On solution of the light scattering problem for a spheroid in the TM and TE modes when using spheroidal bases

Light scattering by spheroids plays an important role in various applications. The most efficient algorithm to calculate the optical properties of spheroids implies the field expansions in a special spheroidal basis, but its application is limited by three problems – difficulties of computations of the spheroidal functions of complex argument, the absent transition to the standard $T$-matrix, and loss of precision in complicated calculations for one case of the incident radiation polarization (TE mode). The first two difficulties have been recently overcome to a large extent, and in this work we solve the last problem – by using $T$-matrix transformations, we find the way of expressing the TE mode solution through the more simple and stable TM mode solution. Numerical calculations performed by us demonstrate that the suggested approach improves the result accuracy by several orders, accelerates solution in several times and significantly extends its applicability range (up to the diffraction parameters exceeding 100).