# What are the 'types' of parametric down conversion?

I'm looking at photon entanglement, and everywhere in the literature there's a reference to 'type-II' parametric down conversion as a source of entangled photon pairs. I know what parametric down conversion is, and I understand the resulting entangled state of type-II parametric down conversion.

I'm guessing there's some sort of 'type-I' out there, but I can't seem to find a good comparison between the two. What is type-I parametric down conversion? Does it have anything to do with pairs of entangled photons? I feel like I'm missing something.

As opposed to type II phase matching that produces orthogonally polarized photons in parametric down conversion (PDC), the type I PDC process produces identically polarized photons in the output signal and idler modes (labels $s$ and $i$ below).
Normally the output state from type I PDC is not entangled: to get the required phase matching in the nonlinear material, the pump polarization must be fixed. Both the PDC photons may then either be horizontally or vertically polarized. An often-used trick is to employ two similar nonlinear crystals (placed one after the other with their optic axes orthogonal) and sending a pump with a $45^{\circ}$ polarization. If the crystals are thin enough to simultaneously lie inside the coherence length of the pump, and losses between the first and second crystal are negligible, then a pump photon is equally likely to excite the PDC process in either of the two crystals. In that case, the output state may be approximated as $\propto |H_s,H_i\rangle + e^{i \phi}|V_s,V_i\rangle$ which is an entangled state. The relative phase $\phi$ is a function of the phase matching, thickness of the crystals, etc.