Diffraction of a Gaussian beam in a three-dimensional smoothly inhomogeneous medium: an eikonal-based complex geometrical-optics approach

Abstract

We present an ab initio account of the paraxial complex geometrical optics (CGO) in application to scalar Gaussian beam propagation and diffraction in a 3D smoothly inhomogeneous medium. The paraxial CGO deals with quadratic expansion of the complex eikonal and reduces the wave problem to the solution of ordinary differential equations of the Riccati type. This substantially simplifies the description of Gaussian beam diffraction as compared with full-wave or parabolic (quasi-optics) equations. For a Gaussian beam propagating in a homogeneous medium or along the symmetry axis in a lenslike medium, the CGO equations possess analytical solutions; otherwise, they can be readily solved numerically. As a nontrivial example we consider Gaussian beam propagation and diffraction along a helical ray in an axially symmetric waveguide medium. It is shown that the major axis of the beam's elliptical cross section grows unboundedly; it is oriented predominantly in the azimuthal (binormal) direction and does not obey the parallel-transport law.

Gaussian beam evolution in longitudinally and transversely inhomogeneous nonlinear fibres with absorption

“…Hence, it is worth to deal with an orthogonal coordinate system, which would be locally rotationless with respect to the central ray. The unit vectors (l, e 2 , e 2 ) of such system are determined by the relations [6,10] e 1 = ncosq + bsinq, e 2 = bcosq -nsinq,…”

“…where q is the rotation angle, obeying the equation [10]. In what follows we refer to the set (l, e 2 , e 2 ) as to the parallel transport basis.…”

“…A surprising feature of CGO is its ability to describe Gaussian beam (GB) diffraction in both ray−based and eikonal−based approaches. Recently, CGO method has been applied to describe GB evolution in inhomogeneous media [9][10][11] and nonlinear media of Kerr type [11,12] including graded−index [13] and nonlinear fibres [14]. It is shown in Ref.…”

Gaussian beam evolution in nonlinear inhomogeneous fibres

“…A surprising feature of paraxial complex geometrical optics (PCGO) is its ability to describe the diffraction of a Gaussian beam (GB) [1]- [6]. Recently, the PCGO method has been generalized for nonlinear media of the Kerr type reproducing the basic results of classical nonlinear optics [7] and opening the way for further generalizations.…”

“…(10) Solution (10) reflects the general feature of PCGO, which in fact deals with Gaussian beams. The general form of the Gaussian beams in the 3D inhomogeneous media, as well as a general Riccati equation for complex curvature parameter B can be found in [6] and in the review paper [3].…”