Eigenmodes of a two-dimensional elliptical laser resonator

TLDR: In this paper, the authors presented a version of a Green function for interior field of cylindrical elliptic cavities or screens. But the problem of finding the Green function is not solved.

Abstract: It is well-known that the Fresnel - Kirchhoff diffraction theory imposes boundary conditions on a field strength and a normal derivative simultaneously. It is also known that these requirements contradict each other. A. Sommerfeld has shown that for elimination of these conflicts of the diffraction theory, it is necessary and enough to select a Green function so that the function or its normal derivative on a boundary surface are equal to zero. But such Green function is known, however, for practically unique case of diffraction on a plane screen. According to the Sommerfeld, this Green function is formed by a point source and its mirror image in a plane screen. If the point source and its image are antiphased, the Green function is identically equal to zero on all surface of the screen. If the point source and its image are cophased, the normal derivative of the Green function is equal to zero on the screen. The conflicts of the Fresnel - Kirchhoff theory are eliminated in each case. In the work, we present one of possible versions of a Green function for interior field of cylindrical elliptic cavities or screens. At the solution of diffraction problems in such fields for simplification of the equations everyone usually transfers Cartesian coordinates to elliptical coordinates. The interrelations between coordinates are featured by multivalued functions, therefore are available real field of the existence of coordinates and set "dummy" fields overlapped on real field. This circumstance allows to design for an elliptic cavity a Green function satisfying to the Sommerfeld conditions. For this purpose, in an interior point of real field of a cavity we put the point source featured by the Hankel function H10 (kr) of the first kind and of the zero order, and in the relevant conjugate points "dummy" field we put the point source featured by the Hankel function H20 (kr) of the second kind and of the zero order. The Green function is formed as the total sum of the aforesaid cophased point sources. We note that argument r for the Hankel function in "dummy" field is necessary to take by negative. It is easy to verify that such Green function is equal to zero and its normal derivative is not equal to zero on a boundary surface of an elliptic cavity.