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

If A = A−1, and Alp, l, p = 1, 2, . . . , N(k) + 1, are the block-matrix components of A, the recurrence relation for T -matrix calculation read as

The procedure is repeated until all N particles are exhausted. At each iteration step, only a [N(k) + 1]-scatterer problem needs to be solved, where N(k) usually is much smaller than N. Thus, we can keep the dimension of the problem manageable even when N is very large. The geometric constraint of the T -matrix method requires that the N(k) particles completely reside inside the spherical shell between Rcs(k − 1) and Rcs(k). However, for a large number of particles and small size parameters, numerical simulations certify the accuracy of the recursive scheme even if this geometric constraint is violated.

As mentioned before, the T 11(−ksr0l) matrices are used to translate the incident field from the global coordinate system to the lth particle coordinate system, while the T 11(ksr0l) matrices are used to translate the scattered fields from the basis of the lth particle to the global coordinate system. A recursive T -matrix algorithm using phase shift terms instead of translation matrices has been proposed by Auger and Stout [4].

2.7 Composite Particles

A composite particle consists of several nonenclosing parts, each characterized by arbitrary but constant values of electric permittivity and magnetic permeability. The null-field analysis of composite particles using the addition theorem for regular and radiating vector spherical wave functions is equivalent to the multiple scattering formalism [189]. The translation addition theorem for radiating vector spherical wave functions introduces geometric constraints which are not fulfilled for composite particles. Several alternative expressions for the transition matrix have been derived by Str¨om and Zheng [219] using the Q matrices for open surfaces (the interfaces between the di erent parts of a composite particle). In the present analysis we avoid the use of any local origin translation and consider a formalism based on closed-surface Q matrices.

2.7.1 General Formulation

To simplify our presentation we first consider a composite particle with two homogeneous parts as shown in Fig. 2.8. We assume that the surfaces S1c = S1 S12 and S2c = S2 S12 are star-shaped with respect to O1 and O2,

140 2 Null-Field Method

P

n1

M1 r1 r19

Fig. 2.8. Geometry of a composite particle

respectively. The main reason for introducing the star-shapedness restriction is that we want to keep the individual constituents reasonably simple. A novel feature is the appearance of edges, which are allowed by the basic regularity assumptions of the T -matrix formalism [256]. A composite particle can be treated by the multiple scattering formalism under appropriate geometrical conditions. The boundary-value problem for the composite particle depicted in Fig. 2.8 has the following formulation.

Given the external excitation Ee, He as an entire solution to the Maxwell equations, find the scattered field Es, Hs and the internal fields Ei,1, Hi,1 and Ei,2, Hi,2 satisfying the Maxwell equations

on S2, and

on S12, and the Silver–M¨uller radiation condition for the scattered field (2.3).

The Stratton–Chu representation theorem for the incident and scattered fields in Di, where Di = Di,1 Di,2 S12 = R3−Ds, together with the boundary conditions (2.154) and (2.155), lead to the general null-field equation

g (ks, r, r ) = g (ks, r1, r1) , g (ks, r, r ) = g (ks, r1, r1 ) ,

restricting r1 to lie on a sphere enclosed in Di,1 and taking into account the integral form of the boundary conditions (2.156)

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