Spatial Symmetry and Conservation of Orbital Angular Momentum in Spontaneous Parametric Down-Conversion

Directly contradictory arguments coexist regarding the conservation rule of orbital angular momentum in spontaneous parametric down-conversion. We analytically show how this rule is decided by spatial symmetry. We discover that the down-converted photon pairs can carry non-negligible extrinsic orbital angular momentum in the degrees of relative-movement freedom due to spatial symmetry breaking, leading to nonconservation of total orbital angular momentum in type-II down-conversion. Also, we demonstrate that the traditional technique does not measure the extrinsic orbital angular momentum.