Acoustic and electromagnetic scattering analysis using discrete sources VII systems of functions in electromagnetic theory

Acoustic and electromagnetic scattering analysis using discrete sources VII systems of functions in electromagnetic theory Acoustic and electromagnetic scattering analysis using discrete sources VII systems of functions in electromagnetic theory Acoustic and electromagnetic scattering analysis using discrete sources VII systems of functions in electromagnetic theory Acoustic and electromagnetic scattering analysis using discrete sources VII systems of functions in electromagnetic theory Acoustic and electromagnetic scattering analysis using discrete sources VII systems of functions in electromagnetic theory

VII SYSTEMS OF FUNCTIONS IN ELECTROMAGNETIC THEORY In this chapter we will analyze complete and linear independent systems of functions for the Maxwell equations We will construct complete systems in £tan('S') and in the product spacefi^anC'^)-Complete systems in ^^anl-^) will be used to solve the exterior Maxwell and the impedance boundaryvalue problems, while complete systems in Xl^an('S') will be employed to solve the transmission boundary-value problem We begin our analysis by presenting some fundamental results on the completeness of the localized spherical vector wave functions To preserve the completeness at irregular frequencies, linear combinations of these functions will be considered We then pay attention to the systems of localized vector multipoles In Chapter we will apply these results to axisymmetric geometries by taking into account the polarization of the external excitation We will then proceed to analyze the completeness properties of the systems of distributed sources We start with the spherical vector wave functions and vector multipoles distributed on a straight line Our analysis is based on the addition theorem for spherical wave and vector wave functions The next sections concern the completeness of the system of magnetic and electric dipoles and the system of vector Mie potentials with singularities distributed on auxiliary closed and open surfaces These 137 138 CHAPTER VII SYSTEMS OF FUNCTIONS IN ELECTROMAGNETICS functions are suitable for analyzing the scattering by particles without rotational symmetry The last section of this chapter deals with the linear independence of these systems COMPLETE SYSTEMS OF FUNCTIONS The completeness properties of the systems of discrete sources are of primary interest since they provide a means for approximating the exact solutions to the scattering problems For instance, the set of radiating spherical vector wave functions is known to be complete in /^tanC*^)- Consequently, any radiating solution to Maxwell equation can be approximated uniformly in closed subsets of Dg and in the mean square sense on by a sequence of linear combinations of spherical vector wave functions In this section we will present these basic results for localized and distributed sources 1.1 Localized spherical vector wave functions and vector multipoles We begin our analysis by defining our notations The independent solutions to the vector wave equations VxVxX-fe2x = o (7.1) Mii?.(x) = V«ii?,(x) X X, Nii^Jx) = i v X M^i^Jx), (7.2) can be constructed as where n = 1,2, , m = —n, ,n, and in spherical coordinates the uj^^ are the spherical wave functions The specific forms of the spherical vector wave functions are ' PJr^icose) dPi^'(cose) sm N M (X) = | n ( n + ifJ^Mp^r^cosd) ^TrTM^'ikr)] AP]r\cose) aO ^ pjrmp e P^^'(cos0) sm ^ (7.3) pjm(p where {er^eg^e^) are the unit vectors in spherical coordinates The superscript *r stands for the regular spherical vector wave functions while the superscript '3' stands for the radiating spherical vector wave functions It is useful to note that for n = m = we have MJJ? = NQJ? = Mj„^, N^ IS an entire solution to the Maxwell equations and M^^, N^^ is a radiating solution to the Maxwell equations in R^ - {0} 139 COMPLETE SYSTEMS OF FUNCTIONS The spherical vector wave expansion of the dyadic ^I is of basic importance in our investigation It is 5(x,y,fc)I=^5; £ D„ n—\ m^^n \ [M3 (y)Mi,„(x) + Ni_(y) |x| (7.4) X < [ML^„(y)ML(x) + N!._(y)NL(x)] [ -f Irrotational terms, |y( < |x| where the normalization constant Dmn is given by 2n + l (n-|m|)! (7.5) •^mn — 4n(n + l) (n+|m|)!" Using the calculation rules for dyadic functions and the identity ag = a-^I we find the following simple but useful expansions VxX[a(y)g(y,x,fc)] = : i ^ ; J2 A 71=1171=—n {[a(y).Mi„„(y)lNU(x) + [a(y) • Ni„„(y)j MlJx)} , |y| > |x| (7.6) X < {[a(y)-ML„„(y)lN^„(x) + [a(y) • Nl^Jy)] M3.„(X)} , |y| < |x| and Vx X Vx X (a(y)p(y,x,/^)] = ^ E E ^' n=rl Tn=—n [ {[a(y).Mi„„(y)lMj„„(x) + [a(y) • Ni„„(yj] Nj„„(x)} |y| > |x| (7.7) {[a(y).Mi^„(y)lM^„(x) [ +[a(y).Ni^„(y)]NUx)},|y| 0} are complete in Cl^^{S) Here { a j } ^ ^ is a basis ofN ( ^ J 4- M) andmM = dim7V(il + Al) Proof: We consider (a) Let ya*.(nxM^„)d5 = 0, s (7.8) ya*(nxN^„)d5 = 0, for n = 1,2, , m = - n , ,n, and a € C^^ni'^)- Consider the vector field £ = {j/k)V X V X Aa' with density a' = n x a* For x € £>[, where D[ is the interior of a spherical surface S^ enclosed in i?i, we have fc2 ^w = ~v^ S ^^ /a'(y)-Mi^„(y)d5(y) Mj„„(x) n = l m——n + y"a'(y)-Ni„„(y)d5(y) Ni,„(x) Ls (7.9) The closure relations show that £ = m Di, Application of theorem 2.2 given in Chapter finishes the proof of (a) COMPLETE SYSTEMS OF FUNCTIONS 141 The second part of the theorem corresponding to the casefc^ l ) (n X N L ) , n = 1,2, , m = - n , ,n/ k ^ (T{Di)} , (7 18) is also complete in ^?an('^)- "^^ prove this assertion we debut with ya*.(ij + M)(nxMDdS = 0, (7.19) |a'.(iJ + Al)(nxNDd5 = 0, 143 COMPLETE SYSTEMS OF FUNCTIONS for n = 1,2, , m = —n, ,n, and a G£tan('^)- Using the definition of the adjoint operator with respect to the L^ bilinear form we rewrite the closeness relations as (7.20) n = 1,2, , m = - n , ,n, where a' = a* The completeness of the radiating spherical vector wave functions in £tan('S') yields {^I-\- M^) a' = almost everywhere on S Therefore ( | l ~ M) (n x a') = almost everywhere on and employing the same arguments as in theorem 2.2 of Chapter we receive n x a ' ' ^ ^ n x a o € C^^ndi"^)- Using this and the fact that k ^ (T{Di) we deduce that V x Anxa;, vanishes in A - Theorem 2.2 of the precedent chapter may now be used to conclude Next we will analyze complete systems of vector functions in the product space £?an(5') = C^a,n{S) X >C?an(5') We recall that £'^{S) is a Hilbert space endowed with the scalar product and obviously £tan('S') is a subspace of £^(5) THEOREM 1.2: Let S be a closed surface of class C^ with unit outward normal n Then the system of vector functions n X M^i, -7\/^nxN^L \ / n X N^i n = 1,2, , m = —n, ,n} is complete in S't&ni'^)' ^^^ radiating spherical vector wave functions are defined with respect to the wave number kg, while the regular spherical vector wave functions are defined with respect to the wave number ki Proof: It has to shown that for tions / aJ.(nxM^J,)+a| (iO € £tan('S') *^he closure rela- -j./-^nxN^J, d5 = 0, / (7.21) al- (n X N^i„) + a2- - j ^ n x M^J, dS = 0, 144 CHAPTER VII SYSTEMS OF FUNCTIONS IN ELECTROMAGNETICS n = 1,2, , m = —n, ,n, gives Si '-^ and a2 ^ on Setting = n X aj and ig = n x £2 we rewrite (7.21) as /(5^N!:L+jY^5i.M^jJ„„)d5 = 0, (7.22) |(ai.Mii„+jYgai.Nii„)d5 = 0, s n = 1,2, , m = —n, ,n Prom (7.6) and (7.7) we find that '^''•^t +^ ^ ' ' ' ^ ' ' - ^ \ = O^"^' (7.23) Application of theorem 2.6 given in Chapter finishes the proof of the theorem The same technique can be used to prove the completeness of the system n X MrtiL J ,^nxN'^\ \ / n X N^i M jJtl^nxM^ n = 1,2, , m = —n, , n} We now turn our attention to the general null-field equations for the exterior Maxwell boundary-value problem The following theorem states the equivalence between the null-field equations formulated in terms of spherical vector wave functions and the general formulation (6.103) THEOREM Jinx 1.3: Let hg solve the set of null-field equations hs)'{n X M ^ , ) d = - j / ( n x eo) • (n x N ^ ) d5, (7.24) Jinx s h , ) (n X N ^ ^ ) d = -J J (^ x eo) • (n x M ^ d S , s for n = 1,2, , m = —n, , n Then hg solve the general null-field equation (6.103) and conversely Proof: The proof follows from the vector spherical waves expansion of the electric field = - V X Aeo + ^ V X V X A,,, (7.25) COMPLETE SYSTEMS OF FUNCTIONS 145 inside a sphere enclosed in Dj The unique solvabihty of the null-field equations (7.24) follows from the completeness of the system of vector functions given by theorem 1.1 Let us consider the general null-field equations for the transmission boundary-value problem We have the following result THEOREM 1.4: Let e and h solve the set of null-field equations: J{[nx{e-eo)]-{nxMl„) s +j y / ^ [ n x ( h - h o ) ] ( n x N D } d / = 0, {[nx(e-eo)]-(nxN^„) + j y j [ n x ( h - h o ) ] ( n x M L ) } d = 0, I [(n X e) • (n X M ^ ) + j ^ (n x h) • (n x N ^ ) ] d5 = 0, y " [ ( n x e ) - ( n x N j „ „ ) + i y ^ ( n x h ) ( n x M U ] d = 0, (7.26) n = 1,2, , m = —n, ,n, where the radiating spherical vector wave functions are defined with respect to the wave number kg, while the regular spherical vector waveJunctions are defined with respect to the wave number ki Then e and h solve the general null-field equations (6.113) and conversely Proof: The proof of the theorem is provided by the spherical vector wave expansions of the electric fields ^ = V X A|_,„ + ^ V X V X A|_-,^ (7.27) and Si = VxAi-\- ^ V X V X Ai (7.28) inside a sphere enclosed in Di and outside a sphere enclosing Dj, respectively 146 CHAPTER VII SYSTEMS OF FUNCTIONS IN ELECTROMAGNETICS The completeness of the system of vector functions given by theorem 1.2 impUes that the particular null-field equations are uniquely solvable in £?an(5) Let us now investigate the completeness properties of the systems of magnetic and electric vector multipoles In Cartesian coordinates they are defined by Mj^3„p(x) = i v X («ii?.(x)ep) , Nklpi^) = ^ V x Ml^„^{x), (7.29) where p = 1,2,3, n = 0,1, , m = —n, , n, and Cp denote the Cartesian unit vectors Note that Mj^^^p, ^Innp ^^ ^^ entire solution to the Maxwell equations and Mj^^p, 'N^^p ^s a radiating solution to the Maxwell equations in R - {0} Accounting of the series representation of the Green function and the identity (Vx X (a(y)y(x,y,fc))] • Cp = (a(y) x Vyp(x,y,fc)) • Cp = - a ( y ) - [ep x Vyp(x,y,fc)] = [Vy x (epp(x,y,fc))] • a(y), x 7^ y, (7.30) we find the expansion [Vxx(a(y)^(x,y,fc))].ep = ^ ] X^ P,mn n=Om=—n a(y) • Mi„„p(y)uJ„„(x), |y| > |x| ( ^^'^^^ a(y)-ML„„JyR„(x), |y| •••, (^max —!)• The latter implies that Q;mn„,ax = for all m = — rimax? •••j^max and arguing as above we can show step by step that all coefficients are zero T H E O R E M 2.32: Let S he a surface of class C^ and let n denotes the unit outward normal to S Then the system n X M!;! \ / n X N^ii 1,2, , m = ~n, is linear independent in £?an('S')- The radiating spherical vector wave functions are defined with respect to the wave number kg, while the regular spherical vector wave functions are defined with respect to the wave number ki 188 CHAPTER VII SYSTEMS OF FUNCTIONS IN ELECTROMAGNETICS Proof: To prove the theorem we debut with for some constants a^^^ and /JJ^;*^, n = 1,2, , rimax? ^ = —^, • ing the electromagnetic fields ^^ = E E < n M L + / ? m n N L , « = 7 — V X £, (7.183) and ^i = - E E "mnM;„„ + i9J„„Ni,„, W, = - — V X Si (7.184) J*JoMt n = l m=—n it is readily seen that (7.182) leads ionxiEgEA = and n x \Hs ~ Wi) = on ^ y the uniqueness^n the transmission boundary-value problem we obtain 5^ = in Ds and £^j = in D^ Theorem 2.31 now completes the proof The following theorem states the linear independence of a finite collection of spherical vector wave functions with different origins THEOREM 2.33: Consider S a surface of class C^ and let {xop} ^ be a finite collection o poles located inside Di Replace in theorem 2.31 the localized spherical vector wave functions M^J^ and N^J^, n = 1,2, , m = —n, , n, by the multiple spherical vector wave functions M^^p ^'^^J^npiP = 1,2, ,P, n = 1,2, , m = —n, ,n, respectively Then, the resulting systems of vector functions are linearly independent in C^g^j^{S) Proof: Let us prove (a) Consider E E E «mnpn(y) X Mlnpiy) p=l n=l m=—n ^^ Pmnp^iv) X ^mnp(y) = 0, y G 5, (7.185) LINEAR INDEPENDENT SYSTEMS OF FUNCTIONS 189 for some constants amnp and /?^^p, p = 1,2, , P, n = 1,2, , nP^^, m = —n, , n It is readily seen that P ^max n S S S «mnpA