fundamentals of optical waveguides.pdf

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front cover

table of contents

front matter

Preface to the First Edition

Preface to the Second Edition

body

1 Wave Theory of Optical Waveguides

1.1. WAVEGUIDE STRUCTURE

1.2. FORMATION OF GUIDED MODES

1.3. MAXWELL’S EQUATIONS

1.4. PROPAGATING POWER

Chapter 1 REFERENCES

2 Planar Optical Waveguides

2.1. SLAB WAVEGUIDES

2.1.1. Derivation of Basic Equations

2.1.2. Dispersion Equations for TE and TM Modes

2.1.3. Computation of Propagation Constant

2.1.4. Electric Field Distribution

2.1.5. Dispersion Equation for TM Mode

2.2. RECTANGULAR WAVEGUIDES

2.2.1. Basic Equations

2.2.2. Dispersion Equations for Expq and Eypq Modes

2.2.3. Kumar’s Method

2.2.4. Effective Index Method

2.3. RADIATION FIELD FROM WAVEGUIDE

2.3.1. Fresnel and Fraunhofer Regions

2.3.2. Radiation Pattern of Gaussian Beam

2.4. MULTIMODE INTERFERENCE (MMI) DEVICE

Chapter 2 REFERENCES

3 Optical Fibers

3.1. BASIC EQUATIONS

3.2. WAVE THEORY OF STEP-INDEX FIBERS

3.2.1. TE Modes

3.2.2. TM Modes

3.2.3. Hybrid Modes

3.3. OPTICAL POWER CARRIED BY EACH MODE

3.3.1. TE Modes

3.3.2. TM Modes

3.3.3. Hybrid Modes

3.4. LINEARLY POLARIZED (LP) MODES

3.4.1. Unified Dispersion Equation for LP Modes

3.4.2. Dispersion Characteristics of LP Modes

3.4.3. Propagating Power of LP Modes

3.5. FUNDAMENTAL HE11 MODE

3.6. DISPERSION CHARACTERISTICS OF STEP-INDEX FIBERS

3.6.1. Signal Distortion Caused by Group Velocity Dispersion

3.6.2. Mechanisms Causing Dispersion

3.6.3. Derivation of Delay-time Formula

3.6.4. Chromatic Dispersion

3.6.5. Zero-dispersion Wavelength

3.7. WAVE THEORY OF GRADED-INDEX FIBERS

3.7.1. Basic Equations and Mode Concepts in Graded- index Fibers

3.7.2. Analysis of Graded-index Fibers by the WKB Method

3.7.3. Dispersion Characteristics of Graded-index Fibers

3.8. RELATION BETWEEN DISPERSION AND TRANSMISSION CAPACITY

3.8.1. Multimode Fiber

3.8.2. Single-mode Fiber

3.9. BIREFRINGENT OPTICAL FIBERS

3.9.1. Two Orthogonally-polarized Modes in Nominally Single-mode Fibers

3.9.2. Derivation of Basic Equations

3.9.3. Elliptical-core Fibers

3.9.4. Modal Birefringence

3.9.5. Polarization Mode Dispersion

3.10. DISPERSION CONTROL IN SINGLE-MODE OPTICAL FIBERS

3.10.1. Dispersion Compensating Fibers

3.10.2. Dispersion-shifted Fibers

3.10.3. Dispersion Flattened Fibers

3.10.4. Broadly Dispersion Compensating Fibers

3.11. PHOTONIC CRYSTAL FIBERS

Appendix 3A Vector wave equations in graded-index fibers

Appendix 3B Derivation of equation (3.219)

Chapter 3 REFERENCES

4 Coupled Mode Theory

4.1. DERIVATION OF COUPLED MODE EQUATIONS BASED ON PERTURBATION THEORY

4.2. CODIRECTIONAL COUPLERS

4.3. CONTRADIRECTIONAL COUPLING IN CORRUGATED WAVEGUIDES

4.3.1. Transmission and Reflection Characteristics in Uniform Gratings

4.3.2. Phase-shift Grating

4.4. DERIVATION OF COUPLING COEFFICIENTS

4.4.1. Coupling Coefficients for Slab Waveguides

4.4.2. Coupling Coefficients for Rectangular Waveguides

4.4.3. Derivation of Coupling Coefficients Based on Mode Interference

4.4.4. Coupling Coefficients for Optical Fibers

4.4.5. Coupling Coefficients for Corrugated Waveguides

4.5. OPTICAL WAVEGUIDE DEVICES USING DIRECTIONAL COUPLERS

4.5.1. Mach–Zehnder Interferometers

4.5.2. Ring Resonators

4.5.3. Bistable Devices

4.6. FIBER BRAGG GRATINGS

Appendix 4A Derivation of Equations (4.8) and (4.9)

Appendix 4B Exact Solutions for the Coupled Mode Equations ( 4.26) and ( 4.27)

Chapter 4 REFERENCES

5 Nonlinear Optical Effects in Optical Fibers

5.1. FIGURE OF MERIT FOR NONLINEAR EFFECTS

5.2. OPTICAL KERR EFFECT

5.2.1. Self-phase Modulation

5.2.2. Nonlinear Schrödinger Equation

5.3. OPTICAL SOLITONS

5.3.1. Fundamental and Higher-Order Solitons

5.3.2. Fiber Loss Compensation by Optical Amplification

5.3.3. Modulational Instability

5.3.4. Dark Solitons

5.4. OPTICAL PULSE COMPRESSION

5.5. LIGHT SCATTERING IN ISOTROPIC MEDIA

5.5.1. Vibration of One-Dimensional Lattice

5.5.2. Selection Rules for Light Scattering by Phonons

5.6. STIMULATED RAMAN SCATTERING

5.7. STIMULATED BRILLOUIN SCATTERING

5.8. SECOND-HARMONIC GENERATION

5.9. ERBIUM-DOPED FIBER AMPLIFIER

5.10. FOUR-WAVE MIXING IN OPTICAL FIBER

Chapter 5 REFERENCES

6 Finite Element Method

6.1. INTRODUCTION

6.2. FINITE ELEMENT METHOD ANALYSIS OF SLAB WAVEGUIDES

6.2.1. Variational Formulation

6.2.2. Discretization of the Functional

6.2.3. Dispersion Equation Based on the Stationary Condition

6.2.4. Dispersion Characteristics of Graded-index Slab Waveguides

6.3. FINITE ELEMENT METHOD ANALYSIS OF OPTICAL FIBERS

6.3.1. Variational Formulation

6.3.2. Discretization of the Functional

6.3.3. Dispersion Equation Based on the Stationary Condition

6.3.4. Single-mode Conditions of Graded-index Fibers

6.3.5. Variational Expression for the Delay Time

6.4. FINITE ELEMENT METHOD ANALYSIS OF RECTANGULAR WAVEGUIDES

6.4.1. Vector and Scalar Analyses

6.4.2. Variational Formulation and Discretization into Finite Number of Elements

6.4.3. Dispersion Equation Based on the Stationary Condition

6.5. STRESS ANALYSIS OF OPTICAL WAVEGUIDES

6.5.1. Energy Principle

6.5.2. Plane Strain and Plane Stress

6.5.3. Basic Equations for Displacement, Strain and Stress

6.5.4. Formulation of the Total Potential Energy

6.5.5. Solution of the Problem by the Stationary Condition

6.5.6. Combination of Finite-Element Waveguide and Stress Analysis

6.6. SEMI-VECTOR FEM ANALYSIS OF HIGH-INDEX CONTRAST WAVEGUIDES

6.6.1. E-field Formulation

6.6.2. H-field Formulation

6.6.3. Steady State Mode Analysis

6A Derivation of Equation (6.59)

6B Proof of Equation (6.66)

Chapter 6 REFERENCES

7 Beam Propagation Method

7.1. BASIC EQUATIONS FOR BEAM PROPAGATION METHOD BASED ON THE FFT

7.1.1. Wave Propagation in Optical Waveguides

7.1.2. Pulse Propagation in Optical Fibers

7.2. FFTBPM ANALYSIS OF OPTICAL WAVE PROPAGATION

7.2.1. Formal Solution Using Operators

7.2.2. Concrete Numerical Procedures Using Split-step Fourier Algorithm

7.3. FFTBPM ANALYSIS OF OPTICAL PULSE PROPAGATION

7.4. DISCRETE FOURIER TRANSFORM

7.5. FAST FOURIER TRANSFORM

7.6. FORMULATION OF NUMERICAL PROCEDURES USING DISCRETE FOURIER TRANSFORM

7.7. APPLICATIONS OF FFTBPM

7.8. FINITE DIFFERENCE METHOD ANALYSIS OF PLANAR OPTICAL WAVEGUIDES

7.8.1. Derivation of Basic Equations

7.8.2. Transparent Boundary Conditions

7.8.3. Solution of Tri-diagonal Equations

7.9. FDMBPM ANALYSIS OF RECTANGULAR WAVEGUIDES

7.10. FDMBPM ANALYSIS OF OPTICAL PULSE PROPAGATION

7.11. SEMI-VECTOR FDMBPM ANALYSIS OF HIGH- INDEX CONTRAST WAVEGUIDES

7.11.1. Quasi-TE Modes

7.11.2. Quasi-TM Modes

7.11.3. Polarization Splitter Using Silicon-on-Insulator (SOI) Waveguide

7.12. FINITE DIFFERENCE TIME DOMAIN (FDTD) METHOD

Chapter 7 REFERENCES

8 Staircase Concatenation Method

8.1. STAIRCASE APPROXIMATION OF WAVEGUIDE BOUNDARY

8.2. AMPLITUDES AND PHASES BETWEEN THE CONNECTING INTERFACES

8.3. WAVELENGTH DIVISION MULTIPLEXING COUPLERS

8.4. WAVELENGTH-FLATTENED COUPLERS

Chapter 8 REFERENCES

9 Planar Lightwave Circuits

9.1. WAVEGUIDE FABRICATION

9.2. N×N STAR COUPLER

9.3. ARRAYED-WAVEGUIDE GRATING

9.3.1. Principle of Operation and Fundamental Characteristics

9.3.2. Analytical Treatment of AWG Demultiplexing Properties

9.3.3. Waveguide Layout of AWG

9.3.4. Gaussian Spectral Response AWG

9.3.5. Polarization Dependence of Pass Wavelength

9.3.6. Vernier Technique for the Center Wavelength Adjustment

9.4. CROSSTALK AND DISPERSION CHARACTERISTICS OF AWGs

9.4.1. Crosstalk of AWGs

9.4.2. Dispersion Characteristics of AWGs

9.5. FUNCTIONAL AWGs

9.5.1. Flat Spectral Response AWG

9.5.2. Loss Reduction in AWG

9.5.3. Unequal Channel Spacing AWG

9.5.4. Variable Bandwidth AWG

9.5.5. Uniform-loss and Cyclic-frequency (ULCF) AWG

9.5.6. Athermal (Temperature Insensitive) AWG

9.5.7. Multiwavelength Simultaneous Monitoring Device Using AWG

9.5.8. Phase Error Compensation of AWG

9.5.9. Tandem AWG Configuration

9.6. RECONFIGURABLE OPTICAL ADD/DROP MULTIPLEXER (ROADM)

9.7. N×N MATRIX SWITCHES

9.8. LATTICE-FORM PROGRAMMABLE DISPERSION EQUALIZERS

9.9. TEMPORAL PULSE WAVEFORM SHAPERS

9.10. COHERENT OPTICAL TRANSVERSAL FILTERS

9.11. OPTICAL LABEL RECOGNITION CIRCUIT FOR PHOTONIC LABEL SWITCH ROUTER

9.12. POLARIZATION MODE DISPERSION COMPENSATOR

9.13. HYBRID INTEGRATION TECHNOLOGY USING PLC PLATFORMS

Chapter 9 REFERENCES

10 Several Important Theorems and Formulas

10.1. GAUSS’S THEOREM

10.2. GREEN’S THEOREM

10.3. STOKES’ THEOREM

10.4. INTEGRAL THEOREM OF HELMHOLTZ AND KIRCHHOFF

10.5. FRESNEL–KIRCHHOFF DIFFRACTION FORMULA

10.6. FORMULAS FOR VECTOR ANALYSIS

10.7. FORMULAS IN CYLINDRICAL AND SPHERICAL COORDINATES

10.7.1. Cylindrical Coordinates

10.7.2. Spherical Coordinates

Chapter 10 REFERENCES

Index

Fundamentals of Optical Waveguides

Fundamentals of Optical Waveguides Second Edition KATSUNARI OKAMOTO Okamoto Laboratory Ltd Ibaraki, Japan AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Academic Press is an imprint of Elsevier

Academic Press is an imprint of Elsevier 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA 525 B Street, Suite 1900, San Diego, California 92101-4495, USA 84 Theobald’s Road, London WC1X 8RR, UK This book is printed on acid-free paper. Copyright © 2006, Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone: (+44) 1865 843830, fax: (+44) 1865 853333, E-mail: permissions@elsevier.co.uk. You may also complete your request on-line via the Elsevier homepage (http://elsevier.com), by selecting “Customer Support” and then “Obtaining Permissions.” Library of Congress Cataloging-in-Publication Data Application Submitted British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 13: 978-0-12-525096-2 ISBN 10: 0-12-525096-7 For information on all Elsevier Academic Press publications visit our Web site at www.books.elsevier.com Printed in the United States of America 05 7 06 09 07 08 10 9 8 6 5 4 3 2 1 Working together to grow libraries in developing countries www.elsevier.com | www.bookaid.org | www.sabre.org

To Kuniko, Hiroaki and Masaaki

Contents Preface to the First edition Preface to the Second edition 1 Wave Theory of Optical Waveguides 1.1 Waveguide Structure 1.2 Formation of Guided Modes 1.3 Maxwell’s Equations 1.4 Propagating Power 2 Planar Optical Waveguides 2.1 Slab Waveguides 2.1.1 Derivation of Basic Equations 2.1.2 Dispersion Equations for TE and TM Modes 2.1.3 Computation of Propagation Constant 2.1.4 Electric Field Distribution 2.1.5 Dispersion Equation for TM Mode 2.2 Rectangular Waveguides 2.2.1 Basic Equations 2.2.2 Dispersion Equations for Ex 2.2.3 Kumar’s Method 2.2.4 Effective Index Method pq and Ey pq Modes 2.3 Radiation Field from Waveguide 2.3.1 Fresnel and Fraunhofer Regions 2.3.2 Radiation Pattern of Gaussian Beam 2.4 Multimode Interference (MMI) Device 3 Optical Fibers 3.1 Basic Equations 3.2 Wave Theory of Step-index Fibers 3.2.1 TE Modes 3.2.2 TM Modes 3.2.3 Hybrid Modes xiii xv 1 1 2 7 10 13 13 13 16 19 22 25 27 27 29 31 37 41 41 43 46 57 57 58 58 62 63 vii

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