Analysis of the method of generalized separation of variables in the problem of light scattering by small axisymmetric particles

In the problem of light scattering by small axisymmetric particles, we have constructed the Rayleigh approximation in which the polarizability of particles is determined by the generalized separation of variables method (GSVM). In this case, electric-field strengths are gradients of scalar potentials, which are represented as expansions in term of eigenfunctions of the Laplace operator in the spherical coordinate system. By virtue of the fact that the separation of variables in the boundary conditions is incomplete, the initial problem is reduced to infinite systems of linear algebraic equations (ISLAEs) with respect to unknown expansion coefficients. We have examined the asymptotic behavior of ISLAE elements at large values of indices. It has been shown that the necessary condition of the solvability of the ISLAE coincides with the condition of correct application of the extended boundary conditions method (ЕВСМ). We have performed numerical calculations for Chebyshev particles with one maximum (also known as Pascal’s snails or limaçons of Pascal). The obtained numerical results for the asymptotics of ISLAE elements and for the matrix support theoretical inferences. We have shown that the scattering and absorption cross sections of examined particles can be calculated in a wide range of variation of parameters with an error of about 1–2% using the spheroidal model. This model is also applicable in the case in which the solvability condition of the ISLAE for nonconvex particles is violated; in this case, the SVM should be considered as an approximate method, which frequently ensures obtaining results with an error less than 0.1–0.5%.