Effective SU(2) gadget: holonomic walk on higher-order Poincaré sphere

TL;DR

This work introduces an effective SU(2) gadget for the higher-order Poincaré sphere (HOPS) by combining three coaxial q-plates into an effective waveplate with tunable retardance $\\delta_e$ governed by the relative offset $\\Delta\\alpha$. Under the holonomy condition $q=\\eta$, the gadget enables continuous, holonomic polarization evolution on the HOPS, with the rotation axis lying in the equatorial plane and the input/output beams preserving their topological charge. The three$q$-plate configurations $q^{Q}q^{H}q^{Q}$, $q^{Q}q^{Q}q^{H}$, and $q^{H}q^{Q}q^{Q}$ are shown to be equivalent to a single effective $q$-plate with tunable $\\delta_e$ and a fixed offset $\\alpha_e$, enabling point-to-point navigation on the HOPS. The theory bridges effective-waveplate formalism with higher-order singular optics, offering a pathway to deterministic control of structured light, vector-vortex beams, and topological optical fields, with potential experimental realization via metasurfaces or liquid-crystal platforms.

Abstract

In a manner commensurate to the SU(2) gadget for the Poincaré sphere, which involves a combination of two quarter-wave plates and one half-wave plate regardless of their sequential order, an analogous construct for the higher-order Poincaré sphere had long remained elusive. To address this, we recently demonstrated, by modifying Euler-angle parameterization, that an optical gadget consisting of two quarter-wave $q$-plates and one half-wave $q$-plate, each endowed with the same topological charge $q$, operates as a viable SU(2) gadget for the higher-order Poincaré sphere, contingent upon the fulfillment of the holonomy condition. This work presents the controlled navigation on the higher-order Poincaré sphere through the proposed architecture, formulated under the concept of an effective waveplate and thus can be referred to as an effective SU(2) gadget. The notion of an effective waveplate refers to a coaxial arrangement of multiple waveplates that, under specific constraints, can act as a single waveplate. In this gadget, the relative alignment of the offset angles of the constituent $q$-plates emerges as the decisive parameter governing systematic navigation on the higher-order Poincaré sphere. This study bear direct relevance to the deterministic control and engineering of structured light, encompassing polarization singularities, vector vortex beams and topological optical fields. Moreover, these results holds potential applications in the contemporary research frontiers in singular optics, spin-orbit photonics and quantum communication.

Figures (4)

• Figure 1: (Color online). The polarization distribution on the higher-order Poincaré spheres for $\eta = -2$, $-1$, $0$, $+1$, and $+2$ is displayed. Here, $2\gamma^{(\eta)}$ and $2\chi^{(\eta)}$ represent the longitude and latitude coordinates, respectively. The red and blue color coding indicate right circular polarization (RCP) and left circular polarization (LCP), respectively. Additionally, the vortex phase embedded with the RCP and LCP for the respective values of $\eta$ (except for $\eta = 0$) is also provided.
• Figure 2: (Color online). Figure showing (a) the HOPS of order $\eta=1$ with four circles lying on its surface around the $S_{1}^{(1)}$-axis, labeled $1$–$4$, and (b) the HOPS of order $\eta=2$ with four circles lying on its surface around the $S_{2}^{(2)}$-axis, labeled $1$–$4$. Each circle contains eight points separated by an angular interval of $\pi/4$.
• Figure 3: (Color online). Figure illustrating the polarization transformation on the HOPS of order $\eta = 1$ through the effective $q^{Q}q^{H}q^{Q}$ gadget. The input beam, chosen on the $S_{1}^{(1)}$–$S_{3}^{(1)}$ plane (shown here), undergoes a complete $2\pi$ rotation aroud the $S_{1}^{(1)}$ axis on the HOPS as $\Delta \alpha = \alpha_{Q} - \alpha_{H}$ is varied from $\pi/2$ to $0$. To achieve rotation around the $S_{1}^{(1)}$ axis, $\alpha_{Q}$ is set to $\pi/4$, effectively making the offset angle zero and aligning the rotation axis with the $S_{1}^{(1)}$ axis. For each point on the circles of the HOPS shown in Fig. \ref{['hops_01']}a, the corresponding HOPS beam is presented here.
• Figure 4: (Color online). Figure illustrating the polarization transformation on the HOPS of order $\eta = 2$ through the effective $q^{Q}q^{H}q^{Q}$ gadget. The input beam, chosen on the $S_{2}^{(2)}$–$S_{3}^{(2)}$ plane (shown here), undergoes a complete $2\pi$ rotation around the $S_{2}^{(2)}$ axis on the HOPS as $\Delta \alpha = \alpha_{Q} - \alpha_{H}$ is varied from $\pi/2$ to $0$. To achieve rotation around the $S_{2}^{(2)}$ axis, $\alpha_{Q}$ is set to $\pi/2$, making the effective offset angle $\pi/4$ and consequently aligning the rotation axis with the $S_{2}^{(2)}$ axis. For each point on the circles of the HOPS shown in Fig. \ref{['hops_01']}b, the corresponding HOPS beam is presented here.