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

for ν = 1, 2, . . . , N . Inserting (2.7) into (2.16) yields

where s = [fνN , gνN ]T is the vector containing the expansion coe cients of the scattered field. Combining (2.10) and (2.17) we deduce that the transition matrix relating the scattered field coe cients to the incident field coe cients, s = T e, is given by

In view of (2.11)–(2.14), we see that the transition matrix computed in the particle coordinate system depends only on the physical and geometrical characteristics of the particle, such as particle shape and relative refractive index and is independent of the propagation direction and polarization states of the incident and scattered fields. In contrast to the infinite transition matrix discussed in Sect. 1.5, the transition matrix given by (2.18) is of finite dimension, and the often used appellation approximate or truncated transition matrix is justified.

2.1.2 Instability

Instability of the null-field method occurs for strongly deformed particles and large size parameters. A number of modifications to the conventional approach has been suggested, especially to improve the numerical stability in computations for particles with extreme geometries. These techniques include formal modifications of the single spherical coordinate-based null-field method by using discrete sources [49, 109], di erent choices of basis functions and the application of the spheroidal coordinate formalism [12, 89], and the orthogonalization method [133, 256]. For large particles, the maximum convergent size parameter can be increased by using a special form of the LU-factorization method [261] and by performing the matrix inversion using extended-precision [166]. For strongly absorbing metallic particles, Gaussian elimination with backsubstitution gives improved convergence results [174].

Formulation with Discrete Sources

The conventional derivation of the T matrix relies on the approximation of the surface fields by the system of localized vector spherical wave functions. Although these wave functions appear to provide a good approximation to the solution when the surface is not extremely aspherical, they are disadvantageous when this is not the case. The numerical instability of the T -matrix calculation arises because the elements of the Q31 matrix di er by many orders of magnitude and the inversion process is ill-conditioned. As a result, slow convergence or divergence occur. If instead of localized vector spherical

90 2 Null-Field Method

functions we use distributed sources, it is possible to extend the applicability range of the single spherical coordinate-based null-field method. Discrete sources were used for the first time in the iterative version of the null-field method [108, 109, 132]. This iterative approach utilizes multipole spherical expansions to represent the internal fields in di erent overlapping regions, while the various expansions are matched in the overlapping regions to enforce the continuity of the fields throughout the entire interior volume.

In the following analysis we summarize the basic concepts of the nullfield method with distributed sources. The distributed vector spherical wave functions are defined as

where {zn}∞n=1 is a dense set of points situated on the z-axis and in the interior of S, ez is the unit vector in the direction of the z-axis, n = 1, 2, . . . , m Z, and l = 1 if m = 0 and l = 0 if m = 0. Taking into account the Stratton– Chu representation theorem for the incident field in Di we rewrite the general null-field equation as

and derive the following set of null-field equations:

$%

The formulation of the null-field method with distributed vector spherical wave functions relies on two basic results. The first result states that if ei and hi solve the set of null-field equations (compare to (2.20))

92 2 Null-Field Method

The expressions of the distributed vector spherical wave functions with the origins located in the complex plane are given by (B.31) and (B.32).

Various systems of vector functions can be used instead of localized and distributed vector spherical wave functions. Formulations of the null-field method with multiple vector spherical wave functions, electric and magnetic dipoles and vector Mie potentials have been given by Doicu et al. [49]. Numerical experiments performed by Str¨om and Zheng [218] demonstrated that the system of tangential fields constructed from the vector spherical harmonics is suitable for analyzing particles with pronounced concavities. When the surface fields have discontinuities in their continuity or any of their derivatives, straightforward application of the global basis functions provides poor convergence of the solution. Wall [247] showed that sub-boundary bases or local basis approximations of the surface fields results in larger condition numbers of the matrix equations.

Orthogonalization Method

To ameliorate the numerical instability of the null-field method for nonabsorbing particles, Waterman [256, 257] and Lakhtakia et al. [133, 134] proposed to exploit the unitarity property of the transition matrix. To summarize this technique we consider nonabsorbing particles and use the identity Q11 = Re{Q31} to rewrite the T -matrix equation

as

QS = −Q ,

where Q = Q31 and S = I + 2T . Applying the Gramm–Schmidt orthogonalization procedure on the row vectors of Q, we construct the unitarity matrix

where M is an upper-triangular matrix with real diagonal elements and M4 = M (M )−1. Taking into account the unitarity condition for the S matrix (cf. (1.109)),