The Geometry of Paraxial Vector Beams

Abstract

This work unveils a novel and fundamental connection between structured light and topological field theory by showing how the natural geometrical setting for paraxial vector beams is that of a $SU(2)$ principal bundle over $\mathbb{R}^{2+1}$. Going beyond the usual high-order Poincaré sphere approach, we show how the nonseparable structure of polarisation and spatial modes in vector beams is naturally described by a non-Abelian Chern-Simons gauge theory. In this framework, we link the Chern-Simons charge to spin-orbit coupling, and we propose a simple way to experimentally detect the presence of non-Abelian phases through Wilson lines. This new insight on vector beams opens new possibilities for realising and probing topological quantum field theories using classical optics, as well as it lays the foundation for implementing topologically protected classical and quantum information protocols with structured light.

Figures (2)

• Figure 1: Pictorial representation of the possible choices of compactification for a paraxial VB. (a) One point compactification at infinity ($\mathbb{R}^{2+1}\cup\{\infty\}\cong S^3$. This is the most common compactification strategy, resulting in identifying the beam with $S^3$, here represented through its Hopf fibration over the Poincaré sphere, where each point on the sphere corresponds to a great circle on $S^3$nakaharamandelWolf. (b) Rayleigh range compactification ($\mathbb{R}^{2+1}\cong S^2\times S^1$. Here, the $z$-direction is compactified into a circle by identifying the endpoints of the propagation interval $[-nz_R,nz_R]$. At each point over the circle $S^1$ (red line, labelled by $z$) we define a Poincaré sphere, identified using the spherical angles $(x,y)\cong(\theta,\varphi)$ using one point compactification at infinity on the transverse plane. (c) Vortex compactification ($\mathbb{R}^{2+1}\cong S^2\times S^1$). This procedure is similar to that in (b), but here we compactify the azimuthal direction $\varphi$ into the circle $S^1$ (red line, labelled $\varphi$) by identifying the endpoints of the interval $[0,2\pi]$. This results in a circle $S^1$ threading a vortex line (blue line), corresponding to the phase singularity carried by the beam. As before, at each point along the $\varphi$-circle we associate a Poincaré sphere identified by the radial and propagation coordinates $(r,z)$ of the beam, using one point compactification on the $(r,z)-$plane.
• Figure 2: Pictorial representation of the topological classification of VBs, defined as in Eq. \ref{['eqs2']} (a). The black ellipse contains 3D topologically nontrivial VBs, for which $\,athcal{S}_V[M]\neq 0$, and the Wilson holonomy is in general matrix-valued. An example of VBs belonging to this class are the hopfions hopfionsMark. The red ellipse contains instead 2D topologically nontrivial VBs, possessing $\mathcal{S}_V[M]=0$, but a nonzero skyrmion-like number $Q_{2D}$. Examples of these VBs are skyrmionic beams ref24 or Poincaré beams ref23. The blue ellipse contains instead the 1D topologically nontrivial VBs, for which both $\mathcal{S}_V[M]$ and $Q_{2D}$ are zero, but they possess nonzero winding number $Q_{1D}$. Representatives of this class are vortex beams, beams carrying polarisation singularities, and VBs that can be defined on a HOPS. Finally, the green ellipse contains topologically trivial VBs, characterised by all the topological charges $\mathcal{S}_V[M]$, $Q_{2D}$, and $Q_{1D}$ being zero, and hence by a trivial Wilson holonomy. This is the class, where uniformly polarised beams live.