208 Pages
English

Information-theoretic aspects of fiber-optic communication channels [Elektronische Ressource] / Bernhard Walter Göbel

-

Gain access to the library to view online
Learn more

Subjects

Informations

Published by
Published 01 January 2010
Reads 20
Language English
Document size 2 MB

¨ ¨TECHNISCHE UNIVERSITAT MUNCHEN
Lehrstuhl fu¨r Nachrichtentechnik
Information-Theoretic Aspects of
Fiber-Optic Communication Channels
Bernhard Walter G¨obel
Vollst¨andiger Abdruck der von der Fakult¨at fu¨r Elektrotechnik und Informationstechnik
der Technischen Universit¨at Mu¨nchen zur Erlangung des akademischen Grades eines
Doktor-Ingenieurs
genehmigten Dissertation.
Vorsitzender: Univ.-Prof. Dr.-Ing. E. Steinbach
Pru¨fer der Dissertation: 1. Univ.-Prof. Dr.-Ing. N. Hanik
2. Univ.-Prof. Dr. sc. techn. G. Kramer
DieDissertationwurdeam16.06.2010beiderTechnischenUniversit¨atMu¨ncheneingereicht
unddurchdieFakulta¨tfu¨rElektrotechnikundInformationstechnikam21.10.2010angenom-
men.“Our humble art, on the other hand, is not a dweller in gilded
monuments to amusement with crimson curtains lit by gas lamps;
ours is not caparisoned in sumptuous costumes, painted in alluring
colors, festooned with oil and tempera canvas flats. No, our stage
is a humble thread of metal stretched beneath the dark and cold
ocean wave; ours is a cord lying on the frigid and muddy bottom
of the sea. Our art manifests itself in the tiny galvanic flickers of
light in a squalid, cramped, and dark room; a flicker with no more
nuanceoraffectionthanonoroff, positiveornegative, leftorright,
dot or dash, yes or no. But that binary essence is about to join
continents, unite nations and unseat tyrants through the spread of
truth; it will permit monarchs to converse with presidents, mother
with son, lover with beloved, you with me, no matter where on
earth either of us may dwell; we shall converse as intimately or as
grandly as we might right here, sitting face-to-face.”
John Griesemer, Signal & Noisev
Preface
I thank Professor Norbert Hanik for giving me the opportunity to work on this thesis as
his assistant at TUM’s Institute for Communications Engineering (LNT). I have always
felt privileged in this position, and I thoroughly enjoyed the freedom and the possibilities
he granted me in research, teaching and numerous other activities. His advice, support,
encouragement and patience helped me and my research work prosper.
IamgratefultoProfessorGerhardKramerforactingastheco-refereeofmydissertation
and for his valuable and extensive advice that helped improve important parts of it.
Working at LNT is a highly rewarding thing to do, partly because of the pleasant social
atmosphere. Among the many colleagues whose company I enjoyed in and out of the
office and on many trips abroad, I would like to mention my office-mate Leonardo Coelho
as well as Florian Breyer, Stephan Hellerbrand and Christian Kuhn. Professor Gu¨nter
S¨oder and Johanna Weindl have helped me improve this thesis by carefully proofreading
the manuscript. Over the years, I have had the pleasure of working with a number
of excellent students. Among them was Maxim Kuschnerov, some of whose results are
included in Chapter 4.
I was lucky and honored to spend last year’s summer at Bell Laboratories on Crawford
Hill. I am thankful to Ren´e-Jean Essiambre for his invitation and his warm hospitality.
Working with and learning from him, Peter Winzer, Gerard Foschini and others was an
exciting experience for me. (So was playing football with the Bell Labs staff right next to
the famous horn antenna, although I am afraid that the excitement was on my side only.)
Parts of Chapters 3 and 6 contain results of our joint research project.
Iamfortunatetohaveafamilythathasencouragedandsupportedmeatalltimes. This
work is dedicated to my parents in deep gratitude for roots and wings. The final word is
for Johanna. It was a pleasure to share three of my LNT years with you, and it is a gift
to share my life with you.
Mu¨nchen, June 2010 Bernhard G¨obelvii
Contents
1 Introduction 1
2 Introduction to fiber-optic communications 5
2.1 Light wave propagation in single-mode fibers . . . . . . . . . . . . . . . . . 6
2.1.1 The general wave equation . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.2 Material polarization . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.3 The nonlinear Schroedinger equation . . . . . . . . . . . . . . . . . 12
2.1.4 Raman scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1.5 Brillouin scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1.6 Effect of light polarization on the nonlinear propagation . . . . . . 27
2.1.7 Solving the nonlinear Schroedinger equation . . . . . . . . . . . . . 32
2.2 Optical fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.2.1 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.2.2 Fiber effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.2.3 Length scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.3 Optical amplification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.3.1 Erbium-doped fiber amplifiers . . . . . . . . . . . . . . . . . . . . . 43
2.3.2 Distributed amplification . . . . . . . . . . . . . . . . . . . . . . . . 45
2.4 Other components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.4.1 Photodetection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.4.2 Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.4.3 Modulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48viii Contents
2.4.4 Other optical components . . . . . . . . . . . . . . . . . . . . . . . 50
2.4.5 Electrical components . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3 Elements of information theory 53
3.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.1.1 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.1.2 Mutual information . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2 Channel capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2.1 Introduction and calculation . . . . . . . . . . . . . . . . . . . . . . 57
3.2.2 Important channel types . . . . . . . . . . . . . . . . . . . . . . . . 58
3.3 A polar decomposition of mutual information . . . . . . . . . . . . . . . . 63
3.4 Decomposition of the AWGN channel . . . . . . . . . . . . . . . . . . . . . 66
3.4.1 Gaussian input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.4.2 Phase-modulated input . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.4.3 Discrete input constellations . . . . . . . . . . . . . . . . . . . . . . 71
3.5 Introduction to directional statistics . . . . . . . . . . . . . . . . . . . . . . 77
3.5.1 Trigonometric moments . . . . . . . . . . . . . . . . . . . . . . . . 77
3.5.2 Circular distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.6 Partially coherent channels . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.6.1 Input optimization and information rate calculation . . . . . . . . . 84
3.6.2 Spectral loss induced by white phase noise . . . . . . . . . . . . . . 86
3.6.3 Noncoherent channels. . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4 Capacity of PMD-impaired channels 93
4.1 Introduction to polarization mode dispersion . . . . . . . . . . . . . . . . . 94
4.1.1 Short fiber segments . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.1.2 Long fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95Contents ix
4.2 Coherent receiver structures . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.3 Impact of PMD on the channel capacity . . . . . . . . . . . . . . . . . . . 100
4.3.1 Channel transfer functions . . . . . . . . . . . . . . . . . . . . . . . 100
4.3.2 Noise statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.3.3 Capacity results for first-order PMD . . . . . . . . . . . . . . . . . 101
4.3.4 Extension to higher-order PMD . . . . . . . . . . . . . . . . . . . . 103
4.4 The interplay of PMD and fiber nonlinearities . . . . . . . . . . . . . . . . 104
4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5 Nonlinear propagation of a single field 109
5.1 Introductory remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.1.1 The character of nonlinear impairments . . . . . . . . . . . . . . . . 109
5.1.2 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.1.3 A frequency-domain view on capacity . . . . . . . . . . . . . . . . . 111
5.1.4 Line coding schemes for fiber nonlinearity reduction . . . . . . . . . 112
5.2 System model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.3 Self- and cross-phase modulation . . . . . . . . . . . . . . . . . . . . . . . 114
5.4 Four-wave mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.4.1 Estimate of the FWM power . . . . . . . . . . . . . . . . . . . . . . 118
5.4.2 Capacity calculation . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.4.3 Phase-matched FWM. . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.4.4 FWM in standard single-mode fibers . . . . . . . . . . . . . . . . . 128
5.5 Nonlinear signal-noise interaction . . . . . . . . . . . . . . . . . . . . . . . 132
5.5.1 Channel capacity including nonlinear signal-noise interaction . . . . 132
5.5.2 Channel capacity with compensation of fiber nonlinearities . . . . . 134
5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6 Capacity limits of WDM systems 139
6.1 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140x Contents
6.2 Time-domain analysis of the channel capacity . . . . . . . . . . . . . . . . 140
6.2.1 Nonlinear WDM effects. . . . . . . . . . . . . . . . . . . . . . . . . 140
6.2.2 Phenomenological channel model . . . . . . . . . . . . . . . . . . . 142
6.2.3 Dispersion-managed links . . . . . . . . . . . . . . . . . . . . . . . 146
6.2.4 Uncompensated links . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.2.5 Effect of spectral loss . . . . . . . . . . . . . . . . . . . . . . . . . . 151
6.3 Frequency-domain analysis of the channel capacity . . . . . . . . . . . . . 153
6.3.1 Effect of inter-channel nonlinearities in WDM systems . . . . . . . 155
6.3.2 Capacity results for different link lengths . . . . . . . . . . . . . . . 157
6.3.3 Nonlinear signal-noise interaction in WDM systems . . . . . . . . . 158
6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
7 Conclusions 161
A Calculation of the number of four-wave mixing products 165
A.1 The number of FWM products . . . . . . . . . . . . . . . . . . . . . . . . 165
A.1.1 Degenerate FWM products . . . . . . . . . . . . . . . . . . . . . . 166
A.1.2 Non-degenerate FWM products . . . . . . . . . . . . . . . . . . . . 167
A.2 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
B Notation, symbols and abbreviations 171
Bibliography 180