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.

Figure 1

Demonstration of the role of spatial symmetry in determining OAM conservation in a type-II SPDC process. In the three cases where $l_c=10\mu \mathrm{m}$, 0.1 mm, and 0.5 mm, spatial symmetry breaking progressively causes higher probability that OAM is not conserved. (a) Typical azimuthal-angle dependence of $W(\Delta k_z)l_c^{-1}$, which is a measure for the degree of spatial symmetry breaking of the Hamiltonian. (b) The (normalized) probability of generating down-converted photon pairs as a function of extrinsic OAM $l_{-}\hslash$ per pair in the degrees of relative-movement freedom. When the medium thickness is $l_c=0.5\mathrm{mm}$, the symmetry-breaking induced OAM nonconservation dominates over OAM conservation, in which case there is a chance of $<15\%$ that the OAM carried by the photon pairs is conserved, i.e., $l_{-}=0$.

Figure 2

Illustrating how the extrinsic part of OAM ($l_{-}\hslash$ per pair) carried by photon pairs in the degrees of relative-movement freedom escapes detection in the traditional OAM measurement scheme [3], which only measures the intrinsic portion of OAM ($l_{+}\hslash$ per pair) in the degrees of freedom of c.m. movement. In the traditional scheme, two sets of optical devices are located at conjugate positions ($\pm {\mathbf{p}}_0$) on the down-conversion rings (thin solid circles). However, in the degrees of relative-movement freedom, detection of one photon by one set of detection devices centered at ${\mathbf{p}}_0$ (donutlike marker) projects its twin into a mode (indicated by the dashed circle) centered also at ${\mathbf{p}}_0$, which is out of detection scope for the other set of devices centered at $-{\mathbf{p}}_0$ (cross).