  Orbital angular momentum (OAM) conservation plays an important role in
shaping and controlling structured light with nonlinear optics. The OAM of a
beam originating from three-wave mixing should be the sum or difference of the
other two inputs because no light-matter OAM exchange occurs in parametric
nonlinear interactions. Here, we report anomalous OAM conservation during
parametric upconversion, in which a Hermite-Gauss mode signal interacts with a
specially engineered pump capable of astigmatic transformation in a crystal,
resulting in Laguerre-Gaussian mode sum-frequency generation (SFG). The anomaly
here refers to the fact that the pump and signal carry no net OAM, while their
SFG does. We show that the lost OAM with the opposite sign that maintains OAM
conservation in the system is hidden in the residual pump. This unexpected OAM
selection rule improves our understanding of OAM conservation in parametric
nonlinear systems and may inspire new ideas for controlling OAM states via
nonlinear optics, especially in quantum applications.
