• Sergey E. Savotchenko, MIREA – Russian technological university (Moscow, Russia), Sergo Ordzhonikidze Russian state university for geological prospecting (Moscow, Russia)
• Nadezhda O. Afanasyeva, Sergo Ordzhonikidze Russian state university for geological prospecting (Moscow, Russia)

This paper presents a simulation of the spatial redistribution of optical field intensity in a planar waveguide structure formed by the interface between a nonlinear medium and a medium with a graded refractive index. A generalized dependence of permittivity on the distance from the contact surface of the media was chosen as a model for the gradient medium. A medium with a linear optical effect (Pockels effect), in which permittivity is linearly dependent on the amplitude of the electric field, was chosen as a model for the nonlinear medium. The study identifies critical operating modes in which the maximum intensity of the guided mode can be deliberately located within any of the constituent media or precisely at their interface by adjusting the optical parameters of the system that determine the critical value of the effective refractive index. The main result of the study is proof of the existence of a critical effective refractive index regardless of the choice of nonlinearity model, at which the field maximum is precisely localized at the interface. It is shown that the value depends solely on the parameters determining the profile of the graded refractive index, which ensures the dispersion relation. Particular emphasis is placed on identifying how each of the gradient profile parameters individually affects the magnitude of the transition threshold. Аanalysis revealed that the critical value of the effective refractive index varies monotonically with changes in the characteristic spatial scale and dielectric parameters of the graded layer. It is shown that the nature of this influence is determined by the sign and magnitude of the corresponding dielectric constants, allowing for the targeted formation of the desired dependence of the critical refractive index on the optical parameters of a planar waveguide structure. Unlike most studies, which are limited to numerical modeling of specific configurations, this study proposes a rigorous analytical approach that allows for the derivation of explicit criteria for the existence of a critical regime. The potential use of the obtained results for calibrating experimental setups, adjusting phase-matching conditions for second-harmonic generation, and estimating acceptable parameter tolerances in integrated circuit manufacturing is also discussed. The results provide a theoretical basis for the development of methods for controlled light localization in hybrid waveguide structures by varying their material and geometric parameters or the operating wavelength.

mathematical modeling, nonlinear equations, optical waveguide, nonlinear medium, boundary value problem

2026-09-03