Dependence of beating dynamics on the ellipticity of a Gaussian beam in graded-index absorbing nonlinear fibers

Abstract

Using the collective variable approach technique, we analyze propagation of elliptical Gaussian beams in nonlinear waveguides with a parabolic graded-index (GRIN) profile. We considered both saturable and cubicquintic models to describe the nonlinearity, taking into account both linear and nonlinear absorption. For lossless media, we construct diagrams, which define regions of self-focusing and self-diffractive beam propagation for both models in GRIN waveguides and compare them with those for nongraded waveguides. The widths of the propagating elliptic beam exhibit an oscillatory pattern, similar to the "breathing" and "beating" behavior found in nongraded media. Two types of beating oscillations are observed in both models. We calculate the dependence of L beat /L br , the ratio of the "beating" to "breathing" oscillation periods, on the beam ellipticity ρ and the GRIN index g. We find that there is a remarkable difference in this dependence between saturable and cubic-quintic media: in the saturable model, L beat /L br is a monotonic function of ρ, whereas in the cubic-quintic model, it is characterized by singularities, which correspond to transitions between the types of beat oscillations. For lossy media, we discuss the difference between the breathing behavior in nongraded and GRIN waveguides.

Analytic methods to find beating transitions of asymmetric Gaussian beams in GNLS equations

“…x k k 2 is the Spence or dilogarithm function [1]. For both CQ and SAT cases we observe that A 2 a x a y is conserved [23,22,21], and we write A 2 a x a y = 4E (4E is the beam energy), and use this to eliminate A(z) . Furthermore, φ(z) evidently plays no role in determining the other functions and can be computed by a simple quadrature once the other functions have been found.…”

“…In the sequence of papers [23,22,21] a variational approach was taken to investigate the propagation of asymmetric (elliptic) Gaussian beams in nonlinear waveguides, with cubic-quintic and saturable nonlinearities and a parabolic graded-index (GRIN) profile, as described by suitable generalized nonlinear Schrödinger equations (GNLSEs). The beam widths in the two transverse directions to the direction of propagation were found to obey a set of ordinary differential equations which can be identified as the equations of motion of a point particle in certain rather complicated, but tractable, 2d potentials.…”

“…The structure of this paper is as follows. In the next section we review the relevant models from nonlinear optics and the collective variable approximation to obtain equations for the propagation of beam widths, and present the main findings of papers [23,22,21] and some further numerical results. In section 3, we develop our method of integrable approximation for small oscillations in a 2 degree-of-freedom Hamiltonian system near a fixed point close to 1 : 1 resonance.…”

“…Here, modulo suitable normalizations [30,21], ψ is the strength of the electric field, z is the longitdinal coordinate, x, y are transverse coordinates, and Q, α, g are parameters.…”

“…The term −g(x 2 + y 2 )ψ reflects the graded-index nature of the fibre, that the refractive index n falls with distance r from the center of the fibre according to the law n 2 = n 2 0 − Gr 2 ; the physical significance of this is explained in [47,14,21].…”