Optical fiber communications__principles_and_practice__3rd_edition_john_m_senior
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Optical fiber communications__principles_and_practice__3rd_edition_john_m_senior Optical fiber communications__principles_and_practice__3rd_edition_john_m_senior Presentation Transcript

  • Optical Fiber Communications Optical Fiber Communications Principles and Practice Third Edition JOHN M. SENIOR This highly successful book, now in its third edition, has been extensively updated to include both new developments and improvements to technology and their utilization within the optical fiber global communications network. The third edition, which contains an additional chapter and many new sections, is now structured into 15 chapters to facilitate a logical progression of the material, to enable both straightforward access to topics and provide an appropriate background and theoretical support. Key features • An entirely new chapter on optical networks, incorporating wavelength routing and optical switching networks • A restructured chapter providing new material on optical amplifier technology, wavelength conversion and regeneration, and another focusing entirely on integrated optics and photonics • Many areas have been updated, including: low water peak and high performance single-mode fibers, photonic crystal fibers, coherent and particularly phase-modulated systems, and optical networking techniques • Inclusion of relevant up-to-date standardization developments • Mathematical fundamentals where appropriate • Increased number of worked examples, problems and new references Third Edition Professor John Senior is Pro Vice-Chancellor for Research and Dean of the Faculty of Engineering and Information Sciences at the University of Hertfordshire, UK. This third edition of the book draws on his extensive experience of both teaching and research in this area. CVR_SENI6812_03_SE_CVR.indd 1 Cover image © INMAGINE www.pearson-books.com JOHN M. SENIOR This new edition remains an extremely comprehensive introductory text with a practical orientation for undergraduate and postgraduate engineers and scientists. It provides excellent coverage of all aspects of the technology and encompasses the new developments in the field. Hence it continues to be of substantial benefit and assistance for practising engineers, technologists and scientists who need access to a wide-ranging and up-to-date reference to this continually expanding field. Optical Fiber Communications Principles and Practice Third Edition JOHN M. SENIOR 5/11/08 15:40:38
  • OPTF_A01.qxd 11/6/08 10:52 Page i Optical Fiber Communications
  • OPTF_A01.qxd 11/6/08 10:52 Page ii We work with leading authors to develop the strongest educational materials in engineering, bringing cutting-edge thinking and best learning practice to a global market. Under a range of well-known imprints, including Prentice Hall, we craft high quality print and electronic publications which help readers to understand and apply their content, whether studying or at work. To find out more about the complete range of our publishing, please visit us on the World Wide Web at: www.pearsoned.co.uk
  • OPTF_A01.qxd 11/6/08 10:52 Page iii Optical Fiber Communications Principles and Practice Third edition John M. Senior assisted by M. Yousif Jamro
  • OPTF_A01.qxd 11/6/08 10:52 Page iv Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: www.pearsoned.co.uk First published 1985 Second edition 1992 Third edition published 2009 © Prentice Hall Europe 1985, 1992 © Pearson Education Limited 2009 The right of John M. Senior to be identified as author of this work has been asserted by him in accordance with the Copyright, Designs and Patents Act 1988. All rights reserved. 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 or otherwise, without either the prior written permission of the publisher or a licence permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, Saffron House, 6–10 Kirby Street, London EC1N 8TS. All trademarks used herein are the property of their respective owners. The use of any trademark in this text does not vest in the author or publisher any trademark ownership rights in such trademarks, nor does the use of such trademarks imply any affiliation with or endorsement of this book by such owners. ISBN: 978-0-13-032681-2 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data Senior, John M., 1951– Optical fiber communications : principles and practice / John M. Senior, assisted by M. Yousif Jamro. — 3rd ed. p. cm. Includes bibliographical references and index. ISBN-13: 978-0-13-032681-2 (alk. paper) 1. Optical communications. 2. Fiber optics. I. Jamro, M. Yousif. II. Title. TK5103.59.S46 2008 621.382′75—dc22 2008018133 10 12 9 8 7 6 11 10 09 5 4 08 3 2 1 Typeset in 10/12 Times by 35 Printed and bound by Ashford Colour Press Ltd, Gosport The publisher’s policy is to use paper manufactured from sustainable forests.
  • OPTF_A01.qxd 11/6/08 10:52 Page v To Judy and my mother Joan, and in memory of my father Ken
  • OPTF_A01.qxd 11/6/08 10:52 Page vi
  • OPTF_A01.qxd 11/6/08 10:52 Page vii Contents Preface Acknowledgements List of symbols and abbreviations Chapter 1: Introduction 1.1 1.2 1.3 Historical development The general system Advantages of optical fiber communication References Chapter 2: Optical fiber waveguides 2.1 2.2 2.3 2.4 2.5 Introduction Ray theory transmission 2.2.1 Total internal reflection 2.2.2 Acceptance angle 2.2.3 Numerical aperture 2.2.4 Skew rays Electromagnetic mode theory for optical propagation 2.3.1 Electromagnetic waves 2.3.2 Modes in a planar guide 2.3.3 Phase and group velocity 2.3.4 Phase shift with total internal reflection and the evanescent field 2.3.5 Goos–Haenchen shift Cylindrical fiber 2.4.1 Modes 2.4.2 Mode coupling 2.4.3 Step index fibers 2.4.4 Graded index fibers Single-mode fibers 2.5.1 Cutoff wavelength 2.5.2 Mode-field diameter and spot size 2.5.3 Effective refractive index xix xxiii xxxii 1 1 5 7 10 12 12 14 14 16 17 20 24 24 26 28 30 35 35 35 42 43 46 54 59 60 61
  • OPTF_A01.qxd 11/6/08 10:52 Page viii viii Contents 2.6 2.5.4 Group delay and mode delay factor 2.5.5 The Gaussian approximation 2.5.6 Equivalent step index methods Photonic crystal fibers 2.6.1 Index-guided microstructures 2.6.2 Photonic bandgap fibers Problems References Chapter 3: Transmission characteristics of optical fibers 3.1 3.2 3.3 Introduction Attenuation Material absorption losses in silica glass fibers 3.3.1 Intrinsic absorption 3.3.2 Extrinsic absorption 3.4 Linear scattering losses 3.4.1 Rayleigh scattering 3.4.2 Mie scattering 3.5 Nonlinear scattering losses 3.5.1 Stimulated Brillouin scattering 3.5.2 Stimulated Raman scattering 3.6 Fiber bend loss 3.7 Mid-infrared and far-infrared transmission 3.8 Dispersion 3.9 Chromatic dispersion 3.9.1 Material dispersion 3.9.2 Waveguide dispersion 3.10 Intermodal dispersion 3.10.1 Multimode step index fiber 3.10.2 Multimode graded index fiber 3.10.3 Modal noise 3.11 Overall fiber dispersion 3.11.1 Multimode fibers 3.11.2 Single-mode fibers 3.12 Dispersion-modified single-mode fibers 3.12.1 Dispersion-shifted fibers 3.12.2 Dispersion-flattened fibers 3.12.3 Nonzero-dispersion-shifted fibers 64 65 71 75 75 77 78 82 86 87 88 90 90 91 95 95 97 98 98 99 100 102 105 109 110 113 113 114 119 122 124 124 125 132 133 137 137
  • OPTF_A01.qxd 11/6/08 10:52 Page ix Contents ix 3.13 Polarization 3.13.1 Fiber birefringence 3.13.2 Polarization mode dispersion 3.13.3 Polarization-maintaining fibers 3.14 Nonlinear effects 3.14.1 Scattering effects 3.14.2 Kerr effects 3.15 Soliton propagation Problems References Chapter 4: Optical fibers and cables 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 Introduction Preparation of optical fibers Liquid-phase (melting) techniques 4.3.1 Fiber drawing Vapor-phase deposition techniques 4.4.1 Outside vapor-phase oxidation process 4.4.2 Vapor axial deposition (VAD) 4.4.3 Modified chemical vapor deposition 4.4.4 Plasma-activated chemical vapor deposition (PCVD) 4.4.5 Summary of vapor-phase deposition techniques Optical fibers 4.5.1 Multimode step index fibers 4.5.2 Multimode graded index fibers 4.5.3 Single-mode fibers 4.5.4 Plastic-clad fibers 4.5.5 Plastic optical fibers Optical fiber cables 4.6.1 Fiber strength and durability Stability of the fiber transmission characteristics 4.7.1 Microbending 4.7.2 Hydrogen absorption 4.7.3 Nuclear radiation exposure Cable design 4.8.1 Fiber buffering 4.8.2 Cable structural and strength members 140 141 144 147 151 151 154 155 158 163 169 169 170 171 172 175 176 178 180 181 182 183 184 185 187 190 191 194 195 199 199 200 201 203 203 204
  • OPTF_A01.qxd 11/6/08 10:52 Page x x Contents 4.8.3 Cable sheath, water barrier and cable core 4.8.4 Examples of fiber cables Problems References Chapter 5: Optical fiber connections: joints, couplers and isolators 5.1 5.2 5.3 5.4 5.5 5.6 5.7 Introduction Fiber alignment and joint loss 5.2.1 Multimode fiber joints 5.2.2 Single-mode fiber joints Fiber splices 5.3.1 Fusion splices 5.3.2 Mechanical splices 5.3.3 Multiple splices Fiber connectors 5.4.1 Cylindrical ferrule connectors 5.4.2 Duplex and multiple-fiber connectors 5.4.3 Fiber connector-type summary Expanded beam connectors 5.5.1 GRIN-rod lenses Fiber couplers 5.6.1 Three- and four-port couplers 5.6.2 Star couplers 5.6.3 Wavelength division multiplexing couplers Optical isolators and circulators Problems References Chapter 6: Optical sources 1: the laser 6.1 6.2 Introduction Basic concepts 6.2.1 Absorption and emission of radiation 6.2.2 The Einstein relations 6.2.3 Population inversion 6.2.4 Optical feedback and laser oscillation 6.2.5 Threshold condition for laser oscillation 206 207 212 213 217 217 219 222 230 233 234 236 241 243 244 247 249 251 254 256 259 264 269 280 283 287 294 294 297 297 299 302 303 307
  • OPTF_A01.qxd 11/6/08 10:52 Page xi Contents xi 6.3 Optical emission from semiconductors 6.3.1 The p–n junction 6.3.2 Spontaneous emission 6.3.3 Carrier recombination 6.3.4 Stimulated emission and lasing 6.3.5 Heterojunctions 6.3.6 Semiconductor materials 6.4 The semiconductor injection laser 6.4.1 Efficiency 6.4.2 Stripe geometry 6.4.3 Laser modes 6.4.4 Single-mode operation 6.5 Some injection laser structures 6.5.1 Gain-guided lasers 6.5.2 Index-guided lasers 6.5.3 Quantum-well lasers 6.5.4 Quantum-dot lasers 6.6 Single-frequency injection lasers 6.6.1 Short- and couple-cavity lasers 6.6.2 Distributed feedback lasers 6.6.3 Vertical cavity surface-emitting lasers 6.7 Injection laser characteristics 6.7.1 Threshold current temperature dependence 6.7.2 Dynamic response 6.7.3 Frequency chirp 6.7.4 Noise 6.7.5 Mode hopping 6.7.6 Reliability 6.8 Injection laser to fiber coupling 6.9 Nonsemiconductor lasers 6.9.1 The Nd:YAG laser 6.9.2 Glass fiber lasers 6.10 Narrow-linewidth and wavelength-tunable lasers 6.10.1 Long external cavity lasers 6.10.2 Integrated external cavity lasers 6.10.3 Fiber lasers 6.11 Mid-infrared and far-infrared lasers 6.11.1 Quantum cascade lasers Problems References 309 309 311 313 317 323 325 327 328 330 332 333 334 334 336 339 339 342 342 344 347 350 350 354 355 356 360 361 362 364 364 366 369 371 372 376 378 381 383 386
  • OPTF_A01.qxd 11/6/08 10:52 Page xii xii Contents Chapter 7: Optical sources 2: the light-emitting diode 396 7.1 7.2 7.3 7.4 7.5 Introduction LED power and efficiency 7.2.1 The double-heterojunction LED LED structures 7.3.1 Planar LED 7.3.2 Dome LED 7.3.3 Surface emitter LEDs 7.3.4 Edge emitter LEDs 7.3.5 Superluminescent LEDs 7.3.6 Resonant cavity and quantum-dot LEDs 7.3.7 Lens coupling to fiber LED characteristics 7.4.1 Optical output power 7.4.2 Output spectrum 7.4.3 Modulation bandwidth 7.4.4 Reliability Modulation Problems References Chapter 8: Optical detectors 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 Introduction Device types Optical detection principles Absorption 8.4.1 Absorption coefficient 8.4.2 Direct and indirect absorption: silicon and germanium 8.4.3 III–V alloys Quantum efficiency Responsivity Long-wavelength cutoff Semiconductor photodiodes without internal gain 8.8.1 The p–n photodiode 8.8.2 The p–i–n photodiode 8.8.3 Speed of response and traveling-wave photodiodes 8.8.4 Noise 396 398 405 406 407 407 407 411 414 416 419 422 422 425 428 433 435 436 439 444 444 446 447 448 448 449 450 451 451 455 456 456 457 462 468
  • OPTF_A01.qxd 11/6/08 10:52 Page xiii Contents xiii 8.9 Semiconductor photodiodes with internal gain 8.9.1 Avalanche photodiodes 8.9.2 Silicon reach through avalanche photodiodes 8.9.3 Germanium avalanche photodiodes 8.9.4 III–V alloy avalanche photodiodes 8.9.5 Benefits and drawbacks with the avalanche photodiode 8.9.6 Multiplication factor 8.10 Mid-infrared and far-infrared photodiodes 8.10.1 Quantum-dot photodetectors 8.11 Phototransistors 8.12 Metal–semiconductor–metal photodetectors Problems References Chapter 9: Direct detection receiver performance considerations 9.1 9.2 9.3 9.4 9.5 9.6 Introduction Noise 9.2.1 Thermal noise 9.2.2 Dark current noise 9.2.3 Quantum noise 9.2.4 Digital signaling quantum noise 9.2.5 Analog transmission quantum noise Receiver noise 9.3.1 The p–n and p–i–n photodiode receiver 9.3.2 Receiver capacitance and bandwidth 9.3.3 Avalanche photodiode (APD) receiver 9.3.4 Excess avalanche noise factor 9.3.5 Gain–bandwidth product Receiver structures 9.4.1 Low-impedance front-end 9.4.2 High-impedance (integrating) front-end 9.4.3 The transimpedance front-end FET preamplifiers 9.5.1 Gallium arsenide MESFETs 9.5.2 PIN–FET hybrid receivers High-performance receivers Problems References 470 470 472 473 474 480 482 482 484 485 489 493 496 502 502 503 503 504 504 505 508 510 511 515 516 522 523 524 525 526 526 530 531 532 534 542 545
  • OPTF_A01.qxd 11/6/08 10:52 Page xiv xiv Contents Chapter 10: Optical amplification, wavelength conversion and regeneration 10.1 Introduction 10.2 Optical amplifiers 10.3 Semiconductor optical amplifiers 10.3.1 Theory 10.3.2 Performance characteristics 10.3.3 Gain clamping 10.3.4 Quantum dots 10.4 Fiber and waveguide amplifiers 10.4.1 Rare-earth-doped fiber amplifiers 10.4.2 Raman and Brillouin fiber amplifiers 10.4.3 Waveguide amplifiers and fiber amplets 10.4.4 Optical parametric amplifiers 10.4.5 Wideband fiber amplifiers 10.5 Wavelength conversion 10.5.1 Cross-gain modulation wavelength converter 10.5.2 Cross-phase modulation wavelength converter 10.5.3 Cross-absorption modulation wavelength converters 10.5.4 Coherent wavelength converters 10.6 Optical regeneration Problems References Chapter 11: Integrated optics and photonics 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 Introduction Integrated optics and photonics technologies Planar waveguides Some integrated optical devices 11.4.1 Beam splitters, directional couplers and switches 11.4.2 Modulators 11.4.3 Periodic structures for filters and injection lasers 11.4.4 Polarization transformers and wavelength converters Optoelectronic integration Photonic integrated circuits Optical bistability and digital optics Optical computation Problems References 549 549 550 552 554 559 563 565 567 568 571 575 578 581 583 584 586 592 593 595 598 600 606 606 607 610 615 616 623 627 634 636 643 648 656 663 665
  • OPTF_A01.qxd 11/6/08 10:52 Page xv Contents xv Chapter 12: Optical fiber systems 1: intensity modulation/direct detection 12.1 Introduction 12.2 The optical transmitter circuit 12.2.1 Source limitations 12.2.2 LED drive circuits 12.2.3 Laser drive circuits 12.3 The optical receiver circuit 12.3.1 The preamplifier 12.3.2 Automatic gain control 12.3.3 Equalization 12.4 System design considerations 12.4.1 Component choice 12.4.2 Multiplexing 12.5 Digital systems 12.6 Digital system planning considerations 12.6.1 The optoelectronic regenerative repeater 12.6.2 The optical transmitter and modulation formats 12.6.3 The optical receiver 12.6.4 Channel losses 12.6.5 Temporal response 12.6.6 Optical power budgeting 12.6.7 Line coding and forward error correction 12.7 Analog systems 12.7.1 Direct intensity modulation (D–IM) 12.7.2 System planning 12.7.3 Subcarrier intensity modulation 12.7.4 Subcarrier double-sideband modulation (DSB–IM) 12.7.5 Subcarrier frequency modulation (FM–IM) 12.7.6 Subcarrier phase modulation (PM–IM) 12.7.7 Pulse analog techniques 12.8 Distribution systems 12.9 Multiplexing strategies 12.9.1 Optical time division multiplexing 12.9.2 Subcarrier multiplexing 12.9.3 Orthogonal frequency division multiplexing 12.9.4 Wavelength division multiplexing 12.9.5 Optical code division multiplexing 12.9.6 Hybrid multiplexing 673 673 675 676 679 686 690 691 694 697 700 701 702 703 708 708 711 715 725 726 731 734 739 742 748 750 752 754 756 758 760 765 765 766 768 771 777 778
  • OPTF_A01.qxd 8/18/09 11:36 AM Page xvi xvi Contents 12.10 Application of optical amplifiers 12.11 Dispersion management 12.12 Soliton systems Problems References Chapter 13: Optical fiber systems 2: coherent and phase modulated 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 Introduction Basic coherent system Coherent detection principles Practical constraints of coherent transmission 13.4.1 Injection laser linewidth 13.4.2 State of polarization 13.4.3 Local oscillator power 13.4.4 Transmission medium limitations Modulation formats 13.5.1 Amplitude shift keying 13.5.2 Frequency shift keying 13.5.3 Phase shift keying 13.5.4 Polarization shift keying Demodulation schemes 13.6.1 Heterodyne synchronous detection 13.6.2 Heterodyne asynchronous detection 13.6.3 Homodyne detection 13.6.4 Intradyne detection 13.6.5 Phase diversity reception 13.6.6 Polarization diversity reception and polarization scrambling Differential phase shift keying Receiver sensitivities 13.8.1 ASK heterodyne detection 13.8.2 FSK heterodyne detection 13.8.3 PSK heterodyne detection 13.8.4 ASK and PSK homodyne detection 13.8.5 Dual-filter direct detection FSK 13.8.6 Interferometric direct detection DPSK 13.8.7 Comparison of sensitivities 778 786 792 802 811 823 823 827 830 835 835 836 840 843 845 845 846 847 850 851 853 855 856 859 860 863 864 868 868 871 873 874 875 876 877
  • OPTF_A01.qxd 11/6/08 10:52 Page xvii Contents xvii 13.9 Multicarrier systems 13.9.1 Polarization multiplexing 13.9.2 High-capacity transmission Problems References 886 889 890 894 897 Chapter 14: Optical fiber measurements 905 14.1 Introduction 14.2 Fiber attenuation measurements 14.2.1 Total fiber attenuation 14.2.2 Fiber absorption loss measurement 14.2.3 Fiber scattering loss measurement 14.3 Fiber dispersion measurements 14.3.1 Time domain measurement 14.3.2 Frequency domain measurement 14.4 Fiber refractive index profile measurements 14.4.1 Interferometric methods 14.4.2 Near-field scanning method 14.4.3 Refracted near-field method 14.5 Fiber cutoff wavelength measurements 14.6 Fiber numerical aperture measurements 14.7 Fiber diameter measurements 14.7.1 Outer diameter 14.7.2 Core diameter 14.8 Mode-field diameter for single-mode fiber 14.9 Reflectance and optical return loss 14.10 Field measurements 14.10.1 Optical time domain reflectometry Problems References Chapter 15: Optical networks 15.1 Introduction 15.2 Optical network concepts 15.2.1 Optical networking terminology 15.2.2 Optical network node and switching elements 15.2.3 Wavelength division multiplexed networks 15.2.4 Public telecommunications network overview 905 909 910 914 917 919 920 923 926 927 930 932 934 938 941 941 943 943 946 948 952 958 962 967 967 969 970 974 976 978
  • OPTF_A01.qxd 11/6/08 10:52 Page xviii xviii Contents 15.3 Optical network transmission modes, layers and protocols 15.3.1 Synchronous networks 15.3.2 Asynchronous transfer mode 15.3.3 Open Systems Interconnection reference model 15.3.4 Optical transport network 15.3.5 Internet Protocol 15.4 Wavelength routing networks 15.4.1 Wavelength routing and assignment 15.5 Optical switching networks 15.5.1 Optical circuit-switched networks 15.5.2 Optical packet-switched networks 15.5.3 Multiprotocol Label Switching 15.5.4 Optical burst switching networks 15.6 Optical network deployment 15.6.1 Long-haul networks 15.6.2 Metropolitan area networks 15.6.3 Access networks 15.6.4 Local area networks 15.7 Optical Ethernet 15.8 Network protection, restoration and survivability Problems References 979 980 985 985 987 989 992 996 998 998 1000 1002 1004 1007 1008 1011 1013 1023 1028 1034 1038 1041 Appendix A Appendix B Appendix C Appendix D Appendix E 1051 1052 1053 1055 The field relations in a planar guide Gaussian pulse response Variance of a random variable Variance of the sum of independent random variables Closed loop transfer function for the transimpedance amplifier Index 1056 1057 Supporting resources Visit www.pearsoned.co.uk/senior-optical to find valuable online resources For instructors • An Instructor’s Manual that provides full solutions to all the numerical problems, which are provided at the end of each chapter in the book. For more information please contact your local Pearson Education sales representative or visit www.pearsoned.co.uk/senior-optical
  • OPTF_A01.qxd 11/6/08 10:52 Page xix Preface The preface to the second edition drew attention to the relentless onslaught in the development of optical fiber communications technology identified in the first edition in the context of the 1980s. Indeed, although optical fiber communications could now, nearly two decades after that period finished, be defined as mature, this statement fails to signal the continuing rapid and extensive developments that have subsequently taken place. Furthermore the pace of innovation and deployment fuelled, in particular, by the Internet is set to continue with developments in the next decade likely to match or even exceed those which have occurred in the last decade. Hence this third edition seeks to record and explain the improvements in both the technology and its utilization within what is largely an optical fiber global communications network. Major advances which have occurred while the second edition has been in print include: those associated with low-water-peak and high-performance single-mode fibers; the development of photonic crystal fibers; a new generation of multimode graded index plastic optical fibers; quantum-dot fabrication for optical sources and detectors; improvements in optical amplifier technology and, in particular, all-optical regeneration; the realization of photonic integrated circuits to provide ultrafast optical signal processing together with silicon photonics; developments in digital signal processing to mitigate fiber transmission impairments and the application of forward error correction strategies. In addition, there have been substantial enhancements in transmission and multiplexing techniques such as the use of duobinary-encoded transmission, orthogonal frequency division multiplexing and coarse/dense wavelength division multiplexing, while, more recently, there has been a resurgence of activity concerned with coherent and, especially, phase-modulated transmission. Finally, optical networking techniques and optical networks have become established employing both specific reference models for the optical transport network together with developments originating from local area networks based on Ethernet to provide for the future optical Internet (i.e. 100 Gigabit Ethernet for carrier-class transport networks). Moreover, driven by similar broadband considerations, activity has significantly increased in relation to optical fiber solutions for the telecommunication access network. Although a long period has elapsed since the publication of the second edition in 1992, it has continued to be used extensively in both academia and industry. Furthermore, as delays associated with my ability to devote the necessary time to writing the updates for this edition became apparent, it has been most gratifying that interest from the extensive user community of the second edition has encouraged me to find ways to pursue the necessary revision and enhancement of the book. A major strategy to enable this process has been the support provided by my former student and now colleague, Dr M. Yousif Jamro, working with me, undertaking primary literature searches and producing update drafts for many chapters which formed the first stage of the development for the new edition. An extensive series of iterations, modifications and further additions then ensued to craft the final text.
  • OPTF_A01.qxd 11/6/08 10:52 Page xx xx Preface to the third edition In common with the other editions, this edition relies upon source material from the numerous research and other publications in the field including, most recently, the Proceedings of the 33rd European Conference on Optical Communications (ECOC’07) which took place in Berlin, Germany, in September 2007. Furthermore, it also draws upon the research activities of the research group focused on optical systems and networks that I established at the University of Hertfordshire when I took up the post as Dean of Faculty in 1998, having moved from Manchester Metropolitan University. Although the book remains a comprehensive introductory text for use by both undergraduate and postgraduate engineers and scientists to provide them with a firm grounding in all significant aspects of the technology, it now also encompasses a substantial chapter devoted to optical networks and networking concepts as this area, in totality, constitutes the most important and extensive range of developments in the field to have taken place since the publication of the second edition. In keeping with a substantial revision and updating of the content, then, the practical nature of the coverage combined with the inclusion of the relevant up-to-date standardization developments has been retained to ensure that this third edition can continue to be widely employed as a reference text for practicing engineers and scientists. Following very positive feedback from reviewers in relation to its primary intended use as a teaching/ learning text, the number of worked examples interspersed throughout the book has been increased to over 120, while a total of 372 problems are now provided at the end of relevant chapters to enable testing of the reader’s understanding and to assist tutorial work. Furthermore, in a number of cases they are designed to extend the learning experience facilitated by the book. Answers to the numerical problems are provided at the end of the relevant sections in the book and the full solutions can be accessed on the publisher’s website using an appropriate password. Although the third edition has grown into a larger book, its status as an introductory text ensures that the fundamentals are included where necessary, while there has been no attempt to cover the entire field in full mathematical rigor. Selected proofs are developed, however, in important areas throughout the text. It is assumed that the reader is conversant with differential and integral calculus and differential equations. In addition, the reader will find it useful to have a grounding in optics as well as a reasonable familiarity with the fundamentals of solid-state physics. This third edition is structured into 15 chapters to facilitate a logical progression of material and to enable straightforward access to topics by providing the appropriate background and theoretical support. Chapter 1 gives a short introduction to optical fiber communications by considering the historical development, the general system and the major advantages provided by this technology. In Chapter 2 the concept of the optical fiber as a transmission medium is introduced using the simple ray theory approach. This is followed by discussion of electromagnetic wave theory applied to optical fibers prior to consideration of lightwave transmission within the various fiber types. In particular, single-mode fiber, together with a more recent class of microstructured optical fiber, referred to as photonic crystal fiber, are covered in further detail. The major transmission characteristics of optical fibers are then dealt with in Chapter 3. Again there is a specific focus on the properties and characteristics of single-mode fibers including, in this third edition, enhanced discussion of single-mode fiber types, polarization mode dispersion, nonlinear effects and, in particular, soliton propagation.
  • OPTF_A01.qxd 11/6/08 10:52 Page xxi Preface to the third edition xxi Chapters 4 and 5 deal with the more practical aspects of optical fiber communications and therefore could be omitted from an initial teaching program. A number of these areas, however, are of crucial importance and thus should not be lightly overlooked. Chapter 4 deals with the manufacturing and cabling of the various fiber types, while in Chapter 5 the different techniques to provide optical fiber connection are described. In this latter chapter both fiber-to-fiber joints (i.e. connectors and splices) are discussed as well as fiber branching devices, or couplers, which provide versatility within the configuration of optical fiber systems and networks. Furthermore, a new section incorporating coverage of optical isolators and circulators which are utilized for the manipulation of signals within optical networks has been included. Chapters 6 and 7 describe the light sources employed in optical fiber communications. In Chapter 6 the fundamental physical principles of photoemission and laser action are discussed prior to consideration of the various types of semiconductor and nonsemiconductor laser currently in use, or under investigation, for optical fiber communications. The other important semiconductor optical source, namely the light-emitting diode, is dealt with in Chapter 7. The next two chapters are devoted to the detection of the optical signal and the amplification of the electrical signal obtained. Chapter 8 discusses the basic principles of optical detection in semiconductors; this is followed by a description of the various types of photodetector currently employed. The optical fiber direct detection receiver is then considered in Chapter 9, with particular emphasis on its performance characteristics. Enhanced coverage of optical amplifiers and amplification is provided in Chapter 10, which also incorporates major new sections concerned with wavelength conversion processes and optical regeneration. Both of these areas are of key importance for current and future global optical networks. Chapter 11 then focuses on the fundamentals and ongoing developments in integrated optics and photonics providing descriptions of device technology, optoelectronic integration and photonic integrated circuits. In addition, the chapter includes a discussion of optical bistability and digital optics which leads into an overview of optical computation. Chapter 12 draws together the preceding material in a detailed discussion of the major current implementations of optical fiber communication systems (i.e. those using intensity modulation and the direct detection process) in order to give an insight into the design criteria and practices for all the main aspects of both digital and analog fiber systems. Two new sections have been incorporated into this third edition dealing with the crucial topic of dispersion management and describing the research activities into the performance attributes and realization of optical soliton systems. Over the initial period since the publication of the second edition, research interest and activities concerned with coherent optical fiber communications ceased as a result of the improved performance which could be achieved using optical amplification with conventional intensity modulation–direct detection optical fiber systems. Hence no significant progress in this area was made for around a decade until a renewed focus on coherent optical systems was initiated in 2002 following experimental demonstrations using phasemodulated transmission. Coherent and phase-modulated optical systems are therefore dealt with in some detail in Chapter 13 which covers both the fundamentals and the initial period of research and development associated with coherent transmission prior to 1992, together with the important recent experimental system and field trial demonstrations
  • OPTF_A01.qxd 11/6/08 10:52 Page xxii xxii Preface to the third edition primarily focused on phase-modulated transmission that have taken place since 2002. In particular a major new section describing differential phase shift keying systems together with new sections on polarization multiplexing and high-capacity transmission have been incorporated into this third edition. Chapter 14 provides a general treatment of the major measurements which may be undertaken on optical fibers in both the laboratory and the field. The chapter is incorporated at this stage in the book to enable the reader to obtain a more complete understanding of optical fiber subsystems and systems prior to consideration of these issues. It continues to include the measurements required to be taken on single-mode fibers and it addresses the measurement techniques which have been adopted as national and international standards. Finally, Chapter 15 on optical networks comprises an almost entirely new chapter for the third edition which provides both a detailed overview of this expanding field and a discussion of all the major aspects and technological solutions currently being explored. In particular, important implementations of wavelength routing and optical switching networks are described prior to consideration of the various optical network deployments that have occurred or are under active investigation. The chapter finishes with a section which addresses optical network protection and survivability. The book is also referenced throughout to extensive end-of-chapter references which provide a guide for further reading and also indicate a source for those equations that have been quoted without derivation. A complete list of symbols, together with a list of common abbreviations in the text, is also provided. SI units are used throughout the book. I must extend my gratitude for the many useful comments and suggestions provided by the diligent reviewers that have both encouraged and stimulated improvements to the text. Many thanks are also given to the authors of the multitude of journal and conference papers, articles and books that have been consulted and referenced in the preparation of this third edition and especially to those authors, publishers and companies who have kindly granted permission for the reproduction of diagrams and photographs. I would also like to thank the many readers of the second edition for their constructive and courteous feedback which has enabled me to make the substantial improvements that now comprise this third edition. Furthermore, I remain extremely grateful to my family and friends who have continued to be supportive and express interest over the long period of the revision for this edition of the book. In particular, my very special thanks go to Judy for her continued patience and unwavering support which enabled me to finally complete the task, albeit at the expense of evenings and weekends which could have been spent more frequently together. John M. Senior
  • OPTF_A01.qxd 11/6/08 10:52 Page xxiii Acknowledgements We are grateful to the following for permission to reproduce copyright material: Figures 2.17 and 2.18 from Weakly guiding fibers in Applied Optics, 10, p. 2552, OSA (Gloge, D. 1971), with permission from The Optical Society of America; Figure 2.30 from Fiber manufacture at AT&T with the MCVD process in Journal of Lightwave Technology, LT-4(8), pp. 1016–1019, OSA (Jablonowski, D. P. 1986), with permission from The Optical Society of America; Figure 2.35 from Gaussian approximation of the fundamental modes of graded-index fibers in Journal of the Optical Society of America, 68, p. 103, OSA (Marcuse, D. 1978), with permission from The Optical Society of America; Figure 2.36 from Applied Optics, 19, p. 3151, OSA (Matsumura, H. and Suganuma, T. 1980), with permission from The Optical Society of America; Figures 3.1 and 3.3 from Ultimate low-loss single-mode fibre at 1.55 mum in Electronic Letters, 15(4), pp. 106–108, Institution of Engineering and Technology (T Miya, T., Teramuna, Y., Hosaka, Y. and Miyashita, T. 1979), with permission from IET; Figure 3.2 from Applied Physics Letters, 22, 307, Copyright 1973, American Institute of Physics (Keck, D. B., Maurer, R. D. and Schultz, P. C. 1973), reproduced with permission; Figure 3.10 from Electronic Letters, 11, p. 176, Institution of Engineering and Technology (Payne, D. N. and Gambling, W. A. 1975), with permission from IET; Figures 3.15 and 3.17 from The Radio and Electronic Engineer, 51, p. 313, Institution of Engineering and Technology (Gambling, W. A., Hartog, A. H. and Ragdale, C. M. 1981), with permission from IET; Figure 3.18 from High-speed optical pulse transmission at 1.29 mum wavelength using low-loss singlemode fibers in IEEE Journal of Quantum Electronics, QE-14, p. 791, IEEE (Yamada, J. I., Saruwatari, M., Asatani, K., Tsuchiya, H., Kawana, A., Sugiyama, K. and Kumara, T. 1978), © IEEE 1978, reproduced with permission; Figure 3.30 from Polarization-maintaining fibers and their applications in Journal of Lightwave Technology, LT-4(8), pp. 1071–1089, OSA (Noda, J., Okamoto, K. and Susaki, Y. 1986), with permission from The Optical Society of America; Figure 3.34 from Nonlinear phenomena in optical fibers in IEEE Communications Magazine, 26, p. 36, IEEE (Tomlinson, W. J. and Stolen, R. H. 1988), © IEEE 1988, reproduced with permission; Figures 4.1 and 4.4 from Preparation of sodium borosilicate glass fibers for optical communication in Proceedings of IEE, 123, pp. 591– 595, Institution of Engineering and Technology (Beales, K. J., Day, C. R., Duncan, W. J., Midwinter, J. E. and Newns, G. R. 1976), with permission from IET; Figure 4.5 from A review of glass fibers for optical communications in Phys. Chem. Glasses, 21(1), p. 5, Society of Glass Technology (Beales, K. J. and Day, C. R. 1980), reproduced with permission; Figure 4.7 Reprinted from Optics Communication, 25, pp. 43–48, D. B. Keck and R. Bouilile, Measurements on high-bandwidth optical waveguides, copyright 1978, with permission from Elsevier; Figure 4.8 from Low-OH-content optical fiber fabricated by vapor-phase axial-deposition method in Electronic Letters, 14(17), pp. 534–535,
  • OPTF_A01.qxd 11/6/08 10:52 Page xxiv xxiv Acknowledgements Institution of Engineering and Technology (Sudo, S., Kawachi, M., Edahiro, M., Izawa, T., Shoida, T. and Gotoh, H. 1978), with permission from IET; Figure 4.20 from Optical fibre cables in Radio and Electronic Engineer (IERE J.), 51(7/8), p. 327, Institution of Engineering and Technology (Reeve, M. H. 1981), with permission from IET; Figure 4.21 from Power loss, modal noise and distortion due to microbending of optical fibres in Applied Optics, 24, pp. 2323, OSA ( Das, S., Englefield, C. G. and Goud, P. A. 1985), with permission from The Optical Society of America; Figure 4.22 from Hydrogen induced loss in MCVD fibers, Optical Fiber Communication Conference, OFC 1985, USA, TUII, February 1985, OFC/NFOEC (Lemaire, P. J. and Tomita, A. 1985), with permission from The Optical Society of America; Figures 5.2 (a) and 5.16 (a) from Connectors for optical fibre systems in Radio and Electronic Engineer (J. IERE), 51(7/8), p. 333, Institution of Engineering and Technology (Mossman, P. 1981), with permission from IET; Figure 5.5 (b) from Jointing loss in single-loss fibres in Electronic Letters, 14(3), pp. 54–55, Institution of Engineering and Technology (Gambling, W. A., Matsumura, H. and Cowley, A. G. 1978), with permission from IET; Figure 5.7 (a) from Figure 1, page 1, Optical Fiber Arc Fusion Splicer FSM-45F, No.: B- 06F0013Cm, 13 February 2007, http:// www.fujikura.co.jp/00/splicer/front-page/pdf/e_fsm-45f.pdf; with permission from Fujikura Limited; Figure 5.7 (b) from Figure 4, page 1, Arc Fusion Splicer, SpliceMate, SpliceMate Brochure, http://www.fujikura.co.jp/00/splicer/front-page/pdf/splicemate_ brochure.pdf, with permission from Fujikura Limited; Figure 5.8 (a) from Optical communications research and technology in Proceedings of the IEEE, 66(7), pp. 744–780, IEEE (Giallorenzi, T. G. 1978), © IEEE 1978, reproduced with permission; Figure 5.13 from Simple high-performance mechanical splice for single mode fibers in Proceedings of the Optical Fiber Communication Conference, OFC 1985, USA, paper M12, OFC/NFOEC (Miller, C. M., DeVeau, G. F. and Smith, M. Y. 1985), with permission from The Optical Society of America; Figure 5.15 from Rapid ribbon splice for multimode fiber splicing in Proceedings of the Optical Fiber Communication Conference, OFC1985, USA, paper TUQ27, OFC/NFOEC (Hardwick, N. E. and Davies, S. T. 1985), with permission from The Optical Society of America; Figure 5.21 (a) from Demountable multiple connector with precise V-grooved silicon in Electronic Letters, 15(14), pp. 424–425, Institution of Engineering and Technology (Fujii, Y., Minowa, J. and Suzuki, N. 1979), with permission from IET; Figure 5.21 (b) from Very small single-mode ten-fiber connector in Journal of Lightwave Technology, 6(2), pp. 269–272, OSA (Sakake, T., Kashima, N. and Oki, M. 1988), with permission from The Optical Society of America; Figure 5.20 from Highcoupling-efficiency optical interconnection using a 90-degree bent fiber array connector in optical printed circuit boards in IEEE Photonics Technology Letters, 17(3), pp. 690– 692, IEEE (Cho, M. H., Hwang, S. H., Cho, H. S. and Park, H. H. 2005), © IEEE 2005, reproduced with permission; Figure 5.22 (a) from Practical low-loss lens connector for optical fibers in Electronic Letters, 14(16), pp. 511–512, Institution of Engineering and Technology (Nicia, A. 1978), with permission from IET; Figure 5.23 from Assembly technology for multi-fiber optical connectivity solutions in Proceedings of IEEE/LEOS Workshop on Fibres and Optical Passive Components, 22–24 June 2005, Mondello, Italy, IEEE (Bauknecht, R. Kunde, J., Krahenbuhl, R., Grossman, S. and Bosshard, C. 2005), © IEEE 2005, reproduced with permission; Figure 5.31 from Polarization-independent optical circulator consisting of two fiber-optic polarizing beamsplitters and two YIG spherical lenses in Electronic Letters, 22, pp. 370–372, Institution of Engineering and
  • OPTF_A01.qxd 11/6/08 10:52 Page xxv Acknowledgements xxv Technology (Yokohama, I., Okamoto, K. and Noda, J. 1985), with permission from IET; Figures 5.36 and 5.38 (a) from Optical demultiplexer using a silicon echette grating in IEEE Journal of Quantum Electronics, QE-16, pp. 165–169, IEEE (Fujii, Y., Aoyama, K. and Minowa, J. 1980), © IEEE 1980, reproduced with permission; Figure 5.44 from Filterless ‘add’ multiplexer based on novel complex gratings assisted coupler in IEEE Photonics Technology Letters, 17(7), pp. 1450–1452, IEEE (Greenberg, M. and Orenstein, M. 2005), © IEEE 2005, reproduced with permission; Figure 6.33 from Low threshold operation of 1.5 μm DFB laser diodes in Journal of Lightwave Technology, LT-5, p. 822, IEEE (Tsuji, S., Ohishi, A., Nakamura, H., Hirao, M., Chinone, N. and Matsumura, H. 1987), © IEEE 1987, reproduced with permission; Figure 6.37 adapted from Vertical-Cavity Surface-Emitting Lasers: Design, Fabrication, Characterization, and Applications, Cambridge University Press (Wilmsen, C. W., Temkin, H. and Coldren, L. A. 2001), reproduced with permission; Figure 6.38 from Semiconductor laser sources for optical communication in Radio and Electronic Engineer, 51, p. 362, Institution of Engineering and Technology (Kirby, P. A. 1981), with permission from IET; Figure 6.47 from Optical amplification in an erbium-doped fluorozirconate fibre between 1480 nm and 1600 nm in IEE Conference Publication 292, Pt 1, p. 66, Institution of Engineering and Technology (Millar, C. A., Brierley, M. C. and France, P. W. 1988), with permission from IET; Figure 6.48 (a) from High efficiency Nd-doped fibre lasers using direct-coated dielectric mirrors in Electronic Letters, 23, p. 768, Institution of Engineering and Technology (Shimtzu, M., Suda, H. and Horiguchi, M. 1987), with permission from IET; Figure 6.48 (b) from Rare-earth-doped fibre lasers and amplifiers in IEE Conference Publication, 292. Pt 1, p. 49, Institution of Engineering and Technology (Payne, D. N. and Reekie, L. 1988), with permission from IET; Figure 6.53 from Wavelength-tunable and single-frequency semiconductor lasers for photonic communications networks in IEEE Communications Magazine, October, p. 42, IEEE (Lee, T. P. and Zah, C. E. 1989), © IEEE 1989, reproduced with permission; Figure 6.55 from Single longitudinal-mode operation on an Nd3+-doped fibre laser in Electronic Letters, 24, pp. 24–26, IEEE (Jauncey, I. M., Reekie, L., Townsend, K. E. and Payne, D. N. 1988), © IEEE 1988, reproduced with permission; Figure 6.56 from Tunable single-mode fiber lasers in Journal of Lightwave Technology, LT-4, p. 956, IEEE (Reekie, L., Mears, R. J., Poole, S. B. and Payne, D. N. 1986), © IEEE 1986, reproduced with permission; Figure 6.58 reprinted from Semiconductors and Semimetals: Lightwave communication technology, 22C, Y. Horikoshi, ‘Semiconductor lasers with wavelengths exceeding 2 μm’, pp. 93–151, 1985, edited by W. T. Tsang (volume editor), copyright 1985, with permission from Elsevier; Figure 6.59 from PbEuTe lasers with 4–6 μm wavelength mode with hot-well epitaxy in IEEE Journal of Quantum Electronics, 25(6), pp. 1381–1384, IEEE (Ebe, H., Nishijima, Y. and Shinohara, K. 1989), © IEEE 1989, reproduced with permission; Figure 7.5 reprinted from Optical Communications, 4, C. A. Burrus and B. I. Miller, Small-area double heterostructure aluminum-gallium arsenide electroluminsecent diode sources for optical fiber transmission lines, pp. 307–369, 1971, copyright 1971, with permission from Elsevier; Figure 7.6 from High-power single-mode optical-fiber coupling to InGaAsP 1.3 μm mesa-structure surface-emitting LEDs in Electronic Letters, 21(10), pp. 418–419, Institution of Engineering and Technology (Uji, T. and Hayashi, J. 1985), with permission from IET; Figure 7.8 from Sources and detectors for optical fiber communications applications: the first 20 years in IEE Proceedings on Optoelectronics, 133(3), pp. 213–228,
  • OPTF_A01.qxd 11/6/08 10:52 Page xxvi xxvi Acknowledgements Institution of Engineering and Technology (Newman, D. H. and Ritchie, S. 1986), with permission from IET; Figure 7.9 (a) from 2 Gbit/s and 600 Mbit/s single-mode fibretransmission experiments using a high-speed Zn-doped 1.3 μm edge-emitting LED in Electronic Letters, 13(12), pp. 636–637, Institution of Engineering and Technology (Fujita, S., Hayashi, J., Isoda, Y., Uji, T. and Shikada, M. 1987), with permission from IET; Figure 7.9 (b) from Gigabit single-mode fiber transmission using 1.3 μm edgeemitting LEDs for broadband subscriber loops in Journal of Lightwave Technology, LT-5(10) pp. 1534–1541, OSA (Ohtsuka, T., Fujimoto, N., Yamaguchi, K., Taniguchi, A., Naitou, N. and Nabeshima, Y. 1987), with permission from The Optical Society of America; Figure 7.10 (a) from A stripe-geometry double-heterostructure amplifiedspontaneous-emission (superluminescent) diode in IEEE Journal of Quantum Electronics QE-9, p. 820 (Lee, T. P., Burrus, C. A. and Miller, B. I. 1973), with permission from IET; Figure 7.10 (b) from High output power GaInAsP/InP superluminescent diode at 1.3 μm in Electronic Letters, 24(24) pp. 1507–1508, Institution of Engineering and Technology (Kashima, Y., Kobayashi, M. and Takano, T. 1988), with permission from IET; Figure 7.14 from Highly efficient long lived GaAlAs LEDs for fiber-optical communications in IEEE Trans. Electron Devices, ED-24(7) pp. 990–994, Institution of Engineering and Technology (Abe, M., Umebu, I., Hasegawa, O., Yamakoshi, S., Yamaoka, T., Kotani, T., Okada, H., and Takamashi, H. 1977), with permission from IET; Figure 7.15 from CaInAsP/InP fast, high radiance, 1.05–1.3 μm wavelength LEDs with efficient lens coupling to small numerical aperture silica optical fibers in IEEE Trans Electron. Devices, ED-26(8), pp. 1215–1220, Institution of Engineering and Technology (Goodfellow, R. C., Carter, A. C., Griffith, I. and Bradley, R. R. 1979), with permission from IET; Figures 7.19 and 7.23 were published in Optical Fiber Telecommunications II, T. P. Lee, C. A. Burrus Jr and R. H. Saul, Light-emitting diodes for telecommunications, pp. 467–507, edited by S. E. Miller and I. P. Kaminow, 1988, Copyright Elsevier 1988; Figure 7.20 from Lateral confinement InGaAsP superluminescent diode at 1.3 μm in IEEE Journal of Quantum Electronics, QE19, p. 79, IEEE (Kaminow, I. P., Eisenstein, G., Stulz, L. W. and Dentai, A. G. 1983), © IEEE 1983, reproduced with permission; Figure 7.21 adapted from Figure 6, page 121 of AlGaInN resonant-cavity LED devices studied by electromodulated reflectance and carrier lifetime techniques in IEE Proceedings on Optoelectronics, vol. 152, no. 2, pp. 118–124, 8 April 2005, Institution of Engineering and Technology (Blume, G., Hosea, T. J. C., Sweeney, S. J., de Mierry, P., Lancefield, D. 2005), with permission from IET; Figure 7.22 (b) from Light-emitting diodes for optical fibre systems in Radio and Electronic Engineer (J. IERE), 51(7/8), p. 41, Institution of Engineering and Technology (Carter, A. C. 1981), with permission from IET; Figure 8.3 from Optical Communications Essentials (Telecommunications), McGraw-Hill Companies (Keiser, G. 2003), with permission of the McGraw-Hill Companies; Figure 8.19 (a) from Improved germanium avalanche photodiodes in IEEE Journal of Quantum Electronics, QE-16(9), pp. 1002–1007 (Mikami, O., Ando, H., Kanbe, H., Mikawa, T., Kaneda, T. and Toyama, Y. 1980), © IEEE 1980, reproduced with permission; Figure 8.19 (b) from High-sensitivity Hi-Lo germanium avalanche photodiode for 1.5 μm wavelength optical communication in Electronic Letters, 20(13), pp. 552–553, Institution of Engineering and Technology (Niwa, M., Tashiro, Y., Minemura, K. and Iwasaki, H. 1984), with permission from IET; Figure 8.24 from Impact ionisation in multi-layer heterojunction structures in Electronic Letters, 16(12), pp. 467–468, Institution of Engineering and Technology (Chin, R.,
  • OPTF_A01.qxd 11/6/08 10:52 Page xxvii Acknowledgements xxvii Holonyak, N., Stillman, G. E., Tang, J. Y. and Hess, K. 1980), with permission from IET; Figure 8.25 Reused with permission from Federico Capasso, Journal of Vacuum Science & Technology B, 1, 457 (1983). Copyright 1983, AVS The Science & Technology Society; Figure 8.29 Reused with permission from P. D. Wright, R. J. Nelson, and T. Cella, Applied Physics Letters, 37, 192 (1980). Copyright 1980, American Institute of Physics; Figure 8.32 from MSM-based integrated CMOS wavelength-tunable optical receiver in IEEE Photonics Technology Letters, 17(6) pp. 1271–1273 (Chen, R., Chin, H., Miller, D. A. B., Ma, K. and Harris Jr., J. S. 2005); © IEEE 2005, reproduced with permission; Figure 9.5 from Receivers for optical fibre communications in Electronic and Radio Engineer, 51(7/8), p. 349, Institution of Engineering and Technology (Garrett, I. 1981), with permission from IET; Figure 9.7 from Photoreceiver architectures beyond 40 Gbit/s, IEEE Symposium on Compound Semiconductor Integrated circuits, Monterey, California, USA, pp. 85–88, October ( Ito, H. 2004), © IEEE 2004, reproduced with permission; Figure 9.14 from GaAs FET tranimpedance front-end design for a wideband optical receiver in Electronic Letters, 15(20), pp. 650–652, Institution of Engineering and Technology (Ogawa, K. and Chinnock, E. L. 1979), with permission from IET; Figure 9.15 published in Optical Fiber Telecommunications II, B. L. Kaspar, Receiver design, p. 689, edited by S. E. Miller and I. P. Kaminow, 1988, Copyright Elsevier 1988; Figure 9.17 from An APD/FET optical receiver operating at 8 Gbit/s in Journal of Lightwave Technology, LT-5(3) pp. 344–347, OSA (Kaspar, B. L., Campbell, J. C., Talman, J. R., Gnauck, A. H., Bowers, J. E. and Holden, W. S. 1987), with permission from The Optical Society of America; Figure 9.23 Reprinted from Optical Fiber Telecommunications IV A: Components, B. L. Kaspar, O. Mizuhara and Y. K. Chen, High bit-rates receivers, transmiters and electronics, pp. 784–852, Figure 1.13, page 807, edited by I. P. Kaminow and T. Li, Copyright 2002, with permission from Elsevier; Figure 10.3 from Semiconductor laser optical amplifiers for use in future fiber systems in Journal of Lightwave Technology 6(4), p. 53, OSA (O’Mahony, M. J. 1988), with permission from The Optical Society of America; Figure 10.8 from Noise performance of semiconductor optical amplifiers, International Conference on Trends in Communication, EUROCON, 2001, Bratislava, Slovakia, 1, pp. 161–163, July (Udvary, E. 2001), © IEEE 2001, reproduced with permission; Figure 10.17 from Properties of fiber Raman amplifiers and their applicability to digital optical communication systems in Journal of Lightwave Technology, 6(7), p. 1225, IEEE (Aoki, Y. 1988), © IEEE 1988, reproduced with permission; Figure 10.18 (a) from Semiconductor Raman amplifier for terahertz bandwidth optical communication in Journal of Lightwave Technology, 20(4), pp. 705–711, IEEE (Suto, K., Saito, T., Kimura, T., Nishizawa, J. I. and Tanube, T. 2002), © IEEE 2002, reproduced with permission; Figure 11.2 from Scaling rules for thin-film optical waveguides, Applied Optics, 13(8), p. 1857, OSA (Kogelnik, H. and Ramaswamy, V. 1974), with permission from the Optical Society of America; Figure 11.7 Reused with permission from M. Papuchon, Y. Combemale, X. Mathieu, D. B. Ostrowsky, L. Reiber, A. M. Roy, B. Sejourne, and M. Werner, Applied Physics Letters, 27, 289 (1975). Copyright 1975, American Institute of Physics; Figure 11.13 from Beam-steering micromirrors for large optical cross-connects in Journal of Lightwave Technology, 21(3), pp. 634–642, OSA (Aksyuk, V. A. et al. 2003), with permission from The Optical Society of America; Figure 11.23 from 5 Git/s modulation characteristics of optical intensity modulator monolithically integrated with DFB laser in Electronic Letters, 25(5), pp. 1285–1287, Institution of Engineering and
  • OPTF_A01.qxd 11/6/08 10:52 Page xxviii xxviii Acknowledgements Technology (Soda, H., Furutsa, M., Sato, K., Matsuda, M. and Ishikawa, H. 1989), with permission from IET; Figure 11.24 from Widely tunable EAM-integrated SGDBR laser transmitter for analog applications in IEEE Photonics Technology Letters, 15(9), pp. 1285–1297, IEEE (Johansson, L. A., Alkulova, Y. A., Fish, G. A. and Coldren, L. A. 2003), © IEEE 2003, reproduced with permission; Figure 11.25 from 80-Gb/s InP-based waveguide-integrated photoreceiver in IEEE Journal of Sel. Top. Quantum Electronics, 11(2), pp. 356–360, IEEE (Mekonne, G. G., Bach, H. G., Beling, A., Kunkel, R., Schmidt, D. and Schlaak, W. 2005), © IEEE 2005, reproduced with permission; Figure 11.27 from Wafer-scale replication of optical components on VCSEL wafers in Proceedings of Optical Fiber Communication, OFC 2004, Los Angeles, USA, vol. 1, 23–27 February, © IEEE 2004, reproduced with permission; Figure 11.28 from Terabus: terabit/secondclass card-level optical interconnect technologies in IEEE Journal of Sel. Top. Quantum Electronics, 12(5), pp. 1032–1044, IEEE (Schares, L., Kash, J. A., Doany, F. E., Schow, C. L., Schuster, C., Kuchta, D. M., Pepeljugoski, P. K., Trewhella, J. M., Baks, C. W. and John, R. A. 2006), © IEEE 2006, reproduced with permission; Figure 11.29 from Figure 2, http://www.fujitsu.com/global/news/pr/archives/month/2007/20070119-01.html, courtesy of Fujitsu Limited; Figure 11.33 (b) from Large-scale InP photonic integrated circuits: enabling efficient scaling of optical transport networks, IEEE Journal of Se. Top. Quantum Electronics, 13(1) pp. 22–31, IEEE (Welch, D. F. et al. 2007), © IEEE 2007, reproduced with permission; Figure 11.33 (c) from Monolithically integrated 100-channel WDM channel selector employing low-crosstalk AWG in IEEE Photonics Technology Letters, 16(11), pp. 2481–2483, IEEE (Kikuchi, N., Shibata, Y., Okamoto, H., Kawaguchi, Y., Oku, S., Kondo, Y. and Tohmori, Y. 2004), © IEEE 2004, reproduced with permission; Figure 11.35 Reused with permission from P. W. Smith, I. P. Kaminow, P. J. Maloney, and L. W. Stulz, Applied Physics Letters, 33, 24 (1978). Copyright 1978, American Institute of Physics; Figure 11.38 from All-optical flip-flop multimode interference bistable laser diode in IEEE Photonics Technology Letters, 17(5), pp. 968–970, IEEE (Takenaka, M., Raburn, M. and Nakano, Y. 2005), © IEEE 2005, reproduced with permission; Figure 11.42 from Optical bistability, phonomic logic and optical computation in Applied Optics, 25, pp. 1550–1564, OSA (Smith, S. D. 1986), with permission from The Optical Society of America; Figure 12.4 from Non-linear phase distortion and its compensation in LED direct modulation in Electronic Letters, 13(6), pp. 162–163, Institution of Engineering and Technology (Asatani, K. and Kimura, T. 1977), with permission from IET; Figures 12.6 and 12.7 from Springer-Verlag, Topics in Applied Physics, vol. 39, 1982, pp. 161–200, Lightwave transmitters, P. W. Schumate Jr. and M. DiDomenico Jr., in H. Kressel, ed., Semiconductor Devices for Optical Communications, with kind permission from Springer Science and Business Media; Figure 12.12 from Electronic circuits for high bit rate digital fiber optic communication systems in IEEE Trans. Communications, COM-26(7), pp. 1088–1098, IEEE (Gruber, J., Marten, P., Petschacher, R. and Russer, P. 1978), © IEEE 1978, reproduced with permission; Figure 12.13 from Design and stability analysis of a CMOS feedback laser driver in IEEE Trans. Instrum. Meas., 53(1), pp. 102–108, IEEE (Zivojinovic, P., Lescure, M. and Tap-Beteille, H. 2004), © IEEE 2004, reproduced with permission; Figure 12.14 from Laser automatic level control for optical communications systems in Third European Conference on Optical Communications, Munich, Germany, 14–16 September (S. R. Salter, S. R., Smith, D. R., White, B. R. and Webb, R. P. 1977), with permission from VDE-Verlag GMBH;
  • OPTF_A01.qxd 11/6/08 10:52 Page xxix Acknowledgements xxix Figure 12.15 from Electronic circuits for high bit rate digital fiber optic communication systems in IEEE Trans. Communications, COM-26(7), pp. 1088–1098, IEEE (Gruber, J., Marten, P., Petschacher, R. and Russer, P. 1978), © IEEE 1978, reproduced with permission; Figure 12.35 from NRZ versus RZ in 10–14-Gb/s dispersion-managed WDM transmission systems in IEEE Photonics Technology Letters, 11(8), pp. 991–993, IEEE (Hayee, M. I., Willner, A. E., Syst, T. S. and Eacontown, N. J. 1999), © IEEE 1999, reproduced with permission; Figure 12.36 from Dispersion-tolerant optical transmission system using duobinary transmitter and binary receiver in Journal of Lightwave Technology, 15(8), pp. 1530–1537, OSA (Yonenaga, K. and Kuwano, S. 1997), with permission from The Optical Society of America; Figure 12.44 Copyright BAE systems Plc. Reproduced with permission from Fibre optic systems for analogue transmission, Marconi Review, XLIV(221), pp. 78–100 (Windus, G. G. 1981); Figure 12.53 from Performance of optical OFDM in ultralong-haul WDM lightwave systems in Journal of Lightwave Technology, 25(1), pp. 131–138, OSA (Lowery, A. J., Du, L. B. and Armstrong, J. 2007), with permission from The Optical Society of America; Figure 12.58 from 110 channels × 2.35 Gb/s from a single femtosecond laser in IEEE Photonics Technology Letters, 11(4), pp. 466–468, IEEE (Boivin, L., Wegmueller, M., Nuss, M. C. and Knox, W. H. 1999), © IEEE 1999, reproduced with permission; Figure 12.64 from 10 000-hop cascaded in-line all-optical 3R regeneration to achieve 1 250 000-km 10-Gb/s transmission in IEEE Photonics Technology Letters, 18(5), pp. 718–720, IEEE (Zuqing, Z., Funabashi, M., Zhong, P., Paraschis, L. and Yoo, S. J. B. 2006), © IEEE 2006, reproduced with permission; Figure 12.65 from An experimental analysis of performance fluctuations in highcapacity repeaterless WDM systems in Proceedings of OFC/Fiber Optics Engineering Conference (NFOEC) 2006, Anaheim, CA, USA, p. 3, 5–10 March, OSA (Bakhshi, B., Richardson, L., Golovchenko, E. A., Mohs, G. and Manna, M. 2006), with permission from The Optical Society of America; Figure 12.73 from Springer-Verlag, Massive WDM and TDM Soliton Transmission Systems 2002, A. Hasegawa, © 2002 Springer, with kind permission from Springer Science and Business Media; Figure 13.7 from Techniques for multigigabit coherent optical transmission in Journal of Lightwave Technology, LT-5, p. 1466, IEEE (Smith, D. W. 1987), © IEEE 1987, reproduced with permission; Figure 13.15 from Costas loop experiments for a 10.6 μm communications receiver in IEEE Tras. Communications, COM-31(8), pp. 1000–1002, IEEE (Phillip, H. K., Scholtz, A. L., Bonekand, E. and Leeb, W. 1983), © IEEE 1983, reproduced with permission; Figure 13.18 from Semiconductor laser homodyne optical phase lock loop in Electronic Letters, 22, pp. 421–422, Institution of Engineering and Technology (Malyon, D. J., Smith, D. W. and Wyatt, R. 1986), with permission from IET; Figure 13.32 from A consideration of factors affecting future coherent lightwave communication systems in Journal of Lightwave Technology, 6, p. 686, OSA (Nosu, K. and Iwashita, K. 1988), with permission from The Optical Society of America; Figure 13.34 from RZ-DPSK field trail over 13 100 km of installed non-slope matched submarine fibers in Journal of Lightwave Technology, 23(1) pp. 95–103, OSA (Cai, J. X. et al. 2005), with permission from The Optical Society of America; Figure 13.35 from Polarization-multiplexed 2.8 Gbit/s synchronous QPSK transmission with real-time polarization tracking in Proceedings of the 33rd European Conference on Optical Communications, Berlin, Germany, pp. 263–264, 3 September (Pfau, T. et al. 2007), with permission from VDE Verlag GMBH; Figure 13.36 from Hybrid 107-Gb/s polarization-multiplexed DQPSK and 42.7-Gb/s DQPSK transmission
  • OPTF_A01.qxd 11/6/08 10:52 Page xxx xxx Acknowledgements at 1.4 bits/s/Hz spectral efficiency over 1280 km of SSMF and 4 bandwidth-managed ROADMs in Proceedings of the 33rd European Conference on Optical Communications, Berlin, Germany, PD1.9, September, with permission from VDE Verlag GMBH; Figure 13.37 from Coherent optical orthogonal frequency division multiplexing in Electronic Letters, 42(10), pp. 587–588, Institution of Engineering and Technology (Shieh, W. and Athaudage, C. 2006), with permission from IET; Figure 14.1 (a) from Mode scrambler for optical fibres in Applied Optics, 16(4), pp. 1045–1049, OSA (Ikeda, M., Murakami, Y. and Kitayama, C. 1977), with permission from The Optical Society of America; Figure 14.1 (b) from Measurement of baseband frequency reponse of multimode fibre by using a new type of mode scrambler in Electronic Letters, 13(5), pp. 146–147, Institution of Engineering and Technology (Seikai, S., Tokuda, M., Yoshida, K. and Uchida, N. 1977), with permission from IET; Figure 14.6 from An improved technique for the measurement of low optical absorption losses in bulk glass in Opto-electronics, 5, p. 323, Institution of Engineering and Technology (White, K. I. and Midwinter, J. E. 1973); with permission from IET; Figure 14.8 from Self pulsing GaAs laser for fiber dispersion measurement in IEEE Journal of Quantum Electronics, QE-8, pp. 844–846, IEEE (Gloge, D., Chinnock, E. L. and Lee, T. P. 1972), © IEEE 1972, reproduced with permission; Figure 14.12, image of 86038B Optical Dispersion Analyzer, © Agilent Technologies, Inc. 2005, Reproduced with Permission, Courtesy of Agilent Technologies, Inc.; Figure 14.13 (a) from Refractive index profile measurements of diffused optical waveguides in Applied Optics, 13(9), pp. 2112–2116, OSA (Martin, W. E. 1974), with permission from The Optical Society of America; Figures 14.13 (b) and 14.14 Reused with permission from L. G. Cohen, P. Kaiser, J. B. Mac Chesney, P. B. O’Connor, and H. M. Presby, Applied Physics Letters, 26, 472 (1975). Copyright 1975, American Institute of Physics; Figures 14.17 and 14.18 (a) Reused with permission from F. M. E. Sladen, D. N. Payne, and M. J. Adams, Applied Physics Letters, 28, 255 (1976). Copyright 1976, American Institute of Physics; Figure 14.19 from An Introduction to Optical Fibers, McGraw-Hill Companies (Cherin, A. H. 1983), with permission of the McGraw-Hill Companies; Figure 14.26 Reused with permission from L. G. Cohen and P. Glynn, Review of Scientific Instruments, 44, 1749 (1973). Copyright 1973, American Institute of Physics; Figure 14.31 (a) from EXFO http://documents.exfo.com/appnotes/anote044-ang.pdf, accessed 21 September 2007, with permission from EXFO; Figure 14.31 (b) from http://www.afltelecommunications.com, Afltelecommunications Inc., accessed 21 September 2007, with permission from Fujikura Limited; Figure 14.35 from EXFO, OTDR FTB-7000B http://documents.exfo.com/ appnotes/anote087-ang.pdf, accessed 21 September 2007, with permission from EXFO; Figure 14.36 from EXFO P-OTDR, accessed 21 September 2007, with permission from EXFO; Figure 15.1 from Future optical networks in Journal of Lightwave Technology, 24(12), pp. 4684–4696 (O’Mahony, M. J., Politi, C., Klonidis, D., Nejabati, R. and Simeonidou, D. 2006), with permission from The Optical Society of America; Figures 15.14 (a) and (b) from ITU-T Recommendation G.709/Y.1331(03/03) Interfaces for the Optical Transport Network (TON), 2003, reproduced with kind permission from ITU; Figures 15.25 and 15.26 from Enabling technologies for next-generation optical packetswitching networks in Proceedings of IEEE 94(5), pp. 892–910 (Gee-Kung, C., Jianjun, Y., Yong-Kee, Y., Chowdhury, A. and Zhensheng, J. 2006), © IEEE 2006, reproduced with permission; Figures 15.30, 15.31 and 15.32 from www.telegeography.com, accessed 17 October 2007, reproduced with permission; Figure 15.34 from Transparent optical
  • OPTF_A01.qxd 11/6/08 10:52 Page xxxi Acknowledgements xxxi protection ring architectures and applications in Journal of Lightwave Technology, 23(10), pp. 3388–3403 (Ming-Jun, L., Soulliere, M. J., Tebben, D. J., Nderlof, L., Vaughn, M. D. and R. E. Wagner, R. E. 2005), with permission from The Optical Society of America; Figure 15.46 from Hybrid DWDM-TDM long-reach PON for next-generation optical access in Journal of Lightwave Technology, 24(7), pp. 2827–2834 (Talli, G. and Townsend, P. D. 2006), with permission from The Optical Society of America; Figure 15.52 from IEEE 802.3 CSMA/CD (ETHERNET), accessed 17 October 2007, reproduced with permission; Figure 15.53 (a) from ITU-T Recommendation G.985 (03/2003) 100 Mbit/s point-to-pint Ethernet based optical access system, accessed 22 October 2007, reproduced with kind permission from ITU; Figure 15.53 (b) from ITU-T Recommendation Q.838.1 (10/2004) Requirements and analysis for the management interface of Ethernet passive optical networks (EPON), accessed 19 October 2007, reproduced with kind permission from ITU; Table 15.4 from Deployment of submarine optical fiber bacle and communication systems since 2001, www.atlantic-cable.com/Cables/ CableTimeLine/index2001.htm, reproduced with permission. In some instances we have been unable to trace the owners of copyright material, and we would appreciate any information that would enable us to do so.
  • OPTF_A01.qxd 11/6/08 10:52 Page xxxii List of symbols and abbreviations A A21 Ac a ab(λ) aeff ak am(λ) B B12, B21 BF Bfib BFPA Bm Bopt Br BT b C Ca Cd Cf Cj CL C0 CT CT constant, area (cross-section, emission), far-field pattern size, mode amplitude, wave amplitude (A0) Einstein coefficient of spontaneous emission peak amplitude of the subcarrier waveform (analog transmission) fiber core radius, parameter which defines the asymmetry of a planar guide (given by Eq. (10.21)), baseband message signal (a(t)) effective fiber core radius bend attenuation fiber integer 1 or 0 relative attenuation between optical powers launched into multimode and single-mode fibers constant, electrical bandwidth (post-detection), magnetic flux density, mode amplitude, wave amplitude (B0) Einstein coefficients of absorption, stimulated emission modal birefringence fiber bandwidth mode bandwidth (Fabry–Pérot amplifier) bandwidth of an intensity-modulated optical signal m(t), maximum 3 dB bandwidth (photodiode) optical bandwidth recombination coefficient for electrons and holes bit rate, when the system becomes dispersion limited (BT(DL)) normalized propagation constant for a fiber, ratio of luminance to composite video, linewidth broadening factor (injection laser) constant, capacitance, crack depth (fiber), wave coupling coefficient per unit length, coefficient incorporating Einstein coefficients effective input capacitance of an optical fiber receiver amplifier optical detector capacitance capacitance associated with the feedback resistor of a transimpedance optical fiber receiver amplifier junction capacitance (photodiode) total optical fiber channel loss in decibels, including the dispersion– equalization penalty (CLD) wave amplitude total capacitance polarization crosstalk
  • OPTF_A01.qxd 11/6/08 10:52 Page xxxiii List of symbols and abbreviations c ci D Dc Df DL DP DT d dfdn do E Ea EF Eg Em(t) Eo Eq e F Ᏺ Fn Fto f fD fd fo G Gi (r) Go Gp GR Gs Gsn g xxxiii velocity of light in a vacuum, constant (c1, c2) tap coefficients for a transversal equalizer amplitude coefficient, electric flux density, distance, diffusion coefficient, corrugation period, decision threshold in digital optical fiber transmission, fiber dispersion parameters: material (DM); profile (DP); total first order (DT); waveguide (DW), detectivity (photodiode), specific detectivity (D*) minority carrier diffusion coefficient frequency deviation ratio (subcarrier FM) dispersion–equalization penalty in decibels frequency deviation ratio (subcarrier PM) total chromatic dispersion (fibers) fiber core diameter, hole diameter, distance, width of the absorption region (photodetector), thickness of recombination region (optical source), pin diameter (mode scrambler) far-field mode-field diameter (single-mode fiber) near-field mode-field diameter (single-mode fiber) fiber outer (cladding) diameter electric field, energy, Young’s modulus, expected value of a random variable, electron energy activation energy of homogeneous degradation for an LED Fermi level (energy), quasi-Fermi level located in the conduction band (EFc), valence band (EFv) of a semiconductor separation energy between the valence and conduction bands in a semiconductor (bandgap energy) subcarrier electric field (analog transmission) optical energy separation energy of the quasi-Fermi levels electronic charge, base for natural logarithms probability of failure, transmission factor of a semiconductor–external interface, excess avalanche noise factor (F(M)), optical amplifier noise figure Fourier transformation noise figure (electronic amplifier) total noise figure for system of cascaded optical amplifiers frequency peak-to-peak frequency deviation (PFM–IM) peak frequency deviation (subcarrier FM and PM) Fabry–Pérot resonant frequency (optical amplifier), pulse rate (PFM–IM) open loop gain of an optical fiber receiver amplifier, photoconductive gain, cavity gain of a semiconductor laser amplifier amplitude function in the WKB method optical gain (phototransistor) parametric gain (fiber amplifier) Raman gain (fiber amplifier) single-pass gain of a semiconductor laser amplifier Gaussian (distribution) degeneracy parameter
  • OPTF_A01.qxd 11/6/08 10:52 Page xxxiv xxxiv C gm g0 gR Cth H H(ω) HA(ω) HCL(ω) Heq(ω) HOL(ω) Hout(ω) h hA(t) heff hFE hf(t) hout(t) hp(t) ht(t) I Ib Ibias Ic Id Io Ip IS Ith i ia iamp iD id idet if iN in is iSA List of symbols and abbreviations gain coefficient per unit length (laser cavity) transconductance of a field effect transistor, material gain coefficient unsaturated material gain coefficient power Raman gain coefficient threshold gain per unit length (laser cavity) magnetic field optical power transfer function (fiber), circuit transfer function optical fiber receiver amplifier frequency response (including any equalization) closed loop current to voltage transfer function (receiver amplifier) equalizer transfer function (frequency response) open loop current to voltage transfer function (receiver amplifier) output pulse spectrum from an optical fiber receiver Planck’s constant, thickness of a planar waveguide, power impulse response for optical fiber (h(t)), mode coupling parameter (PM fiber) optical fiber receiver amplifier impulse response (including any equalization) effective thickness of a planar waveguide common emitter current gain for a bipolar transistor optical fiber impulse response output pulse shape from an optical fiber receiver input pulse shape to an optical fiber receiver transmitted pulse shape on an optical fiber link electric current, optical intensity background-radiation-induced photocurrent (optical receiver) bias current for an optical detector collector current (phototransistor) dark current (optical detector) maximum optical intensity photocurrent generated in an optical detector output current from photodetector resulting from intermediate frequency in coherent receiver threshold current (injection laser) electric current optical receiver preamplifier shunt noise current optical receiver, preamplifier total noise current decision threshold current (digital transmission) photodiode dark noise current output current from an optical detector noise current generated in the feedback resistor of an optical fiber receiver transimpedance preamplifier total noise current at a digital optical fiber receiver multiplied shot noise current at the output of an APD excluding dark noise current shot noise current on the photocurrent for a photodiode multiplied shot noise current at the output of an APD including the noise current
  • OPTF_A01.qxd 11/6/08 10:52 Page xxxv List of symbols and abbreviations isig it iTS J Jth j K KI k kf kp L La LA Lac LB Lbc Lc LD Lex Lmap L0 Lt Ltr ᏸ l la l0 M Ma Mop Mx m ma N xxxv signal current obtained in an optical fiber receiver thermal noise current generated in a resistor total shot noise current for a photodiode without internal gain Bessel function, current density threshold current density (injection laser) −1 Boltzmann’s constant, constant, modified Bessel function stress intensity factor, for an elliptical crack (KIC) wave propagation constant in a vacuum (free space wave number), wave vector for an electron in a crystal, ratio of ionization rates for holes and electrons, integer, coupling coefficient for two interacting waveguide modes, constant angular frequency deviation (subcarrier FM) phase deviation constant (subcarrier PM) length (fiber), distance between mirrors (laser), coupling length (waveguide modes) length of amplifier (asymmetric twin-waveguide) amplifying space (soliton transmission) insertion loss of access coupler in distribution system beat length in a monomode optical fiber coherence length in a monomode optical fiber characteristic length (fiber) diffusion length of charge carriers (LED), fiber dispersion length star coupler excess loss in distribution system dispersion management map period constant with dimensions of length lateral misalignment loss at an optical fiber joint tap ratio loss in distribution system transmission loss factor (transmissivity) of an optical fiber azimuthal mode number, distance, length atomic spacing (bond distance) wave coupling length avalanche multiplication factor, material dispersion parameter, total number of guided modes or mode volume; for a multimode step index fiber (Ms); for multimode graded index fiber (Mg), mean value (M1) and mean square value (M2) of a random variable safety margin in an optical power budget optimum avalanche multiplication factor excess avalanche noise factor (also denoted as F(M)) radial mode number, Weibull distribution parameter, intensity-modulated optical signal (m(t)), mean value of a random variable, integer, optical modulation index (subcarrier amplitude modulation) modulation index integer, density of atoms in a particular energy level (e.g. N1, N2, N3), minority carrier concentration in n-type semiconductor material, number of input/output ports on a fiber star coupler, number of nodes on distribution
  • OPTF_A01.qxd 11/6/08 10:52 Page xxxvi xxxvi NA NEP Ng Nge N0 Np n ne neff n0 nsp P Pa PB Pb Pc PD Pdc Pe PG Pi Pin Pint PL Pm Po Popt Pout Pp Ppo Pr PR PRa(t) PS Ps Psc Pt List of symbols and abbreviations system, noise current, dimensionless combination of pulse and fiber parameters (soliton) numerical aperture of an optical fiber noise equivalent power group index of an optical waveguide effective group index or group index of a single-mode waveguide defined by Eq. (11.80) number of photons per bit (coherent transmission) refractive index (e.g. n1, n2, n3), stress corrosion susceptibility, negative-type semiconductor material, electron density, number of chips (OCDM) effective refractive index of a planar waveguide effective refractive index of a single-mode fiber refractive index of air spontaneous emission factor (injection laser) electric power, minority carrier concentration in p-type semiconductor material, probability of error (P(e)), of detecting a zero level (P(0)), of detecting a one level (P(1)), of detecting z photons in a particular time period (P(z)), conditional probability of detecting a zero when a one is transmitted (P(0|1)), of detecting a one when a zero is transmitted (P(1|0)), optical power (P1, P2, etc.) total power in a baseband message signal a(t) threshold optical power for Brillouin scattering backward traveling signal power (semiconductor laser amplifier), power transmitted through fiber sample optical power coupled into a step index fiber, optical power level optical power density d.c. optical output power optical power emitted from an optical source optical power in a guided mode mean input (transmitted) optical power launched into a fiber input signal power (semiconductor laser amplifier) internally generated optical power (optical source) optical power of local oscillator signal (coherent system) total power in an intensity-modulated optical signal m(t) mean output (received) optical power from a fiber mean optical power traveling in a fiber initial output optical (prior to degradation) power from an optical source optical pump power (fiber amplifier) peak received optical power reference optical power level, optical power level threshold optical power for Raman scattering backscattered optical power (Rayleigh) within a fiber optical power of incoming signal (coherent system) total power transmitted through a fiber sample optical power scattered from a fiber optical transmitter power, launch power (Ptx)
  • OPTF_A01.qxd 11/6/08 10:52 Page xxxvii List of symbols and abbreviations p q q0 R R12 R21 Ra Rb Rc RD REdB Rf RL ROdB Rt RTL r re rER, rET rHR, rHT rnr rp rr rt S Sf Si(r) Sm(ψ) S/N S0 St s T Ta Tc TD xxxvii crystal momentum, average photoelastic coefficient, positive-type semiconductor material, probability density function (p(x)) integer, fringe shift dimensionless parameter (soliton transmission) photodiode responsivity, radius of curvature of a fiber bend, electrical resistance (e.g. Rin, Rout); facet reflectivity (R1, R2) upward transition rate for electrons from energy level 1 to level 2 downward transition rate for electrons from energy level 2 to level 1 effective input resistance of an optical fiber receiver preamplifier bias resistance, for optical fiber receiver preamplifier (Rba) critical radius of an optical fiber radiance of an optical source ratio of electrical output power to electrical input power in decibels for an optical fiber system feedback resistance in an optical fiber receiver transimpedance preamplifier load resistance associated with an optical fiber detector ratio of optical output power to optical input power in decibels for an optical fiber system total carrier recombination rate (semiconductor optical source) total load resistance within an optical fiber receiver radial distance from the fiber axis, Fresnel reflection coefficient, mirror reflectivity, electro-optic coefficient. generated electron rate in an optical detector reflection and transmission coefficients, respectively, for the electric field at a planar, guide–cladding interface reflection and transmission coefficients respectively for the magnetic field at a planar, guide–cladding interface nonradiative carrier recombination rate per unit volume incident photon rate at an optical detector radiative carrier recombination rate per unit volume total carrier recombination rate per unit volume fraction of captured optical power, macroscopic stress, dispersion slope (fiber), power spectral density S(ω) fracture stress phase function in the WKB method spectral density of the intensity-modulated optical signal m(t) peak signal power to rms noise power ratio, with peak-to-peak signal power [(S/N)p–p] with rms signal power [(S/N)rms] scale parameter; zero-dispersion slope (fiber) theoretical cohesive strength pin spacing (mode scrambler) temperature, time, arbitrary parameter representing soliton pulse duration insertion loss resulting from an angular offset between jointed optical fibers 10 to 90% rise time arising from chromatic dispersion on an optical fiber link 10 to 90% rise time for an optical detector
  • OPTF_A01.qxd 11/6/08 10:52 Page xxxviii xxxviii List of symbols and abbreviations TF Tl Tn T0 TR TS Tsyst TT Tt T∞ t tc td te tr U V Vbias Vc VCC VCE VEE Veff Vopt Vsc v va vA(t) vc vd vg vout(t) vp W We Wo w X x Y y Z Z0 fictive temperature insertion loss resulting from a lateral offset between jointed optical fibers 10 to 90% rise time arising from intermodal dispersion on an optical fiber link threshold temperature (injection laser), nominal pulse period (PFM–IM) 10 to 90% rise time at the regenerator circuit input (PFM–IM) 10 to 90% rise time for an optical source total 10 to 90% rise time for an optical fiber system total insertion loss at an optical fiber joint temperature rise at time t maximum temperature rise time, carrier transit time, slow(ts), fast (tf) time constant switch-on delay (laser) 1/e pulse width from the center 10 to 90% rise time eigenvalue of the fiber core electrical voltage, normalized frequency for an optical fiber or planar waveguide bias voltage for a photodiode cutoff value of normalized frequency (fiber) collector supply voltage collector–emitter voltage (bipolar transistor) emitter supply voltage effective normalized frequency (fiber) voltage reading corresponding to the total optical power in a fiber voltage reading corresponding to the scattered optical power in a fiber electrical voltage amplifier series noise voltage receiver amplifier output voltage crack velocity drift velocity of carriers (photodiode) group velocity output voltage from an RC filter circuit phase velocity eigenvalue of the fiber cladding, random variable electric pulse width optical pulse width depletion layer width (photodiode) random variable coordinate, distance, constant, evanescent field penetration depth, slab thickness, grating line spacing constant, shunt admittance, random variable coordinate, lateral offset at a fiber joint random variable, constant electrical impedance
  • OPTF_A01.qxd 11/6/08 10:52 Page xxxix List of symbols and abbreviations z zm zmd α A αcr αdB αfc αi αj αm αN α0 αp αr β B βc β0 β2 βr Γ γ γp γR Δ Δf ΔG Δn δE δf δH δλ δT δ Tg δ Ts δ Tg ε xxxix coordinate, number of photons average or mean number of photons arriving at a detector in a time period τ average number of photons detected in a time period τ characteristic refractive index profile for fiber (profile parameter), optimum profile parameter (αop), linewidth enhancement factor (injection laser), optical link loss loss coefficient per unit length (laser cavity) connector loss at transmitter and receiver in decibels signal attenuation in decibels per unit length fiber cable loss in decibels per kilometer internal wavelength loss per unit length (injection laser) fiber joint loss in decibels per kilometer mirror loss per unit length (injection laser) signal attenuation in nepers absorption coefficient fiber transmission loss at the pump wavelength (fiber amplifier) radiation attenuation coefficient wave propagation constant gain factor (injection laser cavity) isothermal compressibility proportionality constant second-order dispersion coefficient degradation rate optical confinement factor (semiconductor laser amplifier) angle, attenuation coefficient per unit length for a fiber, nonlinear coefficient resulting from the Kerr effect surface energy of a material Rayleigh scattering coefficient for a fiber relative refractive index difference between the fiber core and cladding linewidth of single-frequency injection laser peak–trough ratio of the passband ripple (semiconductor laser amplifier) index difference between fiber core and cladding (Δn/n1 fractional index difference) phase shift associated with transverse electric waves uncorrelated source frequency widths phase shift associated with transverse magnetic waves optical source spectral width (linewidth), mode spacing (laser) intermodal dispersion time in an optical fiber delay difference between an extreme meridional ray and an axial ray for a graded index fiber delay difference between an extreme meridional ray and an axial ray for a step index fiber, with mode coupling (δ Tsc) polarization mode dispersion in fiber electric permittivity, of free space (ε0), relative (εr), semiconductor (εs), extinction ratio (optical transmitter)
  • OPTF_A01.qxd 11/6/08 10:52 Page xl xl ζ η ηang ηc ηD ηep ηext ηi ηint ηlat ηpc ηT θ θB Λ Λc λ λB λc λ0 μ ν ρ ρf σ σc σm σn σT τ τ21 τE τe τg τi τph τr τsp Φ φ ψ List of symbols and abbreviations solid acceptance angle quantum efficiency (optical detector) angular coupling efficiency (fiber joint) coupling efficiency (optical source to fiber) differential external quantum efficiency (optical source) external power efficiency (optical source) external quantum efficiency (light-emitting devices) internal quantum efficiency injection laser internal quantum efficiency (LED) lateral coupling efficiency (fiber joint) overall power conversion efficiency (optical source) total external quantum efficiency (optical source) angle, fiber acceptance angle (θa) Bragg diffraction angle, blaze angle diffraction grating acoustic wavelength, period for perturbations in a fiber, optical grating period, spacing between holes and pitch (photonic crystal fiber) cutoff period for perturbations in a fiber optical wavelength Bragg wavelength (DFB laser) long-wavelength cutoff (photodiode), cutoff wavelength for single-mode fiber, effective cutoff wavelength (λ ce) wavelength at which first-order dispersion is zero magnetic permeability, relative permeability, (μr), permeability of free space (μ0) optical source bandwidth in gigahertz polarization rotation in a single-mode optical fiber spectral density of the radiation energy at a transition frequency f standard deviation (rms pulse width), variance (σ 2) rms pulse broadening resulting from chromatic dispersion in a fiber rms pulse broadening resulting from material dispersion in a fiber rms pulse broadening resulting from intermodal dispersion in a graded index fiber (σg), in a step index fiber (σs) total rms pulse broadening in a fiber or fiber link time period, bit period, signaling interval, pulse duration, 3 dB pulse width (τ (3 dB)), retarded time spontaneous transition lifetime between energy levels 2 and 1 time delay in a transversal equalizer 1/e full width pulse broadening due to dispersion on an optical fiber link group delay injected (minority) carrier lifetime photon lifetime (semiconductor laser) radiative minority carrier lifetime spontaneous emission lifetime (equivalent to τ21) linear retardation angle, critical angle (φc), photon density, phase shift scalar quantity representing E or H field
  • OPTF_A01.qxd 11/6/08 10:52 Page xli List of symbols and abbreviations ω xli ω0 ∇ angular frequency, of the subcarrier waveform in analog transmission (ωc), of the modulating signal in analog transmission (ωm), pump frequency (ωp), Stokes component (ωs), anti-Stokes component (ωa), intermediate frequency of coherent heterodyne receiver (ω IF), normalized spot size of the fundamental mode spot size of the fundamental mode vector operator, Laplacian operator (∇2) A–D a.c. ADCCP AFC AGC AM AMI ANSI AOWC APD AR ARROW ASE ASK ASON ATM AWG BCH BER BERTS BGP BH BHC BHP BLSR BOD BPSK BXC CAPEX CATV CCTV CDH CMI CMOS CNR CO CPFSK CPU analog to digital alternating current advanced data communications control procedure (optical networks) automatic frequency control automatic gain control amplitude modulation alternate mark inversion (line code) American National Standards Institute all-optical wavelength converter avalanche photodiode antireflection (surface, coating) antiresonant reflecting optical waveguide amplified spontaneous emission (optical amplifier) amplitude shift keying automatic switched optical network alternative test method (fiber), asynchronous transfer mode (transmission) arrayed-waveguide grating Bose Chowdhry Hocquenghem (line codes) bit-error-rate bit-error-rate test set Border Gateway Protocol buried heterostructure (injection laser) burst header cell (optical switch) burst header packet (optical switch) bi-directional line-switched ring (optical networks) bistable optical device binary phase shift keying waveband cross-connect (optical networks) capital expenditure common antenna television closed circuit television constricted double heterojunction (injection laser) coded mark inversion complementary metal oxide silicon carrier to noise ratio central office (telephone switching center) continuous phase frequency shift keying central processing unit
  • OPTF_A01.qxd 11/6/08 10:52 Page xlii xlii List of symbols and abbreviations CRZ CSMA/CD CSP CSRZ CW CWDM D–A DB dB DBPSK DBR D–IM DC d.c. DCC DCF DDF DF DFB DFF DFG DGD DGE DH DI DLD DMS DOP DPSK DQPSK DS DSB DSD DSF DSL DSP DSTM DUT DWDM DWELL DXC E/O EAM ECL EDFA EDWA chirped return to zero Carrier Sense Multiple Access with Collision Detection channelled substrate planar (injection laser) carrier-suppressed return to zero continuous wave or operation coarse wavelength division multiplexing digital to analog duobinary (line code) decibel differential binary phase shift keying distributed Bragg reflector (laser) direct intensity modulation depressed cladding (fiber design) direct current data control channel (optical networks) dispersion-compensating fiber dispersion-decreasing fiber dispersion flattened (single-mode fiber) distributed feedback (injection laser) dispersion-flattened fiber difference frequency generation (nonlinear effect) differential group delay dynamic gain equalizer double heterostructure or heterojunction (injection laser or LED) delay interferometer dark line defect (semiconductor optical source) dispersion-managed soliton degree of polarization differential phase shift keying differential quadrature phase shift keying dispersion shifted (single-mode fiber) double sideband (amplitude modulation) dark spot defect (laser) dispersion-shifted fiber digital subscriber line, asymmetrical (ADSL), very high speed (VDSL) digital signal processing dynamic synchronous transfer mode device under test (fiber measurement) dense wavelength division multiplexing dots-in-well (photodiode) digital cross-connect electrical (or electronic) to optical conversion electro-absorption modulator emitter-coupler logic erbium-doped fiber amplifier erbium-doped waveguide amplifier
  • OPTF_A01.qxd 11/6/08 10:52 Page xliii List of symbols and abbreviations EH EIA ELED ELH EMFA EMI EMP EPON erf erfc ESI ESTI ETDM EYDFA FAST FBG FBT FC FDDI FDM FEC FET FFT FM FOTP FPA FSAN FSK FTTB FTTC FTTCab FTTH FWHP FWHM FWM FXC GbE GC-SOA GCSR GEM GEPON GFF GFP GI GMPLS GPON traditional mode designation Electronics Industries Association edge-emitting light-emitting diode extended long haul erbium micro-fiber amplifier electromagnetic interference electromagnetic pulse Ethernet passive optical network error function complementary error function equivalent step index (fiber) European Telecommunications Standards Institute electrical time division multiplexing erbium–ytterbium-doped fiber amplifier field assembly simple technique (optical connector) fiber Bragg grating fused biconical taper (fiber coupler) fiber connector, ferrule connector Fiber Distributed Data Interface frequency division multiplexing forward error correction field effect transistor, junction (JFET) fast Fourier transform frequency modulation Fiber Optic Test Procedure Fabry–Pérot amplifier Full Service Access Network frequency shift keying fiber-to-the-building fiber-to-the-curb fiber-to-the-cabinet fiber-to-the-home full width half power full width half maximum (equivalent to FWHP) four-wave mixing (nonlinear effect) fiber cross-connect gigabit Ethernet gain clamped-semiconductor optical amplifier grating-assisted codirectional coupler with sampled reflector (laser) gigabit passive optical network encapsulation method gigabit Ethernet passive optical network gain flattening filter generic frame procedure (network protocols) guard time insertion Generalized Multiprotocol Label Switching gigabit passive optical network xliii
  • OPTF_A01.qxd 8/28/09 10:16 AM Page xliv xliv List of symbols and abbreviations GRIN GTC GVD HB HBT HDB HDLC HDTV HE HEMT He–Ne HF HFC HV IF ILD IM I3O IEC IEEE IET IFFT IGA ILM INCITS IO I/O I and Q IP ISDN ISI IS–IS ISO ITU-T JET JIT LAN LB LC LDPC LEC LED LH LLC LO LOA graded index (rod lens) gigabit passive optical network transmission convergence group velocity delay (fiber) high birefringence (fiber) heterojunction bipolar transmitter high-density bipolar High-level Data Link Control (network protocol) high-definition television traditional mode designation high electron mobility transistor helium–neon (laser) high frequency hybrid fiber coaxial high voltage intermediate frequency injection laser diode intensity modulation, with direct detection (IM/DD) ion-implanted integrated optics International Electrotechnical Commission Institute of Electrical and Electronics Engineers Institution of Engineering and Technology inverse fast Fourier transform induced-grating autocorrelation (fiber measurement) integrated laser modulator International Committee for Information Technology Standards (ANSI) integrated optics input/output inphase and quadrature (coherent receiver) integrated photonics, Internet Protocol integrated services digital network, broadband (BISDN) intersymbol interference Intermediate-System-to-Intermediate-System (optical network protocol) International Organization for Standardization International Telecommunication Union – Telecom sector just-enough-time (optical network protocol) just-in-time (optical network protocol) local area network low birefringence (fiber) Lucent connector, local connector low-density parity check codes long external cavity (laser) light-emitting diode long haul logical link control (LAN) local oscillator linear optical amplifier
  • OPTF_A01.qxd 8/28/09 10:16 AM Page xlv List of symbols and abbreviations LOC LP LPE LR-PON LSP LSR LWPF MAC MAN MBE MC MCVD MEMS MESFET MFD MFSK MG-OXC MI MISFET MMF MMI MOSFET MOVPE MPLS MPO MQW MSM MTP MT-RJ MUSE MZI MZM NdDFA NDF Nd : YAG NIU NODG NOLM NRZ NT NZDF NZ-DSF O/E OADM OBPF OBS large optical cavity (injection laser) linearly polarized (mode notation) liquid-phase epitaxy long-reach passive optical network label-switched path (optical networks) label-switching router (optical networks) low-water-peak fiber Medium Access Control (LAN), isochronous (I-MAC) metropolitan area network molecular beam epitaxy matched cladding (fiber design) modified chemical vapor deposition micro-electro-mechanical systems, optical (OMEMS) metal Schottky field effect transistor mode-field diameter (single-mode fiber) multilevel frequency shift keying multi-granular optical cross-connect (optical networks) Michelson interferometer metal integrated-semiconductor field effect transistor multimode fiber multimode interference (optical coupler) metal oxide semiconductor field effect transistor metal oxide vapor-phase epitaxy Multiprotocol Label Switching multi-fiber push-on (fiber connector) multiquantum well metal–semiconductor–metal (photodetector) multifiber termination push-on (fiber connector) mechanical transfer registered jack (fiber connector) multiple sub-Nyquist sampling encoding Mach–Zehnder interferometer Mach–Zehnder modulator neodymium-doped fiber amplifier negative dispersion fiber neodymium-doped yttrium–aluminum–garnet (laser) networking interface unit nonlinear optical digital gate nonlinear optical loop mirror nonreturn-to-zero network termination nonzero-dispersion fiber nonzero-dispersion-shifted fiber optical to electrical (or electronic) conversion optical add/drop multiplexer optical bandpass filter optical burst switch(ed) xlv
  • OPTF_A01.qxd 11/6/08 10:52 Page xlvi xlvi List of symbols and abbreviations OC OCDM OCDMA OCh OCS OCWR ODN ODU OEIC OFDM OFSTP OLS OLT OMEMS OMS ONT ONU OOK OPEX OPM OPS OPU OSPF ORL OSI OTDM OTDR OTN OTS OTU OVPO OWC OXC PAM PANDA PASS PBC PBG PBS PC PCF PCM PCS PCVD PCW PD optical carrier (SONET) optical code division multiplexing optical code division multiple access optical channel (network protocols) optical circuit switch(ed) optical continuous wave reflectometer optical distribution node optical channel data unit (optical networks) optoelectronic integrated circuit orthogonal frequency division multiplexing, coherent optical (CO-OFDM) Optical Fiber System Test Procedure optical label switch(ed) optical line termination (optical networks) optical micro-electro-mechanical systems (integrated optics) optical multiplexing section (optical networks) optical network termination optical network unit on–off keying (equivalent to binary amplitude shift keying) operational expenditure optical power monitor (fiber measurement) optical physical section (optical networks) optical channel payload unit (optical networks) Open Shortest-Path First optical return loss Open Systems Interconnection optical time division multiplexing optical time domain reflectrometry optical transport network optical transmission section (optical networks) optical channel transport unit (optical networks) outside vapor-phase oxidation optical wavelength converter optical cross-connect pulse amplitude modulation polarization maintaining and absorption reducing (fiber) phased amplitude-shift signaling (line codes) polarization beam combiner photonic bandgap (microstructured fiber) polarization beam splitter physical contact (fiber connector) photonic crystal fiber pulse code modulation plastic-clad silica (fiber) plasma-activated chemical vapor deposition planar convex waveguide (injection laser) photodiode, photodetector
  • OPTF_A01.qxd 8/28/09 10:16 AM Page xlvii List of symbols and abbreviations PDF PDFA PDM PFM PFVBE PF-POF PHASAR PHY PIC PIN–FET PLC PLL PM PM–AM PMD PMMA POF POLMUX PoLSK PON POP PoS POTDR PPM PPP PR PSK PTT PWM QAM QC QD QD-SOA QPSK RAPD RCE RC-LED RDS RFD RIN RLD rms RN RNF RO xlvii probability density function praseodymium-doped fiber amplifier polarization division multiplexing pulse frequency modulation perfluoro-butenylvinyl ether perfluorinated plastic optical fiber phased array (integrated optics) physical layer (network reference model) photonic integrated circuit p–i–n photodiode followed by a field effect transistor planar lightwave circuit phase-locked loop phase modulation, polarization maintaining (fiber) phase modulation–amplitude modulation polarization mode dispersion (fiber), physical medium dependent (FDDI) polymethyl methacrylate plastic optical fiber polarization multiplexing polarization shift keying passive optical network, telephony (TPON), broadband (BPON), asynchronous (APON) points-of-presence (optical networks) packet over SONET polarization optical time domain reflectrometer pulse position modulation Point-to-Point Protocol partial response (line codes) phase shift keying post, telegraph and telecommunications pulse width modulation quadrature amplitude modulation quantum cascade (laser) quantum-dot quantum-dot semiconductor optical amplifier quadrature phase shift keying reach through avalanche photodiode resonant cavity enhanced (optical source/photodiode) resonant cavity light-emitting diode relative dispersion slope (fiber) reserve-a-fixed duration (optical switch) relative intensity noise (injection laser) reserve-a-limited duration (optical switch) root mean square remote node (optical networks) refracted near field (method for fiber refractive index profile measurement) relaxation oscillation
  • OPTF_A01.qxd 11/6/08 10:52 Page xlviii xlviii List of symbols and abbreviations ROADM RS RTM RWA Rx RZ SACM SAGCM SAM SAT SAW SBS SC SCM SDH SDM SEED SFF SFP SG-DBR SHF SIU SLA SLD SLED SMA SMC SMF SML SMT SNI SNR SONET SOA SOI SOL SOP SOS SPAD SPE SPM SQW SRS SSG-DBR SSMF ST reconfigurable optical add/drop multiplexer Reed–Solomon (line codes) reference test method (fiber) routing and wavelength assignment (optical networks) receiver return-to-zero separate absorption, charge and multiplication (avalanche photodiode) separate absorption, grading, charge and multiplication (avalanche photodiode) separate absorption and multiplication (avalanche photodiode) South Atlantic (optical fiber cable) surface acoustic wave stimulated Brillouin scattering subscriber connector (fiber) subcarrier multiplexing synchronous digital hierarchy space division multiplexing self-electro-optic device, symmetric (S-SEED) small form factor (fiber connector) small form pluggable (fiber connector) sampled-grating distributed Bragg reflector (laser) super high frequency subscriber interface unit semiconductor laser amplifier superluminescent diode surface emitter light-emitting diode subminiature A (fiber connector) subminiature C (fiber connector) single-mode fiber separated multiclad layer (injection laser) station management (FDDI) service network interface signal-to-noise ratio synchronous optical network semiconductor optical amplifier silicon-on-insulator (integrated optics) silicon-on-liquid (integrated optics) state of polarization silica-on-silicon (integrated optics) single-photon-counting avalanche photodetector synchronous payload envelope (SONET) self-phase modulation (nonlinear effect) single quantum well stimulated Raman scattering superstructure grating distributed Bragg reflector (laser) standard single-mode fiber straight tip (fiber connector)
  • OPTF_A01.qxd 11/6/08 10:52 Page xlix List of symbols and abbreviations STM STS TAG TAT TAW TCP TDFA TDS TDM TDMA TE Te-EDFA TEM ThDFA TIA TJS TM TOAD TRC TTL TWA Tx UDP UHF ULH UNI UTC VAD VC VCO VCSEL VHF VIFO VOA VPE VPN VSB VT WADD WAN WBC WBS WC WCB WDM WIXC synchronous transport module (SDH) synchronous transport signal (SONET) tell-and-go (optical network protocol) transatlantic (optical fiber cables) tell-and-wait (optical network protocol) Transmission Control Protocol thulium-doped fiber amplifier time domain sampling time division multiplexing time division multiple access transverse electric tellurium–erbium-doped fluoride fiber amplifier transverse electromagnetic thorium-doped fiber amplifier Telecommunication Industry Association transverse junction stripe (injection laser) transverse magnetic terahertz optical asymmetrical demultiplexer time-resolved chirp (lasers) transistor–transistor logic traveling wave amplifier transmitter User Datagram Protocol ultra high frequency ultra long haul user network interface unitraveling carrier (photodiode) vapor axial deposition virtual concatenation voltage-controlled oscillator vertical cavity surface-emitting laser very high frequency variable input–fixed output (wavelength conversion) variable optical attenuator vapor-phase epitaxy virtual private network vestigial sideband (modulation) virtual tributary (SONET) wavelength add/drop device wide area network waveband cross-connect (optical networks) waveband switching (optical networks) wavelength converter wavelength converter bank wavelength division multiplexing wavelength exchange cross-connect (optical networks) xlix
  • OPTF_A01.qxd 11/6/08 10:52 Page l l List of symbols and abbreviations WKB WRA WSS WXC XAM XGM XPM ZD ZMD ZWP Wentzel, Kramers, Brillouin (analysis technique for graded fiber) wavelength routing assignment (optical networks) wavelength selective switch wavelength cross-connecting (optical networks) cross-absorption modulation (nonlinear effect) cross-gain modulation (nonlinear effect) cross-phase modulation (nonlinear effect) Zener diode zero material dispersion (fiber) zero water peak (fiber)
  • OPTF_C01.qxd 11/6/08 10:52 Page 1 CHAPTER 1 Introduction 1.1 Historical development 1.2 The general system 1.3 Advantages of optical fiber communication References Communication may be broadly defined as the transfer of information from one point to another. When the information is to be conveyed over any distance a communication system is usually required. Within a communication system the information transfer is frequently achieved by superimposing or modulating the information onto an electromagnetic wave which acts as a carrier for the information signal. This modulated carrier is then transmitted to the required destination where it is received and the original information signal is obtained by demodulation. Sophisticated techniques have been developed for this process using electromagnetic carrier waves operating at radio frequencies as well as microwave and millimeter wave frequencies. However, ‘communication’ may also be achieved using an electromagnetic carrier which is selected from the optical range of frequencies. 1.1 Historical development The use of visible optical carrier waves or light for communication has been common for many years. Simple systems such as signal fires, reflecting mirrors and, more recently, signaling lamps have provided successful, if limited, information transfer. Moreover, as early as 1880 Alexander Graham Bell reported the transmission of speech using a light beam [Ref. 1]. The photophone proposed by Bell just four years after the invention of the telephone modulated sunlight with a diaphragm giving speech transmission over a distance of 200 m. However, although some investigation of optical communication continued in the early part of the twentieth century [Refs 2 and 3] its use was limited to mobile,
  • OPTF_C01.qxd 11/6/08 10:52 Page 2 2 Introduction Chapter 1 low-capacity communication links. This was due to both the lack of suitable light sources and the problem that light transmission in the atmosphere is restricted to line of sight and is severely affected by disturbances such as rain, snow, fog, dust and atmospheric turbulence. Nevertheless lower frequency and hence longer wavelength electromagnetic waves* (i.e. radio and microwave) proved suitable carriers for information transfer in the atmosphere, being far less affected by these atmospheric conditions. Depending on their wavelengths, these electromagnetic carriers can be transmitted over considerable distances but are limited in the amount of information they can convey by their frequencies (i.e. the information-carrying capacity is directly related to the bandwidth or frequency extent of the modulated carrier, which is generally limited to a fixed fraction of the carrier frequency). In theory, the greater the carrier frequency, the larger the available transmission bandwidth and thus the information-carrying capacity of the communication system. For this reason radio communication was developed to higher frequencies (i.e. VHF and UHF) leading to the introduction of the even higher frequency microwave and, latterly, millimeter wave transmission. The relative frequencies and wavelengths of these types of electromagnetic wave can be observed from the electromagnetic spectrum shown in Figure 1.1. In this context it may also be noted that communication at optical frequencies offers an increase in the potential usable bandwidth by a factor of around 104 over high-frequency microwave transmission. An additional benefit of the use of high carrier frequencies is the general ability of the communication system to concentrate the available power within the transmitted electromagnetic wave, thus giving an improved system performance [Ref. 4]. A renewed interest in optical communication was stimulated in the early 1960s with the invention of the laser [Ref. 5]. This device provided a powerful coherent light source, together with the possibility of modulation at high frequency. In addition the low beam divergence of the laser made enhanced free space optical transmission a practical possibility. However, the previously mentioned constraints of light transmission in the atmosphere tended to restrict these systems to short-distance applications. Nevertheless, despite the problems some modest free space optical communication links have been implemented for applications such as the linking of a television camera to a base vehicle and for data links of a few hundred meters between buildings. There is also some interest in optical communication between satellites in outer space using similar techniques [Ref. 6]. Although the use of the laser for free space optical communication proved somewhat limited, the invention of the laser instigated a tremendous research effort into the study of optical components to achieve reliable information transfer using a lightwave carrier. The proposals for optical communication via dielectric waveguides or optical fibers fabricated from glass to avoid degradation of the optical signal by the atmosphere were made almost simultaneously in 1966 by Kao and Hockham [Ref. 7] and Werts [Ref. 8]. Such systems were viewed as a replacement for coaxial cable or carrier transmission systems. Initially the optical fibers exhibited very high attenuation (i.e. 1000 dB km−1) and were therefore not comparable with the coaxial cables they were to replace (i.e. 5 to 10 dB km−1). There were also serious problems involved in jointing the fiber cables in a satisfactory manner to achieve low loss and to enable the process to be performed relatively easily and repeatedly * For the propagation of electromagnetic waves in free space, the wavelength λ equals the velocity of light in a vacuum c times the reciprocal of the frequency f in hertz or λ = c/f.
  • Figure 1.1 The electromagnetic spectrum showing the region used for optical fiber communications OPTF_C01.qxd 11/6/08 10:52 Page 3 Historical development 3
  • OPTF_C01.qxd 11/6/08 10:52 Page 4 4 Introduction Chapter 1 in the field. Nevertheless, within the space of 10 years optical fiber losses were reduced to below 5 dB km−1 and suitable low-loss jointing techniques were perfected. In parallel with the development of the fiber waveguide, attention was also focused on the other optical components which would constitute the optical fiber communication system. Since optical frequencies are accompanied by extremely small wavelengths, the development of all these optical components essentially required a new technology. Thus semiconductor optical sources (i.e. injection lasers and light-emitting diodes) and detectors (i.e. photodiodes and to a lesser extent phototransistors) compatible in size with optical fibers were designed and fabricated to enable successful implementation of the optical fiber system. Initially the semiconductor lasers exhibited very short lifetimes of at best a few hours, but significant advances in the device structure enabled lifetimes greater than 1000 h [Ref. 9] and 7000 h [Ref. 10] to be obtained by 1973 and 1977 respectively. These devices were originally fabricated from alloys of gallium arsenide (AlGaAs) which emitted in the near infrared between 0.8 and 0.9 μm. Subsequently the above wavelength range was extended to include the 1.1 to 1.6 μm region by the use of other semiconductor alloys (see Section 6.3.6) to take advantage of the enhanced performance characteristics displayed by optical fibers over this range. In particular for this longer wavelength region around 1.3 μm and 1.55 μm, semiconductor lasers and also the simpler structured light-emitting diodes based on the quaternary alloy InGaAsP-grown lattice matched to an InP substrate have been available since the late 1980s with projected median lifetimes in excess of 25 years (when operated at 10 °C) for the former and 100 years (when operated at 70 °C) for the latter device types [Ref. 11]. Hence the materials growth and fabrication technology has been developed specifically for telecommunication applications and it is now mature [Ref. 12]. Moreover, for telecommunication applications such lasers are often provided with a thermoelectric cooler together with a monitoring photodiode in the device package in order to facilitate current and thus temperature control. Direct modulation of commercial semiconductor lasers at 2.5 Gbit s−1 over single-mode fiber transmission distances up to 200 km at a wavelength of 1.55 μm can be achieved and this may be extended up to 10 Gbit s−1 over shorter unrepeated fiber links [Ref. 13]. Indeed, more recent research and development has focused on 40 Gbit s−1 transmission where external laser modulation is required using, for example, a Mach–Zehnder or an electroabsorption modulator (see Section 11.4.2) [Refs 14, 15]. This aspect also proves useful in the first longer wavelength window region around 1.3 μm where fiber intramodal dispersion is minimized and hence the transmission bandwidth is maximized, particularly for single-mode fibers. It is also noteworthy that this fiber type quickly came to dominate system applications within telecommunications since its initial field trial demonstration in 1982 [Ref. 16]. Moreover, the lowest silica glass fiber losses to date of 0.1484 dB km−1 were reported in 2002 for the other longer wavelength window at 1.57 μm [Ref. 17] but, unfortunately, chromatic dispersion is greater at this wavelength, thus limiting the maximum bandwidth achievable with conventional single-mode fiber. To obtain low loss over the entire fiber transmission longer wavelength region from 1.3 to 1.6 μm, or alternatively, very low loss and low dispersion at the same operating wavelength of typically 1.55 μm, advanced single-mode fiber structures have been commercially realized: namely, low-water-peak fiber and nonzero dispersion-shifted fiber. Although developments in fiber technology have continued rapidly over recent years, certain
  • OPTF_C01.qxd 11/6/08 10:52 Page 5 The general system 5 previously favored areas of interest such as the application of fluoride fibers for even longer wavelength operation in the mid-infrared (2 to 5 μm) and far-infrared (8 to 12 μm) regions have declined due to their failure to demonstrate practically the theoretically predicted, extremely low fiber losses combined with the emergence of optical amplifiers suitable for use with silica-based fibers. An important development, however, concerns the discovery of the phenomenon of photonic bandgaps which can be created in structures which propagate light, such as crystals or optical fibers. One particular form of photonic crystal fiber, for example, comprises a microstructured regular lattice of air holes running along its length (see Section 2.6). Such ‘holey’ fibers have the unusual property that they only transmit a single mode of light and hence form an entirely new single-mode fiber type which can carry more optical power than a conventional one. A further class of photonic bandgap fiber is defined by a large hollow core in which the light is guided. Such air guiding or hollow-core optical fibers could find application in photonic bandgap devices to provide dispersion compensation on long-haul fiber links or for high-resolution, tunable spectral filters [Ref. 18]. Nevertheless, even without the commercial availability of photonic bandgap devices, the implementation of a wide range of conventional fiber components (splices, connectors, couplers, etc.) and active optoelectronic devices (sources, detectors, amplifiers, etc.) has also moved to a stage of maturity. High-performance, reliable optical fiber communication systems and networks are therefore now widely deployed within the worldwide telecommunication network and in many more localized communication application areas. 1.2 The general system An optical fiber communication system is similar in basic concept to any type of communication system. A block schematic of a general communication system is shown in Figure 1.2(a), the function of which is to convey the signal from the information source over the transmission medium to the destination. The communication system therefore consists of a transmitter or modulator linked to the information source, the transmission medium, and a receiver or demodulator at the destination point. In electrical communications the information source provides an electrical signal, usually derived from a message signal which is not electrical (e.g. sound), to a transmitter comprising electrical and electronic components which converts the signal into a suitable form for propagation over the transmission medium. This is often achieved by modulating a carrier, which, as mentioned previously, may be an electromagnetic wave. The transmission medium can consist of a pair of wires, a coaxial cable or a radio link through free space down which the signal is transmitted to the receiver, where it is transformed into the original electrical information signal (demodulated) before being passed to the destination. However, it must be noted that in any transmission medium the signal is attenuated, or suffers loss, and is subject to degradations due to contamination by random signals and noise, as well as possible distortions imposed by mechanisms within the medium itself. Therefore, in any communication system there is a maximum permitted distance between the transmitter and the receiver beyond which the system effectively ceases to give intelligible communication. For longhaul applications these factors necessitate the installation of repeaters or line amplifiers
  • OPTF_C01.qxd 11/6/08 10:52 Page 6 6 Introduction Chapter 1 Figure 1.2 (a) The general communication system. (b) The optical fiber communication system (see Sections 12.4 and 12.10) at intervals, both to remove signal distortion and to increase signal level before transmission is continued down the link. For optical fiber communications the system shown in Figure 1.2(a) may be considered in slightly greater detail, as given in Figure 1.2(b). In this case the information source provides an electrical signal to a transmitter comprising an electrical stage which drives an optical source to give modulation of the lightwave carrier. The optical source which provides the electrical–optical conversion may be either a semiconductor laser or light-emitting diode (LED). The transmission medium consists of an optical fiber cable and the receiver consists of an optical detector which drives a further electrical stage and hence provides demodulation of the optical carrier. Photodiodes (p–n, p–i–n or avalanche) and, in some instances, phototransistors and photoconductors are utilized for the detection of the optical signal and the optical–electrical conversion. Thus there is a requirement for electrical interfacing at either end of the optical link and at present the signal processing is usually performed electrically.* The optical carrier may be modulated using either an analog or digital information signal. In the system shown in Figure 1.2(b) analog modulation involves the variation of the light emitted from the optical source in a continuous manner. With digital modulation, however, discrete changes in the light intensity are obtained (i.e. on–off pulses). Although often simpler to implement, analog modulation with an optical fiber communication system is less efficient, requiring a far higher signal-to-noise ratio at the receiver than digital modulation. Also, the linearity needed for analog modulation is not always provided by semiconductor optical sources, especially at high modulation frequencies. For these reasons, analog optical fiber communication links are generally limited to shorter distances and lower bandwidth operation than digital links. * Significant developments have taken place in devices for optical signal processing which are starting to alter this situation (see Chapter 11).
  • OPTF_C01.qxd 11/6/08 10:52 Page 7 Advantages of optical fiber communication 7 Figure 1.3 A digital optical fiber link using a semiconductor laser source and an avalanche photodiode (APD) detector Figure 1.3 shows a block schematic of a typical digital optical fiber link. Initially, the input digital signal from the information source is suitably encoded for optical transmission. The laser drive circuit directly modulates the intensity of the semiconductor laser with the encoded digital signal. Hence a digital optical signal is launched into the optical fiber cable. The avalanche photodiode (APD) detector is followed by a front-end amplifier and equalizer or filter to provide gain as well as linear signal processing and noise bandwidth reduction. Finally, the signal obtained is decoded to give the original digital information. The various elements of this and alternative optical fiber system configurations are discussed in detail in the following chapters. However, at this stage it is instructive to consider the advantages provided by lightwave communication via optical fibers in comparison with other forms of line and radio communication which have brought about the extensive use of such systems in many areas throughout the world. 1.3 Advantages of optical fiber communication Communication using an optical carrier wave guided along a glass fiber has a number of extremely attractive features, several of which were apparent when the technique was originally conceived. Furthermore, the advances in the technology to date have surpassed even the most optimistic predictions, creating additional advantages. Hence it is useful to consider the merits and special features offered by optical fiber communications over more conventional electrical communications. In this context we commence with the originally foreseen advantages and then consider additional features which have become apparent as the technology has been developed. (a) Enormous potential bandwidth. The optical carrier frequency in the range 1013 to 1016 Hz (generally in the near infrared around 1014 Hz or 105 GHz) yields a far greater potential transmission bandwidth than metallic cable systems (i.e. coaxial cable bandwidth typically around 20 MHz over distances up to a maximum of 10 km) or even millimeter wave radio systems (i.e. systems currently operating with modulation bandwidths of 700 MHz over a few hundreds of meters). Indeed, by the year 2000 the typical bandwidth multiplied by length product for an optical fiber link incorporating fiber amplifiers (see Section 10.4) was 5000 GHz km in comparison with the typical bandwidth–length product for coaxial cable of around 100 MHz km. Hence at this time optical fiber was already
  • OPTF_C01.qxd 11/6/08 10:52 Page 8 8 Introduction Chapter 1 demonstrating a factor of 50 000 bandwidth improvement over coaxial cable while also providing this superior information-carrying capacity over much longer transmission distances [Ref. 16]. Although the usable fiber bandwidth will be extended further towards the optical carrier frequency, it is clear that this parameter is limited by the use of a single optical carrier signal. Hence a much enhanced bandwidth utilization for an optical fiber can be achieved by transmitting several optical signals, each at different center wavelengths, in parallel on the same fiber. This wavelength division multiplexed operation (see Section 12.9.4), particularly with dense packing of the optical wavelengths (or, essentially, fine frequency spacing), offers the potential for a fiber information-carrying capacity that is many orders of magnitude in excess of that obtained using copper cables or a wideband radio system. (b) Small size and weight. Optical fibers have very small diameters which are often no greater than the diameter of a human hair. Hence, even when such fibers are covered with protective coatings they are far smaller and much lighter than corresponding copper cables. This is a tremendous boon towards the alleviation of duct congestion in cities, as well as allowing for an expansion of signal transmission within mobiles such as aircraft, satellites and even ships. (c) Electrical isolation. Optical fibers which are fabricated from glass, or sometimes a plastic polymer, are electrical insulators and therefore, unlike their metallic counterparts, they do not exhibit earth loop and interface problems. Furthermore, this property makes optical fiber transmission ideally suited for communication in electrically hazardous environments as the fibers create no arcing or spark hazard at abrasions or short circuits. (d) Immunity to interference and crosstalk. Optical fibers form a dielectric waveguide and are therefore free from electromagnetic interference (EMI), radio-frequency interference (RFI), or switching transients giving electromagnetic pulses (EMPs). Hence the operation of an optical fiber communication system is unaffected by transmission through an electrically noisy environment and the fiber cable requires no shielding from EMI. The fiber cable is also not susceptible to lightning strikes if used overhead rather than underground. Moreover, it is fairly easy to ensure that there is no optical interference between fibers and hence, unlike communication using electrical conductors, crosstalk is negligible, even when many fibers are cabled together. (e) Signal security. The light from optical fibers does not radiate significantly and therefore they provide a high degree of signal security. Unlike the situation with copper cables, a transmitted optical signal cannot be obtained from a fiber in a noninvasive manner (i.e. without drawing optical power from the fiber). Therefore, in theory, any attempt to acquire a message signal transmitted optically may be detected. This feature is obviously attractive for military, banking and general data transmission (i.e. computer network) applications. (f) Low transmission loss. The development of optical fibers over the last 20 years has resulted in the production of optical fiber cables which exhibit very low attenuation or transmission loss in comparison with the best copper conductors. Fibers have been
  • OPTF_C01.qxd 11/6/08 10:52 Page 9 Advantages of optical fiber communication 9 fabricated with losses as low as 0.15 dB km−1 (see Section 3.3.2) and this feature has become a major advantage of optical fiber communications. It facilitates the implementation of communication links with extremely wide optical repeater or amplifier spacings, thus reducing both system cost and complexity. Together with the already proven modulation bandwidth capability of fiber cables, this property has provided a totally compelling case for the adoption of optical fiber communications in the majority of long-haul telecommunication applications, replacing not only copper cables, but also satellite communications, as a consequence of the very noticeable delay incurred for voice transmission when using this latter approach. (g) Ruggedness and flexibility. Although protective coatings are essential, optical fibers may be manufactured with very high tensile strengths (see Section 4.6). Perhaps surprisingly for a glassy substance, the fibers may also be bent to quite small radii or twisted without damage. Furthermore, cable structures have been developed (see Section 4.8.4) which have proved flexible, compact and extremely rugged. Taking the size and weight advantage into account, these optical fiber cables are generally superior in terms of storage, transportation, handling and installation to corresponding copper cables, while exhibiting at least comparable strength and durability. (h) System reliability and ease of maintenance. These features primarily stem from the low-loss property of optical fiber cables which reduces the requirement for intermediate repeaters or line amplifiers to boost the transmitted signal strength. Hence with fewer optical repeaters or amplifiers, system reliability is generally enhanced in comparison with conventional electrical conductor systems. Furthermore, the reliability of the optical components is no longer a problem with predicted lifetimes of 20 to 30 years being quite common. Both these factors also tend to reduce maintenance time and costs. (i) Potential low cost. The glass which generally provides the optical fiber transmission medium is made from sand – not a scarce resource. So, in comparison with copper conductors, optical fibers offer the potential for low-cost line communication. Although over recent years this potential has largely been realized in the costs of the optical fiber transmission medium which for bulk purchases has become competitive with copper wires (i.e. twisted pairs), it has not yet been achieved in all the other component areas associated with optical fiber communications. For example, the costs of high-performance semiconductor lasers and detector photodiodes are still relatively high, as well as some of those concerned with the connection technology (demountable connectors, couplers, etc.). Overall system costs when utilizing optical fiber communication on long-haul links, however, are substantially less than those for equivalent electrical line systems because of the low-loss and wideband properties of the optical transmission medium. As indicated in (f), the requirement for intermediate repeaters and the associated electronics is reduced, giving a substantial cost advantage. Although this cost benefit gives a net gain for longhaul links, it is not always the case in short-haul applications where the additional cost incurred, due to the electrical–optical conversion (and vice versa), may be a deciding factor. Nevertheless, there are other possible cost advantages in relation to shipping, handling, installation and maintenance, as well as the features indicated in (c) and (d) which may prove significant in the system choice.
  • OPTF_C01.qxd 11/6/08 10:52 Page 10 10 Introduction Chapter 1 The reducing costs of optical fiber communications has provided strong competition not only with electrical line transmission systems, but also for microwave and millimeter wave radio transmission systems. Although these systems are reasonably wideband, the relatively short-span ‘line of sight’ transmission necessitates expensive aerial towers at intervals no greater than a few tens of kilometers. Hence, with the exception of the telecommunication access network (see Section 15.6.3) due primarily to current first installed cost constraints, optical fiber has become the dominant transmission medium within the major industrialized societies. Many advantages are therefore provided by the use of a lightwave carrier within a transmission medium consisting of an optical fiber. The fundamental principles giving rise to these enhanced performance characteristics, together with their practical realization, are described in the following chapters. However, a general understanding of the basic nature and properties of light is assumed. If this is lacking, the reader is directed to the many excellent texts encompassing the topic, a few of which are indicated in Refs 19 to 23. References [1] A. G. Bell, ‘Selenium and the photophone’, The Electrician, pp. 214, 215, 220, 221, 1880. [2] W. S. Huxford and J. R. Platt, ‘Survey of near infra-red communication systems’, J. Opt. Soc. Am., 38, pp. 253–268, 1948. [3] N. C. Beese, ‘Light sources for optical communication’, Infrared Phys., 1, pp. 5–16, 1961. [4] R. M. Gagliardi and S. Karp, Optical Communications, Wiley, 1976. [5] T. H. Maiman, ‘Stimulated optical radiation in ruby’, Nature, 187, pp. 493–494, 1960. [6] A. R. Kraemer, ‘Free-space optical communications’, Signal, pp. 26–32, 1977. [7] K. C. Kao and G. A. Hockham, ‘Dielectric fiber surface waveguides for optical frequencies’, Proc. IEE, 113(7), pp. 1151–1158, 1966. [8] A. Werts, ‘Propagation de la lumière cohérente dans les fibres optiques’, L’Onde Electrique, 46, pp. 967–980, 1966. [9] R. L. Hartman, J. C. Dyment, C. J. Hwang and H. Kuhn, ‘Continuous operation of GaAs–Gax Al1–x As, double heterostructure lasers with 330 °C half lives exceeding 1000 h’, Appl. Phys. Lett., 23(4), pp. 181–183, 1973. [10] A. R. Goodwin, J. F. Peters, M. Pion and W. O. Bourne, ‘GaAs lasers with consistently low degradation rates at room temperature’, Appl. Phys. Lett., 30(2), pp. 110–113, 1977. [11] S. E. Miller, ‘Overview and summary of progress’, in S. E. Miller and I. P. Kaminow (Eds), Optical Fiber Telecommunications II, pp. 1–27, Academic Press, 1988. [12] E. Garmire, ‘Sources, modulators, and detectors for fiber-optic communication systems’, in M. Bass and Eric W. Van Stryland (Eds), Fiber Optics Handbook, pp. 4.1–4.80, McGraw-Hill, 2002. [13] D. A. Ackerman, J. E. Johnson, L. J. P. Ketelsen, L. E. Eng, P. A. Kiely and T. G. B. Mason, ‘Telecommunication lasers’, in I. P. Kaminow and T. Li (Eds), Optical Fiber Telecommunications IVA, pp. 587–665, Academic Press, 2002. [14] R. DeSalvo et al., ‘Advanced components and sub-system solutions for 40 Gb/s transmission’, J. Lightwave Technol., 20(12), pp. 2154–2181, 2002. [15] A. Belahlou et al., ‘Fiber design considerations for 40 Gb/s systems’, J. Lightwave Technol., 20(12), pp. 2290–2305, 2002.
  • OPTF_C01.qxd 11/6/08 10:52 Page 11 References 11 [16] W. A. Gambling, ‘The rise and rise of optical fibers’, IEEE J. Sel. Top. Quantum Elecbron, 6(6), pp. 1084–1093, 2000. [17] K. Nayayama, M. Kakui, M. Matsui, T. Saitoh and Y. Chigusa, ‘Ultra-low-loss (0.1484 dB/km) pure silica core fibre and extension of transmission distance’, Electron. Lett., 38(20), pp. 1168–1169, 2002. [18] K. Oh, S. Choi, Y. Jung and J. W. Lee, ‘Novel hollow optical fibers and their applications in photonic devices for optical communications’, J. Lightwave Technol., 23(2), pp. 524–532, 2005. [19] M. Born and E. Wolf, Principles of Optics (7th edn), Cambridge University Press, 1999. [20] W. J. Smith, Modern Optical Engineering (3rd edn), McGraw-Hill, 2000. [21] E. Hecht and A. Zajac, Optics (4th edn), Addison-Wesley, 2003. [22] F. L. Pedrotti, L. S. Pedrotti and L. M. Pedrotti, Introduction to Optics (3rd edn), Prentice Hall, 2006. [23] F. Graham Smith, T. A. King and D. Wilkins, Optics and Photonics: An Introduction (2nd edn), Wiley, 2007.
  • OPTF_C02.qxd 11/6/08 10:53 Page 12 CHAPTER 2 Optical fiber waveguides 2.1 Introduction 2.2 Ray theory transmission 2.3 Electromagnetic mode theory for optical propagation 2.4 Cylindrical fiber 2.5 Single-mode fibers 2.6 Photonic crystal fibers Problems References 2.1 Introduction The transmission of light via a dielectric waveguide structure was first proposed and investigated at the beginning of the twentieth century. In 1910 Hondros and Debye [Ref. 1] conducted a theoretical study, and experimental work was reported by Schriever in 1920 [Ref. 2]. However, a transparent dielectric rod, typically of silica glass with a refractive index of around 1.5, surrounded by air, proved to be an impractical waveguide due to its unsupported structure (especially when very thin waveguides were considered in order to limit the number of optical modes propagated) and the excessive losses at any discontinuities of the glass–air interface. Nevertheless, interest in the application of dielectric optical waveguides in such areas as optical imaging and medical diagnosis (e.g. endoscopes) led to proposals [Refs 3, 4] for a clad dielectric rod in the mid-1950s in order to overcome these problems. This structure is illustrated in Figure 2.1, which shows a transparent core with a refractive index n1 surrounded by a transparent cladding of slightly lower refractive index n2. The cladding supports the waveguide structure while also, when
  • OPTF_C02.qxd 11/6/08 10:53 Page 13 Introduction 13 Figure 2.1 Optical fiber waveguide showing the core of refractive index n1, surrounded by the cladding of slightly lower refractive index n2 sufficiently thick, substantially reducing the radiation loss into the surrounding air. In essence, the light energy travels in both the core and the cladding allowing the associated fields to decay to a negligible value at the cladding–air interface. The invention of the clad waveguide structure led to the first serious proposals by Kao and Hockham [Ref. 5] and Werts [Ref. 6], in 1966, to utilize optical fibers as a communications medium, even though they had losses in excess of 1000 dB km−1. These proposals stimulated tremendous efforts to reduce the attenuation by purification of the materials. This has resulted in improved conventional glass refining techniques giving fibers with losses of around 4.2 dB km−1 [Ref. 7]. Also, progress in glass refining processes such as depositing vapor-phase reagents to form silica [Ref. 8] allowed fibers with losses below 1 dB km−1 to be fabricated. Most of this work was focused on the 0.8 to 0.9 μm wavelength band because the first generation of optical sources fabricated from gallium aluminum arsenide alloys operated in this region. However, as silica fibers were studied in further detail it became apparent that transmission at longer wavelengths (1.1 to 1.6 μm) would result in lower losses and reduced signal dispersion. This produced a shift in optical fiber source and detector technology in order to provide operation at these longer wavelengths. Hence at longer wavelengths, especially around 1.55 μm, typical high-performance fibers have losses of 0.2 dB km−1 [Ref. 9]. As such losses are very close to the theoretical lower limit for silicate glass fiber, there is interest in glass-forming systems which can provide low-loss transmission in the midinfrared (2 to 5 μm) optical wavelength regions. Although a system based on fluoride glass offers the potential for ultra-low-loss transmission of 0.01 dB km−1 at a wavelength of 2.55 μm, such fibers still exhibit losses of at least 0.65 dB km−1 and they also cannot yet be produced with the robust mechanical properties of silica fibers [Ref. 10]. In order to appreciate the transmission mechanism of optical fibers with dimensions approximating to those of a human hair, it is necessary to consider the optical waveguiding of a cylindrical glass fiber. Such a fiber acts as an open optical waveguide, which may be analyzed utilizing simple ray theory. However, the concepts of geometric optics are not sufficient when considering all types of optical fiber, and electromagnetic mode theory must be used to give a complete picture. The following sections will therefore outline the transmission of light in optical fibers prior to a more detailed discussion of the various types of fiber. In Section 2.2 we continue the discussion of light propagation in optical fibers using the ray theory approach in order to develop some of the fundamental parameters associated with optical fiber transmission (acceptance angle, numerical aperture, etc.). Furthermore,
  • OPTF_C02.qxd 11/6/08 10:53 Page 14 14 Optical fiber waveguides Chapter 2 this provides a basis for the discussion of electromagnetic wave propagation presented in Section 2.3, where the electromagnetic mode theory is developed for the planar (rectangular) waveguide. Then, in Section 2.4, we discuss the waveguiding mechanism within cylindrical fibers prior to consideration of both step and graded index fibers. Finally, in Section 2.5 the theoretical concepts and important parameters (cutoff wavelength, spot size, propagation constant, etc.) associated with optical propagation in single-mode fibers are introduced and approximate techniques to obtain values for these parameters are described. All consideration in the above sections is concerned with what can be referred to as conventional optical fiber in the context that it comprises both solid-core and cladding regions as depicted in Figure 2.1. In the mid-1990s, however, a new class of microstructured optical fiber, termed photonic crystal fiber, was experimentally demonstrated [Ref. 11] which has subsequently exhibited the potential to deliver applications ranging from light transmission over distance to optical device implementations (e.g. power splitters, amplifiers, bistable switches, wavelength converters). The significant physical feature of this microstructured optical fiber is that it typically contains an array of air holes running along the longitudinal axis rather than consisting of a solid silica rod structure. Moreover, the presence of these holes provides an additional dimension to fiber design which has already resulted in new developments for both guiding and controlling light. Hence the major photonic crystal fiber structures and their guidance mechanisms are outlined and discussed in Section 2.6 in order to give an insight into the fundamental developments of this increasingly important fiber class. 2.2 Ray theory transmission 2.2.1 Total internal reflection To consider the propagation of light within an optical fiber utilizing the ray theory model it is necessary to take account of the refractive index of the dielectric medium. The refractive index of a medium is defined as the ratio of the velocity of light in a vacuum to the velocity of light in the medium. A ray of light travels more slowly in an optically dense medium than in one that is less dense, and the refractive index gives a measure of this effect. When a ray is incident on the interface between two dielectrics of differing refractive indices (e.g. glass–air), refraction occurs, as illustrated in Figure 2.2(a). It may be observed that the ray approaching the interface is propagating in a dielectric of refractive index n1 and is at an angle φ1 to the normal at the surface of the interface. If the dielectric on the other side of the interface has a refractive index n2 which is less than n1, then the refraction is such that the ray path in this lower index medium is at an angle φ2 to the normal, where φ2 is greater than φ1. The angles of incidence φ1 and refraction φ2 are related to each other and to the refractive indices of the dielectrics by Snell’s law of refraction [Ref. 12], which states that: n1 sin φ1 = n2 sin φ2
  • OPTF_C02.qxd 11/6/08 10:53 Page 15 15 Ray theory transmission Figure 2.2 Light rays incident on a high to low refractive index interface (e.g. glass–air): (a) refraction; (b) the limiting case of refraction showing the critical ray at an angle φc; (c) total internal reflection where φ > φc or: sin φ1 n2 = sin φ2 n1 (2.1) It may also be observed in Figure 2.2(a) that a small amount of light is reflected back into the originating dielectric medium (partial internal reflection). As n1 is greater than n2, the angle of refraction is always greater than the angle of incidence. Thus when the angle of refraction is 90° and the refracted ray emerges parallel to the interface between the dielectrics, the angle of incidence must be less than 90°. This is the limiting case of refraction and the angle of incidence is now known as the critical angle φc, as shown in Figure 2.2(b). From Eq. (2.1) the value of the critical angle is given by: sin φc = n2 n1 (2.2) At angles of incidence greater than the critical angle the light is reflected back into the originating dielectric medium (total internal reflection) with high efficiency (around 99.9%). Hence, it may be observed in Figure 2.2(c) that total internal reflection occurs at the interface between two dielectrics of differing refractive indices when light is incident on the dielectric of lower index from the dielectric of higher index, and the angle of incidence of
  • OPTF_C02.qxd 11/6/08 10:53 Page 16 16 Optical fiber waveguides Chapter 2 Figure 2.3 The transmission of a light ray in a perfect optical fiber the ray exceeds the critical value. This is the mechanism by which light at a sufficiently shallow angle (less than 90° − φc) may be considered to propagate down an optical fiber with low loss. Figure 2.3 illustrates the transmission of a light ray in an optical fiber via a series of total internal reflections at the interface of the silica core and the slightly lower refractive index silica cladding. The ray has an angle of incidence φ at the interface which is greater than the critical angle and is reflected at the same angle to the normal. The light ray shown in Figure 2.3 is known as a meridional ray as it passes through the axis of the fiber core. This type of ray is the simplest to describe and is generally used when illustrating the fundamental transmission properties of optical fibers. It must also be noted that the light transmission illustrated in Figure 2.3 assumes a perfect fiber, and that any discontinuities or imperfections at the core–cladding interface would probably result in refraction rather than total internal reflection, with the subsequent loss of the light ray into the cladding. 2.2.2 Acceptance angle Having considered the propagation of light in an optical fiber through total internal reflection at the core–cladding interface, it is useful to enlarge upon the geometric optics approach with reference to light rays entering the fiber. Since only rays with a sufficiently shallow grazing angle (i.e. with an angle to the normal greater than φc) at the core–cladding interface are transmitted by total internal reflection, it is clear that not all rays entering the fiber core will continue to be propagated down its length. The geometry concerned with launching a light ray into an optical fiber is shown in Figure 2.4, which illustrates a meridional ray A at the critical angle φc within the fiber at the core–cladding interface. It may be observed that this ray enters the fiber core at an angle θa to the fiber axis and is refracted at the air–core interface before transmission to the core–cladding interface at the critical angle. Hence, any rays which are incident into the fiber core at an angle greater than θa will be transmitted to the core–cladding interface at an angle less than φc, and will not be totally internally reflected. This situation is also illustrated in Figure 2.4, where the incident ray B at an angle greater than θa is refracted into the cladding and eventually lost by radiation. Thus for rays to be transmitted by total internal reflection within the fiber core they must be incident on the fiber core within an acceptance cone defined by the conical half angle θa. Hence θa is the maximum angle to the axis at which light may enter the fiber in order to be propagated, and is often referred to as the acceptance angle* for the fiber. * θa is sometimes referred to as the maximum or total acceptance angle.
  • OPTF_C02.qxd 11/6/08 10:53 Page 17 Ray theory transmission 17 Figure 2.4 The acceptance angle θa when launching light into an optical fiber If the fiber has a regular cross-section (i.e. the core–cladding interfaces are parallel and there are no discontinuities) an incident meridional ray at greater than the critical angle will continue to be reflected and will be transmitted through the fiber. From symmetry considerations it may be noted that the output angle to the axis will be equal to the input angle for the ray, assuming the ray emerges into a medium of the same refractive index from which it was input. 2.2.3 Numerical aperture The acceptance angle for an optical fiber was defined in the preceding section. However, it is possible to continue the ray theory analysis to obtain a relationship between the acceptance angle and the refractive indices of the three media involved, namely the core, cladding and air. This leads to the definition of a more generally used term, the numerical aperture of the fiber. It must be noted that within this analysis, as with the preceding discussion of acceptance angle, we are concerned with meridional rays within the fiber. Figure 2.5 shows a light ray incident on the fiber core at an angle θ1 to the fiber axis which is less than the acceptance angle for the fiber θa. The ray enters the fiber from a Figure 2.5 The ray path for a meridional ray launched into an optical fiber in air at an input angle less than the acceptance angle for the fiber
  • OPTF_C02.qxd 11/6/08 10:53 Page 18 18 Optical fiber waveguides Chapter 2 medium (air) of refractive index n0, and the fiber core has a refractive index n1, which is slightly greater than the cladding refractive index n2. Assuming the entrance face at the fiber core to be normal to the axis, then considering the refraction at the air–core interface and using Snell’s law given by Eq. (2.1): n0 sin θ1 = n1 sin θ2 (2.3) Considering the right-angled triangle ABC indicated in Figure 2.5, then: φ= π − θ2 2 (2.4) where φ is greater than the critical angle at the core–cladding interface. Hence Eq. (2.3) becomes: n0 sin θ1 = n1 cos φ (2.5) Using the trigonometrical relationship sin2 φ + cos2 φ = 1, Eq. (2.5) may be written in the form: 1 2 n0 sin θ1 = nl (l − sin2 φ)--- (2.6) When the limiting case for total internal reflection is considered, φ becomes equal to the critical angle for the core–cladding interface and is given by Eq. (2.2). Also in this limiting case θ1 becomes the acceptance angle for the fiber θa. Combining these limiting cases into Eq. (2.6) gives: 1 2 2 n0 sin θa = (n1 − n 2)--2 (2.7) Equation (2.7), apart from relating the acceptance angle to the refractive indices, serves as the basis for the definition of the important optical fiber parameter, the numerical aperture (NA). Hence the NA is defined as: 1 2 2 NA = n0 sin θa = (n1 − n 2)--2 (2.8) Since the NA is often used with the fiber in air where n0 is unity, it is simply equal to sin θa. It may also be noted that incident meridional rays over the range 0 ≤ θ1 ≤ θa will be propagated within the fiber. The NA may also be given in terms of the relative refractive index difference Δ between the core and the cladding which is defined as:* * Sometimes another parameter Δn = n1 − n2 is referred to as the index difference and Δn/n1 as the fractional index difference. Hence Δ also approximates to the fractional index difference.
  • OPTF_C02.qxd 11/6/08 10:53 Page 19 19 Ray theory transmission Δ= n2 − n2 1 2 2n 2 1 Ӎ n1 − n2 n1 for Δ Ӷ 1 (2.9) Hence combining Eq. (2.8) with Eq. (2.9) we can write: 1 2 NA = n1(2Δ)--- (2.10) The relationships given in Eqs (2.8) and (2.10) for the numerical aperture are a very useful measure of the light-collecting ability of a fiber. They are independent of the fiber core diameter and will hold for diameters as small as 8 μm. However, for smaller diameters they break down as the geometric optics approach is invalid. This is because the ray theory model is only a partial description of the character of light. It describes the direction a plane wave component takes in the fiber but does not take into account interference between such components. When interference phenomena are considered it is found that only rays with certain discrete characteristics propagate in the fiber core. Thus the fiber will only support a discrete number of guided modes. This becomes critical in smallcore-diameter fibers which only support one or a few modes. Hence electromagnetic mode theory must be applied in these cases (see Section 2.3). Example 2.1 A silica optical fiber with a core diameter large enough to be considered by ray theory analysis has a core refractive index of 1.50 and a cladding refractive index of 1.47. Determine: (a) the critical angle at the core–cladding interface; (b) the NA for the fiber; (c) the acceptance angle in air for the fiber. Solution: (a) The critical angle φc at the core–cladding interface is given by Eq. (2.2) where: φc = sin−1 n2 1.47 = sin−1 n1 1.50 = 78.5° (b) From Eq. (2.8) the NA is: 1 1 2 2 2 NA = (n1 − n 2)--- = (1.502 − 1.472)--2 1 --2 = (2.25 − 2.16) = 0.30 (c) Considering Eq. (2.8) the acceptance angle in air θa is given by: θa = sin−1 NA = sin−1 0.30 = 17.4°
  • OPTF_C02.qxd 11/6/08 10:53 Page 20 20 Optical fiber waveguides Chapter 2 Example 2.2 A typical relative refractive index difference for an optical fiber designed for longdistance transmission is 1%. Estimate the NA and the solid acceptance angle in air for the fiber when the core index is 1.46. Further, calculate the critical angle at the core–cladding interface within the fiber. It may be assumed that the concepts of geometric optics hold for the fiber. Solution: Using Eq. (2.10) with Δ = 0.01 gives the NA as: 1 1 2 2 NA = n1(2Δ)--- = 1.46(0.02)--= 0.21 For small angles the solid acceptance angle in air ζ is given by: 2 ζ Ӎ πθ a = π sin2 θa Hence from Eq. (2.8): ζ Ӎ π (NA)2 = π × 0.04 = 0.13 rad Using Eq. (2.9) for the relative refractive index difference Δ gives: ΔӍ n1 − n2 n =1− 2 n1 n1 Hence n2 = 1 − Δ = 1 − 0.01 n1 = 0.99 From Eq. (2.2) the critical angle at the core–cladding interface is: φc = sin−1 n2 = sin−1 0.99 n1 = 81.9° 2.2.4 Skew rays In the preceding sections we have considered the propagation of meridional rays in the optical waveguide. However, another category of ray exists which is transmitted without passing through the fiber axis. These rays, which greatly outnumber the meridional rays,
  • OPTF_C02.qxd 11/6/08 10:53 Page 21 Ray theory transmission 21 Figure 2.6 The helical path taken by a skew ray in an optical fiber: (a) skew ray path down the fiber; (b) cross-sectional view of the fiber follow a helical path through the fiber, as illustrated in Figure 2.6, and are called skew rays. It is not easy to visualize the skew ray paths in two dimensions, but it may be observed from Figure 2.6(b) that the helical path traced through the fiber gives a change in direction of 2γ at each reflection, where γ is the angle between the projection of the ray in two dimensions and the radius of the fiber core at the point of reflection. Hence, unlike meridional rays, the point of emergence of skew rays from the fiber in air will depend upon the number of reflections they undergo rather than the input conditions to the fiber. When the light input to the fiber is nonuniform, skew rays will therefore tend to have a smoothing effect on the distribution of the light as it is transmitted, giving a more uniform output. The amount of smoothing is dependent on the number of reflections encountered by the skew rays. A further possible advantage of the transmission of skew rays becomes apparent when their acceptance conditions are considered. In order to calculate the acceptance angle for a skew ray it is necessary to define the direction of the ray in two perpendicular planes. The geometry of the situation is illustrated in Figure 2.7 where a skew ray is shown incident on the fiber core at the point A, at an angle θs to the normal at the fiber end face. The ray is refracted at the air–core interface before traveling to the point B in the same plane. The angles of incidence and reflection at the point B are φ, which is greater than the critical angle for the core–cladding interface. When considering the ray between A and B it is necessary to resolve the direction of the ray path AB to the core radius at the point B. As the incident and reflected rays at the point B are in the same plane, this is simply cos φ. However, if the two perpendicular planes through which the ray path AB traverses are considered, then γ is the angle between the core radius and the projection of the ray onto a plane BRS normal to the core axis, and θ is the angle between the ray and a line AT drawn parallel to the core axis. Thus to resolve the ray path AB relative to the radius BR in these two perpendicular planes requires multiplication by cos γ and sin θ.
  • OPTF_C02.qxd 11/6/08 10:53 Page 22 22 Optical fiber waveguides Chapter 2 Figure 2.7 The ray path within the fiber core for a skew ray incident at an angle θs to the normal at the air–core interface Hence, the reflection at point B at an angle φ may be given by: cos γ sin θ = cos φ (2.11) Using the trigonometrical relationship sin2 φ + cos2 φ = 1, Eq. (2.11) becomes: 1 2 cos γ sin θ = cos φ = (1 − sin2 φ)--- (2.12) If the limiting case for total internal reflection is now considered, then φ becomes equal to the critical angle φc for the core–cladding interface and, following Eq. (2.2), is given by sin φc = n2/n1. Hence, Eq. (2.12) may be written as: 1 --- A n2 D 2 cos γ sin θ ≤ cos φc = 1 − 2 C n2 F 1 (2.13) Furthermore, using Snell’s law at the point A, following Eq. (2.1) we can write: n0 sin θa = n1 sin θ (2.14) where θa represents the maximum input axial angle for meridional rays, as expressed in Section 2.2.2, and θ is the internal axial angle. Hence substituting for sin θ from Eq. (2.13) into Eq. (2.14) gives: 1 --- n cos φc n1 A n2 D 2 sin θas = 1 = 1− 2 n0 cos γ n0 cos γ C n2 F 1 (2.15) where θas now represents the maximum input angle or acceptance angle for skew rays. It may be noted that the inequality shown in Eq. (2.13) is no longer necessary as all the terms
  • OPTF_C02.qxd 11/6/08 10:53 Page 23 Ray theory transmission 23 in Eq. (2.15) are specified for the limiting case. Thus the acceptance conditions for skew rays are: 1 2 2 n0 sin θas cos γ = (n1 − n 2)--- = NA 2 (2.16) and in the case of the fiber in air (n0 = 1): sin θas cos γ = NA (2.17) Therefore by comparison with Eq. (2.8) derived for meridional rays, it may be noted that skew rays are accepted at larger axial angles in a given fiber than meridional rays, depending upon the value of cos γ. In fact, for meridional rays cos γ is equal to unity and θas becomes equal to θa. Thus although θa is the maximum conical half angle for the acceptance of meridional rays, it defines the minimum input angle for skew rays. Hence, as may be observed from Figure 2.6, skew rays tend to propagate only in the annular region near the outer surface of the core, and do not fully utilize the core as a transmission medium. However, they are complementary to meridional rays and increase the light-gathering capacity of the fiber. This increased light-gathering ability may be significant for large NA fibers, but for most communication design purposes the expressions given in Eqs (2.8) and (2.10) for meridional rays are considered adequate. Example 2.3 An optical fiber in air has an NA of 0.4. Compare the acceptance angle for meridional rays with that for skew rays which change direction by 100° at each reflection. Solution: The acceptance angle for meridional rays is given by Eq. (2.8) with n0 = 1 as: θa = sin−1 NA = sin−1 0.4 = 23.6° The skew rays change direction by 100° at each reflection, therefore γ = 50°. Hence using Eq. (2.17) the acceptance angle for skew rays is: θas = sin−1 A 0.4 D A NA D = sin−1 C cos 50° F C cos γ F = 38.5° In this example, the acceptance angle for the skew rays is about 15° greater than the corresponding angle for meridional rays. However, it must be noted that we have only compared the acceptance angle of one particular skew ray path. When the light input to the fiber is at an angle to the fiber axis, it is possible that γ will vary from zero for meridional rays to 90° for rays which enter the fiber at the core–cladding interface giving acceptance of skew rays over a conical half angle of π/2 radians.
  • OPTF_C02.qxd 11/6/08 10:53 Page 24 24 Optical fiber waveguides Chapter 2 2.3 Electromagnetic mode theory for optical propagation 2.3.1 Electromagnetic waves In order to obtain an improved model for the propagation of light in an optical fiber, electromagnetic wave theory must be considered. The basis for the study of electromagnetic wave propagation is provided by Maxwell’s equations [Ref. 13]. For a medium with zero conductivity these vector relationships may be written in terms of the electric field E, magnetic field H, electric flux density D and magnetic flux density B as the curl equations: ∇×E=− ∇×H= ∂B ∂t (2.18) ∂D ∂t (2.19) and the divergence conditions: ∇⋅D=0 (no free charges) (2.20) ∇⋅B=0 (no free poles) (2.21) where ∇ is a vector operator. The four field vectors are related by the relations: D = εE B = μH (2.22) where ε is the dielectric permittivity and μ is the magnetic permeability of the medium. Substituting for D and B and taking the curl of Eqs (2.18) and (2.19) gives: ∇ × (∇ × E) = −με ∇ ∂ 2E ∂t 2 (2.23) ∇ × (∇ × H) = −με ∇ ∂ 2H ∂t 2 (2.24) Then using the divergence conditions of Eqs (2.20) and (2.21) with the vector identity: ∇ × (∇ × Y) = ∇(∇ ⋅ Y) − ∇2(Y) ∇ ∇ we obtain the nondispersive wave equations: ∇2E = με ∂ 2E ∂t 2 (2.25)
  • OPTF_C02.qxd 11/6/08 10:53 Page 25 Electromagnetic mode theory for optical propagation 25 and: ∇2H = με ∂ 2H ∂t 2 (2.26) where ∇2 is the Laplacian operator. For rectangular Cartesian and cylindrical polar coordinates the above wave equations hold for each component of the field vector, every component satisfying the scalar wave equation: ∇2ψ = 1 ∂ 2ψ υ2 ∂t 2 p (2.27) where ψ may represent a component of the E or H field and υp is the phase velocity (velocity of propagation of a point of constant phase in the wave) in the dielectric medium. It follows that: υp = 1 1 1 = 1 2 2 (με)--- (μr μ0 εr ε0)--- (2.28) where μr and εr are the relative permeability and permittivity for the dielectric medium and μ0 and ε0 are the permeability and permittivity of free space. The velocity of light in free space c is therefore: c= 1 1 2 (μ0ε0)--- (2.29) If planar waveguides, described by rectangular Cartesian coordinates (x, y, z), or circular fibers, described by cylindrical polar coordinates (r, φ, z), are considered, then the Laplacian operator takes the form: ∇2ψ = ∂2ψ ∂2ψ ∂2ψ + 2 + 2 ∂x 2 ∂y ∂z (2.30) ∇2ψ = ∂2ψ 1 ∂ψ 1 ∂2ψ ∂2ψ + + + ∂r 2 r ∂r r 2 ∂φ 2 ∂z2 (2.31) or: respectively. It is necessary to consider both these forms for a complete treatment of optical propagation in the fiber, although many of the properties of interest may be dealt with using Cartesian coordinates. The basic solution of the wave equation is a sinusoidal wave, the most important form of which is a uniform plane wave given by: ψ = ψ0 exp[j(ω t − k ⋅ r)] (2.32)
  • OPTF_C02.qxd 11/6/08 10:53 Page 26 26 Optical fiber waveguides Chapter 2 where ω is the angular frequency of the field, t is the time, k is the propagation vector which gives the direction of propagation and the rate of change of phase with distance, while the components of r specify the coordinate point at which the field is observed. When λ is the optical wavelength in a vacuum, the magnitude of the propagation vector or the vacuum phase propagation constant k (where k = |k|) is given by: k= 2π λ (2.33) It should be noted that in this case k is also referred to as the free space wave number. 2.3.2 Modes in a planar guide The planar guide is the simplest form of optical waveguide. We may assume it consists of a slab of dielectric with refractive index n1 sandwiched between two regions of lower refractive index n2. In order to obtain an improved model for optical propagation it is useful to consider the interference of plane wave components within this dielectric waveguide. The conceptual transition from ray to wave theory may be aided by consideration of a plane monochromatic wave propagating in the direction of the ray path within the guide (see Figure 2.8(a)). As the refractive index within the guide is n1, the optical wavelength in this region is reduced to λ/n1, while the vacuum propagation constant is increased to n1k. When θ is the angle between the wave propagation vector or the equivalent ray and the Figure 2.8 The formation of a mode in a planar dielectric guide: (a) a plane wave propagating in the guide shown by its wave vector or equivalent ray – the wave vector is resolved into components in the z and x directions; (b) the interference of plane waves in the guide forming the lowest order mode (m = 0)
  • OPTF_C02.qxd 11/6/08 10:53 Page 27 Electromagnetic mode theory for optical propagation 27 guide axis, the plane wave can be resolved into two component plane waves propagating in the z and x directions, as shown in Figure 2.8(a). The component of the phase propagation constant in the z direction βz is given by: βz = n1k cos θ (2.34) The component of the phase propagation constant in the x direction βx is: βx = n1k sin θ (2.35) The component of the plane wave in the x direction is reflected at the interface between the higher and lower refractive index media. When the total phase change* after two successive reflections at the upper and lower interfaces (between the points P and Q) is equal to 2mπ radians, where m is an integer, then constructive interference occurs and a standing wave is obtained in the x direction. This situation is illustrated in Figure 2.8(b), where the interference of two plane waves is shown. In this illustration it is assumed that the interference forms the lowest order (where m = 0) standing wave, where the electric field is a maximum at the center of the guide decaying towards zero at the boundary between the guide and cladding. However, it may be observed from Figure 2.8(b) that the electric field penetrates some distance into the cladding, a phenomenon which is discussed in Section 2.3.4. Nevertheless, the optical wave is effectively confined within the guide and the electric field distribution in the x direction does not change as the wave propagates in the z direction. The sinusoidally varying electric field in the z direction is also shown in Figure 2.8(b). The stable field distribution in the x direction with only a periodic z dependence is known as a mode. A specific mode is obtained only when the angle between the propagation vectors or the rays and the interface have a particular value, as indicated in Figure 2.8(b). In effect, Eqs (2.34) and (2.35) define a group or congruence of rays which in the case described represents the lowest order mode. Hence the light propagating within the guide is formed into discrete modes, each typified by a distinct value of θ. These modes have a periodic z dependence of the form exp(− jβzz) where βz becomes the propagation constant for the mode as the modal field pattern is invariant except for a periodic z dependence. Hence, for notational simplicity, and in common with accepted practice, we denote the mode propagation constant by β, where β = βz. If we now assume a time dependence for the monochromatic electromagnetic light field with angular frequency ω of exp( jω t), then the combined factor exp[ j(ω t − βz)] describes a mode propagating in the z direction. To visualize the dominant modes propagating in the z direction we may consider plane waves corresponding to rays at different specific angles in the planar guide. These plane waves give constructive interference to form standing wave patterns across the guide following a sine or cosine formula. Figure 2.9 shows examples of such rays for m = 1, 2, 3, together with the electric field distributions in the x direction. It may be observed that m * It should be noted that there is a phase shift on reflection of the plane wave at the interface as well as a phase change with distance traveled. The phase shift on reflection at a dielectric interface is dealt with in Section 2.3.4.
  • OPTF_C02.qxd 11/6/08 10:53 Page 28 28 Optical fiber waveguides Chapter 2 Figure 2.9 Physical model showing the ray propagation and the corresponding transverse electric (TE) field patterns of three lower order models (m = 1, 2, 3) in the planar dielectric guide denotes the number of zeros in this transverse field pattern. In this way m signifies the order of the mode and is known as the mode number. When light is described as an electromagnetic wave it consists of a periodically varying electric field E and magnetic field H which are orientated at right angles to each other. The transverse modes shown in Figure 2.9 illustrate the case when the electric field is perpendicular to the direction of propagation and hence Ez = 0, but a corresponding component of the magnetic field H is in the direction of propagation. In this instance the modes are said to be transverse electric (TE). Alternatively, when a component of the E field is in the direction of propagation, but Hz = 0, the modes formed are called transverse magnetic (TM). The mode numbers are incorporated into this nomenclature by referring to the TEm and TMm modes, as illustrated for the transverse electric modes shown in Figure 2.9. When the total field lies in the transverse plane, transverse electromagnetic (TEM) waves exist where both Ez and Hz are zero. However, although TEM waves occur in metallic conductors (e.g. coaxial cables) they are seldom found in optical waveguides. 2.3.3 Phase and group velocity Within all electromagnetic waves, whether plane or otherwise, there are points of constant phase. For plane waves these constant phase points form a surface which is referred to as a wavefront. As a monochromatic lightwave propagates along a waveguide in the z direction these points of constant phase travel at a phase velocity υp given by:
  • OPTF_C02.qxd 11/6/08 10:53 Page 29 Electromagnetic mode theory for optical propagation 29 Figure 2.10 The formation of a wave packet from the combination of two waves with nearly equal frequencies. The envelope of the wave package or group of waves travels at a group velocity υg υp = ω β (2.36) where ω is the angular frequency of the wave. However, it is impossible in practice to produce perfectly monochromatic lightwaves, and light energy is generally composed of a sum of plane wave components of different frequencies. Often the situation exists where a group of waves with closely similar frequencies propagate so that their resultant forms a packet of waves. The formation of such a wave packet resulting from the combination of two waves of slightly different frequency propagating together is illustrated in Figure 2.10. This wave packet does not travel at the phase velocity of the individual waves but is observed to move at a group velocity υg given by: υg = δω δβ (2.37) The group velocity is of greatest importance in the study of the transmission characteristics of optical fibers as it relates to the propagation characteristics of observable wave groups or packets of light. If propagation in an infinite medium of refractive index n1 is considered, then the propagation constant may be written as: β = n1 2π n1ω = λ c (2.38) where c is the velocity of light in free space. Equation (2.38) follows from Eqs (2.33) and (2.34) where we assume propagation in the z direction only and hence cos θ is equal to unity. Using Eq. (2.36) we obtain the following relationship for the phase velocity:
  • OPTF_C02.qxd 11/6/08 10:53 Page 30 30 Optical fiber waveguides υp = c n1 Chapter 2 (2.39) Similarly, employing Eq. (2.37), where in the limit δω /δβ becomes dω /dβ, the group velocity: υg = = = −1 dλ dω d A 2π D A −ω D ⋅ = n1 dβ dλ dλ C λ F C λ F −ω A 1 dn1 n1 D −1 − 2πλ C λ dλ λ2 F c c = A dn1D Ng n1 − λ C dλ F (2.40) The parameter Ng is known as the group index of the guide. 2.3.4 Phase shift with total internal reflection and the evanescent field The discussion of electromagnetic wave propagation in the planar waveguide given in Section 2.3.2 drew attention to certain phenomena that occur at the guide–cladding interface which are not apparent from ray theory considerations of optical propagation. In order to appreciate these phenomena it is necessary to use the wave theory model for total internal reflection at a planar interface. This is illustrated in Figure 2.11, where the arrowed lines represent wave propagation vectors and a component of the wave energy is Figure 2.11 A wave incident on the guide–cladding interface of a planar dielectric waveguide. The wave vectors of the incident, transmitted and reflected waves are indicated (solid arrowed lines) together with their components in the z and x directions (dashed arrowed lines)
  • OPTF_C02.qxd 11/6/08 10:53 Page 31 Electromagnetic mode theory for optical propagation 31 shown to be transmitted through the interface into the cladding. The wave equation in Cartesian coordinates for the electric field in a lossless medium is: ∇2E = με ∂2E ∂2E ∂2E ∂2E = + + ∂t2 ∂x2 ∂y2 ∂z2 (2.41) As the guide–cladding interface lies in the y–z plane and the wave is incident in the x–z plane onto the interface, then ∂/∂y may be assumed to be zero. Since the phase fronts must match all points along the interface in the z direction, the three waves shown in Figure 2.11 will have the same propagation constant β in this direction. Therefore from the discussion of Section 2.3.2 the wave propagation in the z direction may be described by exp[ j(ω t − βz)]. In addition, there will also be propagation in the x direction. When the components are resolved in this plane: βx1 = n1k cos φ1 (2.42) βx2 = n2k cos φ2 (2.43) where βx1 and βx2 are propagation constants in the x direction for the guide and cladding respectively. Thus the three waves in the waveguide indicated in Figure 2.11, the incident, the transmitted and the reflected, with amplitudes A, B and C, respectively, will have the forms: A = A0 exp[−( jβx1 x)] exp[j(ω t − βz)] (2.44) B = B0 exp[−( jβx2 x)] exp[j(ω t − βz)] (2.45) C = C0 exp[( jβx1 x)] exp[j(ω t − βz)] (2.46) Using the simple trigonometrical relationship cos2 φ + sin2 φ = 1: 2 β 2 = (n2 k2 − β 2) = − ξ1 x1 1 (2.47) 2 β 2 = (n2 k2 − β 2) = − ξ2 x2 2 (2.48) and: When an electromagnetic wave is incident upon an interface between two dielectric media, Maxwell’s equations require that both the tangential components of E and H and the normal components of D (= εE) and B (= μH) are continuous across the boundary. If the boundary is defined at x = 0 we may consider the cases of the transverse electric (TE) and transverse magnetic (TM) modes. Initially, let us consider the TE field at the boundary. When Eqs (2.44) and (2.46) are used to represent the electric field components in the y direction Ey and the boundary conditions are applied, then the normal components of the E and H fields at the interface may be equated giving: A0 + C0 = B0 (2.49)
  • OPTF_C02.qxd 11/6/08 10:53 Page 32 32 Optical fiber waveguides Chapter 2 Furthermore, it can be shown (see Appendix A) that an electric field component in the y direction is related to the tangential magnetic field component Hz following: Hz = j ∂Ey μr μ0ω ∂x (2.50) Applying the tangential boundary conditions and equating Hz by differentiating Ey gives: − βx1 A0 + βx2C0 = − βx2B0 (2.51) Algebraic manipulation of Eqs (2.49) and (2.51) provides the following results: C0 = A0 A βx1 − βx2 D = A0 rER C βx1 + βx2 F (2.52) B0 = A0 A 2βx1 D = A0 rET C βx1 + βx2 F (2.53) where rER and rET are the reflection and transmission coefficients for the E field at the interface respectively. The expressions obtained in Eqs (2.52) and (2.53) correspond to the Fresnel relationships [Ref. 12] for radiation polarized perpendicular to the interface (E polarization). When both βx1 and βx2 are real it is clear that the reflected wave C is in phase with the incident wave A. This corresponds to partial reflection of the incident beam. However, as φ1 is increased the component βz (i.e. β ) increases and, following Eqs (2.47) and (2.48), the components βx1 and βx2 decrease. Continuation of this process results in βx2 passing through zero, a point which is signified by φ1 reaching the critical angle for total internal reflection. If φ1 is further increased the component βx2 becomes imaginary and we may write it in the form −jξ2. During this process βx1 remains real because we have assumed that n1 > n2. Under the conditions of total internal reflection Eq. (2.52) may therefore be written as: C0 = A0 A βx1 + jξ2 D = A0 exp(2jδΕ ) C βx2 − jξ2 F (2.54) where we observe there is a phase shift of the reflected wave relative to the incident wave. This is signified by δE which is given by: tan δE = ξ2 2 βx1 (2.55) Furthermore, the modulus of the reflected wave is identical to the modulus of the incident wave (| C0 | = | A0 |). The curves of the amplitude reflection coefficient | rER | and phase shift on reflection, against angle of incidence φ1, for TE waves incident on a glass–air interface are displayed in Figure 2.12 [Ref. 14]. These curves illustrate the above results,
  • OPTF_C02.qxd 11/6/08 10:53 Page 33 Electromagnetic mode theory for optical propagation 33 Figure 2.12 Curves showing the reflection coefficient and phase shift on reflection for transverse electric waves against the angle of incidence for a glass–air interface (n1 = 1.5, n2 = 1.0). From J. E. Midwinter, Optical Fibers for Transmission, John Wiley & Sons Inc., 1979 where under conditions of total internal reflection the reflected wave has an equal amplitude to the incident wave, but undergoes a phase shift corresponding to δE degrees. A similar analysis may be applied to the TM modes at the interface, which leads to expressions for reflection and transmission of the form [Ref. 14]: C0 = A0 A βx1n2 − βx2n2 D 2 1 = A0 rHR C βx1n2 + βx2n2 F 2 1 (2.56) B0 = A0 A 2βx1n2 D 2 = A0 rHT C βx1n2 + βx2n2 F 2 1 (2.57) and: where rHR and rHT are, respectively, the reflection and transmission coefficients for the H field at the interface. Again, the expressions given in Eqs (2.56) and (2.57) correspond to Fresnel relationships [Ref. 12], but in this case they apply to radiation polarized parallel to the interface (H polarization). Furthermore, considerations of an increasing angle of incidence φ1, such that βx2 goes to zero and then becomes imaginary, again results in a phase shift when total internal reflection occurs. However, in this case a different phase shift is obtained corresponding to: C0 = A0 exp(2jδH) (2.58) where: tan δH = A n1 D 2 tan δE C n2 F (2.59)
  • OPTF_C02.qxd 11/6/08 10:53 Page 34 34 Optical fiber waveguides Chapter 2 Figure 2.13 The exponentially decaying evanescent field in the cladding of the optical waveguide Thus the phase shift obtained on total internal reflection is dependent upon both the angle of incidence and the polarization (either TE or TM) of the radiation. The second phenomenon of interest under conditions of total internal reflection is the form of the electric field in the cladding of the guide. Before the critical angle for total internal reflection is reached, and hence when there is only partial reflection, the field in the cladding is of the form given by Eq. (2.45). However, as indicated previously, when total internal reflection occurs, βx2 becomes imaginary and may be written as −jξ2. Substituting for βx2 in Eq. (2.45) gives the transmitted wave in the cladding as: B = B0 exp(− ξ2 x) exp[j(ω t − βz)] (2.60) Thus the amplitude of the field in the cladding is observed to decay exponentially* in the x direction. Such a field, exhibiting an exponentially decaying amplitude, is often referred to as an evanescent field. Figure 2.13 shows a diagrammatic representation of the evanescent field. A field of this type stores energy and transports it in the direction of propagation (z) but does not transport energy in the transverse direction (x). Nevertheless, the existence of an evanescent field beyond the plane of reflection in the lower index medium indicates that optical energy is transmitted into the cladding. The penetration of energy into the cladding underlines the importance of the choice of cladding material. It gives rise to the following requirements: 1. The cladding should be transparent to light at the wavelengths over which the guide is to operate. 2. Ideally, the cladding should consist of a solid material in order to avoid both damage to the guide and the accumulation of foreign matter on the guide walls. These effects degrade the reflection process by interaction with the evanescent field. This in part explains the poor performance (high losses) of early optical waveguides with air cladding. 3. The cladding thickness must be sufficient to allow the evanescent field to decay to a low value or losses from the penetrating energy may be encountered. In many * It should be noted that we have chosen the sign of ξ2 so that the exponential field decays rather than grows with distance into the cladding. In this case a growing exponential field is a physically improbable solution.
  • OPTF_C02.qxd 11/6/08 10:53 Page 35 Cylindrical fiber 35 cases, however, the magnitude of the field falls off rapidly with distance from the guide–cladding interface. This may occur within distances equivalent to a few wavelengths of the transmitted light. Therefore, the most widely used optical fibers consist of a core and cladding, both made of glass. The cladding refractive index is thus higher than would be the case with liquid or gaseous cladding giving a lower numerical aperture for the fiber, but it provides a far more practical solution. 2.3.5 Goos–Haenchen shift The phase change incurred with the total internal reflection of a light beam on a planar dielectric interface may be understood from physical observation. Careful examination shows that the reflected beam is shifted laterally from the trajectory predicted by simple ray theory analysis, as illustrated in Figure 2.14. This lateral displacement is known as the Goos–Haenchen shift, after its first observers. The geometric reflection appears to take place at a virtual reflecting plane which is parallel to the dielectric interface in the lower index medium, as indicated in Figure 2.14. Utilizing wave theory it is possible to determine this lateral shift [Ref. 14] although it is very small (d Ӎ 0.06 to 0.10 μm for a silvered glass interface at a wavelength of 0.55 μm) and difficult to observe. However, this concept provides an important insight into the guidance mechanism of dielectric optical waveguides. Figure 2.14 The lateral displacement of a light beam on reflection at a dielectric interface (Goos–Haenchen shift) 2.4 Cylindrical fiber 2.4.1 Modes The exact solution of Maxwell’s equations for a cylindrical homogeneous core dielectric waveguide* involves much algebra and yields a complex result [Ref. 15]. Although the * This type of optical waveguide with a constant refractive index core is known as a step index fiber (see Section 2.4.3).
  • OPTF_C02.qxd 11/6/08 10:53 Page 36 36 Optical fiber waveguides Chapter 2 presentation of this mathematics is beyond the scope of this text, it is useful to consider the resulting modal fields. In common with the planar guide (Section 2.3.2), TE (where Ez = 0) and TM (where Hz = 0) modes are obtained within the dielectric cylinder. The cylindrical waveguide, however, is bounded in two dimensions rather than one. Thus two integers, l and m, are necessary in order to specify the modes, in contrast to the single integer (m) required for the planar guide. For the cylindrical waveguide we therefore refer to TElm and TMlm modes. These modes correspond to meridional rays (see Section 2.2.1) traveling within the fiber. However, hybrid modes where Ez and Hz are nonzero also occur within the cylindrical waveguide. These modes, which result from skew ray propagation (see Section 2.2.4) within the fiber, are designated HElm and EHlm depending upon whether the components of H or E make the larger contribution to the transverse (to the fiber axis) field. Thus an exact description of the modal fields in a step index fiber proves somewhat complicated. Fortunately, the analysis may be simplified when considering optical fibers for communication purposes. These fibers satisfy the weakly guiding approximation [Ref. 16] where the relative index difference Δ Ӷ 1. This corresponds to small grazing angles θ in Eq. (2.34). In fact Δ is usually less than 0.03 (3%) for optical communications fibers. For weakly guiding structures with dominant forward propagation, mode theory gives dominant transverse field components. Hence approximate solutions for the full set of HE, EH, TE and TM modes may be given by two linearly polarized components [Ref. 16]. These linearly polarized (LP) modes are not exact modes of the fiber except for the fundamental (lowest order) mode. However, as Δ in weakly guiding fibers is very small, then HE–EH mode pairs occur which have almost identical propagation constants. Such modes are said to be degenerate. The superpositions of these degenerating modes characterized by a common propagation constant correspond to particular LP modes regardless of their HE, EH, TE or TM field configurations. This linear combination of degenerate modes obtained from the exact solution produces a useful simplification in the analysis of weakly guiding fibers. The relationship between the traditional HE, EH, TE and TM mode designations and the LPlm mode designations is shown in Table 2.1. The mode subscripts l and m are related to the electric field intensity profile for a particular LP mode (see Figure 2.15(d)). There are in general 2l field maxima around the circumference of the fiber core and m field Table 2.1 Correspondence between the lower order in linearly polarized modes and the traditional exact modes from which they are formed Linearly polarized Exact LP01 LP11 LP21 LP02 LP31 LP12 LPlm LPlm (l ≠ 0 or 1) HE11 HE21, TE01, TM01 HE31, EH11 HE12 HE41, EH21 HE22, TE02, TM02 HE2m, TE0m, TM0m HEl+1.m, EHl −1.m
  • OPTF_C02.qxd 11/6/08 10:53 Page 37 Cylindrical fiber 37 Figure 2.15 The electric field configurations for the three lowest LP modes illustrated in terms of their constituent exact modes: (a) LP mode designations; (b) exact mode designations; (c) electric field distribution of the exact modes; (d) intensity distribution of Ex for the exact modes indicating the electric field intensity profile for the corresponding LP modes maxima along a radius vector. Furthermore, it may be observed from Table 2.1 that the notation for labeling the HE and EH modes has changed from that specified for the exact solution in the cylindrical waveguide mentioned previously. The subscript l in the LP notation now corresponds to HE and EH modes with labels l + 1 and l − 1 respectively. The electric field intensity profiles for the lowest three LP modes, together with the electric field distribution of their constituent exact modes, are shown in Figure 2.15. It may be observed from the field configurations of the exact modes that the field strength in the transverse direction (Ex or Ey) is identical for the modes which belong to the same LP mode. Hence the origin of the term ‘linearly polarized’. Using Eq. (2.31) for the cylindrical homogeneous core waveguide under the weak guidance conditions outlined above, the scalar wave equation can be written in the form [Ref. 17]: d2ψ 1 dψ 1 d2ψ + + + (n2k2 − β 2)ψ = 0 1 dr 2 r dr r 2 d φ2 (2.61)
  • OPTF_C02.qxd 11/6/08 10:53 Page 38 38 Optical fiber waveguides Chapter 2 where ψ is the field (E or H), n1 is the refractive index of the fiber core, k is the propagation constant for light in a vacuum, and r and φ are cylindrical coordinates. The propagation constants of the guided modes β lie in the range: n2k < β < n1k (2.62) where n2 is the refractive index of the fiber cladding. Solutions of the wave equation for the cylindrical fiber are separable, having the form: ψ = E(r) G cos lφ J exp(ω t − βz) I sin lφ L (2.63) where in this case ψ represents the dominant transverse electric field component. The periodic dependence on φ following cos lφ or sin lφ gives a mode of radial order l. Hence the fiber supports a finite number of guided modes of the form of Eq. (2.63). Introducing the solutions given by Eq. (2.63) into Eq. (2.61) results in a differential equation of the form: d2E 1 dE G l2 J + (n1k2 − β 2) − 2 E = 0 2 + dr r dr I rL (2.64) For a step index fiber with a constant refractive index core, Eq. (2.64) is a Bessel differential equation and the solutions are cylinder functions. In the core region the solutions are Bessel functions denoted by Jl. A graph of these gradually damped oscillatory functions (with respect to r) is shown in Figure 2.16(a). It may be noted that the field is finite at r = 0 and may be represented by the zero-order Bessel function J0. However, the field vanishes as r goes to infinity and the solutions in the cladding are therefore modified Bessel functions denoted by Kl. These modified functions decay exponentially with respect to r, as illustrated in Figure 2.16(b). The electric field may therefore be given by: E(r) = GJl(UR) = GJl(U) for R < 1 (core) Ki (WR) Ki (W) for R > 1 (cladding) (2.65) where G is the amplitude coefficient and R = r/a is the normalized radial coordinate when a is the radius of the fiber core; U and W, which are the eigenvalues in the core and cladding respectively,* are defined as [Ref. 17]: 1 2 2 U = a(n1 k 2 − β 2)--1 2 2 W = a(β 2 − n2 k2)--- (2.66) (2.67) * U is also referred to as the radial phase parameter or the radial propagation constant, whereas W is known as the cladding decay parameter [Ref. 19].
  • OPTF_C02.qxd 11/6/08 10:53 Page 39 Cylindrical fiber 39 Figure 2.16 (a) Variation of the Bessel function Jl(r) for l = 0, 1, 2, 3 (first four orders), plotted against r. (b) Graph of the modified Bessel function Kl(r) against r for l = 0, 1 The sum of the squares of U and W defines a very useful quantity [Ref. 18] which is usually referred to as the normalized frequency* V where: 1 1 2 2 V = (U2 + W2)--- = ka(n2 − n2)--1 2 (2.68) It may be observed that the commonly used symbol for this parameter is the same as that normally adopted for voltage. However, within this chapter there should be no confusion over this point. Furthermore, using Eqs (2.8) and (2.10) the normalized frequency may be expressed in terms of the numerical aperture NA and the relative refractive index difference Δ, respectively, as: * When used in the context of the planar waveguide, V is sometimes known as the normalized film thickness as it relates to the thickness of the guide layer (see Section 10.5.1).
  • OPTF_C02.qxd 11/6/08 10:53 Page 40 40 Optical fiber waveguides Chapter 2 V= 2π a(NA) λ (2.69) V= 1 2π 2 an1(2Δ)--λ (2.70) The normalized frequency is a dimensionless parameter and hence is also sometimes simply called the V number or value of the fiber. It combines in a very useful manner the information about three important design variables for the fiber: namely, the core radius a, the relative refractive index difference Δ and the operating wavelength λ. It is also possible to define the normalized propagation constant b for a fiber in terms of the parameters of Eq. (2.68) so that: b=1− U2 (β/k)2 − n2 2 = V2 n2 − n2 1 2 = (β/k)2 − n2 2 2n2Δ 1 (2.71) Referring to the expression for the guided modes given in Eq. (2.62), the limits of β are n2k and n1k, hence b must lie between 0 and 1. In the weak guidance approximation the field matching conditions at the boundary require continuity of the transverse and tangential electric field components at the core– cladding interface (at r = a). Therefore, using the Bessel function relations outlined previously, an eigenvalue equation for the LP modes may be written in the following form [Ref. 20]: U Jl±1(U) K (W) = ±W l±1 Jl (U) Kl(W) (2.72) Solving Eq. (2.72) with Eqs (2.66) and (2.67) allows the eigenvalue U and hence β to be calculated as a function of the normalized frequency. In this way the propagation characteristics of the various modes, and their dependence on the optical wavelength and the fiber parameters, may be determined. Considering the limit of mode propagation when β = n2k, then the mode phase velocity is equal to the velocity of light in the cladding and the mode is no longer properly guided. In this case the mode is said to be cut off and the eigenvalue W = 0 (Eq. 2.67). Unguided or radiation modes have frequencies below cutoff where β < kn2, and hence W is imaginary. Nevertheless, wave propagation does not cease abruptly below cutoff. Modes exist where β < kn2 but the difference is very small, such that some of the energy loss due to radiation is prevented by an angular momentum barrier [Ref. 21] formed near the core–cladding interface. Solutions of the wave equation giving these states are called leaky modes, and often behave as very lossy guided modes rather than radiation modes. Alternatively, as β is increased above n2k, less power is propagated in the cladding until at β = n1k all the power is confined to the fiber core. As indicated previously, this range of values for β signifies the guided modes of the fiber.
  • OPTF_C02.qxd 11/6/08 10:53 Page 41 Cylindrical fiber 41 Figure 2.17 The allowed regions for the LP modes of order l = 0, 1 against normalized frequency (V) for a circular optical waveguide with a constant refractive index core (step index fiber). Reproduced with permission from D. Gloge. Appl. Opt., 10, p. 2552, 1971 The lower order modes obtained in a cylindrical homogeneous core waveguide are shown in Figure 2.17 [Ref. 16]. Both the LP notation and the corresponding traditional HE, EH, TE and TM mode notations are indicated. In addition, the Bessel functions J0 and J1 are plotted against the normalized frequency and where they cross the zero gives the cutoff point for the various modes. Hence, the cutoff point for a particular mode corresponds to a distinctive value of the normalized frequency (where V = Vc) for the fiber. It may be observed from Figure 2.17 that the value of Vc is different for different modes. For example, the first zero crossing J1 occurs when the normalized frequency is 0 and this corresponds to the cutoff for the LP01 mode. However, the first zero crossing for J0 is when the normalized frequency is 2.405, giving a cutoff value Vc of 2.405 for the LP11 mode. Similarly, the second zero of J1 corresponds to a normalized frequency of 3.83, giving a cutoff value Vc for the LP02 mode of 3.83. It is therefore apparent that fibers may be produced with particular values of normalized frequency which allow only certain modes to propagate. This is further illustrated in Figure 2.18 [Ref. 16] which shows the normalized propagation constant b for a number of LP modes as a function of V. It may be observed that the cutoff value of normalized frequency Vc which occurs when β = n2k corresponds to b = 0. The propagation of particular modes within a fiber may also be confirmed through visual analysis. The electric field distribution of different modes gives similar distributions of light intensity within the fiber core. These waveguide patterns (often called mode patterns) may give an indication of the predominant modes propagating in the fiber. The field intensity distributions for the three lower order LP modes were shown in Figure 2.15. In Figure 2.19 we illustrate the mode patterns for two higher order LP modes. However, unless the fiber is designed for the propagation of a particular mode it is likely that the superposition of many modes will result in no distinctive pattern.
  • OPTF_C02.qxd 11/6/08 10:53 Page 42 42 Optical fiber waveguides Chapter 2 Figure 2.18 The normalized propagation constant b as a function of normalized frequency V for a number of LP modes. Reproduced with permission from D. Gloge. Appl. Opt., 10, p. 2552, 1971 Figure 2.19 Sketches of fiber cross-sections illustrating the distinctive light intensity distributions (mode patterns) generated by propagation of individual linearly polarized modes 2.4.2 Mode coupling We have thus far considered the propagation aspects of perfect dielectric waveguides. However, waveguide perturbations such as deviations of the fiber axis from straightness, variations in the core diameter, irregularities at the core–cladding interface and refractive index variations may change the propagation characteristics of the fiber. These will have
  • OPTF_C02.qxd 11/6/08 10:53 Page 43 Cylindrical fiber 43 the effect of coupling energy traveling in one mode to another depending on the specific perturbation. Ray theory aids the understanding of this phenomenon, as shown in Figure 2.20, which illustrates two types of perturbation. It may be observed that in both cases the ray no longer maintains the same angle with the axis. In electromagnetic wave theory this corresponds to a change in the propagating mode for the light. Thus individual modes do not normally propagate throughout the length of the fiber without large energy transfers to adjacent modes, even when the fiber is exceptionally good quality and is not strained or bent by its surroundings. This mode conversion is known as mode coupling or mixing. It is usually analyzed using coupled mode equations which can be obtained directly from Maxwell’s equations. However, the theory is beyond the scope of this text and the reader is directed to Ref. 17 for a comprehensive treatment. Mode coupling affects the transmission properties of fibers in several important ways, a major one being in relation to the dispersive properties of fibers over long distances. This is pursued further in Sections 3.8 to 3.11. 2.4.3 Step index fibers The optical fiber considered in the preceding sections with a core of constant refractive index n1 and a cladding of a slightly lower refractive index n2 is known as step index fiber. This is because the refractive index profile for this type of fiber makes a step change at the Figure 2.20 Ray theory illustrations showing two of the possible fiber perturbations which give mode coupling: (a) irregularity at the core–cladding interface; (b) fiber bend
  • OPTF_C02.qxd 11/6/08 10:53 Page 44 44 Optical fiber waveguides Chapter 2 Figure 2.21 The refractive index profile and ray transmission in step index fibers: (a) multimode step index fiber; (b) single-mode step index fiber core–cladding interface, as indicated in Figure 2.21, which illustrates the two major types of step index fiber. The refractive index profile may be defined as: n(r) = 8 n1 r < a (core) 9 n2 r ≥ a (cladding) (2.73) in both cases. Figure 2.21(a) shows a multimode step index fiber with a core diameter of around 50 μm or greater, which is large enough to allow the propagation of many modes within the fiber core. This is illustrated in Figure 2.21(a) by the many different possible ray paths through the fiber. Figure 2.21(b) shows a single-mode or monomode step index fiber which allows the propagation of only one transverse electromagnetic mode (typically HE11), and hence the core diameter must be of the order of 2 to 10 μm. The propagation of a single mode is illustrated in Figure 2.21(b) as corresponding to a single ray path only (usually shown as the axial ray) through the fiber. The single-mode step index fiber has the distinct advantage of low intermodal dispersion (broadening of transmitted light pulses), as only one mode is transmitted, whereas with multimode step index fiber considerable dispersion may occur due to the differing group velocities of the propagating modes (see Section 3.10). This in turn restricts the maximum bandwidth attainable with multimode step index fibers, especially when compared with single-mode fibers. However, for lower bandwidth applications multimode fibers have several advantages over single-mode fibers. These are: (a) the use of spatially incoherent optical sources (e.g. most light-emitting diodes) which cannot be efficiently coupled to single-mode fibers;
  • OPTF_C02.qxd 11/6/08 10:53 Page 45 Cylindrical fiber 45 (b) larger numerical apertures, as well as core diameters, facilitating easier coupling to optical sources; (c) lower tolerance requirements on fiber connectors. Multimode step index fibers allow the propagation of a finite number of guided modes along the channel. The number of guided modes is dependent upon the physical parameters (i.e. relative refractive index difference, core radius) of the fiber and the wavelengths of the transmitted light which are included in the normalized frequency V for the fiber. It was indicated in Section 2.4.1 that there is a cutoff value of normalized frequency Vc for guided modes below which they cannot exist. However, mode propagation does not entirely cease below cutoff. Modes may propagate as unguided or leaky modes which can travel considerable distances along the fiber. Nevertheless, it is the guided modes which are of paramount importance in optical fiber communications as these are confined to the fiber over its full length. It can be shown [Ref. 16] that the total number of guided modes or mode volume Ms for a step index fiber is related to the V value for the fiber by the approximate expression: Ms Ӎ V2 2 (2.74) which allows an estimate of the number of guided modes propagating in a particular multimode step index fiber. Example 2.4 A multimode step index fiber with a core diameter of 80 μm and a relative index difference of 1.5% is operating at a wavelength of 0.85 μm. If the core refractive index is 1.48, estimate: (a) the normalized frequency for the fiber; (b) the number of guided modes. Solution: (a) The normalized frequency may be obtained from Eq. (2.70) where: VӍ 1 1 2π 2π × 40 × 10−6 × 1.48 2 2 an1(2Δ)--- = (2 × 0.015)--- = 75.8 λ 0.85 × 10−6 (b) The total number of guided modes is given by Eq. (2.74) as: Ms Ӎ V 2 5745.6 = 2 2 = 2873 Hence this fiber has a V number of approximately 76, giving nearly 3000 guided modes.
  • OPTF_C02.qxd 11/6/08 10:53 Page 46 46 Optical fiber waveguides Chapter 2 Therefore, as illustrated in Example 2.4, the optical power is launched into a large number of guided modes, each having different spatial field distributions, propagation constants, etc. In an ideal multimode step index fiber with properties (i.e. relative index difference, core diameter) which are independent of distance, there is no mode coupling, and the optical power launched into a particular mode remains in that mode and travels independently of the power launched into the other guided modes. Also, the majority of these guided modes operate far from cutoff, and are well confined to the fiber core [Ref. 16]. Thus most of the optical power is carried in the core region and not in the cladding. The properties of the cladding (e.g. thickness) do not therefore significantly affect the propagation of these modes. 2.4.4 Graded index fibers Graded index fibers do not have a constant refractive index in the core* but a decreasing core index n(r) with radial distance from a maximum value of n1 at the axis to a constant value n2 beyond the core radius a in the cladding. This index variation may be represented as: 1 n(r) = --8 n1(1 − 2Δ(r/a)α )2 r < a (core) 1 --9 n1(1 − 2Δ)2 = n2 r ≥ a (cladding) (2.75) where Δ is the relative refractive index difference and α is the profile parameter which gives the characteristic refractive index profile of the fiber core. Equation (2.75) which is a convenient method of expressing the refractive index profile of the fiber core as a variation of α, allows representation of the step index profile when α = ∞, a parabolic profile when α = 2 and a triangular profile when α = 1. This range of refractive index profiles is illustrated in Figure 2.22. The graded index profiles which at present produce the best results for multimode optical propagation have a near parabolic refractive index profile core with α Ϸ 2. Fibers Figure 2.22 Possible fiber refractive index profiles for different values of α (given in Eq. (2.75)) * Graded index fibers are therefore sometimes referred to as inhomogeneous core fibers.
  • OPTF_C02.qxd 11/6/08 10:53 Page 47 Cylindrical fiber 47 Figure 2.23 The refractive index profile and ray transmission in a multimode graded index fiber with such core index profiles are well established and consequently when the term ‘graded index’ is used without qualification it usually refers to a fiber with this profile. For this reason in this section we consider the waveguiding properties of graded index fiber with a parabolic refractive index profile core. A multimode graded index fiber with a parabolic index profile core is illustrated in Figure 2.23. It may be observed that the meridional rays shown appear to follow curved paths through the fiber core. Using the concepts of geometric optics, the gradual decrease in refractive index from the center of the core creates many refractions of the rays as they are effectively incident on a large number or high to low index interfaces. This mechanism is illustrated in Figure 2.24 where a ray is shown to be gradually curved, with an everincreasing angle of incidence, until the conditions for total internal reflection are met, and the ray travels back towards the core axis, again being continuously refracted. Multimode graded index fibers exhibit far less intermodal dispersion (see Section 3.10.2) than multimode step index fibers due to their refractive index profile. Although many different modes are excited in the graded index fiber, the different group velocities of the modes tend to be normalized by the index grading. Again considering ray theory, the rays traveling close to the fiber axis have shorter paths when compared with rays which travel Figure 2.24 An expanded ray diagram showing refraction at the various high to low index interfaces within a graded index fiber, giving an overall curved ray path
  • OPTF_C02.qxd 11/6/08 10:53 Page 48 48 Optical fiber waveguides Chapter 2 Figure 2.25 A helical skew ray path within a graded index fiber into the outer regions of the core. However, the near axial rays are transmitted through a region of higher refractive index and therefore travel with a lower velocity than the more extreme rays. This compensates for the shorter path lengths and reduces dispersion in the fiber. A similar situation exists for skew rays which follow longer helical paths, as illustrated in Figure 2.25. These travel for the most part in the lower index region at greater speeds, thus giving the same mechanism of mode transit time equalization. Hence, multimode graded index fibers with parabolic or near-parabolic index profile cores have transmission bandwidths which may be orders of magnitude greater than multimode step index fiber bandwidths. Consequently, although they are not capable of the bandwidths attainable with single-mode fibers, such multimode graded index fibers have the advantage of large core diameters (greater than 30 μm) coupled with bandwidths suitable for longdistance communication. The parameters defined for step index fibers (i.e. NA, Δ, V) may be applied to graded index fibers and give a comparison between the two fiber types. However, it must be noted that for graded index fibers the situation is more complicated since the numerical aperture is a function of the radial distance from the fiber axis. Graded index fibers, therefore, accept less light than corresponding step index fibers with the same relative refractive index difference. Electromagnetic mode theory may also be utilized with the graded profiles. Approximate field solutions of the same order as geometric optics are often obtained employing the WKB method from quantum mechanics after Wentzel, Kramers and Brillouin [Ref. 22]. Using the WKB method modal solutions of the guided wave are achieved by expressing the field in the form: 1 Ex = --{G1(r) exp[jS(r)] + G2(r) exp[−jS(r)]} 2 A cos lφD exp( jβz) C sin lφ F (2.76) where G and S are assumed to be real functions of the radial distance r. Substitution of Eq. (2.76) into the scalar wave equation of the form given by Eq. (2.61) (in which the constant refractive index of the fiber core n1 is replaced by n(r)) and neglecting the second derivative of Gi(r) with respect to r provides approximate solutions for the amplitude function Gi(r) and the phase function S(r). It may be observed from the ray diagram shown in Figure 2.23 that a light ray propagating in a graded index fiber does not necessarily reach every point within the fiber core. The ray is contained within two cylindrical caustic surfaces and for most rays a caustic does not coincide with the core– cladding interface. Hence the caustics define the classical turning points of the light ray
  • OPTF_C02.qxd 11/6/08 10:53 Page 49 Cylindrical fiber 49 within the graded fiber core. These turning points defined by the two caustics may be designated as occurring at r = r1 and r = r2. The result of the WKB approximation yields an oscillatory field in the region r1 < r < r2 between the caustics where: 1 4 G1(r) = G2(r) = D/[(n2(r)k 2 − β 2)r 2 − l2]--- (2.77) (where D is an amplitude coefficient) and: S(r) = Ύ r2 1 2 [(n2(r)k 2 − β 2)r 2 − l 2]--- r1 dr π − r 4 (2.78) Outside the interval r1 < r < r2 the field solution must have an evanescent form. In the region inside the inner caustic defined by r < r1 and assuming r1 is not too close to r = 0, the field decays towards the fiber axis giving: 1 4 G1(r) = D exp( jmx)/[l2 − (n2(r)k 2 − β 2)r 2]--- (2.79) G2(r) = 0 (2.80) where the integer m is the radial mode number and: r1 S(r) = j Ύ [l − (n (r)k − β )r ] dr r 2 2 2 2 2 1 --2 (2.81) r Also outside the outer caustic in the region r > r2, the field decays away from the fiber axis and is described by the equations: 1 4 G1(r) = D exp( jmx)/[l2 − (n2(r)k 2 − β 2)r 2]--- (2.82) G2(r) = 0 (2.83) r S(r) = j Ύ [l − (n (r)k − β )r ] dr r 2 2 2 2 2 1 --2 (2.84) r2 The WKB method does not initially provide valid solutions of the wave equation in the vicinity of the turning points. Fortunately, this may be amended by replacing the actual refractive index profile by a linear approximation at the location of the caustics. The solutions at the turning points can then be expressed in terms of Hankel functions of the first 1 and second kind of order -- [Ref. 23]. This facilitates the joining together of the two 3 separate solutions described previously for inside and outside the interval r1 < r < r2. Thus the WKB theory provides an approximate eigenvalue equation for the propagation constant β of the guided modes which cannot be determined using ray theory. The WKB eigenvalue equation of which β is a solution is given by [Ref. 23]: Ύ r2 1 2 [(n2(r)k 2 − β 2)r 2 − l 2]--- r1 dr π = (2m − 1) r 2 (2.85)
  • OPTF_C02.qxd 11/6/08 10:53 Page 50 50 Optical fiber waveguides Chapter 2 where the radial mode number m = 1, 2, 3 . . . and determines the number of maxima of the oscillatory field in the radial direction. This eigenvalue equation can only be solved in a closed analytical form for a few simple refractive index profiles. Hence, in most cases it must be solved approximately or with the use of numerical techniques [Refs 24, 25]. Finally the amplitude coefficient D may be expressed in terms of the total optical power PG within the guided mode. Considering the power carried between the turning points r1 and r2 gives a geometric optics approximation of [Ref. 26]: 1 D= 1 --2 2 4(μ0 /ε0)---PG 2 n1π a I (2.86) where: I= Ύ r2/a r1/a x dx 1 2 [(n2(ax)k 2 − β 2)a2x 2 − l2]--- (2.87) The properties of the WKB solution may by observed from a graphical representation of the integrand given in Eq. (2.78). This is shown in Figure 2.26, together with the corresponding WKB solution. Figure 2.26 illustrates the functions (n2(r)k 2 − β 2) and (l 2/r 2). The Figure 2.26 Graphical representation of the functions (n2(r)k2 − β 2) and (l2/r 2) that are important in the WKB solution and which define the turning points r1 and r2. Also shown is an example of the corresponding WKB solution for a guided mode where an oscillatory wave exists in the region between the turning points
  • OPTF_C02.qxd 11/6/08 10:53 Page 51 Cylindrical fiber 51 two curves intersect at the turning points r = r1 and r = r2. The oscillatory nature of the WKB solution between the turning points (i.e. when l2/r 2 < n2(r)k2 − β 2) which changes into a decaying exponential (evanescent) form outside the interval r1 < r < r2 (i.e. when l2/r 2 > n2(r)k 2 − β 2) can also be clearly seen. It may be noted that as the azimuthal mode number l increases, the curve (l2/r2) moves higher and the region between the two turning points becomes narrower. In addition, even when l is fixed the curve (n2(r)k 2 − β 2) is shifted up and down with alterations in the value of the propagation constant β. Therefore, modes far from cutoff which have large values of β exhibit more closely spaced turning points. As the value of β decreases below n2k, (n2(r)k 2 − β 2) is no longer negative for large values of r and the guided mode situation depicted in Figure 2.26 changes to one corresponding to Figure 2.27. In this case a third turning point r = r3 is created when at r = a the curve (n2(r)k 2 − β 2) becomes constant, thus allowing the curve (l2/r 2) to drop below it. Now the field displays an evanescent, exponentially decaying form in the region r2 < r < r3, as shown in Figure 2.27. Moreover, for r > r3 the field resumes an oscillatory behavior and therefore carries power away from the fiber core. Unless mode cutoff occurs at β = n2k, the guided mode is no longer fully contained within the fiber core but loses power through leakage or tunneling into the cladding. This situation corresponds to the leaky modes mentioned previously in Section 2.4.1. The WKB method may be used to calculate the propagation constants for the modes in a parabolic refractive index profile core fiber where, following Eq. (2.75): Figure 2.27 Similar graphical representation as that illustrated in Figure 2.26. Here the curve (n2(r)k2 − β 2) no longer goes negative and a third turning point r3 occurs. This corresponds to leaky mode solutions in the WKB method
  • OPTF_C02.qxd 11/6/08 10:53 Page 52 52 Optical fiber waveguides G A rD 2 J n2(r) = n2 1 − 2 Δ 1 I C aF L Chapter 2 for r < a (2.88) Substitution of Eq. (2.88) into Eq. (2.85) gives: r2 Ύ r1 1 --- G 2 2 ArD 2 A l2 J 2 1D n1k − β 2 − 2n2k2 Δ − 2 dr = m + π 1 I C aF C rL 2F (2.89) The integral shown in Eq. (2.89) can be evaluated using a change of variable from r to u = r2. The integral obtained may be found in a standard table of indefinite integrals [Ref. 27]. As the square root term in the resulting expression goes to zero at the turning points (i.e. r = r1 and r = r2), then we can write: G a(n1k2 − β 2) l J A 1D − π= m+ π I 4n1k√(2Δ) 2 L C 2F (2.90) Solving Eq. (2.90) for β 2 gives: β 2 = n2k2 1 J G1 − 2√(2Δ) (2m + l + 1) L I n1ka (2.91) It is interesting to note that the solution for the propagation constant for the various modes in a parabolic refractive index core fiber given in Eq. (2.91) is exact even though it was derived from the approximate WKB eigenvalue equation (Eq. (2.85)). However, although Eq. (2.91) is an exact solution of the scalar wave equation for an infinitely extended parabolic profile medium, the wave equation is only an approximate representation of Maxwell’s equation. Furthermore, practical parabolic refractive index profile core fibers exhibit a truncated parabolic distribution which merges into a constant refractive index at the cladding. Hence Eq. (2.91) is not exact for real fibers. Equation (2.91) does, however, allow us to consider the mode number plane spanned by the radial and azimuthal mode numbers m and l. This plane is displayed in Figure 2.28, Figure 2.28 The mode number plane illustrating the mode boundary and the guided fiber modes
  • OPTF_C02.qxd 11/6/08 10:53 Page 53 Cylindrical fiber 53 where each mode of the fiber described by a pair of mode numbers is represented as a point in the plane. The mode number plane contains guided, leaky and radiation modes. The mode boundary which separates the guided modes from the leaky and radiation modes is indicated by the solid line in Figure 2.28. It depicts a constant value of β following Eq. (2.91) and occurs when β = n2k. Therefore, all the points in the mode number plane lying below the line β = n2k are associated with guided modes, whereas the region above the line is occupied by leaky and radiation modes. The concept of the mode plane allows us to count the total number of guided modes within the fiber. For each pair of mode numbers m and l the corresponding mode field can have azimuthal mode dependence cos lφ or sin lφ and can exist in two possible polarizations (see Section 3.13). Hence the modes are said to be fourfold degenerate.* If we define the mode boundary as the function m = f(l), then the total number of guided modes M is given by: lmax M=4 Ύ f(l) dl (2.92) 0 as each representation point corresponding to four modes occupies an element of unit area in the mode plane. Equation (2.92) allows the derivation of the total number of guided modes or mode volume Mg supported by the graded index fiber. It can be shown [Ref. 23] that: Mg = A α D (n ka)2Δ C α + 2F 1 (2.93) Furthermore, utilizing Eq. (2.70), the normalized frequency V for the fiber when Δ Ӷ 1 is approximately given by: 1 2 V = n1ka(2Δ)--- (2.94) Substituting Eq. (2.94) into Eq. (2.93), we have: Mg Ӎ A α D A V 2D Cα + 2F C 2 F (2.95) Hence for a parabolic refractive index profile core fiber (α = 2), Mg Ϸ V 2/4, which is half the number supported by a step index fiber (α = ∞) with the same V value. * An exception to this are the modes that occur when l = 0 which are only doubly degenerate as cos lφ becomes unity and sin lφ vanishes. However, these modes represent only a small minority and therefore may be neglected.
  • OPTF_C02.qxd 11/6/08 10:53 Page 54 54 Optical fiber waveguides Chapter 2 Example 2.5 A graded index fiber has a core with a parabolic refractive index profile which has a diameter of 50 μm. The fiber has a numerical aperture of 0.2. Estimate the total number of guided modes propagating in the fiber when it is operating at a wavelength of 1 μm. Solution: Using Eq. (2.69), the normalized frequency for the fiber is: V= 2π 2π × 25 × 10−6 × 0.2 a(NA) = λ 1 × 10−6 = 31.4 The mode volume may be obtained from Eq. (2.95) where for a parabolic profile: Mg Ӎ V 2 986 = = 247 4 4 Hence the fiber supports approximately 247 guided modes. 2.5 Single-mode fibers The advantage of the propagation of a single mode within an optical fiber is that the signal dispersion caused by the delay differences between different modes in a multimode fiber may be avoided (see Section 3.10). Multimode step index fibers do not lend themselves to the propagation of a single mode due to the difficulties of maintaining single-mode operation within the fiber when mode conversion (i.e. coupling) to other guided modes takes place at both input mismatches and fiber imperfections. Hence, for the transmission of a single mode the fiber must be designed to allow propagation of only one mode, while all other modes are attenuated by leakage or absorption [Refs 28–34]. Following the preceding discussion of multimode fibers, this may be achieved through choice of a suitable normalized frequency for the fiber. For single-mode operation, only the fundamental LP01 mode can exist. Hence the limit of single-mode operation depends on the lower limit of guided propagation for the LP11 mode. The cutoff normalized frequency for the LP11 mode in step index fibers occurs at Vc = 2.405 (see Section 2.4.1). Thus single-mode propagation of the LP01 mode in step index fibers is possible over the range: 0 ≤ V < 2.405 (2.96) as there is no cutoff for the fundamental mode. It must be noted that there are in fact two modes with orthogonal polarization over this range, and the term single-mode applies to propagation of light of a particular polarization. Also, it is apparent that the normalized frequency for the fiber may be adjusted to within the range given in Eq. (2.96) by reduction
  • OPTF_C02.qxd 11/6/08 10:53 Page 55 Single-mode fibers 55 of the core radius, and possibly the relative refractive index difference following Eq. (2.70), which, for single-mode fibers, is usually less than 1%. Example 2.6 Estimate the maximum core diameter for an optical fiber with the same relative refractive index difference (1.5%) and core refractive index (1.48) as the fiber given in Example 2.4 in order that it may be suitable for single-mode operation. It may be assumed that the fiber is operating at the same wavelength (0.85 μm). Further, estimate the new maximum core diameter for single-mode operation when the relative refractive index difference is reduced by a factor of 10. Solution: Considering the relationship given in Eq. (2.96), the maximum V value for a fiber which gives single-mode operation is 2.4. Hence, from Eq. (2.70) the core radius a is: a= Vλ 2.4 × 0.85 × 10−6 1 = 1 2 2 2πn1(2Δ)--- 2π × 1.48 × (0.03)--- = 1.3 μm Therefore the maximum core diameter for single-mode operation is approximately 2.6 μm. Reducing the relative refractive index difference by a factor of 10 and again using Eq. (2.70) gives: a= 2.4 × 0.85 × 10−6 1 = 4.0 μm 2 2π × 1.48 × (0.003)--- Hence the maximum core diameter for single-mode operation is now approximately 8 μm. It is clear from Example 2.6 that in order to obtain single-mode operation with a maximum V number of 2.4, the single-mode fiber must have a much smaller core diameter than the equivalent multimode step index fiber (in this case by a factor of 32). However, it is possible to achieve single-mode operation with a slightly larger core diameter, albeit still much less than the diameter of multimode step index fiber, by reducing the relative refractive index difference of the fiber.* Both these factors create difficulties with single-mode fibers. The small core diameters pose problems with launching light into the fiber and with field jointing, and the reduced relative refractive index difference presents difficulties in the fiber fabrication process. * Practical values for single-mode step index fiber designed for operation at a wavelength of 1.3 μm are Δ = 0.3%, giving 2a = 8.5 μm.
  • OPTF_C02.qxd 11/6/08 10:53 Page 56 56 Optical fiber waveguides Chapter 2 Graded index fibers may also be designed for single-mode operation and some specialist fiber designs do adopt such non step index profiles (see Section 3.12). However, it may be shown [Ref. 35] that the cutoff value of normalized frequency Vc to support a single mode in a graded index fiber is given by: 1 2 Vc = 2.405(1 + 2/α)--- (2.97) Therefore, as in the step index case, it is possible to determine the fiber parameters which give single-mode operation. Example 2.7 A graded index fiber with a parabolic refractive index profile core has a refractive index at the core axis of 1.5 and a relative index difference of 1%. Estimate the maximum possible core diameter which allows single-mode operation at a wavelength of 1.3 μm. Solution: Using Eq. (2.97) the maximum value of normalized frequency for single-mode operation is: 1 1 2 2 V = 2.4(1 + 2/α)--- = 2.4(1 + 2/2)--= 2.4√2 The maximum core radius may be obtained from Eq. (2.70) where: a= Vλ 2.4√2 × 1.3 × 10−6 1 = 1 --2 2πn1(2Δ)2 2π × 1.5 × (0.02)--= 3.3 μm Hence the maximum core diameter which allows single-mode operation is approximately 6.6 μm. It may be noted that the critical value of normalized frequency for the parabolic profile graded index fiber is increased by a factor of √2 on the step index case. This gives a core diameter increased by a similar factor for the graded index fiber over a step index fiber with the equivalent core refractive index (equivalent to the core axis index) and the same relative refractive index difference. The maximum V number which permits single-mode operation can be increased still further when a graded index fiber with a triangular profile is employed. It is apparent from Eq. (2.97) that the increase in this case is by a factor of √3 over a comparable step index fiber. Hence, significantly larger core diameter single-mode fibers may be produced utilizing this index profile. Such advanced refractive index profiles, which came under serious investigation in the early 1980s [Ref. 36], have now been adopted, particularly in the area of dispersion modified fiber design (see Section 3.12).
  • OPTF_C02.qxd 11/6/08 10:53 Page 57 Single-mode fibers 57 Figure 2.29 The refractive index profile for a single-mode W fiber A further problem with single-mode fibers with low relative refractive index differences and low V values is that the electromagnetic field associated with the LP10 mode extends appreciably into the cladding. For instance, with V values less than 1.4, over half the modal power propagates in the cladding [Ref. 21]. Thus the exponentially decaying evanescent field may extend significant distances into the cladding. It is therefore essential that the cladding is of a suitable thickness, and has low absorption and scattering losses in order to reduce attenuation of the mode. Estimates [Ref. 37] show that the necessary cladding thickness is of the order of 50 μm to avoid prohibitive losses (greater than 1 dB km−1) in single-mode fibers, especially when additional losses resulting from microbending (see Section 4.7.1) are taken into account. Therefore, the total fiber cross-section for single-mode fibers is of a comparable size to multimode fibers [Ref. 38]. Another approach to single-mode fiber design which allows the V value to be increased above 2.405 is the W fiber [Ref. 39]. The refractive index profile for this fiber is illustrated in Figure 2.29 where two cladding regions may be observed. Use of such two-step cladding allows the loss threshold between the desirable and undesirable modes to be substantially increased. The fundamental mode will be fully supported with small cladding loss when its propagation constant lies in the range kn3 < β < kn1. If the undesirable higher order modes are excited or converted to have values of propagation constant β < kn3, they will leak through the barrier layer between a1 and a2 (Figure 2.29) into the outer cladding region n3. Consequently these modes will lose power by radiation into the lossy surroundings. This design can provide single-mode fibers with larger core diameters than can the conventional single-cladding approach which proves useful for easing jointing difficulties; W fibers also tend to give reduced losses at bends in comparison with conventional single-mode fibers. Following the emergence of single-mode fibers as a viable communication medium in 1983, they quickly became the dominant and the most widely used fiber type within telecommunications.* Major reasons for this situation are as follows: * Multimode fibers are still finding significant use within more localized communications (e.g. for short data links and on-board automobile/aircraft applications).
  • OPTF_C02.qxd 11/6/08 10:53 Page 58 58 Optical fiber waveguides Chapter 2 1. They exhibit the greatest transmission bandwidths and the lowest losses of the fiber transmission media (see Chapter 3). 2. They have a superior transmission quality over other fiber types because of the absence of modal noise (see Section 3.10.3). 3. They offer a substantial upgrade capability (i.e. future proofing) for future widebandwidth services using either faster optical transmitters and receivers or advanced transmission techniques (e.g. coherent technology, see Section 13.9.2). 4. They are compatible with the developing integrated optics technology (see Chapter 11). 5. The above reasons 1 to 4 provide confidence that the installation of single-mode fiber will provide a transmission medium which will have adequate performance such that it will not require replacement over its anticipated lifetime of more than 20 years. Widely deployed single-mode fibers employ a step index (or near step index) profile design and are dispersion optimized (referred to as standard single-mode fibers, see Section 3.11.2) for operation in the 1.3 μm wavelength region. These fibers are either of a matchedcladding (MC) or a depressed-cladding (DC) design, as illustrated in Figure 2.30. In the conventional MC fibers, the region external to the core has a constant uniform refractive index which is slightly lower than the core region, typically consisting of pure silica. Figure 2.30 Single-mode fiber step index profiles optimized for operation at a wavelength of 1.3 μm: (a) conventional matched-cladding design; (b) segmented core matched-cladding design; (c) depressed-cladding design; (d) profile specifications of a depressed-cladding fiber [Ref. 42]
  • OPTF_C02.qxd 11/6/08 10:53 Page 59 Single-mode fibers 59 Alternatively, when the core region comprises pure silica then the lower index cladding is obtained through fluorine doping. A mode-field diameter (MFD) (see Section 2.5.2) of 10 μm is typical for MC fibers with relative refractive index differences of around 0.3%. However, improved bend loss performance (see Section 3.6) has been achieved in the 1.55 μm wavelength region with reduced MFDs of about 9.5 μm and relative refractive index differences of 0.37% [Ref. 40]. An alternative MC fiber design employs a segmented core as shown in Figure 2.30(b) [Ref. 41]. Such a structure provides standard single-mode dispersion-optimized performance at wavelengths around 1.3 μm but is multimoded with a few modes (two or three) in the shorter wavelength region around 0.8 μm. The multimode operating region is intended to help relax both the tight tolerances involved when coupling LEDs to such single-mode fibers (see Section 7.3.7) and their connectorization. Thus segmented core fiber of this type provides for applications which require an inexpensive initial solution but upgradeability to standard single-mode fiber performance at the 1.3 μm wavelength in the future. In the DC fibers shown in Figure 2.30 the cladding region immediately adjacent to the core is of a lower refractive index than that of an outer cladding region. A typical MFD (see Section 2.5.2) of a DC fiber is 9 μm with positive and negative relative refractive index differences of 0.25% and 0.12% (see Figure 2.30(d)) [Ref. 42]. 2.5.1 Cutoff wavelength It may be noted by rearrangement of Eq. (2.70) that single-mode operation only occurs above a theoretical cutoff wavelength λc given by: λc = 1 2π an1 2 (2Δ)--Vc (2.98) where Vc is the cutoff normalized frequency. Hence λc is the wavelength above which a particular fiber becomes single-moded. Dividing Eq. (2.98) by Eq. (2.70) for the same fiber we obtain the inverse relationship: λc V = λ Vc (2.99) Thus for step index fiber where Vc = 2.405, the cutoff wavelength is given by [Ref. 43]: λc = Vλ 2.405 (2.100) An effective cutoff wavelength has been defined by the ITU-T [Ref. 44] which is obtained from a 2 m length of fiber containing a single 14 cm radius loop. This definition was produced because the first higher order LP11 mode is strongly affected by fiber length and curvature near cutoff. Recommended cutoff wavelength values for primary coated fiber range from 1.1 to 1.28 μm for single-mode fiber designed for operation in the 1.3 μm wavelength region in order to avoid modal noise and dispersion problems. Moreover,
  • OPTF_C02.qxd 11/6/08 10:53 Page 60 60 Optical fiber waveguides Chapter 2 practical transmission systems are generally operated close to the effective cutoff wavelength in order to enhance the fundamental mode confinement, but sufficiently distant from cutoff so that no power is transmitted in the second-order LP11 mode. Example 2.8 Determine the cutoff wavelength for a step index fiber to exhibit single-mode operation when the core refractive index and radius are 1.46 and 4.5 μm, respectively, with the relative index difference being 0.25%. Solution: Using Eq. (2.98) with Vc = 2.405 gives: 1 λc = 1 2 2 2π an1(2Δ)--- 2π 4.5 × 1.46(0.005)--= μm 2.405 2.405 = 1.214 μm = 1214 nm Hence the fiber is single-moded to a wavelength of 1214 nm. 2.5.2 Mode-field diameter and spot size Many properties of the fundamental mode are determined by the radial extent of its electromagnetic field including losses at launching and jointing, microbend losses, waveguide dispersion and the width of the radiation pattern. Therefore, the MFD is an important parameter for characterizing single-mode fiber properties which takes into account the wavelength-dependent field penetration into the fiber cladding. In this context it is a better measure of the functional properties of single-mode fiber than the core diameter. For step index and graded (near parabolic profile) single-mode fibers operating near the cutoff wavelength λc, the field is well approximated by a Gaussian distribution (see Section 2.5.5). In this case the MFD is generally taken as the distance between the opposite 1/e = 0.37 field amplitude points and the power 1/e2 = 0.135 points in relation to the corresponding values on the fiber axis, as shown in Figure 2.31. Another parameter which is directly related to the MFD of a single-mode fiber is the spot size (or mode-field radius) ω 0. Hence MFD = 2ω 0, where ω 0 is the nominal half width of the input excitation (see Figure 2.31). The MFD can therefore be regarded as the singlemode analog of the fiber core diameter in multimode fibers [Ref. 45]. However, for many refractive index profiles and at typical operating wavelengths the MFD is slightly larger than the single-mode fiber core diameter. Often, for real fibers and those with arbitrary refractive index profiles, the radial field distribution is not strictly Gaussian and hence alternative techniques have been proposed. However, the problem of defining the MFD and spot size for non-Gaussian field distributions is a difficult one and at least eight definitions exist [Ref. 19]. Nevertheless, a more general definition based on the second moment of the far field and known as the
  • OPTF_C02.qxd 11/6/08 10:53 Page 61 Single-mode fibers 61 Figure 2.31 Field amplitude distribution E(r) of the fundamental mode in a single-mode fiber illustrating the mode-field diameter (MFD) and spot size (ω0) Petermann II definition [Ref. 46] is recommended by the ITU-T. Moreover, good agreement has been obtained using this definition for the MFD using different measurement techniques on arbitrary index fibers [Ref. 47]. 2.5.3 Effective refractive index The rate of change of phase of the fundamental LP01 mode propagating along a straight fiber is determined by the phase propagation constant β (see Section 2.3.2). It is directly related to the wavelength of the LP01 mode λ 01 by the factor 2π, since β gives the increase in phase angle per unit length. Hence: βλ 01 = 2π or λ01 = 2π β (2.101) Moreover, it is convenient to define an effective refractive index for single-mode fiber, sometimes referred to as a phase index or normalized phase change coefficient [Ref. 48] neff, by the ratio of the propagation constant of the fundamental mode to that of the vacuum propagation constant: neff = β k (2.102) Hence, the wavelength of the fundamental mode λ01 is smaller than the vacuum wavelength λ by the factor 1/neff where: λ01 = λ neff (2.103)
  • OPTF_C02.qxd 11/6/08 10:53 Page 62 62 Optical fiber waveguides Chapter 2 It should be noted that the fundamental mode propagates in a medium with a refractive index n(r) which is dependent on the distance r from the fiber axis. The effective refractive index can therefore be considered as an average over the refractive index of this medium [Ref. 19]. Within a normally clad fiber, not depressed-cladded fibers (see Section 2.5), at long wavelengths (i.e. small V values) the MFD is large compared to the core diameter and hence the electric field extends far into the cladding region. In this case the propagation constant β will be approximately equal to n2k (i.e. the cladding wave number) and the effective index will be similar to the refractive index of the cladding n2. Physically, most of the power is transmitted in the cladding material. At short wavelengths, however, the field is concentrated in the core region and the propagation constant β approximates to the maximum wave number nlk. Following this discussion, and as indicated previously in Eq. (2.62), then the propagation constant in single-mode fiber varies over the interval n2k < β < n1k. Hence, the effective refractive index will vary over the range n2 < neff < n1. In addition, a relationship between the effective refractive index and the normalized propagation constant b defined in Eq. (2.71) as: b= (β/k)2 − n2 β 2 − n2k2 2 2 = 2 2 n1 − n 2 n1k2 − n2k2 2 (2.104) may be obtained. Making use of the mathematical relation A2 − B2 = (A + B)(A − B), Eq. (2.104) can be written in the form: b= (β + n2k)(β − n2k) (n1k + n2k)(n1k − n2k) (2.105) However, taking regard of the fact that β Ӎ n1k, then Eq. (2.105) becomes: bӍ β − n2k β/k − n2 = n1k − n2k n1 − n2 Finally, in Eq. (2.102) neff is equal to β/k, therefore: bӍ neff − n2 n1 − n2 (2.106) The dimensionless parameter b which varies between 0 and 1 is particularly useful in the theory of single-mode fibers because the relative refractive index difference is very small, giving only a small range for β. Moreover, it allows a simple graphical representation of results to be presented as illustrated by the characteristic shown in Figure 2.32 of the normalized phase constant of β as a function of normalized frequency V in a step index fiber.* It should also be noted that b(V) is a universal function which does not depend explicitly on other fiber parameters [Ref. 49]. * For step index fibers the eigenvalue U, which determines the radial field distribution in the core, can be obtained from the plot of b against V because, from Eq. (2.71), U2 = V2(1 − b).
  • OPTF_C02.qxd 11/6/08 10:53 Page 63 Single-mode fibers 63 Figure 2.32 The normalized propagation constant (b) of the fundamental mode in a step index fiber shown as a function of the normalized frequency (V) Example 2.9 Given that a useful approximation for the eigenvalue of the single-mode step index fiber cladding W is [Ref. 43]: W(V) ϴ 1.1428V − 0.9960 deduce an approximation for the normalized propagation constant b(V). Solution: Substituting from Eq. (2.68) into Eq. (2.71), the normalized propagation constant is given by: b(V) = 1 − (V2 − W2) W2 = 2 V2 V Then substitution of the approximation above gives: b(V) Ӎ (1.1428V − 0.9960)2 V2 A 0.9960 D 2 = 1.1428 − C V F The relative error on this approximation for b(V) is less than 0.2% for 1.5 ≤ V ≤ 2.5 and less than 2% for 1 ≤ V ≤ 3 [Ref. 43].
  • OPTF_C02.qxd 11/6/08 10:53 Page 64 64 Optical fiber waveguides Chapter 2 2.5.4 Group delay and mode delay factor The transit time or group delay τg for a light pulse propagating along a unit length of fiber is the inverse of the group velocity υg (see Section 2.3.3). Hence: τg = 1 dβ 1 dβ = = υg dω c dk (2.107) The group index of a uniform plane wave propagating in a homogeneous medium has been determined following Eq. (2.40) as: Ng = c υg However, for a single-mode fiber, it is usual to define an effective group index* Nge [Ref. 48] by: Nge = c υg (2.108) where υg is considered to be the group velocity of the fundamental fiber mode. Hence, the specific group delay of the fundamental fiber mode becomes: τg = Nge c (2.109) Moreover, the effective group index may be written in terms of the effective refractive index neff defined in Eq. (2.102) as: Nge = neff − λ dneff dλ (2.110) It may be noted that Eq. (2.110) is of the same form as the denominator of Eq. (2.40) which gives the relationship between the group index and the refractive index in a transparent medium (planar guide). Rearranging Eq. (2.71), β may be expressed in terms of the relative index difference Δ and the normalized propagation constant b by the following approximate expression: β = k[(n2 − n2)b + n2] Ӎ kn2[1 + bΔ] 1 2 2 (2.111) Furthermore, approximating the relative refractive index difference as (n1 − n2)/n2, for a weakly guiding fiber where Δ Ӷ 1, we can use the approximation [Ref. 16]: n1 − n2 Ng1 − Ng2 Ӎ n2 Ng2 * Nge may also be referred to as the group index of the single-mode waveguide. (2.112)
  • OPTF_C02.qxd 11/6/08 10:53 Page 65 Single-mode fibers 65 Figure 2.33 The mode delay factor (d(Vb)/dV) for the fundamental mode in a step index fiber shown as a function of normalized frequency (V) where Ng1 and Ng2 are the group indices for the fiber core and cladding regions respectively. Substituting Eq. (2.111) for β into Eq. (2.107) and using the approximate expression given in Eq. (2.112), we obtain the group delay per unit distance as: τg = 1G d(Vb) J N + (Ng1 − Ng2) c I g2 dV L (2.113) The dispersive properties of the fiber core and the cladding are often about the same and therefore the wavelength dependence of Δ can be ignored [Ref. 19]. Hence the group delay can be written as: τg = 1G d(Vb) J N + n2Δ I g2 c dV L (2.114) The initial term in Eq. (2.114) gives the dependence of the group delay on wavelength caused when a uniform plane wave is propagating in an infinitely extended medium with a refractive index which is equivalent to that of the fiber cladding. However, the second term results from the waveguiding properties of the fiber only and is determined by the mode delay factor d(Vb)/dV, which describes the change in group delay caused by the changes in power distribution between the fiber core and cladding. The mode delay factor [Ref. 50] is a further universal parameter which plays a major part in the theory of singlemode fibers. Its variation with normalized frequency for the fundamental mode in a step index fiber is shown in Figure 2.33. 2.5.5 The Gaussian approximation The field shape of the fundamental guided mode within a single-mode step index fiber for two values of normalized frequency is displayed in Figure 2.34. As may be expected,
  • OPTF_C02.qxd 11/6/08 10:53 Page 66 66 Optical fiber waveguides Chapter 2 Figure 2.34 Field shape of the fundamental mode for normalized frequencies, V = 1.5 and V = 2.4 considering the discussion in Section 2.4.1, it has the form of a Bessel function (J0(r)) in the core region matched to a modified Bessel function (K0(r)) in the cladding. Depending on the value of the normalized frequency, a significant proportion of the modal power is propagated in the cladding region, as mentioned earlier. Hence, even at the cutoff value (i.e. Vc) only about 80% of the power propagates within the fiber core. It may be observed from Figure 2.34 that the shape of the fundamental LP01 mode is similar to a Gaussian shape, which allows an approximation of the exact field distribution by a Gaussian function.* The approximation may be investigated by writing the scalar wave equation Eq. (2.27) in the form: ∇2ψ + n2k2ψ = 0 (2.115) where k is the propagation vector defined in Eq. (2.33) and n(x, y) is the refractive index of the fiber, which does not generally depend on z, the coordinate along t