Type I parametric down conversion of highly focused Gaussian beams in   finite length crystals
  We study the correlations in wave vector space of photon pairs generated by type I spontaneous parametric down conversion using a Gaussian pump beam. We analyze both moderate focused and highly focused regimes taking special attention to the angular spectrum and the conditional angular spectrum. Simple analytic expressions are derived that allow us to study in detail the dependence of these spectra on the waist of the source and the length of the nonlinear crystal. These expressions are in good agreement with numerical expectations and reported experimental results. They are used to make a systematic search of optimization parameters that improve the feasibility of using highly focused Gaussian beams to generate idler and signal photons with predetermined mean values and spread of their transverse wave vectors. 
Yasser Jeronimo-Moreno and R. J\'auregui
quant-ph physics.optics