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.
 A: 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.
A: It is important to note that extraordinary laser beam splits into two ordinary photons, i.e. polarizations of laser and two downconverted photons are orthogonal. More over, photons produced in two crystals are not entangled.
