TITLE: Ordinary and extraordinary polarization
QUESTION [2 upvotes]: The paper Entangled biphoton source - property and preparation says

A very interesting situation of type-II SPDC is for ‘non-collinear phase matching’. The signal–idler pair are emitted from a SPDC crystal, for example BBO, cut in type-II phase matching, into two cones, one ordinary polarized, the other extraordinary polarized (see figure 10). Along the intersection, where the cones overlap, two pinholes numbered as 1 and 2 are used for defining the direction of the $k$ vectors of the signal–idler pair. One may consider the polarization state of the signal–idler pair as,
$$|\psi\rangle = \frac{1}{\sqrt{2}}(|o_1+e_2\rangle+|e_1o_2\rangle)$$
where $o_j$, $e_j$, $j = 1, 2$ are ordinary and extraordinary polarization, respectively. 


What does it mean by "ordinary and extraordinary polarization"? I haven't heard those terms before.

REPLY [2 votes]: The simplest way of modeling light traveling through a medium is with "ray-optics" which is a method by which you model your beam of light as a series of light rays. Here's a picture to give you an idea of what it looks like if you haven't seen it before:

One of the most basic principles of ray optics is Snell's Law, which tells you how each light-ray bends when they hit a medium. In the attached picture, ray-optics is used to show that light can be focused with a lens. The math of how this lens bends light can be derived from Snell's Law. Ray optics is a simplified model and there are exceptions; one particularly notable exception is birefringence, in which different polarizations of light get bent differently. Sometimes one of these polarizations obeys Snell's law (it acts ordinary) and the other one doesn't (it acts extraordinary). The main idea is simply that two beams travel two different paths along the crystal associated with the polarization.  
I work in an quantum information lab that uses such a crystal for SPDC inside a cavity. Depending on what type of crystal you use, you can go from having entangled light to squeezed light!
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