  We elucidate the consequences of a phase-matching theory that describes
second-harmonic generation of two non-collinear incident light beams that carry
orbital angular momentum (OAM). More specifically, the two incident beams
generate a third that, depending on the incident OAM, may experience a
significantly smaller conversion efficiency in comparison to that based on the
conventional phase-matching theory. This is the case even for incident angles
substantially less than those required for non-conservation of OAM in the
nonlinear interaction. Experiments are performed under different conditions and
are in excellent agreement with the theory. Our results have implications
beyond the specific case studied here of second-harmonic generation, in
particular for parametric down-conversion of photons.
