ISSN 1064-2269, Journal of Communications Technology and Electronics, 2010, Vol. 55, No. 1, pp. 98–105. © Pleiades Publishing, Inc., 2010. Original Russian Text © E.Yu. Al’tshuler, M.V. Davidovich, Yu.V. Stefyuk, 2010, published in Radiotekhnika i Electronika, 2010, Vol. 55, No. 1, pp. 105–112. 98 INTRODUCTION Rectangular dielectric waveguides (RDWs) have been investigated in a number of studies (see, e.g., [1– 26]). These waveguides are used as probes, as compo- nents of filters and resonators, as components of pla- nar integrated circuits and circuits for measuring the dielectric properties of materials, and in other cases. The analysis of RDWs and their irregularities is per- formed using both rigorous and approximate methods: cylindrical-harmonic expansion method [2, 12, 17], finite-difference method [12, 15, 24], partial-region and matching method [6, 16], integral equation (IE) method [17, 18, 25, 26], variational-relation method [5], Marcatili method [3], transverse-resonance method [13], and method of generalized telegraph equations [9]. In numerical implementation of the rig- orous models, moment method, Galerkin method, collocation method, finite-element method, and some other methods were used. With rigorous formu- lations and high accuracy, the dimensions of the algo- rithms obtained by these methods are rather high. Therefore, the problem of sufficiently accurate and algorithmically simple determining the parameters of various modes existing in dielectric waveguides (DWs) remains topical [14]. The solution of this problem is the goal of the present study. Multilayered RDWs with dissipative and active semiconductor layers are promising in controlling their dispersion, attenuation, or gain. They can be analyzed on the basis of the method of volume integral and integro-differential equations (IDEs) containing volume and surface integrals or with the use of a vol- ume singular IE [27]. Volume IDEs are also conve- nient in determining the electric and magnetic fields. In the first case, hypersingular IEs occur with a nonin- tegrable singularity, which correspond to pseudodif- ferential operators [28]. The order of singularity of the kernels of these equations is reduced by several meth- ods with the use of vector integral theorems and Green’s formulas. In this case, integrals occur at the interfaces, which circumstance complicates the mode-analysis algorithms. The coordinate and time dependence of the fields defined by exp(j (ωt – γz) and integration with respect to source-point coordinate z' allow us to reduce three-dimensional equations to two-dimensional IEs. In this case, the hypersingular IE in electric field assumes the form (1) and the magnetic field is defined by the relationship (2) On solving Eq. (1), we can determine at any point in plane (x, y). By expressing the electric field from Maxwell’s equations as =–j (∇ ⊥ – jγ × /(ωε 0 ) and substituting it into (2), this relationship is transformed into an IDE. In the given relationships, ∇ ⊥ = ∂/∂x + ∂/∂y; the vectors with zero subscripts denote the orts of the Cartesian coor- dinate system; symbol ⊗ denotes the tensor product; g = –j /4 is the two-dimen- sional Green’s function in which is the zero- Er ⊥ ( ) ∇ ⊥ j γ z 0 – ( ) ∇ ⊥ j γ z 0 – ( ) k 0 2 + ⊗ [ ] = × gr ⊥ r ⊥ ' – ( )κ ˆ r ⊥ ' ( ) Er ⊥ ' ( ) r' , 2 d S ∫ Hr ⊥ ( ) ∇ ⊥ j γ z 0 – ( ) gr ⊥ r ⊥ ' – ( )κ ˆ r ⊥ ' ( ) Er ⊥ ' ( ) r' 2 d S ∫ × . = Hr ⊥ ( ) Er ⊥ ( ) ε ˆ 1 – z 0 ) Hr ⊥ ( ) x0 y0 r ⊥ r ⊥ ' – ( ) H 0 2 () χ 0 r ⊥ r ⊥ ' – ( ) H 0 2 () RADIO PHENOMENA IN SOLIDS AND PLASMA LM Modes in Semiconductor–Dielectric Plane-Layered Waveguide with Dissipative and Active Layers E. Yu. Al’tshuler, M. V. Davidovich, and Yu. V. Stefyuk Received February 13, 2009 Abstract—With the use of the iteration method and bilinear functional based on two-dimensional volume integral equation, LM modes existing in a plane-layered dielectric waveguide and quasi-LM modes existing in a multilayered rectangular dielectric waveguide are studied. The semiconductor layers are described as a homogeneous single-component dissipative plasma that can exhibit negative conductivity owing to external electric field. The results obtained for a homogeneous single-layer rectangular waveguide are compared to the results obtained with a model of a planar single-layer waveguide. Structures with passive and active semicon- ductor-plasma layers are investigated. It is shown that amplification is possible at frequencies above the cutoff frequency and that the dispersion laws below the cutoff are complicated. DOI: 10.1134/S1064226910010134