The diffraction of plane electromagnetic waves by a slit in an infinitely thin, perfectly conducting screen between two media with different dielectric constants is studied using rigorous diffraction theory. Both the case where the electric vector is parallel to slit and the case where the magnetic vector is parallel to slit are examined. The problem is formulated in elliptic cylinder coordinates and solved in terms of Mathieu functions. The explicit determination of the diffracted wave-expansion coefficients leads to solving infinite systems of complex linear equations. The long-wave (Rayleigh scattering) region is studied in detail. The scattered intensity at infinity, transmission coefficient, and backscatter coefficient are evaluated. Finally, numerical results are presented for some special cases.
RICHARD BARAKAT, "Diffraction of Plane Waves by a Slit between Two Different Media," J. Opt. Soc. Am. 53, 1231-1238 (1963)