A Numerically Stable Formulation of the Spherically Layered Media Theory

The spherically layered media theory is widely used in electromagnetic wave scattering problems. In the theory, the numerical computations of Bessel functions are unstable when the argument is small, the media is lossy, or the function's order is large. This paper addresses problem. We first define the renormalized reflection and transmission coefficients which have more clear physical meanings and ordinary magnitude of values. Then, logarithmic derivative of Racatti-Bessel funtions is used to calculate the renormalized single-interface reflection and transmission coefficients. Finally, recursive formulas of the renormalized multi-interface reflection and transmission coefficients are derived. In the proposed method, the stable computations of the layered media theory are simply reduced to iteratively calculate the ratio of Bessel functions of adjacent orders. The validity and stability of this method have been verified by numerical tests.