Asymptotic approach for stable computations of the spherically layered media theory with large orders and small arguments

The computation of electromagnetic wave scatterings of a layered sphere is a canonical problem. Lorentz-Mie theory is suitable for plane wave incidence whereas spherically layered media theory can deal with arbitrary incident waves. Both theories suffer from the notorious numerical instabilities due to the involved Bessel functions with large order, small argument or high loss. Logarithmic derivative method has been proposed to solve the numerical issues with these theories. In this paper, by employing the equivalence between the asymptotic formulas of Bessel functions for small argument and for large order, the numerical issues with the spherically layered theory under both large order case and small argument case can be solved in a unified manner by canceling out the diverging terms in the asymptotic formulas. The derived stable formulas are simpler and faster than those based on logarithmic derivative method. It is shown that the derived formulas are good approximations to the canonical ones but are more numerically stable. The large lossy issue can be solved similarly.