Файл: Doicu A., Wriedt T., Eremin Y.A. Light scattering by systems of particles (OS 124, Springer, 2006.pdf

R∞

Fig. C.2. Illustration of the domain of integration bounded by the spherical surface SR and a sphere situated in the far-field region

where

∞

Gn (Ks, ks, R) = [g (2Rx) − 1] h(1)n (2ksRx) jn (2KsRx) x2dx.

1

C.3 Computation of the Terms S11nn and S12nn

294 C Computational Aspects in E ective Medium Theory

for fixed values of the indices n and n . This function can be expanded in terms of Legendre polynomials

and therefore,

S11nn = snn .

Analogously, we find that

S12nn = −snn .

D

Completeness

of Vector Spherical Wave Functions

The completeness and linear independence of the systems of radiating and regular vector spherical wave functions on closed surfaces have been established by Doicu et al. [49]. In this appendix we prove the completeness and linear independence of the regular and radiating vector spherical wave functions on two enclosing surfaces. A similar result has been given by Aydin and Hizal [5] for both the regular and radiating vector spherical wave functions with nonzero divergence, i.e., the solutions to the vector Helmholtz equation.

Before formulating the boundary-value problem for the Maxwell equations we introduce some normed spaces which are relevant in electromagnetic scattering theory. With S being the boundary of a domain Di we denote by [39, 40]

Ctan(S) = {a/a C(S), n · a = 0}

the space of all continuous tangential fields and by

Ctan0,α(S) = a/a C0,α(S), n · a = 0 , 0 < α ≤ 1 ,

the space of all uniformly H¨older continuous tangential fields equipped with the supremum norm and the H¨older norm, respectively. The space of uniformly H¨older continuous tangential fields with uniformly H¨older continuous surface divergence ( s) is defined as

while the space of square integrable tangential fields is introduced as

L2tan(S) = a/a L2(S), n · a = 0 .

Now, let S1 and S2 be two closed surfaces of class C2, with S1 enclosing S2 (Fig. D.1). Let the doubly connected domain between S1 and S2 be denoted by Di,1, the domain interior to S2 by Di,2, the domain interior to S1 by Di,

296 D Completeness of Vector Spherical Wave Functions

n1

n2

εi,µi O

Fig. D.1. The closed surfaces S1 and S2

and the domain exterior to S1 by Ds. We assume that the origin is in Di,2, and denote by n1 and n2 the outward normal unit vectors√to S1 and S2, respectively. The wave number in the domain Di,1 is ki = k0 εiµi, where k0 is the wave number in free space, and εi and µi are the relative permittivity and permeability of the domain Di,1. We introduce the product spaces

Ctan(S1,2) = Ctan(S1) × Ctan(S2),

C0tan,α(S1,2) = Ctan0,α(S1) × Ctan0,α(S2) ,

C0tan,α,d(S1,2) = Ctan0,α,d(S1) × Ctan0,α,d(S2) ,

L2tan(S1,2) = L2tan(S1) × L2tan(S2),

and define the scalar product in L2tan(S1,2) by

The Maxwell boundary-value problem for the doubly connected domain Di,1 has the following formulation.

Find a solution Ei, Hi C1(Di,1) ∩ C(Di,1) to the Maxwell equations in

Di,1

× Ei = jk0µiHi , × Hi = −jk0εiEi,

satisfying the boundary conditions

where f 1 and f 2 are given tangential vector fields.

Denoting by σ(Di,1) the spectrum of eigenvalues of the Maxwell boundaryvalue problem, we assume that for f 1 Ctan0,α,d(S1) and f 2 Ctan0,α,d(S2), both