# Numerical calculations of microwave scattering from dielectric structures used in vegetation models

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Num erical Calculations of Microwave Scattering from D ielectric Structures used in V egetation M odels by P a o lo de M a tth a e is L a u re a D egree in E lecrical E n g in e e rin g J u n e 1991, U n iv e rsity o f R o m a “T o r V e rg a ta ” , R o m a , Ita ly A d is s e rta tio n s u b m itte d to th e F a c u lty of th e S chool o f E n g in e e rin g a n d A p p lie d Science of T h e G eo rg e W a sh in g to n U n iv e rsity in p a r tia l fu llfillm en t of th e re q u ire m e n ts for th e d eg ree of D o c to r of Science J a n u a r y 31, 2006 D is s e rta tio n d ire c te d by R o g e r H e n ry L a n g P ro fe sso r o f E n g in e e rin g a n d A p p lie d Science Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 3203663 INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and im proper alignm ent can adversely affect reproduction. In the unlikely event that the author did not send a complete m anuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. ® UMI UMI Microform 3203663 Copyright 2006 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A bstract N um erical models for electrom agnetic scattering from individual cpm onents of a vegetation canopy are studied. T he m odels are based on a surface integral representation of the electrom agnetic fields on th e m aterial bodies representing th e vegetation constituents. Surface integral equations are obtained by using th e equivalence principles, and th eir unknowns are equivalent surface currents. From the equivalent surface currents, the b istatic scattering coefficient is evaluated. T he m ethod of m om ents is applied to solve these integral equations in the case of two-dim ensional objects and of axisym m etric scatterers. T h e p a rticu la r geom etry of these bodies allows a sim plifica tio n of the num erical solution. T he results are validated for a num ber of shapes, for b o th perfectly conducting and hom ogeneous dielectric bodies. T he num erical approach is then com pared to a n a lytical approxim ate m odels for scattering from vegetation com ponents. In order to take into account th e presence of th e ground under a vegetation canopy, a sem i infinite half-space is introduced in th e modeling^ a n d th e surface integral equations are m odified to consider its effects. T h e resulting integral equations contain a dyadic G reen’s function for th e half space. In particular, axisym m etric objects are considered, an d the dyadic G reen’s function is w ritte n in term s of cylindrical waves. T he solution by th e m eth o d of m om ents requires th e calculation of Som merfeld-type integrals, an d a brief discussion of th e issues related to their evaluation is included. T he half-space num erical approach is finally applied to th e problem of m odeling tree tru n k s over a fiat ground. Its results are com pared w ith those of an analytical approach th a t neglects the near field interaction of th e tru n k a n d th e surface of th e ground. ii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Acknowledgm ents My sincere appreciation goes first to my advisor, Prof. Roger H enry Lang. W ithout his guidance, patience a n d su p p o rt th ro u g h my doctoral studies, it would have been impossible for me to com plete this work. I would also like to th a n k Prof. W asyl W asylkiwskyj and Prof. W alter K ahn, for all w hat I have learned in th eir E lectrom agnetics classes, an d extend my appreciation to all the m em bers of the D issertation Com m itee for th eir insightful com m ents and advice. M any thanks also to Dr. Allen Glisson for providing a useful num erical code for com parison of results, and to fellow g rad u ate stu d e n t C uneyt U tk u for useful advice and discussions. Last, b u t not least, I would like to express my g ratitu d e to my fam ily and friends who have supported and encouraged me th roughout the m any years of stu d y leading to the com pletion of this degree, and to th e E C E D epartm ent secretaries D eborah Sw anson and Sariane Leigh for their adm inistrative help. m Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. D edication This d issertatio n is dedicated to my dad A ugusto, and in m em ory of my m om Adele. IV Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C ontents A b stra ct ii A ck now ledgm ents iii D ed ica tio n iv C on ten ts v List o f F igures ix List o f Tables xxii 1 In trod u ction 2 1 1.1 M otivation ........................................................................................................................................ 1 1.2 B a c k g ro u n d ........................................................................................................................................ 2 1.3 O v e r v ie w ........................................................................................................................................... 6 T he surface in tegral eq u a tio n form u lation 9 2.1 T he equivalence p r i n c i p l e ............................................................................................................. 10 2.2 Surface integral equations in unbounded space .................................................................... 16 2.3 O ther forms of coupled surface integral e q u a tio n s ................................................................. 19 2.4 A lternative form for th e dyadic G reen’s f u n c t i o n ................................................................. 21 v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.5 S cattered field and scattering c o e ff ic ie n t.................................................................................. 3 T w ord im en sion al b o d ies 21 27 3.1 T he coupled surface integral equations in two-dim ensions ................................................. 28 3.2 Far-field and scattering w i d t h ..................................................................................................... 30 3.3 Perfect electric conducting two-dim ensional b o d y ................................................................ 32 3.3.1 T M incident wave ............................................................................................................. 32 3.3.2 T E incident w a v e ................................................................................................................. 41 Dielectric two-dim ensional b o d y ................................................................................................. 50 3.4.1 ............................................................................................................. 50 C o n c lu s io n s ....................................................................................................................................... 61 3.4 3.5 TM incident wave 4 B od ies o f R ev o lu tio n 63 4.1 G eom etry of th e p r o b l e m ........................................................................................................... 63 4.2 T he electric field integral equations for a body of r e v o lu tio n .............................................. 65 4.2.1 F o rm u la tio n ............................................................................................................................ 65 4.2.2 Separation of m o d e s ........................................................................................................... 67 4.2.3 M ethod of m om ents s o l u t i o n .......................................................................................... 70 4.2.4 Incident f i e l d ........................................................................................................................ 73 4.2.5 S cattered field in the rad iatio n z o n e ............................................................................ 75 V a lid a tio n .......................................................................................................................................... 77 4.3 4.4 4.3.1 Perfectly electric conducting bodies of r e v o l u t io n .................................................... 78 4.3.2 Homogeneous dielectric body of re v o lu tio n ................................................................. 89 A pplication to scattering from vegetation ............................................................................ 96 4.4.1 F in ite-length dielectric cylinder a p p r o x i m a t io n ....................................................... 97 4.4.2 T apered cylinder approxim ation .....................................................................................100 vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.4.3 R esults an d c o m p a riso n s.....................................................................................................103 4.4.4 C o n c lu s io n s ............................................................................................................................ 116 5 H alf-Space P ro b lem 117 5.1 T he dyadic G reen’s function for an infinite, homogeneous m e d i u m .................................... 118 5.2 T h e dyadic G reen’s function for sem i-infinite m e d ia .................................................................122 5.2.1 5.3 T he dyadic G reen’s function in th e far S cattering from a B ody in a Sem i-Infinite M e d i u m ................................................................128 5.3.1 Expression of Surface Integral E quations for Sem i-Infinite M e d iu m .....................128 5.3.2 Solution of th e Surface Integral E quations for a B O R in H alf-S p ace.................... 132 5.3.3 Expression of “incident” field on th e surface S 5.3.4 C o m p utation of the scattering coefficient 5.4 f i e l d ....................................................125 ..........................................................136 .....................................................................140 E valuation of sp ectral i n t e g r a l s .................................................................................................... 141 5.4.1 In tegration by T rapezoidal Rule ......................................................................................143 5.4.2 Tank T ra n sfo rm a tio n .............................................................................................................. 144 5.4.3 W eighted-Averages M e t h o d ................................................................................................145 5.4.4 V alidation of Integration M ethods 5.5 6 .................................................................................. 147 V a lid a tio n ............................................................................................................................................. 150 5.5.1 Perfectly conducting body of r e v o l u t i o n ............................... 5.5.2 Dielectric body of revolution 151 ............................................................................................ 153 5.6 A pplication to scatterin g from vegetation .......................................................................154 5.7 C o n c lu s io n s ..........................................................................................................................................163 C onclusions 167 B ibliography 171 vii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p en d ix A E lectro m a g n etic fields in u n b ou n d ed m edia A .l 181 E lectrom agnetic fields generated by electrom agnetic s o u r c e s .............................................. 181 A.2 G reen’s theorem for a m ultiply connected r e g i o n ...................................................................184 A.3 D erivation of expression (2.17) 186 A.4 E valuation of surface integral in (2.22) at singularity p o i n t ..................................................188 A. 5 Equivalence of alternative dyadic G reen’s f u n c t i o n ............................................................... 190 B D erivation o f tw o-d im en sion al G reen ’s fu n ction 193 C B od ies o f R ev o lu tio n 195 D C .l Explicit form of operators a p q ....................................................................................................... 195 C.2 Explicit form of operators (3p q ....................................................................................................... 196 C.3 D erivation of operators a pq .......................................................................................................... 199 C.4 D erivation of operators /3pq .......................................................................................................... 201 C.5 D erivation of scattering a m p litu d e s ............................................................................................ 202 H alf-Space P ro b lem 205 D .l Expression of dyadic G reen’s function c o m p o n e n t s ............................................................... 205 D.1.1 Electric dyadic G reen’s f u n c t i o n s ....................................................................................205 D .l.2 M agnetic dyadic G reen’s f u n c t i o n s ................................................................................ 207 D.2 D erivation of scattering a m p litu d e s .................... viii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 208 List o f Figures 2.1 D efinition of surfaces 5 i and 52.................................................................................................... 11 2.2 Equivalent sources configuration.................................................................................................. 13 2.3 E xterior problem ............................................................................................................................... 15 2.4 Interior problem ................................................................................................................................. 17 2.5 Definition of geom etry for scattered field calculation............................................................ 21 2.6 Incident an d scattered plane waves on an o b ject.................................................................... 22 3.1 Geom etry of a tw o-dim ensional b o d y ........................................................................................ 26 3.2 Definition of scatterin g problem for a tw o-dim ensional b o d y ............................................ 28 3.3 (a) D iscretization of curve C. (b) Basis (expansion) functions.......................................... 31 3.4 TM and T E incident waves on a circular cylinder of radius a ........................................... 33 3.5 Surface equivalent current obtained w ith m eth o d of m om ents and exact solution for k{]<a = 0.1 a n d num ber of points N — 12. N um ber of term s in analytical expression sum m ation is M = 10....................................................................................................................... 3.6 34 Surface equivalent current obtained w ith m eth o d of m om ents and exact solution for koa = 1.0 an d n um ber of points N = 20. N um ber of term s in analytical expression sum m ation is M = 15....................................................................................................................... ix Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 35 3.7 Surface equivalent current obtained w ith m ethod of m om ents an d exact solution for koa = 2ir and num ber of points N = 100. N um ber of term s in analytical expression sum m ation is M = 20....................................................................................................................... 3.8 35 Surface equivalent current obtained w ith m ethod of m om ents and exact solution for k^a = 87r and num ber of points N = 500. N um ber of term s in analytical expression sum m ation is M = 40....................................................................................................................... 3.9 36 Scattering w idth obtained w ith m ethod of m om ents an d exact solution vs. koa, for scattering angles 0 = 0° (backscattering) and 0 = 90° (forw ard scatterin g )....................... 36 3.10 Scattering w idth obtained w ith m ethod of m om ents a n d exact solution vs. <j>3, for koa — 0.1, num ber of M OM points N = 12, and analytical expression tru n cated to M = 10 term s...................................................................................................................................... 37 3.11 Scattering w idth obtained w ith m ethod of m om ents an d exact solution vs. <f>s, for koa = 1.0, num ber of M OM points N — 20, and analytical expression tru n cated to M — 15 term s...................................................................................................................................... 38 3.12 Scattering w idth obtained w ith m ethod of m om ents a n d exact solution vs. <f>s, for koa = 2n, num ber of M OM points N = 130, and an alytical expression tru n cated to M = 20 term s...................................................................................................................................... 38 3.13 Scattering w idth obtained w ith m ethod of m om ents a n d exact solution vs. 0 S, for koa = 87r, num ber of M OM points N = 500, and an alytical expression tru n cated to M = 40 te rm s ...........................................................................................................................................39 3.14 Surface equivalent current obtained w ith m ethod of m om ents a n d exact solution for koa = 0.1 and num ber of points N = 12. N um ber of term s in analytical expression sum m ation is M = 10........................................................................................................................ x Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 42 3.15 Surface equivalent current obtained w ith m ethod of m om ents and exact solution for koa = 1.0 and num ber of points N = 20. N um ber of term s in analytical expression sum m ation is M = 15....................................................................................................................... 43 3.16 Surface equivalent current obtained w ith m ethod of m om ents and exact solution for koa — 2n an d num ber of points N = 130. N um ber of term s in analytical expression sum m ation is M = 20....................................................................................................................... 43 3.17 Surface equivalent current obtained w ith m ethod of m om ents and exact solution for koa = 87r and num ber of points N = 500. N um ber of term s in analytical expression sum m ation is M = 40....................................................................................................................... 44 3.18 C om parison betw een scattering w idth obtained w ith m eth o d of moments and exact solution as functions of koa for scattering angles <f>— 0° (backscattering) and <f>= 90° (forward sc atterin g )........................................................................................................................... 45 3.19 C om parison betw een scattering w idth obtained w ith m eth o d of m om ents and exact solution for koa = 0.1 and num ber of points N = 12.. N um ber of term s in analytical expression sum m ation is M —... 10................................................................................................ 46 3.20 C om parison betw een scattering w idth obtained w ith m eth o d of m om ents and exact solution for koa = 1.0 and num ber of points N = 20. N um ber of term s in analytical expression sum m ation is M = ... 15................................................................................................ 46 3.21 C om parison betw een scattering w idth obtain ed w ith m eth o d of m om ents and exact solution for koa = 2 tt and num ber of points N = 100.. N um ber of term s in analytical expression sum m ation is M = 20................................................................................................ 47 3.22 C om parison betw een scatterin g w idth o b tained w ith m eth o d of m om ents and exact solution for koa = 87t and num ber of points N = 500.. N um ber of term s in analytical expression sum m ation is M = 40................................................................................................ xi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 47 3.23 Equivalent electric current, m ethod of m om ents solution and theory, vs. <f>, for a lossless hom ogeneous cylinder w ith er = 2.56, for koa = 1.0 and koa = 5.0. N um ber of points used in th e m ethod of m om ents are N = 20 and N = 100, respectively. . . 51 3.24 Equivalent m agnetic current, m ethod of m om ents solution and theory, vs. <f>, for a lossless hom ogeneous cylinder w ith eT = 2.56, for koa = 1.0 and koa = 5.0. N um ber of points used in th e m ethod of m om ents are N — 20 a n d N = 100, respectively. . . 51 3.25 Equivalent electric current, m ethod of m om ents solution a n d theory, vs. (f>, for a lossy homogeneous cylinder w ith er = 2.56 —jO. 102, for koa = 1.0 and koa — 5.0. N um ber of points used in th e m ethod of m om ents are N = 20 and N = 100, respectively. . . 52 3.26 Equivalent m agnetic current, m ethod of m om ents solution and theory, vs. (f>, for a lossy hom ogeneous cylinder w ith er — 2.56, koa = 1.0 and koa — 5.0. N um ber of points used in th e m ethod of m om ents are N = 20 and N = 100, respectively. . . . . 52 3.27 Equivalent electric current, m ethod of m om ents solution and theory, vs. <j>, for a lossy hom ogeneous cylinder w ith er — 5.0 —j 1.0, koa = 1.0 and koa = 5.0. N um ber of points used in th e m ethod of m om ents are N = 20 and N = 100, respectively. . . 53 3.28 Equivalent m agnetic current, m ethod of m om ents solution and theory, vs. <fi, for a lossy homogeneous cylinder w ith er = 5.0 —jl.O , koa — 1.0 and koa = 5.0. N um ber of points used in th e m ethod of m om ents are N = 20 an d N = 100, respectively. 3.29 Equivalent electric current, m ethod of m om ents solution and lossy homogeneous cylinder w ith er = 60 —j 59.9, . . 53 theory, vs. (j>, for a koa = 1.0 and koa = 5.0. N um ber of points used in th e m ethod of m om ents are N = 20 and N = 100, respectively. . . 54 3.30 Equivalent m agnetic current, m ethod of m om ents solution and theory, vs. <f>, for a lossy homogeneous cylinder w ith er = 60 —j’59.9, koa = 1.0 and koa = 5.0. N um ber of points used in th e m eth o d of m om ents are N = 20 and N = 100, respectively. xii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. . . 54 3.31 S cattering w idth, m ethod of m om ents solution and theory, size param eter koa, for a lossless hom ogeneous cylinder w ith er = 2.56, for koa = 1.0 a n d koa = 5.0. Solid line is backscattering (cf>s = 0°), and dashed line is forw ard scatterin g (<fis = 180°). . 55 a lossy hom ogeneous cylinder w ith er = 2.56 —j'0.102, for koa = 1.0 and koa = 5.0. Solid line is backscattering (<ps = 0°), and dashed line is forw ard scattering {<f>3 = 180°)....................................................................................................................................................... 55 3.33 S cattering w idth, m ethod of m om ents solution an d theory, size param eter koa, for a lossy hom ogeneous cylinder w ith er = 5.0 —j l.0 , koa = 1.0 a n d koa = 5.0. Solid line is backscattering (<j>3 = 0°), and dashed line is forw ard scatterin g (<ps = 180°). . 56 3.34 S cattering w idth, m ethod of m om ents solution and theory, size param eter koa, for a lossy hom ogeneous cylinder w ith er = 60 —j 59.9, koa = 1.0 a n d koa — 5.0. Solid line is b ackscattering (<j>s = 0°), and dashed line is forw ard scatterin g (<f>3 = 180°). . 56 3.35 Scattering w idth, m ethod of m om ents solution a n d theory, vs. scattering angle <ps, for a lossless hom ogeneous cylinder w ith er = 2.56, for koa = 1.0 and koa = 5.0. N um ber of points used in the m ethod of m om ents are N — 20 and N — 100, respectively........................................................................................................................................... 57 3.36 Scattering w idth, m ethod of m om ents solution a n d theory, vs. scattering angle <ps, for a lossy hom ogeneous cylinder w ith er — 2.56 —j 0.102, for koa = 1.0 and koa = 5.0. N um ber of points used in the m ethod of m om ents are N = 20 and N = 100, respectively........................................................................................................................................... 57 3.37 Scattering w idth, m ethod of m om ents solution an d theory, vs. scattering angle <ps, for a lossy hom ogeneous cylinder w ith er = 5.0 —j l .0 , kcja = 1.0 and koa = 5.0. N um ber of points used in th e m ethod of m om ents are N = 20 and N = 100, respectively. xiii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. . . 58 3.38 Scattering w idth, m ethod of m om ents solution and theory, vs. scattering angle <f>s , for a lossy homogeneous cylinder w ith eT = 60 — j59.9, koa = 1.0 and koa = 5.0. N um ber of points used in the m ethod of m om ents are N = 20 and N = 100, re s p e c tiv e ly ........................................................................................................................................ 58 4.1 Body of r e v o l u t i o n .......................................................................................................................... 61 4.2 (a) Choice of discretization points {t*,} on the curve C. (b) Discretized curve C. . . 66 4.3 Definition of expansion functions: (a) (b) P m \ t ) , and (c) Tm(t)........................ 68 4.4 G eom etry of various bodies of revolution................................................................................... 74 4.5 Surface currents vs. t /A on a P E C disk w ith radius a = 1.5A, for 0i = 0°. C om parison between results of th e present m ethod (BO R ), W ilton-G lisson (WG), and M autzH arrington (MH) solutions............................................................................................................. 4.6 76 Surface currents vs. t /A on a P E C washer w ith radii oi = 0.4A and a\ = 1.2A, for 9i = 0°. C om parison betw een results of th e present m ethod (BOR), W ilton-G lisson (W G), and M autz-H arrington (MH) solutions......................................................................... 4.7 76 Surface currents vs. t f A on a P E C open-ended cylinder of radius a = 0.4A and length L = 1.2A, for 0i = 0°. Com parison betw een results of th e present m ethod (BOR), W ilton-G lisson (W G ), and M autz-H arrington (MH) solutions........................................... 77 4.8 Surface currents vs. t j A on a P E C cone-sphere of rad iu s a — 0.2A and length L = 1.2A, for 0i = 0°. C om parison betw een results of th e present m ethod (BOR), W iltonGlisson (W G), a n d M autz-H arrington (MH) solutions.......................................................... 4.9 77 Surface currents vs. t / A on a P E C sphere w ith size p a ra m ete r k^a = 1.0. Com parison between results of th e present m ethod (BOR) an d Mie series (MIE) solution................... 78 4.10 Surface currents vs. t /A on a P E C sphere w ith size p a ra m ete r koa = 10.0. C om par ison betw een results of the present m ethod (BO R) a n d M ie series (MIE) solution. . . xiv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 79 4.11 Surface currents vs. t / A on a P E C finite cylinder of radius a = 0.5A and length L = A, for 6i = 0° (left) and 9{ = 45° (right). Com parison betw een results of the present m ethod (BOR) and W ilton-G lisson (WG) solution................................................ 80 4.12 Surface currents vs. t/X on a P E C finite cylinder of radius a = 0.5A and length L = 2.5A, for 9i = 0° (left) an d 0i = 45° (right). Com parison betw een results of the present m ethod (BOR) and W ilton-G lisson (WG) solution.................................................... 80 4.13 Surface currents vs. t/X on a P E C finite cylinder of radius a = 0.05A and length L = 2.5A, for Qi — 0° (left) and 9t = 45° (right). Com parison betw een results of the present m ethod (BOR) and W ilton-G lisson (W G) solution.................................................... 81 4.14 N orm alized b istatic scattering coefficient vs. 9S for a P E C disk of size param eter koa = 10, for 9i = 0°. C om parison betw een results of th e present m ethod (BOR), and Hodge solutions.......................................................................................................................... 82 4.15 Norm alized b istatic scattering coefficient vs. 9S for a P E C open-ended cylinder of radius a = 0.4A and length L = 1.2A, for 9i = 0°. C om parison betw een results of th e present m ethod (BOR) and W ilton-G lisson (WG) solu tio n .......................................... 82 4.16 Norm alized b istatic scattering coefficient vs. 9S for a P E C cone-sphere of radius a = 0.2A and length L = 1.2A, for plane wave incident at 9{ — 0°. Com parison betw een results of th e present m ethod (BO R) and W ilton-G lisson (WG) solution. . . 83 4.17 Norm alized b istatic scattering coefficient vs. 9S for a P E C sphere w ith size param eter koa = 1.0, for — 0°. C om parison betw een results of th e present m ethod (BOR) and Mie series (MIE) solution........................................................................................................ 83 4.18 Norm alized b istatic scattering coefficient vs. 9S for a P E C sphere w ith size param eter koa — 10.0, for 9i = 0°. C om parison betw een results of th e present m ethod (BOR) and Mie series (MIE) solution........................................................................................................ xv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 84 4.19 N orm alized b istatic scattering coefficient vs. 9S for a P E C finite cylinder of radius a = 0.5A and length L = A, for &i = 0° (left) and Qi = 45° (right). Com parison betw een results of th e present m ethod (BOR) and W ilton-G lisson (WG) solution. . . 85 4.20 N orm alized b ista tic scattering coefficient vs. 9S for a P E C finite cylinder of radius a = 0.5A and length L = 2.5A, for 9i = 0° (left) and 9{ = 45° (right). Com parison betw een results of the present m ethod (BOR) a n d W ilton-G lisson (WG) solution. . . 85 4.21 N orm alized b istatic scattering coefficient vs. 9S for a P E C finite cylinder of radius a = 0.05A and length L = 2.5A, for 9i = 0° (left) and 9{ = 45° (right). Com parison between results of th e present m ethod (BOR) and W ilton-G lisson (WG) solution. . . 86 4.22 Equivalent surface currents vs. t j A on a dielectric sphere w ith koa = 1.0 and ec = 4. C om parison betw een results of the present m ethod (BO R) and Mie series (MIE) solution.................................................................................................................................................. 87 4.23 Equivalent surface currents vs. t j A on a dielectric sphere w ith koa = 1.0 and ec — 4 —j . C om parison betw een results of the present m eth o d (BOR) and Mie series (MIE) solution..................................................................................................................................... 87 4.24 Equivalent surface currents vs. t /X on a dielectric sphere w ith koa = 1.0 and ec = 18 —j6. C om parison betw een results of th e present m eth o d (BOR) and Mie series (MIE) solution..................................................................................................................................... 88 4.25 Equivalent surface currents vs. t / X on a dielectric sphere w ith koa = 10.0 and ec = 4. Com parison betw een results of the present m ethod (BO R) and Mie series (MIE) solution.................................................................................................................................................. 88 4.26 Equivalent surface currents vs. t j A on a dielectric sphere w ith koa = 10.0 and ec = 4 —j . C om parison betw een results of th e present m eth o d (BOR) and Mie series (MIE) solution..................................................................................................................................... xvi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 89 4.27 E quivalent surface currents vs. t/X on a dielectric sphere w ith koa = 10.0 and ec = 18 —j 6 . C om parison between results of th e present m ethod (BOR) and Mie series (MIE) so lu tio n .................................................................................................................................... 89 4.28 N orm alized b ista tic scattering coefficient vs. 83 for a dielectric sphere w ith koa = 1.0 and ec = 4. C om parison betw een results of the present m ethod (BOR) an d Mie series (MIE) so lu tio n .................................................................................................................................... 90 4.29 N orm alized b ista tic scattering coefficient vs. 8S for a dielectric sphere w ith koa = 1.0 and e c = 4 — j . C om parison betw een results of the present m ethod (BOR) and Mie series (M IE) solution......................................................................................................................... 90 4.30 N orm alized b ista tic scattering coefficient vs. 8S for a dielectric sphere w ith koa = 1.0 and s c = 18 — j 6. C om parison betw een results of th e present m ethod (BOR) and Mie series (MIE) solution................................................................................................................ 91 4.31 N orm alized b ista tic scattering coefficient vs. 8S for a dielectric sphere w ith koa = 10.0 and £c = 4. C om parison betw een results of th e present m ethod (BOR) and Mie series (MIE) solu tio n ..................................................................................................................................... 91 4.32 Norm alized b ista tic scattering coefficient vs. 8S for a dielectric sphere w ith koa = 10.0 and £c = 4 — j . Com parison betw een results of th e present m ethod (BOR) and Mie series (M IE) solution ............................................................................................................................. 92 4.33 Norm alized b istatic scattering coefficient vs. 8S for a dielectric sphere w ith koa = 10.0 and ec = 18 — j 6. C om parison betw een results of th e present m ethod (BOR) and Mie series (M IE) solution...................................................................................................... 92 4.34 Geom etry of th e finite-length dielectric cylinder p r o b le m .................................................. 95 4.35 Tapered cylinder m o d e l ............................................................................................................... 97 4.36 Scattering p a tte rn of a c y li n d e r .................................................................................................. 99 xvii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.37 N orm alized b istatic scattering coefficient vs. scattering angle for finite dielectric cylinder w ith L = 10.0A, a = 0.04A, and incident angle 9i — 20°..........................................102 4.38 Norm alized b ista tic scattering coefficient vs. scattering angle for finite dielectric cylinder w ith L = 10.0A, a = 0.4A, eT = 18 — j 6, an d incident angle = 20° (top) and 0i — 80° (b o tto m ).........................................................................................................................103 4.39 A bsolute error vs. incident angle for finite dielectric cylinder w ith a = 0.04A, er = 18 — j 6, L = 5.0A, 3.0A, an d 1.0A, for hh- and w -p o la riz a tio n ..............................................104 4.40 A bsolute error vs. incident angle for finite dielectric cylinder w ith a = 0.4A, er = 18 —j 6, L = 10.0A, 5.0A, and 3.0A, for hh- and iw -polarization........................................... 105 4.41 Norm alized b ista tic scattering coefficient vs. scattering angle for tap ered dielectric cylinder w ith L = 10.0 A, a = 0.7 A, b = 0.1 A, and incident angle 6t = 40°. hh- and vv- polarization (left and right, respectively).................. 108 4.42 Norm alized b istatic scattering coefficient vs. scatterin g angle for tap ered dielectric cylinder w ith L = 10.0 A, a = 0.5 A, b = 0.3 A, and incident angle 0, = 40°. hh- and vv- polarization (left and right, re sp e c tiv e ly )............................................................................. 108 4.43 E rror vs. incident angle for tap ered dielectric cylinder w ith L — 10.0 A, (left) a = 0.7 A, d2 = 0.1 A, a n d (right) a = 0.5 A, b = 0.3A. Solid lines are hh-pol, dashed lines vu-pol........................................................................................................................................................ 109 5.1 Definition of (k, h , v ) for waves traveling forw ard an d backw ards in the ^-direction. 5.2 Far-field in half-space m e d iu m ...................................................................................................... 122 5.3 M aterial body in half-space m edium ............................................................................................ 125 5.4 Exterior problem for half-space m edium ..................................................................................... 127 5.5 Plane waves incident on a body in a half-space m ed iu m ...................................................... 133 5.6 Reflection a n d transm ission of plane wave across a plane b o u n d a ry ............................... 134 xviii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 116 5.7 In teg ratio n p a th of I in the com plex A^-plane............................................................................ 139 5.8 A bsolute error [%] for th e Tank T ransform ation m ethod vs. num ber of integration points N ............................. 5.9 148 S cattering from a vertical P E C open-ended cylinder on a P E C half-space (left), and equivalent problem (right) in free space........................................................................................ 149 5.10 C om parison betw een P E C open-ended cylinder of length L — 0.5A and radius a = 0.1 A on a P E C half-space, and equivalent image problem , for incident angle 9i = 60°. Left: surface electric currents vs. z/X . R ight: norm alized bistatic scattering coefficient vs. 9S.................................................................................................................................... 150 5.11 Surface electric currents vs. t/X on a perfectly conducting cylinder of length L = X and koa = 1.0 located a t a distance d = 0.2A above a hom ogeneous, lossy half-space w ith relative dielectric constant 16 —j 16. Incident angle is 9i= 0°....................................151 5.12 Scattering from a vertical dielectric cylinder over a P E C half-space (left), and equiv alent problem (right) in free space.................................................................................................. 152 5.13 C om parison betw een dielectric cylinder of length L = 0.6A, radius a = 0.1A and a relative dielectric constant 4 —j on a P E C half-space, and equivalent image problem , for incident angle 9i = 45°. E quivalent surface electric currents vs. z/X . Right: norm alized b istatic scattering coefficient vs. 9s .......................................................................... 153 5.14 C om parison betw een dielectric cylinder of length L = 0.6A, rad iu s a — 0.1A and a relative dielectric constant 4 - j on a P E C half-space, an d equivalent image problem, for incident angle 9i = 45°. N orm alized b istatic scatterin g coefficient vs. 9s....................153 5.15 N orm alized scatterin g coefficient cr/i/! vs. 9S. C om parison betw een analytical approx im ation an d num erical solution for a dielectric cylinder of len g th L = 2.5A and radius a = 0.025A. R elative dielectric constant is 4 —j for the cylinder, and 10 —jo for the ground. Incident angle is 9i = 30°................................................................................................... 157 xix Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.16 N orm alized scattering coefficient avv vs. 8S. C om parison betw een analytical approx im ation an d num erical solution for a dielectric cylinder of length L = 2.5A an d radius a — 0.025A. R elative dielectric constant is 4 —j for th e cylinder, a n d 10 —j o for the ground. Incident angle is 0* = 30°............................................................................................. 5.17 N orm alized scattering coefficient (T/jft, us. 9S. C om parison betw een analytical approx im ation an d num erical solution for a dielectric cylinder of length L = 2.5A and radius a = 0.025A. R elative dielectric constant is 4 —j for th e cylinder, and 10 —j 5 for the ground. Incident angle is 6i = 60°............................................................................................. 5.18 N orm alized scattering coefficient a vv vs. 0S. C om parison betw een analytical approx im ation an d num erical solution for a dielectric cylinder of length L = 2.5A and radius a = 0.025A. R elative dielectric constant is 4 — j for the cylinder, and 10 —j 5 for the ground. Incident angle is 0* = 60°............................................................................................. 5.19 N orm alized scattering coefficient 07^ vs. 8S. C om parison betw een analytical approx im ation a n d num erical solution for a dielectric cylinder of length L = 4.0A and radius a = 0.025A. R elative dielectric constant is 4 — j for th e cylinder, and 10 —jb for the ground. Incident angle is 0; = 30°............................................................................................. 5.20 N orm alized scatterin g coefficient a vv vs. 9S. C om parison betw een analytical approx im ation an d num erical solution for a dielectric cylinder of length L = 4.0A and radius a = 0.025A. R elative dielectric constant is 4 — j for th e cylinder, and 10 —jo for the ground. Incident angle is 6i = 30°............................................................................................. 5.21 Norm alized scatterin g coefficient us. 9S. C om parison betw een analytical approx im ation an d num erical solution for a dielectric cylinder of length L = 4.0A a n d radius a = 0.025A. R elative dielectric constant is 4 — j for th e cylinder, and 10 —j'o for the ground. Incident angle is 8\ = 60°............................................................................................. xx Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.22 Norm alized scattering coefficient avv vs. 9S. C om parison betw een analytical approx im ation and num erical solution for a dielectric cylinder of length L = 4.0A and radius a = 0.025A. R elative dielectric constant is 4 — j for the cylinder, and 10 —j 5 for th e ground. Incident angle is Qi = 60°.................................................................................................. 161 5.23 Norm alized scattering coefficient c r^ us. Qs. Com parison betw een analytical approx im ation and num erical solution for a dielectric cylinder of length L = 2.5A and radius a = 0.25A. R elative dielectric constant is 4 —j for th e cylinder, and 10 —j 5 for the ground. Incident angle is 6i = 30°.................................................................................................. 162 5.24 N orm alized scattering coefficient a vv vs. 9S. Com parison betw een analytical approx im ation and num erical solution for a dielectric cylinder of length L = 2.5A and radius a = 0.25A. Relative dielectric constant is 4 —j for th e cylinder, and 10 —j 5 for the ground. Incident angle is Qi = 30°.................................................................................................. 162 5.25 Norm alized scatterin g coefficient a^h vs- Qs- C om parison betw een analytical approx im ation and num erical solution for a dielectric cylinder of len g th L = 2.5A and radius a = 0.25A. R elative dielectric constant is 4 —j for th e cylinder, and 10 —j 5 for the ground. Incident angle is Qi = 60°...................................................................................................163 5.26 Norm alized scatterin g coefficient a vv vs. Qs. C om parison betw een analytical approx im ation and num erical solution for a dielectric cylinder of len g th L = 2.5A and radius a = 0.25A. R elative dielectric constant is 4 — j for th e cylinder, and 10 —j 5 for the ground. Incident angle is Qi = 60°.................. 163 A .l Definition of volum e V and surfaces S \ and S 2 for th e G reen’s theorem ..........................182 A .2 Definition of in teg ratio n dom ain.....................................................................................................186 xxi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List o f Tables 3.1 A ccuracy of M OM com putation of TM scattering w idth for P E C circular cylinder of various values of koa, w ith emax in % ..................................................................................... 40 3.2 Accuracy of M OM com putation of T M scattering w idth for P E C circular cylinder of various values of koa, w ith emax h1 %..................................................................................... 44 4.1 G round d a ta ......................................................................................................................................... 93 4.2 G eom etric param eters used in tap ered cylinder approxim ations........................................... 107 4.3 C om parison of com putation tim e betw een analytical m odel and num erical algorithm for four tap e red cylinders of length L a n d a = 0.6A, b = 0.2A............................................... 112 5.1 Results of the evaluation of I s using th e W eighted-Averages (WA) m ethod, Tanh T ransform ation (T T ), and Com posite T rapezoidal (CT) integration, for various val ues of Z , R , an d k = 18 —6j . I b is th e exact value, e is th e absolute accuracy in %, and T is th e com putational tim e in sec.........................................................................................146 xxii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 1 Introduction 1.1 M otivation In th e field of microwave rem ote sensing, the stu d y of scattering from vegetation is of great in ter est. P articularly w ith th e deploym ent of air- and satellite-born Synthetic A perture R adar (SAR) instrum ents, it is especially im p o rta n t to have reliable models to characterize th e electrom agnetic behavior of vegetation. A vegetation canopy can be considered as a m ultilayered m edium above a ground surface. For example, in a forest, its crown region m ay be chosen as one layer, and its tru n k region as another layer. Furtherm ore, each layer generally contains more th a n one ty p e of scatterer. For instance, a branch could be a scatterer, and a leaf another scatterer. T herefore, each layer is m odelled as an ensemble of individual dielectric objects of different type, size, and orientation. Among th e m ost common com ponents in a vegetated m edium are cylindrical stru ctu res, such as stem s, branches, tru n k s or needles, an d disk-like stru ctu res such as leaves. A linear variation of the radius along the axis of a cylinder, can also be introduced to m odel tru n k s an d branches m ore realistically. A long, curved broadleaf m ay be m odeled by several sm all disks each positioned w ith a different slope to sim ulate the entire leaf. T he electrom agnetic scatterin g from a vegetation canopy can be studied by replacing each layer w ith a random m edium whose sta tistic a l properties are related to th e physical and geom etrical 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. characteristics o f th e vegetation layer. This random m odeling can be done in either a discrete or a continuous fashion. In the discrete case, the layer is viewed as a collection of dielectric particles whose size, position and orientation statistics are given. In the continuous case, the vegetation is replaced by a continuous random m edium w ith a perm ittiv ity th a t is a random process whose m om ents, such as th e m ean and correlation function, are known. In b o th cases, th e scattering from the layer is obtain ed by perform ing an average - either discrete or continuous - of th e values of the scattering coefficient of the individual constituents. T he availability of electrom agnetic m odels for the single scatterers is therefore essential in order to investigate th e scattering from a vegetation canopy. T he approach followed in the stu d y of th e electrom agnetic scattering from an object can be analytical or num erical. T he purpose of this dissertation is to develop num erical m ethods to eval uate the electrom agnetic scattering from individual vegetation com ponents. W hile the num erical m ethods can provide exact solutions, their com putational cost often makes their use unpractical in problem s involving large objects. Therefore, a m otivation for developing num erical m ethods is th a t, particularly in th e case of large scatterers, th ey represent an essential tool to test the validity of com putationally faster approxim ate analytical models. 1.2 Background M odels of vegetation as a canopy composed of individual dielectric objects can be found in Fung [1], Ulaby et al. [2], E ngheta and Elachi [3], and Lang an d Sidhu [4], U laby et al. [5] and Tavakoli [6] have used vertical cylinders, representing th e stalks, and random ly oriented disks, representing the leaves, to m odel a canopy. K aram et al. [7] have represented a forest canopy as a two-layered m edium above a rough interface, while Lang et al. [8] have introduced tap ered cylinders to m odel a forest. C hauan et al. have m odeled a boreal forest [9] an d cornfields [10]. Koh et al._ [11] have considered the scattering and atte n u atio n th ro u g h a forest w ith dense foliage. 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T he problem of electrom agnetic scattering from the individual com ponents of the vegetation canopy has been studied by several authors in the past w ith e ither analytical or num erical m ethods. T he analytical m ethods were th e first to be developed due to lim ited com putational tools available in th e past. Historically, the first class of objects considered for analytical solutions was th a t of infinite cylinders. E xact analytical solutions are available for scattering by perfectly conducting and homogeneous dielectric cylinders, b u t not for other cross sections. W ait [12, 13] was among the first ones to investigate and present an exact solution for scattering from an infinite circular cylinder at oblique incidence. Ruck et al. [14] and B ow m an [15] also gave a sum m ary of exact solutions for b o th conducting and homogeneous dielectric cylinders of infinite length. U nfortunately, exact analytical solutions for the scattering from finite dielectric bodies exist only in the case of the sphere [14, 15, 16]. However, for the scatterin g from objects such as finite-length cylinders, finite-thickness disks, and spheroids, a num ber of approxim ations have been studied. T h e Rayleigh-Gans approxim ation [17] is applicable to tenuous scatterers for which the phase shift across the m axim um dim ension is small. A cquista [18] a n d Cohen et al. [19] extended the use of the Rayleigh-Gans approach to particles w ith low polarizability and slightly larger phase shift across them . Schiffer a n d T hielheim [20] introduced an approxim ation for cylinders w ith one dim ension electrically sm all and shorter th a n th e other, i.e. eith er very th in or flat. Shepherd and H olt [21] applied th e Fredholm integral equation m ethod to the scatterin g of electrom agnetic waves by finite cylinders of circular cross section, b u t th eir m ethod also h ad lim i tatio n s at the increase of th e cylinder radius. K aram et al. [22, 23] used Schiffer and T hielheim ’s approxim ation to m odel dielectric discs and cylinders. Stiles a n d Sarabandi [24] provided a solution for th in dielectric cylinders w ith a broader range of validity, b u t still lim ited to small cross sections, and showed th a t Schiffer and T hielheim ’s approxim ation is a specific case of their solution. Seker and Schneider [25] have delveloped a physical optics ap proxim ation for dielectric cylinders of any size, provided th a t th e length is m uch larger th a n th e radius. 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. An analytical m odel for plane wave scatterin g from th in dielectric circular disks has been de veloped by Le Vine et al. [26, 27, 28] a n d W illis et al. [29]. T h eir m odel is based on a physical optics approxim ation of the internal electric field of the disk. Very recently, Koh et al. [11] have introduced a new approxim ate solution for th in dielectric disks of a rb itra ry size and shape th a t perform b e tte r th a n the Le V ine’s m odel for edge-on incident wave. T he num erical treatm ents of problem s of electrom agnetic scattering from perfectly conducting a n d homogeneous dielectric bodies have usually been based on integral equation form ulations, particularly surface integral equations, th e preferred approach to whose solution has been the M ethod of M om ents (MOM) [30]. O ther approaches based on differential equations, such as the F in ite Elem ent M ethod (FEM ) and the F in ite Difference T im e D om ain (FD T D ), are quite powerful com putational tools, b u t require discretization of the entire co m p u tatio n al dom ain, com pared to th e surface integral equation approach, which requires discretization of th e object surface only. For th is reason they have been used m ainly for problem s involving sm all objects or inhomogeneous bodies. Richm ond [31] developed a num erical m eth o d to com pute th e scatterin g by a dielectric cylinder of arb itra ry cross section, however his approach was based on th e solution of a volume integral equation. T he problem can also be tre a te d using a surface integral equation, as shown in [32]. Raz and Lewinsohn [33] investigated the volume a n d surface integral equation form ulations relevant to the scattering and absorption of electrom agnetic waves by th in , finite, a n d lossy dielectric cylinders. Papayiannakis et al. [34] tre a te d th e problem of scattering from a finite dielectric cylinder w ith dim ensions com parable to th e w avelength of th e incident field, by solving an integral equation containing the free-space G reen’s function over the cylinder volume by th e m ethod of m oments. L ater, the same a u th o r [35] employed a tran sfo rm atio n to reduce th e three-dim ensional integrals in th e equation into tw o-dim ensional ones. Dielectric disks have also been tre a te d using num erical models. C hu and Weil [36, 37] have derived a volume integral equation, and solved it w ith the 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. m ethod of m om ents. Form ulations of the equivalence principle and the derivation of surface integral equations have been presented by several authors, including Poggio an d M iller [38], Glisson [39], L eviatan [40], and Peterson [32], These integral equations can be tre a te d by using th e M ethod of M oments. Theoretically, the M ethod of M oments technique can solve any scattering problem , however its m em ory and com putational requirem ents makes its use prohibitive for large problems, even when only the surface integral equations are considered. For this reason only objects w ith p articu lar geo m etrical sym m etries are norm ally considered. T heir geom etrical properties lead to a sim plification of the problem , w ith consequent reduction of com putational tim e and m em ory requirem ents. One simple class of scatterers includes infinite cylindrical bodies for which all the electrom agnetic quantities have no or only periodical variation along th eir axis. T his bodies can be tre a te d as twodim ensional bodies, as done by Richm ond [31]. A nother specific class of objects includes bodies th a t exhibit a ro tatio n al symmetry. For these axisym m etric scatterers - or bodies of revolutions - th e problem can be reduced to the solution of an integral equation along a curve by the m ethod of m om ents. Solutions of the surface integral equations for bodies of revolutions have been presented by A ndreasen [41], M autz and H arrington [42, 43, 44, 45], and W ilton an d Glisson [46, 47], am ong others. T he problem of scatterin g in an half-space m edium was first investigated by B utler et al. [48, 49, 50] and X u et al. [51, 52] for th e two-dim ensional case of infinite cylinders parallel to the bou n d ary betw een the two half-spaces. In [48], th e current on an infinite conducting strip located on the interface between the half-spaces is com puted. [49, 51] and [52, 50] study the surface currents and the scattering from perfect electric conducting a n d hom ogeneous dielectric infinite cylinders, respectively, parallel to th e half-space interface and located above or below it, or p artially buried in it. In their approach, tw o-dim ensional surface integral equations are solved for the surface currents by the MOM; the co m putations include th e evaluation of Som m erfield’s-type integrals. 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. M ichalski and Zheng considered surface integral equations for bodies of arb itra ry shapes, pro viding the theory for layered m edia [53] and some results for the half-space case [54]. Cui et al. provided the theoretical expressions of the volume integral equations for tw o-dim ensional objects buried u n der m ultilayered m edia [55, 56], and later for th e three-dim ensional objects [57] w ith some num erical results [58] using the MOM. V itebskiy et al. [59, 60, 61] considered axisym m etric perfect electric conductors buried in a half-space. Geng et al. [62, 63, 64], and He et al. [65, 66, 67] have extended their algorithm for a dielectric targ et in a m ulti-layered environm ent. 1.3 O verview T he present C hapter 1 provides a background and m otivation for th e work in this doctoral disser tation. It also gives a brief overview of th e inform ation contained in each chapter. In C h ap ter 2, a surface integral equation form ulation for electrom agnetic scattering is derived sta rtin g from basic electrom agnetic theory and using th e equivalence principle. A lternative forms of surface integral equations are also given. In C h ap ter 3, the surface integral equations are solved by th e M ethod of M oments for the case of two-dim ensional bodies. R esults are given for infinite cylinders of circular cross section, b o th perfect electric conductors and dielectrics, and these are com pared to th e exact analytical solutions. T his chapter is intended to be b o th an exam ple of solution of th e coupled surface integral equations developed in C hapter 2, and a validation of the correctness of such an approach. C hapter 4 illustrates th e application of the coupled surface in teg ral equations to axisym m etric objects. T he M ethod of M om ents solution is used, which exploits th e ro tationl sym m etry of the body to reduce the com plexity of th e problem . T he results are validated using objects of different shapes, b o th perfect electric conducting and dielectric. T his num erical procedure is th en used to m odel common vegetation com ponents, and th e results are com pared to those obtained by approxim ate analytical models. 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 5 extends th e application of the m ethod of m om ents to objects located in an half-space m edium. T he surface integral equation approach is extended to include bodies of revolution located in the upper half space, an d its solution is validated by considering equivalent cases. Finally, this half-space num erical approach is applied to the problem of m odeling vegetation over a flat ground. In C hapter 6 conclusions are draw n and recom m endations are m ade for future work. 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 2 T he surface integral equation form ulation T he purpose of this chapter is to introduce the integral equations th a t will be used to solve scattering problem s in the rest of th is dissertation. T he use of such form ulation provides th e advantage of reducing a three-dim ensional problem to the solution of a set of surface integral equations. In p a rticu la r situations, which will be fu rth er considered in the following chapters, such an approach leads to th e reduction of th e num ber of independent variables from three to two, thus allowing a less complex and m ore efficient num erical com putation techniques. In section 2.1, th e electrom agnetic field generated by an a rb itra ry d istribution of sources in an unbounded space is w ritten as a sum of th e contribution from th e sources inside and from the sources outside a volume. T he second contribution is in th e form of a surface integral over the b o undary surface of th e volume, which allows the definition of equivalent sources on this boundary. In section 2.2, a m aterial body is considered, and th e expression derived in section 2.1 is applied to th e exterior and interior of its b o u n d ary surface. T he resulting two integral equations are m atched on the boundary surface to obtain a system of coupled surface integral equations. Section 2.3 presents alternative form s of these surface integral equations. Finally, in section 2.5, expressions for the scattered fields are given an d im p o rta n t quantities for th e stu d y of electrom agnetic scattering are defined. 8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.1 The equivalence principle In th is section, sta rtin g from M axwell’s equations, and applying equivalence principles, an integral equation relating th e electrom agnetic fields them selves to th eir values on a closed surface shall be derived. From hereafter, a tim e harm onic variation of the electrom agnetic field of the form u = 2 7i-/, (2.1) is assum ed, / being th e frequency of the electrom agnetic field. A n infinite TZ3 space is considered, filled w ith a hom ogeneous and isotropic dielectric m edium w ith p e rm ittiv ity e and perm eability p, given by £ — £q £r P — Po Pr j (^*^) where £o and sT are respectively th e free-space p e rm ittiv ity and the relative perm ittivity, b e tte r known as dielectric constant, of the m edium , po and p r are respectively th e free-space perm eability and the relative perm eability of the m edium. In such a m edium , th e electric and m agnetic fields, E an d H respectively, are related to the prescribed electric a n d m agnetic sources J and M by M axwell’s equations: V x E = —j u j p H — M (2.3) V x H = jtue E + J (2.4) = - (2.5) V -E £ V -H = 0 (2.6) where p is the electric charge density, related to J by th e law of conservation of charge V • J = —juip (2.7) As shown in A ppendix A .l, w hen b o th electric and m agnetic sources exists in a volume V, the 9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. solution to the M axwell’s equations (2.3)-(2.6) for r £ 7£3 is given by: E(r) = - j u n [ G (r, r') ■J(r') d V' H(r) = Jv J K{r,r')-J{r')dV' - j u s - [ K(r, r') ■M(r) dV' (2.8) J (2.9) Jv G (r, r') • M(r') dF' w here th e integration is carried out over the region V where th e sources J and M are located, G(r,r') = g (r,r')I + V g^ r/)-V' (2.10) K(r,r') = V5( r , r ' ) x l (2.11) are th e electric and m agnetic dyadic G reen’s functions, respectively, and ““ ’^ / c j r j ff(r,r') = e 47r|r _ r/[ ' ’ k= . ^ i (2.12) is th e scalar G reen’s function defined in A ppendix A .I. Note th a t, technically speaking, th e dyadic G reen’s function defined by (2.10) is actually a functional o p erato r and not a function. C alculation of the electrom agnetic fields th rough (2.8) and (2.9) requires complete knowledge of th e sources J and M a t any point, which is often not possible. However, if the electrom agnetic fields are already known over a closed boundary, it is th e n possible to determ ine them in the entire space by considering equivalent sources on such b o u n d a ry surface. The equations relating the electrom agnetic fields inside th e m edium w ith their value on th e boundary are called surface integral equations a n d will be derived in the rem inder of this section. T he purpose of th e following derivation is to find a solution for th e electrom agnetic field any where in space. Let Si and S2 be two closed surfaces, w ith S2 surrounding S i,asshown in Figure 2.1, and the sources J and M be known only in the volume V betw een S i and S2, w ith no sources on Si or S2. Let n ' be the norm al u n it vector on the surfaces S i an d S 2, pointing inwards. E quations (2.3), (2.4) can be com bined in th e following two wave equations for E and H: V x ( V x E ) - f e 2E = —ju/j, J —V x M 10 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.13) V (known unknown unknown J ,M * 0 Figure 2.1: D efinition of surfaces Si and 52V x (V x H ) — &2 H = -ju e M + V x J , (2.14) Let E be an electric field th a t satisfies M axwell’s equations (2.3) and (2.4)anywhere in space. T he G reen’s theorem , given in A ppendix A.2, can be applied to the volume V and surfaces Si and S 2 previously defined, w ith P = E, Q = ap (2.15) where a is an arbitrary, constant vector. T he result is th e following: / {E • [V' x (V ' x a <7)] - p a - [V' x (V ' x E)]} dV' = Jv p a x ( V ' x E ) - E x (V 'x a p )]- h 'd S ' (2.16) S 12 where S 12 = S\ U S 2 , th e prim e is used for convenience to indicate th e integration variables, an d the negative sign on th e right side of equation (2.16) is due to th e definition of the norm al u n it vector pointing inwards instead of outw ards as in th e derivation of th e G reen’s theorem from A ppendix A.2. 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Using M axwell’s equation (2.3) and the wave equation (2.13), and perform ing the calculations described in A ppendix A .3, th e equality (2.16) can be w ritten as: E(r) = - a • f Jv ( j u g J + V ' x M ) g(r, r ') + X - ^ V ' g jue dV' + [jug, (H x n ') g + ( n x E ) x V'g + V'g (E ■ n ')] dS' + a • JS ,2 (2.17) T his equation is valid for any a rb itra ry value of a, and, as a result, E(r) = - [ ( j u g J + V ' x M ) g( r, r') + — V'g d V' + Jv L JUE [ j u g (H x n ') g + ( n ' x E ) x V'g + V'g (E • n ')] dS' + (f (2.18) JSin Since M = 0 on the surface S, th e application of Stokes’ theorem yields: lv [ V ' x M(r') ] g( r, r') d V ' = J V ' x [g( r, r') M(r') ] d V' - = / g ( r , r ' ) M ( r ' ) ■h ' d S ' - / = - / [ V ' g x M ( r ' ) ] dV' Jv = ~ [ J [ V' g x M(r') ] dV' [ V ' g x M (r') ] dV' [V'g(r,r') x I ] ■ M ( v ' ) d V ' (2.19) JV Use of the relationship ( 2 . 20 ) and the definition of electric an d m agnetic G reen’s functions (2.10) and (2.11) lead to the equation: E (r) = —j u g [ G ( r , r ' ) - J ( r ') dV' — [ K (r, r ') ■M ( r ') dV' + Jv Jv + I J Sl2 [ j u g (H x n ,) g + ( n / x E ) x V'g — V g ( E ■ n ;)] dS' (2.21) T he right side of (2.21) contains two types of integrals, w hich correspond to the contributions to the electric field of th e known sources in the volum e V an d th e unknow n sources outside V, respectively. T he two volum e integrals represent the field generated by the known sources J an d M inside the volume V. T h e surface integral is the co n trib u tio n to th e field from the unknow n 12 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. V (known J,M £ 0) J ,M = 0 e ,|i e.M- J ,M = 0 Figure 2.2: E quivalent sources configuration. sources located outside th e volume V. Its integrand suggests th a t such unknow n replaced by equivalent sources H x n , n x E and E ■ n on th e surface S u , sources can be which would give the same contribution to th e electric field E(r) inside th e volume V as th e unknown sources. T he intro d u ctio n of these equivalent sources im plies th a t an equivalent problem is indeed con sidered, where th e unknow n sources J a n d M inside S i and outside S 2 are set equal to zero and replaced by equivalent sources J s and M s on th e surface S 12. T he sources J s and M s are surface currents and are defined so th a t they generate th e sam e electrom agnetic fields w ithin the volume V as the unknow n sources. This equivalent configuration is illu strated in Figure 2.2. This result is often referred to as th e equivalence principle. In expression (2.21), r is a generic point in space. In order to derive a surface integral equation from it, the point r m ust be located on the surface S. However, in such a situation the integrand of the surface integral is singular a t r = r'. It is shown in A ppendix A.4 th a t its integration by means of a lim iting procedure leads to th e following resu lt (A .65): (f [jtdfj, (H x n ') g{ r, r') + ( n ' x E) x V '^ r, r') + V g { r , r') (E • n ')] dS' = J S\ 2 13 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. + f = (H x JS12 2 ') 9(r, r') + ( n ' x E) x V'g - V g (E • n ')] dS' (2.22) where the asterisk indicates the principal value of th e integral. S u b stitu tin g (2.22)into (2.21) yields ^ = 2 + f - ju t* f G (r, r') • J(r') d V' - f K (r, r') •M(r') dV' + Jv Jv \ju{* (H x n ') g(r, r') + ( n ' x E) x V'g - V g (E ■n ')] dS' (2.23) JS 12 Note th a t the expression above is valid only for points r on th e surface S\ 2 A n equivalent tan g en tial electric surface current J s and an equivalent surface charge density p s can be defined on S 12 = S i U 52 as, respectively, J s (r') = n'xH (r') (2.24) p s ( r') = e E(r') • n ' (2.25) Since in the equivalent configuration J = M = 0 on S 12 and outside V , then, the law of conservation of charge yields E - n ' = — = - V * ' Js e jlj£ (2.26) Note th a t, since Js depends only on th e surface coordinates, V(. ■J s = V ' ■J s. Using (2.24)-(2.26) as well as the definition of electric an d m agnetic dyadic G reen’s functions (2.10) and (2.11), after some m anipulations equation (2.23) can be w ritten as follows: ^ 2 = -jut* [ Jv -jwp, £ G ( r , r ' ) - J ( r ') d V ' - [ K(r,r') ■M(r') dV' + Jv G (r, r') ■J s(r') dS' — <j> J s 12 K (r, r') ■[E(r') x n '] d S ' , r E S 12 (2.27) JS12 In the following section, expression (2.27) will be used to derive a set surface integral equations for a m aterial body in an unbounded space. 14 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. e,fx Figure 2.3: E xterior problem . 2.2 Surface integral equations in unbounded space An homogeneous, isotropic dielectric body of p e rm ittiv ity e and perm eability /j,. surrounded by free space, is now considered. Sources J , M / 0 axe assum ed to exist in th e free space, but not inside the m aterial body. V is th e volume occupied by th e body, and S is the b oundary surface betw een the dielectric and free space. W ith reference to Figure 2.3, th e surfaces 5 1 an d S 2 previously introduced can be defined such th a t S is inside Si , in w hat is called th e exterior problem . If S 2 expands to infinity while at the same tim e the o th er surface S 1 approaches S, th en the integral equation (2.27) assumes the following form: E (r) ! Jv f Js+ G ° (r,r')'J(r ')d V '- G ° ( r , r ') ■J s (r') dS' - f Js+ [ K ° ( r , r') • M (r ') d V ' + Jv K ° ( r , r') ■ [E (r') x n '] d S ' , r 6 S+ (2.28) where the superscript “0” indicates th e free space G reen’s functions, which have the form (2.10)(2.12), w ith e — £0, and /j, = (j,q (eo being the perm ittivity, and /j,q th e perm eability, of free space), 15 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and S + is th e exterior side of the surface S. An “incident” electric field can be defined as E « (r) = - j u n o f Jv G ° ( r , r /) • J ( r ') d V ' - r GS [ K ° ( r , r ' ) • M (r') d V ' , Jv (2.29) which corresponds to th e electric field induced by th e sources J and M on the surface S, a n d can be determ ined from them . Using expression (2.29), equation (2.28) can be rew ritten as = E « ( r ) - j w / i o r G ° ( r , r ' ) - J ^ r ' ) ^ ' - / * K ° ( r , r ' ) • [E (r') x n '] dS ' , r 6 5 + (2.30) J s + JS+ 2 which is the surface integral equation for the exterior problem . T he interior problem is illu strated in Figure 2.4. In this case, b o th surfaces S\ a n d £2 are assum ed to be enclosed by S, w ith S i —>0 and S 2 approaching S. Since there are no sources inside S 2 , equation (2.24) becomes: = juifi G ( r , r ' ) - J s (r' ) d S ' + j> K ( r , r ') ■ [E (r') x n '] d S ' , r G S - (2-31) where S ~ is the interior side of S, and n " = — n ' is th e norm al unit vector to S ~ . T he electrom agnetic field inside S is uniquely determ ined by the values of the tangential com po nent of the electric field over th e boundary S. Hence, th e relationship betw een the electrom agnetic fields inside and outside S can be found by m atching the tan gential com ponents of E given by (2.30) and (2.31), as it will be shown below. T he tangential com ponent vt of a vector v can be determ ined as follows: Vi = v — n ( n • v) = ( I — n n ) ■v (2.32) A pplying (2.32) b o th sides of equations (2.30) and (2.31), yields = E^(r) - ( I - n n ) - <f G ° ( r , r ') • J s (r') dS' + JS+ 2 — ( I —h n ) - ( f Js+ K ° ( r , r ' ) ■[E (r') x h ' ] d S ' , r GS+ 16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.33) S,|d Figure 2.4: Interior problem . = junil-nh)-^ G ( r , r ' ) ■J s( r ' ) dS' + + { l - h h ) - £ _ K ( r , r ' ) • [E (r') x n ' j d S 1, r e S~ (2.34) Because of th e continuity of th e electric field, E t m ust th e sam e on b o th exterior and interior sides of S. In addition, since th e tan g en tial m agnetic fields are also continuous, then J s on S + m ust be equal to J s on S~, and J s in (2.33) and (2.34) are th e same. Hence, expressions (2.33) and (2.34) constitute a system of coupled surface integral equations th a t m ust be solved together to find the unknow n functions J s, and E on th e surface 5 , given th e tangential incident electric field e J ° on S. Since (2.33) and (2.34) have been derived by m atching th e tan g en tial com ponents of the electric field over the boundary surface S, th ey are called surface tan g en tial electric field integral equations (E FIE ). They are also called ’’b o u n d ary integral equations” in [68]. T hey can be w ritten in term s of equivalent currents an d tan g en tial incident field only, by introducing equivalent m agnetic sources 17 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. on S. Sim ilarly to th e definition of the equivalent surface electric current (2.25), an equivalent surface m agnetic current on S can be introduced as M s (r') = E(r') x n ' (2.35) so th a t the tan g en tial electric field on 5 is given by: Et(r') = M s(r') x n ' (2.36) Hence, using (2.35) and (2.36), the two equations (2.33) and (2.34) m ay be w ritten in th e form: --- = E ^ ( r ) - jufiQ (I - n n)-jf + G°(r,r') • J s(r') dS' + - ( I - n n ) i ’ K°(r,r') • M s {r' )dS' , Js+ = r G S+ (2.37) r G S~ (2.38) j u } f j , { l - h n ) - ^ _ G { r , r ' ) - J s{r ' ) dS' + + ( I - n n ) - j f ’ K(r,r') • M s {r' )dS' , E quations (2.37) and (2.38) involve only J s and M s as unknowns, and therefore from now on will be referred to as th e coupled equivalent surface currents integral equations. However, even though the electric field does not ap p ear explicitly in th em anym ore, they are still considered astangential electric field integral equations. Unless noted otherw ise, these are th e integral equations th a t will be solved in the rem ainder of this d issertation work. 2.3 Other forms o f coupled surface integral equations Coupled tangential M agnetic Field Integral E quations (M FIE) m ay be derived by m atching the tangential com ponents of th e m agnetic field over th e surface S th ro u g h a procedure sim ilar to the one followed to find th e coupled tan gential E lectric Field Integral E quations (2.33) and (2.34). 18 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. However, an easier way is to apply th e duality principle. In particular, substituting: E J £ -*• H -> M (JL H m fi -* - E -* - J -> £ (2.39) in (2.33) and (2.34) yields: ^ = H « ( r ) - ( I - n n ) . | ‘ K ° ( r , r ' ). [Ht(r') x n '] dS' + —jus o ( I - n n ) ' / Js+ = ( - ~~ “ G°(r,r') ■ M s(r') dS', — ^r,r^ ' - jujs{l-hh)-f Js+ r e S+ (2.40) x fi/] dS1 + G(r,r') ■ M s{r')dS', r G S~ (2.41) Note th a t by duality from (2.37) and (2.38), it m ay also be w ritten: n x ^ r) = H Sl)(r) + ( I - n n ) - £ + K°(r,r') ■ J s(r')dS" + - jusQ (I - n n ) - <£ G°(r,r') - M s(r') dS', Js+ nxJj(r) _ r € S+ (2.42) K(r,r') • J s{r')dS' + + juje (I —n n)'j> G(r, r ' ) - M s(r')dS', r € S~ (2.43) This alternative form of th e coupled equivalent surface currents integral equations is useful when the expression of the tan g en tial incident m agnetic field is sim pler th a n the expression of the tangential incident electric field E ) '. In addition to th e E F IE form ulation (2.37),(2.38) and th e M FIE form ulation (2.43),(2.44), other form ulations based on any linear com binations of those two system s of integral equations are pos sible and are generally called C om bined Field Integral E quations (C FIE ). T he accuracy of each form ulation for a given scatterin g problem depends on the geom etry and dielectric properties of the scatterer. In particu lar, it has been found th a t resonances can arise in problem s involving con ducting bodies when using th e E F IE or M FIE form ulation, b u t do not occur if a C FIE form ulation is used [42, 69]. However, even for th e C F IE form ulation, some resonances can still occur [70]. 19 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.4 A lternative form for the dyadic G reen’s function T he integral equations derived in this chapter are based on th e form (2.11) of the dyadic G reen’s function. A different form of the dyadic G reen’s function, i.e., 'G (r,r') = (l-^P jg (r,r') = + gfar') (2.44) is often encountered in th e literature and will be used later in the present work, specifically in C hapter 5 for th e half-space problem. T he m ain advantage in using the operator form (2.11) ra th e r th a n the above form (2.44) is th a t the latte r perform s a double differentiation of the scalar G reen’s function g(r,r' ), This can result into problem s w hen perform ing integration th a t require to evaluation of g{r, r') a t the singular point r ' = r. All the surface integral equations previously derived still apply if G ( r , r ' ) defined by (2.44) is used instead of G ( r , r ') in (2.11). In order to prove it, it is sufficient to show th a t [ G (r,r')-Fir^dV’ = f G (r, r ') ■F ( r ') dV' (2.45) G ( r , r ' ) ■J s (r') dS' = £ G (r, r') • J s (r ') dS' (2.46) Jv Jv for F = J , M , and th a t f Condition (2.45) ensures th a t the electrom agnetic fields generated by electrom agnetic sources are still given by (2.9) and (2.10), while condition (2.46) assures th e equivalence between th e integral equations w ith G ( r , r') and G ( r , r ') . P ro o f of th e equivalences (2.45) a n d (2.46) can be found in A ppendix A.5. 2.5 Scattered field and scattering coefficient Once the equivalent tan g en tial surface currents J s a n d M s on 5 are com puted by solving the coupled integral equations, the scattered electric field generated by th em outside th e surface S can 20 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. J,M = 0 Figure 2.5: Definition of geom etry for scattered field calculation. be calculated. W ith reference to the geom etry in Figure 2.5, since r is not on the surface, th e n the electric field E (r) can be found by applying expression (2.21). T here rare no sources w ithin the volume V, hence th e only contribution to th e field outside 5 comes from th e equivalent sources. Using definitions (2.24),(2.25), and (2.35) into expression (2.21) yields EW ( r) = —jw/J-o j> G ° ( r , r') • J s (r') dS' — 5 K ° ( r , r ') ■M s (r') dS' (2.47) S T he scattered m agnetic field can be found by duality, i.e., H (s)(r) K ° ( r , r ' ) ■J s (r') dS' — jcoeo j> G ° ( r , r ' ) • M 5(r') dS' = s (2.48) s For rem ote sensing problem s, th e knowledge of th e scattered electrom agnetic fields a t large distances from the object is essential. T he general configuration of the problem is illu strated in Figure 2.6. The object - also referred to as a scatterer - is located a t th e origin of a coordinate system . T he scatterer is assum ed to be illum inated by a plane wave representing th e incident electric field, having th e form E w (r) = e ~ J' k o ^ ' r (2.49) T he u n it propagation vector k 8 th a t gives the direction of th e incident plane wave is : kj = —sin cos & x — sin sin fa y — cosfy z 21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.50) Figure 2.6: Incident a n d scattered plane waves on an object. where 6i, fa are th e elevation and azim uth angles of the incident wave, respectively, in a spherical coordinate system centered a t the origin. T he incident field (2.49) generates equivalent currents J s a n d M s on th e surface S th a t can be evaluated using th e surface integral equations derived in the previous sections. T hen, the scattered electrom agnetic fields associated w ith these equivalent currents can be determ ined using (2.47), (2.48). T he far-field, or rad iatio n zone, for a source w ith m axim um size D is defined as the region of space where |r — r '| > 2 D 2/ \ , w ith A being the wavelength. In such region the following approxim ations can be used: k0 |r - r ' ko r —ko r • r ' (2.51) 1 r (2.52) 1 where r is the u n it vector into the direction of th e observation point. T he scalar G reen’s function in th e far-field is therefore (2.53) 22 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. U sing (2.53) into expressions (2.11) and (2.12), th e dyadic G reen’s functions in th e far-field are found to be approxim ately given by G°(r,r') ~ + eA , f - r ' (2.54) 47r r K °(r,r') ~ "”**3 ko - j k 0 ( 4> 9 - 9 4>) — A \ / 47r 47Tr i e ^ ° f ' r' (2.55) S u b stitu tin g th e approxim ations (2.54) and (2.55) into expression (2.47), the scattered electric field E (s)(r) can be w ritte n as a spherical wave E^(r) ~ -jko e, - j f a r An r - f Cof s ( e e + 4>4>)- j s (r') f ' r ' dS' + (4> 0 - 9 4 > y M,(r') e i k° f - r ' dS' (2.56) where the sym bol ~ indicates th a t the approxim ation is valid in th e far-field, and Co = M V £o (2.57) is th e im pedance of free space. A sim ilar expression for th e scattered m agnetic field in the rad iatio n zone can be found using (2.54) and (2.55) into (2.48). T he observation point r is located a t elevation and azim uth angles 9S, (fis, respectively, and defines th e direction of scattering. T he propagation vector of the scattered wave is therefore k s = sinf?s cos cf>s x + sin£?s sin cj)s y + cos9s z (2.58) Note th a t the unit vector r is equal to the u nit propagation vector k s. Let the incident plane wave be polarized in the q direction, i.e., 4 ° = Ef (2-59) and E ^ be the corresponding scattered field. T he bistatic scatterin g coefficient, also called b istatic scattering cross-section, in th e polarization p for this (^-polarized incident field is defined as 23 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A n equivalent way of characterizing th e scattering from an object makes use of the so-called scattering am plitudes. T he dyadic scattering am plitude f is im plicitly defined by the relationship -jkor E W(r) - (2.61) f&.kO-Ej while the scalar scattering am plitudes f pq in the p- and (/-polarization are /p ?(k s,k j) = p s • f (ks,kj) • q . (2.62) From (2.56), (2.59), (2.61) and (2.62), it follows th a t /pg(ks,kj) — 1 j k oCo * Ps ■ £ ( § 6 + 4> 4 > y J ^ (r ') e J fc° E (i) 4?r ( 0 0 - 6 4 > y M ^ (r ') e J k° ^s ' r' dS' r' dS' + (2.63) where J ^ , M i9-* are the equivalent surface currents induced by the (/-polarized incident field (2.49), (2.59). T he scattering am plitudes f pq are related to th e b istatic scattering coefficients apq by the ex pression: <rpq(9$,<f>s;8i,<i>i) = 4 7 r | / p9(ks,ki) |2 24 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.64) Chapter 3 T w o-dim ensional bodies T he integral equation form ulation derived in the previous chapter can be easily applied to twodim ensional problem s. If th e surface S of an infinitely long body in a three dim ensional space is shift-invariant along a p articu lar axis, and all th e field quantities are also invariant, or exhibit the same periodic variation, along th a t axis, th en th e problem reduces to two dimensions on an a rb itrary cross section plane perpendicular to the sym m etry axis. In particular, an infinitely long cylinder S in th e z-direction is considered, a section of w ith in the xy-plane is depicted in Figure 3.1. T he cylinder is assum ed to have perm ittiv ity e and perm eability /i, and to be located in free space (perm ittiv ity £o an d perm eability po). Sources J and M are present somewhere in th e free space, and generate an incident field E j ’ and the equivalent surface currents J( and M s on S. If th e sources are constant or have all th e sam e periodic variation along th e z-coordinate, th en th e problem can be seen as tw o-dim ensional in the xy-plane. T he infinite cylinder can th en be uniquely characterized th ro u g h its generating curve C, i.e., the intersection of S w ith th e xy-plane, which is shown in Figure 3.1. For convenience, an orthonorm al coordinate system ( r , n ) is introduced on the curve C , f being the unit vector tangent to C at the point p , and n th e u n it vector norm al to C a t p . ip is th e angle between cos ip = t • x . 25 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. t and x , i.e., Figure 3.1: G eom etry of a two-dim ensional body. 3.1 T he coupled surface integral equations in two-dim ensions It is shown in A ppendix B th a t for a tw o-dim ensional problem th e coupled integral equations (2.37),(2.38) can be w ritten in the following form: Ms ^ X n = Etl)(p) ~ j ' w f i o ( l - n n ) - ^ + Q-2 d { P i p ') ' M p ' ) d C ' - ( I - n n ) - / * K°2D( p , p ' ) - M a( p ' ) d C ' , Jc+ Ms{p2)X— = + PeC+ (3.1) peC - (3.2) j u t i ( l - h h ) - £ _ G 2D( p , p ' ) . J t ( p ' ) d C ' + + (I-n n )-jf & 2 D( p , p ' ) - M s { p ' ) d C ' , where the dyadic and scalar G reen’s functions have th e expressions: i— i / ~ Q.2 d { p , P ) K 2D( p , p ' ) sM p , p ') n „/\ t i ^ 9 2 d (P i P ) = 92D{p,P)l-^ P V /o o\ (3.3) = V g 2D( p , p ' ) x l (3.4) = -jflfw p-p'l) (3.5) and fJg2^(x) is the Hankel function of th e second kind. T he correspondent functions for free space 9$d> & 2D and ^ - 2V have th e same expressions as (3.3)-(3.5) w ith e = £oM — MoX> k = &o26 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T he equivalent surface current densities J t, M s and the tangential incident electric field E :o on th e surface S can be w ritten in term s of their r - and ^-components: J s(p) = J r ( p ) t + J z(p) z (3.6) M s(p) = M T( p ) f + M z {p) z (3.7) Ef{ p ) = e W(P) t + E${p)z (3.8) S u b stitu tin g expressions (3.6)-(3.8) into equations (3.1),(3.2), and separating the z- and r-com ponents yields the following four equations: E i'H p ) — 7^ - ~ j u p o j> ^+ 9 2 d ( P i p ' ) J z { p ' ) d C ' + dg\ 2D- M T( p' ) d C , ■/ Jcc+ dn' E^(p) = M z ^pl + j u g ,o £ C+ +L 0 % Mr(p) = M^ G C+ A , ( p , P V r ( 0 ' ) - i/cq8 & d r 8 dr' (3.10) . j u p <f 92d ( p , p' ) J z { p' ) dC' + Jc+ - 4 M z (p) 2 cos (-ip —ip') d,C' + d c '' dg2D + Jc+ dn' M r ( p' ) dC' , 0 = (3.9) P \ t 1 d92D 9 Jt i\ t ( + JU„ f c + 9 2 D { p , p ) M p ) - ¥ o ^ r l ? ’ Jc+ dg2D Tc+ dn' M z {p' ) dC', (3.11) p G C+ cos {ip — ip') dC' + p ec+ (3.12) This system of surface integral equations will be solved for some sim ple cases in Sections 3.3 and 3.4. Using the results of A ppendix B, th e scattered electric field generated in a two-dim ensional space by equivalent surface currents J s and M s on th e curve C has th e form: E (s)( -jupo j G °2 D{p, p' ) • U p 1) dC' - c j K °2 D( p , p' ) ■M , { p ’) dC' c 27 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.13) Figure 3.2: D efinition of scattering problem for a two-dim ensional body. 3.2 Far-field and scattering w idth For a two-dim ensional body, th e far field is defined as the region of space where p is large com pared to th e wavelength A, so th a t th e following approxim ation can be used: k0 \ p - p ' \ « P = — P k o p -k o p -p ', (3.14) Using (3.14) together w ith th e large argum ent approxim ation of th e Hankel function [71], in the far field the two-dim ensional scalar G reen’s function (3.5) is approxim ated as: e ’*k °P e ^ 0/5' p 92d{p, p ' ) (3.15) From this approxim ation, and from expressions (3.3) and (3.4), th e dyadic G reen’s functions in th e far-field are obtained: G.2d ( p >p ') ~ (z Z + 0 0 ) K.2 d ( p >p ') ~ ~jk0 ( - e_ ^ o P e i k°P' P z 0 ) ^ e ~ ^ p e i k° P' P 28 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.16) (3.17) Using expressions (3.16),(3.17) into (3.13) yields: E (s)( p ) -j&o C o / ( z z + 0 0 V J s ( p ' ) e ^ ° ^ ' P d,C' + Jc TT&oP + / Jc - z 4>^j ■M s ( p' ) e ^ ° P ' P dC' (3.18) T h e incident electric field is assum ed to be a g-polarized plane wave of the form E«(p) = q< e fco ki ■P (3.19) where the unit p ropagation vector of th e incident wave is now two-dimensional: kj = - cos fa x - sin fa y , (3.20) w ith fa being th e angle of incidence, defined as shown in Figure 3.2. Let be th e corresponding scattered field, w ith u nit propagation vector of in the direction cj)s given by k s = cos 4>s x + sin <f>s y . (3.21) For two-dim ensional objects, a q u an tity called the scattering w idth is introduced in a sim ilar fashion as the scatterin g cross section in th ree dim ensions. It is an equivalent w idth proportional to th e apparent size of th e scatterer in a p a rticu la r direction, and it is defined as apq (ks, £i) = JiHL 2?r P p ^ 0° • IE ?S)(P) • P< (3.22) | Eg*) |2 In other words, if th e power density of th e incident field were m ultiplied by th a t length, it would yield the same power as produced by th e scatterer. T he incident field (3.19) generates equivalent currents be evaluated using th e surface integral equations. and on the surface S th a t can S u b stitu tio n of th e expression (3.18) for the scattered field at a point p = k s in th e far zone, into (3.22) yields: (ka,ki) = 1 h IE,« i 2 4 Ps • (o / ( z z Jc + j> ( 0 z - z 0 ) • 4> 0 ) • J ^ ( p O e ^ ° ( p') eZ ■P dQi 29 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. P dC' + (3.23) T h e scatterin g cross section per length I can be obtained from th e scattering w idth through the relationship [72]: o-pg(k s,k i) 3.3 a ^ ( k s, ki ) = (3.24) Perfect electric conducting tw o-dim ensional body Let the two-dim ensional cylinder be m ade of a perfect electric conductor. Since the tangential electric field is null on th e surface of a perfect electric conductor, th e equivalent surface m agnetic current M s also vanishes. Therefore, only th e exterior problem is to be considered, and the equation to be solved is (3.10), which assum es th e form: E «(p) = j u ^ ( l - h h ) - £ Q.°2D( p , p ' ) - J s ( p ' ) d C ' , p e c (3.25) th a t corresponds to the following two equations for th e z- and r-com ponents: E z H p ) = jupo £ p eC 92d ( p , p ' ) J z { p ' ) d C ' , E^(P) 92D\ P i P ) J t \ P ) k,Q d r dr' cos {ip - t f ) d C ' , peC (3.26) (3.27) whose solution gives th e two com ponents J z and JT of th e induced equivalent surface current. Once J z and J T are found, th e scatterin g w idth can be evaluated by using (3.23) w ith M s(p') = 0: apq (ks, ki) = IE 3.3 .1 1 k0 (i)I2 4 Ps ■ £o j> ( z z — 4> ( p ') e k s ■P dC' (3.28) T M incident wave Let the two-dim ensional body be illum inated from an angle fa by a plane wave whose electric field is directed along the 2-axis. Expression (2.49) reduces to: {cos f a x + sin f a y ) EW(p) = (3.29) Such choice of incident field is denoted as transverse m agnetic (TM ) incident wave, and is illu strated on the left side of Figure 3.4. A TM incident wave induces on th e perfect conductor an equivalent 30 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1/ Lt t - A.t/2 (a) t tn (b) Figure 3.3: (a) D iscretization of curve C. (b) Basis (expansion) functions. surface current J s th a t is also directed along the 2-axis: (3.30) j s(p) = Jz{p) Z Therefore, only th e integral equation (3.26) for th e 2-com ponent is needed, which has th e form: 4 p£C Jc (3.31) Once Jz is determ ined by solving th e integral equation (3.31), th e scattering w idth can be found through expression (3.28). Since for 2-polarized incident electric field the scattered electric field is also directed along th e 2-axis, the only non-zero scatterin g w id th is c r j f , for convenience hereafter indicated as cr™ , which is given by: a 2D (kSJ kj) 1 = IE fcoCo £ J z ( p' ) e i k ° * s ' P' dC' (3.32) i Numerical solution using the m e t h o d of m o m e n ts T he integral equation (3.31) is solved using th e m eth o d of m om ents (MOM ). For this purpose, a curvilinear coordinate t is defined on th e curve C , an d C is divided into N intervals by choosing 31 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. N points t n = (n — 1)A t, (n = 1 , 2 , 3 , N ) on it, as shown in Figure 3.3. R ectangular pulses are selected as basis functions: 1 Pn(t) = At ^ At t n -- — < t < t n ---- — 2 2 elsewhere for < 0 n = 1,2,3,..., N , (3.33) while the testing functions are D irac delta functions: Xn{t) = S { t - t k) , k = 1,2,3,. , . , N (3.34) W ith such choices, the equivalent current density can be approxim ated as follows N J 2(t') = t'zC £ lnP n(0> 71= (3.35) 1 where In Jz{tn) i = n = 1,2,3,..., N (3.36) S u b stitu tin g (3.35) into (3.31), and testing the resulting equation w ith th e functions (3.34) for A: = 1,2, 3, . . . , N leads to th e following system of algebraic equations: N vk where Vk = Zk* = = Y z knln, n= 1 & = 1, 2,3,..., N , E ^ ( t k) , H ^ \ k Q\ p k - p ' \ ) d t ' , (3.37) k = 1,2,3,..., N (3.38) k, n = 1,2,3, ...,1V (3.39) Jtn 4 where tn = At ^ + y i n ~ 1,2,3,..., N (3.40) In the case of a circular cylinder of radius a as in Figure 3.4, th e curve C can be described by the equation p' = a cos (j>' x + a sin ^6' y , <j>' € [0,27t] (3.41) and the curvilinear coordinate is related to the angle <f>as follows: t = a<j> , A t — a A<f> , 27r A <j> = — 32 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.42) TM incident wave TE incident wave Figure 3.4: TM and T E incident waves on a circular cylinder of radius a. so th a t, for k ,n = 1 , 2 , 3 , IV, (3.38) and (3.39) sim plify into: yk = jjW g j ko a cos [(k - 1) A <j>- fa] i A <f> Zk (3.43) H q2^ ^ko 01^2(1 — cos (k — n)A4>]j A, j, .2 j a A(f> —— <1 — j — (3.44) /T * o o A ^ A _ 1 V 4 / 7T for for k = n where 70 = 1.781072418.... T he values of th e current I n are calculated by solving the system of linear equations (3.37) for k , n = 1 , 2 , 3 Fr om them , using (3.35)-(3.36) into (3.32) one can determ ine the scattering w idth a t a scattering angle <j>s: 1 JTM ( 1 \ a 2D W s ) IK (i)l h (q N In e i a C0S ~ (n ~ (3.45) 71=1 Results Here, the solution of th e integral equation for a perfectly electric conducting cylinder w ith TM polarized incident wave is illu strate d and com pared w ith th e exact solution. The incident electric field in (3.48) is chosen incom ing from th e positive z-axis 33 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. = 0) and Two-dimensional PEC Cylinder - TM case, ka = 0.1, N=12 $2 » Theory Method of Moments 135 180 Figure 3.5: Surface equivalent current obtained w ith m ethod of m om ents and exact solution for ko a = 0.1 and num ber of points N = 12. N um ber of term s in analytical expression sum m ation is M = 10. (i) w ith a u nit am plitude £* = Coi i-e -i z Co e j k0 x E « (p ) (3.46) T he analytical expressions for th e equivalent surface current and th e scattering w idth can be found in [15]. T he z-com ponent of the current is a function of th e polar angle <f>as follows M<t>) 2 E^l y ' toirkoa ^ -m cos (mcp) m H ^ { k 0 a) (3.47) while the scattering w idth is a function of difference betw een the scattering and incident angles: 2 EH) m Cm m —0 where , , ?2 )------------ COS [ m (<j)3 - J m \ k o CL) rt I H ^' i*k( o a) 1 for m = 0 2 for r a ^ O ( p i )j (3.48) (3.49) T he infinite sum m ations in (3.48) and (3.49) are lim ited to a num ber M or term s such th a t the rem ainder is neglegible. 34 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Two-dimensional PEC Cylinder - TM case, ka = 1.0, N=20 2.5 « Theory Method of Moments cr 0.5 45 135 180 <t>[cleg] Figure 3.6: Surface equivalent current obtained w ith m ethod of m om ents and exact solution for koa = 1.0 and num ber of points N = 20. N um ber of term s in an alytical expression sum m ation is M = 15. Two-dimensional PEC Cylinder - TM case, ka = 2st, N=130 2.5 Theory Method of Moments 0.5 45 135 180 <t>[deg] Figure 3.7: Surface equivalent current obtained w ith m ethod of m om ents and exact solution for koa = 2-k and num ber of points N = 100. N um ber of term s in an alytical expression sum m ation is M = 20. 35 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Two-dimensional PEC Cylinder - TM case, ka = at, N=500 2.5 Theory Method of Moments cr 45 135 180 Figure 3.8: Surface equivalent current obtained w ith m ethod of m om ents and exact solution for koa = 87r and num ber of points N = 500. N um ber of term s in analytical expression sum m ation is M = 40. Two-dimensional PEC Cylinder - TM case Theory Method of Moments x = 180 ° O) = 0° 1 2 3 4 5 6 7 8 9 10 Figure 3.9: Scattering w id th obtained w ith m eth o d of m om ents and exact solution vs. koa, for scattering angles <f>= 0° (backscattering) and <j>— 90° (forward scattering). 36 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Two-dimensional PEC Cylinder - TM case, l^ a = 0.1, N=12 ..... 2 ------Theory * Method of Moments . , 'io' 0 si ;— x— x— x— x— x— x u> c (0 o w ■03O .N E 45 90 135 Scattering Angle <|> [deg] 180 Figure 3.10: Scattering w idth obtained w ith m ethod of m om ents an d exact solution vs. <fis, for koa — 0.1, num ber of M OM points N = 12, a n d analytical expression tru n c a te d to M = 10 term s. In Figures 3.5 to 3.8, th e equivalent current obtained by solving (3.37) is plotted together the exact solution (3.47) for different values of koa. Since for th e incident wave (3.29) the surface equivalent current is sym m etrical w ith respect to the x-axis, th e values of <p are lim ited to the range between 0° and 180°. In all cases considered, the relative difference betw een the analytical and num erical solution has been found to be less th a n 1%. Figure 3.9 shows th e backscattering and forw ard scattering w idth, <j2D((j)s = 0°) and cr2D((ps = 90°) respectively, as functions of th e size p aram eter koa. T he bistatic scattering w idth a 2D as a function of the scattering angle <f>s is plotted in Figures 3.10 to 3.13 for different values of koa. Again, th e bistatic scatterin g w idth is sym m etrical w ith respect to the x-axis, therefore plots are lim ited to the values <j>3 betw een 0° and 180°. T he relative error on th e scattering w idth estim ated using th e m ethod of m om ents is defined 37 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Two-dimensional PEC Cylinder - TM case, l^ a = 1.0, N=20 Theory Method of Moments “ 7 O) Scattering ^ngle <J)=.[deg] 135 180 Figure 3.11: Scattering w id th obtained w ith m ethod of m om ents and exact solution vs. <f>s , for koa — 1.0, num ber of M OM points N = 20, and analytical expression tru n c a te d to M = 15 term s. Two-dim ensional PEC Cylinder - TM case, l^a = 2n, N=130 — Theory x Method of Moments O) Scattering jungle .[d e g ]135 180 Figure 3.12: Scattering w idth obtained w ith m ethod of m om ents and exact solution vs. <f>3, for koa = 27T, num ber of M OM points N = 130, and an alytical expression tru n c a te d to M = 20 term s. 38 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Two-dimensional PEC Cylinder - TM case, l^a = 871 , N=500 Theory x Method of Moments 2 15 "O § 10 U) Scattering i^ngle <()c.[d e g ]135 180 Figure 3.13: Scattering w idth obtained w ith m ethod of m om ents and exact solution vs. <f>s, for koa = 87t, num ber of MOM points N = 500, and analytical expression tru n c a te d to M = 40 term s. w ith respect to th e exact solution, for a scattering angle (f>s, as follows: ± 5 = 2 D(<f>s) N <4h) = ^ — ~ <T2d (4>s ) /0 7 (3' 50) )— where G2 D is the exact value of the scattering w idth found th ro u g h (3.48) and &2 D is the result (3.45) of th e MOM calculation. Table 3.1 gives th e m axim um error emax, in %, defined as emax = A m ax \e((/>s)\ , (3.51) 0 s e [ O ,2 ir ] for the four values of koa considered, and also provides th e num ber N of points in th e MOM ♦ calculation and the num ber of term s M to which sum m ation (3.47) has been truncated. Note how the difference betw een the two solutions is always very sm all and less th a n 1 3 .3 .2 T E in c id e n t w a v e Transverse Electric (TE) polarization is defined as th e case w hen th e incident electric field is parallel to the transverse xy-plane, as depicted on th e right side of Figure 3.4. In this case, the incident 39 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. koa N M emai 0.1 1.0 27T 87T 12 20 130 500 10 15 20 40 0.8 0.9 0.3 0.2 Table 3.1: Accuracy of M OM com putation of TM scattering w idth for P E C circular cylinder of various values of koa, w ith emax in %. m agnetic field is directed along the axis of the infinite cylinder, i.e., th e z-axis, and can be w ritten as HW(p) = £ t f « e iMcos<fo:r + sin<&y) One could derive th e expression of th e incident field (352) from (3.52), th e n solve (3.26) and (3.27) for the equivalent current J s . In this case th e surface equivalent current has only th e r-com ponent: 3 s {p) = J T( p ) f (3.53) T his would require solving th e second integral equation (3.27) th a t contains a derivative of J s and involves differentiation of th e G reen’s function g\D. A m ajor sim plification of the problem is obtained by applying th e d u ality principle, together w ith th e fact th a t in this case the equivalent current has only th e r-com ponent, to equation (3.26), to obtain H {z \ p ) = JT2P>>~ f c { ^ ^ C0S^ ' + J r ( p ' ) d C 'i P ^ c (3-54) Once J r is evaluated by solving (3.54), the scattering w id th can be found through (3.23), which for the TE-case becomes: rT E /C 02i? (k s ,k i) = ^ — 1f^-2 4 | rr(l) | l-^O I ( 4> z - Z 4> ) ■r 'J t { p ') k° ^ s ' p dC' (3.55) N um erical so lu tio n u sin g th e m eth o d o f m om en ts Here, equation (3.54) by using th e m ethod of m om ents. R ectangular pulses as defined in (3.33) are selected as basis functions to expand th e current J T, an d D irac d e lta functions (3.34) are employed 40 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. as testing functions. W ith such choices, the following system of equations is obtained: N Vk = YhZknln, k — 1,2, 3 ,...,1V (3.56) n —1 where n — 1 ,2 ,3 ,..., N (3.57) k = 1 ,2 ,3 ,..., N (3.58) and, for k, n — 1 ,2 ,3 ,..., N , the coefficients of (3.56) are Z *. = A (3. 59) 1 °J \Pk~Pn\ w ith Skn being th e K ronecker’s delta: Jkn 1 for k =n 0 for k j4 n (3.60) For the specific case of a circular cylinder of radius a, expression (3.59) becomes Jkn l + ( l - 8kn)aA(f> ^ H i 2)(kQa^J2[l - cos (k - n ) A f l) (3.61) 4>]ei k ° a cos W*~{ n- l) A < l > ] (3.62) and th e scattering w id th is given by °2D (4>s) = 1 7 4 1 0 N £ In cos [</>, - ( n - 1)A 71=1 R esu lts In this paragraph, th e solution of the integral equation for a perfectly electric conducting cylinder w ith T E -polarized incident wave is illu strated and com pared w ith th e exact solution. T he incident m agnetic field is chosen w ith a u n it am p litu d e H q and incoming from the positive x-axis, i.e., from an incident angle (fix = 0: H w (p ) = z e i k° x 41 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.63) Two-dimensional PEC Cylinder - TE case, 1 r a = 0.1, N=12 ' 2c 1 . o 11 . O c CO -i o.i cr Q O .i CO = 0.< ........ -----Theory » Method of Moments ..................... C/3 90 [deg] 45 135 180 Figure 3.14: Surface equivalent current obtained w ith m ethod of m om ents and exact solution for koa = 0.1 an d num ber of points N = 12. N um ber of term s in analytical expression sum m ation is M = 10. T he analytical expressions for the equivalent surface current and th e scattering w idth are given in [15]. T he surface current is given by th e infinite sum m ation •W ) ir koa m= 0 (3.64) H $ \ k o a) while the scattering w idth is JTM fi \ a 2D W s ) : kQ ^ i fcoa)- cos [m (cf,s - ^-)] cm J /ov £ ( - ! ) m em (3.65) {k0 a) 771=0 where em is defined in (3.49). T he m ethod of m om ents has been applied to four cases of P E C cylinders w ith different values of koa, for a T E incident wave given by (3.63). T he results are illu strated in Figures 3.14-3.17, where the equivalent current obtained by solving (3.54) is p lotted together to the exact solution as a function of the polar angle <f>. In all cases, th e relative difference betw een the two solutions has been found to be less th a n 1%. 42 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Two-dimensional PEC Cylinder - TE case, a = 1.0, N=20 2 1.4 •§ 0.8 c LU a) 0.6 0.4 — * Theory Method of Moments 0.2 135 45 180 <t>[deg] Figure 3.15: Surface equivalent current obtained w ith m ethod of m om ents and exact solution for koa — 1-0 and num ber of points N = 20. N um ber of term s in an alytical expression sum m ation is M = 15. Two-dimensional PEC Cylinder - TE case, 1^a = 2k, N=130 2 1.4 • § 0.8 cr LU a) 0.6 0.4 Theory Method of Moments 0.2 90 <i>[deg] 135 180 Figure 3.16: Surface equivalent current obtained w ith m ethod of m om ents and exact solution for k o a = 2 k and num ber of points N = 130. N um ber of term s in an alytical expression sum m ation is M = 20. 43 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Two-dimensional PEC Cylinder - TE case, 1^ a = 8k, N=500 • = 0.8 0.6 Theory Method of Moments 45 135 180 <|>[deg] Figure 3.17: Surface equivalent current obtained w ith m ethod of m om ents and exact solution for k^a = 87r and num ber of points N = 500. N um ber of term s in analytical expression sum m ation is M = 40. Figure 3.18 shows the backscattering and forw ard scattering w idth, a 2D(<ps = 0°) and a 2D((j)3 = 90°) respectively, as functions of koa. T he b istatic scattering w id th <r2D vs. the scattering angle < is plo tted in Figures 3.19-3.22 for different values of koa. Table 3.2 gives th e m axim um error emax, in %, for th e four values of koa considered, and also provides the num ber N of points in the M OM calculation a n d th e num ber of term s M to which sum m ation (3.65) has been tru n cated . koa N M tmax 0.1 1.0 2ir Sir 12 20 130 500 10 15 20 40 0.01 0.04 0.1 0.01 Table 3.2: Accuracy of M OM com putation of TM scattering w id th for P E C circular cylinder of various values of koa, w ith emax in %. 44 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Tw o-dim ensional PEC Cylinder - TE case Theory x Method of Moments ay -4 Figure 3.18: C om parison betw een scattering w idth obtained w ith m ethod of moments a n d exact solution as functions of k^a for scattering angles <j> = 0° (backscattering) and 0 = 90° (forward scattering). 45 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Two-dimensional PEC C ylinder-TE case, l^a = 0.1, N=12 -20 S -3 0 ■o § -4 0 -5 0 -6 0 Theory x Method of Moments -70. Scattering Jingle ^ .[ d e g ] 135 180 Figure 3.19: C om parison betw een scattering w idth obtained w ith m ethod of moments and exact solution for k^a = 0.1 and num ber of points N = 12.. N um ber of term s in analytical expression sum m ation is M — 10. Two-dim ensional PEC Cylinder - TE case, l^a = 1.0, N=20 Theory Method of Moments -4 -7 - 8. Scattering ^ng le <|>.[d e g ]135 180 Figure 3.20: C om parison betw een scattering w idth o b tain ed w ith m ethod of moments and exact solution for koa = 1 . 0 and num ber of points N = 20. N um ber of term s in analytical expression sum m ation is M = 15. 46 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Two-dimensional PEC Cylinder - TE case, l^a = 2n, N=130 *o O) -§ -1 0 ■o (13 N -2 0 o -2 5 Theory * Method of Moments -30. 45 90 135 Scattering Angle <|> [deg] 180 Figure 3.21: Com parison betw een scattering w idth o b tained w ith m ethod of m om ents and exact solution for koa = 2n and num ber of points N = 100.. N um ber of term s in analytical expression sum m ation is M = 20. Two-dimensional PEC Cylinder - TE case, l^a = 8n, N=500 O) Theory » Method of Moments -30. Scattering ^ngle i|>c.[d e g ]135 180 Figure 3.22: Com parison betw een scattering w idth o b tain ed w ith m ethod of m om ents and exact solution for koa = 87r and num ber of points N = 500.. N um ber of term s in analytical expression sum m ation is M = 40. 47 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.4 D ielectric tw o-dim ensional body For a dielectric body, b o th th e exterior a n d the interior integral equations have to be considered, and th e unknow n are b o th the surface electric an d m agnetic current. As for the PE C two-dim ensional body, the basic cases to be considered when studying th e scattering from a two-dim ensional di electric body are those w ith TM a n d T E incident waves. H ereafter, only the TM case will be considered. T he T E case can be easily derived using the duality principle. Let the incident electric field be a plane wave directed along th e 2-axis, i.e., be TM -polarized, w ith an expression (3.29). Such incident wave induces on th e perfect conductor an equivalent surface electric current J 6 th a t is also directed along the 2-axis, while th e equivalent surface m agnetic current M,s will have only th e transverse com ponent. Hence, J s(p) = U M s {p) p )z (3.66) = M r (p) t (3.67) Therefore, only equations (3.9) and (3.11) need to be solved: e ^ \ p ) = — + j ^ p o j>c+ <?2£>(P> + £ + 0 _ _ M t(p ) _ + p ') U p ') dC' + cos ^ j g2D{ p , p') U (^ d x ^ c o s^ ' + p M t { p ') dC', p e C+ (3.68) M r {p') dC', p ec~ (3.69) ') d C ' + s*n ^ Once Jz and M r are known, the scattering w idth can be determ ined using (3.23), th a t for this case becomes: where cr™ = a 1® is th e only non-zero scattering w idth. 48 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. N u m erical so lu tio n u sin g th e m eth o d o f m om en ts In addition to ap proxim ating Jz in term s of basis functions as in (3.35), the m agnetic current r com ponent is expanded as N t' e C (3.71) 71=1 S ub stitu tin g (3.35) an d (3.71) into th e system of integral equations (3.69),(3.70), and testin g the resulting equations w ith the functions (3.34) for k = 1 , 2 , . . . , N , leads to th e following linear system of algebraic equations: N N 'y ^-A-kn I n 4" y B k n M -n n =1 ~ Vk (3.72) = 0 (3.73) n=l N y N ] Cfcn I n " b y ( -Dfcn M n n— 1 where n=l Ffe = £ « ( * * ): In — Jzi^n) > (3.74) = 1 ,2 ,3 ,..., N — M r (tn) , n — 1 ,2 ,3 ,..., N (3.75) For the p articu lar case of a circular cylinder of rad iu s a, th e coefficients Akn an d B^n are found to be fcoCo u {2)( a A(j) ----------- H q (kou) U) Akn a^ oCo iU t A; -f— I b <3-76) J\ -2 i l - , - (3.77) where 5kn is the K ronecker’s delta defined in (3.60), and a y 2[1 — cos (k — n)A(j)] (3.78) T he coefficients Ckn an d Dkn have the same expressions as Ak n and Bk n, w ith eo, ^o> ko, and (o replaced by e, ju, k, an d (, respectively. 49 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Once th e system (3.72),(3.73) is solved for I n an d M n , the scattering w idth is given by: N kQ (Oi2 Y {Co I n ~ M n cos [<j>s - (n - 1) A 0 ]} e jko a cos [(f)s — (n - 1)A 0] n =1 (3.79) R e s u lts In this subsection, th e solution of the integral equation for a hom ogeneous dielectric cylinder w ith T M -polarized incident wave is illu strated and com pared w ith th e exact solution. A nalytical expressions for the induced electrom agnetic fields and th e scattering w idth can be found in [73]. T he equivalent surface currents can be found from th e induced tangential fields on the cylinder surface by using definitions (2.25) and (2.35), and are given by: p(i) -jT - oo Y • Jz(<P) = £m [jmiko a) + A m H $ ' (k 0 a) cos (m<f>) (3.80) m —0 oo Y 3 m [Jm(ko a) + A m H $ (kQa) cos (m<f>) (3.81) m=0 and while the expression for th e scattering w idth is: 4 ko Y (-1)™ em am (k0 a) cos [m(<f)s - <f>i)} (3.82) 771=0 T he coefficients am in (3.80)-(3.82) are given by am (ki/fJ-i) Jm(ko o) Jm (ki of) (ki/ni) (ko j Ho) Jm(ko ®) Jm(ki a) (3.83) (kQa) J'm (ki a) - (kQ/no) H $ (kQa) Jm (k\ a) T he system of integral equations (3.68),(3.69) has been solved for a num ber of cylinders of different size param eter koa and relative dielectric constant er . Figures 3.23-3.30 show the equiv alent currents J s and M s vs. the azim uth angle <j>for those cases. T he solid line is the analytical solution (3.82), while th e crosses indicate the M OM values. T h e scattering w idth is p lo tted in Figures 3.31-3.34 as a function of th e size param eter koa, and in Figures 3.35-3.35 as a function of the scattering angle <f>s. 50 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Two-dimensional Dielectric Cylinder - TM case,s. =2.56 « Theory Method of Moments k0a = 5.0 a * 1.0 135 180 <i>[deg] Figure 3.23: E quivalent electric current, m ethod of m om ents solution and theory, vs. (f>, for a lossless homogeneous cylinder w ith sT = 2.56, for k^a = 1.0 a n d k^a = 5.0. N um ber of points used in the m ethod of m om ents are N = 20 and N = 100, respectively. Two-dimensional Dielectric Cylinder - TM case,er =2.56 3.5 x Theory Method of Moments 2 2.5 « 1.5 45 90 135 180 <j>[deg] Figure 3.24: Equivalent m agnetic current, m ethod of m om ents solution and theory, vs. (j), for a lossless homogeneous cylinder w ith eT = 2.56, for k^a = 1.0 an d k^a = 5.0. N um ber of points used in the m ethod of m om ents are IV = 20 and N = 100, respectively. 51 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Two-dimensional Dielectric Cylinder - TM case,sr=2.56 - j0.102 Theory Method of Moments kQa = 5.0 135 180 Figure 3.25: E quivalent electric current, m ethod of m om ents solution and theory, vs. 0, for a lossy homogeneous cylinder w ith er — 2.56 —j'0.102, for k^a = 1.0 a n d k^a = 5.0. N um ber of points used in the m ethod of m om ents are N = 20 and N = 100, respectively. Two-dimensional Dielectric Cylinder - TM case,sr=2.56 - jO. 102 3.5 x Theory Method of Moments 2 2.5 ” 1.5 cr 45 90 <t>[deg] 135 180 Figure 3.26: Equivalent m agnetic current, m ethod of m om ents solution and theory, vs. <fi, for a lossy homogeneous cylinder w ith eT = 2.56, k^a = 1.0 and k^a = 5.0. N um ber of points used in the m ethod of m om ents are N = 20 and N = 100, respectively. 52 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Two-dimensional Dielectric Cylinder - TM case,e = 5.0 - j 1.0 4.5 Theory Method of Moments 2.5 = 1.0 0.5 90 <i>[deg] 135 180 Figure 3.27: E quivalent electric current, m ethod of m om ents solution a n d theory, vs. (f), for a lossy homogeneous cylinder w ith er = 5.0 —j‘1.0, k$a = 1.0 and k^a = 5.0. N um ber of points used in the m ethod of m om ents are N = 20 and N = 100, respectively. Two-dimensional Dielectric Cylinder - TM case,£.= 5.0 - j 1.0 « Theory Method of Moments 0.2 45 135 180 (j>[deg] Figure 3.28: Equivalent m agnetic current, m ethod of m om ents solution and theory, vs. (f>, for a lossy homogeneous cylinder w ith sT = 5.0 —j 1.0, k^a = 1.0 and k^a — 5.0. N um ber of points used in th e m ethod of m om ents are N = 20 and N = 100, respectively. 53 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Two-dimensional Dielectric Cylinder - TM case, s =60.0 - j59.9 — * Theory Method of Moments 135 180 Figure 3.29: Equivalent electric current, m ethod of m om ents solution and theory, vs. tfi, for a lossy homogeneous cylinder w ith er = 60 —j59.9, k0a = 1.0 and k^a = 5.0. N um ber of points used in th e m ethod of m om ents are N = 20 and N = 100, respectively. Two-dimensional Dielectric Cylinder - TM case, s =60.0 - j59.9 0.3 £0.25 c 0.15 Theory Method of Moments = .1,0 .k0a.=.S.O 0.05 135 180 Figure 3.30: Equivalent m agnetic current, m ethod of m om ents solution and theory, vs. </>, for a lossy homogeneous cylinder w ith er = 60 —J59.9, fcoa = 1-0 an d k^a = 5.0. Num ber of points used in the m ethod of m om ents are N = 20 and N = 100, respectively. 54 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Two-dimensional Dielectric Cylinder - TM case,£.=2.56 Theory Method of Moments £ 10 ■a = —< 180' CD Figure 3.31: S cattering w idth, m ethod of m om ents solution and theory, size param eter k$a, for a lossless homogeneous cylinder w ith er = 2.56, for k^a = 1.0 a n d k^a = 5.0. Solid line is backscattering (<f>s = 0°), and dashed line is forw ard scattering (<f>s = 180°). Two-dimensional Dielectric Cylinder - TM case,er=2.56 - j 0.1 Theory Method of Moments = 180' £ 10 o> = 0 Figure 3.32: Scattering w idth, m ethod of m om ents solution and theory, size param eter k^a, for a lossy homogeneous cylinder w ith er = 2.56 — j'0.102, for koa = 1.0 and koa = 5.0. Solid line is backscattering (cf>s = 0°), and dashed line is forw ard scattering (<f>s = 180°). 55 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Two-dimensional Dielectric Cylinder - TM case,£=60.0 - j59.9 Theory Method of Moments = £ 10 180' = 0' Figure 3.33: Scattering w idth, m ethod of m om ents solution and theory, size param eter k^a, for a lossy homogeneous cylinder w ith eT = 5.0 — j l .0 , k§a = 1.0 and k^a = 5.0. Solid line is backscattering (<f>s = 0°), an d dashed line is forw ard scattering ((f)s = 180°). Two-dimensional Dielectric Cylinder - TM case,er= 5.0 - j 1.0 25 Theory » Method of Moments 20 = 0' Figure 3.34: S cattering w idth, m ethod of m om ents solution a n d theory, size param eter koa, for a lossy homogeneous cylinder w ith s r = 60 — j 59.9, k^a = 1.0 and k^a = 5.0. Solid line is backscattering (cf>s = 0°), and dashed line is forw ard scatterin g (<ps = 180°). 56 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Two-dimensional Dielectric Cylinder - TM case, e =2.56 Theory x Method of Moments k„a = 5.0 -15 - 20. 45 90 135 Scattering Angle 0 [deg] 180 Figure 3.35: Scattering w idth, m ethod of m om ents solution an d theory, vs. scattering angle <f>s, for a lossless homogeneous cylinder w ith eT = 2.56, for koa = 1.0 and koa = 5.0. N um ber of points used in the m ethod of m om ents are IV = 20 and N = 100, respectively. Two-dimensional Dielectric Cylinder - TM case, e =2.56 —j0.102 Theory Method of Moments D) -5, k.a = 5.0 -15 -20 Scattering jungle <j>e.[deg] 135 180 Figure 3.36: S cattering w idth, m ethod of m om ents solution and theory, vs. scattering angle <f>s, for a lossy homogeneous cylinder w ith eT = 2.56 —y’0 .102, for koa = 1.0 and koa — 5.0. N um ber of points used in the m ethod of m om ents are N — 20 and N = 100, respectively. 57 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Two-dimensional Dielectric Cylinder - TM case,er=60.0 - j59.9 Theory x Method of Moments ■o O) k.a = 5.0 -6 -8 -10 Scattering ^ngle <j>s.[deg] 135 180 Figure 3.37: S cattering w idth, m ethod of m om ents solution an d theory, vs. scattering angle <j>3, for a lossy homogeneous cylinder w ith sr = 5.0 — j 1.0, koa = 1.0 and koa = 5.0. N um ber of points used in the m ethod of m om ents are N — 20 and N = 100, respectively. Two-dimensional Dielectric Cylinder - TM case,e = 5.0 - j 1.0 Theory x Method of Moments TO -10 = -15 -25 45 Scattering Angle <j> [deg] 135 180 Figure 3.38: S cattering w idth, m ethod of m om ents solution and theory, vs. scattering angle <f>s, for a lossy homogeneous cylinder w ith er = 60 — y59.9, koa — 1.0 a n d koa = 5.0. N um ber of points used in the m ethod of m om ents are N = 20 and N — 100, respectively. 58 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.5 Conclusions In this chapter, the m ethod of m om ents has been used to solve the coupled surface integral equa tions, and detailed results have been given for cylinders of circular cross section. B oth perfectly conducting an d dielectric - lossless and lossy - cylinders have been considered. For perfectly con ducting cylinders, th e cases of TM and T E incident wave has been investigated, while for conducting cylinders only th e case of TM incident wave has been studied. All results have been found to agree very well w ith the theory. T his validates th e use of th e coupled integral equations derived in C hapter 2 as a tool to determ ine th e scattering from a rb itra ry objects located in an infinite m edium. 59 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 4 B odies of R evolution T he previous chapter has dealt w ith infinite cylinders illum inated by an electrom agnetic source th a t is function of the coordinate z. A lthough th e resulting problem is three-dim ensional in n atu re, a Fourier decom position along th e cylinder axis can be used to reduce the problem to a superposition of two-dim ensional problem s m ore su itable for num erical solution. In th is chapter another specific class of three-dim ensional problem s will be considered, th a t can also be tre a te d as a superposition of two-dim ensional problem s. A xisym m etric objects, often called bodies of revolution because they can be obtained by ro ta tin g a plan ar arc around an axis, constitutes such particular class. In m ost cases, vegetation elem ents can be m odeled as axisym m etric objects, therefore the ability to tre a t bodies of revolution is very im p o rta n t in the scattering from vegetation. For them , a Fourier series in the ro tatio n angle can be employed to transform the problem into one involving uncoupled equations for each harm onic. 4.1 G eom etry of th e problem As shown in Figure 4.1, a b o d y of revolution is obtained by ro ta tio n of a planar arc C - called the generating curve - ab o u t a n axis, w hich is hereafter chosen as the z-axis of a C artesian coordinate system. T he surface 5 th u s generated represents the bou n d ary betw een free space and the m aterial body, which is assum ed as having p e rm ittiv ity e = £q an d perm eability /r = mo Md- Any point P 60 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.1: B ody of revolution on S is uniquely identified by a pair of variables (f, 0), where t is th e curvilinear coordinate along the generating curve to which P belongs, and 0 is th e ro ta tio n or azim uth angle m easured from the ajz-plane. A right-handed tria d ( n , 0 , r ) of orthonorm al vectors is defined on S, w ith n and f being the norm al and the tangent u nit vectors to S in th e plane of the curve C. All these definitions are illu strated in Figure 4.1. T he tria d ( r , n , 0 ) can be expressed in term s of ( x , y , z ) as follows: f = s in 0 c o s 0 x + s in 0 s in 0 y + cos0z (4.1) n = cos 0 cos (f>x + c o s 0 s i n ^ y —s i n 0 z (4.2) 0 = —s i n ^ x - f c o s ^ y (4.3) where 0 is the angle betw een r and z , i.e. cos 0 = r ■ z , w ith 0 = 0 if f = z and 0 = n / 2 if 61 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. f = p . The infinitesim al element of area on S is given by d S = pdtd<j) 4.2 4 .2 .1 Let th e (4.4) The electric field integral equations for a b od y of revolution F o r m u la tio n body of revolution be excited by an incident plane wave w ith tangential com ponent (i) E) on the surface S. T he induced equivalent tangential electric and m agnetic currents J 5, M s on th e surface S satisfy th e surface integral equations (2.37)-(2.38), on th e exterior and interior surface, S + and S ~ , respectively : C M s(r) x n = 2 E$ (r) — 2ju>po (I — n h ) ■ Js + - 2 (I - n M ,(r) x n = n ) ■[ Js+ 2jup (I - h - 2 (I - n where r7 /-t0 &o VG°(r, r') x M , (r') d S ' n) ■f , r € S+ (4.5) J s (r ' ) d S ' + G( r, r ' ) I — ^ttt-V V ' k2 s Js- L n )• J s(r') dS' + G0( r , r ' ) I ------ VG (r,r') x M .( r ') d S ' , , r € S~ (4.6) I is the unit dyad, I = x x 4 - y y + ^2 = n n + 0 0 + f f (4.7) , G ° ( r ,r ') and G (r, r ') are the scalar G reen’s functions in the two m edia, - j k |r - r '| , ~ j k o \ r - r '| G °( r , r ') = G ( r ,r ') = 47r |r — r 47r |r — r '| ’ (4.8) ko and k are the wavenum bers, k0 = lu^/ eopo , k = u ^ / e p = kQyJpded , (4.9) and V's ■J s is the divergence of J s on th e surface 5 , X-7> T (r l\ V , - J .(r ) = <K ^ r 9 3 s 62 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.10) T he exterior an d interior surfaces S + and S ~ , respectively, have been introduced in C h ap ter 2: S + is defined as a surface infinitesim ally close to th e surface S and enclosing S itself, and S ~ is defined as a surface infinitesim ally close to the surface 5 and enclosed by S itself. E quations (4.5),(4.6) are expressed in term s of vector quantities. T h e very first step in their solution is to decompose th em into a pair of scalar integral equations for each of their com ponents. (i) T he tangential incident electric field E ) a n d th e equivalent surface currents J s, M s on th e surface S can be w ritten in term s of their r - and ^-com ponents as follows: E ^ ( t , c f > ) f +Ef(t,cj>)4> E « (r) = J s (r) = M s(r) = (4.11) JT(t,<p) t + 4> (4.12) M T(t,<j>) t + M<p(t,<f>) $ (4-13) S u b stitu tio n of these expressions into th e first integral equation (4.5) and separation th e single com ponents yields two integral equations th a t contain only th e r - and ^-com ponents of J s, M s and E j^ . Introducing eight integral operators a pq and f3pq, (p, q = r , (p), these equations m ay be w ritten in com pact form as follows: £ a ' « { J J( t V ) } + £ = £?> (<,« (4.14) q=T,<t> ? = T ,0 + £ q = T ,ip (4.15) q = T ,< t> T he operators a pq and f3vq (p,q = r , <f>) perform integrations of th e functions J q(t', <pr) and Jq( f , </>'},respectively, in th e variables t' and (j)', which axe therefore left indicated in their notation. These operators are defined as: a pT{ J T{t',(f>')} d pq{ M q{t',<4>')} — p • r 'G ° ( r , r ') - = - j u j f j ,o [ = -ju p = - [ p ■V G 0( r ,r ') x q ' M g t f , # ’) dS' Js Js p o [ Js , / _ 0, p • 0 G (r, r ') - k i dp dt' [p'JT( t ' d S ' 1 1 dG° d -2 p' rJ dp dcp_ kg 63 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.16) (4.17) (4.18) M ore detailed expressions are given in A ppendix C. T h e integral equation (4.6) for the interior of th e body can be decom posed into its r - and ^ —com ponents in a sim ilar m anner, and w ritten as £ 7t8 q=T,<j> Y ^ q=T,<t> # ' ) } - £ S'" q=T,<j> « } 4)} - q-r,<p E S** *')} + w here the expressions for the operators 7P9 and 5pq - = o (4.19) = 0 (4.20) (p, q = T,<fr) can be found from the expressions (4.16)-(4.18) for a pq and (3pq by replacing G °(r, r ') , &o, £o>and po w ith G(r, r1), k, e, and p, respectively, and they are also given in A ppendix C. 4 .2 .2 S e p a r a tio n o f m o d e s Expressions (4.14),(4.15) and (4.19),(4.20) are tw o-dim ensional integral equations on th e surface 5 . One could solve such system of equations using th e m ethod of m om ents w ith th e definition of a two-dim ensional grid of points on S. However, this approach would require a large num ber of points and th e evaluation of tw o-dim ensional integrals, a n d would result in a a heavy com putational cost. Fortunately, the axial sym m etry of th e surface of revolution S allows a much sim pler tre a tm e n t of th e problem because it im plies th a t all th e quantities involved in such equations have th e sam e periodic dependence on th e azim uth angle 0, w ith a period equal to 2 tt or an integer m ultiple of it. In particular, since Ep \ Jp and M p (p = r, <j>) are periodic functions in the variable <j>, on the surface S they can be expanded into Fourier series as follows: +oo 4 °M ) = E ® {t)ein* E (4.21) 71 — — OO -boo Jp( t J ) = Y -boo M p(t,cf>) = £ M Pn( t ) e qn<» (4.22) n = —oo n=— oo Using such Fourier m ode expansions into the expressions for oiPq and f3pq results into a series 64 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. expansion for those operators themselves: a T ,W + ' ) } = +00 2* £ (4.23) 71= —00 = +00 27T £ T n q { M qn(t')} (4.24) n=— oo a pq and /3pq are linear operators perform ing curvilinear integrals along C. For example: < T {Jrn ( t ' ) } = (4.25) + c o s , 0 o ( t i t ,, _ P Z { M r [<, # ) } = (4.26) = —j [ [p cos ip sin ip' —p' sin ^ cos ip' — (z — z ') sin ip sin ip'] «/c -^n+i(A O where p' and ip' are the values of p and ipfor t = t'. E xplicit expressions given in A ppendix C. T he expressions (4.25) and (4.26) ^ 2 for all a pq and (3pq are for a T nT and /3£T, aswell as those for the other operators a pq and (3pq, contain th e functions: e°(t, f ) = K° ^ = -i rZ'K - jf C W j e - ’W - W < ! ( # '- * ) (4.27) (4-28> where R is the distance betw een th e two points a t r, r': R = |r — r '| = \ ! p 2 + p'2 — 2pf/ cos(0' — <p) + (z — z ' ) 2 (4.29) T he integrals in (4.27)-(4.28) are defined on the surface S. T h e in teg ratio n in <p' — <p removes the angular dependence, so th e results are functions of t a n d t' only. Expressions (4.27) an d (4.28) can be regarded as the coefficients of th e expansion of the G reen’s function G0(r, r') a n d its derivative into Fourier series on th e surface S : +oo G ° ( r ,r ') = J] G0n { t , t ' ) e j n W - ® v ,r'e S n=— oo 65 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.30) 'N+l 'N+l ’N+l (a) (b) Figure 4.2: (a) Choice of discretization points {f&} on the curve C. (b) Discretized curve C 1 dG°{ r , r ') R dR +00 53 r ,r 'e S (4.31) If (4.21) and (4.22) are su b stitu te d into (4.14)-(4.15) and (4.19)-(4.20), and th e resulting equa tions are integrated side-by-side in th e variable 0 over th e interval [0,27r], the following system of linear equations is o b tained for each single harm onic m ode n = 0, ± 1 , ± 2 ,...: « * „ (* ) 9 = T ,< E in g~T,<f> E q = T ,i (4.32) 4 5 (0 (4.33) q=T,<, E a? «»(<')}+ E 0t, {M,Jt)} + M tM ?=T,< 4 5 (0 <7=r,<? - E C - 444 (4.34) q=T,<t> y . ^’ {v,„(t')) + 4 4 4 = 0 q = T ,< f 66 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.35) 4 .2 .3 M e t h o d o f m o m e n ts s o lu t io n A set of N + 2 points {£*}, k = 0 ,1 ,2 , . .. N + 1, is chosen on the generating curve C as illustrated in Figure 4.2 (a), w ith to and fjv+i being th e end points. The points {£&} subdivide C into N + 1 segments, which represent an approxim ation C of the generating curve, as shown in Figure 4.2 (b). T hen, a second set of N + l points {£& }, k = 1,2, ...N + 1, is chosen on C. T h e generic point t is defined as the m idpoint on the segment [tk-itk]. Following the M OM, th e tangential incident electric field and the tangential surface equivalent currents are approxim ated by th e following sum m ations: N N + l PE${t) = E f n (t) = > N P jrjt) = J 0n(f) = (4.37) m= 1 iv n - r t E M % p W (t) m= (4.36) N + l £ j ^ T m (f) , m = 1 ,(t) = p£> (f) , M tJt) = 1 (4.38) m = 1 where the expansion functions are chosen as P « (t) = P (~ t ^ ) , \ A fm / P ${t) = p Tm{t) m = 1 , 2 , 3 ,..., N { ^ ^ Y = r(^ r )’ (4.39) ™= 1 , 2 , 3 , . . . , N + 1 (4.40) rn = l , 2 , 3 , . . . , N (4.41) and are illustrated in Figure 4.3. T he segm ent lengths A tm and A f~ are defined, respectively, as Atm = t S —i ~ t m , A tm = tm —fm-i , m. = 1 ,2 , . . . , N + 1 (4.42) and P(t) and P(t) are rectangular and trian g u lar pulse functions defined as f 1 P(t) = I 0 for _ I < t < I 2 - C<2 elsewhere , T( t) = f 1 — |*| ' ' I 0 for - 1 < t < 1 , u ■ (4.43) elsewhere 67 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. f A tm lm--^T " f j_ A tn lm~I— tm— A tn tm~~ A.trr. A tn (b) (a) (C) Figure 4.3: D efinition of expansion functions: (a) Pm, ^(t ), (b) 5(2) 1 and (c) Tm (t). T he choice of tria n g u la r pulses for the expansion of th e r-com ponent of the equivalent surface current Jt is necessary due to th e presence of a derivative of JTn{t). T he use triangular pulses T(t) makes possible to express such derivative as a linear com bination of rectangular pulses P{t). In particular, it is easy to see th at: ,(2) P t t X t ) + P%1 iCO A t m+1 Atn N Tn 1 = £ • £m m= 1 dt (4.44) Using th e approxim ating sum m ations (4.36)-(4.38) and (4.44) into equations (4.32)-(4.35) yields: N E N + l O' )A \ j T m + Y , aTn { P£ ) (t' ) } j t + P' 771=1 J m= 1 ,(1) N (4.45) + 771=1 V “ J N E a. — Pn[P '}J t + E rjrr 771=1 ' { P t (0}J t + 771=1 ( p ( 1) 'I N 771=1 N + l + t M l + 771=1 P -E 771=1 r rp (4.47) \ 'N 771=1 f p (l) ) I P ) Hr i V - |- l E ^ P #P 2}J t + E 771=1 = 0 E 7? H 2)(0} J t + J JV N (4.46) 771=1 ' T m (t') 71=1 = E ( + t) 1V+1 “E H ’m } M t - 771=1 { p t d ') } J t + no — 1 68 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Pm m =T 1 i p r 1 J, K m = (4.48) 11 Using the following testing functions on equations (4.45),(4.47) and (4.46),(4.48), respectively, xl(t) = X*(t) = S (t-tj), k = 1 ,2 ,3 ,...,1V (4.49) k = 1,2,3,...,N + 1 (4.50) leads to the following linear system of N p algebraic equations, for each polarization p = N q N q Y Y q=r,(f> aPn ( k i m ) Xm m—1 +Y Y K q( k , m ) M ^ = Vpkn (4.51) = o (4.52) q—T,<t> m=l N q N q Y Y + Y q=r,4> m = l q=T,4> Y < " ( k’™ ) M i m=l for k = 1 , . . . , N p where Np N N + l = for p = t for p = (f> (4.53) T he coefficients a ^ ( k , m ) a n d b%}(k,m) k = 1,..., N p-, m = 1,..., ./V ar ( t , m ) = -jf a^{k,m = ~ [ <*P n { P $ ( 0 } Xi (*) d t , Jc bT nT( k , m _ r P k bT J(K m btT(k, m b^(k,m ^ = km [ OTT I k = 1,..., A/p; 77i = 1,..., N + l X litfdt, (4.54) (4.55) k , m = 1,..., AT (4.56) j c f t * { P $ ( t ' ) } xl(t) dt, k = 1,..., AT; m = 1,..., N + 1 (4.57) jc ^ T\ ^ p ^ x W ) d t , k = 1,..., N + 1; m = 1,..., AT (4.58) Pk A tk k, 771 = 1,..., Af + 1 (4.59) where Skm is the th e K ronecker’s delta. Expressions for th e coefficients bf f(k ,m ) and d ^ ( k , m ) can be obtained from (4.54)-(4.59) by replacing the G reen’s function, ko, eq and p,o. Defining th e 69 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. i 1 vectors ■ ' M {n ‘ M v2n 3 3 ■ J-P n V Pn _ , l Pn = , = p = r,4> , (4.60) H ......... ■ • £ £ ? 1 1 i 1 i i 1 the system of equations (4.51)-(4.52) can be w ritten in th e following convenient m atrix form: ] M Tn + + _ Cd \T<t> J<t>r y Tn (4.61) — y<t>n (4.62) i [A ? ] I T" + 4><t> [ A f ] l Tn + [a ^ \ J<Pn + [ B f M Tn + ±■R Jn [C 7 ] ITn 4- [c ;* ] J4>n - p r M Tn - [ c f ] l Tn + [C ^ ] I*" - [ ° f ; M Tn - d t<i> M <Pn = 0 (4.63) M^n 0 (4.64) 71 = where [A£9] is th e m a trix whose elem ent k , m is apq( k , m ) given by (4.54),(4.55), i.e., [A -n9] = . . _ a pq(Np, 1) . apq(Np, 1) ••• . P ,q = r,< (4.65) a g ( N p, N q) . and so on for [B£9], [B£9] , and [D£9]. 4 .2 .4 I n c id e n t field This section focuses on th e expansion into angular m odes of th e tangential electric field on the surface 5, as given in (4.21). A n expression for th e n -th coefficients E Pl (t) will be derived based on their definition \ /"27T P = T, E ^ [t) = (4.66) Let the body of revolution be illum inated by a uniform plane wave E w (r) = In equation (4.67), ko = uj^/ uqS q (4.67) is th e free space wavenum ber, and k i = — sin 9i cos <pi x —sin di sin fa y — cos z 70 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.68) is the propagation vector of th e incident wave from th e direction (9U fa). H orizontal and vertical polarizations of the incident wave are defined by the direction of th eir u nit polarization vectors: A hi = ^ ^ ------ - = |kj x z| —sin fa x + cos fa y (4.69) Vj = hi x ki = - cos 9i cos fa x - cos 9i sin fa y — s i n ^ z (4.70) T he vector E t can be w ritte n as a linear com bination of hi and Vi as follows: Ei - E f hi + ^ (4.71) Using the definitions (4.1)-(4.3), the r - and ^-com ponents of E ^ can be w ritten as E^ = |E ^ s i n '0 s i n ( 0 — 0i) — [sin-0cos0icos (0 — 0i) + c o s0 sin 0 i]j = \E h^ cos (0 ~ & ) + E V5 cos 0 i sin (0 _ <f>ij\ e (4.73) ftokj-r = —u p c o s f i — v z where the exponent is u = fcosin^i, w ith k° ^ r (4.72) (4.74) v = ko cos 9i . (4.75) Perform ing the integration in (4.66) yields the coefficients of th e n -th mode for the r - a n d 0com ponents of E W : E^ = j n | E ^ sin-0 T+ (up) - E ^ [ - j s in -0 cos 9i T ~( up ) + cos 0 sin#; Jn (up)] | e~:in^ieE z (4.76) E« = f co s9iT+(up)\ e ~ ^ e ? vs [jE ^T -(up) + E ^ where the functions (4.77) are defined as T +( X) = Jn + i ( x) + Jn- lM T ~ ( x ) = Jn+l{x) ~ Jn~l{x) , Li (4.78) Li Of particular interest, due to its simplicity, is the case of a plane wave incident along th e 2-axis, i.e., from an angle 9i = 0. Since u = 0, and Jn (0) ^ 0 only for n = 0, it follows the only m odes 71 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. w ith non-zero com ponents are those w ith n = ± 1 , which have r - and ^-com ponents: 4 .2 .5 sin $ ( j '4 ° + 4 ° ) eT ejk°Z E rl = ^ B f ±i = ± ± ( E P + j E U ) e ^ i h e j k 0z (4-79) (48Q) S c a tte r e d field in th e r a d ia tio n z o n e T he coupled surface integral equations are solved w ith (4.67) as incident electric field. In particular, a polarization q is considered, i.e., q = h,v EW = f ? W q i , T he equivalent surface currents (4.81) th a t are found as solutions are subsequently used to com pute the scattering am plitudes. According to expression (2.63), the scattering am plitudes of th e object generated by th e equiv alent surface currents (k , Kk td) Jfpq{Ks -— and are given by ___ 1 jkoCo ^ p- s Co Js £ ( 0 e + 4> J i 9)(r') k° r ' dS' + ( 0 e - e 0 ) •M ^ (r ') e i k° ^ s ' r' dS' q = h,v (4.82) where k 5 is the propagation vector of th e scattered wave in the direction (9S, <fis), k s = sin cos x + s i n 0 s s i n ^ s y + cos0s z . (4.83) while the associated horizontal and vertical polarizatio n unit vectors are A hs = k x z ------- = |k s x z s i n 0 s x — cos vs = hs x ks = cos y = xv 0 cos 05 x + cos 9S sin 0 S y + sin05 z (4.84) = 0 (4.85) Once the scattering am plitudes fpq are known, th e b istatic scattering coefficients apq can be calculated from them using th e relationship (2.64). 72 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In order to evaluate th e integrals in (4.82), the equivalent surface currents J 5, M s axe w ritten as sum m ations o f Fourier m odes an d approxim ated in term s of basis functions used in th e M OM solution of the integral equations (4.5)-(4.6). As a result, th e scattering am plitude will also have the form of a sum m ation of an infinite num ber of modes, which is tru n c a te d to include only those m odes th a t produce a significant contribution to th e scattering. Introducing the functional operators V pw and Q pw as V pw = j> ( p - w ' ) Jw(r') r dS' , 4>')} = £ ( h - w ' ) M w{ v ' ) e i k^ ' r ' dS' = Q vw { M w(t', $')} = Q hw <f ( v . w ' ) M ffi( r ' ) e j l o ^ Js p = h , v , w = T,<j> V vw { M w(t',4>')} r , d5' = - 1 Co V hw (4.86) (4.87) (4.88) the expression (4.82) for th e scattering am plitudes can be w ritten as, for p ,q — h,v, f pq( k , k ) E = E q 77 + (4.89) w = r ,4 > T he expansion of th e r - and qJ- com ponents of the equivalent surface currents into Fourier series as in (4.22) and approxim ation th rough (4.37)-(4.38) leads to w rite th e operators V pw as: +oo. V pw { J w{t',<p')} = E Nw ’ p — h , v \ w = T,(j) (4.90) n=—oc m=l (n) where the dependence of the solution term s Z™n and of th e coefficients pm, w on the p articu lar direction and polarization of th e incident and scattered wave, respectively, have been explicitly indicated. Substitution of (4.90) into (4.89) and use of the relationships (4.87)-(4.88) between V pw and Q pw allows to w rite th e scatterin g am plitudes as an infinite sum m ation of angular modes, i.e., +oo f pq(k s,ki) = E U ) e ? n*‘ n = —oo 73 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.91) X X disk w asher sphere cone-sphere cylinder Figure 4.4: G eom etry of various bodies of revolution, whose n -th m ode term s are /^(k a.k i) = - j k 0C0 4 tT Nw q i) - i - P ^ U k ^ v , ) w = T ,< t> m = z l L q j) (4.92) ^ (4.93) 47t E (q^ w=T,<j>m=l Explicit expressions for the coefficients p ^ L ( k 3, p s) (p = h ,v , w = r , <f>) are given in A ppendix C. T he b istatic scatterin g coefficient can be found from th e scattering am plitudes using <jp9( k s, k j ) 4.3 = 47t \f pq{9s)\2 , p ,q = h , v (4.94) Validation T he M OM procedure to evaluate the equivalent surface currents and b ista tic scattering coefficient for a body of revolution, described earlier in this chapter, has been im plem ented and tested on a num ber of objects of different shapes and dielectric properties. In this section, in order to validate 74 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. th e correctness of this m ethod, com parisons will be m ade w ith o ther results, b o th num erical and analytical, found in the literatu re. Perfectly electric conducting (PE C ) bodies will be treated first, a n d lossy dielectric bodies will be considered in the subsequent subsection. 4 .3 .1 P e r f e c t ly e le c t r ic c o n d u c tin g b o d ie s o f r e v o lu tio n Bodies of revolution m ade of a perfect electric conductor are considered. The tangential electric field is null on the surface of a perfect electric conductor, hence th e equivalent surface m agnetic current also vanishes on S. O nly th e surface integral equation for exterior problem needs to be solved: Ejl}(r) = jwpo (I - n n ) ■ f G °(r,r')I - Js+ J s (r/) dS' , r e 5+ (4.95) Ko a n d the m atrix form (4.61)-(4.64) of the system of algebraic equations becomes [ a ; t ] i Tn + [ a ^ ] i*» = v T" (4.96) [ a £ t ] r™ + [a**] i*» = v ^ (4.97) Using these equations, th e electric surface current J 3 has been calculated for P E C bodies or revolution of various shapes, as illu strated in Figure 4.4. C om parisons have been m ade w ith values from exact analytical expressions - when available - or w ith num erical results from the literature. An im portant source of d a ta for com parisons has been th e num erical code from Glisson and W ilton [47], also based on th e M OM solution of surface integral equations. The d a ta used for the validation th a t is referred to as W G in the plots has been generated using this code. M autz and H arrington have also stu d ied the num erical solution of th e surface integral equation for bodies of revolution [42, 44, 45]. Figures 4.5-4.8 illu strate the results of th e m ethod of m om ents approach derived in this dissertation, denoted as BO R, com pared to th e values reported in [44] (MH) and th e Glisson and W ilton (W G) solution. 75 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. J - BOR J -WG J -MH O" •o SA?a A 0.5 A qV Jt-Ar 0.5 Figure 4.5: Surface currents vs. t /X on a P E C disk w ith radius a = 1.5A, for 6t = 0°. Com parison betw een results of th e present m ethod (B O R ), W ilton-G lisson (W G ), and M autz-H arrington (MH) solutions. J - BOR 3.5 X J - WG J - MH "(/> J - BOR C CD 5 o c ^ > ’3c • oN J - WG 2.5 J -M H 9 1.5 CO £ o c 0.5 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Figure 4.6: Surface currents vs. t /X on a P E C washer w ith radii ai = 0.4A and oi = 1.2A, for 6i = 0°. C om parison betw een results of th e present m ethod (B O R ), W ilton-G lisson (W G), and M autz-H arrington (MH) solutions. 76 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.5 J - BOR J - WG I 3.5 J - MH cCD J - BOR ♦ 3O c 2.5 o A CO J. - WG J - MH •5 cr 2 d) TJ , 0 ) 1.5 N co E o c 0.5 0.2 0.4 0.6 0.8 Figure 4.7: Surface currents vs. t j A on a P E C open-ended cylinder of radius a = 0.4A and length L = 1.2A, for 6i = 0°. C om parison betw een results of th e present m ethod (BOR), W ilton-G lisson (W G), and M autz-H arrington (MH) solutions. J - BOR J - WG —_ 2.5 J - MH J - BOR J - WG J - MH CT 0.5 0.5 t/X Figure 4.8: Surface currents vs. t/A on a P E C cone-sphere of rad iu s a = 0.2A and length L — 1.2A, for 0j = 0°. Com parison betw een results of th e present m ethod (B O R ), W ilton-G lisson (W G), and M autz-H arrington (MH) solutions. 77 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. JT - BOR Jt -MIE J - BOR <> J - MIE 0.4 40 80 100 elevation angle 9 [deg] 120 140 160 180 Figure 4.9: Surface currents vs. t /A on a P E C sphere w ith size param eter k^a = 1.0. C om parison betw een results of the present m ethod (BOR) and Mie series (M IE) solution. Figure 4.5 shows th e norm alized electric surface currents as a function of the norm alized coor dinate t /X (defined along th e generating arc) for a P E C disk of radius a = 1.5A. T he incident field is a plane wave illum inating th e disks from the positive z-axis, i.e., from an incident elevation angle 9l = 0° Figures 4.6 and 4.7 show th e norm alized electric surface currents along the generating arc for a P E C washer of radii a\ — 0.4A an d a i = 1.2A, and for a P E C open-ended cylinder of radius a = 0.16A and length L = 1.0A, respectively, also for 6t = 0°. T he thickness of the disk, washer, and of the walls of th e open ended cylinder are assum ed infinitesim al. Theses cases can be used as approxim ations of real bodies in which the thickness ten d to zero. Figures 4.8 shows th e electric surface currents for a P E C cone-sphere of radius a = 0.2A and length L = 1.15A, for a plane wave incident from 6l = 0°. For a perfectly conducting sphere, th e exact analytical expressions of th e currents excited by a 78 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.8 I I ! 1 x 1.6 X O 3s. —) 1.4 t 3 C/i 1 ........ -o 1 <3 © © © (9 X 0 .8 0.2 0 P .© . / . .... c 6 ■Ar a O o 0 O Q G ° QC I 20 40 i 60 . Q 0.4 JO/ Q E o \x# r ...... E 0.6 •o 05 N CO : 3 C 05 CO M IC v .0 NO 1.2 O a> •a _ mie VBOR ------- c a? 3 U <D CO - BOR ____ jt - 1 80 100 120 elevation angle 8 [deg] 140 _i 160 180 Figure 4.10: Surface currents us. t / X on a P E C sphere w ith size p aram eter k^a = 10.0. C om parison betw een results of th e present m ethod (BOR) and Mie series (MIE) solution. incident plane wave exist, and can be found in [14, 15]. T hey have the form of Mie series: J (0 ) = A . Y koa jn 2 n + P nx ( CO S0) 1 , n ( n + 1) sint [(£:0a) h n \ k o a ) 1 k0a ^ P n (cos 9) n ( n + 1) | [(jfeoa) ^ ( j f e o a ) ] ' sin0 + 5 f£ (c o s0 ) 89 (k0a) h n \ k o a ) dP^ (cos 9) + (k0a ) h ^ ( k 0a) (4.98) ► (4.99) 99 Figures 4.9 and 4.10 show th e surface currents on the generating curve for two P E C spheres w ith size param eters koa = 1.0 and k^a = 10.0, respectively. T he solid and do tted lines represent the exact value calculated from expressions (4.98)-(4.99), while th e symbols denote the B O R num erical solution, which is in excellent agreem ent w ith th e Mie solution. Only plane wave incidence from 9i = 0°, i.e., along the z-axis, has been considered so far. Since in this case, as noted in Section 4.2.4, only the m odes n — ± 1 are excited, the validity of th e present approach has been proved only for those two m odes. In order to consider other modes as well, the incident wave m ust be a t an oblique angle. For this reason, th e results for a PE C finite-length 79 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.5 2.5 x Jt - BOR O j '- B O R _ J - WG Jt -W G X s) C © © t: <n o ■® 3o c sE a a'O'cro-o-s © o a d o - ■© o N « E o c 0.2 0.4 0.6 0.8 2 1.2 0.2 0.4 0.6 1.2 0.8 t/X Figure 4.11: Surface currents vs. t f A on a P E C finite cylinder of radius a = 0.5A and length L = A, for 9i = 0° (left) and 9{ = 45° (right). C om parison betw een results of th e present m ethod (BOR) and W ilton-G lisson (W G) solution. 1.3 X' ”« 1.6 c ® 3 U | 5 o ® T 3 3 'c 0.8 0.8 I 0.6 0.6 CO ® * ia E o c Jt - BOR J t - WG 0.2 99 0.5 1.S 2.5 o V 0.2 a © 3.5 bor J - WG 0.5 2.5 3.5 t/X Figure 4.12: Surface currents vs. t / X on a P E C finite cylinder of radius a = 0.5A and length L = 2.5A, for 6i = 0° (left) a n d 9{ = 45° (right). C om parison betw een results of th e present m ethod (BOR) and W ilton-G lisson (WG) solution. 80 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.5 X ' ~x " c© 3 2 1.4 O 3 3.5 © © jg © 3(A 3U> o c J - BOR C O) £3 0.6 7© Jt - WG •o Ol © © 0.4 £ o c © £ © £ o c o V B0R J -W G 9 2.5 0.5 t/X 2.5 Figure 4.13: Surface currents vs. t /X on a P E C finite cylinder of radius a = 0.05A and length L = 2.5A, for 9l — 0° (left) and 9i = 45° (right). C om parison betw een results of th e present m ethod (BOR) and W ilton-G lisson (WG) solution. cylinder at two different incident angles 9i will now be included. In Figures 4.11-4.13, th e results of the present m eth o d are com pared to those of th e GlissonW ilton code for incident angles 9i = 0° (left) an d 9i = 45° (right). R adius and length are a = 0.5A, L = A for the cylinder in Figure 4.11, a — 0.5A, L — 2.5A for the cylinder in Figure 4.12, and a = 0.05A, L = 2.5A for th e cylinder in Figure 4.13. From the electric surface currents found by solving th e surface integral equation, th e b istatic scattering coefficient is com puted using expression (4.91) (4.93). E xam ples of results are shown in Figures 4.14-4.21 for th e sam e type of bodies for which th e surface currents have been calculated. In m ost cases, the com parisons are m ade w ith the results of th e G lisson-W ilton code. T he incident angle is , for 9{ = 0°, except for the finite P E C cylinder, w here , an oblique angle 9X = 45° is also considered. Figure 4.14, for a P E C disk of size p aram eter koa = 10, shows a comparison betw een the norm alized bistatic scatterin g coefficient calculated w ith th e present num erical approach, and an analytical solution derived by Hodge [74]. Figures 4.17-4.18 show th e norm alized b istatic scattering coefficient for the two P E C sphere whose currents are given in figures 4.9-4.10. The num erical values 81 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. * o HH - BOR HH - Hodge VV-BOR VV - Hodge e m T3 -10 -15 scattering angle 88S [deg] 60 75 Figure 4.14: N orm alized b istatic scattering coefficient vs. 9S for a P E C disk of size param eter koa = 10, for 6l = 0°. Com parison betw een results of the present m ethod (BO R ), and Hodge solutions. s CD T3 CM cti -5 O -10 HH-BOR x HH-WG - - VV-BOR o VV - WG -15 -20 20 40 80 100 120 scattering angle 0s [deg] 140 160 180 Figure 4.15: Norm alized b istatic scattering coefficient vs. 9S for a P E C open-ended cylinder of radius a = 0.4A and length L = 1.2A, for 9i = 0°. C om parison betw een results of th e present m ethod (BOR) and W ilton-G lisson (WG) solution. 82 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. x o HH-BOR HH - WG VV-BOR VV - WG B CD ■O <3 "6 es -2 -3 -4 20 60 80 100 120 scattering angle 8a [deg] 140 160 180 Figure 4.16: N orm alized b ista tic scattering coefficient vs. 9S for a P E C cone-sphere of radius a = 0.2A and length L = 1.2A, for plane wave incident a t 9t = 0°. C om parison betw een results of the present m ethod (BOR) and W ilton-G lisson (WG) solution. E co 13 -4 x -8 o -10 20 ) 80 100 120 scattering angle 0 [deg] 140 HH-BOR HH - MIE VV-BOR VV - MIE 160 180 Figure 4.17: N orm alized b ista tic scattering coefficient vs. 9S for a P E C sphere w ith size param eter k^a — 1.0, for 0j = 0°. C om parison betw een results of the present m ethod (BOR) and Mie series (MIE) solution. 83 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. o-o- E CD ■ o 03 j; B -10 -15 HH-BOR « HH-MIE - -V V -BO R o VV-MIE -20 -25 3 80 100 120 scattering angle 9 [deg] 40 140 160 180 Figure 4.18: N orm alized b istatic scattering coefficient vs. 9S for a P E C sphere w ith size param eter koa — 10.0, for 9%= 0°. C om parison betw een results of th e present m ethod (BOR) and Mie series (MIE) solution. are com pared w ith th e exact values calculated according to th e following form ulas for the scattering am plitudes [15]: -j fhh(0s) OO n+1 = an p i (cos 0) <9pi(cos0) „ - J Pn sin 9 89 (4.100) OCv dP.%(cos 9) -3& 89 (4.101) n =1 fv v(9s) ~ k Y ,J n+ l wn =1 P n (cos 9 ) sint where the coefficients a n and /3n are ,n 2n + 1 aT -J n{n + l) h (n \ k 0a) 2n + l Pn = f j n {koa) [(k0a) j n {k0a)}' n{n + l) [(k0a ) h ^ ( k Qa)]' (4.102) (4.103) and the bistatic scatterin g coefficients are CTpq{9s) = 4tT\ f p q { 9 s ) \ :‘ (4.104) Given th e good agreem ent betw een the results, th e num erical m ethod derived earlier in this chapter can be considered validated in the case of P E C bodies of revolution. 84 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. E CD ■o N(0 e'l< s K B B -15 -1 0 * HH - BOR H H -W G 0 W -B O R - - VV - WG -15 -20; -10 20 80 100 scattering angle 0 [deg] 120 140 160 x o 180 -25 scattering 100 120 ie 0 [deg] 140 HHHH W W - BOR WG BOR WG 160 180 Figure 4.19: N orm alized b istatic scattering coefficient vs. 9S for a P E C finite cylinder of radius a = 0.5A and length L = A, for Oi = 0° (left) and 0t = 45° (right). C om parison betw een results of the present m ethod (BOR) and W ilton-G lisson (WG) solution. E CQ T 3 CO £ B e -15 -20 -1 0 X -15 O -20 80 100 120 scattering angle 9 [deg] 140 -25 HH - BOR HH - WG W -B O R VV - WG 160 x -30 -35 180 o 80 100 scattering angle 0 [deg] 120 140 H H -B O R H H -W G W - BOR W - WG 160 180 Figure 4.20: Norm alized b istatic scattering coefficient vs. 6S for a P E C finite cylinder of radius a = 0.5A and length L = 2.5A, for 6t = 0° (left) a n d 6t = 45° (right). C om parison betw een results of the present m ethod (BOR) and W ilton-G lisson (W G) solution. 85 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -6 0 1 0 1 20 1----------- — 1--------------- 1 40 60 1 1 1 80 100 120 scattering angle 9g [deg] 1 140 160 1 180 .401---------- >— 1!------'-------------1------------1----------- '----------- 1------------1------------1------------1 0 20 40 60 80 100 120 140 160 180 scattering angle 8 [deg] Figure 4.21: N orm alized b istatic scattering coefficient vs. 0S for a P E C finite cylinder of radius a = 0.05A and length L = 2.5A, for 9i = 0° (left) and Oi = 45° (right). Com parison betw een results of the present m ethod (BOR) an d W ilton-G lisson (WG) solution. 4 .3 .2 H o m o g e n e o u s d ie le c tr ic b o d y o f r e v o lu tio n For an homogeneous dielectric body of revolution, b o th integral equations (4.5) and (4.6) need to be solved sim ultaneously. For th e validation, only the case of th e dielectric sphere is treated. Two different values of th e size p aram eter are considered, i.e., k^a = 1.0 a n d k^a = 10.0, in com bination w ith three different values of th e relative dielectric constant, i.e., ec = 4, ec = 4 —j, and ec = 18—jQ. In Figures 4.22-4:27, th e electric and m agnetic surface currents obtained w ith the present num erical m ethod are com pared to th e results of the G lisson-W ilton code. T he bistatic scatterin g coefficients in the hh and un-polarization are calculated from these equivalent surface currents using (4.92)-(4.94), and are p lo tted in Figures 4.28-4.33. In th e same plots, the exact values of th e b istatic scattering coefficients are also shown. These values are calculated from (4.94) and expressions (4.100)-(4.101), w ith th e coefficients a n and /3„ of the Mie series given by: OLji .n 2n + l — —J j n (k0a) [ka j n (ka )} '- j n (ka) [kQa j n {k0a)}' ~ T — 7T7 7T7 n ( n + 1) hb, (koa) [ ka jn (ka)]' - j n (ka)[k0a h ^ ( k 0a)]' 86 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ( 4 .1 1 ) 5 j 0.3 1.4 S B 0.7 0.6 0.5 ^ c 0.2 0.4: * MX~ BOR Mt - WG 0 0.2 M ^-BO R M - WG 0.05 0.1 0.15 0.2 0.25 t/X 0.3 0.35 0.4 0.45 0.5 0.05 0.1 0.15 0.2 0.25 t/X 0.3 0.35 0.4 0.45 0.5 Figure 4.22: Equivalent surface currents vs. t / X on a dielectric sphere w ith k0a = 1.0 and ec — 4. C om parison betw een results of the present m ethod (BOR) and Mie series (MIE) solution. 0.8 S" 0-7 0.6 0.5 .1 0.4 © 0.6 0.4, 0.2 0.2 0.1 0.05 0.1 0.15 0.2 0.25 t/X 0.3 0.35 0.4 0.45 0.5 0.05 0.1 0.15 0.2 0.25 t/X 0.3 0.35 0.4 0.45 0.5 Figure 4.23: Equivalent surface currents vs. t / X on a dielectric sphere w ith k$a = 1.0 and ec = 4 —j. Com parison betw een results of the present m ethod (BOR) and Mie series (MIE) solution. 87 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.8 x Mt - BOR M - WG 2 " 0.7 0 - BOR M - WG 0.6 0.5 > 0.8 .1 0.4 0.6 0.3 0.4 x 0.2 - BOR Jt - WG o VSOR 0.2 J - WG 0.05 0.1 0.15 0.2 0.25 t/X 0.3 0.35 0.4 0.45 0.5 0.05 0.1 0.15 0.2 0.25 t/X 0.3 0.35 0.4 0.45 0.5 Figure 4.24: Equivalent surface currents vs. t /X on a dielectric sphere w ith k^a = 1.0 and ec = 18 — j 6. C om parison betw een results of the present m ethod (BOR) a n d Mie series (MIE) solution. c M? - BOR hi X' 2.5 c © _ - WG M -B O R * 2.5 _ M - WG 3 & ■ c in © © 3 3 tft c C 3 CT © <0 > 3 CT © ■© o 3 © •o 3 & © (0 > O O 'c C CO o» O) E i ■o ■o N 0.5 cO E N 0.5 o c 0.5 2.5 0.5 2.5 Figure 4.25: Equivalent surface currents vs. t /X on a dielectric sphere w ith k$a = 10.0 and ec = 4. Com parison between results of the present m ethod (BO R) and Mie series (MIE) solution. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.5 -* J t -B O R _____Jt - WG o j’ =S 0.45 - bor _____J - WG D 0.35 0.3 3® 0.25 ......... x x * * x £JJx es° x Mt - 8 0 R o M - BOR M -W G o. Ov °&(S M - WG 0.5 2.5 t/X Figure 4.26: Equivalent surface currents vs. t /X on a dielectric sphere w ith k$a = 10.0 and ec = 4 - j . C om parison betw een results of the present m ethod (BOR) and Mie series (MIE) solution. 0.5 X' ^ c o 0.45 0.4 3 U 0.35 0.3 c JJ> .1 > 3 •0 cr 0.25 o CD "O 3 C cs 03 E ■o N E o c 0.5 2.5 0.5 2.5 t/X Figure 4.27: Equivalent surface currents vs. t / X on a dielectric sphere w ith k^a — 10.0 and ec = 18 - j6 . Com parison betw een results of th e present m ethod (BOR) and Mie series (MIE) solution. 89 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -1 0 # -15 -20 HH x HH - - VV o VV - -25 -30 40 80 100 120 scattering angle 9s [deg] 140 MIE BOR MIE BOR 160 180 Figure 4.28: N orm alized b istatic scattering coefficient vs. 9S for a dielectric sphere w ith k^a = 1.0 and ec = 4. C om parison betw een results of th e present m ethod (BOR) and Mie series (MIE) solution. -1 0 -20 -HH-MIE X HH - BOR - - VV - MIE O W - BOR -25 -30 40 80 100 120 scattering angle 0g [deg] 140 160 180 Figure 4.29: Norm alized b istatic scattering coefficient vs. 9S for a dielectric sphere w ith k^a = 1.0 and ec = 4 —j. C om parison betw een results of th e present m ethod (BOR) and Mie series (MIE) solution. 90 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 T> £ m ■a CM 'B -2 -3 HH - MIE x HH - BOR - -W -M I E o W - BOR - 80 100 120 scattering angle 0s [deg] 20 140 160 180 Figure 4.30: N orm alized b istatic scattering coefficient vs. 9S for a dielectric sphere w ith k^a — 1.0 and ec = 18 — j 6. C om parison betw een results of the present m eth o d (BOR) and Mie series (MIE) solution. HH-MIE * HH - BOR - - VV - MIE o VV - BOR E •a Nas CD 'B -5 -1 0 -15 -20 20 40 80 100 120 scattering angle 0g [deg] 140 160 180 Figure 4.31: Norm alized b istatic scattering coefficient vs. 0S for a dielectric sphere w ith k Qa = 10.0 and ec = 4. C om parison betw een results of th e present m eth o d (BOR) and Mie series (MIE) solution. 91 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. »r -10 -20 HH - MIE x HH - BOR - - VV - MIE O VV - BOR -30 -40 80 100 120 scattering angle 9s [deg] 40 140 160 180 Figure 4.32: N orm alized bistatic scattering coefficient vs. 9S for a dielectric sphere w ith kqa = 10.0 and ec — 4 —j. C om parison betw een results of th e present m ethod (BOR) and Mie series (MIE) solution. HH HH VV O VV x MIE BOR MIE BOR E C Q T J cv -5 -1 0 -15 -20 40 60 80 100 120 scattering angle 9g [deg] 140 160 180 Figure 4.33: Norm alized b ista tic scattering coefficient vs. 0S for a dielectric sphere w ith k^a = 10.0 and ec = 18 —j 6. C om parison betw een results of th e present m ethod (BOR) and Mie series (MIE) solution. 92 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2n + 1 Pn = jnjkoa) [kajn (ka)]' - £rjn{ka) [kQa j n {k0a)]' f ( 4 . 106 ) n in + !) h ^ \ k 0a) [kajn (ka)}f - erj n ( k a ) [ k o a h n \ k o a )]' T he com parisons show th a t th e b o d y of revolution num erical m ethod gives p retty accurate results in th e case of a dielectric object, and only a sm all degradation in accuracy for th e larger sphere w ith higher dielectric constant can be noted. 4.4 A pplication to scattering from vegetation T he num erical procedure developed an d tested for bodies of revolution in this chapter can be applied to stu d y the scatterin g from vegetation com ponents. In microwave rem ote sensing, a vegetation canopy is usually m odeled as a layer of dielectric cylinders and disks placed over a half space representing the ground. In th e case of a forest such cylinders and disks represent tru n k s, branches, needles and leaves, while in th e case of sm aller vegetation such as agricultural crops, they represent stalks and leaves. If they are circular and hom ogeneous, these scatterers can be m odeled as bodies of revolution, Since a com parison betw een num erical calculations a n d analy tical approxim ation has already been done by W illis et al. [29], only cylinders will be considered here. T he cylinders are assum ed circular, homogeneous, an d lossy, w ith dimensions a n d orientations typical of trunks, branches, needles and stalks. T ypical values - based on ground d a ta collected during experim ental - for conifer forest are those given in [9] for hemlock trees in a boreal forest. C orn stalk param eters can be found in [10]. vegetation type vegetation element radius a length L hem lock hem lock hem lock hem lock corn secondary branch prim ary branch sm all tru n k large tru n k stalk 1.8 m m 0.6 cm 3.0 c m 10.0 c m 1.25 c m 16 cm 90 cm 2.5 m 7.5 m 62.5 m Table 4.1: G round d ata. 93 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T he relative dielectric constant ec of vegetation structures greatly depends on their w ater con tent. Tree tru n k and branches are usually drier and therefore have a lower ec th a n corn stalks. For the sake of brevity, in th e following, only one value for the dielectric constant will be used, but other values of ec have also been considered, leading to sim ilar results. T he value chosen here is ec = 18 —j’6, which is consistent w ith ground d a ta m easurem ents and U laby’s em pirical m odel [75]. In ra d a r rem ote sensing, particularly when considering scattering from a layer of dielectric objects, th e interest lies in th e m ain scattering lobes. T his is a consequence of th e large num ber of scatterers in such vegetation media. Since the contributions from the single scatterers are added all together, the relative weigh of th e scattering from th e side lobes becomes negligible. Therefore, when com paring approxim ate and exact scattering coefficients, th e approxim ation is considered acceptable if it agrees w ith the theory inside the m ain scattering lobe and does not produce high scattering elsewhere. It has to be stressed th a t in cases where all scattering of any level m ust be taken into account - such as antenna applications - th e conclusions draw n here do no apply. In the following sections, after a brief description of the analytical models, th e values of the scattering coefficient estim ated by th e approxim ate m ethods will be com pared w ith its exact values found numerically. In particu lar, for various incident angles 6i, 0j, th e bistatic scattering coefficient will be determ ined over a range of scattering angles Qs ,<t>s using b o th m ethods. T he illustration of results is lim ited to th e case of scattering in th e sam e plane cj> = fa of the incident wave, b u t sim ilar conclusions apply for scattering a t different azim uth angles <f>s. 4 .4 .1 F in it e - le n g t h d ie le c tr ic c y lin d e r a p p r o x im a tio n Consider a finitely-long circular dielectric cylinder of radius a an d length L, w ith p erm ittiv ity e = e0 s c (where ec = eT — j s j is th e complex relative p e rm ittiv ity or dielectric constant of the cylinder) and perm eability y, = y,o , located in free space. A C artesian coordinate system (x, y, z) is defined w ith its origin in the center of the cylinder, an d th e z-axis coincident w ith th e cylinder 94 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. z 8 ,|A Figure 4.34: G eom etry of the finite-length dielectric cylinder problem axis, as illustrated in Figure 4.34. T he cylinder is considered being illum inated by a g-polarized uniform plane wave q = h,v. E W (r) = (4.107) T he incident wave (4.107) induces an internal field E?nt inside th e cylinder. T he scattering am plitude for the cylinder is related to the internal field by (see [20] for details): f pq( k s S i ) = fc°2(£4C7r7— [ p s - E?n t( r V fc°k s ' r V ' , p,q = h , v (4.108) V Because the internal fields w ithin a finite-length cylinder are not known exactly, they are approxi m ated by the internal fields inside an infinite cylinder of the sam e radius, orientation and dielectric constant. This assum ption requires th a t the cylinder length be large com pared to its radius, i.e., L > 1. 95 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.109) T he internal electric field for horizontally (TE) or vertically (TM ) polarized incident wave can be found in W ait [13]. T he integration w ithin the cylindrical volume V is carried out in [25]. T he resulting expression for th e scattering am plitudes is k l ( s c - 1) 47r fpqfe-si kj) 7T ■ + j e [ l | n)( q • a x ■ p ) + 4 n)( q • a 2 • p ) - j e ~ ^ s [ 4 n)( q a! • p ) + l \ n\ q • a 2 • p ) ] + + e M * [ 4 n)( q - h i - p ) + 4 n)( q • b 2 • p ) + e where h = L + [ 4 n)( q ■b x • p ) + jJ n)( q ■b 2 • p ) } sin[fco(cos 9i + cos 9S) L / 2] ko(cos9i + cos 9s) L / 2 (4.110) (4.111) a r(n) = JnM ) Jn M ) J P' dp' 0 Q»[ Ai Jyi (^1&) Jn (^1&) ] Af + (4.112) a r(n) = f J'n { \ lP') Jn+liMp') P' dp' 0 (4.113) a r(n) 15 = J J ^ X lP ^ J n -lM p 'd p ' (4.114) 0 a (n) = J J n M ) J n + l M ) P'dp' (4.115) 0 a r(n) = J J n M ) J n - l M ) P'dp' (4.116) 0 and Ai = A2 = ko sin 9S (4.117) kQy/er — cos2 9S (4.118) In addition the following relationships perm it a m ore efficient calculation of the l\H . r(«) 4 _ n ~~ Ai A n + l) r(n) ~~ 96 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.119) = n r(n -l) II j•(—n) J6 7? ''f2? 1 II r(~n) Ji (4.121) (4.122) 1 II r(~ n ) 3 ( 4 . 120 ) ' Ai J3, 3 (4.123) Such expressions are valid for cylinders of any thickness, as long as th e constrain given in (4.109) is satisfied. However, if th e cylinder is very th in , a furth er approxim ation can be applied. It is shown in [25] th a t if the radius a of th e cylinder is sm all com pared to its internal wavelength, a quasi-static approxim ation is utilized to reduce the com plexity of th e scattering am plitude expressions. Here the general case of a cylinder w ith an a rb itra ry radius a and length L satisfying (4.109) will be considered, and th e expressions for 4 .4 .2 f h v , f v h and f vv explicitly given in [25] will be used. T a p ered c y lin d e r a p p r o x im a tio n z3 L Z.2 z, v Figure 4.35: T apered cylinder m odel Consider a dielectric tap e red cylinder of length L a n d radii a and b a t the m ajor and m inor base, respectively, centered at th e origin of a C artesian coordinate system , as shown in Figure 4.35. 97 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Let th e p e rm ittiv ity and perm eability of such body be e = ec£o and p = /tq, respectively, and the axis of the cylinder be coincident w ith th e z-axis. T he sim plest way to m odel such ta p e re d cylinder is to approxim ate it w ith a finite cylinder of same length an d volume. However, as it will also be shown later, th is works well only when the ta p e r is small. In general, it is m ore ap p ro p riate to divide th e tap ered cylinder into a num ber N s of sections, an d th en approxim ate each one w ith a cylinder of like length and volume. This m odel has been introduced by [25], following m easurem ents of tru n k diam eters at different heights. T he approach is illu strated in Figure 4.35 for a num ber of sections N s = 3. T his approxim ation is valid when the ta p e r is small: R = Li > 1. (4.124) In general, th e higher th e factor R, th e larger the num ber of sections N s needed to achieve a good approxim ation. T he scattering am plitude of the m -th cylinder, tra n sla te d by a distance z rn along th e z-axis is: p ,q = h , v , m = l , ..., N s / ^ ( k s ,k i) = (4.125) In (4.125), f p ^ ( k s,ki) is th e scatterin g am plitude of th e cylinder centered a t the origin, and the exponential term accounts for th e phase shift in b o th th e incident and scattered wave produced by th e translation of th e cylinder by z m . Assuming no coupling exists betw een th e cylinders, th e to ta l scatterin g am plitude of the con figuration is the sum of N s term s fp^1' (k >3, kj) as in (4.125), i.e., p,q = h,v / p, ( k s, k i ) = X ) / m ° ( m= 1 (4.126) The individual scatterin g am plitudes / p ^ ( k s,k j) in (4.126) are determ ined using the finite cylinder approxim ation described in th e previous section. T he rationale behind this approxim ation is th e following. C onsider a finite cylinder of length L, and divide it in N s sections of a rb itra ry length. T he scattering am plitudes of all cylindrical 98 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. incident wave forward scattering lobe ^ specular scattering lobe •• scattering / cone Figure 4.36: Scattering p a tte rn of a cylinder sections are estim ated using th e finite cylinder m odel described in th e previous section 4.4.1, and added together according to (4.126), to yield the to ta l scatterin g am plitude of th e configuration. Since the contributions of th e end currents from two contiguous cylindrical sections cancel w ith each other, th e final resu lt is no interaction betw een those sections. T he only rem aining effects in the sum (4.126) are those a t the two term inal sections. T he resulting scattering am p litu d e is therefore the same as th a t provided by th e finite cylinder m odel for the original cylinder of length L. The tap ered cylinder m odel described in this section is a p e rtu rb a tio n of the situ atio n ju st illustrated. If the difference in radius betw een contiguous sections is small, only weak interactions exist between those adjoining cylinders, while th e two term in al sections will experience th e effect of replacing their in tern al fields w ith those of two infinite cylinders, as explained in the previous section. 99 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 .4 .3 R e s u lt s a n d c o m p a r is o n s A n exact solution is needed as a reference to evaluate the accuracy of th e finite cylinder m odel and the tap e red cylinder m odel introduced in th e two previous sections. Since no closed-form exact analytical expressions are known for the scattering from finite or tap e red cylinders, th e only available option is to use a num erical analysis approach. In particu lar, the results of the num erical m ethod developed earlier in this chapter will be considered as the reference to evaluate the accuracy of th e finite cylinder and th e tap ered cylinder approxim ations. To b e tte r quantify the accuracy of the analytical model, a p aram eter epp is introduced to m easure th e error betw een exact an d approxim ate solution over a range of scattering angles 9S for a single incident angle 9t. Let app and app be th e values of th e b istatic scattering coefficient obtained using the m ethod of m om ents (i.e., the exact solution) an d th e analytical m odel (i.e., the approxim ate solution), for polarization pp = hh, vv, respectively. T he absolute error in dB, epp, of th e finite cylinder m odel is defined as follows: i Mpp ePP(9i) = j z — I&Pp(dsm\ 0 i ) ) [ d B ] ~ Vppi9^ , 8 i ) [ d B } \ , m vv m= 1 PP = h h , v v (4.127) where the bistatic scatterin g coefficient is expressed in dB: cfpp\dB\ = 101og10 (cTpp) , app[dB] = 10 log10 (app) , pp = h h , v v T he sum is carried out over a set of m = 1 ,2 ,.., M pv scattering angles 9 ^ (4.128) where the scattering coefficient is no lower th a n 10 d B below th e peak value cr™ax(9i), i.e., such th a t aPM m\ 9i)[dB] > a™ax{9i)[dB] - 10 dB , pp = hh, vv (4.129) T he quantity epp provides an estim ate of the difference - expressed in dB - betw een the approxi m ation and the exact solution over a range of angles w here the level of scattering is high enough. In other words, epp m easures th e area betw een th e two exact and approxim ate d B curves app and app 100 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. vs. 9i, lim ited to th e values 9 ^ of the scattering angle for which app satisfies expression (4.129). T he error could be defined in m any other different ways. Here the definition (4.127) is chosen because of its sim plicity and intuitive m eaning, an d will be used m erely as a tool to com pare how the accuracy changes w ith th e incident angles and the cylinder size. F in ite-len g th , con stan t-rad iu s cylinder In order to investigate th e accuracy of the finite cylinder m odel, in this section a num ber of cases of cylinders w ith different dim ensions, chosen consistently w ith th e ground d a ta in [9] and [10], is considered. Since the finite cylinder m odel assum es th a t th e currents inside the cylinder are the sam e as if the cylinder were infinite, one would expect th a t th e accuracy of th e approxim ation decreases for shorter cylinders w ith a sm aller length-to-radius ratio. However, it will be seen th a t for incident waves sufficiently close to th e norm al to the cylinder axis, th e error is still sm all even in such cases. T he general shape of the scattering p a tte rn of a dielectric cylinder is illu strated in Figure 4.36. T he am plitude of the scattering has its m axim um in a conical region also referred to as the scattering cone. In a section cj) — 4>i of the scattering p a tte rn th ere will be two relative m axim a corresponding to the m ain scattering cone, one in the forw ard direction, th e other one in th e specular direction w ith respect to the cylinder side. In the following, th e norm alized bistatic scatterin g coefficient a / ( n a 2) will be p lotted as a function of the scatterin g angle for a fixed incident angle, in b o th hh- and w -polarization. T he scattering angle 9S in th e plots ranges between 0° an d 180°, therefore only the specular lobe at 9S = 180° —9i will be visible. T he first case exam ined is a cylinder of length L = 10.0A and radius a = 0.04A. For a wavelength A = 60 cm (or / = 500 M H z , in P -band), this corresponds to a tree tru n k 6 m long w ith a diam eter of 4.8 cm. This cylinder is several wavelengths long an d its length and radius satisfy the condition 101 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. HH-POLARIZATION VV-POLARIZATION I(n a2) [dBm] MM PO MM PO -10 -io -20 -2 0 -3 0 -3 0 .c D -4 0 60 80 100 120 140 160 180 60 scattering angle 0g [deg] 80 100 120 140 160 scattering angle 0 [deg] Figure 4.37: N orm alized bistatic scattering coefficient vs. scattering angle for finite dielectric cylin der w ith L = 10.0A, a = 0.04A, and incident angle Qt = 20°. (4.109). T he norm alized bistatic scattering coefficient <7 / (7ra2) is p lotted in Figure 4.37 as a function of th e scattering angle 0S for a n incident angle 0j = 20°, in b o th hh- and wv-polarization. T he solid curve represents the m ethod of m om ents num erical solution (MM), while the dashed curve is obtained using the approxim ate physical optics analytical solution (PO ). T here is very good agreem ent betw een the two solutions except for w -p o la riz a tio n a t angles 0S far from th e specular scattering lobe located at approxim ately 9S = 160°. As the cylinder becomes thicker, i.e., the ratio L / a decreases, one would expect the finite cylinder m odel to lose its accuracy. Nevertheless, it will be seen th a t in such case the approxim ation is still satisfactory for incident angles 0* close to th e norm al to th e cylinder axis. There is a possible intuitive explanation for this behaviour. In the analytical m odel, the finite cylinder is tre a te d as infinite and the effects of its ends are not accounted for. T he co ntribution of such end-on scattering becomes more pronounced in thicker cylinders due to th e larger area of th e ends. However, the ends are alm ost invisible to norm al or quasi-norm al incident waves, and therefore in th a t case their contribution to the scattering is negligible. 102 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 180 HH-POLARIZATION VV-POLARIZATION MM PO ----- MM - - PO E CQ •o "O -1 0 -1 0 -20 -2 0 -3 0 -3 0 0 20 40 60 80 100 120 140 160 0 180 20 40 scattering angle 0 [deg] 60 80 100 120 140 160 180 140 160 180 scattering angle 9s [deg] ----- MM ----- MM - - PO - -P O E C Q *o T3 -10 O -20 -2 0 -3 0 -3 0 0 20 40 60 80 100 120 140 160 180 scattering angle 0g [deg] 0 20 40 60 80 100 120 scattering angle 0 [deg] Figure 4.38: N orm alized b ista tic scattering coefficient vs. scattering angle for finite dielectric cylin der w ith L = 10.0A, a = 0.4A, eT = 18 — a nd incident angle = 20°(top) and = 80° (bottom ). 103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. HH-POLARIZATION VV-POLARIZATION 3.5 absolute error [dB] — L = 5.0 X L = 3.0 X L = 1.0 X 3.5 2.5 2.5 0.5 0.5 incident angle 0. [deg] incident angle 0. [deg] Figure 4.39: A bsolute error vs. incident angle for finite dielectric cylinder w ith a — 0.04A, eT = 18 —j'6, L = 5.0A, 3.0A, a n d 1.0A, for hh- and iw -polarization. Figure 4.38 shows th e norm alized bistatic scattering coefficient vs. the scattering angle for a cylinder w ith the sam e length and p e rm ittiv ity as in Figure 4.37, b u t w ith a radius a = 0.4A, for an incident angle = 20°. A lthough it is still L » a, the plots of th e exact and approxim ate a now differ, especially for w -p o lariza tio n , even in the m ain lobe. T he actu al am plitude of the scattering off the m ain scattering cone is m uch higher th a n w hat predicted by the analytical model. It is reasonable to conclude th a t th e m ain co ntribution to th e scatterin g a t those angles 8S comes from th e ends, while in the m ain lobe it comes from the sides. N onetheless, as it has been discussed earlier, the agreem ent im proves considerably when th e incident angle approaches the norm al to th e cylinder axis. Indeed, for a higher incident angle such as = 80°, shown in the two plots at th e bo tto m of Figure 4.38, th e agreem ent is very good, except for a couple of dB difference a t the peak value of the hh-polarization. In th is case, the ends are alm ost invisible when looking a t the cylinder from the angle 9i, which could explain why th e analytical m odel yields accurate results. As explained previously, w hen tre a tin g rem ote sensing problem s, only the scattering w ithin 10 dB of the peak value is relevant, therefore we are concerned w ith achieving a good approxim ation 104 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. HH-POLARIZATION VV-POLARIZATION — L= 10.0 A L = 5.0 A —■»— L = 3.0 A — — L= 10.0 X L = 5.0 A L = 3.0 A. absolute error [dB] CQ •o 0._1 >_ o 03 =3 30 40 50 60 30 40 50 60 70 incident angle 0. [deg] incident angle 0. [deg] Figure 4.40: A bsolute error vs. incident angle for finite dielectric cylinder w ith a = 0.4A, 18 - j 6, L = 10.0A, 5.0A, and 3.0A, for hh- and w -p olarization. e y = only in the region surrounding the m axim a. T his m eans th a t for th e th in n er cylinder illu strated in Figure 4.37, there is very good agreem ent, while for the thicker cylinder in Figure 4.38, there are some problem s a t 9i — 20° due to th e scattering from the ends th a t are not accounted for in the approxim ate model. Such problem s are m ore pronounced for wu-polarization, a possible explanation for which being the discontinuity in th e perpendicular com ponent at th e cylinder ends, b u t disappear a t &i = 80°. To b e tte r und erstan d how the accuracy changes w ith the cylinder dim ensions as well as w ith the incident angle, the absolute error defined by (4.127) is now com puted. Two different values for th e radius are considered a = 0.04 A to represent a th in cylinder, a n d a = 0.4 A to represent a thick cylinder. In b o th cases, th e absolute errors e^h an d evv are p lo tted versus the incident angle 9i for three different values of th e length, e.g., long, m edium ,and short. T he th in cylinder cases (a = 0.04 A) are shown in figure 4.39. T he cylinder lengths are L = A, 3 A, and 5 A, w ith th e hh- and vv- po larization on th e left and right, respectively. All these cylinders have a -C L, in p articu lar L / a = 25, 75, and 625, respectively. If an absolute error below 2 d B is 105 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. considered as acceptable, th en th e agreem ent between finite cylinder m odel and num erical solution a t hh-polarization is excellent at m ost angles and for m ost lengths. T he only exception is th e shortest cylinder (L = A), perhaps due to resonances. As expected, th e error is lower for longer cylinders because the infinite cylinder assum ption is more correct, a n d decreases as the incident angle increases since th e end-on effect becomes less im portant. Also to be noted is the fact th a t evv is generally higher th a n the corresponding Figure 4.40 illustrates three thicker cylinder cases, w ith a rad iu s a = 0.4 A and lengths L = A, 5 A, and 10 A, i.e., L / a = 7.5, 12.5, and 25, respectively. T he m ost obvious comm ent is th a t th e error in b o th polarizations is m uch higher th a n for the previous cases of Figure 4.38, which is expected given th e lower value of L / a . T here seems to be a threshold sc atterin g angle 9° below which th e error increases as 9{ decreases, and above which the error rem ains fairly constant. Supposedly, those two regions - i.e, 6i < 9° an d 9i > 6° - correspond to situations w here end-on effects respectively have or d o n ’t have a significant effect on the m ain scattering cone. Such threshold value of 0j decreases w ith the cylinder length, and varies betw een 15° for th e longest cylinder, and 25° for the shortest. T he approxim ation can be considered good (i.e., w ith an error lower th an 2 dB) for angles th a t vary between 9 i > 9° = 15° for th e longest cylinder, and 9i > 6° = 40° for the shortest cylinder at uv-polarization. As also seen in th e case of Figure 4.38, th e analytical m odel perform ance is slightly worse at wu-polarization th a n a t h/i-polarization, especially for shorter cylinders. T apered cylinder Two cylinders of length L = 10.0 A and different ta p e r have been considered to illustrate the accuracy of the tap ered cylinder m odel described in section 4.4.2. For each case, th e results of th e num erical code are p lo tte d together w ith the approxim ate solutions obtained using only one cylinder (see also equation (4.108)) an d the tapered cylinder m odel (see also, equation (4.126)) w ith two and four cylinders. T h e param eters a, b and L are defined 106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. in Figure 4.35, a n d am is the radius of the cylinder replacing th e m -th section (m = 1 , 2 , N s ). A sum m ary of th e values of the cylinder lengths and radii used in the approxim ations is given in Table 4.2. A relative dielectric constant ec = 18 —j 6 is assum ed, i.e., th e sam e value used in the previous section for the analysis of th e finite cylinder model. No. of cylinders N s ( L / N s )/A ai /X o2/ A a 3/ A a^jX a = 0.7 A, 6 = 0.1 A 2 1 4 a = 0.5 A, b = 0.3 A 1 2 4 10.0 0.4 10.0 0.4 5.0 0.55 0.25 2.5 0.625 0.475 0.325 0.175 5.0 0.45 0.35 2.5 0.475 0.425 0.375 0.325 Table 4.2: Geom etric param eters used in tap ered cylinder approxim ations. Figure 4.41 shows th e norm alized b istatic scattering coefficient vs. scattering angle for a tap ered dielectric cylinder w ith L — 10.0 A, a = 0.7 A, b = 0.1 A, for a wave incident at an angle 6t — 40°. T his cylinder has a ta p e r factor - as defined in (4.124) - R = 0.06. In these plots, a range of scattering angles 120° < 9S < 240° has been chosen in order to show b o th the specular and the forw ard scattering lobe. Note how the approxim ation w ith a single cylinder is inadequate, p articularly around the spec ular scattering lobe, which is not located a t th e correct angle 9S. Such displacem ent occurs because the slope of the side walls of the tap e red cylinder cannot be m odeled using only one cylinder. T he direction of the forw ard scattering lobe, on th e other hand, is not affected by the taper, an d is correctly estim ated by the approxim ate model. T he use of two stacked cylinders (of length and radii given in Table 4.2) leads to a significant im provem ent. T h e m ain scattering lobe is now closer to its correct location, b u t there is still the problem of a high side lobe around 9S — 150°. T he approxim ation w ith four cylinders brings the m ain lobe even closer to its correct position and more im portant - lowers the side lobes. Em ploying m ore th a n four cylinders still increases the accuracy, b u t the im provem ent is quite small. 107 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. VV-POLARIZATION HH-POLARIZATION 40 40 CM ns X -1 0 -1 0 tapered cylinder 1 cylinder —«—2 cylinders —9—4 cylinders -20 -30 120 140 160 180 200 220 tapered cylinder 1 cylinder —**—2 cylinders —9—4 cylinders -20 -30 120 240 140 160 180 200 220 240 scattering angle 0 [deg] Figure 4.41: N orm alized b istatic scattering coefficient vs. scatterin g angle for tap ered dielectric cylinder w ith L = 10.0 A, a = 0.7 A, b — 0.1 A, and incident angle 9t = 40°. h h - and vv- polarization (left and right, respectively). HH-POLARIZATION VV-POLARIZATION tapered cylinder 1 cylinder —•*—2 cylinders —9—4 cylinders — tapered cylinder - -1 cylinder -«—2 cylinders ■9— 4 cylinders £ ■O E CQ CQ -a cj ns as K K o's -1 0 -1 0 -20 -2 0 -30. 120 140 160 180 200 220 240 140 160 180 200 220 scattering angle 0s [deg] scattering angle 0 [deg] Figure 4.42: Norm alized b istatic scattering coefficient vs. sc atterin g angle for tap ered dielectric cylinder with L = 10.0 A, a = 0.5 A, b = 0.3 A, and incident angle 9{ — 40°. hh- and vv- polarization (left and right, respectively) 108 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 240 1 cylinder, HH 2 cylinders, HH 4 cylinders, HH 1 cylinder, VV - •* - 2 cylinders, VV 4 cylinders, VV 1 cylinder, HH —“— 2 cylinders, HH 4 cylinders, HH 1 cylinder, VV - k - 2 cylinders, VV 4 cylinders, VV absolute error [dB] — m ■a I— o 0 0 3 o CO .Q 0 30 45 60 incident angle 0. [deg] incident angle 0. [deg] Figure 4.43: E rro r vs. incident angle for tap e red dielectric cylinder w ith L = 10.0 A, (left) a — 0.7 A, ct2 = 0.1 A, and (right) a — 0.5 A, b = 0.3A. Solid lines are hh-pol, dashed lines w -p o l. Figure 4.42 illustrates th e case of a tap e red cylinder of th e sam e length and volume as th e cylinder in Figure 4.41, b u t w ith less tap e r, specifically w ith a — 0.5 A, b = 0.3 A, which results in a tap e r factor R = 0.02. T h e wave is incident from an angle 0Z = 40° as in the previous case. As one would expect given th e sm aller tap e r, th e tap ered cylinder m odel works m uch b e tte r in this situation, and even th e one cylinder approxim ation exhibits only a sm all displacem ent in the m ain scattering lobe. In b o th Figure 4.41 a n d 4.42, it is evident th a t th e analytical approxim ation works b e tte r in th e forw ard scattering direction. T his behaviour is probably due to the fact th a t the forward scattering depends more on th e dielectric properties of th e cylinder, which are not approxim ated, while the scattering in th e specular direction is m ore affected by the geometry, which depends on th e m odel used. T he error plots in Figure 4.43 help u n d e rsta n d how th e accuracy of th e various approxim ations changes w ith the incident angle T he definition of the errors thh, and evv is still the sam e as in (4.127). T he first of them is th e long tap e red cylinder of Figure 4.41, w ith L = 10.0A, a — OTA, 109 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. a 2 = 0.1 A, and. R = 0.06. Its absolute d B error is plo tted at the left of Figure 4.43 as a function of the incident angle 9i. As already noted in Figure 4.41, the single cylinder solution does not provide a good approxim ation for th e tap e red cylinder around th e specular scattering lobe. The curves on the left of Figure 4.43 confirm th a t by showing an error betw een 3 d B an d 10 d B a t m ost incident angles in h/i-polarization. Such a large error is m ostly caused by th e m isplacem ent of the specular scattering lobe by th e analytical approxim ation. For the sam e reason, using two cylinders produces a lower - b u t still unacceptable - error a t h/i-polarization. T he approxim ation w ith four stacked cylinders yields th e best results, th e error being below 1.5 d B at th e most angles, except for low incident angles, i.e., 9i < 15°. T he problem of m isplacem ent of th e specular lobe in the approxim ation does not seem to affect the w -p o lariza tio n . In reality, as obvious from Figure 4.41, th e problem is present a t b o th polarizations, b u t since a t uw-polarization the specular scattering lobe is more th a n 10 d B lower th a n the forw ard scattering lobe, it does not enter in the calculation of the error as defined in (4.127). On the right side of Figure 4.43 are the error plots for the cylinder of Figure 4.42, w ith L = 10.0 A, and a = 0.5 A, b = 0.3A, and R = 0.02. T he cylinder has a sm aller tap er, and a good approxim ation is already achieved w ith a finite cylinder of th e same length and volume, w ith no m ajor im provem ent using th e tap ered cylinder approxim ation. O n the other hand, com parison betw een the two plots in F igure 4.43 indicates th a t th e error a t low incident angles is not affected by the taper. T he m ost obvious conclusion is th a t th e end-on effect are predom inant there and cannot be predicted accurately by any of the approxim ate m odels here used. C om parison o f co m p u ta tio n a l tim es In this section, a com parison will be m ade betw een the co m putational tim es of the various m ethods, in order to show the advantage of choosing th e analytical approxim ations over the num erical solution approach. In order to estim ate and com pare c o m p u tatio n tim es of th e different m ethods, the 110 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. following considerations are made. T he num ber of calculations needed to evaluate th e scattering coefficient using the finite cylinder m odel does not depend on the cylinder length and is constant for a fixed vale of its radius. T his happens because the expression used to com pute th e scattering am plitudes is a series sum m ation w ith a num ber of term s th a t increases w ith the radius of the cylinder. In practice this num ber has been chosen large enough to achieve convergence in all cases considered in this comparison. U nder this assum ption, let T p c be defined as the com putation tim e of th e scattering coefficient by the finite cylinder model. T he tap e red cylinder m odel requires - as a first step - th e calculation of th e scattering am plitudes of th e N s individual cylinders th a t approxim ate sections of th e original tap ered cylinder. T he scattering am plitudes are th e n used in expression (4.126) to yield th e scattering am plitude of the entire tap e red cylinder. Therefore, if one neglects th e tim e need to perform such sum m ation, the com putation tim e of the tap ered cylinder m odel will be Tt c = N s T p c , (4.130) which depends on the num ber N s of sections in the approxim ation, b u t not directly on the tap ered cylinder length. On th e other hand, th e m ethod of m om ents involves th e selection of a set of N points on the generating curve C an d requires a num ber of calculation th a t increases as N 2. The num ber of points N varies in p ro p o rtio n to th e length I of th e generating curve C in the representation of the cylinder as a body of revolution, m easured in wavelengths A. For a finite cylinder of length L and radius a, the length I of the generating curve C is I = L + 2a, while I = L + a + b for a tapered cylinder of length L and radii a, b. Therefore, it is reasonable to expect the com putation tim e T m m ° f the m ethod of m om ent algorithm to increase approxim ately w ith the square of the cylinder length. I l l Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Em pirically, th e following approxim ate relationship betw een T m m and T FF are found: T m m = Z M (j ) Tf c = Z M ' TF C . (4.131) where M is th e num ber of modes used in the num erical algorithm . Similarly, if the m ethod of m om ents is used to find the scattering coefficient of a tap ered cylinder, th e use of (4.130)-(4.132) yields T m m = 3A f ( j J 2m „ M TFC = 3 — (L + a + b \2 ------— j Ttc . (4.132) In order to exam ine th e gain in com putational speed, TFc and T m m have been estim ated using expressions (4.130) a n d (4.132) for four tap ered cylinders w ith a = 0.6A, b — 0.2A, and lengths L = A, L = 3A, L = 5A, a n d L = 10A. As an exam ple, a com puter system is considered, on which the finite cylinder p rogram needs a tim e TFc = 5 sec to calculate the scattering coefficients. For all four cases, the num ber of sections in the tap ered cylinder approxim ations is chosen as N s = 4, therefore according to (4.130) the com putation tim e is TFc — 20 sec. T he com putational tim e T m m if th e num erical algorithm has been estim ated through expression (4.132), and listed in Table 4.3, where th e ratio T m m / T t c is also given. A num ber of modes M — 5 has been found to achieve good accuracy in the num erical solution in these p articular cases, and used in (4.132). length L A 3A 5A 10 A Tmm Am in 18 m i n 42 m i n 145 m i n Tm m /T tc 12 54 126 438 Table 4.3: C om parison of com putation tim e betw een an alytical m odel and num erical algorithm for four tapered cylinders of length L and a = 0.6A, b = 0.2A. The values in T able 4.3 clearly show the great gain in com putational efficiency at the cost of some loss of accuracy. Indeed, it is easy to see th a t even for a ra th e r short cylinder of length L = 3A, 112 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. a com putation tim e of T m m = 18 m i n for the num erical approach com pared to T t c = 20 sec for the analytical approxim ation makes th e la tte r preferable when dealing w ith a m edia where thousands cylinders of different sizes are present and estim ation of the scattering coefficient from each cylinder is needed. 4 .4 .4 C o n c lu s io n s A nalytical approxim ations for two different types of finite cylinders have been considered, and their accuracy evaluated. For cylinder sizes and dielectric constant sim ilar to those used to represent vegetation elements, such approxim ations work well in th e m ain lobes, which is w hat is most im p o rta n t w ith w hen studying rem ote sensing from vegetation. W hile the physical optics solution for the finite cylinder works very well and is readily and safely usable in vegetation m odels, th e tap ered cylinder approxim ation requires more a tte n tio n in the choice of the num ber N g of cylinders, in order to reduce th e error due to the displacem ent of the specular scattering lobe. T he results also prove th a t w hen th e ta p e r is significant, the single cylinder approxim ation is not adequate for a tap ered cylinder. As seen in the previous section, th e relative sm all loss in accuracy suffered by the use of such m odels is outbalanced by a trem endous gain in com putational speed. However, more work could be done to try to m odel th e end-on scattering. T he integration of such a m odel into the approxim ations exam ined here could allow th eir use in a broader range of electrom agnetic problems. 113 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 5 H alf-Space Problem T he concepts a n d techniques introduced and developed in th e previous chapters can be used to build more realistic m odels for vegetation th a t account for th e effect of th e ground. A very simple approach is to m odel th e ground and the air above it as two sem i-infinite m edia separated by a plane interface. T his type of configuration will be referred to as an Half-Space Problem . In order to s tu d y this problem , it is m ore convenient to use the G reen’s functions in their dyadic forms. For th is reason, the Section 5.1 of this chapter is dedicated to the derivation of explicit forms for th e com ponents of the dyadic G reen’s functions, b o th electric and m agnetic, in the case of an infinite, homogeneous m edium. Following th e sam e steps, in Section 5.2 the dyadic G reen’s functions for th e half-space problem are introduced, and expressions of their com ponents are derived. Section 5.3 shows how to modify the coupled surface integral equations for the half space problem, an d how to solve them by th e M ethod of M om ents for the case of a B ody of Revolution located in th e u p p er half-space. T he im plem entation of th e BOR-M OM m ethod in the half-space often requires the evaluation of slowly converging integrals. Section 5.4 is devoted to the investigation of some efficient num erical techniques for th eir integration. Results are given and discussed in Section 5.5, and finally the application to electrom agnetic scattering from vegetation is illustrated in Section 5.6. 114 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.1 The dyadic G reen’s function for an infinite, hom ogeneous m edium T h e dyadic G reen’s function for an infinite, homogeneous m edium w ith perm ittiv ity e an d perm e ability fj, relates th e vector electrom agnetic fields to the vector current sources, and can be found from the corresponding scalar G reen’s function using th e alternative form (2.44): G (r,r') = f l + ^ w k2 ) 5 (r,r') (5.1) where k = y/]H is th e wavenum ber of the m edium . A m agnetic dyadic G reen’s function is defined by expression (2.11) as: K (r,r') = V j ( r ,r ') x l (5.2) D epending on th e geom etry of the problem under consideration, it is useful to express the dyadic G reen’s function in different forms. One of such representations makes use of plane waves. As shown in [76], in order to derive it, the equation for the scalar G reen’s function V 2fir(r, r ') + k 2g(r, r ') = - 5 { r - r ') (5.3) is solved using Fourier transform s. T he result is an expression in term s of planes waves: ,• r e - j k 0Z\z - A , 9 ( r ' r ' , = “ 4 ^ / ------- 2 K , ------- 6 J (5'4) where k £ and r £ are tw o-dim ensional vectors defined as k s= and kx ± + k y y , r t = j;x + y y koz = y jk ,2 - k l - k l , X m { k oz} < 0 , (5.5) (5.6) Consider a plane electrom agnetic wave traveling in th e direction of a u nit vector k. A ssociated w ith this electrom agnetic wave is a propagation vector k = k k and two other u nit vectors h and v , defined so th a t th e vector tria d (k, h, v) is orthonorm al: k x z h = —----------, |k x z | „ v = h x k 115 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.7) A h \ \ Figure 5.1: D efinition of (k, h , v) for waves traveling forward and backw ards in th e ^-direction. T he u nit vector h corresponds to th e direction of the h-polarization of th e electrom agnetic wave, while v is in the direction of th e u-polarization. Those vectors are shown in Figure 5.1 for the two cases of plane waves traveling in the positive and negative z-direction, identified by the superscripts + and —, respectively. Since = k k* = kx x + ky y ± koz z = k t ± koz z , (5.8) th en the scalar G reen’s function (5.4) m ay also be w ritten as ^ J - j k + • (r - r') —j------------ dkt for z > z' 47T 2J g(r , r ' ) = - < (5.9) g - j k~ • (r - r') 2 kn dkt for z < z' T he dyadic G reen’s function can be o b tained by differentiation of (5.4) th ro u g h expression (5.1). T he procedure is illu strate d in [76] and yields th e result: j k + ■(r - r') J 2tr2 kl h + h+ + v+ v+ ) ~ 2 J { h - h - T v - v - ) -------- _ dkt --------- d k t for z > z' for z < z' — OO T he dyadic G reen’s function can also be w ritte n in term s of vector cylindrical waves. Such a 116 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. representation is particu larly useful when dealing w ith problem s th a t have a cylindrical or ro tatio n al sym m etry. T he vector cylindrical wave functions are defined as ^jn(kpi k z ,v) = k Z: r) M n {kp, k z ,r) = V x [z ipn (kp,kz ,r)} (5.12) N n {kp, k z ,r) = i v x M ^ ^ r ) (5.13) (5.11) where ipn is the solution of th e scalar wave equation in cylindrical coordinates { } r P% + ^ w w ith + h k + k 2 ) i ’ir) = 0 9 9 kp + kz , = (5-14) (5.15) and is related to the Bessel function of ra-th order J n : ^ n { k p, k z , r) = Jn ( k p p ) e ~ 3 kzZ ~ ^ (5.16) kx = kp cos^fc , ky = kp sin$fc (5-17) Using the transform ation and the m athem atical id entity e - j { k xx + kyy) = e - j k pPzos{<t>-§k) = 4-00 Y , ( - j ) n Jn(kPp ) e ~ i n (^ - $*) (5.18) it is possible to w rite th e vector electrom agnetic plane waves in term s of vector cylindrical waves: +°o h e " j k 'r = - — 5 3 ( - j ) U^ - n ( k p , k z , r ) e ~ i n ®k (5.19) P n =— oo i v e-jk-r = _ +oo Y ( - j ) n $ n (kP, k z , r ) e - 3 n ^ P n=— oo Since 117 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.20) su b stitu tio n of (5.20), (5.21) into (5.9), followed by integration in th e variable yields the cylindrical wave representation of the dyadic G reen’s function: (5.22) koz, r ) + ]NTw(fcp, k oz, r) 1ST— n {kp, M-njkp, k oz, r)3Vl—n {kp, Air k oz, v ) koz kp k oz, r)!VI_n{kpi koz, r ) + ~M-n{kpi 47r fcoz,r )N _ n(fc^,A:oz,r ) koz kp dkg for z > z' dkg for z < z' T he m agnetic dyadic G reen’s function can be obtained w ith a sim ilar procedure by su b stitu tin g the expression (5.4) for the scalar G reen’s function into the definition (5.2).C arrying out the differentiation yields: K (r,r0 ( —j k + x I g( r, r') Z , = for z > z' „ , (5-23) so th a t the following expression is obtained: k K (r,r') = r ~ e - i k + • (r - r ') h + - h + v+) -------- — ---------d k , z > z’ -< (5.24) v . 47T2 J v ’ _ h —h " e ~ 3 k ■(r _ r ') v ) -------- —------------- dkt ' 2 koz _ for z < z' By expanding h and v in term s of vector cylindrical waves by m eans of (5.19) and (5.20), the following cylindrical waves representation of the m agnetic dyadic G reen’s function is obtained: OO k 47T K (r,r') k 47T f / f / ^ A ^ n { k p , koz, r) 1V1_n {kp, A k0Z:T^j A T^/ln ^kp, k 0Z: r)N_^(A:p, koz, v ) koz kp koz j r)]VI_n{kpi k 0zi r ) ^^-nikpi k oz, r )N _ n(fcp, koz, r ) koz kp dkg for z > dkg for z < z (5.25) It is desirable to find an expansion of th e dyadic G reen’s functions into Fourier modes in the azim uth angle <j>. T his type of expansion is indeed essential in order to use the BO R approach developed in the previous chapter. 118 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. S u b stitu tin g expression (5.16) into the definitions (5.12) an d (5.13), the vector cylindrical wave functions can be w ritten explicitly as M n {kp,kz ,v) — jn LP N n {kp, kz , r) = JnikpP) P "b kpJn (kpp) </> , ~ j kz z - jn<p (5.26) - j k pkz J'n (kpp ) p + n k z k J n {kpp) <p - k 2Jn {kpp) z e —J kzz — jruf> (5 27) In this form, the periodic behaviour in th e variable <f>is evident. Using (5.26) and (5.27) yields the Fourier series expansion of th e dyadic G reen’s function: +oo G (r,r') = Y. (5.28) S T he coefficients G n are dyads them selves, and have different expression depending on th e coordi nate system used. Using a unprim ed-prim ed cylindrical coordinate system in which a dyad S can be w ritten as S = ^2 S pq p q ' (5.29) p ,q = p,faz the com ponents of G n have th e form: OO Gp/ ( P,z-,p',z') = - j - J -j koz\z - z'\ g pq, ( k p , k o z ] p , p ' ) 2 koz kpdkp, p,q = p,(/),z (5.30) Similarly, the m agnetic dyadic G reen’s function (5.26) can be w ritten as a Fourier series + °° K (r,r') = , K n ( / > , * ; / / , * ,) e J’n ^ ~ 0 ) (5.31) n=—oo and the com ponents of K „ as . ~ j k0Z\z - z'\ K pq' { p ,z - p ' , z ' ) = ~ L 5.2 J K,iq'{kp,koz-p,p') 2fcoz kpdkp, p,q = p,(f>,z (5.32) The dyadic G reen’s function for sem i-infinite media Let the space be divided into two regions w ith different dielectric properties by an infinite plane, which for convenience will be considered coincident w ith th e xy-plane. Each region is referred to 119 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. as an half-space. T he two m edia are characterized by p e rm ittiv ity and perm eability respectively given by: ea £b for z > 0 Ha f A4 = for z < 0 | lu a z > 0 , n for £ < 0 for , (5.33) Let b o th the sources and the observation points be located only in th e upper half-space z > 0. T he dyadic G reen’s functions in th e u p p er half-space, i.e., for z, z' > 0 have different term s corresponding to th e direct (incident) and reflected electrom agnetic waves: G (* ;„$ * ; z,z')' = & D)(kp, $ k-,z,z') + £ W { k p, $ k]z,z') (5.34) K ( k p, $ k -,z,z') = K ^ ( k p, ^ k - z,z ') + K W ( k p, $ k -,z,z') (5.35) T he direct term s have the sam e expression as for the dyadic G reen’s functions in an infinite m edium (5.10) and (5.24): i G ^ (r,r') r - e _ i k + ' (r _ r 0 ' ''h + h + + v + v + ) ---------—------------ d k t 2 koz 47T2 = j , 47r2 K ('D)(r, r ;) = -< - e ’ (r — r ' ) for z > z' (5.36) v ) ---------—------------ dkt 2 rCor for z < z' j k „ , . v + h + — h + v + ) ------- —------------- dkt 47T2 J 2 ka for z > z' h h + v ** f0r2f<i! Using the cylindrical wave expansion of th e plane waves h an d v , these expressions take the following form: / OO ~ 4 tt r / A ]&ln (kp, K U U ( r , r ') = < f a a f u u / A dkD A for z < z' a z z1 p koz kp -~koz, r)]Vl—n for z > z' (5.3? koz, r ) N _ n (fcp, —k0Z: r ) ^ fepz, r ) — J dk0 A a a j4n (kp, k oz, r ) M _ n (fcp, OQ A & 47T x k oz, r ) M ~ n (kp, k oz, r ) + N n (A;p, —A;oz, r ) N _ n (fcp, k oz, r ) koz kp Ca J k 47t /s. koz kp QQ 47T ^ k oz, r ) -I- ~Nn (kp, koz, r)N _ ri(A:j0, —koz, r ) J G ^ }( r , r ') = j ^ F WLn {kp, k 0z-, r)JVl_rj(A:p, j (5.31 A k0Zj r ) — 3VIn (fcpT~ k oz, r ) N _ 7i(A^, k0zi r ) 120 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. dkp for z < z' T he reflected term s have th e following expression: . i f i W (r . O = - ^ / b K ^ ( r , r ' ) = —^~2 / r T h h + e ~3 k + r h~ e l k_ ' ^ +T„ v + e~^' k + ' r v" --------------- — ^ k_ ' ^ dki (5'40) r r , o+ « —j k+ • r it- „ j k~ ■r' r fl+ ~3 k+ ■r <>- „ j k~ • r' 6 — ---------2 ^ " ------------ ----------- — ** (5.41) where IT and r „ take into account the reflection of the horizontal and vertical plane waves, respec tively. T hey depend on th e dielectric properties of the two m edia, as well as on kp, as follows: _ h Mfr K z - Ma K z e b K z - £a K z r lib K z + Mo K z ’ V ^ £b K z + £a K z ,---------- w ith = y /K ~ K = k oz (5.43) Kz = \jK ~ K = \ ! k oz + ( K - kl) (5.44) Kz and the wavenum ber in th e two m edia given by = uj^JiJ,aea for 2 > 0 kb = ujy/JIb£b for 2: < 0 ka k = { (5.45) In term s of cylindrical waves: ri(R)r - i\ 3 47T J0 OC f h ■hdn(^p) k oz, r)M _ n (fc^, k oz, r ) 4 -1\, N n (kp, k oz, r)N _ n(&p, k oz, r ) J7 k^ kp 9 /e OO ~ f 'E'fl i'fn ( kp, koz,v)ls/L—n (kp, k o z i Y ) — T'v 7S/Ln(kp, k oz,ic')N~n (kp, k oz, Y ) -------------------------------------------- k j , ---------------------------------------------dk> (5-47) A is-(R)f !\ k {T' r ) = ^ J 0 As already m entioned in th e previous section for th e infinite-m edium dyadic G reen’s functions, the half-space dyadic G reen ’s functions (5.38)-(5.41) can also be w ritten as infinite sum m ation of angular modes S l ‘>(r,r') = £ = n=—00 £ [ £ S * p q '] e 3 n=—oo \p,q—p,(ji,z ) for SW = £ W , K W , t = D,R 121 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - « (5.48) where th e dyadic com ponents of the n th m ode coefficients have th e following integral forms: Sd^{p^-,p',z') , - j k 0Z\ z - z '| = -zL j S $ n {kp, k oz;p,p') 2k„ o OO J kp d k p , p, q = p,4>,z (5.49) ■k p d k p , P , q = P,<f>,z (5.50) , - j k 0z { z + z ' ) S ^ n {kp,k o z \P ,p ') - 2 k oz Explicit expressions for Qv^ n , fcP r jn , Qv^ n -, and KF^n are given in A ppendix D. 5 .2 .1 T h e d y a d ic G r e e n ’s fu n c tio n in t h e far field In scattering problem s it is very im p o rta n t to be able to evaluate th e electric and m agnetic fields, generated by electrom agnetic sources, a t large distances from th e sources themselves. For this purpose, it is essential to derive the far-field expressions of th e dyadic G reen’s functions derived in th e previous section. " ./// Figure 5.2: Far-field in half-space m edium W ith reference to F igure 5.2 (left), a point in th e far-field - or rad iatio n zone - is defined by a position vector r such th a t k^r 1 a n d r » r ', w here r ' identifies an a rb itrary source point. If 122 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the source has a m axim um overall dim ension D th a t is large com pared to the wavelength A, the far-field region is comm only taken to exist at distances |r —r '| greater th a n 2 D 2/ X from the source. B oth direct an d reflected term s of th e half-space dyadic G reen’s functions need to be evaluated in the far-field. T he direct term has two different expressions, depending on w hether z > z' 01 z < z 1. However, only th e first case needs to be considered, and Figure 5.2 (right) illustrates why. An observation point r has z > z' if it is not inside an infinite slab of thickness h (equal to the height of th e m ost d istan t source point from the half-space interface). T he m axim um allowed value of the elevation angle for th e observation point not to fall inside th a t slab is 6 — arctan (p/h). In the rad iatio n zone, p becomes very large and 6 approaches tt/2. T his m eans th a t for observations angles 6 < 7r/2, which is th e range of elevation angles considered in half-space problems w ith plane wave coming from th e u p p er m edium , only the case z > z' needs to be considered in the evaluation of the far-field dyadic G reen’s functions. Hence, th e problem is now reduced to the evaluation of expressions (5.36)-(5.37) for z > z' and (5.46)-(5.47). All these expressions have the general form F ( r , r ') j = l ( k p, k oz-v') 6 dkt (5.51) An approxim ation for r > r' for th is generic F ( r , r ') can be found by employing the so-called stationary-phase m ethod. T he exponential k+ •r = kx x + ky y + ^Jk2 — k 2 — k 2 z (5.52) is a rapidly oscillating function of k t = kx x + ky y for very large values of r. T hus the contributions to the integral from various points in th e kx ky plane ten d to cancel each other because th ere is a lack of in-phase add itio n from th e various regions. A n exception is a point where the exponent does not vary w ith sm all changes in k x , ky . Such point is a called a stationary-phase point and is characterized by th e vanishing of th e first derivative of the exponent w ith respect w ith kx and ky , i.e., 123 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. At a stationary-phase point, th e phase of (5.51) does not vary rapidly, and a nonzero contribution would be obtained from this region of th e kx ky plane. In a sm all region surrounding the stationaryphase point k x = k x , k y = k y , th e rem aining, slowly-varying p a rt of the integrand of (5.51) is set equal to its value at kx , ky . T he integral th a t rem ains th en involves only the exponential function, and can be easily evaluated. T h e stationary-phase point is at k + = kx x + ky y + k z z w ith k x = k sin 9 c o s^ > , ky = k s in 9 sin <fr , (5.54) k z = k cos 9 (5.55) T h e result of the integration around th is point is shown [77] to be F ( r , r ') ~ F (7c sin (9 , & co s# ;r') -r (5.56) At the stationary-phase point the plane waves k , h and v become k+ = r , k~ = r — 2 cos 9 z (5.57) h + = h " = - 4> v + = —0 , (5.58) = — 0 + 2 cos 9 p , (5.59) Using the general result (5.56) and expressions (5.57)-(5.59), th e far-field expressions of th e direct term s of the dyadic G reen’s functions are found - as expected - to be identical to the ones for an infinite m edium, i.e, G ^ D^ ( r ,r ') ~ ( d 0 + (j) <fr^) e ^ r K ^ ^ r,^ ) ~ —j k ( <j >0 v 0 <^) / k 4tt r r (5.60) •r 124 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.61) while the reflected term s of the dyadic G reen’s functions in th e far field are J ~ (fv 9 \ G ^ (r,r') K ^ (r,r') where ~ 0 / rp \ / A ^ e i fck“ ' r ' 47v r - j k ( T v 4 > d - T h e 4 > - 2 TV c o s e 4 >p ) a n d r „ are the values of jjj + T h 4 > 4> - 2 TV c o s d 9 p ) — A __j T ^ ------- e j k k - - r ' 47r r (5.62) (5 ^ an d r „ , defined in (5.42), calculated a t the stationary-phase point given by (5.54). 5.3 Scattering from a B o d y in a Sem i-Infinite M edium 5 .3 .1 E x p r e s s io n o f S u rfa ce I n te g r a l E q u a tio n s fo r S e m i-I n fin ite M e d iu m Figure 5.3: M aterial body in half-space m edium . T he coupled surface integral equations for a m aterial body in an homogeneous, infinite m edium have been derived in C hapter 2. T h eir expressions are (2.37) a n d (2.38). These equations can be m odified and applied to an half-space problem . T he problem is defined in Figure 5.3. A n homogeneous m aterial body, w ith perm ittiv ity £4 and perm eability jj,d inside its volume V, is located in th e upper half-space, which has perm ittiv ity £0 and perm eability /j,q. T he lower half-space has p erm ittiv ity eg a n d perm eability n g. 125 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T h e m aterial body can be regarded as being located in a n equivalent homogeneous, infinite m edium w ith th e same p e rm ittiv ity £q, perm eability /xq as the u p p er half-space but w ith th e dyadic G reen’s functions G and K previously derived for the half-space problem . A similar approach has been followed by [78]. T he derivation of the coupled surface integral equations can be carried out for th is equivalent m edium in a sim ilar way as done for th e hom ogeneous, infinite m edium case. T h e exterior problem is illu strated in Figure 5.4. If th e surface S 2 is entirely inside th e upper m edium , as in Figure 5.4(a), th e n (2.27) can be w ritten as f —jui/jiQ [ G HS ( v , r ' ) - 3 { r ' ) d V ' K H5 (r, r ') ■M ( r ' ) d V ' + Jv Jv = 2 -jufxo -joJH0 f G HS (r, r') ■ [ n ' x H (r')] dS' - f K HS (r,r' ) ■ [E (r') x n '] dS' + G HS,(r, r') • [ n ' x H(r')] dS' - <£ K fflS(r, r ') ■ [E (r') x n '] d S ' , r G Si U S 2 (5.64) T he dyadic G reen’s functions G HS and ~KHS for th e half-space m edia consist of direct and reflected term s in the upper m edium , an d a tra n sm itte d term in th e lower m edium : for z > 0 for 2 < 0 for 2 > 0 for 2 < 0 (5.65) (5.66) ’ T he direct and reflected term s are given by expressions (5.37),(5.38) an d (5.47),(5.48), respectively, while the expressions of th e tra n sm itte d term s are not given here because they will not be used in this dissertation. As th e surface S 2 expands, p a rt of it sta rts entering th e lower m edium , as shown in Figure 5.4(b). Considering th e surfaces Si U S ^ , w ith 5 ^ = S a U £a> a n d 5 ^ = S b U Eg in Figure 5.4(c), it is possible to w rite the following two expressions: ^ 2 = - j uf i Q [ G H5 ( r , r ' ) • J ( r ') d V ' - [ K K5 (r, r ') • M ( r ') dV' + Jv Jv -jwn0 f G H S {r,v') • [ n ' x H (r')] dS' - f K H S {r,v') ■ [E (r') x n '] dS' + / Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. J,M * 0 J,M * 0 (b) J,M * 0 Figure 5.4: E xterior problem for half-space m edium . 127 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -juii o f G H S (v, r') ■[ n ' x H (r')] dS' - f G HS {r,r') • [ n ' x H (r')] dS' - JSa -jojfj, 0 f K HS (r,r' ) • [E (r') x f i '] dS' f K ff5 ( r, r ') • [E (r') x n '] dS' j Sa JH Ha JHa for r e S i U S i E (r) j —r~y r r, JS b 2 f JzB G h s {y,y') • [ n ' x H (r')] dS' - f K HS (v,r') J Sb f G H S (r,r' ) • [ n ' x H (r')] dS' - •'S b ■ (5.67) [E (r') x n '] dS' + K HS {r,r') ■ [E (r') x n '] dS' for r e S i (5.68) Note th a t there are no volum e integrals in (5.68) because there are no sources in th e lower m edium . W hen the surfaces E a a n d E # approach the interface z = 0, due to continuity a n d to the opposite direction of th e surface norm al vectors, Vo = —n g (f / G ff5 ( r , r ' ) ' [ n ' x H ( r ' ) ] f K HS (r,r' ) • [E (r') x n '] dS' JT,a = G ffS( r, r ') ■[ n ' x H (r')] d 5 ' - f K HS {r,r') ■[E (r') ■'2s x n '] dS' (5.69) (5.70) while the integrals over th e surfaces S a , S b vanish at infinity. Therefore, in th e lim it of S a , S b expanding to infinity, adding equations (5.68)-(5.68) side by side yields th e following result: = 2 —juifiQ -jufio [ Jv Js+ G H S ( r , r ' ) - J ( r ' ) d F ' - / K F 5 (r, r ') • M ( r ' ) d V ' + Jv G H S (r,r' ) • [ n ' x H (r')] dS' — Js+ K H S (r,r' ) ■ [E (r') x n '] d S ' , r e S + (5.71) O n the other hand, th e in terio r equation depends only on the properties of the m edium inside the surface 5 of the m aterial body, therefore its form is th e sam e as for free-space, i.e., ^ 7^2 = jwfJ-o / Js~ G ( r , r ' ) • [ n ' x H (r')] dS' + <£ Js~ K ( r , r ') • [E (r') x n '] d S r , r 6 S ~ (5.72) Taking the tangential com ponent of b o th sides of equations (5.71)-(5.72), and using th e definitions (2.25) and (2.35) of equivalent surface currents, yields 2 = E^(r) - Js+ G ff5 ( r , r ' ) • J s (r') dS' + 128 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - ( I - n n ) - < T K HS {r,r') • M s ( r' ) dS' , 7s+ = r 6 S+ (5.73) r € 5" (5.74) & H S {*,*')■ M { r ' ) d V ' (5.75) - W < i ( I - n n ) ' | _ G ( r , r ;) • J s(r')dS" + - ( I - n n ) . j f * K ( r , r ' ) . M s (r')(l5' where th e ’’incident field” is defined as = 5 .3 .2 -juM J & HS ( r , r ' ) - 3 { r ,) W ' - J S o lu tio n o f t h e S u rfa ce In te g r a l E q u a tio n s for a B O R in H a lf-S p a c e Let th e surface of th e m aterial body have a ro tatio n al sym m etry around the z-axis, so th a t th e B O R approach discussed in C hapter 4 can be applied. D ividing the half-space dyadic G reen’s functions into th eir direct and reflected term s as in (5.65)-(5.66), the exterior integral equation (5.73) can be w ritte n as M *( 3)X n = E ? \ r ) - jujfxQ( 1 - n h ) - £ + G W ( r , r ' ) ■J s (r’) d S ’ + - ( I - n n ) - < T K ^ ( r , r ' ) - M s (r' ) d S ' + Js+ - j u ^ o ( I - n n)- <f Js+ G p ) ( r , r ') ■J s (r') dS' + - ( I - n n ) i * K (jR)( r ,r ' ) ■ M s {r') d S \ Js+ r GS+ (5.76) D iscretization of the surface integral equations (5.76) a n d (5.74) by m eans of the m ethod of m om ents yields a linear system of equations of th e form: E ^ { ( [ A ^ ] (D) + [ A ^ p ) ) i?« + ( [ B W p ) + [Bpp R)) M «"} = V Pn (5.77) E {[CW ]F»-[DM ]M 9»} = 0 q=T,<f> (5.78) T he elements of th e m atrices [CP9] and [DP9] can be evaluated by th e same procedure and expres sions derived in C h ap ter 4. T he direct term s G p ) and G P have the same expressions as th e 129 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. dyadic G reen’s functions for the half-space. However, the calculation of th e elem ents of [A^9] ^ and [B?9] ^ depends on th e particu lar form of the dyadic G reen’s functions used. The operator form is preferred here because the direct term s of dyadic G reen’s functions are highly singular. T he rem aining term s in (4.70), i.e., E q= t ,4> { [A£9](r) I 9n + [B£9](k) M 9" } (5.79) J 1 are obtained by discretizing th e surface integrals Ls(r) = - j u p 0 (I - n n)- <fi G (iJ)(r,r') ■J s(r') dS' + Js+ - ( I - n n ) i * K ^ )(r ,r ') • M s(r')dS', J s+ r e S+ (5.80) As seen in Section 5.2, the reflected term s of th e dyadic G reen’s functions can expressed in cylindrical coordinates as follows G ^ ( r ,r ') - £ ^ (r.O p q ', K^(r,r')= p ,q= p,4> ,z £ K $ ( r , r') p q ' (5.81) p,q=p,<t>,z Therefore, use of (5.81) and separation of th e r - and ^-com ponents of the incident electric field and surface equivalent currents, i.e., J s(r') = M s {r') = + M T(f,<f>') t 4>' (5.82) ' + M <p{t',<j>') 4>' (5.83) allows to transform th e vector term (5.80) into th e following two scalar term s for p = r , (p L p{t, <p) = -jcuMo £ f G $ ( r , r ' ) J qtf,<l>')dS' + £ a=T,< PJ s [ K ^ ) { v y ) M q{ t ' A ' ) d S ' (5.84) q=r,<t>J S where, G ^ (r ,r ') = f ■ G ^ )(r ,r ')- f ' = sin ip ( p ^ J sin ip' + G ^ ) cos ip'^ + cos ip [ G ^ sin ip' + g [^) cos i G%}{ r,r') = f - G ^ ( r , r ' ) - 4>' = sin ip G ^ j + cos ip g [^! 130 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. p (5.85) (5.86) ^ 5 ( r , r ') = 0 • G (E)(r,r')- f ' = G^p! s in 0 ' + G ^ l c o s 0 ' (5.87) = $ ■ G ^ ( r , r') ■ 0 ' = g JJ) (5.88) where G ^ ? (p, q = p, </>,z) are th e com ponents of th e dyadic G reen’s functions G coordinates, as defined in (5.81). T he expressions for the in cylindrical (p ,q = r, 0) can be obtained from pq' w ith K ^ ) , an d G ^ ) w ith K ^ ) for p, q = p,<f>,z. (5.86)-(5.88) by su b stitu tin g T he cylindrical com ponents of G ^ have the form of Fourier series, for p, q = p, 0, z, and +oo +oo G ^ i r . r 1) = Z = £ n = —oo • n= K ^ ( p , z ; / , z') (5.89) —00 where, from (5.50), 7 r00 GP£ n ( P ’ z ' >p' i z ') = J , e - j M * + z ') — ----------------------------------------------- (5.90) O^ p n {k p ,k oz- p , p ' ) o 02 r° ^■^n(P-> Z i P J ^ ) = _ 4^ J o and th e expressions for Qp^ n an d p —j k o z { z J r z ') ^R,n^p->k0Z\P, p ) k p dkp ^ (5.91) 02 are given in A ppendix D. Since the dyadic G reen’s functions G and K th a t are already in the form of a Fourier series, a m ore straightforw ard approach th a n th e one followed in th e infinite m edium solution of th e integral equations can be employed to decouple the angular m odes in th e term (5.84). E xpanding,the equivalent? surface currents into Fourier serieSnfbr p = r, 0 , , %(f,0) = E J pn( t ) e Jn<P, M p( t,0 ) = Y M Vn{ t W n * 71 = — OO (5.92) 71 = — OO the term (5.84) can be w ritte n as +00 Y Lp (f,0) = C w h G re L Pn(t) = L Pn(t )ejn<P i (5-93) r / -jujpo E / A%n ( t , t ' ) J qn(t') d l ' + E / q=T,(j> q=T,<pJC dl' (5.94) and the integral kernels in (5.94) are 4 ^ n ( i, t') = sin 0 (G p^ n sin 0 ' + G£?n cos 0 ') + cos 0 (G z£ n sin 0 ' + G f£ cos 0 ') 131 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.95) At M = Axi^ G R,n Rn + COsV’ ^ R,n n (5.96) = G%n s m $ + G%n co& tf (5.97) = G * (5.98) T he expressions for the term s Bp*n (p ,q = p, <j>,z) can be obtained from (5.95)-(5.98) by replacing GZ w ith K RnApplying th e m ethod of m om ents as in th e unbounded m edium case, the incident tangential electric field and th e tangential equivalent currents are approxim ated by th e sum m ations (4.37)(4.38). Then, th e use of the testing functions (4.49)-(4.50) leads to the two term s (for p = r,(p): N q N q -jun 0 m)M m > S q=T,<f> m = q=T,(j> m = 1 k= l,...,N p (5.99) 1 T he coefficients ap^ n (k ,m ) are given by The + Ir t m oKn(fe>m ) = - / . A R,n{t k , t ) d t ''t - r n = sini sini/>k [ G ^ ( t fc; i " , f+ ) sim pm + Gp^ n (tk ; i “ , t+ ) cos ipv + c o sijjk [ G ^ ( t A:; t “ , t + ) s i n ^ m + G!^ ( t f c ; t ' , t + ) c o s ^ m] (5.100) i tm~\ — sin 'lf/icG ^ n {tk '1fm_i, fm) + COS IpkGQn (tk \ tm —1>tm) at n ( k ’ m ) at n ( k ’ m ) = / A Z n ^ k ^ ' ) dt ' J t-m = Gt n ( f k ; *m» *m) s i n + <?££(**; f~ , t+ ) COS = [ A t n ^ k J ) dt' ^771— *1 = G Z ^ k >^m-li ^m) (5.101) (5.102) (5.103) where 132 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. G % ( t 0-tA , t B ) T he coefficients b ^ n {k, to) , for p = t ,4> = [ tB Gp*n (t0,t') d£ Jtj4 (5.104) can be found from the expressions for aR,n(k ’m ) by replacing Gp£ n w ith Gp^ n , a n d G ^ n w ith = [ tB K r t ( t 0, t ' ) d t ' (5.105) Therefore, th e elem ents of th e m atrices [A^9] ^ and [B£9] ^ in (5.77) are, for p = T,<j>, 5 .3 .3 [A " ] S = [BJ’l S = - i wf*o (5.106) <5-107) E x p r e s s io n o f “in c id e n t ” field o n t h e su r fa c e S A . A. Figure 5.5: P lan e waves incident on a body in a half-space m edium. As in the previous cases of two-dim ensional bodies a n d bodies of revolution in free space, the coupled surface integral equations are solved considering an excitation in the form of a plane wave incident on the m aterial body. C om parison of the expression for the “incident” electric field in half-space (5.75) w ith the expression for the “incident” electric field in free-space (2.29) suggests th a t, in the half-space problem , th e former needs to take into account th e presence of th e interface betw een the two m edia located a t z — 0. Since the effect of th is plane interface is to produce a reflected field in response to th e electric field incident on it, as illu strated in Figure 5.5, it m eans 133 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. th a t a reflected term m ust be added to th e direct term given by expression (2.49). Hence, the to ta l “incident” electric field on the surface S of the m aterial body will have th e form: -3 h K - r + R qe ~ j h E (0 = £ (0 q j T h e expressions for th e reflection coefficients Rh and R v -r ^ ^5>108j in the horizontal and vertical polarization, respectively, will now be briefly derived. R e f le c te d a n d t r a n s m i t t e d fields a t t h e in te r f a c e b e tw e e n tw o lo ssy m e d ia A_ A_ Figure 5.6: Reflection and transm ission of plane wave across a plane boundary. As shown in Figure 5.6, let the plane z = 0 be th e bou n d ary betw een two lossy m edia w ith perm ittiv ity e0, e* and perm eability p a , p r e s p e c t i v e l y . T he incident electric field be a ^-polarized plane wave of the form Ei = q i E \ e ~ ^ ka^ v 134 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.109) and the reflected and tra n sm itte d electric field also have the form of ^-polarized plane waves: Er = q r ^ e - ^ a ^ +-r (5.110) E* = ^ t E tq e ~ ^ k b ^ ~ ' T' (5.111) T he incident, reflected a n d tra n sm itte d m agnetic fields can be found through th e M axwell’s equation (2.3), which in the case of plane waves becomes: jk x E = —jujfiE (5.112) T he problem of finding th e reflected an d tra n sm itte d waves associated w ith th e incident wave (5.109) is solved by requiring the continuity of th e electric and m agnetic field across the b oundary betw een th e two m edia. In p articular, since th e b o u n d ary surface corresponds to th e xy-plane, the conditions to be enforced are: z xEt = i x ( e * + E r) (5.113) z x H* = z x (H ‘ + H r) (5.114) Two cases are considered, horizontal an d vertical incident electric field. In b o th cases, Snell’s law provides th e reflection a n d transm ission angles: 9r = H orizontal p olarization . 9i , ki, cos Qt = k a cos 9{ (5.115) In the case of ^-polarized incident electric field, th e reflected and tra n sm itte d electric fields are also h-polarized, and th e b oundary conditions (5.113)-(5.114) become: Ei = E^ —r- cos 6t = Ei + El EP1 Er - r 1 cos 9{ — —- cos 9r Qb sa (5.116) (5.117) Sa Using Snell’s law, equations (5.116)-(5.117) can be solved to yield K = R h E{ (5.118) El = (1 + R h) E l (5.119) 135 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where th e reflection coefficient in th e h-polarization is _ Qcosej-C'Cost), (ft COS 6 i V e r tic a l p o la r iz a tio n . + COS Qt ( a In th e case of u-polarized incident electric field, the reflected a n d tra n s m itted electric fields are also u-polarized, while th e m agnetic fields axe /i-polarized. T he bou n d ary conditions (5.113)-(5.114) become: E lh cos Qt = E\ jp t cos Qi - E rh cos 9r jr i = Sb (5.121) jp r + (5.122) S>a Qa Using Snell’s law, equations (5.121)-(5.122) can be solved to yield K = El = RvK (5.123) (l + R u ) E i (5.124) where the reflection coefficient in th e u-polarization is = C . 008 ft Ca - COS Qi a COS 9 , + (& cos Qt S tartin g from expression (5.108) for the “incident” electric field, a n d following a derivation sim ilar to th e one employed in Section 4.2.4, th e p-com ponents of th e “incident” electric field on the surface S can be w ritte n as +oo E?{tA) = +oo £ = n=— oo £ [ 4 ? ( f ) + 4 ? ( f ) ] e jn* (5-126) 7i— — oo where the E p „ \ t ) have th e sam e expressions as the E p } (t ) in (4.76)-(4.77), while thep-com ponents of the reflected term for th e n -th m ode are E ? = j n { Rh E ^ sin ip (up) + - Ry E ^ [—j sin if; cos Qi T ~ (up) + cos ip sinfli J n ( up)] } e = j n [ j R h 4 ° T~ ( u p ) + Ry E $ cos Q, T + ( u p )] e vz e vz e 136 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. n(^>i (5.127) (5.128) w ith u and v given by (4.75), and T * defined in (4.78). 5 .3 .4 C o m p u t a t io n o f t h e s c a tte r in g c o e ffic ie n t Let the body o f revolution be illum inated by a g-polarized pane wave of th e form = q i e ~-7’fc° k ' r (5.129) T his incident electric field generates equivalent surface currents j i 9'1 and th a t evaluated by solving the surface integral equations for the half-space problem w ith the m ethod of m om ents. Using (5.35)-(5.36) an d expressions (5.61)-(5.64) for th e far-field half-space dyadic G reen’s functions, the scattering am plitudes associated w ith these surface currents can be w ritten as the sum of two term s: (5.130) f pq(k s , ki ) = f g \ t a& ) + f W ( k , k ) T he first term is th e direct term corresponding to th e far-field generated by the equivalent currents ; absence of th e lower half-space, an d is given by and Mr(9) [sq> in /W (k„ki) = + ^ ^ ) . j W ( r O e ^ o k s+ T , d 5 / + /?(*) 47r j (<£ d - 9 0 ) ■ M ^ ( r ') e i k^ t - r' d S ' q = h , v (5.131) , while the second te rm is th e reflected p a rt, which accounts for the presence of the lower half-space, has th e following form 1 jkoCo ~ pO) An &q -T -ffiv fs (rv 0 e +rh 4> - 2 FV cos9s d p ) - j ^ ( r ' ) 4> 0 - T h 6 4> - 2 f v cos 9S 4> p ) - M ^ ( r ' ) e ^ k ° ^ ' r ' dS' , , j k o k s • r>dS' -f q = h,v (5.132) In expressions (5.131) and (5.132), k j and k j are the u nit propagation vectors of th e scattered plane waves traveling in th e positive and negative ^-direction, respectively: sin#,, cos fa x + s in # s sin<jf>s y ± cosds z 137 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.133) It is shown in A ppendix D th a t, using (5.131) and (5.132) in (5.130), th e scattering am plitudes can be w ritten as infinite sum m ations of angular modes, +00 / p?(k s,£ i) = £ (5.134) whose n -th m ode term s are, for q = h , v , N, / £ ’( + + ) 4”>(ks,k,.) = -7 ^ § 7 E (5.135) 47T E g w=Tj<p m = l 47T E g w~Tt 0 E = t so J SO J + E m =i 1 (5.136) where the currents depend on th e direction and polarization of th e incident wave: = M Z n(ki, qi) q<) , 5.4 (5.137) Evaluation of spectral integrals T h e integrals th a t need to be com puted when solving th e coupled surface integral equations all have the general form I ( p, z - , p' , z ' ) = 00 —j y / k 2 - k 2 Z f f ( k p;p,p') ... kpdkp o V^ — (5.138) T he factor f k p, p,p') contains Bessel function term s of the type Ju {kpR ),w ith R > 0 and u being an integer. T he w avenum ber k is generally a complex num ber: k = kT — j k j (5.139) A n integral over a sem i-infinite range of th e kind (5.138) is called a Sommerfeld-like integral. A lthough the function / has no singularities in kp and is finite for k p —> oo, perform ing the integration of (5.138) presents some problem s due to the presence of two branch points a t kp = ± k — ±A+ =F j kj. A sim ple way to avoid this issue is to tra n sfo rm th e integral I using th e change 138 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. a Im[kz} Figure 5.7: Integration p a th of I in th e complex fc,-plane. of variable hp dkp —- - - \J k 2 - k j = k z , = -dkz (5.140) T he integral I th u s becomes I{p,z\p',z') = k 2 ~ k f , p , p ' ) e °kzZ dkz (5.141) and the integration is perform ed in the com plex fcz-plane instead of th e /cp-pla:ne. F igure 5.7 shows the integration p a th 7 , which follows an hyperbole from th e point kz = k = fcr —j kj to infinity as kp increases. Since th e integrand is an analytical function in kz , th e integral I can also be evaluated along the p a th 71 U 72, which corresponds to breaking it into two parts: I { p , z \ ( J , z ' ) = j f { y k 2 - kj; p, p') e jkzZ dkz + J f (y /k 2 - k 2z ] p, p') e jk*z dkz 71 (5.142) 72 On 7 x, kz = a — j k j , w ith a G [0, fcr ], so th a t th e first integral on the right side of (5.142) can be w ritten as I a = J f { \ ] k 2 - k 2] p, p') e~jkzZ dkz = e~kjZ J 71 f { ^ k 2 - {a - j k j ) 2-,p,p') e~jaZ d a 0 139 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.143) while settin g k z = —j/3, w ith /3 G [kr , oo], th e integral on 72 becomes Ib = f { \ / k 2 - % p , p ' ) e jkzZ dkz = j f { J k 2 + P2-,p,p')e 132 d/3 (5.144) kj 72 W hile th e integral I a is relatively sim ple to evaluate num erically, I s can be difficult to com pute for some values of p, p \ and Z. P articu larly for values of Z close or equal to zero, the integrand of I b m ay exhibit an highly oscillatory behaviour and decay extrem ely slowly, rendering its num erical integration very dem anding in term s of com putational tim e a n d m em ory requirem ents. Several m ethods have been proposed [79, 80, 81] to overcome this problem . Here, two of those approaches, the Tanh T ransform ation and th e W eighted-Averages m ethod, will be considered and im plem ented, and their perform ance com pared to th a t of a sim pler num erical integration routine based on the trapeziodal rule. 5 .4 .1 I n te g r a tio n b y T r a p e z o id a l R u le A very sim ple way to in teg rate I b is by using th e com posite trapezoidal rule. D espite its simplicity, it will be shown th a t th is m ethod provides fairly accurate results. Since the integral is over an semi-infinite interval, it first needs to be tru n c a te d as follows: Pmax Ib ~ j J f { \ J k 2 + /32] p , p ' ) e ~ 0 z dp (5.145) kj Choosing a suitable value of j3max can be difficult and tim e-consum ing, and represents one of the problem s of using this procedure. A pplication of th e com posite trapezoidal rule th en yields: I b « j A/3 /1 + 2 X ] fp + /ivp (5.146) p= 2 where fp = (5.147) 140 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 .4 .2 T a n h T r a n s fo r m a tio n T he Tanh T ransform ation (T T ) was first introduced by [81] and has been further investigated by Squire [82, 83], Evans et al. [84], and Singh et al. [85]. It is used to evaluate integrals of the general form b I = J f(x)dx (5.148) a where th e integrand f ( x ) m ay be singular in one or b o th of th e end points x = a, x = b. Let g(t) be a function such th a t g(t ) —>■± oo as t —> ±oo. W ith th e su b stitu tio n x = i (b + a) + i (b - a) ta n h [g{t)] z z (5.149) the integral (5.148) m ay be w ritten b I I a 1 \ ( b + a) + \ ( b ~ a) ta n h [g(t)] sech2[g(t)]g'(t) dx (5.150) W ith a suitable choice of g(t), the sech2[g(t)] factor falls off extrem ely rapidly as t —> ± o o and controls th e behaviour of the transform ed integrand even when th e singular values of f ( x ) a t x = a and x = b are being approached. T he integral (5.148) th e n can be evaluated by th e m idpoint rule by setting t = tk = k h , k = 0, ± 1, ± 2 , ± 3 , . . . (5.151) and tru n catin g the infinite sum m ation a t k = ± N . T his yields I = 2 h ( b — a) N £ -2 g{kh) + e - 2gM p 3>(<k h ) (5-152) where 1 i i _ e-2g(kh) x k = x{t k ) = - ( 6 + 0) + - ( 6 - o ) 1 + e_ 2g(^ (5.153) T he selection of grid spacing h and num ber of term s N depends on th e particu lar function g(t) used. A choice th a t has shown a very high level of perform ance [84], and therefore has been used in the present d issertation work, is 141 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. g(t) = sinh(t) . (5.154) For th is choice, th e optim al grid spacing for a given N is [84]; L — ln(L —In L) L = In (7r2iV) (5.155) T he algorithm th a t perform s the evaluation is iterative. It sta rts w ith an a rb itra ry initial value of N and a corresponding value of grid spacing h found using (5.155). T he initial value of the integral is calculated based on (5.152)-(5.153). T he value of N is then increased, usually logarithm ically, a n d new values of h = and I = /W are com puted. T h e procedure continues until the difference betw een th e results at two consecutive steps, i.e., j { k ) _ j(A-l) i= W) (5.156) reaches a required value. 5 .4 .3 W e ig h te d -A v e r a g e s M e th o d T he W eighted-Averages (WA) m ethod is an extrap o latio n m ethod for accelerating th e convergence of Sommerfeld-like integrals. A dditional general inform ation ab o u t extrapolation m ethods can be found in [79], while th e WA procedure is discussed in d etail in [80]. W hen using ex trap o latio n m ethods, the integrals are evaluated as sums of a series of p a rtia l integrals over finite subintervals whose speed of convergence is increased th rough an e x trap o latio n procedure. Let OO be the integral th a t needs to be evaluated, and its integrand f(/3) have, for (3 —> 00, an asym ptotic form (5.158) 142 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where p(P) is a 2g-periodic function such th a t p(P) = ~P(P + q) (5.159) T he integral I can be evaluated using a n “integration-then-sum m ation” procedure, in which I is expressed as an infinite sum of p a rtial integrals over finite subintervals. Defining the m -th partial sum as m Sm = £ [ f ( P) d p , (5.160) <=0 A -i th e integral I can be regarded as equivalent to the lim it I = lim Sm , (5.161) m —>oo a n d estim ated by a p a rtial sum S m , w ith M chosen large enough so th a t th e rem ainder - 'm — OO J f(P)dP, (5.162) is negligible. However, usually th e sequence { S m} approaches I slowly because r m does not decay rapidly w ith increasing m. A solution to th is problem is to transform {5m } into another sequence {S ^ } th a t converges rapidly to I. Such transform ation is said to “accelerate”the convergence and is referred to as an extrap o latio n m ethod since the underlying principle is to ob tain an improved estim ate from th e sequence of approxim ated values. In particular, th e W eighted-Averages m ethod employs weighted m eans of consecutive p a rtial sums, w ith weights selected based on rem ainder estim ates. T his tran sfo rm atio n is linear and defined by 5” = S m "h Vm Sm+l i + ^ /r i eo\ ( 5 ' 1 6 3 ) where rim = r m+ 1 143 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.164) T he tran sfo rm atio n is applied repeatedly, leading to a recursive scheme where th e m -th p a rtia l (k) sum s at the A;-th step Sm are used to find the p a rtial sums a t th e (k + l)- th step, s S + 1 , = S m +i <«,(*) b n Sq(k) _ (5 165) 1 +Vm (k) and rjm has th e following expression: V(m } = e9C + (5-166) T he algorithm sta rts by choosing a num ber M of subintervals and com puting th e integrals over them . Then, th e in itial sequence {Sm'1} is set equal to th e sequence {5m} of p a rtial sum s (5.163) of th e previously com puted m = 1,2, • • •, M sub-integrals. T h e iterative com putation of the sequences {Sm}} for k = 1,2, ■• •, M using (5.165) and (5.166) follows. Every sequence { S ^ } has one less term th a n the sequence { S m ~ 1'>}. Therefore, a t th e M -th step the sequence {Sm}} only has one term , which represents th e best estim ate for the integral (5.157), i.e., I ~ S ^ f \ 5 .4 .4 V a lid a tio n o f I n te g r a tio n M e th o d s A simple case is considered in this section in order to validate th e im plem entation of the Tanh T ransform ation an d W eighted-Averages m ethods and to investigate th eir performance. T he results are also com pared w ith those obtained by the com posite trapezoidal rule. T he following identity [71] is used for validation e —j k ( R 2 + Z 2) VR2 / + Z2 z e ~ jj ki v oozZ Jo(kpR) — —------- k p d k p , R > 0, Z> 0 (5.167) where k — kT — j k j , k oz = ^ k 2 — k 2p (5.168) T he integral on th e right side is a Sommerfeld-like integral th a t can be evaluated over the two p ath s 7 i and 72 as in (5.142). Hence, the identity (5.167) can be w ritten as J r Ib = 1— e ~ j k ( R 2 + z 2) M R ^ k * + f 3 i ) e - P z dS ■ Ia kj 144 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.169) where I A = - j e ~ ki z j J0 R y / k P - i a - j k j ) * e i aZ da (5.170) can be easily num erically integrated, while I b is the slowly converging integral th a t will be evaluated using th e m ethods discussed earlier. In th e integration w ith the com posite trapezoidal rule, (3max is chosen as any value where 3maxZ < \ M k R ) \ 1000 J o { R ^ k 2 + (32m ax) (5.171) Using th e asym ptotic form of the Bessel function J q, this inequality corresponds to -a™ z < \ M m (5.172) 1000 7T p /3 n Once (3-max is found, a num ber N p of po in ts is selected such th a t M > 10 p (3max "F kj (5.173) 7T which assures th a t there are a t least 20 subintervals of integration for each period of oscillation Tp 2tt — (5.174) , of the Bessel function. In order to use the Tanh T ransform ation, th e following change of variable is made: x = e P, d(3 = — dx (5.175) T his leads to write I b as e IB = J j Jq R y J k 2 + {ln x )2 x z 1 dx (5.176) which is in the form (5.148), w ith a = 0, b = e hi , and f{x) - Jo R ^ J k 2 + (l n z ) 2 .z - 1 145 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.177) Finally, th e W eighted-Averages m ethod has been applied to th e evaluation of the integral I s Its integrand has an asym ptotic form m ~ \ f ^ p cos( R l i - i ) e 2/5 • (5-178) therefore com parison w ith expressions (5.158) and (5.159) yields: C = Z, « = I » a = y (5.179) I b has been com puted for several values of Z and R, using all th e integration m ethods pre viously discussed. A value k = 20 — 3 j has been chosen, which approxim ately corresponds to the wavenum ber for a n homogeneous m edium w ith relative p e rm ittiv ity ey = 10 — 3 j . T he case R = 0 has not been considered because I b reduces to a very sim ple form th a t can be integrated analytically. In addition, for Z < 0.001 the in teg ratio n has not been perform ed by the com posite trapezoidal rule because it required too m any points. z R Ib 1.000 2.00 - 0.718 + 0.737; tT T *B I'C BT - 0 .7 1 8 + 0.737; —0.720 + 0.740/ 0.010 + 0.01%' < 1 0 -2 < 1 0~ 2 0.40 0.010 + 0.019/ 0.010 + 0.019/ 1.000 0.08 0.022 + 0.007; 0.022 + 0.007; 0.022 + 0.007; 0.022 + 0.007; 0.100 2.00 -3 6 .4 2 4 + 6.905/ - 36.424 + 6.905/ -3 6 .4 0 1 + 6.897; -3 6 .4 1 6 + 6.914; 0.100 0.40 - 0 .2 0 7 + 1.546/ - 0 .2 0 7 + 1 .5 4 6 / - 0.208 + 1 .5 4 6 / - 0.206 + 1 .5 4 7 ; < 10- 2 < 0.100 0.08 1.908 + 0.833; 1.908 + 0.833/ 1.905 + 0.834; 1.915 + 0.836/ 0.010 2.00 - 5 3 .3 7 8 + 3.211; - 5 3 .3 7 8 + 3.211; - 5 3 .3 5 8 + 3.204; -5 3 .5 3 8 + 2.793; 0.010 0.40 - 1 .2 6 5 + 2.528 j - 1 .2 6 5 + 2.528/ - 1 .2 5 0 + 2.527; 0.010 0.08 - 0 .4 3 6 + 2.346/ - 0.436 + 2.346/ - 0 .5 5 9 + 2.342/ 0.001 2.00 - 5 5 .3 9 8 + 2.570/ - 55.398 + 2.570/ - 5 5 .3 6 1 + 2.543/ - < 0.001 0.40 —1.447 + 2.635/ — 1.447 + 2.635/ - 1 .3 5 8 + 2.621; - < <N 1 O i—I 1.000 0.010 + 0.019/ 0.01 < 10-2 «TT ecr Tw a TTt Tct < 10~2 0.27 1.0 1.1 0.2 1 0 -2 0.24 0.5 0.7 0.2 0.5 0.6 0.1 < N 1 O t-H V fW A - 0.718 + 0.738/ 0.28 0.07 0.03 1.4 3.7 0.8 0.04 0.13 0.7 1.3 0.2 0.11 0.40 0.7 1.1 0.2 0.04 0.84 2.1 18.1 0.5 10~2 - 1 .2 1 4 + 2.527; < 1 0 -2 0.53 1.80 1.6 18.0 0.5 —0.421 + 2.34% < 0.62 1.3 21.2 0.6 5.17 10“ 2 0.08 - 3.8 22.7 - - 3.4 34.6 - 3.01 0.001 0.08 - 1 .5 9 5 + 2.588 j - 1.595 + 2.588/ - 2 .5 4 7 + 2.558/ - < 10~2 31.33 - 4.0 34.7 0.000 2.00 —55.626 + 2.495/ - 5 5 .6 2 6 + 2.495/ - 55.579 + 2.460/ - < 1 0 -2 0.10 - 7.5 26.8 - 0.000 0.40 - 1 .4 6 9 + 2.647; - 1.469 + 2.647; — 1.355 + 2.629/ - < 1 0 -2 3.80 - 7.7 40.0 - 0.000 0.08 —1.740 + 2.616/ - 1 .7 4 0 + 2.616/ - 2 .9 2 4 + 2.577; - < 10~2 37.72 - 14.3 44.9 - o (M 1 1 O H < Table 5.1: Results of the evaluation of I b using the Weighted-Averages (WA) method, Tanh Transformation (TT), and Composite Trapezoidal (CT) integration, for various values of Z, R, and k = 18 —6j. I b is the exact value, e is the absolute accuracy in %, and T is the computational time in sec. T he results are sum m arized in Table 5.1. For each m ethod, th e estim ated value I s of the 146 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. integral is given, together w ith the absolute accuracy e = Ib ~ Ib Ib (5.180) in %, and th eir com putational tim e T in sec. T he exact value of I b calculated from the right side of eq u ation (5.169). T h e following considerations can be m ade. T he W eighted-Averages m ethod shows the best accuracy, followed by the Tanh T ransform ation and the Com posite Trapezoidal rule integration. T he C om posite Trapezoidal rule integration is faster but for sm all values of Z cannot be used due to the large num ber of points needed. T he Tanh T ransform ation perform ance degrades considerably for sm all values of R and Z , w ith th e error and th e com putational tim e b o th increasing. Figure 5.8 b e tte r illustrates this issue. T he plot shows the absolute error as a function of the num ber of integration points N , for Z = 0 and four different values of R . It is easy to notice th a t as R decreases, th e error tends to reach a constant value th a t does not decrease even w hen N is increased. Therefore, the Tanh T ransform ation m ethod does not seem to be of any p articular value for the num erical evaluation of th e integrals here considered. Indeed the algorithm is outperform ed by th e Com posite Trapezoidal rule in term s of speed, and - for sm all values of R and Z - by the W eighted-Averages in term s of accuracy. 5.5 Validation T his section describes a few results of th e application of the num erical m ethod derived earlier in this chapter. The goal is to validate the procedure and prove th a t it provides correct results. As done in the previous chapters, first th e case of P E C bodies is considered, followed by th e case of homogeneous dielectric scatterers. 147 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. VP ' ' k- o i_ CD D ■4< — • a ocn .a cc IQ- 50 100 200 500 1000 2000 N Figure 5.8: A bsolute error [%} for th e Tanh T ransform ation m ethod vs. num ber of integration points N . 5 .5 .1 P e r f e c t ly c o n d u c tin g b o d y o f r e v o lu tio n In the case of perfectly conducting bodies of revolution, there is only one surface integral equation to solve, i.e., the one for th e exterior problem . In order to test th e correctness of th e solution, the sim pler situation of a perfectly conducting half-space is considered first, and the surface integral equations are solved for a P E C open-ended cylinder located on th e half-space interface, as shown on the left of Figure 5.9. T his configuration is equivalent to th e one shown on the right in the same figure. Here, the lower half-space is rem oved and its effect is tak en into account by introducing the image of the P E C open-ended cylinder for z < 0. Therefore, th e equivalent configuration consists of an PE C open-ended cylinder twice as long, centered at th e origin, and excited by th e com bination of the incident field and its ” im age” . T h e currents on th e surface of the open-ended cylinder in the half-space problem m ust be th e sam e as those on th e surface of th e equivalent open-ended cylinder 148 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PEC -L Figure 5.9: S cattering from a vertical P E C open-ended cylinder on a P E C half-space (left), and equivalent problem (right) in free space. for z > 0. T his can be seen in Figure 5.10, left side, where th e currents on an open-ended cylinder of length L — 0.5A and radius a = 0.1A are p lo tted as a function of z / A, together w ith the currents on th e open-ended cylinder of length L = A and radius a = 0.1A in the equivalent configuration. T h e incident angle is Qi = 60°. T he two curves m atch very well, except for points very close to th e interface. The scatterin g coefficients of th e two configurations should also be the sam e for scattering angles in th e u p p e r half-space. O n the right side of Figure 5.10 the norm alized b istatic scattering coefficient is p lo tted for scattering angles Qs betw een 0° and 90°. Also in this case, there is good agreement betw een th e half-space an d th e equivalent im age solutions. To validate the results for P E C bodies of revolution over a dielectric half-space, a case studied by Abdelmageed an d M ichalski [86] is considered. T h e body is a perfectly conducting cylinder of length L — A and k^a = 1.0 located at a distance d = 0.2A above a homogeneous, lossy half-space 149 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 — ■ — t h a lf- s p a c e 4.5 - - r - x e q u iv a le n t — * — 4 h a lf- s p a c e - * - ( ) > e q u iv a le n t 3.5 i 25 /0 -1 0 -15, -20 -25 0.5 -30, -0 .5 —1*—HH half-space HH equivalent - e - VV half-space VV equivalent -0 .2 5 0.25 0.5 20 30 40 50 6( scattering angle 0g [deg] 70 90 Figure 5.10: C om parison betw een P E C open-ended cylinder of length L = 0.5A and radius a = 0.1 A on a P E C half-space, an d equivalent image problem , for incident angle 6t = 60°. Left: surface electric currents vs. z /X . Right: norm alized b ista tic scattering coefficient vs. 6S. w ith relative dielectric constant 16—jlQ . T he incident wave is coming from an angle 6t = 0°. Figure 5.11 shows the com parison betw een th e present half-space m ethod (BOR) and th e A bdelm ageed and M ichalski results (AM ), which axe in good agreem ent w ith each other. 5 .5 .2 D ie le c t r ic b o d y o f r e v o lu tio n In order th e test the validity of the half-space procedure, th e case of a dielectric cylinder on a P E C half-space is considered. T h e geom etry of the problem , illu strated on the left of Figure 5.12, is sim ilar to th e one for th e P E C open-ended cylinder on a P E C half-space, except th a t a homogeneous, lossy cylinder is now considered. Therefore, the sam e considerations m ade for the P E C open-ended cylinder case apply. T he cylinder has length L = 0.6A, radius a — 0.1A and a relative dielectric constant 4 —j , and its b o tto m end is on th e interface. C om parisons between equivalent surface currents and between b ista tic scattering coefficients are given in Figures 5.13-5.14, and show a good agreem ent between half-space num erical solution a n d equivalent configuration num erical solution. 150 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.5 X 0.5 -0-0- e -9 -g rg ^ g r 0.2 0.4 0.6 VX 0.8 Figure 5.11: Surface electric currents vs. t/X on a perfectly conducting cylinder of length L — A and koa = 1.0 located a t a distance d = 0.2A above a hom ogeneous, lossy half-space w ith relative dielectric constant 16 —j l § . Incident angle is = 0°. 5.6 A pplication to scattering from vegetation T he m ethod derived earlier in this chapter can be used to m odel vegetation more realistically. In particular, in this section th e problem of a vertical tree tru n k over a fiat ground is considered. A comm on approach is to m odel the tru n k as a finite-length lossy dielectric cylinder of circular cross section, and the ground as a flat interface separating th e free space from a lossy dielectric m edium representing the soil. T his problem can be easily studied by solving th e surface integral equations for the half-space problem w ith the b o d y of revolution being the dielectric cylinder. T his num erical solution can be used to evaluate th e accuracy of approxim ate analytical m ethods applied to the same problem . A n analytical analy tical approxim ation is considered here, bases upon two assum ptions, i.e., th a t 1. the length of th e cylinder is large com pared to its radius; 151 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PEC -L Figure 5.12: Scattering from a vertical dielectric cylinder over a P E C half-space (left), and equiv alent problem (right) in free space. 2. th e cylinder and th e region on th e ground w here induced currents exist are in th e radiation zone of each other. T he first assum ption allows th e use of the finite-length cylinder m odel discussed in C hapter 4. In addition, assum ing th a t the cylinder and th e th e region on th e ground where induced currents exist are in the far-field region of each other, th e effect of th e ground plane interface is taken into account only by considering the contributions from the m irror image. U nder this assum ption, the incident field illum inating the cylinder is a plane wave of th e form E « = q + (5.181) where the reflection coefficients R q have been derived in Section 5.3.3. T h e to ta l scattering ampli152 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.5 400 1 half-space equivalent surface electric current - e- half-space equivalent i)i half-space d equivalent t t equivalent - e- 350 $ half-space —>*— (j>equivalent — tz x 300 S 250 » 200 " 150 •5 100 0.5 0.2 0.4 0.3 0.5 0.6 0.7 0.2 0.3 0.4 0.5 0.7 t/X Figure 5.13: C om parison betw een dielectric cylinder of length L = 0.6A, radius a = 0.1A and a relative dielectric constant 4 —j on a P E C half-space, and equivalent image problem , for incident angle 0* = 45°. E quivalent surface electric currents vs. z /X . R ight: norm alized bistatic scattering coefficient vs. 9S. /(it a2) [dBm] half-space — equivalent E C TQ J b O' -4 -4 half-space equivalent -10 -1 0 scattering angle 9 [deg] 10 40 ,50 rJ 6( scattering angle 8 [deg] Figure 5.14: C om parison betw een dielectric cylinder of len g th L = 0.6A, radius a — 0.1A and a relative dielectric constant 4 — j on a P E C half-space, and equivalent image problem , for incident angle 0, = 45°. N orm alized b ista tic scattering coefficient vs. 9S. 153 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. e, direct direct-reflected A - A_ e, reflected-direct reflected-reflected tudes fpq of the configuration are the sum of four com ponents: = I > « (£ ,+ ,£ - ) (5.182) 1= 1 As illustrated in Figure 5.6, these com ponents are: 1. direct scattering from th e cylinder, e£ W > : k O = f $ l){k + ,k O (5.183) 2. scattering from th e cylinder, followed by reflection from th e ground e g )(k + k -) = R P(9S) 4 f ) ( k + , k + ) e - ^ 154 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.184) 3. reflection from th e ground, followed by scattering from th e cylinder eg ( k + , k n = / j f ) ( k s- , k r ) R q(ei ) e ~ ^ (5.185) 4. reflection from th e ground, followed by scattering from th e cylinder, and by reflection from the ground eg ) ( k + k - ) = R pids) / i f }(ks- , k + ) (5.186) Rqifii) In expressions (5.183)-(5.186), fpqjl> are th e scattering am plitudes of th e cylinder in free space. T h e u nit propagation vectors of the incident and scattered plane wave are: k^ = —sin 9i cos (pi x —sin 9i sin (pi y ± cosOi i (5.187) k^ = sin 0 s cos ^ i c + sin 0 s sin(ps y ± cosOs z (5.188) T he additional phase shifts i/'i and ips have been introduced to account for th e difference in length between the direct p a th and th e p a th reflected from th e ground. T hey depend on the incidence and scattering angles 6i and 0S, an d are given by ipi = koL cosdi (5.189) ips = k o L c o s 9 s (5.190) T h e bistatic scattering coefficient can be found from th e scattering am plitudes as follows: f W k ^ . k " ) = 4 7r|/pg( k + , k - ) |2 (5.191) R e s u lts a n d c o m p a r is o n s In this subsection, results of th e application of th e previously described approxim ation on th ree cylinders of different dim ensions will be given, a n d com pared to the num erical solutions. An interm ediate approach will also be considered, in which th e free space scattering am plitudes fpqy^ of the cylinders are determ in ed num erically by th e M O M -B O R approach rath er th a n by analytical approxim ation. 155 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T hree hom ogeneous, lossy dielectric cylinders are considered. All th e cylinders have th e same relative dielectric constant 4 — j, and the relative dielectric constant of th e ground is 10 — j 5 for all cases. These values are th e sam e used by Lin and Sarabandi in [87]. Two values have been chosen for the cylinder length, i.e., L = 2.5A and L — 4.0A, and two values for the radius, i.e., a = 0.025A and a = 0.25A. T he results are illustrated in Figures 5.15-5.26, for two incident angles, i.e., 9i = 30° and 9t = 60°. As m entioned above, three separate solutions are plotted. In th e legend and in the following, th ey are referred to as: M O M half-space indicates the result of the application of th e half-space m ethod of m om ents algorithm , considered as the exact reference solution; PO - f approx is th e m odel described by (5.181)-(5.186), w ith f p q ^ given by the approxim ate analytical m odel outlined in Section 4.4.1; M O M + approx is th e m odel described by (5.181)-(5.186), w ith fp q ^ determ ined num erically by th e m ethod of m om ents. In particular, the results in Figures 5.15-5.18 are for th e th in n e r (a = 0.025A), b u t shorter (L = 2.5A) cylinder. Especially when considering the m ain scatterin g lobe, the PO + approx solution is in good agreem ent w ith th e M OM half-space results in th e /i/i-polarization, b u t there are some discrepancies in th e uu-polarization. T hese differences alm ost disappear in the M OM + approx solution. If the cylinder length is increased from L = 2.5A to L = 4.0A, as in Figures 5.19-5.22, the agreement of the P O + approx w ith th e MOM half-space results substantially improves for hhpolarization, even in th e side lobes, while for w -p o la riz a tio n degrades a t sm aller incident angles, such as 9i = 30°. If the length of th e cylinder is kept a t L = 2.5A an d its radius increased by a factor 10 to a = 0.25A, there is no m ore agreem ent betw een th e P O -t- approx a n d th e M OM half-space solutions, 156 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. to S e £ -20 -3 0 MOM half-space o PO + approx - « - MOM + approx -4 0 scattering angle 9 [deg] 70 Figure 5.15: N orm alized scattering coefficient cr^h vs. 9S. C om parison betw een analytical approxi m ation and num erical solution for a dielectric cylinder of length L = 2.5A and radius a = 0.025A. Relative dielectric constant is 4 —j for the cylinder, and 10 —jb for the ground. Incident angle is 9i = 30°. I -1 0 -a CVJ <s S o'g -20 -3 0 MOM half-space o PO + approx - * - MOM + approx -4 0 10 40 50 60 scattering angle 9 [deg] 70 Figure 5.16: Norm alized scattering coefficient a vv vs. 6S. C om parison betw een analytical approxi m ation and num erical solution for a dielectric cylinder of length L = 2.5A and radius a = 0.025A. Relative dielectric constant is 4 —j for th e cylinder, a n d 10 — j5 for the ground. Incident angle is 9i = 30°. 157 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I -10' ;o to -3 0 — MOM half-space — PO + approx —MOM + approx -4 0 SO 40 50 60 scattering angle 0s [deg] Figure 5.17: N orm alized scattering coefficient c r^ vs. 6S. C om parison betw een analytical approxi m ation and num erical solution for a dielectric cylinder of length L = 2.5A and radius a — 0.025A. Relative dielectric constant is 4 —j for the cylinder, and 10 —j 5 for th e ground. Incident angle is Qi = 60°. MOM half-space o— PO + approx - « - MOM + approx & -10 73 S -20 -3 0 -4 0 10 40 50 6< scattering angle 0 [deg] 70 Figure 5.18: Norm alized scattering coefficient avv vs. 0S. C om parison betw een analytical approxi m ation and num erical solution for a dielectric cylinder of length L = 2.5A and radius a = 0.025A. Relative dielectric constant is 4 — j for th e cylinder, and 10 —j5 for th e ground. Incident angle is Qi = 60°. 158 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. MOM half-space —e— PO + approx - « - MOM + approx CM o -20 -3 0 -4 0 10 40 50 6< scattering angle 9s [deg] 70 Figure 5.19: N orm alized scattering coefficient a^h vs. 9S. C om parison between analytical approxi m ation and num erical solution for a dielectric cylinder of length L = 4.0A and radius a = 0.025A. R elative dielectric constant is 4 —j for the cylinder, and 10 —j 5 for th e ground. Incident angle is di = 30°. — MOM half-space — PO + approx —MOM + approx E co ■o O' -2 0 -3 0 0 10 20 30 40 50 60 scattering angle 9 [deg] 70 80 90 Figure 5.20: N orm alized scattering coefficient a vv vs. 9S. C om parison betw een analytical approxi m ation and num erical solution for a dielectric cylinder of length L = 4.0A and radius a = 0.025A. Relative dielectric constant is 4 —j for the cylinder, an d 10 — y*5 for the ground. Incident angle is Qi = 30°. 159 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. MOM half-space o PO + approx - * - MOM ■+■approx E m -o "co -10 e b -20 -30. 0 10 20 40 50 30 60 scattering angle 9 [deg] 70 80 90 Figure 5.21: N orm alized scattering coefficient v s . 9s. C om parison betw een analytical approxi m ation and num erical solution for a dielectric cylinder of length L = 4.0A and radius a = 0.025A. R elative dielectric constant is 4 — j for the cylinder, and 10 —j 5 for th e ground. Incident angle is 9i = 60°. even in th e m ain lobe, as can be seen in Figures 5.23-5.26. In this case, the results of th e P O -tapprox become acceptable only for hh-polarization a t higher incident angles such as 9{ = 60° in Figure 5.25. T he fact th a t, in m ost cases here considered, th e M OM + approx shows a good agreem ent w ith th e M OM half-space solution suggests th a t th e sim plification based on expressions (5.181)-(5.186) is valid, and th a t th e differences are likely due to th e P O analytical approxim ation of the scattering am plitudes f p q l\ 5.7 Conclusions A num erical procedure has been developed to tre a t th e scattering from P E C and homogeneous dielectric objects located in free space above a P E C or hom ogeneous dielectric semi-infinite m edium . T he procedure has been validated, and subsequently applied to investigate a simplified analytical m odel for the scattering from a vertical dielectric cylinder over a dielectric ground. 160 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. — MOM half-space — PO + approx —MOM + approx E m -o c\j -10 O' -20 -3 0 0 10 20 30 40 50 60 scattering angle 9 [deg] 70 80 SO Figure 5.22: N orm alized scattering coefficient avv vs. Qs. C om parison betw een analytical approxi m ation and num erical solution for a dielectric cylinder of length L = 4.0A and radius a = 0.025A. Relative dielectric constant is 4 —j for th e cylinder, and 10 —jb for th e ground. Incident angle is Qi = 60°. T he com parison betw een th e results of such analytical approxim ation and those of the num erical m ethod th a t m odels th e half-space exactly indicates th a t the sim plified analytical model can be successfully used to take into account th e effect of the ground in th e scattering from the elem ents of a vegetation layer. 161 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. E m T J K -10 -20 — MOM half-space — PO + approx —MOM + approx -3 0 0 10 20 30 60 40 50 scattering angle 9 [deg] 70 80 90 Figure 5.23: Norm alized scattering coefficient ahh, vs. 9a. C om parison between analytical approx im ation and num erical solution for a dielectric cylinder of length L = 2.5A and radius a = 0.25A. Relative dielectric constant is 4 — j for th e cylinder, and 10 —j 5 for the ground. Incident angle is 0i = 30°. E m -a -20 MOM half-space —e— PO + approx - ■» - MOM + approx -3 0 0 10 20 30 40 50 60 scattering angle 9 [deg] 70 80 90 Figure 5.24: Norm alized scattering coefficient a vv vs. 9S. C om parison betw een analytical approx im ation and num erical solution for a dielectric cylinder of length L = 2.5A and radius a = 0.25A. Relative dielectric constant is 4 —j for the cylinder, an d 10 —j 5 for the ground. Incident angle is Qi = 30°. 162 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. E m T3 -20 — MOM half-space — PO + approx —MOM + approx -3 0 0 10 20 40 30 60 .50 scattering angle 80 [deg] 70 80 90 Figure 5.25: Norm alized scattering coefficient ahh vs. 9S. C om parison betw een analytical approx im ation and num erical solution for a dielectric cylinder of length L = 2.5A and radius a = 0.25A. R elative dielectric constant is 4 - j for th e cylinder, and 10 —j 5 for th e ground. Incident angle is Oi = 60°. MOM half-space —e— PO + approx - ■» - MOM + approx E co -o “a -1 0 •£. < e -20 -3 0 0 10 20 30 40 50 60 scattering angle 9 [deg] 70 80 90 Figure 5.26: Norm alized scattering coefficient a vv vs. Qs. C om parison betw een analytical approx im ation and num erical solution for a dielectric cylinder of length L = 2.5A and radius a = 0.25A. Relative dielectric constant is 4 —j for th e cylinder, and 10 — j5 for th e ground. Incident angle is 9i = 60°. 163 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 6 C onclusions In this dissertation, the problem of solving surface integral equations for electrom agnetic scattering applications has been studied. T his investigation has focused on th e scatterin g properties of objects used 'to represent individual constituents of vegetation layer m odels. T he approach considered for solving the electrom agnetic scattering problem is num erical an d based on a surface integral equation. N um erical procedures have been developed and applied to stu d y bodies of increasing complexity. Once these num erical m ethods have been validated, th ey have been used to investigate th e accuracy of approxim ate analytical solutions. S tartin g for M axwell’s equation, a pair of tan gential electric field integral equations, valid on the boundary surface of th e body of interest, is derived for the case of bodies located in an unbounded medium. Following this in tro d u ctio n of th e tangential electric field integral equations, the m ethod of mo m ents is used to solve these integral equations in th e case of tw o-dim ensional bodies. R esults and com parison w ith th e exact theoretical expressions are given for th e case of circular perfectly con ducting and lossy hom ogeneous dielectric cylinders. T he good agreem ent w ith the theory confirms the validity of the surface electric field integral equations form ulation. Next, th e case of bodies w ith ro tatio n al sym m etry is considered. Such representation is very useful because m any vegetation elem ents exhibit axial sym m etry, a t least in an ideal case, an d can 164 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. be therefore be m odeled as bodies of revolution. A num erical solution to the tan gential electric field integral equations for this class of objects is presented. T his solution is based on the m eth o d of m om ents an d exploits th e ro tatio n al sym m etry to reduce the surface integral equations to integral equations along th e generating arc of the body of revolution. Results are given for perfectly conducting and lossy homogeneous dielectric objects of several different shapes. C om parisons are m ade w ith the literatu re and, w hen available, w ith the exact solutions, and show th e validity of th e algorithm . T he body of revolution num erical procedure is th en used to stu d y th e accuracy of two analytical models for cylindrical structures, such as tree tru n k s, based on a physical optics approxim ation. Plane wave scattering by dielectric cylinders of finite length and circular cross section is exam ined. T his finite cylinder m odel is based on th e assum ption th a t th e currents inside the cylinder are the sam e as if th e cylinder were infinite. C ylinders w ith a radius th a t varies linearly along the cylinder length - hereafter referred to as tap ered cylinders - are also considered. T apered cylinders are m odelled by a num ber of coaxial finite cylinders stacked on top of each other. B oth these types of objects can be used to m odel tree tru n k s, branches, and stalks. T he results show a good agreem ent of th e analytical approxim ation w ith the num erical solution, here being considered as the exact reference, for rem ote sensing applications where th e low level side-lobe scattering is neglected. For b o th constant-radius and tap ered cylindrical stru ctu res, a good agreem ent w ith th e num erical solution is found in th e region of th e m ain scattering lobe, which is the one of interest w hen considering com plex m edia such as vegetation canopies. However, the accuracy of th e approxim ate solutions decreases as th e angle of the incident wave approaches the end-on angle, and is generally b e tte r for hfi-polarization th a n for un-polarization. W hile the physical optics solution for the finite cylinder works very well and is readily and safely usable in vegetation models, th e tap ered cylinder approxim ation requires m ore a tten tio n in the choice of the num ber N s of cylinders, in order to reduce th e error due to th e displacem ent of the specular 165 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. scattering lobe. T he results also prove th a t when th e tap e r is significant, the single cylinder approxim ation is not adequate for a tap e red cylinder. A brief discussion of th e com putational tim es for th e two approaches is also provided, which shows th a t the com putational cost of th e num erical algorithm increases quadratically w ith the cylinder length, while it rem ains alm ost constant for th e approxim ate solutions. T he im p o rtan t conclusion of th is stu d y is therefore th e following. T he analytical m odels should be preferred to th e m ethod of m om ents approach because they are accurate enough an d have shorter com putation tim e, while the num erical com putation is too heavy for practical purposes but represents a powerful tool to validate analytical models. Finally, the num erical m ethod for tre a tin g axisym m etric bodies in an unbounded m edium has been m odified to stu d y th e sam e type of bodies located above a sem i-infinite m edium . T his con figuration is referred to as th e half-space problem . T he tan g en tial electric field integral equations derived for an object in an unbounded m edium are reform ulated to take into account th e presence of the second m edium . T his derivation implies th e definition of a n equivalent infinite m edium whose properties are described by half-space G reen’s functions. T he num erical approach is based on a series expansion of th e dyadic G reen’s functions in term s of cylindrical wave functions. Such representation of the dyadic G reen’s functions allows a sim plification of tangential electric field integral equations for th e half-space problem in th e case of bodies of revolution. However, this approach requires the evaluation of slowly converging Som merfeld-like integrals, and section is ded icated to the analysis a n d discussion of some m ethods to estim ate these kinds of integral correctly and efficiently. Following th e description of the num erical m ethod to solve the surface equations for the half-space problem , some results are given and discussed. As an exam ple of application of th e half-space body of revolution algorithm , the case of sc at tering from a vertical tree tru n k located on a flat ground is studied. A simplified analytical m odel is introduced, which uses a physical optics approxim ation an d also neglects the near-field interac166 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. tions betw een the tree tru n k located on the ground. The num erical procedure is applied to the sam e configuration, and com parisons are m ade w ith the simplified m odel. T he results analytical approach are found to m atch those given by the num erical m ethod around the m ain scattering direction. M ost of th e differences betw een analytical and num erical solution can be a ttrib u te d to th e use of the physical optics approxim ation, while the near-field interactions are very small. In sum m ary, the end result of th is dissertation has been th e analysis and development of num er ical procedures th a t can be applied to stu d y the electrom agnetic scatterin g from individual vegeta tion elem ents. W hile th eir use for large bodies can be in some cases lim ited by com putational costs, they nevertheless co n stitu te a powerful tool to evaluate approxim ate analytical solutions th a t are faster and more com m only used in electrom agnetic m odeling of vegetation. T hey axe also essential if one wants to study situations in w hich approxim ate analytical m odels lose their validity, such as in th e near-field. Some exam ples of application of these num erical m ethods have been provided in this dissertation, especially for electrom agnetic scattering from cylindrical structures such as tree tru n k s and branches, a n d plant stalks. In particular, th e ap p lication of th e half-space num erical m ethod has shown prom ising results, b u t more work is needed b o th to stu d y its full poten tial and to improve its com putational efficiency. 167 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. B ibliography [1] A. 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[77] R. E. Collin, A n ten n a s and Radiowave propagation. M cGraw-Hill, 1985. [78] J. He, M O M and M L F M A fo r Scattering from Dielectric Target in Layered-M edium E nviron ment. PhD thesis, Duke University, 2000. [79] K. Michalski, “O n th e efficient evaluation of integrals arising in th e somm erfeld half-space p rob lem ,” IE E Proceedings H - Microwaves, A n ten n a s and Propagation, vol. 132, no. 5, pp. 312-317, 1985. 176 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [80] A. M ohsen, “O n th e evaluation of somm erfeld integrals,” IE E Proceedings, vol. 129, no. 4, pp. 177-182, 1982. [81] S. H aber, “T he T a n h rule for num erical integration,” S IA M Journal on Num erical Analysis, vol. 14, pp. 668-685, 1977. [82] W. Squire, “A q u a d ra tu re m ethod for finite intervals,” International Journal for Num erical M ethods in Engineering, vol. 10, pp. 708-712, 1976. [83] W . Squire, “A com m ent on q u a d ra tu re in the presence of end p oint singularities,” International Journal o f Com puter M athem atics, vol. 7, pp. 239-241, 1979. [84] G. Evans, R. Forbes, and J. Hyslop, “T he T a n h tran sfo rm atio n for singular integrals,” In te r national Journal o f Com puter M athem atics, vol. 15, pp. 339-358, 1984. [85] R. Singh and S. Singh, “Efficient evaluation of singular and infinite integrals using the ta n h transform ation,” IE E Proceedings - M icrowaves, A ntennas and Propagation, vol. 141, pp. 464466, dec 1994. [86] A. A bdelm ageed a n d K. M ichalski, “Analysis of EM scatterin g by conducting bodies of revolu tion in layered m edia using the discrete com plex image m eth o d ,” in A ntennas and Propagation Society International Sym posium 1996, Digest, vol. 1, pp. 402-405, June 1995. [87] Y. C. Lin and K. Sarabandi, “E lectrom agnetic scattering m odel for a tree tru n k above a tilte d ground plane,” IE E E Transactions on Geoscience and R em ote Sensing, vol. 33, pp. 1063-1070, 1995. [88] C.-T. Tai, Dyadic Green’s Functions in Electromagnetic Theory. In tex t Educational Publishers, 1900. [89] J. V. Bladel, Electrom agnetic Fields. M cGraw-Hill, 1964. 177 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix A Electrom agnetic fields in unbounded media A .l E lectrom agnetic fields generated by electrom agnetic sources T he M axwell’s equations for a homogeneous, isotropic dielectric m edium w ith p e rm ittiv ity e and perm eability fj. are: V xE = —jc jju H — M (A .l) V x H = jue E + J (A.2) F irst, the case w hen only J ^ 0 and M = 0 is considered, so th a t: V xE = —jwfj,H (A.3) V x H = juje E + J (A.4) In order to solve (A .3),(A .4), an electric vector p o ten tia l A is defined such th a t nH = V x A (A.5) V x (E + j u A ) = 0 (A.6) S ubstitu tio n of (A.5) into (A.3) yields: For this equation to be tru e, it m ust be: E + j'wA = —V $ 178 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (A .7) Once A an d 4> are known, the electric and m agnetic fields can be determ ined from (A.5) and (A .7) as follows: E = - j u A + V4> (A .8) H - —V x A H (A .9) Using (A .9) into (A.4) yields (k = u^fjle): V 2A + A;2 A = V V • A + jfw/je V4> E — fi 3 (A.10) T he vector potential can be chosen such th a t V • A = —j u j g e Q E (A .11) which leads to a sim plification of equation (A.4): V 2A + A;2 A = —fj, J (A .12) O n th e other hand, tak in g th e divergence of ( A .ll) and using M axwell’s equation: V •E = — = e jue (A.13) yields th e following differential equation: V 2$ + fc2 $ = (A. 14) jus T he scalar G reen’s function g( r, r') is th en defined as th e solution of: V 2 g{ r, r') + k2 g{ r, r') = - 5 { r - r') (A.15) It can be shown [88] th at: g(r ’r ' ) = e- j k \ r - r'| 47r|r _ r / |'~ ’ k = 179 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (A.16) Since the vector and scalar p o ten tial are solution of (A. 12) and (A. 14), respectively, th ey can be w ritte n as: A (r) = g f < K r ,r ') J ( r ') d y ' $(*) = I f ~ (A.17) Jv e Jv , V '-•J ( r ') dV' ff(r >r ) — 37we (A .18) 3 V being the volume where th e sources J and M are located. T h e expressions for th e electric a n d m agnetic fields are obtained su b stitu tin g (A.17), (A. 18) in (A.8), (A.9): E ( r) H (r) = - j u g [ g ( r , r ' ) J ( r ,) d V ' + Jv = -jug- = -ju g Jv Jv g{r, r') J ( r ') + ^ g{r,r')l + ^ — V [ g ( r , r ') V '- J ( r ') dV' 3 ue Jv Vg{r, r ') V ' • J ( r ') V s ( r ,r ') V ' dV' J ( r ') dV' (A.19) = V x f g { r ,r ’) J ( r ') d V 1 Jv = = f V x g(r,r') J(r') d V 1 Jv [ [V £r(r,r') x I] • J ( r ') dV' JV (A.20) where I is the unit dyadic. Introducing th e electric an d m agnetic dyadic G reen’s functions as, respectively, G ( r , r ') = s(r,r')I+ ^V j(r,r')V ' (A.21) K(r,r') = V<7(r,r')xl (A.22) the expressions (A.17), (A.18) for th e electric and m agnetic fields can be w ritten as follows: E (r) = H (r) - -ju g [ Jv G ( r , r ' ) • J ( r ') dV' [ K ( r , r ' ) • J ( r ') dV' Jv 180 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (A.23) (A.24) Expressions (A.23), (A.24) are valid if only electric sources J are present. In th e case w hen only m agnetic sources exist, i.e. J = 0 and M / 0 , M axwell’s equations (A .l), (A.2) become: V xE V = x H = -jufx H - M (A .25) jue E (A .26) T h e solution to equations (A.25), (A.26) can be found by duality from th e solution of (A.3), (A.4) w ith the substitutions: E J -> H -> M H M ->■ ->■ -E —J £ —^ j-l fJ* —y £ (A .27) T hus, the expressions for th e electrom agnetic fields generated by m agnetic sources M w hen J = 0 are: E (r) - [ K ( r , v ' ) - M ( r ' ) dA' = (A.28) JA =-ju e H (r) fJ A G ( r , r ' ) • M (r') dA ' (A.29) W hen b o th electric and m agnetic sorces are present, th e electrom agnetic fields can be obtained as superposition of A.23), (A.24) an d (A.28), (A.29): E (r) = —jojjj, [ G( r , r ' ) • J ( r ') dA! — [ K ( r , r ' ) - M ( r ') d A 1 JA H (r) = [ K ( r , r ' ) - J ( r ') d A ' - j u e JA A.2 (A.30) JA [ G(r,r') • M (r ') dA' (A.31) JA G reen’s theorem for a m ultiply connected region Let V be a volume corresponding to a m ultiply connected region bounded by an inner closed surface <Si and by an outer closed surface S 2 , as shown in Figure A .I. T he u n it vector n norm al to the surfaces Si and S2 is pointing into V If P , and Q are any two twice-differentiable vector fields, th en the following relations apply: V • [P x (V x Q)] = (V x P ) ■(V x Q ) - P • [V x (V x Q)] (A.32) V • [Q x (V x P )] - (V x Q) • (V x P ) - Q • [V x (V x P)] (A.33) 181 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. e,p s ,|x Figure A .l: D efinition of volume V and surfaces Si an d S 2 for th e G reen’s theorem . S ubtracting (A.32) from (A.33) yields: V • [Q x (V x P ) - P x (V x Q)] = P • [V x (V x Integrating b o th sides of (A .34) over the volume V yields: Q)] - Q ■[Vx (V x P)] (A .34) [ V ■[Q x (V x P ) - P x (V x Q)1 d V = Jv = [ { P • [V x (V x Q)] - Q ■[V x (V x P )l} dV Jv (A.35) Applying the divergence theorem , the left side of (A.35) S becomes: [ V dQ x(V xP)-Px(V xQ )]dF Jv = = £ [ P x (V x Q ) - Q x (V x P )] • n dS (A.36) S u bstitution of (A.36) into (A.35) gives th e G reen’s theorem : j f [P x (V x Q) - Q x (V x P)] • n d S = = [ {P • [V x (V x Q)] - Q ■[V x (V x P)]} dV Jv 182 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (A.37) A .3 D erivation of expression (2.17) Let V be a volume enclosed by a surface S = Si U S 2 , containing all th e sources J and sources on S, a n d n M , w ith no be the u nit vector norm al to 5 pointing into V as shown in Figure A .I. Using th e G reen’s theorem (A.37) w ith P where a = E ,Q = a g+ , g+( r , = g r ' ) ( r ' , (A.38) r ) is an arbitrary, constant vector, yields: f Jv { • [V' x (V ' x &g+)] - E a = j[ g+ ■[V' x (V ' x dV' = E ) ] } x (V ' x a g+) - a.g+ x (V ' x E E ) ] • n ' dS' (A.39) In equation (A.39) and hereafter, th e prim ed sign is used indicate th e variables of integration. Since there are no sources on th e surface S, then: V' x = - j u g H, E on 5 (A.40) so th a t [ a x (V ' x E ) ] ■ ' = —jug, n ( a x H) • ' = —j u g n • (H x a n ') on 5 (A.41) while vector m anipulations yield [ E x (V ' x a 5 + ) ] • n ' = [ E x Vg+ x ( a ) ] ■ n ' = a ■ [ ( n ' x E ) x V'^+j on S (A.42) Using (A.41)and (A.42), th e right side of equation (A.39) can be w ritte n as: £ [ E x ( V ' x a j + - a s + x (V'xE))] ■ n 'd S ' = = Let g+ — g{ r ' , r ) a • j [ j u g (H x ) g+ + n ( n 'x E ) x V'g+} dS' (A.43) be the scalar G reen’s function, i.e, th e solution of V ' 2 g( r ' , r ) + k 2 g( r ' , r ) = —5( r ' — r ) 183 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (A.44) subject to th e condition th a t g be an outgoing function of r'. V ' 2 g{r', r) = - k 2 g(r', r) - <5(r' - r) T hen (A.45) and V ' x ( V ' x a f f +) = V '[V ' • (a<7+)] - V ,2(a5+) = V'[V'■(a<7+)] - a V V V '[V ' • (ac/+ )] - a.[-k 2 g(v', r) - <J(r' - r)] = (A.46) Using th e wave equation (2.14), hereafter rep eated for convenience, in th e volume V V x ( V ' x E ) = fc2 E - jug, J - V ' x M , in V (A.47) together w ith (A.46), we can w rite th e left side of equation (A.39) as f {E • [V' x (V ' x a 0+)] - a g+ ■[ V x (V ' x E)]} dV' = [ E ■V '[V ' • (a5+)] dV' + Jv Jv - a ■f j _ E ( r ') { - k 2 g+ - S{r ' - r)] + g +[k2 E - jug, J - V ' x M ]} dV ' = = [ E -V '[V -(a 5+ )] d V ' + a- f E( r ') 5 ( r' — r ) d V ' + a . - [ ( j u g J + V ' x M ) g + dV' Jv Jv Jv (A.48) If we now consider th e following identity: V ' • (u F ) = (V ' ■F ) u + F • V 'u u = V'<7+ • a ; = V ' • (ag+), w ith (A.49) F = E (A.50) we can write V ' ■[(a • V'c/+ )E] = (V ' ■E ) V'<?+ • a + E ■[ V V ■(a ff+ )] (A.51) Applying the divergence theorem along w ith (A.52) and expression (A.51), yields (a- V g +) E - h ' d S ' 512 = [ V ' ■[(a • V(gf+ ) E ] dV ' , Jv Su - Sx U S2 184 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (A.52) Now using (A.51) on the right hand side of (A.52) yields -<£ JSi 2 (a •V g +) E ■ n ' d S ' = = [ {(V ' •E) V g • a + E • [V 'V ' • (a g+)}} d V JV -a • [ Jv jue V'g+ d V ' + f E • [V 'V ' • (a.g+)] dV' (A.53) Jv so th a t [ E ■[V 'V ' • (a5+ )l d V = a ■ / V'g+ (E • n ') dS' + a • [ ? - ^ V ' ff+ dV' Jv Jsiz Jv jue (A.54) Also, since r is in th e volume V, th en f E(r') <5(r' - r) d V ' = E(r), Jv r G V (A.55) S u bstitution of (A.54) and (A.55) into (A.48) yields: [ {E ■[V' x (V ' x a.g+)] - a g + • [V' x (V ' x E)]} dV ' = - a • I V'g+ (E ■ n ) dS' + Jv Js 12 + a• [ Jv ju s V'g+ dV' + a • E(r) + a • [ (jug, 3 + V ' x M) g+ dV' Jv (A.56) S ubstitution of (A.43) a n d (A.56) into (A.39) yields: a - E( r ) = - a - [ \(j oj gJ + V' x M) g+ + —r— V'^-1"] d V ' + Jv 1 jue + a - / [ j u g ( H x n ') g+ + ( n ' x E ) x V g + + V'5+ ( E • h ' ) \ d S ' Jsu A .4 (A.57) Evaluation of surface integral in (2.22) at singularity point Let V be the volume and S 12 = S i U 52 the surface defined in Figure 2.1. T he surface integral th a t appears in (2.22), i.e., J Su [jug (H x n ') g( r, r') + ( n ' x E) x V'g(r, r') + V' g(r, r') (E • n ')] dS' has a singular integrand w hen th e observation point r is also on th e (A.58) surface S and corresponds w ith the source point r', i.e., r = r'. T he purpose of th is section is to evaluate expression (A.58) 185 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure A.2: D efinition of integration dom ain. in such p articu lar situation. Since 1 + jkR 47TR2 = 1 + J'fek —r7! _—jk\r —r'l r ~ r' r —r '| V '9 ( r ' r ') = where R = r —r ' R = -jkR f. (A.59) R R then, the integrand of (A.58) can be w ritten as j u g (H x n ') g(r, r ') + ( h ' x E ) x V' g { r, r') + V g { r, r') (E • n ') = -jkR = jw/i(Hx n') 47T.fl [( n ' x E) x R + R ( E • n ') L 1 + jkR 47Ti?2 jkR e- j k R = { j u g (H x n ') R + [ e ( n '-R ) - n ' (E • R ) + R (E ■n ')] (1 + j k R ) } Let us now evaluate th e surface integral (A.58) for r = r ' as th e lim it for 4 irR 2 8 (A.60) —>■ 0 of th e integral over the surface Ss U F d- illu strated in Figure A.2. Using (A.60), [jug (H x n ') g(r, r') + ( n ' x E) x V g { r, r ') + V 'g (r, r ') (E • n ')] dS' = Sl2 = lim <£[j u g (H x n ') g(r, r ') + ( n ' x E) x V ' g ( r, r ') + V ' g ( r, r ') (E • n ')] dS' + 5-40 J Ss + lim I [jug (H x n ') g{r, r') + ( n ' x E) x V'<?(r, r ') + V ' g ( r , r') (E • n ')] dS' 5-4o J r j 186 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. = f [jufi (H x n ') g{r, r;) + ( n ' x E ) x V g { r, r ') + V '5 (r, r ') (E • n ')] dS' + JSU f t r ^ ^ ^ -i ^ fs~jkR n (E ■R ) + R ( E • n ')] (1 + j k R ) } J ^ j g d S ' + lim j [ j u n { H x n ) R + [ E ( n '-R ) - (A.61) r4 where £ f{r,r')dS' denotes the principal value of th e integral of / ( r , r') over the surface S, i.e., the integral is com puted excluding th e singularity in r = r'. Since dS' = R 2 dn ' = |r - r ' | 2 dft' (A.62) lim R = lim (r — r ') = n ' $->o 5— >o and (A.63) v ’ th e n the lim it in (A.61) can be evaluated as follows: lim j> jjWjU ( H x n ' ) R + [ E ( n ' - R ) - h ' ( E - R ) + R ( E - n ')] (1 + j k R ) } dS' = r4 = L[ & '■ & ' E ( r ) I *2 d n ' - L E ( r ) i ^ M r e <A-64) Therefore, for r £ S n = S i U 52, we have <fi[jug (H x n ') g(r , r ') + ( n ' x E ) x V g { r, r ') + V g { r, r ') (E ■ n ')] dS' - J S12 = A.5 ^7^ + f 2 J S12 [jwyu (H x n ') g{r, r') + ( n ' x E ) x V g + V g (E • n ')] dS' (A.65) Equivalence o f alternative dyadic G reen ’s function From the divergence theorem , since th ere are no sources F = J , M on th e boundary surface S, Jv V • [5 (r, r ') F (r')] dV' = ^ g(r, r ') F ( r ') ' • f i dS' = 0, (A.66) therefore 0 = V ' • [<?(r, r') F ( r ') ] d V ' = J ’ V ' g ( r, r ') • F ( r ') d V ' + j ' g{r, r ') V ' • F d V 1 187 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (A.67) th a t is [ V'g(r, r') ■F ( r ') d V ' = - Jv f Jv g(r,r')V -F d V ' (A.68) It follows th a t / Jv F ( r ') d V ' G (r, r ') • F ( r ') dV' IV g{r, r ') F ( r ') d V ' - ~ = [ V V '^ r , r')] • F ( r ') dV' = J v g{r, r ') F ( r ') dV ' - 1 V ^ V '5 (r, r') • F ( r ') d 7 ' = ^ ff(r, r ') F ( r ') d V ' + ^ V ^ = J = [g(r, r ') I + ^ ^ V '] g( r, r') V ' ■F d y ' .F(r') dy' [ G (r, r ') ■F ( r ') dV' (A.69) Jv In addition, since for a closed surface S and a J t tan g en tial to 5 , th e following integral theorem (in [89], page 503, no. 42) holds: f V' • [3(rir') Jt(r')] dS ' = 0 (A.70) then J V ’g M - J t W d S ' and r — ] s G(T,T')-Jt(r')dV' r = IY I - = - J s g ( v , r ' ) V - J t dS' VV'N J t (r ) d V = ^ ff( r , r ') J t ( r ') d V ' - ~ ^ = j g(r, r ') J t ( r ') dV ' - ^ = j g(r, r ') J t ( r ') d F ' + ^ V ^ 5 (r, r ') V ' • J t d V 1 = Js [< ?(r,r')I + V ^ = (A.71) [V V ,j ( r , r ') ] ■J t (r') dV ' V 'S(r, r') ■J t (r') dV' V-] • J t (r') dV' [ G ( r , r ' ) - J t ( r ') d y ' Js 188 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (A.72) Appendix B D erivation of tw o-dim ensional G reen’s function Let S be th e surface of a b o d y of infinite length in the z-direction, and C be its generating curve, as depicted in Figure 3.1. If there is no variation of th e electrom agnetic quantities along the z-axis, th en the coupled surface integral equations on S can be reduced to integral equations on C by defining a tw o-dim ensional G reen’s function. Consider th e interior surface integral equation (2.38). Since dS' = dz' dC' , (B .l) th e two surface integrals in (2.38) can be w ritten as follows: j V g V '] ■J t (r') dS' [<?(r, r') I + p = JJ g(r , r ' ) I V5 V' J t ( r ) dz dC (B.2) C —oo OO J [V 'g(r, r') x I] • M t (r') dS' = 5 j J C —oo [V'g x I] ■M t (r') dz' dC' Since J t( r ') does not depend on z’, it is a function of only th e two-dim ensional vector (B.3) p ' , i.e., J t( r ') = J t ( p ;)- T his allows to write OO J OO 5 ( r , r ' ) I • J t( r ') dz' = J J t {p') OO p- J (B.4) g(r,r') dz' OO V g V ' ■J t {r') dz' = ^ V —oo / V'-Jt(p') (B.5) g(r,v') dz' x I • M t ( p ' ) dC' (B.6) L —oo r [V'ff(r, r') x I] ■M t (r') dS' g(*,r') dz' = j V' C oo j L —oo 189 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T he scalar G reen’s function can be w ritten as [72]: OO j ff(r,r') = 9 (B.7) ( p , p ' \ k z ) e i k z (z ~ z ) dkz —OO where .• g { p , p ’-,kz) = - - f f j ) (fep| p - p '|) , ,— kp = y k 2 ■ (B.8) Using (B.7) and exchanging th e order of integration, yields: OO J OOr g(r,r')dz' J = —OO OO J —OO L g { p , p ' \ k z ) e ^ k z ^z ~ oo = / r 9 dk —OO (p ,p '-,k ,)e ik —oo oo T J L —oo e - j k . z ’ dz' dkz oo {p, p ' ] k z ) e i kzZ 8 {kz ) dkz = J —oo = 9 ( p , p ' \ °) 9 = (B.9) -p'|) Hence, defining a tw o-dim ensional scalar G reen’s function as 92 d { p , p ') = - 3- H Q { ]{k\ p (B.10) - p ' l ) th e surface integral (B.2) becomes / p(r,r')I + ^ V<?V' ■J ;(r') dS" = j 5 ~ 92 d { p , p ' ) l + ^2 V' 3t(p')dC' ( B .ll) C while (B.3) can be w ritte n as the following line integral j [Vg( r , r ' ) x I] • M t (r') dS' 5 J [V,g2 D( p , p ' ) = ’ x I] ■M j ( p ' ) dC' (B.12) C By defining a a tw o-dim ensional scalar G reen’s function for th e free space as 52d(p> p ') = - 7 H o2\ ko\p - P'l) . (B.13) a sim ilar procedure can be applied to the surface integral equation (2.37) to obtain the two dim en sional integral equations for the exterior problem . 190 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix C B odies of R evolution C .l E xplicit form o f operators a ■PQ T he operators a pT (p, q = r , 0) are defined by (4.16)-(4.17): a pr{ J T(t',(p')} = - j a p .o [ Js , 0')} = - jt up o Js p ■r ' G ( r , r ' ) J T(t', (j)') 1 1 5G° d{p'Jr ) kl p' dp 1 1 dG°dJ^ pJ dp d<j>' Using expressions (4.1)-(4.3), the dot products betw een n. 0 , f f • f ' t • n dt' and n 0 dS' dS' (C .l) (C.2) f ' are found to be: = s in ip simp' cos ((f)1 — 0) + cos 0 cos 0 ' (C.3) = sin 0 cos ip' cos (0 ' — 0) — cos 0 sin 0 ' (C.4) = —sin 0sin(< // — 0) (C.5) n •r = cos 0 sin 0 ' cos (0 ' —0) —sin 0 cos ip' (C.6) n • n = cos 0 cos 0 ' cos (0 ' —0) + sin 0 sin 0 ' (C.7) n ■0 = —cos 0 sin (0 ' — 0) (C.8) 0 ■f = s in 0 ; sin ( 0 ' — 0) (C.9) 0 • n = cos ip' sin (0' — 0) (c.io) 0 = cos (0' — 0) ( C. l l ) ■ 0 191 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. S u b stitu tio n of expressions (C .3 )-(C .ll) into definitions (C.1)-(C.2) yields: a TT{ J r ( t ' j ' ) } = -jojfj .0 Js | G °( r , r') [sin ip sin ip' cos (p' 1 - p) + cos ip cos ip'] Jr (t',p') - ,<p')} = j u p QJ^i^G0{r,r')smpsin(p' - p)J${t',p') + dS' (C.12) dS' (C.13) .. .. .. f \nOt^ ^ t u> a'i 1 1 dG°d(p'JT) r{ J T(t',p')} = -jojfj .0 [ \ G° (r,r') sin ip'sin {p' - p ) J T{t',p') - - j — dt' JS I Kq p dp dS' (C.14) a « r / 7 / V ^ M _ a ', { J tj >{t',p')} = - j u n o C.2 J , I G ° ( r , r ' ) c o s ( 0 ' - p)J<fi{t',p') E xplicit form of operators 1 1 SG° <9J<» p' dp dp' dS' (C.15) /3pq T he operators f3pT (p , q = r,p) are defined by (4.18): /3™{M9(f',0 ')} = - / p .V G °(r,r')x q 'M ,(^ V S ' Js (C.16) T he gradient V G °(r, r') can be expressed as o/ _ dG° A , dG° . V G ( r ’r ) = f t T f t + ldG° « (C.17) Since R — |r — r '| = \ J P2 + p '2 — 2pp' cos (p 1 — p) + (z — z ' ) 2 (C.18) the p a rtial derivatives of G° can be w ritten as dG°_ _ dGP_dR _ dCP_ „ dn dR dn dR dG° dr dG° d R dR dr dG° dG°dR d R dp dp dG° dR t (C.19) -VR (C.20) (C.21) where, using the expression (4.8) of the free space G reen’s function, M±_ = dR d ( e - i k°R \ = d R y AirR J 1 -jkp R AirR2 j kn R 192 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (C.22) T he gradient V i? can be calculated in cylindrical coordinates as „ „ dR . 1 dR 2 dR . V R - ~ a ^ p + p a f 4' + & a x (a2 3 ) w ith the p a rtial derivatives of i? being ^ ^ ( V ^ 2 + P'2 “ ZPP1 cos((t>' - (f’) + (z - ^ ^ ( V P2 + P'2 - W z 0 2) cos(^ ' - 0) + (* - ^ ) 2) ( V p 2 + P'2 “ 2PP' cos(^ ' - ^) + (« - * ')2) = - — P ° ° ^ --------— = ~ PP = -----— (C.24) (C.25) (C.26) Therefore: _ n -V i? dR ^ ^ IdR = — n •p + ap p o<p dR , dR . = -7—cos ip —— sin op oz = . „ „ T -V i? R = = — . 3 S i? _ n • 0 + — n ■z oz , ip {[p - p' cos(^' — (j ) ) \ cos ip — (z — z') s in ^ } (C.27) &R _ Id R ^ 2 dR „. — T •p + T • 0 + — T ■Z op p o<p oz dR . , dR sm ip + ——cos ip op dz "4 {[P — p/ cos(0' — <p)] simp + (z — z') cos ip) R (C.28) S ubstitution of (C.27)-(C.28) and (C.25) into expressions (C.19)-(C.20) yields: dG° dn dG° dr 1 dG° P ~ ~ai _ = 1 dG° {[p — p' cos(<p' — (p)\ cos ip — (z — z') sin ip} R dR (C.29) 1 dG° {[p — p' cos(<p' — 0)] s i n ^ + (z — z') cos ip} R dR (C.30) 1 dG° , . f±, - R l R psm {* - V (CU1) Hence, the use of (C.29)-(C.31) into (C.17) yields th e following expression for the gradient VG°: 1 dC® VG° = - — (An n + Ht f + A0 4>) where 193 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (C.32) An = \p ~ p' cos(<f)' —4)} cos ip - (z — z') sinip (C.33) Ar = [p - p' cos(0' - 0)] sin ip + (z - z') cos ip (C.34) ^ = —p'sin(<p'— (p) (C.35) Therefore, for q ' — f ', 0 ', f • VG° x cp • V G ° x q' = q -g {^ n[r • ( n X q ')] + A - [ t = {^n[(T x = [M 4>- = = X q ')] + -A*[ f q + A T[ ( f x t ) - '] + A ^ t • ( 0 X q')]} x q ']} $)■ q')] { A n [ 4> ■( ' = q'] n)- •( f n Xq (C.36) ')]+ A r [4> -(f X «)■ x 1 PC 0 R ~ d R [-A n ( * ■ q ') + X q f). ')] + A^[ (p • ( q'] + 4 4 ( ( ^ 0 X q ')]} X (^). q ' ] } fi ■ q ')] (C.37) where, using (C .3 )-(C .ll) a n d (C.36)-(C.37), A n (4> ■f ' ) — A<f,(n ■f ') = {[p — p 1 cos(<p' - (j))] cosip — (z - z') s in ^ } sin ip' sin (cp' - <p) + +p' sin (<p' — 0)[cos ip sin ip' cos (cp1 — <p) - sin ip cos ip'] = A n (4> • <p') - [p cos ip simp' — p' sin ip cos ip' — (z — z') sin ip sin ip'] sin((p' — cp) (C.38) A ^ { n •ip') = {[p - p'cos(<p' - cp)] cos ip - (z - z') s in ^ } cos (cp' - (p)+ —p' sin (<p' — <p) cos ip sin (<p' — (p) — - p ' c o s i p + pcosipcos{(p' — <p) — (z — z')c o s (( p '- ( p ) - A n( f ■ f ' ) + A r ( n ■ ? ') (C.39) = —{[p — p' cos {<p' — ip)\ cos ip — (z — z') sin-i/’} [sin ip sin ip' cos (cp' — <p) + cosi^cos ip'] + + {[p — p' cos (<p' — (p)] sin ip + (z — z') cos V’} [cos ip sin ip' cos (<p' — <p) — sin ip cosip'] = —[p — p' cos (cp' - <p)] cos ip' + (z — z') sin ip' cos (<p' - <p) 194 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (C.40) —A n ( ■ (p>) + A t ( n • ((> ) = t {[p —p, cos(0' — ^ j c o s V ' — (z — z ^ s i n i p } s i n ^ s i n (<f>' — <p) + — {[p — p' cos ((j) 1 — (p)] smip + {z — z') cost/)} cos ip sin (</>' — cp) = - ( z - z ') sin(</>' - (p) (C.41) S u b stitu tin g (C.38)-(C.41) into (C.37)-(C.38), th en (C.37)-(C.38) into (C.19), the final form of the operators f3pg (p , q = r,cp) is obtained: r { M r {t',<p')} = f 1 dG0 - / — -j-r-[pcos ip simp' - p' s i n c o s tp' — (z - z') sm'ipsimp'] sin (<p' — cp)MT(t', <p') dS' JS (C.42) wit ^ { M f 0 ,p ') } = 1dG® - / z z —fFr[-p'cosip + p c o s tp cos (</>'- <p) - (z - z') cos ( </ / - 0)]M ?i(t/, cp1) dS' J 5 il C b ih (C.43) ^ { M T(t',cP')} = - / J ,5 it CLa x , - p! cos(<p' - (p)] cos ip' + (z - 2:') sin 'ip' cos{<p' - <p)}MT(t', <p') dS' r 1 dG® {Af*(t', </>')} = J s k -j k (z - z ') sin (<£' - W t t f , <t>') dS' C.3 D erivation of operators (C.44) (C.45) a,m n From expressions (C.12)-(C.15) and th e fact th a t, from (C.17), S 0 ° - * . V G° = ^ dr dt (C.46, it follows th a t 27T a rT{ JTn {t')e^n^ } = —jcjpo J J| [sin ip sin ip' cos {cp' — <p) + cos ip cos ip'] G °(r, r ') JTn(t')egn^' + k l dt = - j o j p o ejn ^ dt' 2tt J | ^ sin ip sin ip' J y G°(r , r ' ) cos (</>' - <p) e?n^ ' o 195 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. + + cos ijj cos ip' J G° (r, r') e?n^ o }_ d _ kg dt J ^ dp' p' J T„ ( 0 + t } dt' G° (r, r ') ej n W~®d(P'] (C.47) 2tt aT<l>{ J K ^ t ')ejn4>'} = i UJP o j J o c = sin sin — 0 )G °(r, r') jup,o e-7”^ / [ s i n * J c + 1 jndG °i Jctn {t')ejn(p,p'dt'd(P' kn p' dt ■ 2ir G°{r, r') sin (cp1 — <fi) e^n^ ' ® d(p' + o 27r / G " ( r , r ( ( ' ) *’# ' 277 a - =-ju;poJ J ,<pT (C.48) {G°{r,r ')s mT p' sin {( p' - (p)JTn(t')ejn* ' d(p d^Pdt'Jt^ ijn*')p'dt'd(P' 2tt = —j u p o e ^ J j s i n V / [ J G ° ( r ,r ') sin((p' — </>) e^n^ ' ^d(p' p'JTn(t') + c 2tt ±d_ [ J G °(r, r') kl d(p o 5^ Tn)- } dt' (C.49) O-TT- 0 c 277 j = -jw /io C [ J G°(r, r ') cos (<£' - <£) e?n{# d # + 0 277 1 jn 3 P' d</> J G°{r, r ') ejn(^ '- ^ d 0 '] p' J K {t') dt'd<p' (C.50) Using the definition (4.27), th en 277 J G °(r, r ') cos (0' - <p) e>n^ ~ ^d < p ' = 2tt - ^ ^ ± 9 1 + ^1^). (C.51) = 2iry g ° - i( M O ~ G n + itM ') (C.52) = 2 7 r ^ G ° ( M ' ) = 2 7 r ; n G° ( t , t ' ) (C.53) 0 27T 1 G ° (v ,r ')s m{ < p '- (P )e j n W-®d<P' 277 d<j> I Go( r , r V n(0'■“^ , 196 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. it is possible to w rite a ^ { J qn( t ' ) e ^ ' } = 2 i r a ? c>{Jqn( t ' ) } e ^ , P,q = r,<p, n = 0, ± 1, ± 2 , . . . (C.54) where the a^q operators for th e n -th angular m ode are: a nT{ J r n { t') } = - j w f i o J { [ sin ip sin ip' gn-1 _ T(b r t f an 0 r / / 1/ \ 1 . . . . r [ • $ti v J j — J ^ p o J c T (p\\ _ -i/in /~ f - .// simp jtjt) ~ ^ ^ n + 1 ft) = - j ^ o / [ C ° - l ( f ' t ' ) ^ G° + l ( ‘' l ' ) c C.4 G n+l 1 ^ - G ° +1( i , 0 ^ • j simp a n {•'Tnft)} — JU(j,qJ C ^ + cos ip cos ip' G Q n (t , f') p'jTn(t') + 1 k? d 0 t ) j T , P Jrn { t ) j T dt jtl P d(j>n { t ) dt dcp (C.56) u ^ 2 Crn (i,f J 0 ^ (P ^ ^Tn ) ~1 j±i ( ri g '7 ^ j at (C.57) (C.58) 0 Derivation o f operators /^9 From (C.42)-(C.45), it follows th a t /3TT{M Tn( O e 7‘^ } = /’27r/>1 dC® / / ———■[p cos ^ sin J J R dR o c - = — 0 c i p'dt'd<p' (C.59) /■2vr r l dC® I — - ^ - [ —p 1 cosip + p cos ip cos (<p'- cp) — (z - z') simp cos (cp' - c p ^ M ^ ^ ^ e ^ p'dt'dcp' 0 27r — p 1 simp cos ip1 —[z — z') simp simp'] sin (cp1 — <p)MTn(t') c f ^ ^ Up' ~ p) cos ip — 2 p cos ip sin2 R dR ^ ^ + (z — z') simp cos (cp1 — cp)]M^n (t')e^n(t p'dt'dcp1 (C.60) P*T { M Tn(t')ejn<t>'} = 27T q = — [ [ ^ ~ 77r{[/° — p ' cos [cp' — cp)} cos ip' + (z — z') sin ip' cos(<p' — <p)}MTn (t ' ) e ^ p'dt'dcp' J J R dR 0 c 197 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. // 0 IdT T o ^ R dR ~ P) cos ^ ~~ ^P' cos ^ s^n2 ( ^ o ~ l ~ (z ~ z ') sin V,/ c o s (^ — 4>)\Mrn ( t ' ) e ^ ' p'dt'dcp' V 2 / (C.61) c 2 ir ^ = J j I^(z_*')sin (P' 0 fp)M<t>n( t ' ) e ^ ' p'dt'd<P' (C.62) c Using (4.28), an d defining „o, S" ( M ) 1 to I = f 2* 1 dG ° , / R l R Sm ( (C.63) d W ~ <P) it is possible to w rite Ppg{ M gn( t ' ) e ^ ' } = 2 t t (3pg{ M qn( t ' ) } e ^ , p , q = T,<t>, n = 0, ± 1, ± 2 , . . . (C.64) where the /3pg operators for th e n -th angular m ode are: PT n T { M Tn( t ' ) } = pr® A p cos ip simp' — p' sin i/; cos / (f —(z — z') simp simp1}— f 1) _ jrO —’■----- -— (p; - P) cos ipK°{t, t') - 2pcos ipS°{t, t') + { z - z') sin ip Kjl 1 ^ ^ ^ -+1 (f f r) —-— M Tn(t')p'dt' (C.65) - )■ P t T {Mrn ( 0 } / {p1 —p) cosip'K®(t, t ' ) - 2 p ' cos ip'Sn(t, t') — (z —z 1) simp‘ M Tn{t')p'dt' (C.67) p 'd t'd tf C.5 (C.68) D erivation of scattering am plitudes T h e functional operators V pw are defined in (4.86) as V pw {Jw {t',<p')} = j ( p - w ' ) Jw(r') e i k° r dS' , p = h , v \ w = r,cp 198 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (C.69) T h e use of the Fourier series expansion (4.22) for the equivalent currents Jw(t', <j>') leads to w rite expression (C.69) itself as am infinite sum m ation: { OO 'j OO £ JWn(t)ejn(t,\ = £ V pw{ j Wn( t ) e ^ } n= —OO 77,——CO (C.70) J Since k s - r ' = p sin 9S cos (f>s + z cos 9S (C.71) th en the n -th term of th e series at th e right-hand side of (C.70) is 27r j v pw { JWnIt') e?** } = o c J ( p •w') JWn (t') e j n $ e j sin 9*cos $ + * cos 9*) p'dt'dtf (C.72) T h e dot products in (C.72) are: h — 4> ■f ' — —simp 1 sin {(j)' — <f>s) r ' = v ■ t '— —6 h ■ = v • 4>' = t' (C.73) = —cos 0, sin i//co s (<f>' — <j>s) + s in 9a cos ip' (C.74) — <p ■0 ' = —cos {(j)'- 4>s) (C.75) — 0 ■ 4>' = cos 9S sin (</>' — <j>s) (C.76) Using (C.73)-(C.76) together w ith th e integral representation of th e Bessel function, expressions (C.72) become: V hT{ j Tn{ t ' ) e ^ } = - 2 t r j V n*‘ j sin ^ ' T + ( u sp') JTn(t') e ^ z ' p' dt' (C.77) c 'PVT{ j r n{'t') ejn<t>] = 2 n j n ejn(l>3J [ - j cos9s s m i p ' T ~ ( u sp') + sin 0 s cos ip'Jn (usp')\ JTn(t') e ^ VsZ p'dt' (C.78) c P fc* { j* n ( t V n*} - -2 ttj ne ^ j e ^ ' p ' d t 1 (C.79) c ( 0 ejn*} = 2 where cos 9se ^ j T + ( u sp') (t') e ^ z ' p'dt' Us = k 0 sin 9S , v 8 = k 0 cos 9S . 199 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (C.80) (C.81) S u b stitu tin g the approxim ations (4.37) into (C.77)-(C.81), and evaluating the resulting integral as I m dt' = (tB - t A) f , ( C . 8 2 ) tA th e expressions (C.77)-(C.81) can be w ritten as V pw { j Wn{t') e ^ } = ^ p $ w{ks, p s) l % n { k i,q .i )e jn^ , p = h , v ■ w = t , 4> (C.83) m = l w ith the coefficients p ^ L ( k s, p 5) given by p % l(k 3, h 3) = -27Tjne ^ A t m s m ^ m T ^ ( u sPm) e ^ zm p ^ ( k s, v s) = 2Trjnejn<f>3 A t m [ - j c o s & 3 sinipmT ~ ( u spm) + s i n 8 3 cos'ipmJn{u 3 Pm)] e i VsZrn (C.85) p < $ (k s, h s) = —2-Kjne^n^s A t m pm j T ~ ( u spm ) e ^ VsZm ? £ * ( = 27rjn cosd3 e ^ A t m P m T+ {u sPm ) e ^ Zm 200 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (C.84) {CM ) (C.87) Appendix D Half-Space Problem D .l D .l.l Expression o f dyadic G reen’s function com ponents E le c tr ic d y a d ic G r e e n ’s fu n c tio n s T he electric dyadic G reen’s functions can be w ritten as infinite sum m ation of azim uth modes: +00 G ^ fr .r ') E n= —00 GPi(p->z i p ' i z ') p q ' D , j n(<f> — <j>) for t = D , R (D .l) _p,q=p,<bz For the direct term , th e dyadic com ponents of th e n -th m ode coefficients have th e integral form OU GVD,n(Pi z i P'■>z ') ~ _ 4^r J , - j k 0 z\z - z'\ GP f yn {kP, k 0Z-, p, p ) kp d k p , 2 knz p,q = P,<f>,z (D.2) where the kernels Q ^ n {kp, k oz; p, p') are given by Q%n {kp, koz] p, p') = T + ( k pP) T + ( k pP') + g ^ ( k p , k o z - , p, p' ) = - j T ^ ( k p p ) T - ( k pp ' ) - j ,^ T - ( k p p ) T + ( k p p ' ) kh k np^oz k ^D, n(k P i k oz', P tp ) g*D%( k p , k oz -p, p') k2 knk. ~ = T - ( k pp) T ~ ( k p p 1) T n (k pp) Jn(kpp') G%n{kp,koz-,p,(/) kL for Z > Z1 for z < z ' T + { k pP) T + ( k pp f) J n { k pp) T + {kpp') %pkpz k2 — ^2 (D.6) for z > z' (D.7) J n (kpP ) T + {kpp' ) for z < z ' kn @D,n{kpi k 0 z] Pi P ) (D.4) (D.5) Tn (kpP) Jn (kpp') T - { k pp ) T - ( k p p ' ) + k„k. p^oz (D.3) Jn{kpp) JnikpP ) 201 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (D.8) and the functions T * are defined as T^ x) = Jn+l ( x) ± J n - l {x)_ ' T_ ^ = Jn + l (x) 2 Jn-l(x) ^ 2 T he rem aining th e kernels Q^fn {kp , k oz; p, p') can be found by sym m etry relationships: g i i ( k P, k 0 2 ;p,p') = -g p D^ ( k p, k 0Z- p \ p ) (D.10) &D,n(^P’ ^oz't P) = &D,nftpik 0 z', P , p) (D-l l ) g^ z { k P, k 0Z-,p,p') = g % n (kP, k 0z-,p',p) (D . 12 ) T he dyadic com ponents of the n -th mode coefficients of the reflected term of G have th e form j r° , e - j k o z { z + z') J S ^ n {kp, k oz; p , p ) ^ kp dkp , o °z Gp^ n (P>z ; p'iz ') = p,q = p,(j>,z (D.13) where the kernels g™n (kp, koz; p, p') are given by ' g i ( k p , k 0Z-,p,P') g & ( k p , k oz; p, p') = r ftT + M T n+ M = - |r „ T ; ( v ) T ; M r„ T+( V ) T ~( V ' ) + j r„ T " ( V ) r + ( V ) (d .u ) (D.15) G&( k p >k o z - , P, f / ) = (D.16) g Z ( k P, k oz; p , p ' ) = (D-17) g %n (kp , k oz] p, pf ) = - g g n {kp , k oz ] P, t J) = % T v Jn (kpP) J n (k pP') ^ r v j n (kpP) T + ( k pP') (D .i8 ) (D.19) T he rem aining kernels GV ^ n {kpi k oz] p, p') can be found by sym m etry relationships: g ^ n (kP, k 0Z-,p,pf) = - g ^ p n {kp , k oz- p \ p ) (D.20) g J n ( k P, k o z ; p , p ) = g ^ n (kp, k oz; p , p) (D-21) g & ( k P, k oz ] p, p' ) = g ^ { k p , k oz]P' , p ) (D.22) 202 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. D .1 .2 M a g n e tic d y a d ic G r e e n ’s fu n c tio n s Sim ilarly to G , th e m agnetic dyadic G reen’s functions can also be w ritten in the form: +00 K « ( r ,r ') = for t = D , R (D.23) £ .p,q=p,<bz For th e direct term , th e dyadic com ponents of th e n -th m ode coefficients have th e integral form j 1 K pJ n {p, z ■,p', z') = r° e 1 3 ^oz\z z \ J K,p£ n {kp,koz; p , p ' ) ---------—---------- kp dkp , p,q = p,(j>,z (D.24) where th e kernels K ^ { k pi k oz\ p, p') are given by f - j k oz [T~ {kpp) T+ {kpp') - T + [kpp) T ~ {kpp')] for z > z' j koz [T~ (kpp) T + ( k pP') - T + ( k pp) T - ( k pP')\ for * < z> lCpDPtn(kp, k oz]p,p') (D.25) koz [Tn {kpP) Tn {k pp ) + T + (kpp) k °z 'i Pi P) (kpp )] I - k 0z [Tn {kpp) Tn {k pp') + T + (kpp) T + {kpp')] kp ^ D , n { k pi k °Ti Pi P ) for z > z , for z < z ‘ (D.26) (D.27) (kpp) Jn {kpp ) j koz [T+ {kpp) Tn (kpp') + Tn (kpp) T + [kpp')] I C i i ( k p, k oz]p,p') I - j koz [T+ {kpp) Tn (kpp') + Tn (kpp) T+ {kpp')\ for z > z' , for z < z' (D.28) ^n{kpikoziP j) = - j k p T - { k p p ) J n {kpp') (D.29) &D, n { k p i k o z ; P , p ' ) = 0 (D.30) T he rem aining kernels IC^n {kp, k oz: p, p ' ) can be found by sym m etry relationships: T ^ t n { k p i k oz 'i Pi P ) = — K-P t y n { k p i k oz'i P T' Dln i k P i k oz'i Pi P ) = f c f y n ^ k p i k ° z 'i P ^ D , n { k Pi k ° z 'i Pi P ) = —T i ^ n i k p i k o z ' i P i p ) 1 1 p) (D.31) (D.32) P) (D.33) T he dyadic com ponents of th e n -th m ode coefficients of th e reflected term of K have th e form IC^iikp, koz] p, p') = - j koz [ r h T - { k pp) T + { k pp') - r v {kpp) T~{kpp1)] 203 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (D.34) D .2 IC&ikp, kog;p, p') = koz [r„ T ~ ( k pp) T - { k pP') + Tv T + ( k pP) T+{kpp')} 1Cg n (kp,koz-,p,p') = K%n [kp, k oz; p, p') = - k oz [ r h T + ( k pp) T + ( k pP>) + T v T ~ ( k pP) T - { k pP')\ (D.37) S C ^ ( k p, k 0 Z-,p,p') = j k o z [rh T+{ kpp ) T - ( k pp') + T v T ^ ( k pP) T + ( k pp')] (D.38) £R,n {k P’k 0z-,P,p) = j T v k p Tn {kpp ) J n {kpp ) J n {kp P ) (D.39) JC^n (kp, k 0 Z-,p,p') = T h kp Jn (kpp ) T + ( k pP') (D.40) lCz^ ( k p, k oz]p,p') = j T h kp Jn {kpp ) T - { k pp') (D.41) }C^n (kp, k 0 Z;p,pf) = 0 (D.42) - r v kp T + ( k pP) J n (kpP') (D.35) (D.36) D erivation o f scattering am plitudes Com bining (5.131)-(5.131) in (5.130), th e expressions (4.82) for th e scattering am plitudes can be w ritten as, for q = h, v , f pq( k s , ki) = - 4 ) ^ r Eq ^ ^ 1 ^ ( i ' i ' ) } + ^ “ { M W ( ^ ') } ] (D.43) w = r , <f> where the operators 'Hpw are defined as = an d i P ^ { j ^ ( t ' , 0 ,)} + 5 pw{ j ^ ( t ' ) 0')} C hw{ M w(t', </»')} = j - 7-Lvw{ M w(t', ( j ) ' ) } C vw{ M w{ t ' ^ ' ) } = - 1 Co Co U hw{ M w{t',<j>')} (D.44) (D.45) (D.46) T he operators S pw are defined, for w — T,(j>, as S hw{ J w{t\<t>')} = < f T h ( p - w ^ J ^ e i ^ P ' ^ ^ e - ^ ^ ' d S ' (j)')} ~ r „ [( p -w ') - 2 cos0s ( p -w ')] Jw (r') e i Usp C0S(^ e i VsZ dS' 204 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (D.47) (D.48) where u s — ko sin#,, , vs = ko cos 9S . (D.49) and the dot p ro d u cts are p f' p . = s i n ^ cos (<p' — cps) (D.50) = - s i n ( ^ / - 0 s) (D.51) Using the angular m ode expansion of the currents J w, th e operators S pw {Jw(t', cp')} can be w ritten as OC S ? w { J w( t ' , </>')} = E S ^ { j Wn{ t ) e ^ } (D.52) where S h r{ j rn ( f ) ejn<t>} = J T h sin ip' T + ( u sp') JTri (t') e ~ 2 t r j V n* ^ z 'p'dt' (D.53) c s VT { Jrn ({) ei"*} J = 2ttre? * * ' T v [j cos 9S s in i//T~ {usp') + c + sin # s cos ip1J n (usp')] J T„(t') e ~ 3 VsZ p'dt' (D.54) j Y h j T - { u sp ' ) J ^ { t ' ) e ~ jVsZ' p'dt' (D.55) c $ v*{j<t>n ( 0 e ^ } = - 2 t t f cos 9sejn^ J T v T + ( u sp') J K (t') e ~ ^ z ' p'dt' (D.56) c Com bining (D.52)-(D.56) w ith the expressions of th e operators V pw from A ppendix 4 yields U hT{ j Tn{t')ejn<t>} = 2Kjne ^ J r i ( v s,z') s m i ; ' T + ( u sp ' ) J Tn(t')p'dt' (D.57) c 'H VT{ j r n {t')e:!n<t>} = 2x j ne3n^ J [ - - f - { v s,z') j cos9s sinip'T~{usp')+ c + 7 +{vs ,z') sin # s cos ip' Jn (usp')] JTn(t') p'dt' = - 2i r f e ^ J j ^ v ^ z ^ j T - M J ^ p ' d t ' (D.58). (D.59) c = 2 -K f cosdse ^ J 'y -( v s , z ' ) T + ( u sp ' ) J (l>n(t') p'dt' (D.60) c where 7p { v , z ) = . _ e3vz ± r p e ~ 3 vz , p = h,v. 205 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (D.61) Finally, discretization of the (D.57)-(D.60) using (4.37)-(4.38) yields the expressions for th e coeffi cients hrrjw (k 5, P 5) ■ h g . ( k s , h s) = - 2 -Kjn ejn,p3 A t m ^ { v s, z m ) sini;m T + { u spm ) f t ^ ( k 4, v s) = 2 -Kjn ejn<psA t m [ - j 7 v {vs, z m) cos9s sintpmT - ( u spm)+ + J v ( v s , z m) sin&s cos'4>m Jn (uspm)] (D.62) (D.63) / i ^ ( k s , h s) = 2'Kjn e3n<t>s &t Tnpmj'Y £{v s, z m ) T - ( u spm ) (D.64) ^ = —27Tjin cos 9S ejn^s A t m pm % (vs,zm ) T + {uspm ) (D.65) 4,( ^ 3 , ’Vs) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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