A thin-film waveguide problem with positive Kerr nonlinearity and its TM standing wave solution

Abstract

In the previous work [1], we applied the scattering-type TM standing wave solution to a slab waveguide with negative Kerr nonlinearity. In this work, we analyzed a waveguide problem with positive Kerr nonlinearity. In our formalism, fields are expressed through the value of a dielectric function and a constant of non-flow integral of motion. We formulated the necessary boundary condition for both boundaries with no reference to a film thickness. The condition binds a dielectric function to have two equal or two different values at boundaries leading to the existence of symmetrically and asymmetrically fit wave pieces. Families of such standing wave solutions were constructed. The satisfactory boundary condition is implemented when we start to apply a film thickness 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. Afterward, a calculation of fitted solutions is straightforward.