Chapter One - Gaussian Beam Propagation in Inhomogeneous Nonlinear Media: Description in Ordinary Differential Equations by Complex Geometrical Optics

Abstract

Abstract The method of complex geometrical optics (CGO) is presented, which describes the rotation of Gaussian beam (GB) propagating along a curvilinear trajectory in a smoothly inhomogeneous and nonlinear saturable optical medium. The CGO method reduces the problem of Gaussian beam diffraction and self-focusing in inhomogeneous and nonlinear media to the system of the first-order ordinary differential equations for the complex curvature of the wave front and for GB amplitude, which can be readily solved both analytically and numerically. As a result, CGO radically simplifies the description of Gaussian beam diffraction and self-focusing effects as opposed to the other methods of nonlinear optics, such as the variational method approach, method of moments, and beam propagation method. We first present a short review of the applicability of the CGO method to solve the problem of GB evolution in inhomogeneous linear and nonlinear media of the Kerr type. Moreover, we discuss the accuracy of the CGO method by comparing obtained solutions with known results of nonlinear optics obtained by the nonlinear parabolic equation within an aberration-less approximation. The power of the CGO method is presented by showing the example of N -rotating GBs interacting in a nonlinear inhomogeneous medium. We demonstrate the great ability of the CGO method by presenting explicitly the evolution of beam intensities and wave front cross sections for two, three, and four interacting beams. To our knowledge, the analyzed phenomenon of N -interacting rotating beams is a new problem of nonlinear wave optics, which demands a simple and effective method of solving it. Thus, we believe that the CGO method can be an interesting and effective tool to use to address sophisticated problems in electron physics.