Topological aberration of optical vortex beams: determining dielectric interfaces by optical singularity shifts

We predict the splitting of a high-order optical vortex into a constellation of unit vortices, upon total internal reflection of the carrier beam, and analyze the splitting. The reflected vortex constellation generalizes, in a local sense, the familiar longitudinal Goos-Hänchen and transverse Imbert-Fedorov shifts of the centroid of a reflected optical beam. The centroid shift is related to the center of the constellation, whose geometry otherwise depends on higher-order terms in an expansion of the reflection matrix. We derive an approximation of the amplitude around the constellation as a complex analytic polynomial, whose roots are the vortices. Increasing the order of the initial vortex gives an Appell sequence of complex polynomials, which we explain by an analogy with the theory of optical aberration.