Optical Solitons in Fibers

205 Pages·2003·4.4 MB·English

by  Professor Akira Hasegawa

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ByA.HasegawaandM.Matsumoto
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A. Hasegawa M. Matsumoto
Optical Solitons
in Fibers
Third,RevisedandEnlargedEdition
With91Figures
1 3
Professor Akira Hasegawa Professor Masayuki Matsumoto
#403,19-1 Awataguchi Sanjobocho Department of Communications Engineering
Higashiyama-ku, Kyoto, 605-0035, Japan Graduate School of Engineering
E-mail: [email protected] Osaka University
2-1 Yamadaoka, Suita, Osaka 565-0871, Japan
E-mail: [email protected]
Series Editors:
Professor Takeshi Kamiya Dr. Herbert Venghaus
Ministry of Education, Culture, Sports, Heinrich- Hertz-Institut
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National Institution for Academic Degrees, Einsteinufer 37
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Department of Physics Stanford University
and Measurement Technology Edward 1. Ginzton Laboratory
Materials Science Division Stanford, CA 94305, USA
Linkoping University
58183 Linkoping, Sweden
The first edition was published as Volume 116 of the series Springer Tracts in Modern Physics
The second edition was published as a monograph
ISSN 1437-0379
ISBN 978-3-642-07826-2 ISBN 978-3-540-46064-0 (eBook)
DOl 10.1007/978-3-540-46064-0
Library of Congress Cataloging-in-Publication Data.
Hasegawa, Akira, 1934-. Optical solitons in fibers. -3rd, rev. and enl. ed.! A. Hasegawa, M. Matsumoto.
p. cm. - (Springer series in photonics, ISSN 1437-0379; 9). Includes bibliographical references and index.
1. Solitons. 2. Optical fibers. I. Matsumoto, M. (Masayuki), 1960-II. Title. III.
Springer series in photonics; v. 9. QC174.26.W28 H37 2002 530.12' 4-dc21 2002030556
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Preface
Theopticalsolitoninfiberspresentsabeautifulexampleinwhichanabstract
mathematical concept has produced a large impact on the real world of high
technology field. Its existence was theoretically predicted in 1973 and exper-
imentally demonstrated in 1980. However, attempts to implement solitons
forultra-high-speedcommunicationshavebeenarealchallengeformanysci-
entists who devoted their interests to this purpose. The challenge has been
more fundamental and scientific than technical. For example, the solution of
nonlinear Schro¨dinger equation having periodic variation of coefficients by
means of the Lie transformation (to a homogeneous nonlinear Schro¨dinger
equation) is by itself an interesting theoretical contribution. Timing jitter
of solitons due to amplifier noise and its control and effects of polarization
mode dispersion on soliton transmission are still some other examples. The
discovery of the dispersion-managed soliton is an innovative contribution to
the application of solitons to a real transmission system.
The research on optical solitons also produced a large impact on con-
ventionaloptical-transmissiontechnologies.ThenonlinearSchro¨dingerequa-
tion model for lightwave envelope and the split-step method of the numeri-
cal solution are now widely used as standard techniques in general optical-
transmission analyses. The concept of all-optical transmission, first intro-
duced for optical soliton systems, is now used as standard in most recent
transmission systems.
This book is the third edition published by Springer-Verlag under this
title. The new edition contains many chapters that cover interesting devel-
opments that took place in the last decade, including soliton control, effects
of polarization-mode dispersion, and in particular the dispersion-managed
solitons.
Expenses for the preparation of the manuscript were covered by several
fundings.Oneoftheauthors(M.Matsumoto)especiallythanksInternational
Communications Foundation for its support.
Kyoto, Osaka, Akira Hasegawa
June 2002 Masayuki Matsumoto
Contents
1. Introduction.............................................. 1
2. Wave Motion ............................................. 3
2.1 What is Wave Motion?.................................. 3
2.2 Dispersive and Nonlinear Effects of a Wave ................ 4
2.3 Solitary Waves and the Korteweg de Vries Equation ........ 5
2.4 Solution of the Korteweg de Vries Equation................ 7
3. Lightwave in Fibers....................................... 11
3.1 Polarization Effects ..................................... 11
3.2 Plane Electromagnetic Waves in Dielectric Materials ........ 12
3.3 Kerr Effect and Kerr Coefficient.......................... 14
3.4 Dielectric Waveguides................................... 15
4. Information Transfer in Optical Fibers
and Evolution of the Lightwave Packet ................... 19
4.1 How Information is Coded in a Lightwave ................. 19
4.2 How Information is Transferred in Optical Fibers........... 20
4.3 Master Equation for Information Transfer in Optical Fibers:
The Nonlinear Schro¨dinger Equation...................... 23
4.4 Evolution of the Wave Packet
Due to the Group Velocity Dispersion..................... 25
4.5 Evolution of the Wave Packet Due to the Nonlinearity ...... 26
4.6 Technical Data of Dispersion and Nonlinearity
in a Real Optical Fiber.................................. 27
4.7 Nonlinear Schro¨dinger Equation and a Solitary Wave Solution 29
4.8 Modulational Instability................................. 32
4.9 Induced Modulational Instability ......................... 37
4.10 Modulational Instability
Described by the Wave Kinetic Equation .................. 38
5. Optical Solitons in Fibers................................. 41
5.1 Soliton Solutions and the Results of Inverse Scattering ...... 41
5.2 Soliton Periods......................................... 44
VIII Contents
5.3 ConservationQuantitiesoftheNonlinearSchro¨dingerEquation 44
5.4 Dark Solitons .......................................... 45
5.5 Soliton Perturbation Theory ............................. 49
5.6 Effect of Fiber Loss..................................... 52
5.7 Effect of the Waveguide Property of a Fiber ............... 53
5.8 Condition of Generation of a Soliton in Optical Fibers ...... 57
5.9 First Experiments on Generation of Optical Solitons ........ 58
6. All-Optical Soliton Transmission Systems................. 61
6.1 Raman Amplification and Reshaping
of Optical Solitons-First Concept
of All-Optical Transmission Systems ...................... 61
6.2 First Experiments of Soliton Reshaping
and of Long Distance Transmission by Raman Amplifications 64
6.3 First Experiment of Soliton Transmission
by Means of an Erbium Doped Fiber Amplifier............. 67
6.4 Concept of the Guiding Center Soliton .................... 68
6.5 The Gordon–Haus Effect and Soliton Timing Jitter ......... 71
6.6 Interaction Between Two Adjacent Solitons ................ 73
6.7 Interaction Between Two Solitons
in Different Wavelength Channels......................... 74
7. Control of Optical Solitons ............................... 77
7.1 Frequency-Domain Control .............................. 77
7.2 Time-Domain Control................................... 82
7.3 Control by Means of Nonlinear Gain ...................... 86
7.4 Numerical Examples of Soliton Transmission Control........ 90
8. Influence of Higher-Order Terms ......................... 97
8.1 Self-Frequency Shift of a Soliton
Produced by Induced Raman Scattering................... 98
8.2 Fission of Solitons
Produced by Self-Induced Raman Scattering ............... 99
8.3 Effects of Other Higher-Order Dispersion .................. 100
9. Polarization Effects ....................................... 103
9.1 Fiber Birefringence
and Coupled Nonlinear Schro¨dinger Equations ............. 103
9.2 Solitons in Fibers with Constant Birefringence ............. 106
9.3 Polarization-Mode Dispersion ............................ 111
9.4 Solitons in Fibers
with Randomly Varying Birefringence..................... 115
Contents IX
10. Dispersion-Managed Solitons (DMS) ..................... 123
10.1 Problems in Conventional Soliton Transmission............. 123
10.2 Dispersion Management with Dispersion-Decreasing Fibers .. 124
10.3 Dispersion Management with Dispersion Compensation ..... 127
10.4 Quasi Solitons ......................................... 136
11. Application of Dispersion Managed Solitons
for Single-Channel Ultra-High Speed Transmissions....... 141
11.1 Enhancement of Pulse Energy............................ 141
11.2 Reduction of Gordon–Haus Timing Jitter.................. 144
11.3 Interaction Between Adjacent Pulses...................... 147
11.4 Dense Dispersion Management ........................... 151
11.5 Nonstationary RZ Pulse Propagation ..................... 152
11.6 Some Recent Experiments ............................... 155
12. Application of Dispersion Managed Solitons
for WDM Transmission................................... 159
12.1 Frequency Shift Induced by Collisions
Between DM Solitons in Different Channels................ 159
12.2 Temporal Shift Induced by Collisions
Between DM Solitons in Different Channels................ 161
12.3 Doubly Periodic Dispersion Management ................. 164
12.4 Some Recent WDM Experiments Using DM Solitons........ 166
13. Other Applications of Optical Solitons.................... 169
13.1 Soliton Laser........................................... 169
13.2 Pulse Compression ..................................... 173
13.3 All-Optical Switching ................................... 176
13.4 Solitons in Fibers with Gratings.......................... 180
13.5 Solitons in Microstructure Optical Fibers.................. 184
References.................................................... 188
Index......................................................... 197
1. Introduction
The second edition of Optical Soliton in Fibers was published in 1989 when
the first generation long-distance soliton transmission had just began to
emerge.Thefirstandsecondeditionsintendedtointroducethebasicconcept
and properties of optical solitons in fibers and methods of constructing an
all-optical transmission system by taking advantage of the robust nature of
a soltion.
The soliton is the only stable pulse shape in a fiber with (anomalous)
dispersion and nonlinearity for a useful range of pulse width (1∼50ps) and
peakpower(1∼10mW).However,itwassoonrecognizedthatsolitonshave
their unique problems. First, as the name soliton indicates, it is an exact so-
lution only when it exists well separated from other solitons. Therefore, if a
solitonisdesignatedtocarryonedigitofinformation,itshouldbeseparated
sufficiently apart from adjacent digits. This means that the pulse width is
much shorter than bit periods, thus the system requires a much larger band-
widthcomparedwithalinearpulsehavingthesamebitrate.Second,timing
jitters of solitons, rather than a distortion of the pulse shape, was found to
contribute to the major cause of the bit error. Timing jitter may be induced
eitherbyamplifiernoiseorinteractionswithneighboringsolitonsand/orwith
solitons in other channels in a wavelength-multiplexed system.
By taking advantage of the robust nature of the soliton, various methods
have been proposed (and proven to be successful) to control these timing
jitters. One attempt to reduce the timing jitter quite successfully was made
by reducing the average dispersion as close as to zero by means of dispersion
compensation, since these timing jitters are proportional to the average or
integrated dispersion of the transmission line. Fiber dispersion can be pro-
grammedeitherbyaproperdesignofthewaveguidepropertyorbyconnect-
ing fibers having various values of dispersion. The latter method is generally
called dispersion management.
During the ten years since the publication of the second edition, tremen-
dous progress has been made both in the rate as well as in the distance of
error free transmission. More than 1 Tbit/s rate of error-free transmission
has been achieved over a distance of several thousand kilometers. Dispersion
managementisthekeytechnologythathasledtothisremarkableresultboth
inlinear(orquasi-linear)andinsolitonsystems.Inafiberhavingaconstant
A. Hasegawa et al., Optical Solitons in Fibers
© Springer-Verlag Berlin Heidelberg 2003
2 1. Introduction
anomalous dispersion, a soliton is formed by a balance between the nonlin-
early induced frequency chirp and linearly induced chirp. In a fiber having
periodically varying group dispersion, a nonlinear pulse propagates in such
a manner that the nonlinearly induced frequency chirp (or the self-induced
phase shift) reducing (enhancing) the dispersion induced frequency chirp in
an anomalous (normal) dispersion region. This results in narrower (wider)
pulse width and wider (narrower) spectral width in anomalous (normal) dis-
persionregion.Theaverageeffectivedispersioninsuchaperiodicallyvarying
dispersion system is thus given not by a simple averaged dispersion but that
weighted by spectral width of the pulse that varies in the direction of prop-
agation. This means that a fiber having a periodic dispersion works as if it
were a fiber having an anomalous dispersion even if the average dispersion
is exactly zero. The nonlinear stationary pulse is generated by the balance
between the effective dispersion and nonlinearity on the average analogous
toanidealsoliton.Consequentlysuchanonlinearstationarypulseiscalleda
dispersion-managedsoliton.Thetimingjitterofadispersion-managedsoliton
is significantly reduced compared with that of an ideal soliton with the same
energysincetheaveragedispersioncanbemademuchsmaller.Furthermore,
a dispersion-managed soliton has a pulses shape close to a Gaussian rather
than sech shape of an ideal soliton. This allows narrower pulse spacing for
the same pulse width and results in higher bandwidth efficiency.
Inthisedition,weputmoreemphasisontheapplicationofopticalsolitons
for ultra-high-speed communications and present practical issues related to
theapplication.Chapters2and5aredevotedtotheintroductionofthecon-
cept of solitons, one for water surface and the other for the optical soliton in
fibers.Chapters3and4aredevotedtothefundamentalproblemsofinforma-
tion transfer, which includes the light wave propagation in fibers, derivation
ofthemasterequation,thenonlinearSchro¨dingerequation,discussionofthe
propertyofthemasterequation,andderivationofthesolitonsolution.Chap-
ters 6 to 9 introduce practical issues of ultra-high-speed information transfer
in fibers using soltions, which include the concept of all optical transmission
systems, application of ideal solitons for transmission and related problems,
idea of soliton control, effects of higher-order terms, and the influence of po-
larizationmodedispersion.Chapters10to12aredevotedtotheintroduction
of dispersion managed solitons and their applications. Finally, in Chapter 13
we discuss applications of solitons other than long-distance transmission fo-
cusingonthetopicsofgenerationofshortopticalpulses,solitonswitches,and
solitons in media having spatial periodicity in axial or in radial directions.