1.FOUNDATIONS FORGUIDED-WAVE OPTICS Chin-Lin Chen Purdue UniversityWest Lafayette, IndianaWILEY-INTERSCIENCEA JOHN WILEY & SONS, INC., PUBLICATION

2. FOUNDATIONS FORGUIDED-WAVE OPTICS 3. FOUNDATIONS FORGUIDED-WAVE OPTICS Chin-Lin Chen Purdue UniversityWest Lafayette, IndianaWILEY-INTERSCIENCEA JOHN WILEY & SONS, INC., PUBLICATION 4. Copyright 2007 by John Wiley & Sons, Inc. All rights reserved.Published by John Wiley & Sons, Inc., Hoboken, New Jersey.Published simultaneously in Canada.No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any formor by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except aspermitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the priorwritten permission of the Publisher, or authorization through payment of the appropriate per-copy feeto the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400,fax (978) 750-4470, or on the web at www.copyright.com. 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Neither thepublisher nor author shall be liable for any loss of prot or any other commercial damages, includingbut not limited to special, incidental, consequential, or other damages.For general information on our other products and services or for technical support, please contact ourCustomer Care Department within the United States at (800) 762-2974, outside the United States at(317) 572-3993 or fax (317) 572-4002.Wiley also publishes its books in a variety of electronic formats. Some content that appears in printmay not be available in electronic formats. For more information about Wiley products, visit our website at www.wiley.com.Library of Congress Cataloging-in-Publication Data:Chen, Chin-Lin.Foundations for guided-wave optics / by Chin-Lin Chen.p. cm.Includes index.ISBN-13 978-0-471-75687-3 (cloth)ISBN-10 0-471-75687-3 (cloth)1. Optical wave guides. I. Title.TA1750.C473 2006621.36 92dc22 2006000881Printed in the United States of America.10 9 8 7 6 5 4 3 2 1 5. CONTENTSPreface xvii 1BRIEF REVIEW OF ELECTROMAGNETICS ANDGUIDED WAVES11.1 Introduction11.2 Maxwells Equations 41.3 Uniform Plane Waves in Isotropic Media61.4 State of Polarization 71.5 Reection and Refraction by a Planar Boundary betweenTwo Dielectric Media 11 1.5.1 Perpendicular Polarization12 1.5.2 Parallel Polarization 171.6 Guided Waves 19 1.6.1 Transverse Electric Modes 21 1.6.2 Transverse Magnetic Modes 21 1.6.3 Waveguides with Constant Index in Each Region 22Problems 22References 23 2STEP-INDEX THIN-FILM WAVEGUIDES252.1 Introduction 252.2 Dispersion of Step-Index Waveguides27 2.2.1 Transverse Electric Modes 29 2.2.2 Transverse Magnetic Modes 312.3 Generalized Parameters 32 2.3.1 The a, b, c, d, and V Parameters32 2.3.2 The bV Diagram33v 6. vi CONTENTS2.3.3 Cutoff Thicknesses and Cutoff Frequencies352.3.4 Number of Guided Modes 362.3.5 Birefringence in Thin-Film Waveguides37 2.4 Fields of Step-Index Waveguides 382.4.1 Transverse Electric Modes382.4.2 Transverse Magnetic Modes40 2.5 Cover and Substrate Modes 41 2.6 Time-Average Power and Connement Factors 412.6.1 Time-Average Power Transported by TE Modes 412.6.2 Connement Factor of TE Modes442.6.3 Time-Average Power Transported by TM Modes 45 2.7 Phase and Group Velocities47 Problems48 References49 Bibliography49 3 GRADED-INDEX THIN-FILM WAVEGUIDES51 3.1 Introduction51 3.2 Transverse Electric Modes Guided by Linearly Graded Dielectric Waveguides 52 3.3 Exponentially Graded Dielectric Waveguides573.3.1 Transverse Electric Modes573.3.2 Transverse Magnetic Modes59 3.4 The WKB Method613.4.1 Auxiliary Function 623.4.2 Fields in the R Zone 633.4.3 Fields in the L Zone 643.4.4 Fields in the Transition Zone653.4.5 The Constants673.4.6 Dispersion Relation683.4.7 An Example 68 3.5 Hocker and Burns Numerical Method703.5.1 Transverse Electric Modes723.5.2 Transverse Magnetic Modes74 3.6 Step-Index Thin-Film Waveguides versus Graded-Index Dielectric Waveguides 74 Problems75 References76 7. CONTENTS vii 4 PROPAGATION LOSS IN THIN-FILM WAVEGUIDES 774.1 Introduction 774.2 Complex Relative Dielectric Constant and ComplexRefractive Index 784.3 Propagation Loss in Step-Index Waveguides80 4.3.1 Waveguides Having Weakly Absorbing Materials80 4.3.2 Metal-Clad Waveguides 824.4 Attenuation in Thick Waveguides with Step-Index Proles854.5 Loss in TM0 Mode 884.6 Metal-Clad Waveguides with Graded-Index Proles90Problem90References 90 5 THREE-DIMENSIONAL WAVEGUIDES WITH RECTANGULAR BOUNDARIES 935.1 Introduction 935.2 Fields and Modes Guided by Rectangular Waveguides955.3 Orders of Magnitude of Fields96 5.3.1 The E y Modes 97 5.3.2 The E x Modes 995.4 Marcatili Method100 5.4.1 The E y Modes100 5.4.2 The E x Modes106 5.4.3 Discussions106 5.4.4 Generalized Guide Index1075.5 Effective Index Method109 5.5.1 A Pseudowaveguide112 5.5.2 Alternate Pseudowaveguide113 5.5.3 Generalized Guide Index1145.6 Comparison of Methods 115Problems119References120 6 OPTICAL DIRECTIONAL COUPLERS AND THEIR APPLICATIONS 1216.1 Introduction1216.2 Qualitative Description of the Operation ofDirectional Couplers122 8. viiiCONTENTS 6.3 Marcatilis Improved Coupled-Mode Equations 1246.3.1 Fields of Isolated Waveguides1256.3.2 Normal Mode Fields of the Composite Waveguide1266.3.3 Marcatili Relation 1266.3.4 Approximate Normal Mode Fields 1286.3.5 Improved Coupled-Mode Equations1296.3.6 Coupled-Mode Equations in an Equivalent Form 1306.3.7 Coupled-Mode Equations in an Alternate Form131 6.4 Directional Couplers with Uniform Cross Section and Constant Spacing1326.4.1 Transfer Matrix1326.4.2 Essential Characteristics of Couplers with1 = 2 = 1346.4.3 3-dB Directional Couplers1356.4.4 Directional Couplers as Electrically ControlledOptical Switches 1366.4.5 Switching Diagram139 6.5 Switched Couplers 141 6.6 Directional Couplers as Optical Filters 1446.6.1 Directional Coupler Filters with IdenticalWaveguides and Uniform Spacing 1466.6.2 Directional Coupler Filters with NonidenticalWaveguides and Uniform Spacing 1486.6.3 Tapered Directional Coupler Filters151 6.7 Intensity Modulators Based on Directional Couplers1536.7.1 Electrooptic Properties of Lithium Niobate 1546.7.2 Dielectric Waveguide with an Electrooptic Layer1556.7.3 Directional Coupler Modulator Built on a Z-CutLiNbO3 Plate 156 6.8 Normal Mode Theory of Directional Couplers with Two Waveguides159 6.9 Normal Mode Theory of Directional Couplers with Three or More Waveguides161 Problems164 References1657GUIDED-WAVE GRATINGS 169 7.1 Introduction1697.1.1 Types of Guided-Wave Gratings1697.1.2 Applications of Guided-Wave Gratings 172 9. CONTENTSix 7.1.3 Two Methods for Analyzing Guided-Wave Grating Problems1747.2 Perturbation Theory174 7.2.1 Waveguide Perturbation174 7.2.2 Fields of Perturbed Waveguide 176 7.2.3 Coupled Mode Equations and Coupling Coefcients 178 7.2.4 Co-directional Coupling 180 7.2.5 Contra-directional Coupling 1817.3 Coupling Coefcients of a Rectangular GratingAnExample1817.4 Graphical Representation of Grating Equation 1857.5 Grating Filters187 7.5.1 Coupled-Mode Equations187 7.5.2 Filter Response of Grating Reectors189 7.5.3 Bandwidth of Grating Reectors1937.6 Distributed Feedback Lasers194 7.6.1 Coupled-Mode Equations with Optical Gain194 7.6.2 Boundary Conditions and Symmetric Condition 195 7.6.3 Eigenvalue Equations195 7.6.4 Mode Patterns 199 7.6.5 Oscillation Frequency and Threshold Gain199References 202 8 ARRAYED-WAVEGUIDE GRATINGS2078.1 Introduction 2078.2 Arrays of Isotropic Radiators2088.3 Two Examples 212 8.3.1 Arrayed-Waveguide Gratings as DispersiveComponents 212 8.3.2 Arrayed-Waveguide Gratings as FocusingComponents 2148.4 1 2 Arrayed-Waveguide Grating Multiplexers andDemultiplexers 214 8.4.1 Waveguide Grating Elements215 8.4.2 Output Waveguides 217 8.4.3 Spectral Response 2178.5 N N Arrayed-Waveguide Grating Multiplexers andDemultiplexers 219 10. x CONTENTS 8.6 Applications in WDM Communications221 References2229 TRANSMISSION CHARACTERISTICS OF STEP-INDEXOPTICAL FIBERS225 9.1 Introduction225 9.2 Fields and Propagation Characteristic of Modes Guided by Step-Index Fibers 2279.2.1 Electromagnetic Fields 2279.2.2 Characteristic Equation2309.2.3 Traditional Mode Designation and Fields231 9.3 Linearly Polarized Modes Guided by Weakly Guiding Step-Index Fibers 2359.3.1 Basic Properties of Fields of WeaklyGuiding Fibers 2369.3.2 Fields and Boundary Conditions 2389.3.3 Characteristic Equation and Mode Designation 2399.3.4 Fields of x-Polarized LP0m Modes 2449.3.5 Time-Average Power 2449.3.6 Single-Mode Operation246 9.4 Phase Velocity, Group Velocity, and Dispersion of Linearly Polarized Modes2479.4.1 Phase Velocity and Group Velocity2489.4.2 Dispersion 249 Problems255 References25610INPUT AND OUTPUT CHARACTERISTICS OF WEAKLYGUIDING STEP-INDEX FIBERS 25910.1 Introduction25910.2 Radiation of LP Modes 260 10.2.1 Radiated Fields in the Fraunhofer Zone 260 10.2.2 Radiation by a Gaussian Aperture Field 265 10.2.3 Experimental Determination of ka and V 26610.3 Excitation of LP Modes269 10.3.1 Power Coupled to LP Mode 269 10.3.2 Gaussian Beam Excitation 271 Problems273 References273 11. CONTENTSxi11 BIREFRINGENCE IN SINGLE-MODE FIBERS 275 11.1 Introduction 275 11.2 Geometrical Birefringence of Single-Mode Fibers278 11.3 Birefringence Due to Built-In Stress 282 11.4 Birefringence Due to Externally AppliedMechanical Stress28511.4.1 Lateral Stress28511.4.2 Bending 28911.4.3 Mechanical Twisting 293 11.5 Birefringence Due to Applied Electric andMagnetic Fields29411.5.1 Strong Transverse Electric Fields 29411.5.2 Strong Axial Magnetic Fields294 11.6 Jones Matrices for Birefringent Fibers 29611.6.1 Linearly Birefringent Fibers with Stationary Birefringent Axes 29711.6.2 Linearly Birefringent Fibers with a Continuous Rotating Axis 29711.6.3 Circularly Birefringent Fibers30011.6.4 Linearly and Circularly Birefringent Fibers 30011.6.5 Fibers with Linear and Circular Birefringence and Axis Rotation 303References 30612 MANUFACTURED FIBERS 309 12.1 Introduction 309 12.2 Power-Law Index Fibers 31112.2.1 Kurtz and Striefers Theory of Waves Guided byInhomogeneous Media31112.2.2 Fields and Dispersion of LP Modes 31212.2.3 Cutoff of Higher-Order LP Modes 315 12.3 Key Propagation and Dispersion Parameters ofGraded-Index Fibers31812.3.1 Generalized Guide Index b 31812.3.2 Normalized Group Delay d(V b)/dV31812.3.3 Group Delay and the Connement Factor 32012.3.4 Normalized Waveguide DispersionV [d 2 (V b)/dV 2 ]32112.3.5 An Example322 12. xiiCONTENTS12.4 Radiation and Excitation Characteristic of Graded-Index Fibers324 12.4.1 Radiation of Fundamental Modes of Graded-Index Fibers324 12.4.2 Excitation by a Linearly Polarized Gaussian Beam32512.5 Mode Field Radius327 12.5.1 Marcuse Mode Field Radius 329 12.5.2 First Petermann Mode Field Radius 330 12.5.3 Second Petermann Mode Field Radius331 12.5.4 Comparison of Three Mode Field Radii33212.6 Mode Field Radius and Key Propagation and Dispersion Parameters332 Problems 334 References 33413PROPAGATION OF PULSES IN SINGLE-MODE FIBERS33713.1 Introduction 33713.2 Dispersion and Group Velocity Dispersion 34013.3 Fourier Transform Method 34313.4 Propagation of Gaussian Pulses in Waveguides 345 13.4.1 Effects of the First-Order Group Velocity Dispersion 347 13.4.2 Effects of the Second-Order Group Velocity Dispersion34913.5 Impulse Response 352 13.5.1 Approximate Impulse Response Function with Ignored354 13.5.2 Approximate Impulse Response Function with Ignored35513.6 Propagation of Rectangular Pulses in Waveguides35613.7 Evolution of Pulse Envelope357 13.7.1 Monochromatic Waves 360 13.7.2 Envelope Equation 361 13.7.3 Pulse Envelope in Nondispersive Media 363 13.7.4 Effect of the First-Order Group Velocity Dispersion363 13.7.5 Effect of the Second-Order Group Velocity Dispersion36613.8 Dispersion Compensation366 References 368 13. CONTENTSxiii14 OPTICAL SOLITONS IN OPTICAL FIBERS 371 14.1 Introduction371 14.2 Optical Kerr Effect in Isotropic Media37214.2.1 Electric Susceptibility Tensor 37314.2.2 Intensity-Dependent Refractive Index 374 14.3 Nonlinear Envelope Equation 37614.3.1 Linear and Third-Order Polarizations 37614.3.2 Nonlinear Envelope Equation in Nonlinear Media 37814.3.3 Self-Phase Modulation37914.3.4 Nonlinear Envelope Equation for Nonlinear Fibers 38014.3.5 Nonlinear Schr dinger Equationo 381 14.4 Qualitative Description of Solitons 382 14.5 Fundamental Solitons38514.5.1 Canonical Expression 38514.5.2 General Expression 38614.5.3 Basic Soliton Parameters 38714.5.4 Basic Soliton Properties 387 14.6 Higher-Order Solitons 38914.6.1 Second-Order Solitons38914.6.2 Third-Order Solitons 390 14.7 Generation of Solitons39114.7.1 Integer A39314.7.2 Noninteger A 393 14.8 Soliton Units of Time, Distance, and Power395 14.9 Interaction of Solitons 398References402Bibliography403ABROWN IDENTITY 405 A.1 Wave Equations for Inhomogeneous Media 406 A.2 Brown Identity 407 A.3 Two Special Cases410 A.4 Effect of Material Dispersion410 References 411 14. xivCONTENTSB TWO-DIMENSIONAL DIVERGENCE THEOREM 413AND GREENS THEOREM Reference415C ORTHOGONALITY AND ORTHONORMALITY 417OF GUIDED MODESC.1 Lorentz Reciprocity 417C.2 Orthogonality of Guided Modes 418C.3 Orthonormality of Guided Modes420References420D ELASTICITY, PHOTOELASTICITY, ANDELECTROOPTIC EFFECTS 421D.1 Strain Tensors421D.1.1 Strain Tensors in One-Dimensional Objects 421D.1.2 Strain Tensors in Two-Dimensional Objects 422D.1.3 Strain Tensors in Three-Dimensional Objects 424D.2 Stress Tensors424D.3 Hookes Law in Isotropic Materials426D.4 Strain and Stress Tensors in Abbreviated Indices428D.5 Relative Dielectric Constant Tensors and Relative DielectricImpermeability Tensors430D.6 Photoelastic Effect and Photoelastic Constant Tensors 432D.7 Index Change in Isotropic Solids: An Example432D.8 Linear Electrooptic Effect433D.9 Quadratic Electrooptic Effect 434References435E EFFECT OF MECHANICAL TWISTING ON FIBERBIREFRINGENCE437E.1 Relative Dielectric Constant Tensor of a Twisted Medium 437E.2 Linearly Polarized Modes in Weakly Guiding,Untwisted Fibers440E.3 Eigenpolarization Modes in Twisted Fibers 441References442 15. CONTENTS xvFDERIVATION OF (12.7), (12.8), AND (12.9) 443 Reference445GTWO HANKEL TRANSFORM RELATIONS 447 G.1 Parsevals Theorem of Hankel Transforms447 G.2 Hankel Transforms of Derivatives of a Function 448Author Index449Subject Index 455 16. PREFACEOver the last 50 years, we have witnessed an extraordinary evolution and progressin optical science and engineering. When lasers were invented as light sources in1960, cladded glass rods were proposed as transmission media in 1966; few, ifany, would foresee their impact on the daily life of the modern society. Today,lasers and bers are the key building blocks of optical communication systems thattouch all walks of life. Photonic components are also used in consumer products,entertainment and medical equipment, not to mention the scientic and engineeringinstrumentation. Optical devices may be in the bulk or guided-wave optic forms.Guided-wave optic components are relatively new and much remains to be accom-plished or realized. Thus engineering students and graduates should acquire a basicknowledge of principles, capabilities and limitations of guided-wave optic devicesand systems in their education even if their specialization is not optics or photon-ics. The purpose of this book is to present an intermediate and in-depth treatmentof integrated and ber optics. In addition to the basic transmission properties ofdielectric waveguides and optical bers, the book also covers the basic principlesof directional couplers, guided-wave gratings, arrayed-waveguide gratings and beroptic polarization components. In short, the book examines most topics of interestto engineers and scientists. The main objective of Chapter 1 is to introduce the nomenclature and nota-tions. The rest of the book treats three major topics. They are the integrated optics(Chapters 2 to 8), ber optics (Chapters 9 to 12) and the pulse evolution and broad-ening in optical waveguides (Chapters 13 and 14). Attempts are made to keep eachchapter sufciently independent and self-contained. The book is written primarily as a textbook for advanced seniors, rst-yeargraduate students, and recent graduates of engineering or physics. It is also usefulfor self-study. Like all textbooks, materials contained herein may be found in journalarticles, research monographs and/or other textbooks. My aim is to assemble relevantmaterials in a single volume and to present them in a cohesive and unied fashion.It is not meant to be a comprehensive treatise that contains all topics of integratedoptics and ber optics. Nevertheless, most important elements of guided-wave opticsare covered in this book. Each subject selected is treated from the rst principles. A rigorous analy-sis is given to establish its validity and limitation. Whenever possible, elementaryxvii 17. xviii PREFACEmathematics is used to analyze the subject matter. Detailed steps and manipulationsare provided so that readers can follow the development on their own. Extensionsor generalizations are noted following the initial discussion. If possible, nal resultsare cast in terms of normalized parameters. Results or conclusions based on numer-ical calculations or experimental observations are explicitly identied. Convolutedtheories that cant be established in simple mathematics are clearly stated withoutproof. Pertinent references are given so that readers can pursue the subject on theirown. In spite of the analysis and mathematic manipulations, the emphasis of thisbook is physical concepts. The book is based on the lecture notes written for a graduate course on inte-grated and ber optics taught several times over many years at Purdue University. Iapologize to students, past and present, who endured typos, corrections, and incon-sistencies in various versions of the class notes. Their questions and commentshelped immensely in shaping the book to the nal form. I also like to thank mycolleagues at Purdue for their encouragement, free advice, and consultation. I wishto acknowledge 3 friends in particular. They are Professors Daniel S. Elliott, EricC. Furgason and George C. S. Lee. Finally and more importantly, I like to express my sincere appreciation to mywife, Ching-Fong, for her enormous patience, constant encouragement, and steadysupport. She is the true force bonding three generations of Chens together and themain pillar sustaining our family. Because of her, I am healthier and happier. I amforever indebted to her. CHIN-LIN CHENWest Lafayette, IndianaSeptember 2006 18. 1 BRIEF REVIEW OFELECTROMAGNETICS ANDGUIDED WAVES1.1 INTRODUCTIONThe telecommunication systems are a key infrastructure of all modern societies,and the optical ber communications is the backbone of the telecommunication sys-tems. An optical communication system is comprised of many optical, electrical,and electronic devices and components. The optical devices may be in the bulk,integrated, or ber-optic form. Therefore, an understanding of the operation prin-ciples of these optical devices is of crucial importance to electrical engineers andelectrical engineering students. This book is on integrated and ber optics for opticalcommunication applications. The subject matter beyond the introductory chapter isgrouped into three parts. They are the integrated optics (Chapters 28), ber optics(Chapters 912), and the propagation and evolution of optical pulses in linear andnonlinear bers (Chapters 13 and 14). The main purpose of Chapter 1 is to introduce the nomenclature and notations.In the process, we also review the basics of electromagnetism and essential theoriesof guided waves. The theory of thin-lm waveguides with constant index regions is relativelysimple and complete and it is presented in Chapter 2. Most quantities of interest areFoundations for Guided-Wave Optics, by Chin-Lin ChenCopyright 2007 John Wiley & Sons, Inc. 1 19. 2 BRIEF REVIEW OF ELECTROMAGNETICS AND GUIDED WAVESexpressed in elementary functions. Examples include eld components, the disper-sion relation, the connement factor, and power transported in each region. We alsoexpress the quantities in terms of the generalized parameters to facilitate compari-son. In short, we use the step-index thin-lm waveguides to illustrate the notion ofguided waves and the basic properties of optical waveguides. Many dielectric waveguides have a graded-index prole. Examples includeoptical waveguides built on semiconductors and lithium niobates. While the basicproperties of graded-index waveguides are similar to that of step-index waveguides,subtle differences exist. In Chapter 3, we rst analyze modes guided by linearly andexponentially tapered dielectric waveguides. We obtain closed-form expressions forelds and the dispersion relations for these waveguides. Then we apply the WKB(Wentzel, Kramers, and Brillouin) method and a numerical method to study opticalwaveguides with an arbitrary index prole. So far, we have considered ideal waveguides made of loss-free materials andhaving a perfect geometry and index prole. While loss in dielectric materialsmay be very small, it is not zero. Obviously, no real waveguide structure orindex prole is perfect either. As a result, waves decay as they propagate in realwaveguides. In Chapter 4, we examine the effects of dielectric loss on the prop-agation and attenuation of guided modes and the perturbation on the waveguideproperties by the presence of metallic lms near or over the waveguide regions.The use of metal-clad waveguides as waveguide polarizers or mode lters is alsodiscussed. In practical applications, we may wish to pack the components densely so asto make the most efcient use of the available real estate. It is then necessaryto reduce interaction between waveguides and to minimize cross talks. For thispurpose, it is necessary to conne elds in the waveguide regions. To conne eldsin the two transverse directions, geometric boundaries and/or index discontinuitiesare introduced in the transverse directions. This leads to three-dimensional wave-guides such as channel waveguides and ridge waveguides. In Chapter 5, we examinethe modes guided by three-dimensional waveguides with rectangular geometries.Two approximate methods are used to establish the dispersion of modes guided byrectangular dielectric waveguides. They are the Marcatili method and the effectiveindex method. A detailed comparison of the two methods is presented in the lastsection. Having discussed the propagation, attenuation, and elds of modes guided byisolated waveguides, we turn our attention to three classes of passive guided-wavecomponents: the directional coupler devices, the waveguide grating devices andarrayed waveguide gratings. In Chapter 6, we discuss Marcatilis improved coupled-mode equations for co-propagating modes and use these equations to establish theessential characteristics of directional coupling. Then we consider the switched directional couplers and their applications as switches, optical lters, and modulators. Waveguide gratings are periodic topological structures or index variations builtpermanently on the waveguides. Periodic index variations induced by electrooptic oracoustooptic effects onto the waveguides are also waveguide gratings. These gratingsare the building blocks of guided-wave components. Coupled-mode equations are 20. 1.1 I N T R O D U C T I O N3developed in Chapter 7 to describe the interaction of contrapropagating modes inthe grating structures. Then we use these equations to study the operation of grat-ing reectors, grating lters, and distributed feedback lasers. Arrayed-waveguidegratings are briey discussed in Chapter 8. The transmission and input/output properties of single-mode bers are dis-cussed in Chapters 9 and 10. In Chapter 9, we study the transmission propertiesof linearly polarized (LP) modes in weakly guiding step-index bers with a circu-lar core. For these bers, a rigorous analysis of elds is possible, and we obtainclosed-form expressions for several quantities of interest. We discuss the phasevelocity, group velocity, and the group velocity dispersion of LP modes. For obvi-ous reasons, we are particularly interested in the intramodal dispersion of single-mode bers. The generalized parameters of step-index bers are also used in thediscussion. In most applications, it would be necessary to couple light into and out of bers.Therefore, the input and output characteristics of bers are of practical interest. InChapter 10, we suppose that bers are truncated, and we examine the elds radiatedby LP modes from the truncated bers. We also examine the excitation of LP modesin step-index bers by uniform plane waves and Gaussian beams. Ideal bers would have a circular cross section and a rotationally symmetricindex prole and are free from mechanical, electric, and magnetic disturbances.But no ideal ber exists because of the fabrication imperfection and postfabricationdisturbances. Real bers are birefringent. In Chapter 11, we begin by tracing thephysical origins of the ber birefringence. Then we estimate the ber birefringencedue to noncircular cross section and that induced by mechanical, electrical, andmagnetic disturbances. Lastly, we use Jones matrices to describe the birefringenteffects in bers under various conditions. Most manufactured bers have graded-index proles. Naturally, we are inter-ested in the propagation and dispersion of the modes guided by graded-index bers.In Chapter 12, we concentrate on bers having a radially inhomogeneous and angu-larly independent index prole. Of particular interest to telecommunications is thefundamental mode guided by the graded-index bers. The notion of the mode eldradius or spot size is discussed. All bers, ideal or real, are dispersive. As a result, optical pulses evolve asthey propagate in linear bers. In Chapter 13, we study the propagation and evo-lution of pulses in linear, dispersive waveguides and bers. Three approaches areused to analyze the pulse broadening and distortion problems. The rst approach isa straightforward application of the Fourier and inverse Fourier transforms. Theconcept of the impulse response of a transmission medium is then introduced.Finally, we recognize that elds are the product of the carrier sinusoids and thepulse envelope. The propagation of carrier sinusoids is simple and well under-stood. Our attention is mainly on the slow evolution of the pulse envelope. Alinear envelope equation is developed to describe the evolution of the pulse enve-lope. A general discussion of the envelope distortion and frequency chirping is thenpresented. 21. 4BRIEF REVIEW OF ELECTROMAGNETICS AND GUIDED WAVES In nonlinear dispersive bers, both the pulse shape and spectrum evolve as thepulse propagates. However, if the nonlinear bers have anomalous group velocitydispersion and if the input pulse shape, temporal width, and amplitude satisfy awell-dened relationship, the pulses either propagate indenitely without distortionor they reproduce the original pulse shape, width, and peak amplitude periodically.These pulses are known as solitary waves or optical solitons. Naturally, the formationand propagation of the optical solitons are of interest to telecommunications and westudy these subjects in Chapter 14. A nonlinear envelope equation, often referredto as the nonlinear Schr dinger equation, is developed to describe the evolutionoof pulses on nonlinear dispersive bers. We rely on a simple and straightforwardmethod to derive an expression for the fundamental solitons. From the expressionfor fundamental solitons, we extract the key properties and the basic parametersof the fundamental solitons. Higher-order solitons and interaction of fundamentalsolitons are briey discussed.1.2 MAXWELLS EQUATIONSTo study the waves guided by optical waveguides and bers, we begin with thetime-dependent, source-free Maxwell equations: B(r; t) E(r; t) = (1.1)t D(r; t) H(r; t) =(1.2)t B(r; t) = 0 (1.3) D(r; t) = 0(1.4)where E(r; t), D(r; t), H(r; t), and B(r; t) are the electric eld intensity (V/m), elec-tric ux density (C/m2 ), magnetic eld intensity (A/m), and magnetic ux density(T or W/m2 ), respectively. They are real functions of position r and time t. Althoughsome dielectric waveguides and bers may contain anisotropic materials, most opti-cal waveguides and bers of interest are made of isotropic, nonmagnetic dielectricmaterials. We conne our discussion in this book to isotropic, nonmagnetic materialsonly. For nonmagnetic and isotropic materials, the constitutive relations are B(r; t) = 0 H(r; t)(1.5)and D(r; t) = 0 E(r; t) + P(r; t)(1.6)where 0 (1/36 109 F/m) and 0 (= 4 107 H/m) are the vacuum per-mittivity and permeability. P(r; t) is the electric polarization of the medium [13]. 22. 1.2 M A X W E L L S E Q U A T I O N S5 It is convenient to use phasors to describe elds that vary sinusoidally in time.In the frequency domain, the Maxwell equations are E(r; ) = j B(r; ) (1.7) H(r; ) = j D(r; )(1.8)B(r; ) = 0(1.9)D(r; ) = 0 (1.10)where E(r; ), H(r; ), and so forth are the phasor representation of E(r; t), H(r; t),and so forth and is the angular frequency. In general, E(r; ), D(r; ), H(r; ),and B(r; ) are complex functions of r and . The time-domain eld vectors andthe corresponding frequency-domain quantities are related. For example,E(r; t) = Re[E(r; )ej t ] (1.11)The constitutive relations in the frequency domain are, in lieu of (1.5) and (1.6), B(r; ) = 0 H(r; ) (1.12) D(r; ) = 0 E(r; ) + P(r; ) (1.13) We assume that the elds are weak enough that the nonlinear response of themedium is negligibly small. We will not be concerned with the second- and third-order polarizations until Chapter 14. In the rst 13 chapters, we take the media aslinear media. In simple, isotropic and linear media, P(r; ) is proportional to andin parallel with E(r; ). Then the electric ux density can be written asD(r; ) = 0 [1 + (1) (r; )]E(r; ) = 0 r (r; )E(r; )(1.14)where (1) (r; ) is the electric susceptibility, and r (r; ) is the relative dielectricconstant. In optics literature, we often rewrite the above equation in terms of arefractive index n(r; ): D(r; ) = 0 n2 (r; )E(r; )(1.15) The relative dielectric constant and the refractive index may be functions ofposition and frequency. For example, different waveguide regions may have differentr and n. To nd the waves guided by a waveguide amounts to solving the Maxwellequations subject to the usual boundary conditions. Consider the boundary betweenmedia 1 and 2 as shown in Figure 1.1. Let Ei (r; ), Hi (r; ), and so forth be the eld vectors in region i with an index ni where i = 1 or 2. ni is the unit vector normalto the boundary separating the two media and pointing in the outward direction 23. 6BRIEF REVIEW OF ELECTROMAGNETICS AND GUIDED WAVESMedium 1Index n1^n2 ^ n1Medium 2Index n2Figure 1.1 Unit vectors n1 and n2 normal to theboundary.relative to region i. In the absence of the surface charge density and surface currentdensity, the boundary conditions are as follows: 1. The tangential components of E(r; ) and H(r; ) are continuous at theboundary: n1 E1 (r; ) + n2 E2 (r; ) = 0(1.16) n1 H1 (r; ) + n2 H2 (r; ) = 0(1.17) 2. The normal components of D(r; ) and B(r; ) are also continuous at theboundary: n1 D1 (r; ) + n2 D2 (r; ) = 0(1.18) n1 B1 (r; ) + n2 B2 (r; ) = 0(1.19)For brevity, we will drop the arguments (r; t) and (r; ) in the remainingdiscussion. In other words, we simply write E, H, E, H, and so forth in lieu ofE(r; ), H(r; ), E(r; ), and H(r; ).1.3 UNIFORM PLANE WAVES IN ISOTROPIC MEDIAWaves are labeled as plane waves if the constant phase surfaces of the wavesare planes. If the wave amplitude is the same everywhere on the constant phaseplane, waves are identied as uniform plane waves. Consider uniform plane wavespropagating in an arbitrary direction k in free space. The electric and magnetic eldintensities can be expressed asE = E0 ej kr(1.20) j krH = H0 e(1.21) 24. 1.4 S T A T E O F P O L A R I Z A T I O N 7where k is the wave vector in free space. E0 and H0 are the amplitudes of the electricand magnetic eld intensities, respectively. To determine the relation between vari-ous plane wave parameters, we substitute (1.20) and (1.21) into the time-harmonicMaxwell equations (1.7)(1.10) and obtain |k| = k = 0 0(1.22)and1 H0 =k E0(1.23)0 In (1.23), 0 = 0 /0 is the intrinsic impedance of free space. The Poynting vector is1 |E0 |2 S= Re[E H ] =k (1.24)220where stands for the complex conjugation of a complex quantity. For an isotropic medium with a refractive index n, the wave vector and theintrinsic impedance are nk and 0 /n, respectively. In lieu of (1.23) and (1.24), theeld vectors and Poynting vector in the medium with index n are related throughthe following relations:n H0 =k E0(1.25)0n|E0 |2 S=k(1.26) 20In linear isotropic media, E and D are in parallel. So are B and H. It is alsoclear from (1.23) to (1.26) that E and D are perpendicular to B and H. These eldvectors are also perpendicular to k and S as depicted in Figure 1.2(a). Since electricand magnetic eld vectors are transverse to the direction of propagation, uniformplane waves in isotropic media are transverse electromagnetic (TEM) waves. Theseremarks hold for isotropic media. But for anisotropic media, many statements haveto be modied [see Fig. 1.2(b)] [2, 4].1.4 STATE OF POLARIZATIONIn the last section, we consider elds in the frequency domain. It is often instructiveto examine the elds in the time domain as well. In the time-domain description,we can visualize the motion of a eld vector as a function of time. We refer theevolution of the eld vector in time as the state of polarization. Consider the electriceld at a certain point. We suppose that the electric eld is conned in a plane, 25. 8BRIEF REVIEW OF ELECTROMAGNETICS AND GUIDED WAVES DD and E ES kkB and H B and HS(a) (b)Figure 1.2 E, D, B, H, S, and k vectors of uniform plane waves in (a) isotropic and(b) anisotropic dielectric media.which is taken as the xy plane for convenience. In the time-domain representation,the electric eld at this point is E = xEx + yEy = xEx0 cos(t + x ) + yEy0 cos(t + y )(1.27)where Ex0 and Ey0 are amplitudes of the two components, and they are real andpositive quantities. x and y are the phases of the two components relative to anarbitrary time reference. x and y are unit vectors in the x and y directions. Thecorresponding frequency-domain representation is E = xEx + yEy = xEx0 ej x + yEy0 ej x(1.28) Depending on the amplitude ratio Ey0 /Ex0 and the phase difference =y x , the tip of the eld vector may trace a linear, circular, or an ellipticaltrajectory in a left-hand or right-hand sense. If the two components are in timephase, that is, x = y = , (1.27) can be simplied to E = ( Ex0 + yEy0 ) cos(t + ) x While the length of the eld vector changes as a cosine function, the eld vector points to a xed direction xEx0 + yEy0 . In other words, the tip of E moves alonga straight line as time advances. We refer elds with x = y as linearly polarizedor plane polarized elds or waves. If the two eld components have the same amplitudes and are in time quadrature,that is, Ex0 = Ey0 and = y x = /2, (1.27) becomes E = Ex0 [ cos(t + x ) + y sin(t + x )]x As time advances, the tip of E traces a circular path. If the wave under consid-eration moves in the +z direction, then the E vector rotates in the counterclockwisesense for observers looking toward the source. If our right thumb points to the 26. 1.4 S T A T E O F P O L A R I Z A T I O N9direction of propagation, that is, the +z direction, our right-hand ngers would curlin the same sense as the motion of the tip of the electric eld vector. Thus waveshaving eld components specied by Ex0 = Ey0 and y x = /2 and movingin the +z direction are right-hand circularly polarized waves [3, 57]. Similarly, if Ex0 = Ey0 and = y x = +/2, the eld in the time-domain representation is E = Ex0 [ cos(t + x ) y sin(t + x )]x For waves propagating in the +z direction, the tip of the eld vector traces acircle in the clockwise direction to observers facing the approaching waves. In otherwords, the eld vector rotates in the left-hand sense. The elds with Ex0 = Ey0 and = y x = +/2 are left-hand circularly polarized waves. This is the Instituteof Electrical and Electronics Engineers (IEEE) denition for the right-handednessor left-handedness of the waves. The terminology used in the physics and opticsliterature is exactly the opposite. In many books on physics and optics, waves having = /2 and = +/2 are identied, respectively, as the left- and right-handcircularly polarized waves [3, 57]. In general, (1.27), or (1.28), describes elliptically polarized elds. To elaboratethis point further, we combine (1.27) and (1.28) and obtain (Problem 1) E2E2y Ex Eyx2 + 2 2 cos = sin2 (1.29) Ex0Ey0 Ex0 Ey0 The equation describes a polarization ellipse inscribed into a 2Ex0 2Ey0 rect-angle as shown in Figure 1.3. The shape and the orientation of the ellipse dependon the amplitude ratio Ey0 /Ex0 and the phase difference . The sense of rotationdepends only on the phase difference. The major and minor axes of a polarizationellipse do not necessarily coincide with the x and y axes. Therefore, there is no sim-ple way to relate the major and minor axes of the ellipse specied in (1.29) to Ex0and Ey0 . By rotating the coordinates, (1.29) can be transformed to a canonical formfor ellipses [3, 5]. In the canonical form, the lengths of major and minor axes, 2Emjand 2Emn , are readily identied. We take Emj and Emn as positive and Emj Emn .The shape of the ellipse may also be quantied in terms of the ellipticity:EmnEllipticity = (1.30)Emjor the visibility (VS)Emj Emn 2 2 1 (Emn /Emj ) 22 VS ==(1.31)Emj + Emn 2 2 1 + (Emn /Emj ) 22 The ellipticity and visibility of a polarization ellipse are functions of Emn /Emj . 27. 10BRIEF REVIEW OF ELECTROMAGNETICS AND GUIDED WAVES y e j2E m nq 2Ey0x 2E m n 2Ex0 Figure 1.3 Parameters of elliptically polarized waves. As shown in Figure 1.3, is the angle of the major axis relative to the x axis.We refer as the azimuth of the ellipse. The sense of rotation of elliptically polarized eld is the sense of motion of thetip of E as a function of time. For 0 < < , the tip of E rotates in the sameway as our left-hand ngers curl with the left thumb pointing in the direction ofpropagation. Thus elds with a phase difference 0 < < rotate in the left-handsense. Fields with < < 2 rotate in the right-hand direction. The limitingcases of = 0 or correspond to linearly polarized waves. In summary, the state of polarization can be specied by Ey0 /Ex0 and . It canalso be cast in terms of Emj /Emn , , and the sense of rotation. The transformationfrom Ey0 /Ex0 and to Emn /Emj , , and the sense of rotation is facilitated by thefollowing relations [3]: Emn lr = tan (1.32) Emj 4 2sin 2 = (sin 2)sin (1.33)tan 2 = (tan 2)cos 0 < (1.34)Ey0 = tan , 0(1.35)Ex02 In (1.32), lr is +1 for the left-handed rotation and 1 for the right-handedrotation. Detail derivation for these relations is left as an exercise for the reader(Problem 2). The physical meanings of Ey0 /Ex0 , Emn /Emj , , and are easilyunderstood from Figure 1.3. Although and also have geometrical meaning oftheir own, as shown in Figure 1.3, we merely view and as the two auxiliaryvariables introduced to specify the ratios Ey0 /Ex0 and Emn /Emj . As implied in (1.28), an elliptically polarized eld can be considered as thesuperposition of two orthogonal linearly polarized elds. An elliptically polarized 28. 1.5 R E F L E C T I O N A N D R E F R A C T I O N B Y A P L A N A R B O U N D A R Y11eld can also be viewed as the superposition of two counterrotating circularly polar-ized elds. To demonstrate this point, we rewrite (1.28) as Ex + j Ey Ex j Ey E = xEx + yEy =R+L (1.36)22where x jy x + jyR= and L = 2 2are the basis vectors for right-hand and left-hand circularly polarized elds. In isotropic media, the refractive indices seen or experienced by two orthog-onal linearly polarized eld components or the two counterrotating circularly polar-ized eld components are the same. Thus the phase difference between the twoeld components remains unchanged as waves propagate. As a result, the two eldcomponents change at the same rate and by the same amount. Thus there is nochange of the state of polarization as waves propagate in isotropic media. The situation is quite different for waves in anisotropic media. An anisotropicmedium may be birefringent, dichroic, or both. In birefringent media, different eldcomponents experience different refractive indices and travel with different phasevelocities. Thus, changes as waves propagate in birefringent media. Because ofthe phase difference change, the state of polarization evolves as the waves propa-gate. In dichroic media, the two eld components decay with different rates. ThenEy0 /Ex0 changes as waves propagate. Thus the state of polarization also evolvesas waves propagate. In short, the state of polarization evolves as waves travel inanisotropic media. Further discussion on the subject can be found in [3, 8].1.5 REFLECTION AND REFRACTION BY A PLANAR BOUNDARYBETWEEN TWO DIELECTRIC MEDIAIn homogeneous and isotropic media, uniform plane waves propagate alongstraight-line paths until they impinge upon boundaries. In inhomogeneous media,the rays turn continuously until they reach the boundary. At boundaries, wavesare reected and refracted abruptly. In this section, we consider the reection andrefraction of uniform plane waves at a planar boundary separating the two dielectricmedia having indices n1 and n2 (Fig. 1.4). Uniform plane waves propagate in kinprior to impinging on the boundary. Note that kin = k kin and k = /c is the vacuumwave vector of the same angular frequency. It is convenient to refer various eld components to a plane of incidence, which is dened by a unit vector n1 normalto the boundary and the incident wave vector n1 kin . For the geometry shown inFigure 1.4, the plane of incidence is the xz plane. An arbitrary incident plane wavemay be resolved into two orthogonal polarizations. One polarization has the electriceld normal to the plane of incidence, and the other has the electric eld parallelor in the plane of incidence. In the following discussion, we identify the two eldcomponents as E and E|| , respectively, and treat the two polarizations separately. 29. 12 BRIEF REVIEW OF ELECTROMAGNETICS AND GUIDED WAVES x Medium 2 Index: n2^n1 f2 ktrz kin krf f1 frf ^ n2 Medium 1 Index: n1 Figure 1.4 Reection and refraction by a planar boundary.1.5.1 Perpendicular PolarizationIf the incident electric eld is normal to the plane of incidence, so are the reectedand transmitted electric elds. Since all electric elds are normal to the plane ofincidence, we refer this polarization as the perpendicular polarization. The electricelds are also perpendicular to the direction of propagation; the waves are thetransverse electric waves, or simply TE waves. In some literature, they are alsoreferred to as the s waves where s is the rst letter of senkrecht, the German wordfor perpendicular. With reference to the coordinates shown in Figure 1.4, the incident electric eldis in the y direction. The incident magnetic eld accompanying the incident electric eld is in the direction of kin y. Thus, the incident electric and magnetic elds areEin = yEin 0 ej n1 kin r(1.37) n1 Ein 0 j n1 kin rHin = kin y e(1.38)0where Ein 0 is the amplitude of the incident electric eld. Let n1 krf and n2 ktr be thewave vectors of the reected and transmitted planes wave. Then, the reected andtransmitted elds may be written as, respectively, Erf = yErf 0 ej n1 krf r (1.39) n1 Erf 0 j n1 krf r Hrf = krf ye(1.40)0 Etr = yEtr 0 ej n2 ktr r (1.41) n2 Etr 0 j n2 ktr r Htr = ktr y e (1.42) 0 30. 1.5 R E F L E C T I O N A N D R E F R A C T I O N B Y A P L A N A R B O U N D A R Y 13ktr krffrff1kin^nE E Plane of IncidenceEFigure 1.5 Plane of incidence and the perpendicular and parallel eld components.where Erf 0 and Etr 0 are the amplitudes of the reected and transmitted electricelds. The remaining task is to determine various quantities as functions of kin ,Ein 0 , n1 , and n2 . Let 1 and rf be the angles between the normal n1 and the incident and reectedwave vectors as shown in Figure 1.4. Then, kin and krf are kin = ( cos 1 + z sin 1 )kxkrf = ( cos rf + z sin rf )kxWhen the incident angle is smaller than a critical angle, a term to be introducedshortly, it is possible and meaningful to interpret ktr as a real vector having a real angle 2 relative to the normal n2 . Then ktr can be written as ktr = ( cos 2 + z sin 2 )kx When the incident angle is larger than the critical angle, it is not possible toassociate 2 with a real or physical angle. In the next subsection, we rst discussthe cases where the incident angle is small. Then we discuss the necessary changesfor a large incident angle. 31. 14 BRIEF REVIEW OF ELECTROMAGNETICS AND GUIDED WAVES1.5.1.1Reection and RefractionTo determine Erf 0 and Etr 0 , rf , and 2 , we make use of the boundary condi-tions (1.16) and (1.17). From the continuation of the tangential components of Eand H, we obtain two equations:Ein 0 ej n1 k sin 1 z + Erf 0 ej n1 k sin rf z = Etr 0 ej n2 k sin 2 z(1.43) j n1 k sin 1 zj n1 k sin rf z j n2 k sin 2 zn1 cos 1 Ein 0 e n1 cos rf Er0 e = n2 cos 2 Etr 0 e(1.44)The two equations hold for all values of z if and only if all exponents are same.In other words, n1 sin 1 = n1 sin rf = n2 sin 2 We conclude immediately that the reection angle equals to the incident angle, rf = 1 (1.45)This is the law of reection. We also deducen1 sin 1 = n2 sin 2 (1.46)that is Snells law of refraction or the law of refraction. With rf and sin 2 given by (1.45) and (1.46), (1.43) and (1.44) are simpli-ed toEin 0 + Erf 0 = Etr 0 (1.47)n1 cos 1 Ein 0 n1 cos 1 Erf 0 = n2 cos 2 Etr 0 (1.48) We solve for Erf 0 and Etr 0 in terms of Ein 0 . By dening the reection coefcientof the perpendicular polarization as = Erf 0 /Ein 0 , we obtainn1 cos 1 n2 cos 2 = (1.49)n1 cos 1 + n2 cos 2Equation (1.49) is the Fresnel equation for the perpendicular polarization. Theequation can be cast in several equivalent forms. For example, it can be rearranged asn1 cos 1 n2 n2 sin2 1 21 = (1.50)n1 cos 1 + n2 n2 sin2 1 21 For given n1 , n2 , and 1 , (1.49) or (1.50) can be used to calculate . Typicalplots of the reection coefcient versus the incident angle are shown in Figures 1.6,1.7, and 1.8. When medium 2 is denser than medium 1, that is, n2 > n1 , is real 32. 1.5 R E F L E C T I O N A N D R E F R A C T I O N B Y A P L A N A R B O U N D A R Y 15 1n1 = 1.0n2 = 1.50.5 Reflection Coefficient 0 0.510 15 3045 60 7590 Incident Angle f1 (deg)Figure 1.6 Reection coefcient at a planar boundary between two dielectric media withn1 = 1.0 and n2 = 1.5. 1n1 = 1.5n2 = 1.0Aplitude of Reflection Coefficient ||0.80.6||0.40.2 || 00 15 3045 60 7590 Incident Angle f1 (deg)Figure 1.7 Amplitude of reection coefcient at a planar boundary between two dielectricmedia with n1 = 1.5 and n2 = 1.0. 33. 16BRIEF REVIEW OF ELECTROMAGNETICS AND GUIDED WAVES 90n1 = 1.5n2 = 1.0 75 | Phase Angle | and (deg) | 60 | 45 30 15 0015 3045 6075 90Incident Angle f1 (deg)Figure 1.8 Phase of reection coefcient at a planar boundary between two dielectricmedia with n1 = 1.5 and n2 = 1.0.and negative for all values of 1 , as shown in Figure 1.6. When medium 1 is denserthan medium 2 and if the incident angle 1 is small such that n1 sin 1 n2 , then is real and positive. If the incident angle is large such that n1 sin 1 > n2 , then becomes a complex quantity. More importantly, | | is 1. This is the case ofthe total internal reection to be discussed in the next section. The percentage of power reected by the planar boundary is | |2 , which iscommonly referred to as the power reection coefcient or the reectance.1.5.1.2 Total Internal ReectionWhen medium 1 is denser than medium 2, there exists an incident angle such thatn1 sin 1 = n2 . This is the critical angle mentioned earlier. If the incident angle1 is smaller than the critical angle, then as given in (1.49) and (1.50) is a realquantity. But if the incident angle 1 is greater than the critical angle, becomesa complex quantity. Although (1.46) is still valid, 2 is a complex quantity and wecannot assign a physically intuitive meaning to it. For n1 sin 1 > n2 , n2 cos 2 =j n2 sin2 1 n2 . To select a proper sign for n2 cos 2 , we return to (1.41)1 2and (1.42) and examine the exponents as functions of x and z: n2 ktr r = xkn2 cos 2 + zkn2 sin 2 = j xk n2 sin2 2 n2 + zkn1 sin 112 (1.51) 34. 1.5 R E F L E C T I O N A N D R E F R A C T I O N B Y A P L A N A R B O U N D A R Y 17In terms of x and z explicitly, the transmitted electric eld is 2 2Etr = yEtr 0 exk n1 sin 2 n2 j zkn1 sin 12 (1.52)A corresponding expression can be written for the transmitted magnetic elds.As shown in Figure 1.4, x is positive in region 2. Since the elds in region 2 mustdecay as x increases, the exponent in (1.52) must have a negative real part. This ispossible if and only if we choose the minus sign in (1.51) and (1.52). In other words,n2 cos 2 = j n2 sin2 1 n212(1.53) Using this choice of n2 cos 2 , (1.40) becomes n1 cos 1 + j n2 sin2 1 n212 =(1.54) n1 cos 1 j n2 sin2 1 n212when n1 sin 1 > n2 . Obviously, is now a complex quantity. By writing =| |ej 2 , and we note immediately | | =1(1.55) It means that all incident power is reected by the planar boundary when theincident angle is greater than the critical value. This is known as the total internalreection (TIR). It is the mechanism responsible for the lossless or low-loss wave-guiding in thin-lm waveguides and optical bers. We also deduce from (1.54) thatthe phase angle is a positive angle:n2 sin2 1 n2 12 1 = tan(1.56)n1 cos 1 A plot of as a function of 1 for the case with n1 = 1.5 and n2 = 1.0is shown in Figure 1.8. We can also show analytically that tends to /2 as1 approaches /2. This observation is crucial in analyzing modes guided bygraded-index waveguides and bers.1.5.2 Parallel PolarizationIn the parallel polarization, the incident, reected, and transmitted electric elds arein the plane of incidence. In other words, the electric elds have a component normalto and a component in parallel with the boundary. However, the accompanyingmagnetic elds are normal to the plane of incidence and the direction of propagation.The polarization is referred to as the parallel polarization, and the waves are thetransverse magnetic waves, TM waves, or the p waves and p is the rst letter ofparallel in German. 35. 18BRIEF REVIEW OF ELECTROMAGNETICS AND GUIDED WAVES For the parallel polarization, we could work with the incident electric eldintensity as well. However, it is simpler to express all eld quantities in terms ofthe incident magnetic eld intensity, which is in the y direction in Figure 1.4. Thus,we writeHin = yHin 0 ej n1 kin r (1.57)0 Hin 0 j n1 kin r Ein = y kin e (1.58) n1where Hin 0 is the amplitude of the incident magnetic eld intensity. We express thereected and transmitted elds asHrf = yHrf 0 ej n1 krf r (1.59) 0 Hrf 0 j n1 krf r Erf = y krfe(1.60) n1Htr = yHtr 0 ej n2 ktr r (1.61) 0 Htr 0 j n2 ktr r Etr = y ktre(1.62) n2where Hrf 0 and Htr 0 are the amplitudes of the reected and transmitted magneticeld intensities. To solve for krf , ktr , Hrf 0 , and Htr 0 , we again make use of theboundary conditions (1.16) and (1.17) to rederive (1.45), (1.46). Then we obtainHin 0 + Hrf 0 = Htr 0 (1.63)11 1 cos 1 Hin 0 cos 1 Hrf 0 =cos 2 Htr 0(1.64)n1n1 n2 From these equations, we solve for Hrf 0 and Htr 0 in terms of Hin 0 .1.5.2.1 Reection and RefractionBy dening the reection coefcient of the parallel polarization as|| = Hrf 0 /Hin 0 ,we obtain n2 cos 1 n1 cos 2|| =(1.65) n2 cos 1 + n1 cos 2 This is the Fresnel equation for the parallel polarization. || can also be writtenin several equivalent forms, one of which is n2 cos 1 n1 n2 n2 sin2 1221 || = (1.66) n2 cos 1 + n1 n2 n2 sin2 1221The critical angle introduced in connection with the perpendicular polarizationplays an equally important role in the parallel polarization. || as given in (1.66) is 36. 1.6 G U I D E D W A V E S19real when 1 is smaller than the critical angle, and || can be positive or negativedepending on the incidence angle as shown in Figures 1.6 and 1.7. As shown inthese curves, || vanishes if n1 > n2 and if the incidence angle is n2tan 1 =(1.67) n1 The incidence angle is commonly known as the polarizing or Brewster angle.1.5.2.2 Total Internal ReectionFollowing the same reasoning discussed in the perpendicular polarization, wechoose n2 cos 2 = j n2 sin2 1 n2 when n1 sin 1 > n2 . With this choice of 12n2 cos 2 , we obtain from (1.66)n2 cos 1 + j n1 n2 sin2 1 n2 212|| =(1.68)n2 cos 1 j n1 n2 sin2 1 n2 212when n1 sin 1 > n2 . Again, we note that || is a complex quantity for 1 greaterthan the critical angle. By writing || = | || |ej 2 || , we obtain| || | =1 (1.69)It means that the incident power is totally reected when the incident angle isgreater than the critical angle. The phase term || is a positive angle:n1 n2 sin2 1 n212 || = tan1 (1.70) n2 cos 12As shown in Figure 1.8,|| also approaches /2 in the limit of 1 = /2.1.6 GUIDED WAVESIn this book, we are interested in waves guided by waveguides and bers. Wetake the direction of wave propagation as the z axis. The waveguides and bershave a constant cross section in planes transverse to the z axis. The index prole isindependent of z as well. However, the index n may vary in the transverse directions.To consider waves propagating in the +z direction, we separate the elds into thetransverse and longitudinal components and writeE(r; ) = [et (x, y) + zez (x, y)]ejz (1.71) jz H(r; ) = [ht (x, y) + zhz (x, y)]e (1.72) 37. 20 BRIEF REVIEW OF ELECTROMAGNETICS AND GUIDED WAVESwhere is the propagation constant yet to be determined. The eld quantitieset (x, y), ht (x, y), ez (x, y), and hz (x, y) may be functions of x and y as indicatedexplicitly. But they are independent of z. To proceed, we write = t j z.Substituting these expressions in (1.7) and (1.8), we obtain j z et (x, y) z t ez (x, y) = j 0 ht (x, y) (1.73)t et (x, y) = j 0 hz (x, y) z(1.74) j z ht (x, y) z t hz (x, y) = j 0 n (x, y)et (x, y) 2(1.75)t ht (x, y) = j 0 n (x, y)ez (x, y)2z (1.76) Upon eliminating ht (x, y) from (1.73) and (1.75), we obtain j [t ez (x, y) 0 z t hz (x, y)]et (x, y) =2 2 n2 (x, y)(1.77) 0 0 Similarly, by eliminating et (x, y) from the two equations, we obtainj [t hz (x, y) + 0 n2 (x, y) t ez (x, y)]zht (x, y) =2 2 n2 (x, y)(1.78) 0 0In principle, all transverse eld components can be determined once the longi-tudinal eld components are known. But it is rather difcult to determine ez (x, y)and hz (x, y) if the index in each region is a function of x and y. The differentialequations can be obtained by substituting (1.77) and (1.78) into (1.74) and (1.76).But the differential equations contain ez (x, y) and hz (x, y). In other words, ez (x, y)and hz (x, y) are coupled. In addition, the boundary conditions (1.16) and (1.17) canbe met only when ez (x, y) and hz (x, y) are considered simultaneously. In short, itis not a trivial matter to solve for ez (x, y) and hz (x, y) if the index in each regionvaries with x and y. The same conclusion can be reached by considering the waveequations obtained directly from Maxwells equations (1.7) and (1.8): E(r; ) n2 (x, y) 2 E(r; ) + k 2 n2 (x, y)E(r; ) + =0(1.79)n2 (x, y) H(r; ) + k 2 H(r; ) = 0(1.80) n2 (x, y) If, however, under certain conditions, the differential equations for ez (x, y) andhz (x, y) are decoupled, and boundary conditions can be satised by consideringez (x, y) and its derivative alone, or by considering hz (x, y) and its derivative alone,then we have the elds and propagation constant of TM and TE modes, respectively.Two special cases are considered below. 38. 1.6 G U I D E D W A V E S211.6.1 Transverse Electric ModesConsider waveguides that extend indenitely in the y direction and the index ineach region is a constant independent of y and z. In other words, the index in eachregion is n(x) instead of n(x, y). All eld quantities are also independent of y.Taking advantage of these properties, we drop all partial derivatives with respect toy and obtain from (1.73)(1.76):jey (x) = j 0 hx (x) (1.81) dez (x) jex (x) = j 0 hy (x) (1.82)dxdey (x)= j 0 hz (x)(1.83)dxjhy (x) = j 0 n2 (x)ex (x)(1.84) dhz (x) jhx (x) = j 0 n2 (x)ey (x)(1.85)dxdhy (x)= j 0 n2 (x)ez (x) (1.86)dxNote that (1.81), (1.83), and (1.85) involve only ey (x), hx (x), and hz (x). Notein particular that the electric eld intensity has just one Cartesian component, ey (x).We express the two magnetic eld components in terms of ey (x) and obtainhx (x) = ey (x)(1.87) 0j dey (x)hz (x) = (1.88) 0 dxUsing these two expressions in conjunction with (1.85), we obtain a differentialequation for ey (x):d 2 ey (x) + [k 2 n2 (x) 2 ]ey (x) = 0 (1.89) dx 2By solving ey (x) from the above equation and choosing constants to satisfy theboundary conditions, we have the elds and propagation constant of modes guidedby the waveguide. Since the electric eld is normal to the direction of propagation,the modes are classied as the TE modes.1.6.2 Transverse Magnetic ModesOn the other hand, ex (x), ez (x), and hy (x) alone are involved in (1.82), (1.84),and (1.86). Since there is just one magnetic eld component hy (x), we express 39. 22 BRIEF REVIEW OF ELECTROMAGNETICS AND GUIDED WAVESex (x) and ez (x) in terms of hy (x), and obtain ex (x) = hy (x)(1.90) 0 n2 (x) jdhy (x)ez (x) = (1.91) 0 n2 (x) dxand d 1 dhy (x)2 + k2 2hy (x) = 0 (1.92) dxn2 (x) dx n (x) By solving hy (x), we obtain the elds and propagation constant of the TMmodes.1.6.3 Waveguides with Constant Index in Each RegionIn the case where n in each region is independent of y, z, and x, (1.87), (1.88), (1.90),and (1.91) remain valid. However, the two differential equations (1.89) and (1.92)are simplied further: d 2 ey (x)+ (k 2 n2 2 )ey (x) = 0 (1.93) dx 2 d 2 hy (x)+ (k 2 n2 2 )hy (x) = 0 (1.94)dx 2 Equations (1.93) and (1.94) are two ordinary differential equations with constantcoefcients and they can be solved readily. Detail study of TE and TM modes fordielectric waveguides with constant index in each region are discussed in Chapter 2.In fact, (1.90), (1.91), (1.92), and (1.93) are the starting point for the discussions inChapter 2.PROBLEMS1. Show that from (1.27) and (1.28) thatExsin = cos(t + x ) sin Ex0Ex Eycos = sin(t + x ) sinEx0 Ey0 Then, (1.29) follows immediately.2. Derive (1.32) to (1.35).3. Derive (1.79) and (1.80). 40. REFERENCES23REFERENCES1. R. E. Collin, Field Theory of Guided Waves, 2nd ed., IEEE Press, New York, 1991.2. S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and Waves in Communication Elec- tronics, 3rd ed., Wiley, New York, 1994.3. M. Born and E. Wolf, Principles of Optics, 6th ed., Pergamon, Oxford, 1980.4. D. B. Melrose and R. C. McPhedran, Electromagnetic Processes in Dispersive Media, Cambridge University Press, Cambridge, 1991.5. E. Hecht, Optics, 3rd ed., Addison Wesley, Reading, MA, 1998.6. John Krause and D. Fleisch, Electromagnetics with Applications, 5th ed., McGraw-Hill, New York, 1998.7. U. S. Inan and A. S. Inan, Electromagnetic Waves, Prentice Hall, Upper Saddle River, NJ, 2000.8. C. L. Chen, Elements of Optoelectronics and Fiber Optics, R. D. Irwin, Chicago, 1996. 41. 2STEP-INDEX THIN-FILMWAVEGUIDES2.1 INTRODUCTIONIn the simplest form, an optical thin-lm waveguide is a long structure having threedielectric regions as shown in Figure 2.1. The three dielectric regions are a thickregion with an index ns , a thin layer of an index n , and a thick region with anfindex nc . The thin layer has the largest index and is referred to as the lm region.The thick region with the smallest index is the cover region. The region havingthe second lowest index is the substrate region. Throughout our discussions, weassume that n > ns and nc . Since the lm region has the largest index n , elds aref fmainly conned in this region. The lm thickness h is comparable to the operatingwavelength . In contrast, the cover and substrate regions are much thicker than .We take the two thick regions as innitely thick as an approximation. Before presenting the thin-lm waveguide theory, we describe briey howthin-lm waveguides are made. Probably, the simplest method to make a thin-lmwaveguide is to dip a plate in, or to spin-coat a plate with, polymer or photore-sist. By dipping or coating, a thin layer of polymer or photoresist is formed onthe plate and a thin-lm waveguide is thereby formed. The plate mentioned abovecan be, and usually is, a glass slide. Thin-lm waveguides can also be made byFoundations for Guided-Wave Optics, by Chin-Lin ChenCopyright 2007 John Wiley & Sons, Inc.25 42. 26 STEP-INDEX THIN-FILM WAVEGUIDESx Cover nc0 z nfFilm h nsSubstrateFigure 2.1 Step-index thin-lm waveguide.sputtering one type of glass on another type of glass or by dipping a glass slideinto molten AgNO3 , thereby forming a high-index layer via the Ag ion exchangeprocesses. Numerous techniques exist to form high-index layers on Si, GaAs, InP,or other semiconductor substrates. Although Si is lossy at the visible spectra, it istransparent at 1.3 m or longer wavelengths. While many processes may be used toform waveguide layers on lithium niobate (LiNbO3 ) and lithium tantalate (LiTaO3 )substrates, the in-diffusion process is the most popular method of making LiNbO3and LiTaO3 waveguides. At present, most passive waveguide structures are basedon glass, lithium niobate, lithium tantalate, Si, GaAs, and InP. The main attribute oflithium niobate, lithium tantalate, GaAs, or InP waveguides is that these materialsare electrooptic, and the waveguide characteristics can be tuned by applying static,radio-frequency, or microwave electric elds to the waveguide material. Si wafersand glass slides are widely available and at a reasonable cost. Besides, Si and glasstechnologies are two matured and advanced technologies. The index prole of the photoresistglass or polymerglass waveguides can berepresented by three straight-line segments with abrupt index change at the bound-aries as shown in Figure 2.2(a). Waveguides with such an index prole are known asstep-index waveguides. The index of LiNbO3 , LiTaO3 , semiconductor waveguides,or ion-exchanged glass waveguides changes gradually in the lm and substrateregions, as shown in Figure 2.2(b). In fact, the lm and substrate regions maymerge into one region and the index changes continuously as a function of position.The precise index prole would depend on the waveguide materials and the fabrica-tion processes involved. For example, the index prole of many in-diffused LiNbO3or LiTaO3 waveguides can be approximated by an exponential function. Thesewaveguides are generally referred to as the graded-index waveguides. We restrictour discussions in this chapter to the step-index waveguides only. Discussions ongraded-index waveguides are deferred until Chapter 3. The cross sections of several waveguides are shown in Figure 2.3. In Fig-ure 2.3(a), the waveguide dimension in the y direction is much larger than the lmthickness h in the x direction and the operating wavelength . In studying the wavepropagation in the z direction, we may ignore the eld variation in the y direction 43. 2.2 D I S P E R S I O N O F S T E P - I N D E X W A V E G U I D E S27nn nf nf ns nc ns nch0 x0 x(a) (b)Figure 2.2 Two index proles: (a) step-index and (b) graded-index proles.as an approximation. These waveguides are two-dimensional waveguides. They arealso referred to as thin-lm waveguides, dielectric slab waveguides, or planar waveg-uides. For waveguides shown in Figure 2.3(b), the channel, ridge, or strip width inthe y direction is comparable to h and . They are three-dimensional waveguides.Depending on the waveguide geometry and the fabrication processes, they are knownas channel waveguides, ridge or rib waveguides, and embedded strip waveguides.We will discuss three-dimensional waveguides in Chapter 5.2.2 DISPERSION OF STEP-INDEX WAVEGUIDESTo study waves guided by two-dimensional step-index waveguides [14], we beginwith the time-harmonic (ej t ) Maxwell equations (1.7)(1.10) for dielectric media.For a dielectric medium with an index n, the permittivity is = n2 0 and thepermeability is = 0 . Different regions have different indices of refraction. Thebasic waveguide geometry is depicted in Figures 2.1 and 2.3(a). The lm region ish x 0 and it has the largest index n . The cover region is x > 0 and it hasfthe smallest index nc . The substrate region, x h, has an index between n andfnc . We choose the coordinate system such that waves propagate in the z direction.Since waves propagate in the z direction, a phase factor ejz is present in all terms.Accordingly, we write E and H asE = [ ex (x, y) + yey (x, y) + zez (x, y)]ejz x H = [ hx (x, y) + yhy (x, y) + zhz (x, y)]ejz x Clearly, /z = j. Since the material properties and the waveguide geometryare independent of y, all eld components are also independent of y. In other words,ex and hy and so forth are functions of x only. Then Maxwells equations, (1.7)and (1.8), in the component form can be written as jey (x) = j 0 hx (x)(2.1)dez (x) jex (x) = j 0 hy (x)(2.2) dx 44. 28 STEP-INDEX THIN-FILM WAVEGUIDESx ncyh nfz ns(a) x nc nc h nfynf zns ns Channel Waveguides Ridge or Rib Waveguides ncnf ns Embedded Strip Waveguides (b) Figure 2.3 (a) Two- and (b) three-dimensional waveguides. dey (x) = j 0 hz (x)(2.3) dxjhy (x) = j 0 n2 ex (x)(2.4) dhz (x)jhx (x) = j 0 n2 ey (x)(2.5)dxdhy (x)= j 0 n2 ez (x) (2.6)dx These equations have been derived previously in Chapter 1. They are repeatedhere for convenience. Equations (2.1)(2.6) are naturally divided into two groups:The rst group(2.1), (2.3), and (2.5)involves ey , hx , and hz only. Since theelectric eld is in the y direction that is perpendicular to the direction of propagation,this group of elds is referred to as the transverse electric (TE) modes. The second 45. 2.2 D I S P E R S I O N O F S T E P - I N D E X W A V E G U I D E S 29group(2.2), (2.4), and (2.6)involves hy , ex , and ez . They form the transversemagnetic (TM) modes. We will study each group separately.2.2.1 Transverse Electric ModesA TE mode [3, 4] has an electric eld component, ey , and two magnetic eldcomponents, hx and hz . In particular, the electric eld component is in parallel withthe waveguide surface and perpendicular to the direction of propagation. The twomagnetic eld components can be expressed in terms of the electric eld component.More explicitly, hx (x) = ey (x) (2.7)01 dey (x) hz (x) = j(2.8) 0 dxSubstituting the two expressions in (2.5), we obtain a wave equation for ey : d 2 ey (x)+ (k 2 n2 2 )ey (x) = 0 (2.9)dx 2 The boundary conditions are the continuation of ey , hz , and bx at x = 0 andx = h. In view of (2.7), it is clear that bx is continuous when ey is continuous. Italso follows from (2.8) that hz is continuous at the boundary if dey /dx is continuousthere. In short, all boundary conditions are met if the electric eld component eyand its normal derivative dey /dx are continuous at the boundaries. In the followingsections, we solve ey from (2.9) subject to the continuity of ey and dey /dx at thetwo interfaces. In the lm region (h x 0, n = n ), we expect ey to be an oscillatoryffunction of x. This is possible only if k 2 n2 2 is positive. Upon introducing ff =k 2 n2 2 , we write fey (x) = Ef cos(f x + )(2.10)where Ef and are two constants yet undetermined. For future use, we also note dey (x) = f Ef sin(f x + )(2.11) dxFor the substrate region (x h, n = ns ), we expect ey to decay as x becomesmore negative. This can be true only if k 2 n2 2 is negative. Thus we introduce ss = 2 k 2 n2 and writes ey (x) = Es es (x+h) (2.12) dey (x) = s Es es (x+h) (2.13) dx 46. 30STEP-INDEX THIN-FILM WAVEGUIDESSimilarly, we expect k 2 n2 2 to be negative for the cover region (x > 0,cn = nc ). We dene c = 2 k 2 n2 and write cey (x) = Ec ec x(2.14)dey (x)= c Ec ec x (2.15)dxThe constants Ef , Es , Ec , and the phase in (2.10)(2.15) and the propaga-tion constant are chosen to satisfy the boundary conditions noted earlier. Thecontinuation of ey and dey /dx at x = 0 leads toEf cos = Ec (2.16)f Ef sin = c Ec (2.17) From the two equations, we obtain an expression for :ctan =(2.18)f Similarly, from the boundary conditions at x = h, we haveEf cos(f h + ) = Es(2.19)Ef f sin(f h + ) = f Es (2.20) When the two equations are combined, we obtain a second expression for : stan(f h + ) = (2.21) fEliminating from (2.18) and (2.21), we obtain the dispersion relation or char-acteristic equation for TE modes:c s f h = tan1+ tan1+ m (2.22)f fwhere m = 0, 1, 2, 3, . . . is an integer, and m is known as the mode number. Thesolution of (2.22) with a specic value of m gives the propagation constant of TEmmode. Once is known, three of the four constants can be determined. The fourthconstant, say Ef or Ec , represents the amplitude of the guided mode. Thus the TEmodes guided by a thin-lm waveguide are completely determined. Note that Ec isthe electric eld intensity at the coverlm boundary and Ef is the peak electriceld intensity of the TE mode. Therefore Ef and Ec are of interest. 47. 2.2 D I S P E R S I O N O F S T E P - I N D E X W A V E G U I D E S312.2.2 Transverse Magnetic ModesA TM mode [3, 4] has a magnetic eld component, hy , and two electric eldcomponents, ex and ez . The transverse electric eld component ex is normal tothe waveguide surface and the direction of propagation. In addition, the two electriceld components can be expressed in terms of hy . Specically, we obtain from (2.4)and (2.6) ex (x) =hy (x) (2.23) 0 n2 1 dhy (x)ez (x) = j(2.24)0 n2 dxThe boundary conditions for hy , ez , and dx are met if hy and (1/n2 )/(dhy /dx)are continuous at the boundaries. Following the same procedure used in the lastsubsection in analyzing TE modes, we write hy in the cover, lm, and substrateregions as Hc ec x(2.25)hy (x) = Hf cos(f x + ) (2.26)Hs es (x+h) (2.27)where Hc , Hf , Hs , and are constants to be determined. By matching the boundaryvalues, we obtain the dispersion relation or characteristic equation for TM modes:n2 c 2 nf sf h = tan1+ tan1 2 f +m (2.28)n2 fcns f The mode number m is also an integer. When and three of the four constantsare determined, the TM problem is solved. The fourth constant, say Hf or Hc ,represents the amplitude of the TM mode. In particular, Hc is the magnetic eldintensity at the coverlm boundary and Hf is the peak magnetic eld intensity ofthe TM mode in question. For a given waveguide operating at a given wavelength, n , ns , nc , h, and fare known. can be determined from the dispersion relation (2.22) for TE modesand (2.28) for TM modes. No analytic solution of the two transcendental equations isknown. Numerical techniques are applied to determine . But it is not easy or trivialto evaluate numerically either. To appreciate the complication involved, we intro-duce the effective guide index N such that = kN . For a guided mode, N is betweenn and ns . For most waveguides of practical interest, n and ns are numerically close f fand the index difference (n ns )/n is less than a few percent. Thus N differs fromf fn or ns very slightly. Therefore, an accurate determination of N is difcult. Besides, fthe numerical results are valid for a specic set of waveguide parameters only. Tocircumvent these difculties, Kogelnik and Ramaswany [5] introduced the general-ized parameters. An alternate set of parameters has been introduced by Pandraud 48. 32 STEP-INDEX THIN-FILM WAVEGUIDESand Parriaux [6]. It is customary to use Kogelnik and Ramaswanys a, b, c, d, andV as the generalized parameters. One of the advantages of using their parametersis that b and V are also useful to characterize step-index bers.2.3 GENERALIZED PARAMETERSAs explained earlier, it is rather difcult to evaluate and N numerically. Therefore,it is desirable to have a set of universal curves that are applicable to most waveguidesof interest. These curves have been constructed by Kogelnik and Ramaswamy [5].This is the subject of this section.2.3.1 The a, b, c, d, and V ParametersThree generalized parameters [3, 5, 7] are sufcient to describe all TE modes guidedby three-layer step-index waveguides [3, 5]. The three generalized parameters are(a) the asymmetry measure n2 n2 a= s c(2.29) n2 n2f s(b) the generalized frequency, also known as the generalized lm thickness,V = kh n2 n2fs (2.30)and (c) the generalized guide index N 2 n2b= s (2.31) n2 n2f s The generalized parameters a and b are the differences n2 n2 and N 2 n2s csnormalized with respect to n2 n2 . In other words, these generalized parametersf sare in terms of the differences of index squares rather than the indices themselves. A little manipulation will show that h = kh n2 N 2 = V ff1b (2.32) s h = kh N 2 n2 = V s b (2.33)and c h = kh N 2 n2 = V c a+b (2.34) 49. 2.3 G E N E R A L I Z E D P A R A M E T E R S33 Using these expressions in (2.22), we cast the dispersion relation for TE modesin terms of generalized parameters: a+b b V 1 b = tan1+ tan1+ m (2.35)1b1bFor TM modes, we need one more parameter [5, 7] in addition to a, b, and Vdened above. The extra parameter can be eithern2 c= s(2.36)n2 forn2 d= c= c a(1 c)(2.37)n2 f In terms of generalized parameters, the dispersion relation for TM modes is 1 a+b 1b V 1 b = tan1+ tan1 +m (2.38)d 1b c 1b Except for the presence of 1/c and 1/d, (2.38) is similar to (2.35). In the twotranscendental equations, b spans from 0 and 1. More importantly, the solutionsin terms of b are well apart and distinct. In other words, (2.35) and (2.38) can besolved easily by numerical methods.2.3.2 The bV DiagramFor a given waveguide operating at a specic wavelength, n , ns , nc , h, and fare known. The values of a, c, d, and V may be calculated from the waveguideparameters. For each set of a, V , and c, we determine b numerically from (2.35)for TE modes and (2.38) for TM modes. There may be one or more solutions for b,depending on V , a, and c. Each solution of b corresponds to a guided mode. Thelargest b corresponds to m = 0. Knowing b, the effective index N , the propagationconstant , three amplitude constants, and can be evaluated. Plots of b versus Vfor TE modes with four values of a are depicted in Figure 2.4. Similar plots for TMmodes for c = 0.98, 0.94, and 0.90 are shown in Figure 2.5. The bV curves of TEand TM modes are really very close. To compare curves of TE and TM modes, wesuperimpose plots of TE and TM modes for two values of a and for c = 0.94 inFigure 2.6. The bV curves of TE modes are shown as thick lines and dots and thatof TM modes as thin lines and dots. Obviously, the two sets of curves are reallyvery close. As noted earlier, each solution of b corresponds to a guided mode. As the lmthickness gets thinner, corresponding to a smaller V , b becomes smaller. As b of agiven mode approaches 0, the mode approaches its cutoff. All modes, except the TE0 50. 34 STEP-INDEX THIN-FILM WAVEGUIDES1a=0 0.9a=1 m=0a = 10 0.8a = 100 TEm=1 0.7 0.6 m=2 0.5 b m=3 0.4m=4 0.3 0.2m=5 0.10 0246 8 10 121416 1820V Figure 2.4 bV curves of TE modes guided by step-index thin-lm waveguides.and TM0 modes guided by symmetric waveguides, are cutoff if the lm region issufciently thin. In fact, if the lm thickness is sufciently thin, no mode is supportedby the asymmetric waveguides. A waveguide is symmetric if the cover and substrateindices are the same. A waveguide having a thin glass slide surrounded by air onboth sides is a good example of symmetric waveguides. The fundamental modeof symmetric waveguides has no cutoff. In other words, a symmetric waveguidesupports at least a TE and a TM mode each no matter how thin the lm region is.The TE and TM modes are the TE0 and TM0 modes. We can visualize the shapes of the bV curves by recalling the modes guided byparallel-plate waveguides. A parallel-plate waveguide is simply two large conductingplates separated by a dielectric material of index n and thickness h. Modes guided byparallel-plate waveguides are discussed in many texts on electromagnetism [811].By dening V = knh and b = N 2 /n2 , the dispersion of TE modes guided by aparallel-plate waveguide can be written as 2(m + 1)b =1(2.39) VThe bV characteristic of parallel-plate waveguides as given in (2.39) is plottedas dots in Figure 2.7. For comparison, the bV curves of TE modes guided by a 51. 2.3 G E N E R A L I Z E D P A R A M E T E R S 35 1a=0m=0 0.9a=1a = 10 0.8a = 100 TM m=1 0.7 c = 0.98 0.6m=2 0.5 b m=3 0.4 0.3m=4 0.2m=5 0.1 0 02468 10 12 1416 1820 V (a)Figure 2.5 bV curves of TM modes guided by step-index thin-lm waveguides: (a) c = 0.98,(b) c = 0.94, and (c) c = 0.90.symmetric thin-lm waveguide are depicted in the same gure as solid lines. Thesimilarity between the two sets of curves is obvious. To illustrate the use of the bV diagram, we consider a numerical examplewith n = 1.500, ns = nc = 1.300, and h/ = 3.000. Since the cover and sub- fstrate indices are equal, the waveguide is a symmetric waveguide. For a symmetricwaveguide with V = 14.11, we read from Figure 2.4 that b of TE0 mode is about0.96 (with a little imagination?). Using this value of b, we obtain from (2.31) thatN 1.493. The next mode is TE1 mode for which b and N are 0.85 and 1.472,respectively; b and N for higher-order modes can be obtained in the same manner.2.3.3 Cutoff Thicknesses and Cutoff FrequenciesAs noted previously, the cutoff condition is b = 0. By setting b to zero, we have,from (2.35), the cutoff V value for TEm mode:Vm = m + tan1 a(2.40) 52. 36STEP-INDEX THIN-FILM WAVEGUIDES1a=0 0.9a=1 m=0a = 10 0.8a = 100 TMm=1 0.7 c = 0.94 0.6m=2 0.5 b m=3 0.4 m=4 0.3 0.2 m=5 0.10 0246 8 1012 14161820V(b)Figure 2.5 (continued). In other words, a TEm mode is supported by a thin-lm waveguide if the lmthickness ism + tan1 a h=(2.41) 2 n2 n2f sor thicker. The cutoff V for a TMm mode is, from (2.38), an2 Vm = m + tan1 = m + tan1 f a(2.42)d n2 cSince nc is smaller than n , the cutoff V for TEm mode is slightly smaller thanfthe cutoff V of TMm mode with the same mode number m.2.3.4 Number of Guided ModesIt is clear from (2.40), that m modes may be supported by a thin-lm waveguideTE if V is between m + tan1 a and (m 1) + tan1 a. Similarly, there are m+ TM modes if V is between m + tan1 ( a/d) and (m + 1) + tan1 ( a/d). 53. 2.3 G E N E R A L I Z E D P A R A M E T E R S37 1a=0 0.9a=1 m=0a = 10 0.8a = 100TMm=1 0.7 c = 0.90 0.6m=2 0.5 b m=3 0.4m=4 0.3 0.2m=5 0.1 0 024 68 1012 141618 20V(c) Figure 2.5 (continued). We also note that for symmetric waveguides, the cutoff V values for TEm andTMm modes are simply m. For each increment of V by , there is an additionalTE mode and an additional TM mode. We can also determine the number of modesguided from the bV plots if V is less than 20. For symmetric thin-lm waveguideswith V greater than 20, the number of TE and TM modes supported by the waveg-uide is approximately 2[1 + (V /)]. The numbers of TE and TM modes guidedby asymmetric waveguides are approximately the same as that guided by symmet-ric waveguides. In short, the number of TE and TM modes guided by step-indexthin-lm waveguides is approximately 2[1 + (V /)].2.3.5 Birefringence in Thin-Film WaveguidesAs shown in Figures 2.4, 2.5, and 2.6, the bV curves of TEm and TMm modes of thesame waveguide having the same mode number may be close. But the bV curvesdo not cross or intersect. For the same waveguide and for the same mode numberm, the b value of TMm mode is always smaller than that of TEm mode as shownin Figure 2.6. Thus, the propagation constant of a TE mode is slightly differentfrom that of a TM mode with the same mode number. In short, thin-lm waveguidesmade of isotropic dielectric materials are birefringent. The birefringence of thin-lmwaveguides may be problematic in many applications. 54. 38 STEP-INDEX THIN-FILM WAVEGUIDES1 0.9a=1 m=0 0.8a = 100 TE TMm=1 0.7 c = 0.94 0.6 m=2 0.5 bm=3 0.4 m=4 0.3 0.2m=5 0.10 02 4 6 8 10 12 14 16 1820VFigure 2.6 bV curves of TE and TM modes guided by step-index thin-lm waveguides.2.4 FIELDS OF STEP-INDEX WAVEGUIDES2.4.1 Transverse Electric ModesBy requiring the continuation at the lmcover and lmsubstrate boundaries andmaking use of the dispersion relation (2.22), we can eliminate three of the fourconstants Ec , Ef , Es , and . Then ey (x) given in (2.10), (2.12), and (2.14) may beexpressed in terms of a single amplitude constant. If we keep Ec as the amplitudeconstant, then ey (x) isEc ec xx0c Ec cos f x sin f x h x 0ey (x) = f(2.43) E cos h + c sin h es (x+h) cf f x hc When written in terms of the generalized parameters, ey (x) becomes 55. 2.4 F I E L D S O F S T E P - I N D E X W A V E G U I D E S 391 0.8 m=0Normalized Guide Index b 0.6 m=2 m=1m=3 0.4 0.20051015 20Normalized Thickness VFigure 2.7 Comparison of the bV characteristic of TE modes guided by a symmetric dielectricwaveguide (solid lines) with that of a parallel plate waveguide (dotted lines). Ec eV a+bx/ hx0 V 1 bxa+b Ec cos h1b V 1 bx sinh x 0ey (x) = h (2.44) a+b Ec cos(V1 b) + 1b sin(V1 b) eV b[1+(x/ h)]x h Typical eld distributions of ey (x) of TE0 , TE1 , and TE2 modes of asymmetricwaveguides are presented in Figure 2.8. The vertical dash lines in the gures markthe lmsubstrate and lmcover boundaries. In the lm region, the eld distri-bution of a guided mode is an oscillatory function of x, as expected. In the twoouter regions, the eld decays exponentially from the lmcover and lmsubstrateboundaries. For all guided modes, elds in the cover and substrate regions are con-ned mainly in the thin layers near the lmcover or lmsubstrate boundaries.Beyond the thin layers immediately adjacent to the two boundaries, elds are veryweak unless the mode is close to cutoff. A careful examination of these plots also 56. 40STEP-INDEX THIN-FILM WAVEGUIDES 1 V = 8.0 a = 1.0 TE0 b = 0.8957 ey(x)0.5 02 1.51 0.500.5 1 x/h 1V = 8.0 a = 1.0TE10.5 b = 0.5905 ey(x) 0 0.512 1.51 0.500.5 1 x/h 1 TE2 V = 8.0 a = 1.00.5b = 0.1235 ey(x) 0 0.512 1.51 0.500.5 1 x/hFigure 2.8 Field distributions (ey ) of TE0 , TE1 , and TE2 modes.reveals that elds in the substrate region are stronger than those in the cover region.This is also expected since we have assumed that ns nc .Expressions for hx (x) and hz (x) can be obtained by substituting ey (x) into (2.7)and (2.8). If desired, expressions can also be written in terms of Ef or Es (Prob-lem 5). Most quantities of interest, the time-average power, for example, can also beexpressed in terms of the generalized parameters. We will discuss the time-averagepower in Section 2.6.2.4.2 Transverse Magnetic ModesIn terms of Hc , hy (x) isHc ec xx0 n2 c Hc fcos f x 2 sin f x h x 0 hy (x) = nc f(2.45) n2 Hc cos f h + f c sin f h es (x+h) x h n2 f c 57. 2.6 T I M E - A V E R A G E P O W E R A N D C O N F I N E M E N T F A C T O R S 41 When expressed in terms of generalized parameters, hy (x) is Hc eV a+bx/ h x0 V 1 bx 1 a+b Hc cos h d 1b V 1 bx sin h x 0hy (x) = h(2.46) H cos(V 1 b) + 1 a + b c d 1b sin(V 1 b) eV b[1+(x/ h)]x h The other eld components may be obtained by substituting (2.45) or (2.46)into (2.4) and (2.6).2.5 COVER AND SUBSTRATE MODESIn addition to a nite number of guided modes, a thin-lm waveguide also supportsa continuum of radiation modes. Radiation modes are not conned in a regionnor guided by the dielectric boundaries. They are referred to as the cover (or air)and substrate modes, respectively. For a substrate mode, the total internal reectionoccurs at the coverlm interface, but not the lmsubstrate boundary, recalling thatns > nc . In the cover region, the eld of a substrate mode decreases exponentiallyfrom the lmcover boundary. In the substrate region, the eld is the superpositionof, and the interference between, the incident and reected plane waves. Thus,the eld in the substrate region varies in an oscillatory manner instead decayingexponentially from the lmsubstrate boundary. For the cover mode, there is nototal internal reection at either boundary. Thus a cover mode is a plane wave inone region and crests and troughs in the other region. In other words, a cover modepenetrates into the substrate and cover regions. This is depicted schematically inFigure 2.9.2.6 TIME-AVERAGE POWER AND CONFINEMENT FACTORS2.6.1 Time-Average Power Transported by TE ModesThe time-average power transported by a TE mode in a thin-lm waveguide of unitwidth in the y direction is1P = Re[E H ] z dx dy =|ey (x)|2 dx dy2 2oh0 =1Es e2s (x+h) dx +2Ef cos2 (f x + ) dx + 2Ec e2c x dx 2 2o h0 58. 42 STEP-INDEX THIN-FILM WAVEGUIDES xnc z y hnf ns(a)nc hnfns(b)ncnc hnf nf ns ns (c)Figure 2.9 Rays of (a) guided, (b) substrate, and (c) cover modes.where 1 signies a unit width in the y direction. All integrals can be integrated inclosed form and the result is 2 Essin 2 sin 2(f h + )2 EcP = + Ef h + 2 + (2.47)4o s 2fcThe three terms represent the time-average power transported in the substrate,lm, and cover regions. By making use of (2.16)(2.18) and (2.19)(2.21), we canshow 22 Ec2Ef sin 2Ef +=(2.48) c2f cand2 Es Ef sin 2(f h + ) 2 2Ef =(2.49) s 2f s 59. 2.6 T I M E - A V E R A G E P O W E R A N D C O N F I N E M E N T F A C T O R S43Making use of these relations, we can express Ec and Es in terms of Ef . Finally,we simplify (2.47) and obtain 11 P =E2 h ++ (2.50)4o f s c To appreciate the signicance of the bracketed term in the above expression, weconsider TE modes guided by a parallel-plate waveguide with a plate separation of h.Suppose that the peak eld intensity is Ef . Then the time-average power carried byTE modes of a parallel-plate waveguide of unit width is P = (/4o )Ef h [811]. 2Recall that h is the spacing between the two conducting plates. This expression isvery similar to (2.50). By analogy, we dene the effective waveguide thickness of athin-lm waveguide operating in a TE mode as 11heff = h ++ (2.51) s c We also dene the normalized effective lm thickness as H = kheff n2 n2 .fsIn terms of the generalized parameters,11H =V ++ (2.52) a+bb Figure 2.10 is a plot of H versus V for the rst four TE modes. In all cases,H decreases from a large value near the cutoff to a minimum, and then it increaseslinearly with V . This can be understood as follows. When a mode is near the cutoff,elds are not conned in the lm region. Therefore, the effective lm thickness israther large. In other words, a large fraction of time-average power resides outsidethe lm region if a mode is close to the cutoff. Far above the cutoff, however, Happroaches an asymptote: 1H V + 1 + (2.53)1+a It is evident from Figure 2.10 that H is only slightly larger then V for large V .It means that only a small fraction of time-average power is outside the lm regionwhen a mode is far above the cutoff. In short, elds and time-average power areindeed conned in the lm region for modes far above the cutoff. The minimum value of H depends on the asymmetry measure a and the modenumber m. For example, the minima of H of TE0 mode are 4.93, 4.52, and 4.40for a = 0, 1, and 10, respectively. For higher-order TE modes, the minima of Hare larger, as expected. 60. 44STEP-INDEX THIN-FILM WAVEGUIDES 20m=3Normalized Effective Width H 15 m=2 10 m=1 a=0 a=1 a = 105 m=000 5 10 1520 Normalized Thickness VFigure 2.10 Normalized effective widths of TE modes guided by step-index thin-lmwaveguides.2.6.2 Connement Factor of TE ModesIt is instructive to study the percentage of time-average power contained in eachregion. To quantify the fractional power within the lm region, we dene the con-nement factor f as Time-average power transported in the lm region f= Total time-average power transported by the waveguide From the physical meaning of terms in (2.47), we readily deduce that c s h++ 2 + c f2 2 f + s2 f = (2.54)1 1h+ +c sThe connement factor can also be expressed directly in terms of the generalizedparameters a+b V + b+1+af = (2.55)11 V + + b a+b 61. 2.6 T I M E - A V E R A G E P O W E R A N D C O N F I N E M E N T F A C T O R S 45 Similarly, we introduce s and c to quantify the fractional power containedin the substrate and cover regions: Time-average power transported in the substrate region s = Total time-average power transported by the waveguideTime-average power transported in the cover region c = Total time-average power transported by the waveguide Following the same procedure, we obtain1b = (2.56) s 11 b V + +ba+b 1b = (2.57) c 1 1 (1 + a) a + b V + + b a+b It is simple to show indeed that f + s + c = 1. In Figure 2.11 we plot f ,s , and c of TE0 mode as functions of V . Except for thin-lm waveguides operatingnear the cutoff, f is much larger than s and c . For waveguides operating farabove the cutoff, f approaches 1, and s and c are