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Photodetectors: Devices, Circuits and Applications [2 ed.]
 1119769914, 9781119769910

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Photodetectors

IEEE Press 445 Hoes Lane Piscataway, NJ 08854 IEEE Press Editorial Board Ekram Hossain, Editor in Chief Jón Atli Benediktsson Xiaoou Li Saeid Nahavandi Sarah Spurgeon

David Alan Grier Peter Lian Jeffrey Reed Ahmet Murat Tekalp

Elya B. Joffe Andreas Molisch Diomidis Spinellis

Photodetectors Devices, Circuits and Applications

Second Edition

Silvano Donati University of Pavia

Copyright ©2021 by The Institute of Electrical and Electronics Engineers, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com. Library of Congress Cataloging-in-Publication Data: Names: Donati, Silvano, author. Title: Photodetectors : devices, circuits and applications / Silvano Donati, University of Pavia. Description: Second edition, revised and expanded. | Hoboken, New Jersey : Wiley-IEEE Press, [2021] | Includes bibliographical references and index. Identifiers: LCCN 2020048868 (print) | LCCN 2020048869 (ebook) | ISBN 9781119769910 (hardback) | ISBN 9781119769934 (adobe pdf) | ISBN 9781119769941 (epub) Subjects: LCSH: Optoelectronic devices. | Optical detectors. | Photoelectric cells. Classification: LCC TK8300 .D66 2021 (print) | LCC TK8300 (ebook) | DDC 681/.25–dc23 LC record available at https://lccn.loc.gov/2020048868 LC ebook record available at https://lccn.loc.gov/2020048869 Cover Design: Wiley Cover Image: © Dmitriy Rybin/Shutterstock 10 9 8 7 6 5 4 3 2 1

Contents

Preface Preface to the first Edition Chapter 1 Introduction 1.1 Photodetection Preliminary 1.2 Basic Parameters of Photodetectors References

Chapter 2 Radiometry Calculations 2.1 The Law of Photography 2.2 The Invariants in Free Propagation 2.3 Acceptance and Degrees of Freedom 2.4 Applying Invariance to Problem Solving 2.5 Extension of Invariants References Problems and Questions

xi xiii 1 4 6 8

9 9 11 12 14 17 19 19

Chapter 3 Detection Regimes and Figures of Merit

21

3.1 The Bandwidth-Noise Tradeoff 3.2 Quantum and Thermal Regimes 3.3 Figures of Merit of Detectors 3.3.1 NEP and Detectivity 3.3.2 Background Limit or BLIP 3.3.3 NEP and D* for Single Photon Detection References Problems

21 23 26 27 28 29 30 30

Chapter 4 Photomultipliers 4.1 Photocathodes 4.1.1 Properties of Common Photocathodes 4.1.2 Photocathodes Technology

31 34 37 41 v

vi

Contents

4.1.3 Photocathodes Parameters 4.2 Dynode Multiplication Chain 4.2.1 Dynode Materials and Properties 4.3 The Electron Optics 4.4 Common Photomultiplier Structures 4.5 Photomultiplier Response, Gain, and Noise 4.5.1 Charge Response 4.5.2 Current Response 4.5.3 Autocorrelation Response 4.5.4 Time Sorting and Measurements 4.6 Special Photomultiplier Structures 4.7 Photomultiplier Performances 4.7.1 Types of Photocathodes and Spectral Response 4.7.2 Number of Dynodes and Gain 4.7.3 SER Waveform and Related Parameters 4.7.4 Linearity, Dynamic Range, and Saturation 4.7.5 Resolution in Amplitude Measurements 4.7.6 Dark Current 4.7.7 Bias Circuits 4.7.8 Hysteresis and Drift. Ambient Performances 4.8 Applications of Photomultipliers 4.8.1 Detection of Weak Signals of Moderate Bandwidth 4.8.2 Measurement of Fast Waveforms 4.8.3 Time Measurements 4.8.4 Photocounting Techniques 4.8.5 Nuclear Radiation Spectroscopy 4.8.6 Dating with Radionuclides 4.9 Microchannels and MCP Photomultipliers 4.9.1 The Microchannel 4.9.2 MCP Photomultipliers 4.10. MEMS Photomultipliers References Problems

Chapter 5 Photodiodes 5.1 Introduction and Nomenclature 5.2 Junction photodiodes 5.2.1 Photoresponse of the PN Junction 5.2.2 Electrical Characteristics 5.2.3 Equivalent Circuits 5.2.4 Frequency Response: Extrinsic and Intrinsic Cutoff 5.2.5 PN and PIN Junctions 5.2.6 Schottky Junctions

44 47 49 51 52 54 55 58 67 69 71 72 72 73 74 76 78 79 80 81 83 83 83 85 86 89 91 91 91 97 99 100 101

103 103 105 106 115 118 121 124 129

Contents

5.2.7 Heterojunctions 5.2.7.1 Uni-travelling Carrier Photodiode 5.2.7.2 Multispectral Photodiodes 5.2.7.3 Lattice Matching 5.2.7.4 Lattice Constant Diagram 5.2.7.5 Interfaces 5.2.8 Photodiodes Structures 5.2.8.1 Traditional Structures 5.2.8.2 Advanced Structures 5.2.8.3 Resonant Cavity Enhanced Photodetectors 5.2.8.4 Quantum well Photodetectors 5.2.9 Photodiodes Packaging 5.2.10 Photodiode Specifications and Parameters 5.3 Photodiode Circuits 5.3.1 Circuits for Instrumentation Applications 5.3.1.1 Transimpedence Circuit 5.3.1.2 Dark Current Cancellation Circuit 5.3.1.3 Logarithmic Conversion Circuit 5.3.1.4 Circuit for Low-Frequency Suppression 5.3.1.5 Narrow-Band Response Circuit 5.3.2 Circuits for Fast Pulses and Communications 5.3.2.1 High-Frequency Transimpedance Amplifiers (TIA) 5.3.2.2 Equalization Technique 5.3.2.3 Switched Capacitor Technique References Problems

Chapter 6 Avalanche Photodiode, SPAD and SiPM 6.1 Avalanche Photodiode 6.1.1 Gain of the APDs 6.1.2 Frequency Response and Noise 6.1.3 Experimental Evidence and Deviations 6.1.4 APD Structures 6.1.5 Bandgap Engineered APD 6.1.6 APD Biasing and Requisites 6.2 Single Photon Avalanche Detectors (SPAD) 6.2.1 The APD in Geiger Mode 6.2.2 SPAD Structures 6.2.3 SPAD Quenching 6.2.4 SPAD Performances and Parameters 6.3 Silicon Photomultipliers (SiPM) 6.4 SPAD Arrays 6.4.1 Microlenses for SPAD Arrays

vii

130 131 133 133 134 136 137 137 139 141 141 142 142 145 146 146 153 154 157 159 160 160 165 168 172 173

175 175 177 180 187 187 190 193 195 195 200 202 204 206 210 212

viii

Contents

6.4.2 Applications of SPAD Arrays References Problems

Chapter 7 Phototransistors, Photoconductors and SNSPD 7.1 Phototransistors 7.1.1 Bipolar Phototransistor 7.1.2 The Optocoupler 7.1.3 Unipolar Phototransistors and PhoSCR 7.2 Photoconductors 7.2.1 Photoconduction and Trapping Gain 7.2.2 Photoconductance 7.2.3 Frequency Response and Noise 7.2.4 Phoconductor Types 7.2.5 PV and PC Detectors for IR 7.3 Superconducting Nanowire Single Photon Detector References

Chapter 8 Thermal Detectors and Thermography 8.1 Basics of Thermal Detectors 8.2 Detectivity of Thermal Detectors 8.3 Temperature Measurements and NEDT 8.3.1 Accuracy of Temperature Measurement 8.3.2 Emissivity and Correction of Temperature Measurement 8.3.3 Two-Color Pyrometry 8.3.4 Thermography and Applications References Problems

Chapter 9 Solar Cells 9.1 Electrical Parameters 9.2 Solar Spectrum and Quantum Efficiency 9.3 System Efficiency 9.4 Solar Cell Structures and Materials 9.4.1 Second Generation Materials 9.5 Photovoltaic Systems References Problems Chapter 10 Coherent Detection

216 218 220

221 221 222 225 227 231 232 234 234 236 237 239 244

245 246 251 253 254 257 259 259 262 263

265 266 269 271 271 275 277 282 282 283

Contents

10.1 Direct and Coherent Detection 10.1.1 Introduction 10.1.2 Coherence Factor 10.1.3 Signal to Noise Ratio 10.1.4 Conditions for Coherent Detection 10.1.5 S/N and BER, Number of Photons per Bit 10.2 Coherent Techniques 10.2.1 The Balanced Detector 10.2.2 The Balanced Scheme in Phase Measurements 10.2.3 Examples of Coherent Schemes 10.2.4 Photomixing References Problems

ix

284 284 285 287 289 290 293 293 296 296 298 301 301

Chapter 11 Photodetection Techniques

303

11.1 Detection with Optical Preamplification 11.2 Injection Detection 11.2.1 Injection Gain 11.2.2 Bandwidth and Noise of Injection Detection 11.2.3 Detection of Terahertz Waves 11.3 Non-Demolitive Detection 11.4 Detection of Squeezed States 11.5 Ultrafast (ps and fs) Pulse Detection 11.5.1 Autocorrelation Measurements 11.5.2 FROG 11.6 Detection for Quantum Communications 11.7 Detection for LIDAR References Problems

303 308 309 314 314 316 320 326 327 333 335 340 343 344

Chapter 12 Image Detectors

347

12.1 The Early Imaging Device: the Vidicon 12.2 Charge Coupled Devices 12.2.1 Introduction 12.2.2 Principle of Operation 12.2.3 Properties and Parameters 12.2.4 Image Organization 12.2.5 Output Stage 12.3 Spatial Resolution and MTF 12.3.1 Spatial Transfer Function 12.3.2 MTF Properties 12.3.3 Image Sampling and Moiré

348 349 349 349 351 360 366 368 368 370 372

x

Contents

12.3.4 Applications 12.4 Image Converters and Intensifiers 12.4.1 Introduction 12.4.2 Basic Functions and Gain 12.4.3 Intensifier Generations 12.4.4 Parameters and Performances 12.4.5 Zoom, Gated and X-Rays Intensifiers 12.4.6 Streak-camera Intensifiers References Problems

Appendixes A1 Spectral Ranges and Measure Units A1.1 Nomenclature A1.2 Transmission of Natural Media A1.3 Radiometric and Photometric Units A1.4 Attenuation Units A1.5 Blackbody Radiance A1.6 Luminous and Radiant Sensitivity References Problems A2 Eye Performances A2.1 Visual Acuity A2.2 Chromatic Perception References A3 Noise Revisited A3.1 Shot Noise A3.2 Noise in Resistors A3.3 Noise from Statistical Thermodynamics References A4 Calculations on Photodiodes A4.1 Calculation of the Intrinsic Speed of Response A4.2 Series Resistance A4.3 Calculations on the Transimpedance Circuit A4.4 The Transimpedance Scheme at High Frequencies A4.5 Edge Effects and Guard Ring References A5 New Model of Noise A5.1 Semiclassical Wave model References

Index

375 377 377 378 381 385 388 390 393 394

395 395 395 355 397 399 401 403 404 404 405 405 408 414 414 414 416 417 419 419 419 422 423 424 426 426 427 427 432

433

Preface

A

fter twenty years from the first edition of this book, published by Prentice Hall with the title Photodetectors - Devices, Circuits and Applications, I think that much of the material covered in the book has maintained its pedagogic value and also its scientific validity, especially the parts dealing with the fundamental issues of photodetection. This circumstance has encouraged me to make a second edition of the book, in the hope that it will be well received by the scientific community as a useful tool for teaching the exciting subject of photodetection, like it has been for me on twenty years of courses in my own University and some more years when I have been a Visiting Professor, mostly in Taiwan. In this second edition, I have introduced many improvements, updated the material almost in every page, and added new parts to keep abreast of novel technologies and new concepts recently developed. Some other parts have been shortened, especially the chapter on photomultipliers (PMTs), which I decided to summarize and yet keep because they are still unsurpassed for large-area applications (see Fig.P-1) and for use in scintillation detector, photon counting, and dating with radionuclides. Yet, as novel single-photon detectors, the SPAD, the SiPM, and the SNSPD, are progressing up fast and are already superior to PMTs, especially in the timing performance, I have reserved a suitable number of pages to their description in this second edition. I have also added the discussion of several new photodetectors, like the UTC, the WGPD/TWPD for multi-hundreds of GHz applications, the QWIP for thermal infrared, the LTGaAs for THz detection, and the superconducting nanowire detector (SNSPD). In the chapter on Thermography, I have expanded the temperature measurement by bolometers and added a new part about the emissivity correction. xi

xii

Preface

In Chapter 11, I have added a new section on autocorrelation and FROG measurements on ultrafast pulses, and two sections orienting the reader on the choice of Single-Photon Detectors for the emerging applications to Quantum Communications and LiDARs. I have moved from an Appendix to Chapter 2 the material on Radiometry and invariants, following the advice of those finding the treatment very helpful for problem solving of tricky problems of radiometry, and split the chapter on Photodetection Techniques into one devoted to Coherent Detection, in which I treat also photomixing, and another on Advanced Techniques. I have not expanded significantly the chapter on solar cells because in an introductory textbook like "Photodetectors" we need the basic concepts and shall leave the in-depth treatment to the monographs available on the subject. However, I have updated the chapter with a description on second-generation materials for photovoltaics. Another important novelty, thanks to the suggestion of colleagues teaching the subject, is the selected problems I have added at the end of each chapter, with a total of about one hundred problems. I will provide the booklet of solutions to those using the book for their classes. I constantly tried to keep the character of the book appropriate for students of the Faculty of Engineering where I teach – focussing not only on the principles but also on the design and the engineering application of photodetectors. So, I start each new argument with a description of the working principle of devices, then develop the circuits for their use, and finally illustrate the applications. One distinctive feature of the book, reflecting my research interests, is the constant reference to the noise limits of devices and circuits, those that determine the ultimate performances that are achievable. Thus, I try to leverage on the pedagogic value of the book based on developments of concepts, rather than offering a collection of arguments grouped by technicality. Thus, I think that this book, even better than its first edition, is well suited for forging the young minds of MSc and PhD students in courses in Photonics Engineering, and also in Measurement Science and Electronics. To teachers and instructors using the book as official textbook of their course, in addition to the booklet of solved problems, I will be glad to supply the set of about 400 pdf slides to deliver the classroom lessons. Silvano Donati, email: [email protected] website: www.unipv.it/donati

Preface to the First Edition

T

his book is an outgrowth of the lecture notes of a two-semester course in optoelectronics I started to give at the beginning of my career at University of Pavia to electronic engineers in their final year of the MS curriculum. Through the years, I have rewritten the text several times, enlarged the scope, added data useful for design and included new results coming from my own research or from new findings by others. Thus, the present book has become more similar to a treatise than to the original lecture book. I have tried to use a modern teaching approach, starting with simple ideas and developing a wide range of topics, so that this book can serve as a textbook in basic photoelectronics and be also profitable to the designer engineer looking for practical hints and solutions. Mathematical derivations are kept to a minimum, and readers interested in applications may skip them and go directly to the results that I have tried to make useful by comments and examples. Complementary information and advanced considerations are in reduced character size and can be skipped in a first reading. In Appendixes, I have collected some material either necessary as introductory or of practical relevance for applications. In the paradigm of optoelectronics, light is generated by a source __ frequently a laser __ propagates in a medium or interacts with it, and comes to a detector where the desired information is converted back to an electrical signal. In the signal-to-noise budget, increasing the laser power by a factor K is equivalent to reducing the detector noise by K __ a simple statement whose importance is sometimes understated but one that should always be kept in mind to optimize system performances of the photodetector. Photodetectors are the forerunners of any new frontier in optoelectronics. When an unusual spectral range or a new frequency band is explored, firstly photodetectors establish a tool for handling the optical signal, and then optical components and laser sources follow. Thus, the excitement of the discovery comes more frequently with photodetectors than with the other optoelectronic devices, as illustrated by Figs. P-1 to P-3 just sampling a few of the big achievements of photodetectors. The book covers the very basic background of photodetection that should be mastered by any scientist or designer active in modern optoelectronics. To give an order-of-magnitude idea, in the optoelectronics-oriented curriculum offered at our Engineering Faculty, one course is devoted to lasers, one to photodetectors, and one each to fiber-optics communication and electro-optics instrumentation. xiii

xiv

Preface

This book is structured as follows. All types of photodetectors of practical importance covering the spectral range from UV to far IR are considered, first treating single-point devices and then their image counterparts. For each photodetector, we begin by understanding the principle of operation and then come to discuss parameters of performances, basic characteristics, special features, application circuits with schematic details and design hints, and end up analyzing in special detail noise which is the ultimate limit of sensitivity performance __ the goal to be approached in any well-designed application. Commenting on the list of contents, we start with the photoemissive (i.e., vacuum) devices, historically the first and presently in a technical decline but still unparalleled in allowing the powerful single-photon counting regime actually exploited in several scientific applications. After a chapter on photocathodes and vacuum phototubes, we consider the system aspects of photodetection in Chapter 3, stressing the importance of an internal gain mechanism in photodetectors, and also clarifying the regimes of detection as well as introducing the figures of merit. Chapter 4 is fairly long and is devoted to photomultipliers because of their importance in a variety of applications. We present the basic theory, analyze the response in time and frequency domains, and discuss a number of applications. Chapter 5 deals with semiconductor photodetectors and related devices, including the family of photodiodes, avalanche devices, phototransistors and photoconductors, and solar cells. In addition to physical aspects and electrical characteristics, we treat the application circuits, elucidating the design of front-end circuits and discuss their performances in a number of wellestablished applications, from instrumentation to large-bandwidth communication. Infrared techniques and thermal detectors for non-contact temperature measurements and thermal-image pickup are treated in Chapters 6 and 7. Chapter 8 is devoted to coherent detection and advanced techniques in photodetection, covering topics not usually found in textbooks that demonstrate how photodetection is far from being a completely explored field. Chapter 9 treats image-sensing devices, including vidicon tubes, intensified image tubes, and the CCD family. Thus, the book offers a rather wide coverage of photodetectors from the point of view of devices, circuits, and applications. I have tried to make each chapter self-contained and readable in itself so that it can be useful as a reference for anyone interested in solving his/her particular photodetection problem. In my experience from the Italian edition, the text can be used for a two-semesters course as a whole, or for a one semester with several different choices of arguments or stress on device/circuit/system aspects. I have also used it for a series of seminars to PhD students in an advanced course on of photodetection and noise. I think that the book will be also a useful reference and a help for technical people and professionals involved in design of photodetection systems, i.e., for the engineer and physicist as a guide to choose the best solution and to evaluate the achievable performances in a photodetection problem, and for the designer because of the abundant reference data on actual photodetectors as well as for the practical circuits discussed. For a full understanding of the technical content, the book requires, as a prerequisite, the basic courses in electronics devices and circuits, and the very fundamentals of semiconductors and noise. Of course, many topics and ideas can be gathered as well from the material presented in this book with a more modest background.

Preface

xv

In my scientific career, I have found photodetection a very exciting and rewarding field for study. If I can convey to the reader the same enthusiasm and satisfaction, I will be amply rewarded for my efforts in writing this book. Silvano Donati June, 1999

Figure P-1 The Super-Kamiokande facility employs 11200 giant PMTs (photomultipliers, see Chapter 4) valued about 80 million US$, paving the walls of a 40-m diameter tank of water to probe the most elusive nuclear particles __ neutrinos (by courtesy of Kamioka Observatory, ICRR, University of Tokyo).

Figure P-2 Aboard the Sky Telescope, a 5000x5000 pixel CCD (charge coupled device, see Chapter 12.2) has provided this amazing 10-days integration picture of deep sky (Hubble Deep Field Survey). Faintest spots are 30th magnitude galaxies estimated to be 8 billion lightyears away. By courtesy of R. Williams and the HDF Team (ST ScI), and NASA.

xvi

Preface

Figure P-3 Infrared thermography (see Chapter 8) unveils the blackbody thermal emission in the middle and far infrared (see App.A1), providing a non-contact map of the temperature differences, a powerful diagnostic tool in industrial, biomedical and military applications (by courtesy of Avio - Nippon Avionics Co.).

C

H

A

P

T

E

R

1

Introduction

P

hotodetectors can be broadly defined as those electronic devices yielding an output electrical signal in response to, and as a replica of the input light signal. They are a key element in virtually any optoelectronic system and application, paralleling in importance the role of sources. Indeed, whenever the system performance is described by a signal-to-noise ratio, increasing the source power by a factor K is equivalent to reducing the noise of the photodetector by the same amount. The birth of photodetectors can be dated back to 1873 when W. Smith discovered photoconductivity in selenium, although Nobili and Macedonio Melloni already in 1829 observed that a thermocouple is sensitive to the warm body of a human passing by, what today we would describe as detection of infrared by a thermopile. Progress was slow until 1905, when Einstein explained the newly observed photoelectric effect in metals, after Planck solving the blackbody emission puzzle with the introduction of the quanta hypothesis. Applications and new devices soon flourished, pushed by the dawning technology of television. In 30 years, photoelectric vacuum tubes covering all the fields of detection were developed, including the orthicon __ the father of image pickup devices, the image converters and intensifiers, and the photomultiplier which technically survives today as the easy way to single-photon detection and a key device for probing elusive nuclear particles like solar neutrinos (see Fig.P-1). Worth to mention, in the 1950's Weimer, Forgue, and Goodrich at RCA invented the vidicon, a beautifully simple alternative to the complex orthicon, that soon became the undisputed workhorse of television cameras before

Photodetectors: Devices, Circuits and Applications, Second Edition. Silvano Donati © 2021 The Institute of Electrical and Electronics Engineers, Inc. Published 2021 by John Wiley & Sons, Inc.

2

Introduction • Chapter 1

being superseded by CCDs forty years later. After World War II, semiconductor devices were invented, a result of the improved understanding of solid-state physics along with adequate control of their technology. As a technical fallout, a number of semiconductor photodetectors have been developed with improved performance, spectral coverage, reliability and size. These have made possible new applications in instrumentation and communications. Zworykin and Morton, the celebrated fathers of videonics, on the last page of their legendary book Television (1939) concluded that: ‘when rockets will fly to the moon and to other celestial bodies, the first images we will see of them will be those taken by camera tubes, which will open to mankind new horizons’. Their foresight became a reality with the Apollo and Explorer missions showing the landscape of the moon. More recently (1997), the CCD camera aboard the 2.4-m Hubble Space Telescope delivered a deep-space picture (see Fig.P-2), the result of 10 days integration, featuring galaxies of the 30th magnitude __ an unimaginable figure even for astronomers of our generation. The next effort is the 6.5-m Webb Telescope due to launch in spring 2021. This instrument will feature a 25,00025,000 pixel InSb CCD to detect the middle infrared and observe high redshift sources, looking back even closer to the big-bang age. Thus, photodetectors continue to open to mankind the most amazing new horizons. We can distinguish two classes of photodetectors (Table 1-1) according to the handling of the received power: Single-element photodetectors are those yielding an output signal proportional to the total (or integral) power collected by the active area, and Image photodetectors are those with an active area physically or virtually divided in a string (1-D) or array (2-D) of individual elements (also called pixels from picture element), whose signals can be properly organized to be readable at the electrical output port. Pickup tubes, CCDs, and image converters or intensifiers belong to the image class. Photodetectors are further classified according to the type of optical to electrical conversion effect. The most important is the photoelectric effect, by which a photon (or quantum of energy h ) is absorbed by the material with the release of an electron-hole pair. If the photogenerated electron is further emitted out of the material, becoming available for collection or multiplication, the device is called a photoemission device, or one based on the external photoelectric effect. This is the class of vacuum photodiodes, photomultipliers, and photocathode-based pickup and image intensifier tubes. Instead, if no emission takes place but the photogenerated carriers are available for current circulation in the external circuit, we talk of an internal photoelectric effect or device, and we find all the semiconductor photodetectors: junction and avalanche photodiodes, phototransistors, photoresistances, etc. The above classification has a practical importance: the emitted photoelectron is more valuable than one internal to the material, because it may be better singled out with respect to non-photogenerated electrons, that is, the dark current of the device. In addition, the emitted photoelectron can be multiplied by other electrodes with little added noise, as it actually is by means of the dynodes in a photomultiplier. This is the reason why a photoemission device like the photomultiplier has been the unsurpassed detector of single photons, easily reaching the quantum-limit regime even at very low levels of input power. However, recent developments in SPAD devices (Chapter 6) have much progressed the solid-state counterpart of photomultipliers and added new functionali-

Introduction • Chapter 1

3

ties like the image capability, as well as the superconducting nanowire (Chapter 7) that has unequalled speed of response. Table 1-1 Synopsis of Photodetectors and their Spectral Ranges _____________________________________________________________________ SINGLE ELEMENT

IMAGE

- photoemission devices (or external photoelectric devices)

vacuum photodiode gas photodiode photomultiplier

pickup tubes image intensifiers and converters

- internal photoelectric devices

semiconductor photodiode avalanche photodiode, SPAD phototransistor (BJT, FET) photoresistance

CCDs

vidicon

thermocouple (or photopile) thermistor (or bolometer) pyroelectric

uncooled IR FPA IR vidicon

- thermal detectors

- weak interaction detectors

photon drag, Golay cell photoelectromagnetic point contact diode _____________________________________________________________________ SPECTRAL RANGES

1 m 10 m 100 m 1000 m 0.1 m _____|____________|____________|_____________|_____________|______ __photoemission___ __internal photoelectric effect____ _______________________thermal____________________

(

_____________________________________________________________________ Photoemission devices share the drawbacks of vacuum-tube technology (high-bias voltages, fragility, unsuitability for batch manufacture) and are limited to rather a narrow spectral range of operation, from UV to near infrared. Conversely, internal photoelectric devices cover a much wider spectral range, from deep UV to far infrared, albeit with different semiconductors, each optimal for a -interval not wider that an octave. The big advantage of solid-state components is that they are compact, rugged, reliable, manufactured in batches, and require low bias voltages. Synonymous with photoelectric device (both internal and external) is the term quantum detector, in reference to the conversion mechanism. Still another important class of photodetectors is that of thermal detectors. These detectors are based on a two-step process: radiation is first dissipated in an absorber and the sub-

4

Introduction • Chapter 1

sequent small increase of temperature is measured with a conventional electronic temperature sensor. Though they have a much lower sensitivity as compared with photoelectric detectors, thermal detectors offer a very wide spectral range of operation, covering UV to far IR in a single unit and with a nearly flat response. This feature has used to advantage in power meters and for the easy calibration of detectors in different spectral ranges. Recently, integrated thermal detectors have been exploited to realize third-generation uncooled focalplane arrays for handheld infrared cameras (see Fig.P-3 and Chapter 8). Lastly, a lot of indirect and weak-interaction effects that have been discovered and proposed, especially in the spectral range of the far and extreme IR. The photon drag detector, the Golay cell, and the photo-electromagnetic detector are just few examples. We will not treat them in this book, because of performance or reproducibility not fully satisfactory from the engineering point of view. Instead, new effects based on the superconducting properties of materials have been demonstrated viable, and the SNSPD (Chapter 7) is an example. From the perspective of applications, photodetectors are usually understood, and most frequently used, as the devices for the conversion of a radiant signal into an electrical signal, for data communications or performing measurement, and in this case, operation is in the small-power signal regime. Recently, however, photodetectors have become important also for power conversion or photovoltaics, since solar cells (Chapter 9) are manufactured on large scale with cheap technologies and attain good conversion efficiency, and the cost of photovoltaics has matched the target of 1/ and all (or most of) the available power (and photons) is absorbed and converted in an excited state (normally, a charge carrier).

1.2 BASIC PARAMETERS OF PHOTODETECTORS Let us consider Fig.1.2, the basic scheme of operation of a quantum detector. Here, Vbb is the biasing voltage and R is the load resistance across which we the detected signal is available for further processing. We may express the power collected by the detector as the product of the photon energy h times the rate of arrival of photons F, that is P = h F, and similarly the output current I = e F’ as the product of electrons charge e and rate of charge flowing in the circuit.

Figure 1-2 Circuit for biasing and using a quantum photodetector.

Taking the ratio of the two equations, we find that photodetected current is proportional to collected power: I/P=

= e F’ / h F =

e /h

(1.4)

Introduction • Chapter 1

7

where = F’/F is called the quantum efficiency (average ratio of electrons to received photons) of the detector, and = I/P=

e /hc =

/1.24)

(1.5)

is known as the spectral sensitivity (ratio of current out to power in, in A/W). Quantities and describe the response of the photodetector and are equivalent to each other if is given. Yet, is connected to the photon-to-electron conversion process, while gives the current signal available upon detection. For an ideal quantum detector with =1, we have = 1 A/W at =1.24 m, in the near infrared. Spectral sensitivity increases with , because of the increasing number of photons contained in the same energy at increasing . The increasing trend of with continues until a wavelength is finally reached at which h ≤E, or hc ≤E, or, the photon energy is no more enough to excite the transition E= E2 -E1, and accordingly the photocurrent ceases abruptly. The value of wavelength at which this happens given by hc =E or hc, is called t, the photoelectric threshold of the material and is explicitly given by: t

eV

e hc =

eV

1.24 ( m)

(1.6)

where energy E(eV) is expressed in electron-Volt (eV). To trade photons for electrons, we need to find a material with adequate energy difference, E= E2 - E1, not larger than the photon energy h . Energy E may be either the work function in external photoemission where E=Ew, or the bandgap in internal photoemission where E=EG. Typical exemplary values of energy E and of popular materials are [3]: (i) alkaline antimonides, with Ew ≈1.2__3.0 eV, so that t ≈1__0.4 m (from blue to near-infrared); (ii) silicon with EG=1.1 eV and threshold up to the near-infrared at t =1.1 m; (iii) ternaries semiconductors like InGaAs, with EG≈0.75 eV and threshold up to the 3rd window of optical fibers, at t ≈1.8 m, (iv) InSb with EG ≈0.25 eV, and coverage of the middle-infrared up to t ≈5 m; (v) HgCdTe or PbSnTe with EG ≈0.08 eV, and coverage of the far-infrared up to t ≈16 m. In summary, the ideal diagram of versus is triangular, as indicated in Fig.1-3. In a real detector, however, deviations from the ideal trend are found at both edges of the response curve: indeed, on approaching the threshold t, the material becomes increasing transparent and some photons are lost, and also in the blue region there are surface losses introducing a decrease of efficiency, so that in the end the real trend is the smooth curve drawn in Fig.1.3. A last feature common to all photodetectors is the requirement of an electric field internal to the structure, positioned in the detecting region and useful to remove the generated charge carriers and convey them fast to the collecting electrode. As most electronics components rely on a working principle based on an internal field, or have one to assist operation, it is no surprise that virtually all the electronic components have a photodetector counterpart (see the circuit symbols in Fig.1-3), although not always of the same importance. Indeed, we find a favorable electric field in devices like:

8

Introduction • Chapter 1

σ = I/P (A/W)

threshold η=1

1.0 η=0.5

real response 1.24

Figure 1-3

λt

λ

General response of a quantum detector, for an ideal ( =1) quantum efficiency, and for the real case of a detector with decreasing both close to the threshold of photoemission t and in the blue region.

- vacuum phototube (or vacuum photodiode) as the biasing field of the anode-to-cathode voltage supply; - semiconductor photodiode: the field across the depleted region of base to collector; - photoresistance: the field produced by the applied bias voltage; - photo-MOS: the electric field produced by the gate to drain voltage difference; - photo-FET: the electric field produced by the gate to drain voltage difference; - (bipolar) phototransistor: the electric field at the reverse biased base-collector junction.

Figure1-3. Circuit symbols of commonly used photodetectors, from left to right: phototube (or vacuum photodiode), semiconductor photodiode, (bipolar) phototransistor, photo-FET, photo-MOS, and photoresistance.

References [1] V.C. Coffey: “Vision Accomplished: The Bionic Eye”, Opt. Phot. News, Apr. 2017, p.25. [2] P.W. Milonni, J.H. Eberly: “Laser Physics” John Wiley and Sons, Inc., 2010. [3] S.M. Sze, K.K. Ng : “Physics of Semiconductor Devices”, 3rd ed., Wiley-Interscience, New York, 2006.

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2

Radiometry Calculations

I

n nearly any problem of photodetection we have to calculate the radiant power (or eventually the irradiance) received by a detector at a specified aperture and angular field of view, either in direct sight or through an optical system, when the source power, its emission characteristics, and the geometric parameters of propagation are specified. These calculations of radiometry are greatly simplified, and frequently even avoided by writing directly the result after an inspection of the experiment or the setup, if we are able to use the concepts of invariance of radiance and acceptance that we develop and illustrate with examples in this Chapter.

2.1 THE LAW OF PHOTOGRAPHY Let us start with the case of image formation, that is, of a detector and an optical source placed at the object and image planes of an objective lens as shown in Fig.2-1. Let B indicate the source brilliance (see Sect.A1.3 for the definition), A the area of emission and the solid angle under which the lens is seen by the source; let B' be the brilliance of the radiation received at the image plane, A' the image area conjugated to A, and ' the solid angle of the rays forming the image.

Photodetectors: Devices, Circuits and Applications, Second Edition. Silvano Donati © 2021 The Institute of Electrical and Electronics Engineers, Inc. Published 2021 by John Wiley & Sons, Inc.

10

Radiometry Calculations Chapter 2

For the sake of simplicity, let us assume a condition of paraxial rays, taking identical and ' for all the points of image and source along A and A'.

Figure 2-1 Optical conjugation of object A and image A’ planes through a lens.

We can then write the radiant power P emitted by the source and collected by the lens simply as: P=BA

(2.1)

and similarly, for the power P' received on the detector from the lens: P' = B' A' Now, let us note that '/

'

(2.1') and

' are connected, through the geometry of Fig.2-1, by:

= [( D2/4) /F2] / [( D2/4) /L2] = L2/ F2

(2.2)

We assume here L>>F, so that F=p is the conjugate of L=q in the Newton lens formula 1/p+1/q=1/F; otherwise, we should change F in F(1+m) where m=p/q is the magnification, but the results would be similar. Also, let us note that the lens conjugation gives for the areas A and A’: A' / A = F2 / L2 Therefore it follows, making the product with Eq.(2.2): A'

'= A

(2.2')

We have thus found the conservation of the quantity A that is called acceptance a. Moreover, going back to Eqs.(2.1) and (2.1') and as P'=P [by definition, it has no physical implication underneath], we also get from Eq.(2.2'): B' = B

(2.3)

that is, the brilliance is also conserved. To calculate the illuminance E' on the image plane, let us take into account Eq.(2.3), and write: E' = B'

'=B

'=B

sin2

(2.4)

Chapter 2

11

where is the semi-angle under which the lens is seen from the focal plane (Fig.2-1). For a Lambertian source with unity diffusivity on which an illuminance E is incident, giving a brilliance B = E/ we have: E' = E sin2

(2.4')

In Eq.(2.4), sin is called the numerical aperture of the lens and is denoted by NA. Frequently, the numerical aperture is approximated for small as: sin

= D/2F = 1/2(F/)

(2.5)

F/ is the F-number, the well-known number to indicate the ‘luminosity’ of the lens. Returning to Eq.(2.3), we can express it in terms of NA or F-number as: E' = E NA2 = E / 4(F/)2

(2.6)

Eq.(2.6) is the law of photography, and states that the illuminance on the image plane cannot exceed the illuminance on the object (ideal diffuser), and is smaller than that by a factor NA2 (or 1/4(F/)2). The ratio E'/E depends only on NA (or F/) and not, for example, on the lens diameter. If we finally want to take account of the finite diffusivity 1, or that acceptance varies with aperture: this can-

Chapter 2

19

not be, as is readily shown by considering, for example, the passage of a beam through a medium with n>1, or an ideal blackbody covered by a layer with n>1. (iii) Dependence upon polarization. The above results are for unpolarized light, i.e., a statistical incoherent mixture of all possible states of polarization. To extend them to the polarized case, let us consider that each state of polarization, in a chosen pair of orthogonal states, can carry an equal energy content. Thus, the maximum brilliance allowed for each state can be written as: B = P/2A = (

T4/2

for a blackbody, and in general: B = dP / 2 n2 dA d for a generic situation of propagation. (iv) Validity limits of invariance principles. Of course, the principles of B and a invariance apply to all of the cases of propagation following the rules of geometrical optics in homogeneous and non-homogeneous media. However, as a caution for the reader, let us quote some cases of apparent violation. An aperture increasing the solid angle by diffraction (for example, our pupil when we watch a star) or acting as an area stop, a diffuser re-radiating in a hemisphere what is collected under a small solid angle, or an absorber which attenuates power, are clear examples of violation of the conservation of B or a. Obviously, these cases must be dealt with by separating the sections that conserve and degrade brilliance (or acceptance), and by calculating separately the effects of both, so as to be able to combine them in the final result.

References [1] Burle Industries: “Electro-Optics Handbook” vol. TP-135, Lancaster, PA, 1985. [2] “Radiometry”, Wikipedia, https://en.wikipedia.org/wiki/Radiometry [3] M.Bass (editor): “Handbook of Optics”, vol.1, 3rd ed., Opt. Soc. Am., 2010.

Problems and Questions Q2.1 Going out to take a picture of the newly reported comet – or of the usual night starring sky should you prefer to take with you your telescope - having a 20-cm diameter and a 200-cm focal length - or your camera (having an objective with 40-mm diameter and a 80-mm focal length) ? Explain why.

20

Chapter 2

P2.2. By assuming that sun and planets are blackbodies, calculate the earth and planets temperature by applying the invariance of brilliance (the sun has: Tsun=6,000 K). What happens if the emissivity is not unity ? P2.3 An engineer of road-lighting crossed his head with the idea of illuminating tunnels by optical fibers collecting sunlight. Using the brilliance invariance, calculate for him: i) the power that can be collected from the sun at AM1.5 with a multimode fiber having a 50- m diameter and a 30-deg numerical aperture; and ii) the same, at the maximum of practicable concentration with a lens (specify which). P2.4 Can it be true the old tale of Archimedes devising a mirror solar burner of foe-ships sails? Assuming burning temperature of sail is 300°C, and mirrors held by hands, with 40-cm diameter and 30% reflectivity (for copper), calculate how many mirrors are required to burn sails at 200-m distance (of course, assume a sunny day, at noon, no clouds). P2.5. Consider a Y branching device on a planar substrate, made of equal lossless waveguides up to the branching region, which on its turn is smooth and without any geometrical defect. What is the power out of the lower waveguide of the Y when two equal powers P enter the two upper waveguides of the Y? Second, what is the power out of the two upper waveguides when a power P enters the lower waveguide? Third, what is the phase shift of the fields E1, E2 out of the upper waveguides when a field E enters the lower waveguide (assuming a perfectly symmetrical geometry?) P2.6. To make an endoscope, we want to use a SELFOC fiber (with parabolic index profile, it refocalizes periodically the image) to guide an image of N=500 500 pixel (in the visible, =0.5 m). If the numerical aperture is sin =0.3, what is the minimum diameter required? P2.7. It is probably known to the reader that, when laser light is shed on a diffusing surface, it reradiates light exhibiting a grainy appearance called speckle pattern. This is because the diffuser has destroyed the spatial coherence of the light, at least locally. At a distance z from a diffuser, each speckle grain has a longitudinal (parallel to the z-axis) dimension (say sl) and a transversal (perpendicular to the z-axis) dimension st. By considering that a speckle is an individual bright (or dark) spot inside which no structure can be resolved, or, it is a single-mode, calculate its dimensions applying the -acceptance value of the single mode. P2.8. Should the Greek astronomer Eratostene (200 BC) have had a photometer, he could have succeeded in computing the earth-to-moon distance. Hint how, considering the ratio of brilliances of the moon illuminated by the sun and of the part receiving illumination from the earth (the albedo).

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3

Detection Regimes and Figures of Merit

B

efore starting to describe photodetectors, it is useful to address a system consideration, and ask ourselves the question: "how good is the performance of my photodetector in view of the fundamental limits of sensitivity and speed of response?" The question is a system consideration because is regardless of the actual structure or working mechanism of the device. Noise considerations will also be developed in next Chapters, because noise limits the sensitivity performance we can obtain. Here, we start with the analysis of thermal and quantum regimes and point out the need for an internal gain in photodetectors. Even more important, we introduce the figures of merit, which are frequently used in thermal IR detectors and allow to compare completely different detectors.

3.1 THE BANDWIDTH-NOISE TRADEOFF The problem of how to terminate the photodetector on a suitable load resistor, and trade the performances between bandwidth and noise, is common to any quantum detector yielding emitted electrons (or internal electron-hole pairs) as a response to incoming photons. This is a feature shared by photodiodes, CCD's, photoresistances, vidicon targets, etc., all of which are described from the electrical point of view by a current generator P with a stray capacitance C across it.

Photodetectors: Devices, Circuits and Applications, Second Edition. Silvano Donati © 2021 The Institute of Electrical and Electronics Engineers, Inc. Published 2021 by John Wiley & Sons, Inc.

22

Detection Regimes and Figures of Merit • Chapter 3

Let us then consider the equivalent circuit of a quantum photodetector ending on a load resistor R, as shown in Fig.3-1, and evaluate its bandwidth and noise. Let us assume that the ratio of detected current I to input power P is a constant, = I/P.

Figure 3-1

General circuit of a quantum photodetector (left) and its equivalent circuit including noise generators.

Because of the RC group, the output signal, either in voltage V= P, has a 3-dB high-frequency cutoff given by: B = 1/(2 RC)

P.R or in current I = (3.1)

Added to the signal, we find two noise contributions. One is the Johnson (or thermal) noise (see also Appendix A3) of the resistance R, with a quadratic mean value: i2R = 4kTB/R

(3.2)

where k is the Boltzmann constant and T the absolute temperature. The other noise is the quantum (or shot) noise associated with the discrete nature of the total current I= Iph+Id, the sum of the signal current and of the dark current, and its quadratic mean value is given by: i2I = 2e (Iph+Id) B

(3.3)

where e is the electron charge and B, as in Eqs.(3.1 and 3.2), is the observation bandwidth. The two fluctuations are added to the useful signal, and so the corresponding noise generators are placed across the device terminals as in Fig.3-1. Since the two noises are statistically independent, their quadratic mean values combine to give the total fluctuation i2n as: i2n = 2e (Iph+Id) B + 4kTB/R

(3.4)

From Eqs.(3.1) and (3.4), we can see that bandwidth and noise impose opposite requirements on R: to maximize B one should use the smallest possible R, while to minimize i2n the largest possible R is required. Of course, the same conclusion would apply if, instead of the current I, we had chosen the voltage V across R as the output signal.

3.2 Quantum and Thermal Regimes

23

A quantum photodetector, regarded in its basic form of photon-to-electron converter, cannot avoid this problem: it can have a good sensitivity to small signals using very high load resistances (up to the G ) as is actually done in some instrumentation applications requiring modest bandwidths ( kHz or less), or it can be made fast by using low load resistances (e.g., R=50 ), but at the expense of sensitivity. To circumvent this tradeoff, two approaches are currently used: (i) circuit-level solutions, providing an active two-terminal load or circuit exhibiting less noise than the Johnson limit applicable to any passive device. These may take the form of: a special termination, such as: cold resistance and switched capacitor, or of a system recovery of bandwidth, as the equalization technique; (ii) device-level solutions, providing a mechanism of internal gain that, by amplification of the photodetected current by a factor G (and of the quadratic noise fluctuations by G2) before it is added to the Johnson noise generator, allows the use of a load resistance smaller by a factor G2 at equal sensitivity.

3.2 QUANTUM AND THERMAL REGIMES Let us now evaluate the relative weight of the two terms in Eq. (3.4). In general, we will have a good sensitivity performance when the shot noise is dominant compared to Johnson noise, i.e., for: 2e (Iph +Id) B ≥ 4kTB/R whence the condition: Rmin ≥ (2kT/e)/(Iph+Id), or Rmin ≥ 50 mV/(Iph+Id) [at T=300 K]

(3.5)

At low signal levels, that is, for Iph>n'= /4 : n1 = √(n0 n)

√n

(5.6)

and the corresponding transmission is: T = T0T1 = 1 - 2 [(n1-1)/(n1+1)]2

(5.7)

5.2 Junction Photodiodes

111

With only a single layer, the reflection loss has been greatly reduced (of a factor 5__3 for n=3__4). In Fig.5-5 the total reflection loss R=1-T is plotted as a function of wavelength for an anti-reflection layer on silicon, of either silicon oxide (SiO2) or silicon nitride (Si3N4) __ the latter being better matched to the condition n1=√n. The layer thickness is kept much larger than the wavelength (S>> to avoid interference effects (averaged to zero on the distribution of optical pathlength n1S/cos for an incident beam with a finite angular aperture ).

REFLECTION COEFFICIENT R

6

5

4

3

2

hν (eV)

0.8

1.0

0.6

0.6

0.4

Ge

InAs Si

GaAs InP 0.2 SiO 2 ⏐Si Si 3N 4 ⏐Si

λ/4 0

200

300

400

600

800

1000

1400

2000

WAVELENGTH λ (nm)

Figure 5-5 Reflection coefficient R for vacuum-semiconductor interfaces, untreated (full lines) and single-layer anti-reflection coated (dotted lines).

Also, one can optimize the transmission T at a certain wavelength with a single layer, using the destructive interference of the two reflection contributions at the two interfaces n0|n1 and n1|n. The layer must have n'=√n to give equal reflection amplitudes, and a thickness of S= /4, so that the go-and-return pathlength /2 gives a phaseshift between the reflections at the first and second interfaces. In this way, the two contributions theoretically cancel out at the desired . In practice, the residual reflection can go down to

112

Photodiodes • Chapter 5

>kT/e (hundreds mV) we have from Eq.(5.9): I = - Io - Iph

(5.10)

that is, the photodiode supplies a current proportional to detected power P superposed on a small reverse current Io, for this reason also called the dark current. The bias Vbb is usually selected fairly high, up to about the junction breakdown voltage (dotted line in Fig.5-8), for best performance. In fact, at increasing Vbb, the depletion layer W becomes wider, and, as a consequence, (i) the spectral sensitivity increases, especially near the photoelectric threshold, and (ii) parasitic elements limiting the speed of response, i.e., the series resistance Rs and the junction capacitance Cb, both decrease. Physically, Rs represents the ohmic resistance of undepleted or neutral regions in series with the junction up to the contacts (see Appendix A4.2 for more details), and Cb is the barrier capacitance of the pn junction; their typical trend with Vbb for an abrupt pn junction is shown in the insert of Fig.5-8. In some instrumentation applications one may look for the maximum sensitivity in dc or at low frequencies, to detect very weak signals IphIo) and a constant ideality factor (i.e., n=1) becomes, at T=300 K:

5.2 Junction Photodiodes

117

V = [59.6 mV] Log10(Iph/Io)

(5.12')

In some silicon photodiodes, the conformity of the logarithmic conversion given by Eq.(5.12') can be up to five/six decades with a relative error less than ±1%. Finally, the IV quadrant of the photodiode characteristic is that of the photovoltaic regime, peculiar to solar cells (see Chapter 7), in which, with the voltage V positive and the current I negative (outgoing from the positive terminal), the device works as a generator supplying a power V.I . Dark current and ideality factor. In most applications, Eq.(5.9) provides a sufficiently accurate description of the I/V characteristic of photodiodes, in which Io and n can be taken as experimental parameters of the photodiode at hand, eventually dependent on T and V. Further information on the behavior of such parameters is obtained from the theory of the pn junction [1]. Usually, the current in the junction is analyzed as being contributed by diffusion from the neutral regions of majority carriers (direct flow) and by drift of minority carriers from the depleted region (reverse flow) [1]; in this way one can obtain the well-known Shockley equation: I = Id [exp (eV/kT) - 1]

(5.9')

in which the ideality factor is unity (n=1) and the reverse current Id is given by: Id = A e ni2 [(Dp/LpND)+(Dn/LnNA)] A e ni2 (Dp/LpND)

(for NA>>ND)

(5.13) (5.13')

In Eq.(5.13), A is the active area (junction area) of the photodiode, Dn,p are the minority diffusion constants (electron and hole) and Ln,p are their diffusion lengths, ND,A are the doping concentrations of donor and acceptor in regions n and p, and ni is the intrinsic concentration of charge carriers, explicitly given by: ni2 = NC NV exp -Eg/kT

T3 exp-Eg/kT,

(5.14)

Note that here Id is independent from V. Using Eq.(5.14) in Eq.(5.13) and taking for (D/L) a dependence T from temperature, we obtain: Id

T3+ exp -Eg/kT,

(5.15)

whence a temperature coefficient of the dark current Io=Id: dIo /Io dT = [3+ +Eg/kT] / T ≈ 0.33 [3+ +Eg/kT] (%/°C at 300 K)

(5.16)

Eqs. (5.9'), (5.13) and (5.16) apply at weak current levels in semiconductor junctions with very small defect concentration, or when the intrinsic concentration of charge carrier ni is not too low. Another important contribution is generation-recombination in the depleted region, through defect levels near bandgap middle, which give recombination in direct conduction and pair generation in inverse. This contribution of g-r current is [1]: I = Ig-r [exp (eV/2kT) - 1],

(5.9'')

that is, it has an ideality factor n=2; in addition, the reverse saturation current is: Ig-r = A e ni W / 2

(5.17)

118

Photodiodes • Chapter 5

where W is the width of the depleted region, and =1/(√3kT/m) tNt is the lifetime of the charge carriers, dependent on concentration Nt and cross section t of the g-r levels. The term Ig-r has (through W) a dependence V upon voltage, with =1/2 or 1/3 for abrupt or gradual junctions; its temperature coefficient is, using Eq.(5.14): dIo /IodT = [2+Eg/2kT]/T ≈ 0.33[2+Eg/2kT] (%/°C at 300 K)

(5.16')

The total current in the photodiode is thus the sum of the contributions [Eqs.(5.9') and (5.9'') and of the photogenerated current Iph. Eq.(5.9) can then be interpreted as an approximant of such a sum. In particular, at high reverse bias the dark current is the sum of the two saturation terms [Eqs.(5.13) and (5.17)]: I = - Io = - Id - Ig-r

(5.18)

and its behavior with V and T reflects the relative weight of the two addenda. In particular, the trend will be that of diffusion (n=1) for ni (Dp/LpND)> W/2 , and of g-r (n=2) in the opposite case. In direct or zero bias [see Eq.(5.12)], using Eqs.(5.9') and (5.9''), we obtain an ideality factor n=1 for voltages V > (2kT/e) ln [(W/2 )/(niDp/LpND)], and n=2 otherwise. It is notable that contains the volume concentration Nt of the g-r centers, which can be utilized in previous expressions by introducing the factor FN=N*/Nt, where N*=2ni (Dp/LpND)/[W(√3kT/m) t] has the meaning of a critical concentration: the previous conditions then become FN>1 for the dark current and V>(2kT/e)lnFN for the ideality factor. A final contribution to reverse current comes from surface states, that is, from defects at the interfaces (in particular, that of optical access) which introduce levels in the bandgap. Such a contribution, of importance only in photodiodes with very low reverse current, is formally given by the same expressions of Ig-r [Eqs.(5.9'') and (5.17)] if we substitute, in place of NtW, the defect concentration Ns per unit area. Let us briefly discuss the saturation of photodiode at high levels of photogenerated charge injection, or at high Iph. Saturation has importance in determining the maximum level of signal detectable with linearity (III quadrant), the logarithmic conformity, and the voltage in the photovoltaic mode (IV quadrant). A first cause for saturation is the storage of photogenerated charges collected at the boundary of undepleted regions after drift in the junction. When such a charge (Q= Iph , where is the drift time) becomes comparable with that (Q=CbV) supplied by ionized dopant atoms to sustain the applied voltage V (Cb being the barrier capacitance), the potential distribution is altered, the electric field across the junction is reduced, and reverse fields appear in the undepleted regions, thus impeding any further increase of Iph with increasing power. For a p+n photodiode and photogeneration in the depletion region, this happens at a level of photogenerated current given by: Iph(sat) = A e NA * V / 2W

(5.19)

* =(1/ n+1/ p)-1 is the effective mobility; note the dependence on area A and junction field V/W.

where Moreover, if the generation is in the neutrality region p+ (as, for example, in the UV) the limit is still lower because of the slow diffusion times ( =Ln2/Dn), and we have: Iph(sat) = A e NADn / 2W Ln2

(5.19')

Eqs.(5.19) and (5.19’) set an upper limit for the current detectable in reverse bias (usually in the range 0.1-1 A/mm2). At high currents, one should also take account of the significant voltage drop in undepleted regions and in access contacts, which are summarized by the series resistance Rs. Eq.(5.9) still applies in this case, provided V is changed to the effective voltage V-RsI applied to the junction.

5.2.3 Equivalent Circuits Biasing Scheme From the discussion of the previous Section, the basic biasing scheme of a photodiode is that illustrated in Fig.5-9, in which a load resistance R is connected in series with the device

5.2 Junction Photodiodes

119

anode A up to the supply voltage -Vbb, and the cathode K is grounded. The output voltage is taken across the load resistance R. Of course, one can also reverse the photodiode and the polarity of the Vbb voltage, or connect the load R to ground (as in Fig.5-24): all these circuits have the same behavior (neglecting, for the moment, parasitic effects of the device case). Practical circuits for photodiodes, even very complex ones, can always be brought to the basic scheme to analyze the performance of the device. R -V A

C

bb

Vu

K

(a)

Rs

Iu A

I ph Cg

(b)

Cp

Rp

Vu

R

K C Rs

(c)

4kTB R s i 2nph

2 i nb

Cg

Rp

4kTB ____ Rp

Cp

R

Figure 5-9 (a) Basic scheme for biasing a pn photodiode (note the cathode K connected to the case C and grounded), (b) equivalent circuit for small-signal analysis, (c) equivalent circuit for noise analysis.

Small-signals circuit Following the schematization commonly used in electronics, the small-signal behavior of the photodiode can be represented by means of an equivalent circuit, as shown in Fig.5-9. In our case, the signal is given by the current generator driven by photogenerated current

120

Photodiodes • Chapter 5

Iph= P [see Eq.(5.9)], across which we have the junction capacitance Cg (here coincident with barrier capacitance Cb). Going from junction to access terminals (A and K), we find the series resistance Rs of the undepleted and must also add, in parallel, a dispersion resistance Rp to take account of the finite conductance dI/dV of the junction (to be discussed later). For a typical pn photodiode, Rs and Cb have the values indicated in Fig.5-8; the resistance Rp is usually very high, typically 1__100 M and is also dependent on the signal level Iph and the wavelength . Lastly, Cp is the parasitic capacitance of the output terminal. In photodiodes with a metal case, Cp is the capacitance of the active terminal (A in Fig.5-9), with respect to the case. The case is usually connected to the ground (as in Fig.5-9) or to a fixed voltage to shortcircuit its large stray capacitance (Cc>>Cp) with respect to ground; also, in this way, the case acts as an electrostatic screen, protecting the chip from external electromagnetic interference. For photodiodes in chip format or in a plastic or ceramic case, Cp is the capacitance of the active terminal to ground. As the line going to the load adds to Cp, it is necessary to minimize its length and area. The circuit of Fig.5-9b is an adequate representation up to, say, about 100 MHz. At still higher frequencies, it may be improved by considering other parasitic elements, like the inductance Lp of the connecting line (typically 2nH/cm) and the distributed capacitance of the load. Noise Circuit The equivalent circuit that describes the noise properties of the photodiode is reported in Fig. 5-9c. It is obtained by adding, in that for small signals (Fig.5-9b), the noise generators i2n associated with circuit resistances and with currents, at their points of generation. At first instance, the current noise associated with a physical current I is due to the granular or shot fluctuation, and has a mean square value i2n = 2eI B, where e is the electron charge and B is the measurement bandwidth. The noise associated with a resistance R is given by the Johnson noise, i.e., i2n=4kTB/R in terms of a parallel current source, or, equivalently, by v2n=4kTBR in terms of a series voltage source. All the noise terms i2n are statistically independent from one another, and thus they sum up quadratically (see also Appendix A3). To be noted, both shot and Johnson noises have a white spectrum, or a constant spectral density, = 2eI or 4kTB/R, respectively. We prefer to represent noises by their total mean square values, but it is understood that the di2n/df description is equivalent. In the scheme of Fig.5.9a, the generated currents are the dark Ib and photogenerated Iph currents, whence the terms i2nph=2eIph B and i2nb=2eIb B connected to the input in Fig.5-9c. The Johnson noise of the resistance Rp is connected in parallel as i2n=4kTB/Rp, and that of the resistance Rs is connected in series as v2n=4kTBRs. The equivalent circuit of Fig.5-9c is the basic model for noise, and yields a good approximation in most practical cases. Further possible contributions sometimes found in addition to the physical limits considered (shot

5.2 Junction Photodiodes

121

and Johnson noises) are the 1/f noises at very low frequencies and the excess noises (white spectrum) of resistances and currents. About shot noise of the photogenerated current, let us remark that it can be expressed, more generally, as i2nph=2eIph BF, where F is the Fano factor associated with the statistics of the radiation. In the vast majority of practical cases, the rate of photons arriving on the photodetector follows the Poisson statistics and F=1; less frequently it is F>1 (laser near the threshold, sources with a random excitation, etc.). Also for the dark current I0=Id+Ig-r, one can still write i2nb=2eIdBF, where F=1 for the diffusion term Id and F=2/3 for that of generation-recombination Ig-r. However, the approximation F=1 is always acceptable in practice.

5.2.4 Frequency Response: Extrinsic and Intrinsic Cutoff Going back to the equivalent circuit of Fig.5-9b, we may interpret the small-signal voltages V( ) and currents I( ), superposed on the dc bias, as rotating vectors at frequency . By the aid of the usual electrical circuits method, the output voltage Vu and current Iu are easily found to be: Vu ( ) = Iph( ) {Rp//(1/j Cg)//[Rs+(R//(1/j Cp))]/[1+Rs/(R//(1/j Cp)) Iph( ) Z( ) Iu( ) = Iph( ) Z( ) /R

(5.20) Iph( ) f( )

(5.21)

where the sign // indicates the parallel compounding of impedances [explicitly, Z1//Z2= Z1Z2/(Z1+Z2)], and Z( ), f( ) represent the effective impedance seen by the photodiode and the transfer factor of the photogenerated current Iph( ). Eqs.(5.20) and (5.21) include the two contributions to the frequency response of the photodiode. The first is due to Z( ) or f( ) and is called extrinsic because it comes from the parasitic elements external to the junction. The second, contained in Iph( ), is called intrinsic because it is inherent to the mechanism of collection of the photogenerated charge inside the junction. Let us start with the analysis of the extrinsic cutoff in the two typical cases of practical interest. In instrumentation applications with a modest bandwidth requirement, the photodiode responsivity is maximized using a high value of load resistance R, such that R>>Rs. In this case we have from Eq.(5.20) (or directly by inspection of Fig.5-9b): Vu( ) /Vu(0) = [Iph( )/Iph(0)]/[1+j Cg+Cp](Rp//R)]

(5.22)

and the 3-dB cutoff frequency is: f2 = 1 / 2 (Rp//R) Cg+Cp)

(5.23)

Example: assuming the typical values for a Si photodiode: Rp = 100 M and Cg+Cp= (2+8) pF =10 pF, with R = 100 M we have f2 = 300 Hz, a rather low value yet sufficient in several measurement applications (for example, in power meters).

122

Photodiodes • Chapter 5

For the maximum speed of response, the photodiode is connected to a low load-resistance R, so that the capacitance Cp in Fig.5-9b is short-circuited. This situation is the best possible, although it sacrifices responsivity. Taking REg2 photon dissipation is confined in the narrow p-region at low Eg2, and photo-generated electrons drift through the depleted i-region in nearly the same time, or with very little transit time dispersion, while holes are immediately collected. This structure is called the UTC (uni-travelling carrier) photodiode because only one carrier is involved in the charge transport, the electron in the ppin+ of Fig.5-15. As the paths travelled by the electrons are substantially the same length, the time dispersion is greatly reduced and the intrinsic cutoff frequency fi can reach 100 GHz and more (Fig. 5-16).

Figure 5-16 (top) detailed structure of the layers of a fast UTC photodiode and (bottom) its frequency response (from Ref. [5], courtesy of the IEEE).

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133

Carriers move out of the narrow absorption layer by diffusion, but as this layer is very thin (e.g., 0.05 m like in Fig. 5-16 left) the associated time is very short. Making the device very small (for example, a few m by side, like enough for fiber receivers) so that the extrinsic cutoff fe is larger than fi, the detector reaches intrinsic cutoff performance and a 116-GHz cutoff (see Fig.5-16, right). For vertical light entrance like in Fig.5-15 right, the thin layer Eg2 has a limited absorption and quantum efficiency is low; however, we can use lateral entrance and a travelling wave structure (see Sect.5.2.8) and obtain high .

5.2.7.2 Multispectral Photodiodes Still another possibility offered by hetero-structures is that we may grow, one above the other, several junctions in materials with increasing bandgap Eg1>Eg2 >Eg3 … etc. This multi-spectral photodiode is the candidate for spectrum-resolved photodiodes, as well as high efficiency solar cells (Chapter 9), because in each layer the energy gap Egk is matched to the photon energy h reaching the layer, so the waste h -Egk is minimized, and cell efficiency is much higher than a single junction cell.

5.2.7.3 Lattice Matching To fabricate semiconductor hetero-structures with the usual epitaxial-growth techniques, the condition of lattice matching must be satisfied (Fig.5-17).

a)

b)

c)

Figure 5-17 Lattice matching condition in the growth of epitaxial layers.

134

Photodiodes • Chapter 5

Indeed, when atoms from a liquid or vapor phase are deposited on a monocrystalline substrate to originate a layer of different composition, the spacing of deposited atoms (or lattice constant db) shall be nearly identical to the spacing of the atoms in the substrate (or lattice constant ds), so that the new atoms are naturally accommodated in the same positions of the replaced substrate atoms and no defect is introduced in the grown layer. At the opposite, end, if any lattice mismatch, a number of defects will be generated at the interface between layer and substrate (Fig.5-17 a and b). Indeed, while in the vicinity of the interface the grown atoms are forced to adhere to the substrate atoms (Fig.5-17a), away from the interface they revert to the spacing of their bulk material lattice constant db (Fig.5-17b). Thus, the tolerable mismatch d= ds-db in thick epitaxial layers is very small, at most a thousandth of ds or db. Only if the deposited layer is very thin (a few atomic layers), the deposited atoms can keep the spacing of the substrate atoms (Fig.5-17c), albeit with an internal stress to force them in the new position. In this case there are no defects and the layer is called a strained layer.

5.2.7.4 Lattice Constant Diagram Fig.5-18 (top) shows the lattice constant d of the most commonly used elemental and compound semiconductors as a function of the energy gap Eg, and also (top X-axis scale) as a function of the threshold wavelength s of photoemission corresponding to Eg. Fig. 5-18 (bottom) is the same diagram for high- bandgap materials of interest for the blue and UV wavelength range applications. Lines joining semiconductors indicate compounds with a variable composition; full lines are for direct bandgap materials and dashed lines for indirect bandgap materials. To have the lattice matching and yet be able to vary Eg with composition, we shall move along horizontal lines in Fig.5-18. This rule is mandatory for growing hetero-structures of substantial thickness (say ≥10 nm). On the other hand, for very thin epitaxial layers (up to tens of atoms in thickness), the requirement is relaxed because the substrate atoms force the incoming atoms to assume the available positions of minimum energy, those of the atom places in the substrate lattice (Fig.5-18 c). Then, a mismatch of lattice up to a percent of ds can be tolerable (the order of magnitude is the strain limit l/l of the material). However, the mismatch creates a strong internal mechanical stress in the epitaxial layer, one that can be sustained only for small thicknesses, and this is accompanied by variations (often useful) of electrical and optical properties respect to the bulk material (in particular, of Eg and band alignment). For this reason, we call this film a strained layer and the material is distinguished from the bulk by the prefix s (e.g., s-Ge). In Fig.5-18(top), for the compound GexSi1-x on Si, we point out the difference between strained layers (point line) and bulk layers (dashed line). Strained layers are exploited in the so-called quantum well devices, in which the thickness of the sandwiched layers is very small.

5.2 Junction Photodiodes

Figure 5-18 (top): lattice constant d as a function of Eg for semiconductors of interest in optical communications, VIS-IR sensors, and MIR/FIR thermal imagers; (bottom) the same for semiconductors with high energy gap for blue and UV photodiodes.

135

136

Photodiodes • Chapter 5

Materials for hetero junctions of importance in applications are the following:  the ternary compound III-V GaxAl1-xAs (gallium aluminum arsenide), with intrinsic lattice match on all the compositions (Fig.5-2.17 top) and a direct bandgap from 1.4 to 1.92 eV, compatible with growth on GaAs and Ge substrates for solar cells and photodiodes, and with threshold s from 650 to 900 nm,  the quaternary compound III-V InxGa1-xAsyP1-y (indium gallium phospho-arsenide), compatible with growth on InP (indium phosphide) substrates (horizontal line at the center of Fig.5-2.17 top), and which, depending on composition, yields a threshold s from 1000 to 1800 nm. For the application in the third window of optical fiber, the threshold s =1800 nm (for y=1) is usually chosen; that is, we have a heterojunction photodiode in the ternary InxGa1-xAs between layers of InP,  The ternary compounds II-VI HgxCd1-xTe (mercury cadmium telluride or CMT) and IIVI SnxPb1-xTe (lead tin telluride or LTT), used for detectors in the middle and far infrared, can be grown with good reproducibility in a vast latitude of the composition fraction x on substrates with x=0. These materials go across the Eg=0 line (Fig.5-18 top) at increasing x; that is, they exhibit a transition from semiconductors to semimetals (for which the valence band top overlaps the conduction band bottom),  Gallium nitride and zinc selenide are popular material for photodetectors sensitive to blue and UV-A regions and, even more important, for LED and high voltage transistors.

5.2.7.5 Interfaces When growing epitaxial layers of different compositions, at the interface a band bending is created because of the migration of charges from one material to the other. According to the band alignment, three cases may happen (see Fig.5-19) that are known as in literature as type I, II and III alignments: (I) both electron and holes near the junction migrate to the smaller bandgap material, (II) electron and holes separate, each going to opposite material, (III) one carrier is unaffected and the other favors one material.

Figure 5-19 Alignment of bands in a hetero-junction interface.

Examples of resulting band bending in a type I junction are reported in Fig.5-20. Here, as junction is formed, boundary electrons in n-type (holes in p-type) drift to the opposite material and leave ions depleted; charges of depleted ions bend the bands, upward in conduction band CB (downward in the valence band VB) and attract carriers of opposite polarity creating a double bending (Fig. 5-20). Bending in CB drags the VB (n-type) and

5.2 Junction Photodiodes

bending in VB drags the CB (p-type) while keeping unchanged EG, detailed treatment of band bending can be found in Ref. [6].

137

EC and

E V. A

Figure 5-20 Band bending in type I hetero-junction interface.

Band bending introduces unintended obstacle to the flow of charge carriers and can behave as a barrier to carrier transport, like in a rectifying contact. The effect is more severe at lower doping levels because the width of the spikes is larger, while in junctions of heavily doped layers, tunneling may help overcome the barriers. Also, the junctions depicted in Fig. 5-19 are abrupt and exhibit the maximum bending, whereas making the junction graded, the spikes are washed out. In general, type I alignment is the best arrangement when we wish that both carriers are accumulated in the same narrow region to enhance recombination (like in lasers), while type III is preferable to favor one carrier to transport charge (like in photodetectors).

5.2.8 Photodiode Structures 5.2.8.1 Traditional Structures Fig.5-21 illustrates some examples of common structures of photodiodes fabricated with the standard planar technology. Devices are shown as cross-sections of the circular sensitive area; the optical access window is indicated by arrows. In a normal pn-Si photodiode (Fig.5-21a), the active region is obtained by a diffusion p (or p+) in the n-type substrate. After the oxidation and opening of windows, a ring metallization supplies the anode contact (A); to obtain the cathode contact (K), the chip bottom is first n+-diffused to decrease the series resistance and then a metallization follows. The final passivation with silicon oxide (SiO2) of the optical window is also useful to reduce the reflection loss (Section 5.2.1). Sometimes, a guard ring (G, see Fig.5-21a) is also fabricated around the anode, to divert reverse current contributions coming from the edges; this allows an increase in the bias voltage up to a value close to breakdown voltage (see also Appendix A4). A similar structure is that of a pin-Si photodiode, see Fig.5-21b, where the active layer is intrinsic and the guard diffusions are left to collapse on the anode.

138

Photodiodes • Chapter 5

R G

A A

SiO

2

p+

p

p+

n Al

V

bb

In P p+

-

InGaAs i

InP n

+

InP n+

n+ K

K

(a)

(d)

A

A p

p+

p+

p+

i

i

n+

n+ K

K

(e)

(b)

K

E

B n

n+ n

p n

m

(c)

A

(f)

C

Figure 5-21 Typical photodiodes structures: (a) pn-junction with guard ring, (b) pin junction, (c) Schottky junction, (d) heterojunction pin photodiode; (e) lateral-entrance photodiode, (f) the npn phototransistor.

Fig.5-21c shows a Schottky photodiode of mnn+m (or min+m) structure, with optical access from the semiconductor side. A typical heterojunction pin photodiode is shown in Fig.5-21d. Here, on an n-type InP substrate, first an epitaxial active layer is grown in a compound at low Eg (the ternary InGaAs type i), and on top of it, a second epitaxial p+ layer in InP to present a high Eg region to input photons. The final etching leaves a mesa shape helpful for limiting the field enhancement at the edges. In Fig.5-21e shows an example of lateral-entrance photodiode. This yields an exceptionally large width W of photon absorption, useful for low- materials (like Si, indirect-bandgap and extrinsic detectors), especially near the photoelectric threshold s. A

5.2 Junction Photodiodes

139

limitation of such a structure is the small active area, at least if one starts with a normal wafer thickness. In Fig.5-21f we report the structure of a bipolar npn phototransistor (see Section 7.1), in which the base-collector junction is made available for optical access. Lastly, let us note that for any of the above reported components there are also the complementary ones with the n and p-doping region interchanged. Both types are equivalent in operation principle with the interchange of electron and hole, but not necessarily equivalent in performance, in view of the different mobility, available energy levels, etc.

5.2.8.2 Advanced Structures While the performances of traditional-structure photodiodes are satisfactory at medium and high frequency (say, up to 10…30-GHz), new advanced structures are searched to enhance the frequency performances of photodiodes up to hundreds GHz, in view of applications as the receiver of optical communications at increasing bit rate. Reduced noise is also another requirement, but this is pursued by searching for an internal gain and partially satisfied by the APD (see Chapter 6). To improve the high-frequency performance, we should increase both the extrinsic and intrinsic cutoffs. In a pin structure, this requires decreasing both active area and specific junction capacitance Cb. But, as we can see from Fig.5-13, this is incompatible with a low drift time d (and its dispersion d). Also, a high absorption thickness W for good efficiency is not compatible with low d in a front-entrance structure, and the reason is that photons and charge carriers travel parallel paths in a pn or pin diode. To overcome the trade-off, we shall move to a lateral entrance (Fig.5-21e), in which the optical and electrical signals travel perpendicular. In this way, W and d are no more bound to each other, and the extrinsic cutoff is removed because the device inter-electrode parasitic capacitance becomes part of a distributed electrical transmission line [7,8]. In Fig.5-22 (top), we can see an example of realization of the concept: light enters parallel to the junction plane of the photodiode and the output signal is picked out by the coplanar line connected to the electrodes of the device. This device is called waveguide photodiode or WGPD, and the remote end of the line (Fig.5-2.21 bottom) is matched to 50 , the characteristic impedance, and is biased through the inductance Lb choke. A further step is to allow the electrical signal travelling along the line be continuously supplied by the detected current from a distributed photodiode as in Fig.5-23. This device is called travelling wave photodiode or TWPD, and is fabricated with a periodic structure of several UTCs placed upon an optical planar waveguide carrying the light signal, as shown in Fig.5-23, left. The periodic structure is better suited than a continuous one to satisfy the condition of phase matching necessary to sum up coherently all the contributions along the line. In the TWPD structure of Fig.5-23 using three UTC photodiodes of the type shown in Fig.516 left, the measured impulse response FWHM (full-width half-maximum) was =1.85 ps

140

Photodiodes • Chapter 5

corresponding to a high-frequency cutoff f2= 116 GHz. Without the terminating resistor, the impulse response had =4.59 ps and f2= 56 GHz (Fig. 5-23, right) [5].

Figure 5-22 By incorporating the tiny photodiode PD in a coplanar transmission line (top), we get a WGPD (waveguide photodiode) with improved frequency performance. At the receiving end (bottom), the line is terminated on the characteristic impedance 50 , and the bias is supplied through a decoupling choke Lb.

Figure 5-23 (left) structure of a fabricated travelling wave photodiode TWPD, and (right), the device impulse response with and without the terminating 50- resistor (from Ref. [5], by courtesy of the IEEE).

5.2 Junction Photodiodes

141

Another ultrafast device is the LT-GaAs (low-temperature grown GaAs) photodetector, which is fabricated in a variety of methods, including MBE (molecular beam epitaxy), high energy molecular bombardment, deep level doping, and polycrystalline or amorphous material growth. All these methods are appropriate to create defects in the material, that act as traps and recombination centers for the photogenerated carriers. In this way, the carrier lifetime is shortened to picosecond and below, and thanks to the unaltered high carrier mobility, the device reaches frequency cutoff of several hundred GHz and up to THz. Structures preferred for the LT-GaAs are: the Schottky junctions (or MSM, metal-semiconductor-metal) and photoconductors (Sect.7.2). Also, a modified MSM structure has been demonstrated [9], that incorporates an optical waveguide underneath the LT-GaAs absorbing layer so as to make a TWPD device with a bandwidth of 190 GHz. Extensions of the detection range to =1000__1300 nm, where LT-GaAs has negligible response, have been fabricated with LT-InGaAs grown on a GaAs substrate [10].

5.2.8.3 Resonant Cavity-Enhanced Photodetectors When the photodetector width W is too small to completely absorb the incoming photons, W>Ipm,Iom, Ipr,Ior, and n=1, we have: V = Vm-Vr

156

Photodiodes • Chapter 5

= (kT/e) ln A + Vosm-Vosr + (kT/e) ln( mIor /a rIom)

(5.55)

The first term in Eq.(5.55) is the desired attenuation measurement A, in a logarithmic scale as required in applications - where the absorbance in AU or the dB loss are used (see Appendix A1.4); the other terms are dc components added to the useful signal, irrelevant provided they are stable and constant. In particular, note that the signal V in Eq.(5.55) is no more dependent on light-source power P, contrary to the single channel given by Eq.(5.54) in which Iph = P. It only depends on the ratios of dark currents and of spectral sensitivities, and on the offset differentials, rather than being affected by the absolute values of such quantities. This brings about a good immunity of the log differential circuit to the drift of components. Going back to Eq.(5.55), the logarithmic signal is relatively small, VA=(kT/e)lnA= [59.6 mV] Log10A, or it amounts to 59.6 mV per decade of attenuation. Therefore, we add a gain in the difference stage (with resistances R and r), for example: A2 = R/r = 1 V/59.6 mV = 16.78, so as to bring the output to the standard level of 1V/decade. The third stage performs the temperature correction for scale factor and additive constant by means of a single device, a thermistor, the resistance of which is expressed as RT= RTo[1+ T(T-To)]. Here, RTo=R2 is the ohmic value at To=20°C and T is the temperature coefficient, typically -5 to -7%/°C, for an NTC-type thermistor. Assuming for the moment V0=0, the third-stage gain is: A3 = [RT/(R1+RT)].(1+R1/R2) and we have A3=1 for RTo=R2 (at 20°C). The temperature coefficient of A3 is easily calculated as TR1/(R1+R2). To compensate for the temperature coefficient of the scale factor, given by A= dVA/VAdT= +0.366%/°C (at 20°C) it suffices to let - TR1/(R1+R2)= A and solve for the resistive ratio of the voltage divider as R2/R1=- T/ A-1 and R1=RTo/(- T/ A-1). In practice, with such a compensation one can obtain a residual error of ±0.005%/°C in the scale factor. The dc additive term at the output Vu of amplifier A2, after the scale factor compensation, is, from Eq.(5.55): Vu(dc)=A2 V=[(e/kT)(Vosm-Vosr)+ln( mIor /a rIom)].1V

(5.56)

Typical values of the temperature coefficients of the quantities in Eq.(5.56) are:   

-4 mV/°C for the offset voltage, ±5 mV/°C for the spectral sensitivity, and +140 mV/°C for the dark current.

In the ideal case of identical components, we would have Vosm=Vosr, m= r and Ior=Iom and in this case their contributions would cancel out; in practice, real components may have residual differences on the order of ±10 mV/°C. The corresponding error can be compensated

5.3 Photodiode Circuits

157

by adding a voltage Vo, adjustable in polarity and amplitude, at the terminal common to R2 and RT (Fig.5-28). Then we have at the output a dc term calculated as: Vo [R1/(R1+RT)](1+R1/R2) -Vo R1/R2 whose temperature coefficient is: Vo TR2/(R1+R2). Adjusting Vo can null the thermal drift, and in a real circuit the residual output error Vu is reduced to I0/50 mV, whence the assertion. It is also possible to go beyond f2 with the equalization, although at the expense of a worsening of noise because the voltage contribution vA now increases with frequency. If f3>f2 is the desired new frequency and f2 is the previously calculated maximum, with an equalization to f3, noise is still given by Eq.(5.70) but with the right-hand term multiplied by 1+(f3/3f2)2. This result applies in general, provided that f3,f2>>f1. Thus, the contribution of the voltage noise vA, neglected so far, amounts to a factor 1+(1/3)2=1.11 at f=f2. On the other hand, if we want a large f3, for example f3=10f2, the noise increases by a factor of 12. As already noted, the requisite on R is alleviated if we assume a specific signal level P>I0 as a reference. In this case the previous expressions still hold, provided we change I0 to I0+ P. Lastly, let us remark that the equalization can also be used in connection with any other low-noise circuit method, for an a-posteriori improvement of bandwidth at equal noise or of noise at equal bandwidth (as in the case of the transimpedance circuits of Fig. 5-31). The fundamental limitation of the equalization approach is that the equalizer stage requires a substantial gain A=f2/f1 at the equalized frequency f2, and thus it is necessary to have an active device with a transition frequency fT at least equal to Af2. When f2 approaches fT, there is clearly no reserve of gain available for equalization. Thus, the technique is good and widely applied for frequencies f2 [Eq.(6.8'')]. APD structures with non-uniform multiplication. Let us briefly report the extension of the above results to the case, very often found in practical structures, of a non-uniform multiplication. When the electric field is not a constant in the multiplying region, we must take the ionization coefficients (x) and (x) as a function of position x. By repeating the calculations leading to Eq.(6.2) with the aid of the integration factor exp ∫ ( - )dx, we obtain the gain for pairs injected at x=0 as: Mc(x) = [exp ∫ x-L ( - ) dx'] / [1 - ∫ 0-Ldx' exp ∫ x’-L( - ) dx' ]

(6.16)

an expression that equivalently can be written: Mc(x) = [exp -∫ 0-x ( - ) dx'] / [1 - ∫ 0-Ldx' exp -∫ 0-x’( - ) dx' ]

(6.16')

The condition of infinity gain [Eq.(6.3)] becomes:

∫0-L dx' exp ∫ x'-L( - )dx'' = ∫ 0-L dx' exp -∫ 0-x' ( - )dx'' = 1

(6.3')

In terms of gain for pairs injected at x=0, Eq.(6.16') can also be expressed as: Mc(x) = Mc(0) exp -∫ ( - ) dx'

(6.17)

0-x

to show the similarity of the trend with Eq.(6.5B). From a numerical comparison with Eqs.(6.2) through (6.5), valid for uniform field and , =const, one can find that the diagrams of gain and frequency cutoff still hold with a good approximation using, in place of the ionization ratio / the averaged value: / =∫ 0-L dx / ∫ 0-L dx. Analogously, gain variance M2 and excess noise are still given by Eqs.(6.10) and (6.13) using the weighted value for / : /

= (K2-1)/(K12-K2)

(6.18)

where: K1 = ∫ 0-L

exp -∫ 0-x ( - )dx'] dx / ∫ 0-L

exp -∫ 0-x ( - )dx'] dx

(6.18')

K2 = ∫ 0-L

exp -2∫ 0-x ( - )dx'] dx / ∫ 0-L

exp –2 ∫ 0-x ( - )dx'] dx

(6.18'')

Let us quote a final result, useful for the numerical simulation of cascaded multiplication regions, each having a constant internal field. In a structure with two multiplication regions 1 and 2, with gains M1n(0) and M1p(L) for the injection of electrons at x=0 and of holes at x=L in region 1, and gains M2n(0) and M2p(L) in region 2, the compound Mn(0) gain of the cascade is found as:

6.1 Avalanche Photodiode

Mn(0) = M1n(0) M2n(0)/[1-(M2n(0)-1)M1p(0)]

187

(6.19)

and applying Eq.(6.19) recurrently to a structure with step-wise approximated field distribution, the gain can be calculated. Also, by computing M2=[ Mn(0)]2, one can find that Eq.(6.18) still applies.

6.1.3 Experimental Evidence and Deviations Experimental results match well the general trends of theory findings. The optimal gain is clearly confirmed as the key parameter of both frequency response and noise of M 0= the APD. However, surprisingly, in applications of APDs, gain-bandwidth product and excess noise factor are found serendipitously a little bit better than expected form theory, especially in thin junctions [3,4]. This mitigation of the positive feedback effects has been observed in GaAs [5], Al0.8Ga0.2As [6], and in N2-cooled HgCdTe [7]. Such improved performance has been attributed to a number of effects that regularize the impact ionization, beyond the random exponential path distribution assumed to model the APD and presented in the previous Section. One effect is dead space between successive multiplications, i.e., the nonzero path the carrier shall cover before becoming able to ionize. Another is bandgap boundary effect encountered in heterojunction APDs, in which injected carriers arrive in the multiplication region with an initial energy [5]. The theory incorporating these nonlocal effects is rather involved and will not be reported here. However, in most cases the deviations amount to an improvement of a factor 1.5…2 of the gain-bandwidth product f2DM, Eq.(6.8"), and of the excess noise factor F, Eq.(6.10"), when we are working at a gain exceeding the optimal gain (M> ), whereas below the optimal gain (M< there is no deviation from the results of the old theory. Therefore, we can take the results of the ‘memoryless’ theory reported in previous Section as a conservative estimate of the APD performance.

6.1.4 APD Structures With the normal structure of a pin (e.g., Fig.5-21 b,d) the APD does work, but does not yield good performance, because the depleted region serves both for the photon dissipation and the carrier multiplication. In this case, the length L of the multiplication region should be tied to the absorption length La to get a good quantum efficiency; moreover, the multiplication gain is that of MI for distributed pairs, less than the maximum available Mc(0) [Eq.(6.7)] and, even worse, with a large variance [Eqs.(6.9) and (6.10)]. A desirable structure for an APD would have two separate regions, as in Fig.6-1.8: one of thickness L1 La for the photon absorption and the carrier drift, with a medium/high applied field Ed to reduce transit time, the other with a high field Em for the multiplication of the most efficient ionizing carrier. The ionizing carrier will be injected from the absorption region into the avalanche region, which is of a thickness L adjusted to get the desired gain M=M( L). To obtain a high opti-

188

Avalanche Photodiode, SPAD, and SiPM • Chapter 6

mum gain, materials with a high / ratio are preferred and, in the case of heterostructures, the optical entrance from the high field side is allowed, if a high bandgap material is used there (see Section 5.2.7), which is transparent to incident radiation at the of interest. Not all these requirements can always be satisfied in a practical design, because of material-related limitations or fabrication difficulties. The structures currently employed for APDs are illustrated in Fig.6-1.9, along with the schematic distributions of depleted charge and electric field, which may be compared (Fig.69a) with that of the pin shown as a reference.

Em

P

P

Ed

L1 Figure 6-8

L

z

The desirable field distribution for an APD: left curve is for entrance from the drift region, right curve from the multiplication region.

The simple abrupt n+np junction of Fig.6-9b employed in the first germanium APDs (in which > ) allows to obtain a reasonable quantum efficiency because of the absorption has a thick n-region available. Of the pairs produced in this region, electrons are at once collected by the cathode without multiplication, and holes are injected in the junction where they can contribute to the multiplication gain in correspondence with the peak of the electric field. This structure is not optimal, because the electric field in the multiplication region is not constant, nor can the drift region be made too thick; but it is simple and practicable and is used in many commercially available Ge APDs. A substantial improvement is offered by the n+p+ p+ structure, called reach-through (fig.6-9c) and commonly used in silicon APDs. Here, a first n+p+ junction of small thickness and high field provides the multiplication for the electrons coming from the thick undepleted (i.e., lightly p-doped) region where photons are absorbed. Here, photons can cross the first junction with no loss because of its thinness, and the photogenerated holes in are collected at the anode without multiplication. The name reach-through comes from the way p regions are depleted at increasing bias voltage: first, the p+ region is depleted and from here depletion becomes extended all through the region reaching the substrate p+; this reach-through condition is necessary to set-up the drift field in region . A useful consequence of this mechanism is that the avalanche field at the n+p+ junction is less dependent on the external bias voltage compared, e.g., to an n+np structure.

6.1 Avalanche Photodiode

189

Figure 6-9 The pin structure (a) compared to several commonly used APD structures: (b) the Ge-APD n+np; (c) the popular reach-through structure; (d) the inverted reach-through, and (e) quaternary compound SAM. On the right, the schematic distributions of dopant charges and electric field are indicated.

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Avalanche Photodiode, SPAD, and SiPM • Chapter 6

An interesting variation for Si APDs, that looks like an inverted reach-through is the p+ p+ n+ structure of Fig.6-9d. This offers a more uniform field in the drift and multiplication regions and has the optical access from the drift side. The structure is more critical to fabricate because of the thick epitaxial layer, but it offers a wider multiplication region and allows operation at moderate fields, where Si has the largest / ratio (see Fig.6-1). A further improvement of the reach-through structure is obtained with the use of a heterojunction in compound semiconductors, because one can then choose the materials best suited for absorption and ionization in the two regions. Fig.6-9e shows an example of a heterojunction APD in In0.53Ga0.47As/InP, designed for the detection in the near IR up to 1.8 m as required for fiber-optics communications. Such a structure is called SAM (separate absorption and multiplication), to indicate that the absorption and ionization regions are separated and in different material or composition. Radiation is incident (Fig.6-9e) on the material with higher Eg, the indium phosphide of the substrate, and is dissipated in the thick region in indium-gallium arsenide; electrons are directly collected and holes drift to the p+n+ junction of multiplication. We may remark that the p+n+ n+ structure is complementary to the n+p+ p+ reachthrough of fig.6-9c, as it is > in InP (Fig.6-1) and as it uses the principle of a transparent incident layer (as described in Sect.5.2.7 and Fig.5-15). A problem with the heterojunction is the presence of interface states at the InP/InGaAs boundary, which trap charge carriers. These states are nearly eliminated by sandwiching a layer with gradually varying composition, a structure called SAGM __ separate absorption, graded, and multiplication. An APD based on InGaAs/InP heterostructure has a useful spectral range 0.9__1.7 m, see Fig.5-2.16. It has a good bandwidth performance ( GHz), while the noise figure is relatively high because of the low ionization ratio / available in InP (see Fig.6-1.1). Another interesting compound semiconductor for the APD heterostructure is gallium aluminum antimonide (Ga0.94Al0.06Sb/GaSb) which is grown on GaAs substrates and has a high ionization ratio / 50 (Fig.6-1.1) and a spectral range comparable to In0.53Ga0.47As (see Fig.5-17). The material is however difficult to grow and this hampers the commercial development of such a device.

6.1.5 Bandgap Engineered APDs The excellent potentials of low-noise and large bandwidth of APDs as compared with a pin photodiode, have not yet been fully exploited. Even the best performances, offered by SiAPDs, are limited to a moderate internal gain (50__100 max) and cover a narrow spectral range (0.5__1.0 m). One direction is toward work on new materials, the best candidates being those with a very high ionization ratio / and a high carrier mobility , and an energy gap suitable for

6.1 Avalanche Photodiode

191

the intended spectral range. Compound semiconductors and GaAlSb are an example of this trend, yet the difficulty is that a new material requires development of a new fabrication technology. The many steps like doping, oxidizing, etching, epitaxial growth, etc., are so costly that a single new component hardly justifies them. Another approach is to develop the SAM concept further, in combination with heterostructures. If a single material cannot match both the spectral range and the / requisite, we can try to make a two-layer sandwich of different materials, one optimal for photon dissipation and the other for multiplication. Of course, the two layers shall be lattice- matched (Section 5.2.7), or very thin strained quantum wells. This approach has been attempted with Si as the thick multiplication material, and a Ge/Si strained multilayer as the photon dissipation material. The multilayer is made of alternated Ge and Si layers, thin enough to match the Si bulk lattice constant (Fig.5-18). While Ge absorbs up to 1.7 m, silicon is transparent beyond its threshold s 1.1 m and only serves to keep memory of the lattice constant for the Ge layers. This Ge/Si APD has a p-doped Si region, and an n-doped Ge/Si layer, so that electrons are injected from Si into Ge/Si for multiplication; the optimum gain is that of Si (50__100) and the spectral response is that of Ge ( s 1.7 m), which is adequate for optical communications. Two viable approaches for the synthesis of materials with unusual electronic properties have been demonstrated. Useful for APDs as well as for lasers and other photonic devices, these are the bandgap engineering and superlattice structures proposed by F. Capasso [8]. An example of a bandwidth engineering device is the sawtooth avalanche structure shown in Fig.6-10a. Many (30-50) layers of p-type GaxAl1-xAs are grown one over the other, each with a thickness V through a large Rb, and a low resistance load RL is coupled through capacitor C (top left). When a photon is detected, the SPAD switches to a high-current state and discharges C on RL (top right). So, the sequence of detected photons is replicated with a sequence of pulses. Different from the linear mode of the APD supplying a linear replica of P(t), the Geiger mode supplies a sequence of pulses whose rate is proportional to P(t) (bottom).

The PDP increases at increasing overdrive voltage V above V , with the typical trend shown in Fig.6-15. Because of the PDP, the equivalent quantum efficiency of the device is decreased to eq = PDP with respect to the value of the quantum detection process. Further, as the probability of triggering applies also to thermally generated carriers, at the increasing of the overdrive V also dark current I0 will increases. Therefore, we have to make a trade-off between a high PDP and a low I0 by choosing the appropriate V. Once the avalanche has built up, the device is in a state of high current capability, and it draws all the current permitted by the external circuit, in this case the current of the discharge of capacitor C on the load resistance RL. As this current is much larger than that drawn by Rb from the battery Vbb, now the diode voltage doesn’t increase anymore, and Vak is pinned to V + V. Current I continues to flow until the voltage drop across resistance Rb (Fig.6-14) is such that the cathode voltage Vak=Vbb-RbI falls below V , and the gain returns to finite value. Then, current is no more sustained now and damps off, or, the avalanche is quenched.

198

Avalanche Photodiode, SPAD, and SiPM • Chapter 6

Meanwhile, capacitor C has been partially discharged across the output resistance RL, delivering at the output an exponential pulse, decaying with time constant RLC and with an amplitude given by the excess V voltage, with respect to V , reached during the charging phase [in practice, up to V≈Vbb-V ].

Figure 6-15 Typical PDP for a reach-through structure SPAD as a function of overdrive voltage V in excess of V [for shallow junction, voltages are smaller].

In conclusion, for each detected photon (or thermally generated carrier) in the multiplication region when the SPAD is in the infinity gain state (at Vak≥V ) a pulse is generated at the output with probability PDP. [This is as shown schematically in Fig.6-14, top, for the case PDP=1, while for PDP>ICE0 to optimize hfe and fT (with improvement factors 5-20). The differential stage subtracts the dc collector voltage Vbb-RIBhfe and supplies an output referred to zero (and a complementary output). For noise, in Eq. (7.3) we have now the shot noise of the new term IB to be added to P+ICB0.

Available both in Si for the visible and near IR (threshold at 1 m) and in Ge with response up to 1.8 m, generally in plastic package layouts incorporating a dome lens to increase the acceptance area (from a fraction of mm2 to a few mm2), phototransistors find extensive applications in industrial apparatus where small outlines and maximum circuit simplicity are required, as in proximity alarms, intrusion detectors, item-counters, barcode readers, etc. For high-frequency applications, several structures and materials have been tried to improve the performances of the basic phototransistor with the goal of even surpassing pin and APD photodiodes. However, even the best laboratory prototypes have reached a gainbandwidth product of 50-100 GHz, and, in view of Eq. (7.4), they have a useful cutoff limited to fT 10 GHz at gains hfe 5-10.

7.1.2 The Optocoupler Another important application for the phototransistor is the optocoupler, a component of widespread use in instrumentation for purposes of decoupling, isolation and disturbance suppression. In the optocoupler, a Si phototransistor is mounted in front of an LED (in GaAlAs or GaAsP) as shown in Fig.7-3a. The combination of the two devices permits one to transfer a signal without electrical continuity of the circuit, i.e., realizing the so-called galvanic insulation of the ground lines, but still with a current transfer ratio close to or larger than unity between the two ports. Galvanic insulation is important when connecting a signal from a sender to a receiver that are using different grounds (Fig.7-3c), because between remote or physically different grounds we can find an EMI (electro-magnetic interference) conductive disturbance, due to the electrical returns of other users or instruments or to ground dispersion currents, with amplitudes of Vd ranging for small values (mV's) to unexpectedly high (several Volts).

226

Phototransistors, Photoconductors, and SNSPD • Chapter 7

Figure 7-3 a) an optocoupler mounted in a 4-pin DIP package; b) the current transfer ratio for a single and a Darlington phototransistor; c) the ground loop in a sender-to-receiver connection generates a disturbance at the receiver; d) by breaking the ground loop the optocoupler eliminates the disturbance, and if provided of internal screen also the stray capacitance coupling is removed.

In the direct connection of Fig.7-3c, disturbance is in series to the circuit of the receiver input and thus will be partly superposed to the useful signal. Instead, by breaking the ground loop with the optocoupler, the EMI conductive disturbance is eliminated. The transfer ratio of the optocoupler is the product of LED mission factor =PLED/Iin, optical coupling efficiency opt = Ppht/PLED, and phototransistor spectral sensitivity pht

7.1 Phototransistors

227

=IC/Ppht = Sihfe, where Sihfe is the spectral sensitivity of silicon, the phototransistor mate__ rial. The total transfer efficiency tra = opt pht is typically in the range of 0.5 5 for the single transistor receiver, and can increase to 10...50 in the unit using a Darlington phototransistor (Fig.7-3b) or an amplification function, which are then mounted in a 6__8 pin DIP package. Isolation is enhanced by interposing a thin slab of transparent insulating material (usually glass) between the LED and the phototransistor. In a typical optocoupler, the resistance between input and output attains some 1011 , and the voltage that can be applied is 5 kV rms. Capacitive coupling between input and output – of the order of magnitude of pF – becomes detrimental at frequency of MHz's, and can be reduced by depositing on the insulating slab a thin mesh-shaped metallization (only slightly reducing the coupling efficiency), that breaks the mutual capacitance into two separate Cp and C'p (see Fig.7-3d). In this case the screen will be connected at the receiver ground (Fig.7-3d). Maximum frequency of operation of the basic optocoupler is typically 40__80 kHz, but can exceed the MHz in units with added internal amplifier. About linearity of response, because of the hfe dependence on collector current, the basic optocoupler is adequate for digital signals only, and we can see in Fig.7-3b the typical large variation of the transfer ratio as a function of the drive current to the LED. By using a photodiode for the receiver side and pre-distortion to correct the LED -dependence on input drive current Iin, linearity errors of >1 we have: G

Gb+(

n,p n,p /L

2)P

= G b+ P

(7.9)

whence a linear dependence of the device conductance G upon the detected radiant power P is apparent. This dependence is well matched over several decades by photoconductors, as can be seen from Fig.7-8. Deviations at the low illumination levels are due to the term Gb, and at high levels are due either to the dissipated power or to the series resistance of the access contacts.

7.2.3 Frequency Response and Noise In the time domain, the current impulse response to an individual charge pair photogenerated at t=0 is a rectangular pulse of the type: i(t) = i0 rect(0,t, ), where is the pulse duration, a random variable with mean value n,p. Since the distribution of is a negative exponential, i.e., p( )=(1/ n,p) exp- n,p the mean response-time that is obtained by weighting the rectangular response with the exponential probability turns out to be an exponential of the type i(t) = i0 exp-t/ n,p. Correspondingly, the frequency response of the gain is: M( ) = 1+ [ n,p n,p V/L2] / (1+j = 1+M(0) / (1+j

n,p)

n,p)

(for M>>1)

(7.10)

or, we have a pole at the cutoff frequency f2 given by: f2 = 1/2 n,p

(7.11)

7.2 Photoconductors

235

By comparing this with Eq.(7.8), we can see that the photoconductor has a constant gain-bandwidth product at high gains (M>>1): f2 M = n,p V/2 L2

(7.12)

In other words, gain is obtained at the expense of bandwidth. Thus, in designing the doping with trap states, one must choose dopants that have a time constant n,p=1/2 f2 compatible with the desired frequency performance. Then, the factor n,p/T of excess with respect to the intrinsic drift time n,p will supply the useful photoconductor gain M.

Figure 7-8

Typical voltage/current characteristics of a CdS photoconductor, with the illuminance E as a parameter. Lines at +45 deg are at constant resistance, lines at -45 deg are at constant dissipation. Photoconductance curves are linear up to excessive dissipation, then deviate to lower ohmic values.

Concerning photoconductor noise, first of all we have the shot noise of photogenerated and dark currents, given by i(t)2 =2e(Iph+Igr)B, a term that we shall expect to be amplified by

236

Phototransistors, Photoconductors, and SNSPD • Chapter 7

M2 as a quadratic fluctuation. Moreover, the gain M= n,p/T is a random variable like the trapping time n,p of the individual carrier, and this gives a further noise contribution. The statistical analysis of the process (see below) yields as a result for the current variance: 2= i

2e [1+2(M-1)2](Iph+Igr)B

(7.13)

At high gains (M>>1), Eq.(7.13) shows that the photoconductor has a noise figure F=2 (first term on the right hand). Of course, we should add the Johnson noise of load resistance R, 4kTB/R, plus any preamplifier noise (remembering that their weight is scaled by M2). Noise due to the conductance G, however, need not be considered because it already has been accounted for in Eq.(7.13). Noise analysis. Let us interpret the trapping gain as a statistical process, by assuming first M= /Td where Td and are the drift and trapping times. The photogenerated current Iph produces, in the observation time T, a random number n of carriers with mean n IphT/e and variance n2 n (in the reasonable hypothesis of Poissondistributed times of arrival for the photons). For a single carrier, a multiplication by M= /Td takes place in the photoconductor, where Td is deterministic while is random and has the negative-exponential distribution typical of decay phenomena p( )=(1/ n,p)exp- n,p, where n,p now has the meaning of mean trapping time. Thus, the gain is also a random variable M, and from its distribution p(M)=(1/M) exp-M/M it is easy to compute the mean M =M= /Td and the variance M2. At the photoconductor terminals, we collect N=Mn charge carriers in the time interval T. To calculate mean and variance, we can use the results of Eq.(4.7'') and write: N = M n =M IphT/e, N

2

2

M

M 2 n2 M2 IphT/e + M2 IphT/e = 2 M2 IphT/e

n

By recalling that current is i=N(e/T), in terms of charge carrier number and that variances of the number and of the current are i2 N2(e/T)2, and considering that T=1/2B because of Nyquist theorem, we get: i = MIph ,

i

2 4 e M 2I B ph

(7.13')

Now, because the dark current Igr adds to the photogenerated Iph, in the above expression we must also add Igr. In addition, as gain is more accurately defined as M=1+ /Td, we shall change M to M-1 in the variance i2 of Eq.(7.13'), so that for M=1 there is a residual shot noise Iph+Igr, from which Eq.(7.13) is obtained.

7.2.4 Photoconductor Types Cadmium sulfide CdS is perhaps the most frequently used photoconductor in the visible range, because its spectral response (Fig.7-9) matches the visibility curve V ) of the human eye fairly well (see Appendix A2). It can be used undoped, but it is most frequently prepared by sintering a film doped with a small quantity of CdCl2 and CuCl2. Chlorine acts as a donor, while copper is an acceptor, which compensates the material and supplies trap levels for holes with a typical lifetime p 50 ms. A CdS photoconductor typically has a gain M 104 and a dark current MIgr 1 A/mm2 at bias voltages V 80 V.

7.2 Photoconductors

237

Cadmium selenide CdSe is prepared in a fashion similar to CdS, with the addition of Cl and Cu ( p 10 ms) as dopants; it has a spectral response shifted to the red (Fig.7-9) and a higher dark current (typ.10 A/mm2 at V= 20 Volts).

Figure 7-9 Spectral responses of some common photoconductors, with the relative eye sensitivity V( ) plotted for comparison.

Also very common is cadmium sulfo-selenide Cd(S,Se), with a spectral response and properties intermediate to those of CdS and CdSe. Zinc sulfide ZnS has a response extended to UV and is usually prepared without intentional doping. Two early photoconductors, used especially in vidicons for the visible and the near IR, are lead oxide PbO and lead sulfo-oxide Pb(S,O), the latter reaches a response up to 2 m and is used for near-IR cameras. Photoconductors in Si, Ge and GaAs have been also fabricated, doped respectively with Au, Mn, Si and Cu, to yield spectral responses covering the far and extreme infrared (Fig.57). Other exotic compositions are those of organic photoconductors (i.e., phthalocyanines, polyarylenes and polyacetylenes), which are actively pursued for low-cost applications in reprographic apparatus.

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Phototransistors, Photoconductors, and SNSPD • Chapter 7

7.2.5 PV and PC Detectors for IR As single-element detectors for the middle and far IR, photoconductors can be used as an alternative to photodiodes in the corresponding materials, though generally with less sensitivity. Indeed, the dark current is smaller in a reverse-biased junction than in a homogeneous block of material of the same conductivity. However, as photoconductor fabrication is much easier and its cost is lower, for a long time such detectors have been competitive with respect to photodiodes, and are widely used in many applications. For this reason, an established categorization for the thermal IR photodetectors is that of distinguishing photovoltaic (PV) and photoconductive (PC) detectors [1]. Popular PC detectors are indium antimonide InSb for the MIR, and mercury cadmium telluride HgCdTe or lead-tin telluride PbSnTe for the FIR. Their spectral response is the same as the corresponding PV (or photodiode) detectors shown in Fig.5-7. Even more important, and only available as PC detectors, are the extrinsic Ge:Au (gold- doped germanium) and Ge:Cu (copper-doped germanium) photoconductors for the FIR and EIR (see their spectral response in Fig.5-7). These detectors are called extrinsic because they use a transition between a donor level and the conduction band bottom (with threshold h >Ed-Ec) for photon absorption instead of the more usual intrinsic (interband) transition between valence and conduction band. In the operation of detectors for the middle and far IR, it is imperative to cool the photodetector to low temperatures, so as to keep the dark current reasonably low. In fact, dark current depends on temperature as T3+ exp-Eg/kT, [see Eq.(5.15)], and the exponential is the dominating term. When a material with small bandgap Eg is used to detect a small energy photon h Eg in IR, the only way to keep the term exp-Eg/kT small is to decrease kT appropriately Thus, moving from the visible (h 1 eV) to detectors for the MIR (h 200 meV) and FIR (h 100 meV) wavelength ranges, theoretically we require a decrease of absolute temperature from T= 300 K to 5__10 times less to have nearly the same dark current density. A reasonable choice for cooling is liquid nitrogen (LN) temperature, T=77 K. For EIR detectors (h 20__50 meV), a further decrease is necessary, and usually the detector is cooled at T=4 K (liquid helium, or LH). Note that LN is relatively cheap and safe to use, while LH is much more expensive. For cooling, detectors are mounted in a cryostat filled with LN or LH, or in a more sophisticated closed-cycle refrigerator (for example, Joule-Thomson). As shown in Fig.7-10, in the cryostat we find a Dewar a pair of concentric glass tubes with the internal facing walls metallized to reduce radiative loss. The detector is mounted on a metal bracket submerged in LN, and points outward through a clearance in the Dewar wall. The detector is kept under a vacuum to avoid dew formation; the window may be of Ge, Si, ZnSe or Irtran (all transparent in MIR and FIR) and has the outside face at ambient temperature.

7.3 Superconducting Detectors (SNSPD)

239

With this arrangement, the detector has a few hours of operating time before the Dewar has to be refilled. As a more sophisticate alternative to the Dewar, we may use a closed-cycle refrigerator that eliminates the burden of stored cryogenics to be carried along with the detector. An account on the closed-cycle refrigerator is provided by Ref.[2].

Figure 7-10 Example of a cryostat for cooling IR detectors to liquid nitrogen (LN) temperature (77 K).

7.3 SUPERCONDUCTING DETECTORS (SNSPD) The superconducting state is observed in a number of materials when temperature is lowered below a critical value Tc specific of the material, at 1__10 degrees Kelvin. Then resistance abruptly drops to zero because the electrons become bounded in Cooper's pairs and can move without losses due to scattering through the material, as explained by the theory of Nobel Prize winners Bardeen, Cooper and Schrieffer [3]. The electrons in the Cooper pair are weakly bonded through the lattice acoustic phonon, with binding energy of typ. Ec =0.2__1 meV. Thus, the bond is broken by thermal energy at temperatures T>Tc such that kTc > Ec. The superconducting state is also destroyed when the current density through the material is increased above a critical value Icv. When this happens, the resistance makes a transition

240

Phototransistors, Photoconductors, and SNSPD • Chapter 7

from zero to a finite value, and the dc current Ib fed by the supply into the device is forced to decrease. However, the kinetic inductance of the device Lk prevents this to happen with an opposing LkdIb/dt, and the result is a pulse developed in the circuit. The kinetic inductance is specific to high mobility materials like the superconductor, and is due to the inertial mass of the charge carriers, that takes a finite time to accelerate and thus gives rise to a charge opposing the emf (electromotive force) applied to the material, exactly like in an inductor [4]. The value of Lk is in the range 200 to 800 nH [5,6]. To take advantage of the transition and make a detector, a superconducting nanowire structure is employed. An excellent review of the structures, performances, and applications of these detectors is provided by Natarajan et al. [5]. Typically, the nanowire has a 4-nm thickness and 20__100 nm width and is a simple segment or a meander grown on a sapphire or silica substrate. The total length is tens to hundred m. As the material of the nanowire, NbN (niobium nitride) with Tc=10.5 K is the most commonly employed, but also or NiTiN and YBaCuO have been reported. Now, let us consider what happens when a photon hits the nanowire and is absorbed (Fig.7-11b) [4,5]. As the binding energy is small, 1-meV, photons with wavelengths ranging from the UV/VIS to the EIR have enough energy ( 1-eV to -meV) to excite the transition and break the Cooper pair. Thus, the wavelength range of response is potentially very wide.

Figure 7-11 Sequence of a single-photon detection in a superconducting SNSPD device: (a) the superconducting current flowing into the wire in quiescent condition; (b) a photon is detected and breaks locally a Cooper pair; (c) the superconducting current becomes diverted to the edge (d) a hotspot is formed that grows thanks to Joule heating, up to occupy the whole width of the wire (e); the wire ceases to be superconducting and (f) a pulse is developed at the output; after a recovery time current returns to the initial state (a).

7.3 Superconducting Detectors (SNSPD)

241

The electrons released from the pair can't participate anymore to the superconducting current, and therefore the current flow is diverted to the edges of the broken pair (Fig.7-11c), increasing the local current density. As the device is biased below but close the current threshold, the increase of current density will break additional Cooper pairs, and a hotspot is formed (Fig.7-11c, d). The process continues until the hotspot expands and occupies the full width of the nanowire (Fig.7-11e). So, we have a kind of avalanche multiplication, leading to a macroscopic change of an electrical parameter __ the nanowire resistance __ following the detection of a single photon. Additionally, the Joule dissipation contributes with a positive feedback mechanism to the growth of the hotspot [5]. When the hotspot has grown to the size of the nanowire width (Fig.7-11e), the wire switches from the superconducting with negligible resistance (r 0) to the normal conduction state, and the resistance increases suddenly to a high value, typically R=1...20 k depending on the dimensions of the wire [5]. However, the current through the device cannot decrease as fast, because of the opposing emf developed by the series kinetic inductance Lk. The transient that follows is rather complicate because of the time-varying resistance and of the interaction between electrical and thermal system of the wire, and a full analysis is presented in Ref.[6]. To explain qualitatively the waveforms developed in the device, we refer to the biasing circuit of Fig.7-12, in which the SNSPD is biased by a constant current generator and the pulse following the photon detection is taken by a load RL coupled to the device in ac. Let's now assume that a photon is detected, the hotspot is formed, and the SNSPD resistance suddenly increases from r to R. Current cannot decrease instantly, however, because the kinetic inductance develops an emf large enough to maintain the same current Ib flowing through R. To do this, a voltage VL equal to the voltage drop across R is developed at the resistance switching, or VL= LkdIb/dt = RIb. Thereafter, the transient due to the group Lk-R starts, and current decreases as Ib = (1/Lk)∫ RIb dt = (R/Lk) Ib t. But, as the device is biased at constant current, the decrease Ib is taken by the group C-RL in parallel to the device (Fig.7-12) and it develops the output pulse RL Ib. The pulse is very fast because it is governed by the time constant r = Lk/R. Taking typical values R= 5-k Lk=400-nH we get r =80 ps, the typical value of pulse rise time (Fig.7-12, right). As current decreases and heat is sunk into the device substrate, the hotspot evaporates and the device returns to the superconducting state (Fig.7-11 f and a) after a recovery transient. Indeed, in the superconducting state, we need that current recovers the Ib value, close to the critical current Icv, to become ready for a new r R transition. Now, the kinetic inductance Lk is closed across the series resistance RL+r RL, and the time constant of recovery is f = Lk/RL. This value is much larger than the rise time r because RL shall be kept a low value (i.e., 50 ) to match cable impedance and preserve bandwidth of the electrical detected signal.

242

Phototransistors, Photoconductors, and SNSPD • Chapter 7

Thus, we have typically a fall time f = 8 ns (Fig.7-12). As pointed out in [5], this value can be decreased by inserting a resistance Rs in series to the SNSPD, so as to have f = Lk/(RL+Rs).

Figure 7-12

(left) Typical biasing circuit of the SNSPM: in quiescent conditions, current Ib passes through the device in the superconducting state with r 0. When a photon is detected and the hotspot is developed, the device switches from superconducting to a high resistance R (position 1 to 2). The kinetic inductance Lk reacts developing a voltage RIb and letting current across the SNSPM decrease with time constant Lk/R (top right). As total current is constant, an output pulse RL(Ib-ISNSPM) is generated at the output. Shortly thereafter the hotspot vanishes (bottom right), the switch returns in position 1 and current through the SNSPD recovers to the initial value Ib with a time constant Lk/RL.

In conclusion, the SNSPD is able to detect single photons but one at the time, and like the SPAD, it has a dead time during which it ignores any new incoming photon. Also, the response pulse is just the same for a single photon and multiple simultaneous photons, or there is no PNR (photon number resolution). Like in the SPAD, we can remove the Geiger-mode or PNR limitation by subdividing the sensitive area in a number of individual devices whose outputs are finally summed up. In Ref.[7] this concept is demonstrated for a 10 m 10 m device made by a meander patterning of single elements. About the photosensitive area of the SNSPD, the basic single-element nanowire has a rather small cross section, and therefore the coupling efficiency c [5] to the most favorable

7.3 Superconducting Detectors (SNSPD)

243

input distribution i.e., a focused single-mode signal, is only a few percent. If the nanowire size is increased, other performances are sacrificed. With the meander geometry, 10 m 10 m or 20 m 20 m devices have been demonstrated to reach 25% efficiency at 1550 nm, while maintaining low dark current rate (10 Hz) and a good response (300 ps FWHM) [7]. The device had also marginal detection efficiency (10-4) in the middle infrared, a distinctive feature of SNSPD. A refinement is to incorporate the wire in a resonant cavity so as to enhance the efficiency of photon absorption (see Sect.5.2.8.4). To reach the tiny sensitive area, a single-mode optical fiber can be used, butt-coupled to the SNSPD and coming out from the cryostat to the laboratory temperature environment. The fiber with its transmission window limits the wide wavelength range of the SNSPD, whereas a normal optical access window to the cryostat may pose problems of efficiency because of the difficult optical coupling. Like any other Geiger-mode single photon detector, noise of SNSPD is dictated by the dark current Idark (or dark count rate DCR), the physical origin of which in superconducting wire is still matter of study [8]. In addition to a strong dependence on temperature below Tc (with nearly a tenfold increase for a temperature of T= 2K), there is an even stronger dependence from bias current Ib, with the DCR increasing of three orders of magnitude for Ib going from 0.85 to 0.95 of the critical current Icv [8]. So, it's no surprise that data on DCR reported in literature are widely scattered, from the low 10__102 cps up to 105__106 cps. The timing performance of the SNSPD is exceptionally good, and distinctive of this photodetector, thanks to the very fast ( ps mechanism of Cooper pair breaking and hotspot formation, followed by the fast Lk/R transient governed by the kinetic inductance.

Figure 7-13 The ID28 module has a compact 0.8-K cryostat for mounting up to 16 SNSPD detectors and offers detection efficiency up to >80% from 1300 to 1550 nm, with >1/ . The SC-preamplifier has the further advantage of being able to chop the signal for subsequent amplification (Fig.5-3.15). If we want to go beyond the frequency capabilities of transimpedance or SC op-amps, we may even terminate the pyroelectric on a R=50- load and obtain a high-frequency cutoff f2=1/2 RCt up to hundreds MHz, although at the expense of a poor noise performance.

250

Thermal Detectors and Thermography • Chapter 6

In conclusion, thermopiles and bolometers are intrinsically slow detectors with a response time dominated by the thermal section and limited to milliseconds, whereas the pyroelectric can be very fast if current is taken as the output. Indeed, this is the only thermal detector capable of detecting nanosecond pulses from CO2 and other FIR and EIR lasers. The typical behavior of responsivity versus frequency is shown in Fig.8-4.

Figure 8-4

Frequency response of thermal detectors with thermocouple, thermistor, and pyroelectric readout. For thermocouple and thermistor, f2 is the high-frequency cutoff due to the thermal circuit; for the pyroelectric, the thermal cutoff is moved to the cutoff f2 of the preamplifier, but there is a low-pass cutoff at f1.

On the other hand, a disadvantage of the pyroelectric detector is that its responsivity vanishes at zero frequency, because of the derivative dependence on temperature (pdTd /dt). A remedy for this, when the pyroelectric detector is intended for use as a power meter, is to chop the incoming power by means of a mechanical shutter placed in front of the detector and running at a frequency fch. Doing so, the frequency response of the detector becomes the portion of the diagram in Fig.8-4 starting from fch. Thus, the response is flat from =0 if we use a fch larger than f1(pyr). Single-point thermal detectors are available with large sensitive areas (>cm2) at low cost. They are extensively used in instruments that measure radiant power, taking advantage of

8.2 Detectivity of Thermal Detectors

251

the practically flat wavelength response, extended from 0.3 to 300 m (see, for example, Fig.8-1), as well as in radiometric thermometers. Another field of application for single-element pyroelectric detectors is in IR intrusion alarms, where the zero-frequency suppression is useful to detect the blackbody emission from a person entering the field of view, typically up to a 10-m range, while ignoring slow signals due to ambient temperature fluctuations.

8.2 DETECTIVITY OF THERMAL DETECTORS Let us now consider the intrinsic limit of detectivity expected for an ideal thermal detector, assuming that the thermal response is maximized and that all the parasitic contributions of any other disturbance are made negligible. To maximize the thermal resistance Kt of the dissipation section, we must make the thermal exchange by conduction and convection as small as possible; for example, we will use thin access wires and mount the detector in a vacuum, so that only the unavoidable radiative exchange remains. For a blackbody, the radiated power is P=A T4, with 5.67.10-8 Wm-2K-4 for the Stefan-Boltzmann constant. The power exchanged with the ambient is dP = 4A T3 dT, and from the definition of Kt we have: Kt = dT/dP = 1/(4A T3)

(8.4)

Concerning fluctuations, let us observe that, an in element in thermal equilibrium with the ambient at absolute temperature T, minute temperature fluctuations still occur. These fluctuations are the consequence of the energy exchange mechanism with the external ambient through radiated quanta. The exchange balance is zero on average, but, statistically, when there is a temporary excess of blackbody emitted photons, the temperature has a negative deviation, while when the unbalance is negative, the temperature has a positive deviation. By means of statistical thermodynamics considerations (see Appendix A3.3), one can find that, at thermal equilibrium, the temperature fluctuation has a rms value Tn given by: Tn = [kT2/C]1/2

(8.5)

where as usual k is the Boltzmann constant, and C is the thermal capacitance of the considered element or detector. Using Eq.(8.5) with Eq.(8.4), we can find for the NEP of the thermal detector: NEP = Tn /Kt = [kT2/C]1/2 / Kt = [kT22 B]1/2 √(4A T3)

(8.6) (8.6')

having used Eq.(8.2) in Eq.(8.6') with f2=B. The detectivity D* is accordingly given by:

252

Thermal Detectors and Thermography • Chapter 6

D* = √AB / NEP = 1/ [8

T5]1/2

(8.7)

This expression gives the sensitivity performance of an ideal thermal detector, i.e., the ultimate limit of performance due to the thermodynamic temperature fluctuations. Quantitatively, from Eq.(8.7) the thermodynamic limit is: D*=1.8.1010 W-1 cm Hz1/2 at T=300K, and D*=5.2.1011 W-1 cm Hz1/2 at T=77 K. These are very good values indeed, comparable or even larger than those of same temperature photovoltaic detectors, and in the reach of the BLIP limit. However, such high values are never achieved in practice, because of a number of nonidealities, like heat dispersion, readout noise, etc. Thus, the practically attainable detectivity of thermal detectors is one or two decades less than the theoretical limit and about 1½ decade worse than a good photoemissive detector like HgCdTe or PbSnTe (see Fig.8-5), with a dramatic difference, however: these thermal detectors are operating at ambient temperature instead of requiring cooling at 77 K. Cooling is a hindrance in applications, because requires

Figure 8-5

Detectivity of infrared thermal detectors compared with other detectors.

8.3 Temperature Measurements and NEDT

253

the user to travel with a bottle of LN (liquid nitrogen) to refill the Dewar mounting the detector (Fig.7-10), while Stirling cycle or Peltier cell cooler are generally limited to ≈ 100oC temperature decrease with respect to ambient, too little to reach the 77 K (or -196 oC) and avoid LN for the high-performance photoemissive detectors.

8.3 TEMPERATURE MEASUREMENTS AND NEDT A non-contact thermometer is based on the measurement of the radiant power emitted by a target surface, which is assumed to follow the blackbody (or gray-body) emission law (see Appendix A1.5). A thermal detector is the preferred choice because it is low cost and has a satisfactory performance. The simplified layout of a radiant thermometer is that of Fig.8-6, showing the target under measurement imaged by an objective lens (in IR materials like ZnSe, Si, or Ge) onto the detector. The electrical signal from the detector is proportional to r( ,T), where r( ,T) is the Planck radiance function, is the emissivity of the target surface, and T is the absolute temperature.

Figure 8-6 (top) A thermal detector (TD) used for non-contact measurements of temperature: the field-of-view (fov) and target size are determined by the objective focal length, while the numerical aperture determines the collection efficiency; (bottom) filter response is designed to pass a large fraction of the emission spectra, especially the MIR an FIR.

254

Thermal Detectors and Thermography • Chapter 6

Expanding the Planck function at first order in temperature as r( ,T0+ T) = r( ,T0) + [d r( ,T0)/dT] T, reveals that the deviation of the output signal from the quiescent point r( ,T0) is proportional to the temperature increment T of the target with respect to ambient temperature T0, the quantity to be measured. In the following, we will analyze the theoretical accuracy of the temperature measurement and present illustrative material about typical applications of non-contact thermometry.

8.3.1 Accuracy of Temperature Measurement and NEDT Limit Using the law of photography (Chapter 2), we can write the signal power received by the photodetector aimed at the target at temperature T as: p = r( ,T) NA2 A

,

(8.8)

where r( ,T) is the spectral radiance (W/cm2sr m) given by Eq.(3.22), A is the detector area and is the width of spectral response of the filter or detector response. As already noted above, the useful signal is the variation of received power p= dp/dT) T, which duplicates as a photocurrent variation i = dp/dT) T. Using Eqs.(8.8) and (3.22), p is written as: p = (dp/dT) T= dr( )/dT] T =

NA2 A

NA2 A

r( ) T/T

(8.9)

where we the dimensionless factor , the thermal contrast, is defined as: = (h /kT) [exp h /kT /(exp h /kT-1)]

(8.10)

In MIR and FIR it is h >kT (because h is of the order of 100__200 meV, while kT is 26 meV at T=300 K), and so, the term in square brackets in Eq.(8.10) is nearly unity, and can be simplified to: hc/ kT= 47.88( m) / ( m)

(at T=300 K)

(8.10')

From Eq.(8.9) we can see that the useful thermal signal is proportional to the fractional bandwidth / of the detector and to the radiance r( ) at the working wavelength. For this reason, the FIR should be preferable because the useful signal is larger than in MIR. But, to draw a conclusion about thermal measurement sensitivity, noise has also to be taken into account. The noise of the best possible detector aiming at a scene at the average temperature T is set by the background limited detectivity DBLIP (Sect.3.3.2). In terms of current, noise is given by:

8.3 Temperature Measurements and NEDT

in(bg) =

NEP =

255

NA√AB/DBLIP

= NA√[2 e r( )

(8.11)

AB]

(8.11')

where r( ) is calculated at T'=T+ T T for small T, and Eq.(3.24) has been used to get Eq.(8.11') from Eq.(8.11). Because i= p, the signal-to-noise ratio S/N can be calculated from Eq.(8.9) and Eq.(8.11') with the following result: S/N = i/in(bg) =

T/T) NA √[

Ar( )

/2eB]

(8.12)

Letting S/N=1 in Eq.(8.12) as a marginal working condition, we can find the minimum detectable temperature difference T or NEDT (Noise-Equivalent Differential Temperature), as: NEDT = (T/ NA) √[2eB/ = T (2e/

r( )

A]

) DBLIP (1/NA) √(B/A)

(8.13)

= 2kT2 DBLIP (1/ NA) √(B/A)

(8.13')

Eq.(8.13') is obtained from Eq.(8.13) recalling that e/kT. Commenting on Eq.(8.13), we can see that the best thermal sensitivity is achieved in the spectral range where DBLIP is minimum, that is, near to the blackbody peak m (see Fig. A1.5). This is simply the consequence of the useful signal being maximum where DBLIP is a minimum. Therefore, this argument demonstrates that, for a BLIP-limited photodetector, the FIR is better suited than the MIR or the EIR for non-contact thermal measurements. If the detector is not BLIP-limited (D**>I0q I+Id+IR, gives the quantum limited S/N as: (S/N)2hom/q = 4 2I /2eB

(10.12)

Note that, different from direct detection, in coherent detection the quantum-limited condition applies to the local oscillator amplitude rather than to the signal amplitude. This is an important difference, because by making the local oscillator is large enough, I0>>I+Id+IR, the quantum limit is always reached, even at weak signal levels. Example. At 10 GHz or beyond, a fast pin photodiode will certainly be used matched to a 50 load to keep parasitics low, and will then have IR=(50mV)/R=1 mA as the term dominating all other noise sources; the quantum regime is attained for I0 (>>I0q=1mA) =10mA, corresponding to P0 10 mW. As a general remark, coherent detection will provide a substantial improvement when the required bandwidth is very wide and/or detectors have a poor noise performance in direct detection.

Heterodyne detection is treated by the same arguments, with the difference that now the beating signal is at the frequency - 0. Using Eqs.(10.3) and (10.5), we can write the useful signal as: Iph = ( A/2Z0) 2EE0 cos [( - 0)t+ - 0] 2√(I I0 ) {cos[( - 0)t+

2√(I I0) cos [( - 0)t+

- 0] cos r

sin[( - 0)t+

r 0]

- 0] sin r

and having again sin r = 0 we get: Iph

2√(I I0 ) cos [( - 0)t+

- 0] cos r

(10.13)

Eq.(10.13) shows that in heterodyne detection the phase condition does not apply (different from homodyne), because - 0 only affects the unimportant phase of the carrier at frequency - 0, on which the beating signal is impressed. From the application point of view, this is a substantial advantage compared with homodyne detection. Regarding the S/N ratio, by repeating the arguments leading to Eq.(10.12) and taking into account that the rms value of cos [( - 0)t+ - 0] is 1/√2, we get for the heterodyne: (S/N)2het = [( A/2Z0) 2 EE0 . (1/√2)]2/{2e [( A/2Z0)(E2+E02)+Id]B+4kTB/R} = 2 2I I0 / [2e(

0+Id)B

+ 4kTB/R]

(10.14)

From Eq.(10.14) we can see that there is a modest penalty, a factor 2 (or 3 dB), in the signalto-noise ratio compared with homodyne. For I0>>I0q = I+Id+IR, we have as the quantum limit of the heterodyne: (S/N)2het/q= 2

2I

/ 2eB

(10.15)

10.1 Direct and Coherent Detection

289

In the quantum limits given by Eqs.(10.10), (10.12) and (10.15), the key dependence is on I/2eB. The multiplying factors 4 and 2 of Eqs.(10.12) and (10.15) missing in Eq.(10.10) do not necessarily imply a better performance of coherent detection with respect to direct detection, because they depend on the definition of S/N. To get a more general definition of S/N ratio, it is better to use the quality factor Q defined as: Q = [S(1)-S(0)]2 / [N2(1)+N2(0)]

(10.15')

This factor accounts for an eventual noise N(0) at the zero signal level. It can be shown that the Q-factors of direct and homodyne detection coincide (and for heterodyne Q is 3 dB worse).

10.1.4 Conditions for Coherent Detection To implement a coherent detection system, we have to satisfy the following conditions, already hinted above:  phase matching of signal and local oscillator (for homodyne), otherwise the beating is reduced by a factor cos( - 0), 

phase coherence, otherwise the useful signal is reduced by cos r

.

In addition, we must take account of all the possible causes of incomplete superposition of signal and local oscillator, and specifically:  superposition of E and E0 on the photodetector with spatial coherence (i.e., with the same modal distribution), otherwise the beating will be reduced by a factor: sp = ∫ A 

E(x,y) E0*(x,y)dxdy /[∫A E0(x,y) 2dxdy ∫A E(x,y) 2dxdy]1/2

(10.16)

superposition of E and E0 with the same state of polarization (or SOP), otherwise the signal will be reduced by a factor: E . E 0/ E

E0 = pol

(10.17)

where E.E0 is the product of Jones matrixes of signal E and local oscillator E0, and .. indicates the modulus of vectors. Of course, the previous expressions [Eq.(10.3') and Eqs.(10.11) through (10.15)] can now be generalized by substituting in them: sp pol

(10.18)

At the experimental level, the requisite of spatial coherence implies that the beams of signal and local oscillator must reach the photodetector active surface with the same wavefront curvature and propagation vector. This requires in practice that both beams have a single mode spatial distribution and are aligned in angle within their diffraction limit. The polarization requisite calls for the use of a polarization controller, which may be fixed (or tuned only

290

Coherent Detection • Chapter 10

once) for signals with a stable SOP, or require a dynamical control when the signal has SOP fluctuations (as, for example, after a propagation in normal optical fibers).

10.1.5 S/N and BER, Number of Photons per Bit The S/N ratio is the most immediate parameter for describing signal quality. Together with bandwidth and dynamic range, it provides an adequate characterization for analog signals. However, system performances can also be described by other parameters (always connected to S/N). In digital transmissions, for example, it is common to specify the bit error rate or BER, defined as the relative frequency of errors in recognition of transmitted symbols. As a guide, one can take BER=10-9 10-11 for telephone line signals, and 10-15 for interconnection of computers. In the following, we calculate the number of photons-per-bit necessary to reach a specified BER at the photodetector, and show how the result is dependent on the type of detection as well as on modulation and coding (for more details, see Ref. [5]). In a digital transmission, we transmit a sequence of discrete levels impressed on a variable of the electrical signal, the simplest example being the binary transmission with the levels zero and one, impressed on optical power amplitude as an on-off-keying (or OOK). Suppose that we transmit N photons for the “1” and zero photons for the “0”. The error probability for equiprobable “1” and “0” is the half-sum of the probabilities of wrong recognition, that is, P(0 1) of detecting “0” when the “1” has been transmitted, and P(1 0) of detecting “1” when the “0” has been transmitted: BER = ½ [P(0 1) + P(1 0)] In an ideal direct detection (no dark current, noiseless amplifier, etc.) there is only the unavoidable contribution of the fluctuation of the number of photons transmitted by the source, N as an average for the symbol “1” but varying from sample to sample according to the Poisson distribution: p(n) = N e-N/n! while the symbol “0” is transmitted with n=0 photons and no fluctuation. Thus, P(1 0)=0 as no photon can be received if none is transmitted. The threshold S of discrimination of the received signal minimizing the BER= ½ P(0 1) is therefore the 1-photon level (the lowest possible to recognize n≠0), see Fig.10-3. We then find P(0 1) = p(n=0) or BER=e-N. Taking BER=10-9, it is easy to find N= 20 as the average number of photons for the “1” symbol. If the symbols “0” and “1” are equiprobable in the message __ as it is reasonable to assume in practice __ we have the performance of N=10 photons per bit to identify the ideal direct detection. As we have considered only the Poisson (or shot) noise, this is the quantum limit performance. From Eq.(10.10'), the corresponding signal-to-noise requirement is found as (S/N)2=10 (or 10 dB). Translated in an optical power, the associated signal is P=10 h /T, where T=1/2B is the bit period connected to system bandwidth B.

10.1 Direct and Coherent Detection

291

For a homodyne detection of binary signal modulated in amplitude or ASK (amplitudeshift keying), we again transmit N photons as an average for the “1”, and zero photons for the “0”. The situation is now different, however, because noise is dominated by the local oscillator and is the same for both symbols.

Figure 10-3

Probability distributions in binary transmission and choice of decision threshold in ideal direct detection (left) and in coherent detection (right).

Thus, we have P(0 1) =P(1 0) and will place the optimum threshold at half the mean signal amplitude (Fig.10-3), S=N/2. Assuming for the signal amplitude a Gaussian distribution (a good hypothesis for moderate signals), with mean N (for “1”) and 0 (for “0”) and a variance 2 N for both, one has: BER = erfc S/ N = erfc N/2 N where erfc(x)=(2 )-1/2∫ x- exp -x2/2 dx is the standard complementary error function. By substituting in the BER the mean signal N=2(I.I0)1/2T/e=2(NsN0)1/2 and its standard deviation N=(2eI0/2T)1/2T/e=N01/2, where Ns and N0 are the mean number of signal and local oscillator photons, and T is the bit period, we get: BER = erfc √Ns For BER=10-9 the required mean number of photons per bit is found Ns=36. It is now easy to repeat the same calculation for the heterodyne detection of a binary message ASK-modulated in amplitude, by noting that the signal is a half [Eq.(10.13)] while N is unchanged, to obtain the bit error rate as BER= erfc√(Ns/2), whence we need Ns=72 for a BER= 10-9. In the case of homodyne detection of a binary message phase-modulated PSK (phaseshift keying), we transmit the carrier and phase-shift it by 0 or for the two symbols. Therefore, the detected signal doubles in amplitude (it changes sign for “1” and “0”) but is the same amplitude as ASK, whence BER = erfc 2√Ns, and the mean number of photons per bit

292

Coherent Detection • Chapter 10

becomes Ns=9. Again, for a heterodyne detection of a PSK-modulated binary message, we can find BER= erfc√ 2Ns, and a mean number of photons per bit Ns=18. Analogously, for a homodyne detection of binary message modulated with a 4 PSK (four-phase PSK) in which four phases (0, ± /2, and ) are coded, we have the same noise and the same signal amplitude in both the phase-quadrature channels, obtaining the double of the bit transmitted, so that the mean number of photons per bit is Ns=4.5. Further cases of interest should be now easy for the reader. As a comment, the above results show that an ideal direct detection has a performance surpassing many of the coherent ones and nearly equal to that of the homodyne PSK. From practical point of view, however, the situation is quite different. Compared to the theoretical limit, the best results obtained in direct detection at the high end of bandwidths are about 2 decades off the 10 photons/bit theoretical limit, and the reason is that noises other than signal shot noise (e.g., load resistance noise, preamplifier noise, and dark current noise) are not at all negligible. Coherent detection schemes, on the other hand, fairly well approach their theoretical limits (typically within 3-5 dB), thus outperforming direct detection.

Figure 10-4 Sensitivity performances of receivers for fiber-optics communications, expressed in dBm (dB respect to 1 mW, left scale) as a function of transmission bit rate 1/T (T=bit period). While direct-detection receivers (crosses) are far from the quantum limit of 10 ph/bit, coherent receivers (circles) approach their limits closely. Broken line and squares are for optically preamplified receivers at 1550 nm. Symbols indicate actual products.

10.2 Coherent Techniques

293

As an illustration, we plot in Fig.10-4 the sensitivity performances of coherent receivers for the II and III windows of optical fibers (1300 and 1500 nm). These state-of-the-art records were obtained by telecom companies in the years 85-90's, when coherent systems were developed with the aim of longer regeneration spacing over direct-detection systems in longhaul span-by-span transmission. However, coherent systems proved to be difficult to implement and more expensive, not justifying the span increase (roughly a factor of 1.5) respect to direct detection. Then, the advent of the optical amplifier in the mid '90s shelved span-byspan regeneration and coherent systems. But in 2010 the situation reversed, when fully digital coherent systems were introduced. These are capable of an extraordinary operation, that is, they can correct chromatic and polarization dispersion and non-linearity of the fiber propagation, thanks to the availability of the complex signal, the output of the coherent detection, which supplies a phase and a quadrature signal as well as the two polarizations. In this way, the channel efficiency is increased by 22=4 and a 40-Gb/s receiver can be made by 10 Gb/s receivers compatible with the processing speed of CMOS-integrated circuits performing the corrections, and in four-phaseshift keying (or DPQPSK), today reaching 160 Gb/s. Results in Fig.10-4 are also representative of expected performances in general applications from the visible to the NIR, with small-area photodetectors and good preamplifier design.

10.2 COHERENT TECHNIQUES 10.2.1 The Balanced Detector In implementing coherent detection, the basic scheme of Fig.10-1 is seldom used, because of the attenuation introduced by the beamsplitter, both on the signal (with S/N degradation) and on the local oscillator (with increase of required power).

Figure 10-5 Scheme of the balanced detector (photodiodes include the front-end preamplifier).

294

Coherent Detection • Chapter 10

Thus, unless the detector is very expensive or critical, it is preferable to use a pair of photodetectors placed in a symmetrical arrangement relative to the beamsplitter, as indicated in Fig.10-5, a scheme called balanced detector. Let R and T=1-R be the beamsplitter reflection and transmission coefficients for powers; then, for field amplitudes the corresponding coefficients will be √R and √(1-R). Recalling the /2 phaseshift (see box) introduced by the beamsplitter, we have for Fig.10-5: E1 = E(r)+E0(t),

E2 = E(t)+E0(r)

(10.19)

and, by substituting terms, we obtain: E1 = √R E exp i( + - /2) + √(1-R) E0 exp i( 0+ )E2 = √(1-R) E exp i( + ) + √R E0 exp i( 0+ - /2)

(10.20)

PHASE SHIFT OF THE BEAMSPLITTER At a beamsplitter, the continuity condition of electric fields at the separation boundary requires that the incident Ei is always the sum of reflected Er and transmitted Et fields: Ei = Et + Er (I) where the underlines indicate rotating vectors. Also, in a lossless beamsplitter power P is unchanged upon splitting and, as P is proportional to E2, we have: E i2 = E t2 + E r 2 (II) To have both equations satisfied, the three vectors must lie on a right-angle triangle, as shown in the figure below. Then, the angle - or phase shift - between reflected Er and transmitted Et vectors is /2 irrespective of the actual splitting ratio, while the angle between incident and transmitted fields increases from 0 to /2 as Et decreases from Ei to 0 (or, R goes from 0 to 1). We can then write, for the lossless beamsplitter:

Et = √(1- R) Ei ei ,

Er = √R Ei ei(

)

(III)

For a lossy beamsplitter, Eq.(I) still applies, while (II) holds with the ≥ sign; then point P in the figure shifts internal to the circle and the Er Et phase shift becomes larger than /2 (of an angle p/2√[R(1-R)] where p is the loss).

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295

The photogenerated currents at the two detectors are accordingly: I1 = ( A/2Z0) E1 2 A/2Z0) [RE2 (1-R) E02+2√R(1-R)E0E cos( - 0- /2)] 2 I2 = ( A/2Z0) [(1-R)E RE02+2√R(1-R)E0E cos( - 0+ /2)] (10.21) and from here we can see that the beating term is maximized for √R(1-R) =max, or for R=½. Assuming that a 50% beamsplitter is used, and writing Eq.(10.21) in terms of currents corresponding to fields [I=( A/2Z0) E2], we have: I1 = ½ (I0+I)+√(I0I) sin( - 0), I2 = ½ (I0+I)-√(I0I) sin( - 0)

(10.22)

whence an output S, after the subtraction stage (Fig.10-5): S = I1-I2 = 2√(I0I) sin

- 0)

(10.23)

Because of the phase shift of the beamsplitter, signal and local oscillator must now be placed in quadrature [sin( - 0)=1] at the beamsplitter input (so that they become in-phase at the detectors). Note that (i) S is equal to the full beating signal Eq.(10.3') of the superposition of E and E0, without any loss due to finite beamsplitter transmission and (ii), the dc or continuous component due to the local oscillator is cancelled, and also the common-mode fluctuations and the associated excess noises are suppressed. This is a definite improvement gained with the balanced scheme. The shot noises of the two components ½ (I0+I) in Eq.(10.22) are statistically uncorrelated, and therefore, in the subtraction, their variances must be added, giving as a total: N2 = S2 2.2eB½ (I0+I) = 2e(I0+I) B Thus, the S/N for an amplitude measurement is equal to the ideal case given by Eqs.(10.11') and (10.12), with a quantum limit (S/N)2hom= 4 2I /2eB. Analogously, we can analyze the heterodyne detection and obtain again the result Eq.(10. 23) but with the factor - 0 replaced by ( - 0)t+ - 0.



















 Figure 10-6 Balanced detection scheme with dc subtraction at the amplifier input.

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In conclusion, the balanced scheme readily supplies an ideal (that is, lossless) superposition of signal and local oscillator, and also suppresses the dc component and related drifts. For practical purposes, the balanced detection circuit can be implemented with two lownoise preamplifiers converting the detector photocurrents to voltage signals followed by a main subtraction amplifier, as shown in Fig.10-6. If the dc component is high, to avoid preamplifier saturation, using the scheme of Fig.10-6 may be advantageous, because it performs the subtraction directly at the input terminal of the main transimpedance amplifier (see Section 5.3); however, the scheme requires a pair of photodiodes packaged with inverted polarities.

10.2.2 The Balanced Scheme in Phase Measurements In interferometric measurements, information is carried by the phase difference = - 0 associated with the optical pathlength - 0=k L, which is in turn related to a physical quantity to be measured. From Eq.(10.23) we can see that the output S of the balanced detector allows us to go back to the phaseshift signal, and that for small the dependence is linear: =S/2√(I0I). To evaluate the accuracy in the measurement of let us observe that a deviation S in amplitude gives a deviation = S/2√(I0I) in phase, so that by squaring and averaging, we 2 2 = S2 /4(I I). By inserting the variance 2 find 2= 0 S for S in this expression, we get: 2=

2e(I0+I) B/4(I0I)

(2eB/I)/4 = 1/[4(S/N)dir/q2] = 1/4N

(10.24)

showing that the phase variance is inversely proportional to the amplitude S/N ratio of the direct detection, or, also, to the number of detected photons.

10.2.3 Examples of Coherent Schemes The design schemes of coherent detection take very differing shapes in different application areas. Here, we present just two general examples. The basic scheme of a heterodyne coherent detection with polarization diversity is reported in Fig.10-7. With this scheme, a major problem in applications __ polarization fading due to propagation __ is solved. To this end, signal and local oscillator beams are divided in orthogonal polarized components by the Glan cubes, and the two pairs of outputs are recombined and detected by the two balanced detectors. So, whatever the state of polarization of the input signal beam, at least one of the detectors will have a beating with pol≠0. By summing the outputs, the polarization fading is thus eliminated, and we obtain a detected signal smaller than the ideal one ( pol=1) by a factor of 3 dB at the most. After the summation, the demodulator of Fig.10-7 brings the signal back in the base band as the useful output. The average frequency in the output, taken by the frequency-voltage (f/V)

10.2 Coherent Techniques

297

converter, is used as a feedback signal to control the local oscillator frequency, so as to keep f0 - fs dynamically locked to the desired intermediate frequency f. In fiber-optics transmissions, this scheme is implemented with a narrow-line tunable laser, typically a DBR (Distributed Bragg Reflector) diode laser, and the beamsplitters are usually made by a fiber-coupler (50/50 normal, or PSC__polarization splitter couplers). Stateof-the-art performances of such a system are those already reported in Fig.10-4.

 

 !

    

 



 !





   

  



   



 



 

Figure 10-7 Scheme of a heterodyne detection receiver with polarization diversity.

For the detection of weak signals from remote targets, a typical heterodyne scheme employing a two-frequency source is reported in Fig.10-8. This scheme finds application in instruments such as displacement interferometers, sub-micrometer vibrometers, and coherent telemeters. The source is a frequency-stabilized laser, either Zeeman-splitted or frequency-shifted by an external modulator, so that it supplies a main beam at and a second beam shifted to + , where is small compared with and stable in time. The beam at is projected onto the remote target through a collimating telescope, and the returning echo collected by the same telescope is directed, through a beamsplitter, to the balanced detector with the local oscillator beam shifted to frequency + by means of an acousto-optical modulator. To ensure an adequate coherence factor, the linewidth of the laser should be less than the inverse c/2L of the time delay of the echo from distance L. Sometimes, an optical circulator is used in place of the beamsplitter (Fig.10-8) to avoid the 50% loss of a normal beamsplitter. This allows a recovery of 6 dB of signal amplitude in the double passage but, as the circulator is a rather expensive component, it is only used when the extra cost is justified by the application.

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Coherent Detection • Chapter 10

       

     

     

          



 

    





 



Figure 10-8 Scheme of a heterodyne receiver of weak optical echoes.

10.2.4 Photomixing Beating of two coherent waves of different frequencies on a photodetector is the basic scheme of a heterodyne receiver used for the detection of a weak signal, as we have seen in last Section. Now, if the same scheme is used for the beating of two waves, at frequency f1 and f2, both strong like a local oscillator, we get the scheme called photomixing, a technique to supply a substantial power at the photodetector output at the frequency difference f1-f2. Photomixing is a technique to build a generator, by optical technologies, at microwave frequencies, roughly in the range of ten to hundreds GHz, and most recently up to THz. For this feature, it is called a microwave photonics technique. It offers potentially good stability, narrow linewidth, and compact size with respect to traditional microwave sources. The basic scheme of photomixing is shown in Fig.10-9 and it comprises two lasers with fiber-coupled (or guide) output, a fiber coupler, and a high-frequency, high current photodiode. To implement the scheme, we need lasers that oscillate on well-defined frequency, so that the frequency difference, f1-f2, is stable and narrow line. So, we will usually employ either DFB lasers or external-cavity grating lasers, providing typical linewidths of 10__100 kHz. About the photodetector, we need a high-frequency, high current unit like those available from a UTCx or TW structure (see Sect.5.2.8) [6]. About the electrical power supplied by the photodiode, we can write it as: Pel = R ( Popt)2 where R is the load resistance of the photodiode and Popt =I is the photodetected current. The quadratic dependence of Pel from Popt tells us that, surprisingly, the electrical output power can even be larger than optical power arriving at the photodiode. The break point between PelPopt is at the value Pel=50 Popt2 or Popt=13-dBm, for a spectral

10.2 Coherent Techniques

sensitivity unitary

1 A/W and R

299

as shown in Fig.10-10, and a little bit more for sub-

Figure 10-9 Basic scheme of photomixing for the generation of a microwave tone by beating two laser fields at on a photodetector.

The typical output electric powers that can be obtained at 30 GHz by photomixing of two DFB laser outputs is shown by the dots in Fig.10-10 [7]. Photomixing is a promising approach to develop a PIC (Photonics Integrated Circuit) capable of working as a generator of microwave power.

Figure 10-10 Electrical power of the output microwave oscillation supplied by the photodiode in the photomixing scheme. Dots are an experimental result obtained with two DFB lasers at 1500 nm beating their 30-GHz frequency difference on a discrete 60-GHz UTC photodiode [7].

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Coherent Detection • Chapter 10

In an early example of such a PIC (Ref. [8]), the longitudinal modes of a ring laser were amplified by an active waveguide and beat on a waveguide photodetector; by changing the ring diameter (from 860 to 430 and 290 m) and hence the longitudinal mode spacing c/2nL, the authors were able to obtain microwaves tones at 30, 60, and 90 GHz, albeit of modest power (-10, -20 and -30 dBm) [8]. The bottleneck of photomixing is the maximum current that fast photodiodes can supply at high frequency. Improvements in the saturation current have been steady in recent years, and from the sub-mA level of low-power photodiodes of the years '90, we arrived the 100 mA at 11__30 GHz obtained with improved UTC structures [6], and the more recently 37 mA at 110 GHz and 17 mA at 270 GHz [10]. Another important application of photomixing is the testing of frequency response of high-frequency photodiodes. Indeed, using the basic configuration of Fig.10-9 and scanning the wavelength of oscillation of one of the two laser, the beating frequency will scan across the electrical frequency range, and the output supplied by the photodiode will be just the frequency-dependent response of the device. Thus, without requiring any modulated source to excite the photodiode, and just an electrical spectrum analyzer to display the photodiode output, we can obtain the frequency response up to 70 GHz as shown in Fig.10-11. To scan the wavelength range of = 0.5__1 nm (which corresponds to cover a frequency __ response range 70 140-GHz at 1500 nm), we can use the thermal dependence of wavelength ( t = d /dT= 0.1__0.3 nm/oC typ.) in either DFB or Fabry-Perot lasers, and act for example on the Peltier-cell regulating the diode laser temperature or, for Fabry-Perot lasers, use the current dependence on wavelength ( t= d /dI= 0.2 nm/mA typ.) and act on the bias of the diode laser (but minding this will change also the emitted power).

Figure 10-11 Examplary frequency response measured on a commercial 60-GHz photodiode by the photomixing circuit of Fig.10-9 [7].

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301

References [1] H.A. Haus: “Electromagnetic Noise and Quantum Optical Measurements”, Springer, Berlin 2000. [2] B.M. Oliver: “Thermal and Quantum Noise”, Proc. of the IEEE, vol.53 (1965), pp. 436454; also reprinted in: M.S. Gupta (Editor): “Electrical Noise: Fundamentals and Sources”, IEEE Sel. Reprint Series, New York, NY, 1977. [3] L. Mandel and E. Wolf: “Optical Coherence and Quantum Optics”, Cambridge University Press, Cambridge, U.K., 1995. [4] M. Birk, et al.: “Coherent 100 Gb/s PM-QPSK Field Trial” IEEE Comm. Mag. vol.48 (2010), pp.52–60. [5] P. J. Winzer, D.T. Neilson, A. R. Chraplyvy: “Fiber-optic Transmission and Networking: the Previous 20 and the Next 20 Years”, Opt. Expr., vol.26 (2018) pp.24190-24239. [6] A. Beling, X. Xie, J. C. Campbell: “High-power, High-linearity Photodiodes” Optica, vol.3 (2016) pp.328-338. [7] M. Norgia, G. Giuliani, S. Donati, R. Miglierina, T. Tambosso: “Simultaneous Optical and Electrical Mixing in a Single Fast Photodiode for Demodulation of mm-Wave Signals” Proc. SPIE, vol. 5466 (2004), pp. 72-79. [8] G.A. Vawter, A. Mar, V. Hietala, J. Zolper, J. Hohimer: “All Optical Millimiter-Wave Signal Generation using Integrated Mode-Locked Semiconductor Ring Laser and Photodiode”, IEEE Phot. Techn. Lett., vol. 9 (1997), pp. 1634-36. [9] J.-W. Shi, F.-M. Kuo, Mark Rodwell, J. E. Bowers: “Ultra-High Speed (270 GHz) Near-Ballistic Uni-Traveling-Carrier Photodiode with Very-High Saturation Current (17 mA) under a 50-ohm Load”, IEEE Photonic Soc. 24th Ann. Meet., 2011, paper MC-2.

Problems P10-1 A weak signal, P=1 nW (or, respectively, 1 pW, 1 W) is detected by a coherent homodyne scheme, using a silicon photodiode with =1A/W, Idark= 0.1 A, terminated on a 1-k load. Find: (i) the minimum local oscillator strength that is required; (ii) the signal amplitude that is obtained, as compared with direct detection; and (iii) the S/N ratio and its improvement with respect to direct detection. Assume =1 and B=1 GHz. P10-2 A phase-modulated signal, of the type: E0 exp i[ t+ +2ks(t)] is detected by a homodyne coherent scheme. Write the expression of the output signal and guess the likely regime of detection. P10-3 How large is the signal associated with a N=10-photon-per-bit rate, the one giving a BER=10-9? Consider bit rates R= 1Mbit/s, and 1Gbit/s. Assume =1.24 m, and a spectral sensitivity = 0.10 A/W. What is the quantum noise and the S/N ratio of this signal ? P10-4 Given the data of Probl.10-1, what is the theoretical minimum an advantage in using coherent versus direct detection?

that would be tolerable to still have

P10-5 A coherent detection uses a non-ideal local oscillator, that has: a circular polarization instead of the linear one of the signal, a mode size (Gaussian) twice as wide as that of the signal, and a phase rms deviation = 0.5 rad. Calculate the total coherence factor and the S/N penalty.

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P10-6 Evaluate the number of photons per bit to achieve a BER=10-9 in a SOPSK (state-ofpolarization-shift-keying) transmission with homodyne detection, where the ‘1’ is transmitted with a given state of polarization (e.g., a linear one) polarization (for example, same as that of the local oscillator) and the ‘0’ with the polarization state orthogonal to that of the ‘1’. P10-7 Again consider a SOPSK coherent detection, and assume using a balanced detector with an input Glan beamsplitter dividing the incoming polarization. What is the number of photons per bit to achieve a BER=10-9 ? P10-8 As optical receivers for a 100 Gbit/s transmission rate are developed and available for system use, which would be their expected sensitivity (in W or dBm) ?

C

H

A

P

T

E

R

1 1

Photodetection Techniques

I

n this chapter we describe advanced photodetection techniques that represent nonconventional treatment of the optical signal or new approaches based on special properties of the electromagnetic optical field. We start with optical preamplification, the most mature technique, and then describe injection detection, an extension of coherent detection, to continue with non-demolitive detection and detection of squeezed states that exploit second quantization principles. Finally, we consider two important applications: measurement of ultrafast pulses and use of single-photon detectors in LIDAR and in quantum communications.

11.1 DETECTION WITH OPTICAL PREAMPLIFICATION Optical preamplification of weak signals before their photodetection has become a realistic option since the advent of optical amplifiers, which are largely employed for the repeaterless regeneration of long-haul transmission systems using optical fibers. The optical amplifier is a traveling-wave device in which the signal increases its power at the expense of the active medium, properly pumped through an external source to make available new photons by the stimulated emission mechanism (for more details, see Ref. [1]).

Photodetectors: Devices, Circuits and Applications, Second Edition. Silvano Donati © 2021 The Institute of Electrical and Electronics Engineers, Inc. Published 2021 by John Wiley & Sons, Inc.

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To analyze optical preamplification, let us refer to Fig.11-1 where the system is modeled by means of an ideal block with a radiant power gain G, receiving an input Ps and giving an amplified replica GPs at the output to the photodetector, which in its turn will supply a photodetected current Iph = GPs. In addition to gain, the optical amplifier also gives at its output a dc component superposed to the signal power even at Ps=0. This is called ASE (amplified spontaneous emission).

Figure 11-1 Functional scheme of an optically preamplified detection.

The ASE is generated by spontaneous emission, a second path to atoms’ de-excitation competitive to stimulated emission, and is distributed along the active medium crossed by the signal. Like the signal, the ASE itself is amplified by the active medium and arrives at the amplifier output with a power ASEout, found as: ASEout = nsp (G-1) h

(11.1)

0

where 0 is the gain linewidth of the active medium, and nsp=n2/(n2-n1) the inversion factor involving the populations n2 and n1 of the upper and lower levels. Eq.(11.1) holds for each spatial mode and a single polarization state. For N modes of arbitrary polarization, the ASE is increased by a factor 2N. We can schematize the preamplification with the equivalent circuit of Fig.11-2, where a noiseless amplifier block G receives the input signal Ps and the input-equivalent ASE component plus the eventual noise. Dividing Eq. (11.1) by the gain, we have: ASEi = ASEout /G = nsp [(G-1) /G] h

0

h

0

(11.1')

or, the input-equivalent ASEi amounts to one photon per Hertz of the gain linewidth 0 when the medium is fully inverted (nsp=1), and for G>>1. For example, we have ASEi= 0.6 W for 0=5 THz (corresponding to 0= 40 nm at =1500 nm). The mean ASE power is a dc component and can be easily filtered out from the ac signal component. Yet the ASE is also accompanied by a shot-noise fluctuation due to the discrete nature of photons, which falls in the signal baseband and thus cannot be eliminated. The ASE shot noise, in terms of quadratic power referred to the input, is: 2

ASEi =

2 h ASEi B = 2 (h )2

0

B

(11.2)

11.1 Detection with Optical Preamplification

where B is the electrical band of observation (different from ewidth).

305 0,

the active medium lin-

Figure 11-2 Equivalent circuit of an optical preamplifier.

In the optical amplifier, there is a further noise contribution [1] because of the discrete nature of photons, by which the number of photons can increase only by integers along the amplifier. The length between successive multiplications of a photon is a random variable, exactly like the avalanche multiplication of charge carriers in an avalanche photodiode. Thus, it is not surprising that the shot noise associated with the signal becomes greater, as in an APD with >> (see Section 6.1.4) by a theoretical factor F=2 [1,2] and in practice F>2 because of several non-idealities. Alternatively, the excess noise can be explained by a field description in terms of a beating between the ASE and signal field fluctuations [2] (see also App.A5). Of course, both descriptions lead to the same result and are an example of the particle-wave duality of quantum mechanics applied to the optical amplifier. They are a consequence of the Heisenberg uncertainty principle (discussed later in Section 11.4), which can no longer be neglected at optical frequencies as it is at electrical frequencies. Thus, the noise figure of the optical amplifier has the general trend of Fig.11-3, showing a range of optimal performance as a function of signal power, from a minimum determined by the ASE to a maximum corresponding to incipient saturation. From the phenomenological point of view, we can schematize as in Fig.11-2 the excess noise by means of a noise generator of quadratic power 2h (F-1) PsB, added to the signal 1 and the ASE shot-noises 2h Ps+ASEi)B. We have therefore at the output of the optical amplifier: Pu = GPs + G ASEi 2

Pu =

2F G2 h Ps B + 2 h G2ASEi B

(11.3)

__________________________________________________________________________ In some textbooks, two more terms due to the beating of ASE with the signal Ps and with itself, are written in the variance expression [see App. A5]. We prefer to include these terms in the excess factor F. 1

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Photodetection Techniques • Chapter 11

Figure 11-3 Typical diagram of gain G and noise figure F of an optical amplifier as a function of input-signal power (data for EDFAs at =1550 nm).

Correspondingly, the current photodetected at the amplifier output will have terms coming from Eq.(11.3) in addition to the usual dark current Id noise and load resistance R noise, whence: Iph = (GPs + G ASEi) = G Is+ G IASEi 2

Iph=

2Fe G2 Is B + 2e G2 IASEi B + 2e IbB + 4kTB/R

(11.3')

where Is and IASEi denote the photocurrents that signal Ps and ASEi powers would give at the photodetector without the amplifier. The second of Eq.(11.3') shows that optical preamplification gives an internal gain G with a satisfactory figure of noise F (thus, with the benefits discussed in Section 3.2), provided the signal is larger than the ASE, that is: Is > IASEi /F

or

Ps > ASEi /F

(11.4)

and also that GIs>> Ib+ 2kT/eR. When this condition is met, we have: (S/N)2 = Is /2FeB

(11.4')

that is, we have only a penalty equal to F, the noise figure of the optical amplifier, compared to the quantum limit. On the contrary, for small signals Is0.95) from electrons to photons found in a semiconductor laser working at low temperatures: if the laser is fed by a constant current __ i.e., through an ordered flux of electrons __ the photon flux replicates the electron statistics and is subpoissonian (squeezing factors down to F=0.2 have been obtained). For a phase-quadrature squeezing, a source that has been experimentally demonstrated is the parametric laser based on an OPO (optical parametric oscillator), where the gain diversity for the two phases due to the pump is used to generate the squeezing. For the direct detection of a squeezed-state signal with squeezing factor F, by means of an ideal detector with =1, the signal to noise ratio is: (S/N)2 = n 2/ 2n N / F

(11.29)

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Photodetection Techniques • Chapter 11

Figure 11-14 Squeezed-state sources: (a) number-phase, obtained by biasing through a constant current a high- laser (cooled at 77 K); (b) phasequadrature, obtained by pumping an optical parametric oscillator (an OPO in Lithium Niobate crystal) with photons of double energy.

The same result is found for the squeezed phase of a phase-quadrature. For the other phase, as a conjugate variable, we have (S/N)2 = FN. In practice, the improvement of S/N by squeezing is difficult to obtain, because of the attenuation suffered by the radiation propagating from source to detector, so far neglected. An attenuation encountered by a squeezed-state radiation with an initial factor F changes the squeezing factor to a new value F ', given by: F ' = 1 - (1-F)

(11.30)

and the result, plotted in Fig.11-15, shows that F is degraded more severely at the smaller values of [14]. Also, the finite photodetector quantum efficiency has the same effect as attenuation, and can be modeled accordingly, as shown in Fig.11-15. Physically, the attenuation corresponds to random absorptions in the photon sequence shown in Fig.11-12, with a survival probability for each photon; when 10% from 600 to 1000 nm. The dark current of the PMT is exceptionally low, down to a few counts/s cm2 for the S-24 (Fig.4-11), the best of any other SPD. There is no dead time in the PMT response, and afterpulses (on scale of s) are only found in aged tubes with the vacuum contaminated by gas leaked through the tube walls. The single photon response (Sect.4.5.2) SER or SPM, is a pulse with typical FWHM of 2...3ns (Fig.4-3) that can be reduced to 0.5__1-ns in MCP units (Sect.4.9 and Fig.4-27). Linearity of response and photon number resolution is flawless in PMTs (Fig.4-29) and this is advantage when Eve attempts tampering by sending a large photon packet to jam the receiver. Last, acceptance of PMTs is quite large, several cm2•str, so there is no problem of coupling to the incoming light signal, either from free-space or fiber-guided. SPADs are the single-photon version of APDs. APDs cannot be used as an SPD because in the linear mode the gain is insufficient to detect single photons (Sect. 6.1.1 and 6.2). In the Geiger mode, the SPAD cannot respond to multiple photons, and has a relatively long dead time (typ. from 10 ns to 1 s, see Sect.6.2.1), especially when affected by afterpulses, that is better to cutoff by extending the dead time rather than have them retrigger the device. The waveform is quite fast, with typical FWHM of 0.2 to 1 ns and the timing jitter is even faster,

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Photodetection Techniques • Chapter 11

usually 150 __250-ps FWHM (although in QKD application, it is the duration of the waveform that counts, not the timing accuracy). Dark current is rather high, typ. 6-30•106 counts/s•mm2 and only thanks to reducing the area to a small 10 ( m)2 and controlling stray contributions (Sect.6.2.1) it can be brought to a reasonable 20__100 counts/s. The total quantum efficiency •PDP (Sect.6.2.1) rarely exceeds 40% and depends an overvoltage (Fig.6-2.4) which affects not only the PDP but also the dark current (Fig.6-2.10). The SPAD spectral response is rather narrow in wavelength, because of the shallow junction, and typically ranges from 400 to 550-nm FWHM for a silicon SPADs (it becomes better in thick junctions at the expense of increased response time). Ternary and quaternary compounds allow to move the response to the NIR but exhibit increased dark current. Thus, SPADs are generally outperformed by PMTs with the exception of the smaller size and rugged structure. Si-PMT share many of the SPAD parameters because they are nothing else that an array of SPDs. Thanks to the array structure, they relieve many of the SPAD limitations, in particular the dead time and the multiple photon detection capability (Sect.6.3 and Fig.6-3.2). The efficiency of Si-PMT is somehow decreased (of 30% typ.) because of the area occupied by the quenching circuit of each SPAD pixel, but the fill-factor loss can be recovered by a microlens array (Sect.6.4.1). As the sensitive area (and the acceptance) are increased to the mmsize, also dark current is larger, typically 1k __10k cps. Time waveform, spectral response and effective efficiency are the same of the SPADs used in the array of the Si-PMT, so basically the same values quoted above. In conclusion, Si-PMT is definitely an improved SPD respect to the SPAD, and a good alternative to the PMT in quantum SPD applications. Superconducting Nanowire SPM. This device has superior timing performance, down to 30-ps FWHM, and may be tailored to attain an 80% efficiency from 1300 to 1500-nm (Sect.7.3). The photosensitive surface of the single device is rather small, but when arranged in an array, the SNSPM can reach an acceptance of several mm2•str, good enough for efficient collection of beams after free-space or out of optical fibers. The problem of SNSPM is the operation at cryogenic temperature, a requirement that calls for a sophisticated and rather bulky closed-cycle refrigerator (Fig.7-3.3) or the need of a reservoir of the expensive LHe. Other SPMs. Prompted by the several recently developed applications of single-photon detectors, several new approaches have been proposed in literature, some based on semiconductor quantum dots and charge-trapping effects, others on cryogenic temperature transitions [26]. One or more reported performance, particularly efficiency and dark rate, look quite good, yet these devices are still in their technical infancy and need much development to reach the state of well-engineered detectors to be used in applications deployed in the field.

11.7 DETECTORS FOR LIDAR LIDAR is the acronym for Laser Identification, Detection and Ranging, and historically this term has been used to identify instruments intended for remote sensing of atmosphere constituents, especially pollutants [15].

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More recently, as the LIDAR employs a pulsed laser and a time-of-flight measurement, the term LIDAR (or also, its new equivalent LiDAR) has been increasingly used to denote rangefinders, also called telemeters, especially in the context of collision avoidance and automatic driving of cars. Laser rangefinders are extensively treated in Chapter 3 of Ref.[15], where the interested reader can find the full theory of the measurement and the detailed design of the electronic circuits to develop rangefinders or LIDARs. Here, we will just mention the basic parts of a LIDAR, so as to focus on the requirements of the detector and suggest the possible candidates for this instrument. In the following, we refer explicitly to the requirements of the LIDAR detector intended for automotive application. A rangefinder LIDAR, in its simplest form, consists of a laser illuminator, a scanning beam optical system, a low-noise detector, and a time-of-flight measurement circuit. The time-of-flight measurement is carried out either: (i) as a measurement of the timedelay between the light pulse sent out s(t) and its received replica, s(r-T), where T=2L/s is the time delay containing the distance L information, or (ii) as a phase shift measurement between the sine-wave modulated illumination sin t, and its returned replica sin t-T), with the phase shift = T= 2L/s containing the distance to be measured. The two approaches, pulsed and sine wave have been demonstrated equivalent [15] in terms of SNR when the average power and the measurement time are the same. So, the sine-wave modulation may be preferred for laser safety reasons because it delivers a steady, almost constant (low) power instead of a (large) short pulse of the time-delay LIDAR. Further, the detector choice is strongly conditioned by the choice of the laser wavelength selected for the LIDAR, in consideration of eye safety and of the sources available at low cost, given that the application is a consumer-market one. To the detector we require a good signal-to-noise ratio at the maximum range to be covered, what is equivalent to have the lowest possible NEP. This condition doesn't necessarily call for an SPD even though the SPD can be used successfully. Thus, the performances of importance for a LIDAR detector are: - quantum efficiency and wavelength range of response (to maximize detected signal), - reasonably low dark current (smaller than the returning signal), - reasonably low or no dead time (to avoid delays in sending the next outgoing pulse), - fast waveform response or SER and timing resolution (impacting distance resolution), - linearity of response to multiple photons (as opposed to the Geiger-mode), - large acceptance to maximize the received signal back from the target. This wish list for LIDAR detectors is similar to that of last Section, with some difference, however. The main one is that we do not want a Geiger-mode SPD, because the returning optical signal collected by the receiver is a packet of photons, not a single photon, so all the photons collected after the first one shall not be wasted. In other words, the distance error is L=2c t and the timing error t in a linear detector improves as the inverse square root of the number of detected photons 1/√Nph, while it is fixed (at Nph=1) in a Geiger-mode SPD. So, the performance of the former easily surpasses that of the latter at Nph>>1, that is, at shorter distances.

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Also, dark current is mistaken for useful pulses in the SPAD, giving false measurements that shall be sorted out from good ones and discarded. Second, we need a detector with a quantum efficiency matched to the laser wavelength which, on its turn, is chosen so as to comply to the laser safety standards [27]. Third, the detector and the laser source shall be low cost to enable the consumer application of an instrument for the automotive. These requirements restrict the choice of the laser to two possible candidates: GaAs at 905 nm and InGaAsP at 1550 nm. At these wavelengths, the maximum permissible power is 1.05 and 10 mW, respectively, from the international IEC standards [27]. So, good candidates for LIDAR detectors are: (i) PMT, (ii) APD, and (iii) Si-PMT. The PMT has not a good response at the above wavelengths (see Fig.4-9). APDs can cover both wavelengths, with Si at 905 nm and InGaAsP at 1550 nm. The optimal gain (Sect.6.1.1) is larger in silicon than in the quaternary (Fig.6-4.1) and the noise is accordingly better (Sect.6.1.2) for operation at 905 nm, so the Si-APD is preferred. The Si-PMT outperforms the APD in minimum detectable signal if the wavelength of response covered 905 nm, but unfortunately the response is centred at 400__550 nm like the single SPAD of the array (Fig.6-2.9). However, efforts are under way to extend the response to the red. The other wavelength option is not preferred because a multi-element SPAD PMT, though feasible, has not yet been demonstrated as an engineered product and would be more expensive. An example of an automotive LIDAR embedded in the headlight of a car is illustrated in Fig.11-24 [28].

Figure 11-24

A LIDAR fitting in the headlight of a car, emitting 2-W optical power in a 12 by 12 deg field-of-view and using APD as a detector can detect a pedestrian up to a range of 40-m [28] (by courtesy of the IEEE).

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References [1] E. Desurvire: “Erbium Doped Fiber Amplifiers”, 2nd ed., J. Wiley and Sons, New York, NY, 2002. [2] S. Donati, G. Giuliani: “Noise in an Optical Amplifier: Formulation of a new Semiclassical Theory”, IEEE J. of Quant. Electr., vol. QE-33 (Sept.1997), pp.1481-1488. [3] K.-T. Shiu, S. S. Agashe, S. R. Forrest: “A Simple Monolithically Integrated Optical Receiver Consisting of an Optical Preamplifier and a p-i-n Photodiode”, IEEE Phot.Techn. Lett., vol. 18 (2006), pp. 956-958. [4] M.B. Spencer, W.E. Lamb: “Laser with a Transmitting Window”, Phys. Rev. A5 (1972) pp. 884-891; “Laser with External Injection”, Phys. Rev. vol.A5 (1972), pp. 891-897; see also: K. Petermann: “Laser Diode Modulation and Noise”, Kluwer Acadamic Press, Norwell, MA, 1988. [5] V. Annovazzi Lodi, S. Donati: “Injection Modulation in Coupled Laser Oscillators”, IEEE J. of Quant. Electr., vol. QE-16 (1980), pp. 859-864. [6] S. Donati: “Laser Interferometry by Induced Modulation of the Cavity Feld”, Journal of Applied Physics, vol. 49 (2), 1978, pp. 495-498, and: S. Donati, M. Sorel: “A PhaseModulated Feedback Method for Testing Optical Isolators Assembled in the Laser Package”, IEEE Phot.Techn. Lett., vol. 8 (1996), pp. 405-07. [7] R. Lang and K. Kobayashi, “External Optical Feedback Effects on Semiconductor Injection Laser Properties”, IEEE J. Quant. Electr., vol. QE-16 (1980), pp.347-355. [8] R. Adler: “A Study of Locking Phenomena in Oscillators”, IRE Proc. Waves and Electrons, vol.34 (1946) pp.351-357. [9] S. Donati: “Developing Self-Mixing Interferometry for Instrumentation and Measurements”, Laser Photonics Rev. vol.6 (2012), pp. 393–417; see also: IEEE Trans. Instr. and Meas., vol. 45 (1996), pp. 942-946; IEEE J. Quant. Electr., vol. QE-33 (1997), pp. 527-531. [10] S. Donati: “Responsivity and Noise of Self-Mixing Photodetection Schemes”, IEEE J. Quant. Electr., vol.47, 2011, pp.1428-1433. [11] P.Dean, Y.L.Lim, A.Valavanis, R.Kleise, M.Nikolic, S.P.Khanna, M.Lachab, D.Indjin, Z.Ikonic, P.Harrison, A.D.Rakic, E.H.Lindfield, G.Davies: “Terahertz Imaging through Self-Mixing in a Quantum Cascade Laser”, Opt. Lett. vol.36 (2011), pp.2587-2589; see also: “Self-Mixing Interferometry with Terahertz Quantum Cascade Laser”, IEEE Sensor J., vol.13 (2013), pp.37-43. [12] V. Braginski, F. Khalili: “Quantum Measurements”, Cambridge Univ. Press, 1992. [13] R. Slusher, B. Yurke: “Squeezed Light for Coherent Communications”, IEEE J. Lightw. Techn. vol. LT-8 (1990), pp. 466-477; Y. Yamamoto, H.A. Haus: “Measurements and Information Capacity of Optical Quantum States”, Journal of Modern Phys., vol. 58 (1986), pp. 1001-1020. [14] S. Donati, V. Annovazzi, S. Merlo: “Squeezed States in Direct and Coherent Detection”, Journal of Opt. and Quant. Electr. vol. 24 (1992), pp. 285-301.

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[15] S. Donati: “Electrooptical Instrumentation”, Prentice Hall, 2004. [16] D.J. Bradley, G.H.C. New: “Ultrashort Pulse Measurements”, Proc. IEEE, vol.62 (1974), pp.313-344. [17] S. Franco: “Design with Operational Amplifiers and Analog Integrated Circuits”, 4th ed., McGraw Hill, 2015. [18] J.-C. Diels, J. J. Fontaine, I. C. McMichael, F. Simoni: “Control and Measurement of Ultrashort Pulse Shapes with Femtosecond Accuracy”, Appl. Opt., vol.24 (1985), pp.1270-1282, see also J.-C. Diels, W. Rudolph: “Ultrashort Laser Pulse Phenomena”, Elsevier, 2006, and K.L. Sala, G.A. Kenney-Wallace, G.E. Hall: “CW Autocorrelation of Picosecond Laser Pulses”, IEEE J. Quant. Electr., vol. QE-16 (1980) pp. 990-996. [19] W.K. Frederick: “Hilbert Transforms: Vol.1”. Cambridge University Press, 2009; see also: A. Mecozzi: “A Necessary and Sufficient Condition for Minimum Phase and Implications for Phase Retrieval”, IEEE Trans. Inform. Theory, vol.13 (2014), pp.1-9. [20] S. Toenger, R. Makalito, J. Ahvenjarvi, P. Ryczkowski, M. Nahi, J. Dudley, G. Genty: “Interferometric Correlation Measurement of Supercontinuum by Two-Photon Absorption in GaP detector”, Proc. CLEO Europe 2019, DOI: 10.1109/CLEOEEQEC.2019.8871936. [21] R. Trebino: “Frequency-resolved Optical Gating: the Measurement of Ultrashort Laser Pulses”, Kluwer Academic, 2002; see also D.J. Kane, R. Trebino: “Single-shot Measurement of the Intensity and Phase of an Arbitrary Ultrashort Pulse by FrequencyResolved Optical Gating”, Opt. Lett. vol.18 (1993), pp.823-825. [22] M.A. Nielsen, I.L. Chuang: “Quantum Computation and Quantum Information”, Cambridge University Press, 2010. [23] N.D. Mermin: “Is the Moon There when Nobody Looks? Reality and Quantum Theory”, Physics Today, vol.38 no.4 (1985), pp.38-47, see also: C.H. Bennett: “Quantum Information and Computation”, Physics Today, vol.48 no.10 (1995), pp.24-30. [24] J. Lin et al.: “Entanglement-based Secure Quantum Cryptography over 1,120 km”, Nature, vol.551 (2020), DoI 10.1038/ s41586-020-2401-y. [25] A. Boaron et al.: “Secure Quantum Key Distribution over 421 km of Optical Fiber”, Phys. Rev. Lett., vol.121 (2018), DoI 190502. [26] R.H. Hadfield: “Single-Photon Detectors for Quantum Information Applications”, Nature Phot., vol.3 (2009), pp.696-705. [27] IEC Laser Safety Standard 60825-1, 2014 3rd edition class1 laser product. See also: D.H. Sliney, M. Wolbarsht (editors): “Safety with Lasers and Other Optical Sources”, Springer 1980. [28] C.N. Liu, Y.P. Chang, H.K. Shi, H. Pin, K. Li, Z. Pei, S. Donati, W.H.Cheng: “LIDAR Embedded Smart Laser Headlight Module using a Single Digital Micromirror Device for Autonomous Drive”, Proc. CLEO 2020, paper ATu3T.2.

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Problems P11-1 An optical amplifier with a gain G=103 is intended for use as a preamplifier at =1500 nm. Calculate: the output ASE, the input-equivalent ASE, the NEP (noise-equivalent- input) for the full band. width available (40 nm) and for a B=10 GHz electrical bandwidth. What is changed if we filter the amplifier output with a narrowband filter with =0.5 nm ? P11-2 A signal from a narrow-line DBR laser around 1500 nm is fed in another similar laser, detuned of f=20 GHz from the first. The laser gain is c = 200 cm-1, loss is c =10 cm-1, cavity length is L=200 m, mirror facets reflectivity is R=0.3, and power emitted is Pout= 1mW. Calculate the heterodyne injection gain and the signal amplitude for a detected power PS=1pW. P11-3 In a heterodyne or homodyne injection detection, what is the minimum signal that can be detected by a 1-mW semiconductor laser with the parameters of problem P11-2 ? Evaluate it both for a small detuned signal f injected in the laser and for a weak echo at the same frequency of the laser. P11-4 Could the QND scheme of detection be implemented with a pump at the same frequency of the signal? How shall the setup of Fig.11-9 be modified? P11-5 What is the squeezing factor we can obtain from a diode laser converting the injected electrons into photons with an efficiency =0.95 ? What is the squeezing factor when this laser signal is launched in a fiber? Q11-6 Why the squeezed radiation actually improves the interferometer readout while it does not help improving in data transmission and detection ? P11-7 Show that the term √I1I2 =E1E2 of the photodetected sum < (E1+E2) noise, different from terms I1 and I2.

2

> doesn't carry the shot

Q11-8 What is the difference between the (auto)correlation measurements described in Sect.4.5.3 and the autocorrelation measurements of Sect. 11.5.1 ? Are they not the same or if any, what is the conceptual difference of the two ? Q11-9 A spectrum analyzer has measured a wavelength distribution I ) for the emission of a modelocked laser. An autocorrelation measurement has provided the I( ) frequency distribution. How are the two related ? P11-10 In an intensity autocorrelation pulse measurement, the FWHM of the curve is obtained with a stroke s=300 m. Find the width of the wavelength spectrum of the pulse, assuming a transform limited Gaussian waveform. P11-11 An autocorrelation measurement is carried out by scanning the moveable mirror of Fig.11-17 along a stroke of s=2 mm in a scanning period of 1 s. Calculate the maximum time pulse duration that can be measured and the bandwidth requirement for the acquisition of the fringe signal by the photodiode. P11-12 An autocorrelation interferogram (like in Fig.11-18) has been taken from a mode-locked laser at =1.06 m, and it has 100 fringes at FWHM (half-width half-maximum). The correlation waveform is a Gaussian. Calculate the time FWHM of the pulse assuming it is transform limited.

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Image Detectors

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n this Chapter, we first describe image detectors, i.e., the CCDs and the family of image intensifiers/converters, and then introduce the function describing the resolution, that is, the modulation transfer function and associated concepts, including the undersampling (or Moiré) effects, and then illustrate some useful application of this otherwise disturbing feature. With the term image detector we understand the photodetectors that contain a single or double multiplicity of individual picture elements (or pixels), organized in a linear array or an image array. These detectors can be divided into two classes: 

image pickup detectors, in which the individual pixel outputs are organized serially in a single electrical signal, suitable for transmission or processing;  direct-vision photodetectors in which an intensified (or spectrally converted) replica of the input image is produced at an output screen. In both classes, we need a photosensitive surface __ virtually any of those described for single-point detectors are useable, __ and a provision to sort out the individual image pixels. In addition, in pickup detectors we need an arrangement for the sequential readout of the photogenerated charge, and, equally important, an integration of the photogenerated charge during the time between successive readouts, i.e., the frame period Tf.

Photodetectors: Devices, Circuits and Applications, Second Edition. Silvano Donati © 2021 The Institute of Electrical and Electronics Engineers, Inc. Published 2021 by John Wiley & Sons, Inc.

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Reason for integration is that the pixel readout time is tp=Tf/N, where N is the number of pixels per frame, and thus reading the photogenerated charge only during tp, the signal would be N times smaller than that available during Tf, with an unacceptable degradation of the S/N ratio. In direct-vision detectors, we need an optical conjugation between input and output surfaces __ usually provided by electron optics in tube technology devices __ and a mechanism for an internal gain in charge number or energy, so that the output replicates the input image with a brilliance gain.

12.1 THE EARLY IMAGING DEVICE: THE VIDICON Image-taking devices have been developed since the early 1930s with the technology of vacuum tubes, and have been the enabling components for the development of modern television pickup and broadcasting technology of today [1,2]. In particular, the vidicon has been the workhorse of TV pickup cameras and a key device to develop the television broadcasting service on about 60 years up to the come of age of CCD. We will just give a brief account of this component because it has become obsolete; the reader interested to learn the technical details of vidicon and other device of the vacuumtube family, as well as their history, may consult Refs. [3,5]. The vidicon is a vacuum tube, in which most of the volume is occupied to build up the fine spot electron beam that reads the photosensitive target. The target is deposited on the inner surface of the optical access window and is a photoconductor (Sect.5.9) with one surface facing the electron beam used for charge accumulation, and the other being used for biasing and sending out the video signal. The signal is made up by the charges photodetected in the target and accumulated on the internal surface during a frame period. They are read in a raster fashion by the electron beam. Sensitivity of the vidicon was very good, thanks to the internal gain provided by photoconductivity (Sect.5.9.1). Cost is limited because the thin photoconductive layer [typically PbO, Cd (S,Se)] is deposited by a cheap evaporation or sputtering technology, different from the CCD which is a semiconductor epitaxial device (and started from being more expensive but then improved in the learning curve by orders of magnitude). Vidicon formats range from ½ to 2-inches in diameter, and tube length typ. from 10 to 25cm, with the ½ inch good for surveillance, and the 2 inches for high quality image pickup. Much of the overall size and weight are however much larger, due to the heavy focussing and deflection coils wound around the tube (and also current hungry). Since the years '80s, the field of application of the vidicon has been increasingly eroded by the advent of CCD – much lighter and compact but initially more expensive. Up to the year 2000, vidicons were still unsurpassed for the better price in high-performance applications like high-definition (HD) TV-studio pickup, operation in spectral ranges extended to IR or UV, or at very low-light levels (the so called LLL-TV). Nowadays, all new video cameras are designed with a CCD, even for the ultra-high-definition (UHD) 4K 120 or 240 fps units (where 4K stands for 3840 2160 pixels and fps = frame per second).

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12.2 CHARGE-COUPLED DEVICES Charge-coupled devices or CCDs are semiconductor devices for image pickup. Compared to their vacuum-tube counterparts, they have the typical advantages of small size and weight, superior reliability, better endurance and immunity to mechanical stresses and electrical interference, and an easier direct interfacing and integration of the ancillary circuits.

12.2.1 Introduction Devised at the beginning of the 1970s as a new approach to implementing integrated shiftregisters memories, CCDs have soon found serendipitously in image pickup their most significant and high-volume application area. For the invention of the CCD, W.S. Boyle and G. E. Smith have been awarded the Nobel Prize in Physics in 2009. Following the strong development efforts devoted to them since the years 1980, CCDs have nicely followed the so-called "learning curve" of electronic devices over several decades of improved overall performance – especially number of pixel per square mm – while reducing unit cost, thus opening the way for mass applications, such as the hand-held video camera (or camcorder), based on silicon technology. More recently, the concept of charge storage of photogenerated carriers and readout for sequential shift has been also demonstrated in other materials, for example InSb (the CCD of next-generation Space Telescope) and also in microbolometer (the focal-plane-arrays of room-temperature IR imagers).

12.2.2 Principle of Operation The CCD is basically a MOS (metal-oxide-semiconductor) - or an MIS (metal-insulatorsemiconductor) - structure with multiple gates, which, working in the weak inversion regime, can detect, store, and shift along the semiconductor-oxide interface packets of photogenerated charges to collection at a (drain) output electrode. Referring to a p-type substrate MOS (Fig.12-1a), it is known that when the gate voltage is positive with respect to the substrate, an n-channel is generated under the gate, because the positive charges of the electrode push away the positive charges of the p semiconductor, thus creating a depletion at the oxide-semiconductor interface. In the CCD, the channel is used as a potential well, spatially localized under the electrode (Fig.12-1b), and controlled by the voltage applied to the electrode. Thus, it can store the photogenerated charge of minority carriers (electrons) __ the integration operation required in an image detector. Letting the radiation enter from the substrate side as shown in Fig.12-1c, where the common negative contact is obtained with a thin p+-doped region, the photon dissipation is mainly in the p-region (note the p+pn structure), and the photogenerated charges are separated by drift because the substrate is depleted under the applied voltage (Fig.12-1c); thus, the hole is collected by the negative electrode, and the electron reaches the potential well under the oxide where it stops because there is no electrode to collect it.

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Figure 12-1 a) Basic structure of the CCD; b) temporal sequence of the potential wells for charge storage and shift when the phases are driven as indicated in b'); c) structure showing the photogeneration in the pregion, the shift and the collection at the drain; the distribution of charge density and field is indicated on the left.

To carry out the scanning of picture elements, the gates (or phases ) are connected three by three and brought high sequentially (Fig.12-1b'), one after the other, with a partial overlap in time. As illustrated in Figs.12-1b and 12-1b', the charge initially present in the

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well under 1 (up to time t=t1) is spread by diffusion (see also below), also under 2 when 2 is raised between t1 and t2; then, when 1 is lowered between t2 and t3, the charge falls completely under 2, and in the same way is found under 3 at t5. To help the charge transfer, one can use an overdrive in the phase voltages (as shown in the last line of Fig.12-1b), so that the newly formed well is a little deeper than the adjacent one. Once they reach the last electrode, the charges in the channel well are collected in the drain (Fig.12-1c) as in a standard MOS, and constitute an output current organized serially in charge packets representative of the pixel sequence. Of course, for an n-type substrate we can repeat the same considerations valid for the psubstrate, by interchanging the sign of voltages and charges. Note that the elemental shift cycle requires three phases, because two of them are not enough to identify the desired shift direction for the charge diffusion. For N pixels, 3N electrodes are thus necessary. Actually, by appropriately conforming the electrode geometry and the waveforms of phase drives, one can eventually succeed in making a quasi-two-phase shift device; however, in practice this arrangement does not fully satisfy all the system requirements.

The structure illustrated in Fig.12-1 is a single line of an image device, and as such it is called a linear array CCD (with N typically ranging from 64 to 2048 pixels), used in several instrumentation and industrial control applications. In addition, this structure is the starting point for the image scanning described in next Sections.

12.2.3 Properties and Parameters Let us now analyze the properties of the potential well in more detail. As indicated in Fig.12-2, already at zero voltage (VG=0) we find a modest band bending in the p region, due to the presence of positive charges Qox trapped at the oxide-semiconductor interface, from which we get a (band-bending) potential ox. By applying a moderate positive voltage to the gate (VG>0), holes are repelled from the interface, and a depleted layer is generated, i.e., a potential well under the oxide capable of collecting the minority carriers (electrons). The potential well depth s at the oxide-semiconductor interface, Fig.12-2 a2), can be calculated from the simple electrostatic balance of charges if we assume a complete depletion of the layer, and is given by: s=

VG - ( ms - ox) - e(WNA+N)/Cox

where: is the difference of work functions of metal and semiconductor, Qox/Cox is the contribution due to the residual charges in the oxide, NA (cm-3) is the doping concentration of the p-region, ms

ox=

(12.1)

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Figure 12-2 a) Potential diagrams of the MOS junction in a CCD at several applied voltages; and b) structure of a CCD with buried channel.

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N (cm-2) is the density of minority carriers, Cox= ox/wox (Fcm-2) is the oxide specific capacitance, and wox its thickness. The thickness W of the depletion layer (or well extension) in the same hypothesis is [see Eq.(5.32)]: W = [2 s s/eNA] 1/2

(12.2)

where s is the dielectric constant of the semiconductor. The applied voltage VG drops across the series of two capacitances, namely, the oxide capacitance Cox= ox/wox and the depletion capacitance Cs= s/W=[e sNA/2 s]1/2. Because the doping NA is low and the oxide thickness is small (wox 100nm), we have in practice Cs