Optimal collinear Gaussian beams for spontaneous parametric down-conversion

I investigate the properties of spontaneous parametric down-conversion (SPDC) involving co-linear Gaussian spatial modes for the pump and the photon collection optics. Approximate analytical and numerical results are obtained for the peak spectral density, photon bandwidth, pair collection probability, heralding ratio, and spectral purity, as a function of crystal length and beam focusing parameters. I address the optimization of these properties individually as well as jointly, and find focusing conditions that simultaneously bring the pair collection probability, heralding ratio, and spectral purity to near-optimal values. These properties are also found to be nearly scale invariant, that is, ultimately independent of crystal length. The results obtained here are expected to be useful for designing SPDC sources with high performance in multiple categories for the next generation of SPDC applications.