A Unified SU(2) Framework for Vector Beam Transformations and Complex Beam Shaping

TL;DR

This paper introduces a unified SU(2) framework for designing optical elements that transform structured light fields, enabling complex beam shaping and vector beam transformations with explicit constructive methods.

Contribution

It provides a systematic, constructive approach to design doubly inhomogeneous waveplates for structured light manipulation using SU(2) operations, unifying various problems in a single framework.

Findings

Explicit construction of doubly inhomogeneous waveplates (d-plates)

Condition for exact implementation of prescribed transformations

Application to quantum channels on orbital angular momentum

Abstract

We present a constructive framework for designing transformations between structured light fields using birefringent optical elements, formulated in terms of SU(2) operations on polarization. Within this framework, transformations between vector beams are treated as spatially varying SU(2) operations, leading to a direct procedure for designing doubly inhomogeneous waveplates (d-plates) that implement the desired mapping. We identify a condition under which a single element implements a prescribed transformation exactly, including the global phase, and provide an explicit prescription for constructing the corresponding doubly inhomogeneous waveplate (d-plate) when this condition is satisfied, along with its realization using a finite sequence of singly inhomogeneous plates, including a QHQ configuration. Within this formulation, a broad class of problems in structured light can be treated within a single framework, including vector beam transformations, spin-orbital dynamics, and complex beam shaping. Crucially, the same SU(2) operations directly realize quantum channels on the orbital angular momentum degree of freedom, with polarization serving as a physical ancilla. These results establish a unified and explicitly constructive route to complex beam shaping and vector beam transformations based on SU(2) parameter synthesis, and provide a systematic foundation for designing next-generation photonic elements for structured light and spin-orbit information processing.