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Optical vortex light engendered with integer units of orbital angular momentum (OAM) may be involved in frequency upconversion. Second harmonic generation is usually forbidden in isotropic media due to parity constraints, but it becomes allowed by six-wave mixing. Here, we present a rigorous quantum analysis for the case of a Laguerre-Gaussian input beam comprising photons endowed with a single unit of OAM. Such a process gives rise to the novel entanglement of orbital momentum in two emergent photons; it transpires that the mechanism delivers a harmonic output whose polarization is essentially parallel to the incident radiation. This investigation ascertains the character of the emission, both under forward propagation and back-reflection geometries, and identifies in detail the form of distribution in the entangled total orbital momentum. A distinctive conical spread, originating from the entangled …