Plasmons and Diffraction of an Electromagnetic Plane Wave by a Metallic Sphere
The di®raction 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 field 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 field, which we solve. In the non-retarded limit (Coulomb interaction) we get the branch of "spherical" (surface) plasmons at frequencies ω = ωpsqrt(l/(2(l/ + 1)), where ωp is the (bulk) plasma frequency and l = 1, 2,.... When retardation is included, for an incident plane wave, we compute the field inside and outside the sphere (the scattered field), the corresponding energy stored by these fields, Poynting vector and scattering cross-section. The results agree with the so-called theory of "effective 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 field and continuity of the normal component of the electric displacement at the sphere surface, as well as the conservation of the energy flow 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.