Surface area dependent corrections in the theory of black body radiation

Abstract

For a certain class of boundary geometries, the solutions of the electromagnetic vector wave equation in a finite lossless closed cavity can be classified into E- and H-type modes. We show that each kind of solution contributes a surface area dependent term to the corresponding asymptotic eigenvalue densities, D E and D H. These surface terms are equal, but have opposite signs. This constitutes a new proof of the well known vanishing of the first order correction of the complete mode density D E + D H = D. We present a formula for the difference of the E- and H-type densities, D E − D H , which is stricly valid in the asymptotic limits of infinite frequency or infinite cavity volume. We show that this formula is in good agreement with computed eigenvalue spectra obtained from the first 10 6 modes for several cavity geometries, i.e. the cube, square prisms, cylinders, sectors of cylinders, the sphere, and sectors of the sphere. Furthermore, we study the surface correction of the time auto-correlation functions of the electric and magnetic field components.