The Mie series solution for the scattering of an electromagnetic plane wave by a homogeneous sphere is introduced. Though ‘exact’, ‘a mathematical difficulty develops which quite generally is a drawback of this “method of series development”: for fairly large particles … the series converge so slowly that they become practically useless’ (Sommerfeld 1954). What is still worse, numerical studies reveal the occurrence in Mie cross sections of very rapid and complicated fluctuations, known as the ‘ripple’, that are extremely sensitive to small changes in the input parameters. They are related to orbiting and resonances, which may also be detected through their contribution to nonlinear optical effects, such as lasing, that are observed in liquid droplets.

The Mie solution

We consider a monochromatic linearly polarized plane electromagnetic wave with wave number k incident on a homogeneous sphere of radius a and a complex refractive index (relative to the surrounding medium) N = n + iκ, where n is the real refractive index and κ is the extinction coefficient (Jackson 1975). The time factor exp(–ωt) where ω = ck is the circular frequency, is omitted.

Taking the origin at the center of the sphere, the z axis along the direction of propagation of the incident wave and the x axis along its direction of polarization, the incident electric field is E0 = exp(ikz) ◯. In spherical coordinates, the components of the scattered electric field at large distances are of the form (Bohren & Huffman 1983)