The propagation of guided electromagnetic waves in dielectric waveguides has received considerable attention in the engineering literature [l, 5-7. 13, 141. The practical importance of these waves in both communications using optical fibers and in integrated optical circuitry using thin film technology is well known [5-7, 131. The dielectric waveguides used in integrated optics are asymmetric slabs which constitute the simplest optical waveguides. The configuration for such a waveguide consists of three-dimensional Euclidean space decomposed into a horizontal dielectric slab of finite thickness sandwiched between two infinite half spaces; a true waveguide is formed when the ratios of the phase speeds of the half spaces to that of the slab are greater than one. These ratios are the reciprocals of the refractive indices found in [S]. Besides the direct applications in integrated optics, the study of slab waveguides and their properties is often useful in gaining an understanding of the waveguiding properties of more complicated dielectric waveguides. The analysis of this problem is usually carried out using the methods of geometrical (ray) optics and/or a reduction, based on various simplifying assumptions, to finding solutions for the one-dimensional reduced wave equation [S, 6, 13 J. This analysis yields the electromagnetic field as an orthogonal direct sum of TE (transverse electric) and TM(transvese magnetic) radiation modes and TM, TE guided modes which propagate in the slab and decay away from it. The complete set of generalized eigenfunctions obtained below from the spectral analysis of the Maxwell operator in an appropriate Hilbert space are of exactly this same type. This of course is to be expected. The spectral analysis of a similar waveguide problem-electromagnetic waves over a dielectric clad perfect conductor-has been carried out in 31 0022-247X/85 S3.00