Numerically Stable Calculations of the Spherically Layered Media Theory

The electromagnetic (EM) scattering of a layered sphere is a canonical problem. Mie theory is suitable for plane wave incidence cases, whereas spherically layered media theory (SLMT) can deal with arbitrary incident waves. Both Mie theory and SLMT suffer from numerical instabilities due to the involved spherical Bessel functions when the order is large, the argument is small, or the medium is lossy. The logarithmic derivative method had been proposed to solve this numerical issue with Mie theory successfully, while the numerical issue with SLMT has not been solved fully so far. Computations of reflection and transmission coefficients are the key part of SLMT. In this article, we first define the renormalized reflection and transmission coefficients, which enjoy the feature of having an ordinary level of magnitude. Then, borrowing the idea of the logarithmic derivative method, the expressions for the renormalized canonical reflection and transmission coefficients as well as other terms in the theory are rearranged. Recursive formulas for the product or division of Bessel functions with some common combinations of order and argument are derived. Numerical tests show that the proposed approach, validated by the full wave numerical method, is more stable than the conventional formulation.