Fully Stable Formulations of the Spherically Layered Media Theory Using Scaled Bessel Functions

The spherically layered media (SLM) theory has wide applications for electromagnetic wave scattering analysis. Due to the involved Bessel functions, the traditional formulations of the SLM theory suffer from numerical overflow or underflow when the Bessel function’s order is large, the argument is small, or the argument has a large imaginary part. Recently, by arranging the Bessel functions in ratio forms and computing these ratios with recursive formulas, these numerical issues have been solved. However, the iteration direction of some ratios is backward, i.e., from large order to small order, which is inconvenient in coding. Another method proposed recently for the numerical issues with SLM theory is to use the small-argument asymptotic formulas of Bessel functions and cancel out the divergent factors. This method can be carried out simply in forward manner, but it only solves the first two issues mentioned above while the third issue remains unsolved for this method. In this article, the Bessel functions in the traditional formulation of the theory are replaced by the scaled Bessel functions which have good numerical properties for high-loss media. As a result, the numerical breakdown issue of the SLM theory in high-loss case can be solved. Then, asymptotic formulas for the scaled Bessel functions are derived and applied to the theory to solve the numerical breakdown problem when the arguments of the scaled Bessel functions are small or the functions’ order is large. By this way, fully numerically stable formulations of the SLM theory are obtained, and the series summation is performed directly without any backward iterations. Numerical tests show that the proposed approach can work properly when the media are very high lossy, the arguments are very small, or the scaled Bessel functions’ order is very large.