Direct recursion of the ratio of Bessel functions with applications to Mie scattering calculations

The primary goal of this paper is to describe a method to directly calculate the ratios between the Bessel functions of the first and second kind. This new method is introduced and discussed under the context of Mie scattering calculations. Past work on Mie calculations has mostly relied on calculating the Bessel functions separately, then taking the ratio. This technique fails when the order of the functions becomes appreciably larger than the argument. To solve this problem, we developed a stable method to directly calculate the ratio. When applied to Mie scattering calculations, the new method improves the control of the desired accuracy and enables calculations with higher precision. Since many other wave propagation problems also involve the ratios of Bessel functions and our method provides reliable and accurate calculations over a wide range of conditions, we expect it to find applications beyond Mie calculations.