A novel geometric phase for optical beams

TL;DR

The paper reframes optical geometric phases as holonomy on Hopf-fibration structures, connecting the Pancharatnam phase to a line-bundle picture on $S^3\to S^2$ and then introducing a novel geometric phase for vortex beams in nanostructure–beam interactions. It develops dual geometric perspectives: a differential-geometry view via fibre bundles and connections, and a mode-structure view using Hermite-Gaussian and Majorana representations, linking physical phase to topological objects such as roots on the Majorana sphere. The main contributions are (i) a concrete description of a new geometric phase for beams under interaction with $ fold$-fold symmetric nanostructures and vortex TAM changes, (ii) explicit connections to SU(2) actions on Jones vectors and to Hopf-bundle geometry, and (iii) methods to realize mode-selective phase control with simple nanostructures as alternatives to spatial light modulators. The findings illuminate the shared geometric framework of Pancharatnam and vortex-beam phases and highlight practical implications for beam discrimination and control in nanostructured optics, while acknowledging open questions about the universality and uniqueness of the underlying connections.

Abstract

In this paper, we provide an accurate description of geometric phases emerging in simple optical systems -- nanostructures (metaatoms) interacting with vortex beams. We show that this interaction leads to a new class of geometric phase for optical beams, which is different from the geometric phases commonly discussed in structured-light optics. We compare this setting to the usual description of geometric phases for beams and show that the underlying geometry is different.

Figures (5)

• Figure 1: Poincaré sphere and examples of nanostructures for a geometric-phase experiment. (a) Depending on the rotation $\beta$ of the structure, the Poincaré sphere rotates around different axes. The path corresponds to the case where incident RCP light is converted to LCP light upon transmission through such a structure. (b) Schematic of the setup illustrating geometric phase. (c) Numerically calculated phase acquired upon rotation of a rectangular nanoparticle under excitation by a fundamental Gaussian beam.
• Figure 2: Illustration of the circle bundle over the sphere $S^2$. Fibers F are topologically circles $S^1$, the base manifold is a sphere $S^2$. The horizontal lift $\gamma$ of the curve $\gamma_0$ via a connection is shown. The tangent space to the total space $E$ at a point $p \in E$, $T_pE$ is shown, together with its decomposition as the direct sum of a horizontal and vertical subspace. Horizontal subspaces are determined by a connection form. The holonomy $\phi$ of the connection is the geometric phase.
• Figure 3: (a) Geometry of the experiment involving the transmission of vortex beams with total angular momentum projection $m$ through a prism with $\mathfrak{n}$-fold rotational symmetry. This setup is fully analogous to the one depicted in Fig. \ref{['mainscheme']}, but different $\beta$ is required. (b) Numerically calculated phase acquired by the rotation of nanoparticles with $D_{3h}$ and $D_{6h}$ symmetries under excitation by a vortex beam ($m$ = 3), the condition $2m=\mathfrak{n}\nu$ is satisfied. In both cases 2$\pi$ phase accumulated by $\pi/3$ rotation.
• Figure 4: A frequently used illustration of a geometric phase which, however, is not a Pancharatnam-Berry phase and has a different geometry; it is equal to the full solid angle rather than half of it.
• Figure 5: Stereographic projection establishes a one-to-one correspondence between the points on the sphere and those on the plane.