Nonlinear-Emission Photonic Glass Fiber and Waveguide Devices 1108418457, 9781108418454

This book presents a comprehensive introduction to the design of compact and broadband fiber and waveguide devices using

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Nonlinear-Emission Photonic Glass Fiber and Waveguide Devices
 1108418457, 9781108418454

Table of contents :
Contents
Preface
1 Fundamental Mathematics of Nonlinear-Emission Photonic Glass Fiber and Waveguide Devices
1.1 Introduction
1.2 Newton Iteration Algorithm for Nonlinear Rate Equation Solution
1.2.1 Single-Variable
1.2.2 Multi-Variable
1.3 Runge–Kutta Algorithm for Power-Propagation Equation Solution
1.3.1 Single-Function
1.3.2 Multi-Functions
1.4 Two-Point Boundary Problem for Power-Propagation Equations in a Laser Cavity
1.4.1 Principle
1.4.2 Shooting Method and Relaxation Method
References
2 Fundamental Spectral Theory of Photonic Glasses
2.1 Introduction
2.2 Judd–Ofelt Theory
2.3 Transition Probability and Quantum Efficiency
2.4 Fluorescence Branch Ratio
2.5 Homogeneous and Inhomogeneous Broadening of Spectra
References
3 Spectral Properties of Ytterbium-Doped Glasses
3.1 Introduction
3.2 Formation Region of Yb2O3-Containing Glasses
3.3 Laser Performance Parameters of Ytterbium-Doped Glasses
3.3.1 Minimum Fraction of Excited State Ions
3.3.2 Saturation Pump Intensity
3.3.3 Minimum Pump Intensity
3.3.4 Storage-Energy and Gain Parameters
3.4 Spectral Properties of Yb3+-Doped Borate Glasses
3.4.1 Compositional Dependence of Spectral Properties
3.4.2 Dependence of Spectral Properties on Active Ion Concentration
3.5 Spectral Properties of Yb3+-Doped Phosphate Glasses
3.5.1 Compositional Dependence of Spectral Properties
3.5.2 Dependence of Spectral Properties on Active Ion Concentration
3.6 Spectral Properties of Yb3+-Doped Silicate Glasses
3.6.1 Compositional Dependence of Spectral Properties
3.6.2 Dependence of Spectral Properties on Active Ion Concentration
3.7 Spectral Properties of Yb3+-Doped Germanate Glasses
3.7.1 Compositional Dependence of Spectral Properties
3.8 Spectral Properties of Yb3+-Doped Telluride Glasses
3.8.1 Compositional Dependence of Spectral Properties
3.8.2 Dependence of Spectral Properties on Active Ion Concentration
3.9 Dependence of Spectral Property and Laser Performance Parameters on Glass System
3.9.1 Dependence of Spectral Property on Glass Systems
3.9.2 Dependence of Laser Performance Parameters on Glass Systems
3.10 Dependence of Energy-Level Structure of Yb3+ on Glass Systems
3.11 Cooperative Upconversion of Yb3+ Ion Pairs
3.11.1 Cooperative Upconversion Luminescence
3.11.2 Concentration-Quenching Mechanics
3.11.3 Concentration Dependence of Luminescence Intensity
3.12 Fluorescence Trap Effect of Yb3+ Ions in Glasses
References
4 Compact Fiber Amplifiers
4.1 Introduction
4.2 Level Structure and Numerical Model
4.3 Dependence of Gain and Noise Figure on Concentrations
4.4 Doping Concentrations with Short-Length High Gain
References
5 Photonic Glass Fiber Lasers
5.1 Introduction
5.2 Fundamental Physics of Fiber Laser
5.2.1 Lasing Conditions of Laser
5.2.2 Threshold Gain
5.2.3 Phase Condition and Laser Modes
5.2.4 Population Inversion Calculation
5.3 Numerical Models of Rare-Earth-Doped Fiber Lasers
5.3.1 Configuration and Power-Propagation Equations of Fiber Laser
5.3.2 Output Power of a Two-Level Fiber Laser
5.3.3 Output Power of a Three-Level Fiber Laser
5.3.4 Output Power of a Four-Level Fiber Laser
5.3.5 Output Power of Yb3+-Doped Fiber Laser
References
6 Broadband Fiber Amplifiers and Sources
6.1 Introduction
6.2 Pr3+-Tm3+-Er3+-Co-Doped Fiber System
6.2.1 General Rate and Power-Propagation Equations with Two Wavelength Pumps
6.2.2 Gain Characteristics with 980nm Pump
6.2.3 Gain Characteristics with 793nm Pump
6.2.4 Gain Characteristics with Double Pumps
6.3 Gain Characteristics of Pr3+-Er3+-Co-Doped Fiber System
6.3.1 Rate and Power-Propagation Equations
6.3.2 Dependence of Gain on Fiber Parameters
6.4 WDM Transmission System Cascaded with Tm3+-Er3+-Co-Doped Fiber Amplifiers
6.4.1 WDM System with Single Pump
6.4.2 WDM System with Dual Pumps
References
7 Photonic Glass Waveguides for Spectral Conversion
7.1 Introduction
7.2 Theoretical Model and Spectral Characterization
7.2.1 Theoretical Model
7.2.2 Spectral Characterization
7.3 Doubly-Doped System
7.3.1 Energy Transfer Model
7.3.2 Quantum Efficiency of Photonic Glass Waveguide
7.4 Triply-Doped System
7.4.1 Energy Transfer Model
7.4.2 Quantum Efficiency of Photonic Glass Waveguide
7.5 Performance Evaluation of sc-Si-Solar Cell with Photonic Glass Waveguides
References
8 Photonic Glass Waveguide for White-Light Generation
8.1 Introduction
8.2 White-Light Glasses
8.2.1 Tm3+-Tb3+-Eu3+-Co-Doped System
8.2.2 Yb3+-Er3+-Tm3+-Co-Doped System
8.3 Emission-Tunable Glasses
8.3.1 Tb3+-Sm3+-Dy3+-Co-Doped System
8.3.2 Tm3+-Yb3+-Ho3+-Co-Doped System
References
Appendix 1 Matlab Code for Solving Nonlinear Rate and Power-Propagation Equation Groups in Co-Doped Fiber Amplifiers or Fiber Sources
Appendix 2 Matlab Code for Solving Power-Propagation Equations of a Laser Cavity with Four-Level System
Index

Citation preview

Nonlinear-Emission Photonic Glass Fiber and Waveguide Devices A comprehensive introduction to the design of compact and broadband fiber and waveguide devices using active-ion-doped photonic glasses. Combining cutting-edge theory with new applications, it shows how the complementarity of emission spectra of different active ions can be used in broadband fiber amplifiers and optical fiber communication, and describes how the quantum cutting of active ions can improve the match between the solar spectrum and the responsiveness of silicon cells. Mathematical modeling is used to predict the performance of photonic fiber and waveguide devices, and experimental data from glass doped with rare-earth ions is included. Offering unique insights into the state of the art of the field, this is an ideal reference for researchers and practitioners, and invaluable reading for students in optoelectronics, electrical engineering, and materials science. Chun Jiang is a professor in the Department of Electronic Engineering at Shanghai Jiao Tong University. Pei Song is an assistant professor in the School of Mathematics, Physics and Statistics at Shanghai University of Engineering Science.

Nonlinear-Emission Photonic Glass Fiber and Waveguide Devices CHUN JIANG Shanghai Jiao Tong University

PEI SONG Shanghai University of Engineering Science

University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781108418454 DOI: 10.1017/9781108290074 © Shanghai Scientific & Technical Publishers 2019 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2019 Printed and bound in Great Britain by Clays Ltd, Elcograf S.p.A. A catalogue record for this publication is available from the British Library. Library of Congress Cataloging-in-Publication Data Names: Jiang, Chun, 1965- author. | Song, Pei, 1979- author. Title: Nonlinear-emission photonic glass fiber and waveguide devices / Chun Jiang (Shanghai Jiao Tong University), Pei Song (Shanghai University of Engineering Science). Description: Cambridge ; New York, NY : Cambridge University Press, 2019. | Includes bibliographical references. Identifiers: LCCN 2018045346 | ISBN 9781108418454 (hardback) Subjects: LCSH: Photonics–Materials. | Glass. | Optical wave guides. | Nonlinear optics. Classification: LCC TA1522 .J53 2019 | DDC 621.36/5–dc23 LC record available at https://lccn.loc.gov/2018045346 ISBN 978-1-108-41845-4 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Contents

Preface 1

2

3

Fundamental Mathematics of Nonlinear-Emission Photonic Glass Fiber and Waveguide Devices 1.1 Introduction 1.2 Newton Iteration Algorithm for Nonlinear Rate Equation Solution 1.2.1 Single-Variable 1.2.2 Multi-Variable 1.3 Runge–Kutta Algorithm for Power-Propagation Equation Solution 1.3.1 Single-Function 1.3.2 Multi-Functions 1.4 Two-Point Boundary Problem for Power-Propagation Equations in a Laser Cavity 1.4.1 Principle 1.4.2 Shooting Method and Relaxation Method References

page ix

1 1 1 1 3 4 4 6 7 7 7 9

Fundamental Spectral Theory of Photonic Glasses 2.1 Introduction 2.2 Judd–Ofelt Theory 2.3 Transition Probability and Quantum Efficiency 2.4 Fluorescence Branch Ratio 2.5 Homogeneous and Inhomogeneous Broadening of Spectra References

10 10 10 12 13

Spectral Properties of Ytterbium-Doped Glasses 3.1 Introduction 3.2 Formation Region of Yb2O3-Containing Glasses 3.3 Laser Performance Parameters of Ytterbium-Doped Glasses 3.3.1 Minimum Fraction of Excited State Ions 3.3.2 Saturation Pump Intensity 3.3.3 Minimum Pump Intensity 3.3.4 Storage-Energy and Gain Parameters

16 16 16 17 17 18 18 18

14 15

v

vi

Contents

Spectral Properties of Yb3+-Doped Borate Glasses 3.4.1 Compositional Dependence of Spectral Properties 3.4.2 Dependence of Spectral Properties on Active Ion Concentration 3.5 Spectral Properties of Yb3+-Doped Phosphate Glasses 3.5.1 Compositional Dependence of Spectral Properties 3.5.2 Dependence of Spectral Properties on Active Ion Concentration 3.6 Spectral Properties of Yb3+-Doped Silicate Glasses 3.6.1 Compositional Dependence of Spectral Properties 3.6.2 Dependence of Spectral Properties on Active Ion Concentration 3.7 Spectral Properties of Yb3+-Doped Germanate Glasses 3.7.1 Compositional Dependence of Spectral Properties 3.8 Spectral Properties of Yb3+-Doped Telluride Glasses 3.8.1 Compositional Dependence of Spectral Properties 3.8.2 Dependence of Spectral Properties on Active Ion Concentration 3.9 Dependence of Spectral Property and Laser Performance Parameters on Glass System 3.9.1 Dependence of Spectral Property on Glass Systems 3.9.2 Dependence of Laser Performance Parameters on Glass Systems 3.10 Dependence of Energy-Level Structure of Yb3+ on Glass Systems 3.11 Cooperative Upconversion of Yb3+ Ion Pairs 3.11.1 Cooperative Upconversion Luminescence 3.11.2 Concentration-Quenching Mechanics 3.11.3 Concentration Dependence of Luminescence Intensity 3.12 Fluorescence Trap Effect of Yb3+ Ions in Glasses References

46 51 53 53 57 59 60 63

4

Compact Fiber Amplifiers 4.1 Introduction 4.2 Level Structure and Numerical Model 4.3 Dependence of Gain and Noise Figure on Concentrations 4.4 Doping Concentrations with Short-Length High Gain References

65 65 66 67 72 72

5

Photonic Glass Fiber Lasers 5.1 Introduction 5.2 Fundamental Physics of Fiber Laser 5.2.1 Lasing Conditions of Laser 5.2.2 Threshold Gain 5.2.3 Phase Condition and Laser Modes 5.2.4 Population Inversion Calculation 5.3 Numerical Models of Rare-Earth-Doped Fiber Lasers 5.3.1 Configuration and Power-Propagation Equations of Fiber Laser

74 74 74 74 75 76 76 80 80

3.4

19 19 22 23 23 26 28 28 32 34 34 36 36 39 43 43

Contents

5.3.2 5.3.3 5.3.4 5.3.5 References 6

7

8

Output Output Output Output

Power Power Power Power

of a Two-Level Fiber Laser of a Three-Level Fiber Laser of a Four-Level Fiber Laser of Yb3+-Doped Fiber Laser

Broadband Fiber Amplifiers and Sources 6.1 Introduction 6.2 Pr3+-Tm3+-Er3+-Co-Doped Fiber System 6.2.1 General Rate and Power-Propagation Equations with Two Wavelength Pumps 6.2.2 Gain Characteristics with 980nm Pump 6.2.3 Gain Characteristics with 793nm Pump 6.2.4 Gain Characteristics with Double Pumps 6.3 Gain Characteristics of Pr3+-Er3+-Co-Doped Fiber System 6.3.1 Rate and Power-Propagation Equations 6.3.2 Dependence of Gain on Fiber Parameters 6.4 WDM Transmission System Cascaded with Tm3+-Er3+-Co-Doped Fiber Amplifiers 6.4.1 WDM System with Single Pump 6.4.2 WDM System with Dual Pumps References

vii

81 83 84 85 90 91 91 92 92 96 99 105 131 131 134 139 140 141 143

Photonic Glass Waveguides for Spectral Conversion 7.1 Introduction 7.2 Theoretical Model and Spectral Characterization 7.2.1 Theoretical Model 7.2.2 Spectral Characterization 7.3 Doubly-Doped System 7.3.1 Energy Transfer Model 7.3.2 Quantum Efficiency of Photonic Glass Waveguide 7.4 Triply-Doped System 7.4.1 Energy Transfer Model 7.4.2 Quantum Efficiency of Photonic Glass Waveguide 7.5 Performance Evaluation of sc-Si-Solar Cell with Photonic Glass Waveguides References

145 145 146 146 148 148 149 152 159 159 163

Photonic Glass Waveguide for White-Light Generation 8.1 Introduction 8.2 White-Light Glasses 8.2.1 Tm3+-Tb3+-Eu3+-Co-Doped System 8.2.2 Yb3+-Er3+-Tm3+-Co-Doped System

177 177 178 178 185

171 174

viii

Contents

8.3

Emission-Tunable Glasses 8.3.1 Tb3+-Sm3+-Dy3+-Co-Doped System 8.3.2 Tm3+-Yb3+-Ho3+-Co-Doped System References

Appendix 1 Matlab Code for Solving Nonlinear Rate and Power-Propagation Equation Groups in Co-Doped Fiber Amplifiers or Fiber Sources Appendix 2 Matlab Code for Solving Power-Propagation Equations of a Laser Cavity with Four-Level System Index

194 194 205 214

219 225 228

Preface

Luminescence of transition-metal ions and rare-earth ions has important applications in optoelectronic devices and systems including fiber amplifiers, fiber lasers, and fiber sources. With advances in integrated photonic devices and broadband and compact fiber-optic devices, it is necessary to make active fiber devices that have short interaction length and have broadband gain and emission spectra by using high-concentration active ion-doping and multi-active ion-doping techniques. For low-concentration doped-fiber devices, the dependence of emission intensity on excitation power generally is a linear relationship. However, in highly doped and multiply doped fiber devices, the relation is not linear and but nonlinear, due to the interaction between rare-earth ions such as upconversion, cross-relaxation, energy transfer, and so on. In this book, thus, the active fiber and waveguide devices including high concentration-doping and multi-rare-earth-doping are defined as nonlinear-emission photonic fiber and waveguide devices. This book consists of eight parts as follows: Chapter 1 introduces the fundamental mathematics of nonlinear-emission photonic glass fiber and waveguide devices. In the design and analysis of the photonic glass fiber and waveguide devices, one of most important tasks is to solve a multi-variable rate equation group and power-propagation equation group. The methods introduced in this chapter are Newton iteration, Runge–Kutta algorithms and their combination as well as solution of two-point boundary problem, which are effective numerical techniques for highly doped or co-doped fiber amplifiers, fiber sources, and fiber laser systems. Chapter 2 introduces the fundamental of spectral theory of photonic glasses. In this chapter, spectral properties of rare-earth-doped glasses, including absorption and emission cross-sections, spontaneous emission transition probability, fluorescence branch ratio and quantum efficiency, and homogeneous and inhomogeneous broadening of fluorescence spectra and their calculating methods, are summarized. Chapter 3 systematically reports the spectral properties and laser performance parameters of ytterbium-doped glass systems. Ytterbium ions can be used as sensitizers to other active ions due to simple level structure and strong absorption coefficient, ytterbium ion-doped fiber can be used as a high-power fiber laser system for industrial processing, and ytterbium ion-doped glass waveguides can be used as a spectral converter because its emission wavelength matches well with the spectral responsivity of single-crystal silicon solar cell. ix

x

Preface

Chapter 4 presents the modeling and numerical results of compact ytterbium-erbiumco-doped fiber amplifiers, which are supposed to obtain higher internal gain and higher gain per unit length in fiber amplifiers using numerical solutions of rate and evolution equations of signal, pump power, and amplified spontaneous emission. Chapter 5 introduces the fundamentals of lasers and a numerical model of ytterbiumdoped-glass fiber systems, which can be widely used in industrial processing. Nonlinear interaction between high-concentration active ions or the co-upconversion effect will degrade these system performances. Our numerical model considers the nonlinear transition in the high-concentration doped system and may be used to calculate the threshold power and output power. Chapter 6 proposes several schemes for all-wave fiber transmission systems, including doubly doped fiber amplifiers such as erbium-thulium-co-doped and erbium-praseodymium-co-doped fiber amplifiers and triply doped fiber amplifiers such as erbium-thuliumpraseodymium-co-doped fiber amplifiers, and presents their numerical models and calculates the dependence of gains at different wavelength on fiber and pump parameters. Chapter 7 introduces the spectral conversion mechanisms of multi-rare-earth-codoped glass waveguides including the setting up of rate-equations and powerpropagation equations model of several co-doped systems. These kinds of photonic glass waveguides are simple applications of spectral downconversion and quantum cutting for enhancing performance of c-Si solar cells. The power-conversion efficiency and quantum conversion efficiency of the co-doped systems are analyzed and the enhanced performances of a sc-Si solar cell model are evaluated. Chapter 8 establishes photonic glass waveguide systems for white-light generation and presents their numerical models. White-light generation has important applications in lighting and display areas. In this chapter, the energy level, electron transition process, and numerical models are proposed and the fluorescence intensity of the system is calculated. Optimal active concentrations are proposed to enable system to emit red, blue, and green light, which are mixed to generate white light. Some topics in this book appear as color reprints of authors’ published articles taken, with permission, from various journals, including Journal of Solid State Chemistry, Journal of Luminescence, Material Letters, IEEE Journal of Quantum Electronics, Journal of Optics Society of America, Applied Physics B, IEEE Photonics Journal, and Applied Optics, and so on. Finally, the writing of this book would not have been possible without the help and encouragement of our many colleagues. It is our great pleasure to thank these people. Especial thanks are given to Mr. Wenbin Xu for his contribution to the part of Chapter 8. Special thanks are given to Ms. Li Jin for her contribution to the part of numerical technique, and to Ms. Wenhui Xu and Mr. Yaming Lin for their contribution to the parts of Chapter 6. We also thank our editors for their encouragement of this project. We thank you, the reader, for your time and effort spent reading this book. Though we tried to cleanse this text of conceptual and typographical errors, we apologize in advance for those that have slipped through. No book is perfect and we can only improve the text with your comments and suggestions.

1

Fundamental Mathematics of Nonlinear-Emission Photonic Glass Fiber and Waveguide Devices

1.1

Introduction In the design and analysis of nonlinear-emission photonic glass fiber and waveguide devices, one of the most important aspects is to solve the multi-variable rate equation and power-propagation equation groups. In many cases, the equation groups are nonlinear. Currently, there is no commercial software that can be directly used to solve multi-variable nonlinear equation groups. The methods introduced in this chapter include the Newton iteration algorithm and Runge–Kutta algorithm for the initialvalue problem and the algorithm for the two-point boundary problem, which are effective numerical techniques for highly doped and co-doped fiber amplifiers and fiber laser systems as well as photonic glass waveguide systems for spectral conversion and white-light generation.

1.2

Newton Iteration Algorithm for Nonlinear Rate Equation Solution

1.2.1

Single-Variable [1] A nonlinear equation group can be linearized into a linear equation group. At each iteration step, a new linear equation group can be obtained and solved, this method is called linearization method. For example, for a two-dimensional nonlinear population rate equation group: dN 1 ¼ aN 31 þ bN 1 N 2 þ cN 22 þ d dt dN 2 ¼ aN 31  bN 1 N 2  cN 22  d; dt

ð1:1Þ

N1 þ N2 ¼ N where N1 and N2 are the population numbers at the lower level and upper level of a twolevel laser system with ion–ion nonlinear interaction, respectively. In steady condition, dN1/dt = dN2/dt = 0, the equation group is nonlinear, its solution has no analytical form. Let us consider the solution of N1. Equation (1.1) is simplified as Equation (1.2), where x represents N1: f ð xÞ ¼ 0

(1.2) 1

2

Fundamental Mathematics of Nonlinear-Emission Photonic Glass Fiber and Waveguide Devices

If x* is the true root of Equation (1.2) and x0 is an approximation of x*, then the linear function through the point ðx0 ; f ðx0 ÞÞis: LðxÞ ¼ a0 ðx  x0 Þ þ f ðx0 Þ

(1.3)

0

where a0 ¼ f ðx0 Þ 6¼ 0. If LðxÞ  f ðxÞ, then the root of Equation (1.2) can be approximately replaced with the root of LðxÞ ¼ 0, and the new approximate root x1 is: 1 x1 ¼ x0  f ð x0 Þ (1.4) a0 The linear function through the point ðx1 ; f ðx1 ÞÞ is: LðxÞ ¼ a1 ðx  x1 Þ þ f ðx1 Þ

(1.5)

If LðxÞ  f ðxÞ, then the root of Equation (1.1) can be approximately replaced with the root of LðxÞ ¼ 0, and the new approximate root x2 is: 1 x2 ¼ x1  f ð x1 Þ (1.6) a1 0

where a1 ¼ f ðx1 Þ 6¼ 0. Generally: xkþ1 ¼ xk 

1 f ðxk Þ, ak

ak 6¼ 0,

k ¼ 0, 1, . . .

(1.7)

0

where ak ¼ f ðxk Þ 6¼ 0. Iteration Equation (1.7) is a linearization method for solving Equation (1.2); its geometric idea is to plot a line through the point ðxk ; f ðxk ÞÞ, and the crossing point of the line with the x-axis is considered the new approximation of the root of Equation (1.2), as shown in Figure 1.1. Actually, with different ak , a different iteration method can be obtained. The method 0 with a ¼ f ðx0 Þ is called the Newton iteration algorithm. If we plot a tangent of y ¼ f ðxÞ 0 at the point ðxk ; f ðxk ÞÞ and let xkþ1 be a root of LðxÞ ¼ f ðxk Þðx  xk Þ þ f ðxk Þ ¼ 0, then: xkþ1 ¼ xk 

f ðxk Þ , k ¼ 0, 1, 2, . . . 0 f ð xk Þ

(1.8)

When k is sufficiently large, the root xkþ1 will approach the true root x∗ of Equation (1.2).

Figure 1.1 Finding procedure of root of nonlinear equation using linearization method.

3

1.2 Newton Iteration Algorithm for Nonlinear Rate Equation Solution

1.2.2

Multi-Variable For the equation with n variables x1 , x2 , . . . , xn : f ðx1 ; x2 ; . . . ; xn Þ ¼ 0

(1.9)

the solution of the single-variable Equation(1.2) can be used to solve the n-variable ∗ ∗ equation. It is assumed that x∗ (1.9), and 1 ; x2 ; . . . ; xn are the true roots of Equation  ∗ ∗ ðx10 ; x20 ; . . . ; xn0 Þ are approximations of the true roots x∗ ; x ; . . . ; x . For simplifica1 2 n tion, let us use X to represent the n-dimensional variables ðx1 ; x2 ; . . . ; xn Þ, and use X 0 to represent ðx10 ; x20 ; . . . ; xn0 Þ and X k to represent ðx1k ; x2k ; . . . ; xnk Þ. The linear function through ðX 0 ; f ðX 0 ÞÞ is: Lk ðX Þ ¼ Ak ðX  X k Þ þ f ðX k Þ

(1.10)

where Ak is an n-order nonsingular matrix; if Lk ðX k Þ  f ðX k Þ, one can use the solution of linear equation group (Equation 1.10) as the approximation of the true root of Equation (1.9). Let: Lk ðX Þ ¼ Ak ðX  X k Þ þ f ðX k Þ ¼ 0

(1.11)

Then X ¼ X kþ1 is a new approximation of the true root of Equation (1.9), that is: X kþ1 ¼ X k 

f ðX k Þ , k ¼ 0, 1, 2, . . . Ak

(1.12)

This procedure becomes linear iteration method of the nonlinear equation group f ðX Þ ¼ 0. Generally, a different iteration method with different Ak is used. The Newton 0 0 iteration method uses A0 ¼ f ðX 0 Þ, Ak ¼ f ðX k Þ and for f ðX Þ ¼ 0: X kþ1 ¼ X k 

f ðX k Þ , k ¼ 0, 1, 2, . . . 0 f ðX k Þ

(1.13)

This method is a simplified Newton method.

Figure 1.2 Finding procedure of root of multi-variable nonlinear equation using Newton iteration

method.

4

Fundamental Mathematics of Nonlinear-Emission Photonic Glass Fiber and Waveguide Devices

1.3

Runge–Kutta Algorithm for Power-Propagation Equation Solution

1.3.1

Single-Function [2] The Runge–Kutta algorithm is used to numerically solve differential equations. It is known to be very accurate for a wide range of problems. Consider a single-variable problem: dy ¼ f ðx; yÞ, a  x  b dx

(1.14)

y ð aÞ ¼ y 0 with initial condition yðaÞ ¼ y0 , and suppose that yn is the value of function at the variable xn . The Runge–Kutta formula takes yn and xn to calculate an approximation of ynþ1 at xnþh by using a weighted average of approximate values of f ðx; yÞ within the interval (xn , xnþh ). yðxÞ can be expanded using Taylor series: 0

yðxÞ ¼ yðx0 Þ þ ðx  x0 Þy ðx0 Þ þ 0

ðx  x0 Þ2 00 ðx  x0 Þ3 000 y ð x0 Þ þ y ð x0 Þ þ . . . 2 6

00

(1.15)

000

where y ðxÞ ¼ dy=dx, y ðxÞ ¼ d2 y=dx2 , y ðxÞ ¼ d3 y=dx3 . Equation (1.14) is inserted into Equation (1.15) and Equation (1.16) is obtained: 0

yðxÞ  yðx0 Þ þ ðx  x0 Þy ðx0 Þ ¼ yðx0 Þ þ ðx  x0 Þ f ðx0 ; y0 Þ

(1.16)

The numerical approximation of Equation (1.16) is: y1 ¼ y0 þ hf ðx0 ; y0 Þ

(1.17)

where h ¼ x1  x0 is a step length, y2 ¼ y1 þ hf ðx1 ; y1 Þ, and generally: yk ¼ yk1 þ hf ðxk1 ; yk1 Þ

(1.18)

Thus, yk can be obtained from y0 and f ðx0 ; y0 Þ, and this method is called the Euler method. According to the mean value theorem of differentials: 0

yðxkþ1 Þ  yðxk Þ ¼ y ðξ Þðxkþ1  xk Þ

(1.19)

Equation (1.20) can be obtained from Equation (1.18): yðxkþ1 Þ ¼ yðxk Þ þ hf ðξ; yðξ ÞÞ

(1.20)

where h ¼ xkþ1  xk , f ðξ; yðξ ÞÞ ¼ S∗ is the mean slope of yðxÞ in the range ½xk ; xkþ1 . If f ðxk ; yðxk ÞÞ  f ðxk ; yk Þ ¼ S1 is an approximation of S∗ , then ykþ1 ¼ yk þ hS1 , and this is the first-order Euler equation. Assume that f ðxkþ1 ; yðxkþ1 ÞÞ  f ðxk þ h; yk þ hf ðxk ; yk ÞÞ ¼ S2 , and the weighted average values of S1 and S2 are used as the approximation of S*, then:

1.3 Runge–Kutta Algorithm for Power-Propagation Equation Solution

1 ykþ1 ¼ yk þ hðS1 þ S2 Þ, 2 S1 ¼ f ðxk ; yk Þ,

5

ð1:21Þ

S2 ¼ f ðxk þ h; yk þ hS1 Þ This is the second-order Euler equation. Generally, if there are m slope values, such as S1 , S2 , S3 , . . . , Sm in the range ½xk ; xkþ1 , then: ykþ1 ¼ yk þ hðα1 S1 þ α2 S2 þ . . . þ αm Sm Þ, S1 ¼ f ðxk ; yk Þ, S2 ¼ f ðxk þ β2 h; yk þ γ2 hÞ

(1.22)

... Sm ¼ f ðxk þ βm h; yk þ γm hÞ This is a general form of the Runge–Kutta algorithm. Where 0  λk  1, yk þ γk h is an approximate value of yðxk þ βm hÞ, and αk , βk , γk are undetermined coefficients. For the third-order Runge–Kutta algorithm: 1 ykþ1 ¼ yk þ ðS1 þ 4S2 þ S3 Þh 6 S1 ¼ f ð x k ; y k Þ   (1.23) 1 1 S2 ¼ f xk þ h; yk þ S1 h 2 2 S3 ¼ f ðxk þ h; yk  S1 h þ 2S2 hÞ For the fourth-order Runge–Kutta algorithm: 1 ykþ1 ¼ yk þ ðS1 þ 2S2 þ 2S3 þ S4 Þh 6 S1 ¼ f ð x k ; y k Þ   1 1 S2 ¼ f xk þ h; yk þ S1 h 2 2   1 1 S3 ¼ f xk þ h; yk þ S2 h 2 2

(1.24)

S4 ¼ f ðxk þ h; yk þ S3 hÞ To run the simulation, it can be started with y0 , and y1 can be found using the above formula, then y1 is plugged in to find y2 , and so on. To solve the power-propagation equations of rare-earth-doped fiber and waveguide systems, y represents pump power, signal power, or amplified spontaneous emission power, and x represents the propagation distance z.

6

Fundamental Mathematics of Nonlinear-Emission Photonic Glass Fiber and Waveguide Devices

1.3.2

Multi-Functions With multiple functions, the Runge–Kutta algorithm is similar to the above equations, except that the functions become a vector. Suppose there are m functions y1 , y2 , . . . , ym , each of which varies with same variable x. Suppose further that there are m coupled differential equations for these m functions: dy1 ¼ f 1 ðx; y1 ; y2 ; . . . ; ym Þ dx dy2 ¼ f 2 ðx; y1 ; y2 ; . . . ; ym Þ dx ..................... dym ¼ f n ðx; y1 ; y2 ; . . . ; ym Þ dx

ð1:25Þ

Note there are no derivatives on the right-hand side of those equations, and there are only first derivatives on the left-hand side. These equations can be summarized in vector form as: dY ¼ F ðx; Y Þ dx

(1.26)

where Y ¼ ðy1 ; y2 ; . . . ; ym Þ. Next, let us label the states ym , ymþ1 , which are separated by the interval h of variable x. That is, ym is the value of the function at the variable xm , and y1m is the value of the first function y1 at xm : Y m ¼ ðy1, m , y2, m , . . . , ym, m Þ

(1.27)

Y mþ1 ¼ ðy1, mþ1 ; y2, mþ1 . . . , ym, mþ1 Þ

(1.28)

To compute the state at a short length h and put the results into xm+1, the fourth-order Runge–Kutta algorithm does the following [2]: h Y mþ1 ¼ Y m þ ðS1, m þ 2S2, m þ 2S3, m þ S4, m Þ 6

(1.29)

where: S1, m ¼ F ðxm ; Y m Þ,   h h S2, m ¼ F xm þ ; Y m þ am , 2 2   h h S3, m ¼ F xm þ ; Y m þ bm , 2 2 S4, m ¼ F ðxm þ h; Y m þ hcm Þ The new vector Ym+1 gives the values after variable x has passed the small length h.

1.4 Two-Point Boundary Problem for Power-Propagation Equations in a Laser Cavity

7

To solve power-propagation equations of rare-earth-co-doped fiber and waveguide systems, Y represents pump power, signal power, and amplified spontaneous emission power and x represents propagation distance z. Matlab codes of the Newton iteration method and Runge–Kutta algorithm are attached in the Appendix.

1.4

Two-Point Boundary Problem for Power-Propagation Equations in a Laser Cavity

1.4.1

Principle [3] In solid and fiber lasers, the pump power, lasing power, and amplified spontaneous emission power-propagation equations form a coupled differential equation group. The pump powers at the two ends of the laser cavity are known and the lasing power at the output end is unknown; thus, this problem is a standard two-point boundary problem. We need to solve a set of m coupled first-order differential equations, meeting m1 boundary conditions at starting point z1 , and another set of m2 ¼ m  m1 boundary conditions at the final point z2 . The differential equation group can be written as follows: dy1 ¼ f 1 ðz; y1 ; y2 ; . . . ; ym Þ dz dy2 ¼ f 2 ðz; y1 ; y2 ; . . . ; ym Þ dz .....................

ð1:30Þ

dym ¼ f m ðz; y1 ; y2 ; . . . ; ym Þ dz At z1 , the solution is supposed to meet: B1i ðz1 ; y1 ; y2 ; . . . ; ym Þ ¼ a1 , a2 , . . . , am1 , i ¼ 1, . . . , m1

(1.31)

At z2 , the solution is supposed to meet: B2j ðz2 ; y1 ; y2 ; . . . ; ym Þ ¼ b1 , b2 , . . . , bm2 , j ¼ 1, . . . , m2

1.4.2

(1.32)

Shooting Method and Relaxation Method [3] Shooting and relaxation methods are usually used to solve the two-point boundary problem. One of methods for two-point boundary problem solution is the method which is used to solve numerically the differential equation of initial value problem and the nonlinear algebraic equation. This method is called shooting method. In the shooting method, we can choose values for all dependent variables at one boundary; these values must be consistent with any boundary conditions. The ordinary differential equations are solved by integrating and arriving at the other boundary by using the initial-value method. In general, the difference from the desired boundary value can be found, then we meet a multidimensional root-finding problem, which can be solved by using the

8

Fundamental Mathematics of Nonlinear-Emission Photonic Glass Fiber and Waveguide Devices

Newton iteration algorithm introduced in the previous section: the variables are adjusted at a boundary point to reduce the difference at the other boundary points. If the differential equations are integrated to follow the trajectory of a shot from gun to a target, then picking the initial conditions corresponds to aiming in Figure 1.3. The shooting algorithm provides a systematic approach to taking a set of ranging shots that allow us to improve our aim systematically.

Figure 1.3 Schematic diagram of shooting algorithm for solving m coupled differential equations. Trial integration meeting the boundary condition at one endpoint is started. The difference from the desired boundary condition at the final endpoint is used to adjust the starting conditions, until the boundary conditions at the two endpoints are finally met.

Figure 1.4 Schematic diagram of relaxation algorithm for solving m coupled differential equations. A set of initial values is guessed that meets the differential equations and boundary conditions. An iteration process is used to adjust the function to make it close the true solution.

References

9

The relaxation algorithm uses a differential approach. The differential equations become finite-difference equations on a mesh of points that covers the range of the integration. A trial solution consists of values for the dependent variable at each mesh point, not meeting the desired finite-difference equations, nor necessarily even meeting the required boundary conditions. The iteration, which is called relaxation, consists of adjusting all values on the mesh in order to bring them into closer agreement with the finite-difference equations and simultaneously with the boundary conditions shown in Figure 1.4. In Matlab, the function bvp4c( ) usually is used to solve the two-point boundary problem described with the equations (1.30)-(1.32). It is assumed that S(z) is an approximate solution of the equations (1.30)-(1.32) and is a continuous function and cubic polynomial at each subrange [zn,zn+1] in the range 0=z0= norm(fx(x, ne,nt,np,p)) w = w/2; xx = w*x0'+x; end X = xx; x1 = fx(x, ne,nt,np,p); x2 = -dfx(x, ne,nt,np,p); x3 = inv(x2); x0 = x3*x1'; end s = x0'+x; return

A1.4

Code for Variation of Gain with Fiber Length Ne = 1e26; Nt = 1e26; Np = 1e26; L = 1; Ps10 = 1e-6; Ps20 = 1e-6; Ps30 = 1e-6; Pase10 = 0; Pase20 = 0; Pase30 = 0; [z,P] = ode45(@fun,[0,L],[Pp10;Pp20; Ps10;Ps20; Ps30; Pase10;Pase20; Pase30], ne,nt,np), 'AbsTol',ne, nt,np);

Appendix 1

223

G1 = 10*log10(P(length(P),3)/ Ps10); G2 = 10*log10(P(length(P),4)/ Ps20; G3 = 10*log10(P(length(P),5)./ Ps30); plot(z,G1,z,G2,z,G3); function f = fun(z,p,ne,nt,np); n = NewtonIterate ([1 1 1 1 1 1 1 1 . . .. . .. F = [(Gama1 * (n(j1)*sigma_j1i1 alpha1) * P(1); (Gama2 * (n(j2) * sigma_j2i2 - n(i2) * P(2); (Gama3 * (n(j3) * sigma_j3i3 - n(i3) * P(3); (Gama4 * (n(j4) * sigma_j4i4 - n(i4) * P(4); (Gama5 * (n(j5) * sigma_j5i5 - n(i5) * P(5); (Gama6 * (n(j6) * sigma_j6i6 - n(i6) * P(6) +2*dltav * hvs1 * n(j6)*sigma_j6i6; (Gama7 * (n(j7) * sigma_j7i7 - n(i7) * P(7) + 2*dltav * hvs2 * n(j7) * sigma_j7i7; (Gama8 * ((n(j8) * sigma_j8i8 alpha8) * P(8) + 2*dltav * hvs3 * n(j8) * sigma_j8i8 ];

A1.5

1]*1e26,ne,nt,np,p); n(i1) * sigma_i1j1) * sigma_i2j2) - alpha2) * sigma_i3j3) - alpha3) * sigma_i4j4) - alpha4) * sigma_i5j5) - alpha5) * sigma_i6j6) - alpha6)

* sigma_i7j7) - alpha7)

n(i8) * sigma_i8j8) -

Code for Variation of Gain with Active Ion Concentration ne = [0:1:30] * 1e26; nt = 1e26; np = 1e26; len = length(ne); L = 1; Ps10 = 1e-6; Ps20 = 1e-6; Ps30 = 1e-6; Pase10 = 0; Pase20 = 0; Pase30 = 0; G1 = zeros(1,len); G2 = zeros(1,len);

224

Appendix 1

G3 = zeros(1,len); for count = 1:len [z,P] = ode45(@fun,[0,L],[Pp10;Pp20; Ps10;Ps20; Pase10;Pase20; Pase30], 'AbsTol', ne(count), nt,np); G1(1,count) = 10 * log10(P(length(P),3)./Ps10); G2(1,count) = 10 * log10(P(length(P),4)./Ps20); G3(1,count) = 10 * log10(P(length(P),5)./Ps30); end

Ps30;

plot(ne,G1,ne,G2,ne,G3); function f = fun(z,p,ne,nt,np); n = NewtonIterate ([1 1 1 1 1 1 1 1 1] * 1e26,ne,nt,np,p); . . . . . .. f = [(Gama1 * (n(j1) * sigma_j1i1 - n(i1) * sigma_i1j1) alpha1)* P(1); (Gama2 * (n(j2) * sigma_j2i2 - n(i2) * sigma_i2j2) - alpha2) * P(2); (Gama3 * (n(j3) * sigma_j3i3 - n(i3) * sigma_i3j3) - alpha3) * P(3); (Gama4 * (n(j4) * sigma_j4i4 - n(i4) * sigma_i4j4) - alpha4) * P(4); (Gama5 * (n(j5) * sigma_j5i5 - n(i5) * sigma_i5j5) - alpha5) * P(5); (Gama6 * (n(j6) * sigma_j6i6 - n(i6) * sigma_i6j6) - alpha6) * P(6) + 2*dltav * hvs1 * n(j6) * sigma_j6i6; (Gama7 * (n(j7) * sigma_j7i7 - n(i7) * sigma_i7j7) - alpha7) * P(7) +2*dltav * hvs2 * n(j7) * sigma_j7i7; (Gama8 * ((n(j8) * sigma_j8i8 - n(i8) * sigma_i8j8) alpha8) * P(8) + 2*dltav * hvs3 * n(j8) * sigma_j8i8 ];

Appendix 2 Matlab Code for Solving PowerPropagation Equations of a Laser Cavity with Four-Level System

% Code of solution of coupled differential equation group by using approach to two-point % boundary value problem: function Pout = fiberlaser global R1 R2 Ppa Ppb sgmpa sgmpe sgmsa sgmse gms gmp N ap as Psst Ppst mu kapa elta bta Pth % parameter value lambdas ; lambdap; tau; sgmpa; sgmpe; sgmsa; sgmse; Ac; N; ap; as; gms; gmp; R1; R2; L; c = 3e8; h = 6.626e-34; fs = c/lambdas; fp = c/lambdap; Psst = h * fs * Ac/( gms * (sgmse+sgmsa) * tau); Ppst = h * fp * Ac/( gmp * (sgmpe+sgmpa) * tau); bta = (sgmpe + sgmpa) * gmp /((sgmse + sgmsa) * gms) * ((N * gms * sgmsa + as) * L + log (1 / ((R1 * R2)^0.5))) (N * gmp * sgmpa + ap) * L; Pth = ((N * gms * sgmsa + as) * L + log(1/((R1 * R2)^0.5)))/ (1 - exp(bta)) * fs / fp * Psst; Ppa = 0:1:10; Ppb = 0; len = length(Ppa); for I = 1:len Ppa = (I - 1)*1; OPTION = bvpset('Stats','ON'); solinit = bvpinit(linspace(0,L,10),[Ppa Ppb 50 Ppb]); sol = bvp4c(@f,@fsbc,solinit); y = [sol.y]; x = [sol.x]; Pout (i) = y(3,end)*(1 - R2); end 225

226

Appendix 2

Ppa = 0:1:10; figure (1); set (findall (gcf,'type','line'),'linewidth',2) plot (Ppa,((Pout-Pth)+abs(Pout-Pth)) * 0.5,'k-*'); set (findall (gcf,'type','line'),'linewidth',2) xlabel ('pump power (W)'); ylabel ('laser power (W)') figure (2); subplot(1,1,1) plot(x,y(1,:),'r.-',x,y(2,:),'b*-',x,y(3,:),'g',x,y (4,:),'m-'); title ('Pump and laser powers'); legend ('Pp + (z)','Pp - (z)','Ps + (z)','Ps - (z)'); xlabel ('Position z (m)'); ylabel ('Power (W)'); % function for rate and power-propagation equations function dy = f(x,y) global sgmpa sgmpe sgmsa sgmse gms gmp N ap as Psst Ppst dy = zeros(4,1); % parameter value input ap; as; r; c = 3e8; lambdas; lambdap; c = 3e8; fs = c/lambdas; fp = c/lambdap; h = 6.626e-34; N3a = N * (1 – 2 * r) * (sgmpa /(sgmpa+sgmpe) * (y(1) + y(2)) / Ppst + sgmsa / (sgmsa + sgmse)* (y(3) + y(4)) / Psst)/((y (1) + y(2)) / Ppst + 1 + (y(4) + y(3)) / Psst); N3b = N * 2 * r * (sgmpa / (sgmpa + sgmpe)*(y(1) + y(2)) / Ppst + sgmsa / (sgmsa+sgmse) * (y(3) + y(4)) / Psst) / ((y (1) + y(2)) / Ppst + 1 + (y(4) + y(3)) / Psst + (y(1) + y(2)) * ap / (h * fp) + (y(4) + y(3)) * as / (h * fs)); % if ion-ion pairs are not considered N3 = N3a + N3b; % if ion-ion pairs are considered N3 = N3a; dy(1) = (-gmp * (sgmpa * N - (sgmpa + sgmpe) * N3) - ap) * y(1); dy(2) = -(-gmp * (sgmpa * N - (sgmpa + sgmpe) * N3) - ap) * y(2);

Appendix 2

227

dy(3) = (gms * ((sgmsa + sgmse) * N3 – sgmsa * N) - as) * y(3); dy(4) = -(gms * ((sgmsa + sgmse) * N3 – sgmsa * N )-as) * y(4); % boundary condition function res = fsbc(y0,yL) global R1 R2 Ppa Ppb res = [y0(1) - Ppa yL(2) - Ppb y0(3) - R1 * y0(4) yL(4) - R2 * yL(3)];

Index

4f-4f transition, 10 4f-5d transition, 10 absorbed photon, 13 absorption coefficient, 13 absorption cross-section, 12, 13 absorption cross-section peak, 19 absorption cross-section spectrum, 35 absorption loss, 18 absorption spectrum, 22, 60 absorption wavelength, 16 active ion, 16 active ion concentration, 32 all-wave fiber, 91 alumina–silica glass fiber, 65 amplified spontaneous emission power, 5, 7 amplified spontaneous emission power-propagation equation, 7 amplified spontaneous emission (ASE), 67, 87, 181 analytical form, 1 attenuation coefficient, 147 background loss, 67 backlight source, 177 backward lasing power, 80 backward pump power, 80 band-shape luminescence, 10 bit error rate (BER), 141 bismuth-doped silica fiber, 92 Boltzmann constant, 13 Boltzmann effect, 51 borate glass, 16 boro-abnormal phenomenon, 19, 21 boundary condition, 7, 9 cooperative energy transfer (CET), 149 charge carriers, 145 coarse wavelength division multiplexing (CWDM), 91 coherent photon density, 75

228

(CRI), 177 complex core-shell structure, 177 concentration quenching, 16, 53 conservation equation, 67 conversion efficiency, 148 cooperation upconversion, 80 cooperative conversion, 53 cooperative dipole-dipole energy transfer, 145 cooperative luminescence, 59 cooperative upconversion emission probability, 55 co-upconversion coefficient, 96 co-upconversion effect, 74 coupled differential equation, 6, 67 coupled differential equation group, 7 covalence band, 43 color rendering index (CRI), 177 critical angle, 146 cross-relaxation, 57 cross-relaxation coefficient, 67, 150 crystal field, 10, 51 crystal field splitting, 51 crystalline silicon, 145 c-Si semiconductor, 145 c-Si solar cell, 145 different iteration method, 2 differential equation, 4, 67 differential equation group, 7 display system, 177 dopant concentration, 55 doping concentration, 18 double-clad fiber, 80 double-frequency, 54 down-convert, 177 enhancement factor (EF), 147 Einstein coefficient, 74 electric dipole, 10 electric quadrupole, 12 electrical-dipole interaction, 10

Index

electrical-dipole transition, 10 electron mass, 10 emission cross section, 15, 17, 20–1, 28, 34 emission peak wavelength, 28 emission spectrum, 60 emission wavelength, 16 emitted photon, 13 energy conversion efficiency, 145 energy efficiency, 65 energy gap, 13, 51 energy level, 13, 150 energy-level lifetime, 55 energy transfer efficiency, 91 energy-level diagram, 149 energy-level structure, 18 energy-spectrum line, 14 energy transfer (ET), 16, 159 erbium- and ytterbium-co-doped silica-based amplifiers, 65 erbium-doped fiber amplifier, 91 erbium-doped silica-based amplifier, 65 erbium-doped waveguide, 65 Euler equation first-order Euler equation, 4 second-order Euler equation, 5 Euler method, 4 excited level, 17–18 Fabray-Perot cavity, 75 fiber, 1 fiber amplifier, 1, 10 fiber laser, 10, 74 fiber laser system, 1 fiber source, 10 fill-factor, 172 finite-difference equation, 8, 9 fluorescence branch ratio, 10, 13–14 fluorescence branch wavelength, 13 fluorescence effective linewidth, 19 fluorescence emission intensity, 12 fluorescence intensity, 40, 181 fluorescence lifetime, 13, 18–19, 23, 40 fluorescence trap, 60 fluoride glass, 16, 91 fluorophosphate glass, 16 forward lasing power, 80 forward-pump power, 80 Fresnel equation, 146 Fuchbauer-Ladenburger theory, 12 gain coefficient, 18, 74 gain parameter, 18 Gaussian function, 14–15 geometry idea, 2 germanate glass, 45 glass system, 16

ground level, 18 ground state absorption cross section, 77 heavy-metal oxide glass, 91 high-gain, 16, 65 highly-doped material, 65 high-power fiber laser system, 16 homogeneous, 15 incident angular, 146 inhomogeneous, 15 initial condition, 4 initial state, 10 initial value method, 7 input intensity, 11–12 input solar power, 150 integrated absorption area, 19, 21, 24, 28, 34 internal gain, 65 Judd-Ofled theory, 43 Lambert law, 12 laser material, 13, 18 lasing photon energy, 77 lasing photon frequency, 80 lasing power, 7, 77 lasing wavelength, 75 laser diode (LD), 131 least square method, 11 light-emitting diode (LED), 92, 177 line width, 12 linear equation group, 1, 220 linear function, 2 linear transition, 74 linearization method, 1–2 line-shape luminescence, 10 long-range disorder, 12 Lorentz function, 14–15 lower-level, 1 Luminescence, 10 luminescence branch ratio, 13 m coupled first-order differential equations, 7 magnetic-dipole, 11 maximum output power, 172 mean value theorem of differentials, 4 melting-casting method, 16 metastable level, 18 mono-chromatic light, 177 multidimensional root-finding problem, 7 multiple function, 6 multi-variable, 1 multi-variable nonlinear equation, 1 n-dimensional variable, 3 near-infrared (NIR), 91, 159

229

230

Index

near-infrared quantum cutting (NIQC), 145, 159 Nephelauxetic effect, 46 Newton iteration, 220 Newton iteration algorithm, 1–2, 8 Newton iteration method, 7, 96 niobo-silicate, 54 nonlinear, 1 nonlinear equation group, 1 nonlinear interaction, 1 nonlinear-emission, ix, 1 nonradiation transition probability, 13, 45 nonradiation transition rate, 150 nonradiative transition rate, 67 n-order nonsingular matrix, 3 numerical approximation, 4 numerical integration, 67 numerical model, 74 numerical technique, 1 n-variable equation, 3 odd-order term, 10 open-circuit voltage, 172 optical fiber amplifier, 91 optical fiber communication system, 65 optical signal–noise ratio (OSNR), 141–2 optoelectronic device, 10, 177 oscillation strength, 10 output intensity, 11–12 output light power, 150 output power, 74 overlap factor, 80 overlapping factor, 60 oxide glass, 45 parity-prohibition, 10 partition function, 13 PC1D, 152 power conversion efficiency (PCE), 157 power efficiency (PE), 148 phonon energy, 13, 45 phosphate glass, 16, 45 photon intensity, 147 photon number, 74, 147 photon number density spectrum, 148 photonic glass, ix–x, 1, 10, 147–8, 152, 163, 171, 173 photonic glass waveguide, 1 planar spectral converter, 145 planar waveguide amplifier, 65 Planck constant, 10, 80 population inversion, 51, 76 population number, 1, 18 power density spectrum, 148 power-propagation equation, 1, 5 propagation distance, 7 propagation equation, 7

pump absorption coefficient, 65 pump absorption cross section, 18, 55 pump energy fluency, 55 pump intensity, 18 pump mode field, 67 pump photon density, 18 pump photon energy, 77 pump photon frequency, 80 pump power, 5, 7, 68, 77 pump rate, 77 quadratic function, 59 quantum conversion efficiency (QCE), 157 quantum cutting, 145 quantum efficiency (QE), 10, 13, 147–8 quantum-dot-doped fiber, 92 quasi-four-energy-level, 51 radiation lifetime, 13 radiation transition probability, 12–13, 19, 24, 26, 28, 34 Raman amplifier, 91 rare-earth (RE) ion, 10, 146 rare-earth-co-doped fiber system, 7 rare-earth-doped fiber, 92 rare-earth-doped fiber system, 5 rare-earth-doped glass, 10 rare-earth-doped fiber laser, 74 rare-earth-doping, 65 rate equation, 1 rate equation group, 1 reciprocal method, 12–13, 45 red shift, 19 refractive index, 12, 13, 19, 146 Relaxation algorithm, 9 resonant absorption coefficient, 17 resonant ET, 152 Runge–Kutta algorithm, 1, 4–7, 96, 225 fourth-order Runge–Kutta algorithm, 5, 6 third-order Runge–Kutta algorithm, 5 saturation laser power, 85 saturation pump intensity, 23 saturation pump power, 82, 85 scattering loss, 87, 146, 150 semiconductor laser, 16 sensitizer, 16 shooting algorithm, 8 shooting method, 7, 9 short length, 6 short-circuit, 172 short-length, 16, 65 short-range order, 12 signal mode field, 67 signal power, 5, 7 silica fiber, 65

Index

silicate glass, 16, 28, 45 simplified Newton method, 3 single wavelength excitation, 91 single-crystal silicon, 16 single-variable, 3 slop efficiency, 83 solar cell, 16 solar spectrum, 145 spectral conversion, 1 spectral conversion layer, 171 spectral converter, 16, 145 spectral downconverters, 145 spectral intensity function, 14 spectral response, 145 spectral width, 75 spontaneous emission, 150 spontaneous emission rate, 67 spontaneous emission transition probability, 10, 14 spontaneous transition rate, 150 steady condition, 1 steady pump condition, 55 steady state, 18 steady state condition, 83 step length, 4 stimulated emission mechanism, 75 stimulated emission rate, 78 stimulated transition rate, 67 storage energy parameter, 18 sub-level, 51 Taylor series, 59 telluride glass, 16 temperature, 13

terminal state, 10 threshold power, 18 thulium-doped fiber amplifier, 91 total transition probability, 12 transition frequency, 10 transition probability, 11 transition-metal ions, 10 trial solution, 9 two-dimensional nonlinear population, 1 two-point boundary problem, 1, 7 undetermined coefficient, 5 unit-length gain, 65 upconversion, 53 upconversion coefficient, 73 upconversion luminescence, 54 upconvert, 177 upper-level, 1 ultraviolet (UV), 159 velocity of light, 10 waveguide, 1 waveguide amplifier, 16, 65 wavelength division multiplexing (WDM), 91 white light, 177 white light generation, 1 white light-emitting diode (W-LED), 194 Ytterbium ion, 16 zero-line energy, 13

231