Chapter One

Gaussian Beam Propagation in Inhomogeneous Nonlinear Media

Description in Ordinary Differential Equations by Complex Geometrical Optics

Pawel Berczynski1 and Slawomir Marczynski2     1Institute of Physics, West Pomeranian University of Technology, Szczecin 70-310, Poland     2Faculty of Mechanical Engineering and Mechatronics, West Pomeranian University of Technology, Szczecin 70-310, Poland

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.

Keywords

rotating Gaussian beams interacting in nonlinear medium; self-focusing; light diffraction; complex geometrical optics

1. Introduction

In the traditional understanding, geometrical optics is a method assigned to describe trajectories of rays, along which the phase and amplitude of a wave field can be calculated via diffractionless approximation (Kravtsov & Orlov 1990; Kravtsov, Kravtsov, & Zhu, 2010). Complex generalization of the classical geometrical optics theory allows one to include diffraction processes into the scope of consideration, which characterize wave rather than geometrical features of wave beams (by diffraction, we mean diffraction spreading of the wave beam, which results in GB having inhomogeneous waves). Although the first attempts to introduce complex rays and complex incident angles started before World War II, the real understanding of the potential of complex geometrical optics (CGO) began with the work of Keller (1958), which contains the consistent definition of a complex ray. Actually, the CGO method took two equivalent forms: the ray-based form, which deals with complex rays—i.e., trajectories in complex space (Kravtsov et al., 2010; Kravtsov, Forbes, & Asatryan 1999; Chapman et al. 1999; Kravtsov 1967)—and the eikonal-based form, which uses complex eikonal instead of complex rays (Keller & Streifer 1971; Kravtsov et al., 2010; Kravtsov, Forbes, & Asatryan 1999; Kravtsov 1967). The ability of the CGO method to describe the diffraction of GB on the basis of complex Hamiltonian ray equations was demonstrated many years ago in the framework of the ray-based approach. Development of numerical methods in the framework of the ray-based CGO in the recent years allowed for the description of GB diffraction in inhomogeneous media, including GB focusing by localized inhomogeneities (Deschamps 1971; Egorchenkov & Kravtsov 2000) and reflection from a linear-profile layer (Egorchenkov & Kravtsov 2001). The evolution of paraxial rays through optical structures also was studied by Kogelnik and Li (1966), who introduced the concept of a very convenient ray-transfer matrix (also see Arnaud 1976). This method of transformation is known as the ABCD matrix method (Akhmediev 1998; Stegeman & Segev 1999; Chen, Segev, & Christodoulides 2012; Agrawal 1989). The eikonal-based CGO, which deals with complex eikonal and complex amplitude was essentially influenced by quasi-optics (Fox 1964), which is based on the parabolic wave equation (PWE; Fox 1964; Babi? & Buldyrev 1991; Kogelnik 1965; Kogelnik & Li 1966; Arnaud 1976; Akhmanov & Nikitin 1997; Pereverzev 1993). In the case of a spatially narrow wave beam concentrated in the vicinity of the central ray, the parabolic equation reduces to the abridged PWE (Vlasov & Talanov 1995; Permitin & Smirnov 1996), which preserves only quadratic terms in small deviations from the central ray. The abridged PWE allows for describing the electromagnetic GB evolution in inhomogeneous and anisotropic plasmas (Pereverzev 1998) and in optically smoothly inhomogeneous media (Permitin & Smirnov 1996). The description of GB diffraction by the abridged PWE is an essential feature of quasi-optical model. It is a convenient simplification, nevertheless it still requires solving of partial differential equations. The essential step in the development of quasi-optics was done in various studies that analyzed laser beams by introducing a quasi-optical complex parameter q (Kogelnik 1965; Kogelnik and Li 1966), which allows for solving the parabolic equation in a more compact way, taking into account the wave nature of the beams. The obtained PWE solution enables one to determine such GB parameters as beam width, amplitude, and wave front curvature. The quasi-optical approach is very convenient and commonly used in the framework of beam transmission and transformation through optical systems. However, modeling GB evolution by means of the quasi-optical parameter q using the ABCD matrix is effective for GB propagation in free space or along axial symmetry in graded-index optics (on axis beam propagation) when the A,B,C, and D elements of the transformation matrix are known. Thus, the problem of GB evolution along curvilinear trajectories requires the solution of the parabolic equation, which is complicated even for inhomogeneous media (Vlasov & Talanov 1995). In fact, the description of GB evolution along curvilinear trajectories by means of the parabolic equation is limited only to the consideration of linear inhomogeneous media (Pereverzev 1998; Vlasov & Talanov 1995; Permitin & Smirnov 1996). In our opinion, the eikonal-based form of the paraxial CGO seems to be a more powerful and simpler tool involving wave theory, as opposed to quasi-optics based on the parabolic equation, and even the CGO ray-based version based on Hamiltonian equations. The problem of Gaussian beam self-focusing in nonlinear media was usually studied by solving the nonlinear parabolic equation (Akhmanov, Sukhorukov, & Khokhlov 1968; Akhmanov, Khokhlov, & Sukhorukov 1972). The abberrationless approximation enables to reduce the nonlinear parabolic equation to solving the second-order ordinary differential equation for Gaussian beam width evolution in a nonlinear medium of the Kerr type, but the procedure is complicated. Because of the general refraction coefficient, the CGO method presented in this paper deals with ordinary differential equations; it does not ask to reduce diffraction and self-focusing descriptions starting every time from partial differential equations. The well-known approaches of nonlinear optics, such as the variational method and method of moments, demand that the nonlinear parabolic equation gets solved by complicated integral procedures of theoretical physics, which can be unfamiliar to engineers of optoelectronics, computer modeling, and electron physics. It is worthwhile to emphasize that the variational method and method of moments have been applied to model Gaussian beam evolution in nonlinear graded-index fibers (Manash, Baldeck, & Alfano 1988; Karlsson, Anderson, & Desaix 1992; Paré & Bélanger 1992; Perez-Garcia et al. 2000; Malomed 2002; Longhi & Janner 2004). Moreover, analogous solutions can be obtained by the CGO method in a more convenient and illustrative way. The CGO method deals with Gaussian beams, which are convenient and appropriate wave objects to model famous optical solutions (Anderson 1983; Hasegawa 1990; Akhmediev 1998; Stegeman and Segev 1999; Chen, Segev, & Christodoulides 2012) propagating in nonlinear optical fibers (Agrawal 1989). The CGO method presented in this paper has been applied in the past to describe GB evolution in inhomogeneous media (Berczynski and Kravtsov 2004; Berczynski et al. 2006), nonlinear media of the Kerr type (Berczynski, Kravtsov, & Sukhorukov 2010), nonlinear inhomogeneous fibers (Berczynski 2011) and nonlinear saturable media (Berczynski 2012,...