Existence of Optical Vortices

Yisong Yang,Ruifeng Zhang

doi:10.1137/120894105

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 Schrödinger 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 saddlepoint 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 of solution.

Highlights

Vortices have important applications in many areas of modern physics including condensed matter systems, particle interactions, cosmology, and superfluids
Research on vortices in optics has a long history and was initiated in as early as 1964 by Chiao, Garmire, and Townes [5] who explored some conditions under which a light beam can produce its own waveguide and propagate without spreading
They described such phenomenon as self-trapping, attributed it to light propagation in materials whose dielectric coefficient increases with field intensity in the context of high-intensity light beams such as lasers, predicted marked optical and physical effects, and suggested the occurrence of optical vortices

Summary

Introduction

Vortices have important applications in many areas of modern physics including condensed matter systems, particle interactions, cosmology, and superfluids. A fundamental prototype situation is when the light waves are described by a complex-valued wave function governed by nonlinear Schrodinger equations [1, 7, 17, 18, 20, 21, 26, 28]. These theoretical studies provide a broad range of interesting analytic problems related to the existence and properties of optical vortices for mathematical investigation.

Vortices via constrained minimization

Vortices as saddle points