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

134 2 Null-Field Method

gradient methods and iterative approaches [74, 95, 198, 244, 271]. Originally all methods were implemented for spherical particles, but Xu [274] recently extended his computer code for axisymmetric particles.

In the following analysis, we derive (2.144) and (2.145) by using the superposition T -matrix method. The field exciting the lth particle can be expressed as

N

Eexc,l (rl) = Ee (rl) + Es,p (rl) ,

p=l

where Ee is the incident field and Es,p is the field scattered by the pth particle. In the null-field method, transformation rules between the expansion coe cients of the incident and scattered fields in di erent coordinate systems have been derived by using the integral representations for the expansion co- e cients. In the superposition T -matrix method, these transformations are obtained by using the series representations for the electromagnetic fields in di erent coordinate systems. For the external excitation, we consider the vector spherical wave expansion

Ee(r) = aµM 1µ (ksr) + bµN 1µ (ksr)

µ

and use the addition theorem

Similarly, for the field scattered by the pth particle, we consider the series representation

Es,p (rp) = fp,µM 3µ (ksrp) + gp,µN 3µ (ksrp)

µ

and use the addition theorem

to derive

Es,p (rl) = flp,ν M 1ν (ksrl) + glp,ν N 1ν (ksrl)

ν

with

Thus, the field exciting the lth particle can be expressed in terms of regular vector spherical wave functions centered at the origin Ol

where the expansion coe cients are given by

and as usually, el = [al,ν , bl,ν ]T. Further, using the T -matrix equation sl = T lel, we obtain

p=l

and since (cf. (B.74) and (B.75))

Tν31µ (ksrlp) = Tµν31 (ksrpl)

and

N

Es(r) = Es,l (rl)

l=1

whence, using the addition theorem

138 2 Null-Field Method

T matrix of the previous system of n − q components. In this section we use the recursive T -matrix algorithm to analyze electromagnetic scattering by a system of identical particles randomly distributed inside an “imaginary” spherical surface.

Let us consider Ncs circumscribing spheres with radii Rcs(k), k = 1, 2, . . . , Ncs, in increasing order. Inside the sphere of radius Rcs(1) there are N(1) particles having their centers located at r(1)0l , l = 1, 2, . . . , N(1), and in each spherical shell bounded by the radii Rcs(k − 1) and Rcs(k) there are N(k) particles having their centers located at r(0kl ), l = 1, 2, . . . , N(k). Obviously, N = N(p) is the total number of particles and Rcs(Ncs) is the radius of the circumscribing sphere containing the particles. For simplicity, we assume that the particles are identical and have the same spatial orientation. The scattering geometry for the problem under examination is depicted in Fig. 2.7.

At the first iteration step, we compute the system T -matrix T (1) of all particles situated inside the sphere of radius Rcs(1). At the iteration step k, we compute the system T -matrix T (k) of all particles situated inside the sphere of radius Rcs(k) by considering a [N(k) + 1]-scatterer problem. In fact, the transition matrix T (k) is computed by using the system T -matrix T (k−1) of the previous ,kp=1−1 N(p) particles situated inside the sphere of radius Rcs(k − 1) and the individual transition matrix T p of all N(k) particles situated in the spherical shell between Rcs(k − 1) and Rcs(k). In this specific case, the blockmatrix components of the global matrix A are given by

AN(k)+1,l = −T (k−1)T 31 ksr(0kl ) , l = 1, 2, . . . , N(k) .

Fig. 2.7. Illustration of the recursive T -matrix algorithm for many spheres