Applications of Nonlinear Fiber Optics OPTICS AND PHOTONICS (Formerly Quantum Electronics) Series Editors PAUL L. KELLEY Tufts University Medford, Massachusetts IVAN P. KAMINOW Lucent Technologies Holmdel, New Jersey GOVIND P. AGRAWAL University of Rochester Rochester, New York Recently Published Books in the Series: Jean-Claude Diels and Wolfgang Rudolph, Ultrashort Laser Pulse Phenomena: Fundamentals, Techniques, and Applications on a Femtosecond Time Scale Eli Kapon, editor, Semiconductor Lasers I: Fundamentals Eli Kapon, editor, Semiconductor Lasers II: Materials and Structures P. C. Becker, N. A. Olsson, and J. R. Simpson, Erbium-Doped Fiber Amplifiers: Fundamentals and Technology Raman Kashyap, Fiber Bragg Gratings Katsunari Okamoto, Fundamentals of Optical Waveguides Govind P. Agrawal, Nonlinear Fiber Optics, Third Edition A complete list of titles in this series appears at the end of this volume. Applications of Nonlinear Fiber Optics GOVIND P. AGRAWAL The Institute of Optics University of Rochester OPTICS AND PHOTONICS San Diego San Francisco New York Boston London Sydney Tokyo This book is printed on acid-free paper. Copyright c 2001 by ACADEMIC PRESS All rights reserved. No part of this publication may be reproducedor transmitted in any form or by any means, electronic or mechanical, including photocopy, record- ing, or any information storage and retrieval system, without permission in writing from the publisher. Requests for permission to make copies of any part of the work should be mailed to: Permissions Department, Harcourt, Inc., 6277 Sea Harbor Drive, Orlando, Florida 32887-6777. Explicit permission from Academic Press is not required to reproduce a maximum of two figures or tables from an Academic Press chapter in another scientific or research publication provided that the material has not been credited to another source and that full credit to the Academic Press chapter is given. Academic Press A Harcourt Science and Technology Company 525 B Street, Suite 1900, San Diego, California 92101-4495, USA http://www.academicpress.com Academic Press Harcourt Place, 32 Jamestown Road, London NW1 7BY, UK http://www.academicpress.com Library of Congress Catalog Card Number: 00-111105 International Standard Book Number: 0-12-045144-1 PRINTED IN THE UNITED STATES OF AMERICA 000102030405ML987654321 For Anne, Sipra, Caroline, and Claire Contents Preface xiii 1 Fiber Gratings 1 1.1 Basic Concepts . . 1 1.1.1 Bragg Diffraction . . 2 1.1.2 Photosensitivity . . . 3 1.2 Fabrication Techniques . . . 5 1.2.1 Single-Beam Internal Technique 5 1.2.2 Dual-Beam Holographic Technique . . . 6 1.2.3 Phase Mask Technique 8 1.2.4 Point-by-Point Fabrication Technique . . 10 1.3 Grating Characteristics . . . 11 1.3.1 Coupled-Mode Equations . . . 11 1.3.2 CW Solution in the Linear Case 14 1.3.3 Photonic Bandgap, or Stop Band 15 1.3.4 Grating as an Optical Filter . . . 17 1.3.5 Experimental Verification . . . 20 1.4 CW Nonlinear Effects 22 1.4.1 Nonlinear Dispersion Curves . . 23 1.4.2 Optical Bistability . 25 1.5 Modulation Instability . . . 27 1.5.1 Linear Stability Analysis 28 1.5.2 Effective NLS Equation 30 1.5.3 Experimental Results 32 1.6 Nonlinear Pulse Propagation 33 1.6.1 Bragg Solitons . . . 34 1.6.2 Relation to NLS Solitons 35 vii viii Contents 1.6.3 Formation of Bragg Solitons . . 36 1.6.4 Nonlinear Switching 40 1.6.5 Effects of Birefringence 42 1.7 Related Periodic Structures . 44 1.7.1 Long-Period Gratings 45 1.7.2 Nonuniform Bragg Gratings . . 47 1.7.3 Photonic-Crystal Fibers 51 Problems . . 54 References . . 55 2 Fiber Couplers 62 2.1 Coupler Characteristics . . . 62 2.1.1 Coupled-Mode Equations . . . 63 2.1.2 Low-Power Optical Beams . . . 66 2.1.3 Linear Pulse Switching 70 2.2 Nonlinear Effects . 71 2.2.1 Quasi-CW Switching 72 2.2.2 Experimental Results 74 2.2.3 Nonlinear Supermodes 77 2.2.4 Modulation Instability 79 2.3 Ultrashort Pulse Propagation 83 2.3.1 Nonlinear Switching of Optical Pulses . . 83 2.3.2 Variational Approach 85 2.4 Coupler-Paired Solitons . . . 89 2.5 Extensions and Applications 93 2.5.1 Asymmetric Couplers 93 2.5.2 Active Couplers . . . 96 2.5.3 Grating-Assisted Couplers . . . 98 2.5.4 Birefringent Couplers 101 2.5.5 Multicore Couplers . 102 Problems . . 105 References . . 106 3 Fiber Interferometers 112 3.1 Fabry–Perot and Ring Resonators . . . 112 3.1.1 Transmission Resonances . . . 113 3.1.2 Optical Bistability . 116 3.1.3 Nonlinear Dynamics and Chaos 118 Contents ix 3.1.4 Modulation Instability 120 3.1.5 Ultrafast Nonlinear Effects . . . 122 3.2 Sagnac Interferometers . . . 124 3.2.1 Nonlinear Transmission 125 3.2.2 Nonlinear Switching 126 3.2.3 Applications 131 3.3 Mach–Zehnder Interferometers 138 3.3.1 Nonlinear Characteristics 139 3.3.2 Applications 141 3.4 Michelson Interferometers . 142 Problems . . 144 References . . 145 4 Fiber Amplifiers 151 4.1 Basic Concepts . . 151 4.1.1 Pumping and Gain Coefficient . 152 4.1.2 Amplifier Gain and Bandwidth . 153 4.1.3 Amplifier Noise . . . 156 4.2 Erbium-Doped Fiber Amplifiers 158 4.2.1 Gain Spectrum . . . 159 4.2.2 Amplifier Gain . . . 161 4.2.3 Amplifier Noise . . . 164 4.3 Dispersive and Nonlinear Effects 166 4.3.1 Maxwell–Bloch Equations . . . 166 4.3.2 Ginzburg–Landau Equation . . 168 4.4 Modulation Instability . . . 171 4.4.1 Distributed Amplification . . . 171 4.4.2 Periodic Lumped Amplification 173 4.4.3 Noise Amplification 174 4.5 Optical Solitons . . 177 4.5.1 Autosolitons 177 4.5.2 Maxwell–Bloch Solitons 181 4.6 Pulse Amplification 184 4.6.1 Picosecond Pulses . 184 4.6.2 Ultrashort Pulses . . 189 Problems . . 193 References . . 194 x Contents 5 Fiber Lasers 201 5.1 Basic Concepts . . 201 5.1.1 Pumping and Optical Gain . . . 202 5.1.2 Cavity Design 203 5.1.3 Laser Threshold and Output Power . . . 206 5.2 CW Fiber Lasers . 208 5.2.1 Nd-Doped Fiber Lasers 208 5.2.2 Erbium-Doped Fiber Lasers . . 211 5.2.3 Other Fiber Lasers . 215 5.2.4 Self-Pulsing and Chaos 216 5.3 Short-Pulse Fiber Lasers . . 218 5.3.1 Physics of Mode Locking . . . 219 5.3.2 Active Mode Locking 220 5.3.3 Harmonic Mode Locking 223 5.3.4 Other Techniques . . 227 5.4 Passive Mode Locking . . . 229 5.4.1 Saturable Absorbers 229 5.4.2 Nonlinear Fiber-Loop Mirrors . 232 5.4.3 Nonlinear Polarization Rotation 236 5.4.4 Hybrid Mode Locking 238 5.4.5 Other Mode-Locking Techniques 240 5.5 Role of Fiber Nonlinearity and Dispersion 241 5.5.1 Saturable-Absorber Mode Locking 241 5.5.2 Additive-Pulse Mode Locking . 243 5.5.3 Spectral Sidebands . 244 5.5.4 Polarization Effects . 247 Problems . . 249 References . . 250 6 Pulse Compression 263 6.1 Physical Mechanism 263 6.2 Grating-Fiber Compressors . 266 6.2.1 Grating Pair 266 6.2.2 Optimum Compressor Design . 269 6.2.3 Practical Limitations 273 6.2.4 Experimental Results 275 6.3 Soliton-Effect Compressors . 280 6.3.1 Compressor Optimization . . . 281 Contents xi 6.3.2 Experimental Results 283 6.3.3 Higher-Order Nonlinear Effects 285 6.4 Fiber Bragg Gratings 287 6.4.1 Gratings as a Compact Dispersive Element 287 6.4.2 Grating-Induced Nonlinear Chirp 289 6.4.3 Bragg-Soliton Compression . . 291 6.5 Chirped-Pulse Amplification 292 6.6 Dispersion-Decreasing Fibers 294 6.6.1 Compression Mechanism 295 6.6.2 Experimental Results 296 6.7 Other Compression Techniques 299 6.7.1 Cross-Phase Modulation 299 6.7.2 Gain-Switched Semiconductor Lasers . . 303 6.7.3 Optical Amplifiers . 305 6.7.4 Fiber Couplers and Interferometers . . . 307 Problems . . 308 References . . 309 7 Fiber-Optic Communications 319 7.1 System Basics . . . 319 7.1.1 Loss Management . 320 7.1.2 Dispersion Management 323 7.2 Stimulated Brillouin Scattering 326 7.2.1 Brillouin Threshold . 326 7.2.2 Control of SBS . . . 328 7.3 Stimulated Raman Scattering 330 7.3.1 Raman Crosstalk . . 330 7.3.2 Power Penalty . . . 332 7.4 Self-Phase Modulation . . . 335 7.4.1 SPM-Induced Frequency Chirp 335 7.4.2 Loss and Dispersion Management 338 7.5 Cross-Phase Modulation . . 340 7.5.1 XPM-Induced Phase Shift . . . 340 7.5.2 Power Penalty . . . 342 7.6 Four-Wave Mixing 344 7.6.1 FWM Efficiency . . 345 7.6.2 FWM-Induced Crosstalk 346 7.7 System Design . . 349 [...]... the subject of nonlinear fiber optics An attempt was made to include recent research results on all topics relevant to the field of nonlinear fiber optics Such an ambitious objective increased the size of the book to the extent that it was necessary to split it into two separate books, thus creating this new book Applications of Nonlinear Fiber Optics The third edition of Nonlinear Fiber Optics deals... fundamental aspects of the field This book is devoted to the applications of nonlinear fiber optics, and its use requires knowledge of the fundamentals covered in Nonlinear Fiber Optics Please note that when an equation or section number is prefaced with the letter A, that indicates that the topic is covered in more detail in the third edition of of Nonlinear Fiber Optics xiii xiv Preface Most of the material... fields of fiber optics and optical communications This volume may be a useful text for graduate and senior-level courses dealing with nonlinear optics, fiber optics, or optical communications that are designed to provide mastery of the fundamental aspects Some universities may even opt to offer a high-level graduate course devoted solely to nonlinear fiber optics The problems provided at the end of each... Nonlinear fiber optics plays an increasingly important role in the design of such high-capacity lightwave systems In fact, an understanding of various nonlinear effects occurring inside optical fibers is almost a prerequisite for a lightwave-system designer While preparing the third edition of Nonlinear Fiber Optics, my intention was to bring the book up to date so that it remains a unique source of comprehensive... Since the publication of the first edition of Nonlinear Fiber Optics in 1989, this field has virtually exploded A major factor behind such tremendous growth was the advent of fiber amplifiers, made by doping silica or fluoride fibers with rare-earth ions such as erbium and neodymium Such amplifiers revolutionized the design of fiber-optic communication systems, including those making use of optical solitons... govern propagation of two copropagating waves inside optical fibers The two major differences are: (i) the negative sign appearing in front of the ∂ Ab ∂ z term in Eq (1.3.11) because of backward propagation of Ab and (ii) the presence of linear coupling between the counterpropagating waves governed by the parameter κ Both of these differences change the character of wave propagation profoundly Before... grating The shape of the reflected and transmitted pulses will be quite different than that of the incident pulse because of the splitting of the spectrum and the dispersive properties of the fiber grating If the peak power of pulses is small enough that nonlinear effects are negligible, we can first calculate the reflection and transmission coefficients for each spectral component The shape of the transmitted... grating of the same periodicity as that of the phase mask The chief advantage of the phase mask method is that the demands on the temporal and spatial coherence of the ultraviolet beam are much less stringent because of the noninterferometric nature of the technique In fact, even a nonlaser source such as an ultraviolet lamp can be used Furthermore, the phase mask technique allows fabrication of fiber... components—fiber-based gratings, couplers, and interferometers—that serve as the building blocks of lightwave technology In view of the enormous impact of rare-earth-doped fibers, amplifiers and lasers made by using such fibers are covered in Chapters 4 and 5 The last three chapters describe important applications of nonlinear fiber optics and are devoted to pulse-compression techniques, fiber-optic communication systems,... presence of nonlinear effects in optical fibers Optical amplifiers permit propagation of lightwave signals over thousands of kilometers as they can compensate for all losses encountered by the signal in the optical domain At the same time, fiber amplifiers enable the use of massive wavelength-division multiplexing (WDM) and have led to the development of lightwave systems with capacities exceeding 1 Tb/s Nonlinear . complete list of titles in this series appears at the end of this volume. Applications of Nonlinear Fiber Optics GOVIND P. AGRAWAL The Institute of Optics University of Rochester OPTICS AND PHOTONICS San. Simpson, Erbium-Doped Fiber Amplifiers: Fundamentals and Technology Raman Kashyap, Fiber Bragg Gratings Katsunari Okamoto, Fundamentals of Optical Waveguides Govind P. Agrawal, Nonlinear Fiber Optics, . 285 6.4 Fiber Bragg Gratings 287 6.4.1 Gratings as a Compact Dispersive Element 287 6.4.2 Grating-Induced Nonlinear Chirp 289 6.4.3 Bragg-Soliton Compression . . 291 6.5 Chirped-Pulse Amplification