The Condition of Applicability for the Extended Boundary Conditions Method for Small Multilayer Particles

We have examined an analog to the extended boundary conditions method (EBCM) with the standard spherical basis, which is popular in light scattering theory, with respect to its applicability to the solution of an electrostatic problem that arises for multilayer scatterers the sizes of which are smaller compared to the wavelength of the incident radiation. It has been found that, in the case of two or more layers, to determine the polarizability and other optical characteristics of particles in the far-field zone, the parameters of the surfaces of layers should obey the condition max{σ₁^(j)} < min{σ₂^(j)}. In this case, appearing infinite systems of linear equations for expansion coefficients of unknown fields have a unique solution, which can be found by the reduction method. For nonspheroidal particles, this condition is related to the convergence radii of expansions of regular and irregular fields outside and inside of the particle, including its shells—R₁^(j) = σ₁^(j) and R₂^(j) = σ₂^(j). In other words, a spherical shell should exist in which expansions of all regular and irregular fields converge simultaneously. This condition is a natural generalization of the result for homogeneous particles, for which such a condition is imposed only on expansions of the “scattered” and internal fields—R₁ < R₂. For spheroidal multilayer particles, which should be singled out into a separate class, the EBCM applicability condition is written as max{σ₁⁽¹⁾, σ₁⁽²⁾, …, σ₁^(J−1), σ₁^(J)} < min{σ₂⁽¹⁾, σ₂⁽²⁾, …, σ₂^(J−1)} and parameters σ₂^(j) of the surfaces of shells are not related to corresponding convergence radii R₂^j of irregular fields. Numerical calculations for two-layer spheroids and pseudospheroids have confirmed completely theoretical inferences. Apart from the EBCM algorithm, an approximate formula has been proposed for the calculation of the polarizability of two-layer particles, in which the polarizability of a two-layer particle is interpreted as a linear combination of the polarizabilities of homogeneous particles that consist of the materials of the shell and core proportionally to their volumes. The range of applicability of this formula is wider than that for the EBCM, and the calculation error is smaller than 1%.