Numerically Stable Calculations of the Spherically Layered Media Theory

Abstract

The electromagnetic scatterings of a layered sphere is a canonical problem. Mie theory is suitable for plane wave incidence case, whereas spherically layered media theory can deal with arbitrary incident waves. Both Mie theory and spherically layered media theory 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. Logarithmic derivative method had been proposed to solve this numerical issue with Mie theory successfully, while the numerical issue with spherically layered media theory has not been solved fully so far. Computations of reflection and transmission coefficients are the key part of spherically layered media theory. In this paper, we first define the renormalized reflection and transmission coefficients, which enjoys the feature of having an ordinary level of magnitude. Then, borrowing the idea of 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 full wave numerical method, is more stable than the conventional formulation.