TM-(transverse magnetic)guided waves, propagating in a lossless, nonmagnetic three-layer structure (substrate-film-cladding) are studied. Two types of the dielectric permittivities (I and II) are analyzed. All three media of the waveguide with the permittivity of type I are assumed to exhibit a local Kerr-like tensorial nonlinearity. Maxwell's equations in this case are reduced to an exact differential equation leading to a first integral, relating two electric field components so that one component can be eliminated. The other one can be found by integration. Combination of the first integral with the boundary conditions leads to an exact analytical dispersion relations (expressed in terms of integrals) establishing a link between the parameters of the problem (in particular, thickness of the film, the propagation constant of the travelling wave, the electric field components at the interface substrate-film). The film thickness and the propagation constant satisfying the dispersion relation (by given electric field component at the boundary substrate-film)are associated to the possible modes travelling through the waveguide. Numerical evaluation of the corresponding power flow derived using of Maxwell' equations and the first integral processes straightforwardly, without known wave solutions at first. The waveguide with the permittivity of type II consists of the film with the dielectric function depending on the field intensity (Kerr-type nonlinearity) as well as on the transverse coordinate (spatially varying permittivity) situated between the linear, isotropic substrate and cladding. The problem in this case is reduced to a system of two integral equations. Using the Banach fixed-point theorem it is shown that the solutions of Maxwell's equations exist in form of a uniformly convergent sequence of iterations. The conditions of the Banach fixed-point theorem are derived and used to estimate the quality of the approximation. The exact dispersion relation is derived. Results of numerical evaluation of the dispersion relation and field solutions are presented in the first approximation. Solutions of the dispersion relation, the field components and the power flow obtained using the method for the permittivity I are compared with these found using an integral equation approach (the permittivity II but without the coordinate dependence) - the consistency is remarkably good. The proposed methods seem to be applicable to permittivities more general as considered.

The reflection and transmission of an electromagnetic TE-polarized plane wave at a dielectric film between two linear semi-infinite media (substrate and cladding) is considered. All media are assumed to be homogeneous in x- and z- direction, isotropic, and non-magnetic. The permittivity of the film is assumed to be characterized by a continuously differentiable function of the transverse coordinate and the field. To obtain solutions of Maxwell´s equations that satisfy the boundary conditions the problem is reduced to a Helmholtz equation, which is transformed to a Volterra integral equation for the field intensity inside the film. The Volterra equation is solved by iteration subject to the appropriate boundary conditions. The (iteration) solutions for the linear case and for the nonlinear case are expressed in terms of a uniformly convergent series and a uniformly convergent sequence, respectively. The uniform convergence is proved using the Banach Fixed-Point Theorem. The condition for its applicability leads to a condition for the parameters of the problem. By iterating the Volterra equation an approximate solution for the intensity inside the film is presented. The mathematical basis of the procedure is outlined in detail. Using an approximate solution, the phase function,the phase shifts on reflection and transmission, the reflectivity and the absorptance are determined.Further iterations of the Volterra equation are possible.Semianalytical and numerical examples illustrate the main features of the approach. The method is succesfully applied to different permittivity functions (real, complex, Kerr-like and saturable nonlinear). The agreement between the approximate analytical solutions and numerical solutions is satisfactory. It seems that the method proposed can serve as a means to optimize certain parameters of the problem (material and/or geometrical) for particular purposes.