Mie scattering

Statement

Let a homogeneous, isotropic, non-magnetic sphere of radius $a$ lie in a homogeneous non-magnetic medium, with complex refractive index $m$ relative to the medium. Assume $\operatorname{Re} m > 0$ and $\operatorname{Im} m \ge 0$. Let it be illuminated by a monochromatic plane wave with wavenumber $k$ in the medium and time factor $e^{-i\omega t}$, and put $x = ka$. Let $\psi_n(\rho) = \rho\, j_n(\rho)$ and $\xi_n(\rho) = \rho\, h_n^{(1)}(\rho)$ be the Riccati–Bessel functions built from the spherical Bessel function $j_n$ and the spherical Hankel function $h_n^{(1)}$. Then the scattering problem has a unique solution. Its scattered field is a series over $n \ge 1$ of vector spherical wave functions with coefficients given by $$a_n = \frac{m\, \psi_n(mx)\, \psi_n'(x) - \psi_n(x)\, \psi_n'(mx)}{m\, \psi_n(mx)\, \xi_n'(x) - \xi_n(x)\, \psi_n'(mx)}, \qquad b_n = \frac{\psi_n(mx)\, \psi_n'(x) - m\, \psi_n(x)\, \psi_n'(mx)}{\psi_n(mx)\, \xi_n'(x) - m\, \xi_n(x)\, \psi_n'(mx)}.$$ The scattering and extinction cross-sections are $$C_{\mathrm{sca}} = \frac{2\pi}{k^2} \sum_{n=1}^{\infty} (2n + 1)\left( \abs{a_n}^2 + \abs{b_n}^2 \right), \qquad C_{\mathrm{ext}} = \frac{2\pi}{k^2} \sum_{n=1}^{\infty} (2n + 1) \operatorname{Re}(a_n + b_n).$$