Abstract
In this paper we consider the process of the second harmonic generation in a gradient waveguide, taking into account diffraction and relatively weak temporal dispersion. Using the slowly varying envelope approximation and neglecting the dispersion of the nonlinear part of the response of the medium we obtain the system of parabolic equations for the envelopes of both harmonics. We also derive integrals of motion of this system. To solve it numerically we construct a nonlinear finite-difference scheme based on the Crank-Nicolson method preserving the integrals. Primarily, we focus our investigations on the processes of a two-component light bullets generation. We demonstrate that the generation of a coupled pair is possible in a planar waveguide even at normal group velocity dispersion.
Highlights
Propagation of waves in a homogeneous boundless medium is an idealization which is rarely found in nature
The way to derive the equation describing the interplay between dispersion, diffraction, inhomogeneity, and nonlinearity was described in detail in [3] for Kerr nonlinearity
Consecutive processes of second harmonic generation (SHG) and two-component light bullet formation are in the focus of the second series of our numerical simulation
Summary
Propagation of waves in a homogeneous boundless medium is an idealization which is rarely found in nature. By linear effects we mean diffraction and normal dispersion stretching a wave packet As it has been already mentioned above, waveguides demonstrate a remarkable ability to support multi-dimensional soliton structures in the media with Kerr nonlinearity [1,2,3,4]. Using the slowly varying envelope approximation and neglecting dispersion of the nonlinear part of the medium response, we develop a governing system of equations for SHG in a waveguide with transverse inhomogeneity. We prove this system possesses two motion integrals.
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