,. 1.0.69 26- I. 090.! .HHHII. “all? M4! 4.31.»... X9. 7. 129;? ( .fl 1" X‘ln‘l’. I. .1! null? #9.. r. L fir»... . 3.... mummunmmWW 3 1293 00 1 This is to certify that the dissertation entitled SCATTERING OF ARBITRARILY-POLARIZED EM WAVES BY A DISCONTINUITY IN A GROUNDED DIELECTRIC SHEET AND PROPOAGATION OF EM PULSES EXCITED BY AN ELECTRIC DIPOLE IN CONDUCTING MEDIA presented by Jiming Song has been accepted towards fulfillment of the requirements for Ph.D. degree in Electrical Engineering MM; Major professor Date 3//L/73 MS U is an Affirmatiw- Action/Eq ual Opportunity Institution 0-12771 LIBRARY 1 University Michigan State PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. DATE DUE DATE DUE DATE DUE We” \J§\J ix . l\ MSU is An Affirmative Action/Equal Opportunity Institution “chem {Ina-p. 1 SCATTERING OF ARBITRARILY-POLARIZED EM WAVES BY A DISCONTINUITY IN A GROUNDED DIELECTRIC SHEET AND PROPAGATION OF EM PULSES EXCITED BY AN ELECTRIC DIPOLE IN CONDUCTING MEDIA By liming Song A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Electrical Engineering 1993 ABSTRACT SCATTERING OF ARBITRARILY-POLARIZED EM WAVES BY A DISCONTINUITY IN A GROUNDED DIELECTRIC SHEET AND PROPAGATION OF EM PULSES EXCITED BY AN ELECTRIC DIPOLE IN CONDUCTING MEDIA By Jirning Song There are two parts in this thesis. In Part I, the two-dimensional problem of the inter- action between an arbitrarily-polarized electromagnetic (EM) wave and a dielectric dis- continuity in a grounded dielectric sheet is analyzed. An electric field integral equation (EFIE) is derived based on the dyadic Green’s function for the field produced by induced polarization currents in the discontinuity region. Impressed electric fields consist of either plane waves incident from space above the dielectric sheet or surface waves supported by that sheet; both TE and TM polarizations are considered. The EFIE is solved using Galer— kin’s method with pulse and sinusoidal/piecewise—sinusoidal (SPS) basis functions. Com- putation of the inverse Fourier transform representation of the Green’s function is performed by integration along. the real axis, along the branch cuts directly, and along the branch cuts with a variable change. It is found that the excitation of the surface-wave mode reduces the back scattered radiation field for TIE-polarized plane wave incidence. The reflected power for both 113- and I'M-polarized surface-wave incidence is calculated, and it is found that the numerical results agree very well with those of existing studies. The EFIE is also solved iteratively using successive approximations. These approximate results agree very well with the direct MoM (Method of Moments) solutions to the EFIE. In Part II, the exact solutions are given for the transient EM fields excited by an elec- tric dipole antenna with an impulsive current in a conducting medium. There exists an optimum waveform for the antenna current which can generate an EM pulse with a maxi- mum intensity at a particular distance from the antenna. It is found that the EM fields of an EM pulse excited by an antenna in a conducting medium can be divided into two parts. The first part is an impulse wave which propagates with the speed of light (1/ Ju_e) and decays exponentially. The second part builds up gradually and propagates slowly, and more importantly this part attenuates as an inverse power of distance which is a much slower rate than an exponential decay. Copyright by JIMING SONG 1993 To my wife Weiqun ACKNOWLEDGEMENTS I would like to express my heartfelt appreciation to my major professor, Dr. Kun-Mu Chen, for his sincere guidance and encouragement throughout the course of this work. I am also indebted to Dr. Dennis P. Nyquist and Dr. Edward J. Rothwell for their generous support and constructive suggestions during the period of this study. A special note of thanks is due Dr. Byron C. Drachman for his special assistance and concern to me. I owe a great deal to my parents for their love and continuous support. I must ac- knowledge my wonderful wife, Weiqun. Her love, sacrifice, constant encouragement and support have been invaluable in the successful completion of my graduate study. The financial support of K. C. Wong Education Foundation Ltd. (Hong Kong) is grate- fully acknowledged for making my study in USA possible. This research was supported in part by Boeing Defense and Space Group under Con- tract No. 13-225383, and by Naval Underwater Systems Center under Contract No. N66604-90—C-0685. USIOI nit,» Mu. TABLE OF CONTENTS LIST OF FIGURES .................................................................................................. x PART I SCATTERING OF ARBITRARILY-POLARIZED EM WAVES BY A DISCONTINUITY IN A GROUNDED DIELECTRIC SHEET Chapter 1. Introduction .......................................................................................... 1 Chapter 2. Problem Description and Green’s Function ....................................... 6 2.1 Introduction and Geometry ............................................................................ 6 2.2 Problem Decomposition and Electric Field Integral Equation (EFIE) _ __ .............................................. 8 2.3 Electric Field without the Discontinuity ....................................................... 9 2.4 2-D Dyadic Green’s Function for a Grounded Dielectric Material Sheet .............................................................. 11 Chapter 3. Propagation Mode Spectrum in Dielectric Slab Waveguide ............ 15 3.1 Introduction .................................................................................................... 15 3.2 Contour Defamation .................................................................................... 19 3.3 The Spatial and Eigenfunction Representations of Dyadic Green’s Function in Source Region .............................................. 24 3.4 Summary of Green’s Function ...................................................................... 29 Chapter 4. Calculation of Scattered Field and Method of Moments (MOM) Solution to the EFIE ........................... 33 4.1 Introduction .................................................................................................... 33 4.2 Evaluation of the Electric Field of Surface-wave Modes ............................. 34 4.3 Asymptotic Evaluation of the Far-zone Field ............................................... 37 4.4 The Scattered Power and Back-scattering Width ...... - -- - - - 48 4.5 MOM Solution to the EFIE ........................................................................... 52 Chapter 5. Narrow Gap Approximations - - .... . ..... - 56 5.1 Introduction ..................................... 56 5.2 Integral Equation and Successive Approximations ....................................... 57 vii 5.3 Narrow Gap Approximations for TE-polarized Plane Wave Incidence ................................................................................... 61 5.4 Narrow Gap Approximations for I'M-polarized Plane Wave Incidence ................................................................................... 67 5.5 More Numerical Results for a Narrow Air Gap ............................................ 74 Chapter 6. Pulse Galerkin’s Solutions for TE-polarized Wave Incidence ........... 83 6. 1 Introduction .................................................................................................... 83 6.2 Pulse Galerkin’s Solutions ....................................................... 83 6.3 Numerical Integration along Real Axis ........................................................ 91 6.4 Numerical Integration along Branch Cuts .................................................... 97 6.5 Comparison of the Integration Methods ........................................................ 103 Chapter 7. Pulse Galerkin’s Solutions for TM-polarized Wave Incidence .......... 108 7.1 Introduction .................................................................................................... 108 7.2 Pulse Galerkin’s Solutions ............................................................................ 108 7.3 Numerical Integration along Real Axis ........................................................ 117 7.4 Numerical Integration along Branch Cuts .................................................... 123 7.5 Comparison of the Integration Methods ........................................................ 127 Chapter 8. Sinusoidal and Piecewise-sinusoidal (SPS) Galerkin’s Solutions for TM-polarized Wave Incidence ..................................... 130 8.1 Introduction .................................................................................................... 130 8.2 Sinusoidal Functions and Piecewise-sinusoidal Functions ............................ 131 8.3 SP8 Galerkin’s Solutions .............................................................................. 137 8.4 Numerical Integration along Real Axis ........................................................ 150 Chapter 9. Numerical Results ................................................................................. 155 9.1 Introduction ................................................................................................... 155 9.2 Numerical Results for Surface-wave Incidence ............................................. 156 9.3 Numerical Results for TE-polarized Plane Wave Incidence .......................... 164 9.4 Numerical Results for I'M-polarized Plane Wave Incidence ......................... 170 9.5 Numerical Results for the Lossy Dielectric Sheet ......................................... 187 Chapter 10. Conclusions .......................................................................................... 198 Appendix A: Electric Field without the Discontinuity for TIE-polarized Plane Wave Incidence ......................................... 200 Appendix B: Electric Field without the Discontinuity for TM-polarized Plane Wave Incidence ........................................ 203 Appendix C: 2-D Green’s Function for TE Excitation of Grounded Dielectric Material Sheet ................................................ 206 Appendix D: 2.1) Green’s Function for TM Excitation of Grounded Dielectric Material Sheet ................................................ 215 Bibliography ....... - .......................................................................................... 226 viii PART II PROPAGATION OF EM PULSES EXCITED BY AN ELECTRIC DIPOLE IN CONDUCTING MEDIA Chapter 1. Introduction Chapter 2. EM Pulses Excited by an Electric Dipole Antenna with an Impulsive Current in Conducting Media 2.1 Exact Solutions for the EM Fields of Impulse Radiation Maintained by an Electric Dipole Antenna in Conducting Media .............. 2.2 Some Special Cases and Discussions 2.3 Several Computational Methods and Numerical Results Chapter 3. EM Pulses Radiated by an Electric Dipole Antenna with Various Excitation Currents 3.1 EM Fields of EM Pulses Generated by an Electric Dipole with Arbitrary Excitation Currents 3.2 The EM Fields of an EM Pulse Excited by an Electric Dipole with a Heaviside Step-Function Excitation Current 3.3 Optimal Excitation Current 3.4 Numerical Results of EM Fields of EM Pulses Excited by an Electric Dipole with Various Excitation Currents Chapter 4. EM Fields of EM Pulses Excited by an Electric Dipole Antenna with a Fixed ‘Ibtal EMP Energy in Seawater 4.1 Parameters of an Electric Dipole Antenna Generating an EM Pulse with a Fixed EMP Energy 4.2 Numerical Results of the EM Fields of an EM Pulse Generated by an Electric Dipole Antenna with the Optimum Excitation Current and with a Finite EMP Energy in Seawater Chapter 5. Effects of Antenna Size on the Propagation of EM Pulses Excited in Seawater Chapter 6. Interaction of an EM Pulse with a Conducting Cylindrical Shell in Seawater 6.1 The EM Fields in the Interior Space of the Shell 6.2 EM Fields at the Cylinder Axis 6.3 Numerical Results Chapter 7. Conclusions Bibliography - 232 235 235 243 253 269 269 27 l 277 280 286 286 289 295 301 301 313 314 2.1. 3.1. 3.2. 4.1. 4.2. 4.3. 4.4. 5.1. LIST OF FIGURES PART I SCATTERING OF ARBITRARILY-POLARIZED EM WAVES BY A DISCONTINUITY IN A GROUNDED DIELECTRIC SHEET Problem description and decomposition Branch cut and integration contour deformation on the top Riemann sheet (R3 {1’1} > 0) The path of integration along the branch cut on the top Riemann sheet (Re {p1} > 0). Im {p1} < 0 in the shadow region and Im {pl} > 0 in other region Polar coordinates in the x-z plane Mapping of the two Riemann sheets of the C plane onto a strip of the o = o +jn plane. The crosshatched regions are the proper Riemann sheet (top sheet) The original integration path and the steepest-descent path Deformation of contour C into steepest-descent contour (SDC) ....................... 18 22 39 43 45 62 The explanation of the resonance phenomenon when 6‘. -) 90° ....................... Normaliwd back scattering width as a function of frequency at various inci- dence angles 0, when a TE plane wave is incident upon a grounded sheet with a narrow air gap. Lines - First-order approximation, Points - MoM solutions ofEFIE. (ch = ed, = 1, £2, = 1.1, 2a = 0.06”, r = 1”, Nz = 2 to 8, and Nx = 5 to 30) Normalized back scattering width as a function of frequency at the incidence angle of 0‘ = 60° when a TE plane wave is incident upon a grounded sheet with a narrow air gap. Dashed line -- First-order approximation, Solid line - X 5.4. 5.5. 5.6. 5.7. 5.8. 5.9. 5.10. Second-order approximation, Points -- MoM solutions of EFIE. (£1, = 84, = l, 82' = 1.1, 2a = 1”, t = l”,Nz = 5 to 20, and N, = 5 to 20) Normalized back scattering width as a function of frequency at the incidence angle of 6‘ = 60° when a TE plane wave is incident upon a grounded sheet with a narrow air gap. Dashed line -- First-order approximation, Solid line -- Second-order approximation, Points - MOM solutions of EFIE. (ch = ed, = 1,22, = 2, 2a = 0.15”, t = 0.,N5” = 5 to 10, and N, = 5 to 20) Normalized back scattering width as a function of frequency at various inci- dence angles at when a TM plane wave is incident upon a grounded sheet with a narrow air gap. Lines - First-order approximation, Points - MoM solutions ofEFIE. (81, = ed, = 1, £2, = 1.1, Za = 0.06”, = 1”, N1 = 2 to 8, ande=5 to 30 Normalized back scattering width as a function of frequency at the incidence angle of 0, = 60° when a TM plane wave is incident upon the grounded sheet with a narrow air gap. Line -- First—order approximation, Points -- MoM so- lutions ofEFIE. (£1, = ed, = 1, £2, = 1.1, Zn = 0.3”, t = 1”, N2 = 3 to 8, ande= 5 to 30) Normalized back scattering width as a function of frequency at various inci- dence angles at when a TE plane wave is incident upon a grounded sheet with a narrow air gap. Lines - First-order approximation, Points - MoM solutions ofEFIE. (81' = ed, = 1, £2, = 1.1, 2a = 0.1", t = 1”, N2 = 2 to 8, ande = s to 30) Normalized back scattering width as a function of frequency at various inci- dence angles 91 when a TM plane wave is incident upon a grounded sheet with a narrow air gap. Lines -- First-order approximation, Points —- MoM solu- tions ofEFIE. (cl: -,1 £2 = 1.1, 2a = 0.1”, t = 1”, “(it Nz=2to8, andN =5t0 30) Normalized back scattering width as a function of incidence angle at at vari- ous frequencies when a TE plane wave is incident upon a grounded sheet with a narrow air gap. (£1, = e = 1, 52, = 1.1, Zn = 0.06”, t = 1”, dr Nz = 2 to 8,ande= 5 to 30) Normalized back scattering width as a function of incidence angle at at vari- ous frequencies when a TE plane wave is incident upon a grounded sheet with anarrowairgap. (£1, tz-dr -1,e2 =1.1, 2a=0.1”,t=l”, Nz =2to8,andN —5 to 30) 68 69 73 75 76 77 78 79 5.11. Normalized back scattering width as a function of incidence angle 91 at vari- ous frequencies when a TM plane wave is incident upon a grounded sheet with a narrow air gap. (21, = ed, = 1, £2, = 1.1, 2a = 0.06”, t = 1”, Nz = 2 to 8,andN, = 5 to 30) 5.12. Normalized back scattering width as a function of incidence angle 9i at vari— ous frequencies when a TM plane wave is incident upon a grounded sheet with a narrow airgap. (ell, = ed, = l, 82, = 1.1, 2a = 0.1”, t = 1”, Nz = 2 to 8,andN, = 5 to 30) 6.1. Applying Pulse Galerkin’s Method to the EFIE in the discontinuity region. 6.2. Integration along the real axis 6.3. The path of integration along the branch cut on the top Riemann sheet when region 1 is lossless 6.4. TEl surface wave E; is incident upon the discontinuity. There is a transmit- ted TELsurface wave E}, a reflected TE] surface wave 3",, and radiated waves E,“ 6.5. The numerical total power normalized by the incident power with TE] surface wave incidence vs. the width of the discontinuity. It is computed by three in- tegration methods: along the real axis, along the branch out with/without a variable change. (£1, = ed, = 1, £2, = 4, t/A.o = 0.25, N, = 13 and 1vz = 4 to 17) 6.6. Convergence check on the reflected power computed with difl‘erent N, and N 2 when a TB] surface wave is incident upon an air gap in the grounded sheet. (an = ed, = 1, 2,, = 2.1316, 2am0 = 1,and k0: = 25) 7 .1. Convergence check on the reflected power computed with difl‘erent N, and N , calculated by integration along real axis and integration along branch out when it IMO surface wave is incident upon an air gap in the grounded sheet. (61, = ed, = 1, £2, = 2.1316, 2a/i\0 = 1, and k0: = 2.5) 8.1. Extend E, (x, z) and E, (x, z) to periodic functions of x at a given 2 .............. 8.2. Applying sinusoidal and piecewise-sinusoidal (SPS) Galerkin’s Method to EFIE in the discontinuity region 8.3. Piecewise-sinusoidal functions 8.4. Integration along real axis in the SPS Galerkin’s method 81 82 84 94 100 104 105 107 129 133 135 136 152 8.5. 9.1. 9.2. 9.3. 9.4. 9.5. 9.6. 9.7. 9.8. Convergence check on the transmitted power computed with different N, and N, by sinusoidal and piecewise-sinusoidal (SPS) Galerkin’s method and pulse Galerkin’s method when a TMO surface wave is incident upon an air gap in the . groundedsheet. (e1, = ed, = 1,15,, = 2.1316,2a/1.0 = 1,and k0: = 2.5) Comparison of the reflected power computed with MOM when a TEl surface wave is incident upon an air gap in the grounded sheet (solid line) and that computed by Shigesawa and Tsuji [19] by the mode-expansion method when a TE] surface wave is incident upon an abruptly ended grounded sheet (dashed line). (2,, = 2.1316, k0: = 2.5, N, = 10, and N, = 2 to 100) The reflected power as a function of tllt when a T131 surface wave propagates through a step discontinuity in the grounded dielectric sheet computed with MoM (points) and by the mode-expansion method (line) [19]. (2,, = 2.1316) The radiated power, transmitted power, and reflected power normalized by the incident power as functions of the width of the discontinuity when a TE] sur- face wave is incident upon an air gap in the grounded sheet. (8,, = 5 , k0: =1,N, = 8,andN, = 2 to 50) Comparison of the reflected power computed with MoM when a TMO surface wave is incident upon an air gap in the grounded sheet (solid line) and that computed by Shigesawa and Tsuji [19] by the mode-expansion method when a TMO surface wave is incident in an abruptly ended grounded sheet (dashed line). (2,, = 2.1316, Ito: = 2.5, SPS Galerkin’s method with N, = 4 and N, = 2 to 62) - Amplitudes of the reflection coefficient no and the coupling coemcient a, to 1M2 mode computed by SPS Galerkin’s method (points) and by Neuman se- ries [14] (lines) as functions of kot when a TM, surface wave is incident in an abruptly ended grounded sheet for 6,, = 20 Amplitudes of the reflection coefficient a2 and the coupling coefficient on to TMO mode computed by SPS Galerkin’s method (points) and by Neuman se- ries [14] (lines) as functions of k0: when a TM2 surface wave is incident in an abruptly ended grounded sheet for 82’ = 20 The radiated power, transmitted power, and reflected power normalized by the incident power as functions of the width of the discontinuity when a TMO sur- face wave is incident upon an air gap in the grounded sheet. (e,= -,5 kot = 1, SP8 Galerkin’s method with N, = 3 and N, = 3 to 20) .............. Comparison of the normalized total electric field amplitude |E2 Eel (dashed line) and the incident field |E‘ y/r:(,| (solid line) when a TE-po plane xiii 154 157 158 159 161 162 163 165 9.9. 9.10. 9.11. 9.12. 9.13. 9.14 . 9.15. 9.16. 9.17. wave is incident upon a narrow gap in a grounded sheet at various incidence anglesfli. (E = e = 1,82 = 4520/10 = 0.01,” = 1, into = 0.125’ and if, = 12) ' z 166 The comparison of the normalized total electric field amplitude [Eb/E0] (dashed line) and the incident field IE'zy/Eol (solid line) when a TE-polarized plane wave is incident upon a narrow gap in a grounded sheet at various inci- denceanglesei. (g = E = ,1, 82 = 4520/11) = 0.01,” =1, into = 0.25 and N; = 235 ' ' z 167 Spatial intensity distribution of the normalized total electric field in the dis- continuity when a TE-polarized plane wave is incident upon the grounded sheet normally (0, = 0°). (6 = g r = 1, 6,, = 4, 2a/7t0 = 0.25, N, = 13, min = 0.125 and it, = ‘12) 168 Spatial intensity distribution of the normalized total electric field in the dis- continuity when a TE-polarized plane wave is incident upon the grounded sheet normally (0,. = 0°). (8 = g -.- 1, 8,, = 4, 2a/7to = 0.7, N, = 17, int, = 0.25 and N; = 12’) 169 Radiation patterns of the far field when a TE-polarized plane wave is incident upon the grounded sheet at various incidence angles 91- (e 1' = g r = 1, 2,, = 4, 211/10 = 0.25, N, = 134/10 = 0.125 and N, = 123 ................ 171 Radiation patterns of the far field when a TE-polarized plane wave is incident upon the grounded sheet at various incidence angles 91- (g I, = 8,, = 1, 8,, = 4, 2a/A.o = 0.7, N, = 17, t/A.o = 0.25 and N, = 12) .................... 172 Normalized back scattering width as a function of the thickness t/ 1.0 of the discontinuity when a TE-polarized plane wave is incident upon the grounded sheet at various incidence angles 91- (811' = g = 1, 8,, = 4, 2a/21.0 = 0.25, N, = 13,and N, = 4 to l 173 Normalized equivalent scattering width of the surface-wave mode excited as a function of the thickness t/71.0 of the discontinuity when a TE-polarized plane wave is incident upon the grounded sheet at various incidence angles 91- (3" = "3,, = 1, 8,, = 4, 2a/7to = 0.25, N, = 13, and N, = 4 to 14) 174 Normalized back scattering width as a function of the width 2a/A0 of the dis- continuity when a TE-polarized plane wave is incident upon the grounded sheet at various incidence angles 91- (3" = e r = 1, 8,, = 4, into = 0.125, N, = 13. and N, = 4 to 173 175 Normalized back scattering width as a function of the width 2a/Ao of the dis— continuity when a TE-polarized plane wave is incident upon the grounded xiv 9.18. 9.19. 9.20. 9.21. 9.22. 9.23. 9.24. sheetatvariousincidence angles0i. (air 3,, = 1, £2, = 4, t/i.o = 0.25, N, = 13,andN, = 4 to 17) 176 Normalized equivalent scattering width of the surface-wave mode excited as a function of the width 2a/A.0 of the discontinuity when a TE-polarized plane wave is incident upon the grounded sheet at various incidence angles 0,. (err = 3,, = 1, £2, = 4, t/A0 = 0.25, N, = 13, and N, = 4 to 17) ...... 177 The comparison of the normalized amplitude of x-component of the total elec- uic field |E2,/Eo| (points) at the center of a narrow gap (2 = 0) and the elec- tric field in a zero-width gap IE'b/Eol (solid line) when a TM-polarized plane wave is incident upon the gap in a grounded sheet at various incidence angles 0,. (e,’=gd =1,e2=1.,12a=0.l”,t=1",f=18GHz,Pu1se Galerkrn’s method with N, = 3 and N, = 30) 179 The comparison of the normalized amplitude of z-component of the total elec- tric field |E2,/Eo| (points) at the center of_ a narrow gap (2 = 0) and the elec- tric field In a zero-width gap | (eh/sh) 15', ,/E,,| (solid line) when a TM polarized plane wave is incident upon the gap in a grounded sheet at various incidenceangles0,. (gl 3, =1,e2,=1.,12a=0.1”,t=1”, f- - 18GHz, Pulse Galerkrn’ s method with N, = 3 and N, = 30) ................ 180 Spatial intensity distribution of the x-component of the normalized total elec- tric field in the discontinuity when a I'M-polarized plane wave is incident upon the grounded sheet normally (0 = 0°). (5 = 3, = 1, e2: —,4 2a/A.o = 0.7, t/71.o = 0.,25 PulseGalerkin’ s method wrth N, = 10 and N, = 025) 181 Spatial intensity distribution of the z-component of the normalized total elec- tric field in the discontinuity when a I'M-polarized plane wave is incident upon the grounded sheet normally (0. = 0°). (g = 1, 2,: —4, 2a/7to = 0.,7 t/ito = 0. 25, Pulse Galerkin’ 8 method with N, -10 and = 025) 182 Radiation patterns of the far-zone field when a 'IM-polarized plane wave is incident upon the grounded sheet at various incidence angles 0,. 1' = 3,, =1, 82 = 4, 2a/7t0 = 0.,7 t/A.o = 0.25,PulseGa1erkin’s me,thodwnhN =10andN,=025) 183 Normalized back scattering width as functions of the thickness t/7i.o of the discontinuity when a TM-polarized plane wave is incident upon the grounded sheet at various incidence angles 0,. (g = 1, e2: —,4 211/710 = 0.125, Pulse Galerkin’s meth‘od =striati't' N, = 8 and N, = 4 to 25) 184 XV 9.25. 9.26. 9.27. 9.28. 9.29. 9.30. 9.31. 9.32. 9.33. Normal'md equivalent scattering width of the surface-wave modes excited as functions of the thickness t/ 10 of the discontinuity when a I'M-polarized plane wave is incident upon the grounded sheet at various incidence angles 0,. (alr— _ g, = 1, e, = 4, 2a/7to = 0.125, Pulse Galerkin’s method with N,=8andN,=41025) 185 Normalized back scattering width as a function of the width 2a/ 710 of the dis- continuity when a I'M-polarized plane wave is incident upon the grounded sheet at various incidence angles 0,. (3,, = ed, = 1,32,: 4, t/lo = 0.,125 SPSGalerkin’smethodwithN, = 3 andN, = 3 to 40) 186 The loci of the z-axis prepagation constant of B, for the TE, surface-wave mode along a lossy sheet (8,, = 1, 62, = 4 -j0.4) 190 Normalized back scattering width as a function of the width 2a/2.o of the dis- continuity when a TE-polarized plane wave is incident upon the grounded sheet at various incidence angles 0,. Solid line: 82 = 4, Dashed line: e2r=4 j-0..4 (8, ”=5, =1, t/Ao =0.,125 N, = 13, and N, = 4 to 17) 192 Normalized back scattering width as a function of the width 2a/2.0 of the dis- continuity when a TE-polarized plane wave is incident upon the grounded sheet at various incidence angles 0,. Solid line: 82, = 4, Dashed line: t'.2,=4-j0..4(g1 =5, =1, t/l0 =0.,25 N, = 13, and N, = 4 to 17) 193 Normalized back scattering width as a function of the thickness t/A.o of the discontinuity when a TB plane wave is incident upon the grounded sheet at various incidence angles 0,. Solid line: 62, = 4, Dashed line: a, = 4 -j0. 4. (s =3,=1,2a/Ao=0.,,25N=13,andN,=4to 14) .................. 194 Normalized back scattering width as a function of the thiclmess t/Ao of the discontinuity when a TE-polarized plane wave is incident upon the grounded Sheet at various incidence angles 0,. Solid line: 22, = 4, Dashed line: 82'=4-j0"4(81=ed =1, 2a/7t0 =05, N, = 13, and N, = 4 to 14) 195 Normalized back scattering width as functions of the thickness t/ 1. of the discontinuity when a TM-polarized plane wave is incident upon the grounded Sheet at various incidence angles 0,. Solid line: 82 = 4, Dashed line: $1,, = 4- -j0..4 (31 = 1, 2a/A0 = 025, PulseGalerkin’s method WrthN,=10andl:=N “J to 25) 196 Normalized back scattering width as a function of the width Za/Ao of the dis- continuity when a I'M-polarized plane wave is incident upon the grounded xvi A.1. B.l. C.l. DJ. 2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7. sheet at various incidence angles 0,. Solid line: 22, = 4, Dashed line: e , = 4—j0.4. (8,, = £4, = l, t/l.0 = 0.25, SPS Galerkin’s method with Af,=5andN,=3 to 40) 197 TE-polarized plane wave incidence - - 201 I‘M-polarized plane wave incidence -- 204 Geometrical configuration 3 (i) = 91, (x, z) to: 2.13 TB excitation ............ 207 Geometrical configuration. 7 (i) = 21, (x, z) + 21, (x, z) for 2-D TM exci- tation 216 PART II PROPAGATION OF EM PULSES EXCITED BY AN ELECTRIC DIPOLE IN CONDUCTING MEDIA An EM pulse generated by an electric dipole antenna in conducting media. 237 Integration path for inverting Laplace transform to obtain time domain electric field and magnetic field of an EM pulse propagating in conducting media. 255 The detail of the integration path Cb indicated in Figure 2.2 257 Comparison the numerical results (using FFT algorithm) of the electric field (0-c0mponent) maintained by an elecuic dipole with an impulsive current with the corresponding exact solution and the late-time approximation (t it r/v and t » to) of the exact solution 262 Comparison the numerical results (using FFT algorithm) of the electric field (0-component) maintained by an electric dipole with an impulsive current with the corresponding exact solution of the same quality at the late-time ap- proximation (t it r/ V and t lb 10). (1'=0.2 m) 264 Comparison the numerical results (using FFI‘ algorithm) of the electric field (0-component) maintained by an electric dipole with an impulsive current with the corresponding exact solution of the same quality at the late-time ap- proximation (t » r/v and t » to). 0:] m) 265 Electric and magnetic fields maintained by an electric dipole with an impul- sive current at a distance of 10 m in seawater 266 xvii 2.8. 2.9. 3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 4.1. 4.2. 4.3. 5.1. 5.2. Ebctric and magnetic fields maintained by an electric dipole with an impul- sive current at a distance of 100 m in seawater 267 Electric and magnetic fields maintained by an electric dipole with an impul- sive current as functions of normalized time (a2 = our ) in seawater ............ 268 Electric and magnetic fields maintained by an electric dipole with a step-func- tion excitation current as functions of normalized time (02 = ourz) in seawa- ter 274 The amplitudes of Fourier transforms of electric and magnetic fields main- tained by an electric dipole with an impulsive current at a distance of 100 m in seawater 276 The total energy of the 0-component of electric field maintained by an electric dipole antenna excited by an impulsive current in seawater 279 Various forms of excitation currents for an electric dipole including rectangu- lar, exponentially decaying, Gaussian and optimum (designed for a distance of 100 m) excitation 283 The 0 components of electric fields maintained by an electric dipole with var- ious excitation currents at a distance of 100 m in seawater 284 The 0 components of electric fields maintained by an electric dipole with var- ious excitation currents at a distance of 10 m in seawater 285 The waveforms of the elecuic field of an optimally excited EM pulse at the distance of 10 m, 50 m, and 100 m away from the antenna plotted as functions of time 290 Intensity of the maximum electric field as a function of distance, which is ex- cited by a dipole antenna with optimum current excitation (with the waveform of Figure 3.4) and input EM pulse energy of 1 Megajoule 291 Poynting vectors maintained by an electric dipole in seawater at two different instants: (a) t is small: (b) t is large (compare to the duration of the excitation current) 294 The electric field of the EM pulse generated by an antenna of finite length with a step-function excitation current in seawater 296 The normalized electric field of the EM pulse excited by cylindrical antennas of various lengths with a stop-function excitation current in seawater in the broadside direction as functions of the distance from the antenna .................... 300 xviii 6.1. 6.2. 6.3. 6.4. An EM pulse is incident upon a conducting, cylindrical shell in seawater ........ 302 Electric field at the cenual axis of an infinite cylindrical conducting shell in- duced by an incident EMP as a function of shell thickness. The EMP has an electric field parallel to the cylindrical axis and is excited by a dipole antenna which is located 100 m away from the shell and is driven by an optimum exci- tation current with the total EMP energy of 1 Megajoule 309 Electric field at the central axis of an infinite cylindrical conducting shell in- duced by an incident EMP as a function of shell outer radius. The EMP has an electric field parallel to the cylindrical axis and is excited by a dipole antenna which is located 100 m away from the shell and is driven by an optimum exci- tation current with the total EMP energy of 1 Megajoule 310 Incident electric field of an EMP and induced electric field at the central axis of an infinite cylindrical conducting shell as functions of time. The EMP is excited by a dipole antenna which is located 100 m away from the shell and is driven by an optimum excitation current with the total EMP energy of 1 Megajoule 312 PART I SCATTERING OF ARBITRARILY-POLARIZED EM WAVES BY A DISCONTINUITY IN A GROUNDED DIELECTRIC SHEET CHAPTER 1 INTRODUCTION Scattering of an electromagnetic (EM) wave by a conducting body can be controlled or modified by coating the body with a dielectric sheet. However, when there is a discon- tinuity in the dielectric sheet, undesirable EM wave scattering can occur. If the interaction of an incident EM wave with a discontinuity in a grounded dielectric sheet is determined, and the scattering of that EM wave excited by the discontinuity can be analyzed accurately, it may be possible to mitigate the undesirable scattering due to the discontinuity. It may also provide information useful for the selection of an appropriate dielectric coating for the purpose of controlling the FJVI wave scattering by a metallic body. On the other hand, the problem of surface wave scattering from discontinuities or gaps along dielectric waveguides is also very important, because accurate knowledge of scattered waves permits the design and development of optical and millimeter-wave com- Ponents. A discontinuity problem arises in many forms such as in the splicing of two di- electric waveguides or in inter-device coupling of millimeter and optical integrated circuits. The theory of planar waveguides may be found in [ll-[4]. There are discrete surface- WaVe modes and continuous radiation modes supported by dielectric waveguides. Be- cause of the importance of dielectric waveguide discontinuities for many applications in optical and millimeter-wave circuits, various analytical methods for the study of disconti- n‘lities along dialecuic-slab waveguides have been developed [5]-[31], and most of them 2 assume that fundamental TE and TM surface-wave-mode incidence in their studies. A few results have been reported for higher order surface-wave incidence and plane wave in- cidence. A literature review covering the analysis of discontinuities in planar waveguides was given by Liu and Chew [30] recently. One difficulty encountered in the analysis of planar waveguide discontinuity problems is the treatment of the radiation modes, which have a continuous spectrum. Various types of mode matching methods have been applied. For small steps, Marcuse [5] derived a simple approximation; Clarricoats and Sharpe [6] ne- glected the continuous radiation modes; Hockman and Sharpe [7] proposed the use of a first-order variational solution which neglected the backward radiation loss at the step. For large abrupt step discontinuities, [16] introduced a metallic enclosure to the original waveguide structure in order to discretize the radiation spectrum; the Laguerre transform has been used in [8], [l9]; and [15] approximated the waveguide structure in the trans- verse direction so as to discretize the continuous radiation modes. In addition to the various types of mode matching methods, variational approaches have been applied successfully. Morishita et a1. [12] formulated a least-squares boundary residual method. The Ritz-Galerkin (RG) variational approach has been used by Rozzi [10] to analyze a step discontinuity. A successive iteration method has been proposed by Gelin et a1. [13], [14], and has been improved by Morita [31] recently. Hirayama and Ko- Shiba [24] have developed a combination of the finite element method (FEM) and bound- any element method (BEM). Chung and Chen [20]-[22] proposed a partial variational Drillciple for the analysis of the waveguide discontinuity problem. In addition to the above methods, the beam-propagation method (BPM) [25]-[29] has been Proposed to calculate the propagation of waveguide modes through waveguide dis- c°ntinuities Recently, Liu and Chew [30] proposed an eigenmode propagation method (BPM), 3 One of the most powerful and commonly used techniques to analyze planar waveguide discontinuity problems is the integral equation approach [32]-[38]. An electric field inte- gral equation (EFIE) can be derived based on the dyadic Green’s function for the electric field produced by induced polarization currents in the discontinuity region [351-[38]. The EFIE is solved by the Method of Moments (MoM) [321-[34]. This method has several ad- vantages over other existing approaches, viz. (1) the impressed electric field can consist of either plane waves incident from space above the dielectric sheet or surface waves sup- ported by that sheet, (2) it is conceptually exact, and (3) it is believed to be computational- ly less cumbersome than the finite-element and related methods. Recently, Moore and Ling [39] [40] proposed to analyze the electromagnetic scatter- ing from an isolated conductor-backed dielectric gap using a boundary integral approach and a spectral integral approach. To facilitate numerical implementation, the specular so- lution and the surface wave contribution are removed from the boundary integral equation. Consequently the discretization domain of the infinite structure can be reduced to a local- ized region near the gap. Numerical diffraction coefficients and surface wave launch coef- ficients are extracted from the scattered field. In this thesis, the'two—dimensional (2-D) problem of arbitrarily-polarized electromag- netic wave interactions with a dielectric discontinuity in a grounded dielectric sheet is ana- lyzed, An electric field integral equation (EFIE) is derived based on the Green’s function for the field produced by induced polarization currents in the discontinuity region. Imlil'essed electric fields consist of either plane waves incident from space above the dielecu-ic sheet or surface waves supported by that sheet; both TE and TM polarizations are considered. The EFIE is solved using Galerkin’s method with pulse and sinusoidal! pie(Blithe-sinusoidal (SPS) basis functions. Computation of the inverse Fourier transform rePrehentation of the electric Green’s function is performed by integration along real axis, along the branch cuts directly, and along the branch cuts with a variable change. The fence,“ power for both TE- and I'M-polarized surface wave incidence is calculated, and 4 it is found that the numerical results agree very well with those of existing studies. The EFIE is also solved iteratively using successive approximations. First, both the zeroth- and the first-order approximations (leading terms of the Neumann series) for the total elec- tric field in the discontinuity region are derived when a TE- or I'M-polarized plane wave is incident upon a narrow gap in grounded sheet. Then, the first- and second-order approxi— mations ‘0 the scattered field are obtained by using the steepest-descents method. These approximate results agree very well with the direct MoM solutions to the EFIE. ’Ihere are ten chapters in this part. Chapter 1 gives an introduction for this research and a literature survey, Chapter 2 presents a description of the problem to be solved and the derivation of the EFIE- Details on the derivation of the dyadic Green’s functions and the electric field in the mnermrbed dielectric sheet with re- and TM-polarized plane wave incidence are given in Appendices. In Chapter 3, the complex analysis is applied to the spectral integral representation of the Gwen’s function to identify the propagation-mode spectrum in the dielectric slab wa"egtlide. Then, those Green’s functions are rewritten for the field in a source-flee re- gion in terms of integrals along branch cuts and residues at surface-wave poles. Finally. some, Observations on the Sommerfeld-integral representation of the dyadic Green’s func- 6°“ in the source region will be discussed. Chapter 4 presents the derivations of formulas to calculate the surface waves excited, and the far-zone field is obtained by using the saddle-point method Then, MOM is applied ‘0 t1’iIIISform the integral equations into a matrix equation. In Chapter 5, the EFIE is solved iteratively by using successive approximations. First, the zerOtlt- and the first-order approximations for the total electric field in the discontinuity logic, are derived when a TE- or TM-polarized plane wave is incident upon a grounded sheet With a narrow gap. Then, the first- and second-order approximations to the scattered 5 field are obtained by using the steepest-descents method. These approximate results agree very well with the direct MoM solutions to the EFIE. The EFIE is solved numerically using Galerkin’s method in Chapter 6 for TE illumina- tion. Computation of the inverse Fourier transform representation of the electric Green’s function is performed by integration along the real axis, along the branch cuts directly, and along the branch cuts with a variable change. It is found that the integration along the branch 01118 directly requires the least computation time, and allows a separation of the field into bound surface-wave modes and radiation modes. For M illumination, the EFIE is solved using Galerkin’s method with pulse and sinu- somaypiecewise-sinusoidal (SP8) basis functions in Chapters 7 and 8, respectively. Some numerical techniques are also discussed in these two chapters. Chapter 9 presents the numerical results for both TE- and TM-polarized surface-wave madame and compares them with those of existing studies. Next, the electric field distri- butions, radiation patterns, and back scattering width are quantified for TB- and TM—polar- imd plane wave incidence. F'mally, the back scattering width for a lossy dielectric sheet is calculated, In the final chapter, Chapter 10, the research accomplishments are summarized. Con- clusions are drawn from the collected results of this part. CHAPTER 2 PROBLEM DESCRIPTION AND GREEN ’8 FUNCTION 2.1. INTRODUCTION AND GEOMETRY In this chapter the Electric Field Integral Equation (EFIE) will be derived. First, the problem is described and decomposed by using the principle of superposition. Then, the electric field without the discontinuity and the Green’s function are found to form the EFIE. Some details will be given in the appendices. The configuration of electromagnetic (EM) fields scattered from the discontinuity in a grounded dielectric sheet is described geometrically in Figure 2.1(a). Region 1 is a half- 8pace dielectric medium with permittivity e, and permeability 110 (it is usually air). Re- gion 2 represents dielectric sheet of complex permittivity £2 and permeability 1.10 with a discontinuity region of permittivity ed and permeability 110. Region 3 is a perfect conduc- 101'. When a TE- or I'M-polarized plane wave is incident on the sheet, if there is no discon- tinuity, the scattered EM field is the reflected field which is easily derived. If there is a dis- continuity, the problem becomes quite complicated. Since this problem can not be solved ahalytically, an Electric Field Integral Equation (EFIE) will be derived in the next several sE=Ctiorls, and is solved numerically in the following chapters. .. \\\\\\\\ \\\\\\\\\ .0 \\\\\\§\\\\\\\\\\\\\\\\\\\\\\ if. M Flgure 2.1. Problem description and decomposition. 2.2. PROBLEM DECOMPOSITION AND ELECTRIC FIELD INTEGRAL EQUATION (EFIE) The total electric field E can be decomposed into the electric field is“ without the dis- continuity and the scattered electric field is“ maintained by the equivalent current i" in the discontinuity region (Figure 2.1) by using the principle of superposition. Then 3 = iii 4» ii” (22.1) If the x-axis is normal to the horizontal sheet interfaces while the waveguiding z-axis is parallel to them, then when a plane wave is incident (the propagation direction is in x-z plane), a y-invariant electric field is excited by the induced current. The scattered electric field E: can be written as: 4: _ .eq 15 (1,2) = I G (x,zlx',z') OJ (x',z')dx’dz' (2.2.2) LCS where LCS designates the longitudinal cross section of the discontinuity region and 5 (1:, 2| 1', z') is the electric dyadic Green’s function. The equivalent current .7“ is identified by adding and subtracting the displacement current of the unperturbed waveguide in the Ampere’s-law Maxwell equation [harmonic time dependence exp (10):) implied but suppressed throughout] to obtain ini (x, z) = .7. (x, z) +10) [ed (1:. z) ~82] E (x, z) +jc082§ (x, z) a‘ neg .3 = J (x,z) +J (x,z) +jco£2E (x, 2) (22.3) where 3' (1,2) is the impressed electric current, which maintains an impressed incident .3 a field a (x, 2) without the discontinuity, and the induced current 1“ (x, z) , nonvanishing only in the discontinuity region, is expressed in terms of the total electric field E in that region as 9 3‘9 (x, z) = 10) [ed (1, z) - 6211: (x, z) = jco5n2 (x, z) EOE (x, 2) (22.4) where 5.30:, z) = [ed (x, 2) -e2] /e0 = ed, (x, z) -ez, (22.5) with 84, = 84/60, and £2, = 82/80. Rearranging i = 3,445.: leadstothe EFIE: E(x,z) -jcoeo I 5112 (x',z')C—i(x,zlx',z') 03(x',z')dx'dz' = E'(x,z) (2.2.6) LCS In the next two sections, the electric field 3‘ without the discontinuity and the dyadic Green’s function E (x, zl x', z') will be derived. 2.3. ELECTRIC FIELD WITHOUT THE DISCONTINUITY It is easy to find the elecuic field E; when there is no discontinuity. The details are given in Appendices A and B for TE- and I'M-polarized plane wave incidence, tespective- ly. The final results are E l' I] . .1 : Regionl: 0x' - ' = 2.4.15 senor x)_ {__1 “I. ( ) pi = $3-192 1': 1,2 (2.4.16) 5, = plcosh (p21) +p2(81/€2) sinh (p2!) (2.4.17) 5,, = pzcosh (pzt) +p1sinh (p21) (2.4.1:) A. = 0 is the eigenvalue equation of TM-even modes, and 5,, = 0 is the eigenvalue equation of TE-odd modes. ‘ . pp» 1. Lit.» 12:? up; 14 Note that to properly ensure waves which decay as they propagate, the appropriate branch of the square root which determines p 1 from (2.4.16) is that which demands Re {pi} >0 Im {p5} >0 1' = 1,2 (2.4.19) where Re {...} designates the real partofthe quantity within the braces, Im {...} desig- natestheimaginary part. These Green’s functions satisfy the reciprocity theorem, i.e. G (x,zlx‘,z') = G (x',z°lx,z) 22“" 2251: (2.4.20) a=x.y.z B=x.y.z It is clear that all Green’s functions are functions of (z - 2’). These Green’s functions can be thus rewritten as 65203 (x, 2] x', z') = 612013 (xlx', z - 2') (2.4.21) a=x.y.z B=x.y.z i= 1.2 Therefore, EFIE in Section 2.2 in the discontinuity region is rewritten as: .3 k -— .5 .5! 32(1, 2) -j—9- I 5112 (x', z') 022 (xl x', z - z') 0 82(1', 2') dx'dz' = E; (x, 2) (2.4.22) oLCS where *o = Jean.) (2.4.23) 1"0 = «Jpn/ea (2.4.24) The electric field without the discontinuity and the dyadic Green’s functions have been derived and the EFIE is formed. These Green’s functions are expressed as Sommerfeld in- tegrals along the real axis in the C-plane. In the next chapter, the complex analysis is ap- plied to rewrite these Green’s functions in terms of integrals along branch cuts and residues of surface-wave poles. CHAPTER 3 PROPAGATION MODE SPECTRUM IN DIELECTRIC SLAB WAVEGUIDE 3.1. INTRODUCTION In this chapter, the complex analysis is applied to the spectral integral representation of the Green’s function to identify the propagation-mode spectrum in dielectric slab waveguide. Then, these Green’s functions for the field in a source-free region will be re- written in terms of integrals along branch cuts and residues at surface-wave poles. Finally, some observations on the Sommerfeld-integral representation of the dyadic Green’s func- tion in the source region will be discussed. The functions p 3 = C2 - k? (1' = 1, 2) are double-value functions because, in taking the square root of a number, two values are possible. Then, ikl, :th are branch points. . . . —R . After some algebraic mampulatron. G 22 (xl x', z - 2') can be rewrrtten as m " G§m(xlx'.z-z') = 723] {pllsinhlpzflx-x'l-IH +sinh1p2(x+x'+:)11 81 C2 C(z’z') —E-p2[cosh [p2 (lx-x'I-r)] +cosh [p2 (x+x'+t)]] }——.—dC (3.1.1) 2 41tp2A¢ cfmolxzz-z') = I {pllsgn (x-x') cosh [p2(lx-x’l-t)] -cosh [p2(x+x'+t)]] —ee 15 16 El --P2[sgn(1‘1)3inh[Pz(lx- Il-F)]"$inh[P2(1+1 +3)“ }4—5——2dc nguhlx'J-Zi) = I{p1[sgn(x—x')cosh[p2(Ix—x'l-t)] +cosh[p2(x+x'+t)]] _1e e—i-pzlsgnh X')sinh[pz(|x- x'l-t)]+sinh[p2(x+x' +t)]] }T—2dc m " ngu(x|x',z-z') = "7:! {Pll’SiRhIPzflx-‘X'I-IH +sinh [P2(X+I"H)]] p2 JCQ- z‘) +—p2[cosh [p2(lx- x'I-r)] -cosh[p2(x+x‘ +r)]] } —d§ (3.1.4) 4nA¢ c§”(x|x'.z-z') = 21211,] {pgl-Sinhlpzflx-x'l-IH +sinh1p2(x+x'+r)11 JUz-z') +p1[coshp2(t-Ix-x'l) -coshp2 (1+x +x')] } .. dc (3.1.5) 41tp2Ah The integrands in the Green’s functions Emma: -z') and 522(xlx',z -z') are even functions of p2. Hence, 21:1:2 are removable branch points and only branch points at ikl are significant. This observation can be generalized to a n-layered environment (n=3 in this problem, the third layer is the perfect conductor). Only :l:kl and 21:]: n are branch points, as discussed by Chew [44]. A double-valued function is represented on the complex plane with the help of Rie- mann sheets [44]. No Riemann sheets can be assigned to a single double-valued com- l7 plex function. On one of the Riemann sheets, pl assumes just the opposite sign to the value on the other sheet. The branch cut separates these sheets. In the integrations along the real axis to calculate the Green’s functions 512(xlx',z-z') and 522(xlx',z-z'), Re {pi} >0, Im {pi} >o,.- = 1,2 are chosen to satisfy the radiation condition that leads to outward-propagating and attenuating waves. Let the integration contour reside on the top Riemann sheet, then Re { p1} > O and In: {pl} > 0 along the real C axis on the top Riemann sheet. When the contour is de- formed to the infinite semicircle in the upper half plane (UI-IP) (not crossing the branch cut), Re {p1} > 0 should be chosen to force the integrand of (712 (xl x', z - z') vanish on the infinite semicircle. Hence, it is expedient to define a t0p Riemann sheet on the com- plex C plane on which R e {p l} > 0. The corresponding bottom Riemann sheet is then de- fined by Re {p1} < 0, and the branch cut that separates these sheets is defined by Re {p1} = 0. To find such a branch cut, first assume that *1 = 121,412”, C = Cr+jCi so that *1 and C are both complex numbers. Then pi = C2 4? = (C3 - C?) - Uzi,- kf.) +12 (CA-41,41.) (3.1.6) InorderforRe {p1} = 0, the conditionsarethat 10103} = 0 Re {pi} <0 (3.1.7) whichleadsto C5. = kirk“ C3-C? O). liCO L'siz 19 3.2. CONTOUR DEFORMATION Using the relations I ch““"’r(c> = j acii'mluc) when f(C) is an even function of C, and I dcl““"’r(o = sgn(z-z') j ch"""'r(§) (3.2.1) (3.2.2) when f(C) is an odd function of C, the integrations in 5:2 (xl x', z - 2') can be rewritten 1n " -. ngap(xlx',z-z') = 723 IgaB/m zldC where g = _§2 {e-pzlx-r|+e-p,(2:+x+x') p -p e /e _ -2 1 .21 2: p"cosh[p2(r+x)]cosh[p2(t+x')]} ¢ __ A 1 1C . -p,|x—x'l -p,(2:+x+x') 3xz' sgn(z-z)z;{sgn(x-x)e -e p -p 8 /€ - -2 1 £2 ‘ 22 p’tcosh[p2(r+x)]sinh[p2(r+x')] } ¢ , 1C , -p,lx-x'l -p2(2r+x+x') gu=sgn(z-z)2—n{sgn(x-x)e +e 21’1’1’2‘31/32 13 ¢ e"='s11111 [1.2 (1 +x)] cosh [pz (1 +20] } (3.2.3) (3.2.4) (3.2.5) (3.2.6) tare - mu F; —p lx-x'l -p (2t+x+x') 822 = I; {e 3 “'6 2 Pl-P231/52 2 z e A ””"sinh [p2(t+x)] sinh[p2(r+x')] } (32.7) 2 k2 —p,lx-x'l -p,(2:+x+x') -— {e -e 3” + 25%;”.111111 [p2 (1 +1)] sinh [,12 (1 +1')] } (3.2.3) It Since the 8a]! are even functions of p2, only the :11:1 are branch points. The integra- tion contour is chosen as shown in Figure 3.1 using the branch cut which was introduced in the last section. The integrand in (3.2.3) is analytic and Cauchy’s theorem for the con- tourintegralleadsto [gag/("’“dt; = 4! +1 +1 nap/{"’“dc (32.9) where c. designates the infinite semicircle, C b designates the contour along the branch cut, and C p designates the small circle around the surface wave pole at -Cp, which is the solution of the eigenvalue equation of TM-even modes or TE-odd modes. In any source-free region, 1: at x' and z ¢ 2', it can be shown that the gap exponentially decay when |C| -) co and the integral along the infinite semicircle C_ vanishes. Thus 1811111292”qu = 0 (32.10) and IguagClz-z'ldg = _ gangClz-z'ld§_JgaBJCIz-z'ldg (3.2.11) 21 The original integral is equal to integration along the branch cut augmented by the sur- face-wave pole integral. The branch out integral represents the continuous radiation spec- trum, while the pole contribution represents the discrete surface-wave modes. Applying the residue theorem gives the integration around the surface-wave poles as lsaplg'bt'd; = -2rthe.r {gape’c'z'fi} lg: _( (32.12) ' ' where Res {...} designates the residue of the quantity within the braces at the given point. Let’s discuss the integral along the branch cut. There are two components to the inte- gration path along the branch cut as shown in Figure 3.2. One is the right-lower part from jun to "kr in the complex C plane. Another is the left-upper part from "kr to joo. :5, = 11,11" and pi = (cf—cf) — ($42,) <0 along the branch cut, such that [21 = :tjp, where p > 0. Thus the following conditions prevail: i) On the right-lower part, C,C,~‘k1,"11—> 0*, then arg {pfi} a 1: and arg {p1} art/2, where arg{...) designates the phase angle. Therefore, p1 = jp is chosen. ii) On the left-upper part, C,C‘ - k1,ku—) 0’, then arg {pf} 5 -n and arg {P1} 5 -1t/2. Therefore, p1 = -jp is chosen. Along the integration path following the branch cut, C2 = C3 '.' C1? + zjcrci = Pi + kIr ‘ in + 2jk1,ku = k: ’ P2 (312113) To satisfy the radiation condition, Re {C} < 0 and In: {C} > 0 (32,14) should be chosen. Let g = for, then a2 = pz-kf (32.15) Im{C} Im {p1} <0 b Im {p1} >0 Cb C _ 1 > > > 0 k1, Re{C} C Im {p1} >0 Im {P1} <0 Figure 3.2. The path of integration along the branch cut on the top Riemann sheet (Re {P1} > 0). Im {p1} < O in the shadow region and Im {P1} > 0 in other region. “7331': 5110:. 23 and Re {01} >0, Im {01} >0. Changing variable C to p leads to 11C = Mo: = japdp (3.2.16) Then the integration along the branch cut can be written as o e. (Iz-z'l _ 'Clz-z'l 'CIz-z'l jP 51'“:an 4’; ‘ {lgaflg lp1=jp+£8apel I“ =-jp} Edp 10 Edp (3.2.17) “80132;”-qu °'—-u' [80,5202 - z'l Ip 1="jP P1=jP] ZC'Z’“ j—pd 3.2.18 i8“ |p1=-ip 0! p ( ) where a = .lp’ -Icf (32.19) Hence, the original integration along the real axis can be written as the sum of a contri- bution from the surface-wave-mode pole and a contribution from the continuous spectrum radiation modes in source-free region, i.e. IguBJCk-tldC = 21thes {gas/CIZ—z'lfi .- c=-c, it.) [8 “MCI: — 2" L. = -J'p - “32m - 21 In =Jp]%pdp (322°) .537 1?: 24 3.3. THE SPATIAL AND EIGENFUNCTION REPRESENTATIONS OF DYADIC GREEN’S FUNCTION IN SOURCE REGION The dyadic Green’s function in source region has been studied by numerous workers [45]-[54], some of them write the Green’s function as the sum of principal volume integral and a correcting term, which is determined solely from the geometry of the principal vol- ume. Recently, the principal-volume method of expressing the spatial representation of the dyadic Green’s function was reviewed by Chew [55]. The Green’s function in the Sommerfeld-integral representation derived in the last chapter is valid in source region. But, the complex analysis in the last section is invalid when x = x' and z = z' , since the integrand becomes unbounded when |C| -) co. Thus, a delta singularity may be missed on the right side of (3.2.20) at points in source region, and the equation should be IsapJCIZ-zoldc = C111gr5(13"13')+2101?”{8111(1fi'z"z.l}l;=_c _ 2C1: - z'l [Clz- z 'l 3.3.] {[81:11 L'hj‘, son I '14-;- dp ( ) for all (‘5 and 3', where p = 21: + 22, and can is an unknown constant. One way to find the constant is by integrating over variables x', z' in a small cross section S 5 which sur- rounds the field point (x, z) , i.e. Cap = slimo dx' dz' {Igap/C'z-z'ldC-anRes {gainful-2"” 408 t=-g + {Papjflz-z'llpl =-jp_ M‘JCIz- z 'pll =jp]J£ _dp } (3.3.2) The surface integral of a single term on the right side of the above equation is depen- dent on the geometry of the small surface. Therefore, it seems that C as is nonunique. However, a combination of these integrals still yields a unique value. Hence, the Green’s function in (3.3.1) is unique. 25 Since it is needed do integrals of all Green’s function terms to find the constant, it is not elementary and the procedure is dependent on the radiation mode spectrum. Thus, Chew’s [44], [55] Fourier representation of Green’s function in unbounded region will be used to find some relations, then, the delta singularity term in source region can be ob- tained using these relations. The electric field is expressed in terms of the dyadic Green’s function by E (2) = -j(r)|.t di'E (F, 1') 03(1') (3.33) In an unbounded region, the dyadic Green’s function is the solution of VxVxE (1, 1') 413,5 (2, 1') = i8(;—?') (3.3.4) where i= £2+99+ 22 (3.3.5) Fourier transforming (3.3.4) in the 3-D spatial F variable yields -2 xi x501, 1') 413502, P) = ie‘j’m' (3.3.6) with i = it, + 91, + 212. The above can be inverted to yield [56] 11?, (11 -13) e'j‘m' (3.3.7) E (k. r') = where 22 = k: + k; + 23. Consequently, the dyadic Green’s function in space domain can be written as .. ”2 -kk i. ;,_;. G(r,r) (21:)3“ I j k: —-(——H:_": ——2 ‘ ’ (3.3.8) where d)? = drxdkydrz. The above Fourier representation of 5 (P, F') is a nonconvergent integral in the classi- cal sense because the magnitude of the integrand tends to a constant when lil -) co. Chew [55] divided the integrand into two terms, one term is a constant and its inverse Fourier transformation is a delta function. The integral of the second term can be evaluated using Jordan’s lemma and Cauchy’s theorem. Consequently, -2... M " ikz 4020 2, _. G(r,r') = -§-;-5(r-r ')+Lu2 I Id'i, —°—— elk (" ' ”A" " (3.3.9) *0 -- -- *0‘0: ‘ 1. I2 2 2 a A A Where k: = ka+2kzg dk‘ = dk yde’ kox = ko-ky-kz, r: = yy+zz, and in = fkmsgn (x -x') +9k2+£rz (3.3.10) The Green’s function in y-invariant 2—D problem is = I 5 (;,?')dy' (3.3.11) wherefi = £x+2z,anditis notedthat IJ*’("'"’dy' = 21:80,) (3.3.12) Thus, _ . " it 4050 k - G(§.P')= §;-5(l5 -P')+—Idk —°—2——e" ‘2 ”I'm" " (3.3.13) -k0 kokox where to, = Jig-r3, and in = fkmsgnOc-x’) +212. + 221.3,, + fir}, - (£2 + if) kotkzsgn (x -x') (3.3.14) refit—3m?) = -££6(fi-B') + IdCengz-“g- (3.3.15) - 22:2 -213.»ng - (22 + 22))pngn (x -x') sea (2 — z') _,.,_,.. 8 = 47W e (3.3.16) It is noted that k2 is changed to C, and to: is changed to jp, where p = 2’? -k%. On the other hand, interchanging the roles of x and z in (3.3.13), the dyadic Green’s function can written as IkO-zkoioejk(1:-1r')-alr.- z'l 5613') =-—§5(p- p')+z-12Idk (3.3.17) "0 tom where or = Iii-13,, and £0 = ka-i-fjasgn (z -z'). Similarly, after some operations, (3.3.17) becomes .0 AA AA 2 AA 2 AA AA . I O -xxa2+zzkx+yyko- (xz+zx)1kxasgn (x -x ) sgn (z -z) 7, 'x_x.|_a|2_2.| + Idkx 21 . 41m “’ (3.3.18) .. . e. "- I - . 11: = -z25(p-p) - I e’“ ‘ (=20, idkx (3.3.19) "' p=-jk Hence, an important formula is obtained, -225(§-3')+ Ich‘"""§ = -225(p’- B)- -Ig ‘2‘" " —co 1*, ,2; ,0,“ —dr, (33.20) =-jkx 28 In fact, the second term on the right side of the above equation is the integral along the branch cut of p . Now, let’s apply this formula to our problem. First, the Sommerfeld-integral represen- tation of the dyadic Green’s function is rewritten as _ 1112 622(xlx', z-z') =xx—5(p- p')+622(x|x', z-z') 111 (J a " ’ - ' .. _ = 2.2.{2525 (p— p) -Id dCe’C" "[gli-gfi} (3.3.21) where _ fiCz-f‘p§+iik§- (16 z+zx)ip2ngn(x- x")sgn(z-z) -p|x XI :1 - 3 (3.3.22) 41tp2 and E2 is a second part of the integrand of the Green’s function. 51 and £2 are not even functions of p2, but 3.1+ £2 is even in p2. Applying complex analysis on E1 and £2 sep- aratelyyields ianhlxflz-z') =i£5(f5- 6')- Id dCJCIz z'g-l -Id§JCIz “52 ”x .- (lz-z'l L94 =1.“ dex'i" ’81} pg: fr: (1 p2 = -jkx —. = -jp dp {E C=l¢ a P1=’jP jkx — 'Clz—z'l p2 = .1k3 dP + 2am: { [51 + £2] J"“"°'} | (3.3.23) c=-c, Since {1+}: is an even function of p2, then 29 ”1"; '2‘“ —dk = 0 (3334) (lz-z’l (gjjaz-Jdki-Ig 2e’ 2' ’1‘: Therefore, 622(xlx’, z- z') =££T :5(p- p')‘I-jl:|—22 Igd‘“ “4C —oo =22—5(P -p')——+ “:21thes{g€mz “”‘,"; _j__;122£[-(IC|1- zll p = -19 " EJCIz-z'l Ir. =JPJJEpdp (33°25) 3.4. SUMMARY OF GREEN’S FUNCTION 622(15le z- z') =fij3 22W P)+5R (xlx'J-Z') =223m—28(p— p)+5’° ‘(xlx',z—z')+6§,“(x|x',z-z') (3.4.1) where the superscript “R” designates the integration along real axis, “Pole” designates the contribution from the surface wave which is the integration about surface-wave poles, and “Rad” designates the contribution from radiation modes arising from integration along branch cuts. The contribution by term 22—5 (:5- -p ') is the same as the delta singularity term in the eigenfunction representation of the dyadic Green’s function in rect- angular waveguides and cavities derived by some authors [451-[47], [49]. 52101 x', z - 2') was given in Chapter 2, and 30 _.i_1'|2}CpPICOS [qz (t+x)] cos [qz (t +1 ')] 22.1-02-5] 6”" ' 3.4.2 22,,qu z Z): *2 424'. (c, )sinw) ( ) 6312(2'2,’ PI.) _ sgn(z_ )J_112CPPICOSqu(t+x)]sin[42(t+x')] _,-;.z- 75.513) *2 A; (C? )Sin (42‘) GP°"(x|x', Pl) _ _sgn(z_ )Lflchmsinlqzo-HHcos{qz(t+x')] 20205.4) "2 A' (C )sin(qzt) Gigghlx'J-z') = iflzlqulsin [42 (1+1)]Sin [020+x )] e-jg'lz- z'l (3.4.5) I‘2 13' ”(C )8111 (421) nghlx'J-z') = -jk2'q nzsin[q2(t+x)]sin[q2(t+x)]e -j§,lz ll (3.4.6) A' ,.(§, )sm (42:) where (I, is the solution of 152m) = 0 or 5,. (C) = 0, and p1 = JCfi-ki (3.4.7) q; = kg-C: (3.4.8) 252 (C) = plcos (qzr) —q2(el/ez) sin (qzt) (3.4.9) 08(q t) 8 sin (4 t) 8 A' (g) = ;{—— 2 [l+—1p1t:I+ 2 [p,:+—‘]} (3.4.10) P1 32 42 ' 52 -. cos ((121) 82 2 £1 2 A¢(§p) ,= l; p——-2—- k2-k21+plt —p1+-e—q2 (3.4.11) p1q2 el 2 When p1 = p2 = p0 005( )8 a At 2pm) _Cp°°5‘1__22_‘ 5': (l-EIIkIU-plt) +p1r§:(l+éi):l (3.4.12) P142 31 5.0:) =j142c°5(42‘) +p1sin(qzr)1 (3.4.13) A'h(§)= .j_C-(1+pl,) [q2sm(q2t) -plcos(q2t)] (3.4.14) - 1C (1+P1') A' = p . 3.4.1 Mp) P18111012!) ( 5) —mw The five components of 622 (xl x‘, z - z‘) are in a C _. 6:2:(xlx'.z-z')= TE0..—p2?I:-q—-:Plcos[(1204-11)]cos[q2(t+1r‘)]e-°llz zldp (3.4.16) 1n ( ) - - GR“ nu'hlx, z-z') = 8311(2- z')-l]p anp Cos[q2(t+x)]sin[q2(t+x')]e-alz “do 20 (3.4.17) )1“ -- 0‘2'5‘2’.(xlx'.z-z') = -sgn(z- ’12: sintqzmxncostq.(z+x')1e‘°‘" "dp (3.4.18) _1'l ”Pq C “(9) . __ . 6:24:2(xlx'J-z') = —2k2-1_—22_—1tasmlqz(t+x)]smlqz(t+x')]e-alz z'dp (3.4.19) ”PC (P) . -° 03;},(xlx', z-z) = -jk211 n2] Thug 42 sin[q2(t+x)]sm[q2(t+x')]e_alz "dp (3.4.20) 1Mwm 2 2 -k k < <00 a = p24} = { p 1 ‘ 9 (3.4.21) 1 lei-p O all z' , and “-” to represent the surface-wave field propagating along the -z direc- tion forz < all z'. 131.1..1: The electric a: 51’, (x. 2) 5;? (x. z) field of excited TE-odd surface-wave modes has y-component only It 0 — = J I 6"2 (xi, Z.) Gf;;;£2ydx'd2' = A: $1.“ (q2‘)€ plxeiilcf'. (427) "OLcs - jka 2 I 0 Pole . . __ j: . xjfi'z " — I 5" (’12 )GZZnyZydx dz - A). 8111 [420-11)]: (42.8) 0 Les where t k: 2 . U U . ' ' fig 2' I C A]: = - . 8n (x,z)b'2y(x,z)sm[q2(t+x)]e ' dxdz (42.9) A',‘ (gr) 81110121) LCS and p 1' 42’ and 5". (C1)) were given in the last chapter. It is noted that the electric fields in two regions satisfy the boundary condition at _ :31: £2, “go—Ely 3:0 (42.10) We: The elecuic fields of excited TM-even surface-wave modes in both regions have x-, 2- components it Bi? (1.7.) = —9 I 8112 (x',z') [GPOI‘EZx-l-Gfgglizz] dx'dz' 1sz ”as = A, (”5’91 (4.2.11) it 5:? (x,z) = 1T0 I 5n2(x',z') [szogb‘ui-Gfggb‘h] dx'dz' 0Lcs 1C - - = =1=—£Af e ”"5““ (42.12) P1 uzl: _ ”‘0 5 2 . . Pole Pole . . 521 (1,2) -' — I n (x ,Z ) [02221sz+ 02222522] d1 dz , 0Lcs :t 'C e . 3E] z = —.——s1n[ (1+x)]e ' (4.2.13) 313(42‘) 42 fl: 3;? (11.2) = —-o I 5112 (x',z') [Gpno‘l‘th-l-Ggggb‘u] dx'dz' "OLcs :1: N; A = p ‘ cos [q2(1+1:)]e*fl;'z (42.14) :F——.-—— 92 313(42‘) 37 where A: = 8-0 m I5n2(x'.z'){111,008[qz(t+x')]sz(x'.z') £2 5'. (CF) LCS -J'qzsin [92 (t + x') ] Eh (x'. z') } em'z'dx'dz' (42.15) and p1, qz, and 5" (Cr) were given in the last chapter. Thus, the electric fields of TM-even surface-wave modes excited in the two regions are 1 WC _ - 3:: (x. z) = A: {375 +2} e Nem’z (42.16) 1 Ant A: 4:ij . -p x $111 E; (x. z) = W {fi-aE—cos [q2 (t+x)] +2srn [q2 (t+x)] } e 'e ' (42.17) and they satisfy the boundary condition 3.2;: L: o = 3:2: 1‘: 0 (42.18) 4.3. ASYIWPTOTIC EVALUATION OF THE FAR-ZONE FIELD In order to calculate the far-zone field more efficiently, the saddle-point method (also known as the method of steepest descent) of approximate integration will be used [1] [44]. Let’s write 61", (xl x', z - 2') as , , i112 Glzap (II x 9 Z - Z ) = k—zlafl (4.3.1) where Ian = I gap(§) {11.51;de (4.3.2) 38 with c2 - - a..(§> = - - cosh tpz(r+x')1e"“ 21tA¢ jpzc ° 1 ’jCz' 8,1“) = - smh[pz(t+x)]e 21m, jplc o Tia. gum = .. coshlp2(r+x)]e 2m, ' gum) = ”“7? sinh 1p2(:+x'>1 i” 21111. (C) k; sinhlp 1+ ')1"“' = -—:— t x e 8” 21M» 2 (4.3.3) (4.3.4) (4.3.5) (4.3.6) (4.3.7) To implement the saddle-point method, the integral of (4.3.2) must first be placed in an ap- propriate standard form by transforming to polar coordinates in both the (x, z) and (p, C) planes (where p1 = jp). Let x = r6089 2 = rsinO C = -klsin¢ p1 = jklcoso ¢ = a+m e-pggfiz = e-jkircoe (¢-0) with polar coordinates as defined in Figure 4.1. (4.3.8) (4.3.9) (4.3.10) (4.3.11) (4.3.12) (4.3.13) 39 (x. 2) 0 2 Figure 4.1. Polar coordinates in the x-z plane. Eqs. (4.3.10) and (4.3.11) represent a mapping of the complex C plane into a strip of the complex 4) plane. The two sheets of the Riemann surface map into a connected strip of with 21: along the c axis. From (4.3.10)-(4.3.12), it is found that C = B+ja = -klsin(0'+j11) _ (4.3.14) [3 = —klsinocoshn (4.3.15) 01 = -k,cosasinhn (4.3.16) pl = jp = klsinosinhn +jk1cosocoshn (4.3.17) This mapping is illustrated in Figure 4.2. On the proper sheet of the Riemann surface (top sheet), Re {pl} > 0, and hence sinosinhn > 0. The four quadrants of the proper and improper branches of the C plane map into the regions designated Pi and Ii (1 = l, 2, 3, 4), respectively, in Figure 4.2. The original contour of integration along the real C axis denoted by C passes from (a = 0.3 = -°°)to(a=0, B = co). Since C $ 3'.- a a ‘ \ path along the branch cut Figure 4.2. Mapping of the two Riemann sheets of the C plane onto a strip of the 4) = a +jn plane. The crosshatched regions are the proper Riemann sheet (top sheet). 41 (a.=O,B=-oo) H (o=x/2,n=oo) (a=0,B=oo) H (o=-x/2,n=-oo) the integral in (4.3.2) now becomes -l/2 -j- . In“ = _ I klcos¢gap (-1,sin¢) c"""°°‘ (44)“, u/2 +j- Let’s consider the following generic integral 10.) = 1" (t)c"“’m C Iff(r) hasastationary point at: = ro,implyingf'(ro) = 0,then (‘ "’ to) 2 [(1) -f(zo) - 2 f"('o) = %I-‘R22(”25) = %FR2cos (s + 25) + éI-‘stin (s + 28) where Mr.) = n” t-to = R25 (4.3.18) (4.3.19) (4.3%) (4.3.21) (4.3.22) (4.3.23) (4.3.21) On the path of s + 25 = in, the function f(r) - [(10) is purely real and negative, hence ems) ~ e- Ari/2 + mto) (4.3.25) Notice that along this path, cm" has a constant phase, hence, it is also called the con- stant-phase path. Furthermore aim) has a maximum at R = 0, or t = to, and is expo- nentially small along the path passing away from R = O, or away from r = to. But along 42 the path 3 + 25 = O, 211:, (M0) - (Ami/2 ”fl“ becomes exponentially large. Therefore, the function e110) , when A —> oo, looks like a saddle at the point r = to on the complex r plane. Hence, the constant-phase path, on which the function all“) descends steeply away from its value at the saddle-point, is known as the steepest-descent path. With the change of variable in (4.3.24), (4.3.21) becomes I (71.) = cm“) ”511100 +12”) {”‘,/’43 (4.325) C The path of integration can be deformed from C to the steepest-descent path and paths A and B in Figure 4.3. The contributions to the integral over the paths A and B are vanish- ingly small at infinity. However, in deforming the contour, the contributions due to any singularities enclosed between two paths (captured during the path deformation) need to be included. This leads to tot) = rp-t-cm") ”'5 I h (10+Reja)e-”Rz/2dk (4.3.27) - where [p is the contribution from the singularities enclosed between the original integra- tion path and the steepest-descent path. When A. —) co, (”RI/2 is exponentially small away from R = O on the steepest-de- scent path. This irnplies that most of the contribution to the integral is from around R = 0. Hence " _ 2 1(1) ~rp+h ("pay“) +15! r ”Ra/2r” = 1p+ ’)‘—:2 (to)cm'°’ +15 (4.3.28) —. In the integral of (4.3.20), 43 Re{t} F s+25= Figure 4.3. The original integration path and the steepest-descent path 7t = k‘r f(¢) = —1008(¢-0) f’(¢) =Jsin(¢-6) f"(¢) = 1cos(¢-9) From f'(¢o) = 0, the saddle-point is obtained on = 0 Thus, f"(9) =1 F =1 3 = n/z 0n the steepest-descent path, 1'! fl = —_i._ arr{¢}=5=-§t 4 2 N13 (4.3.29) (4.3.30) (4.3.31) (4.3.32) (4.3.33) (4.3.34) (4.3.35) 44 From Figure 4.4, it is clear that 8 = -3rt/ 4 should be chosen. Thus, the integral (4.3.20) becomes [an " ugh + age (4.3.36) where 12;" is the contribution from the integration along the poles which are enclosed be- tween the original integration path and the steepest-descent path, and lilac is the contribu- tion from the integration along the steepest-descent contour with [we = k cosO (-k sine) if} 0"" V4) (43 37) (1B 1 3 a3 1 k1 r ' There are two kinds of the singularities, the surface-wave poles of finite number (poles on the proper Riemann sheet) and the leaky modes of infinite number (poles on the im- proper Riemann sheet) [1]. When 0 is small, no singularities captured. When 9 is large, - rcosO _ePl since the field of the surface-wave mode will exponentially decay as e-p" - , then, if 0 at 1t/2, e'p'm'a —) 0, as r —> oo. Thus, it is not needed to consider the contribu- tion from the surface-wave mode in the far-zone field for [0| < 1t/ 2. It is assumed that a leaky mode pole C L could be captured. The electric field of the leaky mode can be written as if = ALEM-IPLx-Ktz ___ ALEOe-jk'rm‘“‘-e) ___ Alice-hull: (UL-0) sinhnLe-jkgcoc (oL-O) 0031111,, (4.333) where 4;" = ”ML = “Sinwfimfl (43.39) ..L Since (CL-9) <0 and nL<0 for any leaky pole captured, then 51-90, as r-roo. This is also true for 6 = in/ 2. Thus, the far-zone field (|9| < u/2) can be approximated entirely by the radiation field arising from the integration along the steepest—descent path. 45 in Figure 4.4. Deformation of contour C into steepest-descent contour (SDC). 46 ‘8' jko E; (x, z) = — I Sn: (1', 2') 5:200 (xl x', z - z') 0 E; (x', z') dx'dz' (4.3.40) oLCS where the superscript “sr” represents the radiative component of the scattered field, and 111 I 21: _- - Giff” (xl x', z - z') = —2k1cosegap(-klsin0) —e ’(k'r “4) (4.3.41) *2 k 1’ Leaky mode poles arise from solutions of the eigenvalue equation which occur on the improper Riemann sheets, they are also called improper modes [ll-[3], [57]-[59]. The corresponding EM fields do not satisfy the radiation condition at lxl = on. They are not included in a complete spectral representation used to describe a general field in the guid- ing layer and its surround. They are parts of an approximation to the continuous radiation spectrum. But, some of these modes can plan an important part in the asymptotic descrip- tion of the radiation field when the distance between source and field points is not very large. The EM fields of leaky modes decrease exponentially along the interface direction (2 direction in our problem), and increase exponentially along the direction which is nor- mal to the interface, but they decrease exponentially along any radius from the source within their domain of existence. Thus leaky modes exist near source and near the inter- face only. Leaky modes lead to a net power flow away from the interface, which is why they are called leaky modes. Cassedy and Cohn [60] calculated the amplitude of the electric field of the leaky wave due to a linesource above a grounded dielectric slab, and found that the amplitude de- creases as the distance from the source increases when 6 is near :trt/ 2. They also verified the existence of the leaky-wave mode experimentally. Thus, the EM fields of leaky modes are negligible in the far-zone field, and that field can be calculated by the saddle point method. Let’s simplify the formula to calculate the far—zone field 47 131.1..1: ”‘0 2 soc 2“ -j(k r—x/4) I" _ 1 o I 1 = l Ely(x,z) - 710438)) (x,z)Gunyzydx dz 3,,(6) _k1re (4.3.42) This is a cylindrical wave polarized in the y-direction, and where 2 k c030 . . 3,,(9) = 11°“ 5r2(r',z')szy(r',z')sin[qz(r+r')]c"‘"““°dr'dr' 21tAh(-klsin6) LCS (4.3.43) with 42 = ,lkfi-lfsinze (4.3.44) 15,, (-k1sin0) = jqzcos (qu) -k1cos0sin (qzr) (4.3.45) jk £126.11) = —o I 5n2(x',z') [fogEu+Gf€sz2z]dx'dz' 1"Ores 21: - .. = B, (0) cosO [IT—re “k" "4) (4.3.46) 1 It E;;(x,z) = —2 I 51:2 (x',z') [GfgSEZx+Gf2DxEEZz] dx'dz' flows 2 .. - = -a,(e)sine ’31,: 1“" "4’ (4.3.47) . 1 so jkfcosfi 2 B‘ (9) = — _, 5n (x',z') {jk1sin0cos [qz (t+x')]EZx(x',z') 32 22:11, (418319) was + qzsin [42(1 + x')] 8210', z') }¢jk'z.modx'dz' (4.3.48) with 42 = [645,129 (4.3.49) - . el . Ac (—k1srn0) = jklcosecos(q21) - e—qzsrn (qzt) (4.3.50) 2 Combining (4.3.46) and (4.3.47) yields -' k -lt/4 e1(1' ) _..rr 21C . E; (r, 0) = B, (9) n {icose-i‘srnfi} (4.3.51) 1 This is a cylindrical wave polarized in the O-direction. If the electric field if of the surface-wave mode is calculated separately, the total far- zone field is .3: .533: .537 [51(1, 2) = El (x, z) +51 (r, 9) (4.3.52) forr large and IO] 5 1t/2. 4.4. THE SCATTERED POWER AND BACK-SCATTERING WIDTH The power of the surface-wave modes and radiation modes can be calculated by using the Poynting’s theorem as follows E l. i. . '1 : A: sin[qz(.t'l-t)]e:‘:j§"z -t’) = 2 sm{(x-y)tl _sm{(x+y)t]} (5.4.14) (Jr-y)! (1+y)t From the definition, the first-order approximation of the equivalent scattering width of the discontinuity associated with the scattered radiated field is found to be :1 2 E1 (r: e) 2 lf(e’9‘)|2 Anzcosecoseikzat 51 2 = ‘ 4 ( ) (5.4.15) 50 0(9) = Eng” "1 C.(9)C.(9,-) E; where the relation .. 2 , |A¢ (-k,sine)| = (81q29/82)28in2 (qzot) + Itfcoszeeos2 (qzot) = 1:in (6) (5.4.16) has been used. It is assumed that region 1 and the discontinuity region are air (81 = ed = 0) throughout in the following numerical results. The MoM solutions of EFIE are calculated by the pulse Galerkin’s method given in Chapter 7. Figure 5.5 shows the normalized back scattering width calculated by the first-order ap- proximation and the MoM EFIE solutions as functions of frequency at various incidence angles 9.. when a TM-polarized plane wave is incident upon a grounded sheet with a nar- row air gap (2a = 0.06”). It is observed that two methods agree with each other very well. It is also observed that the normalized back scattering width becomes almost zero at some discrete frequencies when 6‘ = 50°. This phenomena can be explained from the behavior of the term f(0‘., 9..) . Since the air gap is very thin (2a/2t0 = 0.09 at f = 186112), then the second term sin (2k1asin0‘) / (2klasin0i) in f(0., 0‘.) is almost unity. After some simplifications, the first term of f (0‘, 0‘.) (ed = 81) can be reduced to 73 "' l O I r I I I I I \ ? .’_.-‘-°-’d I _ 2 O - ‘ 1"" - , [’1‘ Are/'4 E - 3 0 ’ 2+ 4" /"/ -- ' ’ '1 0° ° A” ,4?” 80° I: 68 _ 4 0 .. .59.»... '2’- 1'1. -‘ ’ 4 ? x‘JIflJ’J-h’. ' 'r a“- a “13'. .19.” ' ’l , . U a ‘ . fl . .a' '-a I — S O I I", a". o B I I " (dB) "‘1. P 6° .' .. I a - 6 O f 2‘ ’ . .1. J— ¥ 3 o . o..o' ...!.......O.I.o a co co . 00....... e I . E I P N o g I R l U '2 . Figure 5.5. Normalized back scattering width as a function of frequency at various inci- dence angles 91 when a TM plane wave is incident upon a grounded sheet with a narrow air gap. Lines -- First-order approximation, Points -- MoM solu- tions of EFIE. (cl, = ed, = 1, £2, = 1.1, 2a = 0.06”, t = 1”, ”2:2 to 8, ande=5 to 30) 74 sinzejl (429, q...) - (e,/e,) (qze‘/k1)2f2(q29‘.q29‘) = an; + 1) sinzo‘. - "3+ [n3 - (n: - 1) sin20 :1 sin (2k2rcos0r) / (2kzicose,) }/2 (5.4.17) where n; = 82/81. Let the right-hand side of the above equation equal to zero, then sin (2k2tcos6r) _ n; - (n; + 1) sinze, 2k21COSGr - n; _ (n; _ l) sinZei (5.4.18) When 0‘ increases from 0° to 90°, the right-hand side of the above equation decreases from 1 to (ng-ng- 1)/(n;—n§+ 1) . Thus, given 9,. and ez/el, (5.4.18) may have no solution, one solution, several solutions, or infinite solutions for t/ho. Figure 5.6 plots the normalized back scattering width calculated by the first-order ap- proximation and the MoM EFIE solutions as functions of frequency at the incidence angle of 9‘ = 60° with 2a = 0.3”. It is found that the difl'erence between the first-order ap- proximation and the MoM EFIE solutions becomes large as the frequency increases. 5.5. MORE NUMERICAL RESULTS FOR A NARROW AIR GAP Figures 5.7 and 5.8 show the normalized back scattering width calculated by the first- order approximation and the MoM EFIE solutions as functions of frequency at various in- cidence angles 9‘ when a TB and a TM-polarized plane wave is incident upon a grounded sheet with a narrow air gap (2a = 0.1”), respectively. Figures 5.9 and 5.10 replot the normalized back scattering width calculated by the MoM EFIE solutions as functions of incidence angle 6‘. at various frequencies when a TEE-polarized plane wave is incident upon a grounded sheet with a narrow air gap with 20 = 0.06”, and 0.1”, respectively. 75 _10 I I ‘ 1 I I I MoM Solutions of EFIE o First—order Approximation ..-..-- ~15 - i 2.3.... o .. 2 0 - .9", “‘“x ° . . ’9’ ‘\\ o 0 GB ’9’ \~\ . O o t) x; _25 r- If \-~I ’1 (dB) -30 I- I’ .. y! — 3 S 7 \\\. I“ d \\\1‘0/ _40 I l l l l l l Figure 5.6. Normalized back scattering width as a function of frequency at the incidence angle of 0.. = 60° when a TM plane wave is incident upon the grounded sheet with a narrow air gap. Line -- First-order approximation, Points -- MoM solu- tions of EFIE. (eh = ed, = l, 22, = 1.1, Zn = 0.3”, t = 1”, Nz = 3 to 8, andN,,= 5 to 30) 76 Figure 5.7. Normalized back scattering width as a function of frequency at various inci- dence angles at when a TB plane wave is incident upon a grounded sheet with a narrow air gap. Lines -- First-order approximation, Points -- MoM solutions ofEFIE. (s1, = ed, = 1, £2, = 1.1, Zn = 0.1”, r = 1”, It]z = 2 to 8, and NJIF = 5 to 30) (dB)_50 -60 a \ \ ' ---.-" -7O .- ‘.--..---~ N Ht. .3. p m on H O H N H p H m H (I) Figure 5.8. Normalized back scattering width as a function of frequency at various inci- dence angles as when a TM plane wave is incident upon a grounded sheet with a narrow air gap. Lines -- First-order approximation, Points -— MoM solutions of EFIE. (£1, = 64, = 1, £2, = 1.1, 2a = 0.1”, r = 1”, Nz = 2 to 8, ande=5 to 30) 78 -10 9I I I fir I I I I7 53161:: :kz'flrfi'f-I 33:5ij ‘5: amgfiflwaw. -2o fig,,f:;.-;..::..1:-.t=--§-.a k s... ~\;‘Q\ . “41' ... "5.....- ‘M. L...“ “ma-H...” ::... 3.3.2: ’T‘ -30 . "nan-~90"O-"G’"a""a""am ”a ""8 ""a '.‘B :i. ..j\\; -T--. . . .7,“... I; -40 P 4 '* ~+---.._\‘\ -50 .- (dB) -60 . -70 1. -80 91 (degrees) Figure 5.9. Normalized back scattering width as a function of incidence angle at at various frequencies when a TB plane wave is incident upon a grounded sheet with a narrow air gap. (81, = ed, = l, 82' = 1.1, 2a = 0.06”, r = 1”, N2 = 2 to 8,ande = 5 to 30) (dB) 9: (degrees) Figure 5.10. Normalized back scattering width as a function of incidence angle 6i at vari- ous frequencies when a TB plane wave is incident upon a grounded sheet with a narrow air gap. (81, = ed, =1, 62, = 1.1, 2a = 0.1”, t =1”, Nz = 2 to 8,ande = 5 to 30) 80 Figures 5.11 and 5.12 replot the normalized back scattering width calculated by the MoM EFIE solutions as functions of incidence angle 6‘. at various frequencies when a TM-polarized plane wave is incident upon a grounded sheet with a narrow air gap with 2a = 0.06”, and 0.1”, respectively. Comparing Figure 5.2 with Figure 5.7, Figure 5.5 with Figure 5.8, Figure 5.9 with Fig- ure 5.10, and Figure 5.11 with Figure 5.12, it is found that the shape of curves are almost identical in the two corresponding figures, in which only 20 is changed from 0.06” to O. l ”. In the first-order approximations of the back scattering width for both TE- and I'M-polarized plane wave incidence, the term involves 2a is [ (2koa)2sin (2klasin6i) / (Zklasinfii) 12. Because the air gap is thin, the term sin (Zklasinei) / (2k1asin65) is nearly equal to 1. [sin (2klasin0i)/(2k1asin6i) = 0.85, when 9‘. = 90°, 2a = 0.1”, and f = lBGHz] Thus, the back scattering width is approximately proportional to (2koa) 2. But when the air gap is not thin, the term sin (2k1asin65) / (2klasin9i) is small, and the curve shape of the back scattering width vs. frequency or incident angle will be changed, as shown in Figure 5.3. In this chapter, the total electric field in the discontinuity region is approximated with the electric field in a zero-width gap, and then the first— and the second-order iterative ap- proximationsto the far-zone field for a narrow air gap are calculated. The approximate re- sults agree very well with the MoM EFIE solutions. The first-order approximation to the far-zone field is expressed in a closed form in terms of elementary functions, and the sec- ond-order approximation is expressed as an integral representation. However, the MoM requires calculation of the matrix elements by integration followed by solution of linear equations. Thus, the approximate method is much computationally efficient than the MoM for a narrow gap. 81 ’10 ,. . L8' I I I I I I I -20 -30 . .. “a . “x; -40 -50 (dB) -60 —7O -80 0 10 20 30 4O 50 6O 70 80 90 9i (degrees) Figure 5.11. Normalized back scattering width as a function of incidence angle Oi at vari- ous frequencies when a TM plane wave is incident upon a grounded sheet with a narrow air gap. (81, = 24, = 1, £2, = 1.1, 20 = 0.06”, t = 1”, N1 = 2 to 8,ande = 5 to 30) 82 "' 1 0 " §}-§Q’:xfl:y fl” 7 .-. .4.“..‘M‘: xii}: ‘\ at“? “Ali _ 2 0 -mI--Zt:1;3r:;‘b‘::\ . .,,..J'v-ox. -.\ a . (dB) 91 (degrees) Figure 5.12. Normalized back scattering width as a function of incidence angle at at vari- ous frequencies when a TM plane wave is incident upon a grounded sheet with a narrow air gap. (81, = ed, = 1, £2, = 1.1, 2a = 0.1”, r = 1”, Nz = 2 to 8,ande = 5 to 30) CHAPTER 6 PULSE GALERKIN’S SOLUTIONS FOR TE-POLARIZED WAVE INCIDENCE 6.1. INTRODUCTION In Chapter 4, the Method of Moments (MoM) is applied to transform the electric field integral equation (EFIE) into matrix equations for general basis and testing functions. Here, a pulse Galerkin’s implementation is applied to establish the matrix elements A?" and the forcing vectors Fm for TIE-polarized wave incidence. First, formulas to calculate the matrix elements and forcing term will be derived when both basis and testing functions are 2-D pulse functions. ‘Ihen, computation of the inverse Fourier transform representa- tion of the electric Green’s function is performed by integration along real axis, along the branch cuts directly, and along the branch cuts with a variable change. Finally, these inte- gration methods will be compared with each other by checking the power conservation. 6.2. PULSE GALERKIN’S SOLUTIONS It is assumed that the cross section of the discontinuity region is rectangular (lzl S a, -t S x S 0) (Figure 6.1) and the permittivity ed is a constant. For TIE-polarized wave inci- dence, the EFIE in the discontinuity region becomes k A 20 0 . 52,042) -J' on" I I Guy,(xlx'.z-z')152,(x'. z')dx'dz'=E'2,(x. z) -r (6.2.1) o-a for lzlSa,-thSO 83 AS a (x... z.) x 81! no x=0 z Nx 52’ "a x=~t I‘V-“l z=-a NZ 2:0 Figure 6.1. Applying Pulse Galerkin’s Method to the EFIE in the discontinuity region. 85 where An2 = ni-ng = (ed-82) /80 (62.2) In Figure 6.1, the discontinuity region is partitioned into N = N‘Nz equal rectangular elements AS" centered at (x, z"), where N, Nz are total numbers of partitions along 2 and 2, respectively. The partitioning is defined such that 11:1 (3 0 N I jf(x.z)dxdz = 2 j f(x,z)dxdz (62.3) _, as, -a From Figure 6.1, it is found that AS”: 1’"le (62.28) L (n) =n— 111m")v }Nz N: J and n is from 1 to N, then Lx (n) has NJr different values from 1 to N, and L2 (n) has Nz difl‘erent values from 1 to ”2' Thus 2 -Z = be—"l' = 0,1,...,Nz-1 (62.29) 39 Ix -x.l = ”Ax =0,1,...,N,-1 (6.2.30) xm+xn+21 s = A1 = 1,2, ...,2Nx-l (6.2.31) There are Nz 1’s and 2NJr s’s. Therefore, only 2N1”; integrations are needed to com- pute. Evidently, much computer time can be conserved. For example, if Nx = N z = 10, then N2/2 = 5000, but 21mlz = 200. SUMMARY: 2313.5" = m = 1,2, ...,N (62.32) n=1 kgzlzmx)’ 2 |zm- znl |x -x| |z,,, -z,,| 2t+x +x Alynyn = shin-_T—An [Byy( Az ' Ax )- -Byy( Ax n] (6.2.33) where 3’70.» = ISACUZ’KMC (62.34) with g (C) _ {[1-cos(§Az)]JUN/(CA2)2 I = 1, ...,Nz-l (6.2 35) I (1+jCAz-JCAZ)/(CAZ)2 I = 0 173m = 1 2+—L——3-{1.t“"=‘”"-l--p———2 p‘ e-p"[cosh(p2Ax) -1]} (62.36) (pzAX) (mm) A). f? (g) = COSh (pzAx) -1 {e'p’SAx-fle-P’tcosh (pzsAx)} (62.37) (1224203 A, for: = 1,2, ...,2Nx- 1. The forcing vector F2»! is computed in the same way to be 2n2cos6isin [k2 (xm + t) c036,] FY" - otlzcose‘sin [kzrcoser] -jnlcosefcos [kztcoser] sin (cosOrszx/ 2) sin (sineiklAz/Z) 1,14,“. cosOrszx/Z sine iklAz/2 (“'38) for TIE-polarized plane wave incidence, and . sin (q Ax/2) sin (C Az/2) . Fm = Eosm [q2 (t + xm)] 2 P 6’9" (62.39) qux/ 2 CpAz/ 2 for TE-odd surface-wave mode incidence. Since the unknown electric field in the discontinuity region is expanded into a set of 2- D pulse functions, then the amplitudes of the excited surface wave and radiation wave are obtained by substituting the pulse functions in (4.5.13) and (4.5.14). kgAxAzAnz sin (qux/Z) sin(CpAz/2) N A: = -. a sin [q2(r+x )]efl('z' A'),(C,) sin (421) qux/2 CpAz/ 2 2:1 "' " (62.40) B (e) jkfikIAxAzAn’cose sin (4262/2) sin (sinOkIAz/2) " ' 2n5h(_k‘smg) qux/Z sin6k1A2/2 N at a o 2 aynsin [q2(t+x,,)] e’ ""m (62.41) II=1 where [1", (Q) and 5,, (-k1sin6) are given in Chapter 4. 91 6.3. NUMERICAL INTEGRATION ALONG REAL AXIS It is noted that f? (C) in (6.2.36) and (6.2.37) is an even function of C, and g, (C) can be written as the sum of an even function and an odd function of C, i.e. (C) {[1 "”5 “AZ” (WWW) tisin(Icziizn/(ciiz)2 1 = 1, ...,Ng’l g = I {ll-cos(CAz)]+j[CAz-sin(CAz)]}/(CAz)2 [=0 (6.3.1) Since the integral from -oo to on in (6.2.34) is symmetric, then it can be rewritten as B”(I.s) = 137(0):”(04: (6.32) 0 where gm) = 2cos(lCAz) [l—cos(CAz)]/(CAz)2 1= 0,1,...,Nz-—1 (6.3.3) Since 13" (C) involves hyperbolic sine and cosine functions, it is prone to numerical difficulty. As the integration variable in (6.3.2) increases toward infinitely, both the sinh and cosh functions become unbounded. In addition, it is diffith to ascertain the conver- gence properties of the integral. Both of these problems can be overcome if the integrands are written in terms of exponential functions. After some algebraic manipulation, the re- sults become 1?“); 1 -1"-p2~{l+(l-e‘P’A‘)p2—.-Ele”’(2"m} (6.3.4) (PzAx)2 (P2A1)3 A111 _ 2 Pa) = “"m’ (“WW ‘ 2624203 _P2~ Pt [e-p,[2t- (:+1)Ax] + e—p3[2t+ (:—1)Ax] Aral ] } (6.3.5) 92 where A“ = 2Ahe-p" = pz (1 + :4”) + p1 (1 - e4”) (6.3.6) CONVERGENCE OF THE MoM MATRIX ELEMENT INTEGRALS Before undertaking the numerical integrations of equation (6.3.2), it is quite helpful to anticipate the rate of convergence of the integrals. Of interest is the behavior of the inte- grand g: (01%;) as c-2 .,, m;2 s = o f? (C) -> { l/C3 s = (6.3.7) {cum/C3 otherwise where (:2 1. Then [1 - cos (CAz)] cos (lCAz) /C4 s = 0 8; (0137“) '9 [l - cos (CAz)] cos (lCAz)/Cs s = 1 (6.3.8) [1 - cos (CAz)] cos (lCAz) [cw/C5 otherwise Hence, the integration I gf (C) f? (C) dC will convergence quickly. 0 INTEGRATION THROUGH SURFACE-WAVE-POLE SINGULARITIES Since the presence of the loss in region 2 makes k2 a complex number, then all the sur- face-wave poles of the integrand are not along the real axis. But, if the loss becomes small or vanishes, the surface-wave poles will shift to the real axis, and thus reside within the ‘ domain of integration. Therefore, pole extraction will be used to deal with the surface- wave-pole singularities. In (6.3.2), the integrand becomes infinite when 93 5,,(0 = pzcosh (p2!) +plsinh (p21) = 0 (6.3.9) which leads to TB surface-wave poles at C = iCp. Since P.- = JCz-krz i = 1.2 (6.3.10) and p2 is changed to P2 = 142 42 = Jki-c’ (6.3.11) Using above definition, 51. (C) = 0 becomes 5,.(c) = jlqzcosmzr) +p1sin (4201 = 0 (6.3.12) Rearranging leads to This is the well-known TE-odd mode eigenvalue equation in dielectric slab waveguide. When the frequency is greater than the cut 03' frequency, it has proper root. The roots of (6.3.12) or (6.3.13) can be numerically obtained by the Newton-Raphson method, which requires the derivative of ti 1. (c) with respect to g A", (C) = fighIC) = A (1 +P1‘) [423m (92’) “P1305 (920] (6.3.14) P192 The function Ah (C) can be approximated by a Taylor’s series of first degree near C = iCp. Hence it can be written as 5,, (29) = 0 (6.3.15) 5,. (C) 23',(ic,) (c: 9.) (6.3.16) Let’s consider the integral (Figure 6.2). Cpr"8 C1""..5 er-a k2’+8 1.1%” x c Figure 6.2. Integration along the real axis. Cm f(C)d (6.3.17) .,,-.A, (0" where C, = C" +ij, Cpr and Cpi are the real and imaginary parts of CF, respectively. 5 is a small number. Making use of the approximation for the 3,, (c) in (6.3.16), and f(C) 5f(C,) ,the integralisevaluatedas ("+8 :I: a-f-—(§)d him") C—C-ic ~2j ..flg") aunts— (6.3.18) 15A(C)dC A'). (C,) q". 5 P A'). (C,) P‘ Therefore, the integration of (6.3.2) from the vicinity of the surface-wave pole is cfl+5 p 28 ‘,(C )1” (C, ) I 87(C)f"(C)dC =1 ’ , ”“28— (6.3.19) Cfl-O All(cp) pi where j 1- -cos (qux) cos (qzsAx) P _ f? (C)= Ax (4211):)” sin (qzt) (6.3.20) Using (6.3.14) and (6.3.15) leads to 95 .. jC 1+p,r A' :1: = —". 6.3.21 "( (P) pt 810(92‘) ( ) The result of (6.3.19) is right only when C, i is small, i.e. for small loss or lossless di- electric. If CF, is large, since 5 is small, then Cfl+8 I s: (C)fi’(C)dC-287 (c,,)f.”(c,,)6 (6.3.22) C',-8 INTEGRATION IN THE VICINITY OF [:2 Let’s consider the integrands of (6.3.2) in the vicinity of k2 = k2, +jk2, (Figure 6.2) 5+8 I 87 (01‘:y (C) (K (6323) 19,-; From (6.2.36) and (6.2.37), it is found that C = 1:2(p2 = O) is a branch point, and p2 oc- curs in its denominator. But the behavior of the x-relative part of integrand in (6.2.19) as C —) k2 (p2 -) 0) is .,,. (0 + 00" (PzAx)'1(_e-p,(zr+x.+x.) +P2:p1‘-p,r all: (”Mr T {’mlP2(’.‘x.)l+0flhlpz(21+x_+x.)]}} Ir Ir ”I? (*2) _|x-_x_l '1? (k2) “Mm (6.3.24) 3— AI 3: Ax where I: _ s Ple :2 _ I? (C) - 5+1—Ip—‘t7 s- 1,2,...,2Nx 1 (5.3.25) 1%"(0 = -1/6 (6.3.26) 96 Then, C = 162(1), = 0) does not lead to a zero in the denominator of the integrand of 5+8 5+8 (6.2.19). The integral I g;(§)fi’(§)dc canbereplaced by I gf(c)fi"(§)dc. 1,4 5-6 It does not change the final result in (6.2.22). This leads to 5+8 kz,+8 I 87(C)f?(C)dC=> I 87(C)I".’"(C)dC 1,4 k,,-8 ~2sg;(k2)f?*(k2) for small 1:, (6.3.27) If k2, is large, the integration of (6.3.2) can be replaced by 5+ 8 I x:(C)£’(C)dC~2sgi(k2,)£’(k2,) (6.3.28) k3; 8 SUMMARY The integration of (6.3.2) along the real axis shown in Figure 6.2 can be written as the sumoffiveterms - (,4 C,.+8 g,- -5 It,” .. Isi(C)fl’(C)dC={I + ;I + I + I + I }67(C)f,”(C)dC o 0 -8 t +8 19,-6 2,,” (6.3.29) The second and fourth terms are discussed above. Rombcrg integration method [66] is used in the first and third terms. The fifth term is computed by using the convergence form [(6.3.4) and (6.3.5)]. The convergence is found to be relatively rapid. 97 6.4. NUMERICAL INTEGRATION ALONG BRANCH CUTS In the last section, the matrix elements are expressed as the integrals along the real axis. There are two ways to express the matrix elements in terms of integrals along branch cuts and residues at surface-wave poles. One is substituting Guy, (I! x'. z - z') = Gig; (xl x', z - z') + 22"; (:4 x', z — 2') (6.4.1) in (6.2.14) and doing the same simplifications in the last section. Other way is applying complex analysis to (6.2.34), and rewriting the integral along the real axis in terms of inte- grals along branch cuts and residues at surface-wave poles. 'I\vo ways give us the same resultsasfollows: was) = Ig.(C)f.”(C)dC = - I g.(C)fI’(C)dC- I rt(C)fi’(C)dC (6.42) .. c, c, Residues at surface-wave poles are I r, (or? (C) (K = -2anes (rum? (C) 1 |,__, _C C, ' 8)(-C,)flw (-C,) _ 2“,gi(-C,)£”(C,) = —21[j _. J .,,. (6.4.3) A ). ('C,) A ). (C,) where A', (iC,) and If” (Cr) are given in the last section, and [1 -cos(CpAz)]e-jc'W/(CPA2)2 z = 1, ...,Nz-l (6.4.4) 8 (-C ) '= { . I p (1-jC,Az-c""v“‘)/(Cpnz)2 1= 0 Integrals along the branch cut can be written as I r.(C)f:’(C)dC = Is,0a)£”’(p)1§dp (6.4.5) CI 0 98 where _1_ e-GU-I)Az_2e-alAz+e—G(I+I)Az]/(aAz)2 [ 8100) = {2 (e‘°““+aAz- 1) / (aAz)2 o l-cos Ax 137%!” = jCh(P) (Q23 ) cos (qzsAx) s = 0,1, .,,, 2Nx‘1 (92”) with 299 C"(p) = 2 2 22 - 2 qzcos (qzt) +p sm (qzt) and a = Jpz-kf ‘12 =Jl‘g'ki'1’92 Let’s discuss the behavior of the integrand in (6.4.5). When p -) co, C. (p) -> 2 Up I = 0 810a)-*{ l/l)2 l=l [mm/p2 otherwise where c21."l'hus, ’ Cum/p4 1= 0 8:00)fiyb(9) ->< C,,(p)/p5 1: 1 .Cp. (P) (“Mir/ps otherwise (6.4.6) (6.4.7) (6.4.8) (6.4.9) (6.4. 10) (6.4.11) (6.4.12) (6.4.13) 99 It is observed that the integrands of the integral along the branch cuts and along real axis go to zero with a same rate (at least Up4 or 1/ C‘) when the integral variable is large. SPECIAL CASE: Im {k1} = o Ifregion 1 is lossless, then k“ = O and k1 = 1:1,. The equations defining the branch cut become QC: = 0 C3 - C? < ki (6.4.14) Thus, the branch cuts are decomposed to include a contour on the real C axis and a con- tour on the imaginary C, axis as shown in Figure 6.3. By the same argument as given in Chapter 3, pl should be chosen as positive pure imaginary along the right and lower parts and negative pure imaginary along the upper and left parts. E1 'llElCD'l If the integration is along the branch cut directly, then 0 a. J momodc = I g.(§)f.’"(t>dc+ Ig.(ia>fi"(a)jda (6.4.15) In the first term on the right side of above equation, . 1- cos (q Ax) f’.’"(C) =Jc,.(§) 23 cos (qzsAx) s = 0,1,....2Nx—1 (6.4.16) (92“) with 29 q C).(C) = l 2 (6.4.17) (1:0082 (qzt) + qfsin’z (qzt) and 100 Figure 6.3. The path of integration along the branch out on the top Riemann sheet when region 1 is lossless. 101 2 91" "1’ 3:! 2 2 92: 1‘2" In the second term. C has been changed to ja, and where . l-cos (q AI) fi"(a) =1C).(a) 2. “18(4st 8 = 0,1,... 92 with 2 Ch (on) = 2 9142 2 2 . 2 42°05 (92‘) +9131“ (92‘) and ql = Jki‘sz J——2 q2 = [cg-H! BI . 1 IE IE 'I ”.11g Let’s change the variables in the integrals in eq. (6.4.15). §= 414-92 (0 Eref - t 82' l’10 8" “0 82’ 11o X—- k \ 0’ = 00 ‘ Z = -a Z = a x L»! V Figure 6. 4. ml surface wave E,~ is incident upon the discontinuity. There rs a transmitted T131 surface wave E,, a reflected TE] surface wave Ewe. and radiated waves End. 105 1 O 2 S I I I I I I .., Along the .branch cut “3’ 1 . 02 .. With a vanable change 8. 5 1 o 15 - - 35?. .. fa: .. 8/Ir0 = 10‘4 2 1 . 01 - 5 q ‘3’ " 6/k0 = 10" o \ z c:- 1 o 05 - . . = 3 Alon the real axis 0 g E.“ l r- . z t 3 ¢ ¢ ‘ ‘ /" ‘ A ‘—"~‘ Along the branch out directly 0 . 9 9 5 I l l l l l 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 2a / 2.0 Figure 6.5. The numerical total power normalized by the incident power with TB; sur- face wave incidence vs. the width of the discontinuity. It is computed by three integration methods: along the real axis, along the branch cut with! without a variable change. (81, = ed, = 1, £2, = 4, t/ko = 0.25, N, = 13 andNz = 4 to 17) 1% The relative computation times of these methods are Integration along the real axis 100% Integration along the branch cut directly 27% Integration along the branch cut with a variable change (5/k0 = 10") 50% Integration along the branch cut with a variable change (6/k0 = 10") 105% Comparing the curves in Figure 6.5 and above results, it is observed that the integration along the branch cut directly is the best in the three methods. It is also found that the sum of the transmitted power, reflected power and radiated power is almost equal to the inci- dent power (the relative difference between the incident power and the total power com- puted by integration along the branch cut directly is less than 0.02%). Convergence of the MoM solution is checked by comparing results when the partition numbers N; and N z of the discontinuity region are increased. The results are presented in Figure 6.6, where the convergence is evident. 0.07 0.069 0.068 0.067 .066 0.065 Reflected Power 0 0.064 0.063 0.062 107 Nz V §i‘\~\ \_\ O = 00 5 10 15 20 25 30 35 Figure 6.6. Convergence check on the reflected power computed with difl'erent N x and Nz when a TEl surface wave is incident upon an air gap in the grounded sheet. (el, = ed, = 1. e2, = 2.1316, zen.0 = 1,and k0: = 2.5) CHAPTER7 PULSE GALERKIN’S SOLUTIONS FOR TM-POLARIZED WAVE INCIDENCE 7.1. INTRODUCTION In Chapter 4. the Method of Moments (MoM) is applied to transform the electric field integral equation (EFIE) into matrix equations for general basis and testing functions. Here, a pulse Galerkin’s implementation is applied to establish the matrix elements AS}: and the forcing vectors FM for TM-polarized wave incidence. First. formulas to calcu- late the matrix elements and forcing term will be derived when both basis and testing functions are 2-D pulse functions. Then some numerical integral techniques in the con- tour integrations along real axis and along the branch cuts will be discussed. Finally, these methods will be compared with each other by checking their convergencies. 7.2. PULSE GALERKIN’S SOLUTIONS It is assumed that the cross section of the discontinuity region is rectangular (lzl S a, —t S x S 0) (Figure 6.1) and the permittivity ed is a constant. For I'M-polarized wave in- cidence, the EFIE in the discontinuity region becomes .koAnz a 0_ A 3' 22 (I, Z) -1 no 1' £022 (1' X's Z " Z.) . E2 (X‘, Z.) dX'dZ' = 22(1’ Z) (7.2.1) for lziSa,—thSO where 108 109 An2 = Iii-n; = (ed-rave0 (722) The discontinuity region is partitioned into N = MINz equal rectangular elements AS" centered at (xn, z"), where N, Nz are total numbers of partitions along 2 and 2. re- spectively. The partitioning is defined such that i. 0 N f (x. z) dxdz = 2 I f (x, z) dxdz (7.2.3) .g l = 1 AS, From Figure 6.1, it is found that A5,”: [x-xn] zu 8811 (Zm’ln) = { 0 Z,” = Z» 1 (7.2.17) '1 zma 'I'sin2 (ax) (a—b)1t/2 bin(lCAz)l/(CAz)2 I = 1, ...,Nz'l g = l {ll-cos(CAz)]+j[CAz—sin((;Az)]}/(CAZ)2 [:0 (7.3.1) Since the‘integral from —oo to no in (7.2.44) is symmetric, then it can be rewritten as B“(l.s) = 137(C)f,“(§)dc (7.32) 811(1. S) = 187“”? (QdC (73-3) 3110.3) é 1821(C)f:z(C)dC (7.3.4) where gm) = 2cos (1:752) [1-cos(CAz)]/(CAz)2 z= o,1,...,1v -1 (7.3.5) Z gm) =.2jsinutAz) [1 «osmium/(gay z = 1,...,1vz-r (7.3.6) Since 811(1, 0) = B‘1 (0, s) = 0, then no definition for g; (C) is necessary. Since f: B (C) involves hyperbolic sine and cosine functions, it is prone to numerical difficulty. As the integration variable in (7.3.2) to (7.3.4) increase toward infinitely, both the sinh and cosh functions become unbounded. In addition, it is difficult to ascertain the 118 convergence properties of the integral Both of these problems can be overcome if the in- tegrands are written in terms of exponential functions. After some algebraic manipula- tion, the results become 1:2 21_ #7an _ Ax pl-pze /s _ 24, mt) = —-§+55—‘75—{1+(1-e"= ) .. 1 2M“ ’1 (7.3.7) P2 P2 p2 Act _ -p;Ax _ p -p e /t-: _ _ pzAx A21 13‘“) = 0 (7.3.9) 2 'PzAx 2 Wm = 5—11" 1 i-e"=“'1”1’ -' p; 2p2Ax +pl-2281/62 [e-p,[21-(s+1)Ax] +e-p2[21+(:-1)Ax]] } (7.3.10) :1 .. Ax 2 _ (l-ep1 ) -p,(s-1)Ax p -p e la _ _ _ _ _ l 82 l 2[e p,[2t (s+l)Ax]+ep,[2r+(s 1)Ax]] } (7.3.11) cl . -p,Ax 2 ’82“) = {S (1‘8 ) {_e-p2(:-1)Ax '1 P2 2P2” _pl_:2€l/62[e-p,[21-(:+1)Ax] _ e-p,[2t+(s-1)As]] } (7.3.12) c1 where 75“ = 275,721 = pl (1 +e'11’21) +p2(l -e-2p’1)el/ez (7.3.13) 119 CONVERGENCE OF THE MoM MATRIX ELEMENT INTEGRALS Before undertaking the numerical integrations of equations (7.3.2-4), it is quite helpful to anticipate the rate of convergence of the integrals. Of interest is the behavior of the in- tegrand 87(C)!?11(C) as c-w. l/C s = O for QB = xx, zz 1:75 (C) _){ 3:1,2Nx-1 for all aB's (7.3.14) [cw/ C otherwise where c2 1. Then I {1118:2223}[1-905(CAZH/C3 s = 0 for (15 = xx, zz g, (011,111 (C) —> s = 1, 2N, -l for all up's cos (lCAz) __ -cCAx 3 _ {jsin (ICAz)} [1 cos (CA2) ] e / C otherWlse (7.3.15) Since there are some periodic functions in the integrand g, (c) {:11 (c) , it is integrated one by one period when C is large in the program. Using cosx sinx sinxd Ide = _+'1x”'*1 (7.3.16) yields ’0 + 2‘ m to + 2! sinx x I Eidx = M0|:(x +02“) -1]+ m I," sinx de (7.3.17) ‘0 x1" x0 0 x0 x'" When x0 -) co, ( xo ) 2m1t .. 1 ,... ._ (7.3.18) x0 + 27: x0 12D Thus,therightside of(7.3.17)willgotozeroatleastas l/xg'“. Thereisasimilarresult xo+2: ' cos 1 A2 for I fldex. However, {, , (C ) xg {1 }srn(lCAz)} 11 -1105 (CAI-)1 insrm isnotasinglepe- riodic function, the period of cosCAz is 1 times of the period of cos (lCAz) and sin (lCAz) . If they are written in terms of several single periodic functions as 2cos (ICAz) [1- cos (CAz)] = 2cos (lCAz) - cos [ (l- l) CA2] — cos [ (1+ 1)CAz] (7.3.19) 23in(lCAz) -sin[(l— l)CAz] -sin[(l+1)CAz] (7.3.20) 23in (ICAz) [1— cos (CAz)] g, (C) can be expressed as the sum of three single periodic functions, and the integrations can be carried out separately. Hence, the integration I g, (C) 11:11 (C) dC will convergence 0 quickly. INTEGRATION THROUGH SURFACE-WAVE-POLE SINGULARITIES Since the presence of the loss in region 2 makes k2 a complex number, then all the sur- face wave poles of the integrand are not along the real axis. But, if the loss becomes small or vanishes, the surface-wave poles will shift to the real axis, and thus reside within the domain of integration. Therefore, pole extraction will be used to deal with the surface- wave-pole singularities. Let Cp = C” +iji, where C" and CPI are the real and imaginary parts of CF, respec- tively, and CF is a solution of TM-even mode eigenvalue equation A. = 0. After the same analysis in the last chapter, it is found that 121 t +8 3p " 2 ( ) , ( ) I 8r(C)f:p(C)d§ =J' 81 E11 f C11 am;— (7.3.21) CW-s A¢(cp) P1 where C, 1 pl sin2(q2Ax/2) fix, (C9) = -(q_2) sin (42:) qux/Z 11°81‘11”“) (7'3”) - 2 p sm (4 Ax/Z) If” (C,) = - “((1120 quzx/z cosuqux) (7.3.23) 2, _ _J'C, pr sin1(qux/2) , f: (C,) - -q—2- sin(q2t) qz Ax /2 sm(sq2Ax) (73.24) with P1= (iii-k1 (73.25) 2 42 = sz - Ci (73.26) Equation (7.3.21) is valid only when C p i is small, i.e. for small loss or lossless dielec- tric. If Cp‘ is large, 8 is small, then C"+ 8 I s; (0f;111 (C) dC - 281 (£1,911,111 (CW) 8 (7327) t,,- a INTEGRATION IN THE VICINITY OF It2 Let’s consider the integrands of (73.24) in the vicinity of k2 = 1:2, +jk2‘ (Figure 6.2). From (7.2.46—51), it is noted that C = 1:2(p2 = O) is a branch point, and p2 occurs in their denominators. But the behavior of]:113 (c) as t; —> k2(p2 —> 0) is 122 13%,) = 1+ (k,Ax)’{1/3- [t/Ax+£l/(ezple)]}/2 (7.3.28) fiflkz) = -1 (7.3.29) f5‘(k2) = 0. (73.30) ff‘urz) = (szx)2{s— [t/Ax+81/(ezp1Ax)]}/2 (73.31) fi‘Ucz) = 0 (73.32) fizUrz) = -J'k2Ax/ 2 (73.33) Then, C = 1:2(172 = 0) does not lead to a zero in the denominators of the integrands. Thus 12‘” (C) can be replaced by ff" (k2) in the vicinity of k2. SUMMARY The integrations of (7.3.2-4) along the real axis shown in Figure 6.2 can be written as the sum of four terms .. (r-s (”+5 2a,, .. Jr, (073%) dc = { j + j + J + I }r, (0)?” (t) dc (73.34) 0 0 (.,,-5 (”+5 2’5, The second term is discussed above. Romberg integration method [66] is used in the first and third terms. The fourth term is computed by using the convergence form [eqs. (7.3.7-12)] and dividing g, (C) into the sum of three single periodic functions. The con- vergence is found to be relatively rapid. 123 7.4. NUMERICAL INTEGRATION ALONG BRANCH CUTS In the last section, the matrix elements are expressed as the integrals along the real axis. There are two ways to express the matrix elements in terms of integrals along branch cuts and residues at surface-wave poles. One is substituting z?-22(Jrlx'.z-z') = iiT—z5(f5-l3') +53“ (x1 x'.z-z') +agzad(xlx'.z-z') (7.4.1) 2 in (7.2.11) and doing the same simplifications in the last section. Other way is applying complex analysis to (7 .2.44), and rewriting the integral along the real axis in terms of inte- grals along branch cuts and residues at surface-wave poles. 'I\vo ways give us the same resultsasfollows: Ewan) = Ig.(§)ff‘“(c>dc = - jg,(C)r‘."(C)dc- jg,(§)f,‘”(C)dc (7.42) _. c, c, Residues at surface-wave poles are 1mm?" (04: = ~2anes {81 (01?" (C) } I.“C P r,(-c,)f:‘””(-c,) _ urn-gmi‘n-g) = ..2“). ~. I ~ .4.3) A c (“Cp) A; (Cp) where f?” (Cp) is given in the last section, and ' [1-oos(§ Az)]e""’A‘/(§ 13:)2 1= 1,...,1v -1 .. P p z 8; (-C,) - (7.4.4) (1 —j§pAz - fig“) / (CPAZ)2 l = o Integrals along the branch cut can be written as I 81%)}? B (C) 4C = Igrfia)f§’pb(p)j-g-dp (7.4.5) C6 0 124 where . é.[e'“""’A‘-2r'°"“+r'°‘"*"“]/(orAz)2 z= 1,2, ..,Nz-l 8100') = (e'aAz+aAz-l)/(orAz)2 1= 0 (7.4.6) 2 . 2 b _ 41 Ce ((1) sm (qux/Z) f)“ (9) -J 4; qux/z cos(squx) (7.4.7) - 2 b _ . srn (qux/Z) t? (p) - 1C.(a) 42M” 008(sq2Ax) (7.4.3) - 2 2,, _ 41C, ((1) sm (qux/Z) . f: (p) — j 42 (1213/2 sm(sq2Ax) (7.4.9) with 2pqze /e C¢(p) = 2 2 ‘ 2 2 . 2 (7.4.10) p cos (qzr)+(q2t:1/ez) sin (qzt) and or = pz-kf (7.4.11) 42 = Jkg-kfflDZ (7.4.12) Let’s discuss the behavior of the integrand in (7.4.5). When p —-> oo, . 481/82 C, (p) -) 2 2 (7.4.13) 1+ (cl/£2) + [1 - (cl/£2) ]cos(2pt) Up I = o 8, (it!) -> { 1/ p2 1= 1 (7.4.14) {emu/p2 otherwise 125 where c21. Thus, C¢(p)/p2 I: o for aB=xx, zz 81 0a)}? “(9) -+ CAN/pa I=1 (7.4.15) C, (p) [CW/p3 otherwise It is observed that the integrands of the integral along the branch cut go to zero slower than the integrands of the integrals along real axis as shown in (7.3.15). Furthermore, there is an oscillating function in the denominator of C, (p) . The integrands in (7.4.5) cannot be written in terms of several single periodic functions. Thus, the integration along the branch cuts will take more time than the integration along real axis when the integral vari- able is large. SPECIAL CASE: Im{k,} = o Ifregion 1 is lossless, then k1 i = 0 and k1 = 1:1,. The equations defining the branch out become QC: = 0 C3 - C? < *1 (7.4.16) Thus, the branch cuts are decomposed to include a contour on the real C axis and a con- tour on the imaginary C axis. By the same argument as given in Chapters 3 and 6, the inte- gration along the branch cut directly is found to be I g,(o£"(c>dc = j g,(C))f“'(C)dc+ jammfimmma (7.4.17) Co -I:, o In the first term on the right side of the above equation, , _ jc’cxosinzmzAx/z) f? (C) - ‘ q; qux/Z cos (squx) (7.4.18) 126 , sin2 (q Ax/2) 7:1(c)= -jC.(C) Mix/2 cos (squx) CC.(C) sin2(qux/2) , 2" = f: (C) qz qux/z 8m (squx) with 2q1qzel/e2 C¢(C) = 2 2 2 . 2 qlcos (qzt) + (4251/82) srn (qzt) and 91 " i- C2 42 = ’9?" C2 In the second term, C has been changed to jar, and where 2 - 2 i _ .a Ce ((1) srn (qux/Z) f? (a) -J q: (hm/2 cosmzAx) ,. sin2 (q Ax/2) fizm) =—jC¢(a) qui’x/Z cos(sq2Ax) pic, ((1) sin2 (qux/Z) zr' _ . f: (a) - J 92 92 Ax /2 sm (squx) with 291923 /8 Ce ((1) = 1 2 trims2 (qzt) + (4281/82):st (ant) and (7.4.19) (7.4.20) (7.4.21) (7.4.22) (7.4.23) (7.4.24) (7.4.25) (7.4.26) (7.4.27) 127 (11 = kai- <12 (7.4.28) 42 = Jk§+ 0‘2 (7.4.29) Let’s change the variables in the integrals in (7.4.17). In the first term (—Ir1 < C < O) C = ‘ka - 92 (0 < p < k1) (7.4.30) and a = ,/p2 - k} (I:l < p < co) (7.4.31) in the second term (0 < a < co). Substituting (7.4.30) and (7.4.31) into (7.4.17) leads to the integration along the branch out with a variable change I 81(C)f‘rw(§)d§ = Is; (fwffflb (13)];de (7.4.32) Co 0 where g, (for) and if” (p) are given in (7.4.6-9), respectively, and Jpz-kf k1 -2t 4 0 t 21 3 51 (x, z) A Figure 8.1. Extend I?Jlr (x, z) and E2 (x, z) to periodic functions of x at a given 1. 134 2 n=0 1 n21 (8.2.9) Similarly, E2 (2:, z) is an even function of x with a period 4t, so it can be expanded in terms of the Fourier cosine series Ez (x, z) = 2 a”, (2) cos? n=0 where azn(z)= sl—tj‘Ez (x, z)cos—dx "—2r Changing the variable x to y = t + x, the above equation leads to 1 an m: . rm . nu ” -r (8.2.10) (8.2.11) (8.2.12) Since Ez (-x,z) = E, (x, z) = --Ez (2t+x,z), i.e. Ez (t-y,z) = ~13z (t+-y, z) is an odd function of y, then 28111 (rm/2) “2.77 (z) = { Thus, E2 (2:, 2;) can be expanded as (2n-1)1tx Ez(x, z) = 2am“) cos 2t =1 Consequently, the x-dependent basis functions are obtained iEz (y- t, z) sin—dy n = 1,3, 5,... 0 n=0,2,4,.. (8.2.13) (8.2.14) 135 n -1 fix Xm, (x) = cos(_x__)__ (2n,- 1)xx (”‘15) X215“) = cos 2: for nJr = l, 2, ..., Nx. It is clear that the above basis functions satisfy conditions (8.2.5) and (8.2.6). The discontinuity region is partitioned into Nz - 1 equal sub-domain along the z-direc- tion as shown in Figure 8.2. It is observed that X 819 "'0 u1 = 1 2 3 N: X=0 y >2 Nx 82' l‘0 x=—t * ° = °° | I z=-a z=a Figure 8.2. Applying sinusoidal and piecewise-sinusoidal (SPS) Galerkin’s Method to EFIE in the discontinuity region. z": = -a+ (nz—1)Az (8.2.16) where Ar = 2a/(Nz- 1) (8.2.17) 136 Recall that in the solution of Hallen’s integral equation for the current distribution on a lin- ear antenna [68], only Nz - 2 piecewise-sinusoidal functions are needed since the currents are zero at both ends of the antenna, where Nz - l is the number of partitions. However, since the electric field E, (x, z) and Ez (x, z) are non-zero at boundaries z = in, then we can not use just N z - 2 piecewise-sinusoidal functions to expand these fields but should add one half piecewise-sinusoidal function at each end as shown in Figure 8.3. an‘Zn‘ (z) A aIZI (z) 0222 (Z) “N‘ZN‘ (Z) g (H? z1=-a z ZN‘=a Figure 8.3. Piecewise-sinusoidal functions Both Zn: (2) and Zzn‘ (z) are chosen as the piecewise-sinusoidal functions an'(z) = Zzn‘ (z) = Zn‘(z) (8.2.18) From Figure 8.3, it is evident that sin[k(z2-z)] 21(2) = { sin(kAz) 0 otherwise < < ‘1 2 z2 (8.2.19) 137 Sin [k (Z -zy‘_ 1)] ZN, (Z) = { sin (kAz) I'M" < z < z”: (32,”) 0 otherwise sin [Hz -z,.‘- 1)] . [ sin (kAZ) z":- 1 < Z < z": Zn‘ (z) = 4 sin [1: (zn‘+1 - 2)] (8.2.21) sin (kAz) 1». < z < in.“ 0 otherwise for nz = 2, 3, ...,Nz- 1, and where I: will be [to for numerical results in free space. Zn‘ (2) has been normalized by sinkAz to let the maximum is 1. It is observed that since the piecewise-sinusoidal functions are continuous in the do- main lzl < a, then SPS Galerkin’s method may converge faster than pulse Galerkin’s method, because the pulse functions are discontinuous in the same domain. 8.3. SP8 GALERKIN’S SOLUTIONS In this section, the formulas to calculate the matrix elements and the forcing term will be derived by using SPS Galerkin’s method. Substituting the expressions for the basis and testing functions 1n the formulas to calculate A? and FM [(4.5.5) and (4.5.6)] gives vii-4‘1 m (x, z) X”. (x) 2m (2) dxdz (8.3.1) 8“,, Azfi=ti A312 (2)2, (z)dzIXMM(x)X ’(x)dx -f 20 O in 0:112] Idxdzxam, (1)2». (Z) ,1 1022113le 2' Z')Xp,, (x',,)Z (z")dx'dz 0 -a -a -t (8.3.2) where 138 m = (mar-1)Nz+mz (8.3.3) I: = (nx—1)Nz+nz (8.3.4) Note that A :E/ (tAz) and FM / (tAz) are newly defined as 4“” and FM ,respectively, and where A112 = rig-n: = (54732) /t-:o, and 1:“2m (x,z) , 6220,5011 x',z-z') are giv- en in Chapters 2 to 4. Let’s evaluate the above spatial integrals required in spectral-integral representations of MoM matrix elements. The first term of (8.3.2) is evaluated first. o o (m -l)nx (n -l)1tx {Km’ (x) X”, (x) dx = Icos—x—t——cos—-’—t——dx = 55:752.)”, (8.3.5) ° ° (2mx-l)1tx (2nx-1)1tx , IX”, (1:) X“: (x) dx = Icos 2t cos 2: dx = 58171,», (8.3.6) -I 1 sin(2kAz) _ _ '2'[1" 2kAz ] "'1 " "z " I’N‘ sin (2kAz) _ _ _ sin2 (kAz) film (1)2 (z)dz = 1 l- __2_kAz mz - nz - 2,...,Nz 1 8111 (kAz) 2[—sz—-COS (kAZ)] [mt-n2] = l t 0 otherwise 1 s1n(2kAz) ~ sin (kAz) _ —2.{[l 2kAz ]SM,5m,n,+[ kAz cos (kAz)]8lm 11.] l} (837) where 2 m = 5,, - { 1 m>1 (8.3.8) 8m. = {l m = n (8.3.9) 139 (8.3.10) all 3 ll p5 Substituting 62, (xl x'. z - z') = 225 (p‘ - B')jr|2/k2 + ("352 (74 x', z - z') in (8.3.2), where (":32 (11 x'. z - z') is given in (2.4.1013), it is found that A”: (Lt—An: )5MDMR+§2_4 otA __zAnz 1‘“;- "Erma”... I{-e -p,lx- -xl 82 -t -t . - e/e _ _e”1(2'””)+pl p21 22e p"cosh[p2(r+x)]cosh[p2(t+x')]} 5. m —1 1:2: n -—l 1tx' ' cos( x ) cos( x ) (In? (8.3.11) r r r 3 tAz l 'I Afr?» = Dmn+EE—An 2 IdCPngun (C); I{e ”P31 x _ . p -pe/e _ep,(2t+x+x)+ 1 A21 2 2e"’=‘sinh [p2 (t+x)] sinh 1p; (t+x')] } (me-l)1tx (2n, - 1) nx’ dxdx' COS COS 8.3.12 2: 2r ,2 ( ) 4:, = 2%:49: n2 Idmgm”. (of jrsgn(x- x)e ”h" "' -t -r _ e’P2(2”“’) — p1 -p§81/822e'p"cosh [p2 (r+x)] sinh [p2 (t+x')] } cos (:nx-tl) xx cos (2% ;t1) “x. (11:211. (8.3.13) Affm = Afifn (8.3.14) where 140 8”,": [1_______ sin__(____2kAz)]5 8 [sin (kAz) -cos(kAz)]5l,,,‘-,,J,1} (8.3.15) 3..., (C) = j j J‘“ ‘ ’2.. (2)2. (z ) dz“ (8.3.16) (.2822 Now, the integrations with respect to x and x' are evaluated. Using (11 . Iewcos (bx) dx = e [acos (bzx) +2bs1n (bx)] (83.17) a +b yields 0 o I | 1 -p2 X-X‘ I I I, [e {sgn (x -x') } cos (ax) cos (bx )dxdx 0 x 0 = Ive-”(kn cos (bx') dx' :1: JIM-x ) cos (bx') dx'] cos (ax) dx 1 x 0 1 2] ”PM” - e’P’m” [pzcos (bt) — bsin (no) P2+ +p2cos (bx) + bsin (bx) 1F [-p2cos (bx) + bsin (bx)] } cos (ax) dx ——{—-5——12{=Fp2[p2-e"'p [pzcos (at)- asin (at)]] - [pzcos (bt)- -bsin (bt)] =p2+b2. p2+a2 [—-p2 {p — [ pzcos (bt) bsin (1701] } + (1:1: 1)P2[sm2[((bb::))d + sm2[((bb::)) t1] 1-cos[(b-a)t] 1-°°S[(b+“)‘]]} (8.3.18) “(1‘1)b[ 2(b-a) + 2(b-I-a) Then, 141 o o , (Don I I Ie-p2'x-x'lcos———(m— l) nxcos—(n - l)1tx dxdx' - 122:5”st (”2‘)2[1+(‘1)"e.p"l[1+(-1)"”"l 13319) (1’2‘)2+[('"-1)1tl2 [(pzt)2+[(m-1)u1’1[(pzt)’+[(n-1)u171‘" o o . l I Je_p"’-x.'cos (2m-1)nxcos (2n -l)1tx dxdx' Ozz _ _ (hm - ,2 21 2: -t -t p215” _ (pzt)2+ [(2m-1)1t/2]2 (p,t)2- (-1)"*"(2m- 1) (2n- 1)x’/4-p,te"*'[ (2n!- 1) (-1)"'+ (2n- 1) (-1)']z/2 2 2 2 2 (8.3.20) [(Pzt) +[(2m-l)t/2] ”(1’20 +[(2n-l)t/2]] 1° ° l | (m l)1tx (2n l)1tx' 132 _ _ -p2x-x. _ I - - 1 (PM - 12!, la sgn (x x)cos——cos 21 dxdx _ 1 {_ 2(2n- t)2 +P2‘[1+('1)'¢_h‘] 1p,” (-1)"”(2n- oat/21} (pzt)2+[(2n-l)t/2]2 (2n—l)2-4(m-l)2 (p2!)2+[(m-l)l]2 (8.3.21) Using the following integral formulas, o . -1 tsmh t -lt- I cosh [p2 (t+x)] cos de = p; (p, ) 2 (83.22) _. ‘ (p20 + [on-1m 0 - m 2 _1 tsmh t - -l 2m-l u/2 lIcoshIp2(t+x)]cos( m )nxdx = p2 (p22) ( ) ( 3 ‘_, 2‘ (p2!) + [(2m- l)n/2] (83.23) 0 m , -1 tICOSM 1) + (-1) ] %Ismh [p2 (t+x)] cos-LT—t—UE-xdx = pz 2 pz 2 (8.3.24) ., (P21) + [(m- 1M] 142 0 ._ tcosh t ljsinn1p2(t+x)1eos(2m 1) "’dx = 2’” (p’) 2 (8.3.25) ‘_, 2' (p2!) +[(2m-l)1t/2] 0 -p;‘ M - - - t e + -1 life p’(‘+x)cos———(m 1)::de = p2 E ( ) 12 (8.3.26) ’-, ‘ (pzt) +[(m-1)1t] 1° ., n... (2m-l)1tx p.te""+ (—1)"'(2m-1)n/2 - I e ’ cos dx = — 2 2 (8.3.27) ‘_, 2‘ (pzt) +[(2m-1)n/21 the remaining integrations with respect to x and x' can be evaluated. Consequently, it is found that 8 A“ = (1+E9An’)8 D —°t4—A—ZA mu 2 "I: "I" n’ I (ICE,— —:”gm (§)[— ogjfijel’gx (8.3.28) tAz Afn‘n = 0222+ :—:—An2 I dcngm 2 (g) [°o:f.,+¢.2:zu, (8.3.29) 8 1132A xz _ _(_)_ n2 lxz 2x: - - 2 - 2,, _ (p,t)’{-1(-1)"+e"*‘1 1(-1)"+e “‘1 +(1-e ”*‘1 (p,-p,2./e,)/A..1 an - 2 - 2 2 2 (8.3.31) ((102!) +1011 1):11[(p,t) +[(n-l)nll —1p2te" '-+( 1)"'(2m-1)tt/21[pzte'"’z +(-1)'(2n-1)2/21+(p21) (1+e'2")2(p1-pze./ez)/A.t ¢221= "" [(pzt)2 + [(2m-l)1t/2] ’1[(p,t)’ +[(2n-1)u/21 ’1 (8.3.32) t[(-1)"+e""1[ t”+(- -1)" (2 1)1:/2]— t)’ (1- 4'” e/e)/A 2“: "P2 p2 C n- (pg e )(pl- pz 1 21(83.33) [(22:1’+1(m—1)81’11(p,n’+ uzn-ox/zl’l with 143 8“ = 28”” = pl (1 +e'2p") +p2(l -e'2"")8l/e2 (8.3.34) It is observed that the leading terms of (CI/p2) é?” and ”$2; are constant when I; —) too, they can be consequently rewritten as 1 (802182;; = ; {5.}... + 81.3,} (8.3.35) 1 p2¢gzxznx = 7 {813,113+ ¢Liznx} (83.36) with dun: [“2" -I<~- 1)'115..5.._ 12C’r’n+(-11"c""1t1+(-11'*'1 n 2 2 2 2 (8.3.37) (P1!)2 +[(m-1)!l [(Pzi) +[(M-1)Il ”(1'20 +[(n-1)Kll 0’“ [(2M - 1) 1/2] ’5” "" (p,:)’+ [(Zm- l)t/2]2 1w (p,t1’- (-11"“ (2.... 11 (2.- 11894-11217"! (2...-1) (-11"+ (211-11 (-11'1:/21 2 2 2 2 (8.3.38) [(Pz') +[(2fl-1)V211[(P;') +[(2n-1)li/21 l Inverting the constant terms leads to 3 A ”d -‘-’-‘—‘An’ 1 is 6 8.1.3:) £2 41: t m "1.". _eoAZ 2 a a , dzdz'.‘ ((-') --€-2-—An 5.m5..£ iz”‘(2)z"‘(2)755’idw ‘ ‘ - 80A: 2 dZdZ 13.8.5, ”5”] 12..., (212,. (z z'Az)( )6(zz — 1 ioAn’s .—. 2 .. ...,..12... «>2 .z-«» 144 _‘0A250 8339 "E;"»-.nm (..1 Thus, A; and A32 have the new forms A; = (‘1222Dm,,+——An2 IngM (§)[- <11” “11": C zt/pz] (83.40) 824 e 8 oAZA ’43.. = (1+e—ZM2)Dm+—ez:fi 112 IngM (C) [$21232 +p2t¢22238 (8.3.41) It is noted that since all d>'s are even functions of C and the integrals from —oo to co in (8.3.40-41) and (8.3.30) are symmetric, then 8 Az °' A32 = 5,._D....+éfiAnz jdcgfn222(§1[- 81‘” +412" c zt/pz] (8.3.42) 0 A“: 1.1-8.0m.2 D .1—80-‘3—z Anzmjdcg‘ (g) [(1,122 +p 241’ “J (8343) nut 82 mn €241:ng 2 ' ' tAz 44:3. = l—An 2 Idchm”. (C) [0,13 +¢2‘f,_] (8.3.44) where 8.2,. (C) = .uzke{g,,.,.122 = (112123224- ”102222222 (8.3.67) 63222 = (fog; + (112221] (8.3.68) 1.. = [(5112-[(m-11n1118.6.._ p,c’?[1+(¥11'e""1 [1+(-11""1 (222.69) "" W12». ((...- 111:12 [(19202+[(m-1)t121 [0,112+ [(n-11nl2l ((2... - 1) V2125" "" — (“02+ [(201-1)t/2]2 lu- p21{(p21)2- (-1)"'"(2m-1) (2n- 1)::1/4- p,:e""( (2111- 1) (-1)"+ (211- 1) (-1)"] n/2} 2 _ 2 2 _ 2 (8.3.70) [(pzl) +[(2m 1)n/2] ][(pzt) +[(2n 1)1r/2]] 1... _ 1 {_ 2(24-11’ +1:»,rt1+(-11"'e""'1[1121+(-11"""'(2n-11u/21} "" ()p,:)’+[(2u-1)u/2]2 (2n-1)’-4(m-1)2 (p21)2+[(nt-l)t]2 (8.3.71) - _, 2 - 2,“ (p,r1’{-[(-11"+e"‘1((-11'+a""1+(1-e 2’“) (p.—p,e./e,1/A..} (22 22) "' [(11.0% [(m-1)xl2l [0202+ [on-1121’] ' 2 - 2,222 - [p,:e"=‘+ (-1)"(2m-1)x/2] [p,:¢'“'+ (-1)"(2n-1)x/2] + (p,:)’(1+¢'2'=‘) (hazel/spud '"' ' (o2r1’+t(2m-11x/21’1 Io.:1’+[(2n-11n/21’1 ’ (8.3.73) of: = -p2t[ (-1)"+¢""1 [21,157+ (-l)'(2u;1)x/2]2- (11,1120 -e"’:) (p1 -)u,el/1:2)/I1,l (8.3.74) [(Pz') +[(m-1)x] ][(P2‘) +[(2"'1)3/2]] 21 2 2 2 2 81 (C) (k “C ) . [cos(§Az)-cos(kAz)]2+ [216mm -(§/k).1n(mz)]’ 1: o 2lcos((AZ)-cos(kAz)l{IM(CAZ)-cos(kAz)lm(l(Az) 0 1/§ (8.4.1) v? l=QN-l 87‘ (C) -> 3 2 (8.4.2) 1/ C 0 < I < Nz - 1 8:2 (C) -) 1/C4 (8.4.3) It is noted that g? (C) is not a single periodic function, consequently it is written as the sum of several such functions [k2 - c2123? (g) = 1 + cos2 (kAz) + (g/It)2sin2 (kAz) - 2cos (ItAz) cos (can -2 (mt) sin (kAz) sin (CA1) (3.4.4) [It2 - C2]zg;:_1 (C) = cos [ (Nz - 3) CA2] - 2cos (kAz) cos [ (Nz — 2) CA2] 151 + 1cos2(1tAz) - (C/k) 2sin2 (1mm cos [(111z - 1) CA2] + 2 (C/k) sin (kAz) {cos (ItAz) sin [(111z - 1) CA2] — sin 1 (NZ - 2) CA2] } (3.45) 11:2 - (212112).) (1) = sin 1 (NZ - 3) CM - 2cos (kAz) sin 1 (NZ — 2) CA2] ' + 1cos2(1tAz) - (C/k)2sin2 (1mm sin 1 (111z - 1) CA2] —2 (C/k) sin (kAz) {cos (kAz) cos [ (Nz - l) CA2] - cos [ (Nz — 2) CA2] } (8.4.6) 11:2 - c2123,“ (1) = cos 1 (1- 2) CA2] — 3cos (kAz) cos 1 (1- 1) 14:1 + [l + 2cos2 (1412)] cos 111m] - cos (kAz) cos 1 (1 + 1) CA2] -(C/k) sin (kAz) {sin [ (l- l) CA2] - 2cos (kAz) sin [lCAz] + sin [ (H- l) CA2] } (8.4.7) 113-121 zgi‘m = smut-211421 -3008(kAz)sin[(l- mm + 11+ 2cos2(1rAz)] sin [1:112] - cos (kAz) sin 1 (1+ 1) CA2] + (C/k) sin (kAz) {cos [ (I - l) CA2] - 2cos (kAz) cos [lCAz] + cos [ (1+ 1) CA2] } (8.4.8) for! = 1, 2, ...,Nz-Z, and [k2- £212872 (C) = cos [ (I - 2) CA1] -4COS(kAz)008[ (1-1)CAz] + 211+ 2cos2(1tAz)] cos [lCAz] -4cos (kAz) cos 1 (1+ 1) CA2] + cos 1 (1+ 2) CA2] (8.4.9) 1k2- 121 2:72 (C) = sin 1 (1— 2) {Ad -4cos(kAz) sin 1(1- 1)CAz] + 211+ 2cos2 (1412)] sin 111112] - 4cos (kAz) sin 1 (1+ 1) CA2] + sin 1 (1+ 2) CA2] (8.4.10) for l = 0, 1, . . ., N2 - 3. Finally, the periodic integrand terms can be integrated separately for large C. When CA2 issmall,andC~k, 152 cos (CA2) — cos (kAz) - - (c2 - 12) (A2) 2/2 (3.4.11) If it is calculated numerically by using cos (CA2) - cos (kAz) directly, some significant digits will be lost. Thus, cos (CA2) — cos (kAz) = -2sin1 (C + k) A2/2] sin 1 (c - k) A2/2] (3.4.12) and sin (CA2) "' (C/k) sin (kAz) = 2cos [ (C + k) Az/Z] sin [ (C - 1:) A2/2] + (l - C/k) sin (kAz) (8.4.13) willbeused for small C. The points c= 1t, c= 1:2, (p,t)2+1(m,-l)n]2 = o. and (pzt)2+ 1 (211,- 1) 11/2] 2 = 0 do not lead poles of the integrand, even though t-It, p2, (p,t)2+ 1(m,- l) n] 2, and (p,t)2+ 1 (211,- 1) n/2] 2 occur in its denominators. Numerical problems occur, however, if the integration is along real axis exactly. Further- more, if the loss in region 2 becomes small or vanishes, the surface-wave poles will shift to locations along the real axis. Although the pole extraction can be used to deal with the surface-wave pole singularities, the integration will be calculated, for small C, along a de- formed path above the real axis as shown in Figure 8.4. 8 l 0 f k C: *2 Re{C} .1... Figure 8.4. Integration along real axis in the SPS Galerkin’s method. 153 It is important to choose an appropriate 5. If 5 is too small, there are numerical diffi- culties in the vicinities of Cp’ 1:, etc. If 8 is too large, it is impossible to obtain enough sig- nificant digits in calculating cos [lCAz] and sin [ICAz] . By our experience, [52‘ < 10" (8.4.14) and 25/110 > 0.1 (8.4.15) should be chosen using double-precision variables, where 2a is the width of the gap. The above two equations can be combined to 0.1 < 5/110 < 3.0/ a (8.4.16) Let’s consider a grounded dielectric sheet with a gap as shown Figure 2.1. A 'I'Mo surface wave is incident upon the discontinuity from right to left. It is assumed that the di- electric sheet is so thin that only TMO surface-wave mode exists. Then, a transmitted TMO surface wave will propagate to the left, a reflected TMO surface wave will propagate to the right, and radiated waves will be excited in space. Figure 8.5 shows the numerical conver- gence of the transmitted power for TMO surface-wave incidence, calculated by SPS and pulse Galerkin’s methods as the partition numbers N, and N2 of the discontinuity region are increased. It is observed that the SPS Galerkin’s method converges much faster than the pulse method. To achieve the same accuracy, N x and N z in the SPS Galerkin’s meth- od are only about one third of N, and N z in the pulse method for the parameters given in Figure 8.5. Thus, the number of linear equations implicated in the SPS Galerkin’s method is only one ninth of that in the corresponding pulse method. It is also found that the rela- tive difl'erence between the numerical total power and the incident power is about 0.001%. Therefore, SPS Galerkin’s method can save much memory space and computer time. 154 0.676 r r r r . SPS Galerkin's Method *— 0-574 F Pulse Galerkin's Method -+-- ‘ o 672 - NX=4s.‘ - \\ 0.67 - x.“ . g *\ \‘\ a. *~~. 1 ' o \\\ memes-.. . \a\ . 0 . 6 6 6 . \ Transmitted Power 0 m m (I) u—s Figure 8.5. Convergence check on the transmitted power computed with different N“‘ and N1 by sinusoidal and piecewise-sinusoidal (SPS) Galerkin’s method and pulse Galerkin’s method when a mo surface wave is incident upon an air gap in the grounded sheet. (81, = ed, = 1, £2, = 2.1316, 2a/7co = l, and k0: = 2.5) CHAPTER 9 NUMERICAL RESULTS 9.1. INTRODUCTION FORTRAN programs have been written to mplement the MoM solution for the inter- action of EM waves with a discontinuity in a grounded dielectric sheet for both pulse and SPS Galerkin’s methods. The width and thickness are normalized by the wavelength in free space. For the chosen parameters of the dielectric sheet the roots of the eigenvalue equation are found by the Newton-Raphson method [66]. Next, the moment matrix is filled by us- ing integration along the real axis or along the branch cuts, then the linear equations are solved for the total electric field in the discontinuity region by using a standard numerical technique. Finally, the excited surface wave and the far-zone field are calculated. Since the equivalent scattering width has the dimension of length, it is normalized by the wave- length in free space. The numerical results for both TE- and I'M-polarized surface-wave incidence will be presented and compared with those of existing studies. Then, the electric field distribu- tions, radiation patterns, and back scattering width for TB- and TM-polarized plane wave incidence will be given in Sections 9.3 and 9.4, respectively. Finally, the back scattering width for a lossy dielectric sheet will be calculated. 155 156 9.2. NUMERICAL RESULTS FOR SURFACE-WAVE INCIDENCE It is assumed that region 1 and the discontinuity region are air (£1, = ed, = 1) throughout in the following numerical calculations. There are some results for TB- and I'M-polarized surface-wave incidence to compare different integration methods and check the convergence. Now, more numerical results for surface-wave incidence will be given. Let’s consider a grounded dielectric sheet with a gap as shown Figure 6.5. If a TEI surface wave is incident upon the discontinuity from right to left. a transmitted TE] sur- face wave will propagate to the left, a reflected TEl surface wave will propagate to the right, and a radiated wave will be excited in space. Figure 9.1 shows the reflected power, indicated by the solid line, when 8 TE; surface wave is incident upon an air gap in the grounded sheet. It is observed that when the width of the air gap increases, the reflected power oscillates and converges to a constant. When the width is infinite, the problem be- comes that of power reflected by an abruptly ended grounded sheet. This case was studied by Shigesawa and Tsuji [19] by the mode-expansion method and their results are indicated by the dashed line. The asymptotic value of our results for an infinite width gap agrees very well with reference [19]. Figure 9.2 plots the asymptotic reflected power of the MoM solution of EFIE and the reflected power computed by Shigesawa and Tsuji [19] by the mode-expansion method when a T131 surface wave propagates through a step discontinui— ty in a grounded sheet as a function of the thickness ratio. Some deviations are observed only when the power reflection is very small. In Figure'9.3, the radiated power, transmitted power, and reflected power normalized by the incident power are plotted as functions of the width of the discontinuity for k0! = l and 82' = 5. It is found that the sum of the normalized transmitted power, re- flected power and radiated power is almost equal to 1. 157 ' b\\‘\ “SEQ xz-t .. O O (I) l Reflected Power 0 O 01 0.04 - 211% . C = co oi . . . . O 1 2 3 4 5 Figure 9.1. Comparison of the reflected power computed with MoM when a TE; surface wave is incident upon an air gap in the grounded sheet (solid line) and that computed by Shigesawa and Tsuji [19] by the mode-expansion method when a, TEI surface wave is incident upon an abruptly ended grounded sheet (dashed line). (c2, = 2.1316,]:01 = 2.5.111, = 10, and N2 = 2 to 100) 158 1:40.01 ': 5 i i g . a, . 0.001 r \4 . '- 0 0001 b ‘ ‘ ‘ ' ‘ 0 0.2 0.4 0.6 0.8 1 tllt Figure 9.2. The reflected power as a function of tllt when a TB; surface wave propagates through a step discontinuity in the grounded dielectric sheet computed by MoM (points) and by the mode-expansion method (line) [19]. (£2, = 2.1316) 159 1 '\ T I I I 4“ Radi at ed Power —— "1, Reflected Power ------- 0 . 8 - “x, ' Transmitted Power - ----- - H '\, o x, 3 °\.\ 0 x E O o 6 '- ‘\\ .1 % .\_\\‘ E o .4 - ' \ _, C K." 2 ......... o .2 - ............................................... . 0 l ' l L 0 0 . 2 0 . 4 0 . 6 0 . 8 1 2a/7cO Figure 9.3. The radiated power, transmitted power, and reflected power normalized by the incident power as functions of the width of the discontinuity when a TB] sur- face wave is incident upon an air gap in the grounded sheet. (82, = 5, hot =1,N, = 8,ansz = 2 to 50) 160 Since only TM-even surface-wave modes can be excited in a grounded sheet, a TM- even surface wave incidence will be considered. Figure 9.4 shows the reflected power, in- dicated by the solid line, when a TMO surface wave is incident upon an air gap in the grounded sheet. It is observed that when the width of the air gap increases, the reflected power oscillates and converges to a constant. When the width is infinite, the problem be- comes that of power reflected by an abruptly ended grounded sheet, which was studied by Shigesawa and Tsuji [19] by the mode-expansion method and their results are indicated by the dashed line. The asymptotic value of our results for an infinite width gap agrees very well with reference [19]. Thus, to study the power reflected by an abruptly ended ground- ed sheet for surface-wave incidence, we can first calculate the power reflected by a finite width gap with MoM is calculated, then, find the asymptotic value for an infinite width 83P- When the thickness of the dielectric sheet increases or the permittivity increases, more TM-even surface-wave modes can be excited. When TMO surface wave is incident from right to left in an abruptly ended grounded sheet (see the geometry in Figure 9.5), a reflect- ed TMO surface wave will propagate to the right, a radiated wave will be excited in space, and TM-even higher order surface-wave modes will be excited and propagate to the right if the higher order modes can be supported by that sheet. Figure 9.5 plots the amplitudes of the reflection coefficient a0 and the coupling coemcient (12 to 1M2 mode computed by SPS Galerkin’s method (points) and by Neuman series [14] (lines) as functions of 1101 when a mo surface wave is incident in an abruptly ended grounded sheet for £2, = 20. It is found that our results agrees very well with reference [14]. Gelin et a1. [14] derived coupled integral equations on discrete and continuous wave amplitudes of the modal fields at the discontinuity. These coupled integral equations are solved by an iterative proce- dure, namely the Neuman series. Figure 9.6 shows the amplitudes of the reflection coeffi- cient a2 and the coupling coefficient a0 to TMO mode computed by SPS Galerkin’s method (points) and by Neuman series [ 14] (lines) as functions of 1101 when a TM; surface I61 0.1 I 4 . . . . . 0.09 "n .4 0.08 - ‘— . 433:: 1 ~ ,...: °" ° ““\\\‘2 ' '8 0.05 - 8,11 W . g 0.04 - n n /\ l 0 ‘\\\ x=-t . “0'03“ AAA/\NAAAAA‘ 2”“ UVVVVVVVVVVV- 0.01 i U U U . 0 1 1 . n L . . 0 1 3 4 5 7 8 Figure 9.4. Comparison of the reflected power computed with MoM when a TMO surface wave is incident upon an air gap in the grounded sheet (solid line) and that computed by Shigesawa and Tsuji [19] by the mode-expansion method when a TM surface wave is incident in an abruptly ended grounded sheet (dashed line). (32, = 2.1316, k0: = 2.5, SP8 Galerkin’s method with N, = 4 and N2 = 2 to 62) l . r 1 r r 1 r 0.9 - ' 0.8 - . 0.7 - - 430-5 _ i laol €05 - _‘NN TMO .. ‘ \\ TM - <3: ‘ ° 0.2 - 0.1 - 0 . . Figure 9.5. Amplitudes of the reflection coefficient a0 and the coupling coefficient a2 to TMZ mode computed by SPS Galerkin’s method (points) and by Neuman se- ries [14] (lines) as functions of kot when a mo surface wave is incident in an abruptly ended grounded sheet for £2, = 20. 163 0.9 kw ’ / \ ‘{\\\§{‘ /// ////// . (S = 00 Figure 9.6. Amplitudes of the reflection coefficient a2 and the coupling coefficient an to 'I'Mo mode computed by SPS Galerkin’s method (points) and by Neuman se— ries [14] (lines) as functions of [(01 when a TM2 surface wave is incident in an abruptly ended grounded sheet for 82’ = 20. 164 wave is incident in an abruptly ended grounded sheet. It is also found that our results agree very well with the reference [14]. It is noted that power conservation in [14] is veri- fied to an accuracy of better than 1%, while our results on power conservation has an accu- racy of better than 0.1% for the parameters given in Figures 9.5 and 9.6. In Figure 9.7, the radiated power, transmitted power, and reflected power normalized by the incident power are plotted as functions of the width of the discontinuity for [tot = 1 and £2, = 5. It is also found that the sum of the normalized transmitted power, reflected power and radiated power is almost equal to 1. In this section, the computer programs are tested by using surface wave incidence, and find that our results agree very well with those of existing studies. The problem of surface wave scattering by a discontinuity along a dielectric waveguide is very important, because a good understanding of this problem is essential in the design and development of optical and millimeter-wave components. A discontinuity problem arises in many forms such as in the splicing of two dielectric waveguides or in inter-device coupling of millimeter and optical integrated circuits. In the next two sections, numerical results for TE- and TM-polarized plane wave inci- dences will be given. 9.3. NUMERICAL RESULTS FOR TE-POLARIZED PLANE WAVE INCIDENCE It is clear that when the discontinuity is narrow (Zn/2.0 <1 1), the total electric field in the discontinuity should converge to the incident field for TIE-polarized plane wave inci- dence. This phenomenon is shown in Figures 9.8 and 9.9. Figures 9.10 and 9.11 show the spatial intensity distribution of the normalized total electric field in the discontinuity when a TE-polarized plane wave is incident upon the 165 1 '\ 1 F l y "‘1. Radiated Power — \ Reflected Power ------- 0.8 - "~. Transmitted Power ------- - 0 ON Normalized Power 0 Ira 0.2 Figure 9.7. The radiated power, transmitted power, and reflected power normalized by the incident power as functions of the width of the discontinuity when a TM, sur- face wave is incident upon an air gap in the grounded sheet. (82, = 5, kbt = l, SPS Galerkin’s method with N, = 3 and N2 = 3 to 20) 166 Normalized Electric Field 0.8 - 4 0.6 - .. 0.4 - 1 0.2 1- .4 0 r n r 1 1 -0.12 -o.1 -0.08 -0.06 -o.04 -0.02 0 Figure 9.8. Comparison of the normalized total electric field amplitude IEzy/Eol (dashed line) and the incident field lag/EDI (solid line) when a TE-polarized plane wave is incident upon a narrow gap in a grounded sheet at various incidence angles 0,. (81, = 8.1, = 1, c2, = 4, zit/1.0 = 0.01, 111z = l, 1/1.0 = 0.125 and N, = 12) 167 Normalized Electric Field Figure 9.9. The comparison of the normalized total electric field amplitude IEZy/EOI (dashed line) and the incident field Iggy/EDI (solid line) when a TE-polarized plane wave is incident upon a narrow gap in a grounded sheet at various inci- dence angles 91- (51’ = 5‘” = 1, £2, = 4, 2a/}co = 0.01, N, = l, 1/1.o = 0.25 and N, = 24) I68 Figure 9.10. Spatial intensity distribution of the normalized total electric field in the dis- continuity when a TE-polarized plane wave is incident upon the grounded sheet normally (0, = 0°). (811' = ed, = 1. 82, = 4. 20/3.o = 0.25. 1v, = 13,1/1to = 0.125 and N, = 12) 169 Figure 9.11. Spatial intensity distribution of the normalized total electric field in the dis- continuity when a TE-polarized plane wave is incident upon the grounded sheet normally (0, = 0°). (31' = gdr = 1, 82' = 4, 2a/l.0 = 0.7, N, = 17.1/1.o = 0.25 and N, = 12) 170 sheet normally (0, = 0°). It is found that the field distribution is symmetric with respect to the 2 = 0 plane. Figures 9.12 and 9.13 show the normalized radiation patterns of the radiated far field when a plane wave is incident upon the discontinuity at various incidence angles 0 1' The normalized back scattering width and the equivalent scattering width of the excit- ed surface-wave mode as functions of the thickness 1/10 of the discontinuity are shown in Figures 9.14 and 9.15, respectively. Comparing these two figures, it is observed that if 1 is less than a critical value 1‘, no surface-wave mode is excited and the back scattering width increases as the thickness increases. On the other hand, if t is greater than the critical val- ue 1c, the surface-wave-mode is excited and the back scattering width decreases slowly as the thickness increases. This is expected since only TE-odd modes can be excited in a grounded sheet. Since the cut-ofl' thickness of the 'I'Ell mode is [1] ten 11 = —— n = l, 3,5, (9.3.1) To 4Je2ru2r-elru’lr it is observed that the lowest TE-odd mode is the TB] and the cut-ofl' thickness is 1,1/3.o = 0.1443375 for the parameters given in Figures 9.14 and 9.15. Figures 9.16 and 9.17 show the back scattering width as functions of the width of the discontinuity for t/l.o = 0.125 and t/A.0 = 0.25, respectively. When t/lo = 0.25, the TE; surface-wave mode can be excited. The normalized equivalent scattering width of the excited TE! mode is shown in Figure 9.18. 9.4. NUMERICAL RESULTS FOR I'M-POLARIZED PLANE WAVE INCIDENCE It was found that when the discontinuity of the dielectric sheet is narrow (Zn/71.0 41 1), the total electric field in the discontinuity converges to the electric field in a zero-width 171 00 0. 30° 4 0. 4 0. 60° 4 .8 o. 4 5 1‘: 0. 4 I-fl On 5 0. . o. 4 . = 85° 0. 4 0. 20 40 60 80 0(degrees) Figure 9.12. Radiation patterns of the far field when a TE-polarized plane wave is inci- dent upon the grounded sheet at various incidence angles 91- (glr = 8.1, = 1, c2, = 4, 2a/1.0 = 0.25, N, = 13, 1/110 = 0.125 and N, = 12) 172 85° -80 -60 -40 -20 0 20 40 60 80 0 (degrees) Figure 9.13. Radiation patterns of the far field when a TE-polarized plane wave is inci- dent upon the grounded sheet at various incidence angles 01. (31' = ed, = 1, 82, = 4, 221/10 = 0.7, N, = 17, t/l.0 = 0.25 and N, = 12) 173 0.0001 le-OS 13-06 1/240 Figure 9.14. Normalized back scattering width as a function of the thickness t/A.0 of the discontinuity when a TE-polarized plane wave is incident upon the grounded sheet at various incidence angles 0,. (g = 1, 82' = 4, 2a/3. = 0.25,N = 13,andN = 4 to 14) 0 z 1 11‘ = edr c>1 49 174 3S 1 I I I I 9,. = 0° .3 - 4 .25 ~ 4 30° .2 '- - 15 - 4 60° .1 - - .05 - - o. 1 l 1 O 0 05 0 1 0 15 0.2 0.25 0 3 1/140 Figure 9.15. Normalized equivalent scattering width of the surface-wave mode excited as a function of the thickness t/A.0 of the discontinuity when a TE-polarized plane wave is incident upon the grounded sheet at various incidence angles 01. (3" = 3,, = 1, £2, = 4, 2a/A.o = 0.25, N, = 13, and N, = 41014) 175 1'0 V U f r f r :1 4° Figure 9.16. Normalized back scattering width as a function of the width 2a/ 1.0 of the discontinuity when a TE-polarized plane wave is incident upon the grounded sheet at various incidence angles 91- (51’ = 3,, = 1, 82' = 4, (”11) = 0.125,N, = 13,and N, = 4 to 17) 176 10 I I j I i I I b F 1 r .. o r . 1 r i s 300 GB 0.11- - X; E P 0.01? i p 0.001 I l l l 1 j 0 0 1 0 2 0 3 0.4 0 5 0 6 0 7 20 To Figure 9.17. Normalized back scattering width as a function of the width 2a/24o of the discontinuity when a TE-polarized plane wave is incident upon the grounded sheet at various incidence angles 91- (g 1’ = 3,, = 1, 8,, = 4, 1/10 = 0.25,N, = 13,and N, = 4 to 17) 177 {a Figure 9.18. Normalized equivalent scattering width of the surface-wave mode excited as a function of the width 2a/3.0 of the discontinuity when a TE-polarized plane wave is incident upon the grounded sheet at various incidence angles 91- (air = ed, = 1, 8,, = 4, t/lt0 = 0.25,N, = 13,and N, = 4 to 17) 178 gap which was defined in Chapter 5. This phenomenon is shown in Figures 9.19 and 9.20. These two figures plot the normalized amplitudes of the x- and z-components of the total electric field (points) at the center of a narrow gap (2 = 0) and the electric field in a zero- width gap (solid line) when a TM-polarized plane wave is incident upon the gap in a grounded sheet at various incidence angles 0,, respectively. It is noted that the x-compo- nent of the incident electric field becomes zero when a TM-polarized plane wave is inci- dent upon the sheet normally (0,. = 0°). Figures 9.21 and 9.22 show the spatial intensity distributions of the x- and z—compo- nents, respectively, of the total electric field in the discontinuity normalized by the ampli- tude of the incident electric field when a TM-polarized plane wave is incident upon the sheet normally (0 1' = 0°). It is found that the field distribution is symmetric with respect to the 2 = 0 plane, and the x-component of the total electric field is not zero. Figure 9.23 plots the normalized radiation patterns of the radiated far-zone field when a plane wave is incident upon the discontinuity at various incidence angles 0,.. The normalized back scattering width and the equivalent scattering width of the excit- ed surface—wave mode as functions of the thickness 1/240 of the discontinuity when a TM- polarized plane wave is incident upon the grounded sheet at various incidence angles Oi are shown in Figures 9.24 and 9.25, respectively. Comparing these two figures, it is ob- served that as the sheet thickness increases the equivalent scattering width of the excited surface-wave mode increases while the normalized back scattering width does not de- crease. In the last section, it is found that the excited surface wave will significantly alter the magnitude and pattern of the scattered field for a TE-polarized plane wave incidence. Figure 9.26 shows the back scattering width as a function of the width of the disconti- nuity for t/l.0 = 0.125 when a TM-polarized plane wave is incident upon the grounded sheet at various incidence angles 91- 179 H 0‘ .1 I r I I 4 )- 2 r 1 ' e. = 85° I ‘ ‘ . l . ) 60° ' . 30° - A 1» -1.4 -1.2 -l -0.8 -0.6 -0.4 -0.2 0 x/l. o H 1-’ ox I N ormah'zed Electric Field 0 (I) 0 1b l O [\J T 0 Figure 9.19. The comparison of the normalized amplitude of x-component of the total electric field Isz/Eol (points) at the center of a narrow gap (2 = 0) and the electric field in a zero-width gap lEih/Eol (solid line) when a TM-polarized plane wave is incident upon the gap in a grounded sheet at various incidence angles 9i. (elf = 841’ = l, 62’ = 1.1, 20 = 0.1", I = 1”, f = IBGHZ, Pulse Galerkin’s method with N, = 3 and N, = 30) 180 2.5 T I 1 1 l I 1 Normalized Electric Field -0.8 -0.6 -0.4 -0.2 0 x/7tO Figure 9.20. The comparison of the normalized amplitude of z-component of the total electric field IEzz/Eol (points) at the center of a narrow gap (2 = 0) and the electric field in a zero-width gap I (82/6,) Biz/EDI (solid line) when a TM- polarized plane wave is incident upon the gap in a grounded sheet at various incidence angles 91- (glr = gdr = 1, 8,, = 1.1, Zn = 0.1", 1 = 1", f = lSGI—Iz, Pulse Galerkin’s method with N, = 3 and N, = 30) 181 Figure 9.21. Spatial intensity distribution of the x-component of the normalized total elec- tric field in the discontinuity when a TM-polarized plane wave is incident upon the grounded sheet normally (0,. = 0°). (5" = 8dr = 1, 8,, = 4, 211/10 = 0.7, 1/1.0 = 0.25, pulse Galerkin’s method with N, = 10 and N, = 25) 182 Figure 9.22. Spatial intensity distribution of the z—component of the normalized total elec- uic field in the discontinuity when a TM-polarized plane wave is incident upon the grounded sheet normally (0,. = 0°). (5" = ed, = 1, 8,, = 4, 2a/24.0 = 0.7, t/ic0 = 0.25, Pulse Galerkin’s method with N, = 10 and N, = 25) Amplitude O 0301-2 00000000 Ol—‘Nbdth'ImQ 1 L 183 30° 60° ' 9.=0° -20 0 20 40 60 80 -80 ~60 -40 0(degrees) Figure 9.23. Radiation patterns of the far-zone field when a TM-polarized plane wave is incident upon the grounded sheet at various incidence angles 0, (51' = 5", = 1, £2, = 4, 2a/A0 = 0.7, t/l.0 = 0.25, Pulse Galerkin’s 'method with N, = 10 and N, = 25) 184 l r ............ ”I”. ..... .+ 5 /’l //”'°"“W:~\ ,,,,,,,, x ’ / GB 0.1 g ’1’, _z’. "" ITO E f, .1"! . I" .4". O 01 E. {I {I ___0 =00 3 . ’ 1’ -----e,. = 30° 0.001 p1,!” 1’ / ----- e,=60° ”1"), 1 0.0001 1.1" 1‘11“ ‘1 1.1 15 .’ le-OS r u '1' le_06 l l 4 l l l 0 0.1 0.2 0.3 0.4 0.5 1/71.o Figure 9.24. Normalized back scattering width as functions of the thickness 1/ 10 of the discontinuity when a TM-polarized plane wave is incident upon the ground- ed sheet at various incidence angles Bi. (3" = 3,, = 1, 8,, = 4, 2a/1.o = 0.125, Pulse Galerkin’s method with N, = 8 and N, = 4 to 25) 18$ 0.35 . . . . . TMo _ 9,. = 0° 0 .3 - TMo ----- 9,. = 30° 0 . 25 - 2 """" 9.- = 0° TM2 ___... 9.- : 30° 0’s .. 2 """"" e = 60° 1; 0 2 0.15 - ,1” , .. ’1’ i, \ ," 'l O . 1 ,. /0 III." \:-\::‘.\ f. {x’. .1 l; .,,/o \\-:.\ 0”,. O . 05 1' " I!" ' ‘ O 1 1 1 ‘51:. -------- r o o .1 o 2 o .3 o 4 o 5 t/lo Figure 9.25. Normalized equivalent scattering width of the surface-wave modes excited as functions of the thickness t/Ao of the discontinuity when a I'M-polarized plane wave is incident upon the grounded sheet at various incidence angles 91. (cl, = ed, = 1, £2, = 4, 2a/71.o = 0.125, Pulse Galerkin’s method witth = 8ande = 4 to 25). 186 100 : I T I _l 10; 0.01 0.001 { 0.0001 ‘ ‘ ' ‘ 2a/k Figure 9.26. Normalized back scattering width as a function of the width 211/10 of the discontinuity when a 'I'M-polarized plane wave is incident upon the ground- ed sheet at various incidence angles Si. (81, = ed, = 1, £2, = 4, 't/lo = 0.125, SPS Galerkin’s method with NJlr = 3 and Nz = 3 to 40) 187 9.5. NUMERICAL RESULTS FOR THE LOSSY DIELECTRIC SHEET The various surface-wave mode fields are orthogonal to each other, as well as orthogo- nal to the continuous radiation modes. The different spectral components of continuous modes are orthogonal too. The orthogonality properties may be established using the rec- iprocity theorem with the hypothesis that the total radiation field satisfies the radiation condition [1] [69] [70]. Now, let’s discuss the orthogonality properties for power in the lossy dielectric slab. The total radiation electric field (only TE modes will be discussed) can be written as E” (x, z) = IA (p) e, (x, p) a.” (p) zdp (9.5.1) 0 Hence e, (x, p) satisfies 8’ 2 2 _ {a—z+ [k (x) -B (p)]}e,(x.p) - 0 (952) x where {.ka—p’ 0 (95.14) z 2 (”"0 £ 132(9) - B,” 1 It is observed that if both regions are lossless [8" (x) = 0], then P; = 0 because [3 (p) at B: . The surface-wave modes and radiation modes are then orthogonal in the power relation. Thus the total scattered power is the sum of the radiated power and the surface-wave power. On other hand, if e" (x) at 0, i.e. one or both regions are lossy, then P: at 0. If e" (x) is small and the surface-wave mode is not near the branch cuts, P, ~ 0, and they are almost orthogonal in power. But, since the surface-wave mode may be near the branch cuts, i.e. 52 (p) - 3:2 is small for some p’s, then P2 :6 0. Therefore, if the dielectric slab is lossy, although the surface-wave modes and radia- tion mode fields are orthogonal, the modes are not orthogonal in power. The total power cannot be calculated as the sum of the radiated power and the surface-wave power. Figure 9.27 plots the loci of the z-axis propagation constant of 51 for the T51 surface- wave mode along a lossy sheet. It is observed that the T51 mode becomes a fast surface wave (Re {51} < k1 = 1:0) when the thickness decreases. Shevchenko and Richmond et. al. discuss this in [59] [71]. 190 Imiflrl/ko. 0.144 ‘ ' ' -o.02 1- 0.15 . -0.04 I -0.06 -0.08 fast -0-1 ' surface wave -0.12 Figure 9.27. The loci of the z-axis propagation constant of Bl for the TB] surface- wave mode along a lossy sheet (151 , = 1, £2" = 4 -j0.4). 191 The normalized back scattering width as functions of the width 2a/B\.0 of the disconti- nuity for 82, = 4 and £2, = 4 -j0.4 are shown in Figures 9.28 and 9.29 when a TE-po- larized plane wave is incident upon the discontinuity in a grounded dielectric slab with t/l.0 = 0.125 and t/A0 = 0.25, respectively. The back scattering width of the lossy di- electric slab is less than that of the perfect dielectric slab. Figures 9.30 and 9.31 show the normalized back scattering width as functions of the thickness 1/71.0 of the discontinuity for 82, = 4 and £2, = 4 —j0.4 when Za/A.0 = 0.25 and 2a/A.0 = 0.5 , respectively. It is observed that the back scattering width of the lossy dielectric slab is less than that of the perfect dielectric slab. Figure 9.32 shows the normalized back scattering width as a function of the thickness t/A.o of the discontinuity for £2, = 4 and £2, = 4-j0.4 when a I'M-polarized plane wave is incident upon the discontinuity in a grounded dielectric sheet with 2a/2e0 = 0.25 . It is observed that the back scattering width of the lossy dielectric sheet is less than that of the perfect dielectric sheet. Finally, the normalized back scattering width as a function of the width 2a/2.o of the discontinuity for £2, = 4 and £2, = 4 -j0.4 is shown in Figure 9.33 when t/l.o = 0.25. The back scattering width of the lossy dielectric sheet is found to be less than that of the perfect dielectric sheet. 192 10, 1 I I I l I :1 e? 0.1 Vifi v vvvvv' 0.7 Figure 9.28. Normalized back scattering width as a function of the width 2a/A.0 of the discontinuity when a TEL-polarized plane wave is incident upon the grounded sheet at various incidence angles 91- Solid line: 82, = 4, Dashed line: 22, = 4-j0.4. (err = edr = 1, t/Ao = 0.125, N, = 13, and N2 = 4 to 17) 193 10: I I I U I V / Iv.” 1 ' e = 0° / / 1 I ‘ /’ I. 3 b o > / 1 GB 0.1:- x; 0.01;- F b 0.001 1 l l I 4 l 0 O 1 O 2 0 3 0.4 0.5 O 6 0 7 20 To Figure 9.29. Normalized back scattering width as a function of the width 2a/2eo of the discontinuity when a TE-polarized plane wave is incident upon the grounded sheet at various incidence angles 6,. Solid line: 82, = 4, Dashed line: a2, = 4—j0.4. (5" = ed, = 1, t/3\.0 = 0.25, I)!Jr = 13, and N2 = 4 to 17) 194 1 1 1 I I N - or “ax-33‘»-.. 6‘ O / A\ “ \ M :“~. -- O ‘ 0.1 p/ a '- 30 1 .r 1 “260° 1 o 01 E- 1 0.001 " “a i- 1: X; 0.0001 { 1 1 le-OS i 1 ’ 1 18-06 1 1 1 L 1 o “ 0.05 0.1 0.15 0.2 0.25 0.3 t/k0 Figure 9.30. Normalized back scattering width as a function of the thickness t/l.o of the discontinuity when a TB plane wave is incident upon the grounded sheet at various incidence angles Oi. Solid line: 62' = 4, Dashed line: a2, = 4-j0.4. (51' = ed, = 1, 2a/2eo = 0.25, N2 = 13, and N, = 4 to 14) 195 10 "'q '1"" U I 1e-O6 V V'V'V‘ jjvt“ V V 1e-07 0 0.05 0.1 0.15 0.2 0.25 0.3 t/ 1.0 Figure 9.31. Normalized back scattering width as a function of the thickness t/l.0 of the discontinuity when a TE-polarized plane wave is incident upon the ' grounded sheet at various incidence angles 61. Solid line: 62, = 4, Dashed line: 82, = 4-j0.4. (err = 3‘" = 1, 2a/3.0 = 0.5, Nz = 13, and N, = 4 to 14) 196 1 ,...... e = 00 I” 'M"-..-."“~. - —. l 1' I , {ll/Oi = 30° --- --~..-,...:““‘-----._ ..... fi » (,..---. . -...--__ “‘32:: : /I/ “~ ~~ o'aov‘” “a o , 1 _r w" I _. 0 To . 9,. _ 60 0.01 r 0.001 g \’ ; 0.0001 g 0 O . l O . 2 O . 3 0 . 4 O . S r/A.o Figure 9.32. Normalized back scattering width as functions of the thickness t/Ao of the discontinuity when a I'M-polarized plane wave is incident upon the ground- ed sheet at various incidence angles Oi. Solid line: 82, = 4, Dashed line: '82, = 4-j0.4. (3" = ed, = 1, 2a/A.0 = 0.25, Pulse Galerkin’s method 1111111111z =10andN,= 4 to 25) 197 100 = 10 0.01 Figure 9.33. Normalized back scattering width as a function of the width 2a/2to of the discontinuity when a I'M-polarized plane wave is incident upon the ground- ed sheet at various incidence angles 9i. Solid line: 82, = 4, Dashed line: '62, = 4"j0-4'(31r = ed, = l, t/JL0 = 0.25, SPS Galerkin’s method with Nx=5ansz= 3 to 40) CHAPTER 10 CONCLUSIONS The scattering of I'M-polarized EM waves by a dielectric discontinuity in an infinite grounded dielectric sheet has been analyzed. An electric field integral equation (EFIE) has been derived based on the Green’s function for the electric field produced by induced polarization currents in the discontinuity region. The EFIE is solved numerically using Galerkin’s method for TE-polarized wave inci- dence. Computation of the inverse Fourier transform representation of the electric Green’s function is performed by integration along real axis, along the branch cuts direct- ly, and along the branch cuts with a variable change. It is found that the integration along the branch cuts directly requires the least computation time, and allows a separation of the field into bound surface-wave modes and radiation modes. Numerical calculations of the scattering pattern and the back scattering width are performed when a TEE-polarized plane wave is incident upon the discontinuity in the grounded sheet. It is found that the discon- tinuity causes the excitation of surface-wave modes in the sheet and the excitation of the surface-wave mode reduces the back scattered radiation field. The EFIE is solved using Galerkin’s method with pulse and sinusoidal/piecewise-sinu- soidal (SPS) basis functions for TM-polarized wave incidence. The latter afi'ord a great savings in memory requirements and computation time. It is found that the excitation of the surface-wave modes do not reduce the back scattered radiation field for I'M-polarized plane wave incidence. On the other hand, the EFIE is also solved iteratively by using suc- 198 199 cessive approximations (the Neumann series). Both the zeroth- and the first-order approx- imations for the total electric field in the discontinuity region are found when a TE- or TM-polarized plane wave is incident upon a grounded sheet with a narrow gap. Then, the first- and second-order approximations to the scattered field are calculated by applying the steepest-descents method. _These approximate results agree very well with the direct MoM solutions to the EFIE. ‘ It is found that the back scattering width is a minimum when (Zn/ho) sine: = n/ 2 for I'M-polarized plane wave incidence, and for 'I'E-polarized plane wave incidence for the thickness t < ‘cr , where ‘cr is the cut-off thickness of the TB] mode. The reflected power for both TE- and I'M-polarized surface-wave incidence is calcu- lated too, and it is found that the numerical results agree very well with those of existing studies. APPENDICES APPENDIX A ELECTRIC FIELD WITHOUT THE DISCONTINUITY FOR TE- POLARIZED PLANE WAVE INCIDENCE From Figure A.l, it is found that the EM field in the absence of the discontinuity con- sists of two plane waves. Then, the electric field in the two regions can be written as: Region 1: 0R2 - e The conditions that E; and H ; are continuous at x=0 give 1' _ i Elz|x=o - Eux=o = 'k ulna -'k no.0 jk line ‘11 l = e 1 2 e 2" r Eocoseill —Rl] '2jBIEo cosersin [kztcoser] Hiy|x=o = ”‘2y|x=o = (13.9) (3.10) 205 E _ E ' 71—0 [l +R1] ejk‘zm‘ = e jk’wo'O'ZBT—‘(3 cos [kztcoser] (#1281110, (13.11) 1 2 The two equations (BJO) and (B.ll) have the solutions: klsinO‘ = kzsine, (13.12) “26°39‘19”“, _ 111cosG‘cos [kztcosfir] +j‘n2cos0rsin [kztcoser] (3.13) nlcoseicos [kztcoser] -j'n2cosfirsin [kztcoser] -j2¢. 1 ’ nlcoseicosfiztcoser] +jfl20089,sin[k2tcos9r] " (3.14) where nzcoser 6 tan¢¢ - Wmlkztcos r] (3.15) SUMMARY: Region 1: O x=0 82' H2 J J x=-t ‘\\ a‘=oo X YL—QZ Figure C.l. Geometrical configuration. .7 (f) = 91’ (x, z) for 2-D TE excitation. Since the current .7 has only a y-component, the Hertzian potential H has only a y-compo- nent. Therefore fi = 9Hy(x,z) (02.5) is = 91¢an (x, z) (c245) - _ . _ a .a H - Jwe[ 23711, (x. z) +233“y (x. 2)] (C21) 3 2 _ _J, (x. z) [5;+3_:+ +k ]r1y (x, z) — 1.008 (02.8) C.3. SPECTRAL-DOMAIN POTENTIAL REPRESENTION Equation (C.2.8) can be solved for the potential by Fourier transforming on spatial variables tangential to the layer interfaces. Axial uniformity along the z-axis [i.e.. e at s (z) and u a: u (z) ] prompts Fourier transformation on that variable. Define the axi- al transform pair F(x, 2) Hf(x, C) as: F(x, 2) = -2-11-‘If(x.C)/€zdc (C.3.l) _- f (x, C) = I F (x, z) e-szz (C.3.2) Taking Fourier transforms of equations (C.2.6) through (C.2.8) yields 3(x, c) = 9k21ty(x, g) (C33) $(x, c) = j08[-2jC1ty (x, ;) +28%):y (x, Q] (C.3.4) 32- 2 _ ..iy (x, C) [3; p ]ny (x, c) _. jam (C.3.5) where p2 = g2 _. 13 (C35) is wavenumber parameter. C.4. DECOMPOSITION INTO PRIMARY AND REFLECTED WAVES Since region 1 is source-free, the homogeneous wave equation 493- 2 n: (x c) - 0 (C41) 31:2 p1 1’ ’ - H hasasolution 1:1, (x, g) = A(§)e“"" (C.4.2) Note that the solution satisfy the radiation condition, i.e. it is an outward travelling, decay- ing potential wave with Re {p1} > 0 and Im {p1} > 0, where P1 = JCZ - ki (C.4.3) 209 and Re {...} designates the real part of the quantity within the braces, Im { ...} desig- nates the imaginary part. The total potential in region 2 is the sum of a primary wave #2” bounded layer region of the layer and the source free reflected wave fig), in the layer. excited by j), in an un- These potentials must satisfy piz— 2‘ _ _jy(x’ C) 53x2 sz 1:5, (x, C) - jmez (C.4.4) r' 82 2" r _..p 3 (Ag) = () (C.4.5) L812 24 2’ where P2 = C2 - "i (C.4.6) ngy is found directly from the homogeneous equation as n;, (x. C) = B (C) e"="+ cm 2"” (C41) Before solving eq. (C.4.4), the general inhomogeneous second order equation a’ 2 _ 7 - v (x. C) — -s (x. C) (C43) fix is considered. Fourier transforming on the x variable gives W (x. C) H )1! (5. C) s (x. C) H 5 (5. C) (C49) 1 ' - W1“) = 7,; IWE. C) e’gxdé (C.4.10) _ 1 ” ~ 1:: s (x. C) - 3;: j s (fi. C) e’ d§ (0.4.11) Substituting (C.4.10) and (C.4.ll) into (C.4.8) yields 210 5(§ C)_ 3(§.§) g2 + p2 =(E -J'p) (é +jp) (C.4.12) \tl.(§ C)= Note that this spectral solution has simple pole singularities at g = ijp. Taking the in- verse Fourier transform yields 3 (§. C) '6: (E -J'p) (5+1?) ‘1: V(xvC) = 51;"! 1.. ' . eltb" 3) fildxs(x’0iT-1p)(§+1p)d§ I s (x', C) gig (xi x') dx' (C.4.13) —oo where g’t (x1 x') is the Green’s function for the forced principal wave Jflx- x') (E-J'p) (5+1?) 82(1)! 1') = 2L I“ d5 (C414) To perform the inverse transform, a deformation of the real line integration is used and Cauchy’s theorem is applied to an upper half plane closure (UHP) for x > x' and to a lower half plane closure (Ll-1P) for x < x'. This yields e-plx-x'l s‘g’ (II x ) = 2p (C415) and with Re {p} >0 and Im {p} > O. The primary wave solution is obtained by comparing eqs. (C.4.4) and (C.4.8) 0]. 193,0. C) = IJ’ 03$ (xl x ")dx (c.4.16) 4 [($82 211 where ‘P:"""l 8‘2 (x' x ) = ___—2p: (C417) and with Re {p2} > 0 and Im {p2} > 0. SUMMARY OF SPECTRAL SOLUTION: any (x, g) = A (C) i” (C418) °i (x'. C) - 1:2). (x, g) = j rim g€(x| x')dx'+3(§)e ”flame” (C419) 2 -! Here A (C) , B (C) and C (C) are unknown coefficients which will be determined by application of the boundary conditions. C.5. IMPLEMENTATION OF BOUNDARY CONTIONS TO DETERMINE UNKNOWN SPECTRAL AMPLITUDES The boundary conditions that [5‘y = 0 at the surface of conductor and Ey, 1‘1z are con- tinuous at the interface between two regions give Ey|x=_, = 0—)ey|x=_‘ = 0-91r2y|x=_‘ = 0 (C11) E" g = 12),] _ —-)ey| + = eyl _ akin” _ = kinz | (C32) x=0 x=o x=0 x=0 x-O ’1 81:” 31:2) Hzlflo 0' = H‘lx=o' --)hz|x=0 = hz|x=o’ #815; P0 = 823— 1:0“:53) Substituting (C.4.18) and (C.4.l9) allows us to solve for the spectral amplitudes ,r A (1;) = Ftp (C.5.4) A k 2 e p C(C) = -_L—. —‘ -‘—l (C55) 2e PI‘A k2 Czpz 212 0 - l —- ’g' B(§) = -e’2""[C(t;) + I 1’ (LC) ‘ p dx'] (C56) 4 1(082 2p2 where 01)“ “DP e p" . . F= J; jmez sinh [p2 (x + t) ] dx ‘ (C.S.7) A- k‘ 2 h ”+611": inh t) C53 -— k2 COS (p2 'e—z-pzs (p2 ( . ) C.6. POTENTIAL SOLUTION IN EITHER REGION Substituting the spectral amplitudes (C.5.4)-(C.S.6) into the transform domain repre- sentation of the potentials (C.4.l8) and (C.4.l9) gives 0 . J (133;) , , 1:1,(x, g) = LI) yjmez 312(11):”): (C.6.l) °)', (x'. C) (:2, (x. C) = _j‘ jmez 322(11 x')dx' (C.6.2) where e-p‘xsinh ( '+r)] 812(xl x') = [p3 x (C63) P2A 82204 x') = 352(41):) +g§2(61x') (C.6.4) e-pglx-x'l 35'; (II I') = T (C.6.5) r , -p2(x+x'+2t) [(181171 e mum =-e +[(;-)- J] (x+r)1sinh[p2(x+:)1 2 ezpz pzA (C.6.6) 213 C.7. SPACE-DOMAIN FIELDS AND GREEN ’8 FUNCTIONS Applying the relation between the electric field and Hertzian potentials, the electric field in the spectral domain is obtained . 2 -jk1nl€1 o . e1, (1. C) = k1n1y(x.§) = Ti1y("'§)312(’1")d" (C.7.l) .3 0 an (x, g) = 1:;ny (x, g) = -jk2n2 j j, (x', g) 3,, (1| x')dx' (C72) .3 where ll- n, = 35 i = 1.2 (0.73) i Taking the inverse Fourier transform of e, (x, C) and noting that jy is the Fourier trans- form of Jy yields -jk n8 C(z- z') sum) =—‘——‘—ljdx' 14221 (x 2)].(:u(x1x'2)‘i 4C -r = I J). (x': Z.) 612 (11 I" Z - Z.) dx'dz' (CJA) L03 0 C(z-z) E2,(X.z) = -1k2nzjdx'jdz',1(x z) I 822041) dC .3 _u = I Jy (x', z') 022 (11 x'. z - z') dx'dz' (C.7.5) LCS where LCS designates the longitudinal cross section of the discontinuity region, and -jk1nle 1 eJC(2('.- z') Tjg 812011) G12 (xl x'. z - z')= dC (C.7.6) C (z- z ') d; (0.7.7) 022(xlx'. z- z') = -jk2fl2I822(XIx'2) 214 C.8. SUMMARY OF THE GREEN ’8 FUNCTIONS jklnlel' ep'xsinh [p2 (x' + 012“?!) 612(XIX, Z- Z) = "' 8 Ia 2 - dC 21tp2A Cumin-z') = G’n(x1x'.z-z') +G§2(XIx'.z-z') "-p,lx- x’lJCu-z’) 47tp2 dC 0:; (XIX, z- Z) =‘Jk2n2-. " -p.(x+x'+2r) C(z-z') e e’ (K 032 (XI 1', z - z') = -jk2'r|2 {- —oo 47th 2P2 ,3 C( ') +I[(:_ :2)_ :1P1]eP J 2:2 sinh[p2(x+t)]sinh[pz(x'+t)]dfi} 2np2A p, = ,Icz-k,’ i=1,2 A - (k1)2cosh(p t) + 81p] sinh (p t) k2 2 82122 2 When Ill = “2 = no, the above Green’s functions can be rewritten as e-xp'sinh[ 2( +t)]eiC(Z- Z) 012(31132-2) = -jle|1Ie p22:A d; h G .' . . ((2 Z) -p3lx- -x'l_ 22(2(1):.2-2) =—)k2n,J 4p dCte P2‘ P1 +2— {P’sinhtpzuwnsinhlp2(x'+t)1 } Ah where Ah = pzcosh (pzt) +plsinh (pzt) e-pz (x + x’ + 2:) (C.8.1) (C82) (C83) (C.8.4) (C.8.5) (C.8.6) (C.8.7) (C.8.8) (C.8.9) APPENDIX D 2-D GREEN ’8 FUNCTION FOR TM EXCITATION OF GROUNDED DIELECTRIC MATERIAL SHEET In deriving the Green’s function for TB excitation in Appendix C, the electric Hertzian potential in each region is found first, then the EM fields and the Green’s function. Here, the Green’s function for TM excitation is derived by a direct field approach. D.I. NOTATION Some words about notation here might be helpful. As a convention, upper case letters denote space domain quantities, while their transform domain counterparts are designated by lower case letters. The symbol j denotes the elementary imaginary number. D.2. TRANSFORM DOMAIN MAXWELL EQUATIONS AND GENERAL GUIDED WAVE THEOTY Consider the structure depicted in Figure D]. The current source .7 (i) = 21’ (x, 2) +212 (1:, z) radiates into the two regions, and Maxwell’s equations are: V03 = p/e ( VxE=-jtottf1 Vxfi=3+jme£ VOE=O ‘ v (D.2.l) 215 216 —eo 81’ "I on <— —> x=0 92' I42 x=-t A ,. 0' = on X YL—OZ Figure D.l. Geometrical configuration. .7 (i') = 2]; (x, z) + 212(x, z) for 2—D TM excitation. Maxwell’s equations can be rewritten in spectral domain by Fourier transforming on spatial variables tangential to the layer interfaces. Axial uniformity along the z-axis [i.e., e at e (z) and u at u (z) ] prOmpts Fourier transformation on that variable. Define the axi- al transform pair F (x, z) (-) f (x, C) as: F(x,z = 711: 1 f(x, g) 2921:; (D22) f (x, C) = IF(x,z) e-szz (D23) Taking Fourier transforms of equations (D2. 1) yields 217 (V‘+2j§) 0? = p/e ‘ (V,+2i§) x3 = -jmu7: .. . > (D.2.4) (Vt-t-ng) xh =j+jmeé (V, + 2);) of) = o where - _3_ i. V, - iax+9ay (D.2.5) The EM fields are written as the sum of transverse and longitudinal components , + 2ez (D26) + 2hz ° 8 3'1) 1» 3"» (u The transverse components are derived from the longitudinal components by . l . . . . 2. ‘ e: = --5 DCV,ez +1wuz X VJ!z flown] 1p > (D.2.7) h, = -—2 [-ja)22 x V‘ez +j§Vthz -j§2 xj,] P 1 where p2 = C2 __ [(2 (D.2.8) is a wavenumber parameter and k = axle—u (13.2.9) The longitudinal components of EM fields satisfy the following Helmholtz equations erz-p’e. = Mg; (V,+2iC) ~i+imujz} mm) v3):z -p2hz = -2 . [(V,+ 2K) Xi] 218 n.3. EM FIELDS OF Y-INVARIANT TM MODES For y-invariant TM fields hz = () 3— = o and ,ay i=2j.+2j. E =2ex+2ez} h=5>hy the transverse components are related to ‘2 by 1 . ac: . . ‘ p ) 1 . 38. .. hy = ? 0&85; +ng) 4 where e z is the solution of wave equation 2 . .2. at. 2 C31. 1P1. _E—pez=__a__ 3x (1)8 x (08 D.4. DECOMPOSITION INTO PRIMARY AND REFLECTED WAVES Since region 1 is source-free, the homogeneous wave equation 2 for total field elz has a solution can. C) = A (C) 6”" (D.3.l) (0.3.2) (D.3.3) (D.4.l) (D.4.2) Note that the solution satisfies the radiation condition, i.e. it is an outward travelling, de— caying wave with Re {p1} > 0 and Im {p1} > 0, where Pr = ng’ki (D.4.3) 219 and Re{...} designatestherealpart ofthe quantity within the braces, Im{...} desig- nates the imaginary part. The total EM fields in region 2 are the sum of a primary wave excited by j“ and jz in an unbounded region of that sheet and source-free reflected waves in the same sheet. They satisfy (j: z-e, (x C) __ __c_aj._jp’j. 44 .3):2 p22,. - 0853 0’8 (1)..) r- 2 - 237-1,; 42mg) = o (0.45) .31: .4 where P2 = Cz'ki (D.4.6) e32 is obtained directly from the homogeneous equation as 2;, (x, C) = B(§)e"*‘+0(c)!=’ (13.47) In Appendix C, it is found that the general inhomogeneous second order equation a’_ 2 _ [513 p ]W(x.C) - -S(x.C) (MS) has a solution \V (x, C) = I s (x', Q) g’é (x1 x’) dx' (D.4.9) .- where gf (x1 x') is the Green’s function for the forced principal wave e-plx-x'l 2p and with Re {p} >0 and [m {p} >0. g‘E(xIx') = (D.4.10) 220 The primary wave solutions are obtained by comparing (D.4.4) and (D.4.8), and some operations leadto ° -p,lx- —'xl 220‘, C) 1082 I [Iszlxsgn (x x)+p7,)zl e—IT-dx' 0 ’lex- -x'l j (‘0 e5.0%)=-1m—82[[--CJ',.+1'Cp;iz~°>gn(x-—x')] 2p: dx'- ’jmez -p,lx xl ”(3‘20 = I UCi +P2izsgn(x- x')] 21): dx' where = (-1 :::I and with Re {p2} >0 and Im {p2} >0. SUMMARY OF SPECTRAL SOLUTION: an (x. C) = MC)?” ' e2,(x. C). = as, (x. C) +B(C) e”="+C(C) a” e2,(x.C) = e2,(x.C)+’E[B(C)e"”- can”) jwe - h2,(x. C) = h’z’, (x. C) - p—z’ [8 (C)e ”"— C(C)e"="1 (D.4.ll) (D.4.12) (D.4.13) (D.4.l4) (D.4.15) (D.4.16) (D.4.l7) (D.4.18) (D.4.19) (DAN) Here A (C) , B (C) and C (C) are unknown coefficients which will be determined by application of the boundary conditions. 221 D.5. IMPLEMENTATION OF BOUNDARY CONDITIONS TO DETERIVIINE UNKNOWN SPECTRAL AMPLITUDES The boundary conditions that 31 = 0 at the conductor surface and E2, [1,, are confin- uous at the interface between two regions give E Ix- _‘ =0"‘2z'.=-. = 0 (D55) Bilge: = 1| =0_ +231]wa = ezz|x=0' (D52) Hy|x=o + = y|x=o. -9 h2y|x=o . = h2y|x=o- (D53) Substituting equations from (D.4.15) to (D.4.20) allows us to solve for the spectral ampli- tudes ’V‘+ V‘ A(C) = J; zip: zplpz (D.5.4) B(C)e”p”+C(C)e"” -_-. [-jCV; +P2V; lpze-peumr) - e/ _P P: 1 ezucvfipzvpl’zfl Sinh [PAH-1)] (D55) A B(C)ep2x- C(C)ep’x= [_ jCV’ +p2V']p2e'P2(21-l-x) p-pe/e +1212 (iCVj; + p2 V2) pze-p"cosh [p2 (t + x) ] (D.5.6) where V;=Ij°x (x2 C)32 -pzxd . \ jcor:2 223' * 05.7) x' - Vz-= -}jz ( C)e x j(1)e2 2p2 , -l Vi: }Jx(x C) (3031‘ [P2 (3 +t)]dx -1 1(082 pz 0 > (D.5.8) I120: C) smhlpz (x +t)]dx —: jam:2 p2 A = plcosh (p2!) +p2 (cl/£2) sinh (p20 (D.5.9) D.6. SPACE-DOMAIN FIELDS AND GREEN’S FUNCTIONS Substituting the spectral amplitudes (D.5.4)-(D.5.6) into the transform domain repre- sentations of the EM fields (D.4.15)-(D.4.20) gives ’Pt‘0 1 p e . . . . . . . e,z =.ij2 ‘5 j {JQxcosh[p2(t+x)]+pzjzsmh[p2(t+x)]}dx (D.6.1) -t cl, = L—Eeu (D52) I 'I £2; = -2jmezI{[iCi Sgn(x- x')+p;iz]ep” x _ . - e/e -[-j§i,+p,i,1e Wm" +2”1 ”2. ‘ ’sinhtpzunn e “'[igixcosh [p2 (t+x')] +p3izsinh [p2(t+x')]] }dx' (D.6.3) e2,=- [{l- C1,‘+1C10;Isgn(xx)]e‘""’E " 2jcoezp2_ - . p -p e /e - [Czix+iszizlep’(2'””)-2 ‘ E‘ 2005blP2(‘+x)] - . . . . . . jx e P2: [_ CZchosh [p2 (t+x')] +j§pzjzsmh [p2 (t+x )]] }dx —j(nez (0.6.4) These results are rewritten in Green’s function form as 0 31 (x. C) = IE1; (:1 x') -i (x'. C) dx' -t 0 22 (x. C) = fin (x1 x') -}°(x'. C) we -t where 1 ‘WC’ hl (t+ ')] ' = r— .. cos x 812,,(11x) }(082 A P2 1 “w ' ) inhtp (t+ ')] I = _ _ x 8125:2041” jmez A ( 1sz s 2 l [w ' ) shl (t+ ')] I = — — x 8123(xIx) jmez A (Jcpl CO [’2 1 (.,,, )inhl (t+ ')1 I = _ — x gnuhlx) 1.0082 5 ( P1P: s p; 8 (x " x.) C2 -p,|x-x'l -p,(2:+x+x') O = _ + — e/e _ —2p1 pg 1 2e p"cosh[p2(t+x)]cosh[p2(t+x')] } A u C . -p2|x-x'|_ -p,(2t+x+x') 822xz(11x) = -2mez{sgn(x-x)e e —p a /t-: _ . . _sz .2. ‘ 2e ”cosh [p2(t+x)]s1nh [p2(t+x)] } ' A t C . —p Ix-x'l -p,(2t+x+x') 822u(IIX) — -2m£2{sgn(x—X)e 2 +e p1 ”P281 _, leze-P"sinh[p2(t+x)]cosh [p2(t+x')] } (D.6.5) (D.6.6) (D.6.7) (D.6.8) (D.6.9) (D.6.10) (D.6.ll) (0.6.12) (D.6.13) 224 p2 {e --p,lx -x'_l e—p,(2:+x+x’) 82222041.) = 2jm82 p-pe/e _ +2‘ 31 2e?" A sinh [p2 (t+x)] sinh [p2 (t+x')] } (D.6.l4) Taking the inverse Fourier transform of 2 (x, Q) and noting that} is the Fourier transform of ? yields E, (x, z) = I 512 (x, 21 x', z') . .7(x', z') dx'dz' (D.6.15) LCS E; (x, z) = I 622 (x, 2] x', z') 0 .7 (x'. z') dx'dz' (D.6.16) LCS where LCS designates the longitudinal cross section of the discontinuity region, and C(z- -z') 512(x.zlx'.z') = I 312(2):) dC (D.6.17) [C(2z- z') I 822(11 X') dC (D.6.18) 522 (x: 2' I'D Z.) D.7. SUMMARY OF THE GREEN’S FUNCTIONS Glm(x,zlx', z') = —m—:I—§2 cosh [p2(t+x')]tmz UdC (D.7.l) ' j‘n .- ‘Pr‘. . . _ . 012310;, zlx', z') = 1:2 I Z—ngpzsrnh[p2(t+x')]e'((z “(1C (D12) _. n Gnu (x, zl x', z')= 1122!: —.:;Pj§p1cosh [p2 (t+x' ) ] (ICU z )dC (D13) -. 225 012u(x,zlx',z') = $ng n-L.plp2sinh (”(t+-1:311!“z “d; (D14) 1 1 _ jn252( e-pzlx—x'l ~p,(2:+x+x') GZM(xsZ|x'!Z) - — +k_2 WIT—“C p2e2{ +e p —p e /e _ _ , -2 ‘ 2‘ 2e p”cosh[p2(t+x)]cosh[p2(t+x')] }2‘“ "dz; (D.7.5) 022:2 (x, zlxt’ Z!) = Jk_2—. -LC “{sgn (x- x .) e -p,lx-x'l _ e-p,(2t+x+x') _21’1 ’Pzer/ 52(1),: A cosh [p2(t+x)]sinh[p2(t+x')] }J““"’d§ (D.7.6) 022” (x! 2' x" z.) = L222... ..j—S' ”{sgn (x- x ) e -p,|x-x'l -p,(2t+x+x') +8 p—pe/e +21 212 ,5 e"=‘sinh[p2(1+x)]cosh [p2(t+x')] }J“""’dc (13.7.7) 1 1 jnZ- p2 -p,lx-x'| -p,(2r+x+x') 022u(x’2|xsz)- k—z-u4n{ -e - e /e _ _. 2121 p“ 2e P"sinh[p2(t+x)]sinh[p2(t+x')] }J‘“ “at; (D.7.8) A where P.- = C2-k12 i= 1,2 (D79) [1. “1 = —‘ i = 1,2 (D.7.10) 81 A = plcosh (p21) +p2(81/£2) sinh (p21) (13.7.11) BIBLIOGRAPHY [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] BIBLIOGRAPHY R. 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PART II PROPAGATION OF EM PULSES EXCITED BY AN ELECTRIC DIPOLE IN CONDUCTING MEDIA CHAPTER 1 INTRODUCTION The purpose of this research is to analyze the propagation characteristics of electro- magnetic (EM) pulse in conducting media, and to understand the effects of the pulse waveform, the antenna dimensions and other factors on the EM pulse propagation. The interaction of an EM pulse with a conducting shell in seawater will also be studied. The results of this study may lead to some applications of EM pulses in an underground or sea- water environment. It is well known that a sinusoidal EM wave suffers a strong attenuation of exponential decaying nature when it propagates in a conducting medium. However, an EM pulse excited by an antenna in a conducting medium may not attenuate as rapidly and may prop- agate over a moderate distance from the antenna because it contains a wide band of low frequency components. Existing studies on the propagation of a transient EM wave in a conducting medium are either incomplete or approximate in nature [ll-[6]. In this part of the thesis, the exact solutions for transient EM fields excited by an elec- tric dipole antenna with an impulsive current in a conducting medium are found. Thus, by convolution, the propagation of EM pulses excited by the antenna with currents of arbi- trary waveforms can be evaluated. The optimum waveform for the antenna current which can generate an EM pulse with a maximum intensity at a particular distance from the antenna has been found. It is found that the EM fields of an EM pulse excited by an antenna in a conducting medium can be divided into two parts. The first part is an impulse 232 233 wave which propagates with the speed of light (ll lit—e) and decays exponentially. The second part of the EM field builds up gradually and propagates slowly. More importantly, this second part of the EM field attenuates as an inverse power of distance (between the third and the fifth inverse power of distance depending on the waveform of the antenna excitation current), which is a much slower rate than exponential decay. This wave behaves somewhat like a “diffusion wave”. Computational methods for obtaining numerical results are discussed. EM pulses excited by a dipole antenna with various antenna currents in a conducting medium are obtained. The optimum antenna current which generates an EM pulse with a maximum intensity at a particular distance is derived. Numerical results are presented when the con- ducting medium is seawater. There are seven chapters in this part. Chapter 1 gives an introduction for this research. Chapter 2 presents the derivations of the exact time domain solutions for the electric and magnetic fields of the impulse radiation of an electric dipole antenna in conducting media. Some special cases and several computational methods are discussed. In Chapter 3, the EM pulses in conducting media generated by an electric dipole an- tenna excited by various excitation currents are studied. The optimum waveform for the antenna current which can generate an EM pulse with a maximum intensity at a particular distance from the antenna is derived. Chapter 4 presents an estimation of the maximum strength of an EM pulse which can be excited by a dipole antenna with a given finite EM pulse energy. In Chapter 5, the effects of antenna size on the propagation of EM pulses excited in conducting media is studied. 234 How well an EM pulse can penetrate into a conducting shell in seawater is discussed in Chapter 6. In the final chapter, Chapter 7, the research accomplishments are summa- rized. CHAPTER 2 EM PULSES EXCITED BY AN ELECTRIC DIPOLE ANTENNA WITH AN IMPULSIVE CURRENT IN CONDUCTING MEDIA In this chapter the electromagnetic (EM) pulse radiated by an electric dipole antenna which is excited by a pulse antenna current and immersed in conducting media of infinite extent will be studied. Since the nature of the EM pulse radiated by the antenna in con- ducting media is dependent on the antenna excitation current, the EM pulse radiated by the antenna with an impulsive current will be determined. This EM pulse may be called the impulse radiation of a dipole antenna in conducting media. After this impulse radiation is obtained, the EM pulses generated by the same antenna with difl'erent antenna excitation currents can be easily obtained through the convolution of the impulse radiation with the antenna excitation currents. First, the exact solutions in the time domain for the electric “and magnetic fields of the impulse radiation of“ an electric dipole antenna in conducting media will be derived. Then some special cases will be reduced. Finally, several computational techniques will be dis- cussed. 2.l. EXACT SOLUTIONS FOR THE EM FIELDS OF IMPULSE RADIATION MAINTAINED BY AN ELECTRIC DIPOLE ANTENNA IN CONDUCTING MEDIA Consider an electric dipole antenna immersed in the conducting media of infinite ex- tent and excited by an impulsive current. The dipole antenna located at the origin of a 235 236 spherical coordinate system and oriented in the z-direction has a small length dl and is ex- cited by a current If (t) where I is the amplitude and f (t) the temporal function of the an- tenna current as shown in Figure 2.1. Constant conductivity 0‘ and permittivity e and the free-space permeability u are assumed for the conducting media. Our aim is to find the electric and magnetic fields radiated by the dipole antenna using Laplace transfonn techniques. The current density of the dipole antenna can be expressed as i¢(i.t) = 25(i)ldlf(t) (2.1.1) 01' 7,0, s) = 25(i')lle(s) (2.12) in Laplace transform, and where _ -rt F(s) .. If(t) e dr} (2.1.3) 17(3) fiffl) The electric and magnetic fields maintained by the dipole antenna are related to the an- tenna current by Maxwell’s equations (in Laplace transform) as follows: Vx E (i', s) #1117 (i‘, 3) (2.1.4) mer, s) = 0E0, s) “1:20, s) +J,(r, s) = s(e+o/s)E(r, s) +1.0, s) = .1630, s) +3.0, s) (2.1.5) where e" = a + GA is the complex permittivity, and is (r, s) , f1 (r, s) are the single- sided Laplace transforms of the electric and magnetic fields, respectively. 237 EM Pulse rm dll .y Figure 2.1. An EM pulse generated by an electric dipole antenna in conducting media. 238 The electric and magnetic fields can be expressed as the vector potential A (7", s) and the scalar potential Cb (i', s) : FI(i',s) = fiv x20, s) (2.1.6) E(i‘,s) = -V r/v, i.e. bl, (2.1.29), (2.1.37) and (2.1.39) can be normalized to «nar ? mot—71'5“ ‘=) 51,0, t) I 11115215 E'("‘)’cos_e_1dz‘r(””) = —(1-x/2+R’/3)e 5 .. 11 r . e9 (r, t) 3 Meg (r, t) k2 = T(l-R/2+Rz/8)e --R(T 1/2)+ R‘ 250 02 -[1+Y(S)r]e'7(’)' 231+7(3)r 41(r)r 2 1+.1't0 021+Y(8)r+‘f(3)r2—1(1)1 4 1+.s'10 £61L111tlzlllw/zne-Rr/2 (2.2.40) (2.2.41) (2.2.42) (22.43) (22.44) (2.2.45) 3(1-1/2)+ R‘ } ¢.R(T-1/2)12[J12- 1(R/2)]dt 4.? 1 (2.2.46) Te-R(T-t/2)Iz[12 (12_ 1(R/2)ld +16 251 +_ ”a Hm 111%.[1-12L/ 1(R/2)] -./T2-111[JT2—l(R/2)]] 16¢ The late time approximation (1 » r/v and t » to) can be simplified to Now, let’s discuss the waveform of the late time approximation. When 1 = a2/4 = our2/4, 39 (r, t) = 0 When a2 2 t = = 0.0650 1' , 9+ 15 2 29 (r, t) has a maximum of 3.981, and when a2 2 t = = 0.3856 r , 9-1717 “ Ea (r, 1) reaches a minimum of -0.140. 0n the other hand, at 1 = a2/10 = 0.101112, 2, (r, 1) has a maximum of 1.831, and i1... (1, 1) has a maximum of 0.915. (2.2.47) (2.2.48) (2.2.49) (2.2.50) (2.2.51) (2.2.52) (2.2.53) (2.2.54) (2.2.55) 252 For seawater with e,=80, 11:110 and 0:4 s/m, To = e/o = 1.77xlo"°s (2.2.56) v = c/ e, = 3.35xlo"(m/s) (22.51) Atthe distance of =1 m, a2 = 5.031110%, whenat mi km, a2 = 5.03 s. The maximum point of 39 (r, 1) occurs at r = 0.065a2, thus, the propagation velocity v0 of this maximum point in the waveform can find to be 1 3r V0 - a? - m (2.2.58) For seawater, at the distance of r=100 m from the antenna, vo=1.53x10‘mls and at 1:1 km, vo=l.53x103mls. This implies 111111 the maximum point of the z, (r, 1) compo... nent of the EM pulse propagates with a velocity inversely proportional to the conductivity and the distance from the antenna. The exact time domain solutions for the electric and magnetic fields maintained in conducting media by an electric dipole antenna with an impulsive current are given in sec- tion 2.1. Mathematical expressions for these EM fields are quite complicated. In this sec- tion, the exact solutions are reduced to simple forms in lossless and low loss medias and approximated for the wavefront structure and late time waveforms. From these results, there are the following observations. The EM fields of the EM pulse excited by an electric dipole with an impulsive current in conducting media consist of two parts. The first part is an impulse wave with a wavefront structure which propagates with the speed of light (1 ME) and decays exponentially. This part of the EM fields is negligibly small except at an immediate vicinity of the antenna and can be ignored for practical purposes. The sec- ond part of the EM fields is the late time waveform. It, builds up slowly and propagates slowly with a decreasing speed, which is in the same order as the acoustic speed in seawa- 253 ter at a distance of 1000 m away from the antenna. More importantly, this part of the EM fields decays as an inverse power of distance (the order depends on the shape of the excita- tion current and it is the fifth for the impulsive current), which is a much slower rate than the exponential decay. This wave behaves somewhat like a “diffusion wave” in nature. In fact, this is the only significant EM fields which can be measured at a distance from the an- tenna when the conducting medium is seawater. 2.3. SEVERAL COMPUTATIONAL METHODS AND NUMERICAL RESULTS In this section, several computational methods will be discussed and compared with each other. Since the EM fields have singularities at t = r/v for the case of the impulsive radiation, the fields at r > r/v will be computed. For simplicity, only the 0—component of electric fields is discussed in some cases. 0W Section 2.2 presents the normalized exact time domain solutions for the electric and magnetic fields maintained in conducting media by an electric dipole antenna with an im- pulsive current. Equations (2.2.45), (2.2.46) and (2.2.47) can be used to compute these fields. However, the mathematical expressions for these fields are quite complicated. For a given time t, it is needed to integrate some functions from r/v to 1. These functions in- volve modified Bessel functions with real arguments, and they converge sloWer than an FFI‘ algorithm which will be inu'oduced later. At the end of this section, the results com- puted by these methods will be presented. In this part, only the Momponent of electric fields will be discussed for brevity. 254 The Laplace transform of the normalized O-component of electric fields is given in (2.2.42) as r5 a2 .. 11 _1+y(s)r+v2(s)r2 «((1)1 Ce (r: S) !— ine—I'dl em as S): 1..-310 (2.31) where 2 V2 1 V2 __ 1 1/2 7(s) = [sits +ucs] =;[s2 +s/to] - ;[s (HI/10)] ‘ (2.3.2) Taking the inverse Laplace transform of 29 (r, s) to obtain the time domain solution 29 (r, t) YICldS ...,- 29(131) = 5% I 290,3) e"ds (2.3.3) “vi" Since 39 (r, s) has a branch out between 3 = 0 and s = -l/to due to 7(s) , an inte- gration path over the closed contours in the complex s-plane as indicated in the Figure 2.2 will be considered. Because 39 (r, s) does not have any pole within the contour, then . “0+1." i390,” eflds = I “(r,s): ds+ JCOU'J)‘ ds-I- éfeo (r, 3): ds = 0 (2.3.4) C (lo-j. C. CD based on Cautby’s integral theorem. First, let’s show that the second term of the right hand side of (2.3.4) vanishes. - 2 1+ + 2 - I e9(r, s) ("ds = a_ 7(3),. 72(3), e 7(')”"ds (2.3.5) c 4 - l-t-st0 Let s = p140 (2.3.6) 255 Go+j°° Clo-1'” Figure 2.2. Integration path for inverting Laplace transform to obtain time do- main electric field and magnetic field of an EM pulse propagating in conducting media. 256 when p —-) no, -70).”, = —-$[s(s+l/‘to)]m+st --5 2° 1+533 ”2+ 11° - vP p10 P . —j0 . r 0 e 0 ”“"v'pd (1+i—p-fi)+ptd r r . r . (t-;)pC056-m;+j(t-;)psme Now, (to—>0, n/25|9| Sn and cosGSO. Therefore, when t>r/v, Re{-'y(s)r+st} = (t—r/v)pcosG-)—oo and e—1(s)+sr_)0 Then IEO (r, s) euds = 0 C. Substituting the above result into (2.3.4) yields 1 a0.)- 1 290,1) = — I 39(r,s)e"ds = -2_1tj I39(r,s)e“ds 2 . njao-j- c, The path Cb indicated in the Figure 2.3, consists of 7 parts. Cb = C2!“ b","+C§§+ 1‘2"+CZ§+ ba3w+cb4 Let (2.3.7) (2.3.8). (2.3.9) (2.3. 10) (2.3.11) (2.3.12) 257 0W b1 Figure 2.3. The detail of the integration path Cb indicated in Figure 2.2. s = pe’° and 3+ l/ro = p'e’o' then, 1 , ' ' m) = ;Jfi!“°*°”" The integrations along C2? + :1" + C3 + (1102" + C54 are easily carried out: 2i“). j 29(1, s) e"ds+ 1 29(1, s) .121.) = o 11 C17 -1— I? (r s) e’tds-i- I E (r s) e"ds - —a—2¢-m° 21:] ° ’ ° ’ J " 410 1: do? 27:7 I39 (r, s) e’tds = 0 Cu (2.3.13) (2.3 . 14) (2.3.15) (2.3.16) (2.3.17) (2.3.18) 258 Now, let’s take greater care for the integrations along the branch out between s = 0 and s = -1/10. For egg, sfrom -l/‘to to 0, then 6 = 1r, 9' = 0, p' = l/to-p and ds = -dp. Thisleads to 7(3) = éJp(1/1:o-p) [”2 = éJpU/‘to-p) (2.3.19) For :3”, sfromOto -1/ro, then 0 = -1t, 0' = 0, p' = l/‘to-p and ds = -dp. Thisyields 1 ~1- - 7(s) = ;fi(l/ro-p)e 2 = -%Jp(l/‘to-p) (2.3.20) Therefore, 2 -1 0 '1/‘0 290,1) = _a_¢ ‘0-_1_{ I Ep(r,s)e"ds+ I 39(1', s)e"ds} 41:0 219' -l/ro o 2 __{_ 2 1/10 -pr ..2. 1., a_ _e - .. 2 2 - 5 - -410. +411 {Emmi p(l/to p)r/v]sln[va(l/to 11)] -5 Jp (1/1o - p) cosL-C Jo (1/10 - p)] }dp (2.3.21) With the change of variables of y = pt, p = y/t, dp = dy/r, above equation can be rewritten as 29(1', 1) -.f. ”‘0 .. 1:0 - = 4070‘ +3“; “I T151757“[l-y(t/1:o-y)r2/(vt)2]sm[£-J(t/TO‘Y))’] -thj(t/10-y) ycos[5J(1/co- y) y] }dy (2.3.22) 259 This integration will have the major contribution from the region where y is small. Now, let’s derive the late time waveform from the integration around the branch cut. When t » to, (2.3.22) can be reduced to -e'-’[(1-ya 2/t) sinJya2 /t-Jya2 /tcos(/ya2 / tde 30 (r. t)= 4amo (2.3.23) Let yaz/t = 1:2 (2.3.24) then dy = 2xdx1/a2. (2.3.23) becomes Ea (r, t) = fliIe"‘z‘/“z[ (1 -x2) sinx-xcosx] xdx (2.3.25) The series expression is used to compute above integral. x2n+l (2.3.26) sinx: 20(1) (—x——-—2n +1)! . 0' 11 x2" cosx = -1 2.3. Ex ) (211)! ( 27) x2n+1 11 x2» (1- x2)sinx- xcosx= (1- “2 (1)"—(T— +1)! 12 (1)( = 13 2114-3 ("'1’ 1)2 4‘20 (_1)x ___—(21146)! (2.3.28) " ‘i " ( 1)2 - __ __2_ a, _ n 2(n+2) "'1' e9(r,t) — “£2 a; ( 1) x ———(2n+3)!dx (2.3.29) Using the relation " 2 -1 1! Ix2"e’”2dx = Lit—2— J2 (2.3.30) 2u+lan yields n+2 - ,.(n+1) (2n+3)!!a 2"“) =‘2J“§0“ ’ (2n+3)!—2m—[T] n+2 = €423”; <-1>"'%.%1[:—-i] I g”; l—'—_| bb N N S l ' M8 A 1:. V a l——'l N“... ;__l 4. fl“... 11M! A as: 3 r—1 51“» i \—-——I 50 a2 .. _‘L[£2.] [211]]? (2331) " fit 4t 4t " This result is the same as the late time approximation from the exact time domain solu- tions given in Section 2.2. Since conducting media are lossy, 39 (r, .1) doesn’t have any poles in the right half plane and on the imaginary axis (it has been checked from the expression given in Section 2.2), then inverse Laplace u'ansform can be changed to inverse Fourier transform. 05+!" 39 (r, r) = 21?] I 390,3) eflds “off“ I NIH “3] 9 (him) eiwdco (2.3.32) 261 = Inverse Fourier Transform of 29 (r, jm) . Since 39 (r, t) should be a real function of t, then 2' 9 (r, jto) = Ea (r, -jm) . Hence, the infinite integral can be converted into a semi-infinite integral as Ego-,2) = -:ERe{JZo(r,jm)dem} (2.3.33) 0 IV). W The inverse Laplace transform is changed to an inverse Fourier transform. The inte- gral of (2.3.33) can be evaluated to obtain the EM fields at a particular time :. However, if the values of the EM fields at many t’s are needed to obtain the waveforms of the fields, FFI’ algorithm (Fast Fourier Transform [12]) is the best method for this purpose. Thus the inverse FFI‘ will be used to evaluate the integral. Let’s present some numerical results now. The conducting medium is seawater throughout in the following numerical calculations. The O—component of electric fields of the EM pulse generated by an electric dipole with an impulsive current are computed using the exact solutions, FFI’ algorithm and the late time approximation at a distance of l m from the antenna. The results are shown in Figure 2.4. It is observed that the numerical results computed with FFT algorithm are very accurate and close to the exact solutions. The numerical integration around the branch cut has been evaluated also. However, it is found that the numerical integration around the branch cut suffers very slow conver- ' ;0(1) 262 4 o 8 . . . . r 3-5 0.75 3 0.7 2.5 30m 0.55 2 0.5 1.5 0.55 1 0.5 °~5 0.45 o 0.4 “0.5 1 l l l 1 1 1 1 0.35 l l l 1 1 1 2 3 4 5 6 7 a 9 10 4.4 4.5 4.5 4.7 4.9 4.9 5 t/ (r/v) (r= l m) t/ (r/v) (r= l m) c (r) ldlsinO; (1) = o o xuo’r’ Late-timeperiod ------ Exactsolution ............... FFl‘algorithm Figure 2.4. Comparison the numerical results (using FFI‘ algorithm) of the electric field (O-component) maintained by an electric dipole with an impul- sive current with the con'esponding exact solution and the late-time approximation (r » r/v and t » to) of the exact solution. 263 gence, and, the total computation time for the branch cut integration is much greater than that using FFI' algorithm. Figures 2.5 and 2.6 show the comparison of the numerical results of the O-component of electric field with the corresponding exact solution of the same quantity under the con- dition of t )b r/v and t » to at r=0.2 m and l m, respectively. An excellent agreement be- tween numerical results and the exact solutions is observed in Figure 2.6 (r=-.1 m). It is noted that the EM pulse at r = r/v, which may be significant only at an immediate vicin- ity of the antenna, is very difficult to obtain by the numerical integration because of its del- ta function behavior. For any other points away from the antenna this impulse wave is negligibly small. The late time waveforms of the electric and magnetic fields maintained by an electric dipole with an impulsive current in the late time period (t » r/v and t » to) are given in Figures 2.7 and 2.8. The plots of waveforms for these fields at a distance of 10 m are shown as functions of time in Figure 2.7. The same fields at a distance of 100 m are shown in Figure 2.8 as functions of time. Comparing these two figures, it is observed that the waveforms of electric and magnetic fields change as they propagate away from the an- tenna. From the expressions of the late time waveforms, it is found that the EM fields depend on t/ (ourz) only. Figure 2.9 shows the EM fields as functions of the normalized time [t/ (ourz) ]. Now that the numerical integration method by FFI' algorithm and the late time ap- proximation for the EM pulse have been found to be valid and accurate, FFI‘ algorithm will be used to evaluate the numerical integration and the late time approximation will be used in the further analysis. 3 ’ — Exact solutiOn (at ‘ 30 u) late-time period) 2.5 - 0 0 0 Numerical results I t/ (r/v) (1' = 0-2 m) Figure 2.5. Comparison the numerical results" (using FFT algorithm) of the electric field (O-component) maintained by an electric dipole with an impulsive current with the corresponding exact solution of the same quality at the late-time approximation (t » r/v and t » 1:0). (r=0.2 m) 265 4 r f , - - r Tfi 3.5 r- .. 3 ’ — Exact solution (at ‘ late-time period) _ 2.5 ~ 0 o o Numerical results -« ‘00) 2 " cl 1.5 " «I ' (t) ldlsinO- m e 8 C. 1 " o xuozr’ " 0.5 h . 0 ................................................... -o.5 - - ------t - - - ----- 1 e - W..- 1 10 100 1000 t/ (r/v) (r: l m) Figure 2.6. Comparison the numerical results (using FFI' algorithm) of the electric field (O-component) maintained by an electric dipole with an impul- sive current with the corresponding exact solution of the same quality at the late-time approximation (t » r/v and t » to). (r=] m) 4 I V I U r T F I U 3.5 r- - - IdlsinO- ‘ ‘0 e (I) = «(1) 3 '- ° '5 4 sho’ (r) IdleosO- U) c = a, 2‘5 _ ' xtrozrs ‘l ’ 1:11an- 2 h,(0 = ,Iw) . xuor 1.5 a 1 d 0.5 ' 0 '3W _0 5 1 1 4 1 g 1 1 1 1 0 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 0.001 t (sec) Figure 2.7. Electric and magnetic fields maintained by an electric. dipole with an im- pulsive current at a distance of 10 m in seawater. 267 ¢o() ¢,() J90 0.02 Figure 2.8. Electric and magnetic fields maintained by an electric dipole with an im- pulsive current at a distance of 100 m in seawater. Idlsinfii. (I) Idleoso- - e,(!) [:11an- 1w) '0.01 0.1 1 10 Figure 2.9. Electric and magnetic fields maintained by an electric dipole with an im- pulsive current as functions of normalized time a2 = our-2 in seawater. CHAPTER 3 EM PULSES RADIATED BY AN ELECTRIC DIPOLE ANTENNA WITH VARIOUS EXCITATION CURRENTS In the last chapter, the EM pulse generated by an electric dipole antenna excited by an impulsive current, Or the impulse radiation of an electric dipole in conducting media, is derived. In this chapter, the EM pulses in conducting media generated by an electric di- pole antenna excited by various excitation currents will be studied. Since the EM pulse can be obtained by convolving the impulse radiation with the antenna excitation current, the convolution theorem will be used to determine the EM pulse excited by an electric di- pole with a step excitation current. However, it is difficult to use the convolution theorem directly to compute the EM pulse excited by an electric dipole with an arbitrary excitation current. Thus, FFI‘ algorithm will be used to compute the EM pulses excited by an elec- . tric dipole with other excitation currents. 3.1. EM FIELDS 0F EM PULSES GENERATED BY AN ELECTRIC DIPOLE WITH ARBITRARY EXCITATION CURRENTS The Laplace transforms of the electric and magnetic fields of an EM pulse maintained by a current source 2, (F, t) = 28(i)1dlf(t) are given by (2.1.12), (2.1.13) and (2.1.14), and can be rewritten as H (as) = Jd—‘[1+y(s)r]e‘7“"siner(s) = ”Ismail (r, 3) (3.1.1) . 4m-2 1|:r2 ¢ 270 I .. 5,633) = Id, 3[1-t-7(.r')r]e-7(')'c0801"(s) = d1co:°£,(r,s) (3.1.2) 21t(0'+se)r nar E . _ Id! 2 -y(a)r . e(r,s) - [l+y(s)r+y’(s)r le srn0F(s) 41:(cr+.i'€.)r3 = Idlflgelmm) (3.1.3) ROI“ where F(s) is the Laplace transform off(t) , and i1,(r, s) = 5%)- (1+-{(3) r] (“‘)' = 125.0, 017(3) (3.1.4) 0 ~ _ 17(3) 1+1(S)r 4(1), _ l- Er(f, S) - T-Tl—S'To—e - 36,17, S)F(S) (3.1.5) - _ Fm1+1r+1r’(s)r2 so), _ 1.. " Eg(r,s) — T 1+"o e - :1—229(r,s)F(s) (3.1.6) where a2 = our2 (3.1.7) in, (r, s) , Z, (r, s) and Ea (r, s) are the normalized Laplace transforms of the EM fields of an EM pulse excited by an electric dipole with an impulsive current and are given in (2.2.40), (2.2.41) and (2.2.42). Applying the convolution theorem, the EM fields in the time domain are: ‘ . IdlsinG- H¢(1‘-, 1) = Ih¢(f',1)f(t-t)d‘t = 2 H,(r,t) (3.1.3) , . 3,0,1) = je,(i,t)f(1-t)dt = Idl°°ieii,(r,t) (3.1.9) o nar 271 t . 500,!) = Ie°(1",1:)f(t-r)dr = “181291.590, t) (3.1.10) 0 nor and .. '- dr H,(r, 1) = Ih¢(r,t)f(t-‘t)—2- (3.1.11) 0 a ' d1: 2,0,1) = I?,(r,t)f(t-1)-—5 (3.1.12) 0 0 ~ ' d1 5,0,1) = [2,0, 1:)f(t-1:)-—2 - (3.1.13) a 0 where In, (r, t), E,(r, t) and 39 (r, r) are the normalized EM fields in the time domain excited by an electric dipole with an impulsive current. 3.2. THE EM FIELDS OF AN EM PULSE EXCITED BY AN ELECTRIC DIPOLE WITH A HEAVISIDE STEP-FUNCTION EXCITATION CURRENT The Heaviside step-function excitation current is expressed as: 1 t>0 f(t) - “(fl - {0 ‘50 (32.1) F (s) = U: (322) 'I'hus,tbeEMfieldsoftheEMpulsearez I ~ - d1 H¢(r,t) = Ih¢(r,1)—2 (32.3) 0 a ' d'c 3,0,1) = [gum—2 (32.4) 0 a ‘V. 272 - ' dr 56(r1t)=Izo(raT)'3 a 0 02.5) Now, let’s discuss the late time waveform of the EM pulse. Equations (2.2.48), (2.2.49) and (2.2.50) give the late time waveforms of the normalized EM fields of the EM pulse excited by an electric dipole with an impulsive current. Then, at r » r/v and r » to, the EM fields of the EM pulse excited by an electric dipole with a step-function excitation current are: .,, t s _— 11,0. o = 3. [1;] . «i: 150 Substituting (3.2.8) and (3.2.9) into (3.2.6) yields ilq,(r,t) = 71- I xze’edx n a/(Lfi) (3.2.6) (3.2.7) (3.2.8) (3.2.9) 273 __1 ' ..2 i": -fil:j dx+2—a];e ] MM) 1 — _ - .2.... _ 4[I erf(2J;)+ TZerf'b— f )] (3.2.10) Similarly, (3.2.4) and (3.2.5) can be evaluated as ~ 5 'idr 5,0,1) fibT] I" =%[l- ”71—72) +zferf' (— z—ffi) ] (3.2.11) ~ 2 I a 5" 45 +13 3 '1‘; d1 3“” = TE! [27%] 222 [2 7] a _ l .. 1. L L - 4{1 erf(2fl) + (2./t+2[2,/i] )erf'(-—-} ft) (3.2.12) The plots of the waveforms for these EM fields [(3.2.10), (3.2.11) and (3.2.12)] are shown in Figure 3.1 as functions of the normalized time, t/az. It is noted that the same results have been reported by J. R. Wait [1] [13] with the ap- proximation that the displacement current is negligible compared with the conduction cur- rent in conducting media. The total current density in seawater is given by i = (64-32) E (32.13) When the frequency is 0.9 GHz, the displacement current jars: is equal to the conduction current 63 in amplitude. Thus, when the frequency is lower than 0.1 GHz, the displace- 274 0.5 I I I I r I 1* - .......................................... .......... -- c" 'O -0 o o O O 0.45 - ’.\ ldlrinO - - 2” t I 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 t/a Figure 3.1. Electric and magnetic fields maintained by an electric dipole with a step- function excitation current as functions of normalized time (a2 = ourz) in seawater. 275 ment current in seawater becomes negligible. Under this condition the propagation con- stant then becomes 'f (s) = 32811 + sap. ~ 3011 (3.2.14) The normalized Laplace transforms of the EM fields of the EM pulse excited by an electric dipole with a step-function excitation current are: H. (r, s) = 74-1} (1 + Jsopr) {m ' (32.15) E, (r, s) = 21} (1 + Jsour) {350—11 ' (3.2.16) E9 (1', s) = 41} (1 + Jsour + sourz) [”le ' (32.17) To invert these Laplace transforms is easier than to invert the Laplace transforms of the EM fields without neglecting the displacement current like what is done in the last chapter. Wait [4] calculated the transient magnetic field of a step-function excited electric di- polein a homogeneous conducting earth. His results showed in a quantitative manner that the displacement current influences only the early portion of the waveform. The late time approximation and the neglect of the displacement current lead to the same physical effect. As time grows, the low frequency component of the EM pulse be- ' comes more dominant. Thus the displacement current can be ignored. Physically, the conducting medium behaves like a low-pass filter to an EM pulse. The spectra of the EM fields of an EM pulse at a distance of 100 m in seawater excited by an electric dipole with an impulsive current are shown in Figure 3.2 as functions of frequency. It is observed that i the 0-component of the electric fields at 200 Hz is reduced to be less than -3dB, relative to the d. .c. value. 276 10 V V T VVVfiT v v TTTT'r' ' ' v ' ""I v ' ' r"": ’ I39 Umll ‘ 1 __;:‘.=—.—: __________ 1: I l O 1 g- \\ ‘g : \ t P \ ‘ r \ 3 P \\ 1 o 01 \x f \ F \ * , [(11an - ~ . 0.001 ? I‘o‘J‘”)I = mag;l‘00“)l l‘r (1‘0” \\ 1 I: (ion - Miguel; (jm)l 15¢ Um)’ \\ I l ' xuo’r’ ' \‘ 0-0001 { ldlsinO - , ; 12.02): = thwml _ r l le-OS L 4 W1 - - w-.. . - MU--. - - “---, 0.1 1 10 100 1000 f (Hz) (=100 m) Figure 3.2. The amplitudes of Fourier nansforms of electric and magnetic fields maintained by an electric dipole with an impulsive current at a distance of 100 m in seawater. 277 3.3. OPTIMAL EXCITATION CURRENT In this section, an optimal excitation current for an electric dipole to maintain the max- imum 0—component of the electric field in seawater will be synthesized. The excitation current f (r) should be causal, and it assumed that f (t) is of finite du— ration. Then m) = 0 for KC and or}, (3.3.1) To compare the EM fields excited by various excitation currents, it should be assumed that the total energy of the EM pulse is the same in each case: .. r, (fwd: = (fwd: = A 0 0 where A is a constant. Now, let’s estimate fie (r, t) at given (r, T), where T>To. IEO(T,T)| = T dr Parana-07 0 a V2,- V2 [122-22] 1 T S —2 [I229 (r, 1) d1] ‘1 0 1/2 T = -l-,B‘”(r. 1) (flow a 0 1/2 iysmo. T) a where (3.32) (3.3.3) (3.3.4) 278 T B (r. T) = 1320 (r. 1) d1 (3.3.5) 0 The well-known Cauchy-Schwarz inequality 8 X xiyi' i=1 .[i..,2]”’[i..,2]“ i=1 i=1 has been used in (3.3.3). If f0) = CEo(r.T-t)u(t) (3.3.7) where C is a constant, then the inequality of (3.3.3) can be replaced by an equality. Thus, for given r and T, in (r, T) will be a maximum when the waveform of the excitation cur- rent is the inverse of the impulse radiation. To satisfy (3.3.2), C should be A c2 _ m (3.3.8) and _ 1x2 Ea (r. T) = 9280.1) = “78‘” (r. 1') (339) a . a . "' . . B (r: T) . When T mcreases, 59 (r, T) mcreases also. Figure 3.3 plots m as a function of normalized time 'r/(ourz). It is observed that at r = W = 212/4, [2,(r, W) = 0], B(r,W) = 0.992130...) and 13:90.11!) = 0.9961230, on). Thus, f(t) = C29(r,W-t)u(t)u(W-t) (3.3.10) will be used as the Optimal excitation current. Since W depends on the distance r, we can only design an optimal excitation current for maintaining a maximum EM fields at a '5 279 T 30.2) = (330.1)“ 1 0 3037) 0.6 " B(r.-) L L l L l l l 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 T/(ourz) Figure 3.3. The total energy of the O-component of electric field maintained by an electric dipole antenna excited by an impulsive current in seawater. 280 particular distance. In the next section, the EM pulse excited by an electric dipole with the optimal excitation current at r=100 m and that with other excitation currents will be pre- sented. 3.4. NUMERICAL RESULTS OF EM FIELDS OF EM PULSES EXCITATED BY AN ELECTRIC DIPOLE WITH VARIOUS EXCITATION CURRENTS Four excitation currents will be considered, (1) optimum excitation current, (2) Gauss- ian excitation current, (3) Exponentially decaying excitation current and (4) rectangular excitation current, each having the same amplitude and the total energy as the optimum excitation current. non E": In the last section, it has shown that the optimum excitation current of the electric di- pole should have a waveform which is the inverse of the impulse radiation waveform. Then, the optimum excitation current to generate a maximum 0—component of the electric field at a distance of 100 m in seawater should have a form: W - _ 1 W 5” W -—w-, f(r) .. 0.9955(W-t) [W 1]e u(t)u(W 1) (3.4.1) where w = 112/4 = (aur2)/4 = 1.26x10‘2s (3.42) V and the coefficient of f (t) is chosen so that f (t) has a maximum of l at W-t=a2/(9+./4—l). Then w . A = If (t) dt = 3ms (3.4.3) o 281 The other excitation currents should have the same amplitude and the total energy as that of the optimum excitation current. ID- [3 . E . . C T‘s—w—“l f(r) =e where W: 3m: ./2_1c it is found that (3.4.4) has a maximum at r = 5W with f(5W) = and a. co 1 2 If(tMt-n [[2157] dr = JEW = 3ms 0 —co m).E I’o n D o E 0| I. c I m) = e'wum where W=6ms (3.4.8) has a maximum f (0) = 1 and me=¥=m (3 .4.4) (3.4.5) (3.4.6) (3.4.7) (3 .4.8) (3.4.9) (3.4.10) 282 1 0 < r < W f0) - {o 150,12W (3.4.11) where W = 3mg (3.4.12) These four excitation currents, as shown in Figure 3.4, are then used to convolve with the impulse radiation of the electric dipole to produce four E9 fields at a distance of 100 m. FFT algorithm is applied to compute these fields. The tour E9 fields produced by these four difl'erent excitation currents at a distance of 100 m are shown in Figure 3.5. It is ob- served that the optimum excitation current (designed for a distance of 100 m) indeed pro- duces a maximum ~ E9 field as compared with the E9 fields produced by other excitation currents. If the same optimum excitation current (designed for a distance of 100 m) is used to generate the E9 field at a distance of 10 m, the generated E9 field is not larger than the E9 fields produced by three other excitation currents. This phenomenon is depicted in Figure 3.6. This indicates that an optimum excitation current will generate a maximum EM field at a specified distance only. To generate a maximum EM field at a difl'erent distance, a dif- ferent excitation current will be needed. 283 (1) """ '5 A’ fit (3) (4) ‘ g l \ \\/ ’/ 1" \\ (l) Rectangular excitation I ,’ (2) Exponentially decaying ‘. ‘ ,’ excitation i I . en‘Ir’ (3) Gaussian excitation l. (4) Optimum (for 100 m) excitation ‘ 0.01 0.015 0.02 Figure 3.4. Various forms of excitation currents for an electric dipole including rect- angular, exponentially decaying, Gaussian and Optimum (designed for a distance of 100 m) excitation. 0 25 f I r I I I /\ (3) 1', \ (4) 1111:1119 - 0.2 - ‘ I 590) = 59(3) - (1)_> ,r-,‘,’ xar’ excitation excitation ( l) Rectangular excitation (3) Gaussian excitation (4) Optimum (for 100 m) (2) Exponentially decaying . _O.05 l l l _L 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 Figure 3.5. The 0 components of elecuic fields maintained by an electric dipole with various excitation currents at a distance of 100 m in seawater. 285 0.35 r 1 1 1 <2) ..., ...,..- i (1) 5.0) = Eco) 0.3 l"... * $ ’ ‘0') -t 0.25 , I/\\ (1) Rectangularexcitation . ‘s E I \ l o 2 _ ,’ \ (2) Exponentially decaying ‘ \ . I \ o o t . 5 , excrtauon ‘. 5 I x: I . . . °~15 1' ( ) g.‘ [I (3) Gaussran excrtatron ‘ - ' 7.~ 5“" 0.. - .," ‘ (4) Optimum (for 100m) . excitation 0.05 " o """"""" -0... . -o.1 ;' + ‘ ‘ 0 0.005 0.01 0.015 0.02 t (see) Figure 3.6. The 0 components of elecuic fields maintained by an electric dipole with various excitation cunents at a distance of 10 m in seawater. CHAPTER 4 EM PULSES EXCITED BY AN ELECTRIC DIPOLE ANTENNA WITH A FIXED TOTAL EMP ENERGY IN SEAWATER 4.1. PARAMETERS OF AN ELECTRIC DIPOLE ANTENNA GENERATING AN EM PULSE WITH A FIXED EMP ENERGY To estimate the maximum strength of an EM pulse which can be excited by a dipole antenna with a given finite EMP (EM pulse) energy, the following assumptions are made. First, the antenna current has the optimum excitation waveform f ( t) with a duration of about 12 ms as shown in Frgure 3.4, which is designed to yield a maximum electric field at a distance of 100 m from the antenna. Secondly, the total energy of the EM pulse propa- gating across a spherical surface of 1 m radius surrounding the antenna has 1 Megajoule. In this section, we aim to find the parameter Idl, the product of the antenna current and the antenna length, of the electric dipole. The optimum excitation current to generate a maximum E9 (the O-component of the electric field) at a distance of 100 m from the antenna has a waveform which is given in chapter 3: T _ 1 T m T 7:. [(1) - 0995 151771] ['77: l]. u(t)u(W 1) (4.1.1) where 02 O r2 T = ‘—°° = u = 1.26xlo'2sec (4.1.2) 4 4 r=100»: 286 287 The total energy of the optimum excitation current is proportional to: - r A = (fwd: = (fwd: = 3m: 0 o The EM fields can be obtained by applying the convolution theorem. This leads to IdlsinGEoUt r, t) = Idlsinejeeu 1)f(t—t)—E (4.1.3) nor31r<3r30 Efi(;9 ‘) = 0' Since 2. (r, t), i, (r, t) have a duration of about 10a} and of = 5.03x10“sec « r at r = 1m, then f(t) can be considered as a constant in the duration when r S 1m. Thus, IdlsinG nor3 590,1) ~ d2 f(t)IEo(r 01 r .. MIST“? a, (r, 1) dr nar arO _Idl e .. sir; f__(z) W H. = 0 nor a, =‘Idls 0 1113“ t) ‘ (4.1.4) 4uar3 Similarly. , Id! 0 H, (r, t)= 3‘“ f( 1) (4.1.5) 4111'2 On application of (4.1.4) and (4.1.4), the r-component of Poynting vector is found to be 2 . 2 P,(i‘,r) = £90, 01190, t) = (Id!) 2s1115 ofo) (4.1.6) 161: Or The total power propagating across a spherical surface of radius r is easily carried out: 288 I 2 ( 401:0) (4.1.7) Gnar I W(r,t) = §p,(i,r)ds = [9,0, t)21rr2sin0d6 = 0 Because of the presence of the l/r3 factor in (4.1.7), the power radiated from an electric dipole in dissipative media behaves singular near 1' = 0. In practice, the dipole antenna is of finite size, thus it is reasonable to compute the total radiated power as the power prop- agating across the spherical surface of 1 m radius surrounding the antenna. By integrating the total power with time, the total energy propagating across the spher- ical surface of radius r surrounding the antenna is obtained as: =1W(" r)dt= (”"32th ”M = (”DA (4.1.8) 61cm 0 61:01-3‘4 It 1s assumed that the total energy is 1 Megajoule, E0: 1x10‘s Joules, when r = 1m andA= 3ms, then 1/2 61:07350 Idl = A = 1.58x10’ (A - m) (4.1.9) The amplitude of the total elecuic field is obtained from its r- and 0- components as 1/2 -2 ..2 V2 - = [5: + 53] = I 413 [sin2059 + 00665,] (4.1.10) Rd? The maximum elecuic field can be calculated as ””121 for liolslil nor E =( (4.1.11) m“ Id, |——-E for IE9| > IE4 t “Or 289 At a given particular distance, E9 (r, r) and E, (r, t) are computed. Then Em is de- termined from the values of IE9 (r, t) In.“ and IE, (r, t) In”. 4.2. NUMERICAL RESULTS OF THE EM FIELDS OF AN EM PULSE GENERATED BY AN ELECTRIC DIPOLE ANTENNA WITH THE OPTMUM EXCITATION CURRENT AND WITH A FINITE EMP ENERGY IN SEAWATER In the last section, two assumptions are made to estimate the EM fields of an EM pulse which is excited by an electric dipole antenna with a given finite EMP energy in seawater. First, the antenna current has the optimum excitation waveform which is designed to yield a maximum elecuic field at a distance of 100 m from the antenna. Secondly, the total en- ergy of the EM pulse propagating across a spherical surface of l m radius surrounding the antenna has 1 Megajoule. Then the parameter Id! of the electric dipole antenna for genera ating such an EM pulse is calculated. With this information, the real values of the EM fields of the EM pulse can be calculated, rather than the normalized values of the EM fields as computed in the last two chapters. First, the waveforms of an optimally excited EMP are calculated as it propagates away _ from the antenna.‘ When this EMP propagates away from the antenna, its amplitude de- creases, its waveform changes and its propagation velocity is reduced. To depict the be- havior of this propagating EMP, the waveforms of the EMP at the distances of 10 m, 50 m, and 100 m away from the antenna are plotted as functions of time in Figure 4.1. It is noted that the scale for the amplitude of the electric field of the EMP indicated in the vertical co- ordinate is different for each plot. The maximum electric field of the EM pulse propagating away from the antenna is cal- culated as a function of distance from the antenna and the results are shown in Figure 4.2. The waveform of the EM pulse changes greatly as it propagates away from the antenna, and the strength of its elecuic field attenuates roughly as 1/13 between 10 to 100 m range 290 3.5 I 3 .1 2.5 . E 2 I (V/m) 1.5 q 1 u 0.5 q 0 -005 l J k 0 0.005 0.01 0.015 0.02 t (sec) (a) r= 10m 0.1 I I I 0.08 - 4 E 0.06 l- - (Vlm)0-04 . 4 0.02 - . o ............................................................................... l l l 0 0.005 0.01 0.015 0.02 t (see) (b) r=50m 0.03 I A1 I 0.025 b .. E 0.02 - a 0.015 - . - lm (V )0.01 r . .. 0.005 - - o L ...................................................................... -0.005 ‘ L - 1 ‘ 0 0.005 0.01 0.015 0.02 t (sec) (c) r=100m Figure 4.1. The waveforms of the electric field of an optimally excited EM pulse at the distance of 10 m, 50 m, and 100 m away from the antenna plotted as func- tions of time. O 0 ES: 32... oEooE 552.52 ..o 5.423.: 0 le- FlEuro 291 10 v V V V V V V v I ' AAAAMJ J AAA-AA A E a 1 2 fig ,0 0.1 1 .5 . 5 8 1 El . g 0.01 1 .3 1 95 0.001 : 1 .5.? : a 1 3 0 0001 .s - . i 1 141-05 ‘ - L - g - - er ‘ - 10 100 300 r(m), Distance from Antenna Figure 4.2. Intensity of the maximum electric field as a function of distance, which is excited by a dipole antenna with optimum current excitation (with the waveform of Figure 3.4) and input EM pulse energy of 1 Megajoule. 292 and as Mrs between 100 to 1000 m range, where r is the distance from the antenna. As in- dicated in Figure 4.2, the suength of the electric field at 100 m from the dipole antenna is in the order of mV/m. This intensity appears small because an energy of 1 Megajoule pos- sessed by an EM pulse is rather small and the loss suffered by the pulse in seawater is ex- tremely great. It is pointed out that if the EM pulse were to attenuate exponentially, the field intensity at a distance of 100 m from the antenna would be completely negligible. An electric field of mV/m level may not be strong enough to damage an electronic device, but it is very strong to be detected by any sensing devices. Another observation is pointed out that if the duration of the excitation current wave- form is narrowed to the order of nanosecond, the power density of the EM pulse will be greatly enhanced in the vicinity of the antenna. However, because of the narrow wave- form of the EM pulse, it will attenuate roughly as 1/r5 instead of Ur" as the case of the op; timum excitation waveform. As the result, the field intensity of this EM. pulse at the distance of 100 m will also be in the same order of mV/m. Next, let’s discuss Poynting vector P which represents the power flow of the EM pulse. 2 p = 2x11 = (If), [sin20E9(r, t) 1?,(r, t)r-cosesinei',(r, 1) 12,0, t) 0] 1‘ Or ‘ _ (Id!)2 .. [15 a + 1290] (4.2.1) £2075 r where 13,0, 0, t) = sin20E9 (r, 1):}, (r, t) (42.2) is l' ...,.l 1' 293 139 (r, 0, t) = -sin0cosGE, (r, t) i1, (r, t) (42.3) From Figure 2.9 in chapter 2, it is found that 1'), (r, t) = £2, (r, t) 2 0 for t>>to and t>>r/v. This leads to: 131,0, 1) = éEArJ) (42.4) and .. - 1-2 E,(r, t)H¢(r, t) = -2-E ,(r,t) 20 (4.2.5) Since 29 (r, r) changes the sign at t = 02/ 4 from positive to negative, and if f (t) is a fi- nite duration function, E9 (r, t) changes the sign also when t is large (compared with the duration of excitation current). Thus, forsmall t, Em, 1) 12,0, t) >0 and I2,>0 for large t, 17:90, t)i1,(r, 1) <0 and P,<0 When P, < 0 at large r, it means the energy flows in the -? direction. Physically, it can be interpreted as follows. As the EM pulse propagates away from the antenna, the in- duced conduction current in conducting media will generate secondary EM pulses which scatter in both + 2 and —? directions. At large r, a scattered EM pulse propagates back toward r = 0 . The Poynting vectors showing this phenomenon are approximately depict- ed in Figure 4.3. 1*. .9un.d=-S_‘. i r _‘ .... L (b) t is large Figure 4.3. Poynting vectors maintained by an elecuic dipole in seawater at two difi'er- ent instants: (a) t is small: (b) t is large (compare to the duration of the ex- citation current). CHAPTER 5 EFFECTS OF ANTENNA SIZE ON THE PROPAGATION OF EM PULSES EXCITED IN SEAWATER So far it has assumed the antenna to be short and of elecuic dipole type. Since the-EM pulse excited by an antenna in seawater depends on the waveform of the antenna excita- tion current, it may also depend on the antenna size. It is to be determined whether an EM pulse excited by a long antenna will attenuate slower than that excited by a short dipole antenna. To study the EM pulse excited by a long antenna in seawater, it is assumed that the an- tenna is a cylindrical antenna of length L located along the z-axis as shown in Figure 5.1. It is also assumed that the displacement current in seawater is negligible and only the late time waveform will be considered. To simplify the analysis, it is assumed that the antenna is excited by a step-function excitation current with an amplitude of I amperes. Under these assumptions, components of the electric field of the EM pulse in Laplace transform at the point (r; 2) can be given as: _ B a2 Ez(r,z,s) _ Ij[F(R) +mQ(R)]dl (s. 1) A . B E,(r, 2,3) = II[a-l%:rQ(R)]dl (5. 2) A where 295 296 Figure 5.1. The electric field of the EM pulse generated by an antenna of finite ' lengtb with a step-function excitation current in seawater. 297 —Jotu R PM) = Tie—T— (5.3) e-Jt'x'u-r n Q(R) = m— (5.4) and W R = [(l-z)2+r2] (5.5) Wait [13] obtained the electric field of the EM pulse in time domain by inverting these Laplaceu'ansformsas: E (. .t) - ,. ‘ 'f = ezi—alzfi’e ‘3’ [erf[B(B-z)l -erf[B(A-z)]] _ +1401) -M(B) }u(t) (5.6) E,(r.z.t) _ 1 “__I— - m[N(A)‘N(B)1u(t) (5.7) where Ma) = %{l-erf[BR(l)l +BR(I)erf'[BR(l)]} ' (3.3) N0) = éU-erleRUH +0120) erf'tBRum (3.9) 2 2 V7 RU) = [(l-z) H] (5.10) and 52 = 93 (5.11) 4t 298 The electric field of the EM pulse excited by an antenna of length L with step—function excitation current in seawater at a distance r from the antenna in the broadside direction (0 = 90°) can be reduced from eqs. (5.6) and (5.7) with 2:0, and A = -B = —L/ 2. This leads to: E ,t _ , 2 Z(Ir ) = _I11t_6{432e(p)erf(%fll‘) +£[l—erflflR) +BRerf'(BR)] }u(t) (5.12) E '(r’t) = o (5.13) I where R = [rz-t- (L/2)2]Vz (5.14) In the limit of L —> on or for an infinitely long antenna, the electric field of the EM pulse becomes: Ez(r,t) _ (Br)2 " -——7 3 ["3" u (t) (5.15) I nor . This expression implies that the peak of the EM pulse u'avels away from the antenna and attenuates with a rate of l/r’. On the other hand, in the limit of L —) 0 or an electric dipole case, the elecuic field of the excited EM pulse is given as Ez(r.t) _ _ L {1— MW) + tar+ 2 (0031 erf'(Br) 1 u (t) (5.16) I 4rtar3 'Tf'f"‘f"".‘§ 3.1-.2: 299 This result is the same as the electric field of EM pulse excited by an electric dipole with step-function excitation current in conducting media given in chapter 3 with 0 = 90°. It also shows that the propagated EM pulse attenuates with a rate of l/r3. Based on these findings, an EM pulse excited by an antenna of finite length with a step-function excitation current should attenuate with a rate somewhere between 1/ r2 and UP. It is pointed out that this attenuation rate is only true when the antenna is excited by a step-function antenna current. This type of excitation current is not practical and it is adopted merely to simplify the analysis. A numerical example is given in Figure 5.2 for the antenna is in seawater, where the normalized electric field of the excited EM pulse in the broadside direction, —Ez/I, is shown as a function of the distance r for three antennas: (1) an infinitely long antenna, (2) an antenna of 20 m long and (3) an antenna of l m long, each carrying a step-function ex- citation current with an amplitude of I amperes. It is shown in Figure 5.2 that the excited EM pulse attenuates like 1/ r2 for antenna (1) and like 1/r3 for antenna (3). For antenna (2), the EM pulse attenuates like l/r2 for r<10 m and like 1/r3 for r>10 m. The ampli- tude of the EM pulse for the antenna (2) for the case of r>10 m is simply 20 times higher ' than that of antenna (3) because the former is 20 times longer than the latter. It is impor- tant to observe that by increasing the antenna length from 1 m to 20 m, it does not reduce the attenuation rate of the excited EM pulse u'aveling beyond the distance of r>10 m. The only change is the increase in the amplitude ofthe EM pulse due to the increase of antenna length. In conclusion, increasing the antenna length does not seem to reduce the attenuation rate of the excited EM pulse; it can only increase the amplitude of the EM pulse. 300 0 1 . . . . , . - r . . - . - l i 1 1 1 1 q l i 1o-05 \\ ‘~.3 I r ‘x )4 : I L x . r E: \\\\I’2m 1 111-07 1 i l \ 1 > 1.1-1m \\\: h N > 1 10-08 - ‘ “““‘ ‘ ‘ ‘ 1-- 1 10 100 r (m) Figure 5.2. The normalized elecuic field of the EM pulse excited by cylindrical antennas of various lengths with a step-function excitation current in seawater in the broadside direction as functions of the distance from the antenna. CHAPTER 6 INTERACTION OF AN EM PULSE WITH A CONDUCTING CYLINDRICAL SHELL IN SEAWATER _ In this chapter, the interaction of an EM pulse with a conducting cylindrical shell in seawater will be studied. A propagating EM pulse is assumed to be incident upon an infi- nite conducting cylindrical shell with an outer radius b and a shell thickness (b - a) as shown in Figure 6.1. We aim to determine the penetration of EM fields into the interior space of the shell. 6.1. THE EM FIELDS IN THE INTERIOR SPACE OF THE SHELL Consider an infinitely long, conducting cylindrical shell in seawater, with the geome- u'y as shown in Figure 6.1. A cylindrical coordinate system is chosen with the z axis coin- ciding with the axis of the cylindrical shell. The whole space is divided into three regions: Region 1, the interior space of the shell, is assumed to be free space. Region 2, the shell, is assumed to be a metallic conductor. Region 3, the exterior space of the shell, is seawa- 161'. The incident EM pulse is generated by an elecuic dipole antenna which is located at a distance of 100 m away from the cylinder and is driven by an optimum excitation current with the total EMP energy of l Megajoule. The optimum excitation current is designed to yield a maximum electric field at the surface of the cylinder. For simplicity, the electric field of the incident EMP at the vicinity of the shell is assumed to be a plane wave and po- 301 conducting shell, :9, 11, 0'2 12“ 3 2 1 [1 +2— ' > ‘n-b—r Region 1: (05950) ”N interior space, £9 11. 0:0 r/ CDN‘N Region 2: (a S p S b) f-J Region 3: (p 2 b) seawater, £3. [.1, 03 Figure 6.1. An EM pulse is incident upon a conducting, cylindrical shell in seawater. 303 larized in the direction of the cylinder axis. The incident plane wave strikes the shell sur- face at the instant of t = 0, and the waveform of the incident electric field can be written as: Ei(§,r) = 2E[t— (x+b) /vo]u[t- (x+b) /v0] (6.1.1) where E (t) is the waveform function of the incident electric field, 14 (t) is the Heaviside step-function and Va is the propagating velocity of the EMP in seawater. i . The Laplace transform of E (5.1) is: Mb) (6.12) . i E (:5, s) = 2E(s) [7° where 70 = s/vo and E (s) is the single-sided Laplace u-ansform of the E (t) . In cylindrical coordinates, for the TM-polarized excitation, the incident electric field has only a z-component. 3‘ (15. S) = 252(1). 0. 3) (6.1.3) The magnetic field is decomposed into p and (9 components: 93(15):) = MUM») +¢H;(po°93) (6.1.4) 1 i The components of II can be related to E through Maxwell’s equations. Since the incident electric field has only a z-component, then the scattered electric field also has a z-component only due to the simple geometry. The scattered electric field Ez satisfies the wave equation: V215z - 1’5, = 0 (6.1.5) where. 304 72 = 32112 + sue (6.1.6) Once E, is known, the magnetic fields I? can be obtained from Maxwell’s equations as: 1 13E, ”9 ' ‘5555 (6.1.7) H 1 BE, 618 '9 - :YTIFP' ( . . ) To facilitate the process of matching boundary conditions, the incident fields are ex- panded into infinite cylindrical wave functions: 5: .-. E(s)e'7°”c'7°p°°'° = E(s)c"’°" 2 (—l)"5n1, (10p) cos (1141) (6.1.9) 11:0 . 70:1 11;: —°E(sc) “’20“ l) 51;,(yop)cos(ntp) (6.1.10) I”l3 To derive eqs. (6.1.9) and (6.1.10) the following relation was used: ° = 2 511’» (2) cos (n0) (6.1.11) = 0 where 1,,(2) denotes the first kind of modified Bessel function and Neumann’s number is 8 = { (6.1.12) The EM fields including the scattered fields in the three regions are expressed using the following cylindrical harmonic expansions: EM fields in region 3 (p > b): 305 Ez .-.- 1533(1) em” 2 (—l)"6nan (s) K, (73p) cos (no) n=0 ‘1’ H = H", + 315(3) e-7°b (-l)"5nan (s) K‘n (13p) cos (mp) n=0 where K, ( z) is the second kind of modified Bessel function. EM fields in region 2 (a < p < b): E. = 2(1).“ 2 (-1)"6,,1b,. (s11, (12p) +c.(s)K,.(72p>1 cos (no) n=0 H _ 7: 40b q, - mE(s)e n=0 (-1)"5,. lb, (s)1‘, (721)) + 6,. (s) K'. (729)] cos (MP) EM fields in region 1 (p < a): E2 = as)?“ Z (-1)"5ndn(s) 1, (mp) cos (mp) n=0 Ho _ 71 flab - EELS) C (-1)"5,,d,. (s) 1', (no) cos (mp) = 0 Here the complex wave numbers are defined as: 7: = fizuefisud‘ i= 1,2,3 The free-space permeability u. is assumed for the three regions. Four independent boundary conditions on the two interfaces ale: (6.1.13) (6.1.14) (6.1.15) (6.1.16) (6.1.17) (6.1.18) (6.1.19) (6.1.20) 306 Hvlp ,_ i = lep g a * (6.1.21) E‘Ipeo- = Ez|p=b § (6.1.22) lep g 1; = H¢|p 3 b . (6.1.23) From eqs. (6.1.20)-(6.l.23), four simultaneous equations for the four unknowns, an (s) , b" (s) , cu (s) and d" (s) , are obtained by enforcing boundary conditions and us- ing the orthogonality of cosine functions. The matrix form of these equations is F ”Kn (73b) In (72b) Kn (72b) 0 - ran (5')- F 1,.(Yob) - 43K. (1312) 121‘. (1212) 12K). (72”) 0 b» (i) = 701". (70b) 0 1,, (12a) K, (720) -I. (11a) 6.. (S) 0 _ 0 121‘,(Yza) YzK‘JYza) '711'11 (ml 31"“). - 0 . (6.1.24) The solutions of these linear equations give (I, (s) as: 1 1101‘. (70b) K, (7312) - 73!. (101:) K‘. (13b) 1 a D” (3) (6.1.25) d" (s) = where D. (s) = [1,1, (121») K”. (13b) - 721'. (73b) K. (7311)] [1,1, (1,0) x'. (1,4) - 111'. (71 a) K. (12a) 1 4131!. (12b) K'. (73b) - 12F. (12b) K. (7311)] 11,1. (1,0) l‘. (1,4) - 1,1'. (1(a) 1.. (1,4)] (5.1.25) The following Wronskin W{K,, (z) , In (2)} = K, (2) I“ (z) - K'n (2)1,I (z) = l/z (6.1.27) has been used to simplify (6.1.25). Therefore, the EM fields penetrated into the interior space of the shell are given in eqs. (6.1.17), (6.1.18). (6.1.25) and (6.1.26). 307 6.2. EM FIELDS AT THE CYLINDER AXIS To determine how well an EM pulse can penetrate into the conducting shell, the elec- tric field induced by the EM pulse at the cylinder axis will be calculated. At the cylinder axis, p = 0, since [11] l n = O = 62.1 1,.(0) { 0 ">0 ( ) and I'o(z) = 11(2) (6.2.2) K‘o (z) = -K1(z) (6.2.3) then, the electric field at the cylinder axis is simplified to: E (s) e-yob [7310 (70b) K1 (73b) ‘1' 7011 (70”) K0 (73b)] 0 W) (6.2.4) Ez (s) = where D (s) = [7310(7212) K1 (13b) +7211 (7212) K0 (1312)] [1210116616 (720) +7111 (710) KO (720)] + [731(1) (721mg (13b) - 72K) (72b) K0 (1312)] [7210 (11a) I, (12a) - 7111 (1161011261 (6.2.5) The time domain solution of Ez (t) can be obtained by taking inverse Laplace trans- form of El (3) . 6.3. NUMERICAL RESULTS Sections 6.1 and 6.2 present the derivations of the Laplace transform of the electric field at the axis of a conducting, cylindrical shell excited by an electric dipole antenna 308 which is located at a distance of 100 m away from the shell and is driven by an optimum excitation current with the total EMP energy of l Megajoule. The Laplace transform of the electric field given in (6.2.4) and (6.2.5) involves the second kind of modified Bessel functions. K0 and K1 go to infinity as s —> 0 [11]. But. it can be shown that l?z (s) does notgotoinfinityass—90,and Ez(~')l,=o = 15(8) |,=o (6.3.1) This means that when an electric field penetrates into a conducting shell. the dc-com- ponent of the electric field at the center of the shell in the direction of the cylinder axis is the same as the dc component of the incident electric field. Thus. no singularity is at s = 0, the FFT algorithm can be used to invert Ez (ire) to obtain the time domain solu- tion. In the calculation, the conducting shell is assumed to be made or iron. Figure 6.2 shows the electric field (normalized to the incident electric field) at the cen- tral axis of the shell, with an outer radius of 27 cm (about 105-inch), induced by the inci- dent EMP as a function of the shell thickness. From the figure, if the shell thickness is ll 8-inch (about 3.2 mm), the electric field at the central axis of the shell is about ll10 of the incident electric field in intensity. The magnitude of the penetrated EMP into the shell in- terior is about 10% of that of the incident EMP. in Figure 6.2. it also shows that the pene- tration of an EMP into the shell can increase or decrease if the shell thickness is reduced or increased. Figure 6.3 shows the elecuic field (normalized to the incident electric field) at the cen- tral axis of the shell, with a shell thickness of 3 mm (about ll8-inch), induced by the inci- dent EMP as a function of shell outer radius. From this figure, for an outer radius of 10.5- inch (about 27 cm), the induced electric field at the central axis of the shell is again about 10% of the incident electric field, the same result as shown in Figure 6.2. A somewhat B field at central axis/incident B field 309 o 45 r . T r r r r r r 0.4 - . 0.35 - ~ 0.3 - . 0.25 - . 0.2 - . 0.15 ~ . 0.1 - b=27cm q 0.05 - .. 0 1 n 1 r r l 1 n L o 1 2 3 4 5 6 7 a 9 10 Shell thickness. [(b-a) in mm] Figure 6.2. Electric field at the central axis of an infinite cylindrical conducting shell induced by an incident EMP as a function of shell thickness. The EMP has an electric field parallel to the cylindrical axis and is excited by a dipole antenna which is located 100 m away from the shell and is driven by an optimum excitation current with the total EMP energy of 1 Megajoule. 310 o 8 T 1 I U I I I I I I § 0.7 - a: In H 5. 0.6 - g g o 5 - E 0.4% c: o 0 H '3 0.3 - P. o u: m 0.2 - ba=3mm 0.1 - o L 1 P l 1 1 l l l l 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0. Outer radius of shell, ((2 in m) Figure 6.3. Electric field at the central axis of an infinite cylindrical conducting shell induced by an incident EMP as a function of shell outer radius. The EMP has an electric field parallel to the cylindrical axis and is ex- cited by a dipole antenna which is located 100 m away from the shell and is driven by an optimum excitation current with the total EMP en- ergy of l Megajoule. SS 311 interesting result from Figure 6.3 is that the penetration of EMP can be enhanced if the outer radius of the shell is reduced. Figure 6.4 shows the waveforms of the elecuic field of incident EMP and the induced electric field at the central axis of the shell as functions of time. The dimensions of the shell are indicated in the figure. It is observed that the waveform of the incident electric field is quite different from that of the induced electric field inside the shell. This phenom- enon is due to the fact that as the incident EMP penetrates through the conducting shell it suffers a strong attenuation and dispersion. Higher frequency components of the penetrat- ed EM fields suffer stronger attenuation and dispersion. In summary, for a rough estimate when an EMP is incident on a conducting cylindrical shell (with a diameter of 21-inch and a thickness of lI8-inch) in seawater, the intensity of penetrated EMP at the central axis of the shell is about 10% of that of the incident EMP. Electric field (vlrn) 312 0.003 0.0025 incident B field 'I . . 0.002 0.0015 0.001 induced B field b _ 27 cm 0.0005 -0.0005 ‘ Figure 6.4. Incident electric field of an EMP and induced electric field at the cen- tral axis of an infinite cylindrical conducting shell as functions of time. The EMP is excited by a dipole antenna which is located 100 m away from the shell and is driven by an optimum excitation current with the total EMP energy of l Megajoule. CHAPTER 7 CONCLUSIONS The propagation of an EM pulse in conducting media has been studied. The exact so- lution for an EM pulse excited by a dipole antenna with an impulsive current has been de- rived. There exists an optimum waveform for the antenna current to excite an EM pulse with a maximum strength at a particular distance. It is found that a properly excited EM pulse can propagate slowly in seawater with a much lower attenuation rate than the expo: nential decay, and this EM pulse behaves like a “difi'usion wave”. The penetration of an EM pulse into a conducting shell in seawater has also been studied. The findings of the present study may lead to some applications in an environment such as seawater or underground. 313 BIBLIOGRAPHY [1] [2] [3] [4] [5] [6] 171 ’ [8] [9] [ICU [11] IBIIBI;I()(}I!AKIWEII{ J. R. Wait, “Transient Electromagnetic Propagation in a Conducting Medium,” Geophysics, vol.16, pp.213-221, 1951. ' J. R. Wait, “A Transient Magnetic Dipole Source in a Dissipative Medium,” J. Appl Physics, vol.24, pp.340-341, 1953. P. I. Richards, ”Transient in Conducting Media,” IRE Trans. Antennas Pmpag., AP-6. pp.178-182. 1958. J. R. 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