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Andrew Ellis
Mariia Sorokina
edited by
Optical
Communication Systems
Limits and Possibilities
Published by
Jenny Stanford Publishing Pte. Ltd.
Level 34, Centennial Tower
3 Temasek Avenue
Singapore 039190
Email: editorial@jennystanford.com
Web: www.jennystanford.com
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Optical Communication Systems: Limits and Possibilities
Copyright © 2020 Jenny Stanford Publishing Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any
form or by any means, electronic or mechanical, including photocopying,
recording or any information storage and retrieval system now known or
to be invented, without written permission from the publisher.
For photocopying of material in this volume, please pay a copying fee
through the Copyright Clearance Center, Inc., 222 Rosewood Drive,
Danvers, MA 01923, USA. In this case permission to photocopy is not
required from the publisher.
ISBN 978-981-4800-28-0 (Hardcover)
ISBN 978-0-429-02780-2 (eBook)
Contents
Preface xiii
1. Modelling High-Capacity Nonlinear Transmission
Systems 1
Hadrien Louchet, Nikolay Karelin, and André Richter
1.1 Introduction 1
1.2 Nonlinear Fibre Propagation: From Single to
Multimode 2
1.2.1 Wave Equation 2
1.2.2 Linear Propagat 4
1.2.2.1 Loss 4
1.2.2.2 Chromatic dispersion 5
1.2.2.3 Birefringence 7
1.2.3 Nonlinear Propagation Effects 9
1.2.4 The Scalar Nonlinear Schrödinger
Equation 11
1.2.5 The Manakov-PMD Equation 12
1.2.6 Extension to SDM Systems Using
Multimode Fibre 14
1.3 Solving the Manakov-PMD Equation 17
1.3.1 Signal Representations 17
1.3.2 Numerical Methods 19
1.3.2.1 The split-step (Fourier) method 19
1.3.2.2 Step-size control 22
1.3.2.3 The coarse-step model 24
1.3.3 Simulation Framework for SDM Systems 25
1.4 Accurate Modelling of System-Level Nonlinear
Impairments 29
vi Contents
1.4.1 Self-Phase Modulation 29
1.4.2 Intra-Channel Cross-Phase Modulation
and Four-Wave Mixing 30
1.4.3 Cross-Phase Modulation 31
1.4.4 Four-Wave Mixing 32
1.4.5 Signal-Noise Interaction 33
1.4.6 Cross-Polarization Modulation 34
1.4.7 Stimulated Raman Scattering 36
1.4.8 The Nature of the Nonlinear Interference
Noise 37
1.5 Guidelines for Modelling High-Capacity
Nonlinear Systems 38
1.5.1 Overview of System Performance
Criteria 38
1.5.1.1 Bit-error-rate 39
1.5.1.2 Signal-to-noise ratio 39
1.5.1.3 System penalty and system
margin 40
1.5.1.4 Error-vector magnitude 42
1.5.2 Estimating the Bit-Error-Rate 43
1.5.2.1 Error-counting 43
1.5.2.2 BER estimation techniques 44
1.5.3 Estimating System Average Performance
and Outage Probability 45
1.5.3.1 System-level components
modelling 45
1.5.3.2 Transmission link modelling 47
1.5.3.3 Deterministic propagation 49
1.5.3.4 Modelling stochastic propagation
0
1.6 Summary and Outlook 52
2. Basic Optical Fiber Nonlinear Limits 63
Mohammad Ahmad Zaki Al-Khateeb, Abdallah Ali,
and Andrew Ellis
2.1 Nonlinear Behavior of Optical Fibers 65
vii
Contents
2.1.1 Kerr Nonlinear Effects in a Single-Span
Transmission System 67
2.1.2 Kerr Nonlinear Effects in a Multi-Span
Transmission System 69
2.2 Noise Accumulation Optical Transmission
Systems 74
2.2.1 Total Nonlinear Kerr Noise 75
2.2.2 Total Linear ASE Noise 80
2.2.3 Total Signal-ASE Nonlinear Noise 81
2.3 Performance of Coherently Detected Optical
Transmission Systems 85
3. Fiber Nonlinearity Compensation: Performance
Limits and Commercial Outlook 95
DanishRafique
3.1 Fiber Nonlinearity Compensation 96
3.2 Digital Back Propagation 98
3.2.1 DBP Performance Scaling 103
3.2.2 DBP Performance Limits 107
3.3 Phase Conjugation 110
3.3.1 Pre-Dispersed PC 111
3.3.2 Comparison of Single-Channel DBP and
PC 112
3.4 Commercial Applications and Perspective 114
4. Phase-Conjugated Twin Waves and Phase-Conjugated
Coding 123
Son Thai Le
4.1 Introduction 123
4.2 General Principle 125
4.2.1 Phase-Conjugated Twin Waves 126
4.2.2 Nonlinear Noise Squeezing 128
4.2.3 Connection between NLNS and PCTW 129
4.2.4 Generalized Phase-Conjugated Twin
Waves 131
4.3 Beneit and Limitation of PCTW 132
viii Contents
4.3.1 SNR and Capacity Gain in PCTW-Based
Transmissions 132
4.3.2 Beneit and Application Range of PCTW 135
4.4 Phase-Conjugated Pilot 140
4.4.1 Principle 140
2
4.5 Phase-Conjugated Subcarrier Coding 148
4.5.1 Principle of PCSC 148
4.5.2 Performance of PCSC 151
4.6 Other Variants of PCTW 156
4.6.1 Temporally Multiplexed PCTW 156
4.6.2 Modiied PCTW 158
4.6.3 PCTW for Multimode and Multi-Core
Fibers 159
4.7 Conclusion 159
5. Information-Theoretic Concepts for Fiber Optic
Communications 165
Mariia Sorokina and Metodi P. Yankov
5.1 Communication Channel 165
5.2 Fiber Optic Communications 167
5.3 Shannon Capacity and Mutual Information 170
5.4 Information-Theoretic Channel Modeling 172
5.5 Numerical Calculations of Lower Bounds on
Shannon Capacity 174
5.6 Probabilistic Shaping 176
5.6.1 Optimization for the Optical Fiber
Channel 178
5.6.2 Probabilistic Shaping of Binary Data 181
5.7 Concluding Remarks 185
6. Advanced Coding for Fiber-Optics Communications
Systems 191
Ivan B. Djordjevic
6.1 Introduction 192
6.2 Turbo-Product Codes 193
ix
Contents
6.3 LDPC Codes 197
6.3.1 LDPC Codes Fundamentals and Large-Girth
Code Design 197
6.3.2 Decoding of Binary LDPC Codes 202
6.3.3 Nonbinary LDPC Codes: Quasi-Cyclic Code
Design and Decoding Algorithms 207
6.3.4 Rate-Adaptive LDPC Coding
Implementations in FPGA 210
6.4 Coded Modulation for Optical Communications 215
6.4.1 Coded Modulation Fundamentals 216
6.4.2 Multilevel Coded Modulation and
Unequal Error Protection 219
6.4.3 Bit-Interleaved Coded Modulation 225
6.4.4 Hybrid Multidimensional Coded
Modulation Scheme for High-Speed
Optical Transport 226
6.5 LDPC Coded Modulation for Optical
Communications Enabling Quasi-Single-Mode
Transmission over Transoceanic Distances
Using Few-Mode Fibers 230
6.6 Concluding Remarks 236
7. Nonlinear Fourier Transform-Based Optical
Transmission: Methods for Capacity Estimation 243
Jaroslaw E. Prilepsky, Stanislav A. Derevyanko,
and Sergei K. Turitsyn
7.1 Introduction 243
7.2 Main Model and Basics of NFT 246
7.2.1 Nonlinear Schrödinger Equation 246
7.2.2 NFT Operations 248
7.3 General Expressions for Noise Autocorrelation
Functions inside NF Domain 250
7.3.1 Perturbed Evolution of NF Spectrum 250
7.3.2 Noise Autocorrelation Functions for the
Continuous Part of NF Spectrum 251
7.4 Capacity Estimates for the Nonlinear Inverse
Synthesis NFT-Based Method 253
xContents
7.4.1 NIS Basics and Continuous Input-Output
Channel Model 253
7.4.2 Discrete Input-Output Model 257
7.4.3 Capacity Estimates for WDM/OFDM NIS
Transmission 258
7.4.4 Applicability of Results 261
7.5 Conclusion 262
8. Spatial Multiplexing: Technology 273
Yongmin Jung, Qiongyue Kang, Shaif-ul Alam,
and David J. Richardson
8.1 Introduction 274
8.2 SDM Transmission Fibres 275
8.2.1 Few-Mode Fibres 277
8.2.2 Multi-Core Fibres 279
8.3 SDM Multiplexers and Demultiplexers 281
8.3.1 Mode MUXs/DEMUXs for Few-Mode
Fibres 281
8.3.2 Fan-In/Fan-Out Devices for Multi-Core
Fibres 283
8.4 SDM Optical Ampliiers 284
8.4.1 Strategies to Minimize Differential Modal
Gain in Few-Mode EDFA 286
8.4.2 Core Pumped 6-Mode EDFA 287
8.4.3 Cladding Pumped 6-Mode EDFA 289
8.4.4 Future Prospects to Increase the Number
of Spatial Modes in FM-EDFAs 290
8.5 Conclusion 292
9. Spatial Multiplexing: Modelling 297
Filipe Ferreira, Christian Costa, Sygletos Stylianos,
and Andrew Ellis
9.1 Introduction 298
9.2 Coupled-Mode Theory for Few-Mode Fibers 300
9.2.1 Coupled-Mode Equations 301
9.2.2 Coupled-Mode Equations Solution for
Two-Mode Fibers 303
xi
Contents
9.3 Semi-Analytical Solutions for Higher-Order
Modes 304
9.3.1 Analytical Expressions for the
Three-Modes Case 306
9.3.2 Analytical Expressions for More Than
Three-Modes 306
9.3.3 Algorithm Complexity 308
9.4 Single-Section Modelling 309
9.5 Multi-Section Modelling 312
9.5.1 Setting Mode Coupling Strength and
Correlation Length 312
9.5.2 Mode Coupling Accumulation over
Transmission Length 314
9.5.3 Polarization Mode Coupling 315
9.6 GD Statistics in Non-Delay-Managed Links 316
9.6.1 GD Standard Deviation and Intensity
Impulse Response 317
9.6.2 GD Probability Density Function and
Maximum GD Spread 320
9.7 GD Statistics in Delay-Managed Links 323
9.8 Nonlinear Propagation Modelling 326
9.8.1 Modiied Split-Step Fourier Method 327
9.8.2 Extreme Coupling Strength Regimes 328
9.8.3 Intermediate Coupling Strength Regime 328
9.8.4 Total Nonlinear Noise: Analytical
Integration 329
9.9 Linear Coupling Impact Nonlinear Noise for
Delay Uncompensated Spans 331
9.10 Linear Coupling Impact on Nonlinear Noise for
Delay Compensated Spans 333
9.11 Manakov Approximation vs. Fully Stochastic
Propagation 336
9.12 Conclusions 342
Index 347
Preface
transmission from dense ultrashort cables in data-centers to
transoceanic distances around the globe, connecting billions of
users and linking cities, countries and continents. It is hard to
overstate the impact
economy, healthcare, public and government services, society, and
almost every aspects of our lives.
online services brings about the escalating pressure on the speed
(capacity) and quality (bit error rate) characteristics of information
for these bandwidth-hungry online services include cloud
computing, on-demand HD video streams, online business analytics
arising from data-center applications, the Internet of Things, and
various other broadband services. It is well recognized nowadays
systems are quickly approaching the limits of current transmission
technologies, many of which were originally developed for
communication over linear channels (e.g., radio). However, optical
Nonlinear effects
Unlike wireless communications, where signal quality can be
enhanced by increasing the optical power at the transmitter, in
signal impairments. Consequently, there is a clear need for the
development
This book gives an overview of the current research by
Nikolay Karelin, and André Richter from VPIphotonics, covers
key requirements and challenges for accurate modeling of
xiv
Chapter 2, Mohammad Ahmad Zaki Al-Khateeb, Abdallah Ali, and
Andrew Ellis from Aston University discuss theoretical models
that can predict the maximum performance and discuss optical
nonlinearity compensation. Chapter 4 is focused on the method
of phase-conjugated twin waves and phase-conjugated coding,
presented by Son Thai Le from Nokia Bell-Labs.
The information-theoretic treatment, focused on channel
models and their limits for estimating transmission throughput
and Metodi P. Yankov from Aston University and Technical
University of Denmark, respectively. In Chapter 6, Ivan B. Djordjevic
from the University of Arizona reviews coding algorithms for
distance. The nonlinear Fourier transform method is described in
Chapter 7 by Jaroslaw E. Prilepsky, Stanislav A. Derevyanko, and
Sergei K. Turitsyn from Aston University and Ben-Gurion University
of the Negev, who discuss challenges and advantages of its
application on communication systems.
Finally, spatial multiplexing is considered here (i) from the
technology perspective in Chapter 8 by Yongmin Jung, Qiongyue
Kang, Shaif-ul Alam, and David J. Richardson from the University
of Southampton, who discuss the prospects of scaling as well
as potential energy and cost savings of this technology and
(ii) from the modeling perspective by Filipe Ferreira, Christian
Costa, Sygletos Stylianos, and Andrew Ellis from Aston University,
To conclude, the book presents a broad overview of the
and prospects.
We wish to thank the authors and the Jenny Stanford
Preface
