Two-dimensional equations for guided electromagnetic waves in dielectric plates surrounded by free space

Abstract

Two-dimensional governing equations of successively higher-order approximations for guided electromagnetic (EM) waves in an isotropic dielectric plate surrounded by free space are deduced from the three-dimensional Maxwell’s equations by expanding the EM vector potential in a series of trigonometric functions of a thickness coordinate in the plate and in exponentially decaying functions of a thickness coordinate in the upper and lower halves of free space. By further satisfying the continuity conditions of the EM field at the interfaces between the plate and free space, a single system of two-dimensional governing equations is obtained. Solutions and dispersion relations are obtained from the two-dimensional approximate equations. Dispersion curves are computed and compared with the corresponding ones obtained from the solutions of the three-dimensional Maxwell’s equations for the transverse electric (TE) and transverse magnetic (TM) waves of the first four modes and for values of the refractive index n̂=1.5, 5, 15. It is shown that the agreement between the approximate and exact dispersion curves is very close for various order of TE and TM waves and for a broad range of the values of n̂. For bounded plates with edges in contact with free space, a uniqueness theorem for the solutions of the system of two-dimensional equations is derived from which the specification of continuity conditions on the components of the two-dimensional H and E fields at the edges are established.