A quasi-analytical direct transient Mie solution for spheres: The EM case

Debye-Mie series solution is one of the most useful tools in time-harmonic analysis of electromagnetic scattering, and has found in extensive applications in both optics and electromagnetics. Their applications range from fields as diverse as light scattering from small particles to analysis of small antennas to photonics to biological applications. However, analytical solutions to scattering from a sphere only exist in the Fourier domain. Deriving their transient analogue is a challenge as it involves an inverse Fourier transform of the spherical Hankel functions (and their derivatives) that are convolved with inverse Fourier transforms of spherical Bessel functions (and their derivatives). Series expansion of these convolutions are highly oscillatory (therefore, poorly convergent) and unstable. Indeed, the literature on numerical computation of this convolution is very sparse.