Models of symmetric three-layer waveguide structures with graded-index core and nonlinear optical liners
Authors: Savotchenko S. E. 
Published: October 2024

Objectives. Determining the patterns of dispersion properties of waveguide modes of the optical range in layered media with distributed optical properties is a both a pressing and significant matter for study. It has fundamental and applied importance in nonlinear optics and optoelectronics. The combination of a nonlinear response and gradedindex distributions of the optical properties of adjacent layers of a layered structure enables the desired values of the output characteristics using a wide range of control parameters to be selected easily. This renders such waveguides the most promising from the point of view of possible technical applications. The aim of this paper is to develop the theory of three-layer planar waveguide structures with a graded-index core and nonlinear optical liners with arbitrary profiles. By doing so it may be possible to find exact analytical solutions to nonlinear stationary wave equations describing explicitly the transverse electric field distribution of waveguide modes.

Methods. The analytical methods of mathematical physics and the theory of special functions applied to nonlinear and waveguide optics are used herein.

Results. The study provides a theoretical description of transverse stationary waves propagating along a symmetrical three-layer planar waveguide structure consisting of the inner graded-index layer sandwiched between nonlinear optical plates. It assumes an arbitrary spatial profile of the interlayer dielectric constant and the nature of the nonlinear response of the liner medium. The mathematical model of this waveguide structure formulated herein is based on nonlinear equations with distributed coefficients. The solutions obtained describe in general terms the transverse distribution of the amplitude of the electric field envelope. The transverse symmetry of the three-layer waveguide structure enables even and odd stationary modes corresponding to symmetric and antisymmetric transverse field profiles to be excited in it. A method was developed for constructing even (symmetric) and odd (antisymmetric) solutions which exist at certain discrete values of the effective refractive index/propagation constant. These discrete spectra were obtained in layers with graded-index linear, parabolic, and exponential profiles. The symmetrical threelayer waveguide structure with inner graded-index layer characterized by parabolic spatial profile and outer liners as Kerr nonlinear optical media is analyzed in detail, as an example of the application of the formulated theory. Analysis of the resulting exact analytical solution indicates that the electric field strength for the fundamental and first-order modes increases with increasing parabolic profile parameter, characterizing the relative change of the dielectric constant in the interlayer, while decreasing for higher order modes.

Conclusions. The theory developed in this paper supports the unambiguous description of the transverse distributions of the stationary electric field in planar symmetrical three-layer waveguides in an explicit analytical form. The results extend the understanding of the physical properties of nonlinear waves and the localization patterns of light beams in distributed media, and may be useful in the design of various optical waveguide devices.