Managing orbital angular momentum in second-harmonic generation

We propose a convenient approach for managing orbital angular momentum (OAM) of second-harmonic photons generated by a quadratic nonlinear interaction under noncolinear type-I phase matching condition. Since the OAMs of two fundamental photons can be flexibly controlled by using computer-generated holograms, we were able to realize arbitrary combination of both OAMs and experimentally confirm the OAM conservation selection rule ${\ensuremath{\ell}}_{2\ensuremath{\omega}}={\ensuremath{\ell}}_{\ensuremath{\omega}}^{\ensuremath{'}}+{\ensuremath{\ell}}_{\ensuremath{\omega}}^{\ensuremath{'}\ensuremath{'}}$ in general cases. In particular, our approach could be used to generate the second-harmonic photon states with fractional and odd-order OAMs.