Controlling Properties of Interfaces in Nonlinear Sandwich-Type Structures with a Defocusing Internal Layer
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lling Properties of Interfaces in Nonlinear Sandwich-Type Structures with a Defocusing Internal Layer S. E. Savotchenko* Shukhov Belgorod State Technological University, Belgorod, 308012 Russia *e-mail: [email protected] Received December 27, 2019; revised January 26, 2020; accepted January 26, 2020
Abstract—A model of a three-dimensional optical structure in which the plane-parallel boundaries have intrinsic nonlinear properties is considered. The internal layer with a finite thickness is an optically transparent medium with defocusing Kerr nonlinearity; on the outside, it is in contact with dielectric linear halfspaces. The mathematical formulation of the model reduces to the nonlinear Schrödinger equation with a positive coefficient of cubic nonlinearity and with a nonlinear self-consistent potential. It is shown analytically that the system contains a nonlinear light wave propagating along the optical layer and localized in dielectric plates. The frequencies of light-field localization in this structure are obtained and the conditions for their existence are determined for different characteristics of the media and the interfaces between them. It is shown that the light field can be localized along the layers for different signs of the nonlinear response of the interfaces between layers of the three-layer structure in the case where one of them is characterized by a focusing nonlinearity; and the other, by a defocusing one. Keywords: localized states, nonlinear Schrödinger equation, planar defect, interface, layered media, nonlinearity coefficient DOI: 10.1134/S1027451020030155
INTRODUCTION Investigation of the variety of properties of nonlinear surface waves is of interest because of their wide application in different technical systems based on the waveguide properties of multilayer heterostructures [1, 2]. Nonlinear surface waves of the optical range propagating along the interfaces between nonlinear media in layered structures, including three-layered ones (so-called “sandwiches”), have been analytically studied by many authors [3–7]. In the given papers, the sought field and its normal derivatives near the interfaces of the layered structure satisfied the conditions for their continuity, which meant that there was no interaction between the wave and the interface as a flat defect. Based on the nonlinear Schrödinger equation with the Kerr nonlinearity, the authors of [8] took into account the interaction between excitations and two flat interfaces between three nonlinear media characterized by one parameter. For theoretical description of the local interaction of nonlinear excitations with the interfaces between layers, they were modeled by the short-range potential in the nonlinear Schrödinger equation, which is written in the following form in the one-dimensional case for the three-layer structure:
U ( x) = U 0{δ( x + a) + δ( x − a)},
where δ(x) is the Dirac delta function, U0 is the intensity of the excitation—interface interaction, and 2а is the distance between the symmetrically located interface
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