We demonstrate electromagnetic quantum states of single photons and of correlated photon pairs exhibiting ``hybrid'' entanglement between spin and orbital angular momentum. These states are obtained from entangled photon pairs emitted by spontaneous parametric down conversion by employing a $q$ plate for coupling the spin and orbital degrees of freedom of a photon. Entanglement and contextual quantum behavior (that is also nonlocal, in the case of photon pairs) is demonstrated by the reported violation of the Clauser-Horne-Shimony-Holt inequality. In addition, a classical analog of the hybrid spin-orbit photonic entanglement is reported and discussed.

Spin-orbit hybrid entanglement of photons and quantum contextuality (vol 82, 022115, 2010)

Abstract Structured light refers to the arbitrarily tailoring of optical fields in all their degrees of freedom (DoFs), from spatial to temporal. Although orbital angular momentum (OAM) is perhaps the most topical example, and celebrating 30 years since its connection to the spatial structure of light, control over other DoFs is slowly gaining traction, promising access to higher-dimensional forms of structured light. Nevertheless, harnessing these new DoFs in quantum and classical states remains challenging, with the toolkit still in its infancy. In this perspective, we discuss methods, challenges, and opportunities for the creation, detection, and control of multiple DoFs for higher-dimensional structured light. We present a roadmap for future development trends, from fundamental research to applications, concentrating on the potential for larger-capacity, higher-security information processing and communication, and beyond. 

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Towards higher-dimensional structured light

The orbital angular momentum of light is distinct from the angular momentum associated with the spin or polarisation state of the light. Thus linearly polarised modes can carry angular momentum. Laguerre-Gaussian laser modes have been identified as a class of modes that can possess such orbital angular momentum. Hermite-Gaussian modes may be converted to Laguerre-Gaussian modes, characterised by two integer indices l and p, by the use of a mode converter. The l index occurs in the azimuthal phase factor eil ϕ in the field distribution of Laguerre-Gaussian beams which possess an orbital angular momentum equivalent to lħ per photon 1.

Second Harmonic Generation and the Orbital Angular Momentum of Light

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Optical angular momentum-based photonic technologies demonstrate the key role of the optical spin–orbit interaction that usually refers to linear optical processes in spatially engineered optical materials1. Re-examining the basics of nonlinear optics of homogeneous crystals under circularly polarized light2,3, we report experiments on the enrichment of the spin–orbit angular momentum spectrum of paraxial light. The demonstration is made within the framework of second-harmonic generation using a crystal with three-fold rotational symmetry. Four spin–orbit optical states for the second harmonic field are predicted from a single fundamental state owing to the interplay between linear spin–orbit coupling and nonlinear wave mixing; three of these states are experimentally verified. Besides representing a spin-controlled nonlinear route to orbital angular multiplexing4, modal vortex light sources5,6, high-dimensional parametric processes7 and multi-state optical magnetization8, our findings suggest that the fundamentals of nonlinear optics are worth revisiting through the prism of the spin–orbit interaction of light. An experimental study of the second-harmonic-generation process in a beta barium borate crystal shows that homogeneous optical crystals can exhibit the rich physics of the spin–orbit angular momentum cascade in the nonlinear optical regime.

Harmonic spin–orbit angular momentum cascade in nonlinear optical crystals

We investigate the nonlinear mixing of orbital angular momentum in type-II second-harmonic generation with arbitrary topological charges imprinted on two orthogonally polarized beams. Starting from the basic nonlinear equations for the interacting fields, we derive the selection rules determining the set of paraxial modes taking part in the interaction. Conservation of orbital angular momentum naturally appears as the topological charge selection rule. However, a less intuitive rule applies to the radial orders when modes carrying opposite helicities are combined in the nonlinear crystal, an intriguing feature confirmed by experimental measurements.

Orbital-angular-momentum mixing in type-II second-harmonic generation

We demonstrate second harmonic generation performed with optical vortices with different topological charges imprinted on orthogonal polarizations. Besides the intuitive charge doubling, we implement arbitrary topological charge addition on the second harmonic field using polarization as an auxiliary parameter.

https://iopscience.iop.org/article/10.1088/1367-2630/16/9/093041/pdf

Arbitrary orbital angular momentum addition in second harmonic generation

We demonstrate polarization-controlled switching of the orbital angular momentum (OAM) transfer in nonlinear wave mixing. By adjusting the input beam geometry, we are able to produce a three-channel orbital OAM, with arbitrary topological charges simultaneously generated and spatially resolved in the second-harmonic wavelength. The use of path and polarization degrees of freedom allows nearly perfect optical switching between different OAM operations. These results are supported by a theoretical model showing very good agreement with the experiments.

Polarization-controlled orbital angular momentum switching in nonlinear wave mixing.

The origin of radial structures generated in nonlinear vortex interactions is investigated. A theoretical model is developed and an experiment is performed to explain how two main parameters govern the process: the mismatch between the topological charge and the corresponding radial power law of the nonlinearly generated beam.

Chiral relations and radial-angular coupling in nonlinear interactions of optical vortices

The interest in tailoring light in all its degrees of freedom is steadily gaining traction, driven by the tremendous developments in the toolkit for the creation, control and detection of what is now called structured light. Because the complexity of these optical fields is generally understood in terms of interference, the tools have historically been linear optical elements that create the desired superpositions. For this reason, despite the long and impressive history of nonlinear optics, only recently has the spatial structure of light in nonlinear processes come to the fore. In this review we provide a concise theoretical framework for understanding nonlinear optics in the context of structured light, offering an overview and perspective on the progress made, and the challenges that remain. 

https://www.oejournal.org/data/article/export-pdf?id=62b2cbe299d8816244ff952c

W. T. Buono

Papers

Orbital-angular-momentum mixing in type-II second-harmonic generation

Arbitrary orbital angular momentum addition in second harmonic generation

Polarization-controlled orbital angular momentum switching in nonlinear wave mixing.

Chiral relations and radial-angular coupling in nonlinear interactions of optical vortices

Nonlinear optics with structured light