[6]Mathematical Principles of Optical Fiber Communications

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Mathematical Principlesof Optical FiberCommunications


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CBMS-NSF REG IONA L CON FERENC E SERIESIN APPLIED M ATHEMATICSA series of lectures on topics of current research interest in applied m athe m atics und er thedirection of the Conference Board of the Mathematical Sciences, supported by the NationalScience Foundation and published by SIAM.G A R R E T T B I R K H O F F , The Numerical Solution of Elliptic EquationsD. V. L I N D L E Y , Bayesian Statistics, A ReviewR. S. V A R G A , Functional Analysis and Approximation Theory in Numerical AnalysisR. R. B A H A D U R . Some Limit Theorems in StatisticsP A T R I C K B I L L I N G S L E Y , Weak Convergence of Mea sures: Applications in ProbabilityJ. L. L I O N S , Some Aspects of th e Optimal Control of Distributed Parameter SystemsR O G E R P E N R O S E , Techniques of Differential Topology in RelativityH E R M A N C H E R N O F F , Sequential Analysis an d Optimal DesignJ. D U R B I N , Distribution Theory fo r Tests Based on the Sample Distribution FunctionSOL 1. R U B I N O W , Mathematical Problems in the Biological SciencesP. D. L A X , Hyperbolic Systems ofConseivation Laws and the Mathematical Theory

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J E R R O L D E. M A R S D E N , Lectures on Geometric Methods in Mathematical PhysicsB R A D L E Y E F R O N , The Jackknife, the Bootstrap, an d Other Resampling PlansM . W O O D R O O F E , Nonlinear Renewal Theory in Sequential AnalysisD. H. S A T T I N G E R , Branching in the Presence of SymmetryR . T E M A M , Navier-Stokes Equations and Nonlinear Functional AnalysisM i KL6s C S O R G O , Quantile Processes with Statistical ApplicationsJ. D. B U C K M A S T E R A N D G. S. S. LuDFORD, Lectures on Mathematical CombustionR. E. T A R J A N , Data Structures and Network AlgorithmsP A U L W A L T M A N , Competition Models in Population BiologyS. R. S. V A R A D H A N , Large Deviations an d ApplicationsK I Y O S I Ii6, Foundations of Stochastic Differential Equations in Infinite Dimensional SpacesA L A N C. N E W E L L , Solitons in Mathematics an d PhysicsP R A N A B K U M A R S E N , Theory an d Applications of Sequential NonparametricsLASZL6 LovAsz, An Algorithmic Theory of Numbers, Graphs an d ConvexityE. W. C H E N E Y , Multivariate Approximation Theory: Selected TopicsJ O E L S P E N C E R , Ten Lectures on the Probabilistic MethodP A U L C. FIFE, Dynamics of Internal Layers an d Diffusive InterfacesC H A R L E S K . CHUI, Multivariate SplinesH E R B E R T S. WILF , Com binatorial Algorithms: An UpdateH E N R Y C. T U C K W E L ! , Stochastic Processes in the NeurosciencesF R A N K H . C L A R K E , Methods of Dynamic an d Nonsmooth OptimizationR O B E R T B . G A R D N E R , The Method of Equivalence and Its ApplicationsG R A C E W A H B A , Spline Models fo r Observational DataR I C H A R D S. V A R G A , Scientific Computation on Mathematical Problems an d ConjecturesI N G R I D D A U B E C H I E S , Ten Lectures on WaveletsS T E P H E N F . McCoRMiCK, Multilevel Projection M ethods fo r Partial Differential EquationsH A R A L D N I E D E R R E I T E R , Random N umber G eneration an d Quasi-Monte Carlo MethodsJ O E L S P E N C E R , Ten Lectures on the Probabilistic M ethod, Second EditionC H A R L E S A . M I C C H E L L I , Mathematical Aspects of Geometric ModelingR O G E R T E M A M , Navier-Stokes Equations and Nonlinear Functional Analysis, Second EditionG L E N N S H A F E R , Pwbabilistic Expert SystemsP E T E R J. H U B E R , Robust Statistical Procedures, Second EditionJ. M I C H A E L S T E E L E , Probability Theory and Combinatorial OptimizationW E R N E R C. R H E I N B O L D T , Methods fo r Solving Systems of Nonlinear Equations, Second EditionJ . M. G U S H I N G , An Introduction to Structured Population DynamicsTAi -P iNG L i u , Hyperbolic and Viscous Conservation LawsM I C H A E L R E N A R D Y , Mathematical Analysis ofViscoelastic FlowsG E R A R D C O R N U E J O L S , Combinatorial Optimization: Packing and CoveringI R E N A L A S I E C K A , Mathematical Control Theory of Coupled PDEsJ . K. S H A W , Mathematical Principles of Optical Fiber Communications


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J. K. ShawVirginia Polytechnic Institute and State UniversityBlacksburg, Virginia

Mathematical Principlesof Optical FiberCommunications

siam.SOCIETY FOR INDUSTRIAL AND APPLIED MATHEMATICSPHILADELPHIA


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Copyright 2004 by the Society fo r Industr ial and Appl ied Mathemat ics .1 0 9 8 7 6 5 4 3 2 1All rights reserved. Printed in the United States of America. No part of this book may bereproduced, stored, or t ransmi t ted in any man ner w i thout the w r i t ten permiss ion of the pub l i sh -er. For information, wri te to the Society fo r Indus t r ia l and Appl ied Mathemat ics , 3600Univers i ty City Science C enter, Philadelphia. PA 19104-2688.Library of Congress Cataloging-in-Publication DataShaw, J. K.

Mathematical pr inciples of optical fiber communicat ions / J. K. Shaw.p. cm.-- (CBMS-NSF regional conference series in appl ied mathemat ics ; 76)

"Sponsored by Conference Board of the Mathemat ica l Sciences; supported by Nat iona lScience Foundat ion."

Includes bibliographical references and index.ISBN 0-89871-556-3 (pbk.)

1 . Optical communica t ions -Mathemat ics . 2. Fiber optics. I. Conference Board of theMathemat ica l Sciences. II. Nat ional Science Foundat ion (U.S.). H I. Title. IV . Series.TK5103.592.F52S53 20046 21 .382 ' 7 5 ' 0151d c22

2004041717

is a registered trademark.


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ContentsPreface ix1 Background and Introduction 1

1 .1 Optical Signals 11 .2 Glass Rods 41 .3 Historical P erspective 7

2 Fiber Modes 92 .1 M axwell 's Equa tions 92.2 Planar Waveguides 1 1

2.2 .1 Even Parity TEM odes: (0, E,,0, //,,0, Hz) 142.2.2 Odd Parity TE Modes: (0, E v ,0 , //,,0, Hz) 162.2.3 Even TMModes: (E,,0, ,,0,ff,.,0)2.3 Circu lar Fibers 1 92 .4 Weakly G uiding Modes an d Pulses 233 Fiber Dispersion and Nonlinearity 273.1 The Nonlinear Schrodinger Equation 273.1.1 Ideal Linear Case 28

3.1.2 Pure No nlinea r Case 293.1.3 Derivation of NLSE 303.2 Interpretations of the NLSE 363.2.1 Perturbation Series 373.2.2 Frequency Generation 383.3 The Linea r Gau ssian Model 383.4 Other Co nsiderations 41

4 The Variational Approach 454.1 Background and Applicat ions 454.1 .1 Lag rangian and Euler Equ ations 464.1 .2 Gaussian Ansatz and W idth Parameter 474.1.3 Com parison w ith Split Step Data 484.1.4 Asym ptotically Linear Pulse Broadening 504.1.5 A sym ptotic Form of the Spectrum 50

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viii Contents

4.2 Dispersion M anagem ent 524.2.1 Dispersion Maps 534.2.2 Dispersion Maps by the Variational Method 534.2.3 The M inu s One-Half Rule 554.2.4 Precompensation and Opt imum Chi rp 584.2.5 Optimum Input Width 594.2.6 Variation al M ethod wi th Loss Term 604.3 Optical Solitons in the Variational Co ntext 614.3.1 Go verning Equ ations in Norm alized Form 614.3.2 No rm al Dispersion 634.3.3 An om alou s Dispersion 634.3.4 Subcase (i): s4.3.5 Subcase (ii):4.3.6 Subcase (iia):4.3.7 Subcase (iib):

5 Optical Solitons 695.1 Backg round in Solitons for the NLS E 695.1.1 Zakharov-Shabat Systems 715.1.2 Inverse Scattering for Zakharov-Shabat Systems 735.1.3 Linkage of the NLSE wi th Zakharov-Shabat Systems . . 745.1.4 Evolution of the Scattering Data as Funct ions of . . . . 755.1.5 Solitons and the Asymptot ics of M ( , r) for Large ... 775.1.6 Well-Posedness of Problem (5.27) 785.2 Purely Im aginary Eigenvalues 795.2.1 Sing le Lobe Po tentials 805.2.2 Com plex Po tentials 815.3 Thresholds fo r Eigenvalue Formation 835.4 R emarks and Sum mary 84

Bibliography 87Index 91

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PrefaceThe s ynergism betw een the World W ide Web and fiber optics is a familiar story to researchersin digital com mu nications. Fibers are the enablers of the rates of inform ation flow that makethe Internet possible. Cu rrently there are transoceanic optical fiber cables tr ans m itting dataat rates in the range of 1 terabit per second (1 Tb/s), or 1012 bits per second. To put thisinto perspective, if one imagines that a typical book m igh t occupy 10 megabits, then 1 Tb/swould be eq uiva lent to tran sm itting 105 books per second, or the conten ts of a respectableunivers i ty library in a few minutes . N o other me dium is capable of this rate of transmissionat such distances.

With the maturing of mobile portable telephony and the emerging broadband accessmarket, greater fiber transm ission capacity will be essential in the early 21st cen tury . Sinc eth e demand fo r more capacity drives th e development of new optics-based technologies,fiber optics therefore rem ains a vibra nt area fo r research. The fact that th e basic technol-ogy is mature means that the open questions are more sharply focused and permit deepermathematical content.What are fibers, and w hy is fiber transmission superior in high bit-rate, lon g-distancecommunications? How is it possible to transm it terabit messages in 1 second across an oceanor continent? Or, for that matter, is this figure actually small relative to some theoreticall imit? These are the kinds of questions taken up in this book. As it turns out, th e answers areusually in equal parts mathematical and physical. Indeed, the development of fiber systemsis one of the most fascinating stories in modern science because it involves the interlinkedand parallel advances of a number of scientific discipline s such as lasers, optical detectors,novel manufacturing techniques, and mathem atical modeling, includ ing wave propagationtheory. Mathematics has been especially critical since the key physical parameters in fiberdesign are determined by intrinsic mathem atical constraints. The purpose of this book is toprovide an account of thi s side of the fiber story, from the basics up to current frontiers ofresearch.To be sure, there is already a great deal of m athem atical research und erw ay involvingoptical fibers. More oen than not, mathe m atical articles on the subject start from an objectsuch as the nonlinear Schrodinger equation an d proceed with analysis from that point,leaving out connections between obtained results and actual physical systems. This may beentirely appropriate, since the omission of physical motivation can simply be a mat te r oftaste. However, the new or aspir ing researcher misses an opportunity to see the context inwhich m athem atical questions arise. Providing such a context is one goal of th is book. Itis written not only fo r both mathematical and engineering researchers who are new to thefiber optics area, but also for experienced investigators who m ay add richn ess to their ow n

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x Preface

backgrounds through a better understanding of context and nom enclature, both phy sical andmathematical.

To o utlin e the book, the first chapter presents an overview of fundamental concepts,including basics of digital communications, light guidance, fiber construction, and somehistory. Fiber modes are discussed in Chapter 2, startin g with Maxw ell 's equations, detail-ing why com m unications fibers need to be single mode and exp lainin g the consequences.Chapter 3 presents the nonl ine ar Schrodinger equ ation, the essential elem ents of its deriv a-tion, and the physical interpretation of its terms. The variational approach to m ode ling pu lsepropagation, which has been a critical tool in the understand ing of nonlinear fiber opticssince the early 1980s, is taken up in the fourth chapter. In Chap ter 5 solitons are discussedfrom the standpoint of inverse scattering. The chap ter and book close with some new resul tson optical soliton formation thresholds.

The chapters are written so as to be as independent as possible. There are occasionalreferences to previous or subsequent material, but these are given mainly to place topicsin context. An effort has been made to keep th e reading lively, useful, and neither tersenor ponderous. The selected topics are representative, rather than comprehensive. Theaims are (1) to provide readers with a means to progress from l imited knowledge to theresearch frontier in the shortest time possible and (2) to offer experienced researchers,who may have a narrower background in either fiber comm unicat ions engineer ing or itsmathematical infrastructure, with sufficient information, terminology, and perspective toengage in interdisciplinary activity.

This monograph is a result of the NSF-CBMS regional conference onMathematica lMethods in Nonline ar Wave Propagation held at North C arolina A & T State University, M ay15-19, 2002. It is a pleasure to thank the conference participants and especially DominicClemence, Guoqing Tang, and the other members of the Mathem atics D epartm ent at A & Tfo r their gracious hospitality. All who may benefit from these pages are also inde btedto the Conference Board of the M athem atical Sciences, SIA M, an d the N atio na l ScienceFoundation.

Arlington. VirginiaJu ly 2003


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Chapter 1Background andIntroduction

This chapter wi l l give an overview of fibers and ho w they are used in optical communicat ions .The first section covers very basic terminology used in digital l ightwave communicat ions ,the second discusses some of the phy sical properties of fibers, and the third gives a historicalperspective.

1.1 Optical SignalsAs in any com m unications netw ork, optical systems have transm itters and receivers. Fibertransmitters are typ ically either semiconductor lasers or lig ht-em itting diodes (LEDs) cou-pled with modulators so as to produce a train of light pulses which represent digital bits.Receivers usually consist of photodetectors, which operate on the principle that a smallelectrical current is generated when a photon is absorbed (photoelectric effect), togetherwith processing electronics. The transmitter and receiver are synchronized so as to launchand detect, respectively, th e pulses of l ight in discrete, predetermined tim ing w indow s, thatis , se t intervals in the t im e variable. The signa l itself serves to synchronize transm it ter andreceiver by feeding the temporal separation of detected pulses into a timing device.In th e binary signal format , w hich is by far the most comm on, th e t ransmit ter laun chesa pulse of l ight into a given w ind ow to represent a "1" bit , whereas no pulse is inserted ifa "0" bit is desired. Likewise, th e photodetector looks into a t iming window and recordsa "1" if a pulse is detected and a "0" otherwise. In practice, there is always some level ofsignal present, w hich may be due to optical noise, spreading from other w indow s, or othereffects. Detectors are therefore decision m akers, and the 0/1 decision is triggered by a l ightintensity threshold. In fiber systems the medium between transmitter and receiver is thethin, flexible glass filam ent, or fiber, that guides the pulses. Len gths of ind ivid ual fibers canrun from centim eters up to hu ndred s of kilometers. Pulse propagation is the term used todescribe th e process of movi ng a pulse through th e m edium from t ransmi t terto receiver.

The format in w hich every 1 bit corresponds to an individual pulse with vanishingtails completely confined to a t iming window is referred to as return-to-zero (RZ). In thenon-retum-to-zero (NRZ) binary format, consecutive 1 bits maintain th e intensity levelwithout dropping to 0 at the interface between the consecutive windows. A string of N1


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2 Chapter 1. Background and Introductionconsec utive 1 NR Z bits surrounded by Os is th us in reality a pulse N t im es as wide as a pulsecorresponding to an isolated 1. The RZ and NRZ formats as described here are examples ofpulse modulation formats, of which there are many and varied. Some modulat ion formatsalter th e phases on strings of pulses, in which case the receiver obviously must be able toextract phase information.

Pulses are distorted during propagation, by both random and determinist ic processes,and the overriding goal of digital communications is to correctly interpret received bitsas they were intended. There are many causes for signal degradation, but the principaldeterministic ones in fiber are attenuation, dispersion, and nonlinear effects. Attenua t ion,or loss, is the process of scattering and absorption of energy by the me dium i tse l f. In fibersth e loss figure is small compared with other systems, which is one of the reasons for thesuperiority of glass. Attenuat ion is measured in the logarithmic decibel, or i//?, scale. Agiven ratio RQ,such as the ratio of output to inp ut light intensity, is converted to dB scaleby the formula

Modern fibers have attenuation figures of about 0 .2dB per ki lometer [A gl] ; that is, RDB =0 . 2 dB /Km . A t 1K m , R0 is thus about 0.955; that is , less than 5% of initial pulse energyhas been lost. To put this into perspective, a d i m m i n g of 5% of l ight in the visible spectrumis undetectable to the hu m an eye; thus, there would be no discernable change in an imageviewed through a pure silica window 1 Km thick. (Silica is silicon dioxide, s i o 2 , the pr inciplematerial used in making glass.) Modern transmitters insert pulses at peak power levels of afew tens of milliwatts (mW; \mW = 10- 6W O per channel, and optical receivers can delectlevels lower than 1%, or 20dB , of that figure. Under such conditions a pulse could bedetected aer traveling over 100Km, with no amplification or boosting of the signal level .In practice, distances between am plifiers vary due to a number of factors. Fiber attenuationis also frequency dependent. The figure of 0 . 2dB /Km is the attenuation m i n i m u m [A g I ] forsilica and occurs at a wavelength of about 1.55 microns (or micrometers, abbreviated /* w ,\fj.m = 10~6m). Note that frequency a) and wavelength A. are related by Xw = 2nc, wherec = 3 x 108m/5 is the vacuum speed of l ight. There are wavelengths w here other m i n i m aoccur, such as dispersion, to be discussed next, but attenuation is the most basic con tribu torto signal degradation, and for this reason communicat ions fibers operate near 1.55/i/w.whichis in the near infrared of the color spectrum, beyond th e visible range. The attenuation versuswavelength curve is rather flat in the vicinity of the m in im um ; thu s a nu m ber of differentisolated wavelengths, or channels, are available for low-loss t ransmission.

It should be noted that losses also occur from bending and splicing. If a bend in a fiberhas a diameter less than a few inches, loss can be sign ificant. There are various methods ofsplicing fibers together. Fusion splicing, for instance, ess entially melts ends together andresults in losses of as low as O.OldB and negligible backscatter [IEC).

Fiber dispersion arises from the fact that the speed of l ight in glass is frequ enc ydependent. Optical pulses have frequency comp onents (colors) that are concentrated a rounda central wave length, or carrier, which, as mentioned, is usu ally near 1.55/im. How ever, acontinuum of wavelengths is present in any real signal, and the fact that dist inct w ave leng thcomponents travel at different speeds means that pulses pull apart, spread, or disperse int ime (unless measures are taken to prevent it, as will be discussed in Chapter 4). In orderto understand pulse spreading, it is convenient to have a measure of pulse width, with


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1.1. Optical Signalsreference to the t ime variable. As the word itself indicates, pulses are short bu rsts of energy.In fibers the energy is in the form of l ight and of in tensi ty |/(z, tl2 where f ( z , t) is theelectromagnetic field envelope at propagated distance z and t ime t. Here \ f ( z , t) l|2 has thephysical dimens ion o f power and is the quant i ty detected in m an y receivers. The total energ yin a pulse is given b y t h e integral /^ |/(z, t)\2dt. Since \ f ( z , t) \2 is norm ally given inm W , then th e total energy has un i t s of picosecond mill iwatts (ps m W ) . A picosecond is10~ 12 seconds. The physical d i m e n s i o n sp s m W are the same (but th e units are different) asthose that appear on a consu m er's electric power bill, nam ely, kilowa tt hours. Oneps m Wis also equal to 1 femtojoule (\fJ = 1015/) in mechanical energy units.Typically, for each distance z, \ f ( z , t) \2 is localized in the neighborhood of a globalm a x i m u m and drops q uic kly to 0 in both t ime (f) direct ions (for RZ pulses) . The t im evariable t can be set to local, that is , t = 0 at the peak or m a x i m u m . For such shapes, whic hare ideally symmetric, it is mean ingfu l to define a. pulse width. A common metric is the h a l f -width-hal f -maximum (HWHM), T = T(z), defined implicitly by \ f ( z , T)| = |f/(z,0)|/2[A g 1 ]; it is the local t ime required for the pulse m odulus to decrease by half. More generally,T can be defined in terms of a reference level other than 1/2. I f f ( z , t) is Gaussian in t,it is cus tomary [Agl ] to define T implic i t ly by the equat ion \ f ( z , T)\ = |/Cz,0)|/>/e.For example, let z 0 and /(O,t) = M e~a ' (compare (3.40) and (4.6) below). Thenthe half -width at z = 0 would be T = l/(a>/2), that is, a2 = \/(2T2). Then /(O,t)m ay be writ ten in the more common form /(O,f) = M e~(t /2 T } [A gl] . Even though th iscorresponds to the reference level 1 l


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4 Chapter 1. Background and Introductionthat the dominant contributionis 0. Some early-generation fibers were designed to operateat 1.3/im. The most recently installed fibers are dispersion-shied fibers [Gof], whose zerodispersion wavelength has been shied by a design and m anuf ac tu r ing process so tha t thedispersion m i n i m u m is also near th e loss minimumat 1 . 5 5 f j . n i .The subject of dispersion will be a factor in essentially every topic in this monograph.This preliminary discussion is meant to provide at this early point an overall sense of itsrole in fiber communications.Turning to nonlinearities, about all that can be said at this introductory stage, be-fore encountering the nonlinear Schrodinger equation, is that fiber nonlinearities cause th eFourier spectrum of a signal to change. By spectrum, or Fourier spectrum, one meansthe support of the Fourier transform of the envelope f(z, t), that is, the set of wavel en g t hswhere th e transform is not negligibly small. In an ideal, linear fiber the spectrum is i n v a r i a n tdu r i n g propagation; no new frequencies are created and none are destroyed. When s igni f -icant nonlinearities are present, the spectrum changes. If new frequencies are generated,then account must be taken of the receiver bandwidth, which is obviously finite, as wellas frequency filters in the system. Frequency components that are outside filter or receiverba nd wid ths are blocked, and consequently portions of the signal are removed. These issueswil l be discussed in more detail in Chapters 3 and 4.

1.2 Glass RodsA communications optical fiber, illustrated in Figure l.l, consists of a glass rod of indexof refraction 2, on the order of 100 or 200 microns in diameter, with a central core regionof fewer than 10 microns in diameter and slightly raised index of refraction |. This issheathed in a plastic or polymer coating for protection. The outer cylinder of the lowerindex of refraction HI is known as the cladding region; th e sheathing is also called th ejacket. Though one t h inks of glass as brittle and prone to shatter, these small-diameterrods of pure silica are strong and flexible. Fibers with a discontinuous index change at thecore-cladding interface are called step index fibers.Ligh t is confined to the core by total internal reflection, which is a consequence of

Figure 1.1. Core, cladding, and jacket of an optical fiber.


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1.2. Glas s Rods 5

Figure 1.2. Snell's law.

Figure 1.3. Total internal reflection in a fiber.

Sn ell 's law. If a light ray strikes the interface between m edia w ith indices of refraction n\and 2 at an angle 9\, as shown in Figure 1.2, then a portion of the light reflects at the incidentangle 0\ and a portion refracts at an angle 62 according to n\ sin 9\ = 2 sin 62 (Snell's law).In Figu re 1.3 (not draw n to scale) a fiber core and cladding are labeled with indices n\ and 2 respectively, where n\ > 2- If in Snell's law B\ is sufficiently close to Tr/2, then 02is forced to be equal to n/2 as in Figure 1.3; specifically, this happens when 9\ attains thecritical angle B c = Arcsin(n2/n1)> at which point all rays are reflected back into the fiber.The figures and discussion here are for planar media, but the same principle applies to raysin a circular core. Since lig ht can be inserted very near the exact center of the core, thenrays im pinge radially on the core-cladding interface and are reflected or refracted as theywould be on s t r ik ing th e tangent line.


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6 Chapter 1. Background and IntroductionThe index of refraction of a medium is the ratio of the speed of light in a vacuum toth e speed of light in the me dium . Fiber cladding usu ally consists of fused silica, which hasan ind ex of about 1.45. The core region is created by doping the silica with germanium

[Gof], which raises th e index to around 1.47. Us ing these value s for ;i| and ; 2 yields acritical angle of C = Arcsin( 1.45/1.47) = 80.5. The index of refraction actually varieswith wavelength; the figure of 1.45 occurs at the standard 1 . 5 5 f j . n i carrier.It is easy to imag ine the challenge of fabricating a glass rod with such a small core andwith such a small increment in index. Indeed, the core diameter is only a few ten- thousandthsof an inch. The cladding outer diameter m ight run , as me ntioned, a few hun dred m icron sor a few thousandths of an inch. The plastic jacket need not be especially thick, since it spurpose is to maintain purity rather than add strength; if exposed to the environment andhandling, an uncoated silica surface can develop m icrofractures that lead to weaken ing . Theentire ensemble is thus quite tiny, on the order of a tenth of a millimeter, but surprisinglystrong; it is not easy to break a fiber by hand. The thickness of the cladding represents acompromise. It should be thick enough to ensure near-total confinement of l ight to the corebut thin enough to maintain flexibility of the fiber. The jacket may consist of two layers:A so inne r layer is desirable as a cushion and also to ease stripping off of the jacket forsplicing and connecting, and the harder outer layer serves as a protector [IEC].If given a choice, ma nufactu rers w ould not make fibers with such small core sizes.However, the small size is an intrinsic mathem atical constraint arising from th e necessityof there being a single eigenvalue for a certain boundary problem. This wil l be discussedin th e next chapter.The standard way to make fiber is to construct apreform, which is a scaled-up versionof a fiber, and to draw it into a strand du rin g a molten stage. In the insid e vapor de po sitio n(IVD) process, the preform starts as a hollow silica tube. In simple terms, germ aniu m gasis pumped into the inner cylindrical space and the entire tube is heated. In some casesthe interior may be placed under a slight vacuum. As in a glass-blowing process, the tubebecomes sligh tly molten and gradually collapses. To prevent bending, th e tube is rotated ona lathe during heating. M eanw hile, the in ner surface of the tube is coated with a germaniumresidue that elevates the index of refraction by about 1%. The cooled, collapsed tube iscalled the preform, which is a solid, transparent rod with a small raised-index core of a fewmicrons. Preforms can vary greatly in size. At research laboratories they can be as sm allas a few centimeters in diameter and a couple of feet in length. Preforms manufactured atthe ma jor fiber providers can be sufficiently large and heavy to require mechanical devicesfor liing and carrying. An outside vapor deposition (OVD) process deposits both core andcladding sequentially on a "bait" ro d [IEC], which is then removed, leavin g th e preform. Ifthe refractive index changes con tinuo usly , the fiber is called a graded index fiber.In the drawing process, which is an impressive science in itself, the preform is feddow nw ard lengthw ise into a furnace and pulled into a thin strand w ith th e core remainingat th e center. As the fiber is pulled it is coated with th e polymer jacket and wrapped onto aspool. The entire drawing, coating, and spooling assembly is called the draw tower. Thefurnace is located at the top, oen 20 feet or more above floor level. The strand of fiberruns throug h a reservoir of liquid polym er for coating, or app lication of the jacke t. There isoen a UV radiation cham ber located below the polymer applicator to accelerate the cur ingand drying of the coating. Aer dropping, d rying , and cooling, the fiber is w oun d onto therotating spool. Obviously, there are a number of delicate operations running simultaneously


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1.3. Historical Perspective 7

and under conditions of precise timing in a draw tower. The speed at which the fiber movesd ur i n g drawing varies according to the tower type. For smaller laboratory towers thedraw speed might be on the order of a meter per second. Industrial draw speeds are oenproprietary information bu t certainly can run in the range of tens of meters per second.The draw speeds are actually controlled by a feedback loop connected to a fiber diametermonitor . That is, the diameter of the drawn fiber can be optically measured and the drawspeed automatically adjusted to increase or decrease the diameter.

1.3 Historical PerspectiveCommunica t ion with electromagnetic waves began in the 1830s with the development ofthe telegraph; the first trans-Atlantic telegraph cable was installed in 1866. The dot-dashalphabet in Morse code is a form of the binary format. The telephone, invented in 1876,originally operated in analog mode; that is, it used continuously varying electrical currentinstead of a fixed number of discrete levels (such as 0 and 1). Whether analog or digi tal ,wire pairs still locally connect most homes and businesses to the larger telecommunicationsnetwork. Once a signal gets out of the home or office and down th e street, or from a cellularphone to the base, it can enter the optical network. The copper wire segment of the networkis referred to informally as the last mile; fiber to the home or office is not widely availablebecause of the expense of optical cable installation. Although th e exact form of the next-generation home and office network is not known, th e network will eventually be mostlyfree of wire. Wireless, cellular, and free-space optical communications are prime candidatesto replace copper in the 21st century.

From the 1940s through the 1970s, coaxial cable and microwave systems dominatedthe long-distance industry. However, it had long been understood that optical frequencieshave an inherent advantage over cable and microwave since th e carrying capacity of anyelectromagnetic communication channel is proportional to the carrier frequency; that is, theavailable bandwidth Aw in a channel is a few percent of the carrier f requency a> [Ag2].Using the figure of 1% [Ag2], a WGHz microwave carrier ca n support a channel withHWHM Aw = \OOMHz (IHz - Is"1 is a standard unit of frequency; \MHz = 106//z,\GHz = \09Hz, \THz = 10l2//z). Fourier transforming to the time domain and usingA w A f = 1 ([Agl]; also see (4.15)) yields a temporal pulse HWHM of T = 10~8s, orlOns, where ns stands for nanosecond, or 10~95. The bit rate / f y , = l/T then becomes108bits/s, or IQQMb/s. A parallel calculation using th e 1.55/^m optical carrier with C D =12.15 x 1014 yields Aw = 1277/z, which is larger than the microwave bandwidthby fiveorders of magni tude. The reasons coaxial cable, microwave, and satellite systems dominatedcommunica t ions between the 1940s and the 1980s were the lack of low-loss fiber and, unt i lthe 1970s, reliable room-temperature lasers. Until 1970, the best grade optical fibers hadloss figures in the range of hundreds of dB/Km. But in 1970 the Corning Corporationdemonstrated a glass with attenuation below 2QdB/Km [Hec]. This is equivalent to at least1 % intensity at 1 Km, which is the loss level of unrepeatered copper wire. At that pointfiber immediately became competitive. Current fibers have loss figures of 0 . . 2 dB /Km , asmentioned earlier, meaning that signals can propagate over l00Km without amplif icationor regeneration. By comparison, the best coaxial cables require repeater stations spaced nomore than a kilometer apart [Ag2].


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8 Chapter 1. Background and Introduction

In 1987 erbium-doped fiber amplifiers became available, which meant tha t opticalsignals could be optically amplif ied; that is, it was no longer necessary to receive, regenerate,and retransmit optical signals. Erbium amplifiers are used because they operate in rangesnear 1.55/zm. With fiber amplifiers, optical signals can be transmittedpractically un l im i t eddistances without electrical intervention. In effect, this means that loss is no longer thedominan t impairment for the 1.55/zm carrier; that is , systems can be designed to mit iga teother deleterious effects such as dispersion and nonlinearity.

Glass filam ents or fibers have a very long history, actually dating back to Roman t imes[Hec]. However, they were regarded as little more than parlor games and decorations un t i llow-loss fibers became available. Th e evolution from novelty to communica t i ons backbonetook years to occur, and in some cases progress had to wait for the development of newtechnologies such as sources (lasers) and detectors operating at specific wave l eng t h s . Infact, according to some writers [Ag2], the present fibers represent at least the fih genera t ion .There are several good accounts of the development of optical fibers. The texts [ Ag 1 ] and[Ag2] of G P.Agrawal contain interesting historical discussions, and the recent book of JeffHecht [Heel is a beaut i ful and detailed history of the subject. References [Sav] and [Den]are historical and general-interest articles on fibers, respectively.


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Chapter 2Fiber Modes

A ll long-distance communications fibers are single-mode fibers, abbreviated SM F in thel i terature. Single mode means that the field propagating in the fiber travels at a singlegroup velocity, w hic h is the princip al part of the wave speed of the pulses. That is, apulse, w hic h is a superposition of a contin uum of frequency components, can be viewedas located and slowly evolving within a moving reference frame, where the speed of theframe is the group velocity. Multimode fields are ones in which the field decomposes intocomponents traveling at different group velocities. The dispersion mentioned in Chapter1 that is caused by different wavelength components traveling at different speeds is calledchromatic or intramodal dispersion and refers to dispersion within a single mode; i.e.,intramodal dispersion refers to the changes in pulse shape within th e m oving time frame. Formultimod e fields, a pulse can have components corresponding to distinct modes with une qualgroup velocities, and thus th e overall change is much greater; this is called intermodaldispersion. Multimode fibers (MMF) are completely undesirable from a long-distancecommunicat ions standpoint, although there are circumstances, such as in sensing and inlocal area networks, where MMF can be used (for example, when the propagation distanceis only a few kilometers [IEC]). This chapter discusses the way in which fiber geometryleads to the phenomeno n of modes, which in turn induces constraints on the physical sizeof the fiber core.In order to keep matters as simple as possible, the main presentation will be given forplanar waveguides and then adapted to circu lar fibers. All of the princ ipal ideas invo lvedin modes of circular fibers are contained in the simpler theory of planar or slab waveguides.The discussion will start at the beginning, namely, Maxwell 's equations and boundarycondit ions at an electrom agnetic interfa ce, then proceed to planar waveguides in section 2.2and circula r fibers in section 2.3. The m odal p atterns obtained in section 2.3 can be greatlysimplified by introducing weakly guiding, or linearly polarized, modes. This is covered insection 2.4.2.1 Maxwell's Equat ionsPulses in fibers are physical realizations of electromagnetic (EM) fields that consist of sep-arate electric and magnetic components and are postulated to satisfy Maxwell's equations,

9


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10 Chapter 2. Fiber Modesnamely [Agl],

where E = E(x, y, z, f) andH = H(x, y, z, t) are theelectric andmagnetic fields, respec-t ively. The field P = P(x, y, z, t) is called thepolarization vector and wil l be discussedpresently. In (2.1) EQ an d ( J L Q are the free-space dielectric permittivity and magnetic perme-ability, respectively, whose specific values [Ish] are unimpor tan t at the moment . One hasthe relationship [Agl] e^no = l^c2, c being the vacuum speed of l ight .

The polarization vector P in (2.1) corresponds physical ly to the response of themedium in which the EM field exists. That is , an incident EM field excites the m ed ium ,which then responds both l inearly and nonlinearly in such a way as to generate a total EMfield. In nonl inear f iber optics the polarization vector is assumed to be the sum of l in ea r andnonl inear contributions, n amely [Agl],

where specific forms of PL(X, y, z, f) and PNL(X, >', z, t) are postulated. The lin ear part ofthe response has the form PL = e\ E, where e\ is a constant associated wi th thepropagatingmedium; in a vacuum, e\ = 0.

Chromatic dispersion and the influence of non l ineari t ies from the term PN L turn out tobe comparable, that is, they have similar magnitudes and can be compared directly, but bothare m uch smaller effects than intermod al dispersion [A gl ]. For this reason, fiber modescan be defined and characterized without regard to the nonl inear term P N L * which wil lconsequently be dropped from P = PL -f PN L in (2.1) in the analy sis of modes [Ag 1 ]: thatis, in (2.1),

in this chapter.Here e e0 + | is the dielectric permitt ivi ty of the silica medium. The term PNLwil l become important in the discussion of the nonl inear Schrodinger equation in Chapter

3, where specific forms for the nonlinear response PN L and a more general form for thelinear term PL are discussed. The form fo r PL w ill specialize to P = e\ E as above wh enthe response t ime of the medium is neglected.

The permitt ivi ty e in a general medium determines the speed of l ight in the m ediu mby the fo rmulae / z0 = 1 /u 2 [Agl], in analogy to the vacuum re la t ionship 0Mo = 1 / f 2 - Theindex of refraction, n, of a medium is the ratio of the vacuum speed of l ight to the speed ofl ight in the medium; that is, n2 = c2/u2. From the above expressions, th is can be wri t tenin th e form


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2.2. Planar Waveguides 11

Figure 2.1. Tangential components at a dielectric interface.

In view of (2.2) th e coefficient e in (2.1) has a discontinuity at the core-cladding interface, oneither side of which the index n takes different values n \ and 7 /2 as in Figure 1.3. It turns outin consequence that th e optical field c an n o t be con t i nuous in all components across such aninterface. Separation of the field into con tinuou s and disco ntin uous com pon en ts is achievedby applyin g interface c ondit ions at the core boundary. In the case of optical waveguides, theappropriate conditions at an interface are cont inui ty of tangential components [Ish, Mar].This boun dary condit ion can be justified on the basis of Stokes' theorem and is derived inm a n y books on vector analysis. The requirement is that th e components of E and H that aretangentia l to the interface be con t i nuous across the interface. Figure 2.1 depicts a genericsurface separating a section of waveguide material into regions with indices of refraction n\an d / i 2- The arrows represent components of either E or H that are tan gen tial to the surface .A t any point on the surface the boundary condition requires that the limits of tangentialcomponents from either side of the surface be the same.

2.2 PlanarWaveguidesNow consider a planar waveguide as illustrated in Figure 2.2. In a slab, planar, or layeredwaveguide the core is a lateral cross section between x = d and x d with indexof refraction n\ sandwiched between regions of index n2 where n\ > n2- The physicald imens ions are such that th e core thickness (2d) is many orders of magnitude smaller thaneither the wid th, measured along the y axis, or the total depth along the x axis. The variablesx and y thus have range oo< x, y < oo. The z axis is the direction of pulse propagation.The pr inc ip le of total in tern al reflection via Snell's Law con fines l ight to the core region asin Figures 1.2 and 1.3. Fields existing in the planar guide are assumed to be uniform in they direction , so tha t all field qua ntitie s have van ishin g partial derivatives w ith respect to they variable (9V = 9y = 0). In general the refractive indices of the regions on either side


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12 Chapter 2. Fiber Modes

Figure 2.2. Planar or slab waveguide.

of the core ca n differ but are kept equal here fo r simplicity.In order to determine modes of (2.1) it wil l be sufficient to find all solutionsof the form

(2.3)where w, the central frequency of the signal, is arbitrary and is to be determined as aneigenvalue parameter. That is, solutions of (2.1) in the form = e(x, v, z)e~'"" andH = h(x, y, z ) e - i w t can be realized as superpositions of solutions of the form (2.3) [Mar].Values of wi l l be determined by applying boundary and interface conditions to solutionsof (2.1) of the form (2.3).Introducing the notation E0(x, y) = (Ex, Ev, Ez\ H0(x, v) = (Hv , Hv, Hz), whereth e subscripts designate x, y, and z components, and substi tuting (2.3) into th e secondequation of (2.1) gives, by definition of the curl vector,

and a similar equation for the curl of E. Equating components in (2.4) and canceling th ecommon exponential yields the three equations

Imposing the uniformity condition d/dy = 0, (2.5) reduces to


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2.2. P l a n a r Waveguides 13Using the other curl equation in (2.1) similarly leads to

Note that equations (2.5) and (2.6) can be regrouped into sets of ind epend ent equations,namely ,

an d

The total field can thus be decomposed as

where the indicated subfields a re self-contained by (2.7) an d (2.8), respectively. The sub-field (E x, 0, E z, 0, H y, 0) is referred to as a transverse magnetic (TM) mode or subfieldbecause th e ma g ne t ic field H has no lo ng i t u d ina l (i.e., z) component . Similar ly, th e sub-field (0, E y, 0, H x, 0, H z) is termed transverse electric (TE).It is worth n ot ing tha t th e second tw o equations in (2.1) have n ot been used and, infact, follow from the first two equat ions for fields of the form (2.3).

A n eigenvalue problem w ill now be solved for the parameter b in the TE case.T E Modes: (0, E,9 0, HX9 0, Hz)Take the derivat ive of the second equation in (2.8) an d substitute from the third eq uationfo r the resulting term 8H z/dx; this gives

and finally


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14 Chapter 2. Fiber ModesReferring to Figure 2.2, the tangentia l com ponents are those in the v and z direct ions.

Thus Ey and Hz are continuous in (2.8). By (2.8) 8Ey/dx is also cont inuous, so that thesolution to (2.10) is smooth across the jump discontinuity in e at the interfaces x = d.Note that H x is also continuous by (2.8) but that d f i z / d x cannot be con tinuo us since e hasa jump.

Note that either change of variables co - a ) or - > leads to the same eq uation(2.10). Thus the original t and z harmonic assumption e~i(tot~^ could be replaced byei(wt-z) w ifa (kgsameresuit (compare [Mar]).

Equation (2.10) is conventionally written in terms of the wave number k defined byk = (o/c = 2 7 T / A . (recalling Xw = 2nc). Using (2.2), and recalling that o A < o = l/c2(below (2.1)), one has n2k2 (e/eo)> 2oMo = w 2e/n 0- In other words (2.10) is the same as

Equation (2.11) can now be solved subject to the physical ly plausible condit ions that th efield Ey should be decaying towards x = 00 and smooth over the t ransi t ions at .v = dbd.This requires that (2.10) should be oscillatory within d < x < d and nonoscillatoryoutside th is interval, the differe nce being determined by the valu e of the index n in core andcladding. It will be convenient to define variables [Ada]

Both quantities in (2.12) must be nonnegative (i.e., n\k2 < 2 < n2/:2) in order for (2 .11)to be oscillatory in the core d d, wherecontinui ty forces Ev = Acos(Kd)e~y(x~d\x > d. That is, the tangen tial compon ent E_vis continuous across jc = d. The other tangential compon ent is //-,which by (2.8) is givenin d < x < d by

and in x > d by


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2.2. P l a n a r Waveguides 15

Figure 2.3. Even TE modes.

Continuity of the tangential component H z is guaranteed by equating (2.14) and (2.15) atx = d, so as to obtain

Noting (2.12) it wil l beuseful to define V2 = K2d2 + y2d2 = k2d2(n ] - n\} andU = Kdso that yd = V V 2 U 2 together with (2.16) yield th e chara cteristic equa tion

The separate sides of (2.17) are graphically il lustrated in Figure 2.3 for V = 0.9, 1.4, 2.5,and 4.0. The solid curve is the graph of y = U tan((7) and the dotted curves are plots ofy = VV2 - U2 for V = 0.9, 1.4, 2.5, and4.0. For every V > 0 there is at least oneintersection, and for V > K there are at least tw o intersections. The key observation is thatif V < TT there is exactly one intersection. The root, call it UQ, relates to through (2.12)as K2 = n}k2 - 2 = Ufi/d2 or 2 - n]k2 - (U%/d2). The value of determines thew a ve speed in the slab guide fo r w av es of the form (2.3). Note that (a is the f requency an dthat k = a)/c = 2n/X, where A is the wavelength. By definition of V, the condition V


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16 Chap te r 2. Fiber Modesis equivalent to (2nd/)^Jn2, n\ < n or

Using typical values of n \ 1.45, 2 = 1 -44 [Gof] to illustrate , (2.18) becomes d < 2.94X.So if the wave length X is on the order of 1.5/zw, then the cond ition V < n says that th ecore of the slab in Figure 2.2 can be no thicker than about 9 microns.Before going on to discuss other types of modes, a point about the unsu i t ab i l i ty ofmult imode waveg uides can now be made. Suppose the electric jield (a similar analys i sholds for the magnetic field) consists of two modes of the form E = EQ(x)e~ i(Mt~^} to rdistinct values of . Components of the field are then superpositions of the form

Indeed, the total field can be expressed in the form of a complete orthogonal eigenfunct ionexpansion involvin g all the modes [Mar]. But assum ing that only the two modes propagate,the previous equation can be w ritten as

At points z along th e guide, where z = $| _ff i * the above expression reduces toe-(a>f-/ ' iz)(yj ( j t ) /2( j t ) ) which is subject to cancellation betw een f\ (x) and /2(.v) depend-in g on the signs of the ind ivi dua l terms. This is the wel l -known phenom enon of destructiveinterference. An optical detector looks for a pulse of light in order to record a "1" state,while the absence of l ight corresponds to the "0" state. D estructive interference can result inholes in the spot of light corresponding to a pulse, or at least diminished intensity sufficientto cause errors. Such interference ca nnot take place if only one mode is present, and th is isthe basic idea behind single-mode fibers.In this same connection it should be mentioned that some optical devices used fo rsensing are designed s o that two modes propagate. Such sensors rely on mo dal interferencepatterns to detect changes in the physical environment of the device. For exam ple, a b im odalfiber can b e tested to determine an equi l ib rium interference pattern between tw o modes. Ifthe fiber is stretched slightly, th e pattern changes, and the change can be used to infer theamount of the displacement from equi l ib r ium. In this way mul t imode fibers can be used todetect, for example, motion or temperature changes.The other mode types will now be treated, but the analys i s is s imi lar and it wil l beenough to discuss jus t th e ma in ideas. First, the odd TE modes wil l b e covered.2.2.2 Odd Parity TE Modes: (0, E,9 0, Hx, 0, Hz)The solution in d < x < d has the form Ey = A sin(Kx) fo r some constant A. Work ingthrough the same analys is as above leads to the characteristic eq ua tio n


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2.2. Planar Waveguides 17

Figure 2.4. Odd TE modes,w h i c h is analogous to (2.17). The separate sides of (2.19) are plotted in Figure 2.4 wi thV = 0.9 ,7T /2, 2.5, and 4.0. The solid curve is the graph of y = U cot(U) and the dottedcurves are plots of > = for V = 0.9, /2 , 2.5, and 4.0. Note tha t there is nointersection unt i l V /2 and at least two intersections aer V = 3 /2. T herefore, thelowest order even T E mode propagates alone for V < n/2.

TM Modes: (E x, 0, Ez, 0, Hy, 0)Now cons ider th e grou ping (2.7). Differentiating th e second equation and substi tut ing forE z/d x from th e third yields th e Sturm-Liouvil le-type equation

in com plete analogy w ith (2.11). D efine K and y as in (2.12) and, follow ing (2.13), considerfirst th e even TM modes.2.2.3 Even T M Modes: (EX9 0, EZ9 0, H,9 0)In th e region d < x < d th e even solution to (2.20) has the form H y = Acos(Kx),whereas H y = A co$(Kd)e~ y(1(~d) fo r x > d; it is sufficient to consider only posi t ive


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18 Chapter 2. Fiber Modesvalues of jc by even symm etry. The relevant tang entia l com pone nts are H y and E- (see

I "1//Figures 2.1 and 2.2), where additionally Ez = 7^- . This case now diverges from(2.14) and (2.15) because the coefficient \/iu>e is discontinuous across ,t = d. In thecore region d


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2.3. Circular Fibers 19

2.3 Circular FibersSome key equations will first be derived in the context of general waveguides and thenspecialized to circular fibers. In this treatment, some technical fine points willbe suppressedin order to avoid gettingbogged down indetails. Forexample, the finalsteps leading to (2.34)below w i l l be omitted, as well as the calculation and classification of its roots. Referencesto these derivations will of course be provided.

It w i l l again be su f f i c ien t to consider harmonic fields of the form (2.3), since th egeneral field is expandable in an orthogonal series of such terms [Mar] for either planar orcircular waveguides.

The separated equations (2.5) and their counterpart continue to hold but cannot bes impl i f ied further as was done for the planar case because d/dy = 0 is no longer assumed.That is, a complete classificationof modes intoTE and TM types does not hold in general; TEand TM modes do exist for circular fibers, but there are other types as well, as wil l be shownpresently. Recall that introducing EQ (X , y) = (EXJ Ey, Ez), HQ(X , >') = (Hx, Hy, Hz) andu s i n g th e second curl equation in (2.1) led to (2.5). Likewise, the first curl equation in (2.1)provides an analogous set of equations i nv o l v i ng the partial derivatives of (Ex, Ey, Ez). Allthe resulting equations written together are [Mar]

w h i c h are valid for fields of the form (2.3) in an arbitrary linear waveguide (recalling th eassumption PNL = 0 in this chapter). Now take the first of (2.23), solve for ia>eEx, andreplace th e term H y by its value in the fih equation in (2.23), giving

Letting K2 = n2k2 2 as in (2.12), but keeping the index n (recall n2k2 = < w 2 e//o)general instead of assuming one of the core or cladding values n\ or 2> the previousequation simplifies to

Similar ly , the components Ey, Hx, Hy can be solved in terms of the longitudinal componentsE z and Hz. Record the four equations as


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20 Chapter 2. Fiber Modes

In order to solve (2.1) it is therefore suff ic ient to find the longi tud ina l components E- andHz. Moreover, there are simplified equations for these components. For E: take the thi rdequation in (2.23) and replace H x and H y by the expressions in the third and fourth equationsin (2.24). Aer s impl i fy ing , one obtains (the equation for H z is derived s imi la r ly)

which are solved subject to interface conditions, as was done wi th planar waveguides insection 2.2.Interestingly, all six components of (, //) satisfy (2.25). To see this, note first that

V E = 0 (PNL = 0 in the modal analysis of this chapter) from (2.1), and so (2.3) togetherwith E0(x, v) = (Ex, Ey, Ez) implies 4^- + + iEz = 0. Taking the part ia l withrespect to x gives

Now take the partial with respect to y of the last equation in (2.23) to obtain

Adding (2.26) and(2.27) one obtains ^ +^ = -ip'^ - JWjUo^, and the firstequation in (2.24) now implies

which is the same as the first equation in (2.25), but for E x instead of E-. Simi la r ly , th eother transverse components satisfy (2.25). Recalling that K 2 n2k2 2, all componentstherefore satisfy

where E in (2.28) stands for any of the components E.v, v, ,, Hxt Hy, H-: (2.28) issometimes called the scalar wave equation.


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2.3. Circular Fibers 21

The solution of the first equ ation of (2.25) w ill be given ne xt. To keep m atters inperspective before going on, the object is to determine values of so that (2.25) fo r E z or//,, in which K 2 = n2k2 2 with n = n\ in the core and n = 2 in the cladding, has asolution with con tinuo us tangential com ponents at the core-cladding interface and decaysin th e cladding region.Since th e waveguide is circular it is natural to use polar coordinates x = r cos(y) and> ' = r s\n( d the cladding index is n = n 2>that is, assum e a step-index fiber. The bounded solution to (2.32) in the core is of the formF(r) = AJv(r) fo r some c onstant A, so that

In th e clad ding region th e bounded and decaying so lut ion is analogously a mul t ip le ofth e modified Bessel function K v [Ada, Agl]. At the core-cladding interface th e tangentialcomponents of E are E z and E^ (see F igure 2 .1), the latter of wh ich can be expressed interms of E z us ing (2.24) aer convert ing th e partial derivatives to polar coordinates [Ada].Aer equat ing th e respective expressions for E t and E v in the core and cladding, a l eng thycalculation yield s a characteristic equ ation in the form [Ada, Agl]

where k co/c = 2n/k and K, y are g iven by (2.12). Equation (2.34) is solved for , th epropagation constant. That is, (2.34) provides values of such that solutio ns of (2.29) ofth e form (2.33) have con t inuou s tangentia l components at the interface, are bounded in theentire wa vegu ide, and are decay ing in the cladding. Each v alue of corresponds to a modeof the fiber.


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22 Chapter 2. Fiber ModesNote that (2.34) simplif ies if v = 0 in (2.31), in which case the field (2.33) is az-

i m u t h a l l y uniform. Obviously, the characteristic equation reduces in t h i s case to two sub-cases, namely,

and

Modes determined by (2.35) have the property that E, = 0 [Ada] and, in analogy withplanar guides, are called transverse electric (TE). (It may seem ironic that E; = 0 despiteth e fact that the eigenvalue problem started by solving for E z in (2.25); recall, however, t h a tthe characteristic equation (2.34) is actually solved for the parameter w i t h o u t referenceto any component.) Modes determined by (2.36) are called transverse magnetic (TM)and satisfy Hz = 0. Each equation has infinitely m a n y roots, whose modes are denotedby TE|, TE2,..., TM (, TM2,..., respectively. U nl i k e the planar case, where the lowestorder TE mode propagates at all wavelengths, the TE| mode can be shown to have a cutof fwavelength given by the condition [Ada]

I

where ./o(co) = 0 in (2.37) and CQ = 2.405 is the first zero of the Bessel func t ion 70-That is, recalling k = w/c = 2x/X, the TEi mode does not propagate u n t i l the condi t ion^ > T v w? ~~ W2 - co is satisfied. Since n\ = 2, thesame condition serves as theap p r ox i ma tecutoff for the TM modes.

For there are no solutions of (2.34) corresponding to modes wi th van i s h i n glongi tudinal component, that is, transverse modes. Modes that are ne i t he r TE nor TM arecalled HE and EH modes, depending on a technical condition associated with th e roots of(2.34) [Ada]. Corresponding to v = 1there is a single mode, the HE| i, t h a t propagates at allfrequencies , where the double subscript notation denotes the first root of (2.34) associatedwith v = 1. It can be shown [Ada] that all other HE and EH modes are ex t inc t under thecondition Consequently, (2.37) becomes th e criterion t h a t the fiberunder consideration is single mode, with only the HEn propagating.

Conver t ing (2.37) to wavelength, where A is fixed, the single-mode condition is there-fore

If n\ = 1.45 and n2 = 1-44, then d < 2.25A.. Taking X = 1.55 gives a constraint on theradius d in the form


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2.4. Weakly Guiding Modes and Pulses 23as the single-mode criterion. This is valid fo r step index fibers, which have the simplestrefractive index profiles. The single-mode conditions fo r graded index fibers, those withcont inuous ly vary ing index,and for dispersion-shied fibers, are in general different [IEC].

The phenomenon of destructive interference, discussed in section 2.2 for planarwaveguides, also holds for fibers. Communications fibers are therefore single mode, whichnecessitates (2.37). Condition (2.38), or more generally 2ndJn2 n\< X C Q , is the pri-mary physical constrainton the core size of optical fibers used in communications.

To summarize the section, light pulses in fibers are realizations of EM fields thatpropagate in modal patterns. The modes are determined by the fiber geometry. Fibers thatcarry communicat ion signals are necessarily single mode, as otherwise interference betweenmult ip le modes can corrupt the signal. In turn, the single-mode condition leads to a physicalconstraint on the core size, whose radius can be no more than a few microns.

2.4 Weakly Guiding Modes and PulsesThe characteristic equation (2.34), although providingan exact theoretical account of fibermodes, is unwieldyand not used in practice. Instead, investigators use a scheme devised byGloge [Glo] which makes an approximation based on the condition n\ = n-i and combinescertain nearly degenerate modes so as to form more simply described pseudomodes referredto as weakly guiding or linearly polarized (LP). Weakly guiding modes will be brieflydescribed in this section, with particular emphasis on the fundamental mode.

Field components corresponding to the modes givenby (2.34) are available in explicitform. For all v, the components in the core region are given, up to a constant multiple,by [Ada]

an d similar expressions for the H fields, where the upper signs () and upper elements inthe brackets correspond to the EH case, and the lower to the HE. Comparing the coefficientsof the bracket expressions in (2.39), the significant distinction is K/k in Et versus n\ in theother terms. The factor K/k can be written K/k = Jn\ (fi/k)2, where f$ 2/k 2 < n\by (2.12), and so K/k < Jn2 n2. Since n\ = n2, it is the case that Jn2 n\ is much lessthan /z i; recall the values i = 1.45, 2= 1.44 used above (2.38). This is theweakly guidingcase, which is defined by the condition n\ = W 2that implies K/k = 0 in (2.39). Therefore,E- = 0 (and similarly Hz =0); that is, the modes are all approximately transverse, in thatthe longi tudinal components are negligible. But there is more: for v = 1 in (2.39), Ex 0in the HE case (lower entry, minus sign). In this approximation th e fundamental mode, HE j i,thus has only one non vanishing electric (and magnetic)field component, namely,E y. The Ean d H fields of the fundamentalHEn mode are therefore approximately linearly polarized;


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24 Chapter 2. Fiber Modesthat is, the longitudinal and one transverse component vanish. In practice, t ransmitters e mi tlinearly polarized light, wh ich therefore rem ains approxima tely linea rly polarized becausethis feature is built into the modal structure.

The conditions Ez = 0 and HZ=Q can be shown to be compatible with (2.24). whichwould seem otherwise to imply that all components vanis h.Since there is (approximately) only one nonzero com ponent in the weakly guid ingcase but the characteristic equation (2.34) is derived on the basis of the two tangent ial com-ponents, then the w eakly guiding analysis m igh t be expected to extend to the chara cteristicequation as well. This is indeed th e case and it can be shown [Ada] tha t in the l imi t n \ - 2.th e interface conditions (continuity of tangential components) reduce to cont inui ty of thenonzero scalar component and its directional d erivativ e in the direction norm al to the core-cladding interface. This directional derivative is sometimes called the normal derivative.The wea kly guiding propagation constant is thus defined as follows. It is a value of fo rwhich (2.28) has a nontriv ial solution E(x, >') that is continuous, with a cont inuous normalderivation, across th e core-cladding boundary a nd decays in the cladd ing region ; com pare[SHW]. In this way a complete, though approximate, modal a nalysis is available that usesonly the scalar wave equation (2.28).The higher order modes associated with (2.34), which will not be of interest in thisbook, share the feature Ez = 0 and Hz = 0. Moreover, modes can be combined to approxi-mately cancel an additional compone nt [Ada]. Specifically, th e combinations H E / + i E H / _ ican be shown to be approximately linearly polarized [Ada]; they comprise th e family of LPmodes. In practice, polarized light launched into a fiber excites one or more LP modes. Asmentioned in the previous section, the condition -Jt\\ n\ > C Q guarantees that onlyth e fundam ental mode propagates. In the LP designation the fundamental mode HEn isdenoted by L P 0i . The com bination of HE 2i, TE oi, and TM0| is called the LP| i pseudom ode;there is a complete family {LP/*} [Ada].In the ana lysis of this and the previous section the orie ntation of transverse coordin ateaxes is arbitrary, assuming that the fiber is perfectly circular. It is the polarizat ion axis ofth e inserted ligh t in a fiber that determines the fiber axes. For the fundamental mode, th econdition E x 0 means that th e axis of polarization becomes th e y axis by definit ion. Itis critical that th e modal structure be compatible with th e predominant form of l ight that isactually used in fibers.Real fibers are not perfectly circular, and if the cross section is not perfectly round ,then th e propagation constant w ill depend on the orientation of axes (which , as ment ioned ,is determined by the polarization axis of the inserted light). Moreover, th e nature of theimperfection in shape changes down the length of the fiber. For example, a fiber may beslightly elliptical and elongated in the vertical direction at the launch point but change toelongation in the horizontal direction further along it s length; th e cross section unde rgoesconstant m inu te change. In fact, th e principal axis of polarization changes somewhat ran-domly d uring transit. C onsequen tly, more tha n one mode propagates, even if a sing le mod eis launched. With multiple modes propagating, one has modal interference and its at ten-dant dispersion. This type of impairment is called polarization mode dispersion (PMD).Although a small effect locally, PMD can become significant over long distances.As mentioned in Chapter 1, optical signals are pulses of durat ion in the range 10 tolOOps. However, the modal structure developed in this chapter is based on the plane wa ve


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2.4. Weakly GuidingModes and Pulses 25

form (2.3), so these d i f fe r i ng views need to be reconciled. At issue is whether th e basicmodal s t ruc tu re ho lds fo r pulses ra ther than cont inuous wave f ields (2.3). A s m i g h t beexpected, th e structure does hold if the pulses as not too short. In fact, if (2.3) is replaced by

where /(f)an d g(t) are func t ions of t ime , then th e r ig ht side of (2.4) becomes

an d a similar expression for the curl of the electric field. The le side of (2.4) will likewisecon t a i n an add i t iona l fac to r of f ( t ) . If M is much greater than /'(0//(0tnen tne termcon ta in ing f'(t) can be neglected in (2.41) and, s ince the common factor f ( t ) cancels ,then (2.4) does n o t change even when (2.40) is assumed. Heurist ical ly, the cond i t ionM/'(f)//(0 holds in practice fo r several reasons. First, the carr ier frequency co is verylarge at optical wavelengths. Moreover, pulses are oen relatively flat in the region of peakm odulus , an d thus the der iva t ive term f ' ( t ) is small . Con sequently, the m odal structureholds at least approximately for the h ighes t in tens i ty port ions of a pulse given by (2.40).The ex tens ion of (2.3) to (2.40) wil l be im por tant when th e no nl in ear Schrodinger equa t ionis considered in the next chapter.To su mmar ize th e section, the fiber m o d e structure extends to pulses. In the weaklygu id ing picture, the E an d H fields of the fundamenta l m o d e HEn may be considered tohave jus t on e n o n v a n i s h i n g c o m p o n en t . That is, they are l inear ly po la rized . Com bina t ion sof th e exact m od es aris ing from (2.34) lead to the f ami ly of LPjk modes . The lowest order( f u n d amen t a l ) H E M is also k n o w n as the LP01 pseudomode.


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Chapter 3Fiber Dispersion andNonlinearity

T he non linea r Schrodinger equation (NLSE) is the fundamenta l m athem at ica l dev ice forana lyz ing non linea r pulse propagation in fibers. In its simplest form th e NLSE includest e rms corresponding to two principal physical effects: dispersion and nonlinearity. Thepurpose of this chapter is to account for these phenomena. In the first section a derivation ofth e basic e quation will be given. Interpretations of its terms will be covered in the secondsection, a discussion of the ideal Gaussian pulse model comes in section 3.3, and section3.4 has some closing remarks.

3.1 The Nonlinear Schrodinger EquationIn its simplest form, th e NLSE is [Agl]

where A = A(z, t) is a function associated with th e electric field E of an optical pulsein a fiber. Here z is physical length along th e longitudinal axis of the fiber and t is t ime,initialized to the pulse center, while fa and y are constants. The purpose of this section isto derive and begin th e analysis of (3.1). Following convention, certain additional terms,which can be important in some si tuations, have been om itted from (3.1) in the interests ofsimplicity. T he most relevant omitted terms will be discussed presently. As it is, however,(3.1) is the un ive rsally accepted basic gove rning equation for pulse propagation in fibers[Agl.NeM].There are higher dim ensional analogs of (3.1) [Bou], but propagation in fibers is aphenomenon that occurs naturally in time and one spatial dim ension.T h e cons tants fa and y have phys ical d imens ions ( t ime) 2 (distance)"1 and(distance)"1 (power) 1, respectively, while |A(z, f) | 2 has dimension (power)+l. Typ-ically, fa is denomina ted in units of ps2/Km, y is in Km ~ lmW~*, and | A ( z , t ) | 2 is inm W \ A(z, t) is not assigned units, which would cancel from (3.1) in any event, and so isdimensionless.

27


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28 Chapter 3. Fiber Dispersion and Non linearityThe initial and bo undary conditions associated with (3.1) are tha t A(0, t) is prescribedand (3.1)is solved subject to | A(z , t ) | -> 0, \t \ oo. Physically speaking, pulses aresmooth and hav e compact support, at least w hen random noise effects are igno red.The NLSE (3.1) is universa l for two reasons: (i) it accurately predicts the behaviorof observed optical pulses and (ii) it can be meaningfully derived. The latter is meant in the

sense that one can start with Maxwell's equations (2.1), which are postulates, hypothesizea physica lly reasonable expression for fiber nonlin earity , and deduce (3.1) aer performinga set of plausible mathematical steps. That philosophy will be followed here and it seemsfitting that some derivation should be given. Other accounts have been provided in [B1W],which is main ly a time domain approach, and in [Fra], which argued in (and in favor of)th e frequency domain. A n excellent derivation has been provided by M enyu k [Men], w hotakes into account coupling between differently polarized modes in an ana lysis based ondominant length scales. The result is a set of coupled equa tions (the Manakov equat ions)which specialize to (3.1) under conditions when polarization effects, loss, and h igh er orderdispersion can be ignored.Before discussing th e origins of (3.1), note first that it is entirely in the t ime domain .It wil l be seen in this chapter that nonlinear effects are more inherently frequency dom ainphenomena; that is , they are more natu rally described in terms of the frequency variable ofthe Fourier transform. Some insight into this can be provided by solvin g (3.1) in the twoideal cases where y 0 (pure linear) and #> = 0 (pure nonlinear), respectively. This isdone next.3.1.1 Ideal Linear CaseIf y = 0, then (3.1)can be solved by Fourier transform. In this case, if A(z.o)) =/f^ A(z, t)ehotdt denotes the transform, then (3.1) is equivalent to

Then \A(z,a))\ = |A(0, w)| for all z\ that is, the Fourier transform has constant mod ulus ,and, in particular, magnitudes of frequency com ponents are not changed during propagation.To comment on nomenclature, C D is called tint frequency variable or frequency com-ponent and is equivalent to wavelength A. through Xw = Inc. In the Fourier transformrelationship f(oj) = F[f(t)](oj) = /^ f(t)eilotdt, the modulus or ampl i tude |/(w)|measures the strength of the frequency component C D . One speaks of time domain wh e ndiscussing /(f) andfrequency domain, Fourier domain, or spectral domain in connect ionwith /(w). The width, time domain width, or sometimes temporal width means the pulsewidth of the function |/(f )! The spectral width refers to the width of |/(a>)|. The spectrumis the set of numerically significant frequency componen ts, that is , values of to where \ f ( w ) \is not negligibly small.It is easy to show that a function A(z, 0 whose transform satisfies (3.2)broadensessentially linearly with z in the time dom ain. Indeed, suppose that f(z,t) is a function

whose solution is


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3.1. The Nonlinear Schrodinger Equation 29that satisfies f(z, w > ) = f(0, w ) ) e l h ( - t o ) z fo r some funct ion b(a)) such as the exponent in (3.2).Let cr(z) and a(i) be the roo t mean square (RMS) t ime and spectral w idths for arbitrary z,that is ,

where / and / denote a Fourier transform pair. The RMSw idth is a standard measure ofthe w idth of a pulse, curve, or distribution [Agl], Then

w here the asterisk denotes the complex conjugate, and since th e transform of T 2 f ( z , T ) is-3 /(z,CD) ,where 3W = d/dco, then Parseval's identity implies

From /(z. co ) = f (0 . wV/)(w)z one obtains

w here the prime denotes different ia t ion w ith respect to t o . Inserting this into (3.4) andintegrating yields

where the coefficients are integrals invo lvin g / and itsderivatives. The squared RMSw idth(3.5), defined in (3.3), corresponding to a solution of (3.1) in the pure linear case (y = 0),is thu s quadratic in z. To summarize, the constant ^2 is responsible for the spreading of apulse in the t ime domain in the pure linear case, wh i l e the spectral w idth is invariant .3.1.2 Pure Nonlinear CaseTo contrast the pure linear case, consider now the opposite extreme in w h i c h fa = 0. Then(3.1) becomes

and, postulating a solution of the form A = Ke l< f and working backwards, it is easy to showthat the un ique solut ion to (3.6) is

Thus the absolute value \A(z , t)\ of (3.7), w h i c h is the pulse shape, does not change w i thz. Altho ug h it can no t be calculated in closed form, the Fourier transform can be intuitively


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30 Chapter 3. Fiber Dispersion and Nonlinearityseen to vary w i t h z because th e imaginary exponent ial in (3.7) has the effect of in t roducingnew frequencies, which causes th e t ransform w id th to spread. To summar ize , in the purel inear case |A(z ,| does not change and in the pure nonlinear case \A(z, 01 does notchange. These simple calculations go a long w ay towards interpret ing th e dispersion (fa)and nonlinear (y ) terms in (3.1); roughly speaking, th e form er causes pulse w id th var ia tionwhi l e the latter causes Fourier transform or spectral vari ation . Th ey also poin t out the d u a lroles of the t ime an d frequency aspects of (3.1).It must be noted, however, that th e association of fa wi th pulse spreading and ywi th spectral broadening is actu ally on ly partly correct because nonlinear i ty also indirectlyinfluences pulse wid th . In fact, if a pulse spectrum is broadened by nonlinearity, then th emismatch in velocities of spectral co mp onen ts becomes larger, thus potent ia l ly caus ing evenmore temporal pulse broadening. But it should be clear that th e der iva t ion of (3.1), w h i c hwil l now be und ertaken, mu st take th e Fourier domain in to account . Th e derivat ion tha tis provided here is somewhat lengthy, because it contai ns some details tha t are miss ing inother sources, and there are some heurist ics. One has the option of read ing t h rough th enext few pages o r accepting (3.1). Th e end of the d erivation is (3.28), w h ich is the spectraldomain equivalent of (3.1).

3.1.3 Derivation of NLSETaking th e curl of the first equation in (2.1), and then us ing th e second equat ion in (2.1),one has

Using the fami liar ident i ty V x (V x ) = V(V E) A2E and the last equation in (2.1),one obtains

Thus the relation eoMo = 1/c2 (section 2.1) implies

where A2E = (A 2v, A 2 E V , A 2j.), E = (E x, Ey, ,), and A2 is theLaplacian operatorA 2 = T^y + ^y 4- - j f a . Equation (3.8) is the usual form o f the no nlin ear w ave equa tio n invector form. There are more general versions based on integral equa tion s and it is possibleto waive the assumption V E = 0 [M en].As in section 2.1 the polarization vector in (3.8) is assumed to have the form P(.v, v, z, t)= PL(X, >', z, t) + PNL(X, >', z, t), but where th e linear and non linear terms are now given


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3.1. The Nonlinear Schrodinger Equation 31

by the more general expressions [Agl, NeM, Men, Mil, BuC]

In the second equation of (3.9) the spatial variables in E have been suppressed anddrid^drshas been written as dr to shorten the formula. The function x ( l ) is called the linear suscepti-bility and satisfies x ( l ) ( j c , y, z, t r) = 0 for t < r. The linear susceptibility represents thel inear response of the medium to an incident field and takes the form of a weighted memoryor delay term. The fact that there can be no response before incidence (t < T) is knownas causality. An ideal, instantaneous linear response is modeled by a constant multipleof adelta funct ion , x ( 1 )C ? ) = const &($), in which case the first of (3.9)becomes PL = e\ E asin section 2.1. The nonlinear susceptibility term involving x(3)C*' > '> z, t\, ti, ^3)likewiseaccounts for a delayed nonlinear material response. For general materials there could be,in principle, terms i nvo l v i ng x(t ) for each integer k. However, x(2) =0 in silica and ther emain in g higher order nonlinear terms are typically disregarded [NeM]. Explicit forms forX ( l ) and x(3) are provided below; these can be heuristically derived [Agl]. Expressions(3.9) are the generally accepted forms for the polarization vector P = PL + PNL in (3.8).

The weakly guiding modal structure discussed in section 2.4 wil l now be incorporatedin to (3.8)-(3.9). An overriding assumption for the rest of this book is that the nonlinear termin (3.9) is sufficiently weak, and pulses are sufficiently long, that the basic weakly guidingmodal structure derived in section 2.4 continues to hold in the nonlinear setting. Moreover,it is assumed that only the fund amen ta l mode HEn propagates; that is, the fiber underconsideration is single mode. In particular, the electric field E has just one nonvanishingcomponent (see the discussion below (2.39)). In this case (3.8) and (3.9) reduce to scalarequat ions for the nonzero component. To simplify notation, the nonzero component willbe denoted by E and the corresponding polarization terms by PL and P#z. respectively, sothat (3.8)and (3.9) become

and

These are the governing equations for the nonzero component. The triple integral in (3.11)denotes integration over three-dimensional space.Impor tant further simplification arise from th e slowly varying envelope approximation(S VEA) [ Ag 1 ], which begins by postulating that the signal under consideration has a central


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32 Chapter 3. Fiber Dispersion and Nonlinearitycarrier frequency W Q and that the distributed frequency components are concentrated near( O Q . Real-valued solutions to (3.10) are then postulated to have the form

where cc stands for the complex conjugate. In (3.12) the most rapid oscillations are ac-counted for by the exponentials; the complex-valued function T(x, y, z, t) is called thecomplex envelope of the carrier wQ. The first simplification of (3.10) occurs when (3.10)is replaced by an equation for T(x, >', z, t). In fact, if (3.12) is substi tuted into (3.10) and(3.11), the resulting equation looks as follows:

The main con tributions to the triple integral in (3.13) come when T\ = f, T2 = t, r = t, sothat there is, rough ly speaking, a coefficient of e~ ia)ot on the right side of (3.13). Proceed ingformally, one may sum up the ways that terms containing e can be obtained on theright side and equate the result to the corresponding expression involving e on the le.There are three ways to obtain th e exponential e t on the right; specifically, they are thedistinct ways to write down the product (Te' ' Te' ' T^e'1 '). Sum ming , cance l inga factor of 1/2, and equating like exponentials yields

Introducing S(x, y, z, t) = T(x, y, z, t)e~lt0f)t (not the same as (3.12)), the previous equa-tion becomes

where t r = (t i\, t ^ t ^3). The spatial variables have been suppressed in (3.14)to simplify notation.Letting S(x, y, z, a> ) = F[S(x, y, z, r)](w) be the Fourier transfo rm of S(x, v. z. t),(3.14) will now be transformed to the frequency domain. The transform of the le side is


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3.1. The Nonlinear Schrodinger Equation 33L e t x ( 1 )( j c , >', z, C D ) = /^ x ( l ) C * > J>z, t)el ) ) in (3.18) is thegeneral medium index of refraction. Theindex is spatia l ly and frequency dependent through the susceptibility term x(l\x, y, z, cS).If the linear response is considered instantaneous and spatially constant, say, in the fibercore, then n ( w ) is constant and k2(w ) in (3.17) reduces to the constant n2k2 that was usedin (2.28).

Turning to the nonlinear part of (3.14), define

th e three-dimensional Fourier transform. It is straightforward but tedious to show that th enonl i ne a r term in (3.14) then transforms to

where the Fourier transform relationship F[S*(t)](a)) = S * ( w ) and the fact that thenonl i ne a r term in (3.14) is a convolut ion expression have been used. Equating (3.17) and(3.19) one has

where the spatial variables have again been suppressed.It w i l l be useful to separate out the transverse and long itudinal variables in (3.20) byp o st u l a t i n g S ( x ,y , z, w) = F(x, y, w ) U ( z , w ) ) e . Substituting, the le side of (3.20)becomes

where A ^ denotes th e transverse Laplacian (x and y variables) and the z subscript denotesa derivative. In the SVEA Uzz is assumed to be smal l in comparison to the other terms


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34 Chapter 3. Fiber Dispersion and Nonlinearityin (3.21) [Agl] and is consequent ly dropped; jus t i f icat ion for the SVEA wi l l be discussedbriefly at the en d of this section. Droppin g U zz leaves (3.21) in the form

Set thefirst erm in (3.22) to 0, A^ -F + (P(w) - 2() are chosen to satisfy the scalar m odal equation (2.28) for each C D . Then (3.22) reducesto s imply the last term, and consequently (3.20) becomes

A

where A/3(


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3.1. The Nonlinear Schrodinger Equation 3 5X ( 3 ) C * v, z, t\, t2, ^3) = constant 5 ( f i , ?2 ? 3), in which case the transform x ( 3 ) i sacons tan tand (3.25) becomes

for another constant K'.The exponentials can be removed from (3.26) byappeal to the power series expansion

of /3(w), which has the form

where o> o is the carrier. The term A/3 ( < w ) = P(u)+p(v u) p(va)) p(a)) s impl i f ies to

because of cancellatio n of the constant and linear terms. Certai nly high er order coef fi cien ts(fo, fa,...) could be used in (3.27), but for now only the quadratic coefficient will be kept.The last step is to let

and substitute into (3.26) to obtain

(K' = constant), where the frequ ency variable has been shifted. The inverse Fourier trans-form of (3.28) is

where y is another constant. Since (3.29) is the same as (3.1), this completes the derivationo f t h e N L S E .The constant y in (3.29) can be determined experimentally by expressing it in terms

of the nonlinear index, n-i [Agl] (not to be confused with the cladding index of refraction,which uses the same symbol), defined by 2 = ycA eff/a)Q, where c is the vacuum speed ofl ight , < wo is the carrier frequency, and A ef is the effective area of the fiber. The nonlinearindex of a fiber can be measured [Agl, App. B]. The constant y iscalled the Kerr coefficient;the Kerr e f f e c t is the term used to describe the dependence of the index of refraction on l ightintensity.

Note that (3.1) is actually the equa tion for the envelope ( slowly va ryi ng) of the electricfield E. Knowing thepropagation constant ft(ca), which can be found from the scalar waveequation (2.28), one obtains E by back-substitution through the changes of variable leadingto (3.29).

Tobriefly motivate theSVEAused in (3.22), start with n\ < ^^ < n\, k (co) = co / c ,as below (3.24). Then (w)) = O(k(w)) , where the symbol 0(*) stands for order of


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36 Chapter 3. Fiber Dispersion and Nonlinearitymagnitude.

[6]Mathematical Principles of Optical Fiber Communications

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