Enhancement of collapse threshold of two Gaussian beams of light with orthogonal polarization in a nonlinear defocusing shifted parabolic graded-index rod lens

Abstract

A semi-analytical treatment of propagation and object–image transformation of two orthogonally polarized Gaussian beams of light passing through a nonlinear defocusing shifted parabolic graded-index rod lens is presented. Fields in the rod lens and imaging formulas through the rod lens are derived in terms of the parameters of two incident beams of light and the rod lens. The investigations show that there are two regimes of propagation in the rod, and collapse powers of two beams in a nonlinear defocusing shifted parabolic graded-index rod lens are much larger than those in a nonlinear shifted parabolic graded-index rod lens. PACS: 42.20; 42.30; 42.65 Erbium-doped fibers whose nonlinear coefficients reach 10−14 m2/W [1] are extensively used in fiber amplifiers and fiber lasers [2]. Intensity-dependent refractive-index properties of nonlinear optical guided-wave materials have many potential applications in all-optical signal-processing devices [3]. We have investigated the field and propagation of two orthogonally polarized Gaussian beams of light passing through a nonlinear shifted parabolic graded-index (GRIN) rod lens [4] and a nonlinear parabolic GRIN rod lens [5]. When the power of one beam of light is changed, the refractive-index profile of the nonlinear rod lens will be changed. The field and propagation of the other beam of light will also be affected because of the variations in the index profile. Therefore one beam of light can affect and control the field and propagation of the other beam of light in the nonlinear rod lens [4, 5]. These properties have many potential applications in the control of light by light. In this paper, we study the field, propagation, and imaging of two orthogonally polarized Gaussian beams of light passing through a nonlinear defocusing shifted parabolic GRIN rod lens. 1 Beam-width and inverse of radius of curvature of the wave front As shown in Figs. 1 and 2, a x-polarized Gaussian beam of light, with an electric field E1 and a waist radius ω01 located at a distance L01 (object distance) from the input plane Fig. 1. The x-polarized beam of two orthogonally polarized Gaussian beams is transformed to a new beam by a nonlinear defocusing shifted parabolic GRIN rod lens whose length is Z. The surfaces of constant index are for A1 > 0 (z = 0) of the rod lens, is transformed to a new Gaussian beam of light with a new waist radius ω31 located at a distance L1 (imaging distance) from the output plane (z = Z) of the rod lens. A y-polarized Gaussian beam of light with an electric field E2 and a waist radius ω02, located at a distance L02 (object distance) from the input plane of the rod lens, is transformed to a new Gaussian beam of light with a new waist radius ω32 located at a distance L2 (imaging distance) from the output plane of the rod lens. According to the ABCD law of Gaussian-beam propagation, at the input plane of the rod, the beam-width radii are given by [6] ω11 = ω01 [ 1 + ( L01 Z01 )2]1/2 ,