Rayleigh approximation for multilayer nonconfocal spheroids

We consider the light scattering by layered spheroids that are small compared to the wavelength of the incident radiation. Simple approximate formulas are obtained for the polarizability of such particles with nonconfocal spheroidal surfaces of layers by reducing infinite matrices in the rigorous solution of the problem to matrices of dimensions 2 $\times$ 2 and 4 $\times$ 4. In the first case, the approximate expression for the polarizability formally coincides with the well-known expression for spheroids with confocal surfaces of layers and, correspondingly, represents an accurate result for such particles. The second case is, in essence, taking into account in the first approximation the effect of nonconfocality of core surfaces and particle layers. The results of numerical calculations carried out for two- and three-layer particles using both approximate expressions and formulas of the rigorous solution showed that, in a wide range of parameters, the relative error of the simpler approximation (2 $\times$ 2) is lower than 1%, while the error of the other approximation (4 $\times$ 4) is smaller than 0.1%. It is inferred that the found approximate formulas are rather accurate and universal, and they can be efficiently used in calculations of the optical properties of small multilayer spheroidal particles.