Topological aberration of optical vortex beams and singularimetry of dielectric interfaces

TL;DR

This paper investigates the complex splitting behavior of high-order optical vortices upon reflection, linking vortex constellation geometry to optical aberrations and providing a new analytical approximation method.

Contribution

It introduces a novel analysis of vortex splitting and constellation formation using polynomial approximations, connecting optical aberration theory with vortex dynamics.

Findings

Vortex splitting leads to a constellation whose geometry depends on higher-order reflection terms.

The centroid of the vortex constellation relates to known beam shifts like Goos-H"anchen and Imbert-Federov.

An approximation using Appell polynomials effectively describes the vortex field around the constellation.

Abstract

The splitting of a high-order optical vortex into a constellation of unit vortices, upon total reflection, is described and analyzed. The vortex constellation generalizes, in a local sense, the familiar longitudinal Goos-H\"anchen and transverse Imbert-Federov shifts of the centroid of a reflected optical beam. The centroid shift is related to the centre of the constellation, whose geometry otherwise depends on higher-order terms in an expansion of the reflection matrix. We present an approximation of the field around the constellation of increasing order as an Appell sequence of complex polynomials whose roots are the vortices, and explain the results by an analogy with the theory of optical aberration.