Uniform Approximation to Mie Resonances

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Abstract
Uniform asymptotic approximations for the positions and widths of Mie resonances are reported. The results improve the accuracy of previous approximations, typically, by several orders of magnitude. They are suitable for fast numerical evaluation, allowing every resonance to be located well within its width, for all resonances of practical interest. The relative errors are very small even for the lowest angular momenta (size parameters) and they decrease at least as fast as the inverse square of the size parameter. Accuracies of the order of parts per billion for typical microsphere sizes are readily attained. A second-order Wentzel–Kramers–Brillouin approximation is also developed. Though far less accurate, it yields useful analytic estimates of many resonance features.