Hi,

If you're only interested in finding the angles so you could order crystals from a crystal supplier, the easiest way is to use SNLO software: http://www.as-photonics.com/snlo

If you're also intersted in searching for the exact phase-matching angles yourself, first you should decide whether you're interested in a collinear solution, in which the pump photon and the two down-converted photons propagate in the same direction, or a non-collinear solution, in which the directions of the pump differs from the direction of the down-converted photons (this naturally coplicates the calculation). Then, you should decide if you're interested in a degenerated case, in which the down-converted photons share the same wavelength, or not.

Then, search for the crystal's Sellmeier's equations, which relate the wavelength to the refractive index (note that there are two equations for uniaxial crystals, one for the ordinary and the other for the extraordinary polarizations). After deciding on the specific wavelengths you're intersted in, calculate no and ne for the wavelengths involved in the process using the Sellmeir equations. All you have to do now is use a computer program to solve the energy and momentum equations for the three photons invloved in the process, depending on the specific process you've chosen (collinear/non-collinear, degenerated/non-degenerated, etc.) . 

For that you should choose an input polarization and define a propagation direction within the crystal. For a uniaxial crystal, only a single angle (usually referred to as theta) is enough. Using this angle you could find the two refractive indices eigenmodes [no, ne(theta)], and use them to solve the momentum conservation equation.

More details (regarding a more general case in which the crystal is biaxial) can be found at:

A. Halevy, E. Megidish, L. Dovrat, H.S. Eisenberg, P. Becker, and L. Bohatý, “The biaxial nonlinear crystal BiB3O6 as a polarization entangled photon source using non-collinear type-II parametric down-conversion,” Opt. Express 19, 20420–20434 (2011).

https://www.osapublishing.org/oe/abstract.cfm?uri=oe-19-21-20420

Another very relevant article is the following:

N. Boeuf, D. Branning, I. Chaperot, E. Dauler, S. Guérin, G. Jaeger, A. Muller, and A. Migdall, “Calculating characteristics of noncollinear phase matching in uniaxial and biaxial crystals,” Opt. Eng. 39, 1016–1024 (2000).

http://opticalengineering.spiedigitallibrary.org/article.aspx?articleid=1076089

I hope it helps,

Assaf

Download the SNLO software for free. 

Thank you so much, Assaf Halevy and Tenio Popmintchev. Answers were really helpful

But, we can use SNLO software for  limited crystals only. What will be the other choices to calculate phase matching angle?

Another option is Phasematch: https://www.nist.gov/services-resources/software/noncollinear-phase-matching-uniaxial-and-biaxial-crystals