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Journal of the Optical Society of America A

Uniqueness of Poincaré beam projections

Susheel Bhardwaj, Mohammad Umar, P. Senthilkumaran

Published Apr 17, 2026• 3 citations

Abstract

A single projective measurement on a superposition state does not provide the full information about the state. Therefore, multiple projections of the state onto a multitude of eigenstates are necessary. In this paper, we have demonstrated that generic Poincaré beams can be uniquely identified through a single projective measurement, although they are superposition states. Polarization optics utilizes different orders of Poincaré spheres as topological models to describe both uniform and spatially varying polarization states. These topological constructs quantify the underlying singularities and symmetries. Any measurement performed using a polarizer collapses the superposition state into one of its eigenstates. The additional features introduced by the orbital angular momentum basis, which vary across different orders of Poincaré spheres, make such projection measurements particularly interesting. In particular, a generic Poincaré beam exhibits the unique property that every possible superposition state produces a distinct signature intensity pattern when passed through a polarizer, a phenomenon explored in this paper.

DOI: 10.1364/josaa.590329

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A single projective measurement on a superposition state does not provide the full information about the state. Therefore, multiple projections of the state onto a multitude of eigenstates are necessary. In this paper, we have demonstrated that generic Poincaré beams can be uniquely identified through a single projective measurement, although they are superposition states. Polarization optics utilizes different orders of Poincaré spheres as topological models to describe both uniform and spatially varying polarization states. These topological constructs quantify the underlying singularities and symmetries. Any measurement performed using a polarizer collapses the superposition state into one of its eigenstates. The additional features introduced by the orbital angular momentum basis, which vary across different orders of Poincaré spheres, make such projection measurements particularly interesting. In particular, a generic Poincaré beam exhibits the unique property that every possible superposition state produces a distinct signature intensity pattern when passed through a polarizer, a phenomenon explored in this paper.

Non-separable superpositions of polarization and spatial mode of a single photon produce a state that has a polarization that depends on the transverse position, and contains all states of polarization represented on the Poincaré sphere. We have done measurements of the space-dependent state of polarization of single photons prepared in distinct 2×2 (qubit-qubit) and 2×3 (qubit-qutrit) non-separable superpositions of Laguerre-Gauss spatial and polarization states. Detection was done by polarimetry of the light projected at distinct locations in the transverse plane. The polarization patterns had a C-point polarization singularity (lemon, star or monstar) at the center of the transverse wavefunction.

Poincaré beams (PBs), characterized by their entanglement between helical wavefronts and vectorial polarization, serve as high-dimensional information carriers in classical and quantum optics. The topological charge (TC) is the fundamental parameter governing their singularity structure; however, characterizing the phase and polarization TCs simultaneously remains a significant challenge. Conventional approaches are often limited to static measurements or decouple these parameters through complex, multi-step processes, failing to capture the real-time dynamics essential for modern optical communications. Here, we demonstrate a robust, single-shot detection system capable of simultaneously resolving phase and polarization TCs for arbitrary states on the hybrid-order Poincaré sphere (HyPS). By leveraging a pair of Pancharatnam-Berry optical elements (PBOEs), we implement a conformal mapping that transforms the full angular momentum eigenstates into spatially separated coordinates. This unique configuration allows for the direct readout of TCs through position encoding, effectively decoupling the spin and orbital angular momentum components. We experimentally validate this method by detecting the dynamic evolution of PBs across eight different HyPSs with component orders up to 20, confirming its capability to distinguish between scalar, vector, and hybrid modes in real-time. This work establishes a versatile framework for the dynamic decoding of structured light, holding profound implications for high-capacity optical communications and information encryption.

We propose a geometric hybrid Poincaré sphere (GHPS) as a unified geometrical framework for describing structured photon states with independently controllable spin angular momentum (SAM) and orbital angular momentum (OAM). Unlike the conventional higher-order Poincaré sphere, in which the SAM and OAM are intrinsically coupled through fixed basis states, the GHPS is constructed by defining its poles as direct products of arbitrary orthogonal bases on the Poincaré sphere (PS) and orbital Poincaré sphere (OPS) and by superposing these pole states. Using numerical simulations, we analyze representative GHPS states and show that the GHPS spherical coordinates govern the amplitude ratio and relative phase between the pole bases. This framework enables spatially inhomogeneous polarization distributions and intensity patterns, including nonseparable structures in which polarization and intensity are intrinsically intertwined, and provides a systematic state-space description for the coherent geometrical control of advanced structured light fields.

Superpositions of orbital angular momentum (OAM) eigenstates form a continuous state space characterized by the complex coefficients of the constituent modes. When such OAM superposition states propagate through a random medium, they produce speckle patterns. The random granular speckle patterns encode the complete information of the original superposition states. This work demonstrates that the complex coefficients of an arbitrary OAM superposition can be retrieved from a single-shot speckle image. The extraction is performed using custom-designed regression model and is validated through both experimental measurements and numerical simulations. The method remains effective for superpositions of high-dimensional OAM eigenstates, including intensity-degenerate states, demonstrating that speckle can serve as a robust information carrier for state characterization.