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

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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.

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