Positive Kerr Nonlinearity in a Thin-Film Waveguide Problem: A Study Associated with TM Standing Wave Solution

Here we demonstrate how the solution of nonlinear Maxwell equations is applied to a thin-film waveguide problem. A periodic scattering type solution we used for this purpose is of a more general form than ones expressed through elliptic integrals of the first and third kinds. In our approach, fields are represented by a constant of a non-flow integral of motion and the value of a dielectric function. Instead of matching field amplitudes at boundaries, we worked through values of this constant as a wave characteristic. So, without taking into account a film thickness, we created the appropriate boundary conditions for both borders. The requirement forces a dielectric function to have two distinct or equal values at borders, resulting in the existence of wave pieces that fit symmetrically and asymmetrically. Such standing wave solutions were built into families. We use a film thickness to apply the satisfying border condition to family data. The family of symmetrically fit wave pieces turned out to have a relatively complex structure due to two singularities in a dielectric function and three regions had to be analyzed separately. To look for matches within the data we presented two simple criteria that completely define the length-form and the number of standing wave solutions within a film. In general, the method can be applied to more complex layered structures.