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A‘I‘E' :L ‘ .t‘4‘1'fill'.“ ' I" ‘1 ‘m' "-l' I‘ " dold'A.‘ Lh'nh'fl This is to certify that the thesis entitled WCOW‘ h smTEGRAL-OPERATOR ANALYSIS OF OPEN-BOUNDARY vxgq- ‘ ‘ -.".‘..\(-\Q\ _) DIELECTRIC WAVEGUIDES \ B or“ 299/ (33$)th ‘\ A presented by Dean Richard Johnson has been accepted towards fulfillment of the requirements for Jul—degree in BBC?» Enqr. WP” ° Major professg Date Jan. 19, 1981 0-7639 s ...__,_.. J——_ V J ~ ——-——. _ llll ll Ill lllllllllllllllfilfllflll L 3 1293 - ft. ;. 4‘11”" ' 11““ .. '. OVERDUE EINES: 25¢ per day per item RETURNING LIBRARY MATERIALS: Place in book return to rem charge from circulation moo: INTEGRAL-OPERATOR ANALYSIS OF OPEN-BOUNDARY DIELECTRIC NAVEGUIDES By Dean Richard Johnson A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Electrical Engineering 1981 ABSTRACT INTEGRAL-OPERATOR ANALYSIS OF OPEN-BOUNDARY ‘ DIELECTRIC HAVEGUIDES By Dean Richard Johnson An integral-operator technique is described as an alternative method to conventional boundary-eigenvalue analysis for EM wave propagation in relatively-general, open-boundary dielectric (millimeter-wave or optical) waveguides. A polarization current is exploited to establish an electric-field integral equation (EFIE) describing the EM field in the heterogeneous cone of an open- boundary waveguide. EFIEs specialized to describe axially- propagating, surface-wave and radiation-mode-wave fields supported by heterogeneous, longitudinally-invariant, open-boundary waveguides of arbitrary cross-section shape are established. Analytical solutions of EFIEs are described to recover the well-known transverse field dependence and propagation characteristics of surface-wave and radiation modes supported by relatively-simple dielectric waveguide structures. Numerical solution of EFIEs describing surface-wave modes (field plots and dispersion character- istics) supported by planar, graded-index, dielectric-slab and step- index, rectangular dielectric waveguides are presented. Finally, an EFIE formulation for excitation of discrete, guided, surface-wave modes and continuous-spectrum radiation modes along open-boundary waveguides is introduced. These results are of general utility for the investigation of propagation, coupling, and scattering phenomenon of EM waves interacting with graded-dielectric surface (open-boundary) waveguides and dielectric devices, such as found in fiber optics and integrated optics. ACKNOWLEDGMENTS The author gratefully acknowledges the generous support and guidance provided by Dr. Dennis P. Nyquist, Professor, Electrical Engineering, Michigan State University, during the course of this work. The author also acknowledges the special assistance provided by Dr. Byron C. Drachman, Associate Professor, Mathematics, Michigan State University. ii ACKNOWLEDGMENTS The author gratefully acknowledges the generous support and guidance provided by Dr. Dennis P. Nyquist, Professor, Electrical Engineering, Michigan State University, during the course of this work. The author also acknowledges the special assistance provided by Dr. Byron C. Drachman, Associate Professor, Mathematics, Michigan State University. ii II. TABLE OF CONTENTS 'LIST OF TABLES LIST OF FIGURES INTRODUCTION ELECTRIC-FIELD INTEGRAL EQUATION DESCRIBING HETEROGENEOUS OPEN-BOUNDARY DIELECTRIC NAVEGUIDES 2.1. Introduction 2.2 Maxwell's Equations Describing EM Field in Millimeter-wave and Optical Media A. Maxwell's Equations for Field in Heterogeneous Dielectric Media 8. Equivalent-Current Description for Heterogeneous Core of Open-Boundary Dielectric Waveguide 2.3 3-d Electric-Field Integral Equation A. Basic Formulation B. Recovery of Nonhomogeneous 3-d Wave Equation From 3-d EFIE C. Identification of Surface-Charge Effects 2.4 EFIE Solution for Scattering of Plane Nave Obliquely Incident Upon Plane Interface iii PAGE vii viii 10 14 14 17 18 23 III. IV. 2.5 Homogeneous EFIE for Natural Surface-Wave Modes Along Heterogeneous Open-Boundary Dielectric Waveguides A. Homogeneous 2-d EFIE Describing Transverse Field Dependence of Natural Surface-Wave Modes Along a Heterogeneous, Open-Boundary Dielectric Waveguide of Arbitrary Cross-Section Shape B. Recovery of Source-Free 2-d Wave Equation From 2-d EFIE C. Homogeneous 1-d EFIE Describing Transverse Field Dependence of Natural Surface-Wave Modes Along a Heterogeneous, Planar, Open-Boundary, Dielectric-Slab Waveguide Which Supports a Field Independent of One Transverse Coordinate EFIE SOLUTIONS FOR NATURAL TE AND TM SURFACE-WAVE MODES SUPPORTED BY STEP-INDEX, DIELECTRIC-SLAB AND CIRCULAR-FIBER WAVEGUIDES 3.1 Introduction 3.2 Planar, Step-Index, Dielectric-Slab Waveguide A. Investigation of Possible TE and TM Eigenmodes B. TE-Mode Field Solutions to y-Component EFIE C. TM-Mode Field Solutions to x-Component EFIE 3.3 Multiple-Step-Index, Dielectric-Slab Waveguide 3.4 Step-Index, Circular-Fiber Dielectric Waveguide A. Investigation of Axially-Symmetric TE and TM Eigenmodes B. TE-Mode Field Solutions to ¢-Component EFIE MOMENT-METHOD NUMERICAL SOLUTION FOR TE AND TM MODES ALONG GRADED-INDEX, DIELECTRIC-SLAB WAVEGUIDES 4.1 Introduction iv 32 33 39 41 45 45 46 46 50 54 58 67 69 73 76 76 4.2 Moment-Method Solutions to Slab-Waveguide EFIE A. EFIES for TE and TM Surface-Wave Modes Along Dielectric-Slab Waveguides Described by Discretized Refractive-Index Profiles . Singular-System Characteristic Equation Describing TE and TM Surface-Wave Modes Along Graded-Index, Dielectric-Slab Waveguides 4.3 Natural-Surface-Wave Modes Supported By Graded-Index, Dielectric-Slab Waveguides A. B. Step-Index, Dielectric-Slab Waveguides Graded-Index, Dielectric-Slab Waveguide: Comparison With Characteristic-Vector Method . Triple-Step-Index, Dielectric-Slab Waveguide . Modal Dispersion of Natural-Surface-Wave Modes Along Power-Law, Graded-Index, Dielectric-Slab Waveguides . Frequency Dispersion of Natural-Surface-Wave Modes Along Power-Law, Graded-Index, Dielectric- Slab Waveguides ‘ INTEGRAL-EQUATION DISCRIPTION AND NUMERICALSOLUTION FOR HYBRID, SURFACE-WAVE MODES SUPPORTED BY STEP-INDEX, RECTANGULAR DIELECTRIC WAVEGUIDES 5.1 Introduction 5.2 Rectangular-Waveguide EFIEs and Their Moment-Method Solution A. Coupled EFIEs Describing Hybrid, Surface-Wave Modes Along Step-Index, Rectangular Dielectric Waveguides . Singular-System Characteristic Equation Describing Hybrid, Surface-Wave Modes Along Step-Index, Rectangular Dielectric Waveguides 5.3 Hybrid, Natural, Surface-Wave Modes Supported by Step-Index, Rectangular Dielectric Waveguides 78 78 82 89 90 91 93 95 159 159 160 I60 163 174 VI. VII. INTEGRAL-OPERATOR FORMULATION OF EXCITATION 0F OPEN- BOUNDARY DIELECTRIC WAVEGUIDES 6.1 6.2 6.3 6.4 Introduction Forced 3-d EFIE and Its Field Solutions EFIE Describing Forced, Radiation-Mode Fields A. Representation of Impressed Field B. 2-d and l-d EFIEs Describing Forced Radiation-Mode Eenggse to Single-Spectral-Component Impressed C. TE-Radiation Modes Along Step-Index, Dielectric- Slab Waveguides Excitation of TE Surface-Wave Modes Along Step-Index, Dielectric-Slab Waveguides CONCLUSION LIST OF REFERENCES APPENDIX A SLABMODE (Version 12.3) souace LISTING APPENDIX B RECTMODE (Version 1.4) SOURCE LISTING APPENDIX C PLOTTING ROUTINES APPENDIX D SLABMODE AND RECTMODE INPUT DATA LISTING vi 194 194 195 200 200 204 208 215 225 229 ‘ 232 239 249 254 Table LIST OF TABLES Page Values of ed for n(x) = now/l-a[G(x+.5)]2 -5[c(x+.5)]3 -Y[G(x+.5)]r with a=1, 5=0, Y=100, G = .081396; comparison of numerical results with those obtained by Bahar. 92 Dependence of TE and TM-mode normalized propagation (phase) constants upon mode number, slab electrical thickness (frequenc ), and refractive-index, profile power (step and quadratic) for high-contrast, dielectric-slab waveguides. 97 Dependence of TE and TM-mode normalized propagation (phase) constants upon mode number, slab electrical thickness (frequenc ), and refractive-index profile power (step and quadratic for low-contrast, dielectric-slab waveguides. . 98 vii Figure 2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 3.1. 3.2. 3.3. LIST OF FIGURES Open-boundary waveguide configuration consisting of a heterogeneous core immersed in an infinite, homogeneous cladding medium. Description of heterogeneous core by equivalent sources immersed in the infinite, homogeneous cladding medium. Pillbox configuration at interface surface where 2(3) is discontinuous, utilized to establish existence of equivalent polarization-charge layer there. Scattering of plane-wave field obliquely incident upon plane dielectric interface. Cross-section geometry and refractive-index characteris- tics of an open-boundary dielectric waveguide, consisting of Ns heterogeneous cross-section subregions and Nc subregion interface-boundary contours, immersed in the infinite, homogeneous cladding medium. Cross-section geometry and refractive-index character- istics of a planar, open-boundary, dielectric slab waveguide, consisting of N planar layers infinite along y, immersed in the infinite, homogeneous cladding medium. Cross-section geometry and refractive-index character- istics of a planar, step-index, dielectric-slab waveguide. Cross-section geometry and refractive-index character- istics of a planar, multiple—step-index, dielectric- slab waveguide. Cross-section geometry and refractive-index character- istics of a step-index, circular-fiber dielectric waveguide. viii Page 13 20 24 34 42 47 59 68 Figure Page 4.1. Partitioning and refractive-index discretization for application of method of moments to the graded-index, dielectric-slab waveguide. 79 4.2. Moment-method numerical solution to EFIE for normalized phase constant and field distribution of TEm modes supported by a high-contrast, step-index, dielectric- slab waveguide. 99 4.3. Moment-method numerical solution to EFIE for normalized phase constant and field distribution of TEm modes supported by a low-contrast, step-index, dielectric-slab waveguide. ' 100 4.4. Moment-method numerical solution to EFIE for normalized phase constant and field distribution of modes supported by a high-contrast, step-index, dielectric- slab waveguide. 101 4.5. Moment-method numerical solution to EFIE for normalized phase constant and field distribution of TMm modes supported by a low-contrast, step-index, dielectric- slab waveguide. 102 4.6. Propagation constants of TEm modes supported by high and low-contrast, step-index, dielectric-slab waveguides, where V = (n? - n3)15 kod = constant = 15.695392, obtained by graphical solution methods. 103 4.7. Propagation constants of TMm modes supported by high and low-contrast, step-index, dielectric-slab waveguides, where V = (n: - hi)15 kod = constant = 15.695392, obtained by graphical solution methods. 104 4.8. Comparison of field distributions of TEm modes on a graded-index, dielectric-slab waveguide obtained by numerical solution of integral-operator equation with those determined by Bahar. 105 4.9. Comparison of field distributions of TM,11 modes on a graded-index, dielectric-slab waveguide obtained by numerical solution of integral-operator equation with those determined by Bahar. 106 ix Figure Page 4.10. Triple-step-index, dielectric-slab waveguide refractive-index profile. 107 4.11. Oscillatory and decaying field characteristics of TE0 and TE modes supported by a triple-step-index, dielec ric-slab waveguide, obtained from moment-method solution of EFIE. . 108 4.12. Oscillatory and decaying field characteristics of TE mode supported by a triple-step-index, dielectric-slgb waveguide, obtained from moment-method solution of EFIE. 109 4.13. Oscillatory and decaying field characteristics of TE3, TE4, and TE5 modes supported by a triple-step-index, dielectric-slab waveguide, obtained from moment-method solution of EFIE. 110 4.14. Oscillatory and decaying field characteristics of THO and TM modes supported by a triple-step-index, dieleciric-slab waveguide, obtained from moment-method solution of EFIE. . 111 4.15. Oscillatory and decaying field characteristics of TMz mode supported by a triple-step-index, dielectric-slab waveguide, obtained from moment-method solution of EFIE. 112 4.16. Oscillatory and decaying field characteristics of TM3, TM4, and TM5 modes supported by a triple—step-index, dielectric-slab waveguide, obtained from moment-method solution of EFIE. 113 4.17. Dielectric-slab waveguide power-law, refractive-index profiles. 114 4.18. Variation of normalized phase constant with mode number (modal dispersion) from moment—method solution to EFIE for field of TE911 modes supported by a dielectric-slab waveguide with igh-contrast, power-law, refractive- index profile. 115 4.19. Variation of normalized phase constant with mode number (modal dispersion) from moment—method solution to EFIE for field of T modes supported by a dielectric-slab waveguide with ow-contrast, power-law, refractive- index profile. 116 Figure 4.20. 4.21. 4.22. 4.23. 4.24. 4.25. 4.26. 4.27. Moment-method numerical solution to EFIE for normalized phase constant and field distribution of TEo modes supported by a dielectric-slab waveguide with high- contrast, power-law, refractive—index profile. Moment-method numerical solution to EFIE for normalized phase constant and field distribution of TE modes supported by a dielectric-slab waveguide wi h low- contrast, power-law, refractive-index profile. Moment-method numerical solution to EFIE for normalized phase constant and field distribution of TE modes supported by a dielectric-slab waveguide wiih high- contrast, power-law, refractive-index profile. Moment-method numerical solution to EFIE for normalized phase constant and field distribution of TE modes supported by a dielectric—slab waveguide wi h low— contrast, power-law, refractive-index profile. Moment-method numerical solution to EFIE for normalized phase constant and field distribution of TEz modes supported by a dielectric-slab waveguide with high- contrast, power-law, refractive-index profile. Moment-method numerical solution to EFIE for normalized phase constant and field distribution of TE modes supported by a dielectric slab waveguide wi h low- contrast, power-law, refractive-index profile. Variation of normalized phase constant with mode number (modal dispersion) from moment-method solution to EFIE for field of modes supported by a dielectric-slab waveguide with igh-contrast, power-law, refractive- index profile. Variation of normalized phase constant with mode number (modal dispersion) from moment-method solution to EFIE for field of modes supported by a dielectric-slab waveguide with ow-contrast, power-law, refractive-index profile. Moment-method numerical solution to EFIE for normalized phase constant and field distribution of TM modes supported by a dielectric-slab waveguide wigh high- contrast, power-law, refractive-index profile. xi Page 117 118 119 120 121 122 123 124 125 Figure Page 4.29. Moment-method numerical solution to EFIE for normalized phase constant and field distribution of TMO modes supported by a dielectric-slab waveguide with low- contrast, power-law, refractive-index profile. 126 4.30. Moment-method numerical solution to EFIE for normalized phase constant and field distribution of TM modes supported by a dielectric-slab waveguide wiih high- contrast, power—law, refractive-index profile. 127 4.31. Moment-method numerical solution to EFIE for normalized phase constant and field distribution of TM1 modes supported by a dielectric-slab waveguide with low- contrast, power-law, refractive-index profile. 128 4.32. Moment-method numerical solution to EFIE for normalized phase constant and field distribution of TM modes supported by a dielectric-slab waveguide wigh high- contrast, power-law, refractive-index profile. 129 4.33. Moment-method numerical solution to EFIE for normalized phase constant and field distribution of TMz modes supported by a dielectric-slab waveguide with low— contrast, power-law, refractive-index profile. 130 4.34. Dependence of normalized phase constant upon slab electrical thickness (frequency dispersion) from moment-method solution of EFIE for field of TEm modes supported by a high-contrast, step-index, dielectric-slab waveguide. 131 4.35. Frequency dispersion data of Figure 4.34 with expanded scaling. 132 4.36. Dependence of normalized phase constant upon slab electrical thickness (frequency dispersion) from moment-method solution of EFIE for field of TEm modes supported by a low-contrast, step-index, dielectric-slab waveguide. 133 4.37. Dependence of normalized phase constant upon slab electrical thickness (frequency dispersion) from moment- method solution of EFIE for field of TMm modes supported by a high-contrast, step-index, dielectric- slab waveguide. 134 xii Figure 4.38. 4.39. 4.40. 4.41. 4.42. 4.43. 4.44. Frequency dispersion data of Figure 4.37 with expanded scaling. Dependence of normalized phase constant upon slab electrical thickness (frequency dispersion) from moment-method solution of EFIE for field of TMm modes supported by a low-contrast, step-index, dielectric-slab waveguide. Dependence of TED-mode field distribution supported by a high-contrast, step-index, dielectric-slab waveguide upon its electrical thickness (frequency), obtained from moment-method solution of EFIE. Dependence of TEo-mode field distribution supported by a low-contrast, step-index, dielectric-slab waveguide upon its electrical thickness (frequency), obtained from moment-method solution of EFIE. Dependence of TMo-mode field distribution supported- by a high-contrast,step-index, dielectric-slab waveguide upon its electrical thickness (frequency), obtained from moment-method solution of EFIE. Dependence of TM -mode field distribution supported I by a low-contras , step-index, dielectric-slab waveguide upon its electrical thickness (frequency), obtained from moment-method solution of EFIE. Dependence of normalized phase constant upon slab electrical thickness (frequency dispersion) from moment-method solution of EFIE for field of T -modes supported by a high-contrast, quadratic- ndex, dielectric-slab waveguide. Frequency dispersion data of Figure 4.44 with expanded scaling. Dependence of normalized phase constant upon slab electrical thickness (frequency dispersion) from moment-method solution of EFIE for field of TEm modes supported by a low-contrast, quadratic-index, dielectric-slab waveguide. xiii Page 135 136 137 138 139 140 141 142 143 Figure Page 4.47. Dependence of normalized phase constant upon slab electrical thickness (frequency dispersion) from moment-method solution of EFIE for field of TMm modes supported by a high-contrast, quadratic-index, dielectric-slab waveguide. 144 4.48. Frequency dispersion data of Figure 4.47 with expanded scaling. 145 4.49. Dependence of normalized phase constant upon slab electrical thickness (frequency dispersion) from moment- method solution of EFIE for field of modes supported by a low-contrast, quadratic-index, die ectric-slab waveguide. - 146 4.50. Dependence of TED-mode field distribution supported by a high-contrast, quadratic-index, dielectric-slab waveguide upon its electrical thickness (frequency), obtained from moment-method solution of EFIE. 147 4.51. Dependence of TEo-mode field distribution supported by a low-contrast, quadratic-index, dielectric-slab waveguide upon its electrical thickness (frequency), obtained from moment-method solution of EFIE. 148 4.52. Dependence of TMo-mode field distribution supported by a high-contrast, quadratic-index, dielectric-slab waveguide upon its electrical thickness (frequency), obtained from moment-method solution of EFIE. 149 4.53. Dependence of TMo-mode field distribution supported by a low-contrast, quadratic-index, dielectric-slab waveguide upon its electrical thickness (frequency), obtained from moment-method solution of EFIE. 150 4.54. Moment-method numerical solution to EFIE for normalized phase constant and field distribution of TEm modes supported by a high-contrast, quadratic-index, dielectric—slab waveguide. 151 4.55. Moment-method numerical solution to EF IE for normalized phase constant and field distribution of TEm modes supported by a low-contrast, quadratic-index, dielectric- slab waveguide. 152 xiv Figure Page 4.56. Moment-method numerical solution to EFIE for normalized phase constant and field distribution of TMm modes supported by a high-contrast, quadratic-index, dielectric- slab waveguide. 153 4.57. Moment-method numerical solution to EFIE for nonnalized phase constant and field distribution of TMm modes supported by a low-contrast, quadratic-index, dielectric- slab waveguide. 154 4.58. Moment-method numerical solution to EFIE for normalized phase constant and field distribution of TEm modes supported by a thick, high-contrast, step-index, dielectric-slab waveguide. 155 4.59. Moment-method numerical solution to EFIE for normalized phase constant and field distribution of TMm modes supported by a thick, high-contrast, step-index, dielectric-slab waveguide. 156 4.60. Moment-method numerical solution to EFIE for normalized phase constant and field distribution of TEm modes supported by a thick, high-contrast, quadratic-index, dielectric-slab waveguide. 157 4.61. Moment-method numerical solution to EFIE for nonnalized phase constant and field distribution of TMm modes supported by a thick, high-contrast, quadratic-index, dielectric-slab waveguide. 158 5.1. Cross-section geometry and refractive-index character- istics of a step-index, rectangular dielectric waveguide. 161 5.2. Partitioning schemes for application of moment method to the step-index, rectangular dielectric waveguide. 164 5.3. Normalized propagation (phase) constants for different modes and normalized waveguide thicknesses. lJ,A integral-operator solutions;——-Marcatilli's numerical solutions of the boundary-value problem; --- Goell's numerical solutions of the boundary-value problem. 176 XV Figure Page 5.4. Dependence of E: hybrid-mode field distribution (dominant field iomponent along x), supported by a high-contrast, step-index, rectangular dielectric waveguide upon its electrical thickness (frequency), obtained from moment-method solution of EFIE. 178 5.5. Dependence of E? hybrid-mode field distribution (dominant field lomponent along y), supported by a high-contrast, step-index, rectangular dielectric waveguide upon its electrical thickness (frequency), obtained from moment-method solution of EFIE. 179 5.6. Dependence of E{ hybrid-mode field distribution (dominant field éomponent along x), supported by a high-contrast, step-index, rectangular dielectric waveguide upon its electrical thickness (frequency), obtained from moment-method solution of EFIE. 180 5.7. Dependence of sf hybrid-mode field distribution (dominant field lomponent along y), supported by a high-contrast, step-index, rectangular dielectric waveguide upon its electrical thickness (frequency), obtained from moment-method solution of EFIE. 181 5.8. Dependence of Efl hybrid-mode field distribution (nondominant field component along x), supported by a high-contrast, step-index, rectangular dielectric waveguide upon its electrical thickness (frequency), obtained from moment-method solution of EFIE. 182 5.9. Dependence of E? hybrid-mode field distribution (nondominant fieid component along y), supported by a high-contrast, step-index, rectangular dielectric waveguide upon its electrical thickness (frequency), obtained from moment-method solution of EFIE. 183 5.10. Dependence of E{ hybrid-mode field distribution (nondominant fieid component along x), supported by a high-contrast, step-index, rectangular dielectric waveguide upon its electrical thickness (frequency), obtained from moment-method solution of EFIE. 184 5.11. Dependence of E! hybrid-mode field distribution (nondominant fieid component along V). supported by a high-contrast, step-index, rectangular dielectric waveguide upon its electrical thickness (frequency), obtained from moment-method solution of EFIE. 185 xvi Figure Page 5.12. Dependence of Eél hybrid-mode field distribution (dominant field component along x), supported by a high-contrast, step-index, rectangular dielectric waveguide upon its electrical thickness (frequency), obtained from moment-method solution of EFIE. 186 5.13. Dependence of E31 hybrid-mode field distribution (dominant field component along y), supported by a high-contrast, step-index, rectangular dielectric waveguide upon its electrical thickness (frequency), obtained from moment-method solution of EFIE. 187 5.14. Dependence of E§1 hybrid-mode field distribution (dominant field component along x), supported by a high-contrast, step-index, rectangular dielectric waveguide upon its electrical thickness (frequency), obtained from moment-method solution of EFIE. 188 5.15. Dependence of E5} hybrid-mode field distribution (dominant field component along y), supported by a high-contrast, step-index, rectangular dielectric waveguide upon its electrical thickness (frequency), obtained from moment-method solution of EFIE. 189 5.15. Dependence of Ex hybrid-mode field distribution (nondominant figid component along x), supported by a high-contrast, step-index, rectangular dielectric waveguide upon its electrical thickness (frequency), obtained from moment-method solution of EFIE. 190 5.17. Dependence of 5% hybrid-mode field distribution (nondominant fieid component along y). supported by a high-contrast, step-index, rectangular dielectric waveguide upon its electrical thickness (frequency), obtained from moment-method solution of EFIE. 191 5.13. Dependence of 5% hybrid-mode field distribution (nondominant fieid component along x), supported by a high-contrast, step-index, rectangular dielectric waveguide upon its electrical thickness (frequency), obtained from moment-method solution of EFIE. 192 5.19. Dependence of E‘Y hybrid-mode field distribution (nondominant figid component along V). supported by a high-contrast, step-index, rectangular dielectric waveguide upon its electrical thickness (frequency), obtained from moment-method solution of EFIE. 193 xvii Figure 6.1. 6.2. Page Excitation of step-index dielectric slab waveguide by single-spectral-component, plane-wave field (slab has infinite extent along y and 2 directions). 209 Excitation of step-index, dielectric slab waveguide by line-source current (slab has infinite extent along y and 2 directions). 216 xviii CHAPTER I INTRODUCTION Open-boundary dielectric waveguides indicate that class of waveguiding structures which exploit the mechanism of total internal reflection to guide electromagnetic (EM) waves along surfaces defining permittivity transition. In its most elementary form, an open-boundary dielectric waveguide consists of a relatively uniform cladding region in which is immersed a dielectric core whose per- mittivity provides a positive contrast relative to the background permittivity of the surrounding cladding medium. The significance of the core is that it represents the fundamental heterogeneity region in the dielectric medium of the open-boundary structure by which a surface-wave field along the core-cladding boundary may be guided. Open-boundary dielectric waveguides are utilized Primarily in those high-frequency applications (millimeter-wave or optical) where performance of conventional closed-boundary wave- guides are excessively degraded by conduction power losses. The advent of laser technology and the subsequent development 0f low-loss glasses have promoted the recent, major research activity in the open-boundary dielectric waveguide and device areas. The POtentials of glass fibers for long-distance optical waveguide communications have been studied [1], and practical fiber-optic communication systems have been developed [2, 3]. It is generally agreed that future fiber system design will be dominated by multi- mode transmission and direct detection (photon-couhting) of surface- wave modes supported by glass fibers whose cross-sectional refractive- index variation is never greater than a few percent [1]. Potentials of integrated-optic devices for single-mode, thin-film signal pro- cessing in fiber-optic systems have also been examined [4, 5]. Many useful individual optical devices such as lasers, switches, modulators, filters, directional couplers, and detectors have already been made, and plans to integrate combinations of these functions on single substrates are proposed to provide optical circuits for fiber-optic conmunication systems. Current electromagnetic research problems involving fiber and integrated optics include investigation of those propagation, coupling, and mode-conversion phenomena of EM waves interacting with graded fibers and dielectric devices [1, 4, 5]. Conventional, boundary-eigenvalue analysis of EM wave propaga- tion along open-boundary dielectric waveguides is based upon the wave equation and associated boundary conditions. Exact solutions exist for EM fields supported by infinite, planar, dielectric-slab waveguides (relevant to both millimeterawave and optical integrated circuits) and infinite dielectric fibers possessing a circular cross-section [6-11]. These exact solutions are obtained only for those dielectric waveguides which possess a spatially-invariant or relative-simple-graded dielectric core. Analytical studies of simple dielectric waveguide structures are useful for determining the fundamental characteristics of EM fields supported by arbitrarily- shaped or graded open-boundary dielectric waveguides. Results of these studies demonstrate that, in general, open-boundary dielec- tric waveguides support a finite number of discrete, confined, surface-wave modes augmented by a continuous-spectrum of nonconfined radiation modes. However, specific studies of waveguides possessing noncircular cross-section shapes, such as the rectangular dielectric waveguide, provide no exact solutions for surface-wave or radiation modes, because the boundary conditions of these finite dimension guides cannot be separated in a convenient manner. Differential- operator-based numerical methods have provided approximate solutions to these problems, as demonstrated by Marcatilli [12] and Goell [13]; however, new methods for the investigation of contemporary problems in open-boundary dielectric waveguides and devices are clearly needed. This dissertation describes an integral-operator technique [14, 15] which represents an alternative to conventional boundary- eigenvalue analysis [6-11] for EM wave propagation in relatively- general, open-boundary dielectric waveguides. The integral—operator technique is based upon the observation that the scattering proper- ties of the heterogeneous core of an open-boundary dielectric wave- gUide may be described by equivalent polarization currents which maintain a secondary-induced field in an unbounded cladding medium. Because the equivalent Current is proportional to the total unknown electric field in the core, superposition of impressed and scattered fields leads to an electric-field integral equation (EFIE), describ- ing the unknown field at each point in the core. Solutions to homogeneous specializations of the EFIE describe the natural, guided, discrete surface-wave fields of the waveguide system, while solutions to single-spectral-component impressed field specializations of the forced EFIE lead to the continuous-spectrum, radiation-mode waves supported by the waveguide. Although this integral-operator method was formulated indepen- dently [14, 15], it is not entirely new since a related, but dif- ferent polarization integral equation was developed some time ago in an obscure paper by Katsenelenbaum [16]. Other investigators, primarily Shaw et al. [17] and Kuester et al. [18] have applied variational techniques to Katsenelenbaum's equation to obtain dis- persion curves for surface-wave modes on dielectric waveguides of arbitrary cross-section shape. This research differs from that pursued by other investigators primarily because of its unique identification of polarization surface-charge effects in the integral equation description and the subsequent computation of eigenmode fields described by the integral-operator formalism. The resulting integral- oDerator technique is readily adaptable to numerical-solution methods (yielding dispersion cUrves and field distributions) and its applicability to contemporary problems involving open-boundary dielectric waveguides and devices appears promising [14, 15, 19, 20]. Chapter II presents the basic analysis leading to the develop- ment and application of the EFIE in its various forms. EFIEs specialized to describe natural, guided, axially-propagating, surface- wave modes supported by heterogeneous,open-boundary dielectric waveguides of arbitrary cross-section shape are presented. Applica- tion of the later specialized EFIEs to recover the well-known, transverse field dependence and propagation characteristics of natural surface-wave modes supported by various simple dielectric structures is presented in Chapter III. Numerical solutions of EFIEs describing TE and TM surface-wave modes supported by planar, dielectric-slab waveguides with arbitrarily-graded core refractive- index profiles are presented in Chapter IV. Chapter V extends the numerical method presented in Chapter IV to study hybrid, surface- wave modes supported by a step-index, rectangular dielectric wave- guide. Chapter VI addresses the problem of excitation of open- boundary dielectric waveguides, and the existence of radiation-mode waves in dielectric waveguide structures is established. Forced EFIES describing radiation modes excited by single-spectral-component impressed fields along heterogeneous,open-boundary dielectric wave- guides of arbitrary cross-section shape are discussed. Coupling of surface-wave modes and localized current sources are also_presented. Conclusions and topics for continuing research in open-boundary dielectric waveguides and devices are discussed in Chapter VII. CHAPTER II ELECTRIC-FIELD INTEGRAL EQUATION DESCRIBING HETEROGENEOUS OPEN-BOUNDARY DIELECTRIC WAVEGUIDES 2.1 Introduction An essential tool in the development of an electric-field integral equation (EFIE) description for the field along open- boundary dielectric waveguide structures is the equivalent current- source representation for heterogeneous core regions. In this representation, the heterogeneity of the core region of an open- boundary waveguide structure, which provides a refractive-index contrast relative to the homogeneous cladding in which it is immersed, is replaced by equivalent sources, immersed in an unbounded region of the homogeneous cladding material, which maintain the same scat- tered electromagnetic (EM) field as does the heterogeneous core when it is excited by incident radiation. The equivalent sources consist of induced-polarization currents and charges which act to augment those which would be present in an unbounded cladding when the core is absent. It is through the expedient of these equivalent sources that the development of an EFIE for the unknown field in the core is facilitated. This chapter presents the basic analysis leading to the devel- opment and application of the EFIE in its various forms. Initially, Maxwell's equations are expressed in a fashion appropriate for the study of the EM field in heterogeneous, millimeter-wave, and optical media, introducing an equivalent-current description for the hetero- geneous core of open-boundary dielectric waveguides. Based upon this system of Maxwell's equations for the unknown field in the waveguide core, the three-dimensional (3-d) EFIE is developed; sub- sequent analysis of this equation reveals that it is an inverse- operator to a 3-d Helmholtz wave equation for the unknown field in the waveguide core, obtained directly from the system of Maxwell's equations described. Specialization of the 3-d EFIE for the identi- fication of polarization, surface-charge effects at the core-cladding interface results in an integral equation which is applied to study the scattering of an obliquely-incident, plane-wave field from a plane interface to further establish the validity of the 3-d EFIE. With open-boundary (millimeter-wave or optical) waveguide applications specifically in mind, the 3-d EFIE is subsequently Specialized to describe natural surface-wave modes supported by two types of waveguide structures. In the first specialization, the 3-d EFIE is reduced to a homogeneous 2-d EFIE describing the transverse field dependence of axially-propagating, natural-surface- Wave modes along a heterogeneous,open-boundary dielectric waveguide 0r arbitrary cross-section shape. In this development, special consideration is given to heterogeneous-core waveguides having arbi- trary contours of jump discontinuities in the core refractive-index Profile. For the second specialization, the 2-d EFIE is further rEduced to a homogeneous 1-d EFIE describing the transverse field dePendence of natural-surface-wave modes along a heterogeneous, ilanar-slab, open-boundary dielectric waveguide which supports a ’ield independent of one transverse coordinate. 2.2 Maxwell's Equations Describing EM Field in FfilTimeter-Wave and Optical Media 1. Maxwell's Equations for Field in Heterogeneous Dielectric MediE’ Open-boundary dielettric waveguides are fabricated from milli- ieter-wave or optical media which are nonmagnetic, linear, isotropic, 1nd generally-heterogeneous, low-loss dielectric materials [1, 2, l, 5, 7], Figure 2.1 illustrates the basic configuration of a hetero- 1eneous,open-boundary dielectric waveguide, consisting of a nonmag- Ietic core with spatially varying complex permittivity 5(F) = :'(F) - jc"(F) immersed in a homogeneous cladding with permittivity :c' The (complex) refractive index is related to the waveguide Termittivities by the relations 5(6) = n2(F)eo and cc = "580‘ Max- tell's equations.describing the EM field (E,H) at any point F'in the waveguide medium are [21] + V'LEE) ' De 1 VXE =qwfi an) «9 +8 . + V x H = J + JweE + V°H = 0 ‘ + here (pe,Je) are the impressed electric source densities, satisfy- + '19 the equation of continuity V-Je = -jwoe, which maintain the ++ A field (E,H) along the dielectric waveguide; they represent a athematical model for the active mi llimeter-wave or optical device CLADDING 51?) = EC ("L glJ'O . CORE n 61?) = 5'1?) - je"(?) F #‘Fo Figure 2.1. Open-boundary waveguide configuration consisting of a heterogeneous core immersed in an infinite, homogeneous cladding medium. 10 regions (e.g., a laser system) which provide the ultimate system excitation. Solutions of the system (2.1) provide the discrete, guided, surface-wave modes and continuous-spectrum radiation modes supported by the waveguide structure. 8. Equivalent-Current Description for Heterogeneous Core of Open-Boundary Dielectric Waveguide Solutions of Maxwell's equations for the unknown fields (E,H) of open-boundary dielectric waveguides are complicated by the spatially-varying permittivity in the heterogeneous core. A superposition integral expression for the EM field in such a structure requires a priori knowledge of the natural modal fields of the system so that an appropriate Green's function can be constructed which incorporates the shape and heterogeneity of the core. However, since solutions for the natural modes supported by the waveguiding structure are of central importance in this work, an alternative formulation leading to the solution for the EM field in open-boundary dielectric waveguides is required. The method which is invoked here involves modeling the effects of the heterogeneous core by equivalent currents immersed in an unbounded cladding; EM processes are subsequently described in a homogeneous medium. Specifically, the equivalent currents maintain the same scattered field as does the heterogeneous core when it is excited by incident radiation. Through this formulation, the EM field main- tained by equivalent sources can be expressed using a Green's func- tion for unbounded homogeneous media. Equivalent currents of this type are formulated by Harrington [21] for the purpose of deriving 11 integral equations for the EM field induced in a dielectric body by incident radiation. To introduce an equivalent-current description for the hetero- geneous core of open-boundary dielectric waveguides, cc is added and subtracted from e on the right side of Ampere's law to yield V x‘H = 33 + jw(6c + cc)E where Ge represents the permittivity con- + + trast between the core and cladding, given as 65(r) = e(r) — cc. The significance of Ge is that it represents the fundamental con- trast heterogeneity in the dielectric median of the open-boundary structure which leads to its ability to guide a surface-wave field. Mathematically,65 leads to the following interpretation: T = jwde represents an equivalent conductivity for the heterogeneous core; + when multiplied by E, it leads to an equivalent polarization current + + + Jeq = ijcE = rE which augments those currents which would be present + in unbounded cladding when the core is absent such that V x H = +8 + + + . J + Jeq + jwecE holds. The affect of Je is to maintain a secondary- q induced field which is identical to the field scattered by the core heterogeneity in the actual waveguide. Invoking a similar procedure with the first Maxwell equation of the system (2.1) leads to the expression VJE = [pe - V-(er)]/ec. The second term on the right hand side of this equation identifies + an equivalent polarization-charge density pe = -V-(65E) which is q also active in describing the heterogeneous core. From the above 4. definitions it is evident that Je and pe satisfy the continuity q 4 equation 12 + + V-J = V°(TE) = -jwpe eq (2.2) q’ which allows the heterogeneous core to be described purely in terms of fundamental, equivalent-induced polarization currents. Figure 2.2 and 3; inside the g g ghost-core of the dielectric waveguide, presently described as a illustrates the electrical configuration of pe homogeneous cladding medium. Note that at any point where c is discontinuous (e.g., the core-cladding boundary, described by con- .9. tour F), Jeq has a jump discontinuity and peq describes a surface polarization-charge layer n . It is shown later that the surface eq charge layer plays an important role in field characterization of the dielectric waveguide. The equivalent sources (9 ) may be combined with the 3 _,. eq’eq impressed sources (pe,Je) to yield the total system of sources + (pt’Jt) which maintain the total EM field in the unbounded region of the homogeneous cladding medium with permittivity cc, where .9. Jt q q' .+ the total EM field maintained by the total source densities (pt’Jt) +8 + e = J + Je and pt = p + pe Maxwell's equations describing immersed in the unbounded, homogeneous cladding region are + V-E = pt/E:c . + + V x E 8 -ijOH + + . + i. -(2-3) VXH=Jt+3wecE + V-H = 0 ‘ This system of Maxwell's equations is in the appropriate form-sought; they describe, through equivalent sources radiating in an otherwise 13 CLADDING , / e<'r’> = so . . .for all r // / / z’ / // fi ",z' “~.\\ > . W V \ ,x 7790 l ‘i \ /x x, 1 _* \ ’ / \ Tleq\ / \ \ . \ // \ \ x / \ \KJ \ // + p99 ’/ z \ 7+1“ // \\\ // Jeq=TE j -+ 428C]:=55‘;7'Jex1 Figure 2.2. Description of heterogeneous core by equivalent sources immersed in the infinite, homogeneous cladding medium. 14 unbounded, homogeneous cladding, the EM field in a generally heter- geneous medium. Total sources (pt,3£) consist of impressed sources (pe’3e) which maintain an incident (excitatory) EM field (E1,Hi) in the uniform cladding, augmented by equivalent sources (peq’jeq) which maintain a secondary-induced EM field (E3,Hs)scattered by the heterogeneous core. By linear superposition, the total EM field maintained by (otfit) is E = 5‘ + ES and ii = W + 1713. It is through application of this system of Maxwell's equations that the EFIE for the unknown field supported by a heterogeneous, open-boundary dielec- tric waveguide is developed. 2.3 3-d Electric-Field Integral Equation A. Basic Formulation The electric field maintained by the total source system (pt,3£) in the homogeneous cladding medium with permittivity cc, described by the system of Maxwell's equations (2.3), can be expressed in terms of the EM potential functions (¢,A) as [9] + + E -V¢ - ij + -v f Siptfi'mmii'mv' -jqu I Jt(‘F')e(F|F')dv' V c V 'Z + + ++ + ++ - %;f-~I; {V'-[Jt(r')]VG(rIr') + kEJ£(r')G(rIr')}dV' ( ) 2.4 + where the relation V-Jt = -ju1pt has been invoked. In the above equations, G(F[F') = exp(-jch)/4nR is the Green's function for the 15 scalar Helmholtz wave equation for an unbounded cladding medium defined by + r + 1“ ') (2-5) (v2 + kfimdl'ri') = -6( where R = I}: - WI and the cladding parameters are kc = «or/176's: = nck0 = wavenunber, Zc = VIE/7:: = intrinsic impedance. The integra- tion is extended over the volume V where pt 7‘ O, 3t 1! 0. The last equation of the set (2.4) expresses the electric field E as a linear integro-differential operation on 3t such that E =2i3t} where Q is the appropriate linear operator. Exploiting the relationship 3t = 33 + Seq in this expression then yields E =2ije}+2{3eq}, showing the explicit field contributions by the impressed and equivalent-induced currents, where "‘i + "'e . . . . . E (r) =26.) } = impressed field maintained by primary excita- + tory current Je, and + E56) =${Jeq} = scattered field maintained by secondary, + equivalent-induced current Jeq' Now, 389 = 1E so ES =9irE}, implying that the scattered field E5 is maintained by the total field E. By linear superposition as noted above, the total field is given as E = E1 + E5 or E - E5 = F, which in terms of the linear operator notation becomes E -.Q’{TE}=Ei. The last expression is a linear-operator equation in the unknown field E; replacing Qby the integro-differential operator used in 4. Eq. (2.4) yields a 3-d EFIE in the unknown field E as 16 + + jz + + + + E(r) +—k—E v {V'°[I(r')E(r')]VG(r|r') c h (2.6) + k§t(‘r')'€(‘r')e(r|‘r')}dv' = Eire) . . for all FEVh where Vh denotes the volume space of the heterogeneous core where r f 0. Eq. (2.6) is known as a vector Fredholm integral equation of the second kind because the unknown E appears both inside and outside the integral operator and the limits of integration are independent of the integration variables [22]. In addition, the EFIE is nonhomogeneous because of the non-zero excitation function 5‘. It is important to note that the domain of definition of this equation is restricted to points Eth, inside the core region; this restriction is necessary in order to insure unique solutions for E throughout the entire domain [23, 24]. To compute E fbr points outside the core region, a known solution for E is first substituted into the integrals of Eq. (2.6). The resultant expression is then evaluated at points in the cladding which leads to an expression for E there. Eq. (2.6) is physically interpreted as a manifestation of the basic equation E -'Es = E1, whereES is given by the integral expression in this equation. Examining the scattered field contribu- tion in Eq. (2.6) more closely,ES is seen to be expressed as an integral operation on terms involving {E = Seq and V-(rE) = -jwo eq’ This reconfirms the previously established result that the scattered 17 field is maintained by equivalent-induced polarization current and charge, lending to a valuable physical interpretation of the com- plicated expression in this equation. 8. Recovery of Nonhomogeneous 3-d Wave Equation From 3-d EFIE The validity of the 3-d EFIE may be demonstrated by recovering + a 3-d Helmholtz wave equation describing the unknown E-field in the core of the open-boundary dielectric waveguide from EFIE (2.6), as follows. The incident field E~i is initially expressed in terms -). of the excitatory sources (De, Je) to yield EN?) = -v fv el—c oe(F')o(?|F')dv' (2.7) - jkac fv 3e(r")s(F|F-)dv'. Eq. (2.7) is subsequently exploited in EFIE (2.6) and the resultant expression is operated on by the 3-d Helmholtz operator 5?: V2 + k: to yield, after interchanging the order of integration and differen- tiation and applying Eq. (2.5), V22+5Zc .++-» ++. ( + kC)E +—k— v f v °['r(r')E(r')][- 0 where n1 > "c' It is desired to apply EFIE (2.17) to recover the z-component of the transmitted and reflected fields of E = xEx(x,z) + 2Ez(x,z). Applying the 3-d EFIE (2.17) to the particular geometry of Figure 2.4, Vh (the core region) is taken to be the half—space volume 2 > 0, Sr is the plane-interface surface 2 = O, and the clad- ding region is the half-space z < O, to the left of Sr. In the core, 0(x) = 01 = constant; therefore sz = O, implying no equivalent volume charge such that the first integral in Eq. (2.17) vanishes. Applying these specializations, as well as the assumed forms for Ei and E, EFIE (2-17) becomes xEx(x,z) + zEz(x,z) 2 co co A'Sé-vf f Ez(x',o)e(F|‘r')dx'dy' k -oo .00 C 211.2]: f: f: [SZEX(x-,z') + 2Ez(x',z')]G('FI-F')dx'dy'dz' (AE AE ) -jkc(-xsin6c+ zcosec) = x + z e Ox Oz . . for ze(0,w) (2.18) 26 2 g 2 _ k2 2) _ 2 2 1 c ‘ k0("1 ' "c ° can immediately be simplified by completing the integral over y' where v = Qa/ax + Ea/az and or k Eq. (2.18) evaluated from [26], as f comer .111, f e R or. R = l? - ?'| = [(x - X')2 + (y - y')2 + (z - z')~2]15 ~21..- K0(J'kc[(x - x1)2 + (z - 2021*) -%%muJo-xoz+u-zwfih mam where K0 is the zero-order modified Bessel function of the second kind and H62) is the zero-order Hankel function of the second kind. The last expression in Eq. (2.19) is recognized as the two-dimen- sional Green's function for an unbounded cladding medium with per- mittivity cc [21]. Using the results of Eq. (2.19) in Eq. (2.18) then yields the following two scalar components EFIE's: x-component: A 2 wk x Ex(x,z) - .J %qf goumqch-mew N D 2 oo 00 ’%EA j;ng¢uqucE-mewu' e-jkc(-xsinec + zcosec) Ox . . for ze(O,m) (2.20) 27 z-component: EZ(X.2)- ——f— ,2 f: E ('x .0)K0(J'k Io - o l')dx 2 oo oo - égL‘ffo f.” EZ(X' .2')l§0(.ikcl3 - 5'|)dX'd2' E e-jkc(-xs1nec + zcosec) Oz . for 22(0,w) (2.21) where I}; - E'I = [(x - x')2 + (z - z')2]é with 3= xx + zi and 5‘ = x'§ + 2'2 defined as 2-d position vectors. Examining Eqs. (2.20) and (2.21), it is observed that the x-component EFIE couples both field components Ex and E2, but the z-component equation is independent in E2. Therefore, Ez may be determined directly from EFIE (2.21). A solution for E2 will be sought by representing that unknown field component, in the region 2 > O, by an inverse Fourier-exponential transform as Ez(x,z) = j: j: E 2(6, Mei“;x + nz)df;dn, z > O (Zn)2 (2.22) where E2 is defined to be the two-dimensional Fourier transform of E2 for z > O, in transform variables E and n. given as 5203.71) = f f Ez(x,z)e"](gx + nz)dxdz . (2.23) It is important to note, in order that the integral over z in Eq. (2.23) bewell-defined, Ez(x,z) for z > O is assumed functionally extended over all z, implying Ez(€.n) has meaning only over the 28 region 2 > 0. Substituting the representation (2.22) of EZ for z > O in the scalar EFIE (2.21) and interchanging the order of inte- gration yields,after evaluation of the integrals over x' and z' oo oo 2 . 1 ~ _ Ak jnz (2102 f... f... Ez(€’"){[l (n2 + 2:2 - Rh] 6 2 2 2 i 2 2 t 2 E +mME-k) '45 -k)2 - + Ak [ 2 C ]e C leJEX dgdn E 2k: - kg + jn(€2- k3,)i e-jkc(-xsin6c + zcosec) Oz . fbr ze(O,m). (2-241- To solve Eq. (2.24) for E2 (which leads to the solution for E2, z > O), E is assumed to be separable in E and n such that z EZ(E,n) = F(E)G(n) where F and G are functions of only one variable. Substituting this expression of E2 in Eq. (2.24) yields (m . -jk cosO z jk sine x -%; -/. F(E)I(€.z)ngxdE = EOze c c e c c (2.25) where °° 2 1(592) ' 'zlfl' f 6(0) “:1 " 2 + :5 :2 e u“ n u- C Ak2 £2 + .in(€2- kg? e-(f;2 - k§)*z an a )i + 2 2 2 . 2 2 2kC - kc + Jn(E --kc (2.26) Multiplying both sides of Eq. (2.25) by exp(-j£'x) and integrating over all x leads to the requirement that 29 -jkccosecz . F(E')I(E',z) = EOze 2n6(E' - kCSInOC) . From this result it is concluded that F(E) must be of the form F(g) = 2nC16(E - kcsinec) (2.27) where Cl is a constant, since I(E,z) is continuous at E = kcsinec. Substituting Eq. (2.27) for F(E) in Eq. (2.25) yields C1I(kcsinec,z)= EOZexp(-jkccosecz). Exploiting Eq. (2.26), with E = kcsinec, in this equation then requires C1 m n2 - kzcosze - Ak2 . c c jnz ‘2?’ G(”)(: 22 2 2 . )" d” -°° n - kccos 8c Ely-(3 0° kcsinzec - ncosOc -jkccosecz _ '2n 2 “m 2 d“+%ze ‘0 2kc -m. kccos 6c + ncosec . . for zE(O,w) (2.28) [The presence of the exponential terms exP(jnz) and exp(-jkccosecz), linearly independent in z in the interval (O,w), in Eq. (2.8) leads to the requirement that the integral expressions associated with each exponential vanish independently. The first integral of Eq. (2.28) is in the form of an inverse Fourier-exponen- tial transform, the vanishing of which requires the associated 2 2 2 2 integrand to vanish as n - kccos 6c = Ak 2 2 i(k1 - kc G(n) must be of the form 6(n) = 2nC26(n + [kg - k , implying ii= sinzecfi. From this observation, it is concluded that 2 2 I csin 6c] ) (2-29) 30 where C2 is a constant and only the negative value of n (correspond- ing to forward (+z) traveling wave solutions) is considered. The desired solution for E7- is then obtained from Eqs. (2.27) and (2.29) as Ez(5.n) = F(E)G(n) = (2n)2CEOZ6(€ - kcsinec)5(n+ [kg - kg 5'" 29 cJ ) (2.30) where C = C1C2/E02. To compute the unknown field component E2 (2 > O).EQ- (2.30) is substituted for E2 in the inverse Fourier transform relation (2.22) and the integrals over E and n are carried out to yield Ez(x,z) a CEOzeJkCS1necxe-J(k§ - k§s1n26c)52, z > 0 (2.31) Eq. (2.31) is the expression for a plane wave; the angle at which it is propagating in the x-z plane may be identified by equating the propagation factor of Eq. (2.31) to the propagation factor describing a plane wave traveling at an angle 61 for z > 0, given as exp[-jk1(-xsin61 + zcos61)]. Equating these expressions then leads to the conclusion that nsine1 = ncsinec, implying that the angle 61 at which the plane wave (2.31) propagates is the trans- mission angle which satisfies Snell's law of refraction [25]. The amplitude coefficient C still remains to be determined and it may be obtained by substituting the result of Eq. (2.29) into Eq. (2.28) and evaluating the integral over n to yield, upon invoking Snell's law, 2k2 coseH[(k1/k )cose1 - cose c] C Akz coseH[(k1/k )cosel- cose c] + 1 (2.32) 31 To verify that Eq. (2.32) gives the correct amplitude coefficient, the well-known transmission coefficient T = IEEI/EOX’ where Et(x,z) == E(x,z) for z > O, is recovered from C as fellows. Expressing T in terms of C gives IEtI n cose IEtl n cose T = X g _l. 1 z = C‘_l 1 (2 33) EOx n cose E0 n cose ' ' C C 2 C C Substituting expression (2.32) in Eq. (2.33) and applying the relation An2 I) 2 2 T = 7‘— cos 61 - cos 6c (2.34) nc c (obtained from Snell's law) yields t IEXI chcosec T a = (2.35) E0x nlcosec + nccose1 which is the correct-known [25] result. Ez for points in the cladding (z < O) can be obtained by sub- stituting the known core field Ez(z > 0) into the integrals of EFIE (2.21) and evaluating the resultant expression for z < O. i 2 E2, is thus identified as the scattered field E:(z < O), maintained Because E2 = E + E2, the field reflected from the plane interface, by equivalent-induced polarization current Jze = rEz iri the q core. From EFIE (2.21) and Eq. (2.31), the z-component of the reflected field for z < 0 may be written as 32 r _ s - Ez(x,z) - Ez(x,z) -£?trEZ} 2 0° jk sine x' _ Ak 3 c c . -+ -*. . - 752— f CEOze K0(chlp " p l)dX 21ikc -m 2 co co jk sine x' -jk case 2' Ak c c 1 1 . . . . +7? OImCEOZe e k0(ikc|B-‘o’ |)dx dz (2.36) where Snell's law has been invoked. Evaluating the integrals over x' and z' in Eq. (2.36) then yields jkc(xsinec + zcosec) r = Ez(x,z) REOze where R is a reflection coefficient defined as R g Ak2 kccoseC .IL. cosec(kccoseg + klcosel) - kc .(2 37) 2 klcose1 kc _kécosec(kccosec + klcosél) ' Substituting expression (2.35) for T and applying Eq. (2.34) in Eq. (2.37) reduces R to the well-known [25] form r r [Ell 3 lExl = nlcosec - nccose1 EOz E0x nlcosec + nccose1 Thus, the integral-operator method leads to the correct reflected and transmitted fields, as obtained by relatively conventional methods. 2.5 Homogeneous EFIE for Natural_§urface-Wave Modes Along Heterogeneous Open-Boundary Dielectric Waveguides Open-boundary dielectric waveguides are structures which are capable of guiding surface-wave modes bounded to the core-cladding 33 interface of the structure. Each surface-wave mode supported by the waveguide propagates as exp(:sz) along the axis of the guide, and decays exponentially away from the interface. These modes are defined as those fields which can exist along an open-boundary wave- guide in the absence of excitation, and thus satisfy the homogeneous specialization of the 3-d EFIE (2.6). For an infinite longitudinally- invariant waveguide structure, a travelling surfaceewave field propagating in the :z direction with phase constant B is expressed as E(F) = 3(3)exp(;j82) where 3 is a 2-d transverse position vector in the waveguide geometry being considered. For such applications, the 3-d EFIE may be reduced to a 2-d or 1-d EFIE for the transverse field dependence 3(5) of axially-propagating, natural,surface-wave modes. The solutions of these EFIEs lead to the natural eigenmode 9. fields en and corresponding eigenvalues B" which describe the propa- gation characteristics of the nth guided surface-wave mode along such structures. A. Homogeneous 2-d EFIE Describing Transverse Field Dependence of Natural Sirface-Wave Modes Along a HeterogeneousLOpen-Boundary Dielectric Waveguide of’Arbitrary cross-Section Shape Figure 2.5 illustrates the cross-section geometry and refractive-index profile of an open-boundary dielectric wave- guide, consisting of Ns cross-section subregions immersed in the infinite, homogeneous cladding; In this configuration, the refrac- tive index varies only as a function of transverse position vector '3. such that n = n(B) where 5 = x2 + yy. n(B) is furthermore assumed to be continuous with respect to‘g within each subregion (AS)l, but 34 CLADDI NG . kc) (ec' nc Figure 2.5. Cross-section geometry and refractive-index characteris- tics of an open-boundary dielectric waveguide, consisting of N5 heterogeneous cross-section subregions and NC subregion interface-boundary contours, immersed in the infinite, homogeneous cladding medium. 35 generally discontinuous at the subregion interface-boundary contours P1, 1 §_i 5.Nc’ describing the interfaces between adjacent sub- regions (I‘Nc describes the core-cladding interface), where Nc denotes the total number of such contours. In the special case where there exist no congruent subregion outer boundaries, the set of boundary contours describing the interfaces between the various regions described become closed; for this specialization rz is chosen to describe the outer boundary of (AS)£, for 1: 2,: Ns = Nc' The cross-section configuration considered here is useful for modeling layered waveguide cores of arbitrary cross-section shape having refractive-index profiles with jump discontinuities at interfaces between layers. The presence of internal jump discontinuities in the refrac- . tive-index profile necessarily leads to the existance of equivalent polarization-charge layers along the contour boundaries Pi’ 1 51 _<_ Nc’ in Figure 2.5. To incorporate these new sources main- taining the scattered field, V'(tE) must be evaluated for the core geometry described by Figure 2.5. To begin, the 3-d EFIE (2.6) is specialized to the homogeneous equation appropriate for source- free natural modes with no excitation (E1 = O), and natural surface- wave field solutions of the form E6?) = 3(5)exp(3jBZ) are assumed. At points 35(AS)£, n(S) is a continuous function of 3, and the com- pulation of V-(iE) follows closely the development of Eq. (2.11) to yield .+ -. e -. V°(TE) = v-(t3e*JBZ) = g VT'Ee+JBZ (2.38) 36 where en'r,and'3 are each generally dependent upon 3. At points along the boundary of (AS)£, described by one or more contours Fi’ Iii i NC, v-(rE) is again computed using the interface configura- tion of Figure 2.3, with the understanding that F = Pi in the present analysis. Therefore Eqs. (2-12) and (2.14) are valid as before (with '* +3-sz . ' + ° E = ee );however, in the limit as 69. + O , the intermediate expression of Eq. (2.12) becomes . A + A ++ A _+‘ -.. gir+-o+ S n‘(TE)dS = [n-(tie:)65 - n-(riei)6S]e+382 (2.39) where h is a unit vector normal (in the right-hand sense) to Pi and where r: and 3: are evaluated just to the right (+6) and left (-fi), respectively, of Pi as shown in Figure 2.3. Equating the results of Eqs. (2.39) and (2.14). in the limit as 62 + D” then yields an expression for De = neq6(5 - 51) along Pi, where 51 locates any q point on r1. Combining this result with the continuity equation (2.2) yields the following expression for V (tE) along Pi as Mr?) = [rim-3:) - flirt-Zane - snail“ (2.40) Eq. (2.38) may be combined with Eq. (2.40) to yield a general expres- + sion for Vo(rE) at any point in the core cross-section as v-(TE) = 171553582) = N 5 C A _ -. [£- v.2 + 21616.???) - animus - 3.)]e"JBZ 6 i=1 i i i i (2.41) 37 Eq. (2.41) may be utilized to reduce the 3-d EFIE (2.6) to a homogeneous 2-d EFIE describing the transverse field variation of natural surface-wave modes along a dielectric waveguide with arbitrary cross-section shape as follows. Substitution of Eq.(2.41) in EFIE (2.6), where E = €(3)exp(¥jsz) and E1 = 0. and applying the spatially-dependent wavenumber quantities exploited in EFIE (2.17) yields f whoa-3(5) 65362655.)“. N 3(3):;sz - V[ S 2 + (AV), k (9') £=1 N cf + i=1 SP - ok2(‘5")h' 56(3) 315582" 6(a)?" )ds] 1 [sk2<3'*)fi' to”) l kC - f 6k2(‘f‘)'e’('5')e+sz'G(F|F')dv' = o v M . for all '5 in cs (2.42) where CS denotes the core cross-sections;i locates points to either side (:3 direction) of the interface boundary contour F1; AV2 and SI.i are, respectively, the longitudinal volune space and surface area defined by subregion (AS)£ and contour P1 along 2. Further simplification of Eq. (2.42) is possible by carrying out the integrals over 2', evaluated from [26] as (I)-_ iii (0+ 38 .. -. , .. __ .4ch f e‘l'JBz 6(Fl'rfi)dzi = _411? f e'i'JBZ 9—?— dZ', R = I? - F'l eiJBZ 1 fixoirlo - (2.43) where 72 = 82 - kg defines a transverse wavenumber parameter depend- ing upon 3. Exploiting Eq. (2.43) in Eq. (2.42) leads to a common e+382 factor in all terms, which may be deleted to yield a 2-d vector EFIE for the transverse-field variation of natural surface- wave modes as N . +. .+.+ ++ 1 - . BA 5 vtk (p.).e(p') + + _ I dsl e(o)- '2—(Vt + (dz)[£2:;1 f(AS) k2(p') Kohlo :3 I) NC +12; 451,2, —[6k (‘5' phi-515+) - 613mm -'€(‘6")1K0(rr6 - anew] - 73,—, [CS 6k2('6')3(‘5')l<0(vl'5 - '5' I)ds' = o . for all ‘5 in cs (2.44) where Vt is the transverse gradient operator. It is clear from this integral-operator equation that the surface-wave fields 3, which experience an asymptotic exponential decay with transverse displace- ment into the cladding, require v > O (lossless media), leading directly to the well-known relation 8 > kc for these waves. Solu- tions to the 2-d EFIE (2.44) lead to the natural eigenmode fields En and corresponding eigenvalues B" which describe the propagation 39 characteristics of the nth surface-wave mode along a heterogeneous, longitudinally-invariant,open-boundary waveguide (Hiarbitrary cross- section shape. In the special case where the refractive index is continuous everywhere within the core cross-section except at the core-cladding boundary described by contour F, Eq. (2.44) reduces to ++ 1 - . A V‘Ek2(g').-€(3') + +1 I eip) - .5 (vt + iez)[fcs a?) romp - o |)dS 2'+. - ¢ Mgi fi'-3(3')Ko(vl3 - '5' |)d)1'] T kc 1 2 l I I I - 5,; [cs .. <5 >56 )KOME - 5 INS = o . . for all '5 in cs. ' (2.45) Eq. (2.45) may be applied to formulate the transverse field dependence of natural surface-wave modes supported by many practical longitudi- nally invariant, open-boundary dielectric waveguide configurations. Examples include the commercially-manufactured heterogeneous circular glass fiber and the asymmetric rectangular dielectric waveguide [1,4]. 8. Recovery of Source-Free 2-d Wave Equation From 2-d EFIE The validity of the homogeneous 2-d EFIE may now be demon- strated by recovering the correct source-free 2-d Helmholtz wave equation, applicable to the waveguide configuration of Figure 2.5, from EFIE (2.44). A source-free 3-d Helmholtz wave equation in a 40 generally heterogeneous region may be expressed as V x V x'E = V(V?E) - VZE = sz. Exploiting the first Maxwell equation of the set (2.1) for V-E in this expression yields (V2 + k2)E = V(V5E) = -V(Vk25E)/k2. Subsequent specialization of the last expression to the configuration of Figure 2.5, where k(E) = k(3) = k£(3), for 35(AS)£; E = E(3)exp($j82) and 8/32 = ij then yields the following 2-d source-free Helmholtz equation -(v.c a ie’z‘nvtkii‘h-‘éi‘rin kid?) 3605),, I: 1 _<_ nS (2.46) (vi - r0313) + sigma-s) = where Y2 = 82 - kg and 6k3 = 3 - kg. Alternately, operation on EFIE (2.44) by the 2-d (transverse) Helmholtz operator $1: = (Vi-Y2), obtained as a specialization of Eq. (2.5) appropriate to the defini- + + tion of the 2-d Green's function 9(DID') as SAM-5|?) = (V: - Y2)9(3|3') = -6(3 - '5') (2.47) where 9(313') = (2D)'1KO(Y[3 - E'I), leads to the following result HTS-3‘ )JdS' + + A S V.k2(5' )°E(E') (vi-rim)- (vtiisz)[ 2: t 2 .. i=1 (AS)g k (p') N C + 12161, ignore-aha- -5rs-+) - ok26'-)a--ao'-)1t-oo-5' nor] a i c - fcs 6k2(5')3(5')[-6(5 - +')1—F—+ l/ ELL—2.... Figure 2.6. Cross-section geometry and refractive-index character- istics of a planar, open-boundary, dielectric slab waveguide, consisting of N planar layers infinite algng y, immersed in the infinite, homogeneous cladding me um. 43 integrals over y' in EFIE (2.44) may now be completed, from [26], as 2'; f” Kohl}? - E'Im' =-,'—Ye""‘ ' X' - (249) Exploiting Eq. (2.49) in Eq. (2.44) reduces the 2-d vector EFIE to a 1-d vector EFIE for the x-dependence of natural surface-wave modes along a layered slab-waveguide as N+1 3(x) +—'2 Z [akzixpexixp ZY C I31 ('7- 2- - . .A'le'xi - 6k (x1)ex(x1)][ysgn(x - xi)x : sz]e N e (x') 2 . A A l "' '21- ; f ’2‘ dk (x )[Ysgn(x- x' )xt sz]e'Y|x:j,|dx' Y 131 (AX)1I( (XI) dxl _ . .. ' v a f X 1 I - .21; f ok2(x')e*(x')e'Y"‘ ' X Idx' = o x ____ N+1 . . for xe(xN+l, x1) . (2.50) It is evident from this integral-operator equation (as in the pre- vious case) that the surface-wave fields 3 along the slab waveguide also require y > O for an exponential decay along x into the clad- ding. Solutions to the 1-d EFIE (2.50) lead to the natural eigen- mode fields;n and corresponding eigenvalues 8n which describe the propagation characteristics of the nth surface-wave mode along a heterogeneous,planar, open-boundary dielectric-slab waveguide. 44 In the special case where the slab waveguide consists of only one layer (continuous core region), between x = a and x = b, Eq. (2.50) reduces to 3m + 47 [5kz(a )ex(a) [vngx - a); : is’iie-YIX - a| Zch - 6k2(b)ex(b)[ysgn(x - b); i j82]e'le ' bl:] b e (X') 2 1 ' 1 dk I " ° " " ' ' I 2? fa k;(x') de§1 [vsgn(x - x )x i 38218 ”X x Idx b . ' 7217 f 'é(x')6k2(x')12'YIX ' x ldX' = 0 a . . for xe( a,b). (2.51) CHAPTER III EFIE SOLUTIONS FOR NATURAL TE AND TM SURFACE-WAVE MODES SUPPORTED BY STEP-INDEX,DIELECTRIC-SLAB AND CIRCULAR-FIBER WAVEGUIDES 3.1 Introduction In section 2.4, the validity of the general, inhomogeneous 3-d EFIE (2.17) was confirmed by recovering a well-known, plane- wave solution to the obliquely-incident, plane-wave scattering problem. In the analysis of that problem, the integral-operator method provided information concerning the coupling of the various components of the unknown field, as well as detailing the field contribution from surface polarization-charge layers at the dielec- tric interface. In much the same way, the validity of the homo- geneous 2-d and l-d EFIEs is confirmed by recovery of the well- known, transverse field dependence for natural surface-wave modes supported by relatively-simple waveguide configurations; these solutions are obtainable by relatively conventional differential- operator methods. In this case, however, the coupling between various components of the unknown natural surface-wave fields, as embedded in the EFIEs, can be used to determine the existence of pure TE or TM natural modes. Furthermore, TM modes are distinguished fundamentally from TE modes in that the TM modes possess a field component normal to the core-cladding boundary and thus are 45 46 associated with a surface polarization-charge layer there; TE modes possess no normal component to the boundary and are thus fundamen- tally simpler due to the absence of any polarization-charge layer. The present chapter applies the homogeneous 2-d and 1-d EFIEs to study the transverse field dependence and propagation charac- teristics of natural surface—wave modes supported by various simple open-boundary dielectric structures. Initially, the 1-d EFIE (2.51) is applied to identify and evaluate TE and TM natural modes along a homogeneous (step—index), single-layer, dielectric-slab waveguide. This analysis is followed by a coupled EFIE solution for natural TE modes along a multiple-step-index, dielectric-slab waveguide, based upon EFIE (2.50). As a final consideration, the 2-d EFIE (2.45) is specialized to study angular-independent, natural,TE surface-wave modes supported by a step-index, circular-fiber dielectric waveguide. 3.2 Planare75tep-Index, Dielectric-Slab Waveguide A. Investi ation of Possible TE and . TM Eigenmodes The geometry and the refractive index characteristics of a planar, single-layer, step-index, dielectric-slab waveguide is illustrated in Figure 3.1. To correspond with previous analysis, the waveguide is assumed infinite along both the y and 2 directions, having refractive-index variations only along x. The core region, defined by the interval (Ax)1 = (-d,O), where d is the core thickness along x, is characterized by a uniform refractive index n = n1; outside this interval is homogeneous cladding, characterized by a 47 x / / z / / n=nc I" / - y / x O I 4—+-- / I :(Ax)1 n-n1> nC d | // )(...§LL___3,E J!.... n-nC Figure 3.1. Cross-section geometry and refractive-index character- istics of a planar, step-index, dielectric-slab waveguide. 48 refractive index nc, where n1 > "c' Applying EFIE (2.51) to the planar-slab structure yields the following system of three scalar- component EFIEs x-component: 2 ex(x) +-§E§[:ex(-d+)e'Y(x + d) + ex(0')er] c (3.1) 2 O I - %%7.~/:d ex(x')e'le ' x 1 dx' = O . . for Xe(-d,0) y-component: 2 O - A5... ' ‘le - X'I ' = ey(x) 2Y- f-d ey(x )e dx 0 (3.2) . . for xe(-d,0) z-component: 2 92(X) ”BEA-£2- [ex(-d+)e']‘(X + d) - ex(0')er] c 2 0 .__4k --rlX-X'| “u. 2y f-d ez(x )e dx 0 (3.3) . . for Xe(-d,0) kg - kg. Examination of Eqs. (3.1) through (3.3) reveals that component Eqs. (3.1) and (3.2) are independent in ex and where Ak2 = ey, respectively, while Eq. (3.3) couples ex to e2. The independence of the component equations for ex and ey is fundamental in the 49 identification of pure TE and TM modes supported by dielectric-slab waveguides, as now shown. For TE modes, ez = O; requiring ez to vanish in EFIE (3.3) then yields ex(-d+)e'Y(x + d) - ex(O-)eYx = O, xe(-d,O) . (3.4) The existence of exponentials exp(i-yx) in Eq. (3.4), linearly inde- pendent in the interval (-d,O), requires the coefficients ex(-d+) and ex(O') to vanish independently for satisfaction of this equa- tion. Since there are no boundary conditions requiring ex to vanish at these points, it is concluded from this observation that ex(x)=() for all x, implying E'= yey(x) for TE modes. This result is consis- tent with EFIEs (3.1) through (3.3), which permit ex = O, ey # 0, e2 = O as a possible surface-wave field polarization. TE modes are consequently established as realizable in the step-index, dielectric-slab waveguide. Possible TM eigenmode fields E = 6(x)e;sz requires h2 = O. From the second Maxwell equation of the set (2.1), h2 = O and B/Dy a 0 require ey = 0. Comparing EFIEs (3.1) through (3.3) reveals that ex f O, ey = 0, ez # 0 represents an allowable surface-wave field polarization. From this observation, it is concluded that '3 = Rex(x) + 2ez(x) for TM modes. It is interesting to note that TM modes possess a field com- ponent which is normal to the core-cladding interface, while TE modes possess no such normal component. This is consistent with the observation that the x and z-component EFIEs incorporate field 50 contributions from surface polarization-charge layers at x = -d and 0, while the y-component EFIE does not. This leads to the funda- mental conclusion that TM modes induce, and are partially supported by, surface polarization-charge layers at the core-cladding boundary; TE modes induce no polarization-charge layers and thus are not maintained by surface-charge sources. Thus, the integral-operator method distinguishes TE and TM modes in a very fundamental manner. 8. TE-Mode Field Solutions to y-Component EFIE Solutions of EFIE (3.2) for the TE-mode field-component ey(x) are obtained [14] by exploiting a Fourier-exponential transform representation for ey(x) in the interval (-d,O) as ey(x) = 3—1, f: tyroeji" dE. xe(-d.0) (3.5) where Ey(g) is the (forward) Fourier transform of ey(x), xe(-d,0) defined as . g 0° -j§x Ey(£;) 1:“ ey(x)e dx . (3.6) In order that the integral in Eq. (3.6) be well defined for all integration points, the unknown core field ey(x), xc(-d,O) is assumed functionally extended into the cladding, implying that Ey(g) has meaning only over the interval (-d,O). This assumption is made in order to be consistent with the requirement that the domain of definition of the EFIE be confined to the core region, as discussed in Section 2.3. 51 To apply the transform-solution technique, EFIE (3.2) is first expressed as ey (x) - — e'YX f: ey (x' )eYX dx' +e "f0 e.y (x')eYx dx] =0 (3.7) Substitution of representation (3.5) for ey(x) in Eq. (3.7) and interchanging the order of integration of g and x' in the resultant expression yields, after carrying out the integrals over x', the following integral expression involving Ey(g) on 2 . “-1 f F- (E) [1 - —-—Ak ] ngx dé: 2“ .00 y Y2 + £2 2 . +55ij 2.4.—ed .2. “Tefm’dwi “'8’ (Y 2 A). ELLE). vx=o +[4"T-/::E ’(E) (v2 H+E)d€]e . for xe(-d,0) . The existence of exponential terms exp(jgx) and exp(iyx), linearly 1 independent in x in the interval (-d,0), in Eq. (3.8) leads to the requirement that the integral expressions associated with each exponential vanish independently for satisfaction of Eq. (3.8). The first integral in this equation is in the form of an inverse Fourier-exponential transform, the vanishing of which requires the 2 associated integrand to vanish as v + E2 = Ak2 or (applying Y2 = 82 - kg, Ak2 = k? - kg) E = iK, where K = (k? - 82)i defines a characteristic transversal wavenumber. The result that g be 52 discretely valued, but ey(x) f 0 for all X€(-d,0) in Eq. (3.5) requires Ey(€) = 2W[A5(€ - K) + 85(5 + K)] where A and B are complex constants. Substitution of this equation in Eq. (3.5) yields the well-known [7] oscillatory field solution (subject to the restric- 2 > 0 and Y2 tions K > 0, implying kc < B < k1) ey(x) = Aexp(ij) + Bexp(-ij) = A'cos(Kx) + B'sin(Kx), for xe(-d,0). Exploiting the expression for Ey(£) in the remaining integral terms of Eq. (3.8) and carrying out the integrals over 6 yields the following functional relationship in the unknown amplitude coefficients A and B A[(Y - J'K)e'(Y + jK)d e'Yx + (Y + 3K)eyx] . (3.9) + B[(Y + .in'<)e'(Y ' 3K)d e'YX + (y - jK)er] = o In order that Eq. (3.9) be satisfied for all Xe(-d,0), the coeffi- cients of linearly independent exponentials exp(iyx) in this expres- sion must vanish independently, yielding the following system of homogeneous algebraic equations for the mode-amplitude coefficients (Y2— j2)e‘j“d (y + jK)ejKd' A o = (3.10) (Y + 3K) (Y2' jK) B 0 from which a relationship of B to A is evident. Nontrivial solutions for the field ey(x), such that A f 0, 8 f 0, require the determinant of the coefficient matrix in Eq. (3.10) to vanish, which leads to the well-known [7] characteristic equation for eigenvalues of natural TE surface-wave modes as 53 tan(Kd) = ——2—2-I'<—2— (3.11) (K - Y ) The characteristic transversal wavenumbers y and K are related to the longitudinal phase constant (wavenumber) B by the definitions 72 = (B2 - k2), K = (k? - 82). Therefore Eq. (3.11) is a transcen-' dental equation in B, the solutions of which lead to the characteris- tic eigenvalues 8n and eigenmodes eyn(x) = An[exp(anx) + (Bu/An)exp(-anx)] of the nth guided TE surface-wave mode supported by the core. Solutions for ey(x) in the cladding are obtained by evaluating the scattered field e;(x) for x > 0 and x < -d, maintained by equivalent polarization currents induced in the core by its field ey(x), x€(-d,0). The expression for the scattered field is obtained from EFIE (3.2), yielding 2 0 a s = AL . -Y|X - X'I . < _ > ey(x) ey(x) ZY Id ey(x )e dx , x d, x 0. (3.12) Consider the evaluation of the cladding field in the region x > 0. For this specialization, Eq. (3.12) reduces to yield an exponentially decaying field ey(x) a ey(0+)exp(-Yx) where 2 o + _ Ak . x' . 9y“) ) ' 77 f4; eyb‘ )eY dx (3'13) and eigenvalue solutions of characteristic Eq. (3.11) are chosen to provide 7 > 0. Substituting the known core field ey(x) = Aexp(ij) + Bexp(-ij) and applying the relationship of B to A, 54 given by the second equation of the homogeneous system (3.10), in Eq. (3.13) then yields A + ._._ _- _ 2 _ 2 -yd jKd _ d-jKd ey(0 ) ZYG _ .110 I: J4YK (K Y )9 (e ) (3.14) + jZYKe-Yd(ejKd + e'jKd)] . Substituting an expression for (K2-Y2) yielded by eigenvalue equa- tion (3.11) into Eq. (3.14) then yields ey(o*) = ~j2KA/(Y - jK). Comparing this result with the expression for the core field evaluated at the boundary x = 0', given as ey(0') = A + B = -j2KA/(Y - jK), leads to the correct result ey(0+) = ey(0'), imply- ing tangential 3’15 continuous at the core-cladding boundary. C. TM-Mode Field Solutions to x-Component EFIE Solutions to the x-component EFIE (3.1) for the TM-mode field- component ex(x) are obtained [14] in a fashion similar to that presented for ey(x), through the application of a Fourier-exponential transfbrm representation of ex(x) in the interval (-d,0), given by ex(x) = 711; j... Ex(g)e35" d5, xe(-d,0) ‘ (3.15) where Ex(g) is defined by the forward transform Ex(§) == foo ex(x)e"jgx dx (3.16) an and has the same restrictions stated previously for Eq. (3.6). EFIE (3.1) differs from EFIE (3.2), applied to analyze TE modes, only in the inclusion of terms which incorporate the effects of 55 surface polarization charge at the boundaries x = 0 and x = -d. Such effects are not present for TE modes, and thus represent a characteristic distinction for TM modes, as discussed earlier. To apply the transform-solution technique, EFIE (3.1) is first rewritten in the form 2 ex(x) + A_k§_ [eX(--d+)e-Y(x + d) + ex(0')er] 2k c 2 x _A_k_ ‘YX I YX' I + 2Y [e I-d ex(x )e dx (3-17) 0 . + eYX f ex(x')e'Yx dx'] = 0 . x Substitution of representation (3.15) into Eq. (3.17) and inter- changing the order of integration over g and x' as before yields, after evaluation of the resulting integrals over x', on 2 . 2 on . . +' 91%.;— f Ex“) [ (Y2- .16) + 1?] e-(Y + JEN dgl e-Yx c Y(Y + 52) k 2 m1 . + Ak I5(5)[Er+.1€) +-1—]dg er-o {Tn— 1:” x vhf + 52) kg . . for xe(-d,0). (3.18) Eq. (3.18) again involves exponentials exp(j£x) and exp(iyx), the linear independence of which demands each integral term associated 56 with these exponentials to vanish independently, to permit its satisfaction. Equating the integral associated with exp(jgx) to 2+€2=Ak2 zero leads to the requirement that y , implying g = iK where K - (kg - 82);, as before. Thus, Ex(€) must be of the form Ex(€) = 2fl[A5(€ - K) + 85(5 + K)] which leads to the well-known [7] oscillatory field solution ex(x) = Aexp(ij) + Bexp(-ij) = 2 2 > 0 such A'cos(Kx) + B'sin(Kx) for x€(-d,0), where K > 0 and Y that kc < B < k1, as before. Exploiting the expression for Ex(€) in the remaining integral terms of Eq. (3.18) and carrying out the integrals over 5 yields A {[(v - jK)/Y - Aka/k§]e*Y * 3"“ e'Yx + [(y + 1K)/v + Akz/kEJer} . (3.19) + B {[(y + jK)/Y - AkZ/k§]e-(Y - JK)d e".Yx + [(Y - jK)/Y + AkzlkEJeYX} = o . The linear independence of exp(iyx) once again requires the coef- ficient of each exponential to vanish independently for satisfaction of Eq. (3.19), leading to the following homogeneous system of equa- tions for amplitude coefficients A and B r- ((1 - in) + 15;) e-jKd (11+ jK) + Akz) €1de fl ~01 Y y 2 kC kc . 2 - 2 11 + QK) Ak (Y - 3K) Ak C C (3.20) 57 from which a relationship of B to A for TM modes is evident. Non- trivial solutions for the field ex(x), such that A f 0, 8 f 0, require the determinant of the coefficient matrix in Eq. (3.20) to vanish, which leads to the well-known [7] characteristic eigen- value equation for TM-modes - ZYK taan - K2(nc/n1)2 - y2(n1/nc)2 . (3.21) The solutions to Eq. (3.21) lead to the characteristic eigenvalues 8n and corresponding eigenmode fields exn(x) = An[exp(anx) + (Bn/An)exp(-anx)] of the nth guided TM surface-wave mode supported by the core. An expression for ex(x) at points in the cladding may be obtained from EFIE (2.51) as follows 2 ex(x) = e:(x) = 95? [sgnx ex(0')e"lel 2kc - sgn(x + d)ex(-d+)e'Y|x + dl:] (3.22) 2 0 A - - ' '+ g: ./:d ex(x')e le x I dx', X < ”d: X > 0- At points in the cladding region x > 0, Eq. (3.22) reduces to yield an exponentially decaying field ex(x) = ex(0+)exp(-yx), where 2 2 0' + - é}— ' + Ak I Yx| 1 ex(0 ) - 33 [ex(0 ) - ex(-d )1 +-2Y— _d ex(x )e dx C (3.23) 58 and eigenvalue solutions of the characteristic equation (3.21) are chosen to insure y > 0. Utilizing the known core field ex(x) = Aexp(ij) + Bexp(-ij) and applying the relationship of B to A, given by the second equation of the homogeneous system (3.20), in Eq. (3.23) then yields ex(0+) = Ae'Yd{jK(n1/nc)2[(ejKd + e'jKd) - 2] + %-[-K2<"c/"1>2 + Y2(n1/nc)2]("l/nc)2Y-1(ejKd - e‘jKd)1 x [(y - 12) + 2(21/nc)2 - 21'1 (3.24) Substitution of the expression for K2(nC/n1)2-'y2(n1/nc)2 from eigen- value equation (3.21) into Eq. (3.24) then yields ex(o+) = , -j2K(n1/nc)2A/[(y - jK) + y(n1/nc)2-y]. Comparing this result‘with the expression for the core field ex(0') = A + B = -j2KA/[(Y - jK) + y(n1/nc)2 - y] evaluated at the boundary at x = 0' leads to the result ex(0+)/ex(0')==(n1/nc)2, which confirms the well-established result that normal component of'g is discontinuous at that boundary by the amount ellec = (n1/nc)2. 3.3 Multiple-Step-Index,Dielectric-Slab Waveguide Figure 3.2 illustrates the geometry and the refractive index characteristics of a planar, multiple-step-index, dielectric-slab- waveguide consisting, in this case, of two homogeneous dielectric layers occupying intervals (Ax)1 and (Ax)2 having thickness (d1,d2) and refractive indices (n1,n2), respectively, immersed.in the 59 ,/' z’ 2"! /’ n=nc F // x=O Y 1 --| 4__T__,// X . / X=-dllL(A )1 n rll [‘2 d1 I/ .... I ;.____12..._ l rl I I / |(AX)2 n=n2 d2" / l | / x=-d I F3 Ll / n=nC n12n2>nc Figure 3.2. Cross-section geometry and refractive-index character- istics of a planar, multiple-step-index, dielectric- slab waveguide. 60 infinite,homogeneous cladding, where n1 3_n2 > nc. An application which may be modeled by this configuration is the thin-film, integrated-optics waveguide, which consists of a thin-film glass waveguide deposited on a silicon substrate [27], where the relative thickness of each component is such that d2 >> d1. The surface- wave fields along the multilayer slab waveguide may be described in terms of the individual fields existing in each layer, which are strongly coupled. In terms of the integral-operator method, this leads to a system of coupled EFIEs describing the eigenmode fields supported by this structure. Application of the general l-d EFIE (2.50) to the double- step—index slab waveguide, with d = d1 + d2, Aki = k? - kg, Akg = kg - kg, results in the following system of three scalar EFIES: x-component: _l__ 2 - yx 2 _2+ _ 2 _ - ex(x) + 2k2 { Aklex(0 )e +2 [Aklex( d1) Akzex( d1)] C 'Y|X+dl x sgn(x + d1)e 1 + Akgex(-d+)e]’(x + d) I 0 -_1_ 2 | I ‘YIX'X'I 1= ZY -/:d 5k (x )ex(x )e dx 0 . . for xe(-d,0) (3.25) y-component: 0 1 ey(x) - %%D-/-d 6k2(x')ey(x')e'le ' x I dx' = 0 . . for xe(-d,0) ‘ (3.26) 61 z-component: Ji-2-YX 2_+ ez(x) 1 2k2 { Aklex(0 )e + [:Aklex( d1) c 'le + d1) ' Akgex('d1)] e +A"SEA-cine)” + d) I 0 _ L 2 I | 'YIX " X'l | = 2Y ~/:d 6k (x )ez(x )e dx 0 . for xe(—d,0).' (3.27) Study of Eqs. (3.25) through (3.27) reveals the existence of indee pendent TE modes having 3'= §ey(x) and TM modes with 3'= fiex(x) + yey(x) as before. Solutions EFIE (3.26) are obtained [15] for the case of TE-modes, by exploiting a Fourier-exponential transform representation fbr ey(x) in each layer, defined in intervals (Ax)1 and (Ax)2 as -_i ‘30 .iEx eylm ~21, f.‘y1(€’e 212;. 22(222)1 (3.28) eyzm =71; f Ey2(n)ejnxdn. x2022), where .. m -J‘€x Ey1(£) - f.” eyl(x)e dx (3.29) a a -jnx d Ey2(n) L” ey2(x)e x 62 and (ey , ey ) are assumed functionally extended outside the inter- 1 2 vals (Ax) and (Ax) , respectively, implying (E , E ) represent 1 2 y1 y2 correct Fourier transforms of ey(x) only over those defining intervals. Applying the above definitions of e and ey to EFIE (3.26) 2 yields initially 2 —d Ak 1 _ 2 . 'YlX X'ld . ey(x) 2y -d ey2(x )e x 2 Ak 0 _ 1 u 'le X l I - ZY -d1 ey1(x )e dx 0 . . for xe(-d,0) (3.30) Expressing Eq. (3.30) appropriately in each interval (Ax)1 and (Ax)2 then yields the following set of coupled EFIES in e and 3’1 e y 2 Akg -dl YIX - X | eyl(X)--2'Y— Id ey2(x )e dX 2 Ak 0 _ 1 . -y|x - x'l . = Y - ey1(x )e dx 0 . . . for xe(Ax)1 2 -d Ak -_.2_ l Y'X ' X I d I ey2(x) 274/:d e (x )e x 2 Ak 0 . - 1 e (x')e"Y|x ' x I dx'= 0 . . . for X€(AX)2 . Y -d ’1 1 (3.31) 63 Simultaneous solution of coupled EFIES (3.31) is initiated by sub- stitution of representations(3.28) into Eqs. (3.31), after appro- priate simplification. Interchanging the order of integration and carrying out the resulting integrals over x' then yields the follow- ing set of coupled integral expressions in Ey (g) and Ey (n) 1 2 if”: (a) [1- “‘1 ]ei€xd 221 -.. yl Y2 + £2 5 2 '(Y + J£)d1 Ak 1 e -yx + [4117 1:51“) (v+J€)1d€]e w ' 1 x + [7432? f... 51(5) (Y '355‘1516Y - + ° Akg w (Y Jn)dl _ e'(Y + Jn)d - _. 6 fix = [4m fm Ey2(n) (Y + J'n) dn] e 0 . for Xe(Ax)1 1 a. Akg jnx Efm5y2(n)[1'T—'§Y M ]e do 2 . Ak °° -(Y + MN ._2. 9 -xx i [4117 [0052(2) (y + in) an)"- 2 e(Y - Jn)d1 Ak “ [1,-3 [:52 W «in? 11].)“ 2 (Y - J€)d1 Ak - ..l. e 1 YX = [My 1:5’1 E (g) (Y ' JET dg]e 0 . for X1»:(AX)2 . . (3.32) 64 The coupled equations (3.32) involve, as in the earlier analysis, exponentials exp(jgx), exp(jnx), and exp(iyx), the linear indepen- dence (over (Ax)1, (Ax)2) of which demands each integral term asso- ciated with these exponentials to vanish independently in each equation to permit their satisfaction. Requirement that the inverse— Fourier transform integrals associated with exp(j£;x) and exp(jnx) vanish 2 2 2 _ 2 1 and Y1 + n - Akz, leads to the requirement that Y2 + 52 = Ak implying E = :K and n = to, where K = (k? - 82)* and o = (k§-Bz)§. Satisfaction of Eqs. (3.28) with eyl(x) f 0, ey2(x) f 0 then requires Ey1(§) = 21r[A6(€ - K) + 86(5 + K)] and Ey2(n) =21r[C6(n-o)+D6(n+o)] implying, through defining Eqs. (3.28) ey1(x) = Aexp(ij)+-Bexp(-ij) = A'cos(Kx) + B'sin(Kx) for xe(Ax)1 and ey2(x) = Cexp(jox) + Dexp(-jox) = C'cos(ox) + B'sin(ox) for-xc(Ax)2 where K2 > 0, o2 2 > 0, and v > 0 (implying kc < B < k2) for oscillatory solutions in either region. In addition to the pure oscillatory solutions described,’ the multiple-step-index waveguide supports TE surface-wave modes which are characterized by exponential decay in (Ax)2. For this 2 2 = 82 - kg > o, implying k < B < k ; thus E (n) = 2fi[C5(n -j01) + 06(7) + ja)] and e (X) = specialization, o < 0 or o =ija, where a Cexp(-ax) + Dexp(ox) for xc(Ax)2. The TE-mode characteristic eigenvalue equation and the rela- tionships between mode-amplitude coefficients A, B, C, and D describing the propagation and field characteristics of surface- wave modes supported by the multilayer dielectric-slab waveguide are obtained by substituting the expressions for Ey (E) and E.y (n) 1 2 65 into the remaining integral terms of the system (3.32) and performing the resulting integrations over 5 and n. For surface-wave modes which are oscillatory in (Ax)1, but exponentially decaying in (Ax)2 (a desired system of modal fields along the integrated-optics wave- guide previously described), these computations yield A[(Y - .i)<)e.(Y + jK)d1 e'Yx + (Y + jK)er] +B[(Y + J'I<)en(Y - jK)d1 e'YX + (Y - jK)er] -C(Y _ a)[e'(Y + a)d1 _ e-(y + a)d]e-yx -D(Y + 0t)[e-(Y - a)d1 - e'(Y ' a)d]e‘Yx = o . Xe(Ax)1 A(v + .1'1<)[e(Y - jK)d1 - 1JeYX +B(v - .1'1<)[e(Y + jK)d1 - 1]eYx - d wuy-m;w*awewx+n+ak“ “’lgq d 'D[(Y + oz)e’(Y ' “)d e'Yx + (Y - 01)e(Y + a) 1 er] = o . xe(Ax)2 . (3233) The linear independence of exp(th) in the intervals (Ax)1 and (Ax)2 requires the coefficient of each exponential to vanish independently 66 for satisfaction of Eqs. (3.33), leading to the following homogeneous system of equations for amplitude coefficients A, B, C, and D _-(Y~JK) (y+jK) (Y,a)[e-(y+a)d (Y+a)[e'(Y'“)d‘ x e'(Y+jK)d1 xe-(Y-vJ'KM1 _ e-(v+a)d1 - e-(Y7a)d1] - (WK) - (22.1.2) 0 0 (“Mefiml (1-1K)ejKd1 - (1+a)e'ad1 - (12a)e°‘d1 0 0 (Y’“)e-ad (Y+a)eOld [' - A 0 J L... (3.34) from which relationships between the various mode-amplitudes are again evident. Non-trivial eigenmode field solutions ey (x) for 1 xe(Ax)1 and ey2(x) for xe(Ax)2, requiring A f 0, B i 0, C f 0, and D f 0, then demand that determinant of the coefficient matrix in Eq. (3.34) vanish, which leads (after considerable algebraic manipula- tion) to the following characteristic eigenvalue equation [15] taan = 2YK[(Y2 + a2)tanhad2 + 2Ya] x {(Y2 + K2)(Y2 - a2)tanhad2 - (YZ - K2)[(Y2 + a?)tanha.d2 + zya]}'1. (3.35) 67 To confirm the validity of characteristic equation (3.35), two specializations of the geometry in Figure 3.2 are considered. In the specialization where d2 = 0, the multiple-step-index slab structure degenerates to a single-layer step-index, dielectric-slab waveguide having core thickness d1; taking the limit as d2 +20 in Eq. (3.35) yields the appropriate characteristic equation (3.11) with d = d1. In the second specialization where d2 becomes large, the multilayer dielectric-slab waveguide becomes asymmetric about the interval (Ax)1; taking the limit as d2 +'w in Eq. (3.35) then gives taan 3 M). 1 K23_ Ya which is the well-established [7] characteristic eigenvalue equation for TE-modes along an asymmetric, dielectric-slab waveguide. 3.4 Stengndex,Circular-Fiber Dielectric Waveguide Figure 3.3 illustrates the geometry and the refractive-index properties of a step-index,circular-fiber dielectric waveguide. The core is characterized by a radius a and a refractive index n a n1 > "c’ immersed in the infinite, homogeneous cladding with n . "c' Surface-wave fields propagating as expG-jsz) in this structure are described by the 2-d EFIE (2.45) for waveguides with continuous cores, as specialized to a circular-cylindrical geometry. Recovery of the well-known, axially-symmetric TE modes is sought to further establish the validity of EFIE (2.45). 68 Figure 3.3. Cross-section geometry and refractive-index character- istics of a step-index, circular-fiber dielectric waveguide. 69 A. Investi ation of Axially-Symmetric TE and TMEjgenmodes Application of EFIE (2.45) to the step-index,circular-fiber 'waveguide configuration yields the following system of scalar- component EFIEs as described in circular-cylindrical coordinates (r. ch. 2) r-component: 2 2n e,.(r;¢) + fk—g-g: f er(a.¢')K0(Y|3-3'l)ad¢' nkc 0 2 a 2n $sz [0 f0 [er(r'.¢')cos(¢- 2') + e¢(r' .2')sin(¢ - ¢')]Ko(vl'5 - E'In'dqa'dr' éo . . for re[0,a), ¢€[0,2n] ‘ (3.36) o-component: 2 2n e¢(r,¢)+ A" %—3- f0 e,(a.¢')Ko(Y|3-3'l)ad¢' ZNkE 3¢ Akz a 2n - 7,”— j; jg) [-er(r' ,¢')sin(¢ ' (9') + e¢(r'.¢')cos(¢ - 2')1K0(YI‘6-'5'I)r'd¢'dr' = o . . for re[0,a), ¢€[0,2n] (3.37) 70 z-component: Akz 2n ‘+ + 2.20.2) 2112—; f (MM-mom. - p'))ad¢' 2nkC 0 Akz a 2n + ‘+ '7fjg.h eJW¢W%Wb-WHW®WW=0 . . for re[0,a), ¢€[0,2n] . (3.38) Examination of Eqs. (3.36) through (3.38) reveals component equa- tions (3.36) and (3.37) couple er(r,¢) with e¢(r,¢), while equation (3.38) couples er(r,¢) with ez(r,¢); these equations then lead to a description of ¢-dependent hybrid surface-wave modes of known existence [7, 9]. In the case of axially-symmetric fields, however, er = er(r), e¢ = e¢(r), and ez = ez(r) with the result that Eqs. (3.36) and (3.37) decouple, as now demonstrated. The o-component EFIE for e¢(r) is coupled to the solution for er(r) via the integral-expressions 2n ‘+ .+ .L smu-¢wqhw-pwmw (sw) and 2n [0 Kohl? - E'Im' (3.40) while the r-component EFIE for er(r) is coupled to the solution for e¢(r) through integral-expression (3.39) only. By the law of + cosines. [3-o'l may be expressed as [p - p'l = r2 + r'2 - 2rr'cos(¢ - ¢') . (3.41) 71 Subsequent application of Eq. (3.41) to expression (3.39) reveals that this integral vanishes, since the associated integrand is 2n periodic and odd about the point ¢' = o. Integral (3.40) is evaluated by exploiting the following general angular-integral formula, ’1n(yr')kn(vr), r' < r 2n + + _ ' f0 Kohlo - p'|)cosn¢'d¢' = 21T ( ‘1n(YY‘)Kn(YY"), r' > r (3.42) derived from [26], where In(z) = exp(-jnn/2)Jn(jz) is the modified Bessel function of the first kind of order n 3_0, to yield an expres- sion independent of ¢. Exploiting these results in EFIES (3.36) and (3.37) decouples those equations, yielding the following system of scalar EFIEs describing the axially-symmetric fields as r-component: 2 2n er(r) + 9L2- e,.(a) 333,-.- f Kohl}? - 3' |)ad¢' anc 0 Akz a 2n + + - ~27;- j; dr'r'erh“) f0 d¢'c05(¢ - ¢')K0(Y|D - 9'!) = 0 . . for re[0,a) (3.43) o-component: Akz a I I I 2‘" I I + +I e¢(r) +7;- 1; dr r e¢(r )1; db cos(¢-¢ )Kohrlo-p l)=0 . . for re[0,a) (3.44) 72 z-component: 211 e 20‘) +38?— e (a) f Kohlo - p ')I adcb' "kc 2 _A2__|_<_ fa dr"rez(r")f2fl d¢'K0(flp-p'|)= . for re[0,a) . (3.45) The existence of axially-symmetric TE and TM eigenmodes is now investigated based upon Eqs. (3.43) through (3.45). Requiring ez = 0 for TE modes in EFIE (3.45) demands er(a) = 0 to satisfy this equation, since the remaining integral is non-vanishing. Because there are no boundary conditions requiring er to vanish at this point, it is concluded that er(r) = 0 for all r, implying '3(r) = $e¢(r) for TE eigenmodes. This result is consistent with EFIEs (3.42) through (3.45), which permit er = 0, e¢ f 0, e2 = 0 as a possible surface-wave field polarization. Requiring h2 = 0 (with 3/3¢ = 0) for TM modes, where E = 6(r)exp($sz) demands, from the second Maxwell equation of the set (2.1) (expressed in circular- cylindrical coordinates), eq, = 0. Comparing EFIES (3.43) through (3.45) reveals that er # 0, e¢ = 0, e2 f 0 represents an allowable surface-wave-field configuration; thus 3(r) = ?er(r) + §ez(r) for a TM eigenmode. These results confirm the known existence of axially-symmetric TE and TM eigenmodes along the step-index,circular- fiber waveguide. They also demonstrate, once again, the charac- teristic property that TM modes are associated with a surface polarization-charge layer while TE modes are not. 73 B. TE-Mode Field Solutions to o-Component EFIE The ¢-component EFIE describing axially-symmetric TE modes 3 = $e¢(r) supported by a step-index,circular-fiber waveguide is given by Eq. (3.44). The angular integral in this equation may be evaluated by exploiting Eq. (3.42) with n = 1 to yield the fol- lowing EFIE describing e¢(r) as e¢(r) - Akz [K1(Yr) er'e¢(r-)11(w)dw a + I1(yr) ‘I. r'e¢(r')K1(Yr')dr'] = 0 r . . for r€[0,a). (3.46) Solutions of EFIE (3.46) are obtained [14] by representing e¢(r), re[0,a), by an inverse Fourier-Hankel transfonm of first order, as 2,0) = f; E¢(2)a1(ar)2da . 2210.2) (3.47) where E¢(§) is the first order (forward) Hankel transform of e¢(r), re[0,a), defined as 1545(5) = [on e¢(r)Jl(£;r)rdr (3.43) and where e¢(r), re[0,a), is assumed functionally extended at points r > a, implying E¢(£) has meaning only for the interval re[0,a). Substitution of the representation (3.47) for e¢(r) in EFIE (3.46) and evaluating the resulting integrals over r', after interchanging 74 the order of integration of g and r', yields the following integral expression in E¢(g) fm E (g) [1 - “‘2 ] J (arm: 0 c) T—Y + a2 1 m J K + J ) + [Ak2_l- E¢(g) (Ea) C(66) 1(Ya) (Ya) 1(66)K0(Ya 0 (Y2 + 52) 22141102) = 0 . . . re[0,a). (3.49) The linear independence of J1(gr) and I1(yr) over [0,a) requires the integral expressions associated with these functions to vanish independently, for satisfaction of Eq. (3.49). The integral- expression associated with J1(§r) is in the form of an inverse- Hankel transform of order one; the vanishing of this integral then demands that the integrand itself vanish, implying y? + 52 = Ak2 or g = K, K = (k? - Bz)§, where only the positive value of K need be considered. The result that g = K be discretely valued, but e¢(r) f 0 for all re[0,a) in Eq. (3.47) requires E¢(§) = Ag'16(§-K), leading to the result that e¢(r) = Adl(Kr) for re[0,a), which is the established [7] correct oscillatory field for kc < B < k1. Sub- sequent substitution of the expression for E¢(§) in the remaining vanishing integral of Eq. (3.49) then requires, for A f 0, satisfac- tion of the established [7] characteristic eigenvalue equation J1(Ka) K1(Ya) (mmdai='wuqno ‘lw’ 75 by which TE-modes along a step-index,circular-fiber waveguide are characterized. Specifically, exploiting definitions K = (kg - 82)3 and y.= (B2 - kg)i in Eq. (3.50) yields a transcendental equation which can be solved graphically or numerically for the discrete surface-wave-mode phase constants 80m of the axially-symmetric TE0m modes. CHAPTER IV MOMENT-METHOD NUMERICAL SOLUTION FOR TE AND TM MODES ALONG GRADED-INDEX, DIELECTRIC-SLAB NAVEGUIDES 4.1 Introduction Very-general, homogeneous 2-d and l-d EFIES, describing the transverse field variation of natural-surface-wave modes along heterogeneous, open-boundary dielectric waveguides of infinite lon- gitudinal extent, were presented in Chapter II. These integral equations were subsequently solved by analytical methods to recover the well-established surface-wave fields supported by homogeneous (step-index) dielectric-slab and circular-fiber waveguides. A 1 similar study of these EFIEs applied to graded-index dielectric waveguides is in general extremely difficult; solutions for the natural-surface-wave fields along heterogeneous, open-boundary dielectric waveguides must be approximated. In this case, the integral-operator method proves to be particularly adaptable to numerical and approximate-analytical methods. In this chapter, EFIE solutions are obtained for TE and TM surface-wave fields along planar, dielectric-slab waveguides with arbitrarily-graded core refractive-index profiles by the well-known moment-method [28] technique. Specifically, moment-method numerical solutions are implemented using pulse functions for expansion of 76 77 unknown fields in the dielectric-slab waveguide core and for repre- sentation of its nonuniform refractive-index profile, and delta functions for testing. Discretization of the core refractive-index profile leads to a set of appropriate l-d EFIEs describing the TE and TM surface-wave fields supported by a multiple-step-index wave- guide. Subsequent field expansion substitution and point matching of the set of EFIES complete their discretization and convert them into a homogeneous system of linear algebraic equations for the unknown expansion coefficients; eigenmode fields in the core are subsequently recovered from the pulse-function expansions. The associated cladding field is finally expressed in terms of the polarization currents and charge layers induced by the known fields in the core. . Moment-method numerical solutions of the EFIEs are computed to document the field distributions and dispersion characteristics of graded-index, dielectric-slab waveguides for cases of both large and small refractive-index contrast. Specifically, TE and TM eigen- modes are identified by numerically locating roots to the singular- system characteristic equation, providing the desired eigenvalues and corresponding eigenmode surface-wave-field distributions. These results are compared with those well-known analytical and approximate solutions of surface-waves propagating along step and graded-index, dielectric-slab waveguides obtained previously by other investigators to establish the credibility of the integral-operator-based numerical analysis. Solutions to the singular-system characteristic equation for TE and TM surface-wave modes supported by a low-contrast, 78 triple-step-index (three-layer), dielectric-slab waveguide are pre— sented. Finally, exhaustive case studies of modal and frequency dispersion in graded-index, dielectric-slab waveguides characterized by integer power-law index profiles are presented to demonstrate the usefulness of the integral-operator method as a practical analysis tool. 4.2 Moment-Method Solutions to Slab-Waveguide EFIE « A. EFIES for TE and TM Surface-Wave Modes Along Dielectric- Slab WaveguidesJ Described by Discretized Refractive- Index Profiles The configuration of a graded-index, dielectric-slab waveguide is described again by Figure 3.1, where n = n(x), n(x) > he, and n(x) is assumed to be a continuous function of x, for xe(-d,0). EFIE solutions for TE and TM surface-wave fields supported by this structure are obtained by the moment method, using pulse functions for expansions of the refractive-index profile and unknown fields, and delta functions for testing (point matching). The pulse-function expansion of fields and refractive index in the core of the dielectric- slab waveguide is implemented using the slab partitioning scheme illus- trated in Figure 4.1(a). Application of this geometry yields the following pulse-function representation of n(x) over (-d,0) as N n(x) = £5; nkpk(x) (421) where Mp = total partition number; "k: n(x = Xk), Xk*= centerpoint; and pk(x) = 1 for x€(Ax)k; = 0 otherwise. The effect of the discretiza- tion process described is to represent the graded-index slab waveguide 79 X =0 lflp+1 ND 0 x O k+1 ° Xk [(AX))< Xk x ' . 3 x2 - XL |(Ax)2 x h d - X1 MAX)1 a. Dielectric-slab waveguide partitioning scheme. Eff—Tim __~>x -d '0 b. Discretized refractive-index profile. Figure 4.1. Partitioning and refractive-index discretization for application of method of moments to the graded-index, dielectric-slab waveguide. 80 as a multiple-step-index slab waveguide consisting of N layers, P as illustrated by Figure 4.1(b). The field supported by such a structure is described by a specialization of EFIE (2.50), given as N +1 EIx)+ 1 @122 (”U-22 e(x")] a"): =1 "kex "k "k-1 x k x x ‘le ‘ Xkl [ysgn(x - xk)xisz]e 2 O 1 ' 7kg f <5112(x')'ef(x')e"le ' x I dx' = 0 -d . . for xe(-d,0) (4.2) where Anfi = ni - n2, n3 = nfip+1 = ng, and 6n2(x) = n2(x) - n2 where n(x) is defined by Eq. (4.1). In the preceding expression, the field is observed to be maintained by discretized induced polariza- tion currents as well as induced polarization-charge layers at the refractive index discontinuity points xk, 1.: ngp + 1. However, there are no contributions from induced-polarization volume charge since Vn2 = o in (Ax)k for 1 _<_ k 5 N in EFIE (2.50). In the special case where nk =pnk.1 for some k, xk does not correspond to a discontinuity point and the associated charge term in EFIE (4.2) must be discarded. This may be expedited through the utilization of the following charge-deletion parameter 0: (asso- ciated with normal field-component ex), defined as 81 1 1f nk # nk 1 I)": " Difn —n k k1 D’l‘=1 (4.3) 0" =1. Np+1 Subsequent multiplication of 0: into the first bracketed quantity of the sum in Eq. (4.2) yields an expression which, when rewritten in terms of two summations, may be simplified by a change of index variable to yield the following final form: N ' -le - x I +( + A . " e(x)+--—17§ 25fl10: ex(xk)[YSgn(x - xk)xiJBz]e k mo . A -YIX - X + I - Di+1 e x(xk+1)[ysgn(x - xk+1)x+ JBz]e k 1 } 2 k 0 . - -9- 6n2(x')3(x')e'YIx ' x I dx' = D 27 -d o for XE(-d’0)o (404) Decomposing Eq. (4.4) into its scalar component EFIEs yields x-component: N -le-X| ex(x) +-§%7 2i; Ani[Dt ex(x:)sgn(x - xk)e k c = x _ 'YIX'X+I - Dk+1 ex(Xk+1)Sgn(X ‘ Xk+1)e k 1 ] k2 0 -y|x-x'| ._ --Jl Jf 6n2(x')ex(x')e dx ' 0 2y . for xe(-d,0) (4.5) 82 y-component: ey (x) --—- k§J[: 5n 2(x')ey (x' )e YIX ' x I dx' = 0 . for X€(-d,0) (4.6) z-component: 'YIX ' XkI e z(x) +..i.. An 2[Dx e (x+)e 2Yn2 23%; k k x k - ’YIX ' x + I D12H ex(xk+1)e k 1 J k D _ o 2 . . -ylx - x'l . _ 2y d 6n (x )ez(x )e dx - 0 . for X€(-d,0) . (4.7) Study of Eqs. (4.5) through (4.7) reveals the existence of indepen- dent TE modes having 3 = yey(x) and TM modes with E= S‘Iex(x) + 2ez(x) as before. Furthermore, these equations demonstrate once again that TM modes are associated with surface polarization-charge layers while TE modes are not. This dependence is evidenced upon setting 0: = o for all k in Eq. (4.5); EFIE (4.5)-describing TM modes then reduces exactly to the form Of EFIE (4.6) for TE modes. 8. Singular-System Characteristic Equation Describing TE and TM Surface-Wave Nbdes[Alonggfiraded¥lnde_, D1electric-Slab Waveguides Moment-method numerical solutions to EFIEs (4.5) and (4.6) for TE and TM modes supported by graded-index, dielectric-slab waveguides are initiated by expanding the unknown fields 83 ea(x)(a = x,y) in a pulse-function representation over the core region as N 2am = k): eakkax) (4.8) where eak is the kth expansion coefficient of the unknown field ea(x). Exploiting Eqs. (4.8) and (4.1) in EFIES (4.5) and (4.6) and inter- changing the order of integration and summation then yields expres- sions of the form N -Y|x - x | £1" Makpkb‘) 251111121an [_nz[°k59"(x'xk)e k -le - x + I - D:+lsgn(x - xk+1)e k 1 1 k2 . - .9. e-YIX ' x I dx'] = 0 (4-9) (AX)k where a = x for TM modes and a = y for TE modes,where D}: = 0 for all k denotes the vanishing charge-deletion function associated with transverse field-component e y’ Operation on Eq. (4.9) by the testing-function operator foo [-]6(x - X£)dx then forces Eq. (4.9) to be satisfied at the discrete matching points X2. 1_§ 2.5 N , thus completing the discretization of the EFIES to P yield N linear algebraic equations in emk as p 84 e + (0‘) =0, 1<£ 0 for which det[AéfiIJ vanishes, thus establishing the propagation constant (Bd)n = [(Yd): + (kcd)2]é associated with the nth guided surface-wave mode. Upon equating (yd) and (yd)n in the expression for Aéfi), any one of the emk is assigned unit amplitude and one of the equations in the system (4.13) is discarded to convert this matrix equation into a nonhomogeneous system of (Np - 1) equations for the remaining (Np - 1) expansion 87 coefficients. For example, assigning eaN = 1 and dropping the p last equation of the system (4.13) yields the nonhomogeneous system of equations - (a) A)? eak " ' Amp ° (4.16) The coefficient matrix of the reduced system of equations is assumed to have a non-zero determinant (i.e. rank[A(“)]= - 1 when = (yd)n), thereby allowing the expansion coefficients to be cal- culated by application of standard matrix methods. Subsequent sum- mation of series (4.8) then yields the field distribution of the nth guided surface-wave mode. The field in the cladding region may be obtained by substitut- ing the known core-field solution into EFIE (4.4) and evaluating the resultant scattered field expression at points x < -d and x > 0 in the cladding. Thus the cladding field is again identified as the scattered field maintained by (discretized) polarization currents and surface polarization-charge layers induced by the fields in the core. Solving for the scattered field in the region x > 0 from EFIEs (4.5) and (4.6) yields ea(x) = e:(x) =.e:(0+)exp(-yx) where + + yx _ yx + e:(0 )= .. ‘1'? £1” k[D:ea(xk)e k ' D(l38 1 '6. T l r l I )- -T.5 ‘00 -003 -002 '001 ‘000 001 0-2 003 w nonnn ZED cooRDINnTE (X/d) 89. ' :19- _l C t 853. 2?, 3 74 3 Figure 4.2. Moment-method numerical solution to EFIE for normalized phase constant and field distribution of TEm modes supported by a high-contrast, step-index, dielectric-slab waveguide. 100 P=0 TE-hDDES NCDRE=1.01 NCLRD=1.0 cvx;17.619 40 PHRTIDN £2! a TE-D 111 .77901 0 TE-l 111 .67506 A TE-Z 111.50353 +-TE-3 111.26322 g x TE-4 110.97913 4.3.3; 1940 .0 00 n I I I EY (VOLTS/HETER) -300 -500 -1.0 “500 100 200 3.0 NORHR IZED X COORDINRTE (Xfld) :10“ :7 o O 0 . [ZED -0 0‘0 wonnnL '4L00 3. 1 1-20 L Figure 4.3. Moment-method numerical solution to EFIE for normalized phase constant and field distribution of TEm modes supported by a low-contrast, step-index, dielectric-slab waveguide. - 101 P=0 Th-MDDES NCORE=1.6 NCLRD=100 d/x32 00 40 PRRTIDN A31 E] TM-o :19 .87479 GTM-I =19-159‘77 A TM-Z :17 .95985 +TM-3 =16.21080 g x rn-4 :14 .03633 3; 5’: 3 :8 7 W \ . (D ‘ 1— _ .1 . E’s I. . X LU D 83. ‘ ._ __ - __ ::€:O.E =0 =0 1 =0 0 o 1 —- 0 2 o 3 9; NORM 1.125 x 00030111an [X/d). ' ' 5%: 2?. 8 3. 6 °.‘ Figure 4.4 Moment-method numerical solution to EFIE for normalized phase constant and field distribution of TMm modes supported by a high-contrast, step-index, dielectric-slab waveguide. 102 P:0 TH-HODES NCORE=1.01 NCL80:1.0 40 marrow fig EITH-O 111.77881 OTl'l-l 111-67438 A TH-Z 111 .50220 +-TH-3 111-26634 g x rn-4 11057735 8 . 3' 6‘ 5 ‘ ~ 2 . . :2 ' , . In? 56‘ (D .— 5‘ o r T l I l '-_*' 1 a -600 ’40 -300 -200 -1-0 -O-0 10° 20° 3 .0 a: NORHH IZED X COORD [NRTE (X/d) I10“ 8% ‘ J C t “a ‘99 2?, 5’: 3‘1 Figure 4.5. Moment-method numerical solution to EFIE for normalized phase constant and field distribution of TMm modes supported by a low-contrast, step-index, dielectric-slab waveguide. - 103 o An=0.6 6n=0.01 6:" g p TE (ed),1 (Bd)m 3 c 0 19.92 111.73 9.1 _ 3 1 19.31 111.67 oo"—\Yd/24-73 2 13.33 111.51 TE0 113/2:233 3 16.33 111.27 c, TE] 4;; 4 14.79 110.93 0‘ U "‘ Yd/2=6.67 152 o- 46 TE Yd/2=5.60 : 3 3 O .A‘ N \ U >- O‘ <5 154 Yd/2=3.90 o “a: ‘2 N- °. 0 6'1 r I r I r r I r W 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 3.0 9.0 26/2 Figure 4.6. Propagation constants of TEm modes supported by high and low-contrast, step-index, dielectric-slab waveguides, where V = (n? - ng)¥ kod = constant = 15.695392, obtained by graphical solution methods. 104 c: An=0.6 An=0.01 3 0 19.33 111.77 0‘ 1 19.22 111.66 co \Yd/2=7.70 g 2 18.02 111.46 TM0 _ +’ 3 16.33 111.20 Yd/2-7.27 :9 TM] y 4 14.19 110.90 ,;' -~\\\\\\\\ +3 Yd/2=6.468 o TM2 00‘ 0 TM Yd/2=5.22 N 16' g E °.J Q’ 9 TM Yd/2=3.30 (gJ 0 “:1 O A. ‘3 O" I I l I I I I j 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 Kd/Z Figure 4.7. Propagation constants of TMm modes supported by high and low-contrast, step-index, dielectric-slab waveguides, where V = (ni - n§)¥ kod = constant = 15.695392, obtained by graphical solution methods. 105 BHHRR TE-HGOES NHHX:1.53 NCLRD:1.528522 CVA:40.107048 30 PHRTIDN Ed 21 TE-D 335 .51864 0 75-1 335 .43477 0 TE-2 335 .34399 + TE-3 335 .26349 g XTE-4 335.19171 . Points from Bahar's data 00.0 v 0.40 . 1 ' 002 -305 -000 001 002 003 ‘ '50 X COORDINRTE (Xfid) Figure 4.8. Comparison of field distributions of TEm modes on a graded-index, dielectric-slab waveguide obtained by numerical solution of integral-operator equation with those determined by Bahar. Y (VOLTSIHETERJ .00” '10.” ORHRLIZED E .11. 0-40 A 4L“) 1 ORHRLIZED EX 01:00L13/HETER ) . . 5‘. 1 -1.80 Figure 4.9. 106 BHHRR TH-HODES NHHX=1653 NCLRD=1.528522 d/X:40 .107046 80 PRRTION m Til-0 385 .51860 0 Tl‘I-l 385 .43466 A Tl'l-Z 385 .34881 + TH-3 385 .26328 X Til-4 385 .19158 Bahar's results for TEm and TMm modes are not distinguishable. 0.2 =0.1 -0.0 0.1 0.2 0.3 ZED X COORDINHTE (X/d) Comparison of field distributions of TMm modes on a graded-index, dielectric-slab waveguide obtained by numerical solution of integral-operator equation with those determined by Bahar. 107 REFRRCTIVE-INDEX PROFILE THREE-LRYER SLRB HRVEGUIDE 0.64 l 8 4.0- I I " I I I I I I I >-‘l-O -O.9 -0-8 '0. -0-8 '0.“ -0.4 -O-3 -O.2 '0-1 -0-0 0.1 0.2 0.3 “’ WRHRLIZED X COORDINHTE (Xflj) 89: " Na“! .3 C t “O 0.2 2°" 3. 71 o ‘9 Figure 4.12. Oscillatory and decaying field characteristics of TE2 mode supported by a triple-step-index, dielectric-slab waveguide, obtained from moment-method solution of EFIE. 1&00 1 '0040 ~00 NORHHL [ZED EY ( VOL TS/HETER l I -0 1 Figure 4.13. 110 THREE-LRYER SLRB HRVEGUIDE TE-HODES KC8| :3 a. a. a. 0 \ . - . = . = . = . - .1 = .0 0.1 0.2 0.3 1... IZED x 00030111015 (X/d) GO m 13% .8 E 58. 15?, -l-20 _l_ Figure 4.14. Oscillatory and decaying field characteristics of TMo and TM1 modes supported by a triple-step-index, dielectric-slab waveguide, obtained from moment-method s0lution of EFIE. 112 THREE-LAYER SLAB HRVEGUIDE 111-110055 K3< B 8 ace. )- In G U .1 C: t 8 253 a. 8 0'. 8. '6 -0-6 Figure 4.20. Moment-method numerical solution to EFIE for normalized phase constant and field distribution of TEO modes supported by a dielectric-slab waveguide with high- contrast, power-law, refractive—index profile. 118 TE-D HODES RS Fl FUNCTION OF INDEX PROFILE NHRX:1.01 NCLHD:1.0 4o PRRTION fig Eleo 111.77901 GDP-:1 11152752 A P=2 111.67262 -+P:3 111.7181? .:- o P:S 111374954 3 «9: 2 .‘1‘. lug t:u \— (D .— 3 :>8 Uéq )- In E: HO“ .1 (I 2 3: 2'5 5'1 0.20 I Figure 4.21. -500 -é.0 -I-O -000 W I00 20° 30° NORNRLIZED X CDORDINRTE (X/d) n10" Moment-method numerical solution to EFIE for normalized phase constant and field distribution of TE modes supported by a dielectric-slab waveguide wi h low- contrast, power-law, refractive-index profile. 119 TE-I HODES RS 9 FUNCTION OF INDEX PROFILE NHRX=1.S NCLFID:1.0 d/x=2 .0 4o PRRTION 8d DP=D 19.31960 0P=I 15.69918 A P=2 17.39379 4-P:3 18.10030 .3 X P=4 18.46054 0.. 0P:5 18067127 3 8: E E ms t 0.. \- (a p- 6‘ 158 “a" )- .lIJ Q 33.. 0—6 .J 3 Z a: 33. a. 3. o' 8. .. °. ' ' I I 0‘s -09‘ -603 ‘002 'OOI '000 O'. o o NORHHLIZED x coonomme (X/d) 1 ° 2 ° 3 Figure 4.22. Moment-method numerical solution to EFIE for normalized phase constant and field distribution of TEl modes supported by a dielectric-slab waveguide with high- contrast, power-law, refractive-index profile. 120 TE-l HODES 95 R FUNCTION OF INDEX PROFILE NHRX=I 001 NCLRD:1.0 d/X:l7 .619 40 PRRTION 8d El P=O 111.67506 OP=1 111.15780 AP=2 111.39021 +P:3 111.49129 ‘3 XP:4 111.54423 ...' 0 P:S 11157570 3 3: E 2 m8 = '4 \~ 0) [— a” =>8 '6’. )- In .538. He. .4 Cl: 2 K 2? ad 3 a" 3 . a I .. L'.‘ -s.o -4.o -§.o -i.o -I.o -o.o 110 2.0 3.0 NORHRLIZEO X COORDINRTE (X/dl I110" Figure 4.23. Moment-method numerical solution to EFIE for normalized phase constant and field distribution of TE1 modes supported by a dielectric-slab waveguide with low- contrast, power-law, refractive-index profile. 121 TE-2 MODES RS Fl FUNCTION OF INDEX PROFILE NHHX=I .6 NCLRD:1.D d/X=Z.O 4o PRRTION fig DJ P=D 18 .30209 O P=I 13 .85868 A P=2 15 .43705 + P:3 16.27724 3 X P:4 IS .77152 ..'.‘ 0 P:S 17 .08992 3: 3:. o a? .‘L" we 56‘ a) .- .J 23 ~ s. “out '1 I I I — » ‘ as -003 -002 "OOI '000 Oct 002 003 U NORI‘IRLIZED X COORDINRTE (X/d) no ' U'. 21‘?“ .J C t ‘58 z?‘ ” 9: 8 JJ I Figure 4.24. Moment-method numerical solution to EFIE for normalized phase constant and field distribution of TEZ modes supported by a dielectric-slab waveguide with high- contrast. power-law, refractive-index profile. 122 TE-2 MODES RS 9 FUNCTION OF INDEX PROFILE Mnnx=1.01 NCLHD=1 00 d/X:17.619 4o PRRTION Ed D P=0 111.50353 0P=1 110.90594 AP=2 111.10964 +P=3 111.220“ 3 x P=4 111.28668 .« o 9:5 111.33034 8. 8 5. E u Be 56' (D .— 3 :>8 ’6’] O I I r '_ “J -100 'D-D 10° 200 3.0 83 NORHRLIZED x COORDINATE (X/d) .104 :i- .J a: 1: 26 6'! 29. I 3‘: 7. 8 :J I Figure 4.25. Moment-method numerical solution to EFIE for normalized phase constant and field distribution of TEZ modes supported by a dielectric slab waveguide with low- contrast, power-law, refractive-index profile. 204M) uLoo l | I‘ZoDD 123 HODRL DISPERSION TH-HODES NHQX=I.6 NCLRD=1.D d/x=2 00 4D PRRTION EJP=O OP=I AP=2 +P=3 XP=4 0P=S Figure 4.26. r T r 1 2 3 MODE NUMBER Variation of normalized phase constant with mode number (modal dispersion) from moment-method solution to EFIE for field of modes supported by a dielectric-slab waveguide with igh-contrast, power-law, refractive- index profile. 112.00 ili.oo 111.60 BETH (8d) 131.40 RLIZED 111 2 NORM 1314m 110.80 1 130.80 124 MODRL DISPERSION TM-MODES NMRX:1.DI NCLRD=I.D d/X:I7-619 4O PRRTION IDP=D (9le AP=2 +P=3 XP=4 0P=S .IID.40 O ‘ Figure 4.27. T f I I 1 2 3 4 MODE NUMBER Variation of normalized phase constant with mode number (modal dispersion) from moment-method solution to EFIE for field of modes supported by a dielectric-slab waveguide with ow-contrast, power-law, refractive-index profile. 125 TM-O MODES 93 F) FUNCTION OF INDEX PROFILE NMRX:I .6 NCLRD=I.D d/X:2.D 4o PRRTION £21 £1on 19.87479 OP=1 17.53133 AP=2 18.65182 +P:3 19.04243 3 XP=4 19.22555 .3: @P:5 19.32734 1.00 l NORMHL I ZED EX (VOLTS/METER) 0:40 0.20 _l_ . .0000 I 006 -004 ’003 -D.2 -D.l -0.0 0:1 0.2 0.3 NORMRLIZED x COORDINATE (X/d) Figure 4.28. Moment-method numerical solution to EFIE for normalized phase constant and field distribution of TMO modes supported by a dielectric-slab waveguide with high- contrast, power-law, refractive-index profile. 126 TM-O MODES RS H FUNCTION OF INDEX PROFILE NMRX=I.DI NCLHD=1.0 d/X:I7 0619 40 PRRTION 13d UP:O III-77881 0P=1 III-52554 AP=2 111.67112 '+P=3 111-71695 g XP=4 “1.73746 .3‘ 0P=5 111 .74858 3 9: a? E tug x: ’d \- co I— 5' .=>S .60.. x u D u fléq .J G t a 33. ad 8 a" a a 6 I r I -800 “‘00 -300 -200 “‘00 -000 1.0 200 300 NORMRLIZED X COORDINRTE (X/d) I10" Figure 4.29. Moment-method numerical solution to EFIE for normalized phase constant and field distribution of TM modes supported by a dielectric-slab waveguide witg low- contrast, power-law, refractive-index profile. 127 TM-I MODES RS Fl FUNCTION OF INDEX PROFILE NMRX=I.6 NCLRD:1.D d/X=2-O ' 4o PRRTION g1 EIP=O 19.16977 0 P=1 14 .71858 A P=2 IS .09953 g X P=4 17 .16409 .1‘ 0 P:S 17 .4027? 3 _y 3 32 P." «as. 55~‘ r' U) ' \ .— a‘ :8. / a. a 1_/- o :8. /~ :9 r e / .1 g .’ :26) i/~ a / a“ . 8 . a. I I I I I I I J ‘008 '0 ‘ 01 002 0.3 “003 “002 -001 ‘000 D NORMRLIZED X COORDINRTE (X/d) Figure 4.30. Moment-method numerical solution to EFIE for normalized phase constant and field distribution of TM1 modes supported by a dielectric-slab waveguide with high- contrast, power-law, refractive-index profile. 128 TM-I MODES RS Fl FUNCTION OF INDEX PROFILE NMRX=1.0I NCLRD=I.D d/le7.519 40 PFIRTION Bd EIP=D III-S7438 OP=I 111.15503 A P=2 “138672 + P:3 111-48783 .3.“ 0 P=S 111-S7249 3 8 E E Ins t 'J \— a) p— 6’ :>8 x u; D 33.. _O —l C t K ‘93 3 6’4 8. . a T I I I I T ' i “‘00 -‘.D -300 -200 '100 -000 ‘00 200 300 Figure 4.31. NORMRLIZED X COORDINRTE (X/d) IIID" Moment-method numerical solution to EFIE for normalized phase constant and field distribution of TM modes supported by a dielectric-slab waveguide wit low- contrast, power-law, refractive-index profile. 1.80 .40 I I-ZO 1:90 1 NORMHL I ZED EX IVOLTS/METER) 0.40 1 128 TM-I MODES 95 R FUNCTION OF INDEX PROFILE NMRX:1.0I NCLno=1.o d/x=17.619 4o PRRTION 3d r11 P=0 111.67438 09:1 111.15503 A P=2 “1.38672 + 9:3 111.48783 Ost III-57249 Figure 4.31. T I I r‘ r I ‘ I -4.0 -3.0 -2.0 -1.0 -0.0 1.0 2.0 3.0 NORMRLIZED X COORDINRTE (X/d) n10“ Moment-method numerical solution to EFIE for normalized phase constant and field distribution of TM modes supported by a dielectric-slab waveguide witR low- contrast, power-law, refractive-index profile. 129 TM-2 MODES RS 9 FUNCTION OF INDEX PROFILE NMRX=I-6 NCLno=1.o d/X=2.D 4o marrow 8d EJP=O 17.95985 09:1 12.63791 AP:2 13.26604 +P:3 13.71864 .2‘ 0 P:5 14 .28400 5’: O O 64 \\ :32 ‘\ 56- \ (D a \ l— \\ s =>8 '0. I ' I I I I I x ' 05 -O ‘ / “003 -002 'OOI -000 001 002 003 m . -/ NORMRLIZED x cooRoINnIE (X/d) tag /4 u; . . fl?) ./ a‘ / - l: ’l :2. 2 Figure 4.32. Moment-method numerical solution to EFIE for normalized phase constant and field distribution of TMg modes supported by a dielectric-slab waveguide wit high- contrast, power-law, refractive-index profile. a? .20 0.00 _L A 0.40 A L NORMRLIgEg EX (VOLTS/METER) - , . -0.00 1 -D 020 Figure 4.33. 130 TM-2 MODES RS R FUNCTION OF INDEX PROFILE NMRX=1.DI NCLRD=I.O d/le7.619 4D PRRTION III P=O OP=I A P=2 -+P:3 )(P:4 ‘Ost 8d 111.50220 110.90203 111.10376 111.2136? 111.28008 111.32371 Ml, . \ "T7 -300 T I ‘200 .100 T -D-D I I-0 NORMRLIZED X COORDINRTE (X/d) 2.0 110“ 3.0 Moment-method numerical solution to EFIE for normalized phase constant and field distribution of TM supported by a dielectric-slab waveguide wit E modes low- contrast, power-law, refractive-index profile. 200.00 s;o.oo 1 1 9L IEEBoBEEgoIgd) 240.00 . .pau 40 .00 1 131 FREQUENCY DISPERSION P:D TE-MODES NCORE:1.6 NCLRD=I.D 4O PRRTION IIIM=O 0M=1 AM=2 +M:3 Figure 4.34. Dependence of normalized phase constant upon slab electrical thickness (frequency dispersion) from moment-method solution of EFIE for field of TEm modes supported by a high-contrast, step-index, dielectric-slab waveguide. 132 FREQUENCY DISPERSION P=O TE-MODES NCORE=1.S NCLRD=I.O 4O PRRTION mnzo 011:1 Ar1=2 +r1=3 gnoo l 37.00 I PC" .3... ORMRLIZED BETH 23.00 23.0 N Z: IE.00 I 7.00 Figure 4.35. Frequency dispersion data of Figure 4.34 with expanded scaling. 160.00 J 140.00 _1 I10' 120.00 1 {salon 00.00 1 60.00 L NORMRL I ZED BETH I 00 133 FREQUENCY DISPERSION P=O TE-MODES NMRX:1.01 NCLRD=1.0 4D PORTION EJM=D OM=I AM=2 +M:3 Figure 4.36. O 9 a. v v t r v I v v v r I v v 1 O 0 36.0 70.0 105.0 140.0 cd/x Dependence of normalized phase constant upon slab electrical thickness (frequency dispersion) from moment-method solution of EFIE for field of TEm modes supported by a low-c ontrast, step-index,m dielectric-slab waveguide. 320.00 «'1 40.00 I 134 FREQUENCY DISPERSION P=O TM-MODES NCORE:1.6 NCLRD=1.D 4O PRRTION EJM=O Ole AM=2 +M=3 no.0!) o 0 Figure 4.37. . . . . . , ,e 4.0 0.0 12.0 10.0 (j/X Dependence of normalized phase constant upon slab electrical thickness (frequency dispersion) from moment- method solution of EFIE for field of TMm modes supported by a high-contrast, step-index, dielectric- slab waveguide. 4| .00 J I Pd) 81.00 28.0 L DRMHLIZED BETH 20.00 N 0.00 ".00 l 135 FREQUENCY DISPERSION P=D TM-MODES NCORE=1.S NCLRD:100 4D PRRTION IDM=O OM=1 AM=2 +M=3 .1.00 Figure 4.38. r u T v v 1.0 2.0 3.0 4.0 Frequency dispersion data of Figure 4.37 with expanded scaling. 160.00 #4 I10' 140.00 120.00 I00.00 J_ 1 00.00 4 O l wonnnuzso 0510 (Ed) 80.00‘ 40.00 4 20.00 1 136 FREQUENCY DISPERSION P:D TM-MODES NMRX=1.0I NCLFID=I.D 4O PRRTION IIIM=D 0M=1 AM=2 +M=3 Figure 4.39. Dependence of normalized phase constant upon slab electrical thickness (frequency dispersion) from moment-method solution of EFIE for field of TMm modes supported by a low-contrast, step-index, dielectric-slab waveguide. 137 TE-D MODES RS 9 FUNCTION OF FREQUENCY P=D NCORE=1.6 NCLno=1.o 4o PRRTIDN 8d md/x=.25 2.04466 Ad/le.o 9.73854 +d/>.=2.o 19.91170 3 di=4.0 40.10176 .3 odxk=6.o 80.36280 +d/X=16.o 160.61419 3 3: E :3 m8 2°. \- (D .— 6‘ . >8 we... )- In D :38. no —I C I: “9 3'. a. a G" S ._ o I I I l —‘ ' ~0.8 -0.4 -0.3 -0.2 -0.1 -0.0 0.: 0.2 0.3 NORMRLIZED x CDORDINRTE (X/d) Figure 4.40. Dependence of TED-mode field distribution supported by a high-contrast, step-index, dielectr1c-slab waveguide upon its electrical thickness (frequency), obtained from moment-method solution of EFIE. 138 TE-D MODES RS 9 FUNCTION OF FREQUENCY P=D NCLRD=1 .0 4o PHRTION fig Dd/Xz2.202441 13.90014 Od/Xz4.404882 27.87855 Ad/X=B.809764 55.85120 +d/X:17.619528 “1.77901 3 x d0. :35 .239056 223 .60778 .3“ Od/Xz70.478111 447.24416 +d/X=140 9956223 894.50424 3 ;4 8 E )3 048 ::u \— 0‘) I... 6‘ =>8 '9" )- IL] E8. 0.06.) _J I: 5' 23. e. 8 O" 3. Q ‘00: Figure 4.41. Dependence of TE -mode field distribution supported by a low-contras , step-index, dielectric-slab 0.3 waveguide upon its electrical thickness (frequency), obtained from moment-method solution of EFIE. 139 TM-O MODES RS 9 FUNCTION OF FREQUENCY P=0 NCOREzl .S NCLRD=10O 4o PRRTIDN fig md/X=.25 1.77993 0d/X:.SD 4.33927 3 xd/>.=4.o 40.08891 03‘ 0d/X:8 .0 SD .35781 9d/X:IS.D 160.81180 O O .11 3 “’1 a? :3 "£80 \N I; l I ll! \ . \ NORMRLIZZEOD EX IVOLTS I \- \:7\. . ‘03s \\ 3 \~ ‘ 5+ \\ x ‘\ a 45!!!!-D-r_._ T I I .4 “503 “002 “001 “000 001 NORMFILIZEO X COORDINRTE (X/d) 0.2 0-3 Figure 4.42. Dependence of TMo-mode field distributiOn supported by a high-contrast, step-index, dielectric-slab waveguide upon its electrical thickness (frequency), obtained from moment-method solution of EFIE. 140 TM-O MODES RS 9 FUNCTION OF FREQUENCY P:0 NCORE=I .01 NCLRD=I .0 4D PRRTION 8d IIId/Xz2 .202441 13 .89930 0d/X=4 .404882 27 .87770 Ad/X=8 .809764 55 .85074 -+CVX=I7.619528 111.77881 +d/X=I4O .956223 894 .5044? 3 _11 8 :4 E :3 0.18 ith Z a s '3 x \ E’s, \ x‘= \ LIJ Q 33.. 0—0 _I C t I an 21. a 3 a" 8 a. I '— -O.S -o 4 0.2 0.3 “003 “602 “$01 “000 OoI NORMRLIZED X COORDINRTE IX/d) Figure 4.43. Dependence of TMo-mode field distribution supported by a low-contrast, step-index, dielectr1c-slab waveguide upon its electrical thickness (frequency), obtained from moment-method solution of EFIE. I00 J 40.00 I (fid) . 0 120.00 1 D BETH 190 BgggMflLIZE A 40.00 141 FREQUENCY DISPERSION P:2 TE-MODES NMRX=1.S NCLRD=1.0 40 PRRTION IDM=0 0M=1 Angz 4.11:3 Figure 4.44. Dependence of normalized phase constant upon slab electrical thickness (frequency dispersion) from moment-method solution of EFIE for field of TE modes supported by a high-contrast, quadratic-1ndex, dielectric-slab waveguide. 142 FREQUENCY DISPERSION P=2 TE-MODES NMFIX=1.6 NCLRD=I.0 40 PRRTION IDM=O 0M=1 AM=2 +M=3 3§.00 49.00 J QJI‘PC” 30.00 A 1.55.9. 8“. .I‘g'ém 1 $0.00 5.00 L 2:0 ' . ' 3.0 ' ' 4.0 d1/X .‘ o .. a Figure 4.45. Frequency dispersion data of Figure 4.44 with expanded scaling. I00.00 I40.00 1 010 l .oan" (éfigLMI l NORMRLIZED BE “L00 40.00 J 143 FREQUENCY DISPERSION P=2 TE-MODES Nnnle 001 NCLRD=1.0 4D PRRTION EIM=O 0M=I AM=2 +M=3 Figure 4.46. Dependence of normalized phase constant upon slab electrical thickness (frequency dispersion) from moment-method solution of EFIE for field of TEm modes supported by a low-contrast, quadratic-index, dielectric-slab waveguide. 144 FREQUENCY DISPERSION P:2 TM-MODES NHHX:I .6 NCLRD=I.D 4D PRRTION IIIMzD OM=1 AM=2 +M=3 Figure 4.47. Dependence of normalized phase constant upon slab electrical thickness (frequency dispersion) from moment-method solution of EFIE for field of TMm modes supported by a high-contrast, quadratic-index, dielectric-slab waveguide. 145 FREQUENCY DISPERSION P=2 TM-MODES NMRX=I.S NCLRD:1.D 4D PRRTION EIM=O 0M=I AM=2 +M=3 40.00 36.00 J 359. 351?...‘119‘1’39... MEL; I391} 1" Y T r V 1' f r 0' Y Y Y ‘r 1 I “000 I00 200 300 ‘00 (d/X ‘ Figure 4.48. Frequency dispersion data of Figure 4.47 with expanded scaling. 146 FREQUENCY DISPERSION P=2 TM-MODES NMRX=I.DI NCLRD=I.O 4D PRRTION EIM=0 0M=1 AM=2 +M=3 A 010' 140.00 I " ‘i‘gl... A NORMRLIZED BET 40.00 J ' I 140.0 Figure 4.49. Dependence of normalized phase constant upon slab electrical thickness (frequency dispersion) from moment- method solution of EFIE for field of TMm modes supported by a low-contrast, quadratic-index, dielectric-slab waveguide. 147 TE-D MODES 95 F1 FUNCTION OF FREQUENCY P=2 Nflnx=106 NCLRD:I.D 4o PnRIIoN _gg 0d/X=.SD 4.20012 Ad/X=1 .0 9.18280 -rd/X:2.0 19.23299 3 di=4.0 39.33323 .3.“ Od/X=B.D 79.52201 +d/X=IS.O 159-91401 3 ;-I 3. ' 32 E 1.1.13 2 °.. \— (D I... S =>3 U6. )- DJ D 213., 0-06 _I C 2 K O 2 '4 a 8 a" 3. .. a I __ fi _— “008 “004 -003 002 003 -0.2 -o.1 -o.o 0.1 NDRMHLIZED x COORDINATE (X/d) Figure 4.50. Dependence of TED-mode field distribution supported by a high-contrast, quadratic-index, dielectric-slab waveguide upon its electrical thickness (frequency), obtained from moment-method solution of EFIE. 148 TE-O MODES RS 9 FUNCTION OF FREQUENCY P:2 nnnx=1.01 NCLRD=1.0 40 PRRTIDN Bd Edna-2.202441 13 .87513 0d /)1=4 .404882 27 .82282 Ad/x-.-8 .809764 55 .76694 +d/x-.-.17.619528 111.67262 g xdl).=35 .239056 223 .48517 .:~ cod/>670 .478111 447 . 10882 +d/>.=140.966223 89485868 2 3: 22 U .- 3g. ~.-T u: '— a‘ =>8 wag )- 1.1.1 Q. 0-09 .1- C I: o: 3% a. a 61 3. a _ .. - “00' “004 “003 002 003 Figure 4.51. “002 “001 “000 001 NORMRLIZED X COORDINRTE IXAd) Dependence of TEo-mode field distribution supported by a low-contrast, quadratic-index, dielectric-slab waveguide upon its electrical thickness (frequency), obtained from moment-method solution of EFIE. 149 TM-D MODES RS 9 FUNCTION OF FREQUENCY P=2 NHRX=I 06 NCLRD=100 4O PRRTION md/X= 025 Od/x= 050 Ad/le .0 +CI/X22 00 8 Xd/Xz4.0 0;. Od/xza00 +d/X=IS 00 3 a “:1 E E 0.18 t '- \0-0 0) I- 6’ =>3 ~06. x Ill-I D #01 FIG .4 C 2 a ?33 a. 3 a" 3 O l “00‘ -0 4 “O03 “002 “001 . ~040 00 NORMRLIZED X COORDINRTE IX/d) t £22 1.66484 3.72127 6.67100 16.66162 38.77875 76.96760 159.34414 002 003 Figure 4.52. Dependence of TMo-mode field distribution supported by a high-contrast, quadratic-index, dielectric-slab waveguide upon its electrical thickness (frequency), obtained from moment-method solution of EFIE. 1.00 1 (VOLTS/I‘IETER) NORflflLgagp EX 1 150 TH-O NOOES 93 R FUNCTION OF FREQUENCY P=2 Nnax=1.01 NCLRO=1.0 40 PRRTION _§_d_ Eld/x=2 .202441 13 .87445 0d/x:4.404882 27 .82150 Ad/X=6 .809764 55 .76540 +d/).=17.619528 111.67112 xd/xzas .239056 223 .48370' od/x=70 .478111 447 . 10733 1» d/>.=140 .956223 894.35714 ._ —. -_— Figure 4.53. 006 -O.4 -003 '002 ‘00‘ ‘000 00‘ 002 003 NORHRLIZEO X COORDINRTE (X/dl Dependence of TMo-mode field distribution supported by a low-contrast, quadratic-index, dielectric-slab waveguide upon its electrical thickness (frequency), obtained from moment-method solution of EFIE. 151 P:2 TE-NODES NHRX:1.8 NCLRO=1.0 d/X=2.0 40 PRRTION fig 01 TE-O 19.23300 0 TE-l 17 .39379 A TE-Z 15.43705 +-TE-3 13.43504 8 8 5’? °< 01- 40 I “0.2 ‘061 -000 001 N RHRLIZEO X COORDINRTE (X/d) .(L00 0 ( 0-2 0.3 ‘OOQO _1_ NORNRLIZEO EY (VOLTS/HETER) 1 Figure 4.54. Moment-method numerical solution to EFIE for normalized phase constant and field distribution of TEm modes supported by a high-contrast, quadratic-index, dielectric- slab waveguide. 152 P=2 TE-HOOES NHRX=1.01 NCLRD=1.0 40 PRRTION fl [5 TE-O 111-67262 GTE-1 111-39021 A TE-Z 111-10964 +-TE-3 110.84321 3; 8 0.40 l lVOLTS/NETER) . 0.00 A. . ‘1 ’00‘0 l ORNRLIZED EY N -1L00 3.0 Figure 4.55. Moment-method numerical solution to EFIE for normalized phase constant and field distribution of TEm modes supported by a low-contrast, quadratic-index, dielectric- slab waveguide. 153 Figure 4.56. P=2 rn-nooes NHQX=1.6 NCLRO:I.0 d/X:2.0 40 PRRTION 3d [I‘m-0 18.66162 0 ”H 16.09963 “ ”-2 13.26604 3 8 o'- 8 O" a? I.“ :2 . £61 (.0 p— 5‘ =>8 ‘60. x LU Q #33 go- 45.3 -O.2 -6 1 _6 0 art ‘-‘”' - go NORHRLIZED x 000120111an 1x/d)' 0.2 0.3 32’. 2 65- O 0 6d Moment-method numerical solution to EFIE for normalized phase constant and field distribution of TMm modes supported by a high-contrast, quadratic-index, dielectric- slab waveguide. 154 F=2 TH-HOOES NNRX=1-01 NCL90=100 40 PRRTION gd ETH-O 111-67112 O TH-l 111-38672 4 "1'2 111-10376 'fTH-3 110-83728 3: 3 LTS/flETERl 0.40 X (V0 .(L00 0 3.0 N RHHLIZED X COORDINFITE (X/d) [ZED E -O.40 1 NORHRL -O.BO l ~l.20 A Fi ure 4.57. Moment-method numerical solution to EFIE for normalized 9 phase constant and field distribution of TMm modes . supported by a low-contrast, quadratic-index, dielectr1c- slab waveguide. 155 Figure 4.58. P=O TE-NOOES NCORE=1.6 NCLRO=1.0 d/X31660 40 PHRTION 8d El TE-O 160.81420 O TE-l 160371340 A TE-Z 160.84363 +-TE-3 160.30572 3 _1 8 .31 6 ' 51 22 :1." us: 5:1 10 h- E3 \ :>8 ~ \ BC. I T r T > “'08 -OJ '0': ’00! -000 0.1 0.2 10.3 “1 NOR LIZED X COORDINHTE (X/d) an: 1a? m. .4 a: -z: ‘53. 12?, 5': 71 O 9 Moment-method numerical solution to EFIE for normalized phase constant and field distribution of TEm modes supported by a thick, high-contrast, step-index, dielectric-slab waveguide. «n40 J 156 P=0 TH-HODES NCURE=1.5 NCLRU=1.0 d/X=16.0 40 PHRTION fihfl EITf‘l-O 160.81180 0 "1‘1 ISO-704.73 5 "1'2 160.52376 +'TH-3 ISO-27033 flfll. 15E}: EX I VOL IS/HE FER ) NOR f9.lfl ‘O 0.0 l— I -0.3 -O.2 -O.l -0.0 ‘ 0.! NOR nLtzeo x cooaotnnrc 1xnd1 I 0.2 0-3 Figure 4.59. Moment-method numerical solution to EFIE for normalized phase constant and field distribution of TNm modes supported by a thick, high-contrast, step-index, dielectric-slab waveguide. 157 P:2 rE-nooes NHRX:1.5 NCLRD=1.0 d/x=1600 40 66611011 _8__q GTE-0 159.91401 OTE-l 156.03404 A TE-2 156.14649 + rs-a 154.26153 3. 8 8 64 32 'AJ 52 561 U) .- a’ =>8 U6 8 0 t 0 3 0 >. C O - o - o ' 02 “001 -0.0 Oct 002 O. u T NORNRLIZEO x 000110111an (X/d) 3 33 2 d _l a: t 86 we 25‘ I «‘3: 74 8 éJ Figure 4.60. Moment-method numerical solution to EFIE for normalized ‘ phase constant and field distribution of T modes supported by a thick, high-contrast, quadra ic-index, dielectric-slab waveguide. . 158 P:2 TH-HODES NHHX=1.5 NCLHO:1.0 d/X:16.0 4O PHRTION Aflg OTI‘l-l 157.0631! 4 Tl‘l-2 153.35250 'rTH-3 151.36195 3 9: 0.40 A EX (VOLTS/HETER) NORHHLIZEO +9.60 ~0J0 A T! -|o80 1 Figure 4.61. Moment-method numerical solution to EFIE for normalized phase constant and field distribution of TMIE modes supported by a thick, high-contrast, quadra ic-index, dielectric-slab waveguide. CHAPTER V INTEGRAL-EQUATION DESCRIPTION AND NUMERICAL SOLUTION FOR HYBRID, SURFACE-WAVE MODES SUPPORTED BY STEP-INDEX, RECTANGULAR DIELECTRIC WAVEGUIDES 5.1 Introduction Planar, dielectric-slab waveguides provide geometrically- simple structures from which fundamental characteristics of guided surface-wave (and radiation) modes supported by open-boundary dielec- tric waveguides may be deduced analytically.’ These simple structures, having one infinite transverse dimension, are not suitable for prac- tical implementation; relatively complicated structures characterized by cores possessing finite transverse dimensions are almost always required [4, 5, 7]. Unfortunately, standard differential-operator techniques provide no exact solutions for surface-wave modes sup- ported by rectangular dielectric waveguides, even for those struc- tures characterized by uniform, step-index profiles. The primary problem presented by this waveguide structure follows from its charac- teristic finite dimensions: tangential-field boundary conditions at the core-cladding interface cannot be conveniently separated, thus rendering a conventional boundary-eigenvalue analysis with differential operators ineffective. However, this problem is cir- cumvented by the integral-operator analysis, which embeds satisfac- tion of these complex boundary conditions in its fundamental 159 160 formulation. This method thus provides a conceptually-exact formula- tion whose solution may be extracted by appropriate numerical methods. In the following development, solutions to a system of two coupled 2-d EFIEs describing hybrid, surface-wave modes supported by step-index, rectangular dielectric waveguides are obtained, again by the moment-method technique. Discretization of the EFIES is accomplished by expanding the transverse field components of the unknown field in a 2-d pulse function representation over the rec- tangular core; subsequent point matching of the resulting EFIES results in two systems of coupled, homogeneous, linear algebraic equations for expansion coefficients of the unknown field components. Hybrid, surface-wave modes are identified by numerically locating roots to a singular-system characteristic equation (discretized representation of coupled Z-d EFIES), providing the desired eigen- values and subsequently the corresponding eigenmode surface-wave- field distributions. These results are compared to those well-known approximate surface-wave mode solutions obtained previously by other investigators. 5.2 Rectangular-Waveguide EFIES and Their Momentéfléthod Solution A. Coupled EFIEs DescribingHybrid,$urface-Nave Modes Along Step-Index, Rectangular Dielectric Waveguides Figure 5.1 indicates the configuration of a step-index, rec- tangular dielectric waveguide having transverse dimensions (a,b) along (x,y), respectively, where n(x) = n1 for xe(0,a), ye(0,b);==nc 161 (O,b) 4'— (a, b) (a, O) N :3 II D Figure 5.1. Cross-section geometry and refractive-index character- ‘ istics of a step-index, rectangular dielectric waveguide. 162 otherwise, and n1 > nc is assumed. Application of EFIE (2.45) to this this geometry yields the following set of three scalar EFIES for rectangular field components ea as 2 b . k 3 | 2 t 2 I £756;- [A ex(a,y )K0(Y[(X'a) + (Y‘y ) Ji)dy e (x,y) + a anc + f0a ey1x'.b)1dy' j: ey1x' .o)Ko(y1<'dy' = 0 . . for xe(0,a), y6(0,b) (5.1) where a = x,y,z; 3/32 = 3J8; and Ak2 = k? - kg = (ni - ng)kg. In Eq. (5.1), single integrals represent effects of equivalent polariza- tion-charge layers at the core-cladding boundary, while the double integral accounts for equivalent polarization current flowing in the waveguide core. Study of scalar EFIES of expression (5.1) reveals that the x and y component equations couple ex and ey while the z-component equation couples ex, ey, and ez. Further study of the latter equation and Maxwell's equations (2.1) confirms that 163 independent TE and TM surface-wave modes cannot be supported by the rectangular guide. Hybrid, natural, surface-wave modes, charac- terized by field polarization ex # 0, ey f D, and ez f 0, satisfying the simultaneous pair of scalar EFIES for ex and e , are thus iden- y tified and defined. B. Singular-System Characteristic Equation Describing Hybrid, Surface-Wave Modes Along_Step-Index, Rectangular DielectricPWaveguides Moment-method numerical solutions to x and y-component equations of expression (5.1) are initiated by expanding the unknown fields ea(x,y) (a = x,y), using basis functions consisting of pulse functions defined by the partitioning scheme illustrated in Figure 5.2(a), over the waveguide core, to yield X N. . y ea(X.y) = "4:31 1 em" pm(X.y). a = X.y (5.2) 3 II where ea = mn'th expansion coefficient; (Nx,Ny) = total partition mn numbers along (x,y); and pmn(x,y) = 1 for (x,y)e(AS)mn;=0 otherwise. Substitution of series (5.2) into the x and y component EFIES of expression (5.1) and interchange of order of integration and summa- tion yields 164 I1531021.__.._. ._.__._}AS> I l I I I l _ _ 4' 0x (1‘ “mm Ay o -p————————n—Y } — -- —|——+—Ir- — —- " ———__ — 4— I l I L65)” I — -+- ' I [(05)]2 101$)in IL _ _ I _ _ I , has)” (08);] 165131] f (“52111 n xm a. Rectangular-core partitioning scheme. Ax 0'0' (88))?” _ '81 (83)mn Av (a. - .:—-——-vn yq-L-EBB 'l (AS) n T- yq | rm 1 i xrpn Xm b. Subpartitioning scheme. Figure 5.2. Partitioning schemes for application of moment method to the step-index, rectangular dielectric waveguide. 165 Ex (1 e p x.y+ n=1 m=1 “m" m" N 2 Y +211)? ["2116wa fw) if: non/[(x - :02 + (y - y'121*)dy' c n Nx J— - | 2 _ 2 D I "T“; Wy [mm B“... K0010 x) + (y b111dx 4. N _ Y _a__ 2 , .21 . em (1101,35: KOMX +0 yum n=1 X ' mi; eyml L011) 6‘3; K0(YE(X - X')2 + y2]*)dX'] Ill “‘2 N, "x 1 11 '12 - 1' n=1 mgl em" f(Ay)n -/(Ax)m K0 Y x - x + (y - y'121*)dx-dy' = o (5.3) where a = x,y. Operation on Eq. (5.3) by the testing-function operator f: f: [.150‘ ' xk. .Y - Yz)dxdy then forces that equation to be satisfied at discrete matching points (Xk, Y2) (centerpoints of (AS)k£) for 1|§_k‘§.Nx, 1 5-2-5-Ny’ thus completing the discretization of the component EFIES to yield two systems of NxNy coupled, linear algebraic equations in eamn as 166 N N 1‘5 2" 1 +b”°‘ e e + a e 01kt n=1 m=1 kJZmn omn kan xmn “a +ck£mn eymn]=0,1_<_k:Nx,1_<_9.:Ny; a=x,y (5.4) u u o o a a ' where coeff1c1ents akflmn’ bktmn’ and ckflmn are defined by the fol- lowing expressions: 2 Ak 2 akRmn 2n 1; )n LAX)!“ K0(YL( k x ) + (Y, - y')ZJ*)dx'dy' (5.5) F 3 2 2 I flAy)n 5'0; K0(Y[(xk 3) + (Y2, y ) J ) y m NX u gk2 a 2 | 2 ' bkgmn a (2: I - LAY)" gua K0(Y[xk+ (Yn‘y ) 1*)dy s m = 1 O,m#lorm7fNx (5.6) K F a .2 2 .' - 'IIAxlmEE Ko(vt(xk-x) +(YR-b) 1*)dx . n - My u 2 0 AK 4 f a I 2 2 D I c = - —— (y[(X-x)+Y]dx,n=1 kILmn mg (mm Bua |<0 k 1 0,nflorn¢Ny. (5.7) K 166 N N 25‘ t + b”“ e + a e e odd n=1 m=1 kflmn omn kan xmn ”01 +ckmeymn] 0,1:ngx,1:£_<_Ny,a=x,y U U (5.4) . 01 01 . where coeffic1ents akflmn’ bkzmn’ and ckfimn are defined by the fol- lowing expressions: 2 Ak 2 akILmn 2" Loy)" 110x)!" WY“ k x) + (Y, - y')2]*)dX'dy' (5.5) r f(A.Y) a301 K0(YE(Xk-a)2+ (Yz'y')2]§)d)". m = N)( n (5.6) = 1 (5.7) u 2 b“ = " l- 3 (vuxzwv -y')21*)dy'.m=1 kRmn 2""‘c Lama—6% k 9. 0, m f 1 or m f Nx K r a .2 21 .' _ — v[(X-x +Y-b dx, -11 v/(Ax)m3u01Ko( 1 11,11) n y cu“ = “‘2 - f ikouux -x')2+v2]*dx' n kimn 20kg (A )m Bua k R ’ 0, n f 1 or n # My . K 166 N N 2% i E ”U“ e + a e e 01kt n=1 m=1 kan omn kan xmn u01 +ckm eymn]=0,1_<_k:Nx,1_52:Ny; 01=x,y (5.4) u u 0 a a 0 where coefficients akfimn’ kamn’ and (=ka are defined by the fol- lowing expressions: 2 Ak 2 akfimn 271' 1;! )n LAX)!“ KO(Y[( k X ) + (v, - y'121*>dx'dy' (5.5) a 2 . 2 I 1 _ ([er araKoWXk-a) +(v,-y 1 ] )dy , m - Nx u b °‘ = k - 3 (YEX2+(Y -y')2]*)dy'. m = 1 kRmn zflkc 1 LAYMEEKO k IL 0,mflormfo (5.6) K r a . 2 2 .' [Mm a; Konuxk- x ) + 0,-61 1*)dx . n = My “ _Akz a .2 2 . ckzm- 2 ( - ’/(Ax)m-a—u-0TKO(Y[(xk-X) +Y£]§dx,n=1 0,nflorany. (5.7) K 167 Subsequent specialization of Eq. (5.4) with e0!mn = e finally xmn’ eymn yields a coupled system of ZNXN‘y (total) linear algebraic equations in 2N N (total) unknown expansion coefficients e x y xmn’ eymn’ 15m . . . if 6 = Ay/NS N5 Y x 6 = 6 < < S yq . 1 q Ny 2 (5.15) _ s W 6xp - 6, 1 < p < Nx -1 - _ S 6, q - q k . . . 1f 6 - Ax/Nx . 6yq= m’qfq' N; J Finally,subpartflfion centerpoints (xg,y:) are given by the relation- - p - (p-l) 5. q - ShlpS 1s“ - xm + [6x(p_1) + 6xp]/2., 2 5 p 5 Nx’ and y": y'gq'l) + [6y(q_1) + dyq]/2 , 2 5 q 5 Ni, where xé=xm- [Ax- 6x1]/2, and yr11 =-Yn - [Ay - 6y1]/2, to complete the subpartitioning process. Application of the subpartitioning scheme described above to evaluation of coefficient akm" (representing effects of equivalent 171 polarization current) in Eq. (5.5) leads to the following representation 2 "s ": akilmn = "%%F )5 alfigmn (5°16) q=1 n=1 where integrals GIEgmn are defined as 5113.... = f Konuxk-xwz + (v,-y')21*)dx'dy' . q P (6y)n (6X)m (5.17) Specialization of GIEEmn to m f k and n f 1 requires no singular- point integration; consequently, Glfigmn can be approximated as pq . p 2 _ q 2 I lefimm = 5Xp5YqKo(YL(Xk- xm) + (Y, y") 1 ) . . for m f k and n f t . (5.18) Specialization of dlfigm tom= k and n = 9. results in the inclusion of the singular-point integration, that being over the center-square subpartition (693:3; whenp = p' and q = q'. Excepting this case, evaluation of integrals alfigkz for p f p' and q f q' may be approxi- mated, as before, to yield GIEgkz s 6xp6yqKo(Y[(Xk - xE)2 + (12 - yg)2]*) (5.19) . . for m = k and n = 2; p f p', q f q' . Integral Glfiggg, on the other hand, is implemented by approximating that quantity by an integral over a circle of equivalent radius 172 . . . . . 2 _ 2: Req centered at the Singularity pOint, whereReq satisfies “Req — 6 ||(65)k,k,ll . to yield p.q. 211 Req Req GIkRkl 3f f rKO(Yr)drd6 = Zirf rK0(Yir)dr . (5.20) O 0 0 Exploiting small-argument approximation K0(Yr) z -£n(yr), for r f-Req’ in Eq. (5.20), expedites the evaluation of that integral to yield p'a' s 2 1 _ Y5 61"sz - 6 [ Rn (00%)] . (5.21) Substitution and subsequent sunmation of expreSsions (5.18), (5.19), and (5.21) in expression (5.16) then yields the numerically- approximated a . kzmn “a ”a Evaluation of coefficients bkfimn and ckzmn (representing effects of equivalent polarization-charge layers) proceeds in a fashion similar to that presented for aklmn’ to yield S ’ Ak2 (X ) if yq 111/[(x, x012+(v,- 172122) Y - x :13 0 n=1 [(xk-xo)2+(v,-yfi) 1* bfizmn = -( . . . for m = 1, Nx 0, otherwise (5.22) 173 F 2 Ak 2 2 2 27? [19,ka - x0) + (v, - 11,, + Av /21 1 1 c by = ( - Kowuxk - x0)2 + (Y, - vn - Ay/212]*)] kzmn . . for m = 1, Nx 0, otherwise (5.23) K r-“—'iz—[ ([(x +11 /212+(v - 121*1 211112 K0Y k ' xm x 11 5'0 c c, g # - Koo/[(xk - xm - Ax /2)2 + (Y, - y0)2]1)] kzmn . . for n = 1, My 0, otherwise I ' (5.24) L s f Akz (Y - ) Nx GXPK1(Y[(xk-x:l)2+(YR-y0)2]§) ansz" yo -1 [(x “2+1 2* c p- k - xm) , ( 2 - yo) ] cklmn = T for n = 1, Ny 0, otherwise (5.25) L where x0 = O for m = 1; = a for m = Nx’ and y0 = 0 for n = 1; = b for n = My. Subsequent to their implementation in matrix expression (5.12), U (1 ”Q kzmn’ bktmn’ and ckzmm are eXpressed in terms of nor- P _ ~ 8 ~ 3 ~ — p malized position variables xk,m Xk’m/a, Y1," Yz’n/a, x xm/a, coefficients a m and ya = yfl/a to yield a coefficient matrix dependent upon known 174 (normalized) waveguide parameters kla and kca, as well as unknown (normalized) transversal wavenumber ya. As in the slab waveguide numerical solution, admissable values of ya are identified as those characteristic values (ya)n for which det[Aij] vanishes (for non- trivial solution vectors [ei] of Eq. (5.14)), thus establishing the (normalized) propagation constant (Ba)n = [(ya)fi + (kca)2]i associated with the nth guided, hybrid, surface-wave mode. Upon equating ya and (ya)n in the expression for Aij’ any one of the e1 (i.e. one of the e , a = x,y) is assigned unit amplitude and amn one of the equations in the system (5.14) is discarded to convert this matrix equation into a nonhomogeneous system of (2N - 1) xNy equations for the remaining (ZNXNy - 1) expansion coefficients. Subsequent solution of [ei] and summation of series (5.2) then yields the field distribution of the nth guided, hybrid, surface-wave mode. 5.3 Hybrid, Natural, Surface-Nave Modes Supported by StepiIndex, Rectangular Dielectric Waveguides Moment-method numerical solutions for hybrid, natural, surface- wave modes along step-index, rectangular dielectric waveguides are obtained by numerically locating roots to the rectangular-waveguide, singular-system characteristic equation, providing the eigenvalues and 2-d eigenfield distributions of the desired natural-surface- wave modes. Root location and field computation is provided by the software-package program RECTMD. Resulting data is plotted by a CALCOMP plotter via subsidiary plotting subroutine CPLOT. Source listing and documentation of these and other auxiliary routines are available in the appendices following this report. 175 Solutions to the rectangular-waveguide. singular-system character- istic equation (5.14) for hybrid, surface-wave modes supported by a step-index, rectangular dielectric waveguide with (n1,nc) = (1.5,1.0), b/a = 0.5, and electrical thicknesses (a/1,b/A) variable, are indi- cated in Figures 5.3 through 5.19. Solutions are obtained by exploiting a (Nx’Ny) = (7,7) partitioning scheme, with subpartition factors (N:,N;) adjusted from (3,3) to (9,9) to provide uniform integration accuracy for increasing waveguide thickness. Partitionings greater than (Nx,Ny) = (7,7) were not considered because of the large computer memory and time usage associated with such systems: however, large partitioning systems may be constructed economically for those weaklybcoupled modes characterized by the independent equation pair [Axx][ex] = [o], [AYY][ey] = [0]. Dependence of normalized eigenvalue Ba upon rectangular-core electrical thickness (frequency) is indicated for hybrid modes E11’ E11’ E21, and E§1 in Figure 5.3, where subscripts rs (r = 1,2 and s = 1) designate the number of modal field extrema along x and y, and superscripts x,y indicate that transverse field component (ex,ey) which is dominant. Comparison is made between this data (squares and triangles) and numerical solutions of those approximated boundary-value problems considered by Marcatilli [12] (solid-lines) and Goell [13] (dashed-lines). Good correspondence is observed for the fundamental E11 and E11 modes, especially for small waveguide thickness (toward cut-off). Less agreement is noted for the higher- order E21 and E§1 modes, especially for large waveguide thickness. The discrepancies observed are due to inadequate partitioning provided 176 2b 2 2 “f'("1 ' "Cy5 Figure 5.3. Normalized propagation (phase) constants for different modes and normalized waveguide thicknesses.1:,A integral-operator solutions: -——-Marcatilli's numerical solutions of the boundary-value problem; —-- Goell's numerical solutions of the boundary-value problem. 177 for the higher-order mode and thick-waveguide specializations, by the integral-operator-based numerical method. Dependence of hybrid modal field distributions upon rectangular- core electrical thickness is indicated in Figures 5.4 through 5.11 for the fundamental E11 and E11 modes, and in Figures 5.12 through 5.19 for the higher-order E21 and E§1 modes. Note the correctly predicted number of extrema for the dominant field components of these modes along x and y. Note also the enhanced field confine- ment achieved for these modes with increasingly thicker waveguides. It is further observed that the cross coupling between the dominant and nondominant field components is negligible (as predicted by Marcatilli [12] and Goell [13]) for the fundamental modes, but con- siderable for the higher-order modes, as indicated by the relative amplitudes of (ex,ey) for these modes. Numerical solutions to the integral-operator equations by the moment-method technique for step- index, rectangular dielectric waveguides is thus demonstrated. NORMHLIZED 0.40 178 E?! HYBRID ”DUES 95 R FUNCTION OF FREQUENCY UUNINHNT FIELD CDNPDNENT RLONG X wa=053 Y:-Sb NCORE:1.5 NCLRDzl-O 7 PHRTION Ag; ma/x=.636666 3.69909 oa/x=.716642 6.24167 Aa/X:I .073313 8.78691 +a/1=1.431083 12.32698 X8/X:1.788854 15.82270 oak-2.146625 19.28179 +a/X=2.862167 26.17536 \\ 1 -D.O Figure 5.4. I 0. I 7 I If 1 0-2 0.6 0.9 1.0 0.3 0.4 0.5 0:8 0.7 NORHRLIZED x COORDINATE 1X/a1 Dependence of E§1 hybridnmode field distribution (dominant field component along x), supported by a high-contrast, step-index, rectangular dielectric waveguide upon its electrical thickness (frequency), obtained from moment-method solution of EFIE. AID 0.8 4 0.70 l A NORMRL I ZED EX (VOLIS/METER 1k§2 (L70 0-54 179 52. HYBRID MODES HS 6 FUNCTION OF FREQUENCY DOMINRNT FIELD coneouenr ALONG v bla=050 X: .58 NCOR6=105 NCLHD:1.0 7~ PRRTION fig Ela/X=.536656 3.59909 Oa/X:.715542 5.24187 Aa/X=1.073313 8.78691 +a/>.=1.431083 12.32698 Xa/X21.788854 15.82270 oa/X=2.146625 19.28179 +a/Xz2.862167 26.17536 / \\ “3.45 0.0 Figure 5.5. T 0-1 0:2 0.3 0.4 NORHRLIZED Y I I 0.8 0.9 I l r 0.6 0.6 0.7 COORDINATE (Y/b) x ' - ' ' 'bution De endence of E hybrid mode field distri (dgminant fieldléomponent along y). supported by a high-contrast, step-index, rectangular dielectric waveguide upon its electrical thickness (frequency), obtained from moment -method solution of EFIE. I. c—O [METER 1 0.70 EY (VOLTS 0.60 J n NDRHRLIZED 0.60 04. 40 180 6?, HYBRID MODES AS a FUNCTION 0F FREQUENCY DDHINRNT FIELD conrouenr ALONG x U8: 06. Y: oSb NCORE:1.S NCLRD=I.D 7 PRRTION g2; Balk: 4536656 3 .47199 Oa/Xzo715542 4.94540 Aalk=1.073313 8.40377 +a/>.=1.431083 11.99492 xa/)1=1.788854 15.54869 oa/Xz2.146625 19.05289 +a/>(=2.862167 26.00481 .0-30 0 O 0 Figure 5.6. I T 0-2 0.8 0.9 1.0 023 014 ch: (is 037 NORMRLIZED x cooaorunie (Xfia) Dependence of E¥1 hybrid-mode field distribution (dominant field component along x), supported by a high-contrast, step-index, rectangular dielectric waveguide upon its electrical thickness (frequency), obtained from moment-method solution of EFIE. R) .00 l HETE 1 (*00 EY (VOLTS/ A l NORHHLIZED 0.40 0.20 l O O 0 Do! Figure 5.7. 181 EH HYBRID MODES RS R FUNCTION OF FREQUENCY DOHINRNT FIELD COHPONENT RLONG Y U5305 o X: .58 NCORE:1.S NCLHD:1.0 7 PRRTIDN Ba ma/x=.63666_6 3.47199 0a/X=.715542 4.94640 Aa/X=l 4073313 8.40377 +a/>.=1.431083 11.99492 xa/X:1.768854 15.54669 cent :2 .146625 19.05289 93A =2.662167 26-00481 012 013 014 016 016 017 NORHHLIZED Y COORDINHIE (Y/bI Dependence of EYI hybrid-mode field distribution (dominant field component along y), supported by a high-contrast, step-index, rectangular dielectric waveguide upon its electrical thickness (frequency), obtained from moment-method solution of EFIE. 1 0.09 0.07 l 0.06 A J NORNRLIIZOED EY (VOLTS/NETER) 0.01 Figure 5.8. 81-0 0.1 0.2 =5 182 E‘x1 HYBRID HODES RS R FUNCTION OF FREQUENCY NONDOHINRNT FIELD COMPONENT RLONO X b/a=-5. Y305b NCDRE:1.S NCLRD=100 7 PHRTION §_a_ (De/12.715542 6.24167 Aa/X:1.D73313 8.78691 X a/X=1.788854 15 .82270 +a/X:2.862167 26.17536 . EEEEEEEEEEE=======¢—— . A v L V K .‘. 0.3 0.4 0.6 0.6 0.7 0.6 0.9 NORHRLIZED X COORDINHTE (X/a) Dependence of E1 hybrid-mode field distribution (nondominant fieId component along x), supported by a high-contrast, step-index, rectangular dielectric waveguide upon its electrical thickness (frequency), obtained from moment-method solution of EFIE. 1. 183 E1] HYBRID MODES 95 F1 FUNCTION OF FREQUENCY NONOOHINRNT FIELD COMPONENT HLONG Y Ua=050 X:-sa A NCORE:I.5 uc100=1.0 7 PRRTION fig 111 a1 = .636666 3.69909 oa/1=.716642 6.24167 Aa/)\:I-073313 8-78691 -Fa/X=1-431083 12-32698 xa/1.=1.788854 15-82270 +3/X:2-862167 26-17536 0.56 0-39 1 TS/HETER) 0J1 (VOL 023 l IZED EY 046 NORHHL O-D? l D-Ol go -0 O-I 0-2 0-6 0-9 1-0 0.3 0.4 0.6 0.6 0.7 NORMRLIZED Y COORDINATE (Y/b) Figure 5.9. Dependence of E? hybrid-mode field distribution (nondominant fieId component along y), supported by a high-contrast, step-index, rectangular dielectric waveguide upon its electrical thickness (frequency), obtained from moment-method solution of EFIE. 184 5?, HYBRID MODES AS 6 FUNCTION OF FREQUENCY NONDOMINANT FIELD COHPONENT ALONG x b/a: .5 0 Y: 05b NCORE:1.S NCLAD=1.D 7 PARTION 4%; ma/x=.636666 3.47199 oa/1=.716642 4.94640 Aa/X:I-O73313 8.40377 X5/X:1-788854 15-54869 Oa/X=2 .146625 19 -05289 +a/X:2-862167 26-00481 0-40 :10“ 0.“) 0-32 A 0.24 1 0-08 NORMHL I ZED EX I VOLTS/HETER ) -00 0.1 0.2 ‘. ‘3187 . - ~ I 44: c: 003 00‘ 005 006 0.7 NORMALIZED X COORDINATE (X/a) -0 J” L Figure 5.10. Dependence of E! hybrid-mode field distribution (nondominant fieId component along x), supported by a high-contrast, step-index, rectangular dielectric waveguide upon its electrical thickness (frequency), obtained from moment-method solution of EFIE. l TS/HETER) 0.04 EU EX (VOL -0004 .0000 o O NORHRL I Z -4LOO . j l— -0062 x.;—- I. 185 5,9, HYBRID MODES 66 A FUNCTION 0F FREQUENCY NONDOMINANT FIELD COMPONENT ALONG Y D/a:-50 x=Osa NCDRE=1.6 NCLAD=1.D 7 PARTION §§ ma/x=.636666 3.47199 oa/x=.716642 4.94640 -ka/X=I-431063 11-99492 X5/X=1 -786854 15.54869 oa/X:2-146625 19.05289' 98/):2-862167 26-00481 _ #- I‘— 0-6 0-9 I-O _ __-_ 0.3 0.4 '0.6 0.6 0.7 NORr' IZED Y COORDINATE (Y/b) Figure 5.11. Dependence of ET} hybrid-mode field distribution (nondominant fie d component along y), supported by a high-contrast, step-index, rectangular dielectric waveguide upon its electrical thickness (frequency), obtained from moment-method solution of EFIE. 186 E5: HYBRID HODES 93 H FUNCTION OF FREQUENCY DOHINDNT FIELD COHPONENT HLONG X D/az.5. Y=0214b "CORE3105 NCLAD:1.0 7 PARTION _B_a_ Dia/x=.694427 6.30437 Oa/X=I.D73313 7.49476 I‘EVX=I-431083 11017833 -ka/X=I.766654 14.84691 ufl I-20 ) 7/ 1 Ex (VOLTS/HETER 0.40 I I -0 0-1 0-2 0.8 1-0 0.3 014 0.. 016 0.7 NORHHLIZED X COOR’INRTE tX/a) NORMRLIZED -0.40 1 ii:; .20 I 1 Figure 5.12. Dependence of 5E1 hybrid-mode field distribution (dominant field component along x), supported by . a high-contrast, step-index, rectangular dielectric waveguide upon its electrical thickness (frequency), obtained from moment-method solution of EFIE. 187 5;. HYBRID H0056 AS A FUNCTION OF FREQUENCY 00HINRNT FIELD COMPONENT ALONG Y we: 050 X: 2143 NCDRE=1.6 NCLAD=1.D 7 PARTION 5%; Ina/1.: .694427 6.60437 Oa/X:1-D73313 7.49478 t+EVX=1-766684 14.84691 8 ,:::7/T 1\::::. 0.72 0.64 NDRMHLIZED EX ( r r I r’ I r’ r’ T‘ r 1 0-4 0 0.8 0-9 '1-0 0.2 0-3 .6 0-6 0.7 NORMRLIZED Y COORDINRTE (Y/b) .0-40 Figure 5.13. Dependence of E61 hybrid-mode field distribution (dominant field component along y), supported by a high-contrast, step-index, rectangular dielectric waveguide upon its electrical thickness (frequency), obtained from moment-method solution of EFIE. 188 E51 HYBRID MODES 95 H FUNCTION OF FREQUENCY DOI‘IINRNT FIELD COMPONENT HLONO X Ua=05| Y=0214b NCORE=I.5 NCLAD:I.0 7 FARTION 5g; ma/1=.694427 6.71033 Aa/)\=I-431083 11.15272 +a/>.=1.788854 14.82981 1 (L00 1 \\ 0.40 l D EY (VOLTS/METER I I I I I I I I I .0 04. 0.3 0.4 0.. 0.6 0.7 0.0 0.9 1.0 NORMALIZED X COOR- NATE (X/a) -0040 NORMRL IZE Figure 5.14. Dependence of [$1 hybrid-mode field distribution (dominant field component along x), supported by a high-contrast, step-index, rectangular dielectric waveguide upon its electrical thickness (frequency), obtained from moment-method solution of EFIE. 189 E51 HYBRID HODES RS 9 FUNCTION OF FREQUENCY DOMINRNT FIELD COMPONENT RLONG Y b/a: .5 . X: .2143 NCORE=1.5 NCLAD=1.0 7 PARTION £3; Elana-894427 6.71033 Oa/xei.073313 7.43913 Aa/Xs1.431083 11.16272 t+EVk:1.788854 14.62981 I VOL I 0.80 NORMRL IZED EY 0-40 I I 0-1 0-2 .(LSO G o O 0.3 014 016 016 017 NORHHLIZED Y COORDINHTE (Y/b) Figure 5.15. Dependence of E§1 hybrid-mode field distribution (dominant field component along y), supported by . a high-contrast, step-index, rectangular dielectric waveguide upon its electrical thickness (frequency), obtained from moment-method solution of EFIE. LTS/I‘IET 190 5;, HYBRID H0056 AS A FUNCTION OF FREQUENCY NONDOMINANT FIELD COMPONENT ALONG x be: '50 Y: 0214b NCORE=I.5 NCLRD=100 7 PARTION 5g; Ina/1:.694427 6.60437 Aa/>.=1.431083 11-17833 -0-0 0-1 0-2 Figure 5.16. NORNRLIZED EY (V0 0 04 0:00 0:12 '1 (C l I 0:3 014 016 0-8 0.7 0-6 0-9 1-0 NORMALIZED x COORDINATE (X/a) Dependence of E? hybrid-mode field distribution (nondominant fie d component along x), supported by a high-contrast, step-index, rectangular dielectric waveguide upon its electrical thickness (frequency), obtained from moment-method solution of EFIE. 0.13 -IO RI Ina] lflET Inn A 191 55‘, HYBRID H0056 AS A FUNCTION OF FREQUENCY NDNDDHINRNT FIELD CDHFDNENT ALONG Y b/a: -5 . X: .2148 NCORE:1.5 NCLRD=1-D 7 PRRTION ma/1=.894427 Oa/X=1 .073313 +a/>.=1.788854 I—° *— (V0 2 "9-.0 . 1250_ 5 - 7 -Jo I:?~ R NOR 1 -0-12 ”0067 L Figure 5.17. I I ' l e I I 0:1 0.2 (L3 CHI 0,: ,, 0. NORMALIZED Y c00\" X £13 5.60437 7-49478 11 . 17833 14 .84691 II~J O O c: «3 Dependence of E2} hybrid-mode field distribution d (nondominant fie component along y), supported by a high-contrast, step-index, rectangular dielectric waveguide upon its electrical thickness (frequency), obtained from moment-method solution of EFIE. .0-00 1 -002‘ l NORMRL I ZED-GETS I VOLTS/ME TER ) 1132 -0000 Figure 5.18. 192 E51 HYBRID MODES RS A FUNCTION OF FREQUENCY NONDOMINHNT FIELD COMPONENT BLONG X b/az-S. Y:.214b NCORE=1.S NCLRD=1.0 7 PRRTION _B_§ Ina/1:.894427 5.71033 Oa/X=I.O73313 7.43913 Aa/x=1.431063 11.16272 -+av1=1.766664 14.62961 12 013 014 016 0.6 0.7 NORHHLIZED X COORDINRTE IX/a) Dependence of 5% hybrid-mode field distribution (nondominant fieId component along x), supported by a high-contrast, step-index, rectangular dielectric waveguide upon its electrical thickness (frequency), obtained from moment-method solution of EFIE. 193 551 HYBRID H0056 AS A FUNCTION OF FREQUENCY NONDOMINANT FIELD COMPONENT ALONG Y W8: '5 I x: .2148 NCORE=1.5 NCLAD=1.0 7 PARTION 5g; ma/x=.094427 6.71033 (pa/1:1.073313 7.43913 ABl).=l.431083 11.15272. .15“ I I I r 1 I I— I T l 441 0.1 042 0.3 . 1.5 (L8 006 0.9 1.0 0-7 NI:.- 4 : /COORDINRTE IY/b) NORMHLIZED EX (VO T .jpoo :gde ‘0024 Figure 5.19. Dependence of E§ hybrid-mode field distribution (nondominant fieId component along y), supported by a high-contrast, step-index, rectangular dielectric waveguide upon its electrical thickness (frequency), obtained from moment-method solution of EFIE. CHAPTER VI INTEGRAL-OPERATOR FORMULATION 0F EXCITATION OF OPEN-BOUNDARY DIELECTRIC NAVEGUIDES 6.1 Introduction A nonhomogeneous integral-operator equation describing the total unknown E field supported by an open-boundary dielectric wave- guide has been presented. Guided, natural-surface-wave modes charac- terized by a discrete eigenvalue spectrum and exhibiting highly- localized transverse confinement have been identified from homogeneous specializations of this EFIE for various waveguiding structures. This finite set of surface-wave modes, however, does not evidently form a complete orthogonal set of functions by which an‘E field excited along an open-boundary dielectric waveguide by an arbitrary source may be expanded. Specifically, a system of non-confined, radiation-mode waves, not satisfying the radiation condition and having a continuous eigenvalue spectrum, is required to augment the set of discrete surface-wave modes in order to provide mode completeness [7, 9-11]. Radiation modes are maintained by single- spectral-component impressed fields originating from sources at infinity and thus may be described as forced solutions to the non- homogeneous EFIE. Thus, an excitation theory for open-boundary dielectric waveguides is necessitated. 194 195 This chapter introduces a theory for excitation of discrete, guided, surface-wave modes and continuous-spectrum radiation modes along open-boundary dielectric waveguides, based upon the integral- operator formulation. Initially, the general nonhomogeneous 3-d EFIE, describing the total field maintained at any point in the core of the waveguide, is reexamined when the excitatory field is nonzero. An EFIE describing radiation modes supported by a dielec- tric waveguide of arbitrary cross-section is then presented. This formulation is subsequently specialized to describe TE radiation modes along a step-index, dielectric-slab waveguide. Finally, the excitation of discrete TE surface-wave modes along a step-index slab waveguide by a localized source is considered. 6.2 Forced 3-d EFIE and Its Field Solutions Total-field response E(F) along an open-boundary dielectric waveguide due to impressed field E1(F) maintained by excitatory currents Se, is described by the general, forCed 3-d EFIE (2.17) expressed as ++ l2+| ++' +4- E(r) + v[ - IVH V " $2};er ) G(r|r')dV' 2+. + f—J—sz" fi'-E(F')G(F|F')d$'] s k C - f 6RZIF'1'EG'16IFI‘F'1dv- =66?) vh . . for all 7th (6.1) 196 where notation 5k2(F) = k2(F) - kg (f 0 for FeYh; = 0 otherwise) is applicable and a continuous refractive-index core is assumed. In general, total-unknown field E is described in terms of a system of zero, one, or more discrete-eigenvalue, surface-wave modes EB(F) augmented by a superposition of continuous-eigenvalue, radiation- mode waves E£(F,§) as 506:) = Z AnEn(?) ‘ n .. * (6.2) ERG!) = ff A(§)Ec(?,6)d3p , where 6 represents a vector transform variable and where Eb and'ER denote, respectively, the discrete surface-wave component and con- tinuous-spectrum radiation-mode component fields of E. That Eb and ER constitute appropriate component fields of'E may be demon- strated by exploiting the following general 3-d completeness condition E75 *1 E755 +',§d3 .1320 2": n(r),,(r1+f.fmf c:(r1c1r )0 (r mm) where?= RR + 99 + 22 is the unit dyadic, which must be satisfied by the system of Eh and E; in order that it represent a complete orthogonal system of functions in which E may be expanded [9, 23]. Post-dot multiplying Eq. (6.3) by‘E and integrating the result over 197 all r' space yields, after appropriate interchange of integration and summation order, + + + + + + + Zn: < ,En>5n(‘F) + fff EC(F,§)d3o = 5(F) (6.4) where < > denotes a 3-d inner-product operation, defined for the fields EE(F), E§(P) by the relation (Ep,5q> = [U Ep(F')-Eq(F')d3r' . The relationship between E, Eb, and Eh is made evident by identify- ing the amplitude coefficients An and A(§) of Eqs. (6.2) as + '6' .5 + An .. and Ma) = to yield, from Eq. (6.4) ++ + + + ++ 3 E = A E A D E ,0 d D . 6.5 (r1§n,(r1+f£ (1,01 ( 1 from which the desired relationship E = Eb +1Ek is established. It has been demonstrated in Section 2.5 that for an infinite, longitudinally-invariant waveguide structure a system E: of travelling surface-wave fields propagating in the :z direction with discrete phase constants:Bn may be expressed as '+1-+ ¢+- $‘8 z ED(r) = Zn: ane.n(3)e J n (6.6) where 3 is the 2-d transverse position vector and where (1) refers to waves propagating in the :2 directions. Fields 3n(3) are 198 formally obtained as natural-mode solutions to the homogeneous 2-d EFIE (2.45); this equation requires the surface-wave fields to experience an asymtotic exponential decay with transverse displace- ment in the cladding, with y > 0, where Y2 = 82 - kg defines the transverse-wavenumber parameter depending upon discrete eigenvalue 8 = B". In a similar fashion, it is demonstrated in the subsequent section that for an infinite, longitudinally-invariant structure,, the radiation field E: consists of a continuum of radiation-wave modes propagating in the :2 direction with continuous phase con- stants 8(6) as E;(?1= f f 4*(7.151e*(‘5.151e"‘~18‘5)z c125. . (6.71 where E defines a 2-d vector transform variable and is related to '5 through the relation 6 =‘E T 8(5)? such that E = [El and 52 . kg - 32 . -42 > o. The condition that 52 = -Y2 > 0 or Y =i05. where E > 0, leads to the conclusion that radiation modes do not display the asymptotic exponential decay in the cladding, charac- teristic of surface-wave modes: radiation modes are thus nonconfined and subsequently fail to satisfy the radiation condition [10, 11]. A second observation, which follows immediately from the first, is that radiation modes are necessarily forced responses; i.e., they are described by forced integral equations, since no nontrivial solutions to EFIE (2.45) exist when y is imaginary. Finally, for the specialization g =3jy where Y > 0 (i.e., when surface-wave modes do exist) then study of the fundamental equation B==(k(2;--€2)é 199 reveals that 8n > kc > 8(6) for all n and 5. This observation is com- patible with the result that system of Eh and EE form a nondegenerate, linearly-independent, and orthogonal (i.e., complete) system of functions by which any field may be expanded [23]. Excitation of fields Eb and Eh along the open-boundary dielec- tric waveguide is provided by the impressed field E1 maintained by excitatory currents 35, as described by EFIE (6.1). In general, a localized current source will excite both component fields of 1E; these fields experience an asymtotic decay with increasing dis- tance from the source and their satisfaction of the radiation con- dition is well-known [9, 10]. However, a constituant spectral- component mode of the radiation field fails to satisfy the radiation condition individually, as described earlier, because it exhibits a standing-wave distribution in the core and cladding which may be regarded physically as arising from the reflection of a single- spectral-component impressed field of infinite energy originating, apparently, at remote locations from the waveguide [8]. The single- spectral-component impressed field, in fact, represents a single- spectral-frequency contribution from the total impressed field originating from a localized current source. The resulting spectral- component radiation mode, as well as the spectral-component impressed field, thus represent spectral field-densities which have meaning only under superposition (integration) and, therefore, represent no physically realizable EM fields to which the radiation condition may be applied. In the next section, the problem concerning 200 excitation of radiation-mode waves along dielectric waveguides is addressed more completely, while coupling to surface-wave modes is addressed in a subsequent exposition. 6.3 EFIE Describing:forced, Radiation-Mode Fields A. Representation of Impressed Field Because radiation modes arise from the irradiation by impressed spectral-component fields of uniform extent, it is appropriate to represent the physical excitatory field EI(F) in terms of a continuous expansion involving the superposition of these single- spectral-frequency waves. In a rectangular coordinate system, the impressed field E1 maintained by excitatory currents 32, immersed in the unbounded cladding medium characterized by a constant wave- number kc’ may be represented by the following 2-d plane-wave expan- sion (inverse Fourier transform) over the (x,y) plane as [30] m . . + 5161 = (2i)? f f .17‘(5.z1e3'5'p d2; , (6.8) 71' where E - 5x R + 8y? and (5x,§y) represent those transform variables related to (x,y). The z-dependence ofE1 is incorporated in the hybrid, vector-amplitude spectrumd;j(§,z) as a separate parameter, where}1 is related to the impressed field through the following Fourier transform relation £5.21 = ff 151(26214‘53'3 42o . (6.91 201 That Eqs. (6.8) and (6.9) represent a valid system of Fourier trans- form representations over the (x,y) and (Fx’gy) planes may be evi- denced by the satisfaction of the following completeness condition for plane waves (#2- ff eJ'E-(p-o') (125 = 5(3- _ '51) . (5,10) fl . from which Eqs. (6.8) and (6.9) may be obtained by appropriate operations. 1 e +'e Impressed field E maintained by excitatory sources (0 ,J ) immersed in the unbounded cladding medium is described by the Helmholtz equation (2.9), with dkz = 0, expressed in modified form as (172 + log)?" -'j(zc/kc)[7(v-3e) + 1:559] .9436} where .9’denotes an appropriate source-system operator. Substitution of representa- 2 2 t + 32/8z2, then yields tion (6.8) in this expression, with V = V m 2 . + (1) f [:7 (5.1+J‘(5.21(v§+k§1] e359 425621361. (6.11) Transverse Laplacian operating on the exponential in Eq. (6.11) yields VEEXPIJ-E'S) - -€2eXP(J’E°3) to give (2 ) fiffI ‘2 + 82(5)]0‘ (5.011331502501511 413"} (6.12) 1T 202 where 82(5) = kg - g2 defines a longitudinal phase constant (wave- number) that establishes the previous nomenclature. Finally, mul- tiplication of Eq. (6.12) by exp(—jE'53) and integrating the result over all cross-sectional space 3 then yields (after appropriate change of variables) the following inhomogeneous, forced wave equation 2 a 2 e -J’E- o .4 gm } d2 6.13 [322 + B (9] (E 2)= If e o ( ) describing the vector-amplitude spectrum.;?. Solutions to Eq. (6.13) describingezj in a source-free region are obtained from its homogeneous specialization [32/3221-82]43i=0 to yieldedi (E,z) = a i(E)exp(+j8(£)z) where a 1defines a vector- amplitude coefficient dependent only upon transform variables (Ex,§y) and where (i) refers to waves propagating away from the excitatory source system in the iz direction with phase constant 8(5). Substitution of this result in integral¥expressions (6.8) and (6.9) then yield +ie+ g 1 a +1: 35-? 2 E (r) W]! a (E)e d g (5.14) iii-"(15) -- f f E‘*(‘6.z)e'jm dzp (6.15) + - A where o = E4B(€)z defines a characteristic propagation wavevector 4.. with magnitude (5| =|(E2 + 82)El= kc' Thus, E1: is observed to 203 consist of a continuous superposition of single-spectral-component plane waves 621(FJD:=3Ei(E)exp(j§53) having an implicit exp($j8(€)z) propagation dependence, prOpagating in a direction defined by wave- vector 5. The polarization ofE:i may be determined by calculating the divergence of Eq. (6.14) and recalling that V 31*(F) = 0 in a homogeneous, source-free medium to yield 311:5 = 0, describing the appropriate polarization constraint for the spectral-plane-wave fields. In a circular-cylindrical coordinate system, the plane-wave inverse-Fourier transform representation (6.14) may be expressed in terms of an infinite series of inverse-Hankel transforms [30,31], as now demonstrated. Spatial and frequency variable pairs (x,y) and (6x,§y) are coordinate transformed to circular-cylindrical representations (r,¢) and (§,w) to yield 5?? = gr cos(w-¢)3B(€)z in Eq. (6.14). Spectral-amplitude components a;i(E) = a;i(£,w) (d = x,y,z) are furthermore assumed separable in 5 and w such that a;t(g,w) = f:(£)g:(w) where f: and g: are functions of the appro- priate single variables and where g:(w) is necessarily Zn-periodic in w. g:(w) is further expressed by the following exponential Fourier series 920») = R}; c; em (5.15) where 2n . cg, =71; f0 gym-W d1» (6.17) 204 defines the Fourier coefficients cil. Substitution of these results in Eq. (6.14) then leads to the desired expression for field com- ponent Eli involving an infinite series of inverse—Hankel transforms of f:(§), Zn-periodic in ¢ as ._ °° _ . . °° _ ; E;+(F) =-%; 22;” egg-1%”?~ f;(§)e J8(5)2J£(€r)€d€ 0 (6.18) where a = x,y,z and where . -2 2n . . J£(u)=-(J2)1—r— f e3"c°59 e319 de (6.19) 0 describes an integral representation of the Bessel function. In +i1 this form, E is observed to consist of a superposition of radial and angular-dependent cylindrical spectral-component modes E;:(;.E)= (211)"1 c:£(-j)2exp(-j£¢)5f:(€)dg(€r)exp(‘18(5)2). representing radially standing waves satisfying polarization constraint 3ii55==0, for a = x,y,z. Conversion to circular-cylindrical components a = r,¢,z completes the description of these waves. 8. Z-d and l-d EFIES DescribingForced Radiation-Mode Response to Single-Spectral-Component ImpressediFields Continuous-spectrum, radiation-mode field response E:(F,E) along an open-boundary dielectric waveguide, due to single-spectral- component impressed field EZE(F,E) is described by the general 3-d EFIE (6.1), specialized to the case of single-spectral excitation as 205 + | 2 +I + 5:622) + v[ - f l'é—ér—l E33" .6601?“ W Vh k (r') 2-+. . f £12,111 a' £32. ,gmmr )dS'] S k C - fv 6k2(‘F')E:(F'.E)G(FIE')dV' =Eli(F.E) h . . . for $th . (6.20) In general, impressed field E1i(F ,E) may be expressed as E:i(F,E) = 3E(3,E)eXp(¥jB(g)z), which describes a single-spectral-frequency wave travelling along the :2 direction with phase constant 8(5), having transverse field dependence 3‘(3,§). Further specialization of EFIE (6.20), when the excitatory source system is infinitely remote along 2 and where a longitudinally invariant medium is assumed, leads to the requirement that the radiation-mode field response also exhibit an exponential propagation dependence along 2 as E:(3,E) = 3(3,E)exp(3j8(€)z) to obtain satisfaction of this equation. Appropriate superposition of these waves yields the radiation field described by integral (6.7). Exploiting these representations in EFIE (6.20) leads to a common integral over 2' in that equation which may be evaluated in a fashion similar to integral (2.43) to yield 206 foo e;j8(€)2' G(F|?1)dzl = e+j8(€)z 2111-(+j€ Io _ D II) 538“” (:41) (137-Hal}? - ‘5' I) (6.21) where the plus sign in the argument of K0 is chosen to provide outward-travelling scattered waves. Substitution of this result in EFIE (6.20) results in a common exp(3jB(E)z) factor throughout, the subsequent cancellation of which leads to the following inhomo- geneous, 2-d vector EFIE, counterpart to Eq. (2.45), for the trans- verse field dependence of radiation-mode fields as V'k2(p') (+' o') . 2'+. - 7} 9%, if? 6' 56(3' .Ewgzhals - '6' I my] C +1} fCS 5126? 13(3' .Ewgz’ma - 6'11ds' -- 316.2) em + (v +js(€)z) [7} fm: - 6(6' .E)H((,2)(al‘5-‘5' l)dS' . for all Bees . (6.22) It is evident from this integral equation that the radiation-mode fields 3, which exhibit a standing-wave distribution with transverse displacement into the cladding, require a > 0 (lossless media), leading directly to the well-known relation 8(6) < kc for these waves. Solutions to the Z-d EFIE (6.22) lead to the spectral- component field 3(3,E) corresponding to continuous eigenvalue 8(a) 207 which describes the propagation characteristics of spectral-component radiation modes having spatial frequency 2. EFIE (6.22) con- sequently provides a complete description of continuous radiation modes supported by heterogeneous,longitudinally-invariant, open- boundary dielectric waveguides of arbitrary cross-section shape. Radiation modes along planar, open-boundary, dielectric- slab waveguides are excited by single-spectral-component impressed plane-wave fields which are independent of one transverse coordinate. Assuming y-invariant fields, the integrals over y' in EFIE (6.22) may be completed, via integral (2.49) with y = jlgxl, to reduce the 2-d vector EFIE to the following inhomogeneous, 1-d vector EFIE, counterpart to Eq. (2.51), which describes the x-dependence of radiation-mode fields along slab waveguides as 1 2 o A — 6k (a)e (a.€ )[JIE lsng(x - a)x A -J'|€ IIx-al : JB(I€xl)z]e " 3(X.€x) + A A or. x-b - ak2(b)ex(b.ax)uIaxlsgnu-b)xiJBUEXHZJE Ix” I] b e x(X ,Ex) I 321%;rLT H. dkczr§x)[1l€Isgn(x-x)x+38(|€”3] -jlgx| lx -x' I dx' . e b -.1'|€ le-X'I +- 1'5 x - Egg—'1 f 3(X' ,gx)6k2(x')e X dX' = A‘(€x)e X X a . for xe(a,b) (6.23) 208 where planar core-cladding interfaces are located at x = a,b and XE is an appropriate normalization coefficient. It is clear from this integral equation (as in the previous specialization) that radiation-mode fields 3 supported by slab waveguides also exhibit a standing-wave distribution along x into the cladding. Solutions to the 1-d EFIE (6.23) lead to the spectral-component field 3(x,§x) corresponding to eigenvalue B(|€x|) which describes the propagation characteristics of the Exth spectral-frequency radiation mode along a heterogeneous, planar, open-boundary dielectric-slab waveguide. C. TE-Radiation Modes Along Step-Index, Dielectric-Slab Waveguides Specialization of EFIE (6.23) to describe radiation modes excited by impressed, single-spectral plane-wave fields along a step-index,dielectric-slab waveguide having cbre-cladding boundaries at x a 1d, with n = 111 for xc(-d,d) and n =_ nc otherwise (Figure6.1) yields the following system of three scalar-component EFIES x-component: 2 . ex(x’€) + ":7 [ex('d+.£)e'algl("+d) +'ex(d‘.€)e3|€|(x-d)] C 2 d . , . - j—glffl' -d ex(X':€)e-J|€Hx-x I dx' = Al(§)ngx . . for xe(-d,d) (6.24) 209 (6 . nc. kc) i jgx-sz KAy(E)€ e )3} 1 x \\\:, //// \ / \ / M \v/ . ' ' C (flew—x L____... _2_..___ (e1,n1,k‘) C1(§)eJ°'x e‘JBZ 92 X=-d g-jBZ \. ' \ 6(5) = (k? - (32(511/2 \71‘ Figure 6.1. Excitation of step-index dielectric slab-waveguide by single-spectral-component, plane-wave field (slab has infinite extent along y and 2 directions). 210 y-component: d 2 0 O o _ Ak . -J|€llX-X'| . = 1 ng ey(X.£) m g _d ey(x .€)e dx Ay(E)e . . for Xe(-d,d) (6.25) z-component: 2 . + .95.. _ + -J|€l(x+d) ez(x.€) _ 6(Ial)zlglk§ [ex( d .€)e .. ex(d"g)ej lgl (X-d)] 2 d . . . . - fig _d «22(x'.(:)e'3|‘5||""x I dx' = A;(a)e~l’5x . . for xe(-d,d) (6.26) 2 where Ak2 é k1 - k2 c deleted for convenience. Examination of Eqs. (6.24) through (6.26) as before and where subscript x of Ex has been reveals that component Eqs. (6.24) and (6.25) are independent in ex and ey, respectively, while Eq. (6.26) couples ex to ez. By arguments similar to those established for Eqs. (3.1) through (3.3) in the discrete-mode specialization, it is concluded that independent TE and TM radiation modes exist along the dielectric-slab waveguide with 3 = yey(x,€) for TE-radiation modes and 3 = iex(x,£) + Eez(x,E) for TM-radiation modes. Solutions to EFIE (6.25), describing spectral component TE- radiation modes along a step-index, dielectric-slab waveguide 211 excited by impressed spectral, plane-wave fields, are initiated by exploiting a Fourier-exponential transform representation for ey(x,g) in the internal (-d,d) as ey(x.e:) = % f... 561.219“ an, xe(-d.d) (6.27) where Ey(n,§) represents the appropriate forward Fourier transform of the functionally extended ey. Substitution of representation (6.27) into EFIE (6.25) and interchanging the integration over space and spatial frequency yields, after further simplification, co 2 , E ( ,5) [1 - Ak ] eJnx dn f... y n (n2 - IEIZ) + 2 ¢» ' - d . aka- I: Ey(n,g) [eif‘n-Iigy] dnl eJIEIX Akz 1 w -j(n+|€|)d _- g [m I” Eyhhg) [eh] + [an ]d0 9 Jl IX i jgx 2nAy(E)e . . for xe(-d,d) (6.28) The linear independence of exp(JnX) with exp(j£x) and exp(ij|§|x) over xe(-d,d) in Eq. (6.28) requires the first integral term of Eq. (6.28) to vanish independently for satisfaction of this equa- tion, implying n2 - IEI2 = Ak2 or (applying 52 = kg - 82. 212 Ak2 = kg - kg) n = to, where 0(6) = [kg - 32(|g|)]i defines a charac- teristic transversal wavenumber dependent upon spectral frequency 5. Satisfaction of Eq. (6.27) with ey f 0 then requires Ey(n,g) = 2n[C1(§)6(n - o) + Cz(§)6(n + 0)] which leads to the well-known [7, 32] radiation core-field solution ey(x,g) = C1(§)exp(jox) + C2(g)exp(-jox) for xe(-d,d) and B(|g|) < kc. Substituting the above required expression for Ey(n,g) into the remaining integral terms of Eq. (6.28) and carrying out the ‘ integrals over n yields the following expression involving unknown mode-amplitude coefficients 61(5) and C2(§) as e3'€'*[c1(2)(o + (2|)ejcd - 62(€)(o - (2|)e‘j0d1 - e'jlilxtc1(a)(o - lane'j"d - c,(a)(o + l€|)ej°d] = ZIEIA;(E)ej|€ld ejEX . . for xa(-d,d) . (6.29) Specialization of Eq. (6.29) to positive spectral frequencies 5 > 0 and exploiting linear independence of resulting complex exponentials provides a nonhomogeneous system of equations in 61(5) and 62(5) as ((o + a)ej°d -(o - a)e‘5“d‘ (61(5)‘ (5;(g)1 = dejgd _w-zwdw-w+aw“d, 5gp) . (o . (6.30) 213 Subsequent solution of Eq. (6.30) for the mode-amplitude coefficients yields c1(€) = ci(g)n;(g) and C2(g) = Cé(g)A;(g) where ci(a)= -2A’la(o + €)exp[j(£ + o)d]. and 05(5) = -2A'1€(o-E)exp[j(£-o)d]. and where A = (o - £)2exp(-j20d) - (o + g)2exp(j20d) is the deter- minant of the coefficient matrix in Eq. (6.30). Alternatively, replacing E by -g in Eq. (6.29) specializes that equation to negative spectral frequencies, for g > 0, and results in the following non- homogeneous system of equations ”(o + me“ -(o - e)e‘j°d1 ”c1(-a)1 ' o 1 = -2£;ejgd _(o - med“ -(o + .5)er - (ogre). 1(2). (6.31) having solutions C1(-€) = Ci(-E)A;(-£) and C2(-E) - C§(-§)A;(-g) where Ci(-E) = Cé(E) and C§(-€) = Ci(§). Combining these results, an impressed field of the form i a i jEx i _ -j§x ey(X.£) IEAy(€)e + Ay( €)e 1 = h§(g)cosgx + jA§(g)singx , (5.32) e _ i i _ o = i _ i _ where Ay(€) — iIAy(€) + Ay( 5)]and Ay(€) 1[4y(€) Ay( 6)] results in the excitation of the TE radiation mode 214 e e (Xi) = (fl) ejl};d ' Gem” cosox Y ce(€) (6.33) A°(€) - o ., j ("’17-”) eJEEd - a (2)] gm c (2:) where Ce(E) = [1 + (v/E)Zsinzod]i, C°(E) = [1 + (v/E)2coszod]i, v2 = Akz, ae(€) = tan'1[(o/E)tanod], and a°(E) = tan'1[(E/o)tanod]. Comparison of this result, specialized to impressed fields even along x (i.e., when A;(-€) = A;(€)), with Rozzi [32], reveals that A;(E) =(2/n)*exp[-j(§d - ae)] to excite an even radiation-mode spectral component satisfying normalization condition co [0 ey(x.e)ey(x.a')dx = 6(2 - 2') . (6.34) Alternatively, specialization of Eqs. (6.32) and (6.33) to the case of odd excitatory fields (A;(-€) = -A;(E)) requires A;(E) = (2/n)iexp[-j(Ed - a°)] to excite an odd radiation-mode spectral component satisfying normalization condition (6.34). As a final calculation, total field ey(x,E) = e;(x,E) + e;(x,€) in the cladding for x > d is determined and compared with Rozzi's results. Substituting representation (6.33) of ey(x,E) into the integral of EFIE (6.25), utilizing the forcing function given by Eq. (6.32), and carrying out the integral over x' yields 215 ey(x.a) = e;(x,a) + ej d. (6.35) Specialization of Eq. (6.35) to the case where A;(-E) = 2A;(g) then gives ey(x,g) = (2/")%COS[£(X - d) + a9] for even modes, and gives ey(x,E) = (2/n)5sin[E(x - d) + do] for odd modes, which are the correct cladding fields established by Rozzi. These results confirm the forced integral equation describing TE radiation modes as well as the impressed field which excites these modes along a step-index, dielectric slab waveguide. 6.4 Excitation of TE Surface-Have Modes Along Step-Index, DielectriceSlab Waveguides In this final development, the integral-operator method is applied to recover the well-known, amplitude-coefficient expressions of surface-wave modes excited along dielectric-slab waveguides by impressed excitatory electric currents. Specifically, coupling of TE surface-wave modes and line-current sources for step-index slab waveguides is considered. Figure 6.2 illustrates the excitation 216 (60' nc'kc) : ]e(x,z)=)7l5(x-x")8(z-z") i i .. _ My ., + x = d amEYm, amEym VA , \ AA I (ei’nl’kl) L_—__—-—_———- '—>Z Xa-d Figure 6.2. Excitation of step-index, dielectric slab waveguide by line-source current (slab has infinite extent along y and 2 directions). 217 of a step-index dielectric-slab waveguide, having core thickness 2d and uniform refractive index n1 > "c’ by a 2-d current source 39(x,z) . 916(x - x")6(z - 2"). Application of EFIE (6.1) to the configuration of Figure 6.2, assuming y-invariant fields, leads to the completion of the y-integral in that equation to yield Ey(x,z) - ff 6k2(x')Ey(x',z')G(x,z|x',z')dx'dz' = E;(x,z) (6.36) where excitation field E1 y is defined by E;(x,z) = -chkc ff J;(x',z')G(x,z|x',z')dx'dz‘ , (6.37) 5k2(x) = kf - kg for xe(-d,d); = 0 otherwise, and G(x,z[x',z') = G(x',z'|x,z) = (2n)'1K0(jkc[(x-»x')2 + (z - z')2]*) defines the Z-d Green's function over the (x,z) plane. An appropriate field representation for Ey maintained by J; is Za'E' (x,z) + E' (x,z) . . . for z < z" r n n yn yR Ey(x,z) = ( + + + u ( ganEynk’z) + EyR(x,z) . . . for z > z (6.38) 218 where Ejn = nth natural surface-wave field of amplitude a: for z 2 z", and Ei . .Y R wave mode Eih is further defined by the homogeneous specialization of Eq. (6.36) = total radiation field for z 2 2". Natural, surface- E:m(x,z) - ff 6k2(x')E:m(x',z')G(x,z|x',z')dx'dz' = 0 ’” (6.39) or its adjoint E;m(x',z') - JK/fl 6k2(x)E:m(x,z)G(x',z'|x,z)dxdz = 0. (6.40) substitution of expressions (6.37) and (6.38) in EFIE (6.36) then yields the following expanded EFIE t 2 i 2n: anEyn(x,z) + EyR(x,z) 1:! 6k2(x') [ZnZaiEjnU'JW + Ei (x',z'):] G(x,z|x',z')dx'dz' YR + = -chkc ff J;(x' ,z')G(x,z|x' ,z')dx'dz' . (5-41) e y operating on each term of expanded EFIE (6.41) by the integral A relationship between a; and J can now be determined by operator 219 ff dxd26k2(x)E:m(x,z) (6.42) as follows. Operation on the right-hand side of Eq. (6.41) by operator (6.42) and interchange of order of integration of primed and unprimed variables and exploitation of adjoint equation (6.40) yields . 2 ‘T' e l u l l I 1 -Jchc ff dxdzdk (x)Eym(x,z) ff Jy(x ,z )G(x,z|x ,2 )dx dz . e 1 u '7' I I 1 I -chkc ff Jy(x ,z )Eym(x ,2 )dx dz ° :- II II -3chcIEym(x ,z ) . (6.43) In a similar fashion, operation on terms involving E: on the left- R hand side of Eq. (6.41) by operator (6.42) yields, after integration- order interchange and adjoint-equation application, 2 f i If dxdzék (x)Eym(x,z) [EyR(x,z) 2 2 - ff 6k (x')EyR(x' ,z')G(x,z|x' ,z')dx'dz'] .. ff 6k2(x)Ejm(x,z)EjR(x,z)dxdz - ff 6k2(x')E:m(X',Z')E;R(X'.Z')dX'dZ' E 0 . (5-44) 220 Finally, operation on the (remaining) terms involving E?" on the left-hand side of Eq. (6.41) by operator (6.42) yields that vanishing result obtained for E: when n f m; however, operation on the integral R term involving Eim results in an ill-behaved integrand for which exchange of integration order over 2 and z' is precluded [33], to yield ff dxd26k2(x)E:m(x,z) { [Za:E;n(x,z)] _m n .f[ 6k2(x') [§a:E;n(x',z')] G(x,z|x',z')dx'dz'l ("g") +f7 dxdzdk2(x)E:m(x,z) '[a;E:m(x,z)] If 6k2(x') [a;E;m(x',z')] G(x,z|x',z')dx'dz'l _ 2" iijz ijz a"I [eym(x) j: e e dz on I: dx6k2(x)eym(x) j: dx'6k2(x')eym(x') 1:: dz ememzeJBmZ oo .'8 x f eJmu G(x-x'|u)du] z-z" ' as ijB z -jB z + m m am [eym(x) L" e e dz + 221 m m ijB z -jB z - f dx'6k2(x')eym(x') f dze m e m z-z" iju x f e G(x - x' Iu)du] I (6.45) Integrals of Eq. (6.45) are completed by exploiting parts integration and Leibnitz's Rule [33] 4(2) 11(2) 3 + 1;“) Ef(z,u)du to yield (1 indicates that $z directed operator applied in Eq. (6.42)) 328ml" -j28mz [e ‘e 11 jZBm . . for + z" ijB z jB z . m m _ l1m ( .I. e e dz - z +‘m '°° z"-z...for'-' (6.46) ’ z - z" . . . for '+' w ijB z -jB z . .l‘ e m e m dz = zl1mon ( z" -j28mz" -j28mz e - e ‘ . . . . for '-' JZBm (6.47) m th z jB z w -jB u f dz e m e m f e m G(x - x' Iu)du -m Z'Z" -j28mz -Ym|x-x'| jZBmz" -vax-x'| _<_. )(e )+ <____. )(e ) 328m zYm 328m ZYm lim ( = . . . for '+' z + co ‘ lex-x'l ymlx-x'l m -jB u z i—— + z" _e____ +f ue m G(x-x' Iu)du 2v,“ arm _. . . for '-' (6.48) w :jB z -jB z z—z" jB u f dze m e m f e m G(x-x'lu)du oo e -Ym|x-x'| -Ym|x-X'| .. .18 u V 2(e ZYm )-z"(e zYm )- Laue m(:‘.(x-x'|u)du = lim ( . . . for '+' z+oo ( -j28mz -ym|x-x'l -j28mz" -Ym|x-x'| -(§—?___) (e > + (£_7____) (e ) 238m 2Ym J28111 2Ym . . for '-' . (6.49) Substitution of expressions (6.46) through (6.49) in Eq. (6.45) and subsequent application of EFIE (3.2) in this result yields an expression involving only a; or a$, corresponding to operations on Eq. (6.41) by operator (6.42) with E§m or E;m, respectively. 223 Equating this result to the right-hand side of Eq. (6.43) then yields l+ a; [on dx6k2(x) eym(x) m 2 (m iiju f dx'6k (x')eym(x') f ue G(x - x' Iu)du = -chkcI E;m(x",z") - (5.50) X Subsequent completion of the integrals on the left-hand side of Eq. (6.50) leads to the result I E; (X",Z") a 4 (6.51) m é£:§m e (x)dx describing amplitude of mth (i z-directed) TE surface-wave modes, H where ZIn = chc/Bm is the associated mode wave impedance, along a step-index, dielectric-slab waveguide, maintained by line-source current I at (x",z"). Generalization of expression (6.51) for arbitrary current distributions by superposition of line-source responses finally provides the known general result [9-11] e . . 3 . . . . f Jy(x ,z )Eym(x ,2 )dx dz am = - d” m (6.52) 2 2 eym(x)dx 224 describing amplitude of mth (i z directed) surface-wave mode excited by impressed excitatory electric currents. If surface-wave amplitude coefficients (6.52) are exploited in a field expansion for E, a compact field representation is obtained in terms of Green's function ejy as (2:refers to waves propagating in the :2 directions) i __ e | | i I I I I i Ey(x,z) ff J.y (x ,z )ny(x,zlx ,2 )dx dz + EyR(x,z) with ' sienlz-2'| eyn(x )eyn(x)e G: s '9' =" yy(x zlx z ) z; This result establishes the coupling between J; and the total guided- wave component field, from which the guided-power flow excited by J; can be calculated. Further research on the excitation of E: R is required. CHAPTER VII CONCLUSION An integral-operator technique, representing an alternative to conventional boundary-eigenvalue analysis for EM wave propagation in relatively-general, open-boundary dielectric waveguides has been presented. An equivalent polarization current was exploited to describe the scattering properties of the heterogeneous core of an open-boundary dielectric waveguide. From this description, an electric-field integral equation (EFIE) was established and sub- sequently specialized to describe axially-propagating, surface-wave and radiation modes supported by heterogeneous, longitudinally- invariant, open-boundary dielectric waveguides of arbitrary cross- section shape. Recovery of the well-known transverse field depen- dence and propagation characteristics of surface-wave and radiation modes supported by simple dielectric structures from the EFIES was obtained. Numerical solution of EFIEs describing surface-wave modes supported by planar, graded-index, dielectric-slab and step-index, rectangular dielectric waveguides was described. Finally, excita- tion of discrete, guided surface-wave modes and continuous-spectrum radiation modes along open-boundary dielectric waveguides was con- sidered. These topics have confirmed the validity of the integral- operator method and have demonstrated its potential for the 225 226 investigation of contemporary problems involving open-boundary dielectric waveguides. Major contributions provided by this dissertation toward research in the open-boundary dielectric waveguide and device areas center about the EFIE formulation and its subsequent applications. First and foremost, an EFIE formulation for EM wave propagation along heterogeneous dielectric structures based upon an equivalent-current description of heterogeneous cores has been established. Secondly, exploitation of boundary, polarization- charge layers for classification of TE, TM, and hybrid surface-wave and radiation-mode fields along open-boundary dielectric waveguides of arbitrary shape and heterogeneity have been identified. Thirdly, a new conceptually-exact theory (compared to differential-operator- based methods) for EM wave propagation along rectangular dielectric waveguides has been defined. Finally, a new conceptual method for study of radiation modes excited along inhomogeneous dielectric structures of complex Shape has been established. Possible future research topics extending from the integral- operator technique described include investigation of those propaga- tion, coupling, and scattering phenomena of EM waves interacting with graded dielectric waveguides and devices. First, extensions of the numerical solution methods considered in Chapter V are pro- posed to investigate the characteristics of hybrid, surface-wave and radiation modes supported by graded-index waveguides having arbitrary cross-section shape. Of this classification of problems, 227 investigation of hybrid, surface-wave and radiation modes along graded circular and rectangular fibers as well as along overlayed, rectangular-dielectric strips (appropriate for optical integrated circuits) are of immediate interest. Secondly, a rigorous integral- operator-based formulation of waveguide excitation is proposed to investigate the coupling of incident radiation to open-boundary dielectric waveguide structures. The initial formulation provided by Chapters II and VI have established a forced 3-d EFIE to describe the excitation of surface-wave and radiation modes along infinite open-boundary dielectric waveguides; however, more research is required to derive the coupling coefficients of modes supported by the open-boundary structure from this equation. Finally, exten— sion of the EFIE formulation to study the scattering, radiation, and mode conversion of longitudinally embedded heterogeneities (dielectric devices) along open-boundary dielectric waveguides is proposed. It has been demonstrated [20] that the scattering and radiation properties of a device discontinuity along a dielectric waveguide may be described by an equivalent polarization current which flows in excess of the equivalent polarization currents main- tained by an unperturbed waveguide system. Superposition of the scattered field maintained by the device discontinuity and the field maintained by the unperturbed waveguide system leads to a volume EFIE describing the total unknown field of the discontinuous waveguide system. Extension of the integral-operator technique for investigation of future research topics in open-boundary dielectric waveguides and devices is thus delineated. 228 Open-boundary dielectric waveguides indicate that class of waveguiding structures which guide EM waves along surfaces defining permittivity transition. Open-boundary dielectric waveguide struc- tures are finding increasing utilization in those millimeter-wave or optical applications where conventional closed-boundary wave- guides are not suitable. The integral-operator technique provides a powerful alternative method for study of propagation, coupling, and scattering of EM waves along open-boundary dielectric structures. Its potential as a viable research tool for the investigation of contemporary problems in open-boundary dielectric waveguides and devices is substantial. LIST OF REFERENCES LIST OF REFERENCES [1] D. Gloge, “Optical Fiber Theory: Opportunities for Advancement Abound," Radio Science, vol. 12, no. 4, pp. 479-490, July- August 1977. [2] I. Jacobs and S.E. Miller, "Optical Transmission of Voice and Data," IEEE Spectrum, vol. 14, no. 2, pp. 33-41, February 1977. [3] N. Mokhoff, "Communications and Microwave," IEEE Spectrum, vol. 18, no. 1, p. 43, January 1980. [4] J.R. Whinnery, "Status of Integrated Optics and Some Unsolved Problems," Radio Science, vol. 12, no. 4, pp. 491-498, July- August 1977. [5] H. Kogelnik, "An Introduction to Integrated Optics," IEEE Trans. Microwave Theory Tech., vol. MTT-23, no. 1, pp. 2-15, January 1975. [6] N.S. Kapany and J.J. Burke, Optical Waveguides, New York: Academic Press, 1972. [7] D. Marcuse, Theory of Dielectric Optical Waveguides, New York: Academic Press, 1974. [8] D. Marcuse, Light Transmission Optics, New York: Van Norstrand, 1972, Chapter 9. [9] R.E. Collin, Field Theory of Guided Waves, New York: McGraw- Hill, 1960, Chapter 11. [IO] V.V. Shevchenko, Continuous Transitions in Open Waveguides, Boulder, CO: Golem Press, 1971. PP. 22-48, 91-116. [11] D.P. Nyquist, D.R. Johnson, and S.V. Hsu, "On the Orthogonality and Amplitude Spectrum of Radiation Modes on Open-Boundary Waveguides," to be published in J. Opt. Sci. Amer., vol. 71. no. 1, January 1981. [12] E.A.J. Marcatilli, "Dielectric Rectangular Waveguide and Directional Coupler for Integrated Optics," Bell Syst. Tech. J., vol. 48, no. 9. PP. 2071-2102, September 1969. 229 [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] 230 J.E. Goell, "A Circular-Harmonic Computer Analysis of Rec- tangular Dielectric Waveguides," Bell Syst. Tech. J., vol. 48, no. 9, pp. 2133-2160, September 1969. D.R. Johnson and D.P. Nyquist, "Integral-Operator Analysis of Dielectric Optical Waveguides--Theory and Application," 1978 National Radio Science (USNC/URSI) Meeting, University of Colorado, Boulder, CO, Digest p. 104, November 1978. D.R. Johnson and D.P. Nquist, "Numerical Solution of Integral- Operator Equation for Natural Modes along Heterogeneous Optical Waveguides," 1979 National Radio Science (USNC/URSI) Meeting, University of Colorado, Bounder, CO, Digest p. 89, November 1979. 8.2. Katsenelenbaum, "On the Propagation of Electromagnetic Waves along an Infinite Dielectric Cylinder at Low Frequencies," Dokl. Akad. Nauk SSSR, vol. 58, no. 7, 1947 (in Russian). C.B. Shaw, B.T. French, and C. Warner, "Further Research on Optical Transmission Lines," Sci. Rept. No. 2 (AD-652501), Autonetics Division of North American Aviation, Anaheim, CA, 1967. R.C. Pate and E.F. Kuester, "Fundamental Propagation Modes on a Dielectric Waveguide of Arbitrary Cross Section," Sci. Rpt. No. 45, prepared for U.S. Army Research Office under Contract No. DAA629-78-C-0173 by Electromagnetics Laboratory, University of Colarado, Boulder, CD, February 1979. S.V. Hsu and D.P. Nyquist, "Integral-Operator Analysis of - Coupled Dielectric Optical Waveguide System-~Theory and Applica- tion," 1979 National Radio Science (USNC/URSI) Meeting, Univer- sity of Washington, Seattle, WA, June 1979. S.V. Hsu and D.P. Nyquist, "Integral Equation Formulation for Scattering from Obstacles in Dielectric Optical Waveguides," 1979 National Radio Science (USNC/URSI) Meeting, University of Colorado, Boulder, CO, Digest p. 90, November 1979. R.F. Harrington, Time-Harmonic Electromagnetic Fields, New York: McGraw-Hill, 1961, pp. 125-127. F.B. Hildebrand, Methods of Applied Mathematics, Englewood Cliffs, NJ: Prentice-Hall, 1965, Chapter 3. I. Stakgold, Green's Functions & Boundary Value Problems, New York: Wiley-Interscience, 1980. (241 [251 [261 [27] [28] [29] [301 [311 [32] [331 231 S.G. Mikhlin, Integralgguations, New York: MacMillan (Pergamon Press), 1957, pp. 7-15. E. Hecht and A. Zajac, Optics, Reading, MA: Addison-Wesley, 1975. PP. 71-94. I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals Series and Products, New York: Academic Press, 1965. K.A. James , R.R. August, and J.E. Coker, "Silicon, Monolithic Optical Integrated Circuits for Laser System Applications," Radio Science, vol. 12, no. 4, pp. 529-535, July-August 1977. R.F. Harrington, Field Computation by Moment Methods, New York: MacMillan, 1968, Chapters 1 and 7. E. Bahar and 8.5. Agrawal, "Application of Generalized Charac- teristic Vectors to Problems of Propagation in Clad Inhomo- geneous Dielectric Waveguides," IEEE Trans. Microwave Theory Tech. vol. MTT-27, no. 4, pp. 345-352, April 1979. J.W. Goodman, Introduction to Fourier Optics, New York: McGraw- Hill, 1968, Chapters 2 and 3. A. Papoulis, Systems and Transforms with Applications in Optics, New York: McGraw-Hill, 1968, Chapter 5. T.E. Rozzi, "Rigorous Analysis of the Step Discontinuity in a Planar Dielectric Waveguide," IEEE Trans. Microwave Tech., vol. MTT-26, no. 10, pp. 738-746, October 1978. J. Olmsted, Advanced Calculus, New York: Appleton-Century- Crofts, 1961, Chapters 9 afid’14. APPENDICES APPENDIX A SLABMODE (Version 12.3) SOURCE LISTING PRODRAR ELADNDIINPUT,OUTPOTI C C TNIO PROORAN CONPOTEE TNE CNARACTERIOTIC PROPAGATION (PHASE) C CONSTANTD (EIOENVALOES) AND, SUDSEOUENTLT, TNE TD EIOENFIELD C DIETRIDUTIONO OF TE AND IN NATURAL SURFACE NAVE RODEO ALONG C ORADED INDEX, DIELECTRIC OLAD UAUEOOIDED C C AUTHOR: DEAN R. JONNSON. PN.D. STUDENT, NICNIOAN STATE UNIVERSITY C CURRENT VERSION: 3LADNODE|2.3 (TRIO) C C INTEOER OOEOE LOGICAL EVONLY,NDONLT.NDPLOT,NDIDP,FDIOP,PEAVE CONNOR lPLTDOF/IDUF(513) COINON lPDATA/TYI127,6),IZ(127,7) CONNON lEVALOE/NNODE.EVDANN(IO).EVRAPNITOI,EODETN(IOI CONNON lEVCONP/ZDETNIIO) CONNON IPOEITN/NP,N8PLOT.NODOII,R(127),NDEL(I23) INPOT DECTION READ A CONTROL CARI OPECIFINO TNE NUNDER OF SETS OF DATA TO DE ANALTIZED DY THE PROORAN READ I.N8ET I FORNATIOX,IT) Oflflflflfl O'DO'DO'D READ A CONTROL CARD DPECIFINO NODAL DISPERSION, FREOUENCY DISPERSION, NODAL FIELD CONPARIDION, AND R-INDEX PROFILE CONPARIDION PLOTS READ 2.NDIOP,FDIDP.NCONP,NCONP 2 FORNATIAX,LI,7I,L|,7X,IT,7R.II) READ A CONTROL CARD DPECIFINO TNE PLOT DOEUE NONDER AND LINE PRINTER PAPER DAVE CONTROL READ 7,00EOE,POAVE 7 FORNATIAR,IT,7R,LT) 000 C IF PLOTTINO ID DEDIRED, OPEN TNE PLOT FILE IFIOOEUE.OT.5T OO TO D CALL PLOTSIIDUF,3|3,00EUE) CALL PLINITIIZO.) 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APPENDIX C PLOTTING ROUTINES DUDROUTINE PLOTITIFCN,ISIZ,INIT,LAST,YSCALE,YNANE,XNANE) C NOTE: THIS IS A LINE PRINIER PLOTTING ROUTINE C PLOIIT PRODUCED A GRAPH OF A ONE DINENSIDNAL FUNCTION OF DIIE ISII DETUEEN C A DPECIFIED INITIAL AND FINAL ELENENT. TSCALE GIVES THE T-AXIS SCALINO C THE X-AXIS SCALE IS AUTOMATICALLY ADJUSTED TO GIVE A FULL PAOE OUTPUT C INANE AND XNANE ARE THE RESPECTIVE LADELD TO THE I AND I AXIED C INTEGER DLANN,DOT,X.LINE.YNANE,XNANE DINENSION LINE(I2I) DINENSIDN FCNIISIZ) DATA DLANN,DOI,X/IN ,IN.,INXI C C C PRINT A LINE OF DOTD UNICN UILL DE IHE Y-AXIS, AND TAIIS NANE DO I I'I,I2I,I I LINEII)8DOT PRINT 2,LINE.TNANE 2 FORNAT(|NI,I2IAI,2N,AIO) C C DLANN OUT TNE LINE DO I III,12I,I 3 LINEII)-DLANN C C PUT A DOT IN LINEIAII, TO PRODUCE TUE HURTZONAL AXIS LINEIAI)'DOT C C CDNPUIE PROPER NUNDER OF DPACED DETUEEN EACN LINELX-ANIS SCALE) NDPACEIDO.OIFLOAT(LAST-INITOII IFINSPACE.LE.O) NSPACE'I IIEIAIIVL! PLDI not ten to A I-INII,LA8T,I ocuECL to: a INVALID rcn stusut IFIAIStYSCALEOFCN(I)).DI.60.0) so to an .conrur: INTEGER LOCATION VARIILE L-vscaLsarcu(x)ost.s . over an I IN tut sELtcttn LOCATION LINE(L)-X vaxut 40,L1NE 40 FORHAI(IHO,121AI) c OREPLACE x 011" a ILANK IN LINE LInEtL)-ntaux c 0P0! nacn not on axxs 1! cos: 11 ans DLANKEI LIIE¢61)-IOI c osuxr «spaces a: ID a: J-a,usracz.1 PRINT 02 a2 ronnar<1u ) 43 contxnut 4 coutxuut c -rnxur x-axts Ian: PRINT 3.xnauc s FORIA1(IMO,SIX,AIO) RETURN [II ""060 DUDROUTINE PDATAIXAXLTN,YANLTN,XINIT,XLADT,ITICK,HTICK,IDEC. C LINTYP, C NCAPN,ICAPN,LCAPN,ITYPC,CAPNCR. C NCURV,IDEFN,LDEFN,ITYPD,DEFNCR,IXANOT,IYANOT) C C PDAIA READS IN PLOT INFORNATION INCLUDIND DRAPHINO CONTROLS. 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Le mug-au .e.. .md. .¢s poaLe. u .uppodL u. o_ «maeau .9 .m..:. ugh .3...uu.. u ox=u. =-a .u_.a_.caou-~ «max». .dun. 4¢.og La ouagau ._oa. _.a.e: o u .pm.4..»_x~..:.=u..zpu-.~_ms.u.m_.x..»._.4.oc u-._=o..=. .3“ ..=_ua .n-....o...=—Jx¢x...4L dd.u bed. pum- a. mu..3.¢ u u u u=._..ou on ~x..~.~x.. _x....~... .n.......»..L a..u .u. a» xx. .9. p..a o ..-~..»~z.a...._.a»‘z.._L.-.»»...-.~.u._. dice au....~.~x.. ._._.....~..~ .~.nx.-~x ...~..u..x .....a¢d.~. ......a...- ....=u...._ on a. .ua.=u us. a... 000 .4ug..-._»..o..=_.x.p.....o...~..o......x¢ adcu .9»...........no....u..g..a«dx u .xuupx.p_z.x..o.4ua....-..o.¢.~.m_x¢x...._~.. dd.o .u~_¢ as» a... u u u u=.__.ou .. dug..._.~.._.._..d..» ._._E.._...._.._u..... ...:.=u.._.. .. .. o.olu~u¢x .o.o.uo.p—.~».u~ au-.\._x—»numqu¢u ......w.» .n...__.~» ._.«.=~4~¢_.¢.u..uu ddcu u...a...~.¢ d4¢gm.....¢ u=.~_xou N. .mam.4»..q._.»»._x.=..umx.d» .Jdcgm»..a....»..._=...d.:a» _.~z._.._ a. a. .a.-_m¢d.~z .s.._.._... ..:.=u....q N. a. sou...unuucca» ouuo.—naacza» :pax¢xx.»~s~unpa¢ax. - Anna u:z~—zeu o. .¢.~.z——:~a.~.apmcau.—.m—a- ..>-=ux.—-— o. o. .v poa‘ us— udcuo m=3—_zou an an.-ae...m°.» a...me.. .n «n a. ea .....mo.. .n a. on .._..uxuu..._ ..~..L»p_..o.._.¢.¢..................u.g=g dice ...._...~..g.__... .«x—od..o..~....uug................de.:»a dacu .o..¢.~=_....~ ..2».4....._.~.zuua..c.............do.=.. “ago .a.¢.-....uuad . «spa. .. . .apna .._._o..=_.4... .....u.4 . _s_.. ..-...._-............o.x.do.=.. ad.” ..»-=g....~ «— .- n....o.~ _. o. .- ....~.a.=g...~ ..e__.._uu. no.3» pg... uaaupzau «a n~.ou°s»umcs» o.—nao‘~ .u «a an on .QQOAmetn DUI-DU u u u 252 .....\3.... .... .o............a........33\3.....\ .e.... .o..3..3. a........ .e.... 3...3.3.. ...............3... 3...3.._. .o...U3............_............o_.3...........3.... 3......_. .o...U3..u..._.u......o......u.....3..u...u....3..u. 3...3.... .u3......3... ....... u u nupmuh mm4~uoxm ma suntan us» :— acaou u— pun: .auanuon. nae—cc: u so. auntaz mac: Manama chm. nun—d3xmo- no cyan—u a»... .unaat u u .pumxvgonoe- unuhaounao .3. ..=... .3.....3__3..e......3............_.3.3_ . ....................3.....3..._...=u.. . ...3..u.u...........3..u..3..._.3.... u ..».3........3...3.3u.....m......3..........3..3........ d... ..3.=. .e........ .93...... p... 00 ._.3.....o3.....u3..........3...........3.=u.3 . ..............3.....3..._.3.... u ..»_3...u.._..o....3._.........._3..........3........... d... . .u.._. .3. .......u .... .... .humx.-oxzu:¢=uax amp—ca. u. or auaxau zo—mcu.a—- no no aunts: us» u~=.:eu DU 90 .33.»... a .....3.... .o........._3_... .a...3.......... ...3.... ~ .. .5....3... .._..3._.. u a. .u...» 3.... ..o_:. .3. .3. x..... .3..33.._. .3. .. .....3... .3. ..3...u u puns-._.x—ocd - on.~.:——a— —.o...-— u on unwed use suns— uu—d¢~——u~ u .o......o.....3.........3.3...... .o.... .o..o...3. 3...3.... ....._.....o_.3.... .e....... .....3.........3... 3...3u._. .o...u3...............o............3......._.3..._ 3........ .._.3.3......_.u................o.........o....... 3......_. .93......3... ....... bums-bum. can oz—pm—uu annex me aunts: map >- nuxnzxuhu- an gunned. au>¢=u no uu-xaz pmuox¢d asp .c-nxca sage - no argue; he cunts: ask a» acacu a— nun: .uu-xaa mac: he made: sucu so. .:¢3:-.¢anzca cu99 - mumxu: «pun nuuuacxcex he usuao 1 area. .muacu .puax..¢-c. mannaecnam nau Isa—us .3.....3..3..~..q.~.......3...3......3.._ . ..........._..3....3.....3..._.3.=.3u ...3.......__.......3..u.....u....... u ..3.....u.....u....3u.......................3..3....o... d... ...... ..... .ed. ...3......3.3...u.................3.....3.=.3. ..u3........_.3.....3.....3..u. . ....3....m...3...3.3u..............3.3......3..3.3...... .... .u.... ... .......u .... ..u. .........3.... . ....3.......3_ ..3.=u..... . .. ..... .3. .._.. ....._..3. uuapun .o.ud.’-aua.u~ ....3_...........3. u...3.._. .....3.........3... 3...3.... .o...u............._..o_.3...........3..._ 3...3.._. .e..3.3.....o_........o......u..._.3......~....3.... 3...3.._. .o...... 3...3.... ............s.~....3. 3...3.._. 009090” DU 00 DU 253 .3. 3.3... ....3.3._._....o..3.......3.......... . ..u.................3.....3..._....3 . ..................u.3.....3.....3.... . ...—.....u...3.....3.........3...3...3......3....u.....u .... .u.... 3......... ....3 .... ...3......3.3...u3..........3.....3.....p... u ...3..u.u.....3..u.....u_.3.... . ....3...u....3.....3.—.3....ax...3...3.................. .... ...... .3. .4...3.. p... ..u. 77.3: — 13.7; . .- «3.2:. 4:03 a: 3.133 DUI-D APPENDIX D SLABMODE AND RECTMODE INPUT DATA LISTING SLADND INPUT DECN EXANPLEI1I DEARCHIND FOR EIOENVALUED; UNIFORN PROFILE NDETID NDISPtF FDIDPIF NCONP'D NCONPDO OUEUE-9 PSAUE'F TE'I DINUAUI40.1070457 NDNN-(1.SJ ,0.0 ) NDIC'(1.5205223 ,0.0 I EUONLYIF NDONLY'F NDPLOT-F POENIT NRPIDO NPCIZO XCOREl-.30 DPLDTIT c1-o.o c2-o.o c3-o.o ca-o.o ua-o nzso na-o nus-2 14.70 .0: o: 9.90 .03 03 7272 tt-r linuav-4o.1q7oa57. Hill-(1.53 .o.o ) unxc-ct.5293223 ,o.o 1 evoqu-r nIoILv-r uanor-r PBEN-t NRP'DO NPC'2D XCONE'-.30 DPLOT'T C1'0.0. C2'0.0 C3'0.0 C4-0.0 N1-0 N230 N330 NRC-1 . 14.7D .03 03 7272 SLADND INPUT DECN EXANPLEIDI CONPUTIND EIDENFIELDSIEIBENUALUES ASSUNED NNOUN), NODAL FIELD CONPARISION PLOT PERFORNED; PROFILES INTERNALLY GENERATED NSETIA NDISP-F FDIDP-F NCDNP'3 NCONP'D OUEUEID PDAUE'F TEIF DINUAUI17.619527D4 NDXN'11.01 ,0.0 I NDNCII1.0 ,0.0 1 EUONLIIF NDONLY'T NDPLOTIF PCEN-T NRPI40 NPC'ID XCDRE'-.30 DPLDTIT C100.0 C2I0.0 C3'0.0 C480.0 N1'0 N2-0 N380 NNODE'D 13.444011 14.669101 13.294911 11.145077 7.745094‘ 3634 TEIF DINUAUI!7.A|9S27D4 NDXNII1.01 ,0.0 1 NDXCII1.0 ,0.0 1 ENONLTIF NDONLI-T NDPLOT-F PDEN'T NIP-40 NPC'10 XCORE'-.50 SPLOTIT C1-0.0 C2-0.0 C330.0 C4'0.0 N130 N2I1 N310 NNODE84 11.409307 254 255 — nu... o. - nu—o. o. — nu.o. o. su—oasn on.nou-oux ou-usx onus-x ..3... ........ ....3... .....o.. . o... ......... . .... .o.......3 .....3.3._. .... s-uooos nnuauao .uscous onsxous unso-. mason-x cosmos a— new. u4~uo¢s “om—ccuxuo whoa. u._.ous nuns—nuonsuocsu- one nouns ..xaoxx ouzammo «waaoozwo—u.modu~szuo_u con—astou .nouastoxu sum. boson neocon .uu—u23opooov nu ounuacccon . \u. uses—ocean a own—dogma: — no no a nus s no no - 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