Elliptical Gaussian beam propagation in inhomogeneous and nonlinear fibres of Kerr type

In this paper complex geometrical optics (CGO) is applied for the evolution of elliptical Gaussian beam (GB) propagating in nonlinear media of Kerr type. Instead of solving commonly accepted Nonlinear Schrodinger Equation, we propose equations of geometrical optics: complex eikonal equation and complex transport equation. Eikonal equation let us derive immediately the ordinary differential equations for principle widths omitting this way complicated variational process used in nonlinear optics. From the transport equation we obtain first order ordinary differential equation for complex amplitude evolution and conservation principle for energy flux in the GB cross-section. We prove in this paper that principle curvatures of the wave front can introduce into the description the additional ellipticity of the beam, which can effectively limit collapse effect. Moreover, we discus the evolution of elliptical Gaussian beam in nonlinear inhomogeneous fibre and we present conditions for stable propagation in the region without collapse effect. Under these conditions GB approaches asymptotically to stationary solutions when parameters of the fibre changes along propagation distance. We discus in this paper also the influence of initial wave front curvatures on Gaussian beam evolution in nonlinear medium of Kerr type and in nonlinear inhomogeneous fibre. We demonstrate high ability and effectively of CGO method as compared with commonly accepted methods of nonlinear optics such as variational method approach and method of moments. Moreover, we compare obtained results with solutions of Nonlinear Schrodinger Equation for ordinary and rotating elliptical beam propagating in nonlinear medium of Kerr type.