time around 1 ms. The cascaded structure of FP filters that consists of several dielectric layers can be also tuned, either thermally or electronically, with the tuning range of up to 40 nm [61, 62, 64]. The acousto-optic tunable filters (AOTF) are very promising candidates for different applications related to the optical wavelength selection [65, 66]. The operation of the AOTF is also based on the Bragg reflection. The refractive index grating is imprinted by generating acoustical frequencies through the transducer, which is driven by an external RF signal, as shown in Figure 2.19. The acoustic frequencies form a standing wave, which determines the character of the refractive index change. The maximums in the refractive index coincide with the peaks of the acoustic wave, while the minimums coincide with the nodes of the standing wave. The acoustic transducer is applied to a waveguide structure based on a highly bi-refringent material. There are two polarization modes of the incoming signal, commonly referred as TM and TE modes, which “see” different refractive indexes during the propagation through the material. The coupling, or the energy exchange between TM and TE modes, occurs if the refractive indexes satisfy the Bragg condition nTM = nTE ± , whereis the wavelength of the optical signal,  is the period of the Bragg grating created by an acoustic wave, while nTM and nTE are the refractive indexes associated with TM and TE modes, respectively. In case when lithium-niobate is used for MZ structure, which is the most often case, the

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difference between nTM and nTE is ~ 0.07, which leads to  = [21.43—22.86] m for the period of the created Bragg grating, and for 1500 nm <  < 1,600 nm.

Input 1

Output 1

TE

Input 2

Input TE polarizer

TM+TE

Output TM polarizer

TM

TM

TE

Refractive index gratings

TM+TE

Output 2 Output polarizer

Figure 2.19 The scheme of an acousto-optic tunable filter.

Since AOTF is a special type of MZ filters the power from the Outputs 1 and 2 can be found by using Equations (2.50) and (2.51) and inserting parameters that are AOTF specific [90]. By doing so, we have that

 sin 2  / 2 1  4    2  2   0 20     Pout,1    Pin,1   ;    2 l nTM  nTE  1  4   0   2    

(2.56)

where l is the length of the interferometer arms where the acousto-optic effect is applied. The energy exchange in AOTF is unidirectional since the optical signal energy at the wavelength is being transferred from TE to TM mode. Therefore, this scheme also needs the TE-mode polarizer at the front of the filter, and the TM-mode polarizer at the filter’s end. The length of the interaction determines the bandwidth of the filter, and it is narrower if the length is longer. However, the tuning speed is decreased with an increase in the interaction length, which means that there are different designs that can be tailored for specific applications. The tuning range of the acousto-optic tunable filters is over 100 nm, while the tuning speed can be less than 10 s. It can be found from Equation (2.56) that 3-dB bandwidth of the AOTF is 3  2.4.5 Optical Multiplexers and Demultiplexers Optical multiplexers and demultiplexers are used to either combine several distinct wavelength channels into a composite signal or split a multichannel WDM signal into its channel constituents. In a general case, the same device can be used in both roles, and just the direction of the signal will determine what function is

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performed. There are several types of optical multiplexers that are commonly used today, and they based either on diffraction, or on the interference effect [67] [68]. We can say that diffraction-based demultiplexers are in essence Bragg filters that usually employ some angularly dispersive element as diffraction grating. The incident light signal is reflected from the grating and dispersed spatially into a number of wavelength components, which are then focused by some lens and introduced into individual optical fibers. The diffraction grating should be properly designed to create a wavelength-specific reflection angles, as explained in Section 2.4.1.3. MZ#31

1 5

MZ#21 MZ#32

1 o8

3 7

MZ#1

MZ#33

2 6

MZ#22

MZ#34

4 8

(a)

Waveguide arms 1 2 3 4

Coupler

Coupler (b)

1 2 3 4

Figure 2.20 Optical multiplexers: (a) Mach-Zehnder filter chain, and (b) Arrayed Waveguide Gratings (AWG).

The operation principle of an optical multiplexer based on diffraction grating is the same as the principle of optical demultiplexer. In fact, the optical multiplexer can play the role of demultiplexer and vice versa. What is needed is just to switch the roles of the input and output ports. The focusing lens in diffraction-based demultiplexers is usually the graded-index rod (GRIN-rod) since it is much more suitable from the design perspective. Furthermore, there is a possibility that the GRIN-rod can be integrated with the diffraction grating. Another possibility to simplify design is to use concave diffraction gratings, so that there is no need for a focusing lens. The concave diffraction grating can be

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made on the planar waveguide structure and eventually integrated with planar waveguides that serve as input or output ports. However, there is a practical issue as to how to couple multiplexers with optical fibers for a larger number of optical wavelengths. Optical multiplexers based on the interference effect use optical couplers and optical filters to combine two or more wavelength channels into a composite signal. The two commonly used types of optical multiplexers that are based on interferometric effect are the dielectric thin-film filter multiplexers and the Arrayed Waveguide Gratings (AWG) [61]. The Arrayed Waveguide Gratings (AWG) are widely used for wavelength multiplexing and for wavelength routing purposes. The AWG multiplexer is a generalized version of the Mach-Zehnder modulator, which consists of two optical couplers interconnected by optical waveguides, as shown in Figure 2.20(b). Optical waveguides, which form multiple arms of Mach-Zehnder interferometer, have distinctly different lengths in order to introduce a phase shift between corresponding optical signals. In the optical demultiplexer function, there is just one input and several output ports. The interferometric process that governs demultiplexing is the same process associated with the signal splitting in a standard version of the Mach-Zehnder interferometer, which forces each output wavelength to take just one output port. The optical multiplexer functionality is obtained by switching the direction of the optical signals shown in Figure 2.20b. In this capacity there are multiple inputs and just one output that accommodate a composite optical signal. The branch-and-three structure Mach-Zehnder interferometers, shown in Figure 2.20(a), can serve as an alternative to the AWG design that was just described. However, the structure from Figure 2.20(a) will have a higher insertion loss and a nonflat response in the wavelength passband, as compared to the AWG design. In addition, it is much easier to realize AWG as an integrated waveguide structure placed on the waveguide substrate. The waveguide substrate material used for AWG manufacturing is usually silicon, while either pure silica or silica mixed with some dopants can be used as the waveguide structure. The AWG structure with multiple input and output ports can also serve as a wavelength router with predetermined routing paths. The routing patterns can be established by proper adjustment of the Mach-Zehnder parameters in the AWG interferometric structure, and that will be discussed in Chapter 8. In the second major type of optical multiplexers, we have a cascade of FabryPerot filters, in which each of them is constructed with multiple-layer dielectric thin films, as shown in Figure 2.21. Each filter contains several resonant cavities to flatten the passband and to provide the steeper slope of the passband edges. As for the filter cascade shown in Figure 2.21, each of individual filters selects a different wavelength from the composite signal. For example, the first filter passes just one wavelength, and directs the rest to the second filter in the cascade, where another wavelength is selected before the rest is directed to the third filter, and so on, as shown in Figure 2.21(a).

5

4

Resonator 1

7

6

Resonator 2

8

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Resonator 3

Optical Components and Modules

3

2

Output fibers

1

(a)

(b)

Figure 2.21 Optical multiplexers based on: (a) the thin-film Fabry-Perot filter, and (b) thin-film filter design.

Optical multiplexers described above can provide relatively small cross-talk between neighboring optical channels, and reasonably flat top of the passband. In addition, they are relatively stable with respect to the temperature changes and insensitive to the polarization state of the incoming optical signal. Typical values of parameters that characterize the commonly used optical multiplexers are summarized in Table 2.2. Table 2.2 Typical values of optical multiplexer parameters Parameter AWG FP thin film Insertion loss in dB Crosstalk attenuation in dB Channel spacing in GHz

[3.5 – 6] [25 – 45] [12.5 – 200]

[0.3 – 0.8] [15 – 25] [50 – 200]

2.5 OPTICAL MODULATORS In a general case, an optical signal generated by the light source still needs to be modulated by an information data signal. The modulation can be done either in parallel with the signal generation or afterwards. Accordingly there are two distinct modulation schemes: direct modulation used when modulation signal current is part of the overall current flowing through either laser or LED, and an external modulation of the optical carrier after his generation in semiconductor laser. Internal modulation includes the impact to the optical signal still in process of optical-wave generation, and, therefore, it is simultaneous with the light source excitation. External modulation means that there is some altering of generated referent wave after it leaves laser cavity in a device known as external optical modulator. External modulation is based on the different physical phenomena, such as: electro-optic effect, acousto-optic effect, and magneto-optic effect. The

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electro-optic effect found the wider application so far, and its basic principles will be explained later in this section. 2.5.1 Direct Optical Modulation The direct modulation process involves a modulation current that is added on top of DC bias current. The level of the DC bias current in lasers is usually chosen in such a way that both logical levels of the modulation current, which correspond to “0” and “1” bits, are above the laser threshold current. In some cases the level corresponding to “0” bits might be lower than the threshold value, but such a scheme is more suitable for lower-speed operations. (As for modulation of light emitting diodes, the modulator design is more flexible, and the main goal is to achieve the highest output power for “1” bits under specified conditions.) Direct modulation is characterized by the extinction ratio Rex, which is

Rex  1 / rex 

P1 P0

(2.57)

where P0 is the power associated with the “0” bite, and P1 is the power related to the “1” bite, as illustrated in Figure 2.22(a). In an ideal case, the extinction ratio would be indefinitely large. However, in reality most sources and modulators generate a nonzero optical power output for zero bits and the extinction ratio takes a finite value. The optical power carried by the “0” bits will increase the probability that bits will be mistaken for the “1” ones at the decision point. The signal-to-noise ratio can be increased by increasing the extinction ratio, but at the cost of additional penalties in the modulation speed and the laser frequency chirp, as discussed below. The parameter rex =1/Rex =P0 /P1, which is also introduced by Equation (2.57), is used more often than Rex in different calculations, such as those related to the receiver sensitivity degradations done in Chapter 4. Initial electrical pulse P1 Rex=P1/P0

Bias Current

P0

Optical pulse

Amplitude, frequency

Optical power

Output optical pulses

Adiabatic chirp Transient chirp

Input modulation pulses

time

I0 Ib I1 (a)

(b)

Figure 2.22 Direct modulation of laser diodes: (a) extinction ratio, and (b) frequency chirp.

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We already mentioned in Section 2.3.1 that the P-I curve, which represents changes of the output light signal with the direct current through laser diode, is not quite stable and changes with aging and temperature. Both effects, the aging and temperature changes, cause the P-I curve degradation and decrease in the output power, as well as the change in the extinction ratio. That degradation will consequently degrade the signal-to-noise ratio at the receiving side. Therefore, a permanent monitoring of the output power, which is accompanied by the feedback control of the temperature and bias current, is needed. Direct modulation can be efficiently used for different modulation bit rates up few tenths of gigabits if fast DFB lasers are used. The transmission distance is determined by the point at which the signal degradation becomes high, while the bit error ratio exceeds a prescribed level. The signal degradation is mainly caused by chromatic dispersion, which is the effect directly proportional to the laser frequency chirp and dispersion properties of the optical fiber. Namely, a finite spectral linewidth can be attached to each individual longitudinal mode of a multimode Fabry-Perot laser—please refer to Equations (2.36) and (2.37). The same is applicable for the remaining mode of single-mode lasers (DFB, DBR). The linewidth of individual longitudinal modes is between 10 MHz and 100 MHz, which is by a factor 5 to 40 greater than the linewidth of an individual longitudinal mode in gas lasers. The linewidth structure in semiconductor lasers was originally explained by Henry in [43] by introducing the linewidth enhancement factor, which is proportional to the factor (1+2chirpThe parameter chirp is an amplitude-phase coupling parameter in traduced in Equation (2.19) and (2.20), which determines the ratio between the refractive index change and the gain change. The chirp factor can also be defined for the external optical modulators having much smaller values, which will be discussed in next section. The finite value of the linewidth occurs due to due to fluctuations in the carrier density caused by the spontaneous emission process. The fluctuations in carrier density will produce fluctuations in the refractive index and fluctuations in the optical signal phase. Therefore, the random fluctuations in carrier density will be transferred to the random fluctuations of the frequency, which leads to the frequency noise and spectral linewidth enhancement. This is exactly what happens during an amplitude modulation of the optical signal, since the change in carrier density during the amplitude modulation causes the causes a fluctuations in instantaneous frequency around the steady-state value 0. Such instantaneous frequency shift during modulation process can be calculated by using formula derived in [52]

 chirp  d

  (2.58)  ln P(t )   P(t ) 4  dt   where P(t) is the time variation of the output optical power and  is a constant related to the material and design parameters. Parameter can vary in the range from 1 THz/W to 10 THz/W.

 (t ) 

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The first term in the brackets on the right side of Equation (2.58) is called the transient (or instantaneous) chirp, while the second term defines adiabatic (or steady-state) frequency chirp. There is an offset between the adiabatic and transient chirp, as shown in Figure 2.22(b), which is sometimes used in direct modulation schemes for their partial mutual cancellation. Generally, the temporary shape of the modulation electrical signal plays an important role in the total value of the frequency chirp, as we can see from Equation (2.58). Please notice from Figure 2.22(b) that the leading edge of the pulse undergoes frequency up-shift (known as the blue shift), while the trailing edge shifts towards lower frequencies (the red shift). The terms “blue” and “red” are related to wavelengths rather than frequencies, since a higher-frequency region corresponds to a lower wavelength region (the blue part of the spectrum), while a lower-frequency region corresponds to a higher wavelength region (the red part). As a summary note, we can say that the frequency chirp interacts with the chromatic dispersion in the optical fiber, thus contributing to the deviation of the pulse shape. The character of the pulse change during the propagation through an optical fiber will be discussed in Chapter 3. 2.5.2 External Modulation of Optical Signals The key devices used as external modulators are the Mach-Zehnder modulator (MZM) and electro-absorption modulator, and we will analyze their characteristics in this section. However, before that, we will explain the basic principles of the electro-optic effect since it found a wider application in the design of advanced external modulators. 2.5.2.1 Electro-Optical Effect as a Base for External Modulation The majority of external modulation methods used in optical communications systems is based on the fact that the refractive index in crystals is dependent on both the geometrical parameters of the crystals and the parameters of the external electric field. If an anisotropic medium is exposed to the external electric field, the density of the electrical energy in the medium is expressed as [1] (2.59) where E is the electric field vector and D is the electric induction vector. The components of the vectors D and E are connected by relation

Dk    km Em

(2.60)

m

where are components of the dielectric permittivity tensor, and subscripts k and m denote any pair of reference Cartesian coordinates x, y, and z. Accordingly, the density of the electrical energy in the crystal can be expressed as

Optical Components and Modules



1 2

  k

km

Ek Em

87

(2.61)

m

In general, the tensor of dielectric permittivity has just six independent components since = [1]. In addition, Equation (2.61) can be simplified by corresponding coordinate transformation with the introduction of three new axes called principal dielectric axes, so it becomes (2.62) In new coordinate system, the tensor of dielectric permittivity takes a diagonal form, which leads to the following relations: Dx=xEx; Dy=yEy; Dz=zEz. By using these relationships, as well as the relationships between the refractive index and dielectric permittivity (nx=√ , ny=√ , nz=√ ), Equation (2.62) becomes 2

Dy Dx2 Dz2 X 2 Y2 Z2    2  2  2 1 2 2 2 2nx 2n y 2nz nx n y nz

(2.63)

Equation (2.63) is a canonic form of an ellipsoid with main the axes covering the coordinates X, Y, and Z, which are defined as: X = Dx/(2 )1/2, Y = Dy/(2 )1/2, and Z = Dz/(2 )1/2. It is known as a principal ellipsoid, while planes with corresponding coordinates are called principal planes. The shape of the ellipsoid depends on its semi-axes defined by corresponding refractive indexes along the axes. If all refractive indexes are equal (nx = ny = nz = n), the crystal is known as isotropic. If we have a case in which nx = ny = n1 and nz = n2, the crystal is known as a one-axis dielectric. Finally, if all refractive indices are different, the crystal is known as two-axis dielectric. There are two types of rays in the light signal propagating in the one-axis medium: ordinary and extraordinary. The refractive index experienced by the ordinary ray equals nx = ny = no, and the refractive index of the extraordinary ray is nz = ne. This denotation with indices “o”, and “e” points out to the character of a ray. The ordinary ray is polarized perpendicularly to the plane cutting the crystal axis (since we have a two axis crystal case), and the extraordinary tray is polarized parallel to that plane. These two rays have different phase velocities along the specified direction, which causes birefringence effect [1]. The birefringence effect is the base for the operation of electro-optical and magneto-optical modulators, due to the fact that refractive index of the principal ellipsoid depends on the external electric or magnetic fields. The principal ellipsoid is deformed under the influence of the external field and changes its position with respect to the main axes or with respect to direction of ray propagation. A similar effect can also appear under the impact of the acoustic

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wave caused by mechanical force, in which case it is acousto-optic or photo elasticity effect. Next we examine the electro-optical effect that can take either linear or a quadratic form. Equation (2.63) can be also rewritten as (2.64) where aj0 (j = 1,2,3) are the reciprocal values of the refractive index squares associated with corresponding axes. The subscript “0” means that coefficients are related to the case without any external field. However, if there is an external electric field, the ellipsoid will be deformed and the main axes of a newly formed ellipsoid will drift from the axes of an initial ellipsoid. The equation of the principal ellipsoid can be now written as (2.65) In linear electro-optic effect, which is also known as the Pockel’s effect [1], the difference aj – aj0 can be expressed as a linear function (2.66) It is assumed that coefficients aj0 for j > 3 are equal zero. The coefficients rjn (j=1,2,3,…6; n=1,2,3) form a 3-by-6 dimension matrix of electro-optical coefficients. Some of them can be equal to zero. A number of crystals have been examined for potential applications in optical communication systems and networks. It has been shown that the most suitable ̅ are ones that belong to the ̅ and 3m crystal groups [1]. So far, the wider application has been found for the crystals from ̅ group (GaAs-based semiconductors) and 3m group (lithium niobate LiNbO 3 and similar crystals). For crystals from the ̅ group, there are just several rjn coefficients that are different from zero. In addition, coefficients a10 and a20 are equal, which leads to equation (

)

(

)

(2.67)

If the electric field acts along the Z-axis, which corresponds to the optical axis of the crystal, only Z-component of the electrical field remains, so Equation (2.67) becomes (

)

(2.68)

Finally, the canonical form of ellipsoid can be obtained with the new coordinates obtained by rotation around the Z-axis for /4, while the axis Y stays unchanged, which leads to

Optical Components and Modules

(

)

(

)

89

(2.69)

The cross-section of a principal ellipsoid with the plane Z=0 is a circle if there is no outside electric field. However, in the presence of the electrical field, the cross-sectional area is an ellipse with the semi-axes (√

)

(2.70)

(√

)

(2.71)

The incident optical wave is split into two components due to the birefringence effect. One of them is polarized along the X’-axis and has phase velocity equal c/ , while the other is polarized along Y’-axis and has the phase velocity c/ . Consequently, it is possible to obtain the phase modulation of the light polarized along either one of these two axes. The phase delay of the outgoing optical wave versus incoming one can be evaluated as (

)

(2.72)

where V=E/d is the voltage that is applied over the crystal having depth d,  is wavelength of the light signal, and l is the length of the light path along the optical axis of the crystal. The phase delay of the signal polarized along the Y’ axis can be determined in a similar way and it has the same value but the opposite sign of the delay along the axis X’. The phase shift between two waves polarized along the X’ and Y’ axes can be expressed as (

)

(2.73)

The phase difference expressed by Equation (2.73) becomes zero if there is no external electric field. At the same time, the polarization of output light from the crystal is the same as the polarization of the light at the crystal input. The phase difference gradually increases with the increase in the applied voltage. The resultant vector of the electric field at the crystal output will draw an ellipse, which will deform into a circle for , while for = 0 the ellipse becomes the line. In such a way, polarization of an input optical wave is changed proportionally with the voltage, which means that polarization modulation is achieved. Polarization modulation can be converted to amplitude modulation if the output signal is passed through an analyzer. It is important to outline that expressions (2.72) and (2.73) are related to crystals from the ̅ group (GaAs semiconductor based). The same kind of expression is valid for crystals from the 3m group (lithium niobate LiNbO3

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crystal). The only difference is that the coefficient r33 is left instead of the coefficient r63. Accordingly, Equation (2.73) takes the form (

)

(2.74)

The typical values of parameters for LiNbO3 crystal from are l ~ 1 cm, d ~ 10 m, n0 ~ 2.27, r33 ~ 31 x 10-12 m/V. The quadratic electro-optical effect, or Kerr’s effect [1], is characterized by square-law dependence of the refractive-index in crystal from the intensity of the applied electric field. Under the impact of external field, the crystal is transformed from an isotropic form to the one-axial form with the refractive indexes given as (2.75) (2.76) If the light is polarized along the x (or y) direction and propagates along the z-axis, then the phase shift between incoming and outgoing waves can be expressed as (

)

(2.77)

When the light is polarized under the angle /4 with respect to the z-axis and travels in y direction, then there will be two waves propagating along the crystal. These two waves have equal amplitudes different polarizations along the x- and yaxes, respectively. The phase shift between these two waves acquired is (

)

(

)

(2.78)

The resultant vector of the electric field at the crystal output will draw an ellipse, which will deform into a circle for = /2, while for = 0 the ellipse becomes the line. 2.5.2.2 Mach-Zehnder (MZ) and Electroabsorption (EA) Modulators As mentioned, the external modulation takes place after the light generation. In this case, the laser diode generates a continuous wave (CW) signal, which serves as an input to the external modulator. The amplitude modulation, or the intensity modulation, is done by switching between two logical levels within the modulator, all under the impact of a digital modulation voltage. The phase modulation can also be performed by an external modulator structure by using electro-optical effects described above. In addition, both amplitude and phase modulation can be applied together as part of the more complex advanced modulation schemes that will be discussed in detail in Chapter 5. The external modulation process is more

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complex than the direct modulation one, but can provide a significant advantage since it can enable the increase in both the modulation bite rate and transmission distance [69–72]. The main configuration of external optical modulators is shown in Figure 2.23. The plain phase modulator in Figure 2.23(a) is the simplest structure that operates on an electro-optical effect. The phase shift between input and output signals is given by Equation (2.73) if the linear electro-optical effect is utilized, or by Equation (2.77) if the Kerr effect is employed to induce the phase shift. The phase shift is determined by the properties of the crystal and its dimensions (length and width). In addition, the electro-optical effect can be induced in the crystals built within the Mach-Zehnder (MZ) interferometric structure, as shown in Figure 2.23(b). Several Mach-Zehnder modulators can be combined to form a more complex structure as illustrated in Figure 2.23(c), where three MZ modulators are employed. Some of materials that are the most suitable for electro-optical effect employment for both the plane phase modulators and the MZ modulators are: lithium niobate (LiNbO3), indium phosphate (InP), gallium arsenate (GaAs), and some polymer materials. The refractive index change occurs relatively fast and in accordance with the changes in the electric field applied, which enables a high-speed operation.

Modulation Data (RF) Voltage V

Bias voltage (DC) Modulated output

CW input

Depth d Length l Bias contacts

Modulation contacts (a)

Bias current

PM 1 Output

Input MZ 3 MZ 2

-U bias (& Modulation)

+ p-type

p-type

n-type

n-type

The light beam to optical fiber

MZ 1

(b) Bragg grating

+U bias + (& Modulation) CW laser (c)

EA modulator (d)

Figure 2.23 External optical modulators: (a) Phase modulator, (b) Mach-Zehnder modulator, (c) Electroabsorption modulator, and (d) Advanced modulator.

The Mach-Zehnder modulator is a planar waveguide structure deposited on the substrate. There are two pairs of electrodes applied to the waveguide, the first

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one for DC bias voltage, and the second one for high-speed AC voltage that represents the modulation data signal. The electrodes for AC and DC voltages are deposited along the interferometer arms. There are a various combinations how to apply these voltages, in terms of numbers and the physical positions of the electrodes. The physical layout is extremely important since the way of how an electrical field is applied has a significant impact to the modulation characteristics of the Mach-Zehnder modulator [69]. The continuous light wave coming from the laser is being equally split between two arms of the modulator. These portions are eventually combined again at the modulator output. The modulation principle of the MZ modulator is the following: any applied voltage to the modulator arms increases the refractive index of the material, and slows down the light speed, thus effectively retarding the phase of the optical signal. The phases of two incoming portions will determine the nature of the eventual recombination at the modulator’s output. If the refractive indexes at any given moment are equal, the two light streams will arrive in phase and their amplitudes are added to each other. This situation corresponds to the “1” bits since a high pulse level has been produced at the output. On the other hand, the output signal level will be lower if there is a difference in refractive indexes between two interferometer arms. The extreme case occurs if the difference between the phases equals radians (or 180 degrees) since two signal streams will interfere destructively by canceling each other. It is easy to understand that this case corresponds to “0” bits. Therefore, we can say that by applying the modulation voltage to the Mach-Zehnder interferometer arms, the continuous wave light is being effectively chopped, and a modulation by digital data stream is being performed. The phase shift between two optical streams is proportional to difference in the voltages applied to two waveguide arms. The voltage difference is usually measured with respect to the voltage value V  that is needed to shift the phase by 180 degrees, and switch the output level from space to mark. The modulation curve of the MZ modulator, which is shown in Figure 2.24, can be expressed by the following relation:  V  (2.79)  Pout  Pin cos 2   2V  where Pin and Pout are the incoming and the outgoing signals from the modulator, and V is the total voltage applied (the bias voltage plus the modulation data signal.) We can select either a positive or a negative slope of the modulation curve by choosing a proper value of the DC bias voltage, and that effectively determines the properties of the modulated signal and the value of the induced frequency chirp. As already mentioned, a 180-degree phase shift between the interferometer arms produces the total mutual cancellation of the signals travelling through the MZ interferometer arms. That would produce an indefinite value of the extinction ratio defined by Equation (2.57). However, the situation is quite different in

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practice since the mutual cancellation is not perfect, and the extinction ratio takes some definite value usually in the range around 20 dB. DC bias

DC bias

Voltage V

V

Voltage

V a

b

Figure 2.24 Modulation curves of: (a) Mach-Zehnder modulator, (b) Electroabsorption modulator.

The electroabsorption modulator (EA) is a semiconductor-based planar waveguide that consists of multiple p-type and n-type layers, as shown in Figure 2.23(d). The multitude of layers form multiple quantum wells (MWQ), [71, 72]. The multiple p-n type layer design serves to support the quantum Stark effect [1] more effectively. The layered structure of these devices has some similarity with the laser structures, which means that layered structure design for both the lasers and EA modulators can be done on the same substrate, as indicated in Figure 2.23(d). However, the laser and modulator must be electrically isolated from each other if the integration is done on the same substrate. Although an EA modulator can be a single packaged device that is connected to the laser by a fiber pigtail, integration is a much better practical solution and has a wider application in highspeed communication systems. From an operational perspective, the EA modulators work just opposite to semiconductor lasers. While the laser is forward biased, the EA modulator is reverse-biased, and operates like a photodiode. The EA modulator is practically transparent to the incoming continuous wave optical signal if there is no bias voltage applied to the modulator waveguide structure. That transparency is not an ideal one since some if incoming photons will eventually produce electron-hole pairs, and cause signal attenuation. The signal transparency associated with this case is possible since the energies of incoming photons are lower than the energy bandgap in layered semiconductor structure of the modulator. The situation changes when some bias voltage is applied. The applied voltage can separate electron-hole pairs and generate a photocurrent, which effectively increases the waveguide attenuation and signal losses. Therefore, we can say that output signal from the EA modulator is the highest when there is no voltage applied and decreases with the bias voltage increase. It is worth noting that the modulated optical signal at the EA output is opposite in phase with respect to the modulation digital stream applied through electrodes. It is because “0” level in voltage generates “1” level in output optical signal, and vice versa. A typical modulation curve of the EA modulator is shown in Figure 2.24(b).

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The modulation speed of the EA modulators is comparable with the speed of the MZ modulators, but the extinction ratio is generally smaller, and it is typically around 10–15 dB. The ability to integrate semiconductor lasers with the EA modulators on the same substrate is a big competitive advantage since it reduces the insertion losses, simplifies packaging by making everything almost like a regular laser package, and reduces the total cost of the device. In addition to integration with the lasers, the EA modulators could be integrated with other semiconductor chips, such as semiconductor optical amplifiers and multichip wavelength selectable lasers. It is important to notice that the EA modulator is more resilient to changes in polarization of the incoming light signal than MZ modulator. However, the output optical power from the EA modulator is generally lower than the power at the output of MZ modulator.

Parameter Insertion loss Extinction ratio Modulation bandwidth Chirp factor

Table 2.3 Typical values of optical parameters EA 7–15 dB 10–13 dB Up to 75 GHz –0.2 to 0.8

MZ 4–7 dB 10–50 dB Up to 85 GHz –1.5 to 1.5

In a summary, we outline the optical modulator parameters that are important from the systems perspective: the insertion loss of the modulator that determines the output power; the frequency chirp characterized by chirp factor chirp; the extinction, or contrast ratio, characterized be the ratio of optical powers corresponding to “1” and “0” bites, respectively; and the modulation speed that is characterized by the frequency response in general, and by the cutoff modulation frequency in particular. Typical values of optical parameters related to MZ and EA modulators are shown in Table 2.3.

2.6 OPTICAL AMPLIFIERS Optical amplifiers are used to restore the signal strength at different points along the lightpath by using a stimulated emission process. The restoration can also be done through retransmission by the opto-electical-opto (O-E-O) conversion process, which means the restoration is done at the electrical level. The O-E-O process is generally both wavelength and bit rate specific, which is uneconomical if applied to multichannel optical transmission systems. From an application perspective, optical amplifiers are more flexible since they have a large signal bandwidth and relatively high and adjustable gain coefficient. There are several major applications of optical amplifiers along the lightpath: (1) booster amplifiers at the optical transmitter side that enhance the power level of the output optical signal to bring it to the level needed for transmission by compensating for losses in the optical elements (couplers, multiplexers,

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modulators); (2) in-line optical amplifiers placed as a chain along the lightwave path that compensate for losses incurred during the signal propagation; and (3) optical preamplifiers are used within optical receivers or optical networking elements to increase the optical signal level before opto-electronic conversion or before signal splitting an switching. Optical amplifiers can be constructed by using the physical principle of stimulated light radiation, the same principle that is used in lasers. However, while lasers need some resonant cavity with mirrors to support the lasing regime, optical amplifiers should be designed to suppress such effect. It can be done by effectively eliminating the mirror reflectivity while providing a strong pumping to populate upper energy levels, or by keeping some reflectivity in combination with weaker pumping. There are two main groups of optical amplifiers that are based on design principles mentioned above: semiconductor optical amplifiers (SOA) and fiber doped amplifiers, all discussed in details in references [73–83]. 2.6.1 Semiconductor Optical Amplifiers A semiconductor optical amplifier is a device similar in structure to semiconductor laser. However, the device is designed to operate below the threshold since either direct injection current or facet reflectivity are intentionally kept low. Multiple reflections still occur at the facets of the Fabry-Perot laser resonator, even if the bias current is below its threshold value, but the feedback is not enough to cause the lasing regime. Incoming optical signal is introduced through an input facet. Such devices are called Fabry-Perot (FP) semiconductor optical amplifiers (SOA). Another type of semiconductor optical amplifiers is known as traveling wave (TW), in which an optical signal does not undergo multiple reflections, instead having just a single pass through the cavity [77]. In this case, the facet reflectivity is very low (below 10-4), but the bias current is relatively high. Since semiconductor optical amplifiers can be considered as a special class of lasers, the same analysis of their properties can be applied. Semiconductor optical amplifiers are good candidates for some applications where larger signal bandwidth or/and moderate gain are needed. They are very compact devices that can be easily integrated with other semiconductor structures (modulators, photodiodes, and couplers). However, they suffer from polarization sensitivity, relatively high noise figure, and the signal cross-talk, which is quite relevant for optical multichannel systems and networks. 2.6.2 Fiber Doped Amplifiers The second group of optical amplifiers is based on amplification in optical fibers, which serve as the active medium. These optical fibers can be either regular ones or specialty ones designed for amplification purposes. The specialty optical fibers are produced by adding dopants, such as erbium or praseodymium, to silica-based core to increase the efficiency of stimulated emission. The population inversion is

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done by a strong pumping by another optical signal, while stimulated emission is triggered by an incoming optical signal that needs amplification. The phase and frequency of radiated photons are the same as those of photons in incoming signal. The amplification of the incoming signal occurs since newly generated photons stay within the fiber waveguide structure and begin propagation in the same direction as incoming signal. The best know amplifiers based on specialty optical fibers are erbium doped fiber amplifiers (EDFA), while Raman amplifiers are the best example of amplifiers that utilize regular optical fibers for signal amplification. 2.6.2.1 Erbium Doped Fiber Amplifiers The most important part of EDFA is an optical fiber that is doped with erbium ions (Er3+ ions) [74, 75]. Some additional dopants, such as fluoride or aluminum, are also used to optimize optical fiber amplifier gain profile with respect to specified wavelength band. A general application scheme of erbium doped fiber amplifiers, which is related to in-line amplification, is shown in Figure 2.25. The design shown in this figure is well known as two-stage scheme, where each of two stages is designed differently. The first stage is optimized to provide both high gain and a low noise, while the second stage serves to boost the output optical power. The intermediate stage is commonly used for optical signal conditioning, which may include dynamic gain equalization of different optical channels, dispersion compensation, and optical signal add/drop and cross-connect functions. As an example, the dispersion compensation fiber (DCF) module and the reconfigurable optical adddrop multiplexer (ROADM) can be placed within intermediate stage, as indicated in Figure 2.25. In either case, there is an insertion signal loss that should be compensated for. The optical pumping is introduced through wavelength-selective couplers. Very often the CW semiconductor laser radiating at 980 nm serves as the first pump, while the second stage pump laser radiates at 1,480 nm. The total noise, expressed through the noise figure, of the two-stage amplifier design is calculated by including contributions of both stages—please refer to Chapter 4.

Input signal

Erbium doped fiber Isolator

Isolator Intermediate Stage (DCF, ROADM)

980 nm pump

Optical WDM couplers

Figure 2.25 Erbium doped fiber amplifier scheme.

Output signal

1480 nm pump

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The erbium doped fiber amplifier is produced by incorporating erbium ions into the glass matrix of the core of a silica optical fiber. This process leads to a classical three-level lasing system that is illustrated in Figure 2.26. There are several energy levels that could be eventually used for the electron transitions, but three levels presented in Figure 2.26(a) are the most suitable to support the amplification process. The population inversion of electrons at upper energy levels is created by optical pumping. Such pumping enhances the electron energy, so they are lifted from the ground energy level (4I15/2) to some of the higher energy levels. For example, the upper energy level (4I9/2) can be populated by using the optical pump radiating at 800 nm, and the lower level (4I11/2) can be populated by using the pump at 980 nm, while the optical pump radiating at 1,480 nm can be used to lift electrons from the ground level to the next higher energy level ( 4I13/2). The electrons lifted higher than the (4I13/2) level will soon slide down to the 4 ( I13/2) level through the process of nonradiative decay. It occurs because both upper energy levels are not stable. Nonradiative decay occurs within 1 s time period calculated from the moment when electrons have populated the upper energy state. However, the 4I13/2 level is known as a metastable state, which means that electrons populating this energy level have a relatively long lifetime, which is the time period they spend occupying this energy level. The electron lifetime at the 4I13/2 level is around 11 ms. Such a relatively long lifetime of electrons helps to establish a reservoir of the energy at the 4I13/2 level that can be used for optical signal amplification. Technically speaking, it is better to use the term “energy zone” instead of “energy level” since the discrete energy levels associated with isolated erbium ions have been split into energy zones through the process called the Stark effect [5]. Pump wavelengths

Absorption spectrum 4I

4I

1530 nm

1480 nm

980 nm

800 nm

4I

11/2

13/2

Stimulated emission 4I

a)

Emission spectrum

9/2

15/2

1450

1550

1650

b)

Figure 2.26 Erbium doped fiber amplifiers: (a) energy levels, and (b) absorption and radiation spectrums.

Optical signal amplification is the process of stimulated radiation of the light when electrons drop from the metastable level to the ground energy level. The

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photons radiated follow the frequency, phase, and direction of the incident optical signal, which effectively leads to an amplification of the input optical signal. The difference between individual energy levels within the metastable zone and levels belonging to the ground zone helps to create photons with the emission spectrum covering wavelengths from approximately 1,530 nm to 1,560 nm. The radiation spectrum can be shifted, or effectively broadened, if some other special co-dopants are added to the silica glass. The radiation spectrum is widened if there are co-doping ions since co-dopant atoms create an electric field that enhances the Stark effect. A wide variety of options have been tried to optimize characteristics of the erbium doped fiber amplifiers with respect to spectrum broadening, gain flattening, or gain wavelength shifting. The highest figure of merit was achieved by adding the aluminum and fluoride ions mixed together with erbium ions [81]. The stimulated emission of the light is accompanied with spontaneous emission, when optical radiation occurs randomly without any established pattern. The spontaneous emission is source of the noise in optical amplifiers, and that effect will be carefully examined in Chapter 4. The spontaneous emission is accumulated along the transmission line. In addition, subsequent in-line amplifiers increase the level of the spontaneous emission and it arrives at the receiving side as the amplified spontaneous emission (ASE) noise. The pumping of optical power is provided by high-power semiconductor lasers since they can radiate in the wavelength region from 800 nm to 1,480 nm, which is needed to excite the electrons from the ground to the metastable level. The 980-nm semiconductor pump lasers are widely employed in the first amplifier stage, while pump lasers radiating at 1,480 nm are used in the second stage from Figure 2.26. When the 980-nm pump lasers are used, the process of stimulated emission is achieved through several steps. In the first step, the electron is excited from the ground level (4I15/2) to level (4I11/2). The second step takes place after about 1 s, when electrons slide down from the level (4I11/2) to the metastable (4I13/2) energy level. The electrons can live up to 11 ms at the metastable level before they radiate the energy and fall back to the ground level. The pumps operating at 980 nm have a lower noise figure since the signal and the pump are separated in wavelengths by several hundred nanometers. In addition, they are rather reliable and require a simpler and less expensive WDM coupler. The 1,480-nm semiconductor pump lasers are also widely used since they are readily available and have better reliability than 980-nm pump lasers. In addition, they radiate at the wavelengths where optical fiber attenuation is lower than the attenuation at 980 nm, which makes them more suitable for applications where a remote pumping may be needed. In this case, he population inversion is achieved in just one step since the electrons have been excited directly the metastable energy levels. However, the noise figure related to the 1,480 pumps is higher than the noise figure associated with 980 nm pumps since the signal and pump wavelengths are relatively close to each other. Some additional optical fiber amplifier schemes utilize specialty fibers that contain dopants different than erbium ions, such as praseodymium,

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neodymium, or thulium ions. Such a design is suitable for amplification of the optical signal at wavelengths different that those around 1,550 nm. As an example, praseodymium and neodymium-doped fiber amplifiers can be used in the wavelength region around 1,300 nm, while thulium-doped fiber amplifiers can be used at the wavelength region around 1,450 nm. The total gain of EDFA depends on the concentration of erbium ions in the glass, the total length and cross-section of the doped fiber active area, as well as the pump power. The exact analysis of the stimulated emission in EDFA can be done by using the same approach as in semiconductor lasers, by including all four levels from Figure 2.26. The system of differential equations is then solved numerically. A simplified approach assumes that just two-level analysis can be used, and we will follow this approach described in [74, 75]. The population densities N1 and N2 at two states (4I15/2) and (4I13/2) should satisfy the following equations = (



)

(



) -

(2.80)

(



)

(



) +

(2.81)

where superscripts “a” and “e” stand for emission and absorption, respectively, and subscripts “s” and “p” stand for the signal and pump, respectively. Parameters  and  are cross-section and photon flux, respectively, while 1 is spontaneous lifetime of the electron at the excited state (~ 10 ms). The photon flux is defined as the ratio of the number of photons by the cross-sectional area of the fiber mode  for signal, or  for pump, which is  



(2.82)



(2.83)

where h is the Planck constant, a s and p represent optical frequencies of the signal and pump, respectively. The powers of the pump and signal vary along the doped fiber length since there are three processes involved: stimulated emission, absorption, and spontaneous emission. Assuming that spontaneous emission can be neglected, the following equations can be written for the powers of the signal and the pump  (  (



) 

(2.84) )

(2.85)

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where  and ’ are fiber losses at the signal and pump wavelengths, respectively; and  and  are mode confinement factors. The sign in front of Equation (2.85) takes into account the pumping direction; it is plus for forward direction pumping and minus for backward pumping. Equations (2.80), (2.81), (2.84), and (2.85) can be solved analytically by assuming that fiber losses can be neglected, which is true only for lumped case when fiber length is measured by tens of meters. By making substitutions back to Equation (2.80), a steady state solution can be written as ( )



(2.86)



where  =  =  is the cross-sectional area of the doped portion of the fiber core. This solution can now be utilized to find solutions of Equations (2.85) and (2.86). Equation (2.86) describes the generic case for signal values that are high enough to force amplifier into saturation. However, for smaller signal values, parameter  from Equations (2.80) and (2.81) can be neglected and the total gain of an EDFA having the length La can be expressed as [14] 

[ ∫ (

)]

(2.87)

There is an optimum length of the doped fiber that depends on the pump power level and that should be taken into account when designing EDFA for different applications. 2.6.2.2 Optical Amplifier Gain The optical gain is related to the stimulated light scattering and will occur after an inverse population is achieved through the pumping process. The amplification coefficient that characterize the stimulated emission process is given as [81] g ( ) 

g0

1  Pin / Psat  2 T2 (   0 )

(2.88)

2

where  is frequency of the incident optical signal, 0 is the atomic transition frequency related to the two-level energy diagram, g0 is the value of the amplification peak, Psat is the saturation power, and T2 is the dipole relaxation time that takes subpicoseconds values. The saturation power Psat is a medium specific parameter related to the population relaxation time or time that carriers spend on the upper energy level, which is denoted by 1 in Equations (2.80). As

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we mentioned, the relaxation time can be in the range (0.1–10) ms for commonly used optical amplifiers. The optical signal gain is proportional to the amplification coefficient and the total length La of the amplifier medium. The gain factor G, which determines the level of the optical signal that is being amplified, can be expressed in a generic form as   g 0 La G( )  exp[ g ( ) La ]  exp  2  1  Pin / Psat  2 T2 (   0 ) 

(2.89)

The gain factor, or simply “the gain”, has a maximum G0=exp(g0La) related to the gain peak g0 and the frequency 0. Therefore, the gain depends on the level of the incident optical power, and decreases as incident optical power becomes comparable with the saturation power Psat. This reduction in amplification capability is known as the gain saturation. The amplification process covers effectively some optical frequency bandwidth a, which is defined as the full width at half maximum (FWHM) of the gain function G()  a 

1  ln 2  T2  ln(G0 / 2) 

(2.90)

The output optical power from an optical amplifier is determined by the amplifier gain, and can be expressed as

Pout  GPin

(2.91)

where Pin and Pout are the input and output powers, respectively. The enhancement in the output power can also be expressed as Pout = Pin + G if all parameters in Equation (2.91) are given in decibels. Equations (2.89) and (2.91) can be used to express the gain G in the form  G  1 Pout   G  G0   G Psat  

(2.92)

Therefore, the gain gradually decreases from its maximum value G0 if the amplifier output power approaches the level Psat determined by the population relaxation time. The functional dependence given by Equation (2.92) is plotted in Figure 2.27 for several values of the maximum gain value G0, also known as a small signal gain. Another parameter of practical interest is known as the output saturation power Po,sat, which is defined as the output power where the gain G drops to just

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half of its maximum value. By replacing G with G0/2, and solving Equation (2.92), we have that G0 ln 2 (2.93) Psat  Psat ln 2 G0  2 The saturated value of the gain parameter, which is associated with the saturated optical power, satisfies the following implicit mathematical equation Po,sat 

Gsat  1 

Psat G ln 0 Pin Gsat

(2.94)

Normalized gain G/G0

As we will see in Chapter 4, the saturated values of the gain and optical power are very important when designing a long-haul transmission systems where there are a large number of cascaded amplifiers. However, a small gain plays an important role in case where optical losses should be compensated for. 1.00 0.9 0.8

0.7 0.6 0.5 0.4

G0=15 dB

0.3

G0=20 dB

0.2 0.1 -20

G0=35 dB -10

0 10 Difference between output power and saturated power [dB]

20

Figure 2.27 Gain parameter as a function of the output optical power.

2.6.2.3 Raman and Brillouin Amplifiers The second type of optical fiber amplifiers is based on stimulated Raman or stimulated Brillouin light scattering effects that occur in regular optical fibers. While the Brillouin amplifiers can be used just for low bit rate applications, such as one presented in [84], the Raman amplifiers based on stimulated the Raman scattering (SRS) effect can find much wider practical applications [85, 86]. There are two types of light scattering that can occur in optical fibers. The first one, known as linear or elastic scattering, is characterized by scattered light signal that

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has the same frequency as incident one. We recall that Rayleigh scattering is a classic example of the elastic process. The second type, known as nonlinear or inelastic scattering, is characterized by downshift in frequency of the scattered signal. Both the Raman and Brillouin scatterings are two well-known effects that belong to the second group [4]. In this section we will describe these processes, before describing the characteristics of Raman amplifiers in more detail. Incident signal (pump)

Vibrating molecules

P Stimulated Raman Scattering Process

Backward scattered signal

Incident signal (pump)

Raman scattering

Absorption

S

Vibration band

P S

Forward scattered signal

Energy levels

Vibration in medium density (acoustic waves)

P Stimulated Brillouin Scattering Process

Backward scattered signal

B Vibration band

P B Energy of acoustic phonons with frequency P B

Figure 2.28 Stimulated Raman and Brillouin scattering processes and associated energy diagrams.

Raman scattering occurs when a propagating optical power interacts with glass molecules in the fiber. This results in a transfer of energy from some photons of the input optical signal to vibrating silica molecules and the creation of new photons with lower energy than the energy of incident photons. The incident optical signal is often referred as the pump signal. Newly generated photons form Stokes signal, which is illustrated in the upper part of Figure 2.28. Since the energy of the Stokes photons is lower than the energy of the incident pump photons, the frequency of the Stokes signal will be also lower than frequency of the pump signal. Difference in frequencies, known as the Raman frequency shift R, is expressed as R= P – S , where P is optical frequency of the incident pump signal, and S is the optical frequency of the scattered Stokes signal. Scattered photons are not in phase and do not follow the same scattering pattern, which means that the energy transfer from the pump to Stokes photons is not a uniform process. As a result, there will be some frequency band R that includes frequencies of all scattered Stokes photons. In addition, scattered Stokes photons

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can take any direction, which means that Raman scattering is an isotropic process. That direction can be either forward or backward with respect to direction of the pump signal in an optical fiber. If the pump power is lower than a certain threshold value, the Raman scattering process will have a spontaneous character, which is characterized by a relatively small number of pump photons that will be scattered and converted to Stokes photons. However, if the pump power exceeds the threshold value, Raman scattering becomes a stimulated process, and we are talking about stimulated Raman scattering (SRS). This could be explained as a positive feedback process, where the pump signals interacts with Stokes signal and creates a beat frequency beat= R = ( P - S). The beat frequency then acts as a stimulator of molecular oscillations, and the process is enhanced (amplified). Assuming that the Stokes signal propagates in the same direction as the pump (the forward direction), the following equations can be established [87, 88] g dPP   R A dz  eff

  P    P P   P PP    P S  S 

dPS g  R PP PS   S PS dz Aeff

(2.95) (2.96)

where z is the axial coordinate, gR is the Raman amplification coefficient (gain), R and S are the fiber attenuation coefficients for the pump and the signal, respectively, and Aeff is the effective cross-sectional area of the fiber—please also refer to Equation (3.100). It is also important to outline that in this case the scattered Stokes wave contributes and enhances the incoming signal. The scattered Stokes photons will not have equal frequencies, and they will occupy a certain frequency band. The number of photons corresponding to any specified frequency within by the frequency band will determine the value of the Raman gain related to that frequency. Therefore, the Raman gain is not constant, but is a function of the optical frequency. The spectrum of the Raman gain is related to the width of the energy band of silica molecules and to the time decay associated to each energy state within the energy band. Although it is difficult to find an analytical representation for Raman gain spectrum, it can be roughly approximated by the Lorentzian spectral profile given as

g R ( R ) 

g R ( R ) 1  ( R   R ) 2 TR2

(2.97)

where TR is the decay time associated with excited vibration states, and R is the Raman frequency shift corresponding to the peak of the Raman gain. The decay time is around 0.1 ps for silica-based materials, which makes the gain bandwidth

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to be wider than 10 THz. The Raman gain peak gR(R)= gRmax is roughly between 10-12 m/W and 10-13 m/W for the wavelengths above 1,300 nm. Input signal

Output signal

Isolator Raman gain profile Triangle approximation of the Raman gain profile (filled) Lorentzian approximation of the Raman gain profile

Raman frequency shift

Beam splitter

WDM coupler Pump module

x-polarization

Isolator Beam splitter

Pump module y-polarization

Forward pumping (a)

WDM coupler

Backward pumping

(b)

Figure 2.29 (a) Raman gain profile, and (b) Raman amplifier structure.

The approximation of the Raman gain with Lorentzian curve for silica fibers and a typical shape of the actual gain profile is shown in Figure 2.29(a). The actual gain profile extends over frequency range of about 40 THz (which is approximately 320 nm), with a peak around 13.2 THz. There are also several smaller peaks that cannot be approximated by the Lorentzian curve. They are located around frequencies of 15, 18, 24, 32, and 37 THz [88, 89]. The gain profile can also be approximated by a triangle function, as g R ( R ) 

g R ( R ) R R

(2.98)

This approximation is also shown in Figure 2.29(a). It is important to estimate a threshold value of the pump power above which the Raman scattering takes a stimulated character. The threshold power is usually defined as the incident power at which the half of the pump power is eventually converted to the Stokes signal. The Raman threshold can be estimated by solving Equations (2.95) and (2.96). For that purpose, value gR from Equations (2.95) and (2.96) should be approximated by the peak value gR(R). Accordingly, the amplification of the Stokes power along the distance L can be expressed as [87] g P L PS ( L)  PS 0 exp  R max S 0   2A  eff  

(2.99)

The value PS0 that corresponds to the Raman threshold PRth, as defined above, is

PRth  PS 0 

16 Aeff g R max Leff

(2.100)

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where Leff is the effective length—please also refer to Equation (3.98). The estimated Raman threshold is about 500 mW for typical values of the fiber parameters (for Aeff =50 m2, Leff = 20 km, and gR(R)= gRmax = 7 ·10–13 m/W). The SRS can be effectively used for optical signal amplification, since it can enhance the optical signal level by transferring the energy from the pump to the signal. Raman amplifiers can improve the performance of optical transmission systems by providing an additional optical power margin. However, the SRS effect could be quite detrimental in the dense WDM transmission systems since the Raman gain spectrum is very broad, and energy transfer occurs from the lower-wavelength channels to the higher-wavelength ones. In this situation, the optical fiber acts as the Raman amplifier since longer wavelengths are amplified by using power carried by lower wavelengths, which serve as the multiple Raman pumps. The signal power at the output of the amplifier length L, which is found by solving Equations (2.95) and (2.96), can be expressed as

g P L  g P PS ( L)  PS 0 exp(  S L) exp  R P 0 eff , P   PS 0 exp(  S L) exp  R P 0    A Aeff    P eff

   

(2.101)

where PS0 = PS(0) is the launched signal power, PP0 is the input pump power, while Leff,P is the effective length for the pump signal. The approximation that Leff,P ~ 1/P was used in Equation (2.101), which is quite valid if L>> 1/P. The gain of the Raman amplifier can be obtained from Equation (2.101) as GR 

 g R PP 0 Leff , P   PS ( L)   exp  g R PP 0  exp     A PS 0 exp(  S L) Aeff    P eff

   exp( g 0 L)  

(2.102)

where the amplification coefficient g0 can be expressed as P g0  g R  P0 A  eff

 Leff ,P  1 g R ( ) PP 0    L  L  A P eff 

(2.103)

There are several important conclusions that follow from Equation (2.101). First, the Raman amplification coefficient keeps the same frequency dependence as the Raman gain spectrum. Second, the Raman amplification coefficient is strongly dependent on the fiber type through the effective cross-sectional area parameter. It is stronger for optical fibers having a smaller cross-sectional area, and vice versa. The largest amplification coefficient is in dispersion compensating fibers (DCF), where it is six to eight times higher than in other fiber types. That is the reason why DCF fibers can be effectively used to construct lumped Raman amplifiers.

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The Raman gain increases exponentially with the pump power, while entering into saturation regime after the pump power exceeds the level of approximately 1 W. This can be verified by solving Equations (2.95) and (2.96) numerically. An approximate expression for saturated gain value, obtained in [14], is

GR , sat 

1  0 0  GR  (1  0 )

(2.104)

where 0 

P PS 0 S PP 0

(2.105)

The amplifier gain is reduced by about 3 dB if the power of amplified signal becomes comparable with the input pump power PP0. Since PP0 is relatively high, the amplifier will basically operate in a linear regime with the gain GR,sat ~ GR . The cost-effective design of Raman amplifiers was enabled by the availability of reliable high-power pump lasers. A generic scheme of a Raman amplifier is shown in Figure 2.29(b). The pump power is usually launched into the fiber in opposition to the signal propagation. Two orthogonally polarized pump signals are combined to provide a polarization-independent pumping scheme. As a result, the forward-propagating optical signals are enhanced since they get additional energy through distributed SRS. The optical gain achieved by the Raman amplifier helps to keep the signal further above the noise level. A fairly flat gain profile over a wide range of optical wavelengths can be achieved by combining several pumps operating at different wavelengths (four pump pairs from Figure 2.29(b), to effectively support both signal polarizations). The main advantage of Raman amplifiers is related to the fact that no specialty fiber is required. In addition, the Raman amplification is better suited to handle the impact of different nonlinearities that occur in the optical fiber since it lowers the level of the launched signal power. The Raman amplifiers can be used either as sole amplifiers, or to operate in combination with both erbium doped fiber amplifiers (EDFA) and thulium doped fiber amplifiers (TDFA). Brillouin amplifiers are based on Brillouin scattering, which is a physical process that occurs when an optical signal interacts with acoustical phonons, rather than with the glass molecules. During this process an incident optical signal reflects backward from the grating formed by acoustic vibrations, and downshifts in frequency, as illustrated in lower part of Figure 2.28. The acoustic vibrations originate from the thermal effect if the power of an incident optical signal is relatively small. In this case the amount of backward scattered Brillouin signal is also small. If the power of the incident optical signal goes up, it increases the material density through the electro-strictive effect [89]. The change in density enhances acoustic vibrations and forces the Brillouin to

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take a form of stimulated Brillouin scattering (SBS). The SBS process can also be explained as a positive feedback mechanism, in which the incident optical signal (or the pump) interacts with the Stokes signal and creates a beat frequency equal beat = B = P – S. This scattering process is the same in nature as the one mentioned in Section 2.4.4, which refers to the acousto-optical filters. However, there is no external electric field applied in this case, since the electric field at the beat frequency beat is created from the inside rather than by applying an external microwave transducer. The parametric interaction among pump, Stokes signal, and acoustical waves requires both energy and momentum conservation. The energy is effectively conserved through the frequency downshift, while the momentum conservation occurs the through backward direction of the Stokes signal. The frequency downshift is expressed by the Brillouin shift B, which is given as

B  2nPVA / c

(2.106)

where n is the refractive index of the fiber material, VA is the acoustic wave velocity, c is the light speed in vacuum, and P is the optical pump frequency. Equation (2.106) can also be rewritten as  B 2nV A (2.107)  2 P where the relation P=2 c/P was used. By inserting typical values of parameters (VA=5.96 km/s, P =1550 nm, n = 1.45) in Equation (2.107), the frequency shift becomes fB=11.5 GHz. This frequency shift is fiber material dependent and can vary from 10.5 GHz to almost 12 GHz for different fiber materials [19]. The SBS process is governed by the following set of coupled equations [87] fB 

dPp

gB PP PS   P PP Aeff

(2.108)

dPs g   B PP PS   S PS dz Aeff

(2.109)

dz



where PP/Aeff and PS/Aeff define the intensities (power over cross-sectional area) of the pump and the Stokes signal, respectively, z is the axial coordinate, gB is the Brillouin amplification coefficient, and B and S are the fiber attenuation coefficients of the pump and the Stokes signal, respectively. The scattered Stokes photons will not have equal frequencies, but will be dispersed around within a frequency band. The number of photons corresponding to any specified frequency within the band determines the value of the Brillouin gain with respect to that frequency. The spectrum of the Brillouin gain is related

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to the lifetime of acoustic phonons that can be characterized by the time constant TB. The gain spectrum can be approximated by Lorentzian spectral profile as

g B ( B ) 

g B ( B ) 1  ( B   B ) 2 TB2

(2.110)

where B is the Brillouin frequency shift calculated by Equation (2.107). It is known that not just B, but also the width of the function given by Equation (2.110), will depend on characteristics of the fiber material. In addition, the Brillouin gain is dependent on the fiber waveguide characteristics. The SBS gain bandwidth will be about 17 MHz at P =1.520 nm in pure silica fiber, while it can be almost 100 MHz in doped silica fibers. The typical value of the SBS gain bandwidth is about 50 MHz. The maximum value of the SBS gain gB(B)= gBmax is also fiber material dependent. It takes values between 10-10 m/W and 10-11 m/W for silica-based optical fibers and for wavelengths above 1 m. The threshold value of the pump power, above which Brillouin scattering takes a stimulated character, can be estimated by following the same way as in the SRS case. The threshold power is defined as the incident power at which the half of the pump power is eventually converted to the backward Stokes signal. The incident power that corresponds to the SBS threshold PBth is given as [87]

PBth 

21Aeff

(2.111)

g B max Leff

where Leff is the effective length. The estimated Brillouin threshold is about 7 mW for typical values of optical fiber parameters (for Aeff = 50 m2, Leff = 20 km, and . gBmax = 5 10-11 m/W). Although the SBS effect can be potentially used for optical signal amplification, its narrow gain bandwidth will limit the application area. However, the SBS effect can be quite detrimental in optical transmission systems, since the transfer of the signal energy to the Stokes signal has the same effect as the signal attenuation. In addition, the back-reflected light contributes to the optical noise and could even enter into the resonant cavity of the transmitting laser. Fortunately, there are some methods that can minimize the SBS effect by spreading laser linewidth through bias current dithering [17]—please refer to Equation (3.152). In summary, we can outline that erbium doped fiber amplifiers (EDFA) can effectively cover both the C and L wavelength bands. However, they cannot be used for other wavelength regions, such as the second wavelength window around 1,300 nm, or below 1,500 nm (S-band). The praseodymium and neodymium doped fiber amplifiers are good candidates for application in the 1,300-nm wavelength region, although they still did not find a wider application. As for the

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S-band, it looks that the thulium doped fiber amplifiers (TDFA) can be effectively used. A major obstacle for wider application of optical amplifiers that are based on specialty fibers and operate at wavelength regions below 1,500 nm is relatively high cost and the lack of good pump lasers.

Parameter Operating wavelength in nm Peak gain in dB

Table 2.4 Typical values of optical amplifier parameters Semiconductor Optical Erbium doped fiber amplifiers (SOA) amplifiers (EDFA) 1,280-1,350 1,528-1,610 1,530-1,610 10-25 17-45

Maximum output power in dBm Noise figure in dB

Raman amplifiers 1,200-1,700 10-25

Up to 15

Up to 37

Up to 40

Around 8

5-7

N/A, see Section 4.3.9

Advanced optical amplifiers for high-capacity long-haul transmission systems should provide enough gain for more than 80 optical channels, which means that the aggregate optical power should be in excess of 20 dB. In addition, the noise figure should be as low as possible, while the gain profile should be equalized along the entire amplifier bandwidth. Typical parameters of common optical amplifiers are summarized in Table 2.4.

2.7 PHOTODIODES The main role of the photodiode is to absorb photons of the incoming optical signal and convert them back to the electrical level through a process just opposite to one that takes place in semiconductor lasers. All incoming photons with energy larger than the bandgap of the semiconductor p-n structure can generate the electron-hole pairs in photodiode structure. The electron-hole pairs are being separated by the strong electrical field across the p-n junction that is created by the bias voltage and drift very rapidly toward electrodes. The readers can consult Section 10.4 for more detailed description of p-n junction property. The photocurrent that is generated is consequently amplified and processed by electrical circuits in optical receiver. There are two groups of photodiodes: PIN photodiodes, and the avalanche photodiodes (APD) [91–97]. The detection process within a PIN photodiode is characterized by a probability that each photon will eventually generate an electron-hole pair. As for the APD, each primary generated electron is accelerated by a strong electric field, which than causes a generation of several secondary electron-hole pairs through the effect of impact ionization. This process is random in nature and avalanche-like. There are three major semiconductor materials that are used photodiode manufacturing: (1) silicon (Si), which is used for photodiodes that have the total bandwidth up to 200 nm, centered around 800 nm; (2)

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germanium (Ge), which is which is used for photodiodes that have the total bandwidth up to 400 nm, centered around 1,400 nm; and (3) indium-galliumarsenate (InGaAs), which is used for photodiodes that have a total bandwidth up to 600 nm, centered around a wavelength peak of 1,500 nm. Incoming optical signal

Incoming optical signal

-V

Antireflection coating

-V Antireflection coating

p+-type I-type

p-type I-type

p-type

n-type

n+-type

+V

(a)

(b)

+V

+ Load resistor

Width w I-depletion region

+ + + Hole drift

n

- - x

p

Electron drift (c)

Incoming photons

Figure 2.30 Structure of semiconductor photodiodes: (a) PIN photodiode, (b) Avalanche photodiode (APD), and (c) Electron-hole drifts across the depletion region.

The PIN photodiode is a layered structure with lightly doped I-region (I stands for “intrinsic”) placed between p-type and n-type semiconductor layers, as shown in Figure 2.30(a). Therefore, we can say that p-type sits on I-type, which is placed on the top of the n-type. (The name PIN denotes the nature of this positioning.) The PIN photodiode is reverse-biased, with very high internal impedance, which means that it acts as a current source and generates the photocurrent that is proportional to the incoming optical signal. The photon can give up its energy and excite the electron from the valence to conduction band only if it has energy equal or greater than the energy bandgap of the semiconductor material that is used. The transfer of the energy from the incoming photons to electron-hole pairs, which happens mostly in the I-region, is through the photodetection process. Some of the generated electron-hole pairs will eventually disappear traveling through the semiconductor structure due to the recombination process. We can assume that these carriers move along and pass some distances or diffusion lengths, which are equal to Le and Lh, for electrons and holes, respectively. The carrier lifetimes, which measure duration of the electrons and holes from the time

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they are generated in photodetection process until their recombination, can be represented by e and h, respectively. Carrier lifetimes are related to diffusion lengths by the expressions [17] (

)

(

)

(2.112)

where De and Dh are diffusion constants for electrons and holes. The incoming optical signal is absorbed along the depth distance x in accordance with the exponential law {

( )

[

() ]}

(2.113)

() is the material-dependent absorption coefficient at a specified where wavelength , and Pin is the incident optical power coming to the photodiode () is very strong and this is the main reason that a surface. The dependence particular semiconductor material can be used only over a limited wavelength range, as it is illustrated in Figure 2.37 for commonly used semiconductors. There are some cutoff wavelengths from both the upper and lower sides for each specific material. The upper cutoff wavelength is determined by width of the energy band gap, while the lower cutoff is result of the absorption process due to coefficient () dependence. The lower cutoff case corresponds to a situation in which photons are absorbed almost immediately after they penetrate the photodiode surface while producing electron-hole pairs with a very short recombination time. As a result, these carriers cannot contribute the photocurrent since they cannot be separated and collected by external bias voltage. The total photocurrent generated along the depletion region width w can be expressed as {

[

() ]}

(2.114)

where q is the electron charge equal to 1.6·10–19 Coulombs, h is the Planck constant equal to 6.63·10–34 Js,  is the optical frequency expressed in Hertz (Hz), and w is the absorption depth. There are several photodiode parameters that are important from the systems perspective: the quantum efficiency, photodiode responsivity, photodiode cutoff frequency, and total noise generated during detection process. The quantum efficiency  is defined as the ratio between the number of the electrons detected in the process and the number of the incident photons. This parameter is always lower than 100% and can be expressed as (

)

(2.115)

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Equation (2.115) can be rearranged and rewritten as (2.116) The responsivity R, introduced in Equation (2.116), is the ratio of the output current generated in PIN photodiode, and the incident optical power coming to the PIN. There is the following relationship between the responsivity and quantum efficiency: R

q   h 1.24

(2.117)

where = c/ is the optical wavelength of the incoming signal. 0.90 0.80 0.70 0.60 0.50 0.40 800

900

1000

1100 1200 1300

1400 1500 1600 1700

Wavelength [nm]

Figure 2.31 Responsivities of commonly used photodiodes expressed a function of wavelength.

The photodiode responsivity can be increased by increasing the size of the area at which the incident light falls. On the other side, the increase in size will slow the response process and limit the photodiode bandwidth. In addition, the responsivity is wavelength dependent, as shown by Equation (2.117). Such an unequal response over the wavelength band should be corrected through the subsequent signal processing in order to provide the same signal-to-noise ratio for all optical channels. The responsivity of commonly used PIN photodiodes is in the range (0.4–0.6) A/W for silicon-based PIN photodiodes, (0.5–0.7) A/W for germanium-based PIN photodiodes, and (0.6–0.85) A/W for InGaAs-based PIN photodiodes, as illustrated in Figure 2.31. The frequency response of the PIN photodiode and frequency bandwidth is characterized by the cutoff frequency fc that characterizes the response speed of the PIN photodiode. The frequency fc is inversely proportional to width of the Ilayer and the capacitance of the reverse biased p-I-n structure. The I-layer width can be decreased in order to decrease the total capacitance, but in such a case the responsivity of the photodiode will decrease. The responsivity of PIN photodiodes

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used in high-speed optical receivers is lower than the responsivity of photodiodes used in low-speed optical receivers. The cutoff frequency of a photodiode can be defined as [14] [

(

)]

(2.118)

where tr is the total transit time of the carriers, and RC is the RC constant of the circuit that consists of the photodiode capacitance and resistance of the external load, which is shown in Figure 2.32. To increase the speed of photodiodes, both time constants from Equation (2.118) should be reduced. As an example, the systems operating above 10 Gb/s need both tr and RC to be below 10 ps, while systems operating above 40 Gb/s need both tr and RC to be below 3 ps. The total noise generated during detection process is another important parameter related to photodetection process. The total noise, which contains the dark current, quantum noise, and thermal noise components, will be discussed in Chapter 4. The avalanche photodiodes (APD) are structured in a similar fashion to the one shown in Figure 2.30(b). The structure is optimized to support the amplification of generated electron-hole pairs before they reach the photodiode electrodes through an internal gain achieved by the process of impact ionization. The strong electrical field increases the kinetic energy of electron by accelerating them, so they became capable of generating new electron-hole pairs. The newly generated electrons are further accelerated by the electric field to generate additional electron-hole pairs by the impact ionization. The avalanche breakdown, which might occur if a number of generated electron-hole pairs grow fast and without a real correlation with an incident optical signal level, is prevented by adjusting the bias voltage to be below a critical value that would produce the breakdown condition. The breakdown bias voltage is specified in manufacturers’ product data sheet. The structure of APD is slightly different than the structure of the PIN since an additional layer is added to enhance the impact ionization process. This is the active layer where the avalanche multiplication occurs. The reverse bias voltage applied to APD can vary from tens to hundreds of volts, as opposed to the PIN bias voltage, which is usually only up to 10 volts. In addition to higher bias voltage, n-type and p-type layers are suitably doped to increase the carriers’ density at the junction. The following parameters are used to characterize the APD: (1) responsivity R, which is the ratio between the output current generated in APD photodiode and the input optical power coming to APD, which is about the same as responsively of the PIN photodiode; (2) frequency response of the APD photodiode and frequency bandwidth characterized by cutoff frequency fc, which depend on capacitance of the reverse biased structure; (3) the instantaneous avalanche signal gain M(t), which is a stochastic parameter that fluctuates around some average value ; and (4) the total noise generated during the detection of an incoming optical signal.

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The speed of the ADP can be increased by if the widths of two internal layers (I-layer and p-layer) are decreased in order to decrease the capacitance, but this will also decrease the responsivity of the photodiode. The cutoff frequency of APD is generally lower than the frequency associated with PIN photodiodes, which limits applications of APD bit rates up to 10–15 Gb/s. As for the noise, there is one more noise component associated to detection process in APD that comes in addition to the noise components found in PIN photodiodes. It is the shot noise related to the avalanche amplification and fluctuation of the gain M(t) around its average value. The APD noise parameters will be discussed in more detail in Chapter 4. Typical parameters associated to signal generated in the PIN and ADP, are summarized in Table 2.5. The noise parameters will be introduced in Chapter 4. Table 2.5 Typical values of optical amplifier parameters Parameter PIN Responsivity in A/W 0.7–0.95 Cut off frequency in GHz Up to 75 Internal gain

1

APD 0.7–0.9 Up to 15 Up to 100

Photodiodes used in an optical receiver are followed by the front-end and the preamplifier that converts photocurrent into a voltage signal that is eventually amplified, as shown in Figure 2.32. A photodiode and the frontend with amplifier can be integrated on the same substrate, which is often done in practice [95, 96]. +

+ Front-end amplifier

Optical signal

RL

Optical signal

Equalizer C

RL

Output voltage

Output voltage

C Front-end amplifier

High impedance front end

Transimpedance front-end

Figure 2.32 Optical receiver front-end schemes.

There are two main types of front-end that work together with photodiodes in an optical receiver, and they are high-impedance front-end and transimpedance front-end, as shown in Figure 2.32. The schemes from Figure 2.32 show that there is always a sole load resistor RL and the total equivalent capacitance C. The choice of the load resistance has an impact to overall design of optical receivers, and some trade-offs are included in the design process. Namely, in order to minimize the thermal noise component, the value of the load resistance should be as high as possible. However, the value RL also determines the value of the RC constant RC is Equation (2.118), which means that it should be as low as

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possible since a larger value for RL means that the speed of the photodiode will be limited to lower bit rates. Therefore, the trade-off that a designer should make is related to the balance between the speed and the noise level, or the signal-to-noise ratio. The transimpedance front-end amplifier design decreases the equivalent input load resistance from the photodiode point of view by factor (1+A), where A is the gain inserted by electrical preamplifier, while the total cutoff frequency fc is increased by the same factor. This positive effect is diminished by the fact that the total thermal noise is also higher as compared to the case where a high-impedance design is applied. However, an increase in the total thermal noise is much smaller than the total increase in the operational speed, which means that figure of merit works in favor of transimpedance design and that is the reason that transimpedance front-ends are widely used in high-speed optical communication systems. Another feature that works favorably for transimpedance front-end is its high dynamic range, which means that it can accommodate larger variations of the input optical power. It is because larger variations in input optical power translate into proportionally smaller variations of the front-end output voltage. The reader is advised to look at the literature related to optical receivers to find more details about front-end designs [91, 93].

2.8 PROCESSING OPTICAL COMPONENTS This section provides a description of design and operational principles of optical components used for adjustment of optical signal parameters. That is the reason we can refer to these components as the processing optical components. These components are shown in the lower part of Figure 2.1. More details processing optical components can be found in literature [61, 62, 98–108]. All processing components used in optical transmission systems and networks can be divided in two groups: the active and passive ones. The active components need some voltage either as a power supply or for adjusting the operational regime, while passive components can operate without any external electrical power. The most important active components are: lasers, optical modulators, photodiodes, optical amplifiers, wavelength converters, and optical switches. The main passive optical components are: optical couplers, optical multiplexers, and optical isolators. In some cases an optical component can be either active or passive, such as case for optical filters, as an example. Any optical component is characterized by set of parameters that are important from systems engineering perspective, and they are: the insertion losses, return coupling loses, polarization-dependent losses, channel signal cross-talk, inserted chromatic dispersion, and inserted polarization mode dispersion. There are several technologies that can be used in the components manufacturing process, such as fiber fusing, combination of graded-index (GRIN) rods and optical resonators, employment of planar optical waveguides, and application of

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optical fiber Bragg gratings. The next section contains a description of the operational principles of the most relevant processing optical components. 2.8.1 Components for Coupling, Isolation, and Adjustment of Optical Power An optical coupler is commonly used component that serve to combine or split optical signals at different points along the lightwave path. There are three major manufacturing schemes of optical couplers, which are based on fused optical fibers, planar optical waveguides, and combination of the GRIN roads and filters [61]. The first two schemes are illustrated in Figure 2.33. The fused tapered optical couplers from Figure 2.33(a) are produced when two optical fibers are first striped from claddings, and then two fiber cores are brought together. The fibers are then heated and stretched, which results in a waveguide structure that can exchange the energy between the branches. As a result, the optical power from input 1 is split between the outputs 1 and 2. The same happens with the optical power coming from the input 2. This basic structure is well known 2 x 2 optical coupler. The coupler effectively becomes 1 x 2 type if just one input port is active. The coupler structure with planar optical waveguides is similar to one just described, but with the planar optical waveguides are used instead of optical fibers. Output 1

Input 1

L

Input 2

Output 2

Input 1

Input 2

GRIN Optical filter rod

GRIN rod

Output fiber

(b)

(a)

Figure 2.33 Optical couplers: (a) fused tapered coupler, and (b) GRIN-rod +optical filter.

The optical power coupling capability of a directional coupler is characterized by the coupling coefficient k that is a function of the following: optical signal wavelength, coupling length L (as shown in Figure 2.33(a), cross-section parameters of the optical fibers or planar waveguides that are used, difference in refractive indexes between the waveguide region and surrounding layers, and the proximity of two axes related to either fibers or planar waveguides. When two waveguides are placed in close proximity, the newly combined waveguide structure is configured in a way that enables the signal coupling from one waveguide to the other. The output signals, or the electrical fields of the outputs from ports 1 and 2, can be expressed as function of the input fields as [3] [

( ) ] ( )



[

(

) (

( )

(

) )

][

( ) ] ( )

(2.119)

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where L is the coupling length,  is propagation constant of the modes in each waveguide, and is the coupling coefficient. The coupling coefficient is a function of the width of the waveguides, the refractive indexes of the waveguide structure, and the proximity of the two waveguides. If there is just one input (i.e., input 1), the outputs from the directional coupler can be expressed as [

( ) ] ( )



( )[

(

) (

In 3-dB coupler the split ratio is equation = (2i+1) /4

] )

(2.120)

= 0.5, and the coupling length satisfies the

(2.121)

where i is either a positive integer or zero. It is important to notice from Equations (2.119) and (2.120) that optical coupler splits input power per defined ratio, but also introduces a relative phase shift of /2 among the individual outputs. This relative phase shift plays an important role in the design of more complex elements, such as Mach-Zehnder modulators or hybrids used in coherent detection receiver. It is also important to know that it is not possible to combine two or more signals without inserting any loss, not because of the attenuation in the material, but because of the physics related to combining the individual electrical fields [3]. It is possible to change both the ratio of the optical powers directed to the outputs 1 and 2, and the wavelength content of the signals appearing at these outputs by changing directional coupler parameters. The parameters are usually adjusted in such a way that the peak of the coupling coefficient coincides with a specified wavelength. The coupling region should be protected from any external effects that might cause the change in the coupling coefficient, which is done by a proper packaging. In addition, the temperature control might be applied if necessary. The combination GRIN rod and optical filter can be affectively used to produce an optical coupler in a way illustrated in Figure 2.33(b). The pair of GRIN rod lenses is employed to collimate and transfer the light from the input to the output ports, while the optical filter selects the wavelengths that should be directed to specified port. There are three distinct optical signal flows that can be recognized in this design, and they are related to the input signal, the part that goes through the filter, and the reflected portion of the input signal. Therefore, the characteristics of the filter performance determine the amount and content of the transmitted and reflected optical signals. A variety of optical coupler designs can be achieved by inserting the optical filters with specified characteristics. There are several types of optical couplers used in optical communication systems. They are usually recognized as the optical taps and optical directional couplers. The optical taps are 1 x 2 optical couplers that are used for signal monitoring, while directional optical couplers present 2 x 2 structures that are

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used for power sharing. The typical coupler splitting ratios, between two outputs values are 1/99%, 5/95%, 10/90%, and 50/50%. Please notice that these numbers are the percentage values. Both, the fused fiber couplers and the GRIN rod devices are commonly used as both optical taps and directional couplers. More complex optical coupler structures are known as N x M optical couplers, where N and M are numbers that can be larger than 1 or 2. Such couplers are used either for power distribution among number of users, or for wavelength specific functions in optical systems. The wavelength specific couplers, which are also known as the WDM optical couplers, are a special group of directional 1 x 2 couplers designed for various applications, such as the pump power introduction, the wavelength band separation, the course wavelength multiplexing, and for the introduction and separation of the supervisory channel. Both types of optical couplers mentioned above (based on fused fiber technology and GRIN-rod structure) can be used for wavelength-specific applications. The fused optical fiber couplers are more suitable for applications where wavelengths, which should be split, are not close to each other. A typical example of this is the separation of signals placed around 1,300 nm from signals that are around 1,550 nm, or separation between contents around 980 nm and 1,480 nm. It is important to notice that the PDL in fused fiber couplers can be up to 0.2 dB, and varies with operational wavelength. On the other hand, the GRINrod based couplers devices provide a flat wavelength response across the wavelength band, while the PDL is relatively small (usually less that less than 0.1 dB). Optical isolator is an optical component used to prevent the impact of the back-reflected optical signals. Three main causes of back reflection are optical connectors, Rayleigh back scattering in fiber spans, and stimulated Brillouin scattering. Optical connectors back reflection is caused by the font surface of receiving fiber end within connector, while the Rayleigh and Brillouin scattering processes occur in a distributed manner. Semiconductor laser sources are particularly sensitive to any back reflections since they will cause additional noise and severe degradations of the system performance—please refer to Chapter 4. Furthermore, an optical amplifier needs optical isolator to prevent a lasing effect and to improve the amplifier performance by isolating the amplifier stages. The operation principle of commonly used optical isolators is based on utilizing a nonreciprocal change in the state of polarization of the incoming optical signal that occurs in some materials under the presence of the magnetic field. This effect is also known as the Faraday rotation [1]. First, the polarization is changed for any forward propagating signal that passes through the Faraday material. If there is a backward-propagating light that passes through the Faraday material again, the material will turn the polarization state again, rather than undoing the forward-induced polarization shift. The Faraday material is placed between the polarizer and the analyzer; any signal either forward or backward propagating is supposed to pass through both of them in order to continue its path, as shown in Figure 2.34(a). The polarization state of the forward-propagating signal will be

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rotated by 45o after passing through the Faraday material. The same will happen with the back-reflected backward-propagating signal. Therefore, the total rotation of the backward propagating signal will be 90 o. In such a case, the polarizer will not recognize the polarization state of the back-reflected signal, and it will be denied the further throughput.

Parameter

Table 2.6 Typical values of optical isolator parameters Typical value High performance value

Insertion loss in dB Isolation loss in dB

0.6 35

0.5 45

Polarization dependent loss (PDL) in dB Polarization mode dispersion (PMD) in ps

0.15

0.05

0.07

0.05

The key system parameters of an optical isolator are: (1) the return isolation loss that should be as high as possible; (2) polarization mode dispersion that should be as low as possible since an optical path can contain as much as 20 to 40 isolators; and (3) the insertion loss that should be as low as possible. The typical and high-performance values of optical isolator parameters are shown in Table 2.6. Variable optical attenuators (VOA) play important role in multichannel optical systems and networks, where optical signals might travel through different optical transmission paths before being combined again and processed together. Optical powers of different channels should be equalized at some reference points in order to assure the equal system performance for individual optical channel. The equalization of optical powers is related to a very precise adjustment of the optical signal powers. It is highly desirable that the process of power adjustment can also be remotely controlled. Variable optical attenuators serve that purpose very well, and they can be manufactured as separate optical components or integrated together with other devices, such as optical couplers or optical filters. The operational principle of the commonly used VOA is based on changing either the polarization state of the incoming polarized light or the signal loss in the material. Variable optical attenuators are not quite passive optical devices since some control voltage is needed to change the state of the material that is used. Optical attenuators are characterized by dynamic attenuation range, as well as the increments in attenuation during the adjustment process. The typical VOA dynamic range is between 0 and 40 dB, while it can be adjusted in increments that range from 0.1 to 20 dB [54]. The operation of an optical circulator is similar in nature to the operation of a revolving door in terms that the signal is supposed to exit at the next port, while propagation is unidirectional. In essence, an optical circulator is a cascade of several optical isolators that form a closed circle, as shown in Figure 2.34(b). The signal can go through just one Faraday rotator, since the entrance to the following

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isolator stage is effectively denied. Optical circulators are very often used within optical subsystems, such as optical add/drop multiplexers and dispersion compensators. Clockwise polarization rotation of forward signal by 45 degrees

Input #2 Faraday rotator

GRIN rod

Output fiber

Input signal

Input #1

Analyzer

Polarizer

Input #3

Clockwise polarization rotation of backward signal by 45 degrees (a)

(b)

Figure 2.34 Principle of (a) optical isolators, and (b) optical circulators.

It important to outline that a hybrid integration of multiple functions into a single device can greatly enhance the performance of the individual optical components. Such integration into a single package reduces the insertion loss, decreases both PDL and PMD, and increases the package reliability. The GRINrod technology that is commonly used for both optical couplers and optical isolators is quite suitable for different types of hybrid optical devices. Some examples of multifunctional optical components are: (1) WDM coupler plus isolator; (2) optical filter plus isolator; (3) WDM coupler plus isolator plus tap coupler; and (4) WDM coupler plus isolator plus bandwidth filter [40]. 2.8.2 Optical Switches Optical switches are optical components used for changing the direction of an optical signal from one lightpath to the other. They can be used for application scaling from the provisioning of lightpaths and protection switching, both of which do not need a fast switching time, to the optical packet switching and optical signal modulation where a high switching speed is required. An optical switching matrix can have basic 1 x 2 or 2 x 2 forms, or may be more complex in terms of the number of input and output ports. In this section we will consider that optical switches are relational ones that serve the purpose of establishing a relationship between specified inputs and the outputs. The relation is dependent on the control signals that are applied to the switch, while it is independent on the contents of the signal that should be switched. Therefore, such switches can be classified as circuit switches, since they perform function on the lightpath (i.e., optical circuit level). This is just opposite of the function of logical or packetbased switches that perform function on the block of the incoming bytes, while the control signal is related to the logical content of the data flow. Packet based

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switches will be discussed in Chapter 8. All optical circuit switches currently in use have the control switching function performed electronically, while the optical signal is being transparently routed from a specified input of the switch to a designated output. There is no relation to the data rate and format of the optical signals. Optical circuit switching, in general, is described by the N x N switching matrix that connects incoming signals at the input ports with outgoing signals from output ports. The following generic equation can be established to express the input-output connection

[

]

[

][

]

(2.122)

where Pin1, Pin2… PinN, and Pout1, Pout2… PoutN are optical signals at the input and output ports, respectively, while Sii (i=1, 2.. N) are connecting coefficients. In an ideal case these coefficients should take the value 1 or 0, where just N, out of the total number of N2 coefficients, would differ from zero. However, there is some insertion loss and signal crosstalk present in most practical cases and coefficients become complex numbers that describe both the change in the amplitude and in phase. Also, it can happen that more than one input signal is looking to be connected to the same output port, in which case we have signal contention. Any large switching matrix can be designed as a unique entity, with a broadcast and select approach by using a gating principle, or to be arranged as a combination of cascaded 2 x 2 basic forms. These two approaches are illustrated in Figure 2.35. Gate #1

Pin1 Gate #2

Pout1

Pin1

Pout2

Pin2

bar

Pout1

Gate #3

Pin2

Gate #4

(a)

bar

Pout2

(b)

Figure 2.35 Basic cross-connect structures: (a) gating (b) cross-bar.

In the N x N gate switch, each input signal is broadcasted in N directions by going through a 1 x N splitter (N=2 in Figure 2.35(a)). The signals then pass through an array of N x N gate elements, after which they are recombined in N combiners and sent to the N outputs. The gate elements can be implemented using optical amplifiers that can switch from the on to the off state and vice versa. In the on state, the amplifier also compensates for coupling losses and losses incurred at the splitters and combiners. Also, gating can be done by using spatial displacement.

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In N x N directional switch, there are a number of cascaded 2 x 2 crossconnect elements. Each of these elements is known as cross-bar switch since it routes optical signals from two input ports to two output ports by taking either one of these two states: cross-state and bar-state, as shown in Figure 2.35(b). In the cross state, the signal changes direction and comes out from different output port, while in the bar state the signal continues its logical flow connecting the ports with the same numbering. In N x N structure, the cross-bar function is repeated a number of times before signal reaches the output port. An N x N switch is made by using multiple basic switch elements from Figure 2.35(a, b). The complexity of the switch depends on the number of basic building blocks that are required. The overall performance is evaluated by the insertion losses of each element, number of signal crossovers, and mutual interaction among individual paths that may happen during propagation through the entire structure. If the paths of two signals cross or interfere with each other, both the power loss and crosstalk may occur. In addition, bigger switch structures are characterized by their contention resolution capability, and they can belong to either nonblocking or blocking categories. The nonblocking feature is related to capability that any unused input port can be connected to any unused output port, thus performing every possible interconnection pattern between the inputs and the outputs. However, if any interconnection pattern cannot be realized, the switch is classified to a blocking category. The majority of applications require a nonblocking capability, which can increase the overall switch complexity. The nonblocking requirement can be relaxed by applying wide-sense nonblocking principle which means that any unused input can be connected to any unused output, but without requiring that any existing connection is rerouted. A strict-sense nonblocking switch allows any unused input to be connected to any unused output without any additional constraints [102]. There are several recognized methods to design large N x N crossconnects, which are described in more detail in [90, 102]: (1) crossbar wide-sense nonblocking architecture requiring N2 of 2x2 switching elements and making the insertion losses that range from [ to (N – 1)], where  is a proprietary attenuation coefficient per single building block element; (2) Clos architecture requiring [4√ ]N3/2 of 2x2 switching elements and making insertion losses that range from [3 to (5√ – 5)]; (3) Spanke architecture requiring 2N of 1 x N switching elements and making equalized insertion losses of [2 Nfor any possible switching combination(4) Benes architecture requiring N·log2N – N/2 of 2 x 2 switching elements, and making insertion losses of [2·log2N – 1] ; (5) Benes-Spanke architecture requiring N·(N – 1)/2 of 2 x 2 switching elements and making insertion losses that range from N/2 to N. Only Clos and Spanke architectures are considered to be strictly nonblocking, while Benes and BenesSpanke architectures require some rerouting of connections to achieve nonblocking status.

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It is important for all optical switches that the extinction ratio, defined as the ratio between the optical powers corresponding to active state (established signal flow) and the passive state (idle state), is as high as possible. In addition, a good isolation is required between lightwave paths traveling between different input and output port pairs. Finally, optical switches should have small insertion loss, and to be insensitive to the polarization state of the incoming optical signal. There are several major technologies used today to manufacture optical switches. Accordingly, several types of optical switches can be recognized: electro-optic switches, thermo-optical switches, mechanical switches, and semiconductor amplifier-based switches. Electro-optical switches are based on optical couplers in which the coupling ratio is changed by the voltage applied to the electrodes. The voltage takes two discrete values enforcing the on and off states of the switch. Electro-optical switches can operate relatively fast and switch the lightpath in less than 1 ns, which makes them good candidates for high-speed optical networking applications. Although the basic 2 x 2 electro-optical switches can be integrated in more complex structures on a single substrate, the integrated structure tend to have relatively high insertion loss. In addition, these switches tend to insert polarization differentiated losses (PDL). Electro-optical switches can be also based on application of the Mach-Zehnder interferometer, where the outside voltage is applied to change the refractive index in the interferometer arms. The phase difference between signals in two arms is changed by varying the refractive index in the arms, which results in constructive or destructive interference at the output ports. Thermo-optic switches are also based on employment of Mach-Zehnder interferometers, where the temperature is applied to change the refractive index in the interferometer arms. The switching process usually takes several milliseconds to select the output port where the constructive interference takes place. Thermooptical switches are relatively simple and cheap, which makes them suitable for network restoration and protection switching applications. Mechanical switches perform lightpath redirection through some mechanical action, such as moving a mirror in and out of the lightwave path. Also, mechanical switch can be realized by using a flexible directional coupler that changes the coupling ratio if it is bent or stretched. The MEMS (micro-electro mechanical system) design is a well-known example of mechanical switch since movable mirrors are used to redirect lightpath. Finally there is a group of optical switches based on semiconductor optical amplifiers (SOA), where the switching process is performed by varying the bias voltage. If the bias voltage is low enough, the device absorbs the incoming optical signal, which corresponds to off-state of the switch. However, the input optical signal is amplified by increasing the bias voltage since the population inversion is achieved, and this situation corresponds to on-state of the switch. The change from the absorption to the amplification state can be done very fast and usually takes less than 1 ns. Although the SOA-based optical switches have amplification capability and can be integrated in a larger switching matrix, the noise

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accumulation and lightpath isolation are of some concern in practical applications. In next section we will describe two technologies that found wider applications in design of various optical switches: liquid crystal-based switching technology [98, 99] and MEMS technology [100, 101]. Liquid crystal switches are based on the employment of liquid crystal cells. These switches operate either by using polarization effects to perform the switching function or by producing multipixel interferometric images (fringes). In the first case, if a voltage is applied to a liquid crystal cell, it will cause the polarization of the light passing through the cell to be rotated. If the cell is combined with the passive polarization beam splitters and combiners, a polarization-dependent switching can be performed. Accordingly, there is similarity in the operation principle of the liquid crystal switch with that of an isolator, which has been described earlier. All elements that are needed (the passive polarization beam splitter, combiner, and switching cell) can be realized by employing a liquid crystal cell array. The amount of the polarization rotation in the liquid crystal cell is directly proportional to the applied voltage. Therefore, in addition to digital control of the polarization, the same approach can be used to realize variable optical attenuator (VOA), which is a very important element to control and equalize signal levels in multichannel environment. Accordingly, VOA can be an integral part of liquid crystal cells array that performs switching function and controls the output power from the switch. The switching time of liquid crystal cells is on the order of a few milliseconds. Input Ray

Output Ray

Frequency Bands

Glass Liquid Crystal Cell CMOS (silicon die)

(a)

(b)

Figure 2.36 LCOS switching technology: (a) single-cell operation, and (b) LSOC cell matrix with frequency bands.

The liquid crystal switching technology based on producing multipixel interferometric images (fringes) is commonly used in wavelength selective switch (WSS) devices. The solid-state WSS device is produced by integrating liquid crystal cells on silicon, which is known as LCOS (liquid crystal on silicon) technology [99]. The liquid crystal cell is modified so it operates as a phase spatial light modulator, as illustrated in Figure 2.36(a). Wavelengths are dispersed across width of megapixel LCOS matrix enabling flexible switching to multiple directions of arbitrary wavelength content, as shown in figure 2.36(b). Also, as a phased-based switching matrix, it allows per channel optical power adjustment.

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Micro-electro mechanical switch (MEMS) is a miniature mechanical device fabricated also on silicon substrate that operates on the light reflection principle. The MEMS contains a number of very small movable mirrors fabricated in silicon. The size of the mirrors can range from a fraction of a millimeter to several millimeters. The mirrors are usually manufactured and packaged as arrays by applying standard manufacturing process of a semiconductor wafer growth. A switching process is performed by deflecting mirrors from one position to another by using known electromagnetic, electrostatic, or piezoelectric methods. The movement can be digital in nature, which means that the mirror can take just two positions or states. Such a switch is known as two-dimensional (or 2-D), and its operational principle is illustrated in Figure 2.37(a). In the original state, the mirror is flat and the light beam is not deflected but follows the original direction. After switching action is taken, the mirror takes the other state or vertical position, thus deflecting the incoming signal. The deflection principle corresponds to a crossbar arrangement well known in the realization of bigger structures that contain N inputs and N outputs. In MEMS manufacturing, the number of input and output ports is usually in the range of 16 to 64.

Micro mirror in “ON” position

ON

OFF

(a)

(b)

Figure 2.37 MEMS technology: (a) 2-D switch, and (b) 3-D switch.

MEMS can be realized with three-dimensional movements of the mirrors, and such a space design is known as 3-D, illustrated in Figure 2.37(b). The mirror can rotate around two axes, thus taking any analog position in the space and deflecting the signal in any given direction. The space movement of the mirrors is enabled by a two-frame design. The inner frame is connected to an outer frame thus allowing a free rotation on two distinct axes and taking a continuous range of angular deflections. The control of 3-D MEMS mirrors is very sophisticated by using precise servo control mechanisms. The 3-D MEMS is suitable to design of a large N x N switch, if two arrays of analog beam steering mirrors are used as in Figure 2.37(b), thus following the

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Spanke architecture [90]. Each array has N mirrors, and each of them is associated with just one input-output port. The signal is switched from any input port to any output port by performing two reflections on its way. This process, if precisely arranged, should not insert any additional cross-talk between the ports. In general, MEMS-based switches have low insertion loss and low cross-talk between individual lightpaths. In addition, their sensitivity to the polarization state of the optical signal is also relatively low. On the other side, the long-term reliability of MEMS-based switches is still of some concern since there is a mechanical movement involved that needs precise alignment at any specific moment. MEMS can switch a wavelength path within several milliseconds. 2.8.3 Wavelength Converters The wavelength converter has become an important component as we enter into a new era of advanced optical networking. Wavelength converters can be used to translate one wavelength to the other for some of the following reasons: (1) to convert the input wavelength to the output one for adaptation purposes, either to improve the overall transmission capability, or to make it more convenient for optical networking; (2) to convert a wavelength within one administrative network domain to a different wavelength within the other administrative domain to facilitate network management and wavelength assignment, as discussed in Chapter 8; and (3) to convert a wavelength within a single network domain to improve utilization of the optical channels, and to prevent lightpath contention. The wavelength conversion can be correlated with the need for full regeneration of the optical signal. A full regeneration, or 3R (reamplification, reshaping, and retiming), is needed when signal quality is compromised and there is no other way to clean the signal and to improve the bit-error-rate. Full regeneration can be done either electrically or by some optical means. Some of the methods applied for wavelength conversion can also be used for the optical signal regeneration. The simplest way for both the wavelength conversion and signal regeneration is to apply the O-E-O (opto-electronic-optic) conversion. In such a scheme, an incoming optical signal is converted to electrical signal by photodiode and then is reconverted back by using another optical source at the wavelength that is different than the one associated with the incoming optical signal. Meanwhile, some electronic signal processing related to 3R functions is commonly done between two conversions. In some cases just one or two of aforementioned functions can be performed (1R or 2R schemes). The O-E-O wavelength conversion and signal regeneration are more traditional approach, which serves the purpose well from the operational point of view. However, both the cost and the complexity of the method are relatively high, and that is the reason why some advanced methods of optical wavelength conversion are needed. Optical wavelength converters should provide high-speed operation and optical signal transparency (in terms of the modulation format and bit rate),

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provide signal reshaping capability and possibly cascade few stages together to perform multihop connections, which are often needed in optical networks. Also, it is required that they provide smaller frequency chirp, polarization insensitivity, and high extinction ratio of the converted optical signal. Gain

1 Optical filter

p (CW)

p

p Input power

1

Conversion scheme

Signal Conversion (inversion)

Figure 2.38 Wavelength converter based on SOA.

There are several methods that could be used for wavelength conversion purposes, which are based either on the employment of optical gating, or on the new frequency generation through nonlinear effects [103–108]. The converters that are based on optical gating include semiconductor optical amplifiers to stimulate either cross-gain modulation or cross-phase modulation as a mean to change the value of an input wavelength. The operational principle of a wavelength converter, which is based on cross-gain modulation (XGM), is illustrated in Figure 2.38. The semiconductor optical amplifier in Figure 2.38 plays the role of an optically controlled gate, whose characteristics are changed with the intensity of an input optical signal. Any increase in the input signal will cause a depletion of the carriers in the active region, which effectively means that the amplifier gain drops. That change occurs very fast and follows dynamics of the input signal on a bit-per-bit basis. If there is a probe signal with lower optical power, it will experience a low gain during “1” bits and a high gain during “0” bits. Since the probe wavelength p is different than the input signal wavelength, the information content will be affectively transferred from the input signal to the probe. It is also possible to use a tunable probe, which will produce a tunable output signal. One should notice that the output signal is out of phase, or inverted, with respect to the input signal. This is because the gain of a semiconductor amplifier structure decreases with the level of the input optical signal. The cross-gain modulation wavelength converter can also operate in two modes, which are known as the counterpropagation mode and copropagation mode. In counterpropagation mode, the input signal and the probe come from different sides of the semiconductor amplifier structure, while the converted signal

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exits just from the input side. This operational scheme does not require any optical filter. In addition, it is possible to generate output signal at the same wavelength as the input one, which means the signal regeneration is achievable with a single stage. However, counterpropagation mode is not suitable for high-speed operations. On the other hand, the copropagation scheme, which is illustrated in Figure 2.38, needs an optical filter at the output to eliminate the remaining portion of the input signal. It is not possible to have an output signal at the same wavelength as the input one, which means that another stage is needed to perform the optical regeneration. However, wavelength convectors based on the copropagation design can operate at higher modulation speeds [108]. Each wavelength conversion is accompanied by optical power penalty, which is dependent on the wavelength converter design and the levels of the input optical signal and probe. It looks that the input signal ranging from –6 dBm to –4 dBm is the most convenient for the SOA-based wavelength conversion since it introduces a minimal power penalty. The power penalty ranges from 0.5 dB to 1.5 dB if the probe level is around –10 dBm. The power penalty also depends on the nature of the conversion process, in such a way that a downward conversion in wavelength introduces lower power penalty than the upward conversion. In summary, wavelength convertors based on cross-gain modulation effect offer some good features such as high-speed operation, polarization insensitivity, and simple implementation. However, they suffer from frequency chirping of the output signal, limited extinction ratio (lower than 10 dB), and limited level of the input signal that can be efficiently converted. A

Carrier density C

B

SOA1

E D

SOA2

phase

Input power

Figure 2.39 MZ SOA wavelength converters.

Wavelength converters based on the optical gating use SOA that stimulates the cross-phase modulation as a way to change the value of input wavelength. These amplifiers are placed in the arms of Mach-Zehnder interferometer, as illustrated in Figure 2.39. Any variation in carrier density due to the input signal variation will change the refractive index in the active regions of semiconductor amplifiers, while the changes in the refractive index will change the phase of the probe signal. The phase modulation induced through this process can be converted to amplitude modulation by the Mach-Zehnder interferometer through the process

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of constructive or destructive interference explained in Section 2.4. The power of a converted optical signal can be expressed as [107]

G  G



 2 G1G2 cos( ) (2.123) 8 where G1, and G2 are the gain coefficients of the amplifiers in arms 1 and 2, respectively, while  is the phase difference between signals at the output of arms 1 and 2. The phase difference is directly proportional to the level of input signal Pin. The maximum output level is achieved if there is no phase shift at all, while the minimum level is obtained when the phase shift reaches a value of (–  radians. On the other hand, the main advantage of the XPM design is that it needs less signal power to achieve the same effect as compared with converters based on cross-gain modulation. It means that a lower signal power can be used in combination with higher probe level to produce the better extinction ratio. There are several design schemes proposed so far to optimize characteristics of the wavelength converter based on the XPM effect. Some of them are all-active versions of semiconductor optical amplifiers and Mach-Zehnder resonators. In addition, wavelength converters based on cross-phase modulation can also serve as 2R (reamplification, reshaping). The 2R functionality is basically the same as one presented in Figure 2.39, except that two stages are needed for conversion to the same optical wavelength as the input one. In summary, we can say that the XPM-based wavelength converters offer the following benefits: fast operation (above 40 Gb/s), high extinction ratio (few decibels higher than the extinction ratio of the input signal), operation at medium input optical powers (ranging from –11 dBm to 0 dBm), high signal-to-noise ratio so they can be cascaded, and are polarization insensitivity. However, their implementation is relatively complex since they need a precise control of the interferometer arms. The total power penalty that occurs due to wavelength conversion is lower than 1 dB. The operation of the XPM-based 3R optical regenerator (reamplification, reshaping, retiming) is illustrated in Figure 2.40. The gating signals in this scheme are introduced through separate input control ports, while the energy contained in the gating pulses determines the beginning and the end of the process [107]. In Figure 2.40, two semiconductor optical amplifiers (SOA) are differently biased to achieve an initial phase difference of radians. The control pulses are applied to the inputs B and C, while the incoming signal is brought to the input A. The destructive interference will happen if there are no control signals at the gates B and C since there is a phase difference of radians between the output ports D and E. Therefore, the outcome does not depend on the signal level at the input A. The time difference t between two control pulses applied at the gates B and C will determine the gating window. The switch-on control pulse will set-up 2 radians phase difference between the outputs D and E of two interferometer arms. That difference will come back to radians after switch-off control signal is Pout  Pprobe

1

2

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applied to the gate C. The phase difference between outputs D and E will fluctuate back and forth between and 2 after each action of the control pulses. The gate will be open for the time period that coincides with the phase difference of 2 radians, while it will be closed whenever the phase difference takes the value of radians. Since the scheme from Figure 2.40 produces the output wavelength different than input one, two stages are needed for full 3R functionality at the specified wavelength. Control Pulses

B

Signal Pulses

A

SOA1

D F E

Control Pulses

C

SOA2

t

t

Figure 2.40 Optical 3R scheme based on SOA and MZ interferometer, after [107].

Another major group of wavelength converters is based on the four-wave mixing (FWM) process, which is intentionally stimulated in some semiconductor structures or in optical fibers. The four-wave mixing process, otherwise explained in Section 3.5.2, involves three optical wavelengths 1, 2, and 3 to produce the fourth one (4). The FWM process, which is quite detrimental from a transmission point of view, is intentionally induced in this case to initiate a wavelength conversion. Such situation can be easily created in a highly nonlinear medium, such as dispersion-shifted fiber (DSF). In addition, the FWM process can be induced if a semiconductor optical amplifier is used as an active nonlinear medium, as shown in Figure 2.41 [108]. probe

Bias

in in probe

Optical filter

out

out

Figure 2.41 Wavelength conversion based on the FWM process.

The FWM process is centered around a strong probe signal, which produces a mirror-like replica of the input wavelengths that interact with the probe. The three wavelengths from Figure 2.41 are connected by the following relation

1/ out  1/  probe  (1/  probe  1/ in )

(2.124)

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where in, out and p are wavelengths related to the input, output, and probe signals, respectively. The wavelength converters based on the FWM process are fully transparent and capable to operate at extremely high speeds, which is the major competitive advantage. These devices are also polarization insensitive. However, they require high power levels to achieve high efficiency, while the full functionality require the tunable pumps and optical filtering of the output signal.

2.9 SUMMARY In this chapter we described characteristics of the optical components that are commonly present along a lightpath, thus performing various functions related to optical transmission and networking photonic networking. The topics about light sources, optical modulators, photodetectors, optical amplifiers, optical filters, optical switches, and optical multiplexers are discussed in more detail. The role of the other components, such as optical isolators, couplers, or circulators is also explained. A basic description of optical fibers is also provided in this chapter, while a detailed analysis of the optical fiber transmission properties is presented in Chapter 3.

PROBLEMS 2.1 A step-index fiber has a core refractive index of 1.45. What is the maximum refractive index ncl of the cladding to have the light entering the fiber-air interface under the angle of /4 degrees to propagate through the fiber? 2.2 The modal dispersion in a step-index multimode fiber is 15 ns/km. Fiber has a refractive index in the core nco=1.50. What is the maximum bit rate to achieve transmission over 20 km? What is the maximum acceptance angle of the incident rays? 2.3 Find the expression for numerical aperture of graded-index multimode fiber with a parabolic index profile. What is the radius where the numerical aperture is just half of its value at the fiber axis? 2.4 An analog electrical signal with bandwidth of 10 MHz and maximum amplitude of 128V is converted to digital signal. The signal is converted to a binary digital stream with 7 bits per digital word. How far can it be transmitted over a silica-based step-index optical fiber having relative refractive index difference between the core and the cladding equal to 1%? 2.5 What would be a composition of the quaternary InGaAsP alloy for making semiconductor lasers operating at = 1,600 nm? 2.6 An InGaAsP laser operating at 1,550 nm has 300 m cavity length, which has an internal loss of 25 cm–1. Find the active-region gain required for the laser

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to reach threshold. Assume that the mode index is 3.3 and the confinement factor is 0.4. 2.7 In Problem 2.6, find the active-region gain required for the laser to reach threshold if one end of the laser cavity ends is coated by dielectric reflector so that reflectivity is now 90%. 2.8 Find the expression for P(I) for a laser operating in a steady state. Use laser rate equations and assume that spontaneous emission is very low. 2.9 An InGaAsP laser has 200-m-long, 1-m-deep and 3-m-wide cavity, with an internal loss of 30 cm–1. It operates in a single mode with the modal index 3.3 and the group index 3.4. Calculate the photon lifetime and the current threshold value if carrier lifetime is 2 ns. Assume typical values of InGaAsP compound parameters. How much power is emitted from one facet when the laser is operated twice above threshold? 2.10 For the laser from Problem 2.10, which operates twice above the threshold, calculate the slope quantum efficiency and the external quantum efficiency. Assume that the internal quantum efficiency is 95%. 2.11 The semiconductor laser is modulated by the current using I(t) = Ib + Imp(t ), where p(t) represents a rectangular pulse of 100 ps duration. Assume that Ib/Ith = 0.9, Im/Ith = 2.5. Solve the rate equations (2.13) and (2.14) numerically assuming that p = 3 ps, e = 2 ns, and Rsp= 2/p. Assume that the total gain G from Equation (2.18), with GN = 104 s–1 and N0= 108, should be modified and multiplied by factor (1 – NLP), where NL = 10–7, to express the modulation conditions [42]. Plot the optical pulse shape and the frequency chirp (assume that chirp=2). Explain the results. 2.12 Find the spectral linewidth and modulation bandwidth of a DFB laser from Problem 2.9 assuming that parameter chirp is equal to 5, and that I/Ith = 2. Also assume that p = 3 ps, e = 2 ns, and Rsp=G·nsp= 2/p. What is the laser modulation bandwidth? 2.13 The LED radiating at 1,310 nm is made of InGaAsP material with the refractive index n = 3.3. Find the output power from the LED if it is biased with the current of 100 mA. Assume that the internal quantum efficiency is 95%. What is the maximum modulation frequency of the LED where the output power is decreased by 3 dB as compared to stationary value? Assume that carrier (electron) lifetime is 2 ns. 2.14 What should be the length and the mirror reflectivities of the Fabry-Perot filter that is used to select N = 40 DWDM channels spaced apart by  =0.4 nm? Assume a 40-Gb/s bit rate, a refractive index of 1.5, and an operating wavelength of 1,550 nm. 2.15 An optical filter is based on the chain of MZ interferometers. How many stages are needed to isolate one of 128 optical channels that are tightly spaced and occupy 30-nm-wide C-band. What is the delay of the fourth interferometer in the chain?

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2.16 What is the active length the acousto-optic filter arms that that is needed to have a full width half maximum (FWHM) passband width of 0.4 nm at an operating wavelength of 1,550 nm? What is the period of the Bragg grating created by an acoustic wave? Assume that nTM – nTE =0.07. 2.17 A DWDM system has 40 wavelengths with the interchannel spacing of 50 GHz. One of the channels needs to be selected by a filter with a 3dB passband of bandwidth of 20 GHz. If FP, MZ cascade, and AOTF are used, please explain what would be the best choice to provide low crosstalk from adjacent channels. What parameters of the optimum filter you would choose? 2.18 A laser diode has a threshold current value ITh=20 mA. The output power for the currents in the range from 0 mA to Ith increases linearly to reach value of –10 dBm at the threshold. The slope of the power-current curve for currents above the threshold value is 0.08 W/A. The laser is modulated by the electrical current containing the string of zeros and ones rectangular pulses. Find the magnitude of the modulating current and the value of a bias current to achieve the extinction ratio of 16 dB (ratio of powers related to “1” and “0” pulses). Explain your results and choice and illustrate it in a diagram. 2.19 In Problem 2.18, the laser is modulated with the pulses Gaussian shape ( )

[

] where P0=4 mW, and T0=20 ps. Assume that the

adiabatic parameter ≈10 THz/W, and the laser chirp parameter is equal chirp=3. Plot the function of instantaneous frequency shift during modulation process by the specified pulse shape in parallel with the plot of the pulse shape. Provide comments on obtained results. 2.20 A lithium-niobate-based Mach-Zehnder modulator is operated at wavelength l=1,550 nm. The length of the electrodes is l=1 cm, and the thickness of the waveguide is d=10 mm. Calculate the voltage needed to cause phase shift of radians. What is the ratio of the output and input powers if modulator is biased with the voltage of 1.5V. 2.21 The saturation power of an erbium-doped fiber amplifier is 16 mW, and amplifier acquires 3 dB of the gain per each mW of pump power. The pump power is set to 8 mW. What is the maximum input power that can be amplified without entering saturation regime? 2.22 The gain profile of an optical amplifier has the FWHM of 500 GHz. What is amplifier bandwidth when it is adjusted to provide 15-dB and 25-dB gain? Assume that there is no gain saturation. 2.23 An EDFA produces 0.5-mW output power when a microwatt signal is present at the input. What would be the output power when a 1-mW signal is incident on the same amplifier? Assume that the saturation power is 12.5 mW. You may use MATLAB or other software to find a solution. 2.24 Find the output power signal of the 10-km-long Raman amplifier that is pumped in the backward direction by using 600 mW of pump power. Assume that signal and pump experience losses of 0.2 dB/km and 0.3 dB/km, respectively, with the parameters Aeff =60 m2 and gR = 6 · 10–14 m/W.

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2.25 The EDFA should be designed to amplify wavelengths between 1,580 nm and 1,610 nm (the L-band) each separated by 50 GHz. Calculate the number of wavelengths that could be amplified within this range and spacing. Calculate the required range in energy transitions to support the entire Lband. 2.26 Two-stage EDFA is used to accommodate losses on the optical link. The gain profile of the optical amplifier placed in one of the stages (denoted as AMP A) has the FWHM of 700 GHz, while the other stage optical amplifier (denoted as AMP B) has the FWHM of 1 THz. One amplifier should provide gain to compensate for the loss in the preceding 100-km-long fiber span (transmission is in C-band), while the other compensates for the loss in the ROADM box, which accounts for approximately two-thirds of the loss at the fiber span. Draw schematics of a two-stage amplifier with ROADM box in between and denote the pump source parameters (power, wavelength) with respect to each other to achieve these gains. How many channels should be amplified if each of them has a bandwidth of 0.1 nm and they are spaced 0.2 nm from each other? 2.27 What is the threshold power for stimulated Brillouin scattering for a 50-km fiber link operating at 1,550 nm and having a loss of 0.2 dB/km. How much does the threshold power change if the operating wavelength is changed to 850 nm, where the fiber loss is 1 dB/km? Fiber has an effective area Aeff = 50 m2 and Brillouin gain gB = 5 · 10–11 m/W over wavelengths in question. 2.28 Explain why in a photodiode only valence band electrons can absorb the incoming photons. What is the reason why that the conduction band electrons cannot absorb the optical signal energy? 2.29 InGaAs Pin photodiode has responsivity of 0.8 A/W at wavelength of 1,550 nm. That responsivity is 0.7 A/W at the wavelength 1,400 nm. Operating at 1,550 nm, a photocurrent of 1 mA has been detected. Calculate the absorption coefficient of the material at wavelengths of 1,550 nm and 1,400 nm. The absorption depth is equal to 10 mm. 2.30 Explain the difference between different front-end designs. What design would you choose if bit rate is 25 Gb/s? What photodiode would be the best match to work with front-end and why? 2.31 Consider the 32 x 32 optical switch can be built by using any of the following designs: (1) crossbar wide-sense non-blocking architecture, (2) Clos architecture, (3) Spanke architecture, (4) Benes architecture; and (5) BenesSpanke architecture. Suppose that each 2 x 2 switch building block has crosstalk suppression of 50 dB and insertion loss of 0.05 dB, while 1x32 building block has a crosstalk suppression of -60 dB and insertion loss of 0.05 dB. What is the overall maximum crosstalk suppression and maximum insertion loss of each architecture? If we wanted an overall crosstalk suppression of 40 dB in a crossbar wide-sense architecture, what should the crosstalk suppression of each switch be?

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2.32 Explain why the wavelength conversion is useful in multichannel optical networks. Calculate the wavelength of the probe in an FWM wavelength converter that would convert an input signal at 1,600 nm to the output signal of 1,400 nm.

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[10] Polyanin, A. D., and Manzhirov, A.V., The Handbook of Mathematics for Engineers and Scientists, London: Chapman and Hall, 2006. [11] Papoulis A., Probability, Random Variables and Stochastic Processes, New York: McGraw Hill, 1984. [12] Proakis, J. G., Digital Communications, 5th edition, New York: McGraw-Hill, 2007. [13] Couch, L. W., Digital and Analog Communication Systems, New York: Prentice Hall, 2007. [14] Agrawal, G. P., Fiber Optic Communication Systems, 4th edition, New York: Wiley, 2010. [15] Cvijetic, M., Coherent and Nonlinear Lightwave Communications, Norwood, MA: Artech House, 1996. [16] Gower J., Optical Communication Systems, 2nd edition, Upper Saddle River, NJ: Prentice Hall, 1993. [17] Keiser, G. E., Optical Fiber Communications, 3rd edition, New York: McGraw-Hill, 2000. [18] Okoshi, T., Optical Fibers, San Diego, CA: Academic Press, 1982. [19] Buck, J., Fundamentals of Optical Fibers, New York: Wiley, 1995. [20] Marcuse, D., Light Transmission Optics, New York: Van Nostrand Reinhold, 1982. [21] Snyder, A. W., and Love, J. D., Optical Waveguide Theory, London: Chapman and Hall, 1983. [22] Li, M. J., and Nolan, D. A., “Optical fiber transmission design evolution,” IEEE/OSA Journ. Ligthwave Techn., Vol 26(9), 2008, pp. 1079-1092. [23] Jeager, R. E. et al., “Fiber drawing and control” in Optical Fiber Telecommunications, Miller S. E. and Chynoweth A. G. (editors), New York: Academic Press, 1979.

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[24] MacChesney, J. B., “Materials and processes for preform fabrication - modified chemical vapor deposition and plasma chemical vapor deposition,” Proc. IEEE, 68(1980), pp. 1181–1184. [25] Izawa T., and Inagaki, N., “Materials and processes for fiber preform fabrication – vapor phase axial deposition,” Proc. IEEE, 68(1980), pp. 1184–1187. [26] Murata, H., Handbook of Optical Fibers and Cables, New York: Marcel Dekker, 1996. [27] Miller, C. M. et al., Optical Fiber Splices and Connectors, New York: Marcel Dekker, 1986. [28] Russel, P. J., “Photonic Crystal Fibers,” IEEE/OSA Journ. Ligthwave Techn., Vol. 24(12), 2006, pp. 4729–4749. [29] Morioka, T, et al., “Enhancing optical communications with brand new fibers,” IEEE Commun. Mag., Vol. 50, no. 2, Feb. 2012, pp. 40–50. [30] Gloge, D., and Marcatili, E., “Multimode theory in graded-index fibers,” Bell Sys. Tech. J., 52, Nov 1973, 1563–1578. [31] Awayi, A., et al, “World First Mode/Spatial Division Multiplexing in Multicore Fiber Using Laguerre Gaussian Mode,” Proc. 2011 European Conf. Opt. Commun. (ECOC), Geneva, Switzerland, paper We.10.P1 [32] Abedin, K., et al., “Amplification and Noise Properties of of an Erbium Doped Multicore Fiber Amplifier,” Opt. Express, Vol. 19, no. 17, May 2011, pp. 16715–16721. [33] Zhu, B., et al., “Seven-Core Multicore Fiber Transmission for Passive Optical Network,” Opt. Express, Vol. 18, no 11, May 2010, pp. 117–122. [34] Cvijetic, M., “Dual mode optical fibers with zero intermodal dispersion,” Optical and Quant. Electron., Vol. 16., 1984, pp. 307–317. [35] Chuang, S. L., Physics of Optoelectronic Devices, 2nd edition, Hoboken, NJ: Wiley, 2008. [36] Chow, W. W., and Kroch, S. W., Semiconductor Laser Fundamentals, New York: Springer, 1999. [37] Ye, C., Tunable Semiconductor Laser Diodes, Singapore, Singapore: World Scientific, 2007. [38] Morthier, G., and Vankwikelberge, P., Handbook on Distributed Feedback Laser Diodes, Norwood, MA: Artech House, 1995. [39] Kressel, H., (editor), Semiconductor Devices for Optical Communications, New York: SpringerVerlag, 1980. [40] Digonnet, M. J., (editor), Optical Devices for Fiber Communications, Bellingham, WA: SPIE Press, 1998. [41] Siegman, A. E., Lasers, Mill Valley, CA: University Science Books, 1986. [42] Agrawal, G. P., and Duta, N. K., Semiconductor Lasers, 2nd edition, New York: Van Nostrand Reinhold, 1993. [43] Henry, C. H., “Theory of the linewidth of the semiconductor lasers,” IEEE J. Quantum Electron, QE-18(1982), pp. 259–264. [44] Chang-Hasnain, C., “Monolithic multiple wavelength surface emitting laser arrays,” IEEE Journal Lightwave Techn., LT-9(1991), pp. 1665–1673. [45] Lee, T. P., “Recent advances in long-wavelength lasers for optical fiber communications,” IRE Proc., 19(1991), pp. 253–276.

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[46] Arakawa Y., and Yariv, A., “Quantum-well lasers – Gain, Spectra, Dynamics,” IEEE Journal Quant. Electron,” QE-22(1986), pp. 1887–1899. [47] Morton, P. A. et al., “25 GHz bandwidth 1.55m GaInAsP p-doped strained multiquantum-well lasers,” Electron Letters, 28(1992), pp. 2156–2157. [48] Osinski, M., and Buus J., “Linewidth broadening factor in semiconductor lasers,” IEEE Journal Quant. Electron, QE-23(1987), pp. 57–61. [49] Suematsu, Y., et al., “Advanced semiconductor lasers,” Proceedings of IEEE, 80(1992), pp. 383–397. [50] Margalit, N. M. et al., “Vertical cavity lasers for telecom applications,” IEEE Communication Magazine, 35(1997), pp. 164–170. [51] Verdeyen, J. T., Laser Electronics, 2nd edition, Upper Saddle River, NJ: Prentice Hall, 1990. [52] Ohstu, M., Frequency Control of Semiconductor Lasers, New York: Wiley, 1996. [53] Kobayashi, K., and Mito, I., “Singular frequency and tunable laser diodes,” IEEE/OSA Journal of Lightwave Techn., LT-6(1988), pp.1623–1633. [54] Lightwave Optical Engineering Sourcebook, 2003 Worldwide Directory, Lightwave 2003 Edition, Nashua: NH, PennWell, 2003. [55] Hong, J., et al., “Matrix-grating strongly gain coupled (MG-SGC) DFB lasers with 34 nm continuous wavelength tuning range,” IEEE Photon. Technol. Lett., 11(1999), pp. 515–517. [56] Kudo, K., et al., “1.55 mm wavelength selectable microarray DFB-LD’s with integrated MMI combiner, SOA, and EA modulator,“ Proc. of European Conf. on Optical Comm., ECOC 2000, Munich, paper TuL 5.1. [57] Libatique, N. J. C., and Jain, K. J., “A broadly tunable wavelength selectable WDM source using a fiber Sagnac loop filter,” IEEE Photon. Technol. Lett., 13(2001), pp. 1283–1285. [58] Coldren, L. A., “Semiconductor Laser Advances: The Middle Years,” IEEE Photonics Society News, Vol. 25, 2011, pp. 4–9. [59] Liu, B., et al., “Wide tunable double ring resonator coupled lasers,” IEEE Photonic Technology Letters, Vol. 14, 2002, pp. 600–603. [60] Special Issue on Multiwavelength Technology and Networks, IEEE/OSA J. of Lightwave Techn., LT-14(1996). [61] Kashima, N., Passive Optical Components for Optical Fiber Transmission, Norwood, MA: Artech House, 1995. [62] Agrawal, G. P., Lightwave Technology: Components and Devices, New York: WileyInterscience, 2004. [63] Bennion, I., et al., “UV-written in fibre Bragg gratings,” Optical Quantum Electronics, 28(1996), pp. 93–135. [64] Kobrinski, H., and Cheung, K. W., “Wavelength tunable optical filters: Applications and Technologies,” IEEE Commun. Magazine, 27(1989), pp. 53–63. [65] G. H. Song, “Toward the ideal codirectional Bragg filter with an acousto-optic filter design,” IEEE/OSA J. of Lightwave Techn., LT-13(1995), pp. 470–481. [66] Cheung, K. W., “Acoustooptic tunable filters in narrowband WDM networks; system issues and network applications,” IEEE J. of Selected Areas in Commun., 8(1990), pp. 1015–1025.

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[67] Takada K., et al., “Low-loss 10-GHz spaced tandem milti-demultiplexer with more that 1000 channels using 1x5 interference multi/demultiplexer as a primary filter,” IEEE Photon. Technol. Lett., 14(2002), pp. 59–61. [68] Takahashi, H. et al., “Transmission characteristics of arrayed nxn wavelength multiplexer,” IEEE/OSA J. of Lightwave Techn., LT-13(1995), pp. 447–455. [69] Wooten, E. L., et al., “A review of lithium niobate modulators for fiber-optic communication systems,” IEEE J. of Select. Topics in Quant. Electron., Vol. 6, 2000, pp. 69–82. [70] Kim, H., and Gnauck, A. H., “Chirp characteristics of dual-drive Mach-Zehnder modulator with a finite DC extinction ratio,” IEEE Photon. Technol. Lett., 14(2002), pp. 298–300. [71] Li, G. L., et al., “High saturation high speed traveling-wave InGaAsP-InP electroabsorption modulator,” IEEE Photon. Technol. Lett., 13(2001), pp. 1076–1078. [72] Mason, B., et al., “40 Gb/s tandem electroabsorption modulator,” IEEE Photon. Techn. Lett., 14(2002), pp. 27–29. [73] Ito, T., et al., “Extremely low power consumption semiconductor optical amplifier gate for WDM applications,” Electron. Letters, 33(1997), pp. 1791–1792. [74] Desurvire, E., et al., Erbium Doped Fiber Amplifiers: Device and System Developments, New York: John Wiley and Sons, 2002. [75] Becker, P. C., et al., Erbium Doped Fiber Amplifiers: Fundamentals and Technology, Boston, MA: Academic Press, 1999. [76] Mikkelsen, B., et al., “High performance semiconductor optical amplifiers as in-line and preamplifiers,” Europ. Conf. on Optical Communications, ECOC’94, Volume 2, pp. 710–713. [77] Mukai, T., et al., “5.2 dB noise figure in a 1.5 m InGaAsP traveling wave laser amplifier,” Electron Letters, 23(1987), pp. 216-217. [78] Mayers, R. J., et al., “Low noise erbium doped fiber amplifier operating at 1.54 m,” Electron Letters, 23(1987), pp. 1026–1028. [79] O’Mahony, M. J., “Semiconductor Laser optical amplifiers for use in future fiber systems,” IEEE/OSA J. of Lightwave Techn., LT-6(1988), pp. 531–544. [80] Stunkjaer, K. E., “Semiconductor optical amplifier based on all gates for high-speed optical processing,” IEEE J. of Selected Topics in Quant. Electron., Vol. 6, 2000, pp. 1428–1325. [81] Desurvire, E., Erbium Doped Fiber Amplifiers, New York: John Wiley, 1994. [82] Miniscalco, W. J., “Erbium doped glasses for fiber amplifiers at 1500 nm,” IEEE/OSA J. of Lightwave Techn., LT-9(1991), pp. 234–250. [83] Clesca, B., et al., “Gain flatness comparison between Erbium doped fluoride and silica fiber amplifiers with wavelength mixed signals,” IEEE Photonics Techn. Letters, 6(1994), pp. 509– 512. [84] Atkins, C. G. et al., “Application of Brillouin amplification in coherent optical transmission,” Electron Letters, 22(1986), pp. 556–558. [85] Mochizuki, K. et al, “Amplified Spontaneous Raman Scattering,” IEEE/OSA J. of Lightwave Techn., LT-4(1986), pp. 1328–1333. [86] Essiambre, R. J., et al., “Design of Bidirectionally Pumped Fiber Amplifiers Generating Double Rayleigh Scattering,” IEEE Photon. Techn. Lett., 14(2002), pp. 914–916.

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[87] Smith, R. G., “Optical power handling capacity of low loss optical fibers as determined by stimulated Raman and Brillouin scattering,” Applied Optics, 11(1972), pp. 2489–2494. [88] Stolen, R. G., and Ippen, E. P., “Raman gain in glass optical waveguides,” Applied Phys. Letters, 22(1973), pp. 276–278. [89] Agrawal, G. P., Nonlinear Fiber Optics, 3rd edition, San Diego, CA: Academic Press, 2001. [90] Ramaswami, R., and Sivarajan, K. N., Optical Networks, San Francisco, CA: Morgan Kaufmann Publishers, 1998. [91] Alexander, S. B., Optical Communication Receiver Design, Bellingham, WA: SPIE Press Vol. TT22, 1997. [92] Nalva, H. S., (Editor), Photodetectors and Fiber Optics, San Diego, CA: Academic Press, 2001. [93] Personic, S. D., “Optical Detectors and Receivers,” IEEE/OSA J. of Lightwave Techn., Vol. 26, 2008, pp. 1005–1020. [94] Yuan, P., et al., “Avalanche photodiodes with an impact ionisation engineered multiplication region,” IEEE Photon. Technol. Lett., 12(2000), pp. 1370–1372. [95] Kuebart, W., et al., “Monolithically integrated 10 Gb/s InP-based receiver OEIC, design and realization,” Proc of European Conf. on Optical Comm., ECOC 1993, TuP6.4, pp. 305–308. [96] Bitter M., et al., “Monolitic InGaAs-InP p-i-n/HBT 40 Gb/s Optical Receiver Module,” IEEE Photon. Technol. Lett., 12(2000), pp. 74–76. [97] Keyes, R. J., Optical and Infrared Detectors, New York: Springer, 1997. [98] Wu, K. J., and Liu, J. Y., “Liquid crystal space and wavelength routing switches,” In Proc. of IEEE LEOS Annual Meeting, 1996, pp. 28–29. [99] Baxter, G., et al., “Highly Programmable Wavelength Selective Switch Based on Liquid Crystal on Silicon Switching Elements," In Proc. of OFC 2006, Anaheim, CA, paper OTuF2. [100] Lin, L. Y., and Goldstein, E. L., “Opportunities and Challenges for MEMS in lightwave communications,” IEEE J. Selected Topics in Quant. Electron., Vol. 8, 2002, pp. 163–172. [101] Ryf, R., et al., “1296-port MEMS transparent Optical Crossconnect with 2.07 Petabits/s switch capacity,” Optical Fiber Conference, OFC 2001, San Diego CA, PD28. [102] Mukharjee, B., Optical Communication Networks, New York: Springer, 2006. [103] Yoo, S. J. B., “Wavelength conversion technologies for WDM network applications,” IEEE/OSA J/ Lightwave Techn., LT-14(1996), pp. 955–966. [104] Deming, L., et al., “Wavelength conversion based on cross-gain modulation of ASE spectrum of SOA,” IEEE Photon. Techn. Lett., 12(2000), pp. 1222–1224. [105] Spiekman, L. H., “All Optical Mach-Zehnder wavelength converter with monolithically integrated DFB probe source,” IEEE Photon. Techn. Lett., 9(1997), pp. 1349–1351. [106] Digonnet, M. J. (editor), Optical Devices for Fiber Communications, Bellingham, WA: SPIE Press, 1998. [107] Ueno, Y., et al., “Penalty free error free all-optical data pulse regeneration at 84 Gb/s by using symmetric Mach-Zehnder type semiconductor regenerator,” IEEE Photon. Techn. Lett., 13(2001), pp. 469–471. [108] Girardin, F., et al., “Low-noise and very high efficiency four-wave mixing in 1.5 mm long semiconductor optical amplifiers,” IEEE Photon. Techn. Lett., 9(1997), pp. 746–748.

Chapter 3 Signal Propagation in Optical Fibers In this chapter we will deal with transmission properties of optical fibers. The waveguide theory is used to find the mode structure and to analyze signal propagation through optical fibers. In this chapter, we analyze the signal attenuation and the properties of transversal modes in multimode optical fibers and signal propagation in single-mode optical fibers. In addition, the most important aspects of signal propagation and coupling among spatial modes will be discussed. The material and methodology used inside this chapter are well documented and described in references [1–41].

3.1 OPTICAL FIBER LOSSES Low signal attenuation in optical fibers is the most important feature that enables long-distance transmission capability. The attenuation, or signal loss, is relatively small in comparison to losses in other transmission media, such as different copper-based cables or free space. A typical attenuation curve of a silica-based optical fiber is shown in Figure 3.1. The optical power P in the fiber is attenuated in accordance with the simple differential equation given as (3.1) where  is the attenuation coefficient, often known as fiber loss. Equation (3.1) reflects the fact that the power decays exponentially between two points along the axes z. The solution of Equation (3.1) for two points characterizing the input and output of the fiber with the length L can be written as =

(

)

(3.2)

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where P1 and P2 are powers measured at the input and output of the fiber. If L is expressed in km, coefficient  should be expressed in km–1. However, as we notice from Figure 3.1, the fiber loss is commonly expressed in dB/km as  

P 10 log10  1  L  P2 

(3.3)

Since optical power at the output is lower than the one at the fiber input, a negative sign appears before the logarithmic function to outline the positive value of the attenuation coefficient . We can see from Equations (3.2) and (3.3) that there is following relationship between numerical values: [dB/km] ≈ 4.434. Also, if we have P1, P2, and  all expressed in decibels, it is P2 = P1 – L

(3.4)

This formula can be generalized if there signal is split along the lightpath by the 1:N coupler before coming to the second point. In such a case it will be P2 = P1 – L – 10·log(N)

(3.5)

Power splitting in accordance with Equation (3.5) is common for passive optical networks (PON) commonly deployed in access network area. As an example, coupler splitting signal to four equal portions will insert additional losses of 10·log(4)= 6 dB. 0.9

Rayleigh scattering limit Infrared intrinsic absorption limit Ultraviolet intrinsic absorption limit

Attenuation  in [dB/km]

0.8 0.7

0.6 0.5 0.4

~0.20dB/km

0.3 0.2 0.1 0.0 1000

1100

1200

1300

1400

1500

1600

1700

1800

Wavelength l in [nm]

Figure 3.1 Wavelength dependence of the losses in typical silica-based optical fibers.

The shape of the curve from Figure 3.1 is determined by contribution of absorption, scattering, and radiation effects the optical signal energy decay. The absorption is dominant factor contributing to the total attenuation. The following physical mechanisms contribute to the total absorption effect: intrinsic absorption,

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143

extrinsic absorption, presence of different material imperfections, light scattering in the fiber, and presence of deviations due to the bending of the fiber axis. Intrinsic absorption is caused by the atoms of basic optical fiber material, which is silica glass. This absorption is a principal factor that defines the attenuation of an optical signal by setting the lower limit for any particular material. The intrinsic absorption is attributed to the perfectly pure base material that does not have any density variations or material inhomogeneity. It is associated with the interaction of incoming photons with electronic absorption bands whenever the energy of an incoming photon is higher than the electronic bandgap of the amorphous glass material. The absorption occurs when photons excite electrons to a higher-energy level through the energy transfer. The energy of the photon gradually decreases with an increase of photon wavelength, which means that a smaller number of photons will be capable to excite electrons to higher-energy levels. Therefore, this type of absorption, commonly associated with the ultraviolet wavelength region, will gradually decrease with the wavelength increase and became negligible for the wavelength region above 1,450 nm, as shown in Figure 3.1. However, for longer optical wavelengths, the intrinsic absorption becomes associated with vibration frequencies of the chemical bonds between the atoms of the basic fiber material. The absorption that occurs at wavelengths longer than 1,450 nm, which belong to the infrared region, is a result of the energy transfer from the electromagnetic field of optical signal to number of bonds in the base fiber material. The absorption loss increases with further increase in wavelengths due to more intense process of energy transfer from photons to chemical bonds. Extrinsic absorption is caused by the presents of impurity ions in the base material. Such ions are the positive ions of some metals (iron, chromium, cobalt, copper) and negative OH ions. The metal ions are present in the original materials in quantities usually around 0.1 ppb (particles per billion). The absorption by these ions occurs due to energy transfer from photons to the electrons associated with the ion subshell. The presence of OH ions is direct result of the use of oxyhydrogen flame that is applied to convert chlorides to the implanted dopants— please refer to Figure 2.3. A few OH particles per billion would cause an attenuation peak of about 20 dB/km. Such a peak can be observed in an older generation of optical fibers at wavelengths around 950 nm and 1,400 nm, since it was very difficult at that time to keep the level of OH ions bellow several ppb. The two attenuation peaks we just mentioned separate three wavelength valleys, where the attenuation is much lower. These three wavelength regions are well known as transmission windows. The absolute minimum of fiber losses occurs around wavelength of 1,550 nm, and for typical silica-based optical fibers its value is around 0.2 dB/km, although newer optical fibers have a minimum of 0.165 dB/km to 0.18 dB/ km. In addition, a significant progress in reducing the content of OH ions to less than 1 ppb practically eliminated the absorption peaks [21]. The imperfections in the fiber material slightly increase the total attenuation effect. The imperfections, such as high-density clusters, missing molecules, or

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even oxygen defects in the glass, occur due to defects in the atomic configuration of the base fiber material. The scattering losses, which are caused by microscopic variations in the fiber material density due to the fiber manufacturing process, are the second dominant factor contributing to the signal attenuation in an optical fiber. The fiber structure of randomly connected glass molecules contains the regions where the molecular density varies from an average value. Any variation in material density will cause variations of the local value of the refractive index that will occur over distances smaller than optical signal wavelength. These variations will cause a phenomenon better known as Rayleigh scattering [17, 18]. The scattering loss decreases in proportion to the fourth power of the signal wavelength, that is l

(3.6)

where the parameter A is a constant ranging from 0.7 m4·[dB/km] to 0.8 m4·[dB/km]. The rapid decay described by Equation (3.6) is the reason why the scattering loss, which is a dominant loss mechanism for wavelengths below 1,000 nm, becomes almost negligible at wavelengths around 1,550 nm. Finally, there is fiber bending and microbending as the third mechanism that contributes to the total signal attenuation. It is associated with the radiation of the signal energy outside of the fiber waveguide region, and occurs when optical fiber is bent. The fiber bending is usually a consequence of the cabling process, or its subsequent employment. Fiber bending is characterized by the curve radius, which is measured by tens of millimeters. It is not recommended to allow fibers to undergo any bending where the bending radius is smaller than couple of centimeters since the attenuation increase can be anywhere from 0.01 dB to almost 3 dB. The value of 3 dB occurs when the bending radius is approximately 1 cm. Microbendings occur if the curve of bending is measured by microns, which usually happens as a result of extrusion of a compressible jacket over the optical fiber. In addition, a number of microbends might appear if an external force is applied to the jacketed fiber. The microbending loss can be evaluated by measuring the total attenuation of an optical fiber before and after the cabling process. Otherwise, it can be characterized by using a statistical evaluation. It is reasonable to assume that for high-quality optical cables inserted microbending losses are lower than 0.1 dB/km. In addition to signal attenuation, there are also optical fiber splices and fiber connectors that insert additional losses in the optical transmission line. Fiber splices can be either permanent or fused, or removable. A typical mean attenuation inserted by a fused optical splice is somewhere between 0.05 and 0.1 dB, while removable mechanical splices insert a loss comparable or slightly above 0.1 dB. However, the optical connectors are designed to be removable, thus

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145

allowing many repeated connections and disconnections. The insertion loss for high-quality single-mode optical connectors should not be higher than 0.25 dB. Optical connectors should be very carefully designed since reflection of the incoming optical signal from the surface of receiving part of the connector can occur and cause an additional noise—please refer to Chapter 4. Such a design can include angled fiber-end surfaces, or some index matching fluid applied at the fiber surfaces to minimize the refractive index change when the optical signal crosses from one fiber to the other. The number of the optical splices and connectors depends on length of the optical transmission line. It is up to transmission line engineer to select where to put the splice and the connector, but the general rule is to use optical splices wherever possible. The number of optical fiber joints (splices and connectors) should be taken into account during the optical transmission system design. The design is greatly simplified if the total connection losses are distributed over the overall transmission line and added to the fiber attenuation. In such a case, the transmission line is characterized by a per kilometer attenuation value.

3.2 WAVEGUIDE THEORY OF OPTICAL FIBERS In Chapter 2 we introduced optical fiber types and estimated the transmission capacity in step-index and graded-index multimode optical fibers by using geometric optics approach. We also mentioned that the congregation of rays can be associated with spatial modes in optical fibers. In this section we will explain the spatial mode properties of multimode optical fibers by using the waveguide theory as presented in [1, 13, 16–28]. 3.2.1 Electromagnetic Field and Wave Equations The electromagnetic field is specified by its electric and magnetic field vectors, usually denoted by E(r,t), and H(r,t), respectively, where r represents the space position vector, while t is the time coordinate. The flux densities of the electric and magnetic fields, usually denoted by D(r,t), and B(r,t), respectively, are given as [1] D=0E+P B=0H+M

(3.7) (3.8)

where 0 and 0 are permeability and permittivity in the vacuum, respectively, and vectors P and M represent induced electric and magnetic polarizations, respectively. The vectors M and P are material specific. The evolution of the electric and magnetic fields in space and time defines the electromagnetic wave. The light propagation in an optical fiber can be evaluated by using the electromagnetic wave theory. Since the fiber does not possess any magnetic

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properties, the vector M becomes zero, while the vectors P and E are mutually connected [13, 19] 

P(r, t )   0   (r, t  t ' )E(r, t ' )dt '

(3.9)



where the parameter  is called the linear susceptibility. In a general case,  is the second-rank tensor, but becomes a scalar for an isotropic medium. Since optical fiber can be considered as an isotropic medium, electric polarization vector P(r,t) from Equation (3.9) will have the same direction as electric field vector E(r,t). Therefore, they have just one component and become scalar functions that can be denoted as P(r,t) and E(r,t), respectively. Equation (3.9) now becomes 

PIS (r, t )   0   (1) (t  t ' )E(r, t ' )dt '

(3.10)



where  (1)(t) is now a scalar function, instead of being the second-rank tensor. Please note that the scalar polarization function was denoted by PIS(r,t), which refers to linear isotropic case. Equation (3.10) is valid for smaller values of the electric field, while the following equation should be used if electric field becomes relatively high [1] P(r,t)= PIS(r,t)+ 0 (3) E3(r,t)

(3.11)

The parameter  (3) is known as the third-order nonlinear susceptibility. It general, there are also nonlinear susceptibilities with i-th order (i = 2,4,5…), but they are either zero in the optical fiber material or can be neglected. The electromagnetic wave is characterized by a change in electric and magnetic fields in space and time, and it is governed by Maxwell’s vectors equations [1]   E  B / t   H  D / t D  0 B  0

(3.12) (3.13) (3.14) (3.15)

where (3.16)    / x   / y   / z denotes the Laplacian operator applied to Cartesian coordinates x, y, and z. The set of Equations (3.7)–(3.8) and (3.12)–(3.15) eventually leads to the wave equation defined as

Signal Propagation in Optical Fibers

    E   0 0

 2E  2P   0 t 2 t 2

147

(3.17)

Equation (3.17) can be transferred from the time to the frequency domain by using the Fourier transform that connects the variables in time and frequency domains. The property of Fourier transforms leads to the following equations E(r, t ) 

P( r ,t ) 

1 2 1 2









~ ~  E(r, ) exp( jt )d; and E(r, )   E(r, t ) exp( jt )dt 

~

 P( r , ) exp(  jt )d

(3.18)

(3.19)



Please note that the superscript (~) above a specific variable denotes the frequency domain. By applying the Fourier transform to Equation (3.17), it becomes ~ ~ ~ ~ ~     E  0 0 2E  0 2P  0 0 2E  0 0 2 ~E

(3.20)

Equation (3.9) has been used in the above relation to express the Fourier transform of the electric field vector by the electric polarization vector. Equation (3.20) can be rewritten as

~ ~  (r,  ) 2E E  c2

(3.21)

where (r,) represents a dielectric permittivity of the transmission medium. It is important to notice that the light speed in the vacuum is defined as c=(00)–1/2, [1]. Dielectric permittivity (r,) from Equation (3.21) is connected to the linear susceptibility by the functional relation

 (r, )  1  ~(r, )

(3.22)

However, it is well known that the function (r, ) can be defined through its real part, which represents the refractive index n of the medium, and the imaginary part associated with the attenuation coefficient (r, ) in the medium, that is

 (r, )  n(r, )  j (r, )c / 2 2

(3.23)

The real (Re) and imaginary (Im) parts of the dielectric constant can be found from Equations (3.22) and (3.23) as

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~1/ 2 n(r,  )  1  Re 

(3.24)

   ~  (r,  )    Im   cn(r,  ) 

(3.25)

Therefore, the refractive index and attenuation coefficient are not constant, but depend on the space position and frequency. The wave equation (3.21) can be simplified if applied to optical fibers by assuming that attenuation in fibers is relatively small and coefficient  can be neglected. In addition, the refractive index can be considered as a parameter that is independent of the spatial position. This approximation is just partially true, but it is justified by the fact that index change occurs over lengths much longer than the wavelength of the signal. After the simplification process, Equation (3.21) takes the form

~ ~ ~ ~ n2 ( ) 2E     E  (  E)  2E  c2

(3.26)

(Please note that the vector identity represented by the first half was used in the above equation, where ( ) , Equation (3.26) can be rewritten as ~ 2~ 2E  n2 ( )k0 E  0

(3.27)

The parameter k0 is the wave number in vacuum, defined as

k0   / c  2 / l

(3.28)

where l is the wavelength. The wave number can be associated with any specific medium, through the propagation vector k(r,) defined as

k (r , )  k 0n(r , ); k z  

(3.29)

The z-axis component kz of the propagation vector is called the propagation constant. The propagation constant, usually denoted by , is one of the most important parameters when considering the wave propagation through an optical fiber. 3.2.2 Optical Modes in Step-Index Optical Fibers The term optical “mode” in optical fiber is related to spatial distribution along the cross-sectional area of the optical fiber. This distribution is not a function of axial coordinate z, but rather a two-dimensional solution of the wave equation (3.27) for specific boundary conditions at core-cladding interface. The solution of wave

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equation for given boundary condition is conveniently expressed through cylindrical coordinate system [r, , z], with coordinate z going along the fiber axis, while r and  define the radial and azimuthal position along the fiber crosssection, respectively. The functional dependence of the electric and magnetic fields of the electromagnetic wave that propagates along the z-axis can be expressed as E(r, , z) and H(r, , z), respectively. If these E and H representation is inserted into Equation (3.26), the lengthy process of derivative calculations and substitutions, [13, 19, 20] leads to the following equations in the frequency domain



(3.30)



(3.31)

The above equations come with just with the z-components of the electrical and magnetic field. The other components (Er, E, Hr, and H) will be determined by using the values for Ez and Hz and going back to Equation (2.26). Please also refer to the basic vector identities in cylindrical coordinates—Equations (10.22) and (10.24). The first step in finding the solution for the wave equation (2.30) is to use the method of separation of variables and express Ez(r, , z) as the product (  )

( )

 ()

( )

(3.32)

Substituting Equation (3.32) into (3.30), the following differential equations can be obtained (3.33) 



(3.34)



(

)

(3.35)

Equation (3.33) has the expected solution Ezz(z) = exp(jz) where  is the propagation constant. Similarly, Equation (3.34) gives back the function  ()

(

)

(3.36)

where parameter m can take just integer values along the azimuthal direction.

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As for Equation (3.35), it is well known form in mathematics with a solution in the form of Bessel functions [9]. If we now assume that optical fiber in question has a step-index profile with core radius a and refractive indexes nco and ncl for the core and cladding regions, respectively, the solution of Equation (3.35) can be expressed as ( (

) )

( ) ( )

(3.37) (3.38)

where Jm and Km are Bessel functions (please refer to Chapter 10.2); A and C are constants, while p and q are expressed as (

)

(

)

(3.39)

After all calculations and replacements, the solutions of equations (3.30) and (3.31) for Ez and Hz components in the core and cladding regions can be expressed as ( (

) )

( ) ( )

( (

) )

( (

) )

(3.40) (3.41)

( (

) )

( ) ( )

( (

) )

( (

) )

(3.42) (3.43)

The radial and azimuthal components in the core region can be expressed as, 

(

)

[

(

)

[

(

)

[

(

)

[

 

]

(3.44) 

 



]

] 

(3.45) (3.46)

]

(3.47)

The solution for the radial and azimuthal components in the cladding region can be expressed in a similar manner, just by replacing p2 by –q2 in Equations (3.44) to (3.47). As a commentary note, factor exp(jz) is denoted as exp(-jz) in some other literature, which is just matter on convenience. From the definition of Bessel functions [9], we can see that Km(qr)→ exp(-qr) as qr → . Since Km(qr) has to be zero as r → it follows that q must be [ ( ) ] positive (q > 0). The relation represents a cutoff condition for guiding modes. The cutoff condition defines the point where a mode is no longer confined in the core region. The second condition imposed to the

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propagation constant can be deduced from the properties of function Jm(pr). Namely, inside the core region, the parameter p must be positive to assure that the function Ezr from Equation (3.35) keeps real values, which leads to the condition ( ) . Therefore, the propagation constant should be in the range (

)

(

)

(3.48)

The solution for propagation constant  of any guided mode can be determined by applying the boundary conditions at the core-cladding interface, which will lead to specifications of the constants A, B, C, and D from Equations (3.40) to (3.43). The boundary conditions require continuity of tangential components Ez, E, Hz, and H of the E and H vectors across the core-boundary interface (for r=a). As an example, the continuity of Ez components from Equations (3.40) and (3.43) leads to a condition (

)

(

)

(

)

(

)

(

)

)

(

(

)

(3.49)

or (

)

(3.50)

The intense mathematical procedure described in [16, 19, 20] leads to set of four differential equations with unknown coefficients A, B, C, and D. A solution to these equations exist only if the corresponding determinant is equal zero, which leads to the following eigenvalue equation for propagation constant 

[

( ) ( )

( ) ( )

] [

( ) ( )

( ) ( )

]

[

][

]

(3.51) where a prime indicates the first derivative of the Bessel function. By solving Equation (3.51) for , it can be found that only discrete values mi (i= 1, 2, 3, …), for any given parameter m and from within the range defined by Equation (3.48), will be allowed. Each discrete value defines a specific mode in step-index multimode fibers. The eigenvalue Equation (3.51) is a complex transcendental one, and it is generally solved by numerical methods. Its solution for any particular mode provides a full characterization of that mode. The field distribution across the cross-sectional area of any specific guided mode does not change during its propagation along the z-axis. As an illustration of the radial distribution of electromagnetic field for fiber modes, a diagram of the Bessel functions for three lower orders (m=0,1,2) is shown in Figure 3.2. In general, the eigenvalue equation produces nonzero values for Ez and Hz, thus giving a hybrid structure of the mode field. Because of that, the electromagnetic field of any specific mode is denoted as HEmi or HMmi, depending

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on whether magnetic or electric field component is the dominant one (H stands for hybrid). The only exception is for m = 0, in which case the mode electromagnetic field distribution has a transverse character. The transverse modes are, therefore, recognized by HE0i or HM0i notations. However, due to transversal character, these modes are often known as transverse-electric TE0i (having Ez=0) or transverse-magnetic TM0i (having Hz = 0) modes. m=0

Bessel function Jm(x)

1.0

m=1

0.6

m=2

0.2

x 4

2

6

8

10

-0.2

-0.6

Figure 3.2 Bessel functions of the first three orders.

To examine the cutoff conditions (i.e., conditions when modes are no longer bounded to the fiber core) for fiber modes, an important parameter the called normalized frequency V is introduced and defined as √

(

)

)

√(

By introducing replacements k0=l and √( (3.52) takes the form  l

)

√(

 l

√

(3.52) )

√

, Equation

(3.53)

The parameter V is also known as V-number or just V-parameter. In addition, another normalized parameter, the normalized propagation constant, is very often used. The normalized propagation constant, known also as the b parameter, is defined as (

)

(

)

(

l ) 

where nmo is known as mode index [13].

(3.54)

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Functional dependence b(V) is very often plotted for first 10 modes or so, [17, 19, 20]. However, a simplified yet accurate approximation, based on the fact that difference  between refractive indices in the core and calling is very small ( ), was proposed in [27]. The group of guided modes with similar b(V) curves is represented by linear polarization (LPli) mode approximation, which is widely used in literature for analysis of optical fiber characteristics. We illustrated b(V) functional dependence for three LPli modes (groups of HEmi, TEmi, HEmi, and TMmi modes) in Figure 3.3 together with their LPli approximation. This figure is an interpolation of numerical results obtained in [23, 24] and serves just for the illustration purposes. We should also mention that mode index l in LP representation is equal: l = 1 for TE and TM modes; l =m+1 for EH modes; and it is l = m – 1 for HE modes. 1.0

HE11

b

Normalized propagation constant b

LP01 0.8

TE01 LP11 LP21

0.6

HE21 0.4

EH11

HE12 HE31

0.2 TM01 0.0

1

2

3

4

LP02

5

6

7

Normalized frequency V

Figure 3.3 b(V) curve for several low-order modes. The plot is based on interpolation of results obtained in [25, 26, 29].

The total number of the modes NSI that can be supported by the step-index multimode fiber can be roughly estimated as the product NSI =Nu, where is the solid acceptance angle closely related to numerical aperture from Equation (2.3), while Nu=2Aco/l2 defines the number of modes per unity solid angle. The factor 2 comes from the fact that two polarizations of the plane electromagnetic wave can arrive at the core cross-sectional area Aco. Since we have that Aco=a2 and =(NA)2 , where NA is the maximum acceptance angle related to Figure 2.7 and Equation (2.3), the total number of modes can be now expressed as

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2a 

2

n



V2 (3.55) l2 2 The cross-sectional distribution of the electromagnetic field (and also the power) of several modes in standard and LP representation is shown in Figure 3.4. We can clearly recognize that the first number in the mode index counts for number of periods of the electrical (or magnetic) field across the azimuthal angle of 2, where the second index counts the number of field maximums across the radial coordinate. N SI 

2 co

 ncl2 

LP01 mode

LP11 ~ TM01 mode

LP21 mode

LP11 ~ TE01 mode

Figure 3.4 Field distributions of several fundamental modes in a multimode optical fiber.

The darker area means a high concentration of the optical energy, while a brighter area means a low energy concentration. As we see, there is a fundamental mode known as the LP01 mode that occupies the central part of the optical fiber, which has an energy maximum at the axis of the optical fiber core. The radial distribution of the power of the LP01 mode can be approximated by the Gaussian curve, as we will see later in Equation (3.59). 3.2.3 Definition of a Single-Mode Regime We can see from Figure 3.3 that for V ≤ 2.405 just a fundamental mode HE11 (or LP01) exists. The value 2.405 is the point where Bessel function J0 becomes 0 as shown in Figure 3.2. Optical fibers with parameters that satisfy the condition

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155

 l

) √( are single-mode ones. A wavelength associated with this limit is known as the cutoff wavelength. A dependence b(V) for fundamental mode HE11 in single mode optical fibers was approximated with the expression [13], [18] b(V) ≈ (1.1428 – 0.9960/V)2

(3.56)

and has been accurate within 0.2% if V is in the range 1.5 < V < 2.5. This condition for parameter V can be used to estimate core radius and refractive index difference of the fiber operating in a specified wavelength range. As an example, it is a ≈ 4 m and  ≈ 0.003 for step-index single mode fibers operating around wavelength l =1,300 nm. Most of the conventional single-mode fibers designed to date have a core radius in the range [4–5] m. The representation of LP modes in [27, 28] assumes that the field change occurs just along orthogonal meridional planes associated with coordinates x and y. Since it assumes linear polarization, one of the components (Ex or Ey) is always set to zero. Assuming that Ey = 0, the following expression is established (

)

(

)

(

)

( (

)

(

( )

)

( )

)

(3.57) (3.58)

where ELP is a constant. The remaining component of the magnetic field, since the other one is also set to zero, is Hy=ncl( 0/0)1/2Ex. Since the ratio of Bessel functions in Equations (3.57) and (3.58) is not convenient for practical calculations, it is often approximated by more practical Gaussian form. Accordingly, the Ex(r) component is expressed as ( )

( )

[

⁄

]

(

)

(3.59)

where w0 is the mode field radius often known as the spot size. The spot size is often compared to the core radius a, and the ratio w0/a can be expressed as the function of the parameter V. The following analytic approximation [19] with an accuracy within 1% for 1.2 < V < 2.4 is quite suitable for practical considerations of the fundamental mode properties w0/a ≈ 0.65 + 1.619V–3/2 +2.879V–6

(3.60)

The mode spot size helps to define the size of the effective core area Aeff as



(3.61)

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The parameter Aeff is widely used in different calculations of optical fiber properties since it helps to measure how well confined the fundamental mode is within the core boundary. The fraction of the optical power that is contained in the core can be found as [13] (

)

∫ |

|

∫ |

|

[

⁄

]

(3.62)

Please note that the optical power is proportional to | | , which is a generic relationship often used in optical communications. Relations (3.61) and (3.62) can be used to evaluate the degree of the mode confinement for different types of single-mode optical fibers. 3.2.4 Modes in Grade-Index Optical Fibers To describe the wave propagation in graded-index optical fibers, we apply a logic similar to that used for the step-index profile analysis. It means that we can again assume that we are dealing with a weak guidance of the modes since the difference in the refractive indexes between any two points along the trajectory is very small. Also, we assume that the angle between the wave position vector r and z-axis is small as well. Consequently, both electrical and magnetic fields have a transversal character and they are perpendicular to each other, which is characteristic of linear polarization of the modes. A diagram of the position vector r, wave vector k, and its components for graded-index optical fiber is shown in Figure 3.5.

 

Figure 3.5 Position vector r, wave vector k (and its components) for skew ray in graded-index fiber.

It is known from [1] that the wave Equation (3.27) can be solved for a parabolic refractive index profile with = 2 from Equation (2.6), but only by

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157

allowing that parabolic index profile is valid for r > a as well. The solution for the electrical field of linearly polarized mode can be written as =

(√

)

The function ()

( )

∑

(

)

[

( )

]

(3.63)

represents Laguerre polynomial [9] defined as

(

)

(

)

(3.64)

Several low-order Laguerre polynomials have the following forms ()

()

()

( )(

)(

)

(

)

(3.65)

Parameter w0 in expression (3.63) is again the mode spot size that measures the scope of the filed confinement in the fiber core area. As we can see, the electric field decay is expressed through the product of Laguerre polynomial and the Gaussian function exp(–r2/ ), and very often the corresponding modes are referred as Laguerre-Gaussian (LG). The distribution of the optical power across the cross-sectional area of the LGlp mode is equivalent to the distribution of the LPl(p+1) mode (i.e., i index associated with the LP mode from Figure 3.4 is i = p+1). Therefore, for a fundamental mode, we have that LG00 is equivalent to mode LP01. The exact waveguide analysis of modal properties in multimode optical fibers is possible only in case of the step-index profile and the parabolic profile under conditions just described above. In all other situations, when refractive profile changes in the fiber core, some approximations are needed. The most widely used analysis of the modal properties in optical fibers with graded-index profile is based on WKB approximation, which is otherwise commonly used in quantum mechanics [7]. The WKB method is used to obtain the asymptotic approximation for the solution of differential equation over a parameter that slowly changes within the specified range. In the case of graded-index optical fibers, that parameter is the refractive index nco(r), which decays only slightly over the distances of ~1 m (which is the wavelength of the optical signal that experiences such an index change). In this case, Equation (3.35) becomes [

( )

⁄ ]

(3.66)

where the refractive-index profile is given by Equation (2.6). It is assumed that the component Ezr is expressed as

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[

( )]

(3.67)

with some coefficient A, which is independent of coordinate r. By substituting (3.67) into (3.66), it becomes (

)

[

⁄

⁄ ]

( )

(3.68)

where primes denote the first and the second derivatives with respect to coordinate r. By using the fact that function n(r) varies slowly over a distance comparable with the wavelength l, the fast converging function Q(r) can be expanded in the power series perl, or in powers of l  , so we have that ( )

(3.69)

where Q0, Q1, and so on, are the functions of r. Substituting the expansion (3.69) into Equation (3.68) and collecting the equal powers of ko, it become { (

)

[

( )

]}

(

)

(3.70) The role of term Add. in Equation (3.70) is to denote all terms of the order (k0)0, (k0)–1,(k0)–2, and so on. The next step is to set to zero the terms that are with equal powers of k0. By doing so for the first two terms in Equation (3.70), we have ∫ [

( )

]

(3.71)

The mode is guided and bound within the fiber core only for real values of Q0, which means that ( ) . For any m that defines the mode rank, there are two values r1(m) and r2(m) that are functions of m, where ( ) becomes equal to zero. The mode associated with index m exists for the values r1 < r < r2, since for any other values of r the function Q0 becomes imaginary, and that indicates the field decay. Values of r1(m) and r2(m) can be associated with the turning points of the helical skew rays propagating through graded-index multimode fiber (please see Figure 2.8), and they are found as a solution of the equation ( ) . The ray congruence propagating under the angle  from Figure 2.8 form a mode only if there is constructive interference with itself and standing wave formation. This requirement leads to the condition that the phase function Q0 must have an integer number l of half-periods between the points r1(m) and r2(m) which is



∫ [

( )

]

(3.72)

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The radial mode number l (l=0, 1, 2…) thus counts for the number of half-periods between the points r1(m) and r2(m). The total number of the guided modes l() bounded inside the fiber can be found by making summation of the values given by Equation (3.72) over parameter m, starting from 0 all the way to mmax, where mmax is the highest order mode for given propagation constant which isstill bounded inside. The value mmax is found as a solution of equation nonlinear equation ( ) Since mmax is relatively large number, the summation can be replaced with an integral, so we have that ( )



∫

( ) [ ( )

∫

( )

]

(3.73)

The factor 4 appears due to the fact that that each combination (l, m) designates a degenerate group consisting of four modes with different polarization. By replacing the order of integration, and assuming that r1=0, Equation (3.73) becomes ( )

∫ [

( )

]

(3.74)

Equation (3.74) can be applied to profile defined by Equation (2.6) to evaluate the total number of modes for nearly parabolic refractive index profile. The upper integration limit r2 is found from the condition that  =k0n(r), which gives back the value [

(

)]

(3.75)

Finally, from Equations (2.6) and (3.75), we have that

( )

(

)

(3.76)

The maximum number NGI of bound modes in graded-index multimode fiber is found by replacing =k0ncl in above relation, which produces the value (3.77) By comparing the numbers given by Equations (3.55) and (3.77), for step-index and graded-index fibers, respectively, we can conclude that the number of the modes supported by graded-index structure is approximately the half of number related to the step-index profile.

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3.3 SIGNAL DISPERSION IN SINGLE-MODE OPTICAL FIBERS Optical pulses propagating through optical fibers will be attenuated due to fiber losses and distorted from their original shape. Generally, that distortion means that their temporary shape has experienced spreading, although the pulse compression during propagation is possible as well if some conditions are met. The major factor for pulse distortion in multimode optical fibers is modal dispersion. The amount of the pulse spreading due to modal dispersion for step and graded index optical fibers was estimated by using geometric optics methods and was given by Equations (2.4) and (2.8), respectively. As for the single-mode fibers, several major factors, such as chromatic dispersion (CD), polarization mode dispersion (PMD), and self-phase modulation (SPM), will contribute to the pulse distortion. If more than one channel is transmitted through a single-mode fiber, the crossphase modulation (XPM) effect will also contribute to the pulse distortion. It is important to mention that these factors are present in the multimode fibers as well, but their impact is much smaller as compared to the impact of modal dispersion. 3.3.1 Modal Dispersion So far, the multimode optical fibers have been usually used for point-to-point signal transmission over shorter distances that are measured by kilometers or tens of kilometers. In such a case, Equations (2.4) and (2.8) can be used to evaluate the transmission capacity if all fiber parameters are known. In addition, the transmission capacity of multimode optical fibers can be evaluated by knowing the bandwidth Bfib that is related to the 1 km of the optical fiber length. This parameter is expressed in GHz·km, or in MHz·km. Optical fiber bandwidth is the one most often specified by the fiber manufacturer and presented in the product data sheet. It is usually measured at wavelengths around 1,310 nm, to make sure that there is no any impact of the chromatic dispersion to the measured results. The bandwidth Bfib varies from several tens of MHz-km for step-index multimode optical fibers, to more than 2 GHz-km for graded-index optical fibers. The bandwidth parameter Bfib can be used to calculate the bandwidth over a specific distance L as [16]

B fib, L 

B fib L

(3.78)

where  is coefficient that can take values that are in the range 0.5 to 1. It is around  = 0.7 for most multimode optical fibers. 3.3.2 Chromatic Dispersion Chromatic dispersion, also known as intramodal dispersion, is often associated with single-mode optical fibers, since there is no intermodal dispersion effect.

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However, it adds to the modal (intermodal) dispersion in multimode optical fibers, although its contribution to the pulse spreading in multimode fibers is much smaller. Chromatic dispersion that occurs in single-mode optical fibers is a result of the fact that the group velocity of the optical signal is function of wavelength. Therefore, the chromatic dispersion would not exist if a monochromatic wave propagates through the optical fiber. Since the optical source is not an ideal monochromatic source, each pulse that carries the information contains a number of spectral components that travel at different velocities through the fiber. The amount of the dispersion is proportional to the spectral width of the optical source. Chromatic dispersion causes pulse broadening since the pulse spreads out of its allocated time, thus causing the intersymbol interference (ISI) as illustrated in Figure 3.6. The pulse broadening due to chromatic limits the total transmission capacity in single-mode optical fibers. The amount of the signal spread outside the time slot will degrade the optical receiver sensitivity, as analyzed in Section 4.3. It is important to mention that chromatic dispersion is also a cumulative effect that increases with optical fiber length. Time slot Original data stream

1

1

0

Pulse spreading Received data pattern

1

Time

Spectral components dispersed in time

Time Intersymbol interference

Figure 3.6 Pulse broadening and intersymbol interference.

The digital pulses in Figure 3.6 provide envelopes in the time domain for the spectral content of the light source. Each spectral component propagates independently through the fiber across the axial coordinate z arriving at different times and thus causing the pulse broadening, as shown in Figure 3.6. A specific spectral component, characterized by the angular optical frequency =2, will arrive at the output end of the fiber after some delay g that is given as [1] g 

L d L d Ll2 d L   vg d c dk 2c dl

(3.79)

where L is the fiber length, l=2c/  =c/ is the wavelength,  is the linear frequency, c is the light speed in the vacuum, and  is the propagation constant. By using Equation (3.54), the propagation constant can be expressed as

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2nmo l

(3.80)

where nmo is the mode refractive index. The parameter vg defined as 1

2c  d   d  vg     2   l  dl   d 

1

(3.81)

is the group velocity defined as a speed at which the energy of an optical pulse travels through the medium. As a result of difference in time delays, the optical pulse disperses after traveling a certain distance before arriving at the output end of the fiber. The following relation characterizes the amount of the pulse broadening due to dispersion

 g 

d g d

 

d g dl

l

(3.82)

where  and l represent the range of frequencies and wavelengths emitted by the optical source, respectively. By inserting Equation (3.81) into Equation (3.82), it becomes  g  L

d 2 L  d d 2   2l l  D  Ll     l2 2 2c  dl d dl2 

(3.83)

The factor

D

1  d d 2   2l  l2 2  2c  dl dl 

(3.84)

is known as the chromatic dispersion coefficient or just chromatic dispersion. It is expressed in ps/(nmkm). There are two factors contributing to the pulse broadening in Equation (3.83): material and waveguide dispersion. The material dispersion Dm arises due to wavelength dependence of the refractive-index in the fiber core. Therefore, the functional dependence nco(l) causes a wavelength dependence of the group delay. The wavelength dependence of the refractive index in silica-based materials is well approximated by the Sellmeier equation [26], given as M  B l2  n(l )  1   2 i 2  i l  li  

1/ 2

(3.85)

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Parameters Bi and li can be determined empirically for a specified material by using interpolation procedure based on the measured results. These coefficients strongly depend on the dopant concentration implanted in the base silica. These coefficients for M=3 in pure silica glass are: B1=0.69617, B2=0.40794, B3=89748, l1= 0.0684 m, l2= 0.1162 m, and l3= 9.8962 m. The waveguide dispersion Dw occurs since the propagation constant  is a function of the fiber parameters (core radius and difference between refractive indexes in fiber core and fiber cladding), and it is also function of the wavelength l The fact is that material and wavelength dispersion are interrelated since the dispersive properties of the refractive-index have an impact to the waveguide dispersion, which makes it more difficult to evaluate them separately. However, a simplified approach is often used in many practical situations, in which the material and waveguide dispersion components are calculated separately, while the total chromatic dispersion is calculated as a sum of these components [29]. By using this approach, Equation (3.84) becomes t g 

 l d 2nco ncl n d 2 ( Vb )  d ( tm  t w )  Ll l  ( Dm  Dw )Ll    V 2 dl cl dV 2   c dl

(3.86)

where ncl is referred to the refractive index in the optical fiber cladding, n is the difference between the refractive indexes at the fiber axis and in the fiber cladding, and V and b are V-parameter and normalized propagation constant that were introduced by Equations (3.53) and (3.54), respectively. 30 2a =10 m

25 20 15

2a =5 m

10 5 0

2a =10 m

-5 -10 -15

2a =5 m

-20

-25 1200

1300

1400

1500

1600

Wavelength [nm]

Figure 3.7 Chromatic dispersion in single-mode optical fibers.

1700

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The dispersion components from Equation (3.86) are shown in Figure 3.7 as functions of wavelength for several values of optical fiber core diameter. The dispersion curves are plotted by using approximations given by Equations (3.56) and (3.85), and for typical values of fiber parameters (V=2.1, = 0.25%). Please note that material dispersion passes through zero in the wavelength region around 1,300 nm (1,270 nm for pure silica core, and 1,310 nm for doped silica). Since material dispersion is not a linear function of wavelength, there is a slope associated with the dispersion curve. That slope is often referred as the chromatic dispersion slope. Waveguide dispersion remains negative in the wavelength region above 1,000 nm, while its absolute value increases with the wavelength. The waveguide component of the chromatic dispersion is much smaller than the material dispersion component if considering the wavelength region between 800 nm and 900 nm. However, they become comparable in the wavelength region around 1,300 nm. The waveguide dispersion can be enhanced to become much larger than material dispersion if a special fiber design is applied. This approach is used to manufacture dispersion compensating fibers (DCF)—please refer to Section 3.4.1. It is possible to reduce the total chromatic dispersion by mutual cancellation of material and waveguide dispersions. A particular wavelength where chromatic dispersion is reduced to zero value can vary, but it is always higher than the wavelength value where the material dispersion curve crosses x-axis. Mutual cancellation of chromatic dispersion components at a specified wavelength can be done through proper selection of dopants (by changing the material dispersion component) or by controlling the waveguide effects through fiber core diameter and the refractive index profile. 30 20 10 0 -10 -20 -30 -40 1.2

1.3

1.4 1.5 Wavelength [m]

1.6

Figure 3.8 Chromatic dispersion of several types of single-mode optical fibers.

1.7

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The former approach was used to produce several types of single-mode optical fibers that are different in design from the standard single mode fibers. The characteristics of different optical fiber types are standardized by International Telecommunication Union (ITU-T) in documents [30–32]. In addition to the standard single-mode fiber (SSMF) there are two other major fiber types, and they are: (1) dispersion shifted fibers (DSF) defined by ITU-T recommendation G.653, with dispersion minimum shifted from 1,310-nm wavelength region to 1,550-nm wavelength region; and (2) Nonzero-Dispersion shifted fibers (NZDSF) defined by ITU-T recommendation G.655, with dispersion minimum shifted from 1,310nm wavelength region to anywhere within the C or L-bands. There are several commercial types of optical fibers from this group that are optimized for the DWDM transmission, such as TrueWave fiber, or LEAF fiber. Chromatic dispersion characteristics of three types of single-mode optical fibers are shown in Figure 3.8. 3.3.3 Polarization Mode Dispersion Polarization mode dispersion (PMD) appears in real optical fibers due to variations in the shape of their core along the fiber length. There is no such effect in an ideal single-mode optical fiber with a perfectly cylindrical core of uniform diameter, which is composed of perfectly isotropic material. Any deviations from an ideal structure will cause birefringence effect and the creation of two distinct polarization modes and it will cause single-mode fibers to effectively become bimodal. PMD distorted optical signal Slower PSP Asymmetric internal stress

y

Ideal cylindrical fiber core

z PSP rotation Deviated elliptic fiber core

x

Faster PSP

PSP rotation

Principal polarization states (PSP) concept

(a)

(b)

Figure 3.9 PMD: (a) ideal and deviated fiber core and (b) pulse deformations and PSP concept.

Two modes that are developed in real optical fibers due to the birefringence effect are related to two orthogonal polarizations of the optical signal. These polarizations see either different core sizes due to elliptical core, or different

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material densities due to asymmetric internal stress, as shown in Figure 3.9(a). These differences are transferred to difference in the refractive indexes, which are associated with two polarization states. The difference in refractive indexes (or birefringence) causes the difference in speeds of two orthogonally polarized modes, and that means that the propagation constants of two modes are also different. The degree of birefringence is measured by the parameter B=|nx–ny|, where nx and ny are the refractive indexes experienced by signal portions polarized along orthogonal axes x and y. The total signal is a vector sum of two polarization modes. The optical fiber will keep the residual birefringence along its length if it does not experience any intrinsic rotation of the birefringence axes x and y, and if there is no any external perturbation applied. This is known as the deterministic birefringence, and it is characterized by welldefined polarization states. These states are also known as the eigenstates, and they do not depend on fiber length or on optical wavelength [34, 35]. The difference in refractive indexes and propagation constants between two polarization modes will lead to a phase shift among them during the propagation process. The phase shift is observed through a differential group delay (DGD) between the slow and fast axis that increases linearly with the fiber length. Consequently, the total width of the optical pulse will be dispersed, as shown in Figure 3.9(b). The two polarization modes will not propagate independently along the optical fiber, but will exchange the energy through the mode-coupling process. The polarization mode coupling is caused by the external perturbation due to bending, twisting, lateral stress, and temperature variations. In this case the degree of birefringence changes randomly along the fiber, and polarization eigenstates are not preserved anymore. In other words, we have a stochastic rotation of the polarization states and a stochastic distribution of energy among them. Due to the random character of the energy exchange, neither the phase difference nor differential group delay between two polarization modes will scale linearly with the distance. It was shown that they would increase in proportion with the square root of the fiber length [34]. The PMD, as a stochastic pulse spreading, is caused by the random process just described. In addition to the stochastic rotation of the polarization states and stochastic distribution of energy between the principal states, different wavelength components will behave differently, which will add to the overall complexity of the polarization mode dispersion. An exact analytical treatment of the PMD effect is rather complex due to the random nature and complexity of the overall process. However, some additional assumptions can be made for more convenient evaluation of PMD effect. Real optical fibers do not generally exhibit well-defined orthogonally polarized eigenstates that are stable with respect to the optical frequency and fiber length. Instead, at any specific optical frequency , there are two orthogonally polarized input states that produce the output polarization states with minimal frequency dependence. These states are known as the “principal states of polarization”

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(PSP) [1]. It is important to note that all other output polarization states will experience frequency dependence. The differential group delay (DGD) between two principal polarization states is the main contributor to the pulse broadening, as shown in Figure 3.9. The differential group delay, also known as first-order PMD, does not include any frequency dependence. The frequency dependence of the PMD effect is included through second-order PMD. In fact, the second-order PMD measures the wavelength dependence of the first-order PMD coefficient. In addition, it measures the rotation of the principal polarization states associated with each individual wavelength. The second-order PMD is often called chromatic polarization mode dispersion since it accounts for frequency dependence of both DGD and PSP. The first-order PMD is characterized by coefficient DP1, expressed in ps/(km)1/2, which is a statistical parameter that varies with time and operating conditions. The total delay between principal polarization states accumulates randomly and in proportion to the square root of the optical fiber length L, so it is [34]  P1  DP1 L

(3.87)

The pulse broadening due to first-order PMD depends on the differential group delay and on the power splitting between two principal polarization states. The following relationship between root mean square (RMS) widths of input and output pulses can be established [34, 35] 2  out   in2  2 P21 (1   )

(3.88)

where out and in are root mean squares of the output and input pulses, respectively, whilerepresents the power splitting of the signal between two principal polarization states, and can vary from 0 to 1. The root mean square (RMS) of a specific pulse having shape A(t,z) along the z-axis is defined in a standard way. The light polarization is often analyzed by using a complex representation and the Jones matrix, which connects inputs and outputs carried on over principal polarization states, so we have that [34]

 Ex ,out   Ex ,in    e j / 2  E   J   E     y ,out   y ,in   1  

 1     Ex ,in     e j / 2   E y ,in 

(3.89)

where J denotes the Jones matrix, Ex,in ~ (Px,in)1/2 and Ey,in ~ (Py,in)1/2 present the input electrical fields of the signal components aligned with polarization states x and y, respectively, and Ex,out ~ (Px,out)1/2 and Ey,out ~ (Py,out)1/2 present the output

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electrical fields ( is delay acquired at the path between input and output). The input and output values can be related to any segment and any length, which is very convenient for statistical analysis by using computer programs and random variable generators. We should also note that the Jones matrix can be used to model the effect of polarization dependent loses (PDL) that arise in some of elements along the lightpath (such as modulators, switches, and so forth).

p(DP1)

P(DP1)

0.02

1.0

0.015

10-2

0.01

10-4

0.005

10-6 DP1

0 0



4 X 10-5

DP1 10-8 0



(a)





(b)

p(DP2) 0.02

0.01

DP2

0 0



(c)

Figure 3.10 (a) The first-order PMD probability density function, (b) the overall first-order PMD probability function, and (c) the second-order PMD probability density function.

There is a general agreement that a statistical nature of the first-order PMD coefficient can be characterized by the Maxwellian probability density distribution p(DP1) given as [35] p( D P1 ) 

2 D P1





3

2

 D P1 2 exp    2 2 

   

(3.90)

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The mean value DP1 from this equation equals DP1 = (8/)1/2. Coefficient can be experimentally determined, and has a typical value of about 30 ps. The probability function expressed by Equation (3.90) is shown in Figure 3.10. This functional curve has a highly asymmetric character and decreases rapidly for an argument larger than 3DP1 . The overall probability PDP1 that coefficient DP1 will be larger than a specified value can be found by performing integration in Equation (3.90), that is

P( DP1 ) 

DP 1

 p( D

P1

)d (DP1 )

(3.91)

0

Function PDP1) is shown in Figure 3.10(b). The probability that actual coefficient -5 is three times larger than the average value DP1 is 4 x 10 . This statistical evaluation means that the first order PMD coefficient will remain larger than three times its average value for approximately 21 minutes per year. For example, if DGD varies significantly once a day, the first-order PMD coefficient will exceed three times the average value once every 70 years, but if it varies once a minute it will exceed three times the average value every 17 days. The character of the distribution expressed by Equation (3.91) has been the reason why three times the average value 3DP1 rather that temporary value of the coefficient DP1, has been used to characterize the first-order PMD. Accordingly, Equation (3.87) takes more practical form  P1  3 DP1

L

(3.92)

The value DP1 is usually measured after the manufacturing process is done, and it expressed in the fiber data product sheet. The measured values can vary from 0.01 ps/(km)1/2 to several ps/(km)1/2. The older fibers that were installed in late 1980s and the early 1990s have a relatively large DP1 coefficient due to imperfections in the manufacturing process, while DP1 is much smaller for newer optical fibers. It is usually lower than 0.1 ps/(km)1/2, while manufacturers claim that typical values are around 0.05 ps/(km) 1/2. The specified value can increase after the installation is done since the cabling process and environmental conditions can add to the original number. The second-order PMD is characterized by coefficient DP2, and occurs due to variations of both differential group delay (DGD) and principal polarization states (PSP) with optical frequency. It can be expressed as [35, 36]  2 2  1 DP1   DP1 S  DP 2      2    2  

(3.93)

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where  is the optical frequency and the vector S determines the position of principal states of polarization along the Poincare sphere [1]—please see Section 10.13. This vector is also known as the Stokes vector. The first term at the right side of Equation (3.93) describes the frequency dependence of the differential group delay, while the second term describes the variations of the principal states of polarization (PSP) and their dependence on optical frequency . The statistical nature of the second-order PMD coefficient in real optical fibers can be characterized by probability density distribution pDP2 given as [38] p( DP 2 ) 

2 2 DP 2 tanh(DP 2 )  cosh(DP 2 )

(3.94)

where is a parameter that can be experimentally determined and ranges around ps. The mean value DP2 of the second-order PMD coefficient can be correlated to the mean value DP1 of the first-order PMD coefficient. It was found that the following approximate relation can be established [38] DP 2 

DP1

2

(3.95)

12

Equation (3.95) is related just to numerical values. (Please recall that the firstorder PMD is expressed in ps/km1/2, while the second-order PMD is expressed in ps/km-nm.) The probability density function given by Equation (3.94) is shown in Figure 3.10(c). It is more asymmetrical and has a larger tail than the function related to the first-order PMD coefficient. The probability that the second-order PMD coefficient will take a value three times larger than the mean value DP2 is still countable. It becomes negligible for arguments that are larger than five times the mean value DP2. Accordingly, we can assume that the actual value the second-order PMD coefficient does not exceed five times of the mean value DP2. Since the second-order PMD scales linearly with the fiber length L, the total pulse spreading due to the second-order PMD effect can be expressed as  P 2  5 DP 2 L

(3.96)

The second-order PMD, or the first term on the right side of Equation (3.93), interacts with the total chromatic dispersion along the fiber length. It can induce either pulse broadening or pulse compression, and could be treated as a component of chromatic dispersion. However, the portion of the second-order PMD that is represented by the second term on the right side of Equation (3.93) is related to rotation of principal polarization states, and cannot be always treated as

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pure chromatic dispersion. This part usually dominates and determines the total character of the second-order PMD. Considering overall PMD effect, we can conclude that the first-order PMD is a dominant part of the polarization mode dispersion that causes the signal broadening and distortion. That is the reason why this term is considered in most analyses of the PMD effect, while the second order term is sometimes neglected. Such an approach is quite beneficial in most cases if the bit rates do not exceed 10 Gb/s. However, the second-order PMD effects are not negligible in high-speed optical transmission systems, and should be included in overall engineering considerations. It is also important to mention that some other components employed along the lightwave path can contribute to the total value of the PMD effect. It is reasonable to assume that an additional PMD in excess of 0.5 ps can be accumulated in different components along the lightpath if the lightwave path exceeds several hundred kilometers. Different compensation schemes have been considered so far in order to compensate for PMD effects in transmission systems with intensity modulation and direct detection (IM/DD) [39]. The original idea of eliminating PMD through the compensation in these systems is abandoned due to PMD complexity and its statistical character. Instead, the attention is concentrated to minimize the distortion effect caused by PMD. Such an approach is known as the PMD mitigation. Most of the compensation schemes in multichannel systems work on a per-channel basis by using digital transfer filters after photodetection process. The situation is quite different if advanced modulation formats are used in combination with coherent detection schemes. In such a case, very effective suppression of PMD effect can be done by intense digital signal processing (DSP), which will be analyzed in Chapter 6. The advanced forward error correction (FEC) schemes also contribute to the elevation of PMD and that will be analyzed in Chapter 7. 3.3.4 Self-Phase Modulation in Optical Fibers The basic properties of optical fibers can be explained assuming that optical fiber acts as a linear transmission medium, which means that: (1) refractive index does not depend on optical power; (2) the superposition principle can always be applied to several independent optical signals; (3) optical wavelength, or carrier frequency, of any optical signal stays unchanged; and (4) the light signal in question does not interact with any other light signal. These assumptions are quite valid if a lower optical power and lower bit rates are employed in an optical transmission system. However, the availability of highpower semiconductor lasers and optical amplifiers, together with deployment of dense WDM technology, created conditions where an optical fiber became a nonlinear medium, with properties just opposite to the ones listed above. Nonlinear effects in optical fibers are not design or manufacturing defects, but can occur regardless and can cause severe transmission impairments. However, in

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some special cases, they may be used to enhance the fiber transmission capabilities. Nonlinear effects are mainly related to single-mode optical fibers since they have a much smaller cross-sectional area of the fiber core. It is well known that any dielectric material gains nonlinear properties if exposed to a strong electromagnetic field [1]. The same conclusion can be applied to the optical fiber, since all nonlinear effects that can appear in the fiber are proportional to the intensity of the electromagnetic field of propagating optical signal. There are two major groups of nonlinear effects related either to the nonlinear refractive index, or to nonlinear optical signal scattering. The effects related to the nonlinear refractive index are based on the Kerr effect [1], which occurs due to the dependence of the refractive index on light intensity. The nonlinear refractive index leads to self-phase modulation (SPM) to the pulse distortion of the propagating optical. The SMP effect is related to a single optical channel. In addition, in multichannel WDM transmission, the nonlinear index can cause interchannel cross-talk due to cross-phase modulation (XPM) and four-wave mixing (FWM) processes. In this section we will examine the SPM effect, while both XPM and FWM effects will be explained later in this section when multichannel transmission is discussed. It is also important to mention that high power in optical fibers also leads to nonlinear scattering effects that are caused by parametric interaction between the light (i.e., photons) and material (i.e., crystal lattice, or phonons). There are two types of nonlinear scattering effects: (1) stimulated Raman scattering (SRS), which leads to energy transfer between different wavelengths; and (2) stimulated Brillouin scattering (SBS), which leads to optical power coupling to backward traveling waves. Both SRS and SBS effects have been explained in Section 2.6.2.3. The effect of a nonlinear index, as well as the effects of stimulated scattering, depends on the transmission length and cross-sectional area of the fiber. The nonlinear interaction will be stronger for a longer optical fiber length and for a smaller cross-sectional fiber core area. That effect will decrease along the transmission line due to a decrease in the optical power. However, the nonlinear effects have a local character, which makes an overall assessment over longer lengths more difficult. It is more practical from engineering perspective to consider the effective length Leff and the effective cross-sectional area Aeff as parameters that characterize the strength of nonlinear effects, which was already done in Section 2.6.2.3. The effective length can be found by assuming that a constant optical power that acts over the effective length Leff will produce the same effect as a decaying optical power P = P0exp(–z) that acts over the physical fiber length L. Therefore, the following relation can be established

Signal Propagation in Optical Fibers L

P0 Leff 

 P( z)dz 

z 0

173

L

 P exp( z)dz 0

z 0

(3.97) where P0 presents an input power to optical fiber and  is the fiber attenuation coefficient. Equation (3.97) leads to the expression 1  exp( L)

(3.98)  The value Leff can be approximated by 1/ if optical transmission is performed over long distances. For example, Leff ~20 km if a long-haul transmission is performed at wavelengths around 1.55 m. The concept of the effective length is illustrated in Figure 3.11. Leff 

Power

P0

Leff (a)

Length L

(Aeff/)0.5

Radius r

(b)

Figure 3.11 Definition of optical fiber parameters: (a) the effective length and (b) effective crosssectional area.

The optical power level will be periodically reset to higher values if there are optical amplifiers along the transmission line, which means that the total effective length for such transmission line will include contributions of multiple fiber spans. If each fiber span has the same physical length, the total effective length will be 1  exp( L) L (3.99) M   l where L is the total length of a transmission line, l is fiber span length that is equal to optical amplifier spacing, and M is the number of fiber spans on the line. As we see from Equation (3.99), the effective length can be reduced by increasing the span length, which means that the number of in-line amplifiers would be reduced. However, if we increase the amplifier spacing, we should also increase the power in proportion to exp(l) to compensate for additional fiber losses. However, any power increase will enhance nonlinear effects. What matters in this case is the product between the launched power P0 and the effective length Leff,t. Since the Leff ,t 

1  exp( L)

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product P0Leff,t increases with the span length l, the overall effect of nonlinearities can be reduced by reducing the amplifier spacing. The impact of nonlinear effects is inversely proportional to the area of the fiber core. This is because the concentration of the optical power per unit crosssectional area, or the power density, is higher for smaller cross-sectional area, and vice versa. Please recall from Section 3.2.3 that optical power in single-mode optical fibers is not uniformly distributed across the core section. It has a maximum at the optical fiber axes, and decays along the fiber diameter. The power still has some nonegligible level at the core-cladding border, and continues its decay through the optical fiber cladding area. Optical power distribution across the cross-section is closely related to the overall refractive index profile. It is, therefore, convenient to introduce an effective cross-sectional core area Aeff by applying the same logic as one used above for the effective length. The effective cross-sectional area can be found by assuming that the effect of constant optical power acting over the cross-sectional area is equal to the effect of decaying optical power acting over the entire fiber radius, as illustrated in Figure 3.11. The effective cross-sectional area is found as  2    rdrd E (r , )   Aeff   r  4 rdrd  E ( r ,  ) 

2

(3.100)

r 

where r and  are the polar coordinates and E(r,is the electric field of an optical signal. nco ncl radius

radius

(b) (a)

radius

(c)

Equivalent step-index profile

radius

(d)

Figure 3.12 Refractive index profiles of single mode fibers used today: (a) standard SMF, (b) NZDSF with reduced dispersion slope, (c) NZDSF with large effective area, and (d) dispersion compensating fiber.

The distribution of the electric field is often approximated by Gaussian function given by Equation (3.59), which then leads to Equation (3.61) connecting

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the effective area Aeff with the mode field radius (mode spot size) w0 by relation  Gaussian approximation cannot be applied to optical fibers having more complex refractive index profile. Such fibers include dispersion-shifted fibers (DSF), nonzero-dispersion shifted fibers (NZDSF), and dispersion compensation fibers (DCF). Several examples of the refractive index profiles related to a different design of single-mode optical fibers are shown in Figure 3.12. A classical single-mode optical fiber has a simple step-index profile, with two distinguished values corresponding to nco and ncl, as shown in Figure 3.12(a). More complex refractive index profiles have been introduced to serve different purposes, such as: (1) to reduce the slope of the chromatic dispersion curve, which might be useful for link dispersion management in some special cases of multichannel optical signal transmission—please see Figure 3.12(b); (2) to increase the effective area of a single mode fiber, which helps to reduce the effect of nonlinearities (the profile from Figure 3.12(c)); (3) to introduce large negative waveguide dispersion that becomes predominant term in the total chromatic dispersion, which can be used to compensate for a positive chromatic dispersion accumulated along the transmission line. They are known as dispersion compensating fibers (DCF), and have refractive index profiles similar to the illustrative example shown in Figure 3.12(d). In all cases a with complex refractive index profile listed above, the effective cross-section area cannot be expressed as . Instead, the mode spot size parameter w0 can be replaced by the parameter wso, which is related to equivalent step-index profile, the concept introduced in [40]. In such a case, the effective area is expressed as 

(3.101)

The concept of the equivalent step-index is based on the assumption that any complex form of the refractive index profile in single-mode optical fibers can be replaced by conventional step-index profile if they produce the same mode spot size. The mode spot size in single-mode optical fibers having an arbitrary refractive-index profile can be calculated by using generic formula [41] 1/ 2

 3  2   r E (r ) dr   w  a 0 1/ 2     r E (r ) dr  0 

(3.102)

where r and a are the radial coordinate and fiber radius, respectively, while E(ris the electrical field of the fundamental mode. Equation (3.102) can be applied to the arbitrary refractive index profile n(r); its mode spot size w can be expressed in a form

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w = a·G(V)

(3.103)

where V is the normalized frequency defined by Equation (3.53) and G(V) is the mode function represented by the right side of Equation (3.102). For the equivalent step-index profile, Equation (3.103) becomes wso = aso·S(Vso)

(3.104)

where wso and aso are the mode spot size and the core radius of the equivalent steep-index profile and S(Vso) is newly defined mode function for the equivalent normalized frequency Vso. To find the parameters wso, aso, and Vso, the following set of conditions should be imposed wso = w aso =X· a Vso= Y·V

(3.105) (3.106) (3.107)

where X and Y are transformational coefficients. These coefficients can be found by establishing the following mathematical functional J= wso – w = a·[X· S(YV)-G(V)] = 0

(3.108)

To find the values of the parameters for the equivalent step-index profile, the functional J should take the minimum value in a specified range [V1 – V2] of the normalized frequency V, which leads to the following conditions 2    J 2 dV  0 X V1

V

(3.109)

V

2    J 2 dV  0 Y V1

(3.110)

By solving Equations (3.109) and (3.110), values of X and Y can be found. As for the boundaries V1 and V2, they should be around the cutoff frequency Vc of the secondary mode. It was assumed in [40] that V1 = 0.7Vc and V2=1.3Vc, and the equivalent step-index parameters were found for a parabolic graded index profile with = 2 from Equation (2.6), finding that for parabolic refractive index profile it is as=0.71a and Vs=0.64V. It was also shown that as < a for the W-shape profile from Figure 3.12(b), while it is as > a for the M-profile shape from Figure 3.12d. The equivalent step-index profiles for these two cases are shown in Figure 3.12(b,d). The refractive index acts as a function of the optical power density if the density is relatively high. That is because the electrical field effectively compresses molecules and increases the refractive index value. The effect where

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refractive index is dependent on the strength of the external electrical field is known as the Kerr effect [1]. The effective value of the refractive index in a silicabased optical fiber can be now expressed as

n( P )  n0  n2

P Aeff

(3.111)

where Peff defines the power density per cross-sectional area and n2 is the second-order refractive index coefficient (or the Kerr coefficient). The Kerr effect is a very fast having the response time of about 10–15 s, and can be considered to be a constant with respect to both the signal wavelength and the polarization state. The typical values of the second-order refractive index coefficient in a silicabased optical fiber are in the range 2.2–3.4 x 10–20 m2/W (or 2.23.4 x 10–8 m2/W). As an example, optical power P = 100 mW that propagates through a standard single mode optical fiber with Aeff = 80 x 10–12 m2 will induce the density =P/Aeff =1.25 x 109 W/m2 and the index change n = n2 = 3 x 10–11. As we can see, the nonlinear portion of the refractive index is much smaller than its constant part n0 (please recall that n0 is around 1.5 for silica-based optical fibers). However, even this small value will have a big impact in some cases of optical signal transmission. The variations in the refractive index due to the Kerr effect will change the propagation constant so it can be now written as

 ( P)  0  P

(3.112)

where n0/l is a linear portion related to linear refractive index n0 and  is a nonlinear coefficient given as  

2n2 lAeff

(3.113)

Therefore, parameter  is not just a function of nonlinear index n2, but also depends on the effective cross-sectional area and signal wavelength. Typical . values of parameter  are in the range from 0.9 to 2.75 (W km)–1 for single-mode optical fibers operating at wavelengths around 1550 nm. Typical values of nonlinear parameters for different types of optical fibers are given in Table 3.1. The propagation constant  will vary along the duration of the optical pulse, since the points along the pulse will see different optical powers. Accordingly, the propagation constant associated with the leading edge of the pulse will be lower than the constant related to the central part of the pulse. The difference in propagation constants will cause the difference in phases associated with different portions of the pulse.

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Table 3.1 Typical values of optical fiber nonlinear parameters – – Effective area Aeff Nonlinear coefficient  [W 1km 1] [m2] at 1,550 nm 80 1.12–1.72 80 1.12–1.72 72 1.23–1.92 101 0.9–1.36 50 1.78–2.75 65 1.37–2.11

Fiber type SMF-28 AllWave LEAF Vascade TrueWave-RS Teralight

The central part of the pulse will acquire phase more rapidly than leading and trailing edges. The total nonlinear phase shift after some length L can be calculated by integrating the propagation constant over the distance z as L

L

0

0

 ( P )    ( P0 )   dz   P( z )dz

(3.114)

The phase shift can be expressed in an explicit form by using Equation (3.97), so it becomes [ P(t )] 

L P0 (t )[1  exp( L)] 2 Leff  Leff P0 (t )  n P (t )  eff  l Aeff 2 0 Lnel

(3.115)

where a nonlinear length Lnel defined in [13] has been introduced. The nonlinear length is given as Lnel 

lAeff 2n 2 P0



1

P0

(3.116)

The time-dependent pulse phase from Equation (3.115) will cause an instantaneous frequency variation SPM(z,t)=(z,t) around the carrier frequency 0=20. The frequency variation (i.e., the frequency chirp) can be found as  (t ) d [ (t )] 1 Leff n2 dP0 (t ) (3.117)  (t )    2 dt l Aeff dt Equations (3.115) and (3.117) describe the nature of the phase and frequency shift due to the Kerr effect. We outlined the fact that the optical pulse power is time dependent, just as a reminder that the position along the pulse determines the instantaneous variation of the optical phase from its stationary value. It is important to notice that the frequency chirp induced by self-phase modulation is not linear in time.

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The frequency chirping due to the self-phase modulation effect can be illustrated by plotting the frequency content under the time form of the pulse envelope, which shown in Figure 3.13. We can see that the leading edge of the pulse undergoes downshift in frequency (or an upshift in wavelength known as a red-shift), while the trailing edge experiences the upshift in frequency (or a downshift wavelength known as blue-shift). The frequency chirp induced by selfphase modulation acts in conjunction with the group velocity dispersion, which can lead either to pulse broadening or to pulse compression.

Figure 3.13 Frequency chirping due to self-phase modulation.

3.4 PULSE PROPAGATION IN SINGLE-MODE OPTICAL FIBERS As mentioned earlier in this Chapter, the optical signal will be distorted while propagating through an optical fiber. Signal attenuation and pulse spreading are the main contributors to pulse distortion. Next we will analyze the propagation process in single-mode optical fibers to get a more comprehensive picture about the impact of the different effects mentioned in previous section, such as chromatic dispersion or self-phase modulation, on fiber transmission properties. Next analysis is related to single-channel transmission, while the multichannel WDM case will be analyzed after that.

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3.4.1 Single-Channel Propagation 3.4.1.1 Impact of Chromatic Dispersion and Source Frequency Chirp The pulse distortion a single-mode optical fiber, most often recognized as pulse attenuation and broadening, will be evaluated by analyzing behavior of each individual frequency component within the incoming optical signal spectrum. It is also convenient to assume that chirped Gaussian-like optical pulses have been generated by the optical modulator to be transmitted through the fiber. Such an approach is adopted in [13, 15, 16] and will be followed in this section. The monochromatic electromagnetic wave propagating along the z-axis can be presented by its complex electric field function as

E ( z, t )  Ea ( z, t ) exp[ j ( ) z ] exp(  j 0 t ))

(3.118)

where Ea(z,t) is the pulse envelope that changes with time t and distance z. Parameter 0=20 is the radial optical frequency of the monochromatic wave, 0 is linear optical frequency, and  is the propagation constant of the monochromatic optical wave. The pulse broadening, which occurs along the optical fiber, can be evaluated by exploring the frequency dependence of the propagation constant Each spectral component within the launched optical pulse will experience a phase shift proportional to z. The pulse spectrum, which is observed at the distance z, is given in the frequency domain as

~ ~ Ea ( z, )  Ea (0, ) exp  j ( ) z 

(3.119)

Please recall that superscript (~) denotes the frequency domain of a specified function. The pulse shape in time domain can be obtained by the inverse Fourier transform of Equation (3.119), that is E a ( z, t ) 

1 2



~

E

a

(0,  ) exp  j ( ) z exp(  jt )d

(3.120)



In general, the exact calculation of the inverse Fourier transform cannot be carried out since the function is not known in most cases. It is useful, therefore, to expand the propagation constant  in a Taylor series around the carrier frequency 0=20. The expansion can be done only if the condition 0  0 is satisfied, which is true even if the bit rate of the optical signal goes up to several terabits. By applying Taylor’s expansion, it becomes  ( )   ( 0 )  (   0 )

d d

   0

(   0 ) 2 d 2  2 d 2

   0

(   0 ) 3 d 3  6 d 3

 ...   0

(3.121)

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The first term on the right side of previous equation is the group delay with respect to a unity length, which was introduced by Equation (3.79). This parameter is often denoted as 1= d/d , which is just the inverse value of group velocity vg—please see Equation (3.81). The parameter 2= d2/d 2 is commonly known as the group velocity dispersion (GVD) coefficient, and determines the extent of the pulse broadening during the propagation. It is easy to relate this parameter with the dispersion parameter D introduced through Equation (3.84). It can be done by using relationships =2cl and = –2cll2 between optical frequency  and wavelength l, so it becomes

D

2c 2 l2

(3.122)

where c is the light speed in the vacuum. The parameters D and 2, which are opposite in sign, have been used to recognize two distinctive wavelength regions, which are: (1) the normal chromatic dispersion wavelength region characterized by D < 0 and 2 > 0, and (2) the anomalous chromatic dispersion region, which is characterized by D > 0 and 2 < 0. Also, the parameter 3= d3/d 3 from Equation (3.121) is known as a differential dispersion parameter, which determines chromatic dispersion slope over a specified wavelength range. This parameter plays an important role if the operation is done at wavelengths where the chromatic dispersion changes the sign (the zero-dispersion region). The expansion given by Equation (3.121) can be used in a concept of slowly varying amplitude A(z,t) of the pulse envelope, which can be introduced to express the pulse field function from Equation (3.118) as [13]

E ( z, t )  Ea ( z, t ) exp[ j ( ) z] exp(  j 0 t )  A( z, t ) exp( j 0 z  j 0 t )

(3.123)

The slowly varying amplitude is the most important parameter from the pulse propagation perspective, and can be found by inserting Equations (3.121) and (3.123) into Equation (3.120), that is,

A( z, t ) 

1 2





j

 j1 z (  0 )  2  2 z (  0 ) ~  A(0, )e 

2

j    3 z (  0 ) 3  6 

e  jt ( 0 ) d

(3.124)



Equation (3.124) can be rewritten in the form of a partial differential equation, just by calculating partial a derivative per axial coordinate z, and by recalling that difference ( – 0) can be considered as a partial derivative of amplitude per time coordinate t. The partial differential equation obtained from Equation (3.124) has a form [13]

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A( z, t ) A( z, t ) j 2  2 A( z, t )  3  3 A( z, t )   1   z t 2 6 t 2 t 3

(3.125)

This equation is the basic one that governs the pulse propagation through a dispersive medium, such as a single-mode optical fiber. It is common to analyze the pulse propagation for a slowly varying amplitude that takes the Gaussian function shape, given as

 t2  A(0, t )  A(0) exp   2   2 0 

(3.126)

where A(0)=A0 is the peak amplitude and 0 =  represents standard deviation of the Gaussian function distribution. The standard deviation is related to the fullwidth at half-maximum (TFWHM =2T0) of the curve given by Equation (3.126) by the relation

TFWHM  2 2 ln 2 0  2.3548 0  2.3548

(3.127a)

If there is time slot T=1/B, where B is the bit rate, at least 95% of the Gaussian pulse energy will be contained in that time slot if the following condition is satisfied.

0  

T 1  4 4B

(3.127b)

The condition above can be used for the evaluation of system transmission characteristics. The frequency spectrum of the modulated optical wave at the fiber input is determined by the Fourier transform of Equation (3.126), which is given as   t2    2 (   0 ) 2  ~ A(0,  )  A0  exp   2  exp  j (   0 )dt  A0 0 2 exp  0  2    2 0  

(3.128)

The spectrum has a Gaussian shape centered on the frequency 0=20. The spectral half-width at the 1/e intensity point is

 0  1 /  0

(3.129)

The pulses that satisfy Equation (3.129) are referred to as transform-limited ones. Equation (3.129) represents the case when there is no frequency chirp imposed on an optical pulse during its generation. However, in most practical cases there is a

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frequency chirp induced during the optical pulse generation or modulation. It can be expressed by an initial chirp parameter C0 in Equation (3.126), which then becomes

 (1  jC 0 )t 2   A(0, t )  A0 exp   2 02  

(3.130)

The chirp parameter C0 defines the instantaneous frequency shift in the chirped Gaussian pulse. As an example, it is associated with a shift given by Equation (2.58) if there is a direct modulation of semiconductor lasers. The instantaneous frequency increases linearly from the leading to the trailing pulse edge for positive values of the parameter C0. The opposite situation occurs for negative values of the chirp parameter since instantaneous frequency decreases linearly from the leading to the trailing edge. The amount of the frequency chirp imposed through the parameter C0 is measured by the frequency deviation  from carrier frequency 0, and can be found by making a derivative of the phase in Equation (3.130), that is,

 

C0

 02

(3.131)

t

The spectrum of the chirped Gaussian pulse can be found by taking the Fourier transform of Equation (3.130), so it becomes   (1  jC 0 )t 2   2 02  ~ A(0,  )  A0  exp   exp( jt )dt  A0   2 2 0   1  jC 0  

1/ 2

 ( 0 ) 2  (3.132) exp    2(1  jC 0 ) 

The spectrum has a Gaussian shape with the spectral half-width at the 1/e intensity point given as

  (1  C02 )1 / 2 /  0   0 (1  C02 )1 / 2

(3.133)

where 0 is the spectral half-width of the chirp-free pulse, given by Equation (3.129). Therefore, the spectral width of a chirped Gaussian pulse is enhanced by factor (1+C02)1/2. A closed-form expression for a slowly varying pulse amplitude can be found by inserting Equation (3.132) into Equation (3.124), and by performing the analytical integration afterwards. The contribution of the term associated with coefficient 3 can be neglected by assuming that the carrier wavelength is far away from the zero-dispersion wavelength region. In addition, we can omit the contribution of the term associated with 1, since it does not

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impact the pulse shape. (Please recall that it contributes just to the pulse delay.) After all these transactions, the expression for the output pulse envelope becomes A( z, t ) 

A0 0

  (1  jC 0 )t 2 j ( z ,t ) (3.134) exp  2   A( z, t ) e 2  0  j 2 z  C0  2 z  2 0  2C0  2 z  j 2  2 z 

where |A(z,t)| and (z,t) are the magnitude and the phase of the complex pulse envelope, respectively. They can be expressed as A( z ,t ) 

A0

1  C  z /     z /   0

 ( z, t )  

2

2 2 0

2 2 2

4 0

1/ 4

  t2 exp  2 2 2 2  2 0  2C0  2 z  2 2 z /  0 

  2 zt 2  z  1 1  tan 1  2 2  2 2 2 2 2  0  C0  2 z    2 z 2   0  C0  2 z 

(3.135)

(3.136)

We can see from Equation (3.135) that the pulse shape remains Gaussian, but with a modified amplitude due to an impact of the chirp parameter. The chirp parameter and the pulse width (expressed though its half-width at 1/e intensity point) change from their initial values C0 and 0, respectively, and after distance z become equal to C ( z )  C0 

(1  C02 )  2 z

(3.137)

 02

 C  z  2   z  2   ( z )   0 1  0 2 2    22   0   0    

1/ 2

 C z  2  z  2      0 1  0    LD   LD    

1/ 2

(3.138)

where LD =02/|2| is the dispersion length, as defined in [13]. The time-dependent pulse phase from Equation (3.136) means that there is an instantaneous frequency variation around the carrier frequency 0, given as

 ( z, t )  

2 z ( z, t )  t 2 t  0  C0  2 z 2   22 z 2

(3.139)

This instantaneous frequency shift is again referred as a linear frequency chirp, since it changes in proportion with the time. There are two factors contributing to the sign and the slope of a linear function presented by Equation (3.139). They are the GVD parameter  2 and the initial chirp parameter C0.

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The frequency deviations across the pulse width due to chromatic dispersion impact is illustrated in Figure 3.14. This figure reflects the case when the initial chirp parameter is zero. As we see, the instantaneous frequencies at the leading edge are lower than the carrier frequency 0=20, if referred to the normal dispersion regime (for D < 0, and 2 > 0). At the same time, the frequencies at the trailing edge are higher than the carrier frequency. Just opposite situation occurs in the anomalous dispersion regime (for D > 0, and 2 < 0).

Figure. 3.14 Unchirped input optical pulses and chirped pulses propagating in single-mode fibers.

The impact of the initial chirp parameter can be evaluated through the ratio

(z)/0, which reflects the broadening of the pulse. It is shown in Figure 3.15 as a function of normalized distance z/LD, where LD =02/|2| is a dispersion-length, the parameter introduced in Equation (3.138). The pulse broadening depends on the sign of the product C02. When C02 > 0, a monotone broadening occurs. Since the broadening rate is proportional to the initial value of the chirp parameter, the smallest width increase occurs for an unchirped pulse. An initial narrowing occurs if C02 < 0, but it is then followed by subsequent almost linear broadening. This situation can happen in one of two following cases: (1) the pulse having positive value of C0 propagates in the anomalous dispersion region (2 < 0 and D > 0) or (2) the pulse having negative value of C0 propagates in the normal dispersion region (2> 0, D < 0). In either case, the frequency chirp introduced by chromatic dispersion counteracts the initial frequency chirp. The pulse undergoes narrowing until these two chirps cancel each other, which can be identified as a minimum at the curve from Figure 3.15. For longer propagation distances these two frequency chirps are out of balance and pulse broadens monotonically. The pulse broadening at the point z = LD =02/|2| is identical to the broadening of a chirpless Gaussian pulse. The pulse

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broadening given by Equation (3.138) does not include a contribution of the higher-order dispersion terms (i.e., term 3 and up). Although this equation can be used in most practical situations, it does not produce good results in some cases where more precise estimate is needed, such as transmission in wavelength region with low chromatic dispersion. 7

2< 0

6 5 4 3 2 1

distance z/LD

0

0

0.5

1.0

1.5

2.0

2.5

Figure 3.15 Pulse broadening factor of the chirped Gaussian pulse.

The exact solution of Equation (3.125) can be obtained even if the term 3 is included, but the pulse shape will not be a Gaussian function any more. Since neither the full-width at half-maximum (FWHM) nor the full-width at the 1/e intensity point (FWEM) can be used to characterize pulse broadening, the width of the pulse can be expressed by its root mean square (RMS) defined as 2     2   t 2 A( z, t ) dt   t A( z, t ) dt       ( z )          A( z, t ) 2 dt A( z, t ) dt            

1/ 2

(3.140)

The amount of pulse broadening at the distance z can be estimated by the variance ratio 2(z)/02, where 02 = 02/2 is determined by the half-width of the input Gaussian pulse [13, 16]   z   2( z )  C 02 z   2 z     2   ( 1  C02 ) 33   1  2 2   4 2  0 2 0   2 0    0  2

2

2

(3.141)

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Equation (3.141) is applicable for all cases where the spectral width of the light source is much smaller than the signal spectral width  0, which was introduced by Equation (3.133). Since this condition is not satisfied in most cases where a direct modulation is applied, Equation (3.141) should be modified to account for the source spectral width. It could be done through a broadening factor c=20s, where 0 is the spectral width of the Gaussian input pulse, while s is the source spectral width measured in GHz. Equation (3.141) now becomes [13] 2

2

  z   z  2 ( z)  C 0  2 z    (1   c2 ) 2 2   (1  C02   c2 ) 33   1  2 2   4 2  0 2 0    2 0   0 

2

(3.142)

Equations (3.141) and (3.142) can be used to analyze chromatic dispersion impact for following two cases that are the most relevant from systems design perspective: (1) when c = 20s >>1, the source spectrum is much larger that the signal spectrum. The effect of the frequency chirp can be neglected here (C0 = 0). In addition, if the carrier wavelength is far away from the zero-dispersion region, the term containing 3 can be neglected as well. In this case Equation (3.142) takes a simplified form

 2 ( z )   02   s2  22 z 2   02   l2 D 2 z 2   02   D2 ,1

(3.143)

If the carrier wavelength is within the zero-dispersion region, the term containing 2 can be neglected, so that Equation (3.142) leads to

 2 ( z )   02 

 s4  32 z 2  4S 2z2   02  l   02   D2 ,2 2 2

(3.144)

Note that wavelength-related parameters l (the source spectral width expressed . in nm), D (chromatic dispersion parameter expressed in ps/nm km), and S=dD/dl 2 (chromatic dispersion slope expressed in ps/nm km) have also been introduced in Equations (3.143) and (3.144); (2) when c = 20c 0), which is referred to as a positive chirp. However, the instantaneous frequency decreases linearly from the leading to the trailing edge for a negative chirp parameter (C0 200 ps/nm·km/dB. The most beneficial use DCF is for chromatic dispersion compensation in multichannel transmission systems. If used in a multichannel or WDM environment, the condition given by Equation (3.174) should be satisfied for each individual channel, which means that Equation (3.174) now becomes

D1 (li ) L1   D2 (li ) L2

(3.175)

where li (i=1,2…M) refers to any optical channel out of the total number of M channels multiplexed together. Since D1(l) is wavelength dependent, as illustrated in Figure 3.8, the accumulated dispersion D1(l)L1 will be different for each individual channel. That puts more stringent requirements on DCF, which should also have a proper negative dispersion slope to satisfy Equation (3.175). The dispersion slope of the DCF can be calculated by assuming that Equation (3.175) is satisfied for a reference channel, which is commonly the central wavelength lc in a composite WDM signal. Dispersion for other channels can be expressed through the referent dispersion and the dispersion slope, which leads to the following set of equations

D1 (lc ) L1   D2 (lc ) L2

(3.176)

[ D1 (lc )  S1 (li  lc )]L1  [ D2 (lc )  S 2 (li  lc )]L2

(3.177)

S 2  S1 ( L1 / L2 )  S1 ( D2 / D1 )

(3.178)

where S1 and S2 are dispersion slopes of the fiber lengths L1 and L2, respectively. The above set of equations governs the chromatic dispersion compensation in multichannel optical transmission. The ratio S/D, often called the relative dispersion slope, should be the same for optical transmission and dispersion compensating fibers. However, it is rather difficult in practice to satisfy Equations (3.176) – (3.178) over a broader wavelength range, and that can lead to imperfect dispersion compensation for some channels, which is illustrated in Figure 3.18. The perfect compensation is achieved for the central wavelength lc, while there is a residual dispersion on both sides of the central wavelength within the transmission bandwidth, as shown in Figure 3.18. The channels with shorter wavelengths are overcompensated, while the channels at longer wavelengths are undercompensated. It is also possible to establish a reference with respect to a

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wavelength different from the central one. In such a case, the picture of residual dispersion will be different. It is easier to find good matching for standard single mode (SMF) fibers, which have S/D ~ 0.003 nm–1, than for dispersion shifted fibers (DSF) and nonzero dispersion shifted fibers (NZDSF) since both of them have much higher S/D ratio (it is usually S/D > 0.02 nm–1 for both DSF and NZDSF). Because of that, the compensation scheme involving NZDSF might require a postcompensation at the receiving side to eliminate the residual dispersion in some of channels. Proper dispersion compensation for all channels can be achieved much more easily if the length L2 is relatively long. If so, the dispersion compensation fibers are not just compensating elements, but become a part of the transmission line having different chromatic dispersion value than preceding session, following a specific dispersion compensation map, as shown in Figure 3.17. Such fibers, which are known as reverse dispersion fibers, are often used in combination with transmission fibers to enhance the overall transmission capability [52]. In general, transmission fibers and DCF can be combined in different manner, often referred to by terms such as precompensation, postcompensation, and overcompensation, to describe the nature of the method used.

0

Resultant (residual) dispersion

lc 1300

1400

1500

1600

1700

Wavelength [nm]

Figure 3.18 Residual dispersion due to imperfect matching between transmission fiber and DCF.

Optical fibers that operate near the cutoff normalized frequency (Vc=2.4) and supports not just fundamental LP01 mode but also the higher-order mode LP11, can be also used for compensation purposes. If used for chromatic dispersion compensation, these fibers are combined with mode converters at the input and

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output of the two-mode fiber section. An incoming optical signal is led through the mode converter that transfers the fundamental mode LP01 to the next highermode LP11. The dual-mode fiber is specially designed so that the dispersion parameter D2 has a large negative value for LP11, and it can be as large as –550 to –770 ps/nm.km [53]. The total length of the dual-mode fiber is approximately 2– 3% of the length of the transmission fiber. The higher-order mode is converted back to fundamental mode after passing through the dual-mode DCF. Mode conversion from the LP01 to LP11 mode, and vice-versa, can be done by a longperiod fiber grating [54], which efficiently couples these modes. Fiber Bragg grating as a device can also be effectively used for chromatic dispersion compensation. Such gratings, which belong to the class of short-period gratings, are well-known as fiber Bragg gratings [54, 55]. The application of fiber Bragg gratings for chromatic dispersion compensation is illustrated in Figure 3.19(a). The gratings are mostly used in combination with optical circulators, which separate the reflected light from the forward-going signal. Fiber grating is a special piece of fiber with chirped Bragg grating imprinted inside. The grating can be imprinted in the core of photosensitive optical fibers by applying a strong UV laser source acting through a special phase mask, or by using special holographic methods [54]. The incident light power causes permanent changes and microvariations in the fiber refractive index, while the index maximums coincide with the diffraction fringes of the diffracted UV light. The refractive index varies along the fiber piece length as  2zG( z )   2z  (3.179)   n c  nv cos n( z )  nc  n v     ( z )  0   where nc is the refractive index in the fiber core, nv (~0.0001) is the magnitude of the index change due to grating imprint, z)=0/G(z) is a variable grating period, 0 is the maximum grating period, and G(z) is a function that characterizes the grating chirp along the fiber piece length. Since the Bragg wavelength varies along the grating length, different wavelengths will be coupled to backward wavers at different places along the grating. The wavelength coupling will occur at the places where the Bragg condition is satisfied locally. Depending on the chirp, which can increase or decrease along the length of the fiber piece, either shorter or longer wavelengths will be coupled first. For example, longer wavelengths will be coupled first in Figure 3.19. Chromatic dispersion in the anomalous dispersion region can be compensated with Bragg gratings having a negative (decreasing) chirp along the length, while the dispersion from normal dispersion region can be compensated by gratings imposing increasing chirp. Please notice that classification to gratings imposing positive and negative chirp is quite irrelevant since each grating can impose any chip if input and output ports are reversed. Therefore, the fiber Bragg grating filters different wavelengths at different points along its length, and introduces wavelength-dependent delays.

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Figure 3.19 Dispersion compensation by using: (a) fiber Bragg gratings and (b) phase conjugator.

The refractive-index variations that have a grating structure of will cause coupling of the forward and backward propagating waves. The coupling occurs only for Bragg wavelengths, that is for wavelengths that satisfy the following Bragg condition

lB  2nc ( z).

(3.180)

The dispersion parameter of the fiber Bragg grating Dgrat can be obtained from round-time delay  between two wavelengths that are coupled at the very beginning and at the very end of the grating. Therefore, it is   Dgratl 

Dgrat 

2nco ,B Lgrat

2 Lgratnco ,B cl

(3.181)

c

(3.182)

where c is the light speed in the vacuum, nco,B is the refractive index in the fiber core, Lgrat is the length of the grating, l=ll is difference in wavelengths that

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were reflected at the opposite ends of the grating. The fiber Bragg grating is, in fact, an optical bandpass filter that reflects the wavelengths from the specified band, which is defined by the grating period, fiber length, and chirping rate. The compensation capability is proportional to the grating length. As an example, the fiber length of 10 cm is needed to compensate for dispersion of 100 ps/nm, while the length of one meter is needed to compensate for dispersion of 1,000 ps/nm. At the same time, the dispersion compensation ability is inversely proportional to the bandwidth, which means that larger dispersion can be compensated over smaller bandwidth, and vice versa. For example, 1,000 ps/nm can be compensated over the bandwidth of 1 nm, while 100 ps/nm can be compensated over the wavelength bandwidth of about 10 nm. Optical Bragg gratings impose some ripples with respect to reflectivity and time delay along the wavelength bandpass. The ripples are associated with the discrete nature of grating imprinting and present one of the biggest problems related to the grating application. This effect can be minimized by the apodization technique, in which the magnitude of the index change, which is given by nv in Equation (3.179), is made nonuniform across the fiber length. The apodization is usually made in such a way that the refractive-index variation has a maximum at the middle of the fiber piece, while it decreases towards its ends. A wider application of the fiber Bragg gratings is restricted by their limited bandwidth, since it is rather difficult to maintain chirping stability over longer lengths. However, they can be used in a number of applications since they have relatively small insertion losses and lower cost. In addition, compensation by fiber Bragg gratings can be combined with other dispersion compensation methods, to enhance the overall compensation ability. Another method for effective dispersion compensation is by using the phase conjugation. The phase conjugation is an all-optical nonlinear dispersion compensation method in which the input optical signal is converted to its mirror image. It is done in such a way that the amplitude stays unchanged, while the output signal spectrum is a complex conjugation, or a phase reversal, of the input spectrum. Accordingly, the Fourier transforms of the input and output signals are related as

Ain ( 0   )  Aout ( 0   )

(3.183)

where Ain and Aout are amplitudes of the input and output pulses, respectively,  is the central optical frequency, while  presents frequency deviation from the central frequency. As we see, the upper spectral components of the input spectrum have been converted to lower spectral components in the output spectrum. The phase conjugators can be used for chromatic dispersion compensation in a way illustrated in Figure 3.19(b). In case where the transmission link is characterized by a uniform chromatic dispersion, the phase conjugation should be performed at the distance z=L/2, where L is the total length of the transmission line. This method is well known as mid-span spectral inversion. However, if transmission

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line is heterogeneous one, the phase conjugator should be placed at the point where

D1 L1  D2 L2

(3.184)

where D1 and L1 are referred to the incoming section length and corresponding chromatic dispersion and D2 and L2 represent the outgoing section and its associated chromatic dispersion, respectively. Equations (3.183) and (3.184) imply that chromatic dispersion accumulated along the first part of the transmission line can be exactly compensated by the second part. This is just partially true since it can be done for the second-order group velocity dispersion, but not for the third-order chromatic dispersion represented by coefficient 3 in Equation (3.149). The main advantage of the phase conjugation method is that it can be utilized not just for chromatic dispersion compensation, but also for compensation of the SPM effect. The phase conjugation can be performed at the middle of the line in case of a homogenous transmission line, while the heterogeneous transmission line requires that the following requirement is satisfied

n2 ( L )( L )L1 L L  n2 ( L )( L )L1 L L 1

(3.185)

2

where n2 is nonlinear Kerr index, L=P(L)/Aeff is the optical power density, P is the optical power, while Aeff is the effective cross-sectional area—please refer to Equations (3.111) to (3.113). An effective application of phase conjugation can be based on several physical effects, such as the four-wave mixing (FWM) process that occurs in optical fibers, in special nonlinear waveguides (for example, LiNbO3 waveguide), or in semiconductor optical amplifiers (SOA). Several lab experiments phase conjugation have been successfully demonstrated to compensate chromatic dispersion over longer distances [56–58]. Numerical simulations showed that the phase conjugation method is feasible over several thousand kilometers if very precise system engineering is used. The phase conjugators can be eventually combined with optical amplifiers to provide both the amplification and chromatic dispersion compensation at the same place. These amplifiers, known as parametric amplifiers show a significant application potential [58, 59]. Chromatic dispersion compensation can be also done by optical filters. The application of optical filters is based on the fact that chromatic dispersion changes the signal phase in proportion to the GDV coefficient 2. Therefore, the chromatic dispersion can be compensated for if there is an optical filter that reverses the phase change. For this purpose, optical filter should have a transfer function just inverse to the transfer function of the optical fiber in question. However, it is very difficult to achieve in practice since there is no such optical filter that can fully satisfy this requirement. It is possible, instead, to use special optical filters for

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partial compensation of chromatic dispersion effect. The transfer function of the filter can be expressed as

1 H fil ( )  H fil ( ) exp[ j( )]  H fil ( ) exp[ j( 0   1   2 2 )] 2

(3.186)

where i=di/d i (i=1,2…) are the derivatives evaluated at the center frequency 0. We should recall that the similar expansion was used for the signal phase— please see Equations (3.121) – ( 3.124). Coefficients with indexes i=0 and i=1 in Equation (3.186) are responsible for an inherent phase shift and for the time delay, respectively, and they are not relevant with respect to the phase change. On the other side, the coefficient with the index i = 2 is the most important one since it is associated with the phase change. This coefficient should be matched with the GVD coefficient by relation 2 = –2L. The amplitude of the optical filter should be |H()|=1, which is necessary to prevent attenuation of any spectral components within the incoming optical signal. Optical filters for chromatic dispersion compensation can be realized as a cascaded structure of either Fabry-Perot, or Mach-Zehnder interferometers. Cascaded structures that contain multiple optical filters can be produced by using planar lightwave circuits (PLC) [60]. Optical filters, which are designed for chromatic dispersion compensation, can bring some additional benefits since they also filter the optical noise and limit its power. In general, optical filters can compensate for more than 1,000 ps/nm, but over relatively narrow bandwidth. The third group of chromatic dispersion compensating schemes, mainly used to eliminate the residual dispersion that remains at the receiving side, is known as dispersion postcompensation. The residual dispersion becomes an issue in WDM systems since it might exceed a critical level for some of channels. The postcompensation should be applied just to specified number of channels, usually per channel basis. However, this approach can be rather cumbersome and expensive if there are a large number of transmission channels. In addition, the employment of dispersion compensating modules with fixed negative dispersion is a static approach that might not suitable for an environment in which the exact amount of inserted chromatic dispersion is not known. It happens due to variations in the group velocity dispersion coefficient that might be caused by temperature, or may appear due to dynamic reconfiguration of lightwave paths in an optical networking environment that changes the total value of accumulated chromatic dispersion. These fluctuations in chromatic dispersion are more critical for high bit rates above 10 Gb/s. Therefore, it is highly desirable to have an adaptive compensation scheme that can be adjusted automatically. Such a scheme can either employ tunable dispersion compensating elements or utilize an electronic equalization of dispersion effects [61–64]. Tunable dispersion compensation that can be remotely controlled and adjusted per channel base is something that would enhance the transmission characteristics and provide benefits from both the cost perspective and the system engineering perspective.

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Adjustable, or tunable, dispersion compensation can be easily understood if we recall that any dispersion compensator can be considered as a filter, which reverses the phase deviation that occurred due to the impact of chromatic dispersion. Therefore, tunability is related to a fine adjustment of the filter characteristics. Several methods of tunable dispersion compensation have been proposed and demonstrated in the lab. Most of them use fiber Bragg gratings as a tunable filter. The controllable chirp that relates to the compensation capability is changed through the distributed heating. By this approach, the achieved dispersion range can be larger than 1,500 ps/nm over subnanometer wavelength band. Generally speaking, any tunable optical filter that follows requirements given by Equation (3.186) could be considered for dynamic dispersion compensation. Finally, postdetection dispersion compensation is done in optical receivers, and it is applied on electrical signals. The suppression of impact of chromatic dispersion is done through a digital signal processing and electronic equalization filtering. In some cases, such as coherent detection schemes described in Chapter 6, in addition to digital filtering the dispersion can be compensated by linear analog filtering methods. A microwave filter that corrects the signal distortion caused by chromatic dispersion should have a transfer function [46],

  j (   IF ) 2  2 L  H ( )  exp   2  

(3.187)

where IF is the intermediate microwave frequency, 2 is the GVD coefficient, and L is the fiber length over which chromatic dispersion has been accumulated. The microstrip lines that are several tens of centimeters long can be used to compensate for chromatic dispersion accumulated over several hundreds of kilometers. The linear filtering method can be used in coherent detection schemes, since the information about the signal phase is preserved. However, the direction detection scheme does not keep track of phase, since a photodiode responds to the optical signal intensity. In this case, some other approaches can be taken, such as changing decision criteria based on the presence of preceding bits. For example, more “1” bits preceding bit in question will mean that there is more intense intersymbol impact to be accounted for, and vice versa. The other approach is based on making a decision only after examining the signal structure consisted of several bits in a row in order to estimate the amount of intersymbol interference that should be compensated. The method is known as maximum likelihood sequence estimation (MLSE) and generally requires fast signal processing and logical circuits operating at higher clock than the signal bit rate [65]. In addition, the longer an examined bit sequence is, the faster the signal processing should be. Therefore, this method is the most effective for moderate bit rates, and for moderate transmission distances.

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Another approach in electronic compensation is to use the digital transversal filters, which are well known from telecommunication theory [11]. In this case, the signal is split in several branches that are mutually delayed and multiplied by weight coefficients. These components are eventually joined again, and the decision is made on a summary signal. Postdetection technique is now considered as the mainstream compensation scheme for advanced high-speed detection schemes [66–68] and will be discussed in details in Chapter 6. Digital backpropagation method looks very promising as way to compensate not only for dispersion effects, but also for the impact of nonlinear effects such as self-phase modulation and nonlinear channel cross-talk. The assumption is that the impact of all linear and nonlinear effects is known at the receiver side and that it is captured in a comprehensive transfer function, obtained from a number of coupled nonlinear Schrodinger equations, of the optical fiber channel. If that transfer function is multiplied by its inverse value, it will eventually conceal the impact of all nonlinear and linear effects [66]. Therefore, there are two important preconditions for the method implementation: (1) the exact form of the comprehensive transfer function should be known, and (2) the exact inverse function should be produced to conceal the impact of original effects. All this means that a heavy digital signal processing is needed to account for all effects that should be compensated. To perform a high-speed signal processing, some parallel processing schemes need to be employed. However, the digital propagation technique is the most comprehensive tool for the compensation in future high-speed transmission systems, and it will find a wider deployment with continuing improvement in the speed of the digital signal processing. In summary, a comparative review of several commercially available modules for a chromatic dispersion compensation is given in Table 3.3. These methods are compared with respect to the figure of merit, which is defined by the insertion loses, dispersion compensation ability, and fine-tuning capability, the frequency band, and sensitivity to polarization.

Method Figure of merit [ps/dB] Mean PMD Bandwidth Tunability

Table 3.3 Optical dispersion compensating schemes Dispersion Fiber Bragg grating compensating fiber 50–300 100–200 0.06–0.1 ps/km0.5 More than 30 nm No

0.5–1.5 ps 0.5–6 nm Yes

Dual mode conversion 50–170 0.05–0.08 ps/km0.5 ~30 nm Yes

As for polarization mode dispersion (PMD) compensation, it becomes very important for high bit rates in an environment where transmission lightwave path can include some sections with relatively high PMD value. The main difference between PMD and chromatic dispersion is in a stochastic nature of the PMD effect, which makes it more difficult to deal with. The PMD compensation

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schemes are generally considered independently from chromatic dispersion schemes, although the same kind of postdetection methods is applied in both cases. Several methods have been proposed and demonstrated so far to compensate for the PMD effect [62, 63, 65]. They can be classified as being related to the receiver side or to the optical transmitter side. In some cases these two schemes can be even combined together. In addition, the PMD compensation can be related to either optical or electrical signals. The basic idea behind the PMD compensation is about either correction and equalization of the delay between two polarization modes, or about changing the polarization state of the incoming optical signal to achieve more favorable detection conditions. The equalization of the PMD effect by optical means is done by using a feedback loop to change the polarization state of the incoming optical signal by some polarization alternating devices. The feedback can be established within the optical receiver side, which provides a relatively fast operation, but with a rather limited dynamic range. However, the feedback can be established all the way to the transmitter side to find and launch signals in the principal polarization state (PSP). Such a method provides a larger dynamic range. However, the operation can be relatively slow since there is transmission delay due to the feedback that is established over longer distances. The PMD equalization can be also done by splitting the incoming optical signal in two polarization modes in accordance with the principal polarization states [63]. Two polarization modes are separated by the polarization splitter and combined together through the polarization combiner. In the meantime, the faster polarization mode is delayed with respect to slower one, which means that the first order PMD is alleviated. The electrical PMD compensation relies on fast electronic signal processing and it is the part of the overall digital signal processing scheme in a digital receiver. It is done through the transversal filter scheme that splits detected electrical signal into multiple parts (branches). Each branch is then multiplied by a weight coefficient and delayed by a specified time amount through the delay lines. All branches are then combined into a unified output signal. In summary, the PMD compensation can be effectively done in an electrical domain and on a per-channel basis. In fact, the approach taken so far is to alleviate the PMD effect enough that its contribution to the sensitivity degradation of the optical receiver is kept under the certain limit. The advanced PMD compensation schemes have become indispensable part of the digital coherent receiver that is now used for high-speed applications.

3.5 MULTICHANNEL PROPAGATION IN OPTICAL FIBERS Multichannel transmission through optical fibers is characterized by number of individual channels multiplexed together and sent over the fiber line. Each

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channel is identified by its carrier frequency i (or carrier wavelength li) and optical power level Pi associated with that frequency. The aggregated signal along the lightpath has a power ∑

(3.188)

where M is the total number of channels. l1 l2 l3

time

l4 l5 T 100 ps

l1 Signal spectrum

l4

l3

l2

ITU-T grid

…

lN

40 Gb/s

10 Gb/s

ITU-T grid

l5

50 GHz

50 GHz

Wavelength l

50 GHz ~ 0.4 nm

Figure 3.20 Multichannel arrangement of individual channels: (a) digital content, and (b) WDM spectrum lineup.

Multichannel arrangement in the spectral domain is known as wavelength division multiplexing (WDM) scheme. The channel lineup along the wavelength coordinate can follow some standardized scheme, defined by spectral slots and channel spacing among the neighboring channels. The ITU-T standard established a grid where wavelength spacing can vary from 12.5 GHz to 200 GHz [69]. The wavelength grid with 50-GHz spacing, which is the most common in the systems deployed to date, is illustrated in Figure 3.20 for 10-Gb/s and 40-Gb/s channels with binary intensity modulation. In general, each channel can be modulated with different bit rate and modulation signal, but it should be designed to be contained within the assigned ITU grid. Multichannel propagation in single-mode optical fibers is characterized by the fact that individual channels may experience different impact from both linear and nonlinear effects in optical fiber. In addition, multichannel transmission will create an environment where multichannel nonlinear effects may present the

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major signal impairment. Next we will describe two nonlinear effects that have an impact to multichannel transmission in single-mode optical fibers, and they are cross-phase modulation (XPM) and four-wave mixing (FWM). 3.5.1 Cross-Phase Modulation Cross-phase modulation (XPM) is another effect caused by the intensity dependence of the refractive index and occurs during the propagation of a composite optical signal through an optical fiber. The nonlinear phase shift of a specific optical channel is affected not just by the power of that channel, but also by the optical power of the other channels. The impact of other optical channels to the channel in question can be evaluated by using Equation (3.115), which is now modified and becomes  m (t )  Leff P0m (t )  2Leff

M

P i m

0i

(t ) 

2 Leff n2 l Aeff

M   (3.189)  P0m (t )  2 P0i (t ) i m  

where m denotes the channel in question, and n2, Aeff , Leff, and M are the nonlinear index, the effective cross-sectional area, the effective fiber length, and number of optical channels, respectively. The factor 2 in Equation (3.189) shows that the cross-phase modulation effect is two times more effective than the self-phase modulation effect [44]. In addition, the phase shift is bit pattern dependent, since just “1” bits will have an impact to the total phase shift. Therefore, the arrangement of “1” and “0” bits, as illustrated in Figure 3.20 will play very important role in the overall effect. In the worst-case scenario, when all channels contribute simultaneously by having “1” bits loaded with power Pm, Equation (3.189) becomes  m (t ) 

2 Leff n2 P0 m (t ) 2 M  1 l Aeff

(3.190)

Equations (3.189) and (3.190) can be used to estimate the cross-phase modulation effect in a dispersionless medium, in which optical pulses from different optical channels propagate with the same group velocity. However, in real optical fibers, optical pulses in different optical channels will have different group velocities. The phase shift given by Equations (3.189) and (3.190) can happen only during the overlapping time. The overlapping among neighboring channels is longer than the overlapping of channels spaced apart, and it will produce the most significant impact to the phase shift. The pulse overlapping and induced frequency shift due to cross-phase modulation is illustrated in Figure 3.21. When overlapping starts, the leading edge of the pulse A experiences decrease in optical frequency, or increase in optical wavelength (known as the red-shift), while the trailing edge of the pulse B

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experiences increase in optical frequency, or decrease in optical wavelength (often called a blue shift). When overlapping finishes, the trailing edge of the pulse A becomes blue-shifted, while the leading edge of the pulse B experiences the red shift. If pulses walk through one another quickly, the described effect on both pulses is diminished since distortion caused by trailing edge undoes distortion caused by the leading edge. This case can be associated with the situations when there is a significant chromatic dispersion, or when interacting channels are widely separated. However, if the pulses walk through one another slowly, the effect that both pulses experience is similar to one illustrated in Figure 3.21.

Figure 3.21 Pulse overlapping and frequency shift due to the XPM effect.

The effect of cross-phase modulation can be done by increasing the spacing between individual channels can reduce the effect of cross-phase modulation. By doing so, the difference in propagation constants between these channels becomes large enough so that interacting optical pulses walk away from each other, and cannot interact any further. That difference will be enhanced if there is a stronger impact of chromatic dispersion. The most unfavorable case will occur in the zero dispersion wavelength region since optical channels will stay together and overlap for significant amount of time. In general, it is very difficult to estimate the real impact of the cross-phase modulation on the transmission system performance just by using Equations (3.189) and (3.190). The impacts of both self-phase and crossphase modulations can be studied more precisely by solving the nonlinear Schrodinger equations, as we will explain shortly. 3.5.2 Four-Wave Mixing (FWM) Four-wave mixing (FWM) is a nonlinear effect that occurs in optical fibers during the propagation of a composite multichannel optical signal. The power dependence of the refractive index in optical fibers will not only shift the phases

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of the signals in individual channels is mentioned above, but will also give a rise to new optical signals through the process known as four-wave mixing. Fourwave mixing is an effect that originates from the third-order nonlinear susceptibility—please see Equation (3.9). The four-wave nonlinear mixing is related to the fact that if there are three optical signals with different carrier frequencies (i, j, andk ; i,j,k=1,…M) that propagate through the fiber, a new optical frequency (ijk=i+j –k) will be generated. The number of possible combinations with three optical frequencies grows rapidly with an increase of the total number M of optical channels propagating together. However, an effective interaction between three frequencies also requires a phasematching, which can be expressed as

 ijk i+jk

(3.191)

where denotes a propagation constant defined as =2n/l, (n is the refractive index, and l is the wavelength). The value i+jk ijk

(3.192)

presents a measure of the phase matching condition for optical waves involved into a four-wave mixing process. An efficient interaction and new wave generation takes place only if  approaches to zero. The phase matching condition can be easily understood if we look at the physical picture of the FWM process. Namely, the FWM process can be considered as mutual interaction and annihilation of two photons with energies hi and hj (h is the Planck constant), resulting in generation of two new photons with energies hk and hijk. The phase matching condition can be considered just as requirement for momentum conservation. The generation of new optical frequencies due to FWM is shown in Figure 3.22, which illustrates the cases where there are two and three interacting optical channels. If there are just two optical channels with frequencies 1 and2, two more optical frequencies will be created, which presents “the degenerate” case shown in Figure 3.22(a) [14]. If there are three optical channels with frequencies 1, 2 and3, the eight more optical frequencies can be created, as in Figure 3.22(b). (If three or more frequencies participate in the process, it is referred as a nongenerate case). The condition for phase matching can be easily understood for the degenerate case from Figure 3.22(a). In this situation we have that 2=1 +, 221=2 +, and 112=1 - . If we use the expansion in the Taylor series for each of the propagation constants in the degenerate case, the phase matching condition becomes 2    (3.193)    2    2 

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where 2 is group velocity dispersion (GVD) coefficient. It is clear from Equation (3.193) that a complete phase matching happens only if 2=0, or at zero chromatic dispersion point. However, a practical phase matching occurs either for very low values of the GVD coefficient or for very narrow channel spacing.

2

112

Optical power

Optical power

1

221



113

3

132 312

123 213

112

231 321

223

332 332



frequency

(a) Degenerate case

2

1

331

frequency

(b) Nondegenerate case

Figure 3.22 Four-wave mixing: a) degenerate case, and b) non-degenerate case.

The newly generated optical frequency will have a power proportional to powers of optical signals involved in the process, the intensity of the nonlinear Kerr effect, and the satisfaction of the phase matching condition. That power can be found as [70] Pijk 

 4 exp( l ) sin 2 ( l / 2)  2 ijk n2 d ijk  1 2 2      1  exp( l )2  3cAeff

2

2

  Pi Pj Pk Leff 2  

(3.194)

where n2 is nonlinear refractive index, Aeff is the effective cross-sectional area, Leff is the effective length, l is the transmission fiber length,  is the fiber loss coefficient, and dijk is degeneracy factor (dijk equals 3 or 6 for degenerate and nondegenerate case, respectively). Equation (3.194) can be simplified by assuming that powers of all channels are equal, which is generally true in advanced WDM systems that employ dynamic optical power equalization along the lightwave paths. In addition, we can insert the phase condition given by Equation (3.193) into Equation (3.194) if we are considering WDM systems with equal channel spacing. Therefore, the power of a newly generated optical wave through the FWM process in the WDM system can be expressed as  2 ijk n2 d ijk Pijk  H ( )  3cA eff 

2

 3 2  P Leff   2  4 exp( l ) sin 2 (l 2 ( ijk / 2 ) 2 / 2)   H ( )  2 1     [  2 ( ijk / 2 ) 2 ]2  1  exp(l )2 

(3.195)

(3.196)

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where P is the optical power per WDM channel and H() reflects the phase matching condition. The FWM process can produce significant signal degradation in WDM systems since several newly generated frequencies can coincide with any specific optical channel. The total number of new frequencies N that can be generated in the process is

N

M 2 ( M  1) 2

(3.197)

where M is the total number of the optical channels propagating through the fiber. As an illustration, 100 WDM channels will produce 495,000 new frequencies. Some of them will be very small and will have a negligible impact to the system performance. However, some of them can have a significant impact, especially if there are several newly generated frequencies that will coincide with a specified WDM channel. Namely, the sum of newly generated optical signals could be fairly significant if compared with the power level of the channel in question. The FWM effect can be minimized by either decreasing the power levels of optical interacting channels, or by preventing a perfect phase matching. The prevention can be done by increasing the chromatic dispersion and the channel spacing. However, the FWM process can be used for wavelength conversion as explained in Section 2.8.3 or for the phase conjunction given by Equation (3.183). The impact of the FWM process to transmission system performance will be evaluated in Chapter 4. 3.5.3 Nonlinear Schrodinger Equation for Multichannel Transmission A nonlinear Schrodinger equation (NSE) given by (3.149) is the fundamental equation in the evaluation of various effects that can occur during the pulse propagation (such as dispersion and selfphase modulation). In addition, the coupled equations can be established to treat more complex effects that are related to the energy exchange between individual optical pulses. In such a case, the total electric field is created by the by the sum of slowly varying amplitudes. It is common to consider interaction among three optical pulses that jointly create the electric field that is proportional to the amplitude

A( z, t )  A1 ( z, t )  A2 ( z, t )  A3 ( z, t )

(3.198)

where A1(z,t), A2(z,t) , and A3(z,t) are associated with three separate signal pulses. The propagation equation [Equation (3.149)] is now converted to three coupled equations, as shown in [13]

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



A1 ( z, t )  2  2 A1 ( z, t ) 2 2 2    A1  2 A2  2 A3 A1  A22 A3* 2 z 2 t A2 ( z, t )  2  2 A2 ( z, t ) 2 2 2 j    A2  2 A1  2 A3 A2  2 A1 A2* A3 2 z 2 t A3 ( z, t )  2  2 A3 ( z, t ) 2 2 2 j    A3  2 A1  2 A2 A3  A22 A1* 2 z 2 t j









  

(3.199) (3.200) (3.201)

The first terms on the right sides of above equations account for the self-phase modulation (SPM) effect, the second and third terms are related to cross-phase modulation (XPM), and the forth terms refer to four-wave mixing (FWM) effect. The impact of third-order chromatic dispersion, which is associated with coefficient 3 in Equation (3.149), has been neglected in Equations (3.199)– (3.201). Generally, the nonlinear Schrodinger equation is solved numerically, although there are some special cases where an analytical solution can be readily found. Such an example is associated with soliton pulses that were discussed in Section 3.4.1.3. As for the numerical solution of the Schrodinger equation, it is often carried by the software package used for modeling and simulation of transmission system characteristics.

3.6 SIGNAL PROPAGATION IN MULTIMODE OPTICAL FIBERS The basic analysis of the signal propagation in multimode optical fibers with step and graded index profiles is often limited just to the calculation of the pulse spreading due to mode dispersion. Accordingly, some basic formulas that evaluate the pulse spreading have been obtained—please refer to Equations (2.4) and (2.8). More exact analysis applied in Section 3.2 can be used to calculate propagation constants of individual modes propagating through multimode optical fiber. In a general case of multimode optical fibers, properties of large number of modes that interact among themselves during propagation should be evaluated. The interaction is known as mode coupling and has been discussed in a number of papers published earlier [74–78]. There are two scenarios for mode coupling analysis that can be applied: (1) for fibers having a large number of propagating modes and (2) for fibers having a number of propagating modes ranging from a few to couple dozen. In this section we will pay attention just to the second case where a fairly limited number of spatial modes can be propagated through the fiber, which corresponds to few-mode fibers (or multicore optical fibers assuming that each core supports just a single mode). Both of these fibers are considered in different transmission and networking scenarios [80–85]. We assume that for these applications at least seven fiber cores will be utilized within multicore optical fiber, while up to 7 modes will be effectively considered in few-mode optical fibers.

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The Equations (3.199)–(3.201) can also be applied in case where limited number of spatial modes is propagating through the fiber. The Nonlinear Schrodinger Equation should be solved numerically under set of initial conditions that describe characteristics of the fiber in question. However, it is also possible to perform some analytical evaluations to estimate the impact of the mode coupling to propagation properties of few-mode and multicore optical fibers. 3.6.1 Mode Coupling in Multimode Fibers Mode interaction, observed through power exchange among them, will have a considerable impact to overall transmission properties of multimode/multicore optical fibers. The analysis of mode coupling in multimode and multicore optical fibers is similar and we will refer to them as fibers supporting a limited number of spatial modes. Mode coupling is caused by the presence of irregularities in the refractive-index profile, which are caused by a deviation of the refractive-index profile from its ideal shape over a given core cross-section. That deviation can be expressed as (  )

(  )

(3.202)

where n0(r) is the regular refractive index profile of a symmetric optical fiber considered so far, while nd(r,,z) describes refractive-index deviations in azimuthal and axial directions. The assumption is that both components at the right side of Equation (3.202) do not change much within the distance that is comparable with the wavelength of the optical signal propagating through the fiber. If so, the wave Equation (3.27) can be rewritten as the set of three independent equations for each of Cartesian coordinates, so for x-coordinate it is

 2 E x  n 2 k0 Ex  0 2

(3.203)

where k0=2/l is the wave number. The solution of Equation (3.203) will be a set of LP modes that are discussed in Section 3.2.2 if the refractive-index has a regular profile dependent just on the radial coordinate. There is no coupling between different modes and polarization in optical fibers having ideal refractiveindex profile. The general solution of Equation (3.203) should be based on the refractiveindex profile defined by Equation (3.202). In such a case the solution is the combination of all LPli modes (groups of HEmi, EHmi, TEmi, and TMmi modes) in accordance with the Figure 3.3, so we have that [78] =∑

( )

( )

(3.204)

where is ( ) the transversal distribution of the electric field of the n-th mode, and Vn(z) is the amplitude coefficient. In the case with the regular refractive-index

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profile, it is Vn(z)~exp{±nz}, where n=n+jn is the complex propagation constant that includes the attenuation n and the propagation constant n. The sign ( ) in the exponential part of Vn(z) means backward propagation, while sign (+) points to the forward direction. ( ) from Equation (3.204) are solutions of transversal The functions wave equation given by





t2 En  n02 k0   n2 En  0 2

(3.205)

where index t stands for transversal. Equation (3.205) is a modification of Equation (3.203) where and dependence on z-coordinate is expressed by factor exp{±nz}. Since functions En experience a rapid decay for a larger value of the parameter r, they can be considered to be orthogonal to each other. If Equation (3.205) is multiplied by the one related to solution Em, that product returns the following equation





Emt2 En  Ent2 Em   n2   m2 Em En  0

(3.206)

If the integration of Equation (3.206) is performed over the fiber crosssectional area, the Green theorem [8] can be applied to the first terms, which leads to 2 r

 E  E m

2 t

0 0

n

2

E   E  Ent2 Em rdrd    Em n  En m rd r r  0



The condition of orthogonality for any two modes with be written as

E

m

En rdrd  0

(3.207)

, for

 , can

(3.208)

In addition, the following condition of orthogonal normalization can be applied

  E E rdrd  1 n

n

(3.209)

Equations (3.202), (3.206), and (3.209) lead to the system of the coupled differential equations

d 2Vm   m2Vm  Vn k02   nd2 (r ) En Em rdrd dz 2 n

(3.210)

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At the same time, the transversal magnetic field of the mode with index n can be expressed as =

( )

( )

(3.211)

On the other hand, by using the Maxwell equations (3.12) - (3.15) for regular refractive index profile, it is

( )

=

(3.212)

By using Equations (3.204), (3.207) and (3.208), the following relation for coefficients Vm and Wm can be established

( )

(3.213)

Also, this equation can be used to transform Equation (3.210) to ( )

( )

∑

(3.214)

where

 mn 

k02 2 m

  n (r ) E E rdrd 2 d

n

(3.215)

m

are mode-coupling coefficients. The amplitude coefficients Vm can be further normalized in Equations (3.213) and (3.214) to eliminate the constants 0 and 0. | Further on, by assuming that | , amplitude coefficients Vm can be represented by coefficients am equal ( )

(

)

(3.216)

Equation (3.214) is now transformed to a differential equation for the slowly varying wave amplitude Am(z), that is ∑

[(

) ]

(3.217)

By applying the iteration method, the first- and second-order iterative solutions of Equation (3.217) can be respectively expressed as

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Am ( z )  Am (0)   An (0)   mn exp  m   n wdw n

(3.218)

0

z

w

Am ( z )  1     mn exp  m   n w  mn exp  n   m u dudw n 0

(3.219)

0

The second-order solution is further simplified by assuming that at the very beginning only a mode with the index m is excited with a normalized amplitude equal 1. It was shown [76, 78] that the second-order iterative solution provides relatively high accuracy for practical considerations. From the mode-coupling perspective, the attenuation of the signal can be neglected in order to further evaluate the coupling conditions. In such a case the complex propagation constant becomes n=jn, which means that . Both n and cmn are now real numbers, and the mode-coupling Equation (3.217) now becomes [ (

∑

 ) ]

(3.220)

The second-order iterative solution of Equation (3.219), for the modes with indexes 1 and n, can be written as z

u

A1 ( z )  1    c1n (u ) exp(  j nu )  cn1 ( w) exp( j n w)dudw n 0

(3.221)

0

z

An ( z )   j  cn1 ( w) exp( j n w)dw

(3.222)

0

where n=n –1. For further calculations, it is necessary to know the character of the refractive index deviations. The easiest way is to expand the (nd)2 term in Equation (3.202) in Fourier series by argument , so it becomes (  )

∑ [ (

)



(

)

]

(3.223)

where q denotes the order of the azimuthal variation. It is logical to assume that ( ) ( ) are changing independently in radial and axial functions directions, so it is (

)

( ) ( )

(3.224)

Equations (3.223) and (3.224) can be used to calculate the coupling coefficients cmn. For any of two modes having fields Em and En, it is ( )

(3.225)

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223

Coefficients Cmn are different from zero only if the difference | | of the azimuthal mode numbers for the modes LPli is | | . The amplitude of the n-th mode caused by its coupling with the first mode over the fiber length L van be calculated as L

An ( L)   jCn1  f ( z ) exp( j n z )dz   jCn1F ( n )

(3.226)

0

where F(n) represents a Fourier transform of the function f(z). The spectral content of spatial frequencies for the function f(z) can be represented by the Fourier series as ( )

∑ [

(

)

(

)]

(3.227)

The contribution of the spatial component Fp to the amplitude An(L) is [78] [



{

( [



)]



] }

(3.228)

Smaller deviations in the axial direction, represented by coefficient Fp, will not cause a significant mode coupling. The more important fact for the mode coupling is if the spatial frequency Ωp of deviations, which is illustrated in Figure 3.23(a), is close to the difference n in propagation constants of modes in question. If so, we have that

 ;

;



(3.229)

where n is the beating length of two coupling modes, which is equal to the period of the spatial irregularity. In the analysis of long optical fibers with smaller grade irregularities, it enough to consider just spatial spectral components having a period very close to the beating length of the coupled modes. The contributions should be evaluated either by using formula (3.228) if n , or formula if n 3.6.2 Mode Coupling in Curved Multimode Optical Fibers The bending of optical fibers can intensify mode coupling between spatial modes. During the bending, spatial frequencies Ωp=2/Lp close to the difference n in propagation constants of modes in question can be induced. The deviations in the geometrical structure due to fiber bending can be transferred to equivalent deviations in the refractive index profile where term from Equation (3.202) can be expressed as [79]

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(3.230) where naxis is the refractive index at the fiber axis and  is the bending radius, as illustrated in Figure 3.23(bc). Now, the coupling coefficients can be found by inserting the function (3.230) into Equation (3.215). The normalized coupling coefficients cnm for bent fibers can be found from Equations (3.211) and (3.215) as

cmn 

k02 2 m

  n E E rdrd   E rdrd   E rdrd  2 d

2 m

n

(3.231)

m

2 n

1/ 2

x

n02

z

n2 Lp=2/p

Lp

(a)

nd2

(b)

(c)

Figure 3.23 Deviations of the refractive-index profile: (a) spatial frequency, (b) curved optical fiber, and (c) equivalent refractive-index of a curved optical fiber.

The coupling coefficients can be calculated for two parabolic refractive index profiles, since there is closed form for the En and Em of the LPli modes given by Equations (3.63) and (3.64). The index m for LPli modes for the parabolic refractive index profile is equal m=2i+l. It was shown in [78] that LPli mode has the coupling connection with the following modes: LPl+1,i mode, LPl–1,i+1 mode, LPl–1,i mode, and LPl+1,i–1 mode. Therefore, only modes having difference in mode index | | are coupled. As for the step-index refractive profile, there is also the closed form for the En and Em of the LPli modes given by Equations (3.57)–(3.59). The index m for LPli modes for the step-index refractive-index profile is also equal to m=2i+l. The LPli mode has the strong coupling connection with the same modes as in the case with parabolic index profile mentioned above. Again, only modes having a difference in mode index | | are strongly coupled. If | | , the coupling coefficients strength decreases proportionally to | | .

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3.6.3 Mode Coupling in Dual-Mode Optical Fibers If the normalized propagation frequency V, given by Equation (3.53), is arranged to support just two LP modes, which are LP01 and LP11, we are talking about dualmode optical fibers. In fact, as we can see from Figure 3.4, there are two flavors of LP11 mode, each with a different polarization. Dual-mode optical fibers are fewmode optical fibers that could be used for both transmission and networking purposes [80, 81]. It is important that dual-mode optical fibers are properly designed in order to provide maximum transmission capacity, but also to have controlled mode coupling. It was proposed in [82] that dual-mode optical fibers should have an optimized refractive index profile that would equalize intermodal dispersion among LP01 and LP11 modes. The refractive-index profile is calculated numerically by solving the set of coupled Schrodinger equations written as d 201(r )  2 1    01  k02ncl2  2  01(r )  V01(r )01(r )  V12 (r )11(r ) 2 dr 4r   2 d 11(r )  2 3    11  k02ncl2  2  01(r )  V21(r )01(r )  V11(r )11(r ) dr 2 4r  

(3.232) (3.233)

where  √ (i, j =0,1) are normalized filed functions of LP01 and LP11 ( ) and ( ) are potential functions that define the mode modes, ( ) and ( ) are potential functions that define the mode confinement, while ( ) ( ), the equation of equal coupling intensity. Assuming that potentials was established, and it is

V01(r )  V12 (r )

11(r )  (r )  V11(r )  V12 (r ) 01 01(r ) 11(r )

(3.234)

Equations (3.232) to (3.234) have been solved by a self-converging iterative procedure. It was found that there is a nearly optimum refractive-index profile that would minimize intermodal dispersion at a specific wavelength, while the fiber core diameter is kept in the range 22–24 m. Dual-mode optical fibers are strong candidates for both transmission and networking applications since the spatial multiplexing and MIMO technique bring both the transmission capacity and networking flexibility.

3.7 SUMMARY In this chapter, we analyzed transmission properties of optical fibers by using the basic principles of waveguide theory and obtained equation that characterize

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guided modes in both multimode and single-mode optical fibers. The nonlinear Schrodinger equation has been used to analyze the impact of both chromatic dispersion and nonlinear phase changes to the pulse shape. The Gaussian pulse shape was utilized to calculate the impact of chromatic dispersion and self-phase modulation. The methods for mitigation of both dispersion and nonlinear effects have been also presented. The analysis of the propagation characteristics of multimode optical fibers considered the mode-coupling case, while the coupling strengths among different modes have been analyzed. Finally, the methodology for analysis of few-mode optical fibers has been presented.

PROBLEMS 3.1 A 1,550-nm optical transmission system uses a 120-km fiber link and requires at least –35 dBm at the receiver side to keep signal to noise ratio above the required level. The fiber loss is 0.2 dB/km. Fiber is spliced every 5 km and the total fiber link has two connectors for source-fiber and fiber photodetector couplings, each with a 1.5-dB loss. Splicing is done in such a way that 95% of the optical power crosses the splice point. What is the maximum power that should be launched into the fiber? 3.2 What is the transmission the distance L over which the optical power will decrease four times for fibers having losses 0.17 dB/km, 10 dB/km and 200 dB/km? Calculate the attenuation coefficient  (in m–1) for all three cases. 3.3 What is the attenuation of the silica-based optical fiber at a wavelength belonging to each end of the visible spectrum? Assume that attenuation is caused mainly by Rayleigh scattering and that attenuation at 1.5 m is 0.2 dB/km. 3.4 Derive the eigenvalue equation (3.51) by applying the necessary conditions to a step-index fiber. 3.5 A step-index multimode fiber in which the refractive indexes of core and the cladding and are 1.5 and 1.4, respectively. The diameter of the core is 50 microns. Estimate the number of modes that are supported by the fiber for a signal at a wavelength of 1,310 nm. What would be the number of modes at wavelengths of 1,550 nm and 850 nm? 3.6 Estimate the number of modes in a plastic step-index multimode fiber with a core diameter of 200 m operating at 850 nm. Assume the refractive index in the core to be 1.55 and that of the cladding to be 1.45. 3.7 Explain how the single-mode step-index optical fiber can become a multimode one without changing any of the fiber parameters. 3.8 A silica-based single-mode fiber has a difference  between the core and cladding index values equal to 0.003, and a core radius a = 4 m. Calculate a cutoff wavelength (associated with the single-mode operation limit.) 3.9 Estimate the spot size of the mode, and the fraction of the mode power inside the core of a silica-based single mode optical fiber operating at the

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wavelength 1,550 nm (nco=1.45). What would be these values if the fiber is operated at wavelength 1,310 nm? What is the effective core area at each wavelength? 3.10 Show that the number of the modes supported by parabolic-index profile multimode optical fiber is just half of the number supported by step-index refractive profile. 3.11 A pure silica core single-mode optical fiber is used for transmission. Calculate the total bandwidth-distance product in the following cases: (1) optical signal carrier wavelength 1,550 nm. A light emitting diode with l=2 nm is used as a source. The difference between refractive indices in the core and cladding is 0.1% and core radius is 7 m; (2) optical carrier wavelength is 1.3 m. The laser source with the linewidth of 1 MHz is used as a source. The same fiber as in the above case is used. 3.12 Rectangle pulses with duration of 10 ps are sent over a single-mode optical fiber having the mean value of the first-order PMD coefficient 0.05 ps/(km)1/2. What is the percentage of the pulse spreading due to the firstorder PMD over a fiber length of 1,600 km. Assume the worst-case scenario. 3.13 An optical pulse at the wavelength 1,550 nm has a Gaussian shape ( ) [

] where P0=7 mW, and τ0=12 ps. The pulse is propagating along

single mode silica-based optical fiber 100 km long. Find the maximum frequency variation (or shift) at the fiber end occurring due to the self-phase modulation (SPM) effect. Assume that the effective area of the fiber is 80 square microns, while the nonlinear refractive index coefficient is 3.0 x 10–20 m2/W. 3.14 Calculate the chromatic dispersion parameters D and 2 of pure silica based fiber core at wavelengths 850 nm, 1,310 nm and 1,550 nm. 3.15 What is the maximum bit rate allowed by chromatic dispersion if the length of a single mode optical fiber is L=100 km, and if the carrier wavelength is 1,320 nm and 1,550 nm? Assume that signal consists of transform-limited, 100-ps (FWHM) input pulses. Also assume that 3= 0.15 ps3/km and 0.05 ps3/km at wavelengths 1,310 nm and 1,550 nm, respectively, while 2= 0.05 ps2/km and –20 ps2/km at wavelengths 1,320 nm and 1,550 nm, respectively. The source spectral width is much smaller than the signal spectrum linewidth. 3.16 An optical signal transmitted in a single mode optical fiber consists of chirped Gaussian input pulses. Assume that the chromatic dispersion slope is zero, and C0 = 2. Find the most favorable conditions in terms of the bit rate dependence from parameters C0, L and 2.Assume that the source has a small spectral width. 3.17 An ptical communication system operating with the bit rate of 10 Gb/s by using chirped Gaussian pulses with the FWHM width equal to 40 ps, and with a chirp parameter C0 = –3. What is the achievable transmission distance, if it is limited just by schematic dispersion? What would be the

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distance if C0 = 3? Assume that source has a small spectral width and that 2= –20 ps2/km at the carrier wavelength of 1,550 nm. 3.18 An optical communication system operating with the bit rate of 10 Gb/s by using unchirped Gaussian pulses with shape

( )

[

] where

P0=5 mW and 0=10 ps. What is the achievable transmission distance limited just by selfphase modulation? Assume that the source has a small spectral width and that transmission is done at the carrier wavelength of 1,550 nm. Assume that the effective area of the fiber is 50 square microns, while the nonlinear refractive index coefficient is 3.0 x 10–20 m2/W. 3.19 The 10-Gb/s optical transmission system is designed with RZ pulses having Gaussian shape with of 30-ps width (FWHM). The prechirp technique is used for chromatic dispersion mitigation so broadening by up to 40% can be tolerated. What is the initial value of the chirp parameter C0 to achieve this goal, and what would be maximum transmission distance at wavelength 1,550 nm? Assume that 2= –20 ps2/km. 3.20 What is the condition in which both SPM and GVD are compensated through mid-span phase conjugation? 3.21 Find the pulse width of the soliton pulse if transmission is done at 1,550 nm and 1,600 nm. Assume that effective area of the fiber core is 80 square microns, 2= –1 ps2/km, the nonlinear refractive index coefficient is 3.0·10–20 m2/W, and maximum launched power is 5 mW. What is corresponding bit rate if the RZ format is applied to prevent soliton self-destruction? 3.22 Calculate the peak power of the input pulse that is needed to ensure that a fundamental managed soliton is preserved in a fiber with 0.22 dB/km loss. The RZ bit rate is 10-Gb/s with 50-km amplifier spacing. Assume that soliton pulse has 0 = 15 ps, for fiber parameters 2= –0.5 ps2/km and = 2W–1/km. Calculate also the average launched power. 3.23 There are 10 optical channels occupying the total bandwidth of 2.5 nm. Transmission is done over 800 km of standard-single mode optical fibers. Dispersion compensation is done periodically by using fiber Bragg gratings. Find the parameters of the grating that can be used for chromatic dispersion compensation. 3.24 Calculate the distance where FWM created component will have a power of 5% of the input power. What other conditions should be met? All channels have equal power of P=2 dBm. Channel spacing is equal to 50 GHz. Assume that 2= –0.5 ps2/km, l= 1,550, and effective area is 50 m2, and fiber loss is 0.2 dB/km. What is the difference between degenerate and nondegenerate cases?

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[10] Polyanin, A. D., and Manzhirov, A.V., The Handbook of Mathematics for Engineers and Scientists, London: Chapman and Hall, 2006. [11] Proakis J. G., Digital Communications, 5th edition, New York: McGraw-Hill, 2007. [12] Couch, L.W., Digital and Analog Communication Systems, New York: Prentice Hall, 2007. [13] Agrawal, G. P., Fiber Optic Communication Systems, 4th edition, New York: Wiley, 2010. [14] Cvijetic, M., Coherent and Nonlinear Lightwave Communications, Norwood, MA: Artech House, 1996. [15] Gower, J., Optical Communication Systems, 2nd edition, Upper Saddle River, NJ: Prentice Hall, 1993. [16] Keiser, G. E., Optical Fiber Communications, 3rd edition, New York: McGraw-Hill, 2000 [17] Okoshi, T., Optical Fibers, San Diego: Academic Press, 1982. [18] Buck, J., Fundamentals of Optical Fibers, New York: Willey, 1995. [19] Marcuse, D., Light Transmission Optics, New York: Van Nostrand Reinhold, 1982. [20] Snyder, A. W., and Love, J. D., Optical Waveguide Theory, London: Chapman and Hall, 1983. [21] Li, M. J., and Nolan, D. A., “Optical fiber transmission design evolution,” IEEE/OSA Journ. Ligthwave Techn., Vol 26(9), 2008, pp. 1079–1092. [22] Russel, P. J., “Photonic crystal fibers,” IEEE/OSA Journ. Ligthwave Techn., Vol. 24(12), 2006, pp. 4729–4749. [23] Gloge, D., and Marcatili, E., “Multimode theory in graded-index fibers,” Bell Sys. Tech. J., Vol. 52, Nov. 1973, pp. 1563–1578. [24] Gloge, D., “Propagation effects in optical fibers,” IEEE Trans. Microwave Theor. Trans., MTT23(1975), pp. 106–120. [25] Rudolph, H. D., and Neumann, E. G. “Approximation of the eigenvalues of the fundamental mode of a step index glass fiber waveguide,” Nachrichtentechnischen Zeitschrift, 29(1976), pp. 328–329.

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[26] Adams, M. J., An Introduction to Optical Waveguides, New York: Wiley, 1981. [27] Gloge, D., “Weakly guided fibers,” Applied Optics, Vol. 10(1971), pp. 2252–2258. [28] Snyder, A. W., “Understanding monomode optical fibers,” Proc. IEEE, Vol 69(1981), pp. 6–12 [29] Marcuse, D., “Interdependence of material and waveguide dispersion,” Applied Optics, Vol. 18(1979), pp. 2930–2932. [30] ITU-T Rec. G. 652 “Characteristics of single mode optical fiber cable,” ITU-T (04/97), 1997. [31] ITU-T Rec. G. 653 “Characteristics of dispersion-shifted ingle mode optical fiber cable,” ITU-T (04/97), 1997. [32] ITU-T Rec. G. 655 “Characteristics of non-zero dispersion shifted single mode optical fiber cable,” ITU-T (10/00), 2000. [33] Heidemann, R., “Investigations of the dominant dispersion penalties occurring in multigigabit direct detection systems,” IEEE J. Lightwave Techn., LT-6(1988), pp. 1693–1697. [34] Kogelnik, H., L. E. Nelson, and R. M. Jobson, “Polarization mode dispersion,” In Optical Fiber Communications, I. P. Kaminov and T. Li (eds), San Diego, CA: Academic Press, 2002. [35] Staif, M., et al., “Mean square magnitude of all orders of PMD and the relation with the bandwidth of the principal states,” IEEE Photonics Techn. Lett., Vol. 12(2000), pp. 53–55. [36] Ciprut, P., et al, “Second-order PMD: impact on analog and digital transmissions,” IEEE J. Lightwave Techn., Vol. LT-16(1998), pp. 757–771. [37] Galtarossa, A., et al, “Statistical description of optical system performances due to random coupling on the principal states of polarization,” IEEE Photon. Techn. Lett., 14(2002), pp. 1307– 1309. [38] Foschini, G. J., Pole, C. D., “Statistical theory of polarization mode dispersion in single mode fibers,” IEEE J. Lightwave Techn., LT-9(1991), pp. 1439–1456. [39] Karlsson, M., et al., “A comparison of different PMD compensation techniques,” Proceedings of European Conference on Optical Communications ECOC, Munich 2002, Vol. II, pp. 33–35. [40] Matsumura, H., et al., “Simple normalization of single mode fibers with arbitrary index-profile,” Proceedings of European Conference on Optical Communications- ECOC, York 1980, paper 103-1-6. [41] Peterman, K., “Theory of microbending loss in monomode fibers with arbitrary refractive profile,” EAU, 30(1976), pp. 337–342. [42] Ramaswami, R., and K. N. Sivarajan, Optical Networks, San Francisco, CA: Morgan Kaufmann Publishers, 1998. [43] Liu, F. et al., “A novel chirped return to zero transmitter and transmission experiments,” in Proc. of European Conference on Optical Communications ECOC, Munich 2000, Vol. 3, pp. 113–114. [44] Agrawal, G. P., Nonlinear Fiber Optics, 3rd edition, San Diego, CA: Academic Press, 2001. [45] Watanabe, S., and Shirasaki, M.,“ Exact compensation for both chromatic dispersion and Kerr effect in a transmission fiber using optical phase conjuction,” IEEE/OSA Journal Lightwave Techn., LT-14(1996), pp. 243–248. [46] Kazovski, L., et al., Optical Fiber Communication Systems, Norwood, MA: Artech House, 1996. [47] Hong, B. J., et al., “Using nonsoliton pulses for soliton based communications,” IEEE/OSA J. Lightwave Techn., LT-8(1990), pp. 568–575.

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[48] Gordon, J. P., and Haus, H. A., “A random Walk of Coherently Amplified Solitons in Optical Fiber Transmission,” Optics Letters, 11(1986), pp. 665–667. [49] Miwa, T., Mathematics of Solitons, New York: Cambridge University Press, 1999. [50] Hasegawa, A., and Kodama, Y., “Signal Transmission by Optical Solitons in Monomode Fiber,” IRE Proc., 69(1981), pp. 1145–1150. [51] Merlaud, F., and Georges, T., “Influence of sliding frequency filtering on dispersion managed solitons,” in Proc. of European Conference on Optical Communications ECOC, Nice 1999, Vol. 1, pp. 228–229. [52] Fukuchi, K., et al., “10.92 Tb/s(273x40 Gb/s) triple band ultra-dense WDM optical-repeated transmission experiment,” in Proc. of Optical Fiber Conference OFC, Anaheim 2001, PD 26. [53] Poole, C. D., et al., “Optical fiber based dispersion compensation using higher order modes near cutoff,” IEEE/OSA Journal Lightwave Techn., LT-12(1994), pp. 1746–1751. [54] Kashyap, R., Fiber Bragg Gratings, San Diego, CA: Academic Press, 1999. [55] Hill, K. O., et al., “Chirped in-fiber Bragg grating dispersion compensators: linearization of the dispersion characteristics and demonstration of dispersion compensation of a 100 km, 10 Gbps optical fiber link,” Electronics Letters, 30(1994), pp. 1755–1757. [56] Jansen, S. L., et al., “10Gbit/s, 25GHz spaced transmission over 800km without using dispersion compensation modules” Optical Fiber Communication Conference, OFC 2004, Los Angeles, Vol. 2, pp. 3–4. [57] Watanabe, S., and Shirasaki, M., “Exact compensation for both chromatic dispersion and Kerr effect in a transmission fiber using optical phase conjunction,” IEEE/OSA Journal Lightwave Techn., LT-14(1996), pp. 243–248. [58] Radic, S., “Parametric Signal Processing,” IEEE Journal of Selected Topics in Quantum Electronics, Vol. 18, 2012, pp. 670–680. [59] Kakande, J., et al., “Detailed characterization of a
fiber-optic parametric amplifier in phasesensitive and phase-insensitive operation,” Opt. Express, Vol. 18, 2010, pp. 4130–4137. [60] Kashima, N., Passive Optical Components for Optical Fiber Transmission, Norwood, MA: Artech House, 1995. [61] Vohra, S. T. et al., “Dynamic dispersion compensation using bandwidth tunable fiber Bragg gratings,” in Proc. of European Conference on Optical Communications ECOC, Munich 2000, Vol. 1, pp. 113–114. [62] Suzuki, M. et al., “PMD mitigation by polarization filtering for high-speed optical transmission systems,” IEEE/LEOS Summer Topical Meetings, Acapulco 2008, pp. 149–150. [63] Bulow, H., et al., “PMD mitigation at 10 Gbps using linear and nonlinear integrated electronic equalizer circuits,” Electron. Letters, Vol. 36, 2000, pp. 163–164. [64] Yonenaga, K. et al., “Automatic dispersion equalization using bit-rate monitoring in a 40 Gb/s transmission system,” in Proc. of European Conference on Optical Communications ECOC, Munich 2000, Vol. 1, pp. 119–120. [65] Bosco, G., et al., “New branch metrics for MLSE receivers based on polarization diversity for PMD Mitigation,” J. Lightwave Technol., Vol. 27, 2009, pp. 4793–4803 [66] Ip, E., “Nonlinear compensation using backpropagation for polarization-multiplexed transmission,” J. Lightwave Technol., Vol. 28, 2010, pp. 939–951

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[67] Dedic, I., “High-speed CMOS DSP and data converters,” in Optical Fiber Communication Conference, OFC 2011, paper OTuN1. [68] Nakazawa, M., “High spectral density optical communication technologies,” in Volume 6 of Optical and Fiber Communications Reports, New York: Springer, 2010. [69] ITU-T, Rec G.694.1, “Spectral grids for WDM applications: CWDM wavelength grid,” ITU-T (06/02), 2002. [70] Shibata, N., et al., “Phase mismatch dependence of efficiency of wave generation through fourwave mixing in a single-mode optical fiber,” IEEE J. of Quantum Electronic, Vol. QE-23, 1987, pp. 1205-1210. [71] Bennion, I., et al., “UV-written in fibre Bragg gratings,” Optical Quantum Electronics, 28(1996), pp. 93-135. [72] Chuang, S. L., Physics of Optoelectronic Devices, 2nd edition, Hoboken, NJ: Wiley, 2008. [73] Agrawal, G. P., Lightwave Technology: Components and Devices, New York: WileyInterscience, 2004. [74] Koeang, P. H., et al., “Statistics of group delays in multimode fiber with strong mode coupling,” Journal of Lightwave Techn., Vol. 29, No 2, pp. 3119–3127. [75] Shemirani, M. B., and Khan, J. M., Principal Modes in Multimode Fiber, Düsseldorf, Germany: VDM Verlag, 2010. [76] Okoshi, T., and Okamoto, K., “Analysis of wave propagation in inhomogeneous optical fibers using a variational method,” IEEE Transactions on Microwave Theory and Techniques, Nov 1974, Vol. 22, pp. 938–945. [77] Marcuse, D., “ Coupled mode theory of round optical fibers,” The Bell System Techn., Journal, Vol. 52, No 6, 1972, pp. 2489–2495. [78] Unger, H., G., Planar Optical Waveguides and Fibers, New York: Oxford University Press, 1979. [79] Peterman, K., “Microbending loss in singlemode W fibers,” Electron. Letters, 12 (1976), pp. 537–538. [80] Morioka, T., et al., “Enhancing optical communications with brand new fibers,” IEEE Commun. Mag., Vol. 50, No. 2, pp. 40–50. [81] Cvijetic, M., et al., “Dynamic multidimensional optical networking based on spatial and spectral processing,” Optics Express, Vol. 20, 2012, pp. 9144–9150. [82] Cvijetic, M., “Dual-mode optical fibers with zero intermodal dispersion,” Optical and Quant. Electron., Vol. 16, 1984, pp. 307–317. [83] Awayi, A., et al., “World first mode/spatial division multiplexing in multicore fiber using Laguerre-Gaussian mode,” Proc. 2011 European Conf. on Optical Communications (ECOC), Geneva, paper We.10.P1. [84] Abedin, K., et al., “Amplification and noise properties of an erbium doped multicore fiber amplifier,” Optics Express, Vol. 19, May 2011, pp. 16715–16721. [85] Zhu, B., et al., “Seven-core multicore fiber transmission for passive optical network,” Optics Express, Vol. 18, May 2010, pp. 117–122.

Chapter 4 Noise Sources and Channel Impairments There are various effects that can cause distortion of optical signal during modulation, propagation, and detection processes, as illustrated earlier in Figure 1.11. In Chapter, 3 we discussed signal distortion caused by optical fiber loss, dispersion, and nonlinear effects. The impairments due to these effects will degrade and compromise the integrity of the signal before it arrives to the decision point in the optical receiver. In addition, there will be corruptive additives to the signal that present a noise created in optical transmission channel, which will also come to the decision point mixed with the signal. The transmission quality is measured by the signal-to-noise ratio (SNR) at the decision point. That ratio determines the receiver sensitivity, which is defined as a minimum optical power needed to keep the SNR at the specified level. Received signal level at the decision point should be as high as possible to keep the distance between the signal and noise and to provide a margin necessary to compensate for other corruptive effects. In this chapter, we describe the characteristics of the noise components in the optical communication channel and evaluate their parameters. We will also evaluate the impairment parameters and determine the degradation of receiver sensitivity.

4.1 OPTICAL CHANNEL NOISE The noise, as a major corruptive addition to the signal, can originate from different places within an optical transmission system, as shown in Figure 4.1. Semiconductor lasers are the source of the relative intensity noise (RIN), laser phase noise, and mode partition noise. Optical fibers are responsible for modal noise generation, while optical splices and optical connectors are the origin of the reflection-induced noise. Optical amplifiers generate spontaneous emission noise, which is subsequently amplified on the line and becomes amplified spontaneous emission (ASE) noise. Also, there is a cross-talk that can be treated as a noise, which is generated mainly in multiplexing and switching network elements

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(optical multiplexers, ROADM, and OXC). Finally, there are several noise components generated during optoelectronic conversion in photodiode, all of them with either thermal or quantum nature. Optical transmitter (laser) Laser intensity noise Mode partition noise Laser phase noise

Optical cable (fiber and splicing) Modal noise Reflection-induced noise

Optical Amplifier #1

ROADM OXC

Spontaneous emission

Channel crosstalk

Optical amplifiers #2 to #N Spontaneous Emission + ASE

Optical receiver (photodiode) Thermal noise Quantum noise Dark current

Figure 4.1 Noise components in an optical transmission channel.

All noise components mentioned above can be divided into ones generated at the optical level (optical noise), and the electrical noise components added to the signal while being in an electrical form. Any optical noise component carries some optical power, which is proportional to the square of its electric field. Since the photodetector generates a photocurrent that is proportional to the incoming optical power rather than to the incoming electric field associated with the power, there will be several side components of the total photocurrent. These components arise since the noise electric field beats against the signal field, against itself, and against the fields of other optical noise components. Although the number of beat components can be substantial, most of them are relatively small and can be neglected. -V (bias voltage)

Hybrid

Photodiode

…

Optical preamplifier

Optical demultiplexer

Optical input

Equalizer

Local oscillator

Channel Equalization Compensation Carrier recovery

Output Symbol signal decision

Front-end and preamplifier

Noise evaluation

Figure 4.2 High-level block diagram of an optical receiver with the noise evaluation point.

The impact of each noise component becomes obvious only after photodetection process in optical receiver. In our analysis we will assume that the optical receiver can be presented by a generic block diagram from Figure 4.2. The optical signal coming from the transmission line is coming to a photodiode to be converted to electrical form and to be processed afterwards. In a case in which coherent detection scheme is applied, incoming optical signal is mixed with the signal from the local laser oscillator (LO) before photodetection takes place, as indicated by the dashed lines. A coherent optical receiver can contain a number of photodiodes (four and more) in advanced detection schemes, which will be discussed in Chapter 6. Every photodiode is followed by a front end and preamplifier and a number of circuits for analog and digital signal processing with

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the role of fully recovering the original signal waveform. The impact of both the noise components and signal impairments can be evaluated at the output of preamplifier, which will be done in this chapter with respect to basic direct detection (DD) scheme. The advanced detection schemes will be discussed in Chapter 6. Electrical signal Amplified Optical signal

Spontaneous emission Crosstalk noise

Amplified intensity noise ASE

ase Spontaneous emission

Spontaneous emission beat noise Intensity noise Thermal noise Quantum shot noise Dark current

Front-end amplifier

ASE noise

Signal-spontaneous beat noise

Photodiode PIN/APD

Intensity noise

Optical preamplifier

Optical signal

Crosstalk noise LO caused beat noise

LO phase noise

LO Intensity noise

Amplified crosstalk noise

Figure 4.3 Noise components coming to the receiver front end.

The noise components coming to the receiver front end can be sorted in a manner shown in Figure 4.3. The optical preamplifier is receiving the intensity noise and spontaneous emission noise from the transmission line. The spontaneous emission noise contains the component originating from last in-line amplifier, and the amplified spontaneous emission originating from the other amplifiers placed along the lightwave path. (The amplified spontaneous emission components from Figure 4.3 are denoted by lowercase letters if they are related to one stage amplification, or by capital letters if they are related to multistage amplification.) An ptical amplifier will enhance all optical inputs in proportion to the amplifier gain. Both signal and noise components are being converted to the electrical level through a photodetection process. In addition to the components due to direct opto-electronic conversion, several kinds of noise beat components have been created. The beating is caused by the interaction among electrical fields belonging to signal, spontaneous emission, and local laser oscillator (if a coherent detection scheme is deployed). Therefore, the total number of the beat noise components that come to the input of the front end could be large, but just few of them will have a significant contribution to the overall noise strength. In a case with direct detection, the most important noise beat components are the signalspontaneous emission beat noise and the spontaneous emission-spontaneous

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emission beat noise. When local oscillator is present, intensity and phase noise components will be added. There are both multiplicative and additive components in the total noise [1, 2]. The multiplicative noise components are produced only if the signal is present, while the additive components are present even if the optical signal is not generated. The multiplicative noise components are:  Mode partition noise generated in multimode lasers due to nonuniform distribution of the total power among longitudinal modes and microvariations in the intensity of each longitudinal mode.  Modal noise, which arises in multimode fibers through the random process of excitation of transversal modes and the power exchange among them.  Laser intensity noise caused by microvariations in the laser output power intensity. This noise is characterized through the relative intensity noise (RIN) parameter.  Laser phase noise caused by microvariations in phase of generated photons. This is the main reason why the output optical signal, which is a collection of individual photons, exhibits a finite nonzero spectral width.  Quantum shot noise, caused by quantum nature of light and the random distribution of the electrons generated during the photodetection process.  Avalanche shot noise, caused by the random nature of the amplification of primary electron-hole pairs through the effect of impact ionization in avalanche photodiodes. The additive noise components are:  Dark current noise generated in photodiodes caused by the generation of the electron-hole pairs due to thermal process.  Thermal noise, also known as Johnson noise, created in the resistive part of the input impedance of an optical receiver.  Amplified spontaneous emission (ASE) noise that is generated by optical amplifiers placed along the lightwave path.  Cross-talk noise that occurs in multichannel WDM systems when another signal interferes with the signal in question. The cross-talk is often analyzed separately since it is different in nature than other noise components. The main sources of the cross-talk noise are optical multiplexer/demultiplexers and optical switching elements (ROADM and OXC). We will analyze the impact of noise components mentioned above while paying more attention to those components that are dominant in different detection scenarios. 4.1.1 Mode Partition Noise All noise components generated in semiconductor lasers are caused by spontaneous emission of photons that accompanies the process of stimulated

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237

radiation in the laser cavity. The spontaneous emission is a stochastic process and has an impact to fluctuations of the output power parameters (intensity, phase, and frequency). The spontaneously emitted photons supplement the coherent optical power by contributing randomly to the signal amplitude and phase, thus causing random perturbations in amplitude and phase of the output power. Accordingly, these fluctuations in intensity and phase of the emitted light are the physical origin of the overall laser noise. Mode partition noise is associated with multimode Fabry-Perot semiconductor lasers, due to uncorrelated emission among longitudinal modes that are confined within the laser spectrum—please refer to Figure 2.10(b). The difference in intensities of any pair of longitudinal modes from Figure 2.10(b) fluctuates randomly, even if we assume that the total output power is constant. These intensity fluctuations will be transferred all the way to the optical receiver since chromatic dispersion in the optical fiber will force all longitudinal modes to travel with different speeds. Consequently, the mode partition noise will be converted to electrical noise and will corrupt the signal at the decision point. The impact of mode partition noise is more relevant for transmission systems where the product B·L is relatively low (B is the signal bit rate, and L is the transmission distance). The intensive study of the mode partition noise was carried out in literature some time ago, in order to estimate the power penalty related to the noise impact [3, 4]. It was concluded that the impact of the mode partition noise could be almost entirely suppressed by satisfying the following condition

BLD   0.075

(4.1)

where D is the chromatic dispersion coefficient and  is the spectral linewidth of the multimode semiconductor laser—please see Table 2.1. The B·L product can be maximized by selecting an operational wavelength within the zero dispersion region, or at the region where chromatic dispersion is lower than 1 ps/km·nm. As an example, the signals with a 1-Gb/s bit rate can be transmitted over about 30 km, while 10 Gb/s signals can be effectively transmitted over 3 km. Semiconductor lasers, such as DFB or DBR, designed to operate in singlemode regime do not produce a mode partition noise. However, the existence of remaining side modes in the laser spectrum may be of some concern. The strength of side modes is characterized by the mode suppression ratio (MSR), which is defined as a difference in powers between the governing longitudinal mode and the most dominant suppressed side-mode, as shown in Figure 2.12(b). We can assume that lasers with MSR > 100 will cause a negligible mode partition noise effect since the power penalty will be lower than 0.1 dB [3]. 4.1.2 Modal Noise In a general case, the total input optical power in multimode optical fibers is nonuniformly distributed among a number of modes. Such modal distribution creates speckle pattern at the receiving side, which contains brighter and darker

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spots in accordance with the mode distribution. The photodiode effectively eliminates the speckle pattern impact by registering the total power that is integrated over the photodiode area. However, if the speckle pattern is not stable in time, it will induce fluctuations in the received optical power. Such fluctuations are referred to as the modal noise, and will be eventually converted to photocurrent fluctuations. The fluctuations in a speckle pattern occur in optical fibers due to mechanical disturbances, such as microbends and vibrations, within optical cables. Also, the splices and connectors will influence the power distribution over transversal modes since they act as spatial optical filters. The modal noise is inversely proportional to the spectral linewidth  of the light source. This comes from the fact that mode interference and speckle pattern changes are relevant only if coherence time tcoh (tcoh ~ 1/) is longer than intermodal dispersion in optical fibers—please refer to Section 2.2.3. This condition is not satisfied if light emitting diodes (LED) are used for signal transmission, since the LED spectral linewidth is quite large. Therefore, it is a good idea to use LED sources in combination with multimode optical fibers whenever possible to avoid the possible impact of modal noise. The situation is quite different if single-mode lasers are used in combination with multimode optical fibers, since the modal noise impact could be quite a serious problem. The impact of the modal noise is higher for smaller number of modes propagating through optical fiber, while the most serious situation occurs if optical power at the receiving side is effectively shared by only several transversal modes, which is the case with few-mode fibers and multicore fibers analyzed in Section 3.6. The numerical solution of coupled Equations (3.199)–(3.201) or Equation (3.226) with the inclusion of the attenuation difference among the modes, can be used to analyze the impact of the modal noise to the performance of the transmission system. From a system design perspective, it is necessary to allocate some power margin P to accommodate the modal noise effect in the case in which single mode lasers are used in combination with multimode optical fibers, but without any advanced modulation/detection scheme in place. The allocated margin, which effectively means that signal optical power should be increased by P to counterattack the impact of the modal noise, should be as high as 1 dB for combination of single-mode lasers and multimode optical fibers. In addition to multimode fibers, even sections of single-mode optical fibers that are up to few meters long can introduce the modal noise, since a higher-order mode can be excited at the fiber discontinuity (connector or splice), and then converted back to the fundamental mode at the next discontinuity. It is good idea, therefore, to use optical fibers that are a little longer than necessary even if the distance is just 1–2 meters. The distance of 5 meters, for example, can effectively eliminate the impact of the modal noise since a higher-order mode cannot reach the second fiber discontinuity. It is important to notice that vertical cavity surface emitting lasers (VCSEL) are often used in combination with multimode optical fibers for very short links (up to several kilometers). Although this combination is a cost-effective solution for gigabit signal rates over very short distances, it is also

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239

a place where modal noise may be a serious factor causing a power penalty even higher than 1 dB [5]. 4.1.3 Laser Phase and Intensity Noise The total power of the quantum noise generated in semiconductor lasers through process of stimulated emission can be evaluated by multiplying the density of the photons representing the noise with the energy of a single photon. The photon density number of the quantum noise is represented by the spontaneous emission factor nsp expressed through the electron populations N2 and N1 at the upper and lower energy levels, as shown in Equation (2.15), so we have that Wlqn  nsph 

N2 h N 2  N1

(4.2)

where Wlqn is the total energy of the quantum noise per longitudinal mode in semiconductor lasers. The total power Plqn of the laser quantum noise is

Plqn  Wlqn  nsp h  

(4.3)

where  is laser spectral bandwidth. In addition to quantum noise, the total noise generated in semiconductor lasers contains the thermal component as well. The thermal component Pltn() of the laser mode is calculated by dividing the Planck’s equation (10.9) by the number of laser eigenmodes Nl,emod= 82/c3, [7], so we have that

Pltn 

h exp( h / k)  1

(4.4)

and

 N h  h Pln   2     N 2  N1 exp( h / k)  1

(4.5)

where Pln() is the total noise generated per laser mode. Since the laser noise is a random process, both the laser phase and intensity noise components can be evaluated by using standard methods of signal analysis [2]. It is necessary to find autocorrelation function, cross-correlation function, and power spectral density with respect to the amplitude and phase of generated light. For that purpose, the electric field of the generated light can be expressed as [8]

E (t )  A0  an (t )exp  j0  n (t )

(4.6)

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where A0 and 0 are the stationary values of the amplitude and the phase, respectively, while an(t) and n(t) are of noisy fluctuations of these values. The phase fluctuations n(t) define the spectrum of frequency noise Sf() since the instantaneous frequency deviations are defined as

 n (t ) 

1 dn 2 dt

(4.7)

The spectral density SF() of the frequency noise is defined as S F ( )  lim

T 

1 Fn ( ) Fn* ( ) T

(4.8)

where Fn() is the Fourier transform of the function n(t). In addition to spectral density function defined by Equation (4.8), it is necessary to find spectral density function of the electric field itself in order to evaluate the phase noise spectrum generated in the laser. The spectral density function can be find as Fourier transformation of the autocorrelation function R() of the electric field [8], and by using Equations (4.6) and (4.8), so we have that R( ) 

E * (t ) E (t   ) A02



 exp jn ( )  exp  n2 ( ) / 2



(4.9)

where n(t) =n(t+) – n(t) defines phase fluctuations. It is assumed that phase fluctuations can be expressed as a Gaussian random process. The Fourier transformation of Equation (4.9) and functional relations (4.7) and (4.8) lead to the following 

n2 ( )   2  4 S F ( ) sin 2  / d

(4.10)

0

where  is the phase noise variance that is often used in system calculations. The spectral density SF() has been evaluated in number of papers, [9–13], for different laser structures. The following expression for InGaAsP laser structure were obtained in [10]

S F ( ) 

2 nsph LR   chirp  R4  1  A02  ( 2   R2 ) 2   2

  2 

  s 2

(4.11)

where chirp is an amplitude-phase coupling parameter defined by Equation (2.20), LR is the frequency bandwidth of the laser resonator, and R and s are the relaxation frequency and depletion constant, respectively, defined as

Noise Sources and Channel Impairments

 R2 

A02G( P0 ) G 2 2 P

(4.12)

1/ 2

 2   s   2 chirp     chirp  1 

241

 A02

G P

(4.13)

where P is the radiated optical power measured by the number of photons involved in the process, P0 is the initial number of photons, and G is the net gain related of the stimulated emission—please refer to Equations (2.13) and (2.18). The laser intensity noise, which has the same nature as the mode partition noise, is associated with single-mode lasers. The intensity fluctuations created at the transmitter side will eventually experience both the attenuation in the optical fiber and amplification through the chain of optical amplifiers. The laser intensity noise will be converted to the electrical noise by a photodiode and corrupt the signal at the decision circuit point. Laser intensity noise can be estimated by the relative intensity noise (RIN) parameter, which is the Fourier transform of the intensity autocorrelation function  defined as 

(  ) 

P( t )P( t   P

(4.14)

2

where

presents the average value of the laser output power measured by number of created photons and P=P(t)

presents small power fluctuations around the average value. It is, therefore RIN ( ) 

1 2



 (t ) exp( j 2t )dt 



1 2



P(t )P(t  



P



2

exp(  j 2 t )dt

(4.15)

The RIN value can be calculated by solving generalized laser rate equations that contain the Langevin noise term FP(t) related to fluctuations of the intensity— please see Equations (2.21) to (2.23). The parameter RIN(), which is usually expressed in decibels per Hertz, has a peak maximum at the relaxation-oscillation frequency R. The parameter of practical interest that measures the impact of the intensity noise is defined as  P( t )P( t  0   rint2      RIN (  )d 2 RIN laserf 2 P   

(4.16)

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where f defines the bandwidth applicable to the intensity noise (which is in fact the bandwidth of an optical receiver) and RINlaser is the parameter that characterizes magnitude of the intensity noise. It is RINlaser ~ –160 dB/Hz for high-quality DFB lasers. The reflection-induced noise is caused by the appearance of a back-reflected optical signal due to refractive-index discontinuities at optical splices, connectors, and optical fiber ends. It has the same nature as the laser intensity noise, and that is the reason why they are often treated together. The power of the reflected light can be estimated by the refection coefficient rref, defined as

 n  nb  rref   a   na  nb 

2

(4.17)

where na and nb are the refractive-index coefficients of materials facing each other. The amount of the reflected light is directly proportional to the coefficient rref. Therefore, it is higher for a bigger difference in refractive indexes, and vice versa. The strongest reflection occurs at the glass-air interface. We can assume that na = 1.46 (for silica) and nb =1 (for the air), which returns the refection coefficient rref ~3.5% (or –14.56 dB). This value can be even higher if optical fiber ends are polished. The amount of reflected light can be reduced below 0.1% if some index-matching oils or gels are used at the fiber-air interface, or if the fiber ends are cut at an angle to deviate the reflected light from the fiber axis. Both methods are extensively used in high-speed optical transmission systems. A considerable amount of back-reflected light can come back and enter semiconductor laser resonant cavity, which would negatively affect the laser operation and lead to the excessive intensity noise at the laser output. That is the main reason why the laser is commonly separated from the optical fiber link by an optical isolator, which will eventually suppress the impact of the reflected light. The relative intensity noise can be increased by as much as 20 dB if the backreflected light exceeds the 30 dBm level. The impact of the reflection-induced noise is not limited just to the laser source, since multiple back-and-forth reflections between optical splices and connectors can be the source of an additional intensity noise. Multiple reflections will eventually create multiple copies of the same signal traveling forward. These copies will be shifted in phase and act as a phase noise. Such phase noise is eventually converted to the intensity noise by chromatic dispersion, and enhanced by optical amplifiers along optical fiber links. In addition to chromatic dispersion, phase noise can be converted to intensity noise at any two reflecting surfaces along the optical fiber links, since they act as the mirrors of the Fabry-Perot interferometer. The end result of conversion of the phase noise to intensity noise will be an increase of the total relative intensity noise. Therefore, it is extremely important to suppress the back reflections along the entire optical transmission line by a careful selection of optical connectors that minimize reflections.

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4.1.4 Quantum Shot Noise The optical signal coming to photodiode contains a number of photons that would generate the electron-hole pairs by the photoelectric effect. The electron-hole pairs are effectively separated by the inverse bias voltage, thus forming a photocurrent. The probability of having n electron-hole pairs at the photodiode during the time interval t is expressed by the Poisson probability distribution [2, 14],

p( n ) 

N neN n!

(4.18)

where N is the mean number of photoelectrons detected during the time interval t, given as t

N

 P(t )dt h 0

(4.19)

The parameters in Equation (4.19) are: isoptical frequency; h is the Planck’s constant; and isthe quantum efficiency, which is the ratio between the number of the electrons detected and the number of the photons that arrived, as presented by Equation (2.115). The Poisson distribution approaches a Gaussian distribution for larger values of the mean N. The mean intensity of the photocurrent, which has been generated by the stream of electrons, is given as

I  i (t ) 

qN qN  t T

(4.20)

where q is the electron charge (q = 1.6 x 10–19 Coulombs). Please notice that it was assumed that the time interval t is equal to the duration T of the either “1” or “0” bits. The actual number of electrons generated during the bit duration will fluctuate around the mean value N due to a random nature of the photodetection process, while the generated photocurrent will fluctuate around the mean value I. Due to the property of Poisson distribution that the variance is equal to the mean value, we have that

n  N 2

N

(4.21)

where the angle brackets indicate averaging function. The instantaneous current during the bit duration T is given as

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i (t ) 

qn T

(4.22)

Equations (4.21) and (4.22) can be used to estimate the fluctuations of the instantaneous current around the mean value. These fluctuations can be expressed through the mean-square value, which is

i

 i (t )  I   2

2 sn

q 2 n(t )  N 

2



T2

q 2 N qI  T2 T

(4.23)

Equation (4.23) presents the power of the quantum shot noise, which is a multiplicative noise component caused by the quantum nature of light. We can correlate the bit duration T with the signal bandwidth f by assuming that f =1/2T as in [15] to get the following relation

i2

sn

 2qIf  S sn ( f )f

(4.24)

The same result as previous one can be obtained by calculating the signal spectral density Ssn(f) through the Fourier transform of the photocurrent correlation function [16, 17]. The spectral density of the quantum shot noise from Equation (3.23) is constant and given as

S sn ( f ) 

d 2 i df

sn

 2qI

(4.25)

Equations (4.24) and (4.25) can be applied just to PIN photodiodes. That is because an internal amplification process in avalanche photodiodes (APD) increases the generated photocurrent and enhances the total quantum noise. The physical background behind this additional noise in APD is related to the fact that secondary electron-hole pairs are generated randomly through stochastic process of impact ionization. It was shown that the avalanche shot noise can be characterized by the Gaussian probability density function, and has the spectrum that is flat with the frequency [15]. The avalanche shot noise power is

i2

sn / APD

 S sn / APD ( f )f  2q M

2

F ( M ) If

(4.26)

where is the average value of the avalanche gain, while F(M) is the excess noise factor, which measures the variations of the instantaneous avalanche gain M around its average value. The amplification process is always noisy since we have that F(M)>1. Therefore, the increase of the noise will be proportionally higher

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245

than the signal enhancement. The excess noise factor in APDs can be expressed as [15, 16] .

 1  F ( M )  k N M  (1  k N ) 2   M  

(4.27)

where parameter kN is known as the ionization coefficient. The ionization coefficient takes the values in the range from 0 to 1, and measures the ability of a carrier to generate other carriers in the avalanche amplification process. This coefficient should be as small as possible in order to minimize the avalanche shot noise that is generated. An approximate form, which is very often used instead of Equation (4.27), is given as F (M )  M

x

(4.28)

The noise coefficient x takes the values in the range 0 to 1, depending on the semiconductor compound used. Typical values of the noise coefficients kN and x for commonly used APD compounds are shown in Table 4.1. Table 4.1 Typical values of optical parameters x k Dark current noise in nA 0.5–0.8 0.3–0.6 Up to 20 1.0 0.7–1.0 50–500 0.4–0.5 0.02–0.04 Up to 10

Semiconductor InGaAs Germanium Silicon

T evaluate the impact of the shot noise, we will assume that noise spectral density for both PIN and APD has a form S sn ( f )  2q M

2 x

I

(4.29)

Example 1: If an optical power of P= –20 dBm falls to PIN photodiode with responsivity R=0.8, M=1, and x=0, it will produce I=8 A and Ssn/PIN ~ 2.6 x 10–24 A2/Hz. Example 2: If an optical power of P= –20 dBm falls to APD with responsivity R=0.8, M=10, and x=0.7, it will produce I=80 A and Ssn/APD ~ 1.29 x 10–21 A2/Hz. 4.1.5 Dark Current Noise The dark current consists of electron-holes pairs, which are thermally created in the photodiode p-n junction and flow through a biased photodiode even if no light is coming to the photodiode surface. These carriers can also get accelerated in

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APD, and can contribute to the avalanche shot noise generation. The dark current noise power is

i2

 S dcn ( f )f  2q M F ( M ) I d f 2

dcn

(4.30)

where Id is the primary dark current in photodiode, while Sdcn is spectral density of the dark current. The typical values of dark current for different semiconductor compounds are shown in Table 2.3. Equation (4.30) can be applied to both PIN photodiode and APD. Example 1: If Id = 5 nA, M =1, and x = 0 for PIN photodiode, we have that Sdcn~ 1.6 x 10–27 A2/Hz. Example 2: If Id=5 nA, M=10, and x = 0.7 for APD photodiode, which will produce the spectral density Sdcn ~ 8 x 10–25 A2/Hz. Time slot 1

Time slot 2

Time slot 3

Time slot 4 time

Photons Dark current

time

Electrons (primary)

time

Electrons (secondary)

Average current

Instantaneous current (signal+noise)

Photocurrent time

Figure 4.4 Noise generation in photodiode.

As we can see, the power of dark noise generated in photodiodes is smaller than the power of the other noise components that can be generated during photodetection. That is the reason why the impact of dark current noise is sometimes neglected. The process of the overall noise generation in the avalanche photodiode is illustrated in Figure 4.4. There are four time slots in Figure 4.4 to illustrate the Poisson statistics and the total noise generation. There is one incoming photon that generates one primary photoelectron within the first time slot. This primary

Noise Sources and Channel Impairments

247

electron will produce several secondary electron-hole pairs (it is seven in Figure 4.4). There is no signal photons captured within the second time slot, but a dark current electron is generated, and it was able to produce a pair of secondary electrons through the ionization process. Next, there are two incident photons within the third time slot that produce a number of secondary electrons. Finally, the time slot number four is similar to the slot number two, with the exception that a smaller number of secondary electrons have been generated. As a result, both the total number of electrons and the generated current flow fluctuate in time around their average value. These fluctuations are associated with the current noise at the photodiode output. 4.1.6 The Thermal Noise The load resistor, which is used to convert the photocurrent to voltage, as shown in Figure 2.32, generates its own noise due to a random thermal motion of electrons. Such noise occurs as a fluctuating current that adds to the generated photocurrent. This additional noise component, also known as Johnson noise [18], has a flat frequency spectrum that is characterized with the zero-mean Gaussian probability density function. This spectral density is expressed in A2/Hz, and given as

S the ( f ) 

4k RL

(4.31)

where RL is the load resistance, is the absolute temperature in Kelvins, while k is the Boltzmann’s constant (k =1.38·10–23 J/K). The thermal noise power contained in the receiver bandwidth f is

i2

the

 S the ( f )f 

4k f RL

(4.32)

The thermal noise can be reduced by using a large value load resistance. Such design, often referred to as the high-impedance front-end amplifier, also increases the receiver sensitivity. On the other side, it limits the receiver bandwidth since the RC constant (C is the capacitance of the circuit) is also increased. Therefore, the high-impedance input requires an equalizer that will boost the high-frequency components and increase the receiver bandwidth. In general, the receiver bandwidth is increased by selecting a smaller value of the load resistance. The design with a low-impedance front-end has a smaller receiver sensitivity than the design with the high-impedance front-end amplifier. As a compromise, the transimpedance front-end, shown in Figure 2.32, is used to achieve both high receiver sensitivity and high-speed operation. The transimpedance front-end design also improves the dynamic range of the optical receiver, which is important

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in cases when significant variations of optical power can occur at the receiving side. The thermal noise generated in the load resistor will be enhanced by electronic components within the front-end amplifier. That noise contribution can be accounted for by the amplifier noise figure NFne, which is the factor that measures the thermal noise enhancement at the front-end output. The total power of the thermal noise that also accounts the contribution of the front-end amplifier is given as

i2

the

 Sthe ( f )f 

4k f  NFne 2  I the NFne RL

(4.33)

The noise figure NFne (the subscript e stands for electrical) can vary from amplifier to amplifier, but for a low-noise front-end amplifier it is around 3 dB. The parameter Ithe, which is expressed in A/Hz1/2, is equivalent to standard deviation of the thermal current. This parameter is usually several pA/Hz1/2. Parameter q in the previous equation represents the electron charge (q = 1.6·10–19 C), k is the Boltzmann’s constant (k = 1.38·10–23 J/K), is the absolute temperature in Kelvins, RL is a load resistance in ohms. By assuming that Ithe=3 pA/Hz1/2, and NFne= 2, we can obtain that Sthe~ 2·10–23 A2/Hz. Therefore, calculated value for spectral density of the thermal noise is at least about two orders of magnitude higher than the one related to the dark current noise. The shot noise component produced in the PIN photodiode will be at the same level as the thermal noise component if optical power of –10.65 dBm (86 W) arrives at the photodiode with responsively R=0.8 A/W. 4.1.7 Spontaneous Emission Noise Signal amplification in an optical amplifier, which is discussed in Section 2.6, is also accompanied by spontaneous emission of the photons. That process is additive, which means that there is no correlation between the signal and the noise generated through the spontaneous emission. The noise induced by spontaneous emission has also a flat frequency spectrum characterized with the zero-mean Gaussian probability density function. The noise spectral density can be written as [19]

Ssp ( )  ( G  1 )NFnoh / 2

(4.34)

where G is the optical amplifier gain, NFno is optical amplifier noise figure that measures the noise increase (the subscript “o” stands for “optical”), h is the Planck’s constant (h=6.63·10–34 J/Hz), and  is the optical frequency. Please notice that we will temporarily carry two notations for frequency, f for the frequency of an electrical signal and  for the frequency of an optical signal.

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However, the variables f and  refer to the same physical parameter, which is expressed in Hertzs. This distinction will be used for few more times (for example, to distinguish optical and electrical filter bandwidths, and optical and electrical signal-to-noise ratios). There is the following relation between the noise figure and the spontaneous emission factor nsp=(N1-N2)/N2 , which is the same parameter as the one defined in Equations (2.15) and (4.2)

NFno 

2nsp ( G  1 )

(4.35)  2nsp  2 G The populations N1 and N2 are related to the number of electrons at the ground level and at the upper energy level, respectively—please refer to Figure 2.26(a). Theoretically, the spontaneous emission factor will become unity if all electrons are moved in energy to the upper level, which is not possible in practice. That is the reason why the spontaneous emission factor will always take values higher than 1. In most practical cases it will be in the region from 2 to 5, which corresponds to 3–7 dB. The effective noise figure of the amplifier chain of cascaded optical amplifiers can be calculated as

NFno,eff  NFno,1 

NFno,2 NFno,3 NFno,k   ... G1 G1G2 G1G2 ...Gk 1

(4.36)

where NFno,eff is the effective noise figure of the amplifier chain that contains the total number of k optical amplifiers. The first amplifier in the chain is the most important one in terms of the noise impact. That is the reason why multistage optical amplifiers should be designed to have the first stage with lower noise figure. Accordingly, any decrease in the effective value of the amplifier noise figure will bring a significant benefit to the overall system performance. The total power of the spontaneous emission noise can be calculated as 2

Psp ( )  2 Esp  2S sp ( ) Bop  (G  1)  NFnohBop

(4.37)

where Esp is the electric field of the spontaneous emission and Bop is the effective bandwidth of spontaneous emission determined either by the optical amplifier bandwidth, or by the optical filter. Please notice that factor 2 in Equation (4.37) accounts for contributions of two fundamental polarization modes that are present at the output of the optical amplifier. 4.1.8 Beating Components of Noise in the Optical Receiver If there is a chain of optical amplifiers, the spontaneous emission noise generated in a preceding amplifier stage is eventually amplified in the following stage, thus

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becoming the amplified spontaneous emission (ASE) noise. In addition, the spontaneous emission noise is also generated at the any specific amplifier in question, which means that the amplifier output is noisier than the input, as illustrated in Figure 4.3. The power of the amplified spontaneous emission noise is being converted from an optical to an electrical level in parallel with the optical signal conversion. The total photocurrent generated at the photodiode output, in the case when optical amplifiers are employed, can be written as

I p  I  inoise  R E G  Esp  isn  ithe

(4.38)

where E=(P)1/2 and Esp=(Psp)1/2 are the electrical fields associated with optical signal power P and amplified spontaneous emission power Psp, respectively, I is the signal current (I=RP), R is photodiode responsivity, G is the amplifier gain, isn is the quantum shot noise component, and ithe is the thermal noise component. The evaluation of noise components in most typical detection scenarios has shown that the three noise components from Equation (4.38) are the dominant ones, and the most relevant from the system design perspective. The amplified spontaneous emission (ASE) is not simply converted to the corresponding electrical noise since there is a beating process between the ASE and the signal electric fields, which results in appearance of several components that can be classified as the beat noise components. It is, therefore, necessary to express the incoming signal and ASE by the corresponding electric fields, as in Equation (4.38), to evaluate these beat noise components. Please notice that Equation (4.38) contains just half of the noise power from Equation (4.37). It belongs to a component of the ASE that has the same polarization with the signal. This is because the orthogonally polarized components cannot beat effectively, and only the component that has the same polarization with the signal is a relevant factor. The total noise current associated with the optical ASE noise occurs due to the beating of the ASE field Esp with the signal field E, and due to beating of the field Esp with itself. The total variance of such a fluctuating current can be found by expressing all electrical fields in Equation (4.38) through the optical power by ( ), and by averaging the field using a general expression √ products over random phases as in [10, 11]. This process leads to the equation i2  i2

the

 i2

sn

 i2

sig  sp

 i2

sp sp

(4.39)

where the first term on the right side represents the power of the thermal noise and the three remaining terms have the following values i2 i2

the



sn, Amp

4k  NFnef RL

(4.40)

 2qR[GP  S sp Bop ]f  S sn, Amp f

(4.41)

Noise Sources and Channel Impairments

i2 i2

sigsp spsp

251

 4R 2GPSsp f  S sigsp f

(4.42)

 2R 2 S sp2 [2Bop  f ]f  S spsp f

(4.43)

The total spectral density of the beat noise components from Equations (4.42) and (4.43) can expressed as S beat ( f )  S sig sp ( f )  S sp sp ( f )

(4.44)

As we can see, the optical amplifier gain coefficient G, the bandwidth of the optical filter Bop, and the bandwidth of an electrical filter of the receiver f will play a key role in the size of the total noise caused by spontaneous emission in optical amplifiers. It is also important to notice from Equation (4.41) that the shot noise is higher when the optical signal is preamplified. Example: Let us assume the following typical values of the amplifier and photodiode parameters: P= –20 dBm, G=100, Fno=3.2 (which is 5 dB), R=0.8 A/W, Bop=0.1 nm, f = 0.5Bop. The spectral densities from Equations (4.41), (4.42), and (4.43) now become Ssig–sp~ 1.05·10–19 A2/Hz, Ssp–sp ~ 0.53·10–22 A2/Hz, and Ssn,Amp ~ 2.5·10–22 A2/Hz, respectively. As we can see, the beating noise components are larger than the shot noise component in PIN photodiodes. Table 4.2 Typical values of spectral densities associated with noise components PIN APD Noise spectral “1” bit “0” bit “1” bit “0” bit density [A2/Hz] Dark current 1.6 x 10–27 1.6 x 10–27 0.8 x 10–24 0.8 x 10–24 Thermal noise Shot noise without preamp Shot noise with preamp Signal-ASE beat noise ASE-ASE beat noise

(M=10)

(M=10)

2 x 10–22 2.6 x 10–24

2 x 10–23 2.6 x 10–25

2 x 10–23 1.28 x 10–21

2 x 10–23 1.28 x 10–22

(M=10)

(M=10)

2.6 x 10–22

2.6 x 10–23

1.97 x 10–20

1.97 x 10–21

(M=5)

(M=5)

1.04 x 10–19

1.04 x 10–20

0.52 x 10–19

0.52 x 10–19

0.54 x 10–22

0.54 x 10–22

2.7 x 10–20

2.7 x 10–20

When considering the impact of individual noise components, we should differentiate two cases that are related to detection of “1” and “0” bits, respectively. That distinction is important with respect to levels of the shot noise, the beating noise components, and intensity-related noise components. The extinction ratio can be used as a measure of the difference in receiving power associated with “1” and “0” bits. The comparison of different noise components that are related to “1” and “0” bits is shown in Table 4.2 for both PIN photodiode and APD. It was assumed that the extinction ratio in accordance with Equation

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(2.57) is 10. As we see from Table 4.2, thermal noise and the shot noise contributions to the total noise power are generally smaller than the contribution coming from the beat noise components. It is possible to reduce the spontaneousspontaneous beat noise by optical filtering, and make it smaller than the signalspontaneous beat noise. 4.1.9 The Cross-Talk Noise Signal cross-talk occurs in multichannel systems when a fraction of an unwanted signal adds to the power of the channel in question, thus acting as a noise. The cross-talk noise can have either out-of-band or in-band nature. The out-of-band or intrachannel cross-talk occurs when the power from a neighboring channel crosses the border between channels and mixes with the power of the specified optical channel, as shown in Figure 4.5(a). This case happens very often when optical filters and optical multiplexers are deployed within an optical transmission system. Optical receiver bandwidth captures the interfering power and converts it to the electrical current. Out-of-band cross-talk is different from the random noise in an advantageous way. Namely, various noise types have amplitude probability distribution with a gradually decaying tail, which determines the decision threshold position. In contrast, in a cross-talk case the undesired cross-talk power that may appear at the decision point is bounded since there are a finite number of contributing sources. The worst case in terms of crosstalk impact would be if all channels are bit-synchronized, and when the channel in question carries “0” bit while all other channels carry “1” bits. However, in reality, the incoming cross-talk optical power is not correlated with the channel signal, which means that the out-of-band noise is incoherent in nature. The out-ofband cross-talk is dominated by two immediately adjacent channels, as illustrated in Figure 4.5(a). The photocurrent produced as a result of conversion of the out-of-band crosstalk to an electrical signal can be considered as a noise, and treated similarly as the way dark current noise was treated. The noise current generated by conversion of the out-of-band cross-talk to the electrical signal is M

icross, out   RX n

(4.45)

nm

where R is photodiode responsivity and Xn is portion of the n-th channel power that has been captured by the optical receiver of the m-th optical channel (i.e. channel in question). We assumed that there are M optical channels altogether, and that the largest contribution to the cross-talk noise level comes from the neighboring channels. The impact of the out-of-band cross-talk can be minimized by optimizing the optical channel spacing and by selecting optical filters with steeper bandwidth characteristics.

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253

The in-band or interchannel cross-talk occurs when a signal at the same optical wavelength as the wavelength of the channel in question comes at the optical receiver to be converted to electrical current. This can happen if there are multiple optical multiplexing/demultiplexing stages between the optical transmitter and optical receiver, or if an optical routing takes the place along the lightwave path. Any separation of optical channels by optical filtering process can cause interchannel cross-talk. That cross-talk can be enhanced by an increased mismatch between multiplexing and demultiplexing filters, which is usually caused by the temperature change or by aging. This will create out-of-band noise, as mentioned above, if demultiplexing is followed by the photodetection process.

1

2 Optical filter

3 Optical filter

4 Optical filter

5

4

5

1

2

2

5

1

3

4

3 Out-of band crosstalk noise

In-band crosstalk noise

a)

b)

Figure 4.5 Cross-talk noise: (a) out-of band and (b) in-band.

However, if the signal continues its propagation along a specified lightpath, the total power will now consist of the signal power and out-of-band cross-talk power. Any additional multiplexing/filtering process that can eventually occur before photodetection will also produce out-of-band cross-talk. It might happen that cross-talk power from the neighboring channel brings back a part of power that originally belonged to the channel in question. Although that portion contains the same data as channel in question, they are not in phase with each other any more since they experienced different delays before reunion occurred. The same process (separation and reunion) might happen in the optical switch due to nonideal isolation between switch ports, as shown in Figure 4.5(b). In-band cross-talk will eventually produce the beat noise components, similar to the case when the ASE noise from optical amplifiers was involved. The only real difference is that the cross-talk, rather than the ASE noise, will beat with the signal and with itself. The total beat photocurrent in the m-th channel is icross,in (t )  R E m (t )  R Em (t ) exp  j mt   m (t )  2

M

 X n En (t ) exp  j nt  n (t )

2

n 1,n  m

(4.46)

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where R is the photodiode responsivity, Em is the electric field of the optical signal in question, and Xn and n are the amplitudes and phases the electric fields associated with in-band cross-talk components. The parameter M denotes the potential number of in-band contributions. The exponential term denotes the coherent nature of the total electric field. Equation (4.46) can be sorted out by using substitution √ and by performing all multiplications, so the total current becomes

icross,in (t )  RPm  2 R

M



n 1,n m

Pn cos m (t )   n (t )  R( Pm  Pcross,in )

(4.47)

As we can see from Equation (4.47) that in-band cross-talk has the same nature as the intensity noise and can be treated as a component of the total the intensity noise.

4.2 DEFINITION OF BER, SNR, AND RECEIVER SENSITIVITY The optical signal that is distorted from the original shape during its travel along a lightwave path is eventually converted to the photocurrent by photodiode in an optical receiver. All acquired impairments will also be transferred to the electrical form. In addition, new noise components as corruptive additives will be generated in optical receiver, as discussed above. Both the distorted signal and corruptive additives will eventually come together to the decision circuit that recognizes logical levels at defined clock intervals, as shown in Figure 4.2. The signal value at the decision point should be as high as possible to keep a favorable distance from the noise level and to possibly compensate for the receiver sensitivity degradations due to impact of other various impairments (such as dispersion, a finite extinction ratio, cross-talk). Signal distortion due to impact of some impairments is illustrated in Figure 4.6(a). In general, distortions are observed through pulse level decrease, pulse shape distortion, phase and frequency shift, or noise additives. The receiver sensitivity degradation due to the pulse level decrease can be evaluated directly, while the evaluation of the pulse shape distortion or pulse phase change have more complex character. In contrast, the impact of the noise is evaluated through the averaged power of the stochastic process. The majority of the topics covered in following sections are related to optical transmission systems with intensity modulation (IM) and direct detection (DD). Accordingly, all discussion serves as a foundation of more advanced topics discussed in Chapters 5, 6, and 7. In this section, we will first evaluate signal-tonoise ratio (SNR) and receiver sensitivity since both of them define the transmission quality. Next, the impacts of other impairments will be quantified through the receiver sensitivity degradation with respect to a reference case. In

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some cases the receiver sensitivity degradation can be considered as the signal power penalty that needs to be compensated for. It can be done by allocating some power margin, which is defined as increase in the signal power that is needed to compensate for the impact of each specific impairment. By increasing the signal power, the SNR is kept at the same level that would exist in a case where no impairment was involved.

Figure 4.6 (a) Pulse distortion and noise, and (b) Gaussian pulse as a referent pulse shape.

The signal increase to compensate for the power penalty could have a real value in some situations, such as those related to the impacts of chromatic dispersion or smaller extinction ratio. An increase in signal power would certainly compensate for the penalty associated with the impairment in question, unless a negative impact of nonlinear effects eliminates all benefits that would be brought with the signal power increase. In contrast, the power margin has a different meaning if it is related to the impact of nonlinear effects. That is because an increase in signal power would not be beneficial since it would also contribute to signal distortion. Regardless, the power penalty serves as a design parameter for the evaluation of the nonlinear impairments and for assessment and optimization of overall transmission capabilities. 4.2.1 Bit Error Rate and Signal to Noise Ratio for IM/DD Scheme The received SNR and the bit-error rate (BER) at the output of the optical receivers are the parameters most commonly used to define transmission quality. In this section we will evaluate these parameters for the fundamental detection scenario where the stream of intensity modulated binary symbols, which occupy the entire slots (i.e., nonreturn to zero—NRZ signals), is converted to an electrical signal by using a direct detection scheme. The other schemes involving the advanced modulation formats and/or coherent detection will be analyzed in Chapters 5 and 6.

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The SNR defines the difference between signal and noise levels at the sampling points. The BER is interrelated with the SNR and defines the probability that a digital signal space (or 0 bits) will be mistaken for a digital signal mark (or 1 bits), and vice versa. The fluctuating signal levels that correspond to 1 or 0 bits can be characterized by corresponding probability density functions, as shown in Figure 4.7. These levels fluctuate around their average values I1 and I0, which are associated with 1 and 0 bits, respectively.

P(1) 1

1

0

Binary “1” level

P(1/0)

I1

Decision threshold

Ith

Binary “0” level

I0 P(0/1)

Additive noise Intersymbol interference

P(0)

Figure 4.7 Probability density functions related to 1 and 0 bits.

These two currents can be expressed as I0=RP0 and I1=RP1, where P0 is the optical power during 0 bit, P1 is the incoming optical power during 1, while R is the responsivity of the photodiode. Any current fluctuations around the average value are associated with the noise. The noise intensity can be characterized by standard deviations 1 and 0, which are related to 1 and 0 bits, respectively. At the same time, the noise power associated with 1 and 0 bits can be characterized by variances (1)2 and (0)2. The current levels at the decision circuit are sampled at moments corresponding to the recovered signal clock, and then compared with some threshold value Ith. Bit 1 is recovered if the sampled value is higher than the threshold, while 0 bit is recovered if the sampled value is lower than the threshold. The decision is correct if it coincides with the situation at the transmitting side. However, an error occurs if 1 bit is recovered when 0 bit was sent. The reason for such an error lies in the fact that fluctuating current at the decision instant was high enough to cross the threshold value, and was recognized as the level associated with 1 bit. Another false decision occurs if bit 0 is recovered when bit 1 was sent. Such a decision has been done since the current fluctuation around average value I1 was relatively high in the negative direction. It eventually went below the threshold level, and when compared to threshold, a false decision was made.

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The BER accounts for both cases of false decision. Accordingly, the total probability of a false decision can be expressed as BER  p(1) P(0 / 1)  p(0) P(1 / 0)  0.5P(0 / 1)  P(1 / 0)

(4.48)

where p(0) and p(1) are the probabilities that 0 and 1 were transmitted. We can safely assume that p(0) = p(1) = 0.5, which applies for a longer data bit stream. The conditional probabilities P(0/1) and P(1/0) are respectively related to cases when 0 was recovered while 1 was sent, and when 1 was recovered while 0 was sent. The probability P(0/1) is represented by the area under the P(1) function that is placed below the threshold level, as shown in Figure 4.7. At the same time, the probability P(1/0) can be identified as the area under function P(0) that lies above the threshold line. The calculation of BER parameter by Equation (4.48) involves both the signal and noise parameters. The signal is characterized by average values I1 and I0, while the total noise is characterized by the standard deviations 1 and 0, which depend on the intensity of different noise components that might contribute to fluctuation of the total current. It is therefore

1 

i12

0 

i02

(4.49) total

(4.50) total

where i1,total and i0,total are the fluctuating currents that are related to 1 and 0 bits, respectively. The statistics of the current fluctuations at the sampling points is rather complex, and an exact calculation of BER is rather tedious. However, there are several fairly good approximations that are used so far to evaluate the BER in optical receivers [20, 21]. The simplest, yet effective method is based on the assumption that both probability functions related to noise are the Gaussian distributions, which are characterized by the mean and standard deviation—please refer to Figure 4.7. The Gaussian model for noise functions leads to the following expressions for conditional probabilities P(0/1) and P(1/0) P(0 / 1) 

P(1 / 0) 

1

1

 I 1  I th   I  I 1 2  1 exp  dI  erfc  2 2 2 1  2    1 2

1

 0 2

   

(4.51)

 I th  I 0  1  dI  erfc   2   0 2 

(4.52)

I th





 exp  I th



I  I 0 2  2

2 0

where erfc(x) is complementary error function, which is defined as [1, 22]

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erfc( x ) 

2





e

 y2

(4.53)

dy

x

Both probabilities from Equations (4.51) and (4.52) depend on the threshold value Ith, which means that the threshold value can be adjusted in order to reduce the probabilities of false detection. The threshold adjustment can be done by equalizing arguments in Equations (4.51) and (4.52), which leads to

I1  I th 2  I th  I 0 2  ln  1   I th  I 0 2 2 12

2 02

   0

2 02

(4.54)

The optimum threshold value obtained from Equation (4.54) is

 1 I 0   0 I1 (4.55) 1  0 which is a valid approximation when 1 is close to 0. The above value for Ith can be inserted in Equations (4.51) and (4.52); the values P(0/1) and P(1/0) can be inserted back to Equation (4.48), so it becomes I th 

BER  BER (Q) 



erfc (Q / 2 ) exp Q 2 / 2  2 Q 2



(4.56)

The approximate expression on the right side of the above equation is reasonably accurate for Q > 4. The Q-factor from Equation (4.56) is defined as I1  I 0 (4.57) 1  0 This parameter is often taken as a direct measure of the SNR, although the exact relationship between the Q-factor and the SNR depends on the detection scheme that is used. The Q-factor can be evaluated experimentally through an eye diagram presented at the oscilloscope screen. The eye diagram is obtained by superposition of several sequences of the received signal form on top of each other. Each sequence is usually several bits long. The eye diagram should be as open as possible and as clear as possible, and it can be very useful to estimate the impact of different impairments. The functional dependence BER(Q) is shown in Figure 4.8, and serves as one of the most important tools that are used in system performance evaluation. This function returns several reference points, shown in Figure 4.8, that can be used in many practical considerations, and they are: Q

  

BER =10–9, which corresponds to Q = 6, or 20 log(6)=15.65 dB BER =10–12, which corresponds to Q = 7, or 20 log(7)=16.90 dB BER =10–15, which corresponds to Q = 8, or 20 log(8)=18.06 dB

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259

The factor 20, rather than 10, was used to calculate decibels value. It is because Qfactor is related to the electrical level and 20·log(Q) measures the power levels. It should be also mentioned that Q2 is sometimes used in association with the optical domain, which is based on the fact that the value 20·log(Q) is equal to 10·log(Q2). 0 -4 -8 -12

-16 -20 4

5

6

7

8

9

Q factor Figure 4.8 BER as a function of the Q-factor.

4.2.2 Optical Receiver Sensitivity The performance of the optical receiver can be evaluated by the parameter PR known as the receiver sensitivity. The receiver sensitivity is defined as the average power needed to achieve BER that is lower or equal to a specified value

PR 

P0  P1 N photonsh   N photons Bh 2 2T

(4.58)

where P1 and P0 are power levels related to 1 and 0 bits, respectively, T is the bit duration, B is signal bit rate, Nphotons is the average number of photons carried by each 1 bit, < Nphotons> = Nphotons /2 is the average number of photons extended over streams of 1 and 0 bits. Receiver sensitivity is very often specified with respect to the three values of BER written above. A minimum number of photons needed to achieve specified BER could be evaluated through Equation (4.18), which can be rewritten as

P( n ) 

n N photons e

n!

 N photons

(4.59)

The above equation gives the probability of generating n electron-hole pairs for the average number Nphotons of photons carried by each 1 bit. The BER of an ideal

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optical receiver can be calculated by using Equations (4.48) and (4.59). The probability P(1/0) will be zero, since it is not possible to generate any electrons if there is no incoming photons. In contrast, there is some probability that zero level will be recognized even if there are incoming photons. That probability can be obtained from Equation (4.59) for n = 0. Accordingly, we have that

PR 

P0  P1 N photonsh   N photons Bh 2 2T

(4.60)

We can establish some initial reference points with respect to the receiver sensitivity by evaluating the sensitivity of an ideal optical receiver for specified bit rates. It is done below in three examples by applying Equations (4.58) to (4.60) to the exemplary high-speed bit rates at the wavelength of 1,550 nm. Example 1: for BER=10–9, Nphotons = 20 or = 10 is needed, which translates to the following sensitivities of ideal receivers:  PR = –54.94 dBm for signal bit rate B =1/T= 2.5 Gb/s  PR = –48.92 dBm for signal bit rate B =10 Gb/s  PR = –42.90 dBm for signal bit rate B =40 Gb/s  PR = –38.92 dBm for signal bit rate B =100 Gb/s Example 2: for BER=10–12, Nphotons = 27 or = 14 is needed, which translates to the following receiver sensitivities:  PR = –53.64 dBm for signal bit rate B = 2.5 Gb/s  PR = –47.62 dBm for signal bit rate B =10 Gb/s  PR = –41.59 dBm for signal bit rate B =40 Gb/s  PR = –37.61 dBm for signal bit rate B =100 Gb/s Example 3: for BER=10–15, Nphotons = 34 or = 17 is needed, which translates to the following receiver sensitivities:  PR = –52.63 dBm for signal bit rate B = 2.5 Gb/s  PR = –46.61 dBm for signal bit rate B =10 Gb/s  PR = –40.59 dBm for signal bit rate B =40 Gb/s.  PR = –36.61 dBm for signal bit rate B =100 Gb/s As we see, the number of photons that are needed to achieve specified BER could be acquired easier for longer bit intervals. That is the reason why receiver sensitivity is better for lower bit rates (better receiver sensitivity is associated with lower power PR). Any specific practical case can be characterized with associated receiver sensitivity, which will be different than one related to an ideal optical receiver. That is because the sensitivity of an ideal optical receiver is determined only by the quantum limit of photodetection, while the impact of any other signal impairments is not included. However, these impacts should be included while considering any practical case. Different impairments will degrade receiver

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261

sensitivity, which is observed through increase in power PR that is required to keep specified BER. The difference between the receiver sensitivities related to an ideal optical receiver and to the practical application scenario can be considered as the optical power penalty associated to that specific case. The biggest contribution to the overall receiver sensitivity degradation comes from the noise side. Since the noise accompanies the signal in all practical cases, it is convenient to evaluate the receiver sensitivity degradation due to noise impact and establish new reference points that are more relevant from a design perspective than the ones related to the ideal receiver case. 4.2.2.1 Receiver Sensitivity Defined by Shot Noise and Thermal Noise The impacts of the thermal and quantum shot noise components should be evaluated first since they are present in any photodetection scenario. It can be done by considering a direct detection scheme with no optical preamplification involved. We can assume that the signal has an indefinite extinction ratio, which means that the power P0 carried by 0 bits can be neglected. In such a case, the receiver sensitivity from Equation (4.58) becomes

PR 

P1 RI 1  2 2

(4.61)

where R is photodiode responsivity given in A/W. We can use values from Table 4.2 to recognize the noise components that are dominant in different detection scenarios, including this one. Some other noise components, including the dark current noise, can be neglected since they have a smaller contribution. Accordingly, one can recognize that the thermal noise dominates for 0 bits in direct detection scenario, while both the thermal noise and quantum shot noise contribute to the total noise for 1 bits. The parameters that define the Q-factor in this detection scenario will have the following values

I1  R M P1  2 M RPR  12  i12

total

 0 2  i02

total

 i12  i2

sn

the

(4.62)

 i2 

the

 2q M

4kNFnef RL

2

F ( M )I1f 

4kNFnef RL

(4.63) (4.64)

where is the average value of photodiode amplification parameter and F(M) is photodiode noise factor. The amplification parameter is always larger than 1 if APD is used, while it equals 1 if the PIN photodiode is employed. The noise factor F(M) has unity value if it is related to the PIN photodiodes, while it higher if it is related to APD and can be evaluated by using Equations (4.27) and (4.28).

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The Q-factor in this case can be expressed as Q

I1

1   0

(4.65)

2 M RPR



 4kNFnef  2 2q M F ( M )2 M RPR f   RL  

1/ 2

 4kNFnef    RL  

1/ 2

It is possible to solve above equation with respect to PR, and it becomes [14] PR 

i2

Q

the

M R



qQ 2 F ( M )f 1  Q(4k  NFnef )1/ 2  qQ 2 F ( M )f    R M  R R RL 

(4.66)

The thermal noise term dominates in Equation (4.66) if the PIN photodiode is used since = F(M) = 1, and this detection scenario is recognized as the thermal-noise limited case [14, 16, 17]. The receiver sensitivity in thermal-noise limited case can be calculated by neglecting the shot noise contribution in Equation (4.66), which leads to PR , PIN 

 theQ R



Q(4k  NFnef )1 / 2 R RL

(4.67)

Therefore, the receiver sensitivity is determined by the receiver bandwidth f, the load resistor RL, and the noise figure of the front-end amplifier NFne. We can compare the thermal noise limited case with an ideal optical receiver case by assuming that f =B, although the receiver bandwidth depends on the modulation format and can range anywhere from 0.5B to B. -10 PIN receiver – thermal noise limited case -20 APD receiver – shot noise limited case -30 -40

Optical receiver with an optical preamplifier

-50

An ideal optical receiver -60 -70 0.1 1

2.5

10

Figure 4.9 Optical receiver sensitivity.

20

30 Bit rate in Gb/s

40

100

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Receiver sensitivity for thermal-noise limited case and sensitivity degradation from the value associated with an ideal receiver are shown in Figure 4.9 for several bit rates of practical interest (i.e., for B=0.1, 1, 2.5, 10, 40, and 100 Gb/s), and for the BER=10–12. The sensitivity that would apply if there were direct detection of signals with bit rate of 100 Gb/s is also denoted in Figure 4.9 just for comparison sake. It was assumed that R=0.8 A/W and RL= 50. Sensitivity degradation is smaller for higher bit rates. That is because in such a case the noise power increases in proportion to square root of the signal bandwidth (bit rate), rather than in proportion to the bit rate. The SNR in the thermal-noise limited case can be calculated as

SNR 

I 12



2 1

 4Q 2

(4.68)

Accordingly, the following SNR values can be associated with Q-factors:  SNR=144 or 21.58 dB for Q=6, which provides BER=10–9  SNR=196 or 22.92 dB for Q=7, which provides BER=10–12  SNR=256 or 24.08 dB for Q=8, which provides BER=10–15 If APD are used, the avalanche amplification will enhance the signal by factor . At the same time, the additional shot noise, which is proportional to the noise factor F(M), will be generated. In this case the shot noise contribution may become comparable or even larger than the thermal noise contribution. It is possible to find an optimum amplification factor opt that will optimize the receiver sensitivity and to minimize the function PR() given by Equation (4.66). For that purpose it is useful to employ Equation (4.27) for F(M). The optimization procedure applied to Equation (4.66) leads to the following value for the APD receiver sensitivity [6, 14]

PR , APD 

qQ 2 [2k N M

opt

 2(1  k N )]f

(4.69)

R

where an optimum value of the gain coefficient is given as 1/ 2

1/ 2

 i2   i2    k  1 the the  (4.70) M opt    N     kN   k N qQf  k N qQf      We can use the noise parameters from Table 4.2 to evaluate the sensitivity of the APD optical receivers. It is easy to show that the APD receiver sensitivity is typically enhanced by at least 5–10 dB as compared to sensitivity of the PINbased optical receivers. This benefit is limited to bit rates up to 10 Gb/s since the APD frequency range prevents them to be deployed in optical receivers operating efficiently at bit rates higher than 10 Gb/s. The receiver sensitivity of the APD-

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based optical receivers with an optimum avalanche gain is also shown in Figure 5.4. Their application area is limited to 10 Gb/s, and that is the reason why a dash line is drown in Figure 4.9 since it serves just for comparison purposes. 4.2.2.2 Receiver Sensitivity Defined by an Optical Preamplifier The receiver sensitivity of IM/DD scheme can be considerably enhanced in a case when an optical amplifier is employed in front of a photodiode. This method, also known as the optical preamplification, is most efficient if it is used in combination with PIN photodiodes, since a combination with APD could introduce relatively high shot noise and diminish the benefit of preamplification. However, if APD is used in combination with optical preamplifiers, the amplification coefficient should be adjusted to relatively low values to suppress the impact of the shot noise. For instance, it might be a good choice to keep lower than 5. Receiver sensitivity can be evaluated by using the same approach that was used previously. We can again assume that there is an indefinite extinction ratio, which means that the power P0 carried by 0 bits can be neglected. In addition, Table 4.2 can be used to estimate the significance of the individual noise components involved in this case. As we see, the beat noise components will be the strongest ones. This applies even if APD is used since the avalanche gain should be adjusted to lower values. For this detection scenario we have again that PR=P1/2, while the noise parameters can be obtained from Equations (4.42) and (4.43). Therefore, it is

I1  RGP1  2GRPR

 12  i12  0 2  i02 Q

I1

 i2

total total

1  0

 i2



sigsp sp sp

(4.71)

 i2

spsp

 8R 2GPR Sspf  2R 2 S 2 sp[2Bop  f ]f

 2 R 2 S 2 sp [2Bop  f ]f

(4.73)

2GRPR [ i2

sig  sp

 i2

sp sp

(4.72)

(4.74)

]1 / 2  [ i 2

sp sp

]1 / 2

The noise related to 0 bits is determined just by spontaneous-spontaneous beat component, since we neglected the power P0 carried by 0 bits. The receiver sensitivity extracted from above equations is PR 





   NF

2S sp f Q 2  Q Bop / f   0.5 G 1

1/ 2

no





 

hff Q 2  Q Bop / f   0.5

1/ 2

(4.75)

In the above equation, we used Equation (4.34) between the spontaneous noise power density and the optical amplifier noise figure to express Ssp as

Noise Sources and Channel Impairments

265

NFnoh (G  1) (4.76) 2 We can conclude from Equation (4.75) that optical amplifiers with a low noise figure should be used to fully utilize benefits brought by optical preamplification. In addition, it is necessary to adjust the optical filter bandwidth to be as close as possible to the signal bandwidth to minimize the noise impact. The receiver sensitivity related to the receiver with an optical preamplifier were again calculated for five exemplary bit rates and for BER=10–12. All results were obtained by assuming that the optical filter has a bandwidth Bop=2f. Obtained results are shown in Figure 4.9 in parallel with results obtained for other cases. The results from this figure prove that optical amplification is very beneficial since receiver sensitivity associated to this detection scheme is the one closest to sensitivity of an ideal optical receiver. S sp 

4.2.2.3 Receiver Sensitivity in Coherent Detection Schemes The analysis applied so far for the calculation of the Q-factor and BER in optical receivers with a direct detection cannot be applied for optical receivers with coherent detection. These receivers will be analyzed separately in Chapter 6. The main reason for that is the fact that signal and noise statistics will be dependent on the modulation scheme that is applied. However, it is possible to estimate the SNR value and receiver sensitivity for a generic case of a coherent detection. A coherent receiver is characterized by the presence of a local laser oscillator, as shown in Figure 4.2. The contribution of the local laser oscillator dominate in creation of major noise components, which are: quantum shot noise, noise due to beating between electrical fields related to ASE and to a local laser oscillator, and the laser intensity and phase noise. Coherent detection can be heterodyne or homodyne in nature. In the first case, there is some difference between carrier frequencies of the incoming optical signal and the local oscillator, while in a homodyne detection scenario the frequency of the local optical oscillator is equal to the frequency of incoming optical signal. In general, the photocurrent generated by the total optical power coming to photodiode can be presented as [8]





I (t )  RP(t )  R PLO  2 PS (t ) PLO cos(S  LO )t   (t )  n (t )

(4.77)

where PS and PLO are the optical powers of the incoming signal and local laser oscillator and S and LO are carrier frequencies of the incoming signal and local oscillator, respectively. The parameter is the phase difference between the incoming signal and the local oscillator, and n(t) presents phase nose component described by Equations (4.9) and (4.10). Since the homodyne detection is just a special case of heterodyne detection (where S = LO), we will analyze just a heterodyne detection scheme. The current corresponding to 1 and 0 bits is the same except in the case when ASK modulation format is used.

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In a heterodyne detection scheme, the power of the shot noise will be enhanced due to contribution of the local oscillator power. The total variance of the shot noise component will depend on the binary state that was sent if ASK modulation schemes are applied, while it irrelevant for FSK and PSK modulation schemes. The total bandwidth f of the optical receiver in heterodyne detection scheme is equal to a doubled value of the receiver bandwidth in direction detection scheme, while a receiver bandwidth in a homodyne detection case is equal to the bandwidth applied in direction detection scheme. However, in advanced modulation schemes with a balanced optical receiver and M-ary modulation formats, the bandwidth of the optical receiver for both heterodyne and homodyne detection become the same and equal to f, since frequency downshift filtering is applied in heterodyne detection. If we assume that binary PSK (BPSK) modulation format is applied, the signal and noise parameters associated with the heterodyne detection can be expressed as



I1  I 0  2R M

PS ( t )PLO

 12  i12

sn,S

 0 2  i02

total

total

 i12

 i12

 i2

total

 sn, LO

(4.78)

 i2

the

 i2

sp LO

 ( RPLO rint ) 2

(4.79) (4.80)

where with the term RPLOrint=2(RPLO)2RINlaser=2(RPLO)2RINLO, we included the impact of the intensity noise in accordance with Equations (4.16) and (4.127). The major contribution to the intensity noise is coming from local laser oscillator. It was again assumed that photodiode produces some gain , while photodiode responsivity is expressed through parameter R. We should outline that the PIN photodiode (=1) is the best fit for coherent detection schemes. The application of APD is limited to cases when gain is kept low ( ) 〈 〉 Components 〈 〉 and 〈 〉 , where 〈 〉 are shot noise components created by signal optical power and local laser oscillator, respectively, while 〈 〉 is the power of the thermal noise component. Component 〈 〉 presents the beating noise due to the interaction of electrical fields of ASE and the local laser oscillator. This component resembles signalspontaneous beat noise in scenarios with direct detection and can be evaluated by applying Equation (4.42) for this specific case, while replacing the product G·P with PLO, assuming that spontaneous emission is coming from an optical preamplifier but from the remote, which means that its power is now equal to Ssp/G. We should also notice that spontaneous-spontaneous beat noise component is small and not relevant any more. Also, the impact of the intensity noise of the local laser oscillator has to be included, as it is done through power of the intensity noise currents given by the

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last terms in Equations (4.79) and (4.80). The parameter rint, defined by Equation (4.16), is applied to local laser oscillator, which power is now dominant. We can assume that the impact caused by intensity noise of the local laser oscillator will be similar to the impact of the transmitter laser intensity noise, but much stronger in magnitude. We can also notice that contribution of the dark current has been neglected. Accordingly, the dominant noise components from Equations (4.79) and (4.80) are calculated as i2

i12 i2

sp LO

 4 R 2 PLO S sp f / G  S spLO f equiv

(4.81)

 2q M F (M ) RPLO  f equiv

(4.82)

4k  NFnef equiv

(4.83)

2

sn,LO

the



RL

where fequiv presents the equivalent receiver bandwidth. In general it is: (1) fequiv =f for heterodyne detection, and (2) fequiv = f for homodyne detection (f is receiver bandwidth that determines the noise bandwidth in direct detection receivers). However, in coherent receivers with M-ary modulation formats and balanced detection both bandwidths become equal to fequiv. The SNR for a heterodyne detection scheme (in this case for the BPSK modulation format) can be found as a ratio of the signal power to the total noise power. By applying Equations (4.78) to (4.83) we have that

SNR 

I12

 12



2R M   2q M  

2

F ( M ) RPLO 

4 R 2 PLO S sp G

PS (t ) PLO



2

 2( RPLO ) 2 RIN LO 

4k  NFne   f equiv RL  

(4.84) We can calculate and compare the power spectral densities of different noise components from Equation (4.84). They are obtained for the following typical values of the optical amplifier, local laser oscillator, and PIN photodiode parameters: PLO= 10 dBm, NFne=2, R=0.8 A/W, RINLO ~ –165 dB/Hz), while parameters related to ASE noise are G=20 dB and NFno=3.2 (which is 5 dB). The spectral densities from Equations (4.41), (4.42) and (4.43) now become SLO-sp~ 1.05·10–20 A2/Hz, Sthe ~ 2·10–23 A2/Hz, and Ssn,LO ~ 2.5·10–21 A2/Hz, and Sint=2(RP)2RINLO ~ 0.4·10–20 A2/Hz, respectively. There are several conclusions with respect to noise components in coherent detection scenario: (1) the impact of the thermal noise is suppressed as long as the power of local laser oscillator is larger than ~ 0.7 mW (in such case shot noise due to impact of local oscillator is approximately 10 times larger than the thermal noise); (2) the impact of beating noise caused by the interaction of spontaneous emission and the electric field of local laser oscillator is dominant as long as the power of a local oscillator is much

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higher than the power of incoming optical signal, which is true in almost all situations; (3) the laser intensity noise is a major factor if RINLO is higher than 160 dB/Hz dB (in which case that impact becomes comparable with the impact of the shot noise, (however in balanced coherent receivers this impact is suppressed by at least 15 dB and becomes comparable with the impact of thermal noise). In conclusion, coherent detection presents a spontaneous emission dominant detection scenario with spectral power density determined by spectral power density of ASE noise, given by Equation (4.76), just enhanced proportionally to the power of a local laser oscillator. In such a case and for = 1, we have the following approximation

SNR 

4 R 2 Ps PLOG 2 Ps G  4 R 2 PLO S sp f equiv NFnoh ( G  1 )  f equiv

(4.85)

where NFno and G are noise figure and gain of optical amplifiers producing ASE. Since coherent detection brings signal gain by mixing with the local oscillator, it is better use PIN photodiode to minimize the total noise (if APD is considered, than gain should be less than 3–5. Also, the employment of an optical preamplifier in combination with heterodyne detection would not serve the purpose due to large noise, and is not considered for practical applications. Since in most coherent detection schemes optical power per bit is equal for both 1 and 0 bits, we can assume that receiver sensitivity PR is equal to , which is then PR 

NFnoh ( G  1 )  f equivSNR

(4.86)

2G

As we already mentioned, the error probability in coherent selection schemes is modulation format specific and should be separately calculated for any case in question, where the impact of the laser phase noise should be also accounted for. As an example, the error probability in the BPSK modulation scheme can be calculated as [8]  1  SNRBPSK cos n  (4.87) BER BPSK  BER ( SNRBPSK )   pn erfc   d (n ) 2  2   where (  ) is the probability distribution function of the random noise phase. It is often assumed that the probability distribution function can be approximated by Gaussian distribution with a variance given by Equation (4.10), that is 

n2  4 S F ( ) sin 2  / d

(4.88)

0

where Sf() is the spectral density function of the frequency noise defined by Equation (4.8). The most relevant advanced modulation and detection schemes will be analyzed in Chapters 5 and 6.

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4.2.3 Optical Signal-to-Noise Ratio The optical signal-to-noise ratio (OSNR) is a very important parameter in monitoring the optical transmission quality. As its name says, the optical noise power is compared with the optical signal power while both of them are still in the optical domain. The main reason for the OSNR evaluation is the optical amplification process of the signal and all side effects of that process. Namely, the optical amplification process is accompanied by the generation of the amplified spontaneous emission (ASE) noise that accumulates along the transmission line. The power of the ASE noise is calculated with respect to a specified optical bandwidth Bop. It can also be measured, which is usually done by optical spectrum analyzers (OSA) that evaluate the power of the ASE noise in an assigned optical bandwidth slot. It is often required to calculate and measure the ratio of the optical signal power and the ASE noise at any specific point along the lightwave path. The OSNR serves as the main constraint in impairment-aware optical routing that will be discussed in Chapter 8. The OSNR at some point along the lightwave path can be defined as

OSNR 

PS PS PS   PASE S sp Bop 2nsph (G  1) Bop

(4.89)

where PS is the optical signal power at that specific point, nsp is the spontaneous emission factor, G is the optical amplifier gain, Bop is the bandwidth of the optical filter, h is the Planck constant, and  is frequency of the optical signal. The factor 2 in the denominator on the right side accounts for two polarization modes of ASE, where each of them carries optical power equal to nsph(G-1)Bop. The optical filter bandwidth, which is equal to the measure slot of the optical spectrum analyzer, is usually declared during the measurement process. For instance, there are a number of measurements in the optical domain that are done within the optical bandwidth equal to 0.1 nm, which is approximately 12.5 GHz if applied to the 1,550-nm wavelength region. The OSNR parameter can also be measured at the receiver entrance point just before photodetection takes place. In such a case, the optical power of the signal can be connected with the receiver sensitivity PR by using the relation PS ~ 2PR, while the OSNR can be expressed by using Equation (4.75) and (4.89), that is OSNR 

2f [Q 2  Q( Bop / f )1 / 2 ] PR  n sp hf (G  1) Bop (G  1) Bop

(4.90)

The OSNR value will be eventually converted to an electrical equivalent that defines the Q-factor and BER. The Q-factor in this detection scenario will be mainly determined by the ASE noise that is eventually converted to beat noise components. The following relationship between the Q-factor and OSNR can be established for NRZ signals if we neglect all other noise contributions except the beat noise components [23]

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Advanced Optical Communication Systems and Networks

2OSNR Bop / f

(4.91) 1  1  4OSNR Equation (4.91) can be further simplified if we assume that there is just one dominant noise term, in this case signal-spontaneous beat noise from Equation (4.72). With this assumption Equation (4.91) becomes

Q

OSNR Bop 2 f

(4.92)

From transmission quality perspective, it is the Q-factor (which is proportional to SNR) that is the most important parameter since it is directly correlated with BER. Accordingly, from a system design and performance monitoring perspective, it became necessary to have a good correlation between the Q-factor and both SNR and OSNR in different detection scenarios. In some situations it can be done by using approximate empiric formulas, such as those given by Equations (4.68) and (4.92), but in many other scenarios it is useful to establish a more precise relationship between the Q-factor and OSNR. The relation between the Q-factor and OSNR is transmission case specific, which means that it should be calculated for input parameters related to transmission scenario in question. For instance, Equations (4.66), (4.67), (4.69), (4.74), and (4.85) can be used to identify the receiver sensitivity for a specific performance requirement that is defined through BER, SNR, or Q-factor. That receiver sensitivity can be used afterwards as a reference to identify OSNR. A more precise correlation among Q-factor, SNR, and OSNR can be established by using numerical calculations, possibly with some measured data inputs. This process can be also performed on a dynamic base. Namely, OSNR can be measured along the lightwave path, while the values for the Q parameter and BER can be calculated for that specific scenario. The calculated values of the Q-factor can then be compared with established reference and used in the system performance monitoring process.

4.3 SIGNAL IMPAIRMENTS Different linear and nonlinear effects discussed in Chapters 2 and 3 will cause optical signal distortion during its propagation along the lightpath. The impact of these signal impairments can be calculated either through the SNR by accounting for the overall signal change, or through optical power penalty related to the receiver sensitivity degradation. The impact of some impairments, such as fiber loss, is very straightforward and can be accounted directly through the SNR. In contrast, the impact of most other impairments is more complex and needs to be evaluated through receiver sensitivity degradation. The precise analysis of the

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impact of various impairments is done by using numerical methods and finding solution of nonlinear Schrodinger equation. However, the purpose of analysis in this section is to evaluate impact of the impairments by using analytical tools, which will lead to closed-form expressions. These expressions are, in essence, approximate in nature, but accurate enough for practical system considerations, especially in field conditions. If the optical power loss were the only cause of the signal distortion, the incoming signal to the photodetector would be expressed as

P2 (t )  P1 (t )  L  c  others

(4.93)

where P2(t), is the power of the signal that passed through different elements (fiber, splices, filters, couplers, switches), P1(t) is the launched power at the beginning of the lightpath, c are splicing and connector losses, and others represent all other insertion losses that might occur along the lightpath. The power P(t) would directly determine the SNR, which means power loss translates directly to the power penalty. The fiber loss coefficient is about 0.2 dB/km at wavelengths around 1,550 nm for single-mode optical fibers based on doped silica. It can be even lower for the PSCF (pure silica core fibers), with the reported loss = 0.1484 dB/km [24]. The loss increases in value up to about 20% for other wavelengths covered by C and L wavelength bands. The attenuation in the S-band is even higher and can be more than 50% above the attenuation value around 1,550 nm. The typical mean attenuation inserted by fused optical splices is somewhere between 0.05 and 0.1 dB, while mechanical splices insert a loss comparable to or slightly above 0.1 dB. Optical connectors are designed to be removable, thus allowing many repeated connections and disconnections. There are several variants of single-mode fiber optical connectors that are commonly used, such as FC, SC, LC, MU, all depending on the shape of the connector and mating sleeve shape [16]. The insertion loss for high-quality single-mode optical fibers should not be higher than 0.25 dB. A special connector design is very often applied to minimize reflection of the incoming optical signal from the surface of receiving part of the connector. Such a design can include angled fiber-end surfaces (FC/APC connector) or perhaps some index matching fluid applied at the fiber surfaces. The matching fluid will minimize the reflection coefficient given by Equation (4.17) by decreasing the refractive-index differences when the optical signal crosses from one fiber to the other. Optical connectors with angled fiber-end surfaces are the most convenient ones to minimize connector return loss, which is a fraction of the optical power reflected back into the fiber at the connection point. In some cases, such as bidirectional transmission through the same fiber, the reflection power should be more than 60 dB below the input level. This requirement is satisfied by using connectors that provide a physical contact of the angled fiber ends.

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The number of the optical splices and connectors depends on the length of the lightwave path and should be taken into account during system design. Optical signal attenuation inserted due to fiber splicing and cabling is often distributed over the entire transmission length and added to the coefficient . Such a distributive approach is very useful in the system engineering process. It is common practice to add an additional fraction of about 10% to the coefficient  to account for the impact of fiber splicing and cabling. This is case when no measurements data are available. It is clear, however, that Equation (4.93) captures only a simplistic reference case. More realistic approach is when Equation (4.93) is expressed as

P2 (t )  P1 (t )  L  c   others   Pi (t )

(4.94)

where a sum of Pi accounts for receiver sensitivity degradation caused by signal impairments. Next, we will discuss the impact of the most relevant impairments to optical receiver sensitivity and evaluate Pi for the most relevant cases. 4.3.1 The Impact of Mode Dispersion in Multimode Fibers The impact of mode dispersion in multimode fibers was evaluated by Equations (2.4) and (2.8) for step-index and graded-index refractive profiles, respectively. It was also mentioned in Chapter 3 that the impact of the intermodal dispersion can be expressed through the fiber bandwidth given by Equation (3.78). The fiber bandwidth is a distance-dependent parameter that can be calculated from the available fiber data for each specified transmission length. It is useful from the system perspective to convert the fiber bandwidth to pulse broadening parameter in multimode optical fibers, and take it as a part of a total system pulse spreading consideration (as presented in Section 4.4.2). The fiber bandwidth can be converted to the pulse broadening time by using the following relation [25]

t fib ,L 

U UL  B fib ,L B fib

(4.95)

where L is the length of the fiber in question, Bfib is the optical fiber bandwidth of a 1-km optical fiber length, Bfib,L is the bandwidth of the specified fiber length, and  is coefficient that can take values in the range from 0.5 to 1. It is around  = 0.7 for most multimode optical fibers. The parameter U in Equation (4.95) represents the fact that the broadening time is also related to the modulation format. It is 0.35 for nonreturn-to-zero (NRZ) modulation formats and 0.7 for return-to-zero (RZ) modulation formats. The broadening time for 1-km fiber length calculated by Equation (4.95) can vary anywhere from 0.5 ns (for graded-index fibers) to 100 ns (for step-index fibers).

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The impact of chromatic dispersion on the system performance is a major factor of the pulse form degradation if transmission is done over single-mode optical fibers. As for multimode fibers, it is generally smaller than the impact of intermodal dispersion. However, it can contribute significantly to the total pulse broadening if transmission is done far away from a zero-dispersion region. 4.3.2 The Impact of Chromatic Dispersion The total impact of chromatic dispersion in single-mode optical fibers is considered in conjunction with several parameters that are related not just to characteristics of optical fiber, but also to the properties of the transmission signal and to characteristics of the light source. The transmission system should be designed to minimize the impact of chromatic dispersion. That impact can be evaluated through the signal power penalty, which in this case presents the signal power increase that is needed to keep the signal-to-noise ratio unchanged. The impact of chromatic dispersion can be evaluated by assuming that the pulse spreading due to dispersion should not exceed a critical limit. That limit can be defined either by the fraction s,chrom of the signal that leaks outside of the bit period, or by the broadening ratio b,chrom of widths associated to the input and output pulse shapes. These parameters donot translate directly to the power penalty since they deal with the pulse shape distortion rather than to the amplitude decrease. The exact evaluation of the power penalty could be rather complicated since it is related to a specific signal pulse shape. Instead, a reasonable approximate evaluation can be done by assuming that the pulse takes a Gaussian shape given by Equation (3.126), while using Equation (3.141) as a starting point. Herewith, we will adopt the approach of using broadening factor b,chrom as a measure of the pulse spreading. This factor has been defined in Chapter 3 and calculated for most relevant transmission scenarios—please see Equations (3.141) and (3.142). The broadening factor can be expressed as

 b , chrom 

 chrom  0 ( B)

(4.96)

wherechromis the pulse root-mean-square (RMS) width at the fiber end, while is the pulse RMS at the fiber input. Please notice that  is expressed as a function of the signal bit rate B=1/T, where T defines the length of the bit interval. If the input has a Gaussian pulse shape, it will be confined within the time slot T if it satisfies the following relation

0 

T 1  4 4B

(4.97)

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where is now related to RMS of the Gaussian pulse. In fact, it was shown in [17] that Equation (4.97) guarantees that almost 100% of the pulse energy is contained within the pulse interval. Any increase of the parameter chromabove the value , will indicate that there is some power penalty associated with that specific case. In contrast, there is no penalty unless the RMS of the output pulse exceeds the RMS of the input pulse. The power penalty could be even negative if an initial pulse compression is observed. We can estimate the power penalty Pchrom due to chromatic dispersion by using the following formula   (4.98) Pchrom  10 log(  b, chrom)  10 log  chrom   ( B )  0  The above formula is relatively simple, but the main task in calculating the power penalty Pchrom is related to the calculation of the broadening factor b,chrom. There are also some other more complex formulas that calculate the power penalty due to chromatic dispersion, such as one presented in [26]. The dispersion penalty given by Equation (4.98) can be calculated by applying Equation (3.141) to determine the broadening factor b,chrom. It is often more appropriate to apply a specific equation from Table 3.2 for the transmission scenario in question. In general, the power penalty is a function of the initial pulse with, source linewidth, initial chirp parameter, chromatic dispersion parameter, and transmission length. Example: Let us consider the most common scenario when high-speed transmission is done out of the zero-dispersion region and when the light source spectrum is much smaller than the signal spectrum. We will calculate the pulse broadening parameter by using Equation (3.145). The results are obtained for the following cases: =17.68 ps,  = 7.06 ps, = 4.41 ps,  = 1.77 ps, which corresponds to high-speed bit rates of 10 Gb/s, 25 Gb/s, 40 Gb/s, and 100 Gb/s, respectively. It is assumed that a single mode optical fiber with chromatic dispersion D=17 ps/(nm-km) is used. The results are shown in Figures 4.10 and 4.11. It is important to mention that bit rate can be translated to symbol rate if a more advanced modulation format is used. As an example, if 100 Gb/s is modulated by applying QPSK format and polarization multiplex, the symbol rate will be 25 Gsymbol/s. We can see from Figures 4.10 and 4.11 that dispersion penalty is dependent on the initial value C0 of the chirp parameter. It can be even positive, as mentioned above, since the pulse undergoes an initial compression. It occurs if a negative initial chirp is combined with a positive chromatic dispersion, and vice versa. The real importance of the results from Figures 4.10 and 4.11 is that they show what amount of the dispersion can be tolerated for a specific dispersion penalty. The dispersion penalty limit can be established by defining a power penalty ceiling. It is commonly 0.5 dB or 1 dB, as shown by the dotted lines in Figures 4.10 and 4.11. In some cases, such as for nonamplified point-to-point

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transmission with bit rates up to 10 Gb/s, even 2-dB power penalty can be tolerated. 8 C0 = - 2

25 Gbaud/s

C0 = 4

Dispersion penalty in dB

6

C0 = 0

4

2

0

-2

-4

0

10

20

30

40

50

60

70

80

90

100

Transmission distance in km

Figure 4.10 Power penalties due to chromatic dispersion impact on 10 Gb/s and 25 Gb/s signal bit/symbol rates. 8

C0 = 0 C0 = - 1

100 Gb/s

6

4

2 1 0

-2

-4 0

1

2

3

4

5

6

7

8

Transmission distance in km

Figure 4.11 Power penalties due to chromatic dispersion impact on 40 Gb/s and 100 Gb/s signal bit/symbol rates.

As an example, if C0 = 0 for 10 Gb/s bit (symbol) rate and for 2-dB dispersion penalty, fiber length corresponding uncompensated chromatic dispersion will be will be ~ 38 km, which translates to dispersion of 608 ps/nm. At the same time, if C0 = 0 for 40 Gb/s bit (symbol) rate and for 0.5-dB dispersion penalty, opticsl

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fiber length corresponding the uncompensated chromatic dispersion (residual dispersion) will be will be ~ 1 km, which translates to chromatic dispersion of ~17 ps/nm. The amount of chromatic dispersion that can be tolerated is also known as the dispersion tolerance. The dispersion tolerance with respect to the 1-dB limit is summarized in Table 4.3 for the cases where no chirp is applied, and for different optical fibers in high bit rate transmission systems. The lengths of two different fiber types (standard SMF and NZDSF from Figure 3.8) that correspond to the dispersion tolerance are also shown in Table 4.3.

Bit (symbol) rate in Gb/s 10 40 100

Table 4.3 Typical values of the chromatic dispersion tolerance Dispersion SMF Fiber length for NZDSF Fiber length for tolerance in ps/nm D=17 ps/km-nm D=4 ps/km-nm 720 42 180 45 2.6 11 7.1 0.42 1.77

The dispersion penalty estimate can be simplified for two extreme cases, when optical source spectrum is relatively wide, and when a chirpless Gaussian pulse propagates through the optical fiber. We can assume that the input Gaussian pulse has an optimum width 20 = (2L)1/2, where 2 is the group velocity dispersion (GVD) parameter, while and 20 represents the full width at 1/e intensity point (FWEM). These two extreme cases can be characterized by following equations derived from Equations (3.141) and (4.98)

  D LB   s ,chrom ,

for the source with a large spectral linewidth

(4.99)

1/ 2

 DL  B    2c 

  s ,chrom,

for an external modulation of a CW laser

(4.100)

where B=1/T is the signal bit rate,  is the source spectral linewidth, L is the transmission distance, and s,chrom is the fraction of the signal that leaks outside of the bit period T. The dispersion penalty limits are recommended in some early standard documents, such as [27], which states that the fraction spreading factor should be up to s,chrom = 0.306 for a chromatic dispersion penalty below 1-dB, and lower than s,chrom 0.491 for a 2-dB dispersion penalty. The impact of chromatic dispersion in IM/DD systems with higher bit rates needs to be suppressed by a proper selection of the parameters related to the pulse shape and optical modulator, and compensated by using proper compensation methods. In most cases chromatic dispersion has to be compensated and suppressed to a level that will cause a minimal power penalty (0.5 dB). The impact of chromatic dispersion, which is observed through the intersymbol interference (ISI), can be effectively canceled in coherent detection schemes by

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using digital filtering methods that require an intense signal procession at the electrical level—please refer to Chapter 6. 4.3.3 Polarization Mode Dispersion Impact The polarization mode dispersion (PMD) effect discussed in Section 3.3.3 can be a serious problem in high-speed IM/DD optical transmission systems. In contrast, in systems with a coherent detection, the PMD effect can be effectively compensated by digital filtering, which will be discussed in Chapter 6. The total pulse broadening due to PMD can be evaluated by using Equation (3.88) for bit rates up to 25 Gb/s (or, better to say, 25 Gsymbol/s) since the second-order PMD term can be neglected. The fraction s,PMD of the signal that leaks outside of the bit period due to the PMD impact can be expressed as

 s ,PMD 

 PMD T

  PMD B

(4.101)

where B=1/T is signal bit rate and PMD is the RMS of the output pulse. The fraction s,PMD should be less than a specified value. The power penalty PPMD related to the PMD effect can be evaluated by using the formula presented in [26], which is

PPMD  10 log(1  d PMD )

(4.102)

where 





d PMD  erfc    2 exp  i 2 2 erf [i  1 ]  erf [i  1 ]

(4.103)

i 1

and



T 0 2 0 

(4.104)

The function erf(x)=1-erfc(x) is well-known error function. The complementary error function erfc(x) has been defined by Equation (4.53). Parameters  and 0 are related to the RMS of the output and input pulses, respectively. If we now use Equations (4.97), (4.101), (4.103), and (4.104) for Gaussian shaped pulses, we can obtain that = 1/(2. Consequently, the parameter dPMD can be defined through the fraction parameter s,PMD as  1 d PMD  erfc  2 s , PMD

2      2 exp   i 2  4  i 1  s , PMD 

  i  1  erf      2 s , PMD

    erf  i  1   2   s , PMD

   

(4.105)

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If we assume that the input pulse has a Gaussian shape, we can again apply Equation (4.97) as a criterion of the pulse spreading limit. We can also assume that the first-order PMD is a dominant effect in most cases. The fraction parameter s,PMD can now be found from Equation (3.88), and it becomes 1 4

 s , PMD  1  2

 P21

 in2



 (1   ) 

1/ 2

  2 1  1  64 2P1  (1   ) 4 T 

1/ 2

(4.106)

where P1 defines the differential delay between two principal polarization states over fiber length L and represents the power splitting of the signal between two principal polarization states. The maximum allowable pulse spreading due to the PMD effect can be determined by following a recommendation issued by ITU-T in [28], which states that the pulse spreading factor s,PMD due to PMD should less be than 0.30 the for optical power penalty to be below 1 dB. Accordingly, Equation (3.92) can be applied, so we have that

 P1  DP1

L  0.1 T

(4.107)

where DP1 presents the mean value of the first-order PMD parameter expressed in ps/(km)1/2. The requirements for the accumulated average first-order PMD, and the actual first-order PMD group delay (expressed as three times the average value), is summarized in Table 4.4 for several high-speed bit rates. The values from Table 4.4 are relevant just for the binary IM/DD scheme. However, they can also be applied to multilevel modulation schemes in which the symbol rate, rather than the bit rate is the relevant parameter. As an example, if a QPSK modulation scheme with a polarization multiplex is applied, the symbol rate will be 25 GSymbol/s, and the actual PMD that can be tolerated is 12 ps (instead of 3 ps valid for a binary scheme, as shown in Table 4.4). From a design perspective of IM/DD systems, the PMD effect becomes a critical factor for bit rates of 10 Gb/s if the first-order PMD is higher than 0.5 ps/(km)1/2, while it becomes dominant for binary 40 Gb/s IM/DD scheme if the fibers have the first-order PMD that exceeds 0.05 ps/(km)1/2.

Bit (Symbol) rate in Gb/s 10 40 100

Table 4.4 Typical values of the first order PMD tolerance The average PMD tolerated, in ps Actual PMD tolerated, in ps 10 2.5 1

30 7.5 3

The impact of the second-order PMD can also be evaluated by using Equation (4.106). In such a case, the first-order PMD parameter P1 should be replaced with the parameter equal to (4.108)  2   P21   P2 2

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where P2 defines the total pulse spreading due to the second-order PMD effect in accordance with Equation (3.96). Finally, in addition to the transmission optical fibers, the PMD effect can also occur in other optical elements along the lightwave path, such as optical amplifiers, dispersion compensating modules (DCM), and optical switches. It is important to include the impact of the pulse spreading in these elements, especially if bit the rate exceeds 10 Gb/s km. The contribution to the pulse spreading due to PMD acquired in different elements can be expressed as



J



i

1/ 2



 PMD ,addit     i2,PMD 

(4.109)  where i,PMD are PMD contributions from optical elements mentioned above. The typical PMD values of optical modules and functional elements within a single span can be found in the corresponding data sheets, and some of them are listed in Table 4.5. As an example, we can calculate from Equation (4.109) that  PMD,addit will be more than 0.5 ps per optical fiber span, assuming that there are three to five optical elements that contribute significantly to the total PMD effect. The generalized formula that includes the entire PMD acquired in transmission fibers and optical modules can be derived from Equations (3.92), (3.96), and (4.109) and written as

3 D

P1

L

  5 D 2

P2

L    i2,PMD  0.3T 2

J

(4.110)

i

Table 4.5 Typical values of PMD in optical modules in a single fiber span Module Actual PMD in ps Optical amplifier 0.15 to 0.3 Dispersion compensating module 0.25 to 0.7 Optical switch 0.2 Optical isolator up to 0.02

4.3.4 The Impact of Nonlinear Effects on System Performance The transmission quality will be degraded due to the impact of nonlinear effects occurring in optical fibers [29–34]. Some nonlinear effects, such as self-phase modulation or cross-phase modulation, can degrade the system performance through the signal spectrum broadening and the pulse shape distortion. Other effects degrade the system performance either through nonlinear cross-talk (in four-wave mixing and stimulated Raman scattering), or by signal power depletion (in stimulated Brillouin scattering). The power penalty associated with nonlinear effects does not have the same meaning as in cases associated with dispersion or attenuation since the impact of nonlinearities cannot be compensated by an eventual increase of the optical signal power. That is because any increase in

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signal power will also increase the impact of nonlinearities. Therefore, the power penalty should be considered rather as a measure of the impacts of nonlinearities. 4.3.4.1 Impact of Stimulated Brillouin Scattering (SBS) In the SBS process the acoustic phonons are involved in interaction with light photons, and that occurs over a very narrow spectral linewidth SBS ranging from 50 to 100 MHz—please recall Equation (2.110). The interaction is rather weak and almost negligible if the spectral linewidth is larger than 100 MHz. The SBS process depletes the propagating optical signal by transferring the power to the backward scattered light (i.e., the Stokes wave). The process becomes very intense if the incident power per channel is higher than some threshold value PBth. The threshold value, which is expressed by Equation (2.111), is estimated to be about 7 mW. The SBS penalty can be reduced by either keeping the power per channel below the SBS threshold, or by broadening the linewidth of the light source. The most practical way of the linewidth broadening is through the signal dithering. By applying the spectral linewidth broadening, Equation (2.111) for the SBS threshold becomes [32, 35]

PBth 

21bAeff   laser  1   g B max Leff   SBS 

(4.111)

where laser is a broadened value of the source linewidth. As an example, the SBS threshold rises to PBth~16 mW = 12 dBm if the source linewidth is broaden to be laser = 250 MHz. This value is good enough to prevent any serious impact of the SBS effect. 4.3.4.2 Impact of Stimulated Raman Scattering (SRS) Since the SRS effect is broadband in nature, its gain coefficient covers a much wider wavelength region than the SBS gain. It was explained in Section 2.6.2.3 that optical channels spaced up to 125 nm from each other could be effectively coupled by the SRS process, possibly in both directions. This is the main reason why the SRS effect can be used to design an efficient optical amplifier, in spite of the fact that the gain peak gR ~ 7 · 10–13 m/W is much smaller than the gain peak associated with the SBS process. The SRS coupling and power transfer occurs from a lower wavelength A to a higher wavelength B in a way shown in Figure 4.12, only if both channels are carrying 1 bits at any specific moment, as illustrated in Figure 4.13.

Noise Sources and Channel Impairments

…

281

…

Shorter wavelengths

Longer wavelengths

B SRS

After transmission

A

…

…

Shorter wavelengths

Longer wavelengths

Figure 4.12 SRS effect as a transfer of optical power between WDM channels.

The power penalty occurs due to depletion of the signal in the originating channel, and can be expressed as PSRS  10 log(1   Raman ) (4.112) where Raman represents the coefficient that is proportional to the leaking power portion out of the channel in question. The coefficient Raman can be calculated as [32, 33] M 1

 Raman   g R i 1

ich PLeff g R ch PLeff M ( M  1)  R 2 Aeff 4R Aeff

(4.113)

whereR ~125 nm is maximum spacing between the channels involved in the SRS process, ch is the channel spacing, P is the power per channel, and M is the number of channels. Equation (4.113) is obtained under the assumption that the powers of all optical channels were equal before the SRS effect took a place. To keep penalty the below 0.5 dB, it should be Raman< 0.1, or

PM ( M  1 )ch Leff  PtottotalLeff  40 W  nm  km

(4.114)

where total is the total optical bandwidth occupied by all channels. Equations (4.113) and (4.114) were derived by using the results obtained in [30, 34], and under the assumption that there was no chromatic dispersion involved. The SRS effect is reduced if there is chromatic dispersion present since different channels travel with different velocities, and the probability of an

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overlapping between pulses decreases. The coefficient Raman expressed by Equation (4.113) should be multiplied by the factor  ~ 0.5 if chromatic dispersion exceeds a critical limit (which is 2.5–3.5 ps/nm-km).

Before transmission

Time slot A

B

Time slot

A

B

Figure 4.13 Bit-pattern dependence of SRS-based power transfer.

As an illustration of the above conditions, we calculated the power penalty associated with the case when chromatic dispersion is present, which is shown in Figure 4.14. The power penalty was calculated for several representative cases by using Equations (4.112) and (4.113). The coefficient Raman was decreased by half to reflect the fact that there is chromatic dispersion present. As we can see from Figure 4.14, the power penalty can be decreased by placing optical channels closer to each other and by decreasing power per channel.

Noise Sources and Channel Impairments

283

Figure 4.14 Power penalties due to SRS.

Since the power decrease will reduce the SNR, it will be beneficial only if the power penalty decreases faster than the power itself. The SRS effect in multichannel system can be neutralized by equalization of the powers associated with individual WDM channels, which is usually done through the dynamic gain equalization applied periodically along the lightwave path. 4.3.4.3 Impact of Four-Wave Mixing (FWM) The FWM effect can be treated as an out-of-band nonlinear cross-talk, and evaluated by using Equation (4.45). However, that negative effect due to FWM cannot be compensated by an optical signal power increase. The out-of-band cross-talk can be measured through the ratio M

 

P

i 1;i  n

i

(4.115)

Pn

where the nominator contains portions of the intruding powers that originate from all channels except the channel in question and the denominator refers to the optical power of channel in question. In the FWM process cross-talk components are generated at optical frequencies ijk=i+j–k. It occurs in situations when three wavelengths with frequencies i, j, and k propagate through the fiber, and a phase matching condition among them is satisfied. The phase matching is defined through the relation between the propagation constants of the optical waves involved in process, which is expressed by Equation (3.192). However, Equation (3.192) commonly takes a more practical form given by Equation (3.193), which is referred to a degenerate case (such as the WDM transmission). The optical power of a resultant new optical wave can be

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calculated by Equations (3.195) and (3.196), while the power penalty related to the FWM impact can be calculated as

PFWM  10  log 1   FWM 

(4.116)

where the cross-talk factor FWM is calculated as M

P

ijk

 FWM  i , j ,k 1(  n )

(4.117)

Pn

It should be FWM < 0.2 if we would like to keep the power penalty below 1 dB. Equation (4.116) gives us a good idea of how to accommodate a targeted power penalty by playing with the GVD parameter, channel spacing, and the optical power per channel. As an illustration, we plotted a family of curves related to five different channel spacings: 12.5 GHz, 25 GHz, 50 GHz, 100 GHz, and 200 GHz, which is shown in Figure 4.15. The calculation was aimed to the WDM system with 80 channels. However, just a limited number of channels was effectively contributing to the level of the cross-talk since the weight coefficient associated with the intensity of interaction between different WDM channels decreases rapidly with the increase of the channel spacing as shown in Figure 4.15. For instance, the impact of the neighboring channels spaced up and down by 50 GHz from the channel in question will be two times stronger than the impact of two channels that are spaced up and down by 300 GHz. For 100-GHz and 200-GHz cannel spacings, the curves in Figure 4.15 were obtained by calculating the impact of channels that are placed within 600 GHz, either up or down from the channel in question, while for 50-GHz, 25-GHz and 12.5-GHz cannel spacing, the curves in Figure 4.15 were obtained by calculating the impact of channels that are placed within 300 GHz. It means that there was the total number of 48 interacting channels for 12.5-GHz channel spacing of 24 interacting channels for 25-GHz channel spacing, 12 interacting channels for 50 GHz channel spacing, 12 interacting channels for 100 GHz channel spacing, and 6 interacting channels for 200 GHz channel spacing. It was assumed that Aeff = 80 m and D=8 ps/nm-km (2 ~ –10 ps2/km). From the practical perspective, it is important to make sure that chromatic dispersion in transmission fiber lies above a critical value, which is 2.5–3.5 ps/nm-km [32]. This is a precondition that might be followed by the optimization procedure that involves the channel spacing selection and optical power per channel adjustment. The FWM effect will severely degrade the system performance if the system operates in a vicinity of a zero dispersion region. That fact explains the reason why the nonzero-dispersion shifted optical fibers (NZDSF) have been introduced.

Noise Sources and Channel Impairments

285

2 25 GHz

Power penalty in dB

12.5 GHz

50 GHz

1.5

1 100 GHz

0.5

200 GHz

0 0

1

2 3 Power per channel in mW

4

5

6

Figure 4.15 Power penalties due to FWM.

4.3.4.4 The Impact of Self-Phase Modulation (SPM) The self-phase modulation (SPM) effect does not cause any cross-talk or power depletion. However, it induces the pulse spectrum spreading, which then interacts with chromatic dispersion and enhances the pulse spreading. The hypothetical power penalty due to SPM can be estimated by using the same approach that was used for chromatic dispersion, that is

PSPM  10 log(  b,SPM )

(4.118)

where b,SPM is the broadening factor due to the SPM effect, which can be calculated by using Equation (3.151) as

 b ,SPM

 2 Leff L 2   SPM  1  0 2 Lnel 02 

1/ 2

2  4 Leff  L2  22   1  2  4   3 3 Lnel  4 0 

(4.119)

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Advanced Optical Communication Systems and Networks P0=15 mW, D= - 17 ps/km.nm

6 5

P0=1 mW, D= 17 ps/km.nm

4

P0=15 mW, D=17 ps/km.nm

P0=2 mW, D=4 ps/km.nm

3

P0=15 mW, D=8 ps/km.nm

2 1 0 1 2

0

10

20

30

40

50

60

70

80

90

100

Line length in km

Figure 4.16 Power penalty due to SPM for 10-Gb/s bit rate.

Equation (4.119) is plotted in Figure 4.16 for 10-Gb/s bit rate, and for several different values of the input power and chromatic dispersion parameter. It was again assumed that the input pulse has a Gaussian shape and that Equation (4.97) can be applied. Therefore, the parameter from Equation (4.119) takes the value  = 17.68 ps. As we can see, even a modest value of input optical power can cause considerable power penalties if transmission is done in the normal dispersion region (negative chromatic dispersion coefficient). In contrast, the SPM effect can help to suppress the impact of chromatic dispersion in the anomalous dispersion region. This possibility is further explored through two special techniques known as chirped RZ coding and soliton transmission, which were discussed in Section 3.4.1.3. 4.3.4.5 Impact of Cross-Phase Modulation (XPM) Effect Cross-phase modulation (XPM) is another nonlinear effect that causes changes in the optical signal phase. This phase shift is pulse-pattern dependent, and it is converted to the power fluctuations in the presence of chromatic dispersion. Therefore, the signal-to-noise ratio will be diminished through the intensity-like noise, which is also pulse-pattern dependent. The XPM effect can be reduced by optical power reduction since the root-mean-square (RMS) of these intensity fluctuations is dependent on the optical power. A rough estimate of the XPM effect was done in [29] by assuming that the total phase shift XPM due to the XPM should be lower than one radian. Therefore, by calculating the total phase shift from Equations (3.189, 3.190), and by assuming that XPM 2 as follows:

Set mˆ  mi if

pk f R  r | mk  is the maximum for k  i

(6.43)

When all symbols occur with the same probability pi=1/M, then the corresponding decision rule is known as the maximum likelihood (ML) rule and can be formulated as:

Set mˆ  mi if

l  m k  is maximum for k  i In terms of observation space, the ML rule can also be formulated as:

(6.44)

Advanced Detection Schemes

425

Observation vector x lies in region Z i if

(6.45)

l  mk  is the maximum for k  i

For an AWGN channel, by using the likelihood function given by Equation (6.37) (which is related to the Euclidean distance squared), we can formulate the ML rule as:

Observation vector r lies in region Z i if

(6.46)

the Euclidean distance r  sk is minimum for k  i From Equation (5.1) in Chapter 5 we know that: r  sk

2

N

2

N

N

(6.47)

N

   rj  skj    rj2 2 rj skj   skj2 j 1 j 1 j 1 j 1  Ek

and by substituting this equation into Equation (6.46), we obtain the final version of the ML rule, which is

Observation vector r lies in region Z i if

(6.48)

D

1 rj skj  Ek is maximum for k  i  2 j 1

The receiver configuration shown in Figure 6.2 is, therefore, an optimum one only for the AWGN channel (such as optical channel dominated by ASE noise), while it is suboptimum for Gaussian-like channels. As for non-Gaussian channels, we need to use the ML rule given by Equation (6.44). Threshold, log(p2/p1)

Observation vector, r

LLR computation

log L(r)

Decision circuit

mˆ Optimum decision

Chose m1 if log L  r   log  p2 / p1   Otherwise choose m2 

Figure 6.5 The log-likelihood ratio receiver.

As an example, for modulation schemes with M=2, we can use the decision strategy based on the rule: m1

 f  r | m1     p2  LLR  r   log  R  log   r | f m    2   p1   R m2

(6.49)

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where LLR() denotes the log-likelihood ratio. Equation (6.49) is also known as the Bayes test [2], while the factor log(p2/p1) is known as a threshold of the test. The scheme of the corresponding log-likelihood ratio receiver is shown in Figure 6.5. When p1=p2=1/2, the threshold of the test is 0, while the corresponding receiver is the ML one. 6.1.4 Error Probability in the Receiver The average error probability of the detection process can be calculated as: M

M

i 1

i 1

Pe   pi P  r does not lie in Z i | mi sent    pi  f X  r | mi dr

(6.50)

Zi

where pi=P(mi) represents the a priori probability that symbol mi is transmitted, and Z i  Z  Z i denotes the complement of Zi (“not Zi decision region”). By using this definition of the complement of decision region, Equation (6.50) can be rewritten as: M

M

Pe   pi 

f X  r | mi dr  1   pi  f X  r | mi dr

Z  Zi

i 1

i 1

(6.51)

Zi

For an equal probability transmission of symbols (pi=1/M), the average error probability can be calculated as Pe  1 

1 M

M

 i 1

Zi

f X  r | mi  dr

(6.52)

In the ML detection, the probability Pe of the symbol error depends only on the relative Euclidean distances between the message points in the constellation. Since AWGN is spherically symmetric in all directions in the signal space, the changes in the orientation (with respect to the both coordinate axes and the origin) do not affect the probability of the symbol error. Therefore, the rotation and translation of signal constellation produce another signal constellation that has the same symbol error probability as the one observed against the minimum Euclidean distance. Let us first observe the rotation operation and denote the original signal constellation by {si}. The rotation matrix R is an orthonormal one (RRT=I, where I is identity matrix). The corresponding constellation obtained after application of rotation matrix is given by {Rsi}. The Euclidean distance between observation vector r and a constellation point from rotated signal constellation, say, Rsi, is given as rrotate  si ,rotate  Rsi  n - Rsi  n  r  si

(6.53)

Advanced Detection Schemes

427

Therefore, the Euclidean distances between constellation points are preserved after rotation since it is only noise dependent. Accordingly, the symbol error probability is invariant to the rotation process. We need also to verify if the noise vector is sensitive to rotation. The variance of rotated noise vector can be expressed as T   E  Rn( Rn)T   E  RnnT RT  E  nrotate nrotate

 RE  nnT  RT 

(6.54)

N0 N RRT  0 I 2 2

 E  nnT 

which means that it does not change during rotation. The mean value of the rotated noise vector is still equal to zero, that is, E  nrotate   E  Rn   RE  n   0

(6.55)

Therefore, the AWGN exhibits spherical symmetric property. Regarding the translation operation, let all constellation points be translated for the same translation vector {sia}. The Euclidean distance between the observation vector and transmitted point si is given as rtranslate  si ,translate  si  a  n   si  a   n  x  si

(6.56)

and does not change with translation. Accordingly, the symbol error probability, as a function of the minimum Euclidean distance, does not change either. However, the average symbol energy changes with translation, and upon translation it becomes M

Etranslate   si  a

2

i 1

2

Since si  a  si M

(6.57)

pi 2

2  2a T si  a , we can rewrite Equation (6.57) as:

M

Etranslate   si pi  2a T  si pi  a i 1 i 1      2

E

2

M

p i 1

i

 E  2a T E  s   a

2

(6.58)

E s

The minimum energy constellation can be obtained by differentiating Etranslate with respect to a, by setting this result to be zero, and by solving for the optimum translate value, which leads to amin  E  s . The minimum energy signal constellation is defined by:

si ,translate  si  E  s 

(6.59)

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The average symbol energy that corresponds to the optimum constellation defined by Equation (6.59) is Etranslate,min  E - E  s  2 . Example 1: The signal constellation of a ternary signal set is obtained by placing the signal constellation points in vertices of an equilateral triangle of side length a, as shown in Figure 6.6 (decision boundaries and decision regions are also shown in this figure). The following basis functions can be used to implement this modulation scheme:  1 , 0t T  1  t    T  0, otherwise 

 1 , 0t T  2 t    T  0, otherwise  S2 s2(0,a(3)/2) a s3(-a/2,0)

=60o

S3

a

s1(a/2,0)

S1

Figure 6.6 Ternary signal constellation.

The signal constellation shown in Figure 6.6 can be expressed in terms of basis functions as s1  t  

a 1  t  2

a s3  t    1  t  2

s2  t  

a 3 2 t  2

The corresponding energies of different signal constellation points are given by: a2 3a 2 E1  E2  , E3  4 4 The average energy of signal constellations is given by 1 5a 2 Eav   E1  E2  E3   3 12 The center of mass of this constellation is given by 1 E  a    a1  a2  a3   0, a / 2 3 3 The minimum energy signal constellation can be obtained by:







Advanced Detection Schemes

429

   a 1/ 2, 1/  2 3    a  E  a     a / 2, a /  2 3    a  1/ 2, 1/  2 3  

a '1  a1  E  a   a / 2, a / 2 3 a '2

2



 

a '3  a3  E  a   0, a 3 / 3  a 0, 3 / 3



The optimum receiver configuration for this case is shown in Figure 6.7. -E1/2

1(t) r1

()dt t=T

r(t)=si(t)+n(t)

()dt t=T

Compute rjsij (j=1,2; i=1-4) r2

Decision -E2/2

Select the largest

-E3/2

2(t) -E4/2

Figure 6.7 The optimum receiver configuration for ternary signal constellation.

The symbol error probabilities for ternary signal constellation are equal to each other because of symmetry, so that the average symbol error probability can be expressed as 2   a  Pe  Pe  s1   Pe  s2   Pe  s3   1  1      2 N    0   2

  6 Eav    1  1        5 N 0   The -function, used above, is defined as a probability that zero-mean Gaussian random variable X of unit variance is larger than u:  1  x2 / 2 1  u  e dx  erfc    u   P( X  u )    u 2 2  2 while the erfc-function was defined by Equation (4.53).  Example 2: Let us apply conclusions to binary signal constellations. We know that the output of matched filter (correlator) r is a Gaussian random variable since the input is a Gaussian random process. In any linear filter the Gaussian input produces the output that is a Gaussian random process as well, but with a different mean value and variance [1, 2]. The statistics of this Gaussian random variable, r, changes depending on whether the transmitted signal was s1 or s2. If s1 was sent, then r is a Gaussian random variable with the mean value m1 and variance 2 given as

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Advanced Optical Communication Systems and Networks

N T m1  E r | S1   E1  12 and  2  0  E1  E2  2 12  , 12   s s1  t  s2  t  dt 0 2

(6.60)

If s2 was sent, then r has a mean value and variance given as: N m2  E r | S2   E2  12 and  2  0  E1  E2  2 12 

(6.61)

2

For an equiprobable symbol transmission, the optimum decision thresholds can be determined in intersection of corresponding conditional probability density functions (PDF). We can now use Equation (6.42) to express the error probability as  E  E2  2 12 Pe Z1 | m2   Pe Z 2 | m1    1 2 N0 

  

(6.62)

The average error probability of now becomes  E  E2  2 12 Pe    1  2 N0 

  

(6.63)

Let us now pay attention to three representative examples of binary communication systems recognized by: antipodal signaling, orthogonal signaling, M

and unipolar signaling. The average symbol energy is given by E  E p ,  k k s k 1

which for an equiprobable transmission of binary symbols becomes Es  0.5E1  0.5E2 . For binary transmission the bit energy Eb is the same as the average symbol energy, that is, Eb=Es/log2M=Es. In antipodal signaling, such as BPSK, the transmitted symbols are: s1  t    s  t  and s2  t    s  t  . In this case, E1 = E2 = Eb, and 12   E , so that the average error probability is given by Pe  





2 Eb / N0 .

In orthogonal signaling, such as FSK, we have that the correlation coefficient 12  0 and E1 = E2 = E. In this case, the average probability of error probability is P    e 

E N 0 

  Eb  , which is about 3 dB worse than in antipodal        N0 

signaling. In unipolar signaling, such as on-off keying, transmitted symbols are represented as: s1  t   s  t  and s2  t   0. In this case, E1 = E, E2 = 0, Eb  E 2, and 12  0 ,

so

that

the

average

error

probability

is

given

by

Advanced Detection Schemes

 E Pe     2N 0 

431

  Eb  , which is about 3 dB worse than in antipodal        N0 

signaling. Let us now establish the relationship between the average error probability and Euclidean distance. From the definition of Euclidean distance, we have that Ts

2

T0

d E2  s1 , s2     s1  t   s2  t   dt   s12  t   2s1  t  s2  t   s22  t  dt 0

0

(6.64)

 E1  E2  2 12 so that the average probability of error can be written as d p , p  Pe    E 1 2   2 N 0  

(6.65)

It can be shown that when E1 = E2 = E, d E  s1, s2  is maximized (while Pe is minimized) when s1  t    s2  t  , which corresponds to antipodal signaling. For antipodal signaling, 12   E , the Euclidean distance is d E  s1 , s2   2 E and the corresponding error probability is Pe  





2 E / N 0 , which is consistent

with expression obtained above. Example 3: M-ary QAM. From Section 5.1.5 we know that M-ary squareQAM signal constellation can be obtained as a two-dimensional Cartesian product of L  M -ary pulse amplitude modulation (PAM) signals:

X  X   x1 , x2  | xi  X  , X   2i  1  L  d ; i  1, 2, , L

(6.66)

The error probability for M-QAM can be expressed as:

Pe  1  Pe2, PAM

(6.67)

as long as for in-phase and quadrature channels we have two independent PAM constellations. The neighboring constellation points in PAM are separated by 2d. To derive the expression of average PAM symbol error probability Pe, PAM , we should first determine the probability of error for the following two symbols: (1) the edge symbol having only one neighbor, and (2) the inner symbol, which has two neighbors. The inner symbol error probability can be expressed as Pe  mi   P  n  d   





2d 2 / N 0 ; i  2,3, , L  1

(6.68)

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Advanced Optical Communication Systems and Networks

On the other hand, the outer symbol error probability is just half of the inner symbol error probability. Accordingly, the average symbol error probability of Lary PAM can be expressed as 2  2d 2  L  N0 

Pe, PAM 

 L2  2d 2  2    N0 L  

 2  L  1  2d 2     N0 L  

   

(6.69)

The average symbol energy of PAM constellations is given by: Es , PAM 





1 L 1 2   2i  L  d 2  L2  1 d 2 L i 1 3

(6.70)

By expressing d2 in terms of the average energy, we can write the average symbol error probability of PAM as: Pe, PAM 

2  L  1 L

 6 Es ,PAM  2  L  1 N0 

 L 1  3 Es ,PAM erfc  2    L   L  1 N0

   

(6.71)

By assuming that all L amplitude levels of in-phase or quadrature channels are equally likely, the average symbol energy of M-ary QAM is simply:

Es  2 Es , PAM 





2 2 2 L  1 d 2   M  1 d 2 3 3

(6.72)

Similarly, by expressing d2 in terms of the average energy, we can write the average symbol error probability of QAM as: M 1  3 Es  Pe  1  1  2   M 1 N M 0  

     

2

 Es   M 1 3  erfc   1  1    2  M  1 N 0   M    

(6.73) 2

For sufficiently high SNRs, we can neglect the 2{} term in the previous equation so that we can write that

Pe  2

 Es  3 M 1 erfc    2  M  1 N 0  M  

(6.74)

Although Equations (6.51) and (6.52) allow us to accurately calculate average symbol error probabilities, it is quite challenging to find a closed-form solution for large signal constellations. In these situations, we use the union bound approximation [1, 4, 22, 25], which gives us an expression that is only the function of Euclidean distances among signal constellation points. This

Advanced Detection Schemes

433

approximation is accurate only for sufficiently high SNRs. Let Aik denote the event that the observation vector r is closer to the signal vector sk than to si, when the symbol mi (vector si) is sent, which means that r  sk  r  si . The constellation point si is detected correctly if r  si  r  sk  k  i, which means that the average error probability can be upper bounded as: M  M  Pe  mi sent   P   Aik    P  Aik   k 1, k i  k 1, k i

(6.75)

The probability of event Aik can be evaluated as:

P  Aik   P  r  sk  r  si | mi   P  si  sk + n  n



(6.76)

In other words, P(Aik) corresponds to the probability that the noise vector is closer to the vector si  sk than to the origin. Given the fact that the i-th noise component is a zero-mean Gaussian random variable of variance N0/2, what matters is just a projection n of the noise vector on line containing vector si  sk . This projection has zero-mean and variance equal to N0/2, so that P(Aik) can be expressed as:  2 1 P  Aik   P(n  si  sk / 2)   e v / 2 dv    dik / 2  N 0 dik

 d    ik  2N 0 

 1  d   erfc  ik  2 2 N 0  

(6.77)

   

The error probability related to symbol mi can be now upper bounded as Pe  mi  

 d 1 M erfc  ik  2 N 2 k 1,k i 0 

 , i  1, 2,..., M  

(6.78)

Finally, the average symbol error probability is upper bounded as M

Pe   pi Pe  mi   i 1

M  d 1 M pi  erfc  ik  2 N 2 i 1 k 1,k  i 0 

   

(6.79)

and this inequality is known as union bound. For equiprobable transmission pi=1/M of symbols the union bound becomes: Pe 

1 2M

 d erfc  ik 2 N i 1 k 1, k  i 0  M

M



   

(6.80)

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Advanced Optical Communication Systems and Networks

By defining the minimum distance of constellation as d min  min i , k dik , we can obtain a looser bound as: Pe 

 d M 1 erfc  min 2 N 2 0 

   

(6.81)

For a circularly symmetric signal constellation, we can use the following bound: Pe 

 d 1 M erfc  ik  2 N 2 k 1,k  i 0 

   

(6.82)

Finally, the nearest neighbor approximation is given as

Pe 

 d erfc  min 2 N 2 0 

M dmin

   

(6.83)

is the number of neighbors at distance dmin from an observed where M d min constellation point. Regarding the bit error probability Pb, when the Gray-like mapping rule [25] is used for sufficiently high SNRs, we can use the following approximation: Pb  Pe / log 2 M

(6.84)

To compare various signal constellation sizes in optical domain, we can use optical SNR (OSNR), introduced by Equation (4.89), instead of electrical SNR. It is convenient to introduce OSNR per bit and per a single polarization state, and define it as [35]

OSNRb 

OSNR Rs ,info  SNRb log 2 M 2 Bop

(6.85)

where M is signal constellation size, Rs,info is information symbol rate, and Bop is referent optical bandwidth (it is commonly assumed that Bop=12.5 GHz, which corresponds to a 0.1 nm spectral width around a carrier wavelength of 1,550 nm). The parameter SNRb = Eb/N0 in Equation (6.85) denotes the SNR per bit of information, where Eb is the bit energy and N0 is a power spectral density (PSD) originating from ASE noise. For convenience, it is assumed that photodiode responsivity is 1 A/W so that N0=NASE, where NASE is the PSD of ASE noise a in single polarization state. The OSNR per single polarization is commonly defined as OSNR=ERs/(2NASEBop), where E is symbol energy and Rs is the symbol rate related to the symbol information rate Rs,info as Rs=Rs,info/Rc (Rc is the code rate).

Advanced Detection Schemes

435

6.1.5 Estimation Theory, ML Estimation, and Cramér-Rao Bound So far our focus was on detection of signal in the presence of additive noise, such as ASE or Gaussian noise. In this section, we will pay attention to the estimation of a certain signal parameter, such as phase, frequency offset, and so on, important from the detection point of view. Let the received signal be denoted by r(t) while and transmitted signal is s(t,p), where p is the parameter to be estimated. The received signal in the presence of additive noise n(t) can be written as:

r t   s t, p   n t  , 0  t  T

(6.86)

where T is the observation interval (symbol duration for modulation schemes). The operation of assigning a value pˆ to an unknown parameter p is known as parameter estimation; the value assigned is an estimate, and the algorithm used to perform this operation is an estimator. There are different criteria used to evaluate the estimator, including the minimum mean-square (MMS) estimate, the maximum a posteriori (MAP) estimate, and the maximum likelihood (ML) estimate. We can define the cost function in the MMS estimate (MMSE) as an integral of quadratic form [2] (6.87)

C MMSE       p  pˆ  f P  p | r  dp 2

   2

where = pˆ -p and fP(p|r) is the a posteriori probability density function (PDF) of the random variable P. The observation vector r in Equation (6.87) is defined by its ri components, where the i-th component is determined by: T

ri   r  t   i  t  dt ; i  1, 2, , N

(6.88)

0

{i(t)} denotes the set of orthonormal basis functions. The estimate that minimizes the average cost function given by Equation (6.87) is known as the Bayes MMS estimate [2, 3]. The estimate that maximizes the a posteriori probability density function fP(p|r) is known as the MAP estimate. Finally, the estimate that maximizes the conditional PDF fr(r|p), which is also known as the likelihood function, is the ML estimate. The MAP estimate can be obtained as a special case of the Bayes estimation by using the uniform cost function C()=(1/)[1rect(/)] as follows:

1 E C      C    f P  p | r  dp     E C    

pˆ  / 2











pˆ  / 2

  f P  p | r  dp 

pˆ  / 2  1 1 1   f P  p | r  dp   (1/  ) 1   f P  p | r     f P  p | r    pˆ  / 2  

(6.89)

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Advanced Optical Communication Systems and Networks

where the approximation above is valid for sufficiently small . The quality of an estimate is typically evaluated in terms of an expected mean value E  pˆ  and a variance of the estimation error. We can recognize several cases with respect to the expected value. If the expected value of the estimate equals the true value of the parameter, E  pˆ   p , we say that the estimate is unbiased. If the expected value of the estimate differs from the true parameter by a fixed value b, that is, if E  pˆ   p  b, we say that the estimate has a known bias. Finally, if the expected value of an estimate is different from the true parameter by a variable amount b(p), that is, if E  pˆ   p  b  p  , we say that the estimate has a variable bias. The estimation error  is defined as the difference between the true value p and the estimate pˆ , namely   p  pˆ. A good estimate is one that simultaneously provides a small bias and a small variance of estimation error. The lower bound on the variance of estimation error can be obtained from the Cramér-Rao inequality, [1, 2], which is E

 p  pˆ    2

(6.90)

1   2 log f r  r | p   E  p 2  

where we assumed that second partial derivative of conditional PDF exists and that it is absolutely integrable. Any estimate that satisfies the Cramér-Rao bound defined by the right side of Equation (6.90) is known as an efficient estimate. To explain the ML estimation, we can assume that the transmitted signal s(t,p) can also be presented as a vector s in signal space, with the i-th component given as T

si  p    s  t , p   i  t  dt ; i  1, 2, , N

(6.91)

0

The noise vector n can be presented in a similar fashion  n1  n  T n   2  , ni   n  t   i  t  dt ; i  1, 2, , N   0    nN 

(6.92)

Since the components ni of the noise are zero-mean Gaussian with PSD equal to N0/2, the components of observation vector will also be Gaussian with mean values si(p). The joint conditional PDF for a given parameter p will be:

fr  r | p  

1

 N 0

N

e  N /2

i 1

2

 j  ri  si  p   / N 0

(6.93)

Advanced Detection Schemes

437

The likelihood function can be defined as   r  t  , p   lim

N 

T fr  r | p  2 T 1 r t s t , p dt s 2  t , p  dt       fr  r  N 0 0 N 0 0

(6.94)

The ML estimate is derived by maximizing the likelihood function, which is done by differentiating the likelihood function with respect to the estimate pˆ and setting the result to zero, so it is [3] T

  r  t   s  t , pˆ  dt 0

s  t , pˆ  0 pˆ

(6.95)

The upper condition is known as the likelihood equation. Example: Let us consider the following sinusoidal signal whose amplitude a and frequency  are known, while the phase is unknown: a cos(t+). Based on the likelihood equation (6.95), we obtain that the ML estimate of phase in the presence of AWGN is  T r  t  cos t  dt    ˆ  tan  0T  r t cos t dt      0   1

(6.96)

6.2 COHERENT DETECTION OF OPTICAL SIGNALS

The coherent detection of optical signals offers several important advantages as compared to direct detection, and they are: (1) improved receiver sensitivity, (2) better frequency selectivity, (3) possibility of using modulation formats with constant amplitude (FSK, PSK), (4) employment of tunable optical receivers, and (5) mitigation of both chromatic dispersion and PMD effects. The local laser, also known as the local laser oscillator, is employed within a coherent optical receiver. We can recognize several detection schemes based on the value of the operating frequency of the local laser with the respect to the frequency of the incoming optical signal, and they are: (1) homodyne detection, in which these frequencies are identical; (2) heterodyne detection, in which the frequency difference is larger than the signal symbol rate, so that all related signal processing upon photodetection is performed at suitable intermediate frequency (IF); and (3) intradyne detection, in which the frequency difference is smaller than the symbol rate but higher than zero. Further on, different coherent detection schemes can be classified into the following categories: (1) synchronous (PSK, FSK, ASK) schemes, (2) asynchronous (FSK, ASK) schemes, (3) differential detection (CPFSK, DPSK) schemes, (4) phase diversity reception schemes, (5) polarization diversity reception schemes, and (6) polarization-division multiplexing schemes. Synchronous detection schemes can further be categorized as residual carrier or suppressed carrier ones.

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Advanced Optical Communication Systems and Networks

6.2.1 Coherent Optical Detection Basics

The basic difference between coherent detection and direct detection schemes is in the presence of a local laser oscillator, as illustrated in Figure 6.8. As we mentioned, coherent detection employs a local laser oscillator so that incoming optical signal and local laser output are mixed in the optical domain (a transparent mirror is shown just as an illustrative example, while an optical hybrid is used in practical applications for signal mixing). np

np

Photodetector

Integrator

(a)

Photodetector

Integrator

nLO

(b)

Figure 6.8 Generic block diagrams of: (a) direct detection scheme, and (b) coherent detection scheme.

Since a photodetector generates the photocurrent in proportion to the total input optical power, the output signal at the output of direct detection integrator can be written as s0,  t   0; 0  t  T

(6.97)

s1,  t   RPS ; 0  t  T

where Ps is the input optical signal power, R is the photodiode responsivity introduced by Equation (2.117), and T represents the bit duration. The subscripts 0 and 1 are used to denote the transmitted bits (zero and one). We can express the photocurrent signal in terms of number of electrons, which can be translated to number np of photons per bit, which is np=RPsT/q (q is an electron charge). The selected signal measured by number of photons per bit can be presented as s0  t   0; 0  t  T s1  t  

np T

(6.98)

; 0t T

The output signal of the corresponding coherent detector integrator for homodyne synchronous detection can be expressed as

s0,1  t  



1  2n p  2nLO 2T

 ; 0t T 2

(6.99)

where nLO denotes the average number of the photons per bit coming from a local laser oscillator, while signs ““ and “+” correspond to transmitted zero bit and one bit, respectively. The optimum coherent detection receiver, which minimizes the

Advanced Detection Schemes

439

bit error probability Pb, is the matched filter (or correlation) receiver having impulse response equal h(t)=s1(Tt) s0(Tt), as discussed in Section 6.1.2. np

Photodetector

s1(t) s0(t)

Matched filter

h(t)=s1(T-t)-s0(T-t)

nLO

Figure 6.9 The synchronous matched filter detector configuration.

The general expression for matched filter output signal, applicable to different modulation formats (such as ASK, FSK, PSK) and shot noise dominated scenario can be written as , s0,1  t   2 R PS PLO cos 0,1  t   0,1  t    nsn  t  ; 0  t  T

(6.100)

where PLO denotes the local laser output signal (PLO>>PS), while 0,1 and 0,1 represent frequency and phase corresponding to 0 and 1 bits. The nsn(t) denotes the shot noise process, which is commonly modeled as zero-mean Gaussian process with power spectral density expressed by Equation (4.25). Herewith, we will express the spectral density of the shot noise due to the impact of local laser oscillator as N 0,  2 RqPLO . Since the bit energy can be expressed as T

, ,2 E ,  E0,1   s0,1  t  dt  2 R 2 PS PLOT , the corresponding signal-to-noise ratio becomes 0

E , 2 R 2 PS PLOT RPS T    np N 0, 2 RqPLO q

(6.101)

Therefore, the SNR in coherent detection scheme equals the number of photons per bit. We can use Equation (6.101) and apply to general expression given by Equation (6.4), so we have that s0,1  t  

2n p T

cos 0,1  t   0,1  t   ; 0  t  T

(6.102)

The correlation coefficient  and the Euclidean distance d between transmitted signals (s0 and s1) are defined as



T

1 s0  t  s1  t  dt E0 E1 0

d 2  E0  E1  2  E0 E1

(6.103)

where Ei is the energy of the i-th (i=0,1) bit. The error probability is related to the Euclidean distance as follows

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Advanced Optical Communication Systems and Networks

 d 1 Pb  erfc  2 N 2 0 

   

(6.104)

Let us now consider ASK, FSK, and PSK modulation schemes and determine the error probabilities in each particular case. For ASK systems the transmitted signals can be expressed in terms of np as s1  t  

2n p T

cos 1t  ; s0  t   0; 0  t  T

(6.105)

By using Equation (6.105), the Euclidean distance squared, defined by Equation (6.21), is found to be d2=np, so that corresponding expression for the bit error probability (or BER) can be obtained as  np  1  Pb  erfc   2  2  

(6.106)

The BER of 10-9 is achieved for 72 photons per bit. For continuous phase FSK (CPFSK) systems, the transmitted symbols can be represented by s1  t  

2n p

s0  t  

2n p

T T

cos 1t  ; 0  t  T

(6.107)

cos 0t  ; 0  t  T

The corresponding correlation coefficient can be expressed as



T   0 2 sin 2 m cos 0t cos 1tdt  , m 1  T 0 2 m 2 / T

(6.108)

where m is the modulation index. The corresponding BER now becomes Pb 

 n p  sin 2 m   1 erfc  1   2 2 m    2 

(6.109)

For m=0.5p (p=1,2,...) the number of required photons per bit to achieve BER of 10-9 is 36.The minimum number np=29.6 is obtained for m = 0.715. For direct modulation PSK (DM-PSK) systems, the transmitted symbols can be represented by

Advanced Detection Schemes

 m t ; 0  t  T /(2m)  cos IF t  1  t   ; 0  t  T ; 1  t    T T   / 2; T /(2m)  t  T

s1  t  

2n p

s0  t  

2n p

441

(6.110)

  m t ; 0  t  T /(2m)  cos IF t  0  t   ; 0  t  T ; 0  t    T T   / 2; T /(2m)  t  T

where IF=|SLO| is the intermediate frequency. The correlation coefficient is obtained by   1/(2 m )  1 , while corresponding error probability can be expressed as  1 1   Pb  erfc  n p 1   2  4m   

(6.111)

For m=1/2 the required number of photons per bit to achieve BER of 10–9 is 36 (the same as for FSK systems), while the minimum np=29.6 is obtained when m  . For FSK systems that employ two oscillators, the transmitted signals are represented by s1  t  

2n p

s0  t  

2n p

where

T T

cos 1t    , 0  t  T

(6.112)

cos 0t    , 0  t  T     m  ; 2 p  m  2 p  1 p  0, 1, 2, ...   m  ; 2 p  1  m  2  p  1

 

(We use [x] to denote the mod  operation.) The correlation coefficient is obtained as    sin  2 m     sin   /  2 m  , while the error probability can be expressed as  n  sin  m   1 Pb  erfc  p 1   2 m   2  

(6.113)

The BER of 10–9 is achieved for np=36/(1+ sin m /m). For instance, it is np=18 when m  0, while np = 36 when m = 1, 2, 3... . The comparison of different FSK modulation formats in terms of required np to achieve BER of 10–9 is shown in Figure 6.10, where the modulation index m is used as an argument.

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Advanced Optical Communication Systems and Networks

Photon number per bit, np

45

CPFSK

40 35

30 FSK 25

20 15 0.0

DM - PSK

0.5

1.0

1.5

2.0

Modulation index, m

Figure 6.10 Comparison of FSK systems.

For PSK systems the transmitted signal can be written by s1  t  

2n p

s0  t  

2n p

T T

cos 1t  ; 0  t  T

(6.114)

cos 1t    ; 0  t  T

The corresponding error probability can be expressed as Pb 

 n  1 erfc  p   2  2  

(6.115)

The BER of 10–9 is achieved for np = 18 in case of heterodyne detection, and np = 9 in case of homodyne detection. The BER curves for different modulation schemes discussed above are plotted in Figure 6.11. We can see that PSK scheme combined with synchronous coherent detection provides the best performance. The comparison of different modulation formats in terms of receiver sensitivity (defined here as required number of photons per bit np to achieve BER of 10–9) is summarized in Table 6. 1.

Bit-error rate, BER

Advanced Detection Schemes 10

-1

10

-3

10

-5

10

-7

10

-9

10

-11

10

-13

10

-15

443

Homodyne PSK Homodyne ASK Homodyne FSK Asynchronous FSK (one filter) Asynchronous FSK (two filters)

2

4

6

8

10

12

14

16

18

20

Number of photons per bit, np [ d B ]

Figure 6.11 BER performance comparison. Table 6. 1 Coherent Detection Receiver Sensitivities in Terms of Required np to Achieve BER of 10–9 Coherent Coherent System Heterodyne Homodyne Super quantum limit 5 ASK Matched filter 72 36 Asynchronous 76 CPFSK m=0.5,1,1.5,... 36 matched filter m=0.715 29.6 FSK m=0 18 9 -two oscillators m=1,2,... 36 -matched filter m=1.43 29.6 FSK m=1,2,... 40 Asynchronous m=1.5 47 DM-PSK m=0.5 36 18 Matched filter m=1 24 12 m=2 20.6 10.3 DM-DPSK m=0.5 61.9 30.9 delay-detection m=1 30.6 15.3 m=2 24.4 12.2 PSK 18 9 Matched filter DPSK 20 10 DPSK IF=1/(2T) 24.7

6.2.2 Optical Hybrids and Balanced Coherent Receivers

So far we considered that a semitransparent optical mirror is used to perform optical mixing before photodection takes place. In practice this operation is performed by a four-port device known as an optical hybrid, which is shown in Figure 6.12. An optical hybrid is an optical coupler that was presented in Figure 2.33 with characteristics described by Equations (2.119) to (2.121). In this case, we assumed that there is control voltage to introduce the phase shift through the phase trimmer.

444

Advanced Optical Communication Systems and Networks Input 1 Output 2 Output 1 Control voltage

Phase trimmer

Input 2

Figure 6.12 Optical hybrid.

Electrical fields at output ports E1o and E2o are related to the electrical fields at input ports E1in and E2in as follows: E1o   E1i  E2i  1  k

(6.116)

E2 o   E1i  E2i exp   j   k

where k is the power splitting ratio, and  is the phase shift introduced by the phase trimmer. Equation (6.116) can also be written in terms of scattering matrix (S) [8] as follows: E  E  s Eo   o1   S  o1   SEi , S   11  s21  Eo 2   Eo 2 

s12   1  k  s22   k

1 k   e k 

(6.117)

 j

In Equations (6.116) and (6.117) we assumed that the hybrid is lossless device, which leads to the following: s11=|s11|, s12=|s12|, s21=|s21|, and s22=|s22|exp(jξ22). Commonly used hybrids in coherent detection schemes are known as -hybrid (designed with ξ=) and /2-hybrid (designed with ξ=/2). The S-matrix of a hybrid can be written as  1 k S  k

1 k    k 

(6.118)

while the S-matrix of a /2-hybrid can be written as

s S   11  s21

s12 s22 e

 j / 2

  

(6.119)

It is easy to conclude that the -hybrid is just the 3-dB coupler (k=1/2). On the other side, if the /2-hybrid is symmetric with sij  1/ L  i, j  , the phase difference between the input electrical fields E1i  E1i , E2i  E2i e ji can be chosen in such a way that the total output ports’ power is maximized. The total output power can be expressed as

Advanced Detection Schemes

E0† E0 

445

2 P1i  P2i  P1i P2i  cos i  sin i   L

(6.120)

where we used † to denote Hermitian transposition (simultaneous transposition and complex conjugation). For equal input powers, the maximum output is obtained for i=/4, which leads to L  2  2 . The corresponding loss is 10log10 ( L / 2)  2.32 dB . The S-matrix can be now expressed as: S

1  1 k  L  k

1 k    j k 

(6.121)

and a parameter k can be optimized with respect to a specific detection scenario, such as a homodyne detection with a Costas loop [6, 8]. The balanced receiver, such as one shown in Figure 6.13 is commonly used to reduce the impact of the laser relative intensity noise (RIN) as well as the impact of crosstalk interferences in multichannel transmission systems. The photocurrents from upper and lower photodetectors in Figure 6.13 can be, respectively, written as 2

1 R( PS  PLO  2 PS PLO cos S )  n1 (t ) 2 1  R( PS  PLO  2 PS PLO cos S )  n2 (t ) 2

i1 (t )  R E1  i2 (t )  R E2

2

(6.122)

where s is the phase of incoming optical signal, and ni(t) (i=1,2) is the i-th photodetector shot noise process with power spectral density (PSD) equal 2 S n  qR Ei . The resultant output current can be written as i

i  t   i1  t   i2  t   2 R PS PLO cos S  n  t 

(6.123)

where n(t)=n1(t)n2(t) is a zero-mean Gaussian shot noise process of PSD S n  S n  Sn  qR( PS  PLO )  qRPLO . 1

2

E1

ES

i1 i=i1–i2

 hybrid ELO

Figure 6.13 Balanced detector.

E2

i2

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Advanced Optical Communication Systems and Networks

6.2.3 Phase, Polarization, and Intensity Noise Sources in a Coherent Optical Detector

In this section we will outline the noise processes due to phase, polarization, and intensity variations in coherent detector; a more detailed description of other optical relevant channel impairments was provided in Chapter 4. Noise due to spontaneous emission causes phase fluctuations in lasers, leading to a nonzero spectral linewidth Δν—please refer to Section 4.1.3. In semiconductor lasers, the linewidth Δν is dependent on the cavity length and the value of the linewidth enhancement factor, and can vary from 100 KHz up to the megahertz range. PSDx()

2Ps /

Ps /



0



Figure 6.14 Lorentzian spectrum of a laser diode.

Herewith we can assume that output from the semiconductor laser can be expressed as x t  

PS e

j 0 t  n  t  

(6.124)

where n(t) is the laser phase noise process, analyzed in Section 4.1.3. We can also assume that corresponding PSD of x(t) can be expressed as Lorentzian curve. For this purpose, we can express Equation (4.11) in a simplified form as 2 2 PS     0   PSDx    1   2         

1

(6.125)

which is shown in Figure 6.14. The laser phase noise, expressed through the spectral linewidth, will cause BER degradation, so the curves from Figure 6.11 (obtained for the case in which phase noise impact was neglected) will get modified. As an example, the impact of the laser phase noise to BPSK performance, calculated by using Equation (4.87), is illustrated in Figure 6.15. As we can see, the effect of phase noise on BER curves is twofold: (1) the BER curves are shifted to the right, and (2) BER floor appears. In design of advanced coherent detection schemes, the impact of the laser phase noise is mitigated by digital signal processing that takes place after

Advanced Detection Schemes

447

photodetection and it does impose a serious limitation if spectral linewidth is measured by hundreds of kilohertz. However, for larger multilevel modulation schemes, such as M-ary QAM, the laser phase noise can cause larger performance degradation. -2

10

-4

  =20

-6

10

( se ca al Ide

-8

10

-10

10

Real case

= 

-12

10

0)

Bit-error rate, BER

10

o

16

o

14

o

12

o

10

o

-14

10

Power penalty

-16

10

=0

-18

10

o

BER floor

-20

10

6

8

10

12

14

16

18

20

22

24

Number of photons per bit, np [ d B ]

Figure 6.15 The laser phase noise influence on BER performance of homodyne binary PSK signaling.

Polarization noise, which comes from discrepancies between the state of polarization (SOP) of incoming optical signal and local laser oscillator, is another factor that causes performance degradation in coherent receiver schemes. To mitigate its impact, several avoidance techniques have been considered. These techniques can be classified as follows: (1) polarization control, (2) employment of polarization maintaining fibers, (3) polarization scrambling, (4) polarization diversity, and (5) polarization division multiplexing. The employment of some of these methods will be discussed throughout this chapter. The polarized electromagnetic field launched into the fiber can be represented as e  t  E  t    x  e jc t ey  t  

(6.126)

where ex and ey represent two orthogonal SOP components, and c is the optical carrier frequency. The received field can be represented by e  t   E S  t   H'  x  e jct  ex  t  

(6.127)

where H is the Jones matrix of birefringence, introduced by Equation (3.89). To match the SOPs of local laser with that of incoming optical signal, an additional transformation is needed, and it can be represented with transformation matrix H, so we have that

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Advanced Optical Communication Systems and Networks

e  t   e  t   E S'  t   H''H'  x  e jct  H  x  e jct , H  H''H' ey  t   ey  t  

(6.128)

The SOP of local laser oscillator can be represented by the Stokes coordinates as SLO=(S1,LO S2,LO S3,LO) (please refer to Section 10.13). The full heterodyne mix is possible only if SLO= SR, where SR is the SOP of the received optical signal. The action of birefringence corresponds to rotating the vector, which represents the launched SOP, on the surface of the Poincaré sphere. This rotation can also be represented in terms of rotation matrices Ri (i=1,2,3) for angles i around axes si. These matrices can be represented as [60]  cos 1 / 2   j sin 1 / 2   cos  2 / 2   sin  2 / 2   R1 1     , R2  2       j sin 1 / 2  cos 1 / 2    sin  2 / 2  cos  2 / 2   (6.129) e  j3 / 2 R3  3     0

0   e j3 / 2 

If we assume that the SOP of local laser oscillator is aligned with the s1-axis (whose Jones vector is given by ELO=[ex(t) 0]T) in the Stokes space, we can establish the relationship between received and local laser SOPs in terms of spherical coordinates (2ξ,2) (see Figure 10.10) as:  ex'  t   cos    sin    e  j  '   ey  t    sin   cos     0   

0  ex  t     e j   0   

(6.130)

E LO

ES

e  j cos       j  ex  t   e sin   

The ratio between the power of heterodyned component (aligned with local laser) and the total power is given by e  j cos   P 1 2  cos   1  cos  2   p( ,  )  het  Ptot e  j cos   2  e j sin   2 2 2

(6.131)

The probability density function of     S R , S LO  is given as [8]: sin  2   4A 2 1cos 2    A2 e 1  2 1  cos  2    2  4  2

PDF ( ) 

(6.132)

  [0,  ]; A  2 R PS PLO ,  2  qRPLO

As for the intensity noise, its characteristics were analyzed in Section 4.1.3. The intensity noise can be characterized by RIN and rint parameters introduced by Equations (4.15) and (4.16), while the impact of the intensity noise to SNR and

Advanced Detection Schemes

449

receiver sensitivity is analyzed in Section 4.2.2.3—please see Equations (4.84) to (4.86). However, in balanced detection receivers, the impact of intensity noise presented by term 2(RPLO)2RINLO in the denominator in Equation (4.84) will be suppressed by factor of more than 15 to 20 dB due to mutual branch cancellation [5] and can be neglected, while contribution from the thermal noise, presented by Equation (4.82) will be doubled. 6.2.4 Homodyne Coherent Detection

Homodyne coherent detection receiver can be designed in either residual carrier or suppressed carrier receiver versions [6, 8]. In scheme with residual carrier, shown in Figure 6.16, the phase deviation between the mark and space bits is less then /2 rad, so that the part of transmitted signal power is used for the nonmodulated carrier transmission, and as consequence some degradation in receiver sensitivity will occur. This situation resembles nonideal extinction ratio in receivers with direct detection, and power penalty can be estimated by using Equation (4.121). Received optical signal

 hybrid

PD 1

+

PD 2

-

Local laser

LPF

Loop filter

Data detector

Data output

PD: photodetector LPF: lowpass filter

Figure 6.16 Balanced loop-based receiver.

The Costas loop and decision driven loop (DDL) [6] based receivers, shown in Figure 6.17, are two alternatives to the receivers with a residual carrier. Both these alternatives employ a fully suppressed carrier transmission, in which the entire transmitted power is used for data transmission. However, at the receiver side, a part of the power is used for the carrier extraction, so some power penalty is incurred with this approach, too. 6.2.5 Phase Diversity Homodyne Receivers

The general architecture of a multiport homodyne receiver is shown in Figure 6.18(a). The electrical fields of incoming optical signal and local laser output signal can be written as [8]

S  t   aEs e

j c t S  t   

L  t   ELO e

j c t LO  t    

(6.133)

where the information is imposed either in amplitude a or phase s. Both the incoming optical signal S and the local laser output signal L are used as inputs to

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Advanced Optical Communication Systems and Networks

N output ports of an optical hybrid, which introduces fixed phase difference k(2/N) (k=0,1,…,N1) between the ports, so that the output electrical fields can be written as Ek  t   N 1/ 2 [ S (t )e

jk

2 N

(6.134)

 L(t )]

Received optical signal PD 1

Data detector

LPF

/2 hybrid PD 2

Delay line

Local laser

Loop filter

Data output

PD: photodetector LPF: lowpass filter

(a) Received optical signal PD 1

+

PD 2

-

PD 3

+

PD 4

-

LPF

Data detector

Data output

Optical hybrid

Local laser

LPF

Loop filter

(b) Figure 6.17 (a) Decision-driven loop-based receiver, and (b) Costas loop-based receiver.

The corresponding photodetector outputs are as follows:

ik  t   R Ek  t   ink  t  

R N

2      PLO  aPS  2a PS PLO cos S  t   LO  t   k   ink  t  N    

(6.135)

where ink(t) is the k-th photodetector shot noise. The outputs are processed by lowpass filters (LPF) and demodulators, which can be realized in different versions. Different versions of demodulators for ASK, DPSK, and DPFSK are shown in Figure 6.18(b). For an ASK scheme we have simply to square photodetector outputs and add them together: N

y   ik2 k 1

while for other schemes the signal is multiplied by its delayed version.

(6.136)

Advanced Detection Schemes S Optical N-port hybrid L

. . . . . .

E1 Ek EN

PD

LPF

Demodulator

PD

LPF

Demodulator

LPF

Demodulator

PD

451

LPF

(a) ASK Squarer DPSK Squarer

Demodulator

Delay, T CPFSK Squarer Delay, 

(b) Figure 6.18 (a) Phase diversity receivers, and (b) demodulator configurations.

6.2.6 Polarization Control and Polarization Diversity in Coherent Receivers

The coherent receivers require matching the SOP of the local laser with that of the received optical signal. In practice, only the SOP of local laser can be controlled; one possible configuration of polarization control receiver is shown in Figure 6.19. Polarization controller from Figure 6.19 is commonly implemented by using four squeezers [8]. Coherent transmitter Local laser

Data

Polarization transformer Receiver

Polarization transformer

Controller

Figure 6.19 Polarization control receiver configuration.

Receiver resilience with respect to polarization fluctuations can be improved if the receiver derives two demodulated signals from two orthogonal polarizations of the received signal, as illustrated in Figure 6.20. This scheme is known as the polarization diversity receiver. In polarization diversity receivers, however, only one polarization is effectively used, which means that spectral efficiency is the same as in receivers without polarization diversity. To double the spectral efficiency of polarization diversity schemes, the polarization multiplexing can be applied [10, 11]. Received signal 3-dB splitter LO

PBS

Photodetector Photodetector

Used for balanced detection

Figure 6.20 Polarization diversity receiver configuration. PBS: polarization beam splitter.

Decision circuit

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Advanced Optical Communication Systems and Networks

6.2.7 Polarization-Division Multiplexing (PDM) and Coded Modulation

In polarization-division multiplexing [10, 11] both polarizations carry independent multilevel modulated streams, which is illustrated in Figure 6.21, which increases overall spectral efficiency. Source channels 1 . . . m

I/Q modulator LDPC encoder 1 R1=k1/n …

Ix

. . .

LDPC encoder m Rm=km/n

m Block interleaver m n

. . . m

LDPC encoder 1 R1=k1/n …

MZM

/2

Qx

DFB 1

MZM

2D mapper (x-pol.)

to SMF

PBS

PBS Iy

. . .

LDPC encoder m Rm=km/n

Block interleaver m n

m

MZM

2D mapper (y-pol.)

MZM

/2

Qy

(a) Extrinsic LLRs (x-pol.) Balanced coherent detector (x-pol.)

From

PBS

rx(Q)

Chromatic

(x-pol.)

dispersion,

Local DFB laser Balanced coherent detector (y-pol.)

APP demapper

ry(I)

PMD and PDL compensation

APP demapper (y-pol.)

ry(Q)

Bit LLRs calculation (y-pol.)

SMF

PBS

Bit LLRs calculation (x-pol.)

rx(I)

LDPC decoder 1 .. .

1

LDPC decoder m

m

LDPC decoder 1 .. .

1

LDPC decoder m

m

Extrinsic LLRs (y-pol.)

(b) Balanced coherent detector From fiber

vI

/2

From local laser

vQ

(c) vI

From

LPF

SMF

cos(IF t) -/2

From local

vQ

oscillator laser

LPF

(d) Figure 6.21 The scheme of the transmission system based on a polarization-division multiplexed scheme: (a) transmitter configuration, (b) receiver configuration, (c) balanced coherent detector architecture, and (d) single balanced detector based on a heterodyne design. DFB: distributed feedback laser, PBS/C: polarization beam splitter/combiner, MZM: Mach-Zehnder modulator, LDPC: lowdensity parity-check, LPF: low-pass filter.

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The M-ary PSK, M-ary QAM, and M-ary DPSK all achieve the transmission of log2M=m bits per symbol. In coherent detection, the data phasor l{0,2/M,..,2(M1)/M} is sent at each l-th transmission interval. We should also outline that in direct detection, the modulation is differential since the data phasor l=l1+l is sent instead, where l {0,2/M,.., 2(M1)/M} value is determined by the sequence of log2M input bits using an appropriate mapping rule. Let us now introduce the transmitter architecture that is optimized to work in combination with polarization division multiplexing and a coherent detection scheme, which means that it includes the employment of forward error correction (FEC) codes, such as low-density parity-check (LDPC) codes, that will be discussed in Chapter 7. If component LDPC codes are of different code rates but of the same length, the corresponding scheme is commonly referred to as multilevel coding (MLC)—please refer to Section 7.9.1. If all component codes are of the same code rate, the corresponding scheme is referred to as the blockinterleaved coded-modulation (BICM)—please refer to Section 7.9.1. The use of MLC allows us to adapt the code rates to the constellation mapper and channel. Let us now explain the configuration of RF transmitter portion corresponding to the x-polarization. In MLC, the bit streams originating from m different information sources are encoded using different (n,ki) FEC codes of code rate ri=ki/n, where ki denotes the number of information bits of the i-th (i=1,2,…,m) component FEC code, and n denotes the codeword length, which is the same for all FEC codes. The mapper accepts m bits, c=(c1,c2,..,cm), at time instance i from the (mn) interleaver column-wise and determines the corresponding M-ary (M=2m) constellation point si=(Ii,Qi)=|si|exp(ji), as shown in Figure 6.21(a). It is evident from Figure 6.21 that two I/Q modulators (with the configuration shown in Figure 5.7) and two 2-D mappers are needed, one for each polarization. The outputs of I/Q modulators are combined using the polarization beam combiner (PBC). The same DFB laser is used as a CW source, with x- and y-polarization being separated by a polarization beam splitter (PBS). The corresponding coherent detector receiver architecture is shown in Figure 6.21(b), while the balanced coherent detector architecture is shown in Figure 6.21(c). The balanced outputs of I- and Q-channel branches for x-polarization at the time instance i can be written as

vI ,i  R Si x

x

 x

 x

vQ ,i  R Si

 L  sin 

  L  cos IF t  i   S , PN  L , PN x

x

x

x

x

 x

 x

 x

IF

t  i  S , PN  L , PN

 

(6.137)

where R is photodiode responsivity, while S,PN and L,PN represent the laser phase noise of transmitting and local laser oscillator, respectively. The IF  S  L denotes the intermediate frequency, and Si x  and L x  represent the incoming

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signal in x-polarization and x-polarization output from local laser oscillator, respectively. Similar expressions hold for y-polarization. The heterodyne receiver can also be implemented based only on a single balanced detector as shown in Figure 6.21(d). For an ASE noise-dominated scenario both heterodyne schemes perform comparably. However, in a shot-noise dominated scenario the heterodyne design with a single balanced detector performs better. In homodyne detection it is set that IF  0. As we can see, this advanced receiver design contains dispersion compensation block, where the chromatic dispersion, PMD, PDL, and other channel impairments are compensated for. This process is known as channel equalization and will be discussed in detail in next section. 6.3 OPTICAL CHANNEL EQUALIZATION 6.3.1 ISI-Free Optical Transmission and Partial-Response Signaling

We will start with the equivalent optical channel model by observing only one polarization state (either x- or y-polarization state), which is shown in Figure 6.22. The function hT(t) denotes an equivalent impulse response of a transmitter obtained as the convolution of pulse shaper impulse response, driver amplifier, and Mach-Zehnder Modulator (MZM) impulse responses, while hc(t) denotes the impulse response of the optical channel. The hR(t) denotes the equivalent impulse response obtained as convolution of impulse responses of photodetector, front end with transimpedance amplifier, and both optical and electrical filters that are employed. The electrical filter can be replaced by matched filter. Finally, n(t) denotes the equivalent noise process dominated by ASE noise. The equivalent impulse response of this system is given by convolution h  t    hT  hc  hR  t  , so that the output of the system can be written as: c  t    bk h  t  kTs 

(6.138)

k

where bk is the transmitted sequence of symbols. By sampling the system output we obtain that c  mTs    bk h  mTs  kTs   bm h  0    bk h  mTs  kTs  k

 bm h  0    bk h  kTs  0 k

k m

(6.139)

ISI term

where the term bmh(0) corresponds to the transmitted symbol an the second term represents the intersymbol interference (ISI) term. The second term can disappear

Advanced Detection Schemes

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if ISI-free optical transmission is achieved, which is possible if the equivalent impulse response satisfies the following condition:

 h  1, k  0 h  kTs    0  0, k  0

(6.140) n(t)

a(t)

Transmitter filter hT(t)

a  t    bk   t  kTs 

b(t)

Receiver filter hR(t)

hc(t)

b  t    bk hT  t  kTs 

k

c(t)

c  t    bk  hT  hc  hR  t  kTs 

k

k

Figure 6.22 Equivalent low-pass optical channel model (single polarization is observed).

If the frequency response of equivalent system function H(f)=FT[h(t)], where FT stands for Fourier transform, satisfies the following condition 



k 



 H f 

k    Ts Ts 

(6.141)

then there will not be ISI at sampling time instances. This claim is known as the Nyquist criterion for zero-ISI, and can be used in design of all channel equalizers. This claim is quite straightforward to prove. Let us start with inverse Fourier transform definition and let us split the integration interval into frequency bands of bandwidth 1/Ts, so we have that h  t   FT

1

H  f  



 H  f e

j 2 ft





 2 k 1

2Ts

j 2 ft df  H  f e

df  

k   2 k 1

(6.142)

2 Ts

By sampling the equivalent impulse response from Equation (6.142), we obtain that  2 k 1



h  mTs   

k  

2 Ts

H  f  e j 2 fmTs df  2 k 1 

(6.143)

2 Ts

Now, by substituting f’=fk/Ts, Equation (6.143) becomes 

h  mTs   

1

1

2 Ts

1

s

2Ts

  Ts e j 2 f mTs df   1

2 Ts



 j 2 f mT0 df     H  f   k Ts  e j 2 f mTs df   H  f   k Ts  e

k  1 2T 1

Ts    2Ts



sin  m  m

2 Ts

k 

1 m  0  0 m  0

(6.144)

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We can see from Equation (6.144) that the ISI term disappeared, which proves that the Nyquist criterion indeed yields to zero-ISI among symbols. The channel with a frequency response satisfying Equation (6.141) with a single term is known as the ideal Nyquist channel. The transfer function of the ideal Nyquist channel has a rectangular form given as H(f)=(1/Rs)rect(f/Rs), while the corresponding impulse response is given as

h t  

sin  t / Ts 

 t / Ts

 sinc  t / Ts 

(6.145)

The ideal rectangular frequency response has sharp transition edges, which is difficult to implement in practice. The frequency response (spectrum) satisfying Equation (6.141) with additional terms (k = 1, 0, 1) is known as raised cosine (RC) spectrum given by:   H RC  f    Ts 2 

Ts   Rs     1  sin  Ts  f      2     

0  f  1    2Ts

(6.146)

1    2Ts  f  1    2Ts

The impulse response of a raised cosine spectrum is given as

hRC  t  

sin  t / Ts  cos   t / Ts 

 t / Ts

1  4 2t 2 / Ts2

(6.147)

The parameter  from Equations (6.146) and (6.147) is known as a rolloff factor and determines the bandwidth required to accommodate the raised cosine spectrum, which is given by B=(1+)/(2Ts). The raised cosine spectrum reduces to the ideal Nyquist channel if =0. It is evident from the raised cosine filter impulse response that in addition to zero-crossings at Ts, 2Ts, …, there are additional zero-crossings at points 3Ts/2, 5Ts/2, …. In Section 6.1 we studied different approaches to minimize the symbol error probability and learned that employment of the matched filter is optimum for ASE noise dominated channels. However, in this section we learned that for ISI-free transmission, the receive filter should be chosen in such a way that the overall system function has the raised cosine form. By combining these two approaches, we can come up with a near-optimum transmit and receive filters. Namely, from zero-ISI condition we require that:

h  t    hT  hc  hR   t   hRC  t 

(6.148)

or in the frequency domain:

H RC  HT H c H R

(6.149)

Advanced Detection Schemes

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The condition for the matching filter can be written as:

H R  f   HT  f  H c  f 

(6.150)

1 0.8 0.6 0.4 

h(t)



0.2 0



-0.2 -0.4 -4

-3

-2

-1

0

1

2

3

4

t / Ts

(a) 1 0.9 

0.8 0.7





0.6 H ( f ) 0.5 0.4 0.3 0.2 0.1 0 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

f Ts

(b) Figure 6.23 Raised cosine function: (a) impulse response, and (b) frequency response.

When these two criteria, given by Equations (6.148) and (6.149), are combined, we obtain that

HT 

H RC , Hc

H R  H RC

(6.151)

This still represents an approximate solution, as two problems get solved independently and then get combined. In our discussion above, we considered the ISI as detrimental effect. However, by inserting the ISI in controllable manner, it is possible to transmit signal at a Nyquist rate of 2B, where B is the bandwidth of the channel. The systems using this strategy are known as partial-response (PR) signaling schemes,

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Advanced Optical Communication Systems and Networks

and they are widely used in magnetic recording and various digital communications [1, 21–23]. The impulse response of PR systems is given as N 1

h  t    wn sinc  t / Tb  n 

(6.152)

n 0

The common PR signaling schemes are listed in Table 6.2. Table 6.2 Common Partial-Response Signaling Schemes. Type class

of

N

w

w

w

w

w

0

1

2

3

4

I

2

1

1

II

3

1

2

1

III

3

2

1

–1

IV

3

1

0

–1

V

5

–1

0

2

Class is also known as:

Duobinary

Modified duobinary 0

–1

For instance, the scheme with N=2 is known as a duobinary scheme, while the scheme with N=3 and weight coefficients 1, 0, 1 is known as modified duobinary scheme. 6.3.2 Zero-Forcing Equalizers

To compensate for linear distortion effects, such as chromatic dispersion, we may use the circuit known as an equalizer, connected in cascade with the system in question, as shown in Figure 6.24(a). The overall frequency response is equal to Hc()·Heq(), where Hc() is the system transfer function, while Heq() is the equalizer transfer function. We assume that balanced coherent detection is used. For distortionless transmission we require that

H c   H eq    e  jt0

(6.153)

so that the frequency response of the equalizer is inversely proportional to the system transfer function, which is

Advanced Detection Schemes

H eq   

459

e  jt0 H c  

(6.154)

A functional block that is well suited for equalization in general, and for chromatic dispersion compensation in particular, is tapped-delay-line (transversal) equalizer, or the finite impulse response (FIR) filter, which is shown in Figure 6.24(b). The total number of filter taps is equal to N+1. As we saw in Chapter 3, the impact of chromatic dispersion can be described by the transfer function H c    exp  j   2 2 / 2  3 3 / 6  L 

(6.155)

where 2 and 3 are group-velocity dispersion (GVD) and second-order GVD parameters, introduced by Equation (3.121) and L is the fiber length. Accordingly, we have to design the FIR equalizer with the frequency response given by Equation (6.155). Hc()

Heq()

Dispersive channel

x[n]

z-1 h[0]

Equalizer

z-1

z-1

h[1]

h[2]

Distortionless transmission system

h[N]

y[n]

(a)

(b)

Figure 6.24 (a) Equalization principle, and (b) tapped-delay-line equalizer (FIR filter).

x’

hxx

x

hxy hyx y’

hyy

y

Figure 6.25 The compensation of polarization-dependent impairments by FIR filters.

The FIR equalizer design can be done by using some of methods described in [14], which are: (1) the symmetry method, (2) the window method, (3) the frequency-sampling method, and (4) the Chebyshev approximation method. The FIR filters obtained by symmetry method have a linear phase so that their discrete impulse response h[n] and system function H(z) satisfy the following symmetry property [14]:

h  n    h  N  1  n  ; n  0,1, , N  1 H  z   Z h  n    z

 N 1

H  z 1 

(6.156)

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Advanced Optical Communication Systems and Networks

where with Z{} presents the z-transform of impulse response h[n]—please refer to Section 10.10.4 for additional details on system function in the z-domain. Therefore, the roots of H(z) must be the same as the roots of H(z-1), which indicates that the roots of H(z) must occur in reciprocal pairs: zk, zk-1. Moreover, for filter coefficients to be real-valued, the roots must appear in complexconjugate pairs: zk, zk-1, z*k, z*k-1. This method is not quite suitable for chromatic dispersion compensation given the fact that the phase frequency response of single-mode fibers, given by Equation (6.155), contains both quadratic and cube terms. The windowing method starts with a specified filter order N (N is typically an even integer), and for a given sampling interval Ts the following steps are taken: (1) set the constant time delay t0=(N/2)/Ts; (2) take the inverse Fourier transform of Heq() to obtain a desired impulse response heq(t); and (3) set h[n]=w[n]heq[nTs], where w[n] is a window of length (N+1). The simplest windowing function is a rectangular window given as

1, n  0,1,..., M  1 wn    0, otherwise

(6.157)

Therefore, the frequency response of a smoothed equalizer can be obtained as the convolution H     H eq    W    , where  =·Ts is the discrete frequency (expressed in rad/sample). For a large filter order having a rectangular window, the Gibbs phenomenon in H() can be noticed [14], which comes from an abrupt truncation of heq[n]. For better results, the tapered windows can be used. The commonly used tapered windowing functions include: Blackman, Hamming, Hanning, Kaiser, Lanczos, and Tukey windows [14]. For instance, the Hanning window is given by the function

w  n 

1 2 n   1  cos  2 N 1 

(6.158)

Tapered windows reduce the Gibbs ringing, but increase the transition bandwidth (they provide better smoothing, but less sharp transition). The design of an FIR equalizer to compensate for chromatic dispersion is a straightforward process if the windowing method is used. As an example, when the second-order GVD parameter can be neglected, the FIR equalizer coefficients become h  n 

 D 2L    cT 2 2   N  jcT 2 N  n ,     n   , N  2  1 exp   j 2 2 2  D L 2  D L   2   2cT 

(6.159)

Advanced Detection Schemes

461

where D is a dispersion parameter introduced by Equations (3.84) and (3.122), which is related to GVD by D =(2c/2)2. The first-order polarization mode dispersion (PMD) can be compensated by using four FIR filters. The output symbols in x-polarization can be determined by [9] N 1

x  k   hxxT x ' hxyT y '   hxx  n  x '  k  n   hxy  n  y '  k  n 

(6.160)

n0

where hij (i,j{x,y}) are responses of FIR filters, each with N taps. The corresponding equation to determine the symbols in y-polarization can be written in a similar fashion. The frequency sampling method is based on a sampling of the frequency response of the equalizer as follows:

H  k   H  

k

 2 k / N

(6.161)

The function H() can be expressed by a discrete-time Fourier transform as N 1

H      h  n  e  j n

(6.162)

n 0

and, since the frequency response is determined from the z-transform by setting z=exp(j), we can easily determine the system function H(z). This is particularly simple to do when the system function is decomposed as follows: H  z 

H k  1  z  N N 1  j 2 k / N 1 1  N e k 0   z H1  z 

(6.163)

H2  z 

where H1(z) represents the all-zero system or a comb filter, with equally spaced zeros located at on the unit circle (zk=ej2(k+)/N, with  = 0 or 0.5), while H2(z) consists of a parallel bank of single-pole filters with resonant frequencies equal to pk=ej2(k+)/N. The unit-sample impulse response h[n] needs now to be determined from H(z). The main drawback of this method is nonexistence of control of H() at frequencies in between points k. In practice, the optimum equiripple method is used to control H(). Namely, the frequency response H() of actual equalizer is different from desired function Hd(). Consequently, the error function E() can be found as the difference between the desired and actual frequency responses:

E     W     H d     H    

(6.164)

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Advanced Optical Communication Systems and Networks

where W() is known as a weighting function. The key idea behind this method is to determine the h[k] by minimizing the maximum absolute error (the minimax criterion), which can be expressed as

hˆ  n  arg min  max E     over h n   S 

(6.165)

where S denotes the disjoint union of frequency bands over which the optimization is performed. For the linear-phase FIR filter design, it is possible to use the Remez algorithm [14]. On the other side, for problems involving both quadratic and cubic phase terms, the Chebyshev approximation method can be used instead. We can conclude from Equation (6.155) that the chromatic dispersion compensator is essentially an all-pass filter with constant magnitude of the response, |H()|=1, 0. An important property of all-pass filters is that zero and poles are reciprocals of one another: H  z   zN

A  z 1  A z 

N

, A  z    a  k  z  k ( a  0  1)

(6.166)

k 0

with response magnitudes H   2  H  z  H  z 1 

z  e j

 1 . Upon factorization, the

system function of an all-pass filter is given by [14]: 1 1 * z 1   k Nc  z   k  z   k  1  1 * 1 k 1 1   k z k 1 1   k z 1   k z  Nr

H z  

1   k  1,

(6.167)

k  1

where Nr is the number of sections with simple poles, while Nc is the number of sections with complex-conjugate poles. By interpreting the chromatic dispersion compensator design as the all-pass filter design, it is expected that the number of taps will be smaller than that needed in FIR filter design (for which the number of taps is a linear function of fiber length). It is evident from Equations (6.166) and (6.167) that the all-pass filter is essentially infinite impulse response (IIR) filter. Additional details on IIR all-pass filter design can be found in [15, 16]. The function of FIR/IIR equalizers described above is to make the ISI to be zero at all instances t=nTs, except for n=0. Because of that, they are also known as the zero-forcing equalizers. However, the problem in their employment is that the ASE noise, chromatic dispersion, and PMD all act together, affecting the behavior of a transmission system in a combined manner, while transversal equalizers ignore the effect of channel noise. This leads to the noise enhancement phenomenon, which can be explained as follows. Let us again consider the model shown in Figure 6.22, but now with the impulse response of receive filter being

Advanced Detection Schemes

463

decomposed into the matched filter impulse response and the equalizer impulse response:

hR  t   hT  t   heq  t 

(6.168)

The Fourier transform of system output, assuming Heq(f)=FT{heq(t)}=1/Hc(f), is given as C  f    H T  f  H c  f   N  f   H T*  f  H eq  f   HT  f   2

N  f  H T*  f  H f c 

(6.169)

N ' f 

The corresponding power spectral density of a colored Gaussian noise N’(f) is given by: N HT  f  PSDN '  f   0 2 Hc  f  2

2

(6.170)

We can see that the ISI is compensated for, but the noise is enhanced unless we have that |HT(f)|=|Hc(f)|. 6.3.3 Optimum Linear Equalizer

The better approach for the receiver design would be to use the minimum-mean square error (MMSE) criterion to determine the equalizer coefficients, which provides a balanced solution to the problem by reducing the effects of both channel noise and ISI. Let hR(t) be the receiver filter impulse response and x(t) be the channel output determined by

x  t    sk q  t  kTs   w  t  , q  t   hT  t   hc  t 

(6.171)

k

where hT(t) is the transmit filter impulse response, hc(t) is the channel impulse response (due to chromatic dispersion and optical/electrical filters), Ts is the symbol duration, sk is transmitted symbol at the k-th time instance and w(t) is the channel noise dominated by ASE noise. The receive filter output can be determined by the convolution of the receive filter impulse response and the corresponding input as

y t  



 h   x  t   d R



By sampling at t = iTs, we obtain that

(6.172)

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y  iTs   i  wi , i   sk k



wi 

 h   w  iT R

s



 h   q  iT R

s

 kTs    d



(6.173)

   d



The signal error can be defined as a difference between the receive sample and the transmitted symbol, which is

ei  y  iTs   si   i  wi  si .

(6.174)

The corresponding mean-square error is 1 E  ei2  2   1 1 1  E i2   E  wi2   E  si2   E i wi   E  wi si   E i si  2 2 2

MSE 

(6.175)

Assuming the stationary environment, Equation (6.175) can be rewritten as MSE 

    1 1 N     Rq  t     0   t     hR  t  hR   dtd 2 2    2 

(6.176)





 h  t  q  t  dt R



where Rq() is the autocorrelation function of q(t), and N0 is the power spectral density of ASE noise. To determine the optimum filter in the MMSE sense, we have to find the derivative of MSE with respect to hR(t) and set it to zero, which returns the equation 





N   R  t     2   t    h   d  q  t  q

0

R

(6.177)



By applying the Fourier transform (FT) of Equation (6.177) and solving it for receive filter transfer function HR(f)=FT[hR(t)], we obtain that HR  f  

Q * f  1  Q * f  N0 N Sq  f   Sq  f   0 2 2   

(6.178)

H eq  f 

where the impulse response of the equalizer is obtained as heq(t)=FT-1[Heq(f)], followed by the windowing method described above. Therefore, the optimum linear receiver, in the MMSE sense, consists of the cascade connection of the matched filter and the transversal equalizer, as shown in Figure 6.26. If the delay

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T is equal to the symbol duration Ts, the corresponding equalizer is known as the symbol-spaced equalizer. If the symbol rate (Rs=1/Ts) is smaller than 2B (B is the channel bandwidth), the equalizer will need to compensate for both channel distortions and the aliasing effect. However, when T is chosen to satisfy the condition 1/T2B>Rs, the aliasing problem is avoided, and equalizer needs to compensate for channel distortions only. This type of equalizer is known as a fractionally-spaced equalizer, and the most common case is the one for which T=Ts/2. Matched filter

x(t)

Transversal (FIR) equalizer

Delay

T

q(-t) c-N[0]

T

T

T

c-N+1 [0]

cN-1 [0]

y(t)

cN [0]

y(nTs)

Figure 6.26 Optimum linear equalizer in an MMSE sense.

6.3.4 Wiener Filtering

We have already discussed in Section 6.1.5 how to estimate certain signal parameters in the presence of additive noise. In this section, we extend that study to the estimation of a random vector s (of length N) from the received vector r. We are specifically interested in linear (unbiased) minimum error variance estimation [2]. Namely, we would like our estimate to be a linear function of a received vector, which can be expressed as

sˆ  Ar  b

(6.179)

where A and b are the matrix and vector, respectively, that need to be determined. The vector b can be determined from the requirement for estimate to be unbiased, which is

E  sˆ  m s = A E r  b = Amr  b   Es

(6.180)

mr

which gives the value

b = m s - Amr The error made by the unbiased estimator will be then expressed as

(6.181)

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 = sˆ  s =  sˆ - m s    s - m s   A  r  mr    s - m s    

(6.182)

A r  m r 

The matrix A can be determined from condition that the trace of the covariance matrix, defined as

Cov     E   

(6.183)

is minimized. By setting the first derivative of the trace of the covariance matrix with respect to matrix A from Equation (6.183) to be zero, we derive the following orthogonality principle:

E  r - mr      0

(6.184)

The orthogonality principle indicates that the error vector is independent of the data. By substituting Equation (6.182) into Equation (6.184), we obtain the following solution for A:

   E  r  m  r  m  

A  C rsT Cr1 ; C rs  E  r  mr  x  m s  Cr

T

(6.185)

T

r

r

where Crs and Cr are corresponding covariance matrices. Now, by substituting Equations (6.181) and (6.185) into Equation (6.179), we obtain the following form of the estimator:

xˆ  C rsT Cr1  r - mr   ms

(6.186)

which represents the most general form of the Wiener filter. The minimum error variance of this estimator is given by:

E     C s  C sr C r1C srT

(6.187)

The Wiener filtering is also applicable in polarization-division multiplexed (PDM) systems. In PDM, the transmitted vector s can be represented as s=[sx sy]T, where the subscripts x and y are used to denote x- and y-polarization states. In a similar fashion, the received vector r can be represented as r=[rx ry]T. We can notice that the only difference is just in dimensionality of vectors s and r, which have twice more components (2N). The dimensionality of matrix A is also higher, 2N2N. 6.3.5 Adaptive Equalization

So far we have assumed that different channel impairments in optical fiber communication systems are time-invariant, which is not completely true,

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especially with respect to the presence of PMD and polarization-dependent losses (PDL). The adaptive filtering offers an attractive solution to address the channel equalization [1, 3, 14, 22]. An adaptive filter has a set of adjustable filter coefficients, as shown in Figure 6.27, which are adjusted based on the algorithms described below. Widely used adaptive filter algorithms are the steepest descent algorithm and the least-mean-square (LMS) algorithm [3, 14, 22]. These algorithms can be used to determine coefficients of transversal equalizer. According to the steepest descent algorithm, we update the k-th filter coefficient wk, shown in Figure 6.27, by making correction of the present value in a direction opposite to the gradient k (in the direction of the steepest descent on the error-performance surface) [3], so it is

1 wk  n  1  wk  n     k  n  ; k  0,1,..., N 2

(6.188)

where the real-valued parameter  determines the speed of convergence. x[n]

z-1

x[n-1]

w0

z-1 w1

… …

x[n-N+1]

z-1

x[n-N] Variable

wN weights

wN-1

… y[n]

e[n] Error signal

d[n] Desired response

Figure 6.27 Adaptive equalizer.

If the error signal e[n] is defined as difference of desired signal (commonly a training sequence in a back-to-back configuration) and the output y[n] of corresponding FIR filter output as N

e  n   d  n    wk  n  x  n  k 

(6.189)

k 0

the gradient k is determined as k n 

 E e 2  n   2 E e  n  x  n  k   2 REX  k  ; k  0,1,..., N wk  n  

(6.190)

where REX[k] is the cross-correlation function between the error signal and adaptive filter input. By substituting Equation (6.190) into Equation (6.188), we obtain the following form of the steepest descent algorithm: wk  n  1  wk  n   REX  k  ; k  0,1,..., N

(6.191)

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Advanced Optical Communication Systems and Networks

It can be shown that for this algorithm to converge the following condition for the parameter  should be satisfied: (6.192)

0    2 / max

where max is the largest eigenvalue of the correlation matrix RX given as  RX  0  RX 1  R R 1   X  0 RX   X  ... ...  R N R X  N  1  X  

... RX  N    ... RX  N  1  , R  l   E  x  n  x  n  l   X ... ...  ... RX  0  

(6.193)

A drawback of the steepest descent algorithm is that it requires the knowledge of gradient k in each iteration. On the other side, the key idea in LMS algorithm . is to approximate the operator of averaging E[ ] in Equation (6.190) by its instantaneous value e[n]x[nk], so that the updated rule is simply:

wˆ k  n  1  wˆ k  n   e  n x  n  k  ; k  0,1,..., N

(6.194)

6.3.6 Decision Feedback Equalizer

Another interesting equalizer is the decision-feedback equalizer (DFE), which is shown in Figure 6.28. The key idea of decision-feedback equalization is to use decision made on the basis of precursors of the channel impulse response to take care of post-cursors. All this is with an assumption that decisions were correct. Let the channel impulse response in discrete form be denoted by hc[n]. The response of the channel, in absence of noise, to an input sequence x[n] is given by: y  n    hc  k  x  n  k 

(6.195)

k

 hc  0 x  n    hc  k  x  n  k    hc  k  x  n  k  k 0 k 0   precursors

postcursors

The first term on the right side h[0]x[n] represents the desired data symbol; the second term is a function of previous samples only (where the channel coefficients are known as precursors), while the third term is a function of incoming samples, with channel coefficients being known as postcursors. The DFE is composed of a feed-forward equalizer (FFE) section, a feedback equalizer section, and a decision device, as shown in Figure 6.28. The feed-forward and feedback sections can be implemented as FIR filters (transversal equalizers), and can also be adaptive. The input to the detector can be written as

Advanced Detection Schemes N FF

N FB

n 1

n 1

469

z  m   f  n  x  mTs  nT    g  n  bˆ  m  n 

(6.196)

where f[n] and g[n] are values of adjustable taps in feed-forward and feedback sections, respectively. The NFF (NFB) denotes the number of taps in the feedforward (feedback) section. Input sequence

+

Feedforward section

x[n]

Transmitted symbol estimate

z[n]

Decision device

-

bˆ  n 

Feedback section

Figure 6.28 Decision-feedback equalizer (DFE).

x[n] T



T



f [-3]

f [-2]

T



f [-1]



f [0]

Decision device

 Error signal

+

accumulators

Ts

g[3]



Ts

g[2]



Ts

g[1]



Figure 6.29 Adaptive decision-feedback equalizer example.

We can see that the feed-forward section is a fractionally-spaced equalizer, which operates with a rate equal to an integer multiple of the symbol rate (1/T=m/Ts, m is an integer). The taps in feed-forward and feedback sections are typically chosen based on MSE criterion, by employing the LMS algorithm. An example of the adaptive decision feedback equalizer is shown in Figure 6.29. Since the transmitted symbol estimate is dependent on previous decisions, this equalizer is a nonlinear functional element. If previous decisions are in error, the error propagation effect will occur. However, the errors do not persist indefinitely, but rather occur in bursts. 6.3.7 MLSD or Viterbi Equalizer

Another very important equalizer is based on the maximum likelihood sequence detection (estimation) (MLSD or MLSE) [24–27]. Since this method estimates the sequence of transmitted symbols, it avoids the problems of noise enhancement and error propagation. The MLSE chooses an input sequence of transmitted symbols {sk} in a way to maximize the likelihood function of received signal r(t). The

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Advanced Optical Communication Systems and Networks

equivalent system model that illustrates MLSD is shown in Figure 6.30. The received signal r(t) can be expressed in terms of a complete set of orthonormal basis functions n(t) as N

r  t    rn  n  t 

(6.197)

n 1

where rn 





k 

L

sk hnk  vn   sk hnk  vn ; k 0

vn 

hnk 

LTs

 h  t  kT    t  dt s

* n

(6.198)

0

LTs

 n  t    t  dt; h  t   h  t   h  t  * n

T

c

0

The functions hT(t) and hc(t) in Equation (6.198) denote the impulse responses of the transmitter and channel, respectively, N denotes the cardinality of basis set, L is the channel memory, and h(t) is the combined channel impulse response (the convolution of transmit and channel impulse responses). Since vn are Gaussian random variables, the distribution of rN=[r1 r2 … rN]T has a multivariate Gaussian character 2 N  L  1    1 exp   p  r N | sL , h t     rn   sk hnk   k 0 n 1   N 0  N 0   

(6.199)

where N0 is the power spectral density of ASE noise. The MLSE decides in favor of the symbol sequence sL that maximizes the likelihood function given by Equation (6.199), so it is 2 L  N  sˆ L  arg max p  r N | s L , h  t    arg max   rn   sk hnk  sL sL k 0   n 1

(6.200)

N N    *   arg max 2 Re   sk*  rn hnk*    sk sm*  hnk hnm  L s n 1 n 1  k  k m  

n(t) Combined channel d(t)

Transmitter filter hT(t)

d  t    sk   t  kTs 

h(t)=hT(t)hc(t) Optical channel hc(t)

r(t)

Matched filter h(-t)

c(t) MLSD

k

Figure 6.30 Equivalent system model to study MLSD (single polarization case).

Estimated data sequence sL

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471

By using substitution u  t   h  t   h*  t  and noticing that the following is valid: N



n 1



 rn hnk*  N

h n 1

 r   h   kT  d  c  k  *

s

(6.201)



h 

* nk nm

 h   kT  h   mT  d  u  mT *

s

s

s

 kTs   u  k  m 



Equation (6.200) can be simplified to take the form     sˆ L  arg max 2 Re  sk*c  k    sk sm* u  k  m sL  k  k m  

(6.202)

Equation (6.202) can efficiently be calculated by the Viterbi algorithm [1]. However, the Viterbi algorithm provides the hard decisions, and as such is not suitable for use with soft decision decoding schemes. To fully exploit the advantages of soft decoding, the soft reliabilities are needed. These reliabilities can be obtained by soft-output Viterbi algorithm (SOVA) [28], BCJR algorithm [29], or Monte Carlo-based equalization [30]. To further improve the BER performance, we can perform the iteration of extrinsic information between the soft equalizer and soft decoder, the procedure known as the turbo equalization [3133], which will be described in Chapter 7. The turbo equalization scheme can be used to simultaneously compensate for chromatic dispersion, PMD, PDL, and fiber nonlinearities, as shown in [34]. If a Gaussian-like approximation is not valid, which is case with the presence of strong fiber nonlinearities, the conditional PDF p  r N | s L , h  t   should be estimated from histograms, by propagating the sufficiently long training sequence. 6.3.8 Blind Equalization

In an MMSE equalizer, the use of training sequences is needed for initial adjustment of equalizer coefficients. However, the adjustment of equalizer coefficients can be done without using the training sequences, and this approach is known as blind equalization (or self-recovery equalization). All blind equalizers can be classified into three broad categories [1]: (1) steepest descent algorithms based, (2) high-order statistics based, and (3) maximum-likelihood approach based. To facilitate the explanation of blind equalization, let us observe a signal polarization state. The output of optical fiber channel upon balanced coherent detection, assuming only either x- or y-polarization, can be expressed as:

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Advanced Optical Communication Systems and Networks K

r n   h  k  a n  k   z n

(6.203)

k 0

where h[n] represents the optical channel impulse response (in either x- or ypolarization), a[n] is transmitted sequence of symbols, and z[n] are samples of the additive ASE dominated noise. The joint PDF of received sequence r=[r1 r2 … rL]T, under the Gaussian noise assumption and for given impulse response vector h=[h0 h1 … hK]T and transmitted sequence a=[a1 … aL], can be written as  1 L f R  r | h, a    N 0  exp    N0 0 0  a1 a 0  2 a1 A   a3 a2 a1      aL aL 1 aL  2

2 r  Ah    0   0  0       I L  K 

(6.204)

When a training sequence is used and the data vector a is known, the ML estimate of a channel impulse response can be obtained by maximizing the likelihood function (6.204) to obtain:

hML  a    A† A AT r 1

(6.205)

In contrast, when optical channel impulse response is known, we can use the Viterbi equalizer, described in Section 6.3.7, to detect the most likely transmitted sequence a. However, when neither a or h are known, we will need to determine them in such a way that the likelihood function from Equation (6.204) is maximized. As an alternative solution, we can estimate h from fR(r|h), which is obtained by averaging over all possible data sequences (MK in total, where M is the signal constellation size). In other words, the conditional PDF can be obtained as MK

 

f R  r | h    f R  r | h, a  P a  k 1

(6.206)

k

The estimate of h that maximizes this new likelihood function from Equation (6.206) can be obtained from the first derivative of fR(r|h) with respect to h, which is set to zero, to obtain:  i 1

h

1

   A  re

 k  l   k † k  r  A h / N0 k     A  A  e P a    k 

k †

k

k l  r  A  h  / N 0

 

P a

k

(6.207)

As the corresponding equation is transcendental, we use an iterative procedure to determine h recursively. Once h is determined, the transmitted sequence a can be

Advanced Detection Schemes

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estimated by the Viterbi algorithm that maximizes the likelihood function from Equation (6.204), or minimizes the Euclidean distance r  AhML 2 :

aˆ  arg min r  AhML

2

(6.208)

a

It is clear that this approach is computationally extensive. Moreover, since h is estimated from average conditional PDF, its estimate is not going to be that accurate as compared to the case when training sequences are used. The better strategy in this case will be to perform channel and data estimation jointly. In a joint channel and data estimation, the process is done by several stages. In the first stage, the corresponding ML estimate of h is determined for each candidate data sequence a(k), which is

     A  A  

hML a 

k

k †

k

1

A

k T

r ; k  1, , M K

(6.209)

In the second stage, we select the data sequence that minimizes the Euclidean distance for all channel estimates:

 

k k aˆ  arg min r  A  hML a   k a 

2

(6.210)

For efficient implementation of joint channel and data estimation, a generalized Viterbi algorithm (GVA), in which the best B (1) estimates of transmitted sequence are preserved, is proposed in [39]. In this algorithm, the conventional Viterbi algorithm up to the K-th stage is applied, which performs an exhaustive search. After that, only B surviving sequences are retained. This algorithm performs well for B=4 and for medium SNR values, as shown in [39]. In stochastic-gradient blind equalization algorithms, a memoryless nonlinear assumption is typically used. The most popular from this class of algorithms is the Godard algorithm [40], also known as the constant-modulus algorithm (CMA). This algorithm has been used for PMD compensation [9, 17, 19], which is performed in addition to the compensation of the I/Q imbalance and other channel impairments. In conventional adaptive equalization, we use a training sequence as desired sequence, so it is d[n] =a[n]. However, in blind equalization we have to generate a desired sequence d[n] from the observed equalizer output based on a certain nonlinear function:  g  aˆ  n  , memoryless case d  n    g  aˆ  n  , aˆ  n  1 , , aˆ  n  m , memory of order m

Commonly used nonlinear functions are: (1) The Godard function, [40], given as

(6.211)

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Advanced Optical Communication Systems and Networks

 

E a  n aˆ  n 3 g  aˆ  n  aˆ  n  R2 aˆ  n   aˆ  n , R2  2 aˆ  n E a  n





4

 

(6.212)

(2) The Sato function, [41], given as g  aˆ  n    sgn  aˆ  n  ,  



E  Re  a  n  



E Re  a  n 

2





(6.213)

(3) The Benveniste-Goursat function, [42], given as g  aˆ  n  aˆ  n  k1  aˆ  n  a  n  k1 aˆ  n  a  n  sgn  aˆ  n  a  n

(6.214)

where k1 and k2 are properly chosen constants, while a  n  is the decision circuit output as shown in Figure 6.31. In the same figure, we also provide the update rule for adaptive equalizer tap coefficients, which is applicable to all three nonlinear functions described above. The Godard algorithm belongs to the class of steepest-descent algorithms, and it is widely used when the training sequence is not available. The Godard scheme is suitable for simultaneous blind adaptive equalization and carrier-phase tracking, as shown in Figure 6.32. With rI[n] and rQ[n] we denoted the in-phase and quadrature components coming from a balanced coherent optical detector. The equalizer output is represented by the discrete-time convolution of the input complex sequence r[n]=(rI[n], rQ[n]) and the equalizer tap coefficients w[n] as aˆ  k  

K

 w  n r  k  n

(6.215)

n  K

This output is multiplied with exp( jˆk ), where ˆk is the carrier-phase estimate at the kth time instance (symbol interval). The error signal can be defined as:





e  k   a  k   exp  jˆk aˆ  k 

(6.216)

which assumes that a[k] were known. The MSE can be minimized with respect to equalizer tap coefficients and carrier-phase estimate to become

ˆ , w   arg min E  a  k   exp   jˆ  aˆ  k   2

k

ˆk , w

k

(6.217)

Advanced Detection Schemes

r  n

Adaptive equalizer

Decision circuit

aˆ  n 

475

a  n 

Nonlinear d [n] function, g()

w  n  1  w  n  

 e  n  r   n

-

+

Error signal

e[n]

Figure 6.31 The generic stochastic-gradient blind equalization algorithm. rI  n  From balanced coherent detector r n Q

Adaptive equalizer

 

aˆ  n 

Decision circuit

a  n 

ˆ

e jk

Carrier tracking

Figure 6.32 The Godard scheme for simultaneous blind equalization and carrier-phase tracking.

The LMS-like algorithm can be used to determine the carrier phase estimate and equalizer tap coefficients, respectively, as

    a k   aˆ k  exp   jˆ 

ˆk  ˆk   Im a  k  aˆ   k  exp jˆk wˆ k 1  wˆ k  w

(6.218)

k

However, since the desired sequence a[n] is not known, the algorithm above will not converge. To solve this problem we can use the cost function that is independent of the carrier phase, defined as p p   C  p   E   aˆ  k   a  k    ,   2





(6.219)

where p is a positive integer (p=2 is the most used value). However, only the signal amplitude will be equalized in this case. Another more general cost function was introduced for the Godard algorithm, defined as [40] 2 p   p D    E   aˆ  k   R p      

(6.220)

where Rp is a positive real number to be determined. The minimization of cost function from Equation (6.225) with respect to equalizer tap coefficients can be done by using the steepest-descent algorithm, which defines the value wi 1  wi   p

dD  dwi p

(6.221)

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After performing a derivation of variance and omitting the expectation operator E{.}, the following LMS-like adaptation algorithm is defined

wk 1  wk   r aˆ  k  aˆ  k   p k

p2

 R  aˆ k   , R p

   E  aˆ  k   E aˆ  k 

p

p

2p

(6.222)

p

As we can see, the knowledge of carrier phase is not needed to determine equalizer tap coefficients. The algorithm is particularly simple for p=2, since

aˆ  k 

p 2

 1. The algorithm from Equation (6.227) will converge if all tap

coefficients have been initialized to zero, but with exception of the center tap that was chosen to satisfy the following inequality:



E a k 

2

w0 



4





2 hmax E  a  k     2

(6.223) 2

where hmax is the channel impulse response sample with largest amplitude. The inequality from Equation (6.223) represents necessary, but not sufficient condition for convergence. 6.3.9 Volterra Series Based Equalization

One important application of Volterra series is in compensation of nonlinear effects [43–50]. The nonlinear compensation can be performed either in the timedomain [43, 50] or the frequency-domain [49, 51]. The frequency-domain is particularly attractive, as the theory behind it is well developed [47, 48]. Expansion in the Volterra series is given as [44] 

y  t    yn  t 

(6.224)

n 0 







yn  t  

   h  , n

1

2

, n  x  t   1  x  t   2  x  t   n  d 1d 2  d n

where x(t) is the input to the nonlinear system, while the n-th term yn(t) represents the n-fold convolution of the input and the n-fold impulse response hn  1 , n  .

, n  , of impulse responses is known as the Volterra kernel of the system. The 0th order corresponds to d.c. component, which is not represented here, while the first order term corresponds to impulse response of the linear system. The first-order impulse response can be expressed in an analytical form, as in Equation (6.159) where the second order The set

h , h   , h  ,  ,, h  , 0

1

1

2

1

2

n

1

2

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GVD can be neglected. The frequency-domain representation of the Volterra series is given by [47]: 



n 1 





Y        H n 1 ,n  X 1  X n  X   1   n  d 1  d n (6.225)

As an example, if H3 denotes the 3-D Fourier transform of kernel h3, it can be estimated as [44]: H3  f , g, h  S xx  f  

S xxxy  f , g , h 

6 S xx  f  S xx  g  S xx  h 





2 1 E X i  f  , S xxxy  f , g , h   E  X i  f  X i  g  X i  h  Yi  f  g  h  T

(6.226)

where X(f) and Y(f) are the Fourier transforms of x(t) and y(t), T is observation interval, and the subscript i denotes the index of record of duration T. The propagation over the SMF is governed by the nonlinear Schrödinger equation (NSE) as described in Chapter 3—please see Equation (3.149). Since we are concerned here with the compensation of chromatic dispersion and nonlinearities, let us observe the following inverse version of NSE that can be obtained from Equation (3.149) by changing the sign of the terms: E  z , t  z

   2  3   Dˆ  Nˆ , Dˆ    j 2 2  3 3 , Nˆ  j E 2 2 2 t 6 t





(6.227)

where E is signal electric field, Dˆ and Nˆ denote the linear and nonlinear operators; and , 2, 3, and  represent attenuation coefficient, GVD, secondorder GVD, and nonlinear coefficient, respectively. By setting that X()=E(,z)=FT{E(z,t)}, the electric field from Equation (6.227) can be presented as [49]: E   , 0   H1   , z  E   , z  

(6.228)

  H  ,  ,    3

1

2

1

 2 , z  E 1 , z  E 2 , z  E   1  2 , z  d12 

where the linear kernel in the frequency-domain is given as H1  , z   e z / 2 j 2

2

z/2

,

(6.229)

while the second-order GVD is neglected. (However, we should notice that neglecting the second-order GVD will not provide accurate results in ultra-longhaul transmission systems.) The third-order kernel in the frequency-domain is given by [49]:

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Advanced Optical Communication Systems and Networks

H 3 1 , 2 ,   1  2 , z    j H1  , z 

 1 2  2 1 1 e   j  2 1   1  2   z  j      z

(6.230)

As we see in Equation (6.230), fiber parameters of opposite sign are used in the design of the nonlinear equalizer based on the Volterra series. As we mentioned, the second-order GVD must be included for ultra-longhaul transmission system studies. Also, the fifth-order kernel term should be calculated as well. Regarding the complexity of this method, the following rule of thumb can be used. The number of operations related for the nth kernel is approximately equal to the n-th power of the number of operations needed for the first-order kernel. 6.4 DIGITAL BACK-PROPAGATION

Another method to simultaneously compensate for chromatic dispersion and fiber nonlinearities is known as digital back-propagation compensation [53, 54]. The key idea of this method is to assume that received signal in the digital-domain can now propagate backwards through the same fiber. This virtual back-propagation is done by using the fiber with parameters just opposite in sign to real fiber parameters (applied in forward propagation). By this approach we will be able in principle to compensate for fiber nonlinearities if there was not signal-to-noise nonlinear interaction. This virtual back-propagation (BP) can be performed either on transmitter side or receiver side, as illustrated in Figure 6.33. In the absence of noise, those two approaches are equivalent to each other. The back-propagation is governed by inverse NSE, given by Equation (6.227). The back-propagation method operates on the signal electric field, and it is, therefore, universal and independent on modulation format. It uses a split-step Fourier method of reasonable high complexity to solve Equation (6.227). Both asymmetric and symmetric split-step Fourier methods [55] can be used. As an illustration, we briefly describe the iterative symmetric split-step Fourier method. Additional details will be given in Chapter 9. The key idea of this method is to apply the linear and nonlinear operators in an iterative fashion. Receiver side back-propagation

SMF

Optical Tx

 ,  2 , 3 , 

Optical Rx

(a)

 ,   2 ,  3 ,   Dˆ  Nˆ

Transmitter side back-propagation

 ,   2 ,   3 ,   Dˆ  Nˆ

SMF

Optical Tx

 ,  2 , 3 , 

Optical Rx

(b)

Figure 6.33 Illustration of digital back-propagation (BP) method: (a) receiver side BP, and (b) transmitter side BP.

Advanced Detection Schemes

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The linear operator, which corresponds to multiplication in the frequencydomain, is expressed as:





exp zDˆ E  z  z , t  

(6.231)

         FT exp      j 2  2  j 3  3  z  FT  E  z  z , t    2 2 6       1

where z is the step-size and FT (FT-1) denotes the Fourier transform (inverse Fourier transform). However, the nonlinear operator, which performs the nonlinear phase “rotation” in the time domain, is expressed as:







exp z Nˆ E  z  z, t   exp  jz E  z  z, t 

2

 E  z  z, t 

(6.232)

where 01 is the correction factor, which is needed to account for the ASE noise-signal nonlinear interaction during propagation. It is evident that the nonlinear operator depends of the magnitude of electric field at location z, which should be determined. The electric field can be found by using the trapezoidal rule, as proposed in [55], so it is z   E  z , t   exp  Dˆ z / 2 exp    N  z ' dz '  exp  Dˆ z / 2 E  z  z , t   z z  ˆ ( z  z )  Nˆ ( z )   N  exp  Dˆ z / 2 exp   z  exp  Dˆ z / 2 E  z  z , t  2  

















(6.233) 2 Since the operator Nˆ ( z )  j E  z , t  is dependent on the output, Equation

(6.227) can be solved in an iterative fashion, as illustrated by the flow-chart in Figure 6.34. The operating frequency range of interest in the BP method must be smaller than the sampling rate, and therefore, the oversampling is needed. Regarding the complexity issues, FFT complexity scales with the factor N·logN, while the filtering and nonlinear phase rotation operations have a complexity proportional to N.

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Advanced Optical Communication Systems and Networks E(z+z, t)

Linear half-step operator exp(-D z/2) Estimate N(z) Nonlinear step operator exp{ - [N(z)+N(z+z) ] z / 2)]

Linear half-step operator exp(-D z/2) Eprev(z, t)

|E-Eprev|/|E|>1/B, calculate  Y2 and compare T / 2 it against  X2 . 6.4. The AWGN signal is passed through the system whose transfer function can be modeled as in Figure 6.45. What is the output noise power? Determine the autocorrelation function of the random noise process at the output of system. Assume that R/L=8 and 1/(LC)=25. R

L +

x(t)

+

C

y(t)

Figure 6.45 The transmission system example.

6.5. Determine the average energy per symbol for 4-ASK assuming that each of the four symbols is equally likely. Derive the expression for symbol error probability. Compare it against QPSK. 6.6. We already studied the 8-QAM signal constellation (see Problem 8 in Chapter 5), shown in Figure 6.46(a), in particular its Gray mapping rule and decision boundaries. (a) Determine the decision boundaries and decision regions for both constellations shown in Figure 6.46. (b) Derive the expression for average error probability of 8-QAM shown in Figure 6.46(a). (c) Derive the expression for average error probability of 8-QAM shown in Figure 6.46(b) and compare it against that of the signal constellation shown in Figure 6.46(a).  

b a

/4

a a/2

a

a a/2

a/2

(b) (a) Figure 6.46 The 8-ary 2-D constellation diagrams: (a) star-8-QAM, and (b) 8-QAM.

Advanced Detection Schemes

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6.7. In this problem we study the performance of the 5-QAM signal constellation shown in Figure 6.47 against that of 5-PSK with constellation points located on circle of radius a. In Problem 10 of Chapter 5 we discussed these two modulations schemes in terms of energy efficiency and robustness to phase errors. Assuming that all constellation points are equally likely. (a) Determine the decision boundaries and decision regions for both 5QAM and 5-PSK. (b) Derive the expression for the average error probability of 5-QAM shown in Fig. 6.47. (c) Derive the expression for the average error probability of 5-PSK and compare it against that of 5-QAM. a a/2 a/2 a/2

Figure 6.47 The 5-QAM signal constellation diagram.

6.8. The cube signal constellation is obtained by placing the signal constellation points in vertices of the cube of side length a. Assuming that all symbols are equally likely, derive the expression for average symbol error probability. Compare it against 8-QAM constellations shown in Figure 6.46. 6.9. Let r(Ts) (Ts is the symbol duration) be the output of the optimum binary receiver. Prove that the following is valid: (a) m1  E r | s1   E1  12 (b) m2  E r | s2    E2  12

(c)

T



0



 2  Var r | m1   Var r | m2   Var   n  t   s1  t   s2  t   dt 

where

N  0  E1  E2  2 12  2

Ts

Ts

Ts

0

0

0

E1   s12  t  dt , E2   s22  t  dt , and 12   s1  t  s2  t  dt 6.10. Let us consider the random process R(t) and its expanded form D

R  t    rj  j  t   Z  t  , 0  t  Ts j 1

where Z(t) is the remainder of noise term. The {j(t)} represents the set of orthonormal basis functions, and rj is the projection of R(t) along the jth basis function. Let Z(tm) be the random variable obtained by observing Z(t) at

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Advanced Optical Communication Systems and Networks

t=tm. Show that the sample of noise process Z(tm) is statistically independent on correlators’ outputs {Rj}: E  R Z  t    0,  j  1, 2,..., D  m   j  0  tm  Ts

6.11. An orthogonal set of signals is characterized by the property that the dotproduct of any pair of signals is zero. Provide a pair of signals that satisfies this condition. Construct the corresponding signal constellations. 6.12. A set of 2M biorthogonal signals can be derived from the set of orthogonal signals by augmenting it with the negative sign signals from orthogonal set. Does the biorthogonal signal construction increase the dimensionality of the original signal set? The construction of the biorthogonal signal constellation for orthogonal signal constellation is developed in Problem 6.11. 6.13. Formulate the signal constellations for the following lines codes: unipolar NRZ, polar NRZ, unipolar RZ, and Manchester line code. For Manchester line code, derive the expression for error probability when the ML rule is applied in the AWGN channel. 6.14. The simplex signals are equally likely, highly-correlated signals. When derived from M orthogonal signals, the correlation coefficient between any two pairs of signals in the set is given by: 1/( M  1), k  l  kl   1, k  l  The simplest way to construct the simplex signal set is to start with an orthogonal signal set, with each element from the set having the same energy, and then create minimum energy signal set. Consider the signal set with constellation points placed on equilateral triangle vertices. Prove that these three signals represent a simplex signal set. 6.15. Let us observe the M-ary cross-constellation described in Chapter 5 (see Figure 5.10). Prove that for sufficiently high SNRs the average symbol error probability can be estimated as Pe  2 1  1/ 2M erfc E0 / N0 , where E0



 



is the smallest symbol energy in constellation. Determine also the union bound of this constellation and discuss its accuracy. Finally, determine the nearest neighbor approximation. 6.16. Let us observe M-ary PSK signal constellation. Derive the expression for the average symbol error probability. Determine also the union bound approximation. Finally, determine the nearest neighbor approximation. Discuss the accuracy of these approximations against the exact expression for average symbol error probability. 6.17. Let us consider a signal constellation that is symmetric with respect to the origin. Assume that there are M signal constellation points in the set, which are equally likely. Using the upper-bound approach, determine the corresponding average symbol error of this signal constellation.

Advanced Detection Schemes

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6.18. Let us reconsider the example from Section 6.1.5. Show how Equation (6.96) has been derived. By using it, deduce how the phase-locked loop can be implemented. 6.19. Derive the orthogonality principle given by Equation (6.184). By using it, derive the most general form of the Wiener filter (6.186). Finally, derive the variance error of the Wiener filter. Show the steps of your derivations. 6.20. This problem is useful in MIMO signal processing and filter design. Let x be transmitted, r be received, and z be noise vectors, which are related by: r=Hx+z, where H is the channel matrix. Derive the Wiener estimate and determine its minimum variance of error. 6.21. In this problem we study the carrier phase recovery by data-aided method. In the data-aided method, the receiver has the knowledge of the preamble of length L, namely al L 1 . Determine the ML estimate of the carrier phase. l 0

Provide the block diagram of this ML phase estimator. 6.22. Let us consider heterodyne coherent detection designs with two balanced detectors and a single balanced detector, as shown in Figure 6.20 and redrawn in Fig. 6.48. Derive vI and vQ outputs in both designs assuming that j s t i S ,PN  where  the incoming optical signal is given by

Si  Ps ai e , s is carrier frequency of transmitting laser, (ai,i) are polar coordinates of 2-D

signal constellation being transmitted at the ith time instance (symbol interval), and S,PN is the laser phase noise of transmitting laser. Assume that local oscillator (LO) laser signal is given by L  PLO e j LOt LO ,PN  , where

LO is the carrier frequency of local laser oscillator and LO,PN is the corresponding laser phase noise. Compare the receiver sensitivities of these two schemes in the following two regimes: (1) ASE noise-dominated scenario, and (2) shot noise-dominated scenario. From

/2

vI

SMF From local

vQ

oscillator laser

vI

From

LPF

SMF

From local oscillator laser

cos(IF t) -/2 vQ LPF

Figure 6.48 The heterodyne coherent detection designs.

6.23. In this problem we study the efficiency of DFT window synchronization. Implement the CO-OFDM system in Matlab or C/C++ to perform the

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following Monte Carlo simulations. The OFDM system parameters should be set as follows: the symbol period of 25.6 ns, the guard time 3.2 ns, the number of subcarriers 256, the aggregate data rate of 10 Gb/s with BPSK format being used, and the laser linewidths of transmit and local oscillator lasers equal to 100 kHz. Plot the timing metric against the timing offset (in samples) for an accumulated chromatic dispersion of 34000 ps/nm and OSNR of 6 dB. Repeat the simulation when QPSK is used instead. Finally, repeat the simulation when the first-order PMD is present for different DGD values: 100 ps, 500 ps, and 1000 ps. 6.24. Here we study the efficiency of Alamouti-type polarization-time (PT) coding in CO-OFDM systems. Design a CO-OFDM system, based on Alamoutitype PT coding, capable of 100Gb/s serial optical transmission over 6,500km of SMF. Perform Monte Carlo simulations in a linear regime, by taking the following effects into account: chromatic dispersion, the firstorder PMD, and laser linewidth of 100 kHz (for both transmitter and local lasers). Use the typical fiber parameters given in previous chapters. Compare this scheme with an equivalent polarization-division multiplexed (PDM) scheme, based on the LS channel estimation, of the same aggregate data rate. Compare and discuss the results. What are advantages and disadvantages of PT coding with respect to PDM? 6.25. Here we study the efficiency of the V-BLAST scheme in CO-OFDM systems. Design a CO-OFDM system, capable of 100Gb/s serial optical transmission over 10,000km of SMF. Perform Monte Carlo simulations in a linear regime, by taking the following effects into account: chromatic dispersion, the first order PMD, and laser linewidth of 100 kHz (for both transmitter and local lasers). Use the typical fiber parameters given in previous chapters. To compensate for PDM and chromatic dispersion, use the V-BLAST approach. Compare this scheme with an equivalent PDM scheme, based on LS channel estimation, of the same aggregate data rate. Compare and discuss the results. What are the advantages and disadvantages of the V-BLAST detection scheme with respect to PDM? 6.26. Repeat Problem 6.25, but now in the nonlinear regime, by solving the NSE for both polarizations. Study the BER performance as a function of the total transmission distance. 6.27. In this problem we are interested in improving the BER performance and the nonlinearity tolerance of V-BLAST-based and conventional PDM, COOFDM systems from Problem 6.26 by using MMSE channel estimation. Plot BER versus total transmission distance for several cases: (1) V-BLAST-LS estimation, (2) conventional PDM-LS estimation, (3) V-BLAST-MMSE estimation, and (4) conventional PDM-MMSE estimation. For MMSE estimation use the first- and second-order interpolation. Discuss the results. 6.28. This problem is related to the derivation of the AMMSE matrix given by Equation (6.300). Let as define the MSE by

Advanced Detection Schemes



 2  A   E Ay  x

2

509

  E Tr  AH  I  x + Az   AH  I  x + Az   †

M Tx

M Tx

By assuming that components of x are i.i.d. zero-mean with a second moment corresponding to the average energy, determine the optimum (in the MMSE sense) linear transformation A. Derive the corresponding PEP. Discuss its asymptotic behavior for: (a) finite MTx, MRx-> ; and (b) MTx-> , MRx-> , while MTx/MRx->a. 6.29. In this problem we deal with the derivation of matrices A and B in ZF-VBLAST. The MSE in the absence of noise is given by: 2 2  2  A, B   E Y  X  E AHX  BXˆ  X . By using the high SNR

 





approximation and the QR-factorization of channel matrix determine the matrices A and B so that the MSE is minimized. Once you determine these matrices, demonstrate that mode coupling is indeed canceled out. Fully describe the detection procedure. 6.30. Here we deal with the derivation of matrices A and B in MMSE V-BLAST. The MSE in the presence of noise is given by:



 2  A, B   E Y  X

2

  E  AY  BX  X   E   AH  L  I 2

M Tx

 X + AZ

2

.

By using the Cholesky factorization of H † H  SNR1 I M  S † S , determine Tx the matrices A and B so that the MSE is minimized. What is the remaining MSE error? Once you determine these matrices, demonstrate that mode coupling is indeed canceled out. 6.31. In this problem we are concerned with 1Tb/s optical transport over 1500 km of few-mode fiber (FMF) by using mode-multiplexed PDM CO-OFDM. For modeling of FMF, use the model described in Section 6.7.1. Design a CO-OFDM system with aggregate data rate exciding 1 Tb/s by employing two spatial and two polarization modes. Use five orthogonal OFDM bands for simulations, with 200 Gb/s in aggregate data rate per band. Perform Monte Carlo simulations for the following cases of interest: (1) V-BLASTLS estimation, (2) conventional PDM-LS estimation, (3) V-BLAST-MMSE estimation, and (4) conventional PDM-MMSE estimation. For MMSE estimation use the first- and second-order interpolation. Discuss the results. References [1]

Proakis, J. G., Digital Communications, Boston, MA: McGraw-Hill, 2001.

[2]

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[3]

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[4]

Jacobsen, G., Noise in Digital Optical Transmission Systems, Norwood, MA: Artech House, 1994.

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[5]

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[10] Djordjevic, I. B., Xu, L., and Wang, T., “PMD compensation in multilevel coded-modulation schemes with coherent detection using BLAST algorithm and iterative polarization cancellation,” Opt. Express, Vol. 16, No. 19, pp. 1484514852, Sept. 15, 2008. [11] Djordjevic, I. B., Xu, L., and Wang, T., “PMD compensation in coded-modulation schemes with coherent detection using Alamouti-type polarization-time coding,” Opt. Express, Vol. 16, No. 18, pp. 1416314172, Sept. 1, 2008. [12] Djordjevic, I. B., Arabaci, M., and Minkov, L., “Next generation FEC for high-capacity communication in optical transport networks,” IEEE/OSA J. Lightw. Technol., Vol. 27, No. 16, pp. 35183530, August 15, 2009. (Invited Paper.) [13] Djordjevic, I. B., Xu, L., and Wang, T., “Beyond 100 Gb/s optical transmission based on polarization multiplexed coded-OFDM with coherent detection,” IEEE/OSA J. Opt. Commun. Netw., Vol. 1, No. 1, pp. 5056, June 2009. [14] Proakis, J. G., and Manolakis, D. G., Digital Signal Processing: Principles, Algorithms, and Applications, 4th ed., Upper Saddle River, NJ: Prentice-Hall, 2007. [15] Tseng, C.-C., “Design of IIR digital all-pass filters using least pth phase error criterion,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., Vol. 50, No. 9, pp. 653656, Sept. 2003. [16] Goldfarb, G., and Li, G., “Chromatic dispersion compensation using digital IIR filtering with coherent detection,” IEEE Photon. Technol. Lett., Vol. 19, No. 13, pp. 969971, July 1, 2007. [17] Ip, E., Pak, A., et. al., “Coherent detection in optical fiber systems,” Opt. Express, Vol. 16, No. 2, pp. 753791, 21 January 2008. [18] Ip, E., and Kahn, J.M., “Digital equalization of chromatic dispersion and polarization mode dispersion,” J. Lightw. Technol., Vol. 25, pp. 20332043, Aug. 2007. [19] Kuschnerov, M., Hauske, F. N., et. al., “DSP for coherent single-carrier receivers,” J. Lightw. Technol., Vol. 27, No. 16, pp. 36143622, Aug. 15, 2009. [20] Djordjevic, I. B., Ryan, W., and Vasic, B., Coding for Optical Channels, New York: Springer, 2010. [21] Proakis, J. G, “Partial response equalization with application to high density magnetic recording channels,” in Coding and Signal Processing for Magnetic Recording Systems, (B. Vasic and E. M. Kurtas, Eds.), Boca Raton, FL: CRC Press, 2005. [22] Haykin, S., Communication Systems, 4th ed., New York: John Wiley & Sons, 2001. [23] Lyubomirsky, I., “Optical duobinary systems for high-speed transmission,” in Advanced Technologies for High-Speed Optical Communications 2007, (L. Xu, Ed.), Research Signpost, 2007.

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[24] Alic, N., Papen, G. C., et. al., “Signal statistics and maximum likelihood sequence estimation in intensity modulated fiber optic links containing a single optical preamplifier,” Opt. Express, Vol. 13, pp. 45684579, June 2005. [25] Goldsmith, A., Wireless Communications, Cambridge, UK: Cambridge University Press, 2005. [26] ColaVolpe, G., Foggi, T., et. al., “Multilevel optical systems with MLSD receivers insensitive to GVD and PMD,” IEEE/OSA J. Lightw. Technol., Vol. 26, pp. 12631273, 2008. [27] Ivkovic, M., Djordjevic, I. B., and Vasic, B., “Hard decision error correcting scheme based on LDPC codes for long-haul optical transmission,” Proc. Optical Transmission Systems and Equipment for Networking V-SPIE Optics East Conference, Vol. 6388, pp. 63880F.163880F.7, Oct. 1-4, 2006, Boston, MA, USA. [28] Hagenauer, J., and Hoeher, P., “A Viterbi algorithm with soft-decision outputs and its applications,” Proc. IEEE Globecom Conf., Dallas, TX, pp. 16801686, Nov. 1989. [29] Bahl, L. R., Cocke, J., Jelinek, F., and Raviv, J., “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Inform. Theory, Vol. IT-20, pp. 284287, Mar. 1974. [30] Wymeersch, H., and Win, M. Z., “Soft electrical equalization for optical channels,” Proc. ICC’08, pp. 548-552, May 19-23, 2008. [31] Douillard, C., Jézéquel, M., et. al., “Iterative correction of intersymbol interference: turbo equalization’” Eur. Trans. Telecommun., Vol. 6, pp. 507511, 1995. [32] Tüchler, M., Koetter, R., and Singer, A. C., “Turbo equalization: principles and new results,” IEEE Trans. Commun., Vol. 50, No. 5, pp. 754767, May 2002. [33] Jäger, M., Rankl, T., et. al., “Performance of turbo equalizers for optical PMD channels,” IEEE/OSA J. Lightw. Technol., Vol. 24, No. 3, pp. 12261236, Mar. 2006. [34] Djordjevic, I. B., Minkov, L. L., et. al., “Suppression of fiber nonlinearities and PMD in codedmodulation schemes with coherent detection by using turbo equalization,” IEEE/OSA J. Opt. Commun. Netw., Vol. 1, No. 6, pp. 555564, Nov. 2009. [35] Djordjevic, I. B., “Spatial-domain-based hybrid multidimensional coded-modulation schemes enabling multi-Tb/s optical transport,” IEEE/OSA J. Lightw. Technol., Vol. 30, No. 14, pp. 23152328, July 15, 2012. [36] Hayes, M. H., Statistical Digital Signal Processing and Modeling, New York: John Wiley & Sons, 1996. [37] Haykin, A., Adaptive Filter Theory, 4th ed., Boston, MA: Pearson Education, 2003. [38] Manolakis, D. G., Ingle, V. K., and Kogon, S. M., Statistical and Adaptive Signal Processing, New York: McGraw-Hill, 2000. [39] Seshadri, N., “Joint data and channel estimation using fast blind trellis search techniques,” IEEE Trans. Comm., Vol. COMM-42, pp. 1000-1011, Feb./Mar./Apr. 1994. [40] Godard, D. N., “Self-recovering equalization and carrier tracking in two-dimensional data communication systems,” IEEE Trans. Comm., Vol. 28 , No. 11, pp. 18671875, Nov. 1980. [41] Sato, Y., “A method of self-recovering equalization for multilevel amplitude-modulation systems,” IEEE Tran. Comm., Vol. 23, No. 6, pp. 679-682, Jun. 1975. [42] Benveniste, A., Goursat, M., and Ruget, G., “Robust identification of a nonminimum phase system,” IEEE Trans. Auto. Control, Vol. AC-25, pp. 385399, June 1980.

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[43] Nowak, R. D., and Van Veen, B. D., “Volterra filter equalization: a fixed point approach,” IEEE Tran. Sig. Proc., Vol. 45, No. 2, pp. 377[78, 89]388, Feb. 1997. [44] Jeruchim, M. C., Balaban, P., and Shanmugan, K. S., Simulation of Communication systems: Modeling, Methodology, and Techniques, 2nd ed., N. York: Kluwer Academic/Plenum Pub., 2000. [45] Nazarathy, M., and Weidenfeld, R., “Nonlinear impairments in coherent optical OFDM and their mitigation,” in Impact of Nonlinearities on Fiber Optic Communication, (S. Kumar Ed.), Springer, pp. 87175, Mar. 2011. [46] Guiomar, F. P., Reis, J. D., et. al., “Mitigation of intra-channel nonlinearities using a frequencydomain Volterra series equalizer,” Proc. ECOC 2011, Paper Tu.6.B.1, Sept. 1822, 2011, Geneva, Switzerland. [47] Peddanarappagari, K. V., and Brandt-Pearce, M., “Volterra series transfer function of singlemode fibers,” J. Lightw. Technol., Vol. 15, pp. 22322241, Dec. 1997. [48] Xu, B., and Brandt-Pearce, M., “Modified Volterra series transfer function method,” IEEE Photon. Technol. Lett., Vol. 14, No. 1, pp. 4749, 2002. [49] Guiomar, F. P., Reis, J. D., at. al., “Digital post-compensation using Volterra series transfer function,” IEEE Photon. Technol. Lett., Vol. 23, No. 19, pp. 14121414, Oct. 1, 2011. [50] Gao, Y., Zgang, F., et. al., “Intra-channel nonlinearities mitigation in pseudo-linear coherent QPSK transmission systems via nonlinear electrical equalizer,” Opt. Commun., Vol. 282, pp. 24212425, 2009. [51] Liu, L., Li, L., et. al., “Intrachannel nonlinearity compensation by inverse Volterra series transfer function,” J. Lightw. Technol., Vol. 30, No. 3, pp. 310316, Feb. 1, 2012. [52] Liu, X., and Nazarathy, M., “Coherent, self-coherent, and differential detection systems,” in Impact of Nonlinearities on Fiber Optic Communication, (S. Kumar, ed.), New York: Springer, pp. 142, Mar. 2011. [53] Ip, E., and Kahn, J. M., “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightw. Technol., Vol. 26, No. 20, pp. 34163425, 2008. [54] Ip, E., and Kahn, J. M., “Nonlinear Impairment Compensation using Backpropagation,” in Optical Fibre, New Developments, (C. Lethien, ed.), Vienna, Austria: In-Tech, 2009. [55] Agrawal, G. P., Nonlinear Fiber Optics, 5th ed., San Diego: Academic Press, 2012. [56] Yang, Q., Al Amin, A., and Shieh, W., “Optical OFDM basics,” in Impact of Nonlinearities on Fiber Optic Communication, (S. Kumar, ed.), New York: Springer, pp. 4385, 2011. [57] Shieh, W., and Djordjevic, Elsevier/Academic Press, 2009.

I., OFDM

for

Optical

Communications,

New

York:

[58] Djordjevic, I. B., Xu, L., and Wang, T., “PMD compensation in multilevel coded-modulation schemes using BLAST algorithm,” Proc. The 21st Annual Meeting of the IEEE Lasers & Electro-Optics Society, Newport Beach, CA, 2008, Paper No. TuP 2. [59] Xu, T., Jacobsen, G., et. al., “Chromatic dispersion compensation in coherent transmission system using digital filters,” Opt. Express, Vol. 18, No.15, pp.1624316257, 19 July 2010. [60] Djordjevic, I. B., Quantum Information Processing and Quantum Error Correction: An Engineering Approach, New York: Elsevier/Academic Press, 2012. [61] Gordon, J. P., and Kogelnik, H., “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Nat. Academy of Science, Vol. 97, No. 9, pp. 45414550, Apr. 2000.

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[62] Tong, Z., Yang, Q., et. al., “21.4 Gb/s coherent optical OFDM transmission over multimode fiber,” in Post-Deadline Papers Technical Digest, Proc. 13th Optoelectronics and Communications Conference (OECC) and 33rd Australian Conference on Optical Fibre Technology (ACOFT), Paper No. PDP-5, 2008. [63] Hsu, R. C. J., Tarighat, A., et. al., “Capacity enhancement in coherent optical MIMO (COMIMO) multimode fiber links,” J. Lightw. Technol., Vol. 23, No. 8, pp. 24102419, Aug. 2005. [64] Tarighat, A., Hsu, R. C. J., et. al., “Fundamentals and challenges of optical multiple-input multiple output multimode fiber links,” IEEE Comm. Mag., Vol. 45, pp. 5763, May 2007. [65] Bikhazi, N. W., Jensen, M. A., and Anderson, A. L., “MIMO signaling over the MMF optical broadcast channel with square-law detection,” IEEE Trans. Comm., Vol. 57, No. 3, pp. 614617, Mar. 2009. [66] Agmon, A., and Nazarathy, M., “Broadcast MIMO over multimode optical interconnects by modal beamforming,” Optics Express, Vol. 15, No. 20, pp. 1312313128, Sept. 26, 2007. [67] Biglieri, E., Calderbank, R., et. al., MIMO Wireless Communications, Cambridge, UK: Cambridge University Press, 2007. [68] Foschini, G. J., “Layered space-time architecture for wireless communication in a fading environment when using multi-element antennas,” Bell Labs Tech. J., Vol. 1, pp. 4159, 1996. [69] Tse, D., and Viswanath, P., Fundamentals of Wireless Communication, Cambridge, UK: Cambridge University Press, 2005. [70] Duman, T. M., and Ghrayeb, A., Coding for MIMO Communication Systems, New York: John Wiley & Sons, 2007. [71] Tarokh, V., Seshadri, N., and Calderbank, A. R., “Space–time codes for high data rate wireless communication: performance criterion and code construction,” IEEE Trans. Information Theory, Vol. 44, No. 2, pp. 744765, Mar. 1998. [72] Biglieri, E., Coding for Wireless Channels, New York: Springer, 2005. [73] Alamouti, S., “A simple transmit diversity technique for wireless communications,” IEEE J. Sel. Areas Commun. Vol. 16, pp. 14511458, 1998. [74] Kogelnik, H., Jopson, R. M., and Nelson, L. E., “Polarization-mode dispersion,” in Optical Fiber Telecommunications IVB: Systems and Impairments, (I. Kaminow and T. Li, Eeds.), San Diego, CA: Academic Press, 2002. [75] Ho, K.-P., and Kahn, J. M., “Statistics of group delays in multimode fiber with strong mode coupling,” J. Lightw. Technol., Vol. 29, No. 21, pp. 31193128, 2011. [76] Zou, D., Lin, C., and Djordjevic, I. B., “LDPC-coded mode-multiplexed CO-OFDM over 1000 km of few-mode fiber,” Proc. CLEO 2012, Paper No. CF3I.3, San Jose, CA, 2012. [77] Shieh, W., and Athaudage, C., “Coherent optical orthogonal frequency division multiplexing,” Electronic Letters, Vol. 42, pp. 587589, May 11, 2006. [78] Djordjevic, I. B., and Vasic, B., “Orthogonal frequency division multiplexing for high-speed optical transmission,” Optics Express, Vol. 14, pp. 37673775, May 1, 2006. [79] Lowery, A. J., Du, L., and Armstrong, J., “Orthogonal frequency division multiplexing for adaptive dispersion compensation in long haul WDM systems,” Proc. Optical Fiber Communication Conference, Paper PDP39, March 5-10, 2006, Anaheim, CA. [80] Shieh, W., Yi, X., et al., “Coherent optical OFDM: has its time come? ” J. Opt. Netw., Vol. 7, pp. 234-255, 2008.

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[81] Schmidl, T. M., and Cox, D. C., “Robust frequency and time synchronization for OFDM,” IEEE Trans. Commun., Vol. 45, pp. 16131621, Dec. 1997. [82] Minn, H., Bhargava, V. K., and Letaief, K. B., “A robust timing and frequency synchronization for OFDM systems,” IEEE Trans. Wireless Comm., Vol. 2, pp. 822839, July 2003. [83] Pollet, T., Van Bladel, M., and Moeneclaey, M., “BER sensitivity of OFDM systems to carrier frequency offset and Wiener phase noise,” IEEE Trans. Comm., Vol. 43, pp. 191193, Feb./Mar./Apr. 1995. [84] Coleri, S., Ergen, M., et. al., “Channel estimation techniques based on pilot arrangement in OFDM systems,” IEEE. Trans. Broadcasting, Vol. 48, No. 3, pp. 223229, Sept. 2002. [85] Zou, D., and Djordjevic, I. B., “Multi-Tb/s optical transmission based on polarizationmultiplexed LDPC-coded multi-band OFDM,” Proc. 13th International Conference on Transparent Optical Networks (ICTON) 2011, Paper Th.B3.3, June 26-30, 2011, Stockholm, Sweden. [86] Zou, D., and Djordjevic, I. B., “Beyond 1Tb/s superchannel optical transmission based on polarization multiplexed coded-OFDM over 2300 km of SSMF,” Proc. 2012 Signal Processing in Photonics Communications (SPPCom), Paper SpTu2A.6, 2012, Colorado Springs, Colorado, USA. [87] Sari, H., Karam, G., and Jeanclaude, I., “Transmission techniques for digital terrestrial TV broadcasting,” IEEE Commun. Mag., Vol. 33, No. 2, pp. 100109, 1995. [88] Moose, P., “A technique for orthogonal frequency division multiplexing frequency offset correction,” IEEE Trans. Commun., Vol. 42, pp. 29082914, 1994. [89] Sandell, M., Van de Beek, J. J., and Börjesson, P. O., “Timing and frequency synchronization in OFDM systems using cyclic prefix,” Proc. Int. Symp. Synchron., pp. 1619, Saalbau, Essen, Germany, Dec. 1995. [90] Zhang, S., Huang, M.-F., et. al., “40x117.6 Gb/s PDM-16QAM OFDM transmission over 10,181 km with soft-decision LDPC coding and nonlinearity compensation,” Proc. OFC 2012 Postdeadline Papers, Paper No. PDP5C.4, 2012, Los Angeles, CA. [91] Liu, X., and Chandrasekhar, S., “Beyond 1-Tb/s superchannel transmission,” Proc. 2011 IEEE Photonics Conference, pp. 893-894, 2011, Arlington, VA.

Chapter 7 Advanced Coding Schemes This chapter represents an overview of advanced forward error correction (FEC) techniques for optical communications. Topics include: codes on graphs, coded modulation, rate-adaptive coded modulation, and turbo equalization. The main objectives of this chapter are: (1) to describe different classes of codes on graphs of interest for optical communications, (2) to describe how to combine multilevel modulation and channel coding, (3) to describe how to perform equalization and soft-decoding jointly, and (4) to demonstrate the efficiency of joint demodulation, decoding, and equalization in dealing with various channel impairments simultaneously. Codes on graphs of interest for the next generation of high-speed optical transport include turbo codes, turbo-product codes, and low-density paritycheck (LDPC) codes. We will describe both binary and nonbinary LDPC codes, and their design and decoding algorithms. To facilitate explanations, we will first provide channel coding preliminaries and short introduction to linear block codes, cyclic codes, BCH codes, RS codes, concatenated codes, and product codes. 7.1 CHANNEL CODING PRELIMINARIES The performance of very high-speed data rates optical fiber communication systems, which are considered for deployment in next generation optical networks [1–3], is significantly degraded due to impact of intrachannel and interchannel fiber nonlinearities, polarization-mode dispersion (PMD), and chromatic dispersion. To deal with those channel impairments, novel advanced techniques in modulation and detection, coding and signal processing should be developed and deployed [1–14]. The state of the art of FEC codes that have been standardized by ITU-T includes different variants of Bose-Ray-Chaudhuri-Hocquenghem (BCH) and Reed-Solomon (RS) codes. The RS(255,239) code in particular has been used in a broad range of long-haul communication systems, and it is commonly considered as the first-generation of FEC [11]. The elementary FEC schemes may be combined together, or concatenated, to obtain more powerful FEC schemes, such

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as the case with RS(255,239)+RS(255,233) combination. Several classes of concatenation codes are also standardized and listed in the ITU-T G 975.1 document. Different concatenation schemes, such as the concatenation of two RS codes, or the concatenation of RS and convolutional codes, are commonly considered as the second generation of FEC [14]. The ITU-T has also defined different sizes of optical channel transport units to better accommodate both the information content and FEC size, as described in Section 1.3.3. Codes on graphs [1–11], such as turbo codes [3] and low-density parity-check (LDPC) codes [1–3], [8–29] have revolutionized communications, and are now becoming standard in a number of applications. The LDPC codes, invented by Gallager in the 1960s [20], have generated a high interest in the coding community in recent years, which has resulted in better understanding of the different aspects of LDPC codes and their decoding process [1–3]. 7.1.1 Channel Coding Principles A typical digital optical communication system, with direct detection, employing channel coding is shown in Figure 7.1. The discrete source generates the information in the form of a sequence of symbols. The channel encoder accepts the message symbols and adds redundant symbols according to the prescribed rule. The channel coding is the act of transforming a length-k sequence into a length-n codeword. The set of rules specifying this transformation is the channel code, which can be represented as the following mapping: C: {M}  {X}, where C denotes the channel coding, M is the set of information sequences of length k, and X is the set of codewords of length n. At the receiving side, the decoder exploits these redundant symbols to determine which message symbol was actually transmitted. The encoder and decoder consider the whole digital transmission system as a discrete channel placed among them. The channel codes can be placed into three broad categories: (1) error detection codes in which we are concerned only with detecting the errors occurred during transmission (examples include automatic request for transmission-ARQ), (2) forward error correction (FEC), where we are interested in correcting the errors occurred during transmission, and (3) hybrid channel codes that combine the previous two approaches. In this chapter we will be discussing only FEC characteristics. The codes commonly considered in optical fiber communications belong to the class of block codes. In an (n,k) block code, the channel encoder accepts information in successive k-symbol blocks and adds n-k redundant symbols that are algebraically related to the k message symbols, thus producing an overall encoded block of n symbols (n>k), known as a codeword. If the block code is systematic, the information symbols stay unchanged during the encoding operation, and the encoding operation may be considered as adding the n-k generalized parity checks to k information symbols. Since the information symbols are statistically independent (a consequence of source coding or

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scrambling), every codeword is independent of the content of the previous codeword. The overall bit rate of the (n,k) block-coded sequence is increased by a factor 1/R, where R is the code rate defined as R=k/n. The code overhead is defined as OH=(1/R-1)100%. The amount of energy that can be saved by coding is commonly described by the coding gain, which is measured by the changes in the ratio (Eb/N0) of the energy per information bit and the spectral power density of the noise in the case when coding is applied, as compared to the case where there is no coding. Discrete memoryless source

Destination Channel decoder

Channel encoder Laser diode

External modulator

Discrete channel

Equalizer + decision circuit

EDFA

Photodetector

WDM multiplexer

EDFA

N spans EDFA

D+

D-

D+ WDM demultiplexer

D-

EDFA

EDFA

EDFA

Figure 7.1 Block diagram of a point-to-point digital optical communication system with direct detection.

Coding Example 1: Repetition Code. In a repetition code each bit is repeated and transmitted n=2m+1 times. For example, for n=3, the bits 0 and 1 are represented as 000 and 111, respectively. On receiver side we first perform the threshold decision for each bit separately, and if the received sample is the above the threshold, we decide in favor of 1; otherwise, in favor of bit 0. The decoder then applies the following majority decoding rule: if in the block containing n bits the number of ones exceeds the number of zeros, the decoder decides in favor of 1; otherwise, the decision is in favor of 0. This code is capable of correcting up to m errors (in a case with n=3 it will be m=1). The probability of error that still remains upon decoding can be evaluated by the following expression:

Pe 

n

n

  i  p 1  p  i

n i

(7.1)

  where p is the probability of making an error on a given position. In Figure 7.2 we illustrate the importance of channel coding by using this trivial example by plotting the bit error probability that remained after decoding against the code rate R=1/n for different values of channel transition probability p. It is interesting to i  m 1

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notice that the target BER of 1015 can be achieved even with this simple code, but the code rate is unacceptably low. Coding Example 2: Hamming Codes. The Hamming codes are single errorcorrecting codes, for which the code parameters (n,k) satisfy the inequality 2nkn+1. In Hamming codes, the (nk) parity bits are located on positions 2j (j=0,1,…,n-k-1), while the information bits are located on the remaining bit positions. To identify the location of an error, we need to determine the syndrome (“checking number”). As an illustration, for a (7,4) Hamming code, the codeword can be represented by p1p2i1p3i2i3i4, where pj (j=1,2,3) are parity bits, and ij (j=1,2,3,4) are the information bits. For information bits 1101, the codeword is obtained as p1p21p3101, while the parity bits are determined as:

p1  i1  i2  i4  1, p2  i1  i3  i4  0, p3  i2  i3  i4  0 Accordingly, the resulting codeword becomes x1x2x3x4x5x6 =1010101. Let’s

assume that the sixth bit was received incorrectly, so we have that the corresponding word is y1y2y3y4y5y6 =1010111. The syndrome computation proceeds as follows:

s1  y1  y3  y5  y7  1  1  1  1  0 s2  y2  y3  y6  y7  0  1  1  1  1 s3  y4  y5  y6  y7  0  1  1  1  1 S  s3 , s2 , s1   1102  6   -3

Bit-error rate, BER

10

-5

10

-2

p=10 -3 p=10 -4 p=10

-7

10

-9

10

-11

10

-13

10

-15

10

0.01

0.1

1

Code rate, R

Figure 7.2 Illustration of the importance of channel coding. The syndrome gives us the location of error, which is position 6 in this example. The located error can simply be corrected by flipping the content of the corresponding bit. The syndrome-check s1 performs parity checks on all bit positions having 1 at first right place (position 20 in binary representation), that is, for: 0012=1, 0112=3, 1012=5, and 1112=7. However, the syndrome-check s2

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performs parity checks on all bit positions having 1 at the second right place (position 21 in binary representation), that is, 0102=2, 0112=3, 1102=6, 1112=7. Finally, the syndrome check s3 performs parity checks on all bit positions having 1 at first left place (position 22 in binary representation): 1002=4, 1012=5, 1102=6, 1112=7. 0/0

00000 s’0

s0

00 0 00

s1

00 0 01

s2

00 0 10

s3

00 0 11

1/0 00001 s’1

p(y1|x1)

x1

p(y2|xi)

…

xi

p(yj|xi) p(y1|xM) p(y |x ) N i p(yN|xM)

…

y2

p(y1|xi)

yj

…

… 1/1

11110 s’30

yN

0/0 1/0

10000 s’16

…

p(yN|x1)

…

…

x2

xM

y1

p(y2|x1)

0/1

11111 s’31 1/1

s29 11 1 01 s30

11 1 10

s31

11 1 11

(a) (b) Figure 7.3 Examples of discrete channels: (a) discrete memoryless channel (DMC), and (b) discrete channel with memory described as dynamic trellis. 7.1.2 Mutual Information and Channel Capacity As we mentioned above, the channel code considers whole transmission system as a discrete channel where the sizes of input and output alphabets are finite. Two examples of such a channel are shown in Figure 7.3. Figure 7.3 (a) shows an example of a discrete memoryless channel (DMC), which is characterized by channel (transition) probabilities. If X={x0, x1, …, xI1} and Y={y0, y1, …, yJ1} denote the channel input alphabet and the channel output alphabet, respectively, the channel is completely characterized by the following set of transition probabilities:

p  y j | xi   P Y  y j | X  xi  , 0  p  y j | xi   1

(7.2)

where i{0,1,…,I1}, j{0,1,…,J1} and I and J denote the sizes of input and output alphabets, respectively. The transition probability p(yj|xi) represents the conditional probability that Y=yj for given input X=xi. Figure 7.3(b) illustrates a discrete channel model with memory [6]. We assume that the optical channel has the memory equal to 2m+1, with 2m being the number of bits that influence the observed bit from both sides. This dynamical trellis is uniquely defined by the combination of previous state, the next state, and the channel output. The state (the bit-pattern configuration) in the trellis is defined as sj=(xj-m,xj-m+1,..,xj,xj+1,…,xj+m)=x[j-m,j+m], where xkX={0,1}. As an example,

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trellis of memory 2m+1=5 is shown in Figure 7.3(b). The trellis has 25=32 states (s0, s1,…, s31), and each of them corresponds to a different 5-bit pattern. To completely describe the trellis, the transition probability density functions (PDFs) p(yj|xj)=p(yj|s), sS can be determined from collected histograms (yj represents the sample that corresponds to the transmitted bit xj, and S is the set of states in the trellis). Unwanted information due to noise, H(XIY)

H(X,Y)

Input information H(X)

H(X)

Optical channel

Output information H(Y)

H(Y) H(X|Y)

I(X;Y)

H(Y|X)

Information lost in channel, H(YIX)

(a)

(b)

Figure 7.4 Interpretation of the mutual information by using: (a) Venn diagrams, and (b) the approach due to Ingels.

One of the most important characteristics of the transmission channel is the channel capacity [30]-[33], which is obtained by the maximization of mutual information I(X;Y) over all possible input distributions: C  max I  X ; Y  , I  X ; Y   H  X   H  X | Y 

 p x 

(7.3)

i

where H(U) =  E(log2P(U)) denotes the entropy of a random variable U and E() denotes the mathematical expectation operator. The mutual information can be determined as I  X ;Y   H  X   H  X | Y   1   p  xi  log 2   p  xi i 1 M

M  N   1    p  y j  p  xi | y j  log 2    p  xi | y j   i 1   j 1

(7.4)

where H(X) represents the uncertainty about the channel input before observing the channel output, also known as entropy, and H(X|Y) denotes the conditional entropy or the amount of uncertainty remaining about the channel input after the channel output has been received. Therefore, the mutual information represents the amount of information (per symbol) that is conveyed by the channel, which represents the uncertainty about the channel input that is resolved by observing the channel output. The mutual information can be interpreted by means of Venn diagram [34] shown in Figure 7.4(a). The left and right circles represent the

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entropy of the channel input and channel output, respectively, while the mutual information is obtained as intersection area of these two circles. Another interpretation is illustrated in Figure 7.4(b) [30]. The mutual information, the information conveyed by the channel, is obtained as the output information minus information lost in the channel. Since for the M-ary input and M-ary output symmetric channel (MSC), we have that p(yj|xi)=Ps/(M1) and p(yj|xj)=1Ps, where Ps is symbol error probability, the channel capacity, in bits/symbol, can be found as  P  C  log 2 M  1  Ps  log 2 1  Ps   Ps log 2  s   M 1 

(7.5)

The channel capacity represents an important bound on data rates achievable by any modulation and coding schemes. It can also be used in comparison of different coded modulation schemes in terms of their distance to the maximum channel capacity curve. 7.1.3 Channel Coding and Information Capacity Theorems We can now formulate the channel coding theorem [3035]. Let a discrete memoryless source, represented by an alphabet S, have the entropy H(S) and emit the symbols every Ts seconds. Let a discrete memoryless channel (DMC) have capacity C and be used once in every Tc seconds. Then, if H(S)/TsC/Tc

(7.6)

there exists a coding scheme by means of which the source output can be transmitted over the channel and reconstructed with an arbitrary small probability of error. The parameter H(S)/Ts is related to the average information rate, while the parameter C/Tc is related to the channel capacity per unit time. For a binary symmetric channel (M=N=2), the inequality (7.6) becomes RC

(7.7)

where R is the code rate introduced above. The information capacity theorem is in literature also known as Shannon’s third theorem [9–30]. The information capacity of a continuous channel of bandwidth B Hz, affected by additive white Gaussian noise (AWGN) of double-sided power spectral density (PSD) equal N0/2 and of bandwidth B, is given by  P  (7.8) C  B log 2 1   [bits/s] N B 0   where P is the average transmitted power. This theorem represents a remarkable result of information theory, because it connects all important system parameters (transmitted power, channel bandwidth, and noise power spectral density) in only one formula.

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7.2 LINEAR BLOCK CODES The linear block code (n,k) satisfies a linearity property, which means that a linear combination of arbitrary two codewords results in another codeword. If we use the terminology of vector spaces, it can be defined as a subspace of a vector space over a finite (Galois) field, denoted as GF(q), with q being the prime power. Every space is described by its basis (a set of linearly independent vectors). The number of vectors in the basis determines the dimension of the space. Therefore, for an (n,k) linear block code, the dimension of the space is n, and the dimension of the code subspace is k. Example 1: The repetition code (n,1) has two codewords x0=(00 … 0) and x1=(11 … 1). Any linear combination of these two codewords is another code word as shown here x 0  x 0  x0 x0  x1  x1  x0 = x1 x1  x1  x0

This operation is a component-wise addition per mod 2, as illustrated in Table 7.1. The set of codewords from a linear block code forms a group under the addition operation, since the all-zero codeword serves as an identity element, and the codeword itself serves as the inverse element. This is the reason why the linear block codes are also called the group codes. The linear block code (n,k) can be observed as a k-dimensional subspace of the vector space of all n-tuples over the binary field GF(2)={0,1}, with addition and multiplication rules given in Table 7.1. All n-tuples over GF(2) form the vector space. The sum of two n-tuples a=(a1 a2 … an) and b=(b1 b2 … bn) forms an n-tuple, and the commutative rule is valid since a+b = (a1+b1 a2+b2 … an +bn)=(b1+a1 b2+a2 … bn +an)=b+a. The all-zero vector 0=(0 0 … 0) is the identity element, while the n-tuple a is an inverse element a+a=0. Therefore, the n-tuples form the Abelian group (see Section 10.11.3) with respect to the addition operation. The scalar multiplication is defined by: a=( a1 a2 … an), F(2). The distributive laws (a+b)=a+b (+)a= a + a,  ,  GF(2) are valid, while the associate law (·)a=·(a) is clearly satisfied. Therefore, the set of all n-tuples is a vector space over GF(2). The set of all codewords from an (n,k) linear block code forms an Abelian group under the addition operation. It can be shown, in a fashion similar to that presented above, that all codewords of an (n,k) linear block codes form the vector space of dimensionality k. There exist k basis vectors (codewords) such that every codeword is a linear combination of these codewords. Example 2: For the (n,1) repetition code, the code book is given by: C={(0 0 … 0), (1 1 … 1)}. Two codewords in C can be represented as a linear combination

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of all-ones basis vectors: (11 … 1)=1·(11 … 1), (00 … 0)= 1·(11 … 1)+1·(11 … 1). Table 7.1 Addition (+) and Multiplication () Rules.

+ 0 1

0 0 1

1 1 0

 0 1

0 0 0

1 0 1

7.2.1 Generator Matrix Let m=(m0 m1…mk-1) denote the k-bit message vector. Any codeword x=(x0 x1 … xn-1) from the (n,k) linear block code can be represented as a linear combination of k basis vectors gi (i=0,1,..,k1) as follows:  g0   g0      g g x  m0 g0  m1 g1  ...  mk 1 gk 1  m  1   mG; G =  1   ...   ...       gk 1   gk 1 

(7.9)

where G is the generator matrix (of dimensions k x n), in which every row represents the basis vector from the coding subspace. Therefore, to be encoded, the message vector m has to be multiplied with a generator matrix G to get the codeword, namely, x=mG. As an example, generator matrices for the (n,1) repetition code and the (n,n1) single-parity-check code are given, respectively, as Gr  11...1

100...01  010...01  Gp   ...    000...11

The code may be transformed into a systematic form by elementary operations on rows in the generator matrix, that is, Gs   I k | P 

(7.10)

where Ik is the unity matrix of dimensions k  k, and P is the matrix of dimensions k x (nk) with columns denoting the positions of parity checks:  p00 p01 ... p0,n k 1    p p11 ... p1,n k 1  P   10  ... ... ...     pk 1,0 pk 1,1 ... pk 1,n k 1 

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The codeword of a systematic code is obtained as x   m | b   m  I k | P   mG , G =  I k | P 

(7.11)

and has the structure as shown in Figure 7.5. The message vector stays unaffected during systematic encoding, while the vector of parity checks b is appended having the bits that are algebraically related to the message bits as follows:

bi  p0i m0  p1i m1  ...  pk 1,i mk 1

(7.12)

where 1, if bi depends on m j pij   0, otherwise 

m0 m1…mk-1 Message bits

b0 b1…bn-k-1 Parity bits

Figure 7.5 Systematic codeword structure. The optical channel introduces the errors during transmission and the received vector r can be represented as r=x+e, where e is the error vector (error pattern) whose components are determined by

1, if an error occured in the ith location ei   0, otherwise  To verify if the received vector r is a codeword one, next we will introduce the concept of a parity check matrix as another useful matrix associated with the linear block codes. 7.2.2 Parity-Check Matrix Let us expand the matrix equation x=mG in a scalar form as follows:

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x0  m0 x1  m1

(7.13)

... xk 1  mk 1 xk  m0 p00  m1 p10  ...  mk 1 pk 1,0 xk 1  m0 p01  m1 p11  ...  mk 1 pk 1,1 ... xn 1  m0 p0, n  k 1  m1 p1,n  k 1  ...  mk 1 pk 1, n  k 1

By using the first k equalities, the last nk equations can be rewritten in terms of the first k codeword elements as follows: x0 p00  x1 p10  ...  xk 1 pk 1,0  xk  0 x0 p01  x1 p11  ...  xk 1 pk 1,0  xk 1  0

(7.14)

... x0 p0,n  k 1  x1 p1,n  k 1  ...  xk 1 pk 1,n  k 1  xn 1  0

The equations presented above can be rewritten through matrix representation as T

 p00 p 10 ... pk 1,0 1 0 ... 0    p p ... p 0 1 ... 0  11 k 1,1  x0 x1 ... xn 1   01 0 ... ... ...    ... x   pk 1,n k 1 0 0 ... 1   p0,nk 1 p1,n k 1 ...  

(7.15)

HT

 xH  0, H   P T

T

I n k 

 n  k  xn

The H-matrix in Equation (7.15) is known as the parity-check one. We can easily verify that G and H matrices satisfy the equation  P  (7.16) P    PP 0  I n k  meaning that the parity-check matrix H of an (n,k) linear block code has rank nk and dimensions (nk)n whose null-space is k-dimensional vector with the basis forming the generator matrix G. Example 1: Parity-check matrices for (n,1) repetition code and (n,n1) singleparity check code are given, respectively, as: GH T   I k

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100...01  010...01  Hr   ...    000...11

H p  11...1

Example 2: For the Hamming (7,4) code, the generator matrix G and paritycheck matrix H are given, respectively, as 1000 |110   0100 | 011  G  0010 |111  0001 |101   

1011 |100  H  1110 | 010     0111 | 001

Every (n,k) linear block code with a generator matrix G and a parity-check matrix H has a dual code, this time having generator matrix H and parity-check matrix G. As an example, (n,1) repetition code and (n,n1) single-parity check code are dual ones. 7.2.3 Code Distance Properties To determine the error correction capability of the linear block code, we have to introduce the concepts of Hamming distance and Hamming weight [31]-[38]. The Hamming distance d(x1,x2) between two codewords, x1 and x2, is defined as the number of locations in which these two vectors differ. The Hamming weight wt(x) of a codeword vector x is defined as the number of nonzero elements in the vector. The minimum distance dmin of a linear block code is defined as the smallest Hamming distance between any pair of code vectors in the code space. Since the all-zero vector is also a codeword, the minimum distance of a linear block code can be defined as the smallest Hamming weight of the nonzero code vectors in the code. We can write the parity-check matrix in a form H=[h1 h2 … hn], where hi presents the ith column in the matrix structure. Since every codeword x must satisfy the syndrome equation xHT=0, the minimum distance of a linear block code is determined by the minimum number of columns in the H-matrix whose sum is equal to zero vector. As an example, the (7,4) Hamming code discussed above has a minimum distance dmin=3 since the sum of first, fifth, and sixth columns leads to zero vector. The codewords can be represented as points in n-dimensional space, as shown in Figure 7.6. The decoding process can be visualized by creating the spheres of radius t around codeword points. The received word vector r in Figure7.6(a) will be decoded as a codeword xi because its Hamming distance d(xi,r)t is closest to the codeword xi. However, in the example shown in Figure 7.6(b), the Hamming distance satisfies the relation d(xi,xj)2t, and the received vector r that falls in

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intersection area of two spheres cannot be uniquely decoded. Therefore, the (n,k) linear block code of minimum distance dmin can correct up to t errors if and only if t 1/2(dmin1) or dmin2t+1 (where · denotes the largest integer smaller or equal to the enclosed quantity). If we are only interested in detecting ed errors, then the minimum distance should be dmined+1. However, if we are interested in detecting ed errors and correcting ec errors then the minimum distance should be dmined+ec+1. Accordingly, the Hamming (7,4) code is a single error correcting and double error detecting code. More generally, Hamming codes are (n,k) linear block codes with following parameters: • Block length: n=2m1 • Number of message bits: k=2mm1 • Number of parity bits: nk=m • dmin=3 where m3. Hamming codes belong to the class of perfect codes, the codes that satisfy the Hamming inequality given as [79, 31] t n 2nk     i 0  i 

(7.17)

This bound gives how many errors t can be corrected with a specific (n,k) linear block code. r

xi t

(a)

r

xji t

xi t

r

xj t

(b)

Figure 7.6 The illustration of Hamming distance: (a) d(xi,xj)2t+1, and (b) d(xi,xj)k0 will not have any zero entry either. The ergodic Markov chains are the most important ones from a communication system point of view. The Markov chain is ergodic if it is possible to move from any specific state to any other state in a finite number of steps with nonzero probability. The Markov chain from Example 1 is nonergodic. It is also interesting to notice that the transition matrix for this example has the following limit

0 1 0  T  lim P   0 1 0  k   0 1 0  k

Example 2: Let us now observe an example of regular Markov chain, which is shown in Figure 9.1(b). The transition matrix P, its fourth and fifth powers, and matrix P limit as k, are given, respectively, as:

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Advanced Optical Communication Systems and Networks

 0 0.25 0 P   0 1 0  0.3750 P 5   0.2500  0.5625

0.75  0.5625 0.0625 0.3750   4 1  , P   0.7500 0 0.2500  ,  0.2500 0.1875 0.5625  0  0.1406 0.4844   0.4444 0.1112 0.4444  0.1875 0.5625  , T  lim P k  0.4444 0.1112 0.4444  k   0.4444 0.1112 0.4444  0.0625 0.3750  We can see that fourth-power has one zero entry, while the fifth power and all higher powers do not have zero entries. Therefore, this Markov chain is both regular and ergodic. We can also notice that the stationary transition matrix T has identical rows. It is evident from Example 2 that for a regular Markov chain the transition matrix converges to stationary transition matrix T with all rows identical to each other:  t1 t T  lim P k   1 k     t1

 tn  t2  tn      t2  tn  t2

(9.24)

In addition, the following is valid:

lim P    lim P   P k  P  T = t1 t2  tn  k

k 

0

0

k 

(9.25)

so we can find stationary probabilities of states (or equivalently solve for elements of T) from equations

t1  p11t1  p21t2    pn1tn t2  p12t1  p22t2    pn 2tn  tn  p1nt1  p2 nt2    pnntn n

t i 1

i

1

For instance, for the Markov chain from Figure 9.1(b), we can write that

t1  t3 , t2  0.25t1 , t3  0.75t1  t2 , t1  t2  t3  1 while the corresponding solution is given by: t1 = t3 = 0.4444, t2 = 0.1112.

(9.26)

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The uncertainty of the source associated with Markov source {S}={S1,…,Sn} when moving one step ahead from an initial state Ai, here denoted as H i1 , can be expressed as n

H i    pij log pij 1

(9.27)

j 1

If the probability associated with state Si is equal to pi, we can obtain the entropy of Markov source by averaging over entropies associated with all states. The uncertainty of moving one step ahead becomes

 

n

n

n

i 1

j 1

H  X   H    E H i    pi H i    pi  pij log pij 1

1

1

i 1

(9.28)

In a similar fashion, the entropy of Markov source for moving k steps ahead from initial states is given by:

    p

H    E H i k

n

k

i 1

H i   k

i 

n

n

n

i 1

j 1

  pi  pij  log pij

 pij k  log pij k 

k

k

(9.29)

j 1

It can be shown that for ergodic Markov sources there is a limit defined as H() = lim H  k  / k . To prove this, we can use the following property (otherwise the k 

subject of Problem 9.3):

H

k 1

 H   H  k

1

(9.30)

By applying this property in an iterative fashion we obtain that

H   H k

k 1

 H   H 1

k 2

 2 H      kH    kH  X  1

1

(9.31)

From Equation (9.31) is evident that:

H  1  H   H X  k  k k

lim

(9.32)

Equation (9.32) can be now used as an alternative definition of entropy of Markov source, which is applicable to an arbitrary stationary source as well. Example 3: Let us determine the entropy of Markov source shown in Figure 9.1(b). By using the definition from Equation (9.28), we obtain that n

n

i 1

j 1

H  X    pi  pij log pij  0.4444  0.25log 0.25  0.75log 0.75   0.1111 1log1  0.4444 1log1  0.6605 bits

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Advanced Optical Communication Systems and Networks

9.2.2 McMillan Sources and Their Entropy McMillan’s description [27] of the discrete source with memory is more general than that of the Markov chain. In this case, we will pay attention to stationary sources. Let S represent a finite source alphabet with the corresponding letters {s1, s2,…,sM}=S. The source emits one symbol at the time instance tk. The transmitted sequence can be represented as X={…,x-1,x0,x1,…}, where xiS. Among all members of ensemble {X}, we are interested only in those having specified source symbols at certain prespecified instances of time. All sequences with these properties create a cylinder set. Example: Let a specified location be defined by numbers -1, 0, 3, and k, with corresponding symbols at these locations equal to x-1=s2, x0=s5, x3=s0, xk=sn-1. The corresponding cylinder set will be given as C1={…, x-2, s2, s5, x1, x2, s0, …, xk= sn-1,…}. Since we observe the stationary processes, the statistical properties of the cylinder will not change if we shift the cylinder for one time unit in either direction (either by T or by T-1). For instance, the time shifted C1 cylinder is given by: TC1={…, x-1, s2, s5, x2, x3, s0, …, xk+1=sn-1,…}. The stationary property for an arbitrary cylinder C can be written as:

P TC  P T 1C  P C

(9.33)

where P{} denotes the probability measure. Let us now specify n letters from alphabet S to be sent at positions k+1, …, k+n. This sequence can be denoted as xk+1, …, xk+n, and there is a total of Mn possible sequences. The entropy of all possible sequences is defined as:

H n   pm  C  log pm  C 

(9.34)

C

where pm() is the probability measure. The McMillan’s definition of entropy of stationary discrete source is given by [27]:

H  X   lim

n 

Hn n

(9.35)

As we can see, the McMillan’s definition of entropy is consistent with Equation (9.32), which is applicable to stationary Markovian sources. 9.2.3 McMillan-Khinchin Model for Channel Capacity Evaluation Let the input and output alphabets of the channel be finite and denoted by A and B respectively, while the channel input and output sequences are denoted by X and

Optical Channel Capacity and Energy Efficiency

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Y. The noise behavior for memoryless channels is generally captured by a conditional probability matrix P{bj|ak} for all bjB and ajA. On the other side, in channels with finite memory (such as the optical channel), the transition probability is dependent on the transmitted sequences up to the certain prior finite instance of time. For instance, the transition matrix for channel described by the Markov process has the form P{Yk=b|…, X-1, X0, X1, …,Xk}= P{Yk=b|Xk}. Let us consider a member x of input ensemble {X}={…,x-2,x-1,x0,x1,…} and its corresponding channel output y from ensemble {Y}={…,y-2,y-1,y0,y1,…}. Let X denote all possible input sequences and Y denote all possible output sequences. By fixing a particular symbol at specific location, we can obtain the cylinder [23]. For instance, the cylinder x4,1 is obtained by fixing a symbol a1 to a position x4, so it is x4,1=…, x-1,x0,x1,x2,x3,a1,x5,… The output cylinder y1,2 is obtained by fixing the output symbol b2 to position 1, namely y1,2=…, y-1,y0,b2, y2,y3,… To characterize the channel, we have to determine transition probability P(y1,2|x4,1), which is the probability that cylinder y1,2 was received if cylinder x4,1 was transmitted. Therefore, for all possible input cylinders SAX, we have to determine the probability that cylinder SBY was received if SA was transmitted. The channel is completely specified by the following: (1) input alphabet A, (2) output alphabet B, and (3) transition probabilities P{SB|SA}=vx for all SAX and SBY. Accordingly, the channel is specified by the triplet [A,vx,B]. If transition probabilities are invariant with respect to time shift T, which means that vTx(TS)=vx(S), the channel is said to be stationary. If the distribution of Yk depends only on the statistical properties of sequence …, xk-1,xk, we say that the channel is without anticipation. If the distribution of Yk depends only on xk-m,…,xk , we say that the channel has finite memory of m units. The source and channel may be described as a new source [C,], where C is the Cartesian product of input A and output B alphabets (C=AxB), and  is a corresponding probability measure. The joint probability of symbol (x,y)C, where xA and yB, is obtained as the product of marginal and conditional probabilities: P(xy)=P{x}P{y|x}. Let us further assume that both the source and the channel are the stationary. Following description presented in [23, 28], it is useful to describe the concatenation of a stationary source and a stationary channel as follows. 1. If the source [A,] ( is the probability measure of the source alphabet) and the channel [A,vx,B] are stationary, the product source [C,] will also be stationary. 2. Each stationary source has an entropy, and therefore [A,], [B,] ( is the probability measure of the output alphabet) and [C,] each have the finite entropies. 3. These entropies can be determined for all n-term sequences x0,x1,…,xn-1 emitted by the source and transmitted over the channel as follows [23]:

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Advanced Optical Communication Systems and Networks

H n  X    x0 , x1 ,..., xn 1 H n Y    y0 , y1 ,..., yn 1 H n  X , Y    x0 , y0  ,  x1 , y1  ,...,  xn 1 , yn 1 

(9.36)

H n Y | X   Y | x0  , Y | x1  ,..., Y | xn 1  H n  X | Y    X | y0  ,  X | y1  ,...,  X | yn 1  It can be shown that the following is valid:

H n  X , Y   H n  X   H n Y | X  H n  X , Y   H n Y   H n  X | Y 

(9.37)

The equations above can be rewritten in terms of entropies per symbol:

1 Hn  X ,Y   n 1 Hn  X ,Y   n

1 1 H n  X   H n Y | X  n n 1 1 H n Y   H n  X | Y  n n

(9.38)

For sufficiently long sequences the following channel entropies exist:

1 1 lim H n  X , Y   H  X , Y  lim H n  X   H  X  n  n n  n 1 1 lim H n Y   H Y  lim H n  X | Y   H  X | Y  n  n n  n 1 lim H n Y | X   H Y | X  n  n

(9.39)

The mutual information also exists and it is defined as

I  X , Y   H  X   H Y   H  X , Y 

(9.40)

The stationary information capacity of the channel is obtained by the maximization of mutual information over all possible information sources:

C  X , Y   max I  X , Y 

(9.41)

using the nonlinear optimization [29, 30]. The results of analysis in this section will be applied in Section 9.4 in evaluation the information capacity of optical channel with memory. Before that, we will briefly describe the adopted model for signal propagation in single-mode optical fibers [31–35], which will also be used in the evaluation of channel capacity.

Optical Channel Capacity and Energy Efficiency

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9.3 MODELING OF SIGNAL PROPAGATION 9.3.1 The Nonlinear Schrödinger Equation (NSE) The propagation over the SMF is governed by the nonlinear Schrödinger equation (NSE), introduced in Chapter 3, which can also be presented in operators form as

E  z, t     2  3  Dˆ  Nˆ E , Dˆ    j 2 2  3 3 , Nˆ  j E 2 z 2 2 t 6 t





(9.42)

where E is the signal electric field; Dˆ and Nˆ denote the linear and nonlinear operators; and , 2, 3, and  represent attenuation coefficient, GVD, secondorder GVD and nonlinear coefficient, respectively. To solve the NSE, the splitstep Fourier method is commonly used [31–33]. The key idea behind this method is to split the fiber into sections, each with length z, and perform the integration of NSE on each section, which leads to the expression





E  z  z , t   exp Dˆ  Nˆ z  E  z, t   

(9.43)

In addition, the Taylor expansion can be used to present exponential term from Equation (9.43) as:









exp Dˆ  Nˆ z   Dˆ  Nˆ n 0



n

z n / n !

(9.44)

Instead of the Taylor expansion, the following two approximations are commonly used in practice:

e Dˆ z / 2 e Nˆ z e Dˆ z / 2 E  z, t  E  z  z , t    Dˆ z Nˆ z  e e E  z, t 

(9.45)

where the first method (the upper arm on the right side of the above equation) is known as the symmetric split-step Fourier method (SSSFM), while the second method (the lower arm on the right side of the above equation) is known as the asymmetric split-step Fourier method (ASSFM). The linear operator in either method corresponds to the multiplication in frequency domain, which is





exp hDˆ E  z, t           FT exp    j 2  2  j 3  3  h  FT  E  z , t    2 6    2  

(9.46)

1

where h is the step-size equal to either z/2 in SSSFM or z in ASSFM, and FT (FT-1) denotes the Fourier transform (inverse Fourier transform) operator. On the

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Advanced Optical Communication Systems and Networks

other side, the nonlinear operator performs the nonlinear phase “rotation” in timedomain, expressed as







exp zNˆ E  z, t   exp jz E  z , t 

2

 E  z, t 

(9.47)

It is evident that the nonlinear operator depends on electric field magnitude at location z, which is not known and should be evaluated. It was proposed in [33] to use the trapezoidal rule and express the electric field function as  E  z  z, t   exp Dˆ z / 2 exp  







z z

 N  z ' dz '  exp  Dˆ z / 2 E  z, t  z

(9.48)

 Nˆ ( z  z )  Nˆ ( z )   exp Dˆ z / 2 exp  z  exp Dˆ z / 2 E  z, t  2  









The iteration procedure that should be applied in accordance with Equation (9.48) is illustrated in Figure 9.2.

E(z, t)

Linear half-step operator exp(D z / 2) Estimate N(z +z) Nonlinear step operator exp{ [N(z)+N(z+z) ] z / 2 }

Linear half-step operator exp(D z / 2) Eprev(z +z, t)

|E-Eprev| / |E|n2

OAM channel N

l=1

l=N-1

1 2 3 4 Coordinates

4N-dim. modulator

(a)

Estimated coordinates 1 2 3 4

OAM mode-demultiplexer OAM channel 1

4N-dim. demodulator

…

OAM channel N

4D Demodulator (polarizationdiversity Rx)

(c) 1

LDPC Decoder D

…

LDPC Decoder 1

…

Bit LLRs Calculation

4N-dimensional APP demapper

…

Local laser

Power splitter

Estimated coordinates Extrinsic LLRs

4N-dimensional demodulator

(b)

4D Demodulator

…

Taper-core

from FMF

Core

n2

…

4D modulator

FMF

l=0

Cladding

Taper-core

…

Power splitter

…

Laser diode

Few-mode fiber (FMF)

OAM mode-multiplexer

D

Figure 9.11 Hybrid 4N-dimensional LDPC-coded modulation scheme: (a) 4N-dimensional transmitter configuration, (b) 4N-dimensional receiver configuration, and (c) 4N-dimensional APP demapper and LDPC decoders.

The N-dimensional signal constellations obtained by either sphere-packing

Optical Channel Capacity and Energy Efficiency

739

method [65] or as a particular instance of EE-OSCD are studied in [66] for use in the few-mode optical fiber system described above for a symbol rate of 25 GS/s. The quasicyclic, girth-10, column-weight-3, LDPC (34665, 27734, 0.8) code is used as a channel code. To precisely estimate the improvement in optical signalto-noise ratio (OSNR) sensitivity with respect to conventional constellations, we performed the Monte Carlo simulations. In this particular instance, codedmodulation is used in combination with polarization-division multiplexing (PDM). Orthogonal OAM modes, properly generated in few-mode fibers, are used as basis functions for N-dimensional signaling. Therefore, 2N degrees of freedom are employed. The BERs of the LDPC-coded ND-OSCDs are evaluated against corresponding M-ary QAM, as shown in Figure 9.12. We can see a superior performance of this scheme with required OSNR below 3 dB, which is confirmation of its energy efficiency. Rs=31.25 GS/s

PDM: QAM:

-2

10

M= 8 M=16 M= 64

-3

Bit-error rate, BER

10

CIPQ:

-4

10

M= 32 ND-OSCD: N=3: M= 8 M=16 M=32 M=64 N=4: M=16 M=32

-5

10

-6

10

-7

10

-8

10

3

4

5

6

7

8

9

10

11

Optical SNR, OSNRb [dB / 0.1 nm] (per bit)

Figure 9.12 LDPC-coded N-dimensional OSCD against LDPC-coded PDM-QAM.

9.7.3 Energy-Efficient Photonic Devices

Considering some general issues related to power consumption and energy efficiency of optical networks, the terms energy efficiency and energy consumption are often mixed up in the literature. Although reducing energy consumption is often required to achieve a high-energy efficiency, minimizing energy consumption should not be always the primary goal. Energy-efficient technologies and concepts are aimed at improving performance and increasing information capacity of the entire network while paying particular attention to the energy consumption, which means providing more functionality while keeping the total energy consumption as low as possible. To obtain a complete picture of this goal, it is crucial to consider the total energy consumption from the system point of view [67, 68], which means that all contributing elements should be taken into account and different concepts should be examined by dimensioning them to provide better energy efficiency ratio (the

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Advanced Optical Communication Systems and Networks

increase in information capacity per energy consumed). For instance, a particular device or technology may provide low power consumption, but at the same time, it may deliver a low performance or its introduction into the system may cause an increased complexity in other subsystems belonging to other network layers or areas. As a consequence, the introduction of such a device or technology in the network could lead to a higher overall network power consumption. Thus, it is beneficial to look at the total power consumption of the entire network and relate it to the network throughput. Consequently, the energy consumption of different network layers and areas should be taken into account, as well as the power consumption of the network itself including the power losses due to inefficiencies of power supply, cooling equipment, and transmission losses in the power grid. Finally, energy efficiency from an optical networking perspective means that advanced optical switching and signal processing systems should be implemented in high-density photonic integration design, and there is still a lot of room for improvement. 9.8 SUMMARY

In this chapter, we described the evaluation of the optical fiber channel capacity by considering the optical channel as a channel with memory. This method consists of two steps, including approximation of probability density functions for energy of pulses and estimation of information capacity. Several models to help with these evaluations have been described, including: (1) linear models suitable for study of various linear channel impairments, and implementation of OFDM and MIMO techniques, and (2) nonlinear propagation models describing intrachannel and interchannel nonlinearities, interaction of ASE noise and signal, and nonlinear interaction of spatial modes. PROBLEMS

9.1 Let an n-dimensional multivariate X=[X1,X2,…,Xn] with a PDF p1(x1,x2,…,xn) be applied to the nonlinear channel with Y=g(X)=[Y1,Y2,…,Yn] denoting the channel output with PDF p2(y1,y2,…,yn). Prove that the output entropy can be determined as:  X , , X n    h Y1 , , Yn   h  X 1 , , X n   E log J  1   Y1 , , Yn   

 X 1 , , X n   is the Jacobian symbol.  Y1 , , Yn 

where J 

9.2 Let the n-dimensional multivariates X=[X1,…,Xn] and Y=[Y1,…,Yn] represent the Gaussian channel input and output, respectively. The input samples are generated from a zero-mean Gaussian distribution with a variance  x2 , while the output samples, spaced 1/2W apart, are generated from Gaussian

Optical Channel Capacity and Energy Efficiency

741

distribution with variance  z2 . Let the input, output, and joint PDFs be denoted by p1  x  , p2  y  and p  x , y  , respectively. By using the mutual information expression:





I X ; Y   p  x , y  log

p  x , y    dxdy p  x  P  y 

derive Equation (9.18). 9.3 The entropy of the Markov source for moving k steps ahead from the initial n

n

i 1

j 1

states is given as H  k    pi  pij k  log pij k  . Prove the following property: k 1 k 1 H   H   H  9.4 Let us observe the Markov source shown in Figure 9.13. Determine the stationary transition matrix and state probabilities. Verify if the source is ergodic. 0.2 S1

0.65 0.8

0.7 0.3

S2

S3

0.35

Figure 9.13 Markov source Example 1.

9.5 For the Markov source from Figure 9.14 determine: (a) Stationary transition and state probabilities. (b) Entropy of this source. 0.7 S1 0.3

0.5 0.5

S3

S2

0.5 0.3

0.5 S4 0.7

Figure 9.14 Markov source Example 2.

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Advanced Optical Communication Systems and Networks

9.6 The Markov state transition matrix is given by: 1/ 3 2 / 3 P  1/ 2 1/ 2 

(a) Determine the state transition diagram. (b) Determine the stationary transition and state probabilities. (c) Determine the entropy of this source (d) Determine how are H(1)and H(2) related to H(3). 9.7 Let us consider the following trivial transition matrix 1 0 0  P  0 1 0  0 0 1  (a) Determine the state transition diagram, (b) Determine the stationary transition matrix. (c) Is this Markov chain regular? (d) Is it ergodic? 9.8 Let us consider the following transition matrix  0 2 / 3 1/ 3  P  3 / 4 1/ 4 0    1/ 4 1/ 4 1/ 2 

(a) (b) (c) (d) (e)

Determine the state transition diagram, Determine the stationary transition matrix. Is this Markov chain regular? Is it ergodic? If the initial probability matrix was [1/3 1/3 1/3], determine the following entropies H(1), H(2), and H(3). Compare H(1)+ H(2) against H(3). (f) Describe a situation when H(1)+ H(2) = H(3). Verify your claim. 9.9 Provide a Markov chain modeling a tennis game. Assume that a tennis player has the fixed probability for winning each point. Assume that deuce is at 30:30. (a) What is the probability for a player to win the game when p=0.75? (b) What is the probability for a player to win the set when p=0.75? (c) Repeat (a) and (b) for p=0.6. 9.10 By using Monte Carlo integration, based on Equation (9.4), reproduce the results shown in Figure 9.4. 9.11 Implement the algorithm described in Section 9.4. Reproduce the results shown in Figure 9.4, and compare them against those from Problem 9.10. 9.12 Plot i.i.d. channel capacity against optical SNR for different memories in forward step of BCJR algorithm, in the presence of both residual chromatic dispersion (11200 ps/nm) and PMD with DGD of 50 ps by observing the NRZ transmission system operating at 10Gb/s. 9.13 Plot the i.i.d. channel capacity in the presence of intrachannel nonlinearities against the number of spans, for dispersion map shown in Figure 9.6, for

Optical Channel Capacity and Energy Efficiency

9.14 9.15 9.16

9.17

9.18 9.19

9.20

9.21

743

different memories in forward step of BCJR algorithm. The span length is set to L=120 km, and each span consists of 2L/3 km of D+ fiber followed by L/3 km of D- fiber. Precompensation of –1600 ps/nm and corresponding postcompensation are also applied. The parameters of D+ and D- fibers, used in simulations, are given in Table 9.1. Assume that the RZ modulation format of a duty cycle of 33% is observed, the extinction ratio is 14 dB, and the launched power is set to 0 dBm. EDFAs with noise figure of 6 dB are deployed after every fiber section, the bandwidth of optical filter (modeled as super-Gaussian filter of eight order) is set to 3Rl and the bandwidth of electrical filter (modeled as Gaussian filter) to 0.7Rl, with Rl being the line rate (defined as the bit rate divided by a code rate). Assume that bit rate is 40Gb/s. Reproduce the results shown in Figure 9.8. Reproduce the results shown in Figure 9.9. Let us now study by the information capacity by using the dispersion map shown in Figure 9.7(a), but replacing EDFAs with a hybrid Raman/EDFA scheme as described in Chapter 4. Study the information capacities in a fashion similar to Figure 9.9. Discuss the improvements with respect to Figure 9.9. Let us now study the information capacity by using the dispersion map shown in Figure 9.7(a), but replacing EDFAs with distributed Raman amplifiers, with typical parameters as described in Chapters 3 and 4. Study the information capacities in a fashion similar to Figure 9.9. Discuss the improvements with respect to Figure 9.9 and Problem 9.16. Repeat Problem 9.17, but taking the quantization effects into account. Observe the information capacities for the following number of bits: 8, 6, 4, and 2. Discuss the results. Implement the model of the MMF transfer function by Yabre [69]. Plot the frequency response for a 100m long PMMA GI-POF for index exponent of 2.1 at a wavelength 850 nm. For fiber model development, use the parameters from [70]. By using power-loading and OFDM, based on Equations (9.19) to (9.21), plot the maximum possible aggregate data rate against SNR for optimum index profiles at following wavelengths: 650 nm, 850 nm, and 1300nm. Assume that the available OFDM signal bandwidth is 25 GHz and the fiber length is 100m. Plot the ergodic channel capacity against SNR for 2 x 2 MIMO over MMF, assuming that MMF can support 500 modes. Assume that channel coefficients follow zero-mean Gaussian distribution. Study different strategies: CSIT, CSIR, and full CSI. Compare the results against the SISO system. Plot the ergodic channel capacity against SNR for different MIMO systems over FMF, assuming that FMF can support 2, 4, and 8 modes. Assume that channel coefficients follow zero-mean Gaussian distribution. Study different

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strategies (CSIT, CSIR, and full-CSI). Compare the results against the SISO system. 9.22 Repeat Problem 9.21, but use the FMF model described in Section 9.6.1. Provide the results for the following MDL values: 0, 10, 15, and 25 dB. Vary the mode differential delay as well. Discuss the results against these from Problem 9.21. 9.23 Repeat Problem 9.22, but use the MIMO-OFDM model described in Section 9.6.2. Compare the results to those obtained in the previous problem. References [1]

Mecozzi, A., and Shtaif, M., “On the capacity of intensity modulated systems using optical amplifiers,” IEEE Photon. Technol. Lett., Vol. 13, pp. 1029–1031, Sept. 2001.

[2]

Mitra, P. P., and Stark, J. B., “Nonlinear limits to the information capacity of optical fiber communications,” Nature, Vol. 411, pp. 1027–1030, June 2001.

[3]

Narimanov, E. E., and Mitra, P., “The channel capacity of a fiber optics communication system: perturbation theory,” J. Lightwave Technol., Vol. 20, pp. 530–537, March 2002.

[4]

Tang, J., “The Shannon channel capacity of dispersion-free nonlinear optical fiber transmission,” J. Lightwave Technol., Vol. 19, pp. 1104–1109, Aug. 2001.

[5]

Tang, J., “The multispan effects of Kerr nonlinearity and amplifier noises on Shannon channel capacity for a dispersion-free nonlinear optical fiber,” J. Lightwave Technol., Vol. 19, pp. 1110– 1115, Aug. 2001.

[6]

Tang, J., “The channel capacity of a multispan DWDM system employing dispersive nonlinear optical fibers and an ideal coherent optical receiver,” J. Lightwave Technol., Vol. 20, pp. 1095– 1101, July 2002.

[7]

Ho, K.-P., and Kahn, J. M., “Channel capacity of WDM systems using constant-intensity modulation formats,” Proc. Opt. Fiber Comm. Conf. (OFC ’02), 2002, paper ThGG85.

[8]

Kahn, J. M., and Ho, K.-P., “Ultimate spectral efficiency limits in DWDM systems,” Optoelectronics Commun. Conf., Yokohama, Japan, 2002.

[9]

Narimanov, E., and Patel, P., “Channel capacity of fiber optics communications systems: WDM vs. TDM,” Proc. Conf. Lasers and Electro-Optics (CLEO '03), 2003, pp. 1666–1668.

[10] Turitsyn, K. S., Derevyanko, S. A., et al., “Information capacity of optical fiber channels with zero average dispersion,” Phys. Rev. Lett., Vol. 91, No. 20, pp. 203901-1203901-4, Nov. 2003. [11] Kahn, J. M., and Ho, K.-P., “Spectral efficiency limits and modulation/detection techniques for DWDM systems,” IEEE Sel. Top. Quantum Electron., Vol. 10, pp. 259–272, March/April 2004. [12] Djordjevic, I. B., and Vasic, B., “Approaching Shannon’s capacity limits of fiber optics communications channels using short LDPC codes,” CLEO/IQEC 2004, paper CWA7. [13] Li, J., “On the achievable information rate of asymmetric optical fiber channels with amplifier spontaneous emission noise,” Proc. IEEE Military Comm. Conf. (MILCOM ‘03), Boston, MA, Oct. 2003. [14] Djordjevic, I. B., Vasic, B., et al., “Achievable information rates for high-speed long-haul optical transmission,” IEEE/OSA J. Lightw. Technol., Vol. 23, pp. 3755–3763, Nov. 2005.

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[15] Ivkovic, M., Djordjevic, I. B., et al., “Calculation of achievable information rates of long-haul optical transmission systems using instanton approach,” IEEE/OSA J. Lightw. Technol., Vol. 25, pp. 1163–1168, May 2007. [16] Ivkovic, M., Djordjevic, I., Rajkovic, P., and Vasic, B., “Pulse energy probability density functions for long-haul optical fiber transmission systems by using instantons and edgeworth expansion,” IEEE Photon. Technol. Lett., Vol. 19, No. 20, pp. 1604–1606, Oct. 15, 2007. [17] Djordjevic, I. B., Alic, N., et. al., “Determination of achievable information rates (AIRs) of IM/DD systems and AIR loss due to chromatic dispersion and quantization,” IEEE Photon. Technol. Lett., Vol. 19, No. 1, pp. 12–14, Jan. 1, 2007. [18] Essiambre, R.-J., Foschini, G., et al., “Capacity limits of information transport in fiber-optic networks,” Phys. Rev. Lett., Vol. 101, pp. 163901–1163901–4, Oct. 2008. [19] Djordjevic, I. B., Minkov, L. L., and Batshon, H. G., “Mitigation of linear and nonlinear impairments in high-speed optical networks by using LDPC-coded turbo equalization,” IEEE J. Sel. Areas Comm., Optical Comm. and Netw., Vol. 26, No. 6, pp. 73–83, Aug. 2008. [20] Djordjevic, I. B., Minkov, L. L., et al., “Suppression of fiber nonlinearities and PMD in codedmodulation schemes with coherent detection by using turbo equalization,” IEEE/OSA J. Opt. Comm. Netw., Vol. 1, No. 6, pp. 555–564, Nov. 2009. [21] Essiambre, R.-J., Kramer, G., et al., “Capacity limits of optical fiber networks,” J. Lightw. Technol., Vol. 28, No. 4, pp. 662–701, Feb. 2010. [22] Cover, T. M., and Thomas, J. A., Elements of Information Theory, New York: John Wiley & Sons, 1991. [23] Reza, F. M., An Introduction to Information Theory, New York: McGraw-Hill, 1961. [24] Gallager, R. G., Information Theory and Reliable Communication, New York: John Wiley & Sons, 1968. [25] Djordjevic, I. B., Ryan, W., and Vasic, B., Coding for Optical Channels, New York: Springer, 2010. [26] Shannon, C. E., “A mathematical theory of communication,'” Bell System Technical Journal, Vol. 27, pp. 379–423 and 623–656, July and Oct. 1948. [27] McMillan, B., “The basic theorems of information theory,” Ann. Math. Statistics, Vol. 24, pp. 196–219, 1952. [28] Khinchin, A. I., Mathematical Foundations of Information Theory. New York: Dover Publications, 1957. [29] Bertsekas, D. P., Nonlinear Programming, Athena Scientific, 2nd edition, 1999. [30] Chong, E. K. P., and Zak, S. H., An Introduction to Optimization, New York: John Wiley & Sons, 3rd edition, 2008. [31] Ip, E., and Kahn, J. M., “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightw. Technol., Vol. 26, No. 20, pp. 3416–3425, 2008. [32] Ip, E., and Kahn, J. M., “Nonlinear impairment compensation using backpropagation,” in Optical Fibre, New Developments, C. Lethien, Ed., In-Tech, Vienna Austria, December 2009. [33] Agrawal, G. P., Nonlinear Fiber Optics, 3rd edition, San Diego: Academic Press, 2001. [34] Djordjevic, I. B., Quantum Information Processing and Quantum Error Correction: An Engineering Approach, AmsterdamBoston: Elsevier/Academic Press, 2012.

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[35] Gordon, J. P., and Kogelnik, H., “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Nat. Academy of Science, Vol. 97, No. 9, pp. 4541–4550, Apr. 2000. [36] Sinkin, O. V., Holzlohner, R., et al., “Optimization of the split-step Fourier method in modeling optical-fiber communications systems,” J. Lightw. Technol., Vol. 21, No. 1, pp. 61–68, Jan. 2003. [37] Rieznik, A., Tolisano, T., et al., “Uncertainty relation for the optimization of optical-fiber transmission systems simulations,” Opt. Express, Vol. 13, pp. 3822–3834, 2005. [38] Zhang, Q., and Hayee, M.I., “Symmetrized split-step Fourier scheme to control global simulation accuracy in fiber-optic communication systems,” J. Lightw. Technol., Vol. 26, No. 2, pp. 302– 316, Jan. 15, 2008. [39] Peric, Z. H., Djordjevic, I. B., et. al., “Design of signal constellations for Gaussian channel by iterative polar quantization,” Proc. 9th Mediterranean Electrotechnical Conference, Vol. 2, pp. 866–869, Tel-Aviv, Israel, 18-20 May 1998. [40] Djordjevic, I. B., Batshon, H. G., et. al., “Coded polarization-multiplexed iterative polar modulation (PM-IPM) for beyond 400 Gb/s serial optical transmission,” Proc. OFC/NFOEC 2010, Paper No. OMK2, San Diego, CA, March 21–25, 2010. [41] Batshon, H. G., Djordjevic, I. B., et. al., “Iterative polar quantization based modulation to achieve channel capacity in ultra-high-speed optical communication systems,” IEEE Photon. Journal, Vol. 2, No. 4, pp. 593–599, Aug. 2010. [42] Ho, K. P., and Kahn, J. M., “Mode-dependent loss and gain: statistics and effect on modedivision multiplexing,” Opt. Express, Vol. 19, No. 17, pp. 16612–16635, 2011. [43] Arnold, D., Kavcic, A., Loeliger, H.-A., et al., “Simulation-based computation of information rates: upper and lower bounds,” Proc. IEEE Intern. Symp. Inform. Theory (ISIT 2003), 2003, pp. 119. [44] Arnold, D., and Loeliger, H.-A., “On the information rate of binary-input channels with memory,” Proc. 2001 Int. Conf. Communications, Helsinki, Finland, June 11–14, 2001, pp. 2692–2695. [45] Pfitser, H. D., Soriaga, J. B., and Siegel, P. H., “On the achievable information rates of finite state ISI channels,” Proc. Globecom 2001, San Antonio, TX, 2001, pp. 2992–2996. [46] Shieh, W., and Djordjevic, I. B., Optical Orthogonal Frequency Division Multiplexing, AmsterdamBoston, Elsevier/Academic Press, 2009. [47] Shieh, W., Yi, X., et al., “Coherent optical OFDM: has its time come?,” J. Opt. Netw., Vol. 7, pp. 234–255, 2008. [48] Bölcskei, H., Gesbert, D., and Paulraj, A. J., “On the capacity of OFDM-based spatial multiplexing systems,” IEEE Trans. Comm., Vol. 50, pp. 225–234, 2002. [49] Wang, J., Zhu, S., and Wang, L., “On the channel capacity of MIMO-OFDM systems,” Proc. International Symposium on Communications and Information Technologies 2005 (ISCIT 2005), pp. 1325–1328, 2005, Beijing, China. [50] Shen, G., Liu, S., Jang, S.-H., and Chong, J.-W., “On the capacity of MIMO-OFDM systems with doubly correlated channels,” Proc. IEEE 66th Vehicular Technology Conference 2007 (VTC-2007), pp.1218-1222, 2007. [51] Goldsmith, A., Wireless Communications, Cambridge: Cambridge University Press, 2005. [52] Winzer, P. J., and Foschini, G. J., “Outage calculations for spatially multiplexed fiber links,” in Proc. Opt. Fiber Commun. Conf. (OFC), Paper OThO5, 2011.

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[53] Winzer, P. J., and Foschini, G. J., “MIMO capacities and outage probabilities in spatially multiplexed optical transport systems,” Opt. Express, Vol. 19, pp. 16680–16696, 2011. [54] Shah, A. R., Hsu, R. C. J., et al., “Coherent optical MIMO (COMIMO),” J. Lightw. Technol., Vol. 23, No. 8, pp. 2410–2419, Aug. 2005. [55] Hsu, R. C. J., Tarighat, A., et al., “Capacity enhancement in coherent optical MIMO (COMIMO) multimode fiber links,” J. Lightw. Technol., Vol. 23, No. 8, pp. 2410–2419, Aug. 2005. [56] Essiambre, R., and Tkach, R.W., “Capacity trends and limits of optical communication networks,” Proceedings of the IEEE, Vol. 100, No. 5, pp. 1035–1055, May 2012. [57] Djordjevic, I. B., “Energy-efficient spatial-domain-based hybrid multidimensional codedmodulations enabling multi-Tb/s optical transport,” Opt. Express, Vol. 19, No. 17, pp. 16708– 16714, 2011. [58] Vereecken, W., Van Heddeghem, W., et al., “Power consumption in telecommunication networks: overview and reduction strategies,” IEEE Comm. Mag., Vol.49, No.6, pp. 62–69, June 2011. [59] Djordjevic, I. B., Xu, L., and Wang, T., “Statistical physics inspired energy-efficient codedmodulation for optical communications,” Opt. Letters, Vol. 37, No. 8, pp. 1340–1342, 2012. [60] Aleksić, S., “Energy efficiency of electronic and optical network elements,” IEEE J. Sel. Top. Quantum Electron., Vol.17, No. 2, pp. 296–308, March–April 2011. [61] Djordjevic, I. B., “Spatial-domain-based hybrid multidimensional coded-modulation schemes enabling multi-Tb/s optical transport,” IEEE/OSA J. Lightw. Technol., Vol. 30, No. 14, pp. 2315– 2328, July 15, 2012. [62] Cvijetic, M., Djordjevic, I. B., and Cvijetic, N., “Dynamic multidimensional optical networking based on spatial and spectral processing,” Opt. Express, Vol. 20, No. 8, pp. 9144–9150, 2012. [63] Wannier, G., Statistical Physics, New York: Dover Publications, 1987. [64] McDonough, R. N., and Whalen, A. D., Detection of Signals in Noise, 2nd. ed., San Diego: Academic Press, 1995. [65] Sloane, N. J. A., et al., “Minimal-energy clusters of hard spheres,” Discrete Computational Geom., Vol. 14, pp. 237–259, 1995. [66] Djordjevic, I. B., Liu, T., et al., “On the multidimensional signal constellation design for fewmode fiber based high-speed optical transmission,” IEEE Photonics Journal, Vol. 4, No. 5, pp. 1325–1332, 2012. [67] Fehratovic, N., and Aleksic, S., “Power consumption and scalability of optically switched interconnects for high- capacity network elements,” Proc. Optical Fiber Communication Conference and Exposition (OFC 2011), Paper JWA84, Los Angeles, CA, March 2011. [68] Aleksic, S., ”Electrical power consumption of large electronic and optical switching fabrics,” Proc. IEEE Winter Topicals 2010, pp. 95–96, Majorca, Spain, 2010. [69] Yabre, G., “Theoretical investigation on the dispersion of graded-index polymer optical fibers,” J. Lightw. Technol., Vol. 18, pp. 869–877, 2000. [70] Ishigure, T., Koike, Y., and Fleming, J. W., “Optimum index profile of the perfluorinated polymer-based GI polymer optical fiber and its dispersion properties,” J. Lightw. Technol., Vol. 18, pp. 178–184, 2000.

Chapter 10 Engineering Tool Box This chapter contains material related to physical phenomena and the mathematical treatment of the topics discussed in previous chapters, and should help the reader to better understand the subjects associated with optical transmission systems and networks. Well established references [1–16] have been mainly used in preparation of this chapter. 10.1 PHYSICAL QUANTITIES AND UNITS USED IN THIS BOOK The physical quantities and physical constants mentioned throughout this book, as well as their units, are presented in Table 10.1 and Table 10.2, respectively. Table 10.1 Physical Quantities and Units Physical quantities Units Quantity Symbol Unit Unit Symbol Length L, l Meter m Mass m Kilogram kg Time t Second s Temperature Kelvin K  Electric current I Ampere A Hertz Hz Frequency f, (if optical) Wavelength Micron m  Force F Newton N Energy E Joule J Power P Watt W Electric charge q Coulomb C Electric voltage V Volt V Resistance R Ohm  Capacitance C Farad F Magnetic flux Webber Wb  Magnetic inductance B Tesla T Electric inductance D Henry H Electric field E Magnetic field H -

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Dimensions m kg s K A 1/s 10–6 m Kg · m/s2 N·m J/s A·s J/C V/A C/V V·s Wb/m2 Wb/A V/m A/m

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Typical value

Electron charge

q

1.61 x 10–19 C

Boltzmann’s constant

k

1.38 x 10–23 J/K

Plank’s constant

h

6.63 x 10–34 J/Hz

Raman gain coefficient at =1.55m

gR

7 x 10–13 m/W

Brillouin gain coefficient=1.55m

gB

5 x 10–11 m/W

Nonlinear refractive index coefficient

n2

2.2-3.4 x 10–8 m2/W

Nonlinear propagation coefficient



Fiber attenuation



0.9-2.75 (W·km) –1 at =1.55 m ~0.2 dB/km at =1.55 m

Photodiode responsivity

R

0.8 A/W

Chromatic dispersion in SMF

D

~17 ps/nm·km at =1.55 m

GVD parameter

2

–20 ps2/km at =1.55 m

Permeability of the vacuum

0

4·10–7 H/m

Permittivity of the vacuum

0

Light speed in vacuum

c=(00)

8.854 · 10–12 F/m –1/2

2.99793 · 108 m/s

10.2 FREQUENCY AND WAVELENGTH OF THE OPTICAL SIGNAL The relationship between frequency  and wavelength of an optical signal is

  c /

(10.1)

where c is the light speed in vacuum. The relationship between the frequency band and the wavelength band (i.e., the spectral width) of an optical signal can be obtained by expanding Equation (10.1) in Taylor series around the central value =c/ of the wavelength band, and by keeping just the two terms of the expansion, that is

  0   

2 (   0 ) c



2  c

(10.2)

This equation can be used to establish a relationship between the wavelength band and the frequency band  for any specific carrier wavelength . For example, frequency band  = 50 GHz corresponds to the wavelength band = 0.4 nm in the vicinity of the wavelength =1.55 m.

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10.3 STIMULATED EMISSION OF THE LIGHT

The process of the emission and absorption of light can be explained by using the basic quantum mechanics model presented in Figure 10.1 [1, 3]. The electrons are associated with their energy level, which means that each electron can be attributed with a specific energy level. E

N2

E2

N2

 B12

E2

E

E1

N1

E1

N1 N

N Thermal equilibrium

Population inversion

Figure 10.1 Energy levels and population inversion.

The population of electrons is associated with the energy band that contains multiple energy levels. The lower energy levels are better populated in normal conditions than higher energy levels. Therefore, in normal conditions it is valid that N1 > N2 if E1 < E2, where N1 and N2 are the number of electrons at the energy levels E1 and E2, respectively. The electrons can move in energy from level to level, by either absorbing the photons (when moving from lower to higher levels), or by radiating photons (while moving from higher to lower levels). The normal conditions are related to thermal equilibrium, where the ratio between the numbers N1 and N2 can be expressed by the Boltzmann’s formula N2  E  E1   h   exp   2   exp    N1 k    k 

(10.3)

where k is the Boltzmann constant,  is the absolute temperature, h is the Planck constant, and  is the optical frequency that is proportional to the difference between the energy levels. Electrons can move from lower to upper energy levels through absorption of the incoming photons, while the movement from the upper to lower energy levels can occur through either stimulated or spontaneous emission, or both of them simultaneously. During the stimulated emission, a downward transition occurs under the influence of the outside photons that penetrate to the region. Newly radiated photons are at the same frequency, phase,

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and polarization as the incoming ones. The spontaneous emission occurs without any outside influence, through a stochastic downward energy transition. The rates of light absorption, stimulated emission and spontaneous emission, all with respect to the two-level system from Figure 10.1, can be written as dN 1,abs dt dN 2,stim dt

dN2, sp dt

 B12  ( ) N 1

(10.4)

 B21  ( ) N 2

(10.5)

 A21N 2

(10.6)

where  is the spectral density of the electromagnetic energy, while coefficients B12, B21, and A21 characterize the absorption, stimulated emission, and spontaneous emission, respectively. The subscripts “abs,” “stim,” and “sp” stand for “absorption,” “stimulated,” and “spontaneous,” respectively. In thermal equilibrium, the upward and downward transitions are equalized, and the following equation is valid A21 N 2  B21  ( ) N 2  B12  ( ) N 1

(10.7)

The spectral density  can be obtained from Equations (10.3) and (10.7) as

 ( ) 

A21 / B21 ( B12 / B21 ) exp(h / k)  1

(10.8)

The coefficients A21, B21, and B12 can be evaluated by comparing Equation (10.8) with the following equation, otherwise known as the Planck’s equation [1]  ( ) 

8h 3 / c 3 exp( h / k)  1

(10.9)

It is therefore A21 

8h 3 B21 c3

B21  B12

(10.10) (10.11)

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Equations (10.10) and (10.11) were first established by Einstein and are known as the Einstein’s coefficients. Spontaneous emission dominates over stimulated emission in a normal situation known as thermal equilibrium. Stimulated emission can become the dominant process only if it overcomes the absorption, which can be achieved if N2>N1. This condition, known as the population inversion, is not a normal state that can be achieved by the thermal equilibrium. The population inversion, which is a prerequisite for stimulated emission of the light, can be achieved through an external process. That means that an external energy, usually in an optical or electrical form, is introduced to excite the electrons and to lift them from the lower to the upper energy level. The scheme presented in Figure 10.1 is the simplest two-level one. However, more complex three- and four-level energy schemes are commonly used in practice. In these schemes the electrons are lifted from the lowest level, often called the ground state, to one of the upper energy levels by skipping the intermediate energy levels. The electrons do not stay at the upper energy level, but rather move to lower intermediate levels through nonradiative energy decays. Therefore, the intermediate energy level, known as the metastable one, serves as a base for population inversion, and the number N2 is commonly associated with the metastable level. 10.4 BASIC PHYSICS OF SEMICONDUCTOR JUNCTIONS

Semiconductors are materials that can easily accommodate the population inversion, but the total picture is more complex than the simplified two-level atomic system presented in Figure 10.1. The atoms in semiconductors are close enough to each other to interact and to shape the distribution of energy levels in a way that is semiconductor compound specific. As for semiconductor properties, only the two highest energy bands, the valence band and the conduction band, are of real importance. These bands are separated by the energy gap, or forbidden band, where no energy levels exist. If electrons are excited by some means (for example, optically or thermally), they move in energy from the valence to the conduction band. For each electron lifted to the conduction band, there is one empty spot in the valence band. This vacancy is called a hole. Both the electrons from the conduction band and holes can move under the impact of an external electric field and contribute to the current flow through the semiconductor crystal. If semiconductor crystal contains no impurities, it is known as an intrinsic material. Such materials are, for example, silicon and germanium, which belong to the IV-th group of elements, and have four electrons in their outer shell. By these electrons an atom makes the covalent bonds with neighboring atoms. In this environment, some electrons can be excited to the conduction band due to thermal vibrations of the atoms in the crystal. Therefore, the electron-hole pairs can be generated by pure thermal energy. There might be an opposite process as well, when an electron releases its energy and

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drops into a free hole in the valence band. This process is known as the electronhole recombination. The generation and recombination rates are equal in thermal equilibrium. The conduction capability of intrinsic semiconductor materials can be increased through the doping process by adding some impurities from either group V or group III of the elements. Group V is characterized by having five electrons in the other shell, while there are just three electrons in the outer shell for the elements from group III. If group V elements are used, four electrons are engaged in covalent bonding, while the fifth electron is loosely bound and is available for conduction. The impurities from group V elements are called donors since they give an electron. They are also known as n-type semiconductors, since the current is carried by electrons (n stands for negative charge). On the other side, by adding atoms from group III of the elements, three electrons will create the covalent bonds, while a hole will be created in the environment where other atoms have four electrons engaged in covalent bonds. The conduction property of the hole is the same as the property of the electron at the conduction level. It is because an outside electron will eventually occupy the hole. That electron will leave the hole at its original place, which will be occupied by some other electron leaving the hole at its original place, and so on. Therefore, the process of occupation and reoccupation is moving along the crystal. The impurities from group III of the elements are called donors, since they accept an electron. They are also known as p-type semiconductors, since the current is carried by holes (p stands for positive charge characterized by the missing electron). By adding donor or acceptor impurities, an extrinsic semiconductor is formed. Each extrinsic semiconductor has majority carriers, which are the electrons in n-type and the holes in p-type, and minority carriers (holes in n-type and electrons in p-type). While both n-type and p-type semiconductors can serve as conductors, the true semiconductor-based device is formed by joining both types in a single continuous crystal structure. The junction between two regions, known as the p-n junction, is responsible for all useful electrical characteristics of semiconductors. The p-n junction is characterized by several important phenomena. First, the holes in p-type will diffuse towards the p-n junction to neutralize the majority of electrons present at the other side of the border. This will leave a small region in p-type that is close to the p-n junction with a smaller number of holes than in the rest of the p-type region. Therefore, the electrical neutrality is effectively disrupted, since bonded electrons are still in place. Consequently, a small negatively charged region close to p-n border is created in p-type semiconductor. Just the opposite will happen at the n-side of the p-n junction, where the electron shortage will occur and a positively charged region will be created.

Engineering Tool Box

Incoming photons Photoelectric effect

Depletion region

p-type

+ - + - + - +_ + + ++ + + + + -+ + - - + - + + Ebias

n-type

_ _ +_ + __+ + + + _ _ ___ + + _ _+ _ _ _ _ + + _ + _ +_ Ebar

Outgoing photons

p-type

Fermi level

_ _ _ _ ___ __ _ ___ _ _ _

+

-

_+ -

Carrier Recombination n-type

_ _ + + __+ + + + _ _ ___ _+ + _ _+ _ _ _ _ + + _ + _ +_

Ebar

+

_

+ + + + +++++++

-

Ebias

Conduction band

_

Depletion region

+ - + -+ + + ++ + + + + -+ + - - + + +

+

_ _ _

755

+

Valence band

+

+

Fermi level

Conduction band

__

_ _ __ _ __ ___ __ _ _ _ _ __

+ ++ + + + + + + + + + + + ++

+ Length

Length

Photodetector mode

+

Valence band

Source mode

Figure 10.2 p-n junction operating in the photodiode mode and in the light source mode.

The carrier’s transition will effectively establish a potential barrier at the p-n junction. The potential barrier can be characterized by the electric field vector Ebar with the direction from n-type to p-type, as shown in Figure 10.2. Any positive charge in n-type, which happens to be close to the p-n junction, is attracted to move towards p-side under the impact of the electric field Ebar. The electric field Ebar will restrict further diffusion of the majority carriers from their native regions (electrons from n-type and holes from p-type). If there is no external voltage applied, this situation will stay, and there is no organized current flow through the junction. The width of the potential barrier and the strength of the electric field Ebar is determined by the concentration of dopants in n-type and p-type semiconductors. The situation will change if an external bias voltage is applied to the p-n junction. If the reverse bias voltage is applied, the external electric field Ebias has the same direction as the internal electric field Ebar and enhances the restriction already imposed. Therefore, the width of the depletion region is increased when pn junction is under reverse bias, and there is no current flow. (The exception is a small leaky current due to the presence of minority carriers in p-type and n-type regions.) If some electrons appear in the depletion region due to thermal activity, they are immediately separated by the electric field Ebias. The end result will always be such that electrons go towards n-type and holes towards p-type. This reverse bias of the p-n junction corresponds to the photodiode mode. The static situation will change if there is the illumination of the depletion region with the

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incoming light signal, since the generation of the carriers (both the electrons and holes) will be initiated. The pairs of electron holes will be created due to the photoelectric effect, as shown in Figure 10.2, but they will be immediately separated by a strong electric field Ebias. The opposite situation occurs if a forward bias voltage is applied to the p-n junction since the external electric field Ebias will be opposite to the internal field Ebar. This situation will result in the reduction in the potential barrier, which means that the electrons from n-type and holes from p-type can flow and cross the p-n junction much more easily. Consequently, the free electrons and holes can be simultaneously present in the depletion region. They can also recombine together and generate the photons due to energy release, as shown in Figure 10.2. The forward biasing corresponds to semiconductor light sources (light emitting diodes and lasers). The rate and the character of recombination will eventually determine the character of the output light signal. Therefore, from the energy perspective, the bias voltage can either enhance the difference in energy bands between p-type and n-type semiconductor compounds, which happens in photodiode mode, or flatten the energy band difference, which occurs in the light source mode. The energy levels for these two operational modes are shown in lower part of Figure 10.2. The energy level called the Fermi level [2] is also shown in Figure 10.2. The meaning of this level will be explained shortly. The conditions related to the light generation and laser emission in semiconductors can be better understood by explaining the conditions for population inversion. The probability that the electron occupies a specified energy level E when the system is in thermal equilibrium is given by the Fermi-Dirac distribution as p( E ) 

1  E  EF exp  k

(10.12)   1 

where  is the absolute temperature, k is the Boltzmann constant, and EF is the Fermi energy level. The Fermi energy level is a parameter that indicates the distribution of electrons and holes in the semiconductor. This level is located at the center of the bandgap in an intrinsic semiconductor that is in the thermal equilibrium, which means that there is a smaller probability that the electron will occupy the conduction band. The probability will be higher if the temperature is increased. The position of the Fermi level varies in different semiconductor types. The Fermi level in n-type semiconductors is raised to a higher position, which increases the probability that electrons will occupy the conduction band. On the other hand, in p-type semiconductors, the Fermi level is lowered to below the center of the bandgap, as shown in Figure 10.2. The probability that electrons will occupy the energies E1 and E2, associated with valance and conduction bands, respectively, can be expressed by using Equation (10.12) as

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1 E  E F ,val     1 exp  1 k   1 pcond ( E2 )   E  EF ,cond  exp 2   1 k  

p val ( E1 ) 

757

(10.13)

(10.14)

The Fermi levels EF,cond and EF,val are related to the conduction and valence bands, respectively. The Fermi level can be moved up, which is relevant to the ntype semiconductors, or moved down in p-type semiconductors, by increasing the dopant concentration. In some cases, the Fermi level can be positioned within conduction band for n-type, or within valence band for p-type semiconductors, if they are heavily doped, or if the population inversion is achieved. The population inversion can be achieved by an electrical current that populates the conduction band with a rate that is higher than the rate at which the band is emptied. The population inversion and position of Fermi levels in valence and conductance bands is illustrated in Figure 10.3. It is done through E-k diagram, where E is the energy and k is the wavespace vector [1]. The situation shown in Figure 10.3 represents direct bandgap semiconductors, where the minimum energy point in the conduction band with respect to parameter k coincides with the maximum energy point in the valence band. Such semiconductors are, for example, GaAs, InP, InGaAsP, and AlGaAs, all of them often used for semiconductor laser manufacturing. There is another type of semiconductors in which the minimum energy point in the conduction band does not coincide with the maximum energy point in the valence band. These semiconductors are known as the indirect bandgap semiconductors. Both silicon and germanium belong to this group. The electronhole recombination in this case cannot be done without an adjustment in momentum (i.e., in the wavespace vector k). It is done through the crystal lattice vibration and creation of phonons or the heat. The efficiency of the light generation will be smaller, since the part of the energy is lost, which is the reason why these semiconductors are not good materials for light sources. Once the population inversion in semiconductor material is achieved, the stimulated emission can take a place. The electrons will fall back in energy to the valence band and recombine with holes. Each recombination should produce a photon of the light signal, due to the energy release. The stimulated emission is achieved if the difference between the Fermi levels EF,con – Ef,val is larger than the energy bandgap E2 – E1. The difference between these two Fermi levels becomes larger with an intensive pumping by the forward bias current that flows through the p-n junction.

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Conduction band EF,cond

Electrons

E2

E1 Holes EF,val

-

-

-

-

- -

-

+ + + + + + + + + + + + Valence band k

Figure 10.3 Energy bands and population inversion in semiconductors.

The condition (EF,con – Ef,val) > (E2 – E1 ), which is needed for stimulated emission, is observed if the direct bias current is higher than some threshold current. Any specific doping of p-type and n-type layers will have an impact on the position of the Fermi levels. Several special doping schemes can be applied to influence the stimulated radiation process (multistructured semiconductor lasers mentioned in Chapter 2.3 are manufactured for this purpose). The difference E = E2–E1 between the energy levels in conduction and valence bands will determine the wavelength and the frequency of the optical radiation. It is expressed through the well-known equation

h  E  E2  E1

(10.15)

where h is the Planck’s constant, and  is the frequency of optical radiation. Therefore, the output frequency is determined by the semiconductor energy bands that are specific for each semiconductor structure. It is also important to notice that the laser regime should be maintained through the proper confinement of the energy in the active region. It is often done through heterojunctions that form a complex waveguide structure in the active region as discussed in Chapter 2. 10.5 BASIC VECTOR ANALYSIS

Both the rectangular system of coordinates (x, y, and z) and the cylindrical system of coordinates (r, and z) will be used to express the basic relations of vector analysis. The corresponding unit vectors can be denoted by ex, ey, and ez, for

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rectangular system, and by er, e, and ez for cylindrical system. Both systems of coordinates are mutually related through relationships cos ;

sin ;

(10.16)

The basic vector operators expressed first in rectangular system of coordinates and after that in cylindrical system are given below: Gradient: +

+

+



(10.17)

+

(10.18)

Divergence: +

+ +



(10.19) +

(10.20)

Curl:

×

(10.21)



×

(10.22)

Laplacian:  +

+ +

(10.23) +

(10.24)

Identities: (

(



(10.25) (10.26)

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10.6 BESSEL FUNCTIONS

The Bessel function of the first kind of order m and argument z, commonly denoted by Jm(z), is expressed as 

 







(10.27)

or  

sin

cos





(10.28)

where m is any integer, and n is a positive integer or zero. Bessel function can be expanded in power series as follows ∑

!

(10.29)

!

For two special cases, where m=0 and m=1, above equation above becomes 2

1

4 1!





2



!

4 2

4 2! 2

! !



6 3

4 3! 2

⋯

⋯

(10.30)

(10.31)

The following recurrence relations are valid 2

(10.32) (10.33)



(10.34)

In addition to class J of Bessel functions, another class called K or modified Bessel functions is recognized and expressed as 

∞ 0

cosh

sinh2

where (·) is known as the Gamma function [14] defined as

(10.35)

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t

(10.36)

For an integer value n, and some fractions around 1 (i. e. 1/2 and 3/2), it is 1

!



1/2

(10.37)

/

 /

3/2

1.77245

(10.38)

0.88632

(10.39)

The following recurrence relations are valid for modified Bessel functions 2

(10.40) (10.41)



where Qm function is given as Qm = ejmKm. Both Bessel and modified Bessel functions can be approximated by the asymptotic expansions for some conditions, such as ≈

, for fixed m (m ≠ -1, -2, -3) and z → ∞





/  



cos

/

1







, for fixed m and abs{z} → ∞

!

⋯

(10.42) (10.43) (10.44)

The above relation is valid for fixed m and large abs{z} → ∞. 10.7 MODULATION OF AN OPTICAL SIGNAL

A monochromatic electromagnetic wave, which is used as a signal carrier, can be represented through its electric field as [10]

E (t )  pA cos(t   )

(10.45)

where A is the wave amplitude,  is the radial frequency, is the phase of the carrier, and p represents the polarization orientation. Each of these parameters (amplitude, frequency, phase, and the polarization state) can be utilized to carry information. It is done by making them time dependent and related the the

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information content. Accordingly, four basic modulation types can be recognized. They are amplitude modulation (AM), frequency modulation (FM), phase modulation (PM), and polarization modulation (PoM). If the information is in digital form, the modulation is referred to as a shift-keying. Therefore, there are the amplitude shift keying (ASK), the frequency shift keying (FSK), the phase shift keying (PSK), and the polarization shift keying (PoSK). 10.8 DIGITAL-TO-ANALOG AND ANALOG-TO-DIGITAL CONVERSION

Although there are a number of signals that are in analog form, digital representation often provides the best way of their processing. One of the reasons is that digital systems have memory capability that would be difficult to implement if the signal has an analog form. In addition, it is easier to perform programmable functions by different software available. Any digital processing requires conversion from the analog-to-digital form, known as analog-to-digital conversion (ADC). At the same time, some digitally processed results may be needed in an analog form by using the digital-to-analog conversion (DAC) process. Both processes are illustrated in Figure 10.4. Analog Processing

Analog signal

ADC

Analog signal

DAC

Digital signal

Digital Processing

Digital signal

Figure 10.4 ADC and DAC processes.

A digital-to-analog converter accepts a digital input and generates analog output in the form of an electrical signal (voltage or current). The n-bit binary digital word Dn-1 Dn-2 …D4D3D2D1D0 will be converted to a voltage 2

2

⋯

2

2

∆

(10.46)

The subscript q emphasizes the fact that we have a discrete or quantized voltage value. The value ∆ is known is the increment or step size because the total

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voltage increases by ∆ when the last significant bit changes from 0 to 1. The minimum voltage value in accordance with Equation (10.46) is 0, while the maximum is (2n – 1)u, which corresponds to a decimal integer equal (2n –1). We have the opposite case during the analog-to-digital conversion, but it is less direct since analog input may take any value within some range, while a digital output can take only a limited number of digital levels. Because of that, any ADC includes round-off and quantizing errors. If we assume that the analog input voltage has a value in the range of ua ∊[0, um] and that we would like to represent it by n-bit binary word, the input range has to be divided into 2n intervals. A quantized value uq is then defined by the midpoint of each interval, and a codeword Dn–1Dn–2 …D4D3D2D1D0 will be assigned to represent each quantized value. Each interval has the size u = um/2n.

(10.47)

The maximum quantization error equals uq = ∣um – uq∣, and will be either u/2 or u, depending on the position of the initial step. As an example, the case where n=2 and um =8 is shown in Figure 10.5. ua uq uq

D1D0

7

1 1

7

5

1 0

5

3

0 1

3

1

0 0

1 0

6.0 4.0 2.0 0.0

11 10 01

u

8.0

00 0

2

4

6

8

ua

Figure 10.5 ADC process of an analog voltage.

As we can see from Figure 10.5, ADC makes a staircase-like approximation of the analog input, with the total of 2n = 4 stair levels. The maximum quantizing error for this specific case is equal u/2 = ±1. The position of the bottom step can be changed to be 0, but then the maximum quantizing error would be u = 2. The only way to minimize the quantizing error is by increasing the number of quantizing levels, which means that n (or the length of the codeword) should increase. If we increase the codeword from 2 bits to 3 bits in the case we just described, the total range will be divided into 2n = 8 intervals and that total quantizing error would be ±0.5u. Since ADC process cannot be continuous in practice, it is important to have a sampling speed that would serve the purpose, which means that ADC should capture the not just the variations in the amplitude, but also variations in the time. The faster changing signals would require a higher sampling rate, and vice versa.

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The sampling or data acquisition process is most often done in digital instrumentation systems (digital scope for example) or during conversion of the row information signals (voice, video), captured by an analog electrical form. Since the variations in time of any signal are characterized by its frequency spectrum, the acquisition is done with a frequency that is comparable with the highest significant frequency in the signal spectrum (the frequency where the portion of the total signal power cannot be neglected). It is important to mention that the ADC acquisition process for an electric signal representing the row information should be done with a minimum sampling frequency fs, which complies with the well-known Nyquist criterion defined as ≥ 2·

(10.48)

where fm is the highest significant frequency in the signal spectrum. It is commonly assumed that fm= Bsig (where Bsig is signal bandwidth). 10.9 OPTICAL RECEIVER TRANSFER FUNCTION

The detection of an optical signal by a photodiode in the optical receiver is followed by amplification and signal filtering. These functions enhance the signal level and limit the noise power. In addition, equalization is usually applied to recover the pulse shape and to suppress the intersymbol interference. Mathematical treatment of these processes is often done in the frequency domain. The photocurrent signal I(t) can be transferred to the frequency domain by Fourier transform as [8, 9]

~ I ( ) 



 I (t ) exp( jt )dt

(10.49)



Please recall that superscript (~) denotes the frequency domain. The inverse Fourier transform converts signals from frequency to time domain as  1 ~ (10.50) I (t )  I ( ) exp(  jt ) d 2  The current signal is converted to a voltage signal at the receiver front-end— please see Figure 2.32. The voltage signal if further amplified by main amplifier and processed by the filter, before it goes to the clock recovery and decision circuits. The output voltage Vout(t) that comes to the decision circuit can be expressed in the frequency domain as [12, 17] ~ I ( ) ~ ~ (10.51) Vout ( )  H F  end ( ) H amp ( ) H filt ( )  I ( ) H rec ( ) Y ( ) where Y) is the input admittance, which is determined by the load resistor and input of the front-end. Transfer functions HF-end(), H)amp, and H()filt are

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referred to as the front end, main amplifier, and filter/equalizer, respectively. The function Hrec() is related to the total transfer properties of the optical receiver. The intersymbol interference from the neighboring pulses is minimized, or possibly removed, if the output voltage signal takes the shape of raised cosine function, that is

   , 1 ~ Vout ( )  1  cos  2  2 B  

for = f