We investigate the non linear 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.

No abstract

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.Besides their intrinsic beauty, optical beams carrying orbital angular momentum (OAM) have proved to be a powerful tool for encoding and processing quantum information. First order Laguerre-Gaussian and Hermite-Gaussian (HG) modes carry the mathematical structure of a qubit [1] and allow for a two-qubit encoding when combined with polarization of a single photon. The interplay between the two degrees of freedom leads to interesting applications, including topological phases [2,3], quantum cryptography [4][5][6], Bell inequalities [7-9], quantum logic gates [10][11][12], and quantum teleportation [13][14][15]. Polarization controlled spatial correlations between entangled photon pairs were first demonstrated in [16,17]. Nowadays, a

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.

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.