PLASMONS AND DIFFRACTION OF AN ELECTROMAGNETIC PLANE WAVE BY A METALLIC SPHERE

Abstract

The difiraction of a plane electromagnetic wave by an ideal metallic sphere (Mie's theory) is investigated by a new method. The method represents the charge disturbances (polarization) by a displacement fleld in the positions of the mobile charges (electrons) and uses the equation of motion for the polarization together with the electromagnetic potentials. We employ a special set of orthogonal functions, which are combinations of spherical Bessel functions and vector spherical harmonics. This way, we obtain coupled integral equations for the displacement fleld, which we solve. In the non- retarded limit (Coulomb interaction) we get the branch of \spherical (surface) plasmons at frequencies ! = !p p l=(2l + 1), where !p is the (bulk) plasma frequency and l = 1;2;:::. When retardation is included, for an incident plane wave, we compute the fleld inside and outside the sphere (the scattered fleld), the corresponding energy stored by these flelds, Poynting vector and scattering cross-section. The results agree with the so-called theory of \efiective medium permittivity, although we do not start the calculations with the dielectric function. In turn, we recover in our results the well-known dielectric function of metals. We have checked the continuity of the tangential components of the electric fleld and continuity of the normal component of the electric displacement at the sphere surface, as well as the conservation of the energy ∞ow and re-derived the \optical theorem. In the limit of small radii (in comparison with the electromagnetic wavelength) the sphere exhibits a series of resonant absorptions at frequencies close to the plasmon frequencies given above. For large radii these resonances disappear.