Existence of Optical Vortices

TL;DR

This paper proves the existence of optical vortex solutions in nonlinear Schrödinger equations, providing mathematical theorems, solution types, and explicit estimates related to light beam properties.

Contribution

It establishes new existence theorems for vortex solutions using variational methods and provides explicit bounds on wave parameters.

Findings

Existence of positive-radial-profile vortex solutions.

Existence of saddle-point vortex solutions with prescribed propagation constants.

Derived bounds for wave propagation constant based on beam power and vortex winding number.

Abstract

Optical vortices arise as phase singularities of the light fields and are of central interest in modern optical physics. In this paper, some existence theorems are established for stationary vortex wave solutions of a general class of nonlinear Schr\"{o}dinger equations. There are two types of results. The first type concerns the existence of positive-radial-profile solutions which are obtained through a constrained minimization approach. The second type addresses the existence of saddle-point solutions through a mountain-pass-theorem or min-max method so that the wave propagation constant may be arbitrarily prescribed in an open interval. Furthermore some explicit estimates for the lower bound and sign of the wave propagation constant with respect to the light beam power and vortex winding number are also derived for the first type solutions.