Acoustic and electromagnetic scattering analysis using discrete sources IX discrete sources method in electromagnetic theory

Acoustic and electromagnetic scattering analysis using discrete sources IX discrete sources method in electromagnetic theory Acoustic and electromagnetic scattering analysis using discrete sources IX discrete sources method in electromagnetic theory Acoustic and electromagnetic scattering analysis using discrete sources IX discrete sources method in electromagnetic theory Acoustic and electromagnetic scattering analysis using discrete sources IX discrete sources method in electromagnetic theory Acoustic and electromagnetic scattering analysis using discrete sources IX discrete sources method in electromagnetic theory

IX DISCRETE SOURCES METHOD IN ELECTROMAGNETIC THEORY This chapter is intended as an introduction into the basic concepts of the discrete sources method (DSM) for solving electromagnetic scattering problems As in the acoustic case we will show that the electromagnetic scattering problem reduces to the approximation problem of the boundary value of the incident field in the L^-norm We not repeat the technical aspects of the method which are fully presented for the acoustic case Our analysis is mainly concentrated on the construction of convergent approximations using the fundamental theorem of discrete approximation For the impedance boundary-value problem we will present a somewhat different approach which we call the D-matrix method since the matrix of the corresponding linear system of equations is dissipative We will establish the dissipativity, the convergence of the approximate solution and the solvability of the linear system of equations using the conservation law of energy Special attention is paid to the discrete sources method with distributed vector multipoles The mathematical foundation of the method is accompanied by results of computer simulations The numerical experiments include comparison with other methods and scattering analysis of concave particles and clusters of particles 203 204 CHAPTER IX DSM IN ELECTROMAGNETICS PROJECTION METHODS In this section we will use theorems 1.1 and 2.2 of the precedent chapter to construct approximate solutions to the boundary-value problems for the Maxwell equations The analysis is valid for any linear independent and complete system of functions To this end let us construct the approximate solution to the exterior Maxwell boundary-value problem as a finite linear combination offieldsof elementary sources If the sources are located in the domain Di the approximate field E^, H ^ will be regular everywhere in the domain £>«, will satisfy the Maxwell equations and the radiation condition at the infinity In this context the representation formula for the exact solution given by (6.79) is also valid for the approximate solution Combining these relations we find the estimate I|E.-E.^IUG + 1|H.-H.^IUG ^^||"xE.-nxE^||^_^ (9.1) in any closed subset G, of Dg According to (9.1) we see that the exterior Maxwell boundary-value problem simplifies to the approximation problem of the tangential components of the incident electric field on the particle surface In the following, we will consider the system of radiating solutions to Maxwell equations {*J^,$J[} with the following properties: the set {n X $J[, n X $^} is linearly independent and complete in £tan('S'), V x ^3 -, ;,$3 g^jjj V X $^ = k'if^ Specifically, the system {*^, 0, and (c) / n X E f 4- n X Eo + An X (n X E f * + n x Ej) , n x ( n x $ ^ * ) ) ^ = 0, (9.5) / n X E f + n X Eo + An X Tn X Ef * -f n x EQ) , « x ( « x * ' * ) > , s = 0' for 1/ = 1,2, , AT and A > Then, by theorem 1.1 of Chapter we have llnxEf-hnxEoll II -^ as AT-• oo; ll2,S thus the approximate solution converges to the exact solution in closed subsets of i?5 Before we proceed to analyze the impedance boundary-value problem we note some observations: The solution to the exterior Maxwell boundary-value problem can be represented by E,(x) = f ; a^M^ (x) + 6^N3 (x), (9.6) the convergence being uniform in any closed subset which belongs to the exterior of the circumscribed sphere 5^ Here // is a complex index containing m and n, that is /i = (m, n) and ^ = 1,2, , when n = 1,2, , and m = —n, , n The coefficients a^ and 6^ are given by a^ = ^ Z ? ^ J [(n X E,) • N i + j (n x H,) • Mi] dS, (9.7) b^ = ^Z)^y[(nxE,)M3+j(„xH,).N|]d5, 206 CHAPTER IX DSM IN ELECTROMAGNETICS where /i = (—m^n) Let E ^ be an approximate solution to the exterior Maxwell problem Then, JlEso-" Efo||2n —> as iV —> oo, where E50 and E ^ are the far-field patterns of the exact and approximate solutions, respectively, and ft is the unit sphere If the system {^^,^} represents the system of localized spherical vector wave functions we see that E K - ^I'lKoIlL +1^ - ^riKolL /x=l (9.8) + E M' W^Ullu + l^'^i' l|N^o|l2,n - a^ JV - 00 Since, for x € fi, •i-j) N3o(x) = n-f-l jm Pi"! (cos g) sin^ ^' d4'"'(cose) d^ ^^ ^jirnp N^„o(x) p{-jr^' dp}rUcose) de p}rUcos0) •eg + jm sin^ r—:;—-e^ (9.9) it follows that \\Mlmn0|l2,n ||N7yi„o j2 _ TT 4n(fi + 1) (n4-|m|)! 2w '2,0 - j ^ n + l in~\m\)\ ^ 7772Jj^' (9.10) Consequently, a^ -^ a^ and 6^ -^ 6^ as iV -4 00, uniformly in /i, where /i = 1,2, , iVo, and NQ is an arbitrary large fixed number The system of equations (9.4) is uniquely solvable even for A = Indeed, assume A: > and set TAT = [a^, ; ^ ] S / i = 1,2, ,iV Multiply each of the first N equations of the homogeneous system by a^*, each of the second N equations by b^*^ and sum the final expressions We obtain (D^Tyv, TN)I2 = j ( n X E f , H f )^ ^ , (9.11) where D^r is the matrix of the resulting system of equations From Gauss' theorem we known that Re J|(nxE,^)Hf*d5Uy Efoi'da; (9.12) PROJECTION METHODS 207 provided that k > Taking the imaginary part of (9.11) we are led to Im (DA^T^V, T;V>,2 = J lEfoI'da; (9.13) Q The above equation shows the dissipativity of the matrix D;v? i.e Im(DivTAr,TAr),2 >Ofor any T^r ^0 Now, let {DNTN,TN)I2 = From (9.13) we see that E ^ = on ft, and further that E ^ = in D^ The linear independence of the system {n X ^ ^ , n X $^} finishes the proof of our assertion The systems of discrete sources can also be used in the boundaryintegral equations treatment of the exterior problem For instance, let the scattered electric field be represented as a surface distribution of magnetic dipoles Es{x) = V X y a(y)^(x,y,A:)d5(y), (9.14) where a solves the integral equation f ^ r + - M ) a = eo (9.15) Fork^a {D^) the operator \l-\-M : Cl^^^{S) -> C^t^^^^{S) is bijective and has a bounded inverse An approximate solution to the integral equation (9.15) can be sought in the form N by using the projection relations given in theorem 3.4 of Chapter Note that the operator M : £?an('S') —^ ^tan('5) is compact and for fc ^ cr [Di), by the Fredholm alternative applied in different dual systems, the operator ^T-\- M : >C^an(«S') -^ ^tan('^) ^^ bijcctive and has a bounded inverse Consequently, the approximate solution E f (x) = V X y a;v(y)5(x,y,fe)d5(y) (9.17) converges to the exact solution in closed subsets of Dg- Furthermore, we have n x E^^ — n x Ef*^ -^ as AT -^ oo, and this conclusion agrees with the completeness result establish for the system of functions { Q l + A^) (n X ^3) Q j ^ ^ - J („ ^ $3^^ M = l,2, /fc^ 0, (9.28) shows the dissipativity of the matrix D^^ The claim (DTVTJV, Tiv)/2 = if and only if T^^ = follows as in observation T H E O R E M 1.2: Let the approximate solution to the impedance boundary-value problem be given by (9.22), where the vector of amplitudes Tjsj solve the system (9.25) Then lim ||E«o-E,^o|l2n = 0, (9-29) where E50 is the far-field pattern of the exact solution E^, Hg Proof: We debut by showing that the sequence H ^ is bounded for any N In (9.24) we multiply the first set of N equations by ja* , the second one by jbl and sum the resulting expressions We get as before (n X E,^ - n X (n X H ^ ) - f, H^)^^^ = (9.30) PROJECTION METHODS 211 Taking the real part of (9.30), using (9.12), and the identity | a ~ b | |a|^ 4- |b|^ - Re (a • b*), we find that = + /|v/RiTnxHf-^nxf (9.31) The above relation shows that the sequence H ^ is bounded for any N Since the system {n X ^l 4-J7n x (n x #^) ,n x ^l+nn i/ = l,2, / R e > } x (n x ^l), is complete in ^tanC*^) there exist the vector fields (9.32) N such that \imJ\e!,\\2S=0 (9.33) and lim ||E30-5iX|U = 0- (9-34) Here, cjv = n x £^ — jnx (n x H^) — f represents the discrepancies of the tangential fields on the surface 5, and £^ is the far-field pattern of the approximate solution £^, H^ Then, we use (9.24) and (9.35) (nxff-7nx(nxKf)-f,*^>2,5 = (€7V,*^>2,5 = {eN,^l\s to derive the following set of equations (9.36) (nx5Ef-7nx(nx6Hf),*3)^^ 212 CHAPTER IX DSM IN ELECTROMAGNETICS for the residual fields SE^ = f f - E f and 6H^ = W f - H ^ Since 6Ef and H ^ are expressed as linear combinations of ^ ^ and ^^, fi = 1,2, ,A/^, we conclude that / [(n X E f ) 6H^* + 7|n x H f p] dS = f CN - m f *d5, s (9.37) s or, equivalently, that /|(5E^oPda;+ / R e | n X ( H f | d Q = S Rel f CN - 6H^*dS' I < s ||6;.||2,5||5Hf||,_^ (9.38) The uniform boundedness of the sequence H ^ with respect to A^ and (9.38) gives The triangle inequality and (9.34) may now be used to conclude From (9.26) we see that in the case of axisymmetric scatterers the surface integrals simplify to one-dimensional integrals along the particle generator The problem decouples over the azimuthal modes and therefore the amount of computer storage required to solve the scattering problem is not excessive high In contrast, for particles without rotational symmetry it is not possible to obtain a separate solution for each azimuthal mode Consequently, the dimensions of the linear systems of equations considerably increases This leads to increased difficulties that are associated with the stability of the solutions Effective solutions of these systems appear to be possible only by means of iterative schemes It is therefore reasonable to analyze the correct solvability of the system (9.25) in order to investigate the applicability of iterative schemes For the system of localized spherical vector wave functions we can prove the following theorem THEOREM 1.3: The system (9.25) is correctly solvable for any fixed N Proof: We have ^ / (9 40) MODEL WITH DISTRIBUTED VECTOR MULTIPOLES ^ (amn COS + jbmn)(^mn n=l 225 {0) = 0, (9.71) 'I'max y ^ Cn sin 9uon{0) = , n=l for all m = 0,1, , mmax and any € [0, TT] From (9.71) it follows that the far-field patterns of the vector fields £im — ^^'ZT ^rnn ^2m = T.IZX bmn {J^%i " A^^na) and £3 = Y^lZX CnA/?„^3 Vanish on fi for all m = 0,1, ,mmax- Hence, the fields 5im, f2m and £z vanish in Dg and also in B? - {z^e^rrZr • Since M^^„^ + A^^f^j, A ^ k i - A/^n2 and yV?„=3 are singular at z^ = Zn^3 we can proceed as in theorem 2.34 of Chapter to conclude In the case of the TE polarization we have Eo(x) = ej,e-'^'^^^'^^^'^^'"^+^^°^^\ Ho(x) = (cOS7e:, - S i n i e j (9.72) ^-jk{pcos^sm^-^zcos^) and the approximate solution read as m=0 n = l + t ^ „ l A e n ( x ) + A/;^„^„i(x)] + X ; C " - A ^ r n ( x ) , X € D n=l (9.73) A similar result as stated in theorem 2.4 holds for the approximate solution (9.73) The far-field pattern E ^ has the components ^U^) = ^ I ] E(-Jrsin(m+lMa^„-jr2{Kp^mi)\G{z,i) V /^s (9.88) for TM polarization and by '^m^M) = ~(-Jr[*/m(fc«psin7) - Jni+2(fc«psin7)]G(2;,7), eXi(^) = (-Jre?[J^(fc,/9sin7)-hJm+2(fc./)sin7)]G(^,7), ^m+i(^) = {-j)'^J^cos'y[Jm{kspsm^) + Jm+2(fc3psin7)]G(2:,7), V /^a ^^+i(^) = (""^')'^i/r- {e?cos7(J^(fc^psin7) Jn,+2(fc5/>sin7)] V "a + 2je^sin7Jr„+i(fc5/^sin7)}G(^,7) (9.89) for TE polarization Here, e^ and e* are the components of the unit vector Cr which is tangent to L For the harmonic not depending on (p we have two separate systems (for either TM or TE polarization) B_ip_i=q_i, (9.90) with a right-hand side vector of length 2jmax and a 2jniax x 2ninax matrix B_i Note, that in this case we have ^"^{v) = ^''iv) = -[;e?cos7Ji(fcapsin7) 4- e^sin7Jo(fc5/)sin7)]G(>2:,7), -jJ^Ji{kspsm-f)G{z,j) V f^8 (9.91) MODEL WITH DISTRIBUTED VECTOR MULTIPOLES 231 for TM polarization and ^''(v) = '^o'^(^) = jJi{kspsin'y)G{z,y), -,/—[je?cos7Ji(A:5psin7) 4-e*sin7Jo(A^5psin7)]G'(2:,7) V ^s (9.92) for TE polarization At first glance we have to consider two kinds of systems (9.86), separately for TM and TE cases However, the fortunate of the proposed scheme is that it is possible to use only one matrix B™ (corresponding to TM case) for both TM and TE polarizations To show this, let us write the matrix equation for the TM polarization as AST rnj _mjr, B:mjn ^s nsip —BZ^ - mjn mjn Mi ^mn mn ^mn L ^mn TM -5 AS