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

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)),

and using the relation Q3 Q3 = I, we deduce that

Because M is upper-triangular with real diagonal elements, it follows that M4 is unit upper-triangular (all diagonal elements are equal 1). Then, (2.24) implies that M4 = I, and therefore

†

S = −Q3 Q3 .

In terms of the transition matrix, the above equation read as

†

T = −Q3 Re Q3

and this new sequence of truncated transition matrices might be expected to converge more rapidly than (2.23) because the unitarity condition is satisfied for each truncation index.

2.1.3 Symmetries of the Transition Matrix

Symmetry properties of the transition matrix can be derived for specific particle shapes. These symmetry relations can be used to test numerical codes as well as to simplify many equations of the T -matrix method and develop e cient numerical procedures. In fact, the computer time for the numerical evaluation of the surface integrals (which is the most time consuming part of the T -matrix calculation) can be substantially reduced. The surface integrals are usually computed in spherical coordinates and for a surface defined by

σ(r) = er − r ∂θ eθ − r sin θ ∂ϕ eϕ

is a vector parallel to the unit normal vector n. In the following analysis, we will investigate the symmetry properties of the transition matrix for axisymmetric particles and particles with azimuthal and mirror symmetries.

1. For an axisymmetric particle, the equation of the surface does not depend on ϕ, and therefore ∂r/∂ϕ = 0. The vector σ has only er - and eθ - components and is independent of ϕ. The integral over the azimuthal angle can be performed analytically and we see that the result is zero unless m = m1.

94 2 Null-Field Method

Therefore, the T matrix is diagonal with respect to the azimuthal indices m and m1, i.e.,

2. We consider a particle with a principal N -fold axis of rotation symmetry and assume that the axis of symmetry coincides with the z-axis of the particle coordinate system. In this case, r, ∂r/∂θ and ∂r/∂ϕ are periodic in ϕ with period 2π/N . Taking into account the definition of the Qpq matrices, we see that the surface integrals are of the form

2π

f (ϕ) ej(m1−m)ϕdϕ ,

As a consequence of the rotation symmetry around the z-axis, the T matrix is invariant with respect to discrete rotations of angles αk = 2πk/N , k = 1, 2, . . . , N − 1. A necessary and su cient condition for the equation

to hold is |m1 − m| = lN , l = 0, 1, . . ., and we conclude that

behavior is usually attended: the relative variations of the di erential scattering cross-sections decrease with increasing the maximum expansion order, attain a relative constant level and afterwards increase. In this case, the main problem of the convergence analysis is the localization of the region of stability. The extinction and scattering cross-sections can also give information on the convergence process. Although the convergence of Cext and Cscat does not guarantee that the di erential scattering cross-section converges, the divergence of Cext and Cscat implies the divergence of the T -matrix calculation.

2.1.5 Surface Integral Equation Method

The null-field method leads to a nonsingular integral equation of the first kind. However, in the framework of the surface integral equation method, the transmission boundary-value problem can be reduced to a pair of singular integral equations of the second kind [97]. These equations are formulated in terms of two surface fields which are treated as independent unknowns. In order to elucidate the di erence between the null-field method and the surface integral equation method we follow the analysis of Martin and Ola [155] and review the basic boundary integral equations for the transmission boundaryvalue problem. We consider the vector potential Aa with density a

and evaluate the tangential components of the curl and double curl of the vector potential on S. For continuous tangential density, we have [39]

where M is the magnetic dipole operator,

For a su ciently smooth tangential density, we also have

where the principal value singular integral operator P, called the electric dipole operator is given by

Considering the null-field equations for the electric and magnetic fields in Di and Ds, and passing to the boundary along a normal direction we obtain

98 2 Null-Field Method

and

respectively. These are four boundary integral equations for the unknowns ei and hi, and we consider two linear combinations of equations, i.e.,

α1 (2.26) + α2 (2.27) + α3 (2.29) and β1 (2.26) + β2(2.27) + β3(2.28) ,

where αi and βi, i = 1, 2, 3, are constants to be chosen. Harrington [97] describes several possible choices, as shown in Table 2.1, and for all these choices we always have existence and unique solvability [155].

The above approach for deriving a pair of boundary integral equations is known as the direct method. In contrast to the null-field method, the direct method considers the null-field equations in both domains Di and Ds, and treats the surface fields as independent unknowns. The indirect method for deriving a pair of boundary integral equations relies on the representation of the electromagnetic fields in terms of four surface fields. Passing to the boundary, using the boundary conditions on the particle surface and imposing two constraints on the surface fields, yields the desired pair of boundary integral equations [97]. It should be emphasized that single integral equations for the transmission boundary-value problem have also been derived by Marx [156], Mautz [157] and Martin and Ola [155]. Another major di erence to the nullfield method is the discretization of the boundary integral equations, which is achieved by using the method of moments [96]. The boundary surface is discretized into a set of surface elements and on each element, the surface fields are approximated by finite expansions of basis functions. Next, a set of

Table 2.1. Choice of constants for the boundary integral equations