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.

Figure 1

Incident and reflected beam geometry. (a) Schematic of the relation between the interface plane and our coordinate frame, showing the incident, virtual and real reflected beam. For each beam the cone depicts a section of constant $\delta$. (b) Intensity profile of an incident Bessel beam. (c) Shifted intensity profile of the real reflected beam for ${\theta }_0=44°$ and $n=2/3$. (d) Schematic of the beam coordinates. The local plane and angle of incidence depend both on $\delta$ and $\alpha$. (e) Beam coordinates for a $\ell =-4$ vortex beam (phase indicated by color wheel) and local resolution into ${\mathbf{e}}_s$ and $\text{}{\mathbf{e}}_p$.

Figure 2

(a) Plot comparing the constellation of vortices obtained from the roots of the local expansion in (8) with a numerical calculation of the phase of the field. The incident field is a Bessel beam with $\ell =-4$ incident at ${\theta }_0=46°$ and a fixed opening angle of $\delta =0.01$. The incident polarization $\mathit{E}$ and the analyzer $\mathit{F}$ are both oriented in the $x$ direction (TM/TM). $\times$ marks the origin and $◯$ the centroid of the vortices. (b) Plot showing the variation of constellations for 45° diagonal incident polarization and different analyzer settings including linear in the $x$ (TM) and $y$ direction (TE), right (RC), and left (LC) circular polarization, as well as 45° and 135° diagonal polarization.

Figure 3

Vortex constellations for increasing positive $\ell$ (a) and negative $\ell$ (b). (c) Contours in Fourier space (conical beam geometry) of the real and imaginary part of ${\mathit{F}}^{*}\mathbf{R}\mathit{E}$ for ${\theta }_0=46°$ and both $\mathit{F}$ and $\mathit{E}$ set to 45° diagonal polarization. (d)–(g) Contour plots for of the real parts of the coefficients in (8) multiplied by the appropriate Fourier factor $\exp (-im\alpha )$ for comparison with $\ell <0$ in (b). (d) $m=1$, (e) $m=2$, (f) $m=3$, (g) $m=4$. [Scaling identical to figure (c)]. The gradient over the contours in (d) corresponds to the shift as indicated by the black line between × and ○ in (b).