LIBRARY MIchIgan State Unlverslty PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE TRANSIENT ANALYSIS OF PLANE WAVE SCATTERING IN A LAYERED MEDIUM By Jungwook Suk A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Electrical and Computer Engineering 2000 “AA A4 I " O __- .t .3 - p v:- "‘ r‘u-‘ — P. a ‘ - ' ~. H‘ ~ L H. “V ‘ '5— v ”v ‘4 -. ABSTRACT TRANSIENT ANALYSIS OF PLANE WAVE SCATTERING IN A LAYERED MEDIUM Bv V J ungwook Suk The transient scattering of plane electromagnetic waves from a dispersive layered medium has been a difficult problem to solve, even though its frequency domain behavior is well known. In this study, analytical transient solutions for the electro- magnetic waves scattered from a multi-layered medium excited at oblique incidence by a uniform plane wave are derived for both TE and TM polarizations. It is as- sumed that each layer has infinite width in space but finite thickness, and isotropic, homogeneous and frequency independent electrical parameters. First, the time-domain reflection coefficient for a single interface in the medium is derived using the inverse Fourier transform of the frequency domain formulation. Then, the overall transient scattered field is found for a layered medium by combining the individual transient reflection coefficients using a series expansion and convolution integrals. The derived expressions are verified by comparison with data measured from laboratory experiments. The results obtained in this study may be used as a basis for material parameter estimation by transient probing. Copyright by J ungwook Suk 2000 To my Father and Mother Tar-”I3 iii" ‘. I , "“f ' "Hr‘y .. with HU..UL‘ ‘I ."“"‘v . , ' “A. MAMA P (U. id..di.,p ox; I v." r o “dy'D‘ LU 5ft ir"“;r, a ‘;\ t H 7 PW. "w. N61. \ 'iJQ-J‘llli Iii-3"} ,. .- H 11 ' V bun-b 'v I? AA.- w" ~ ... ACKNOWLEDGMENTS There are many people who deserve my acknowledgement for this work. First of all, my greatest thanks should be to my advisor Dr. Edward J. Rothwell, who has motivated, trained and encouraged me as well as given invaluable knowledge and advice, to overcome obstacles from the beginning of my degree program to the end of this thesis. I has been so fortunate to study under his guidance. l\»‘Iy special thanks is to Dr. Dennis P. Nyquist for his excellent teaching and precious helps for this work during past years. I also wish to express my thanks to Dr. Kun-Mu Chen and Dr. Byron Drachman, who have provided useful knowledge, and participated in my guidance committee. I appreciate the helps provided by Jeong Seok Lee, Chi-Wei Wu, Ung Sik Kim and Chris Colemann in experiments, and the friendship we have shared during my study. From the other measurement works with Dr. Ahmet Kizlay, I could obtain valuable experiences, and his help was important in writing this thesis. I have been happy to study with the other colleagues in Electromagnetics Lab of Michigan State University. Finally, I would like to say my deep thanks to my parents and sister. Their constant love and care have always encouraged me through my life. L358!!!“ l_'/ 0' Ct.’ W Cs) “—A .4 OJ) I.) IQ I.) ’7'] I.) I.) L; I.) d... ('9 U Q V“ _.A p—r—v r—q ' \ K . H4 TABLE OF CONTENTS LIST OF FIGURES ................................ CHAPTER 1 Introduction ..................................... CHAPTER 2 Interfacial Reflection Coefficients for TE-Polarization .............. 2.1 Introduction ................................ 2.2 Frequency Domain Formulation of Interfacial Reflection Coefficient . 2.2.1 Derivation ............................. 2.2.2 Branch-cuts ............................ 2.2.3 Classification of frequency domain coefficients ......... 2.2.4 Reduction of the interfacial reflection coefficients ........ 2.3 Derivation of Transient Interfacial Reflection Coefficients ....... 2.3.1 The transient forms ........................ 2.3.2 Causality ............................. 2.4 Numerical Examples ........................... 2.4.1 Verification of theoretical expressions .............. 2.4.2 The transient responses for various parameter sets ....... 2.5 Approximation of Interfacial Reflection Coefficients .......... 2.5.1 Case I : P2 < 0 .......................... 2.5.2 Case II : P2 > 0 ......................... CHAPTER 3 Interfacial Reflection Coefficients for TM-Polarization .............. 3.1 Introduction ................................ 3.2 Frequency Domain Formulation of Interfacial Reflection Coeflicient . 3.2.1 Derivation ............................. 3.2.2 Classification of frequency domain coefficients ......... 3.2.3 Reduction of the interfacial reflection coefficients ........ 3.3 Derivation of Transient Interfacial Reflection Coefficients ....... 3.4 Numerical Examples ........................... 3.4.1 Verification of theoretical expressions .............. 3.4.2 The transient responses for various parameter sets ....... vi 48 51 56 62 69 69 69 69 76 79 84 90 90 92 il lilifmi’lf' I? ferrixfid' 4.3 Iraziszvz.’ ll .\"i:::vrir. CHAPTER .3 Eiprfililt'llx , , 3;.- lufndmfi 3.3 Exp-r111»: tn ‘: '- I; ,2.) Canine. I' 3.4 XII-auras; . 'I x "J... 1.3mm B V ”my. l: - am 011?ch T5: I§,\T~' 1~Pp\f\" V w 0' t. .. g 3.,“ ”k. V . . v . T s It ‘ \ SDI bL/IJIU‘.‘ (‘ A .th .rj' LIUGRAPHY (K' CHAPTER 4 Scattering from A Multi-Layered Medium .................... 101 4.1 Introduction ................................ 101 4.2 Formulation of Transient Overall Reflection Coefficient ........ 101 4.3 Transient Propagation .......................... 105 4.4 Numerical Examples of Overall Reflection ............... 107 CHAPTER 5 Experiments ..................................... 120 5.1 Introduction ................................ 120 5.2 Experimental Set Up ........................... 120 5.3 Calibration ................................ 125 5.4 Measurements ............................... 126 5.4.1 Three lossless layer measurements ................ 126 5.4.2 Five lossless layer measurements ................. 129 5.4.3 Lossy layered medium measurements .............. 148 CHAPTER 6 Conclusions ..................................... 162 APPENDIX A Operations of Complex Valued Square Root Functions Using The Branch Cuts 165 APPENDIX B Inverse Fourier Transform Pairs .......................... 167 APPENDIX C Program Source Codes in FORTRAN 77 ..................... 170 BIBLIOGRAPHY ................................. 186 '7' .‘O- ' ‘1' 4- ~ , ,I‘ K anut v re... a . ‘ Ilia”- r ., - "5“" -.J I". #34.“ ‘ .,.. . - Np: \ . h. L l (’1’ (LR It >—-'l H. Figure 1.1 Figure 2.1 Figure 2.2 Figure 2.3 Figure 2.4 Figure 2.5 Figure 2.5 Figure 2.6 Figure 2.7 Figure 2.7 Figure 2.8 Figure 2.8 Figure 2.9 LIST OF FIGURES The geometry for analysis of transient plane wave scattering from a multi-layered medium. ....................... The incident, reflected and transmitted TE-polarized plane wave at an interface. ............................ The brach cut setting. (a) Evaluation of ,/z(w1) and (b) allowed region of branch cuts. ........................ The time relationship of wavefronts. ................ The consideration of the existence of a precursor. ......... (a) Numerical comparison of the derived transient reduced inter- facial reflection coefficient with that from the IFFT (TE polariza- tion) : Mn = #0. 6n = 60. 0n = OHS/m], lin+1 = no. em = 7260. 0",, = 4[U/m], 9,, = 30°, P2 = 0.64 x 1010. ............ (b) Numerical comparison of the derived transient reduced in- terfacial reflection coefficient with that from the IFF T (TE po- larization) : ,u.,, = #0, 6,, = 2.5960, 0,, = 9.73 X 10—3[U/m], 11,,“ = ,uo, 6,,+1 = 1.7060, 0,,“ = 5.60 x 10_2[U/m], 0,1 = 30°, P2 = —0.59 X 1010. ......................... An example of double exponential input waveform. ........ (a) Numerical comparison of the transient reflected electric field waveform for the input waveform shown in Figure 2.6 with the IFFT (TE polarization) : an 2 no, 6,, = 60, 0,, = 0[U/m], un+1 = [10, €n+l = 7260, Un+1 = 4[U/m], 0,1 = 30°, P2 = 0.64 X 1010. (b) Numerical comparison of the transient reflected electric field waveform for the input waveform shown in Figure 2.6 with the IFFT (TE polarization) : an = no, 6,, = 2.5960, 0,, = 9.73 x 10_3[U/m], ”n+1 = #0, 6,,+1 = 1.7060,0,,+1 = 5.60 x 10’2[U/m], 9,1 = 30°, P2 = —0.59 X 1010. .................... (a) The transient reflected electrical field waveform for a unit step excitation : 11,, = [10, 6,, = 60, 0,, = 0, an“ 2 p0, 6,,+1 = 960, Un+1=1.00 X 10‘3[U/m], 6, 2 0°. ................. (b) The transient transmitted electrical field waveform for a unit step excitation 3 [in = #0, 6n = 60, 0n = 0, Mn+1 = #0, €n+1 = 960, Un+1 = 1.00 X 10'3[U/m], 0, = 30°. ................ Time domain reduced interfacial reflection coefficients for various values of permittivity (TE polarization) : an 2 ”n+1 = no, 6,, = 2.5960, 0,, = 9.73 x 10’3, 0,,+1 = 5.60 x 10‘2[U/m], 0,1 = 30°. viii 14 37 39 43 44 45 46 49 50 Flare 2.1L! r- .011 .t-J-irrr .i figurr ”3.13 Prep; 34 . 3‘»- Flsl'e 3.4 T All! I I \lll~1' 3w. T" I ~. I .)_.t , 9‘ 1‘ -.vl \Gl‘ljt-A, // Figure 2.10 Figure 2.11 Figure 2.12 Figure 2.13 Figure 2.14 Figure 3.1 Figure 3.2 Figure 3.3 Figure 3.4 Figure 3.4 Figure 3.5 Figure 3.6 Time domain reduced interfacial reflection coefficients for various values of permeability (TE polarization) : 11,, = 1.5110, 6,, = 2.5960, 0,, = 9.73 x 10‘3, 6,,+1 2 1.7060, 0,,+1 = 5.60 x 10"2[U/m], 6,1 -__—_ 30°. .................................. Time domain reduced interfacial reflection coefficients for various values of conductivity (TE polarization) : 0,, = ,u,,+1 = #0: 6,, = 2.5960, 6,44 = 1.7060, Un+1 = 5.60 X 10'2[U/m], 6,1 = 30°. Time domain reduced interfacial reflection coefficients for various values of incident angle (TE polarization) : 11,, = 1.500, 6,, = 2.5960, 0'" = 9.73 X 10—3, [In-H = [10. 6n+1 Z 1.7060, Orr-H = 5.60 X 10‘2[U/m]. .............................. Transient interfacial reflection coefficient for P; < 0 (TE po- larization) : 0,, = 00, 6,, = 2.5960, 0,, = 9.73 x 10‘3[U/m], ”n+1 = #0, 6n+1 = 1.760, 0n+1 = 5.60 X 10‘2[U/m], 0,1 = 30°, P2 = —0.59 x 1010, and a = 0.37. .................. Transient interfacial reflection coefficient for P2 > 0 (TE polariza- tion) = M. = #0, 6n = to. 0n = OHS/m], Mn+1 == #0. 6n+1 = 7260. 0,,+1 = 4[U/m], 6,1 = 30°, P2 = 0.64 X 1010 and a = 0.65. The incident, reflected and transmitted TM-polarized plane wave at an interface. ............................ Numerical comparison of the derived transient reduced interfacial reflection coeflicient with that from the IF F T (TM polarization) : #n = #0, 6n = 50. 0n = ()[U/mla Mn+1 '—‘ #0, €n+1 =-' 7260, 0n+1 = 4[U/m], 9,, = 30°. .......................... Numerical comparison of the derived transient reflected electric field waveform for the input waveform shown in Figure 2.6 with the IFFT (TM polarization) : 11,, = #07 6,, = 60, 0,, = 0[U/m], 0,,“ = 0,3, 6,,+1 = 7260, 0,,+1 = 4[U/m], 0,1 = 30°. ........ (a) The transient reflected magnetic field waveform for a double exponential excitation : 6 = 1060, ,u = M0, 0 = 2 x 10‘2[U/m], 6,- = 45°. ............................... (b) The transient transmitted magnetic field waveform for a double exponential excitation : 6 = 1060, p = #0. 0 = 2 x 10“2[U/m], 0i = 45°. ............................... Time domain reduced interfacial reflection coefficients for various values of permittivity (TM polarization) : 6,, = 60, 0,, = ,u,,+1 = 00, 0,, = 0, 0,,,1 = 4[U/m], 0,1 = 30°. ................. Time domain reduced interfacial reflection coefficients for various values of permeability (TM polarization) : 6,, = 60, €,,+1 = 7260, ”n = [10,011 =3 0,Un+1= 4[U/m],9,1= 30°. ............ ix 53 54 C11 01 63 l LEW: "' :lf‘l'i’ 3“,- THY. bonfi‘ . 4 , ’\ ' LTD”? 3 \ ‘11:, 13.“ fig)?” «ll The Y" , . P"‘Ir,._1) \‘1vr . KAEAX. v. . lll». with I it ’ P‘ D I V r"- ‘ \II' a \A. '1 J . ,A‘V IIl ~ J T", 5‘"! : ‘I '1 fr "F‘Ir‘ I‘ “341% il \1119 .| U- :r 1 FELT: 4: \f.,; v .‘I‘JI (hi; 'I Ujlw ..“ .2“ 1 . 0 «. 1.“) T" ,. “1,“ TE I J I, #4. : ' I 1) fr ~.J‘Ir. .h‘ «to: . c J} E‘ r'.’ .\‘)04T‘ A I; "T? Figure 3.7 Figure 3.8 Figure 4.1 Figure 4.2 Figure 4.3 Figure 4.4 Figure 4.5 Figure 4.6 Figure 4.7 Figure 4.8 Figure 4.9 Figure 4.10 Figure 5.1 Time domain reduced interfacial reflection coefficients for various values of conductivity (TM polarization) : 6,, = 60, 6,,,1 = 7260, [1,, = [in-+1: [10, (7,, = 0[U/m], 6,1: 30°. ............. Time domain reduced interfacial reflection coefficients for various values of incident angle (TM polarization) : 6,, = 60, 6,,“ = 7260, [1,, = [1,,+1 = [10, 0,, = 0, 0,,+1 = 4[U/m]. .............. The nth layer of a multi-layered environment. ........... Numerical comparison of the derived transient propagation term with that from the IFFT : 11,, = [1.0, 6,, = 7260, 0,, = 4[U/m], 1,, = 0.01[m], 6,1 = 30°. ....................... A double exponential input waveform ............... Numerical comparison of the derived overall reflection from a 3 layered medium with that from the IFFT (TE polarization) : [12 : [10, 62 = 2.5960, 02 = 0[U/m],12 = 0.1[m], 0,1 = 30°. ....... Numerical comparison of the derived overall refleCtion from a 3 layered medium with that from the IFF T (TM polarization) : 11.2 = [10, 62 = 2.5960, 02 = ORE/m], [2 = 0.1[m], 0,1 = 30°. ....... Mutiple reflection from a lossless 5 layered medium (TE polar- ization) : [12 = [10, 62 = 2.5960, 02 = 0[U/m], 12 = 0.1[m], #3 = #0, 63 = 60, 03 = 0[U/m], 13 = 0-1[m]9 #4 = #10, 64 = 213960, 04 = ORE/m], f4 = 0.1[m],6,-1= 30°. ................ Mutiple reflection from a lossy 5 layered medium (TE polarization) : [12 = [10, 62 = 2.5960, 02 = ORB/m], [2 = 0.1[m], [£3 = [10, 63 = 5560, 0’3 2 16.7[U/m], l3 2 0.1[m], [14 = [10, 64 = 2.5960, 04 = 0[U/m], l4 2 0.1[m],6,1= 30°. ................ Mutiple reflection from a lossy 5 layered medium (TM polarization) 2 [12 = [10, 62 = 2.5960, 02 = 0[U/m], l2 2 0.1[m], [13 = [10, 63 = 5560, 03 =3 16.7[0/772], l3 : 0.1[m], [14 Z [10, 64 2 2.5960, 04 = ORB/m], [4 = 0.1[m],0,1= 30°. ................ Indentification of indivisual reflection in overall scattering from a 5 layered medium (TE polarization) : [£2 = [10, 62 = 2.5960, 02 :— 0[U/m], 12 = 0.1[m], [13 = [10, 63 = 5560, 03 = 16.7[U/m], l3 2 0.1[m], [14 = [10,64 = 2.5960, 04 = 0[U/m], [4 = 0.1[m], 0,1 = 30°. Time domain overall reflection for various values of incident angle (TE polarization) : [12 = [10, 62 = 2.5960, 02 = 0[U/m], l2 2 0.1[m], [13 = [10, 63 = 5560, 03 = 16.7[0/771], [3 = 0.1[m], [14 = [10, 64 = 2.5960, 04 = 0[U/m], L, = 0.1[m]. .................. Experimental set up. ......................... 100 114 118 122 '3‘" d‘. I in; ~. ‘ F 131:9 5.9 V‘. r‘.‘ sir? 3.9 {“1“ v ' .~,1 1‘ . . .11 (a l: “I '1 i. Figure 5.2 Figure 5.2 Figure 5.3 Figure 5.4 Figure 5.4 Figure 5.5 Figure 5.5 Figure 5.6 Figure 5.6 Figure 5.7 Figure 5.7 Figure 5.8 Figure 5.8 Figure 5.9 Figure 5.9 Figure 5.10 Figure 5.10 Figure 5.11 Figure 5.11 Figure 5.12 (a) Impulse shaped waveform transmitted from pulse forming net- work. ................................. (b) Spectrum of impulse shaped waveform transmitted from pulse forming network. ........................... Test measurement of background noise level for the arch range. (a) Time domain raw measurement data of a polystyrene plate with 0,1 = 6° (TE polarization) ....................... (b) Frequency domain spectrum of raw measurement data of a polystyrene plate with 6,, 2 6° (TE polarization) .......... (a) Time domain raw measurement data of a PEC plate with 6,1 : 6° (TE polarization) .......................... (b) Frequency domain spectrum of raw measurement data of a PEC plate with 6,1 = ° (TE polarization) ................. (a) Spectrum of intermediate system function ............ (b) Spectrum of intermediate system function after truncation. . . (a) Calibrated trasient scattering from a polystyrene layer without spectrum truncation (TE polarization) ................ (b) Calibrated trasient scattering from a polystyrene layer with spectrum truncation (TE polarization) ................ (a) Transient scattered field from a plexiglass layer with incidence angle 6° (TE polarization) ....................... (b) Transient scattered field from a plexiglass layer with incidence angle 6° (TM polarization). ..................... (a) Transient scattered field from a plexiglass layer with incidence angle 15° (TE polarization) ...................... (b) Transient scattered field from a plexiglass layer with incidence angle 15° (TM polarization) ...................... (a) Transient scattered field from a plexiglass layer with incidence angle 30° (TE polarization) ...................... (b) Transient scattered field from a plexiglass layer with incidence angle 30° (TM polarization) ...................... (a) Transient scattered field from a plexiglass container with inci- dence angle 6° (TE polarization). .................. (b) Transient scattered field from a plexiglass container with inci- dence angle 6° (TM polarization) ................... (a) Transient scattered field from a plexiglass container with inci- dence angle 15° (TE polarization). ................. xi 124 128 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 I». r' :“ titriil figgrr 3.13 hrseoll 9| lig‘flr 71H II‘T‘M ~~ I T” 1‘3:th )1”) r . ,- ...,rfj y,‘ . ,- ,,, fumil.. {fivx_ ‘ . “APP, “1:141. Figure 5.12 Figure 5.13 Figure 5.13 Figure 5.14 Figure 5.14 Figure 5.15 Figure 5.16 Figure 5.17 Figure 5.17 Figure 5.18 Figure 5.18 Figure 5.19 Figure 5.19 (b) Transient scattered field from a plexiglass container with inci- dence angle 15° (TM polarization). ................. (a) Transient scattered field from a plexiglass container with inci- dence angle 30° (TE polarization). ................. (b) Transient scattered field from a plexiglass container with inci- dence angle 30° (TM polarization). ................. (a) Comparison of trasient reflection from a mechanically adhered plexiglass plates with that for a single material (6,, 2 6°, TE po- larization). .............................. (b) Comparison of trasient reflection from a mechanically adhered plexiglass plates with that for an air gap inserted model (6,, = 6°, TE polarization) ............................ —6”/60 curve for water at 20°C given by the Debye model. Trial measurements for trasient scattered field from a 5 lossy lay- ered medium with incidence angle 6° (TE polarization) ....... (a) Transient scattered field from a 5 lossy layered medium with incidence angle 6° (TE polarization). ................ (b) Transient scattered field from a 5 lossy layered medium with incidence angle 6° (TM polarization) ................. (a) Transient scattered field from a 5 lossy layered medium with incidence angle 15° (TE polarization). ............... (b) Transient scattered field from a 5 lossy layered medium with incidence angle 15° (TM polarization). ............... (a) Transient scattered field from a 5 lossy layered medium with incidence angle 30° (TE polarization). ............... (b) Transient scattered field from a 5 lossy layered medium with incidence angle 30° (TM polarization). ............... xii 147 150 152 153 155 156 157 158 159 160 161 ,. . . I 0‘ III [u'tlli ”A Ith " ~ oi . flitil'ilnighmgl ‘ I ‘\‘.rn,v .Y .‘ ,‘ .Lli‘...i. Llpn... I‘M I "1' 1‘" i" v .1‘t1“ 'l'.‘ ' 1n. ‘ .1,” ' ' I '1' ‘V rx “I 'l', . 1'. .51” :;I,Id‘ ““111 1.- '-' " 1»: Audi,“ ‘ v 7., .1 vi. ".1", L ."“‘~hj u d‘ a \E; i, ““rVP‘IIIJL ' E," '- 1" NA. ‘ “’71; ",_ ' t u A-A1A‘T “”J‘| ? rv-a; 111 H", . u “"“A. 1 " , J 11‘. , ‘ t ~.~.;[Y V .2...(1. >131; 1"“ ' A M_‘l *t‘ “ 9'7.“_ 1 (V ‘ Ll 111.11. (113‘ RH. ; . 445.111“ (-lHH-H 1. . ‘ ‘. .\ V11 1, . ”UL “II-:1 D LIL. 7" 6.16:. “do“ -' 6,11, CHAPTER 1 INTRODUCTION The topic of this study is to find an analytical solution for the transient scattering of electromagnetic waves from a multi-layered medium, excited at oblique'incidence by a uniform plane wave. The geometry for the structure of concern is shown in Figure 1.1. It is assumed that each layer of the multi-layered medium consists of a homogeneous, isotropical material, and has frequency independent permeability, permittivity and conductivity. Also, each layer is assumed to have infinite width, but finite thickness. The first layer is assumed to be lossless. Transient solutions are obtained for both polarizations of the incident plane wave, i.e. transverse electic (TE) and transverse magnetic (TM) polarization. Although this research is motivated by a desire to estimate the parameters of each layer in the medium by using its time domain scattered field, the solution to this problem has a wide area of practical applications, such as industrial non-destructive inspection, geophysical probing and subsurface communication. Therefore, it is not surprising that there have been many studies on transient scattering from a lossy medium, or on related topics in the field of electromagnetics. While the frequency domain solution to this problem in terms of Fresnel’s reflection coefficient is already well known, and can be found in many text books and papers, to the best of our knowledge the exact analytical transient solution for an arbitrary input waveform is not available. This is partly because of the difficulty in solving the time domain wave equation directly. Stratton formulated the inverse Laplace transform pair for propagation of elec- tromagnetic wave in a dispersive medium [1], and since then many researchers have tried to solve the scattering problem by formulating the solution in the frequency do- *2 rain and li.I'lI I? ~ .\ " \ 36.93.1101,"- 3516 (iii! 'lic Ii [it‘lil' i1 1‘11! «i' :1 [ml Evm .1er Entire it I Lf'r ,J J“! . l , :rru 'Ii'H'l"',H‘I“ .~ 1{.: ‘Al I . -‘:rhiivi .. d!“ I , ‘ ‘ i . E 'l-..‘IiI‘,I‘ Lubl" ' I tn- . f. o ‘3: .1: .61}. r v,» .. "51‘? 1, f, I P.\.l,, . . , W . . ..~...t§ di‘r‘\t“‘ IL “ l4 ‘4. ltthl I 9 I."‘~ "in Nw‘ {Q 1}}.1; '. 'Ui. n . ”“5““ ri'xl‘l'l EI. ”(-1." .g' .\1 _ .‘ “H 1,11, 1.30,"\ It ‘~v-, 1319'... . nU‘Udhl ‘ ' “‘ E3 EC. ‘3 a “‘I ,‘g‘ \ “' I -" v. ‘J‘k‘TAGI‘ T _ L nth-VII main and then finding the exact analytical inverse transform. Many of these studies use certain types of approximations; e.g., the diffusion approximation obtained by neglecting the displacement current term in Maxwell’s equations [2], or restrictions on the incident angle and material parameters [3]. Other researchers have formulated the problem directly in the time domain, and have employed numerical techniques to solve it [4], [5]. Various contour integral techniques in the Laplace domain had been developed and closed form transient solutions for both polarizations had been obtained for a double exponential and unit step input waveform excitation using in- complete Lipschitz-Hankel integrals (ILHI’s) in [6], [7]. Though these solutions are restricted to specific input waveforms and are valid for a single interface, they may have been the first closed form transient solutions to the problem. Therefore, the results derived in this study are verified by those previous works. In this thesis, the problem is formulated in the frequency domain first, in the same manner as the previous studies. The constant offset value in the derived frequency domain reflection coefficient is then identified and subtracted from the original ex- pression and its inverse Fourier transform is determined. Next, the reduced form of the original reflection coefficient is manipulated so as to use known inverse Fourier transform pairs. This method was developed for a single interface in [8]. This same approach is taken, but extended to treat the more general case of a single interface in a multi-layered medium. Proper branch cuts are derived and applied for this more rigorous solution. The results for a single interface reflection are used to derive the overall transient reflection from a multi-layered medium. Finally, the derived solu- tions are verified by laboratory experiments. In the following chapter, the frequency domain reflection coefficients for a single interface in a multi-layered medium are formulated and classified for TB polarization. The inverse Fourier transform of each classified reflection coefficient is performed to obtain the transient reflection coefficient. The various aspects of the transient forms, l‘ b' !Fl.5—‘ , ‘ p“ .. ‘ . ilk. All‘llng l dim . 1!“ [I"i:‘I‘ i [[T[ L”.' dbl II}. An '11 { (mar thv um» . .. lltlllfl‘flll l't‘fim'l] “on ' on»; T] iygwri )9; I “37 l.-..I ..[..z"l..‘ ' I .1L"" [I ‘11 : "I'l. m “Enl. [HIL‘ . f‘-" "u“. " YLLQJF',“!E(L including causality problems, are discussed, and approximate forms, which might be useful in practical applications, are derived. The same procedures are taken to obtain the transient reflection coefficients for TM polarization in Chapter 3. The transient reflection coeflicients for multi-layered medium are derived in Chapter 4 using series expansion in the frequency domain and applying the convolution theorem. The descriptions for procedures and results of experiments performed to verify the derived formulas are provided in Chapter 5. Hg“ 1.1. «3.13.13. “Ire-d l ’11 a 81 I12 9 £2 9 02 “n ’ 8n ’ O.11 “n+1 ’ 8n+l ’ 011+] “N ’ 8N ’ 0N Layer index 18! 2nd nth n+1lh Nth Figure 1.1. The geometry for analysis of transient plane wave scattering from a multi-layered medium. lNTf. 2.1 lntroduct 3. IU-u - r ‘ " II " I ’ ‘ ”I r . [br;L]fI-gf1l‘t(il A. . . ,I ‘ "v I. ll. lLljllli/ill hm" . . . I ' :v‘Lryol'r 1" V- ‘. .4......'li l.\ "n I“ 'y‘r, I i -Q .. ‘ ‘ I.1...‘."li. ‘1u11.l"l.'.. .,.' ‘1 ' ' ‘ 7m- ... _ 'YI’ y. quA‘ll‘Jaa :Aal‘ ‘b‘; I 0" . f . Hn‘ O V' ,. . ' . 1 ‘ l A ~~A~ U; anus“ u 6~ “ ‘w-I‘ V.‘ ‘ v Alum [yfII ”if 13'3”“ Yvrr I . r. i ~ L.L.‘"‘ . "‘ ~‘ all IL ’ All; a "r‘ ‘ «f._1~".‘- 3.71? n" 1 ”“4 UL. nu. ‘ l. > 1‘. "i 'E~ ‘ 4A1 I . ‘. a L _IL_I\G‘UI-I A... r ;‘._ ‘ . w. d.5~.')lal‘§' 1 _., p h ‘ Eff. . ‘ l 1....".E.:....-[] (1]....41‘ 2") ~ FTEQUQU 1“. “£514” h'l ll Y“ “Sine 2.1 I t} ‘- ”5112—"? 'AI ~i - “ an .. ‘1 CHAPTER 2 INTERFACIAL REFLECTION COEFFICIENTS FOR TE-POLARIZATION 2. 1 Introduction The interfacial reflection coeflicient is defined as the ratio of reflected wave amplitude to incident wave amplitude at an interface between two layers, each of which has infinitely extended depth as shown in Figure 2.1. Because, in this study, the overall transient scattering from a multi-layered medium is derived from a combination of the individual interfacial coefficients for the layers (as discussed in chapter 4), the closed forms of transient interfacial reflection coeflicients for TE-polarized plane waves are found before obtaining the total transient scattered field from the medium. To find the transient interfacial reflection coefficient, the frequency domain interfacial reflection coefficient, which is known as Fresnel’s coefficient, is found first. Then the transient form is obtained using the inverse Fourier transform from a transform table after some algebraic manipulation. Finally, approximate forms of the transient interfacial reflection coefficients are derived. 2.2 Frequency Domain Formulation of Interfacial Reflection Coefficient 2.2.1 Derivation Consider an interface between two homogeneous, isotropic materials as shown in Figure 2.1, where region 11 has time-independent constutive parameters (,un, (man) While region (n+1) has (pn+1,en+1,an+1). A plane wave is assumed to be incident from region 1 onto interface N between the two regions. From Maxwell’s equations, V xE(:1:,z,w) = —jwunH(:c,z,w) (2.1) Region II Reflected \\ IDClI Figure 2.1 ] are. Region 11 : (p1, , 8,. , on) Region n+1 : (um , 8m , on“) \\: Transmitted Wave \ Reflected Wave H. >x Interface 11 Figure 2.1. The incident, reflected and transmitted TE-polarized plane wave at an Interface, "l 'N u'.‘" ‘5‘ '- Ix 21161116: mur ll. m 111w pi [1..- r‘E ‘ ‘ 'VIL [unfit/IR] I firm" i}. M ”-4 I l?‘ (""v . -'._"" J‘ '3 V x H(1:, z,w) = (0,, +jwen)E(;z:, z,w) and the vetor Helmholtz equation for the electric field of region 11 is V2E(r, z,w) — 735(513, z,w) = 0 where the propagation constant for the n-th layer is given by 7721 :jw/l'11(0n+jwevz) : *w2/1n6n +jw/1710n- For TE polarization E 2 9E, and (2.3) can be rewritten as ('32 82 2 (513 + a) Ey(;1:, z,w) — o,‘,,Ey(a:, z,w) = 0. The solution for the electric field is E($, Z, w) : gEo(w)e7Iu-$+7n;z where the components of the Propagation constants are defined by ,2 __ ,2 ,2 flu — 711$ + 7112' (2.2) (2.4) (2.6) (2.7) Only if region 11 is the first region of a multi-layered material and this region is lossless will we speak of an angle of incidence and an angle of reflection. Otherwise, we will only Use 7}”, 7:”, "yr” and 72.4,. For the incident wave, _ Al — I . 771x — llnz _ 71?. C0861” _ ,z‘ _ - . inz—wm—vnsmflm, and. whiir lvl '1 H I. - I V] n vry'l:". IL L“ r-lu‘A4.d‘ \, ‘.’ r77: 7" Tan; 1” ~>:,.'.JI1 r]- and, while for the reflected wave T 7/111: : 7,711 : —",’n COS 67-" (2.10) Al — A/r — A« . (HZ _ In; — In Sln 0171' (2.11) Here 6,,, is the angle of incidence and 9,,“ is the angle of reflection, as shown in Figure 2.1. The magnetic field in region n is obtained by _ 1 _ H(:z:,z,w) = —: VXE(:r,z,w) .7an 1 ABE A0E = — . —:r——y + z——y yea/1,, (92 8:1: qun ' : iii—Eo(w)e'7nzx+7nzz _ éJflEo(w)87nr1+7nzz. (212) qun Jwfln In summary, the fields in region 11 can be expressed as E,(x, z, w) : gE,o(w)e'l'l”I+73=z ' _ ,A Vin '7] I+7‘.z A 771w 7‘ x+'y‘.z H,(:1:,z,w) — .1: _ E,o(w)e "I "~ — z_——E,o(w)e "I "~ (2.13) .7an Jw/‘n Er($,z,w) = gE.,(w)e“r£xI+i£zz H , _ A Vfiz E Watt-F7522 A 71:1: E , vflxx+7fpz 214 ,(r,z,w) — 3:ij ,o(w)e —23;#— ,o(u))e ~ . ( . ) .n .n In region (n+1), (2.3) is V2E(:z:, z,w) — 7§+1E(:r, z,w) = 0 (2.15) .l , s y ‘I 11.1111th pr“, ‘) g- — "‘9 W"; : _ , urn. llli‘ my. ' I I [’a‘. " "1'11 01]“ ' ‘ m-s‘ All.\ V I l ‘ \ V" . t ‘. . \ x' h « At... “an: “i SI 1‘ E, Y‘ 'I" v'.. . Mitt [I]: [l 'I l V A A.- i \"1 War - 1 MIL bI'JI [A ridl. {'33. ‘ , 1- Ina: where the propagation constant is m2 ._ _. 2 .‘ [n+1 — JWl1«n+1(0n+1+Jw€n+1) — —w /11~n+1€n+1+JW/1'n+10n+1° (2'16) Then, the vector Helmhotz equation for the electric field in region (n+1) (i.e., the transmitted electric field) is given by 62 (92 2 (5;); + 5?.) Ey(‘r? va) _ Aflrt+1Ey($’Z’w) : 0 (217) This has the solution Et(:1:,z,w) : g]Et0(w)e"’"+‘"“7"+l'=z (2.18) where the components of propagation constants are defined by 2 2 2 7n+1 Z 7n+1,2: + Alln+1,z' (219) When both regions n and (n+1) are lossless, we may define a transmission angle 9, such that 711+1,x : 71141-1 COS 0t (220) Afn+l,z : 7n+13in0b (2.21) The transmitted magnetic field is given by _ 1 _ Ht($,z,w) = —jwu VxEt(a:,z,w) n+1 1 { ABEy ABEy} = — , —:c + z— qun+1 62' 8:2: E10041) : —'————— {_i7n+1,267n+1.1'17+7n+1.zz + 27n+1,$671z+1.11‘+7n+1,zz} leurH-l ‘ 4‘1— } . I‘. LI. 5.1131111“. nu l1. trim-:11; '1 . \H‘ v . . .. ‘I' ~ le bI-yuhqar‘\ [.1 ‘. WWI H]. . " I ‘-.'ltl_ll'l' l\ 'I‘ ‘L . A~AA. [A 111’ I IL. '1‘" [[ 5.44: )9 . ,- . ‘d [\f‘" {I : i “HIE: Et0(w)€7n+1.xr+7n+1.zz_2 ”H14“ Et0(w)€7n+l,x$+7n+l.:z. J’W'llnH jut/1,,“ (2.22) In summary; the fields in region (n+1) can be expressed as Et(:I:,z,w) = 39E“,(w)e"’"+“’x+7"+1"‘ — o f] +1~ ' - . A," +1..r , . . J HA3), 2,01) : lj_T_n_iEt0(u/I)€7n+l.xl'+7n+l,.Z _ Z . n Et0(w)e’7n+l.r~r+7n+l,-~. Jw/Jn-t-l Jwiun-fl (2.23) To obtain the ratio of the reflected and transmitted electric field amplitudes, two boundary conditions are applied. The first one is the boundary condition for continuity of the tangential components of the electric field at the interface. That is, for all z, or, E.o(w)eriuz + E,,(...»)ei£=z = E,0(w)el"+‘1‘z. (2.24) Then, to satisfy the above boundary condition, the equations 21.57;... = "M1,. (2.25) E100”) 'l' Emu") : Et0(w) (226) must be satisfied. For lossless materials, (2.25) becomes 7,, sin 6,- : 72,, sin 6,. = 7,,“ sin 9, (2.27) 10 fidawflwv: 1'11 l‘ill.:l|li'.' I' ‘ In; EEK St‘t'nlztl lmt; '9.; 0E hr.nrrrfl..t Oi, .. l .. \i J»§. " 51:; q )[ : ~.I_) .] “ I and thus my, sin 19, ‘ = 2.29 712+1 Sin 62'. ( ) which are the well-known Snell’s law of reflection and refraction. From (2.7), (2.25) and considering the direction of propagation, 71in : —A/’:ix : Var (2.30) The second boundary condition requires the continuity of tangential magnetic field on the interface. That is, for all z, or, . - . E..o(w)e*r¥izz : —7”+1”‘ Et0(w)67"+112z. (2.31) Jwfln Jwfln JW/l'nH Using (2.30), this equation simplifies to 27/111 EiO(UJ) _ 7,711: Er0(w) : MEt0(w) (2.32) Hn Mn ”n+1 NOW, multiplying (2.26) by 1&2? and subtracting (2.32) yields (M _ 711,1) Ew(td) + (M + 711$) Er0(w) ——_— 0 (2.33) ”n+1 (‘n ”n+1 #71 Therefore, the frequency domain interfacial reflection coeflicient for TE-polarized 11 q. -,.a:n' '1' u .. -Li (lane \K’aVl’.‘ 1; -~- ‘ :m: is a i"!l"l'a1- ;, r.‘ J i‘ .f... lei Fly”: ~l' v - ‘ V . ; . v u ,. l.‘ \il" “‘- 2.2.2 Branch- rhr 111%?me b I \9 .x Iii:- ‘A 1- .— ~ dim» H1: ulr prop?» .' ~§C ll ,4 . .. plane waves is RTE(w) : Er0(UJ) : (:73: _ 77z+1.$)/ (E + 7n+1,x) ' #11 ”11+ 1 Mn [171+ 1 pn+17m(w)— #n’7n+1,x(w)_ (234) #n+1A/'nx(w) + Hn’)’n+1,x (w) This is a generalized form of the Fresnel’s reflection coefficient for TE-polarized inci- dent and reflect waves at an interface in a layered structure. Note that the form of RTE(w) is the same as that given in [1]. 2.2.2 Branch-cuts Because (2.34) includes square root functions of complex argument, proper branch- cuts (or, square root rules) must be set. To do this properly, it must be assumed that the interface is the n’th interface in a multi-layered material. Since Snell’s law (2.25) holds across each interface, we must have 7M1“, = mm = ... = 712. (2.35) It is assumed that region 1 is lossless and thus '71; = (.02/11618111 0,1 (2.36) where 0,1 is the incident angle of the plane wave in region 1. The sis—components of the propagation constants are 7nx(w) : V “/12: — 7312 7n+1,.(w) = \/73.+1-v3+1,z- (2.37) 12 T1,, ,, :1.‘ lat lull". "‘1 A-l yrrwtr' ‘ ‘ h Wetland: U I ‘k 1 r4«- . 9' ‘ ~341le (‘l \ .1"; sins. flat. in: 11m ' v" i ‘7‘ r". .-., (my "» r! A 1 - “ \.A~(A.L AA D'l'; .1. 9: ..JLGEd\{}L Fifi; T .. ~ . .Ilt .-m:r;pmn+~: fif‘h f «if. LIEQTJW-x, «A r_-r l 17" ‘§‘ ‘ ‘L-",>'-h‘r,, 1 L;\ ",l \ '.AL‘ ll?" g a.» ' ”-4."- n “ Jufi 1". . ) ‘{ I Using (2.35), this becomes 7111(W) : 7121 _ A((122 : \/(_w2#n€n + jwlu'nan) _ (—w2,u1€1 Sl112 63.1) : \/—w2(#nfn — [1161le2 6“) +jw/1'nan 7n+1,x(w) : 7121+1,z _ ’l'fz Z \/—w2(#n+1€n+1 — [£161le2 011)+jw,un+lan+l' (2‘38) The condition that region 1 must be lossless confines the locations of the resulting propagation constants on the complex plane, and contributes to the definition of the branch cuts. The condition for setting branch cuts is the radiation conditon, which says that, by the energy conservation law [9], an isolated wave cannot increase during its propagation. To satisfy this condition, the real part of each component of the prepagation constant must not be positive for both positive and negative frequency. The z-component always satisfies this condition since Rehm} = Re{7,,_1,z} = . . . : Refill} = 0. However, two different branch cuts must be applied to 7m, since the imaginary part inside of the square root has two different signs according to the sign of the frequency variable w, as shown in Figure 2.2 (a). The resulting branch cuts to evaluate a square root of complex value «2(a) = wl) are given by wl > O : éarg[z(w1)] < 63 < arg[z(w1)] wl < 0 : arg[z(w1)] < 03 < $arg[z(w1)] + 7r (2.39) Where 03 and arg[z(w1)] indicate the argument angles of a branch cut line and z(w1) reSpectively, and O S arg[z(w1)] < 27r. The branch cuts of (2.39) should be applied Consistently through the entire derivation procedure, and Appendix A shows the ODerations of square root functions used in this study, which are obtained by applying 13 dag) a), > O a) < O we.) ‘ ‘8: KW Re {2} (2“ Re{z} 85' z(w1) z(w >85 (a) Im{z}1 Im{z} Eargl: Z(w1)] ‘01 > O a), < 0 Re {z} . Re {z} gargldwa] +7! (1)) Figure 2.2. The brach cut setting. (a) Evaluation of ‘/z(w1) and (b) allowed region of branch cuts. 14 itr bran ll m" 2.2.3 Classi: if? frr_-.~:zrl s ., 2*» «My 9 ‘1 103.1 '_'.,'L£.‘Ldlil £5 -\':f:J tilt“ B, 1\ fir ., . I. ‘ a,g.-.1}1fmtr {11.11 .‘r‘,'.l ' Dfi‘J‘HIIJ: ,\ I the branch cuts. 2.2.3 Classification of frequency domain coefficients The Fresnel’s coefficient given by (2.34) can be classified according to the signs of some constant values included within it. Therefore, layer constants may be defined as D" = (unen — M161 sin2 6,1)/p.3, (2.40) 13,, = (In/Mn. (2.41) Note that B" is always positive. In order to compute the square root functions, the appropriate branch cuts must be applied. This depends on the signs of D", B", Dn+1 and 13"“. All possibilities are considered next. Denoting s = jw the terms in (2.34) become ”v ~ ~ ’7” = \/s(D,,s + B") #n M : 5(Du-1-15-+'[3714-1) “n+1 and (2.34) may be rewritten as \/s([)ns + B") - \/3(Dn+1s + 13,1“) \/;(D113 + Bu) + \/3(Dn+13 + Bn-H) RTE(3) = (2.42) Now, the frequency domain reflection coefficient can be classified to four different fOltrns according to which branch cut is needed. This depends on the signs of the layer parameters D" and Du“. The algebraic manipulations with the branch cuts used in the derivation processes are given by (A.1) and (A.2). 15 . l a (1101.4 >l Ii). W'uf} .. 1A1 A 'N A y sir (1)D,,+1> 0 and D... > 0 -\/D nfi(/s+-g—:+ Dn+1\/§ s+-— 3"“ Dn+l _V DnfiVS‘I'g: _ Dn+1f “(611:1 S+%L _\/ Dn___+_l\/S Biz-+1 RTE(S V D11 + Dn+1 by;— \/%n_ \/9+§— (2'43) Dn+1 RTE(S)— Simplifying gives (2) Du“ < 0 and D” < 0 _ij(/anI N3 s— g, 422' an+1lf B I I _IDn+1I :hj IDn |\/§(/s—l—5Lij an+1I\/g :—|Dn+1|. n+1 Simplifying gives ___\/Dn+1 \/S _ Bn+1 RTE(8 IDnI IDnI IDn+1I )_\/__I F . (2.44) S _ ..Bll. + n 1 Bn+1 IDnI DnI \/8— IDn+1I (3) Dn+1 > O and D” < 0 RTE(8)= ij ID" lfV:-IDnI+ ID"+1I\/— S+ID::iI B . ij IDflI I\/- ID I IZDTI+1I\/g 3+ f—D::ll| Simplifying gives /8 IDn__+_I1 n 1 RTE(8)— —IDnI _:_I:Fj\/—'—I I07: I \/8+IDn:1I r Ping’l \/S+ (2'40) IDnI IDn | IDn+1I 16 A -v "h' .— IDA'FPIL ~. Ii l , \m If‘f‘wn v I!" unwilum; LN I2 raIf'Il {If lln‘w II V ' ‘v 'v'IAlf‘.‘_ T0 malp {I33 ’1an '12, (4) 13,,+1 < 0 and [9,, > 0 RTE(8) _ ID" IfV 8 + ITS-LI :FJVID11+1I\/—V:_ID:+1I. —V IDnI\/—V8+—1—-ID:I:I:]. IDn+1I\/— —ID:::11I (2.46) Simplifying gives TE VS +ID I AiJN/D IEI \/5_ ID::II R (s) = . (2.47) B“ n+1 B1i+l V8 + I011 I ¢j\/D ID I \/S— IDn+lI In each of these expressions the upper Sign corresponds to w > 0 and lower sign does to w < 0. Note again that the branch cuts defined in section 2.2.2 have been used through the whole manipulation steps for the square root functions. 2.2.4 Reduction of the interfacial reflection coefficients Observing (2.43)-(2.47), it is realized that there is a constant offset value inside of each frequency domain expression. Since we are supposed to find the transient reflection coefficient by the inverse Fourier transform, and the frequency domain function must be integrable (i.e. ff; [RTE(w)I2dw S 00) for the existence of the inverse Frourier transform, the constant offset value should be extracted first. To make the (2.43)-(2.47) more readable, let D : IDn+1I : #3; IHn+1€n+1 — #161Sin2 9i1I n IDnI #12:“ Iflnén —‘ #161Sin2 gill ~ n0 _ IHn€n — #161 sin2 HilI' 2,5 and take the limit values at infinite frequency. Then, 17 3(1ng > ‘2 an \|l.- Ir.-1 > Bill... (1)Dn+1> O and D" > 0 _ V3+Bn — \/Dn\/S+Bn+1 RTE . _ , 2.49 (5) m+ mm I I , 1— ,/D 1' RTE . = —————'1. 2.50 {fig (5) 1+ \m—n ( ) (2) [2,,+1 < 0 and [2,, < o RTE(9)=V5— "—VD"V8_B"“ (2.51) I V‘s—Bn'l'VDnVS—Bn-H’ 1— ,/D 1' R” = ——-————’1. 2.52 (3) DH, > 0 and [2,, < 0 RTE(S) : VS-BanjVDnVS‘I'Bn—H (253) \/s — Bn :l: j,/D,,,f__s + Bn+1’ . 1$j¢Dn 1 R” = ————————. 2.54 (4) [2,,+1 < o and [2,, > 0 I V3+Bn:F.7-VDnVS_Bn+l, lijI/Dn lim R” s = ———,———-——. (2.56) 4.1—+00 ( ) 1;]N/Dn In these expressions the upper sign corresponds to w > O and lower sign does to w < 0. Next, the obtained constant values have to be subtracted from the original RTE(3). Let’s define RTE(S) = RTE(S) - RTEIS = ijoo), then 18 I}; D“: > II. "V« v . ......uu“ “IR EA. 7': .L , R In 2. 1- 1 It‘- \ I:,,]'.-; ' J‘ ‘43:. 311‘ v (y K ‘A‘ (1)D,,+1> 0 and D" > 0 V8+Bn‘\/l—)—7;V3+Bn+l 1_\/D—n \/s+Bn+\/le/s+Bn+1_1+\/D_n : 2m VS+Bn— V8+Bn+1 (2 57) 1+\/—57_1V8+Bn+\/ET:VS+BTI+1 I RTE“) Eliminating the square roots in the denominator of (2.57), gives RTE(8) _ 2\/D—,, {ME—mHI/s—IB—n-flm} — ”JD: (3+Bn) —Dn(s+B,,+1) 2m x (1" 1911M1 + JZD—n) (1+ EMS + 811) _ (1+ \/D—n)\/S + Bn\/3 + Bu+1_ \/D—n(Bn - 8114-1) Bn“Dan 1 3+( 1—D,, ) (2.58) The .s + 3,, term can be factored out of the first and second term in the numerator of (2.58), giving Vs+Bn ‘ /s+Bn+1} ) (s + Bn){1— s RTE(8) : 2m ( +8" l—Dn 3+(%B_L) 2Dn(Bn — Bn+1) 1 (1— Dn)(1+ V Dn) 3 + (Ln-1?" n+1) D _ 2Bn,/_D,, 1 1 /3 + Bn+1 — 1— D71 8 + (Bn-DriBn+l) S + Bn l—Dn + 2\/ D11 8 1 3 "I" Bn+1 1— 0,, 3 + (Bu—Dump) V s + B,, l—Dn n Bn '— Bn (1— Dn)(1+ (DZ) 3 + (___B,.-u,.3...) 1~Dn 19 where {M ir 0i )139i- F“! priwiure mi.» um D.i < H. .2 D“; < U it 11" i where (A3) is used. Known inverse Fourier transform pairs are available for all terms of (2.59). For the other cases, the quite similar steps are taken, except that the procedure must be done separately for w > 0 and w < O in the cases of Dn+1 > O and Dn < 0, and Dn+1 < 0 and En > 0. These are (2) Du“ < 0 and 13,. < 0 RTELS‘) _—_ VS-B’1— VDnVS—BHH _1_ VD" \/S_Bn+\/Dn\/3—Bn+1 1+\/Dn _QBn\/D—n 1 {1_ S‘Bn+1} 1_ D-n Bil—[)1L811+1 S — Bn 8 + ——_l—D n + 2\/—D_n S {1_ S—Bn-H} ) — Bn—Dan _ 1 Dn 3 + (_T—TrT+—l s 13,, n 2Dfl(Bn-+-1 — Bn) 1 _ - 2.60 (1-— Dn)(1+ «5)... (Wm) ( > 1—n, (3) Dn+1> 0 and D” < 0 RTE(S) : VS_Bnq:jVDnV3+Bn+1-1$j\/Dn ”fix/Em lijx/DT ¥2jB.\/D—n 1 {1_ HEW} 1+Dn 3— (En—Dan+l S—Bn 1+Dn QJx/D: 3 {1_ 3+Bn+1} ¥1+Dns_ (153,.—.I),,19,,+1 S—Bn 1+Dn _ 2Dn(Bn+1+ Bn) 1 (2 61) "/ Bn-Dan 1 ' ' (1+ Du)“ 4:] D") 3 — (—__1+Du + ) (4) [Du—H < 0 and Dn > O RTE“) = V3+Bnij\/Dm/s— B,1+1__1ij\/Dn m7$jmm MIA/1‘); 2'Bm/Dn 1 -3" ¥ J ) {1+ ————-——S +1} 1+ D" 8 + (Bn‘ISIIzJBrH-l S -+- Bn 20 411 “11.251 ‘11“ LEHS. I" (WW; I ;‘ 'lEUrlnlé Hm I) .- I I A .. ‘ my}, «.1 4 14‘s) -)» in.” l: (l I . I ‘ H15 Ur ‘1!)me 1‘ ‘K ""11. 5. v. nlt'uld “(1) N55,“. 2V0” “—311 3F J 8 ){1+ 3——+—1} 1+ 0,, S + (En—Dwail s + 8,, 1+0" + ( +1 +_ ) , (262) (1+ D..)(1:F ND.) .9 + (Ln—11:58....) and (A.4),(A.5), and (A6) are used respectively. In these expressions the upper sign corresponds to w > O and lower sign does to w < 0. RTE(3 = ijoo) is the reflection coefficient obtained at infinitely high frequency, and is equivalent to the coefficient when the medium has no conductivity. Therefore, it depends on the diplacement current term only in the Maxwell’s equation of (2.2), and may be called an aymptotic reflection coefficient, since the value of RTE (s) approaches this (or these) value(s). In contrast, fir-”3(5) exists only when at least one of the two media has some conductivity. Let’s call it a reduced interfacial reflection coefficient. The physical meaning of the two reflection coefficients in transient analysis will be discussed later. 2.3 Derivation of Transient Interfacial Reflection Coefficients 2.3.1 The transient forms The transient forms of the frequency domain interfacial reflection coefficients are given by the inverse Fourier transform. Because the asymptotic reflection coefficients RTE(3 = :tjoo) are constants in the frequency domain, their transforms are simply given as delta-functions multiplied by the constants. Let’s denote it by RLE (t). Then, (1)Dn+1> 0 and D" > 0 R330) 2 F-1{1im RTE(3)} 1 — \fDT. mam (2.63) 21 (2)1)”-E < 0 ( .ls sh=_iv.'ii ln' -.} ‘ . ' V ":'v‘ 9 {U hi" \‘At. l-. T." 5 , “Cinjiwrn‘l ldl (2) [7,,“ < 0 and [2,, < 0 RTE(t) = F-1{lim RTE(3)} w—mo = 1 ‘ mm) (2.64) 1+ \/D,, where F "1 denotes the inverse Fourier transform. (3) D...H > 0 and [3,, < 0 As shown by (2.54), there are two different asymptotic values in this case according to the sign of 6.). Therefore, letting G = M then (2.54) can be rewritten as 1+j\/Dn lim RTE(w) = GU(w) + G*U(—w) (2.65) w—mo where 0" denotes complex conjugate value of G, and U (w) is a frequency domain unit step function defined by U (w) = 1 for w > 0 and U (a2) = 0 for w < 0. From the transform table given in (B2), ROTOEU) = F‘1{GU(w)+G*U(—w)} = $(G + G*)6(t) + 11* (G — GU50) 27rt _ 1—0, 2m 1 _ m6(t)+M{t*6(t)} (2.66) where a(t) * b(t) is the convolution of a(t) and b(t). The convolution of a function with a delta function gives the function itself. Thus, 1 — D 2\/D TE _ " ______” R°° (t) — 1+ an) + 7r(1+ Dn)t' (2'67) Notice that the transient reflection coefficient is a real valued function as it should be in the physical world. 22 llle lldl'nlt 13' with raw ill .Lilu Mg (1“L T; 5. ll. . Tux: film hit". 1 1L llié lll‘.‘;.r 5r. (4) [9,...1 < 0 and [9,, > 0 The transient form for this case is easily obtained by applying the same procedure with case (3). Thus, , 1+D - 2\/D TE _ n _ 11 R00 (t) _ l—Dn (t) ——7r(1+D,.)t' (2.68) The inverse Fourier transforms of the reduced interfacial reflection coefficient can be found directly from the transform table given in Appendix B. Let’s define the following constants P1 = (Bn+1 + Bn)/2 (2-69) P2 = (Bn — Dan+1)/(1_ Dn) (2'70) 5 = (BM — Bn)/2. (2.71) Then for (1) Dn+1 > O and [3,, > O (2.59) is rewritten as 1 — Dn S + P2 V S + B" +2VDn 3 1_ (8+Bn+1 1 ‘— D11 3 + P2 8 + Bn n Bn - n _ 2D ( B +1) 1 (2.72) (1 —Dn)(1+\/lTn)s+P2' This form has two different inverse Fourier transforms according to the sign of P2, and the inverse Fourier transform for each term can be found in (B.3)-(B.4). Thus, 23 when Pg > ll. I r'v ' l WWI -‘ l " ‘- '5' ‘.‘\‘. prrperiy 6f {3... when P2 > 0, 2\/—- e—Pgt {6m P2tu(t) +—— 1 —P2U(t)} * 2Dn(Bn _ Bn-H) ~ 28,, __\/_:€ e—Pi’tu(t) -’ 8—10” 1 ,‘i 0 . u , — [ a {I (36+! (31)} (1)] (1-0.)(1+fi):) ————2\/'D: e“P2'6t 1_ D” (,) * (ewe-Pl: {11(13t) + 10(131)}u(t) + {2B.¢E.‘_ N727}, 1—Dn l—Dn 2D,.(Bn — Bun) ’(1 — D.)<1+ fir.) p} ell-“11(1) ... (—13)e_P“ {1,(131) + Io(,8t)} W) e‘PztuU) (2.73) where In(t) is the first kind modified Bessel function of order n. Using the convolution property of the delta function, (2.73) becomes (Bn —' Bn+1)V Dn fame) = 1_ D e‘Plt {11(61) + 10(61)}u(1) 21)n\/T)—(Bn+1‘_Bn) (Bn _ Bn+1) —P2t. + (1 _ 071),, 2 e u(t) * 6")” {11(61) + 10(13t)} 6(1) — 2MB" ‘ B71“) (1" Dn)(1+ V1711) (Bn — Bn+1)\/lTn 1_De“””{11(13t)+10(3tt)}U() _ (Bn _(1Bn+11;21))2n\/—l)_ne_P2tu(t)/ e—(Pi-ler {110313) + 10(18$)}dl' _ n 0 2‘1)11(13r1-+-1 _ B") —P2tu I +(1— Dn)(1+ x/lTnle (t). (2.74) e‘P2‘u(t) It is necessary to check whether the integral term from time domain convolution e‘Pztu(t)fOe‘(‘P1P2)1x{l(131:()+10(13r)}d2: exists ast—>oo. Let I(t) 2 [0t e‘P'Zte—(Pl’P2)I {11(13x) + IO(,B:1:)}d:1:. When t is large, the integrand varies as e‘0, sincet—$>Oand ,8+P1>0, 24 when P2 > 0. “CHE Irf lb lln‘f WWW Ul llio' 1i. ‘1 lfl‘TH i S laer—J *1 {X ‘ Slate 1 a . V‘ J” when P2 > 0, 12%) z [315ng (t)+12__‘/_ e-P2‘{6<>— aum} =1 _/’ 8—1311 , u _ 2071(811 _ Bn+1) [ ,5 MW) + 10(36} (1)] (1__ 0.1(1 + m) 12%— 8"P2‘5(t (0* (— ‘flle—Plt {11037) + 10(13t)} u(t) + {Li-3):? ,2 5} e-P2‘u(1) * (—,6)e-P1t {11(13t) + 10(8t)} 11(1) 2D,.(Bn — 3.1,) 4,2, _(1— Dn)(1+ \/D—n)e W) (273) e—P'Z’UU) where 1,,(t) is the first kind modified Bessel function of order 11. Using the convolution pr0perty of the delta function, (2.73) becomes (Bn "' Bn+1)\/-1)—n RTEU) = 1_ D e‘Plt{Il(flt) +IO(13t)}u(t) +112Dn\/(Ii—n—(813+)12_Bn) (Bn ‘2Bn+1)e—P2tu(t) * _P1t .1 u _ 2Dn(Bn _ 3"“) 6 MW) + 10(39} (1‘) (1_ Dn)(1+ \/D—n) (Bn jljngjme-Plt {11(61) + 10(13t)} W) _ (Bn —(1BT1D)n§)2n¢E-8_P2tu(t)[) 6—(P1_P2)x{11(13111) + [0(31.)} d.L‘ 2121(«Bn4-1 — Bu) —P2tu . +(1— 11,)(1 + mag (3' (2.7.1) e—P2t'u.(t) It is necessary to check whether the integral term from time domain convolution €“P2tu(t) fot e—(Pl—P‘le {11(52) + 100323)} d1: exists as t —> 00. Let [(15) 2 /0t e"P"e_(P1_P2)“ {11(flx) + 10(13$)}d:r. when t is large, the integrand varies as e’llpl‘P‘Z‘Lfilice‘l’i’t = e-P2(¢-I)e—(5+P1)x for :1: k t, since 1,,(26) z hen P2>0, sincet—1L‘>0and13+P1 >0, 24 ' l-I-l _ llit‘ lfllt‘gldlnl (lo: ,1 “'llt‘ll P: < ll. -o-_ 1 l1 This exllll’fih H.111...“ ii i the integrand decays exponentially. and thus the integral converges between :1: = 0 ~ t. When P2 < 0, the inverse Fourier transform of (2.72) is 2———B"\/D—"e QVD "me—P2t{6(t) l—Dn P2tu(— t)+ )+P«2u(f)} * 2D,,(Bn -— Bu“) (1" Dn)(1+ fl) [—3e"’1‘ {11(31) + 10(13t)} 11(1)] + e"”2‘u(—t) = (Bn‘angl‘m—"e -”1‘{[(3t)+10(3t)}u (t) {_ 2__B:\/D: ESPZ}(_5)€—P2t,u(_t)*e—Pit{11(,3t)+IO(13t)}U(t) + 21),; ("B —Bn+l) e—Pgtu(_t) (l—D )(1+\/—) = (871 — B:+1)\/EI—e—P1t{11(flt)+ 10(Bt)}U(t) 1— Dn +(Bn "an-H) [:71 nl)V neOO—Pgt/ —(P1—P2)r {11(1317)+10(81‘)}dl‘ (1- D71) mar( (,)t0 + 20n(Bn _ Bn+1> (1 - Dn)(1+ x/D_n) e‘Pztu(—t). (2.75) This expression apparently includes non-causal terms in its second and third terms. However, it is shown that these non-causal terms can be removed by reforming the expression, and there are only causal terms remaining in the result. That is, when t<0, .. __ 2 / oo RTE(1) = (3" f“; 1)); D"e-P2‘ e'(P1_P2)x{Il(1317)+Io(/3;1;)}d;z: -' 0 21),, B, — ... _ , +(1 _Bl())(1+\/+ie eP2‘u(—1) (2.76) (1+ Dn)(§n+1— Bu) 2(1— D.) lR€{B}| > 0 to use 6611.4 of [10] given by Let Q = (P1 _ P2) = and check the requirements that Re{a} — °° ..., 3 .1 _ WWW—”l” 277 f0 6 v(, :13) ,1: —— W . (. ) 25 1:018:14 > B" MM)- F": ii lull Brig. < B3 Rt [(11) — li' '( lhtlt’iulf‘. 111 31‘s 1:. ., , '. MK. 11..1:1>.»lvr I 85"; aY‘q'l ’l“ 1AM 111115 EBA [fills Dr > For Bn+1> Bn, - D11 Bn+1 — Bn Bn+1 _ Bn Dn Bn+l _ B71 .- Re{a}—|Re{,3}| 2 (1+ 2&1 D) )—( 2 )2 (1—D ). (2.18) For Bn+1 < B111 1+ Dn)(Bn+1- Bn) + (Bn+1_ Bn) : (Bn+1_ Bn) 2(1—D") 2 1—D.. ' Re{a}-|Re{13}l= ( (2.79) Bn+1>Bn 4: O1 Now, consider the case P2 = B" IPBBHI < 0. Then 1—Dn > 0 <=> B, < Dan+1 < Bn+1 and thus 0 < Dn < 1 ¢# Bn+1 > 8". Also, l—Dn < 0 4:) B,, > Dan+1 > Bn+1 and thus D" > 1 4:) 13,, > Bu“. Using, 0° 1 1'3) 1 f0 (armada: = 3 (—“—, — 1} = fl— — 3, (2.80) ' a2 — 132 {(1+ Dn)2(Bn+1— Bn)2 (Bn+1— Bn)2}% 2 _ , 2 _ a ‘3 4(1- 0,,)2 4 (Bn+1— Bu) V D11 1— Dn and foo e—MI (Br)dr — 1 (2 81) 0 0 ‘ i ‘ a2 _ [32 ' gives RTE“) : (Bn — Bn+1)2Dn\/D;€—P2t (Cl/13) _ 1 + 1 (1 — D")? I 5 0.2 __ [32 0,2 __ [32 26 r- lifietelurc. + 2Dn(Bn -Bn+1) e (1— Dn)(1+ JD?) 2 e—Pgt H 1‘ Dn ((1 + Dn)(Bn+1- B") 2 +1) ‘ (3,.+1 — Bub/TD; 2(1- Dn) (Bn+1—’ Bn) 2 (811 - Bn+1)2Dn\/D—1: 2Dn(Bn - Bn-H) — (BM—1 - 3.1)} x (1 - Du)2 + (1 - Dn)(1+ x/D—n)l = 6 p.1[ 2Dn(Bn+l — Bn)(1- m) + 2Dn(Bn — Bn-H) ] = 0 (I'Dn)(1_\/lTn)(1+\/Eil (l—Dnlu’l‘m) . (2.82) ”flu—t) Therefore, 2.75 consists of ure causal terms, and can be rewritten as when t > 0 7 (Bn _ Bn+1)\/Fn- #56) = ,_ D. {11030 + 10(3)} no (3" _(113:+11)):1):"\/—D—ne—P2tu(t) ft... e-U’I-Pfir {1. (6.1:) + 111327)} dz: z (3" 163N314... {11(31) + 10631)} 11(1) +(Bn —(i3:+g:l))2nme“P2t'u(t) [00° €_(Pl—P2)x{11(1313)+ 10631)} also _ (Bn -(i3:+g:l):”\/D—"6"P2tu(t) fot e—(P1_P2)x {11(,817) + IO(,13;1:)} dz: : (Bn EBnBBJDZG—Plt {11((3t) + 10(13tll U(t) B" — B” 21)?! V Dn ._ . t __ __ . , _( (1 +11)? )2 e Pztu(t)/ 6 (P1 P2)x {11(131‘) + 10(BI)}(11L‘ _ n 0 + 2D,,(Bn+1 — Bn) (1— D.)(1+ «ID—")6 -P2‘u(t) (2.83) 2(1 - Dn) (Bn+1 — Bn)\/b—n was used. Notice Where / e‘lpl‘P2)“: {11(1323) + 10(617)} drr = 0 that (2.83) is exactly same with (2.74). (2) 15"“ < 0 and 13,, < 0 - ZBM—D 1 s—B " TE _ _ n n _ _£t_1 R (3) " 1—D.. s—P2{1 V s-Bn } 27 —' r and when P R1751?! +2\/D s 1_ s — Bn+1 +1— I): 8 — P2 3 "’ Bn 2D.(B. — B.+1) 1 +(1 —D..)(1 + \/D—n) s (2'84) V‘Vhen P; > 0, (2.84) has the inverse Fourier transform (see B.7 as ~ 2Bn\/Dn 2\/ n 1666 = [TIE—epzta—wli—D ”“{6<6 —P-2u<—6} * 2Dn(Bn - Bn+l) (1 — Dn)(1 + {ID—n) [36”1t {11(31) — 10(13t)}u(—t)] — 61264-1) = 21B_‘/1;_/36P”{1 (66 —Io(,3t)} u(— 6 +13{?IB—_%—P12:/;} ePZ’u(—t )* 6P” {11(13t) — IO(,’3t)} u(—t) _ 2Dn(Bn-Bn+1) Pzt, (-t) (1 —D.)(1+f13_)e u ' = (B"+1,_%)‘/P"e ”{I ()6 —Io(66}u (— 6 _ 0 +(Bn+1(1 — 1))Bnl))2:n n\/—D_n eguPt (—t)[ 8(P1—P2)1: {I1(,B.’L') _ 10(18$)}d$ + 20. (B.+1— B.) (1 - Dn)(1+ fl) eP'Ztu(—t) (2.85) and, when P2 < 0, ~ ZBM/Dn 2VD11 RTE“) = [——1-:—D—6P2tu(t) +——-—- 1 _ 6132‘ {5( (1)3” P211(t)} * ,, Plt _ 'U _ _ 2l)11(Bn+1'-Bn) [6e {11(31) 1666} -< 6] (1—D.)(1+./D—,.) (Bn+1_ B n)\/D—n : 1_ Dn 8P1t{11(1’3t)—10(13t)}u(—t) {jig + £132} 66666 * 6”“ {11(31) - 10(1’3t)}u(—t) _ 21)n(Bn-+-1 — B11) (1 — D .)(1 + WT) = (Bn+11:B1)n)\/E;8P1t{11(3t)_ [0(fit)}u(—t) eP2tu(t) ePztu(t) 28 In Comm n and these term 5?)" 011193 \ / Bn _ 2D" n min (t0) _ _( +1(1 _1;n2) VD 8192‘] 6“)1 P2): {11(517) — [0((B$)}d33 2Dn(Bn+1 — B") _(1—Dn)(1+ VD?) emuu). (2.86) In contrast to the Dn+1 > 0, and D. > 0 case, (2.86) has apparent causal terms, and these terms can be converted to pure non-causal terms. When t > 0, (2.86) becomes " Bn _ nVD n _. . RTE“) ___ _( +1(1— B712) 111711): ePgt/O €(P1 P2): {11(51.) _ 10(BI)}dI 2Dn(Bn- Bfl+1) €_Pt + ° , . t . 2.87 ( —D.)( 0+ «if u<) ( ) Letting a: = —-z, the integral term is rewritten as 0 0 / em—lex {11(x336)~10(517)}dz = [8(1) ”I({I ——8z) —-IO( —-,Bz)}dz 0 z _/ e-Z{—11(6z)—10(6z)}d- = _/ e ”’1 P° 61({1 (6(6z)+r0 2.)}dZ 0 (2.88) Then, since t > O RTE“) : ePgt [(Bn+1(;i3n)n))2Dn2n\/—n/OO€_(Pth2)${1103117)+10(513)}d513 + 21971(l3n—Bn+1)) ] (1 — l)71)(1 + E) z 0 (2.89) Consequently, when t < 0, (2.86) becomes RTE“) : (Bu—H " B n)\/m l—D. eP“{I( (6t)— Io(,3t)}u(—t) 29 )3 ' ml) m).)... «' ”11.6% +3.1) '3‘) > ll( l" LSI'H‘J ) ‘1: I If” fr*l(ll 35 . ‘1'. Rut-IE) r j W9 1, (A . Bn+ "‘ Bn 20” D" t ( ,1 ) v eP-gtu(_t)/ 8(P1—P2)1: {1‘5” — IO(,8:L')} dx (1 — D.)2 o. : (B'IEIfgnwzeW {11(32‘) — 1666)} u-(—t) _(Bn+1(;f3rg:l):n\/E6P2tu(_t)[:06(P1—P2)${11(31') _ 1666)} (1.6 (Bn-T—1("1’ _Byg:1)92n\/Eepgtu(_t) [0 6(P1-P2)r {11mm _ 10(31)} (113 : (B”+11:l:5‘3me1)1‘{11(3t)— Io(,3t)} u.(—t) + (B.+1(; €7g:l))2n\/D—rfeP-Ztu(_t) /to 6031—ch {11(513) _ [0(5I)} da: 2Dn(Bn-+-1 - Bn) (1 — Dn)(1+ «TD—")6 P2tu(—t) (2-90) which includes non-causal terms only. (3) Dn+1 > 0 and D. < 0 Using the frequency domain unit step function 0(9) 2 U(jw), (2.61) can be expressed as 6”“(3) = G(s)U(s) + 0*(—s)U(—s) (2.91) where 0(8) : _2]BnVDn 1 {1_ 3+Bn+1} 1+Dn S—Pg s—Bn +2j\/Dn S 1_ S+Bn+1 1+DnS-P2 s-Bn 2Dn(Bn +Bn+1) 1 ”(1 + Dn)(1+j\/‘D:)s — P2' (2'92) If we let g(t) = F"1{G(s)}, then, from (B2), RTEU) = %{g(t) + 9*(t)} + 2%,, * {9(6) — 9*(t)} = 1249(1)} — Wit. * Im{g(t)}. (2.93) 30 Whiz) P1 li'llt’lt *‘T'l , . v When P2 > 0, Bn n ta .1 9m 2_—_j1+\I/)n_e P2 4*) f3(t+) fijflféf ft“) P2U(“t)}€P2 *f3(t) _ an(Bn+Bn+l) Pgtu __ (1+Dn)(1—j\/’DZ)6 ‘( t) (2‘94) where _ 3+Bn+1 f3“) — F1{1 m} B 71 = 8 ‘2— t{11(%‘t)+10(Bz—n t)}U(—t) Bn n 1 Bn B11 ———2+1e-—2*—t{11( 2“ )+Io< 2+1t)}u(t) Bn B77. _11 00 _(Bn 1 8n) B" Bn +—‘“—eBz‘/ e —+2+—I 11 —(t—:1:) +10 —(t—x) 4 ma2:(t,0) 2 2 X {11(83111.) + 10(B;+1$)} d2: (2.95) ~l=°° from (B.10). Now, 2D .(B. + B.+1)€P2, Re{g()}= (1 +13”). u(—6 (2.96) and, Im{g<6} i—fige u(— 3+) f3()+ ff; (t)-qu(-t)}ep""*f3(t) 2D qiDEBmkWH) (297) Finally, for P2 > 0, the transient interfacial reflection coefficient is 6TE<6 = Re{g(6}—;,1-,+Im{g<6} 2D.(B.+B.+1) P, 2(/D. 1 = 2' -—t — - t — (MW e u< ) 1+an3() 7., 31 _2D. ./D.(B .+ B.+1) 6P2tu(——t) * f3(t) +17%; (1( + D .)2 2D". \/D 71+ Bn+1) Pzt 1 (171+ Dn)2 e u( t) >1: 7rt' (2.98) Alternatively, ” 21911 Dn(Bn +Bn 1) 1 TE V + P2 P2 R (t) (1+ D11)2 me tu(—1‘) + {1 — f3(t)} * — * e t ( t) 2\/Dn 1 -1+D. 716M) (299) Similarly, for P2 < 0., 2Dn(Bn + Bn+1) Pt , t = —- 2 2. Re1g< )1 (1+ D"). e u(t) ( 100) and, ’28" —\—_/D n 2V Dn Im{g(t)} 1+ Dn em ”(0 f*3(t) + + anm + P2U(t)} * f3“) 2D an ,,B( n+2Bn+1) Pt 2 .t . 2.1 1 Therefore,the resulting transient interfacial reflection coefficient. for P2 < 0 is given RTEU) = Re{g(t)} — wit * Im{g(t)} _ 2Dn(Bn+Bn-H) P2! 2VD1 — ‘ (1+D.)2 6 33(3) 1+D'.3‘f()*3—3 V811)". ut+<) f*3(t*) +——,— 2D (/ (B .+B.+1) 2. 1 (1+ 071)., GP u(t) * 67—2. (2.102) 32 Almi TV rill. & ll (L l' - H 1 " l..‘m'.; ll Alternatively, _.2D 3/D. B.+B.+1) 1 RTE“) : (1430332) B. P2W)+{1-f3H)}*313*8P2£“(t) D33 -129 3 % 3 f3(t). (2-103) (4) 13.3.1 < 0 and D. > 0 Using frequency domain unit. step function U (an), (2.62) can be expressed as BTE(3) _—. 0(3)U(3) + 0*(—3)U(—3) (2.104) Where 0(3) : _2JB.(/D.' 1 {1+ s—B.+1} 1+Dn 3+P2 S+Bn 2jVDn 3 1+ S—Bn+1 1+DnS+P2 3+Bn 2D.(B. + 8.3.1) 1 +(1+ 03m 4317;) 3 + P3' (2105) If we let g(t) = F‘1{G(s)}, then, from (B.2), BT33) = -;-{g(t)+g (6} +—- +{3— 3*t< )} = Re{g(t)} —- wit * Im{g(t)}. (2.106) When P2 > 0, 3(6 = 93—18%} 333( )+ 33(3) — ijflwm — mane-W + f3(t) 2D“ (B. + 3.3.1) e‘P2tu(t) (2.107) +(1+Dn)(1 43/13:) 33 where _ S—Bl t. = F1 1 ____"_i_. f4” {+V S+Bn} B33 = 521 +W{h( +13)_DH%“3)}DD6 2 2 2 B” e_£2ut B711 B” , (Bn—HBn) glgilt / e_(371+12+311)r {11 [B7f+1(t — 1.)] _ IO [B7+1(t — 1‘)]} mar(t.0) 2 2 B33 Bn X {11(71') — 10(71)} dl‘ (2.108) from (B.11). Now, :122D (811 + Bn+1) —P t 2 t 2.1 and, __,,2B __\__/D ”e _ 2\/D Im{9(tl} 1+Dn Pzt Lit—()*f4(t) —_—7L6({1+Dnt_) P2 W )}e Pzt *f4(t) 2D,l \/D_n(B n+Bn+1) —P-t 2 , l‘, . 2.110 + (1+ Dn)2 e 11( ) ( ) Thus, for P2 > 0, the transient interfacial reflection coefficient is BT33) = Re{g(t)}—%*1m{g(t)} 2D.(B.+B.+1) _ 2, 2f“ 1 (1+D.)2 Pum+ +n1+1l),.f‘lt’"‘)333 +2D n\/—_(B n+Bn+1) — 2t 1 (1+Dn2) e Pu(t)*f4(t)*fi _2D n\/D—n(B n+Bn+1) — 2t, i (1 +0112) 6 P 33(3) + 333' (2.111) 34 Alternatively, “ 21971 V Dn(Bn + B11 ) 1 _ 2 1 _ RTEU) : (1+D)'2 +1 \/D_e Ptu W+{f1)—1}*E*€P2t (t) 2\/Dn 1 1+Dn*7rt*f4() (2112) Similarly, for P2 < 0, 21311(871 + Bn+l)e—P2t Re{g(t)} = — (1+ Dn)2 u(—t) (2.113) and, Im{g(t>} = L—ngfe u—( )*f4(t)-12g{6(t)+qu(—t)}*f4(t) _n2D \/—(B n+Bn—H) — 2t, _ / (1 +0102 6 P u( t). (2.114) Therefore, the resulting transient interfacial reflection coefficient for P2 < 0 is given by Wot) ——— R6{g(t)}-7Tit*1m{9(t)} _ 2Dn(Bn+Bn—+-1) _ 2t 2m 1- 1 “ ‘ (1+Dn2) e P “(_t)+1+Dnt ”(2* m _712D \/—(B n+2Bn+1) -— 2tu1 2Dn \/D_n(B nn+Bn+l) _. 21; 1 .. (1+Dn)2 e P u(—t) =1: E? (2.110) Alternatively, ~T_21)n\/D—n(Bn+25)n+1)l___1___ — gut sarong,” u(—t)D::2+]“_ 131—affirm). (2.116) 35 h the 1 Lau>c1 causal CJKFG]; .3 VV . . [du‘fil 7| r V“ L‘\7l‘\-\‘(. “‘J “ Uni“ E" 0r. {1: ,‘IV M19 r~4 -51: 2.3.2 Causality In the transient forms for the cases of (2), (3) and (4), there are non-causal terms involving u(-t) or %. These are unexpected results, because the convolution of any causal (thus, physically realizable) input waveform with these terms will produce non- causal transient reflected waveform components that appears to disobey Einstein’s causality, which says that nothing can travel faster than the speed of light. Also, it should be noted that this is ‘pure’ non-causality which is different from the non- causality caused by the diffusion approximation referred in [11]. The existence of the non-causal terms depends on the sign of Dn+1 and [3". That is, the non-causality happens when the sign of either one of the two parameters is negative. When the sign is negative, the real part of 73 is always negative by the branch cut; therefore the wave propagates in an evanescent mode, regardless of the existence of conductivity. A good example of this case is total internal reflection at the interface of lossless dielectric material layers. The conventional geometrical ray optics approach can not explain this non- causality. Consider the geometry shown in Figure 2.3, where total internal reflection of an impulse incident plane wave is assumed to occur at the interface between two lossless dielectric layers. The reflected wave packet from the point Q1 arrives at the observation plane at t = 0, at the same moment as the reflection of the incident wave from Q2, because their propagation velocities are the same in this same region, and the lengths of travel along paths P1 and P2 are identical. Consider a surface wave in region 2, which starts at Q1 and propagates along the z-axis with propagation velocity equal to or less then 212,, = fi = —w—' = 9" . Then, the travel time of ’72z ”Viz ’71 sm 9,1 the surface wave is t > i = d 1 8m 6“ = P2 = ——7P2 - Therefore, the d — U22 w SlIl Bil—w (LU/’71 71 SlIl 6n Surface wave arrives at Q2 at the same or later moment at which the ineldent wave impinges on this point. 36 Region 1 Region 2 ; Observation plane I = 0 Wavefront 5 (t) Reflected wave Surface wave Q l Incident wave Wavefront 5 (t - rd) Figure 2.3. The time relationship of wavefronts. 37 ’17», , J—l $5“.‘- Q. y'Hj “‘- 6.‘ ‘9) Next, consider the possible existence of a precursor effect. Any wave front of the spherical wave excited at Q1 that arrives at a point on the observation plane faster than the wave reflected from Q2 might be a precursor and thus a source of the non-causality. From Figure 2.4, consider path 1 with 11 = \/h.2+l§ = 2 \/(d+17)2 + (W) and path 2 with 12 +13 : dsin 6,1 + 371%? To be a pre- 2 cursor, path 1 must be less than path 2. However 2 2 2 _ 2 — 2 :1: " . ' 117 11 (l2 "l" [3) — {(d + x) + (tartan) } (dsm 611 + tangil) 1 1 > — (12 sin2 0,1 tan.20,-1 sin2 (9,1 : (12+1r2+.r2( = c12cos2 02-1 > 0 (2.117) and thus, there is no precursor. In this study, a reasonable explanation of the non-causality has not been found. The non-causality might originate from the impractical assumption of infinite layer width because the incident plane wave interacts with the interface an infinite amount time ago, i.e. as far back as t 2 —oo, although the relationship between this as- sumption and non-causality could not be found using ordinary ray optics approach as explained above. There are several interesting ongoing research activities that might help explain this phenonmenon. In physics, there have been some trials to measure or compute the exact tunneling time of a particle through a potential barrier, which is important in modern microelectronic tunneling devices [12], [13]. It turns out that the tunneling of a particle (or a wavepacket) is quite similar to the transmission of an evanescent electromagnetic wave, e.g. the propagation of a wave having a frequency less than the cut-off frequency in a waveguide [14]. Therefore, there were many experiments performed by using Optical pulses (e.g. [15] and [16]), or microwave propagation in a 38 + 2 Region 1 Region 2 O O O O V Wavefront 5(1) ; Observation plane t=t0 Wavefront 5 (t — td) h=d+x Incident wave Figure 2.4. The consideration of the existence of a precursor. 39 nnmflPWA hihwon anMwhm. nunawmr minal" pan-:1: lll’lt [U betting l . anhnttv: p. havmraba llr per! ‘15.. {anwlmrx dlléllll'dllull of L‘ limit a1 m qeannahm tuehis SUI. hshnll .‘ [T _ .‘ u . HJALll-‘H'Jn l)! Impulse IPV . ,. Flt-1m 19 can. HE'S “PIE ll‘ [. waveguide (e.g. [17]). In these experiments, it has been commonly observed that the pulse peak prop- agation velocities seem to be independent of the thickness of a potential barrier (or an evanescent region, equivalently), which consequently implies so—called “super lu- minal” propagation velocity of the wave modes. For example, if the width of region were to become infinite, then the speed of wave propagation would increase infinitely, therefore faster than the speed of light, which leads to apparent contradiction of the Einstein causality. Although this phenomenon had been theoretically predicted in the past [18], the exact theoretical explanation has yet been given. Many have at- tempted to explain this phenomenon as a reshaping of the pulse [15], [16]. That is, attenuation of the pulse in the barrier shifts the peak of the pulse forward. But little is known about why the barriers attenuate pulses unevenly [19]. Also, there is a question about the validity of ‘peak’ measurement methodology in dispersive media, since it is sometimes hard to define the pulse peak in that case. It should be noted that there is some possiblity that the “super luminal phe- nomenon” of those experiments might be different from the non-causal tail in the impulse response waveform described in this study. In this study the pulse peak point is causal (located at the time origin of t = 0), while the “super luminal” veloci- ties were measured only using the location of the pulse peak, and did not consider the shape of the waveform since very short optical pulse waveforms were used in the past experiments. Neverthless, there might be a relationship between the oberved results and the non-causality in the transient interfacial reflection coefficients, considering the fact that both happen in the same situation, i.e. only in evanescent propagation mode. As a result, only the causal case will be considered in the remainder of this dis- sertation, simply because there is no reliable explanation for the non-causality. The non-causal case can be studied in separate research. Note that the non-causal case is 40 unlikely to on parameter Du - lle first layer - beam it hi» i. sideiing Ilia! >iu‘ coalition of 2 . l 2.4 Numeric: 2.4.1 l’erifice The preriuusly command to iii: The most rim-.1 last Fourier Ir; Itemir‘ined earl: Since the a: in the time dui interfacial reflv comparison of 1. forms obtained repent-9],: p unlikely to occur in a practical situation. That is, to be a non-causal case, the layer parameter D" : (,unen — #161 sin2 6,1)//1.,2, must be negative, and thus #11 61 . —— < —sm26,-1. (2.118) #1 6n The first layer can be any lossless dielectric, but is typically free space in practice because it has to have infinite depth and width. Then, 61 = 60 and #1 = #0- Con- sidering that sin2 6,1 < l and 60 < 6,, always, it can be recognized that satisfying the condition of (2.118) is improbable. 2.4 Numerical Examples 2.4.1 Verification of theoretical expressions The previously derived transient forms for interfacial reflection coefficients may be compared to direct numerical computation to ensure that the transform are correct. The most common method to do this is simply to compute the numerical inverse Fast Fourier transforms (IFFT) from the frequency domain forms. For the reasons mentioned earlier, only the causal case (Dn+1 > 0 and D” > 0) will be discussed. Since the asymptotic interfacial reflection coeflicients appear as delta functions in the time domain, and are thus difficult to express numerically, only the reduced interfacial reflection coefficients are compared. Figure 2.5 (a) and (b) shows the comparison of the derived transient interfacial reflection coefficients with the transient forms obtained by using a 3072 point IFFT, for the P2 > 0 and P2 < 0 cases, respectively. The material parameters for Figure 2.5 (a) are those for an interface of free-space and typical sea water at low frequency as described in [20], while those for Figure 2.5 (b) are for an interface of plexiglass and ethyl alcohol 10GHz, as described in [21]. Note that for the non-magnetic materials, P2 < 0 case happens when 6,, > 6,,“ and (on+1/o,,) > (om/[9,) > 1, or 6,, < 6,,... and (aw/0,,) < (n+1/[3,.) < 1. 41 lht llhl‘.>lt"lti '. iltutitul. urul t i Mi polariz. lo Mil)” tl t‘utflilt'ft’lli. llit an input trun- Where Km ,- ‘iltfillal pr 1],, where K. a the input it figure 2.? input Wall“ and P; < l. well match .ls nit-r pretiuus rt Wat; t Mm. The transient reflection coefficient curves from the two different methods are nearly identical, and thus correctness of the derived transient interfacial reflection coefficients for TE polarization have been verified. To verify the combination of the asymptotic and the reduced transient reflection coefficient, the time domain convolution of the interfacial reflection coefficient with an input waveform is needed, i.e. S(t) = X(t)*RTE(t) = X(t) at {R;E5(t) + RTE(1)} RQEXU) + Emu) * X(t) (2.119) where X(t) denotes the input waveform and S(t) is the resulting reflected waveform. One of the most commonly used input waveforms for this purpose is a double expo- nential pulse waveform, given as X(t) = K (e—at — e-b‘) (2.120) where K, a and b are arbitrary positive constants. Figure 2.6 shows an example of the input waveform where K = 30.71, a = 2 x 109 and b = 4 x 109. Figure 2.7 (a) and Figure 2.7 (b) shows the comparison of the time domain reflected waveforms with the input waveform shown in Figure 2.6 for the two different layer parameter sets (P2 > 0 and P2 < 0). Again, in both cases, the derived expressions and the IFF T results are well matched. As mentioned in chapter 1, exact theoretical expressions are available from the previous research [11] for transient interfactial reflection due to a unit step input waveform. A comparison with those results also helps to verify the work in this study. Figure 2.8 (a) and Figure 2.8 (b) shows that the re—produced results of the 42 11108 i -1] ll [little ..- 4 .._.. alt”..- ...... 1.I . uCO-Ucbnvnvnv Chloe's—flun- -'-0I.-L°u:- awn-033's"- x10 1 I I I I I I I I I o— ———————— — ——————— — 4 E -1- ~ 0 3.3 3: _2_ . _ 3 —Derlved 0 l "lFFT .5. -3. - t = -4 - o a: i -5- _ o a t I 2 -6~ - E 1; — 3 8 a: -8- . _9— . _10 J l l l l l I 1 O 1 3 4 5 6 7 8 9 1O Time[nsec] Figure 2.5. (a) Numerical comparison of the derived transient reduced interfacial reflection coefficient with that from the IF F T (TE polarization) : ,un : no, 6,, = 60, 0n = ORB/771], [[41+] 2 H016n+1 =3 7260,0n+1= 4[U/m], 011 = 30°, P2 = 0.64 X 1010. 43 80 ‘1‘ VA l.lllllll 1| ll 1 h... P l u t. I I'I‘l‘éll l1 1--. lliLilllLi l7 9 . ii. 123.. 4 :w t . aw . 6 Ito-O-fihfloo cO-UO‘uhOE nunnV'hL'UP-u 3‘03“ m ~10 x10 1 I I I I I I I I I Or — — 5 -1- ~ E t -2. . o —Derived g --IFFT 2 '3' ‘ 7% 1 '3 -4 ~ 0: '3 L - .5 -5 (I E z ‘6 . 'U 8 -7 - 3 D 0 I! Time [nsec] Figure 2.5. (b) Numerical comparison of the derived transient reduced interfacial re- flection coefficient with that from the IF FT (TE polarization) : an = #0, 6,, = 2.5960, 0'" = 9.73X 10"3[U/m], ”n+1 1‘— flo, €n+l 21.7060, on+1 Z 5.60x10"2[U/m], 611 = 30°, P2 :2 —O.59 x 1010. 44 0\ Figllrg ll] rill I hiilllillllh] [h :Fllullll H l] .. ii. I h p 0 9 8 7 6 5 4 3 2 1 0 l HE\>H ELOh'>'g Uni-Alp... Input Waveform [Vlm] 10 Time [nsec] Figure 2.6. An example of double exponential input waveform. 10 Amplitude of Reflected W-v- [Vim] f’gwe 27 . .I' . ' " "I for the i! r I \ ’ 1') Ln \ “13‘ ET \ P - 1 ~ r I) . I ~‘ .f . ..., A" ll)‘ o T I I— I I -1- _ -2» - 'g‘ —Derived ; -- IFFT o '3 “ > O z .41 , .9 8 = -5- _ o I! ‘6 - _ e -6 'u 3 i -7- ~ E < -3- — _91. _ _10 L J l l 1 l 1 l 1 0 1 2 3 4 5 6 7 8 9 10 Time [nsec] Figure 2.7. (a) Numerical comparison of the transient reflected electric field wave- form for the input waveform shown in Figure 2.6 with the IF FT (TE polarization) : 11.. = #0, en = 60. 0n = OlU/m], an“ = #0, 6n+1 = 7260, 0n+1 = 4[U/m], 0.1 = 30°, P2 2 0.64 X 1010. 46 .l. . ...... _ ..1Il11. ls..1]|11.1.1 _ ] .1._11ll l1. 2 n m ~11 f. .J . l. ...m . M . 6 Vi ilu l 9 mu 1m m. t 0 n1 29. .a. It‘ll hd 7 F W imrmw .14“ 1P Illt _ 11]] 1 2 - - -111 0 11 ill 1 I m u o HE\>H fl)“; .9 aunt-h h H.- MT- 1 I T I I I l I I I g — Derived -' - - IFFT o .. > a 3 ‘0 _. a a O 2 t .4- - 1: Q- o a _ .— —. u 5 3 .2 Ta. _5. _ E < _7- _ -3— _ _9 l I l 1 l l l l 1 Time [nsec] Figure 2.7. (b) Numerical comparison of the transient reflected electric field waveform for the input waveform shown in Figure 2.6 with the IF F T (TE po- larization) : an = #0, 6n = 2.5960, 0,, = 9.73 x10‘3[U/m], 11,,“ = 110, €n+1 = 1.7060,0’n+1 = 5.60 X 10'2[U/m], 6” = 30°, P2 = —0.59 X 1010. 47 ehtiflt lllit‘fliil' 511111“ a draurtr li ta With 11'] teeth-i131 i~ ‘ . girater El Diff 1“ ‘ “ll "in. 1).)! diff r- MUM], “hit“; 1 i] ' [mi electric fields to a unit step input waveform excitation obtained by using the transient interfacial reflection coefficient and transmission coefficient of this study, are exactly same as those found in Figure 4.5 and Figure 4.4 in [11]. Note that these plots are drawn with the nomalized time axis given by 0n+lt e sinfii2 ' 25H+1 (1- ELLA) #n+1€n+1 at: It can be easily realized that the early time portion of the scattered electrical field waveform is dominated by the high frequency component of the interfacial reflection coefficient, i.e. the asymptotic interfacial reflection coefficient. The late time portion is due to the lower frequency components of the interfacial reflection coeflicient. 2.4.2 The transient responses for various parameter sets It is interesting to study the dependence of the transient reflection ceofficient on the various parameter values, i.e. permittivity, permeability, conductivity and aspect angle. Comparisons of the reduced reflection coefficients are shown in Figure 2.9 - Figure 2.12 which are plotted using semilog scales for easier identification. Note that the first two curves in each plot, except Figure 2.12, are for P2 < 0 cases, so that the effects of parameter change can be observed for both cases of P2 > 0 and P2 < O. (1) Permittivity The curves in Figure 2.9 are plotted for em“ = 1.7, 2.5, 10 and 72. The corre- sponding values of RQE are 0.12, 0.0098, -0.34 and -0.69, respectively. Note that the phase reversal of the asymptotic reflection coefficient happens when en+1 becomes greater than 6,,. It can be seen that the slopes of different RTE (t) curves are inversely proportional to the increasing permittivity values, while the asymptotic reflections are proportional to the increasing permittivity values except the case em“ = 2.5, where the material constants of the two layers approach each other, and therefore 48 l‘ ‘1... t‘ E .I I I I 11w A. d mu. 4. HE\>H a.-.“ 0.1.800.” 5° ouaflnnflhEl‘ Figure 2_ 9:121"; f a ' 9::0: Amplitude of Electrlc Field [Vlrn] I I _l l l l l l l l l l 10 20 30 40 50 60 70 at Figure 2.8. (a) The transient reflected electrical field waveform for a unit step excitation: 11,, = 110, 6,, = 60, on = 0,11,,“ = 110,13,“ 2 960, on+1 : 1.00X10'3[U/m], 5:00. 49 -kryiblifl if]? 1|] 1? l h It 1 1 l J. .0 D 1 1! HE\>H av.-.“- O-Luou-m ho Ava-udu-un-Ed‘ l .8_ I:lgllre 2 1l()n:# ‘) J .02. \ .‘x Amplltude of Electrlc Field [V/m] -2 i i 3 3 s : 10 :iiiiiiiiiiii:i:i:::i.::::31::iiiiiiiiiiiiiiiiiii:iiiIii:iiiiiiiiiii:i7:33;:iiiiiiiiiiiiiii ............... W .................................................................................................. J r... ................................................................................................ 10'3 1 1 1 1 1 1 0 10 20 30 40 50 50 70 at Figure 2.8. (b) The transient transmitted electrical field waveform for a unit step excitation: 11.. = #01611 = 60. 0n = 0.11,.“ = 110.67.“ = 960. 0n+1 = 1.00x10‘3lU/m], 9i = 30°. 50 the overall it int‘reaws who [3: Perruval.. The ('UII'i’S. it. ing values of I are seen to lu- aSjt'rnptutit' 11: fore. the 8111111.: [31' CUIlIlllt_‘{]\' Baause an i111 : sient response: 1 component is in where 0,, : 0.02 (l; Aspect an: The curves at )1)“ corresponding: 1': slope of the mi: component int-r1 dependent on it: 2-5 Approxir. It may be useful 4.1mm C'Oeifit'it-h _ H idllrJll Of {{3de ] '-,( the overall reflection itself is weak. This means that the overall transient reflection increases when the permittivity contrast of both layers is sharper. (2) Permeability The curves in Figure 2.10 are plotted for 11”,“ = 1, 5, 10 and 75. The correspond- ing values of RICE are 0.027, 0.38, 0.51 and 0.79 respectively. The slopes of RTE (t) are seen to be inversely proportional to the increasing permeability values, while the asymptotic reflections are proportional to the increasing permeability values. There- fore, the amount of overall reflection increases with the increasing permeability. (3) Conductivity Because an increase in conductivity causes attenuation in wave propagation, the tran- sient response RTE“) suffers a more rapid decrease with time, while the asymptotic component is independent of the change. This can be seen explicitly in Figure 2.11 where on = 0.01, 0.03, 4 and 10, respectively. (4) Aspect angle The curves shown in Figure 2.12 are plotted for 6,1 2 0°, 30°, 45° and 75°, while the corresponding values of R3,? are 0.11, 0.12, 0.14 and 0.19 respectively. Therefore, the SIOpe of the reduced reflection coeflicient as well as the amplitude of the asymptotic component increases with increasing aspect angle, and the reflection becomes more dependent on its early time portion comming from the aymptotic reflection term. 2.5 Approximation of Interfacial Reflection Coefficients It may be useful to obtain approximate forms of the previously derived transient re- flection coefficients, as long as accuracy is maintained. Because the numerical compu- tation of repeated time domain convolution integrals (needed for obtaining multi-layer scattering as discussed in chapter 4) requires a significant amount of time, the total computation time could be burdensome in real time applications. Also, the process of finding the approximate forms gives some insight into the roles of previously defined 51 o ‘ CI I ‘ 1 110 50 50 1 1 ‘ \ \ - Nfl]. 1 55.51 fif 0.0 .5an fift'qu-hh-°o rho-30°...-“ uflnflv't‘Uf-n n-03n-m 5° -‘U-JflhufllEl.‘ olllPlll tb ifllh] l 15]] ~>1blb11>1bllbll1blllbll1 .1 LPLLifilPl-FIlLlllP] 1 ‘ 0 Amplitude of Reduced Interfacial Reflection Coefficient 10 T I I I I I I i i . \ . . \ \ 8 — epsr=1.7 10 g - - epsr=2.5 3 ;. 1-1- epsr=10 t epsr=72 _ \.‘.‘.. ......"."Oo... "...”... 4 107? .‘h‘K. "“N‘ 106:“ ‘: E °~.. i l TN". . N"; 105; 104 l l l l l l 0 1 2 3 4 S 8 9 10 Time [nsec] Figure 2.9. Time domain reduced interfacial reflection coefficients for various values of permittivity (TE polarization) : 11,, 2 an“ 2 #0: 6,, = 2.5960, on = 9.73 x 10‘“, on“ = 5.60 x 10‘2[U/m], 6,1 = 30°. 10 fi I I I I I i I a u .9 Z .2 [3, ‘3 ° 108 ~ - U E .\ — mur=1 3 g .\ - - mur=5 3 ‘6 1 n.5, '--- mur=10 % L ...}0% 11111 mur=75 1: 107 - T“ - a E ‘1‘ = '5 ‘2“... l6 \‘o t \‘\\ 2 ‘ W .E ‘ ‘ a U 6 1- “3°... _ 8 10 : \ \ {Q-ih. I 1. “o.'o. : 3 ‘ ' "u. 4 8 ‘ 'o". ‘ 1r ‘ ~ s': I- \. o 0 5 ‘U 10 r :3 : :1 'o. E < . 104 I l I l 1 0 3 5 6 8 9 10 Time [nsec] Figure 2.10. Time domain reduced interfacial reflection coefficients for various values of permeability (TE polarization) : an 2 1.5110, 6,, = 2.5960, 0,, = 9.73 x 10‘3, 6n+1 = 1.7060, 0n+1 = 5.60 X 10_2[U/m], 0,1 = 30°. 53 Amplitude of Reduced Interfacial Reflection Coefficient a. 10 _ — Cnd.=0.01[mho/m] ""115 . - - end,=o.03[mholm] "a. 1 -- ~ Cnd.=4[mholm] MW». _ Cnd.=10 mholm 1010, l l "“1. ‘ fan “’45.; 10-15 1 l l l 1 l l l 1 Time [nsec] Figure 2.11. Time domain reduced interfacial reflection coefficients for various values of conductivity (TE polarization) : 11,, = ,u.,,+1 : #0, 6n = 2.5960, en+1 = 1.7050, Un+1= 5.60 X 10_2[U/m], 6n = 30°. 54 10 10 1 1 1 I 1 1 1 r 1 — Angle=0 deg. - - Angle=30 deg. -- 1- Angle=45 deg. Angle=75 deg. 3 d o r T 'TTWYY 10 'V'I 10 Amplitude of Reduced Interfacial Reflection Coefficient —L o Time [nsec] Figure 2.12. Time domain reduced interfacial reflection coefficients for various values of incident angle (TE polarization) : an = 1.5110, 6,, = 2.5960, 0,, = 9.73 x 10‘3, on“ = #01 CHI = 1.7060, on+1 = 5.60 x 10‘2[U/m]. 55 constants in the transient interfacial reflection coefficient expressions. In this section, therefore, the approximate forms for the transient interfacial reflection coefficients (Dn+1 > 0 and Dn > 0 case) are derived, and the accuracy of the approximation method is discussed. There are two different approximate forms since the Dn+1 > 0 and [9,, > 0 case has two different transient reflection coeflicients according to the sign of the constant P2. For each transient reflection coefficient, The approximations for large values of the time variable t and small values of the time variable t are derived. Then they are combined by using a pair of weighting functions to produce a smooth curve. For convenience, the simpler reflection coefficient (for P2 < 0) will be discussed first. 2.5.1 Case I : P2 < 0 (1) Large t approximation For simpler notation, let’s define the constants C _ (Bn _' Bra-+1) V Dn 1 _ (1_ Dn) (Bn _ Bn+l)2Dn V D11 02 2 (1" Dn) 2D,,(Bn+1 — B”) C : 2.121 3 (l—Dn)(1+ VDn) ( ) and a function Q(t) 2 cf” {11031) + 10031)}, (2122) then (2.83) is rewritten as RTEU) : C1Q(t)u(t) + C2u(t) /00 e"P2(t—I)Q(.r)d;r. (2.123) 1 56 Using the change of variables 11 = P2(t — :r), then :1: = —]15‘; + t, and dz = —C}l§i. 2 Therefore :1: = 00 leads to u 2 oo, 2 = t leads to u = 0 and (2.123) becomes RTE(t) = C1Q(t)u(t)—C2—I%u(t)/O e_”Q(—%+t)du = C1Q(t)'u(t)- gum / 6‘"{Q(t- %)-Q(t)]du 02 0° _, . +EQ(t)u(t)/O e (111 (2.124) 22 —%2Q'(t) [000 ue—“du. (2.125) This approximation is possible since for large P2, most of the contribution to the integration occurs when u is small due to 6‘“ term. Also, fooo ire—“din = 1. Finally, for large t, (Bu _ Bn+1)2DnV Dn (Bn '— IDan-H)2 ~TE ~ (B71 _ Bn+l)Bn\/1D_n R (t) ~ (Bn _ Dan+1) Q(t) + Bn—Bn+ B71 D71 —t ‘ ( (B —D1)B +\./)_° P1 {11(flt)+10(13t)}u(tl ‘ (B. — Bn+1)°an/IT. _p.. { 1 }u(t). Q'(t) (Bn _ Dan+1l2 8 (BR + #110315) + Bnlomt) (2.126) (2) Small t approximation Recall that (2.83) originates from (2.75), then RTE(t) = Clo—P“ {11(13t) + 100610} u(t) — C3e_P2tu(—t) —Cge"P2tu(—t) * 67PM {11(fit) + IO(,Bt)} u(t) (2.127) 57 For [P2] >> |P1|, the e‘P'Z‘u(—t) term behaves like a 6—function by its peaking prop- erty. Therefore, RTEU) z Clea—P” {1.031) + 10(80} u(t) — Cge—P” {11(1‘325) + 10031)} u(t) + 0 a: (C1 — C2)e‘P1‘{0 -+- 1} u(t) = Ae_P“u.(t) (2.128) for small t, where A is a constant which will be determined later. FOFIP2]%]P1],—P2Q$P1 and fame) _—. Cle_P1t{Il(/3t)+Io(r’3t)}u(t) —C26+Pltu(—t) * e-Plt {1.031) + 10030} 11(1) + CgeP‘tu(—t) 22 Cle_P1t{0 + l} u(t) — C26P1‘u(—t)* e713” {0 + 1} ’u.(t) + CgeP‘tu(—t) C2 C167P“u(t) — 2P 1 6P” 'l" C3€P1tu(—t) 22 for small t. However, it has already been shown that the exact form does not have any non-causal term. Therefore, RTE“) % Cle_Pltu(t) — g—epltufi) z fie—Pita“) (2.129) 1 For [P2] << [P1], the e‘Pltu(t) term has the role of the 6—function, and fame) z Ole-131%“)+C26_P2tu(—t) +o3e-P2‘u(—1) z AC3e'P‘tu(t) (2.130) where the second and third terms must be discarded to maintain causality. In a conclusion, the small t approximation for the P2 < 0 case is given by RTEU) z Ae‘P‘tu(t). (2.131) 58 til (3) Combined approximation From the large and small t approximations derived above, a direct combination is (Bn _ Bn+1)BnV Dn 123.51....1) = (B_ _B an+1l e—Pltiliwtl'l'lowtllult)— (B(B’_"B‘)Bf:1‘(,Ee—Pl {(8 + %)11(1’3tl"'B 10161)]1111) +Ae-P1‘u(t). (2.132) The most intuitive way to determine the constant A is using the function value of the reduced reflection coefficient at t— — 0, i. e. by letting Rappmgt) i=0 i V i=0 and ~ B, — )B, ./D—,, RTE t ___ ( Bn+l 1 _ (BB: B”; ..)B1:..)/D_ {(0 + 2) + 3"} + A (811 — Bn+lan V Dn = A+ (Bn- .1) an+1) _(B — B...) D «19—. (B + 3B ) (2 133) (Bu— _D an+1)2 4 . RTE (t) _ (BR _ Bn+1) V Dn _ (B71 _ Bn+l)2DnV Dn X 0 exact '— t:0 (1 — Dn) (1 — Dn)2 + 2Dn(13n+l — Bn) (1'— 1311“1 + V Bu) (3" _ 8"“) V D". (2134) (1+ {197)2 To evaluate A, equate (2.133) and (2.134) to give (811 -' Bn+1)\/—1)—n _ (Bn - Bn+1)Bn\/—l)—n (1+ my _ (Bu - Dan+1) 59 _(Bn — Bn+1l )21‘1D VD n (Bn+1+38n) +A (B —D NB...) 4 ‘ So, ‘4 _ (B11 _ Bn+1) V Dn (Bn —- B71+1)Bn‘ /Dn (1 'l' VD )2 ((Bn - D an+l) (Bn — Bn+1) )2nD VD Bn+l + 3Bn) (2135) (En _ Dan+1)24 . Therefore, for P2 < 0, an approximate form is given by (BE);— 33055.? ‘P*‘111+lo<fit>}“ (>— (B. — B..1)2D,,,/E; _B, 1 , , (Ba " Dan+1)2 e {(871 + Z>Il (3t) + B"IO(’3t)} “'(t) {(Bn — Bn+l)\/1)_n _ (Bn — Bn+1)Bn\/D—r: BTEB) z (1+ J17")? (Bn — D an+1) (BB—Bn+1)2Dn\/2D_(Bn+1+3Bn) —P1t Let’s consider the accuracy of the approximate form (2.136). Actually the large t approximation is exact except for the term in (2.125), which is valid when t > u/ |P2| (or |P2|t >> a) is satisfied. Note that the constant power P1 is not involved with the large t approximation. For the small approximation, of course, the 6—function approximation of exponential term contributes to the error, especially when the ratio |P1|/|P2| is not sufficiently big or small. However, the selection of the constant A = BE a) small t approx(t) t—O exact using RTE (t) — ——RTE 0 is t: approx t=0 +RTE large t approx“) t:0 the main source, because the function values from the large t approximation, which should be excluded in the small t interval (including t = 0), is imposed on the early time period. Therefore, the constant A should be given by R73 (1%)] = exact i=0 R75 (t) ("B 3"“) 5’2 ”o.nly However this choice of 4 has trouble small t a rox PP t:0— (1+ /D1))n 60 as well, since it will introduce a discontinuity at the junction of the small t and large t approximations. Therefore, in this study, a pair of weighting functions are introduced to avoid the discontinuity. From observing the derivation process the approximate transient reflection coefficient, it can be realized that the constant P2 behaves like a ‘switch’ which turns ‘on’ and ‘off’ the validity of the large t approximation, as a power factor of an exponential function. Also, note that P2 does not appear in the small t approximaion. In addition to these, the product Pzt provides a normalized time axis. As a result of these observations, a pair of weighting funcions is selected to combine the large and small t approximation functions as In“) 2 (l—e'a'IP'ZIt) W5(t) = ("41’2" (2.137) In this definition, l/VL(t) is a weighting function for the large t approximation, while W S(t) is for the small t approximation. The constant transition factor a determines the transition time from small t approximation to large t approximation. That is, the bigger value of a means an earlier transition from the small t to the large t approximation. Although the choice of a has some flexibility, in this study, the selection method is as follows. First, set the value of reflection coeflicient at the ‘transition’ time as the exact transition reflection coefficient which is given by, for example, 0.5 Egan) t_0. Next find the time corresponding to the function value, say, 25mm. Then a is obtained by letting 6‘0'P2lttm’” = 0.5. In summary, the complete approximation for P2 < 0 is given by Bn _ Bn+1)BnV Dn (Bn _ Dan+l) (Bu _ Bn+1)2DnV Due—Flt (871 _ anBn-H)2 6’” {11(515) + 10(fit)} W) - RTE“) z WL(t) [( {(3. + bum) + 3.10m} um] 61 (Bn - Bn+1)V Due-Pit (1+\/T)_)2 2five—”1311(3) (1D+\/ :2) ' flsz Bn _ lg ”L()P2<1—D>l’3_e P 4D _pt —— 1 B +an —D) {( + + I’VS (t) u.(t) = W's (t) {11030 + 10030} 1>11(3t>+3,.10(3t)}]m<>. (2.138) A example of the transient reduced interfacial reflection coefficient for this case is plotted along the normalized time axis of a|P2|t in Figure 2.13. It can be observed that the largest error occurs in the transition region as expected. Small t approximation shows relatively poorer performance than large t approximation, since it relies on one exponential term only. Also, it is observed that the large t approximation is well matched with the exact reflection coefficient curve in large t interval. 2.5.2 Case II : P2 > 0 (1) Large t approximation Using previous definitions for the constants and the function Q(t), (2.74) is rewritten as RTEU) _—_ C1Q(t)u(t)—C2u(t)/te—P2(t_$)Q(x)dx+C36_P2tu(t). (2.139) Letu=P2(x—t),thenx=Piz+tandd$=%‘:. Also,x=tforu=0andx=0for u = —P2t. Thus (2.139) becomes RTE(t) = C1Q(t)u(t)—C2u(t)/ e"Q(:1: )da:+C3e P“ u(t) = ClQ(t)u(t) — 913—:u(t)/_Pte "Q(t + P—2)du+Cge P2‘u(t) — 0162(333— $333) [Pen {623 + g) — 3(3)} 33 62 — Exact - - Approximate - '-~ Largetapprox. ''''' Small t approx. Amplitude l l l l l l l l l 2 4 6 8 10 12 14 16 18 20 Normalized Time a|P2|t Figure 2.13. Transient interfacial reflection coeflicient for P2 < 0 (T E polariza- tion) : an 2 #0, 6,, = 2.5960, 0,, = 9.73 x 10‘3[U/m], an“ 2 p0, en+1 :2 1.760, 0n+1 =2 5.60 X 10‘2[U/m], 92-1 2 30°, P2 = —0.59 X 1010, and a = 0.37. 63 0 ——C—2-u(t)/ e"Q(t)du + C36_P2'-u(t), (2.140) 0 ~—§3u(t)/ e"Q'(t)du + C3e_P2‘u(t) (2.141) for large t. This can be rewritten as 35.5.0.0 = (Cl—@3333)+%:e-P3‘Qu(t> C2 I 0 u —Pt —— (t)u(t) e du + C38 2 'u(t) 2 P2 —P2t C2 C2 _ 2t — 2t, = (C1 — F)Q(t)-u(t)+—P:e P Q(t)u(t)+C3e P u.(t) 2 +— m) {1 — (1023+ new} 11 1775 = —e—P33{(B,.+1 -;)11<3t)+3,.,10<3t)) (2.142) Q'(t) = —Ple—P“ {110325) + Io(,z’3t)} + 138—Flt { 1(3t)— 11;('3t) + IO(/3t)} Therefore, the large t approximation of transient interfacial reflection coefficient for P2>Ois RTE“) z {(811 - Bn+1)Bn\/D: (Bn —28n+1) Dn \/—n 6_P2t} X (Bn — Dan+1) (1— D1171)(B_D an+1) e—Pit {11((Bn-H — Bn)t) + [0((Bn+1_ Bn)t)}x Lt“) 2 2 _(Bn—’ Bn+1)2 Dn VD Pt. 1— (Pt 1)(? ° P1 (3 -D 3m) { 2 + }e 64 {(Bn + %)Il((Bn+12_ Bn)t) + BnIO((Bn+1— Bn)t)} u“) + 21371(Bn+l — Bn) (1— Dn)(1+ fl) e"P2tu(t). (2.143) (2) Small t approximation It has been shown that (2.74) can be weritten in exacltly the same form as (2.83), and therefore also as (2.75), as long as 6.6114 in [10] is applicable. Let’s consider the required condition for this conversion. As mentioned earlier, to use 6611.4 in [10], Bn+1 > B,, and 0 < D” < 1, or Bn+1 < B" and Du > 1 must be satisfied. But, the given assumption for this case P2 = (B,, —— Dan+1)(1 — D") > 0 does not produce any relation for B,, and Bn+1 that makes 6611.4 applicable, since it is equivalent to either one ofl > D, and B" > Dan+1, or 1 < Dn and B, < Dan+1. On the other hand, considering _ (Bn+1 + Bn) Bn — Dan+l P1 P2 — 2 1 — Dn (1+ Dn)(Bn+l " Bn) 2 . 44 2(1 _ Du) , (2 1 ) if P1 > P2 then 66114 is applicable. Then, RTEU) = Cle—P‘t {11()3t) + 10030} u(t) +Cge—P21u(—t) * e’P“ {Il(,3t) + 10(13t)}u(t) —C3e—P3‘u(—t). (2.145) If the ratio P1/P2 is sufficiently large, then the 6’10 1‘ term, by its peaking property, provides a good approximation of a 6—function. Thus 333(1) 3:: Cle‘P“{Il([)’t)+IO(,Bt)}u(t) +C26’P2'u(t) * 6(t) — C36_P2tu(—t) a: Cle—P‘tufi) + C26_P2tu(—t) — C3e—Pzt'u(—t) (2.146) Previously, it has been proved that RTE“) is purely causal for Dn+1 > 0 and En > 0, and thus the non-causal components may be neglected. Consequently RTE (t) z .46“) 1“11(1) for small t, where .4 is a constant which will be determined later. When P1 < P2, 6611.4 is not applicable. Since P2 > P1, e’PQ‘ 3:: (S(t) and (2.74) is approximated as RTEU) = Cle—P”{Il(,8t)+Io()3t)}u(t) —02€_P2‘U(t) * 6”)“ {11(f3t) + 10(30} ”(I(t) +C3e’P2‘u(—t) C167P11{0 +1}u(t) — 026(1‘) * 6’10” {0 +1}u(t) + C3 - 0 22 22 Ae'Pltu(t) (2.147) where A is a constant. Consequently, the small t approximation for P2 > 0 includes the Ae’P“u(t) term only, and the constant A is obtained by letting R3541) = t:0 Rfrrlfall t approx“) t:0, and ( +1 ) (2.148) (1- Dn)(1 '1' V Dn) As a result, in a way similar to the P2 < 0 case, the approximation of a transient interfacial reflection coefficient for P2 > 0 may be constructed from the combination of the large and small t approximations using the same pair of weighting functions RTE“) z WL(t)X {(Bn —' Bn+1)BnV Dn + (811 _ Bn+1)2DnV Dn 6-132t} X (Bn "' Dan-H) (1_ Dn)(Bn —' Dan-l-l) e—P1t{11((Bn-l-l _ Bn)t) + [0((Bn+1— Bn)t)} 11(1) 2 2 66 Bn—Bn1 QDm/Dn —(18 19119 )2 {1—(P2t+1)ep'zt}e“Pltx n— n n+1 1 (Bn+1- Bn) {(Bn '1’ ;)11( 2 + 21)n(I3n—l-1 — Bn) <1—D.)<1+ «me t) + Bn10((Bn+12— B")t)u(t)} 21)n(Bn+1 _ Bn) (1 — D.)(1+ «me ”‘30) + Ws(t) Btu“) (2.149) where WL(t) and 1175(t) are given in (2.137). An example of the approximation for P2 > 0 is shown in Figure 2.14. In comparison with that shown in Figure 2.13, the combined approximation shows relatively better result in this example. The individual errors from each approximation are cancelled out each other in transition region, and thus the combined approximate function provides a good match with the exact reflection coefficient curve. 67 — Exact - - Approximate ‘ 3-3- Largetapprox. ---- Small t approx. o 'o 3 =9- _ E < -51 - -7 ~ ~ -3 ~ ~ -9 — 4 _10 l l l l l l l l l 0 2 4 6 8 10 12 14 16 18 20 Normalized Time a|P2|t Figure 2.14. Transient interfacial reflection coefficient for P2 > 0 (TE polarization) 1 #n = #0. 6n = 60, 0n = 015/7711, Hn+1 = #0, €n+1 = 7260, 0n+1 = 4115/7711, 911 = 30°, P2 = 0.64 X 1010 and a = 0.65. 68 CHAPTER 3 INTERFACIAL REFLECTION COEFFICIENTS FOR TM-POLARIZATION 3.1 Introduction In this chapter, the transient interfacial reflection coefficient for a transverse magnetic (TM) polarized incident plane wave is derived. For a TM polarized uniform plane wave, the direction of the magnetic field, rather than that of electric field as for TB polarization, is parallel to the interface. By a similar approach to that for TB polarization, the frequency domain formula is derived first, and its inverse Fourier transform is obtained after reducing and classifying the frequency domain reflection coefficients using algebraic manipulation with the previously defined branch cuts. The resulting reflection coefficients for the ratio of incident and reflected electric field are similar to those for TE polarization but more complicated as expected. The theoretically derived transient forms are verified by numreical comparisons with the IFF T of the frequency domain expressions. Finally, the dependence on various parameter sets is discussed. 3.2 Frequency Domain Formulation of Interfacial Reflection Coefficient 3.2.1 Derivation The geometry for the frequency domain formulation is shown in Figure 3.1. From Maxwell’s equations, V xE(:r,z,w) = —jwunH(x,z,w) (3.1) V x H(:1:, 2,33) = (on +jw€n)E‘(;r, z,w) (3.2) 69 Region 11 : (um , 8n , on) A Z Region n+1 : (”n+1 , an“ , 0M1) £11 Transmitted Wave Reflected Wave _. Interface 11 Figure 3.1. The incident, reflected and transmitted TlV’I-polarized plane wave at an interface. 70 and the vetor Helmholtz equation for the magnetic field of region n is V2H($, z,w) — 7311(33, 2,30) = 0 where the propagation constant for the n-th layer is given by A(n Z jw#n(0n+jWEn) = —w2unén +jwun0n. For TM polarization H = 9H3, and (3.3) can be rewritten as 82 82 ( + —) Hy(:z:, z,w) — 73Hy(:1:,z,w) = 0. 51—33 82:2 The solution for the magnetic field is H(:1:, 2,1,5) : gHo(w)eV"‘r+7"‘z where the components of the propagation constants are defined by 2 ,2 2 7n : ”inns + 7712' (3.5) (3.6) (3.7) Only if region n is the first region of a multi-layered material and this region is lossless will we speak of an angle of incidence and an angle of reflection. Otherwise, we will use 71,1, 71,2, 7,2,, and 7'1”. For the incident wave, _ i _ 7nd: — 771:1: — 7n COS 9m _,.i _ - 7112 — (nz — 771 8111 6m, 71 while for the reflected wave 7m = 7.2. = —7.. cos 0m (3.10) ’71” : 71'” : 7” sin 6m, (3.11) Here 6,," is the angle of incidence and 0,1,, is the angle of reflection, as shown in Figure 3.1. The electric field in region n is obtained by _ 1 - E(I,z,w) = ——,——VxH(:r.,z,w) 0n + leun 'w 8H (9H 2 J 1‘"{—e y+2 1’} 7,, Oz 82: 'w H w = J [J’fl 20( ){_i.nl,nze'7nxz+7nzz +£7nxe7n13+7nzz} 7n '1.) 'w = -43—“Z”"zH.(w)eW+3mz + 23—“?"inmewvmz. (3.12) 7n 7n In summary, the fields in region It can be expressed as Ian-(aw) = gH.0(w)e3ia-x+3ézz — 'w i i i Ei(1',z,w) = —:i:J—u—';%EH,O(w)eV"rx+7mz 7n A .0.) 1’71 lil 1 .7: 1 z +zJ—17215H,0(w)e“’"3 +7": (3.13) Hr(x,z,w) = —3)H,0(w)e7331+1533 _ _ A Vfiz '7' 1+7'.z A 71111: Er(a:,z,w) — 2:, H,0(w)e "1 "~ —2. H,0(w) (3.14) qun qun Notice that, unlike with TB polarization, the direction of the reflected magnetic field is reversed from that of the incident field while the electric field maintains the same direction at the interface. 72 In region (n+1), (3.3) is V2H(;r, 2,13) — 711+1H11': 2,12) = 0 (3.15) where the propagation constant is 7121+1 : jwfln+1(0n+1+jWCn+1) : —w2/1'n+1€n+1 +jwltn+1011+1- (316) Then, the vector Helmhotz equation for the magentic field in region n+1 (i.e., the transmitted magnetic field) is given by ('92 62 2 (5.13—2 + 523—2) I‘ll/($3 Z3”) _ 7n+1Hy1$3 2,00) = 0' (3'17) This has the solution HALE, 2,23) = 33H",(w)e*‘+13"""“'+”"‘+1'iz (3.18) Where the components of the propagation constants are defined by 2 2 2 771+1 : 7n+1,:r: + 7n+1,z' (319) When both regions n and n+1 are lossless, we may define a transmission angle 91 such that Afn-l—l,:r Z ’7’n+1C0863 (3.20) 7n+1,z = "/n+1 sin 61- (3.21) 73 The transmitted magnetic field is given by ijn+1 ‘ E,(;r, 2,13) = ———T—V x Ht(:z:, 2,13) 2n+1 jw/Ln-l-l {—11 8H}; MBH y} 'n-+-1 82 81 jw/1W11+1Hto( 2 : 2 ){_ —j:7n+lz 67:14.1 xx+7n+l.' + 2,. 1n“ 167n+1.,zx1:+’7n+lzz} (n+1 : _1 jW——:———ln+l’)n+l MH‘0( )67n+1111‘+’7n+11;z An-H JW/ln-l—l , . +~"—2— [n+1 IHto( Wv)€7n+1.z:€+7n+1..Z_ (3.22) ASH-l In summary, the fields in region (n+1) can be expressed as Ht($, Z,CU) :: gHt0(w)e7n+l,z$+’7n+1,zz — UJ Et($) 2,01) : _ i, j _7n+1/1n+l tho(w )6711+1,,;x$+’7n+1 z 7n+1 UJ +Zj #n+17n+1,erto( )8’7n+1,z1‘+7n+1,;2 (3.23) N7121+1 To obtain the ratio of the reflected and transmitted electric field amplitudes, two boundary conditions are applied. The first one is the boundary condition for continuity of the tangential components of the magnetic field at the interface. That is, for all z, or, t r Hi0(w)67"=z—Hro(w)67'n=z = H,0(w)e7"+1'iz. (3.24) 74 Then, to satisfy the above boundary condition, the equations 772227;.” : A(”+1.3 (325) Hio(w)_Hro(w) : Hto(w) (326) must be satisfied. For lossless materials, (3.25) becomes 7,, sin 0,- = 7,, sin (9, = "fn+1 sin at (3-27) and thus "/71 Sin 6t : 3.29 7,,“ sin 0, ( ) which are Snell’s law of reflection and refraction, consistent with the result derived in Chapter 2. From (3.7), (3.25) and considering the direction of propagation, 71112: = —771ia: : 7711” (3'30) The second boundary condition requires the continuity of tangential electric field on the interface. That is, for all z, or, - up. .- jwu .r jw/L +1 , 1'1 J n / \ ’7 :2 /r n ’7 :2 —- A‘ 11' ’7 ,:z 1.1-,1: A 2 H10(W)€ n — 7111' A2 Hr0((.bl)€ n - I’l1l+1,1' A2 _Ht0((J/‘)e n+1 o n In (n+1 (3.31) 75 Using (3.30), this equation simplifies to it, #7- ,u'n 1 . 7217’an,0(5.0)+ jfynrHro(w) 2’ %7"in+l,rHto(w)- (332) 7n 7n fin-H Alternatively, . 2 l1 +17 +1, ”1' H..(w>+H..\/s(D.s + B.) — (3+ Anna/swam B...) RTM(S) ____ ~ ~ ~ ~ (3 + An)\/S(Dn3 + Bn) + (3 + An+1)\fl(Dn+ls + Bn+l) (3.39) Now, the frequency domain reflection coefficient can be classified to four different forms according to which branch cut is needed. This depends on the signs of the layer parameters D" and Du“. The algebraic manipulations with the branch cuts used in the derivation processes are given by (A.1) and (A2). (1)11,“ > 0 and D” > 0 —(s + An) EMU/3 s + g— + (s + .4n+1)\/D,,,/§(/s + 3a —(S+An) Dn+1\/o§ S+'g:—::—(S+44n+1)v Dnfi‘/3+'gl: ETA/1(8) : Simplifying gives (5 + An) +g:::— (8 + 4,,+1)\/-b%1:—\/SD+ 11.3! (3+An)(/8+g"+1 (8+An+1) \/-.—11—\/3+Tn. RTM(8) = (3.40) 77 (2) [3,,“ < 0 and Du < 0 ij(5+An) an+1|\/§ S-Igflf‘] RTM (S) : ij(3+‘4n) an+1l\/§ 3“ an:i| j5J(3 +An+1> Simplifying gives Bu l '01:] B 3+A s——.—+——s+A F s——.+ ( n) an+1i ( n+1) an-Hl lD'Il +j( 3+An+1) Vanl l\/§1/:—I-gD‘I |D| lx/g -I7;-:I RTM(S) ____ (3) Du“ > 0, and. [9,, < 0 —()8+A Bn+l ibni _ é . (S + A") V8 —an+1l + (8 + 14111.1) an+1l S l—Dzri—l ~ 811 ' an+1l\/_ +l—Dn—++i_l + J( 3 + An+1) (3.41) - B, anl If 3‘35: RTAICS )2 (S "i‘ 471)lDfl+1l\/— S + ID::1I Itj( S + An+1) Simplifying gives (3 + An) 3 +_'_ 871“ i j(3 + Art-+1) anlx/E I—g‘D‘I ID..I \/8_ Ba IDn +1! anI RTM(s) = 'D ' _ . (3.42) n l B (8+.4n)1/3+ID—T—:II $j(8+x4n+1) TDg-al—IJS— l—Dlri—l (4) [9,,“ < 0, and D” > 0 RT“(3)= 8,,)+A IDn+1I If —I—-—,,"II+ 3+ 41.+1)ID..I\/E +IgI (3 + An) iDn+1i \/_ —|D::—'_i_|— 3+ 411+1) iDn l\/— +|fiDJL| Simplifying gives ’8 Bn+1| (3+A")V 8 ID::iii D_L_,L_| +1331; (S)+A"+1:\/:Dn:il\/:—JLS+|D+.I In each of these expressions the upper sign corresponds to w > 0 and the lower sign 78 to w < 0. Note again that the branch cuts defined in section 2.2.2 have been used throughout for the square root functions. 3.2.3 Reduction of the interfacial reflection coefficients Observing (3.40)-(3.43), it is realized that there is a constant offset value inside of each frequency domain expression, as there was with the TE polarization. Therefore, the constant offset value, the asymptotic reflection coefficient, should be extracted first. To make the (3.40)-(3.43) more readable, let Dn = Ell—l— : \/€31+1 lf‘nén '- [1:161 Sin? 6,I| an+1l 6% lfln+1€n+1 _ #15151n2 gill B, z _EL : unan . . anl lflnf-n - #161 sm2 6,I| and take the limiting values at infinite frequency. Then, (1)Dn+1> 0 and [3,, > O (8 + Auk/s + Rm — \/Dn(3 + An.+1)\/s + Bn RT!” ,, z a (D) (s + A,)I/—‘s + 13,,+1 + \/—Dn(s + An+1)\/S_-+ —B,, (2) Dn+1 < O and [3,, < 0 (s + Auk/3 - En“ — \/Dn(8 + An“) s — Bu ETA/Rs) = , (s + An)I/s — Bn+1+ \/Dn(3 + An“) 3 — Bn 1 — VD" lim RTM(s) = —. w—«mo 1+ W/Dn 79 (3.44) (3.45) (3.46) (3.47) (3.48) (3) 13n+1 > O and 13,, < 0 RTM(3) : (8 + An)\/8 + Bn+1ij\/Dn(8 “i" £411.11) 8 — Bn, (3.49) (8 + 44n)\/8 + Bn+1$jVDn<8 + An+1) .S‘ — B" 1d: \/ lim RTM(8) - J D" _ ____ . 3.50 w—mc IZFJE ( ) (4) 13n+1 < 0 and 13,, > O RTM(s) __ (s + An)I/s — Pn+1¥jI/D,,(s + An+1)\/s + B,, (3.51) _ (8 + An)\/8 — Bn+12i2j\/Dn(3 "l” An+1)\/S 'l' Bn3 1 \/l3,, lim RTWS) = L. w—mo 1 :1: ]‘/Dn In these expressions the upper sign corresponds to w > O and lower sign to w < 0. (3.52) Next, the obtained asymptotic reflection coefficient have to be subtracted from the original RTM (3). Let’s denote the asymptotic reflection coefficient as R22,“ and define RTM(5) = RTM(3) — R30“, then (1)13n+1> O and 13,, > O (s + AafiTBn—H— (8 + Awk/D—nm _1— x/D; (8 + Anh/H—BEH (s + An+1)\/D;\/s+_Bn 1+ fD—n 2x/D—n (8+An)\/s—+m— (s+An+1)\/’s‘7rfi 1+flZ 0 and Dn > 0 will be discussed in this study. Assume that 0,, = 0, then An = B,, = O and (3.45) becomes RTMIS) = \/3_V3 'l' Bn+1— \/D—n(3 ‘l' 4n+1l (3 57) fiW'l” J7 0 always, and , Hn-HO'rH-l Bn+1(Bn+1 _ 4DnAn+1) : . 2 l/l'n-HEn-H — #1618111 951' 2 ° 2 _4071.+1 (n+1 “n61! _ #161 SlIl 6“ 2 ' 2 €n+1 6,; fln+1€n+1 - #161$111 921 2 ' 2 Un+1 {MnHEn — 4€n+1lHn€n _ #1613111 gill} - 2 €nlfln+1€n+1 — #161Sln gill (3.67) The definition (3.67) can have any value, even for 13,,“ = 11,, = #1. As a result, b2 —4c can have any sign, and therefore P1 and P2 can be any type of number. Assume that a,,+1 = 0, then An“ = Bu“ 2 O and (3.45) is simplified to (3 + An) — afim (3 + A.) + VITA/EMS + B, RTM(3) = (3.68) Observing this expression carefully, it can be recognized that by replacing An, 13,, and D" with An“, B,,+1 and 1/D,, respectively, RTM(3) for CHI 2 O is equivalent to —RTM (s) for 0,, = 0. That is, the reflection coefficients for both cases are in ‘dual’ relationship. Therefore, it is expected that the transient reflection coefficients for both cases have the same relationship. 3.3 Derivation of Transient Interfacial Reflection Coefficients The transient forms of the frequency domain interfacial reflection coefficients are given by the inverse Fourier transform. Because the asymptotic reflection coefficients R2” 84 are constants in the frequency domain, their transforms are simply given as delta- functions multiplied by the constants. Let’s denote them by R22” (t). For [3,1,1 > 0 and [9,, > 0, RT“(t) = F-1{1im RTM(3)} (,0—>00 1-\/Dn- WOW (3.69) and the resulting transient interfacial reflection coefficient is given by RTMU) = F-1{Iim RTM(3)} + F-1{RTM(S)} w—mo = 1 _ mm) + RTMU). (3.70) 1+\/Dn By the ‘dual’ relationship of an : O and on“ = 0 mentioned earlier, only the transient interfacial reflection coefficients for on = 0 will be derived. Then, what is needed to obtain the transient form for the dual an+1 = 0 case is just taking negative sign of the obtained transient reflection coefficient for on =2 0. The transform pair for each term in (3.63) can be found in Appendix B. Since P1 and P2 can be any number, real or complex, the transient reflection coefficient has five different forms depending on the roots of the frequency domain denominator. Let’s denote P, = 1241932124132} R = Im{P1} = -Im{P2}. (3.71) From (3.64), the constants K1, K2, K3 and K4 are generally complex numbers as 85 well, and using their complex conjugate relationship, let A'lr K1.- K3r K3.- and denote 3— — —’12+—1. Re{K1} = Re{K2} Im{K1} : —Im{K2} Re{K3} 2' Re{K4} Im{K3} = —Im{K4} (3.72) (1) When P1 and P2 are complex numbers, and P, < 0, the transform of (3.63) is given by 2m RTMU) = 2\/~_ 1— (1 — Dn)(1+\/IT,,) n3€iK P1t{6(t))+ 131140)} + K461)” {5(t) + P2u(t)}] * {K1 eplt+K2 ep2t}u( (—3 e‘5‘{11((3t)+ 10(3t>}ut( ) (3.73) Here In(t) is the first kind modified Bessel function of order 11. Each term in this expression can be simplified using (3.71) and (3.72) as Kleplt-l—ngp‘?‘ : Re{K1}eP’+jP't+jIm-{K1}ep’+jp‘t+ Re{K1}eP’+JP“ — jInL{K1}eP'+jP‘t = KlrePrt(ejP,t + e—jPit) +jKliePrt(ejP,-t _ e—jPit.) = ZeP"(K1r cos P,t — K1, sin Pit), (3.74) K3€Plt6(t) + K4epzt6(t) = 1X36“) + K460i) = 2K3r6(t), (3.75) 86 and K3P13P1t+ K4P2€Pzt 2 (K3,. +jK3z.)(p +336) P.+th + (Ksr '— jK3i)(Pr — jP,)eP'_jP*‘ : CW1- Pr( ejplt+ejp,’3t)_1\ MP : (1 _ IDS/ED x/D—n) eP"(K1r cos Pit — K1,- sin P,t)u(t) _4’:_3__D_r_51\)/78-3‘ {1((3t))+ 10(3t)}u (t) —ifi_‘/;(K3.P— K3.P.~e”) W) x jot cos P (t — :r)e—(P'+’9)x{11(1317)+ 10%)} d3: +‘iflfl( K3,.P +K3,P,-)e” u(t) x jot sin 12.3 — 1:)e'(P’+B)I{11(/3$) + 10033} dz : (1_ D:fl \/D_n) eP"(K1r cos Pit — K1,- sin Pit)u(t) _Wflt {11(/3t) + 10(31) } 'u(t) -— gfewn x 87 t / {(K3rPr — K313) COS Pi“ - :17) - (K3,.Pr + ngPz) sin P,(t — 33)} x 0 84’7”” {11(33) + 1333)} 3.1:. (3.77) (2) When P1 and P2 are complex numbers, and P, > 0, the transform of (3.63) is given by RTM (t) — 2 {K1 ePIt+Ku(eP2‘} )+ (l—D n)(1+\/D—) 2%? [K38 Pit {6(1‘) —P1u(—t)} + K4€P2t {5W — P2u("t)}l * 1 (‘73 )eB‘UlUit) +Io(5t)}u (t) _ 4m ,t . (Ix/IX _ —(1 _ Dn)(1+ my? (K1, cosPit + K1,81nP,-t)u(—t)+ 1_ Dn x i [345“) + 810.: {(K3rpr ’ KSiPi) COS Rt — (K3,.Pi + K3,-P,.) sin Ht} x u<—t)] * (wee-5‘ {11(3) + Iowa} um : (1— D4)(f: D )ePrt(K12' sin Pit — Klr COS Pit)u(_t) _4K3rIBV D71 , _ D. e-B‘ {11(3) + 1332:3173) _L 'azep't/ {(K3rPr —K32'Pi) COSPi(t—'CC) 1- Dn maa:(t 0) —(K3,.P. + K333.) sin P,(t — 13)} 5W?” {11(33) + 1333)} das. (3.78) (3) When P1 and P2 are real numbers, and P1 > P2 > 0, the transform of (3.63) is given by RTMU) 2J— Plt P?‘ —(1—DDn)(1+\/—_) {K8 +K€ }U( H 121/; [K3 6P1t{6(t—) P1U(—t)}+K4€P2t{5(t) — P2“(_t)}] 88 *(—B)e—Bt {11(31) + 10(30} 'u(t) _ 2717. _ (1-3(1+,/5;){K18 ”(2 } '( t) 1851085 +K4)8_‘3i {11(Bt) + 10031)} u(t) 2K P n °° ‘ 3 1/3\/D ePlt/x e-(P1+3)x {11(51) + 10(3x)} (1:1: 1 — Dn mar(t, 0) 2K P n 0° _ .- + :2”;le eet/ e”35311333+Io(.8z>}dx. (3.19) "" n max(t0) (4) When P1 and P2 are real numbers, and P1 > 0 > P2, the transform of (3.63) is given by RT1U(t) 2 0,, Pt P2t = (1-D 1%)(ILVD—n){*K1€‘u(—tl+K2€ “(t)” 12:5" [K Mm, p1u,(—t)}+K4eP2t {63) +P2u}l *(-13 (3)6‘5‘ {11(6t) + 10(3)} 3(1) = (1 - 03% J17) {-KleP‘tu(—t) + K26P2‘u(t)} _213\/_(K3+K)e-Bt{1,(3t)+10(8t)}ut() 1 — Dn +2K3P15VD71 eooPlt/ e—(Pi‘f-le {11(51') + 10(fi$)} (1.73 1— Dn max(t0) r t _2A1PgflDxle n “1321 (15)] e—(P2+_8)I {1103513) +10(5I)}dg:. (3.80) — 0 (5) When P1 and P2 are real numbers, and O > P1 > P2, the transform of (3.63) is given by ETA/I(t) 2\/D—n P11 P21 (1*Dn)(1+r) {K18 +ng}u( 89 2\/D,, 1— Dn *(—.3)e-3t {11(,3t) + 10(13’» ’U-(t) 2x/D; : {KleP‘t + ngp'zt} 11(1‘.) (1 — Dn)(1+ \/Dn) 2 \/D,, _, _1‘3_ D (K3 + K4)e 3‘ {11(Bt) + 10(5t)}'u(t) _2K3PIBV Dnepltu 1— Dn 2K4P25an P21 _ 1— D e u [K3eP1‘{3(1)+ 131-3(1)} + K4ep'2t {6(t) + 132-WM] t (t) f 61’1”” {1331) + 1363} dx 0 (t) /0t e‘mfimf {11(1316) + 10(Bx)} dz. (3.81) Notice that, for TM polarization, there are non-causal terms even for the Dn+1 > 0 and Dn > 0 case. 3.4 Numerical Examples 3.4.1 Verification of theoretical expressions The previously derived transient forms of the interfacial reflection coefficients may be compared to direct numerical computation to ensure that the transforms are correct. Again, the numerical IF F T , is used for this purpose. For the reasons explained in the previous chapter, only the causal case will be discussed. Since the asymptotic interfacial reflection coefficients appear as delta functions in the time domain, and are thus difficult to express numerically, only the reduced interfacial reflection coefficients are compared. Figure 3.2 shows the comparison of the derived transient interfacial reflection coefficient with the transient forms obtained by using a 3072 point IF FT for 0,, = 0. The material parameters for Figure 3.2 are those for free-space and typical sea water at low frequency as described in [20]. The transient reflection coefficient curves from the two different methods are nearly identical, and thus the derived transient interfacial reflection coefficient for TM polarization has been verified. 90 E 0 .'§ 3: a 8 : —Derived _3— - g ) --|FFT 0 g l “a 41 ‘ I! | Ta '8 *5 - fl ‘5 i 2 ‘5' ‘ u 8 -7- ~ 3 u 0 1: -3- _ -9» a _10 l l l l l l l l Time [nsec] Figure 3.2. Numerical comparison of the derived transient reduced interfacial re- flection coefficient with that from the IFFT (TM polarization) : 11,, = M0, 6,, = 50, 0n = O[U/m], Hn+1 = #0, €n+1 = 7260.071“ = 4[U/m], 611 = 30°- 91 To verify the combination of the asymptotic and the reduced transient reflection coefficient, the time domain convolution of the interfacial reflection coefficient with an input waveform is needed. i.e. S(t) _—. X(t)*RTM(t) = X(t) * {333(1) + {amps} = RZOMXU) + BTMU) * X(t). (3.32) Here X(t) denotes the input waveform and S(t) is the resulting reflected waveform. The same input waveform given in Figure 2.6 is used for TM polarization. Figure 3.3 shows the comparison of the time domain reflected waveforms with the input waveform shown in Figure 2.6 for on = 0. Again, the derived expression and the IF F T results are well matched. Exact theoretical expressions are available from the previous research [11] for TM polarized transient interfactial reflection due to a unit step input waveform. A comparison with those results also helps to verify the work in this study. Figure 3.4 (a) and (b) show that the re-produced results of the electric fields to a double exponential input waveform excitation as shown in Figure 4.1 in [11], obtained by using the transient interfacial reflection coefficient and transmission coefficient of this study, are exactly same as those found in Figure 5.3 and Figure 5.2 in [11]. 3.4.2 The transient responses for various parameter sets It is interesting to study the dependence of the transient reflection ceofficient on various parameter values, i.e. permittivity, permeability, conductivity and aspect angle. Comparisons of the reduced reflection coefficients are shown in Figure 3.5 - Figure 3.8, which are plotted using semilog scales for easier identification. Note that the curves in each plot are for the on = 0 case. 92 0 r l I I I -1. - ... '2 i E Z _3 —Derived _ g --IFFT a a _4. _ u 9 0 2 -5; . ‘5 tr '6 _5- . 0 1: 2 2-1 -7- - E < _3- T _9- . Time [nsec] Figure 3.3. Numerical comparison of the derived transient reflected electric field waveform for the input waveform shown in Figure 2.6 with the IF FT (TM polariza- tion) 1 pa = #0, 6n = 60, an = OHS/m], ”n+1 = #0, €11+1 = 7260, 0n+1 = 4[U/m], 0n = 30°. 93 . . . ...... . . . . . .... . . . . . ... . . . . . . . . . . . . . . . ... . . . . . . . . . . .... . . . . . ... . . . . . .... . . . . . .... . . . . . ... 07,. ................................................................................................ _. ' . . . . . .... . . . . . .... . . . . . ... . . . . . .... . . . e . .... . . . . . ... . . . . . . . . . . . . . . . ... . . . . . . . . . . . . .... . . . . . .... . . . . . ... . . . . . .... . . . . . .... . . . . . ... . . . . . .... a . . . . . . . . . ... . . . . . ... . . . . . .... . . . . . ... y— ....... . ...... .......................................................................... _. ' . . . . . .. . . . . . . . . . . . ... . . . . . .... . . . . . . . . . . ... . . . . . . .. . . . . . .... . . . . . ... . . . . . . . . ... . . . . ... . . . . . ... u . . . . .... . . . . . ... . . . . . .... . a . . . .... . . . . . ... . . . . . .... . . . . . .... . . . . . ... .— ....... \....... .x. ....... -. ‘.| .......... ' .......... ...-..t ................. ~ ........ - ..'..I.\.'-1 g . . . . . . . . . . .... . . . . . . . . . . .... o . . . . . . . . . ... . . . . . - . . .... . . . . . ... . . . . . . . . . . . . . . . .... . . . . . .... . . . . . ... . . . . . . . . . . . . . . . ... . . . . ..... . . . . . . . . . ... 04.. ..... ... . . . . . . . . . . .... . . . . . ... . . . .. . o . . ... . . . . . ... ' I l n a a .... I l 1 I I ’01- I I c I I on. Amplitude of Magnetic Field [Alm] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . u . . - . . . _ .......................................... . .................................................. . . . . . 1 . . t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . o . . . . . . . . ..........-. ..... ....... 1.. .1. r..'.:.-..-.: ........ '..-..:.'..'_. 10 10 10 10 Time [nsec] Figure 3.4. (a) The transient reflected magnetic field waveform for a double expo- nential excitation : 6 = 1060, ,u = no, a = 2 X 10’2[U/m], t9, = 45°. 94 . . . . . .... . i . . . . . . . . ... . . . . .... . . . . . . . . . . ... . . . . . .... c . . . . .... . . . . . ... —.... . ...... ... ...... .... ...... - . . . .- . . . . .. .. . . . . . ... . . . . ..., . . . . . . . . . . ... . . . .... . . . . , . . . . . ... u - . . . . I l 0.8 06L. ....... _. ..... ..... ........ ., ..... _, . I - c 0 v ln-I a n l C 1 7A.. I I I I I III Amplitude of Magnetlc Fleld [Aim] . . . . . .... . . . . .... . . . . . ... . . . . . .... a . . . .... . . . . - ... . . . . . . . o . . .... . . . . . ... s . . . . .... . . . . . .... . . c . . ... -..... . .................... ........... ............. ..... , ................. _l . - - - . -‘ , . . . . . .... . . . . . ... . . . . . .... . . . . . .... . . . . . ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , a I . . . - - a a a o - - . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - . . . . . . . . . . . . . . o I s - u a . . . . I . . o . _ ....... , ............. . ......................................................................... _ I u . . n 6 - . 1 : - . u I n n . ~ 0 o a n . - . . . . . . - . . . . . . . I . . - o . . . . . . . . . . . . . . . . . . . - - . o u . . . Q . . . . . . 0 i .;:;;;;i .3 LL;3;;.1 10 10 10 10 Time[nsec] Figure 3.4. (b) The transient transmitted magnetic field waveform for a double exponential excitation : e = 1060, u = no, a = 2 x 10‘2[U/m], 6,- : 45°. 95 (1) Permittivity The curves in Figure 3.5 are plotted for em,“ = 1.7, 2.5, 10 and 72. The correspond- ing values of R732” are -0.10, -0.18, -O.47 and -O.76, respectively. It can be seen that the slopes of different R'I‘MU) curves are preportional to the increasing permittivity values as well as increasing asymptotic reflection coefficient R2". This means that. the transient reflection becomes more dependent on its early time portion coming from the asymptotic reflection term when the permittivity contrast of both layers is sharper. (2) Permeability The curves in Figure 3.6 are plotted for pm“ 2 1, 5, 10 and 50. The corresponding values of REC“ are -0.76, -0.53, -0.40 and -0.019 respectively. The slopes of RTMU) are seen to be inversely proportional to increasing permeability values, and to increasing amplitude of the asymptotic reflection coefficient. This indicates that the transient reflection becomes more ‘relaxed’ with increasing values of permeability. (3) Conductivity Because an increase in conductivity causes attenuation in wave prepagation, the transient response RT“ (t) suffers a more rapid decrease with time as conductivity is increased, while the asymptotic component is independent of the change. This can be seen explicitly in Figure 3.7 where 0,, = 0.10, 1, 4 and 10, respectively. (4) Aspect angle The curves shown in Figure 3.8 are plotted for 6,1 2 0°, 30°, 45° and 75°, while the corresponding values of ROTC?" are -0.79, -0.76, -0.72 and -0.38 respectively. Therefore, the slope of the reduced reflection coefficient as well as the amplitude of the asymptotic component decreases with increasing aspect angle, and the transient reflection is less dependent on its early time portion. 96 11 10 : I I I I I I I I I 10_ 10 — epsr=1.7 - - epsr=2.5 --~ epsr=10 epsr=72 A 1 AAA]. Amplltude of Reduced lntertaclal Reflection Coefl‘lclent o o . O- 106 .........u.u..::::o-.-‘ '- 00000- 105 l l l l l l l l l Time [nsec] Figure 3.5. Time domain reduced interfacial reflection coeflicients for various values of permittivity (TM polarization) : 6,, = 60, an 2 an“ : no, on = 0, 0n+1 = 4(6/771], 61'] = 30°. 97 10 10 : I I I I T I I I I : .. i c 4 2 .9 :5 9 ' 0 10 f. 1 o “a, — mur=1 : g \'-'-. - - mur=5 .. *\\,-.. ,_,_ _10 + '5 f \\ mur— 2 i \'\ mur=50 * I .1 ° 8 E 10 3 .2 . 0 (B t e H c " 7 '0 - o 10 : O 3 'u o I: h 0 6 e . 1: 1° : 3 : z . a . E r < 105 1 l l l l I l l l Time [nsec] Figure 3.6. Time domain reduced interfacial reflection coefficients for various val- ues of permeability (TM polarization) : 6,, = 60, en+1 = 7260, an 2 #0, an 2 0, 0n+1 = 4[U/m], 0n = 30°. 98 10 10 : l l T I I T l I l a C 2 2 ’ 1 g 9 i 0 10 E - U T. c a: — Cnd.=0.1[mholm] ; g t; - - Cnd.=1[mholm] l :3 . 13‘ - - -- Cnd.=4[mholm] I- \ - e 8 ‘- Cnd.=10[mholm] E 10 f '.'\\ '5 g ’ "5 : 0 ° \, \ a .0 \ \ t a" .\ \ \ 2 .'. \. \ x c o \ \ \ - 7 ... ‘ + g 10 I- ..o.. \\. ‘ \‘ d 0 . “g. \°\ ‘ ‘ ~~ : ’ .... \ \ ~ ~ 5 'U r . ... -‘ ~ ~ - ~ ~ 0 _ " . ".~ ~ ‘ s -‘ m ..l... -"~.‘ ---—~~ h 00...... ~ ~ ~ ‘1 o 6 0......H... ‘ ~._ -.- 0 — 000000...... ....... .q U 10 : ...°"00.....“ a . 000...... g _D. E < 105 l l l l l l l l 1 Time [nsec] Figure 3.7. Time domain reduced interfacial reflection coefficients for various val- ues of conductivity (TM polarization) : 6,, = 60, en+1 = 7260, pn 2 an“ 2 #0, 0,, = 0[U/m], 6,1 = 30°. 99 10 10 I I I I I I I I I j E f .9 2 z: 3 109 - ‘c’ -— Angle=0 deg. , g - - Angle=30 deg. 3 .- ~ Angle=45 deg. fi 8 ----- Angle=75 deg. 0: 10 f " a . '5 G t 2 C '0 107 r a, . O 3 u o I: ‘6 6 3 1° 3 i E < 105 I l l l l l l l I Time [nsec] Figure 3.8. Time domain reduced interfacial reflection coefficients for various values of incident angle (TM polarization) : 6,, = 60, CHI 2 7260, It” 2 an“ 2 #0, on = 0, on“ = 4[U/m]. 100 CHAPTER 4 SCATTERING FROM A MULTI—LAYERED MEDIUM 4.1 Introduction The overall reflection coefficient is defined as the ratio of the reflected wave amplitude to the incident wave amplitude at the first interface of a muti-layered medium, for which the effect of scattering from the other layers must be considered. In this study, the overall transient scattering from a multi—layered medium is derived from a com- bination of propagation terms for the layers and the individual interfacial coefficients for which closed forms have been derived in the chapters 2 and 3 for each polarization. To obtain the transient formulation for the overall reflection coefficient, its frequency domain formula is derived first using the wave matrix method, and then time domain expressions are found using a series expansion and the convolution theorem. For this derivation, the analytical form of the transient propagation term is found. Finally, the obtained transient expression is verified by numerical computation examples, and various aspects of the formula are discussed. 4.2 Formulation of ’IYansient Overall Reflection Coefficient The overall reflection coefficient for a multi-layered medium (assumed to have N layers) can be computed in the frequency domain using wave matrices [22]. Several other techniques can also be used, but this is the most common technique. Figure 4.1 shows the geometry of the nth layer of a multi-layered configuration. The incident and reflected waves immediately to the left of the (n-l | n) interface are cn and bn respectively. Similarly, the waves immediately to the left and right of the (n | n+1) interface are cu“, bn+1 and cn+2. The overall reflection coefficients immediately to the left of the (n-l | n) and (n | n+1) interfaces are Fn : bn/cn and PHI 2 bn+1/cn+1, 101 i+2 (Il’ti—I’Ei—l’ai—l) i’Ei’Oi) (ILlH-I’EI'H’O'HI) Figure 4.1. The nth layer of a multi-layered environment. 102 respectively. The interfacial reflection and transmission coefficients are R", Tn and R1,“, Tn“. Wave matrices are used here to relate F" to Fn+1- By doing so, we can develop a recursive relation that will ultimately relate the reflection coefficient at the front of the interface (i.e. the first layer) to the reflection coefficients of the underlying layers. Using Collin’s wave matrix result [22], cn and b" are related to an and b... by the following relation On I Pnnl(w) RnPn(w) Cn-H : — (4.1) bn T" RnPn_1(w) Pn(w) bn+1 where Pn(w) = 87”!" is the frequency domain propagation term, and [n is the length of layer 11. Then, the recursive formula for Fn in terms of Fn+1 can be computed using Collin’s wave matrix result as follows 1 —1 bn Tn {RnPn (W)Cn+1+ Pn(w)bn+l} Fn(w) = ”C“ : 1 (4.2) n F{17’1—'1((,¢1)Cn+1‘l' RnPn(w)bn+1} Factoring out Pn'1(w)cn+1 from the numerator and denominator and using PHI 2 bn+1/cn+1 produces the desired frequency domain result : 12.0») +P3Iw)rn+1 0 and the lower sign for w < 0. for dn > 0, and Only the dn > 0 case will be considered due to the causality difficulties mentioned in 105 Chapter 2. Now the frequency domain propagation term is written as Pn(w) : 671171 fits : e—lrn/E 32+dn . (413) To obtain the inverse Fourier transform of (4.13), the transform pair in [23], 6'5 (“pp—"2 4:) 6’856 (t — E) + v . ..(a .2_(;)2).(._;) ...... where Re{p, a} 2 0, is used. When (4.13) and (4.14) are compared, it can be realized that —bfl— > 0 corresponds to p, and 0 = > 0. Then, 2d,. " .—e.(._e) _. e-an—znfln) U 01.. fitlm/CTI; _) ’U 152—011)?" y/t'z-lgdn 2. e—ptll 0\/t2 - (l—n) )u (t — If) —> 63‘2—(:1nr3"t11(2l:;I (#2 — lid”) u(t — lny/dn). (4.15) As a result, the transient propagation term is given by pug) = e-fi‘mu—Im/d.) bnln 2 (1,, bn . +(\/22L_l\i_;_)11 (2d Viz — 13d") u(t — l,,\/d,,). (4.16) 106 Also, it can be easily shown that {13,(3)}" <:> e‘z—«H—i‘imm—II \/d,,) +(klnbn /2\/_n)1 11“ +12./ —I,2,d The transient form (4.17) should be verified by numerical inverse Fourier trans- \/t2— I...)2d,,u) (t—kln\/ci_,,). (4.)17 2d,, form, but it is difficult to compute directly the transient form because it includes the 6-function. Therefore, a modified form obtained by removing the 6-function term is compared with its corresponding frequency domain form. That is, by being modified as em = am — {Rem — 1.1/21:), (418) the corresponding frequency domain transform becomes 13,,(3) = Pn(s) —F{e_fb/”T:’"6(t—ln\/dn)} _ e_ an Sufi-gain —e‘Kb/%i'fl"e’(‘nm)3 :_ (fflvsugfish‘ — e_(§£\/%Z‘+ dnsfl", (4.19) and the transform pair of (4.18) and (4.19) may be compared using numerical inverse Fourier transform. The plot in Figure 4.2 shows an example of such a comparison obtained using a 2048 point IFFT with a 20 nano second interval. The transient form and the IFFT result are well matched, and therefore the correctness of the transient propagation term has been verified. 4.4 Numerical Examples of Overall Reflection The previously derived transient overall reflection form may be verified using nu- merical computation to ensure its correctness. Considering there are 6—function terms included in both the transient propagation and interfacial reflection terms, 107 x10 5" I ‘ 4- _ — Derived - - IFFT Propagation Factor Time [nsec] Figure 4.2. Numerical comparison of the derived transient propagation term with tShat from the IFFT : 11,, = no, 6,, = 7260, on = 4[U/m], ln = 0.01[m], 6,1 = 30°. 108 direct numerical computation of (4.8) may not produce an accetable result. There- fore, the following numerical implementation is used in this study. When assuming P(t) = A6(t — td) + P(t — td) and R(t) = R006(t) + RUE) for example, the convolution becomes Pit) * R(t) = {-460 — ta) + 150—164)} * {120060) + R(t)} = .4R006(t) * 6(t — td) + A6(t — td) * 11(1) +£20.60) * 13(1— 1.)) + Pu — t4) 1. 11(4) = .4R006(t — td) + AR(t — td) + 1200150: — td) + 15(1 — td) 1: R(t). (4.20) The amplitude and position of the 6—function in (4.20) are stored separately for use in subsequent computations. To compute P(t — td) =1: Fl(t), several different nu- merical algorithms are available [24], but the simple discrete convolution algorithm is used in this study, and it turns out that the method provides sufficient accuracy when compared with IFFT results. Detailed programming source codes for numerical implementation can be found in Appendix C. Numerical examples of scattering excited by the input waveform shown in Figure 4.3 from the simplest 3 lossless layer (free space, 10cm thick-plexiglass and free space) cases for both polarizations are shown in Figure 4.4 and Figure 4.5. The double exponential input waveform shown in Figure 4.3 is produced as X(t) = 10.24 x (e"8><1°9‘ — e"16x1°"‘) . (4.21) The permitivity value at 10GHz for the plexiglass layer, 2.5960, is found in [21]. The 1‘ GSults from the transient formulas are compared with those obtained from a 2048 point IFF T with a 20 nano second range, although only the first 10 nano seconds 109 are shown. The comparison shows good agreement in both polarizations. The phase reversed reflected waveforms at each edge are seen in the both plots. To see explicitly mutiple reflections due to time—delaying propagation terms, the scattering from a 5-layered lossless medium (free space, 10cm plexiglass, 10cm free space, 10cm plexiglass and free space) is computed, and shown in Figure 4.6. Be- cause this medium consists of lossless layers, the reflected waveforms are controlled only by the asymptotic reflection coefficients, and there is no attenuation suffered by the reflected waves. Instead, the amplitudes of reflected waves decrease after each reflection because the amplitude of each asymptotic reflection coefficient must be less than unity. The overall reflections from a lossy layered medium are plotted Figure 4.7 and Figure 4.8 for both polarizations respectively. The 5 layered medium is identical to the previous case except that the third layer is distilled water with parameter values taken at 10 GHz, and the equivalent conductivity used to represent the dielectric loss of water at that frequency. As expected, the mutiple reflections suffer severe attenuation, and the reflections after a short period peter out. It may be interesting and practical to identify each reflected waveform in the mutiple reflections shown in Figure 4.6 - Figure 4.8, by associating them with each term in (4.8). The time domain waveform identification makes it possible to remove unwanted signals, for an example, multi-path echos. Figure 4.9 shows the individual reflections in Figure 4.7 due to the first through the fourth term in (4.8). The first peak Obviously is the reflection from the first interface of air to plaxiglass, and depends on the asymptotic reflection coefficient only. Also, notice that the sign of the amplitude iS inverted since the wave is reflected from a electrically denser layer. The second peak comes from the reflection at the plexiglass-water interface. Similary, the sign of amplitude is reversed, but the large reflection occurs because the permitivity profile difference between the two layers is larger than for the previous interface. Now the 110 Amplitude [Vim] 2.5 1.5 * 0.5 ' k l l l l p— — II— 0 1 2 3 4 5 6 7 Time [nsec] Figure 4.3. A double exponential input waveform 111 10 0.8- 4 0.6- _ - — Derived - - IFFT 0.4- 1 0.2 r - Amplitude [Vlm] o -o.2 - 4 .0.4 - . -..] - -0.8 - - -1 1 1 I I I L 1 I 1 Time [nsec] Figure 4.4. Numerical comparison of the derived overall reflection from a 3 lay- ered medium with that from the IF FT (TE polarization) : [12 = 110, 62 = 2.5960, 02 =3 0[U/m], l2 2 0.1[m],6,~1: 300. 112 0.8- « 0.6- - — Derived - - IFFT 0.4- - 0.2 f _ Amplitude [Vlm] o -o.2 - -o.4[ - -o.6- _ -0.8- . '10 1 2 3 4 5 6 7 8 9 10 Time [nsec] Figure 4.5. Numerical comparison of the derived overall reflection from a 3 lay- ered medium with that from the IF FT (TM polarization) : p2 = #0, 62 = 2.5950, 02 = 0[U/m],12 = 0.1[m], 61‘1 ‘-'—" 30°. 113 — Derived - - IFFT 0.4- - 0.2 ~ _ I P N I 1 Amplitude [V/m] r l l i l [ )— — r— p__ _1 l 1 l l l 0 1 2 3 4 5 6 7 8 9 10 Time [nsec] Figure 4.6. Mutiple reflection from a lossless 5 layered medium (TE polarization) - [1,2 = [1.0, 62 = 2.5960, 02 = 0[U/m], [2 = 0.1[m], p3 = [1,0, 63 = (50, 03 = 0[U/m], 13 = 0.1[m],;14 = [10,64 2 2.5960, 04 = ORE/m], 14 = 0.1[m],0,-1= 30°. 114 — Derived 0'5“ - - IFFT 0 - E Z. 0 g -o.5 q i E < -1 - .. -1.5- - _2 M l l l l l l l 1 Time [nsec] Figure 4.7. Mutiple reflection from a lossy 5 layered medium (TE polarization) : #2 = #0, 62 = 2.5960, 02 = 0[U/m], 12 = 0.1[m],/13 = no, 63 = 5560, 03 = 16.7[U/m], [3 = 0.1[m],;14 = M0, 64 = 2.5960, 04 = 0[U/m], l4 2 0.1[m], 6n = 30°. 115 — Derived 0'5 - - IFFT Amplitude [V/m] l o 01 _2 L L l l l l l l l 0 1 2 3 4 5 6 7 8 9 10 Time [nsec] Figure 4.8. Mutiple reflection from a lossy 5 layered medium (TM polarization) : [1.2 = [10, 62 = 2.5960, 02 = 0[U/m],12 = 0.1[m],;13 ‘2 [1.0, 63 = 5560, 03 216.7[0/771], I3 = 0.1[m], [14 = [1.0, 64 = 2.5960, 04 ‘3 ORB/771], l4 2 O.1[m],0,-1= 300. 116 reflected waveform is contributed by the reduced reflection coefficient as well as the asymptotic component. The polarity of the third peak is the same as that of the input waveform. Therefore, there is a possibility that it comes from the reflection at the water—plexiglass interface. However, considering the relatively high permitivity profile of water which makes the waveform propagation much slower than in plexiglass, as well as the heavy attenuation that the wave suffers due to the dielectric loss of water, this possibility must be discarded. Therefore, the third peak originates from three reflections at the interfaces of plexiglass-water, air-plexiglass, plexiglass-water and transmission through the air plexiglass interface. The fourth peak comes from an additional two reflections at the air-plexiglass and plexiglass-water interfaces. Of course, this rigorous geometical ray Optics approach is possible because of the exact timing analysis. The incident angle dependence of the overall reflection is shown in Figure 4.10 for the same 5 layered lossy medium used in Figure 4.9. In the first reflected waveform set, the largest angle produces the largest peak waveform because the asymptotic re- flection coefficient is proportional to angle as described in section 2.4.2. In the second reflected waveform set, the smallest angle produces the largest peak, due to its large reduced reflection coefficient components (see Figure 2.12). The same explanation can be applied to the other reflected waveform sets. Note that the multiple reflections with the largest inciedence angle occur earliest. Although this phenomenon may be intuitively confusing since a longer travel path for a ray inside of a layer is expected for a larger incidence angle, it can be justified by considering the time delay in (4.16), or by the rigorous geometrical ray optics research found in [25]. 117 — 1st Term - - 2nd Term - ‘ '- '- 3rd Term .". ---- 4th Term I 0.5 Amplitude [V/m] 1'9 .— 1.. ._ [— >— .. .— ... Time [nsec] Figure 4.9. Indentification of indivisual reflection in overall scattering from a 5 layered medium (TE polarization) : p2 = no, 62 = 2.5960, 02 = 0[U/ m], 12 = 0.1[m], [1.3 = no, 63 = 5560, 03 =16.7[U/m], 13 = 0.1[m], #4 = no, 64 = 2.5960, 04 = 0[U/m], l4 2 0.1[m],0,-1= 30°. 118 0.5 ~ g; . E 2. o 1, - :3 — 0 degree '5. - - 30 degree E ;.; I:' -- ~ 45 degree _1 75 degree . 4.5% - _2 l L l l l l 1 l 1 Time [nsec] Figure 4.10. Time domain overall reflection for various values of incident angle (TE polarization) : 112 = no, 62 = 2.5960, 02 = ORB/m], [2 = 0.1[m], p3 = #0, 63 = 5560, 03 =16.7[U/m], 13 = 0.1[m], p4 = 110, 64 = 2.5960, 04 = 0[U/m], I4 = 0.1[m]. 119 CHAPTER 5 EXPERIMENTS 5. 1 Introduction The derived formulas in the previous chapters have so far been verified only by numer- ical computation. However, verification by actual experiments may provide additional valuable knowledge such as practical limits in real world applications. In this chap- ter, the description and results of the measurement processes which have been used to verify previously derived expressions are provided. Also, various aspects of the information obtained from the experiments are discussed. The descriptions of the experimental set up and the equipment are given first, and the calibration procedure which is needed to obtain ‘refined’ results from the raw measurement data in order to compare with theory are described. Finally, the experiment results from acutal measurements are provided and discussed. 5.2 Experimental Set Up For experiments, the set up depicted in Figure 5.1 is used. The arch range at Michigan State University consists of two 90° steel rail arcs on to which are attached movable transmitting and receiving antenna mounts. The radius of the rail is 120” (3.05m), and the height of the rail is 47” (1.19m), while the center axes of the antennas are placed at 59” (1.50m) height. Note that although several EM wave absorbers are used to reduce reflection from surrounding environments (e.g. wall, door, or metal rails), this arch range is fundamentally NOT an ‘anechoic’ chamber like those used in frequency domain radiation measurements. A target object (material plates fixed on a metal mounter in this study) is placed at the center of the range. More detailed descriptions on the arch range facility in Michigan State University can be found in 120 [26]. For time domain measurements, Hewelett Packard’s digital sampling oscilloscope HP54750A and its HP54753A time domain transmission / reflectometery (TDT/TDR) plug-in module, providing 20GHz and 18GHz channels, are used. The TDR unit has a built-in step waveform generator that creats pulse trains with 5msec pulse width, 20msec period, and 190mV amplitude. The rising edge of each pulse is used to trigger a Picosecond Pulse Labs (PSPL) 4015B pulse generator. This instrument creates another step using a remote pulse head, which is connected to the input of PSPL 5208 Impulse Forming Network (IF N). The secondary step waveform has a leading edge fall time less than about 15psec, and an amplitude of -9V. An impulse forming network generates the impulse shaped waveform shown in Figure 5.2, using the step as an input. The impulse waveform is fed through connecting cables to a transmitting horn antenna mounted on the arch rail through connecting cables. The plots for the intermidiate waveforms can be found in [27]. The equipment is prone to time-axis drifting, which causes degrading of the measurements. Therefore, pre—measurement warm-up of equipment is required several hours prior to the meaurements. Both the transmitting and receiving horn antennas used for these experiments have a 2 - 18GHz bandwidth, and the polarization of transmitted wave may be changed by rotating the antennas. Dielectric lenses are used to collimate the transmitted spherical wavefront to create an incident plane wave, and also to ensure that a major amount of the transmitted energy is projected on the target. There is a considerable amount of coupling between the two antennas observed in the measured waveform, but most of the coupling can be removed by the calibration process which will be explained later. Several different target objects have been used for the experiments. Polystyrene (u = pg, 6, = 2.55, and a = 0 at 10GHz) and plexiglass (p = no, 6,. =2 2.59, and 0 2 0 at 10GHz) are used for 3 lossless layer (free space, the material and free space) 121 Digital sampling oscilloscope PPL 401513 with 'I'DR/TDT unit f“? HP 54750A ------------------- 4‘ Transmitting antenna Rail Pulse circuit driver Pulse network I .0 I I ,— — .— Remote pulsehead " e . o «t’ Receiving antenna L-.. Dielectric Object lenses 1 ayers Arch Range Figure 5.1. Experimental set up. 122 *AA. rv Amplitude [V] ‘? i i i T #— _4 l l l l l l m l 0 1 2 3 4 5 6 7 8 9 10 Time [nsec] Figure 5.2. (a) Impulse shaped waveform transmitted from pulse forming network. 123 Spectrum 0 10 20 30 40 50 60 Frequency [GHz] Figure 5.2. (b) Spectrum of impulse shaped waveform transmitted from pulse form- ing network. 124 measurements, while a water container made of plexiglass plates is used for 5 lossless or lossy (by filling it with distilled water) measurements. All the plates are 2 feet by 2 feet square in size, and have an approximate thickness of 4.7mm for polystyrene and 5.3mm for plexiglass. Also, an aluminum plate is used as a perfect conductor (PEC) for calibration measurements. 5.3 Calibration Each intermediate component (e.g. cables, antennas) of the experimental set up shown in Figure 5.1 has its own system function due to dispersion, propagation time delay, amplitude attenuation, etc., and each of these changes the original shape of transmitted waveform. Therefore, a calibration procedure is required to isolate and eliminate the effects of the intermediate system functions from the measured wave- form. Let’s denote the original transmitted input waveform by r(t), the impulse response of target object by r(t), the intermediate system impulse response including compo- nent effects and time delays in both transmitting and receiving paths by hsys(t), and the received waveform at the sc0pe by S(t), and denote their Fourier transforms as X (en), R(w), Hsys(w), S (w) respectively. Then, in a measurement, it is expected that and equivalently in the time domain .(t) = 4(4) ... h,,,(t) ... r(t). (5.2) To isolate the unwanted system response H8y3(w), a measurement is performed using an object with a known response. One of the appropriate choices for the object would be a PEC plate, for which the theoretical frequency domain reflection coefficient is 125 -1. Let’s denote the scattering measurement from a PEC plate as C (an) Then C(w) = —X(w)Hsys(w). (5.3) Therefore the intermediate system function is simply given by Hsys(w) = —C(w)/X(w) and the calibrated scattering from a target object is Sea-1(9)) : S(w)/Hsy8(w) = X(w)R(w) (5.4) and scal(t) :2 F“1{Sm,(w)}. There is another factor to be considered in actual calibration procedure. Because of the bandwidth limit of the transmitting and receiving antennas, the measurement data outside of the bandwidth range are unreliable and should be discarded by trun- cating the frequency domain data in outside of the 2-18GHz band and restoring the reduced data size by interpolation. All the data manipulation for the calibration process is executed by using the software WAVECACULATOR, written by Dr. J. E. Ross. 5.4 Measurements 5.4.1 Three lossless layer measurements Using the previously described experimental set up and calibration procedure, actual measurements for several different object layers are performed. For each measure- ment, scattering from an object is measured first, and then the noise signal from the background environment is measured and subtracted from the target signal. A test measurement of the background noise is plotted in Figure 5.3. Due to absorbers used at several critical spots in these experiments, there are no large undesired reflections 126 and the remaining background noise level is usually much smaller than that of the target signal. Also, appropriate time windowing helps remove unwanted strong re- flections from in background objects or mutipath signals. Next, the reflection from PEC plate is measured for calibration purposes and the background noise signal is measured and subtracted again. Ideally, once the system reSponse has been obtained, the calibration measurement does not have to be repeated. But, in practice, it is necessary to repeat it for each measurement, so as to minimize the noise effect from time drifting of the equipment. For the same reason, the time consumed by each measurement must be minimized. Therefore, as a trade off between these considera- tions, 1024 points of data are taken within 10 nano second time range with 256 time averages for each data point. A square polystyrene plate is selected as the first object layer to be measured, because its permittivity (e = 2.5560) has been already verified by a frequency domain measurement at Michigan State University. Figure 5.4 (a) and Figure 5.4 (b) show the time domain object measurement data and its frequency spectrum obtained using a 1024 point fast Fourier transform (FF T), while Figure 5.5 (a) and Figure 5.5 (b) show those of PEC plate measurement data for calibration. These measurements were performed at a 6° aspect angle, which is the closest to normal that is allowed by the experiment system. Note that the amplitude of the reflected wave from the object is quite a bit smaller than that of the transmitted impulse shown in Figure 5.2, so it is plotted on a different scale. Also, it can be recognized that the reflection from the conductor plate is changed significantly from its original transmitted impulse waveform due to the effect of intermediate system function. The system function H3y3(w) is obtained by the method described in previous section 5.2, and shown in Figure 5.6 (a) and (b). The big peaks in the high frequency region ( > 20GHz) come from division errors due to lack of signal content in this band. This fact shows the necessity of truncation of the data spectrum. Finally, the calibrated target responses 127 Amplitude [mV] 0.8 - 0.6 - I _O h I Time [nsec] Figure 5.3. Test measurement of background noise level for the arch range. 128 10 using the system fucntion are shown in Figure 5.7 (a) and (b) for the results without and with the spectrum truncation, respectively. It is obvious that the out—band noise enlarged by division errors produces significant ringing in the calibrated waveform. The calibrated waveform after truncation processing matches well with that from theoretical computation. The slight mismatchs occuring at the third and fourth peaks are mainly due to tilt angle error of the mounted target. The electromagnetic wave propagation velocity inside of polystyrene is W = 1.88 x 108 m/s. Thus, the distance between the third peaks of the measured and theoretical results of about 10 pico seconds corresponds to a two way propagation distance of 1.88mm. Therefore, a slight tilt error or warping of the surface of less than 1mm can make that difference. Similar measurements are performed using a plexiglass (thickness=5.3mm, e = 2.5960) layer for both polarizations and three different incidence angles of 6°, 15° and 30°. The reults are compared with the corresponding theoretical computations and shown in Figure 5.8, Figure 5.9, and Figure 5.10, respectively. Again, all the results show good agreement with the theoretical curves. 5.4.2 Five lossless layer measurements An empty plexiglass container was built to be used as a three lossless layered medium, consisting of three even thickness (5.3mm) layers (plexiglass, free space and plex- iglass). Then, scattering from the five layered object (free space, plexiglass, free space, plexiglass and free space) can be measured. The calibrated results for both polarizations and three different incidence angles are shown in Figure 5.11, Figure 5.12 and Figure 5.13 respectively. All the measurements are well matched with the theoretical results, although they are slightly worse than those from the three layer measurements. This is probably because of the crude construction of the container, With layer thichknesses different from the designed values. But, still mutiple reflec- tions at the layer interfaces can be clearly identified. 129 0.2 l l l l l l l l l I l 0.15 0.05 - . Amplitude [V] o [. I P ..L 0" l l p—- I— Time [nsec] Figure 5.4. (a) Time domain raw measurement data of a polystyrene plate with 0,1 :2 ° (TE polarization). 130 x 10 4 I I F I I 3.5 - . 3 - - 2.5 ~ ~ E E ‘6 2 - - e a. 0) 1.5L 4 1 - - 0.5 - . 0 l l l 0 1O 20 30 40 50 60 Frequency [GHz] Figure 5.4. (b) Frequency domain spectrum of raw measurement data of a POIystyrene plate with 0,1 2 6° (TE polarization). 131 0.2 I I I I I I I I I I l 0.15 I I 0.05 Amplitude [V] O [i 1 -0.05 - J -0.1 ~ - -0.15 - . _0.2 l l l l I l l l I 1 2 3 4 5 6 7 8 9 10 Time [nsec] Figure 5.5. (a) Time domain raw measurement data of a PEC plate with 6,1 = 6° (TE polarization). 132 0.9 r 0.7 r 0.6 - - 0.5 r - Spectrum o.4- — 0.1 l] _ o l 1 l l l 0 10 20 30 40 50 60 Frequency [GHz] Figure 5.5. (b) Frequency domain spectrum of raw measurement data of a PEC plate with 9,1 2 6° (TE polarization). 133 Spectrum 4.5 3.5 2.5 1.5 0.5 T I 10 20 30 40 50 Frequency [GHz] Figure 5.6. (a) Spectrum of intermediate system function. 134 Spectrum 0.2 I I I I I I I I I 1 0.18 I 0.16 I 0.14 1 0.12 I 0.08 0.06 - I 0.04 o l l l l l 1 l l l 0 2 4 6 8 10 12 14 16 18 Frequency [GHz] Figure 5.6. (b) Spectrum of intermediate system function after truncation. 135 20 0.8 I I I T I I — Measured - - Theoretical .>"'.. c u :1 E I 9' I 5 I -0.2 - - -0.4 L 4 I l -0.6 r v _0.8 l l J l l l 0 0.2 0.4 0.6 0.8 1 1.2 Time [nsec] Figure 5.7. (a) Calibrated trasient scattering from a polystyrene layer without Spectrum truncation (TE polarization). 136 0:8 I I I I I I I l 0.6 — Measured - - Theoretical Amplitude [V] I p N I I p h I I _o a: I 4 '0. 8 l l l l l l 0 0.2 0.4 0.6 0.8 1 1.2 Time [nsec] Figure 5.7. (b) Calibrated trasient scattering from a polystyrene layer with spectrum truncation (TE polarization). 137 0.8 T T I I I I 0.6 - - — Measured - - Theoretical l l 0.2 Amplltude [V] o l l l I _O N I J l O J:- 1 I P a: I 1 “0.8 l l l l l J 0 0.2 0.4 0.6 0.8 1 1.2 Time [nsec] Figure 5.8. (a) Transient scattered field from a plexiglass layer with incidence angle 6° (TE polarization). 138 0.8 I I I I I I 0.6- « i J -— Measured - - Theoretical E 0 x o 3 hi; i E < _. -0.6 - . “0.8 l l l l l 1 0 0.2 0.4 0.6 0.8 1 1.2 Time [nsec] Figure 5.8. (b) Transient scattered field from a plexiglass layer with incidence angle 6° (TM polarization). 139 0.8 I I I I I T 0.6 - - — Measured - - Theoretical E .8 / 3 ~ ~ w v =4 E < —I -05 - _ _0.8 l l l l l J 0 0.2 0.4 0.6 0.8 1 1.2 Time [nsec] Figure 5.9. (a) Transient scattered field from a plexiglass layer with incidence angle 15° (TE polarization). 140 0.8 I I I I I I 0.6 ~ - — Measured - - Theoretical E o 'o / 3 1 .vI % z E I -0.2 - I . I l 1 -0.4 _ ,I _ -0.6t - '0.8 J l l l l l 0 0.2 0.4 0.6 0.8 1 1.2 Time [nsec] Figure 5.9. (b) Transient scattered field from a plexiglass layer with incidence angle 15° (TM polarization). 141 0.8 I I I I I I i — Measured - - Theoretical E 0 r U 3 A m __ E E < -0.6 - - _o.8 l l l l l l 0 0.2 0.4 0.6 0.8 1 1.2 Time [nsec] Figure 5.10. (a) Transient scattered field from a plexiglass layer with incidence angle 30° (TE polarization). 142 0.8 I T I I I I 0.6 - - , — Measured - - Theoretical E 0 “O r a m i E < -05 _ 4 _0.8 l l l l l l 0 0.2 0.4 0.6 0.8 1 1.2 Time [nsec] Figure 5.10. (b) Transient scattered field from a plexiglass layer with incidence angle 30° (TM polarization). 143 008 I I r — Measured - - Theoretical _— 0.2 ~ Amplltude [V] O Ik— I p N I ‘ I _O G I 0 0.2 0.4 0.6 0.8 1 1.2 Time [nsec] Figure 5.11. (a) Transient scattered field from a plexiglass container with incidence angle 6° (TE polarization). 144 0.8 0.6 - T 0.4 Amplltude [V] O l — Measured - - Theoretical 0.2 0.4 Time [nsec] 0.6 0.8 1.2 Figure 5.11. (b) Transient scattered field from a plexiglass container with incidence angle 6° (TM polarization). 145 0.8 1 1 0.4 - — Measured - - Theoretical .— 0.2 l Amplltude [V] o k- _ 1 -0.2 l l l -o.4- .' —0.6 " _O. 8 l l l l L l 0 0.2 0.4 0.6 0.8 1 1.2 Time [nsec] Figure 5.12. (a) Transient scattered field from a plexiglass container with incidence angle 15° (TE polarization). 146 0.8 I I I I I I 0.6 ~ . — Measured - - Theoretical Amplitude [V] -o.6~ - _0.8 l l l l l l 0 0.2 0.4 0.6 0.8 1 1.2 Time [nsec] Figure 5.12. (b) Transient scattered field from a plexiglass container with incidence angle 15° (TM polarization). 147 Another measurement that may have practical interest is performed. Two plates of plexiglass are ‘mechanically’ adhered with some pressure, and the total thickness is 9.8mm. Then, the scattering from the object is measured and plotted in Figure 5.14 (a). What is expected is a reflection from a ‘single’ plate of thickness 9.8mm, and is a waveform consisting of two isolated peaks. But the actual measurement is shows quite a different waveform. Therefore, the theoretical reflection from a three layered object (the five layered medium case) is computed under the assumption that there is a small ‘invisible’ air gap of 0.05mm thickness between the two plexiglass plates, and compared with the measured data. As shown in Figure 5.14, the modified model gives a good match with the measuree data. This is a good example of ‘non destructive’ TDR inspection which is being widely used in industry, and suggests the possibility that more accurate information (size and kind) of mechanical defects smaller than the transmitted pulse width can be obtained, since the exact theoretical response can be found for even lossy materials from the theory derived in this study. 5.4.3 Lossy layered medium measurements The plexiglass container used in the previous measurement is filled with distilled water for the measurement of scattering from a lossy five layered object. Note that water is a strong polar material, and therefore has significant dielectric loss at high frequency. The theoretical response is computed by converting this dielectric loss to an equivalent conductivity aeq. Unfortunately, the dielectric loss is a function of frequency, while frequency independent parameters are assumed in this study. The frequency dependence of permittivity which is usually denoted using complex permittivity c = 6’ — jc" where e" = 931, can be modelled by various functions, including Debye equation [28] as shown in Figure 5.15 for distilled water at 20°C. Considering the available distilled water parameters at several frequency points found in [21], the bandwidth range of 2-18GHz, and the maximum dielectric loss frequency in the Debye model, the equivalent constant conductivity is chosen to be 16.70 / m, the 148 0.8 I I 0.6 0.4 — Measured - - Theoretical l 0.2 Am plltude [V] o _0.8 l l 1 l l l 0 0.2 0.4 0.6 0.8 1 1.2 Time [nsec] Figure 5.13. (a) Transient scattered field from a plexiglass container with incidence angle 30° (TE polarization). 149 0.8 1 1 1 1 1 1 0.6 7 .( 0.4 ‘ a 1 1 — Measured l 1 - - Theoretical 0.2 - ' , ' _ .—. l I Z I ' I 9 I l 3 0 - l I — E I I E l I ‘t l l -02 - ‘ . -0.6 - a '0-8 1 l l l l l 0 0.2 0.4 0.6 0.8 1 1.2 Time [nsec] Figure 5.13. (b) Transient scattered field from a plexiglass container with incidence angle 30° (TM polarization). 150 0.8 I I T I I I 0.6 - - 0.4 -fl - I — Measured - - Theoretical 0.2 - r Amplltude [V] o -0.2 - a -0.4 - ~ -0.6 - - “0.8 l l l l I I 0 0.2 0.4 0.6 0.8 1 1.2 Time [nsec] Figure 5.14. (a) Comparison of trasient reflection from a mechanically adhered plexiglass plates with that for a single material (9,-1 2 6°, TE polarization). 151 0.8 I I I I I I I l 0.6 l j l 0.4 — Measured - - Theoretical Amplltude [V] O ”0.8 l l l l l J 0 0.2 0.4 0.6 0.8 1 1.2 Time [nsec] Figure 5.14. (b) Comparison of trasient reflection from a mechanically adhered plexiglass plates with that for an air gap inserted model (61-1 : 6°, TE polarization). 152 Magnitude of Relatlve Permittivity 80 I I I I I I 70 r 60” "' Real - - Imaginary 50 ' 40 r 30 * l 20 I, / / // 10 _ ’l ’/ I ’ ’ 0 __.=—-"T“ I 1 41 I 7.5 8 8.5 9 9.5 10 10.5 logiO( freq. ) Figure 5.15. —e” /60 curve for water at 20°C given by the Debye model. 153 value at 10GHz. Therefore the resulting material parameters for the layers are given by [1.2 = no, (.2 = 2.5960, 02 = OU/m, [2 = 5.3mm, p3 = #0, 63 = 5560, 03 = 16.7U/m, 13 = 5.3mm, p4 = #10, 64 = 2.5960, 04 = OU/m, 14 = 5.3mm. A measurement result for 6,1 2 6° incidence angle is shown in Figure 5.16. As seen in the figure, there is considerable mismatch in the theoretical and measured data. However, by careful observation, it can be realized that there is a constant scaling relationship between the two curves. It turns out that the cause of this mismatch is ‘mechanical’ rather than ‘electrical’. That is, the shape of the water filled container has been changed by water pressure and gravity, and it causes a tilt angle which result in a reduction of the received wave amplitude. This explanation can be justified from the observation that the first reflection peak from air and plexiglass interface, which can not be eflected by the third layer material, has been reduced. By considering this phenomenon, all the measurements for the distilled water container are scaled by the ratio of the first peak amplitudes of the measured data to that of the theoretical result. The scaled measurements are shown in Figure 5.17-Figure 5.19, and all of them show a good match between measurements and theoretical expectations. 154 I — Measured I - - Theoretical l Amplltude [V] ..1 ~ - -1.5 r - _2 1 l l l l l 0 0.2 0.4 0.6 0.8 1 1.2 Time [nsec] Figure 5.16. Trial measurements for trasient scattered field from a 5 lossy layered medium with incidence angle 6° (TE polarization). 155 I — Measured l - - Theoretical Amplltude [V] -1 — - -1.5 - - _2 I l I I I I 0 0.2 0.4 0.6 0.8 1 1.2 Time [nsec] Figure 5.17. (a) Transient scattered field from a 5 lossy layered medium with incidence angle 6° (TE polarization). 156 1.5- - 1- I . , — Measured I - - Theoretical 0.5- I ~ I . I Amplltude [V] o -05 - - -1 - . ..1_5 _ a '20 0:2 0:4 0:6 ole 1 1:2 Time [nsec] Figure 5.17. (b) Transient scattered field from a 5 lossy layered medium with incidence angle 6° (TM polarization). 157 1.5- . .- I - I, — Measured I - - Theoretical _ 0.5~ a 3 I a 0 I i. 5 -0.5- - -1 — . -1.5r . _2 I I I I I I 0 0.2 0.4 0.6 0.8 1 1.2 Time [nsec] Figure 5.18. (a) Transient scattered field from a 5 lossy layered medium with incidence angle 15° (TE polarization). 158 Am plltude [V] O l — Measured — - Theoretical 0 0.2 0.4 0.6 Time [nsec] 0.8 Figure 5.18. (b) Transient scattered field from a 5 lossy layered medium incidence angle 15° (TM polarization). 159 1.2 with 1.5“ - — Measured - - Theoretical Amplltude [V] _2 l l l l l l 0 0.2 0.4 0.6 0.8 1 1.2 Time [nsec] Figure 5.19. (a) Transient scattered field from a 5 lossy layered medium with incidence angle 30° (TE polarization). 160 1.5* ‘ — Measured - - Theoretical Amplltude [V] _2 l l l l l l 0 0.2 0.4 0.6 0.8 1 1.2 Time [nsec] Figure 5.19. (b) Transient scattered field from a 5 lossy layered medium with incidence angle 30° (TM polarization). 161 CHAPTER 6 CONCLUSIONS Transient scattering from a Inulti-layered medium by an oblique uniform plane wave has been studied in this thesis. The frequency domain reflection coefficients were classified according to their different corresponding inverse Fourier transform pairs through careful algebraic manipulation and by using appropriate branch cuts. The exact analytical transient reflection coefficients of a single interface for TB and TM polarizations have been developed from inverse Fourier transforms of the frequency domain reflection coefficients. Also, approximate forms of the reflection coeflicients were suggested for the TB polarization case as an example. It was found that all coef- ficients are causal for certain case. A reasonable explanation for the non causality of other cases could not be found. However, several reports about the occurrence of non causality in theoretical or experimental developments observed by other researchers were introduced. The expressions for the overall transient reflection coefficients were derived using a series expansion and the convolution theorem. The derived interfacial and overall reflection coefficients were verified by numerical computation using the IFF T. Also, actual time domain reflection measurements were performed to compare with the theoretically derived results, using lossless and lossy layers as the objects. All the measured and calibrated results have matched well with computed results based on the derived theory, and therefore verified the correctness of the derived transient expressions. As discussed in Chapter 1, the purpose of this study is to provide fundamental background for parameter estimation of a multi-layered object using the transient scattered field. Except for the unexplained non causal cases, which should be studied separately in the future, the exact theoretical solutions for the transient scattered 162 field have been made available by the results of this study. Because each reflection from each layer has the information about the constitutive parameters, and because their functional relationship with the input waveform has been determined by this study, the next step is to find reliable methods to extract the information from the measured transient waveform. 163 APPENDICES 164 APPENDIX A OPERATIONS OF COMPLEX VALUED SQUARE ROOT FUNCTIONS USING THE BRANCH CUTS The derivation processes of frequency domain expressions explained in Chapter 2 and Chapter 3 require algebraic manipulations of square roots of which arguments are complex numbers. Sometimes, the results of algebraic manipulations using con- ventional negative real axis branch cut are different from those using the branch cuts derived section 2.2.2 from radiation condition. Because the branch cuts from radiation condition are consistantly used in this thesis, it is helpful to introduce selected alge- braic manipulations for derivation of frequency domain expressions using the branch cuts. VVhenD>0andB>0 B VD32+Bs=—\/D\/Ds+B=—\/D\/§1/s+5 (A.1) while for D < 0 and B > 0 +j\/l—D_l\/§1/3 - 1%. : (w > 0) VD32+Bs=\/§\/Ds+ = . —j\/W\/E,/s—,—,%.— : (w<0) (A2) By denoting a square root from the conventional negative real axis branch cut as \/'()a \/3"l'Bn~l~1::t S+Bn+1: .8+Bn+1 (A3) 1/s+B,, 3+Bn s-l—Bn ' where the upper sign corresponds to Bn+1 > 8,, and lower sign does to Bn+1 < B". The relationship between the two different branch cut square roots is needed because its inverse Fourier transform pair in reference material is given by using the negative 165 real axis branch cut. Similarly, 3_Bn+1_i 3_Bn+1_ . S—Bn+1 _ _ A. s—Bn s-Bn S—Bn ( 4) where the upper sign corresponds to Bn+1 < B” and lower sign does to Bn+1 > B". On the other hand, to find its inverse Fourier transform more easily, each of the other fraction of square root terms can be rewritten as VS+Bn+l _ 3+—Bn+1 s—Bn V3 I3+Bn:l Is 5+Bn+1 = “/ A.5 s s—SBn ( ) and 3 "' Bn+1 3 — Bn+1 1/s+B,, S+Bn __ _ 3_Bn+l 5 _— l’ 3 3+8” ./s—Bn+1 / s = — —' . A6 5 8+8” ( ) 166 APPENDIX B INVERSE FOURIER TRANSFORM PAIRS In this appendix, the inverse Fourier transform pairs which are used to derive transient expressions, are provided. Some of these can be directly found from [23], while the others are derived from the given pairs. Assume R(w) = G(w)U*(w) + G(—w)U(—w) where G*(w) is a complex conjugate of C(12), and U(w) is a frequency domain unit step function. By letting g(t) = F‘1{G(w)}, 9*(t) : F‘1{G*(w)} and using the duality theorem for Fourier transform that says, F(t) : F‘1{27rf(—w)} when f(t) = F‘1{F(w)}, 7r6(t) -I- j—t = 27rF—1{U(—w)} 5U) 1 _ _1 T — 55E — 27rF {U(w)} (13-1) and RU) = g(t) * {521) - —.1-}+g*(t)*{-6(2—t) + 5%} Zyrrt J7T = {g(t) + g*(t)} * ég—t) + {g(t) - 9’10} * 531? = gem + gm} + 23,; * {g(t) — rm}. (B2) From the inverse transform pair of 1 4:) e‘Ptu(t) (B 3) 3 + P for P > 0, and —e‘P‘u(—t) for P < 0, its time derivative form is given by 167 ‘ <=> d_t {e’Ptu(t)} s + P : —-Pe_Ptu(l‘) + 8_Pt6(t) : e—Pt {6(t) _ Pu(t)} (BA) for P > 0, and d s :13 ‘i’ a PERM—0} : Pe-P‘u(—t) + e‘P‘rW) = e-Pt {5(1) + Pu(t)} (13-5) for P < O. From 561.0 of [23], 3 + Bn+1 1— {/— s + B" Q B". — Bn Bn — Bn Lt) + 10 (——+1——-t) } U(t), _Bn+1 — Bne___i-__(Bn 12+Bn)t I 2 1 2 2 (B6) and it can be derived that 1 __ . 3 _’ Bn+1 s — Bn 3,, — Bn (8.. 1+Bn> Bn —- 8,, BT, — Bn -—+12—'—6 +2 t{11(—t—1-2——t) — IO (——+1—2———t) } U(—t). (B7) 168 Also, from (A.5), ,_ .' B" B" _ n 1 B11 B11 1/E?iic>6(t)+—2+—le +{1( 2.1,)+,0( 2+1t)}uO .AND. DN >0. An input waveform is obtained from a text data file which contains actual measurement data of transmitting impulse, called "inwave.dat" Text data file "1ayers.dat" is required as an input parameter description for each layer. Text data file "gammax.out" will be output. OOOOOOOOOOOOOOO **************************‘k********************************** CHARACTER*8 TMPS INTEGER ITCONV INTEGER NT,SMAX,LMAX,I,K,NL PARAMETER (SMAx=10250) PARAMETER (LMAleO) REAL MU(LMAX),EN(LMAX),SIG(LMAX),THCK(LMAX),B(LMAX),DN(LMAX) REAL C1(LMAX-l),C2(LMAX-l),C3(LMAX-l),Pl(LMAX-l),P2(LMAX—1) REAL BTA(LMAX-1),D(LMAX—l),CD(LMAX-l) REAL TISMAX),X(SMAX) REAL R(LMAX—l,SMAX),P(LMAX-l,SMAX),GAM(LMAX-l,SMAX) REAL THETAI,G,H,TMAX,DT REAL TMP,PSZERO REAL PI,E0,MUO COMMON /PWRS/C1,C2,C3,P1,P2,BTA,CD COMMON /ARRS/R,P,GAM COMMON /TARRS/T COMMON /INDX/I COMMON /TINDX/K,NT COMMON /LINDX/ITCONV,NL PI=4.*ATAN(1.) E0=l.0E-9/(36.0*PI) MUO=4.0*PI*1.0E-7 PSZERO=1.0E-25 OPEN(5,FILE='layers.dat',STATUS='unknown') OPEN(6,FILE='gammax.0ut',STATUSz'unknown') OPEN(7,FILE='param.buf',STATUSz'unknown') OPEN(8,FILE='inwave.dat',STATUS='unknown') 172 C **************************************** C READ AND COMPUTE CONSTANTS OF EACH LAYER C **************************************** WRITE(*,*) 'Enter the number of layers in <> :' READ(*,*) NL C ******************* C READ INCIDNET ANGLE C ******************* ll FORMAT(A8,F12.4) 12 FORMAT(3X, F12.4,2X, A8,'= ',E12.4) 20 FORMATI/l READ(5,ll) TMPS,TMP THETAI = PI/TMP WRITE(*,12) TMP,TMPS,THETAI WRITE(7,12) TMP,TMPS,THETAI C ************************ C READ MATERIAL PARAMETERS C ************************ DO 21 I=1,NL READ(5,11) TMPS,TMP MU(I)=TMP*MUO WRITE(*,12)TMP,TMPS,MU(I) WRITE(7,12)TMP,TMPS,MU(I) READ(5,11) TMPS,TMP EN(I)=TMP*E0 WRITE(*,12)TMP,TMPS,EN(I) WRITE(7,12)TMP,TMPS,EN(I) READ(5,ll) TMPS,SIG(I) WRITE(*,12)SIG(I),TMPS,SIG(I) WRITE(7,12)SIG(I),TMPS,SIG(I) READ(5,ll) TMPS,THCK(I) WRITE(*,12) THCK(I),TMPS,THCK(I) WRITE(7,12) THCK(I),TMPS,THCK(I) TMP=MU(1)*EN(1)*((SIN(THETAI))**2) DN(I)=MU(I)*EN(I)-TMP BII)=MU(I)*SIG(I)/ABS(DN(I)) WRITE(*,13) DN(I),B(I) WRITE(7,13) DN(I),B(I) l3 FORMAT(5X,'Dnz',ElO.3,2X,'an',E10.3,/) 21 CONTINUE 173 C at****************************************** C COMPUTE LAYER PARAMETERS FOR EACH INTERFACE C *‘k‘k‘k‘k'k'k‘k'k'k'k'k‘k'k'k**************************** DO 22 I=l,NL-1 WRITE(*,l4) I WRITE(7,14) I 14 FORMAT(/,3X,'Interface Index =',I3) DII)=((MU(I)**2)*DN= 0 C ***************************************** REAL FUNCTION FUNCP(U) INTEGER NT,I,K,SMAX,LMAX PARAMETER (SMAX=10250) PARAMETER (LMAX=10) REAL U,DTMP REAL SIO,SI1 REAL T(SMAX) REAL Cl(LMAX-l),C2(LMAX-l),C3(LMAX-I),Pl(LMAX-I),P2(LMAX-l) REAL BTA(LMAX-l),CD(LMAX-l) COMMON /PWRS/C1,C2,C3,PI,P2,BTA,CD COMMON /TARRS/T COMMON /INDX/I COMMON /TINDX/K,NT DTMP=—(P2(I)*T(K)+(P1(I)—P2(I))*U) 181 C C C 0000 FUNCP=SI1(BTA(I)*U,DTMP)+SIO(BTA(I)*U,DTMP) RETURN END ‘k*************************************** INTEGRAND FUNCTION DEFINITION FOR P2 < O *9:************************************** REAL FUNCTION FUNCN(U) INTEGER NT,I,K,SMAX,LMAX PARAMETER (SMAX210250) PARAMETER (LMAX=10) REAL U,DTMP REAL SIO,SIl REAL T(SMAX) REAL Cl(LMAX-l),C2(LMAX-l),C3(LMAX—l),Pl(LMAX—l),P2(LMAX—l) REAL BTA(LMAX-l),CD(LMAX-l) COMMON /PWRS/C1,C2,C3,Pl,P2,BTA,CD COMMON /TARRS/T COMMON /INDX/I COMMON /TINDX/K,NT DTMP=-PI(I)*T(K)+(P1(I)/P2(I)-l.0)*U FUNCN=SIO(BTA(I)*(T(K)-U/P2(I)),DTMP)+ + SII(BTA(I)*(T(K)-U/P2(I)),DTMP) RETURN END **************************************** PRODUCT OF EXPONENTIAL AND MODIFIED BESSEL FUNCTION OF ZEROTH ORDER **************************************** REAL FUNCTION SIO(X,OFST) REAL X,T,Y,OFST T=ABS(X)/3.75 IF (ABS(X).LT.3.75) THEN Y=EXP(OFST) SIO=1.0*Y Y=Y*(T**2) SIO=SIO+3.5156229*Y Y=Y*(T**2) SIO=SIO+3.0899424*Y Y=Y*(T**2) SIO=SIO+1.2067492*Y Y=Y*(T**2) SIO=SIO+O.2659732*Y Y=Y*(T**2) SIO=SIO+0.0360768*Y Y=Y*(T**2) 182 0000 SIO=SIO+0.0045813*Y ELSE Y=EXP(ABS(X)+OFST)/SQRT(ABS(X)) SIO=0.39894228*Y Y=Y/T SIO=SIO+0.0I328592*Y Y=Y/T SIO=SIO+0.002253I9*Y Y=Y/T SIO=SIO-0.00I57565*Y Y=Y/T SIO=SIO+0.00916281*Y Y=Y/T SIO=SIO-0.02057706*Y Y=Y/T SIO=SIO+0.02635537*Y Y=Y/T SIO=SIO-0.01647633*Y Y=Y/T SIO=SIO+0.00392377*Y ENDIF RETURN END **************************************** PRODUCT OF EXPONENTIAL AND MODIFIED BESSEL FUNCTION OF FIRST ORDER **************************************** REAL FUNCTION SII(X,OFST) REAL X,T,Y,OFST T=ABS(X)/3.75 IF (ABS(X).LT.3.75) THEN Y=X*EXP(OFST) SIl=O.5*Y Y=Y*(T**2) SIl=SIl+O.87890594*Y Y=Y*(T**2) SIl=SI1+O.51498869*Y Y=Y*(T**2) SIl=SI1+O.15084934*Y Y=Y*(T**2) SIl=SIl+0.02658733*Y Y=Y*(T**2) SIl=SI1+0.00301532*Y Y=Y*(T**2) SI1=SI1+0.00032411*Y ELSE 183 Y=EXP(ABS(X)+OFST)/SQRT(ABS(X)) SIl=0.39894228*Y Y=Y/T SIl=SIl-0.03988024*Y Y=Y/T SIl=SIl—0.00362018*Y Y=Y/T SIl=SI1+0.00l63801*Y Y=Y/T SIl=SIl-0.0lO31555*Y Y=Y/T SIl=SI1+0.02282967*Y Y=Y/T SIl=SIl-0.02895312*Y Y=Y/T SIl=SI1+0.0l787654*Y Y=Y/T SIl=SIl-0.00420059*Y IF (X.LT.0.0) SI1=-SI1 ENDIF RETURN END 184 BIBLIOGRAPHY 185 BIBLIOGRAPHY [1] J. 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