Frequency-angular spectrum of entangled photon pairs generated by focused pump

Spontaneous parametric down-conversion (SPDC) is a well-established method for generating entangled photon pairs, and their frequency-angular spectrum properties are of significant interest in various applications. Generally, the focused pump can increase the efficiency of single-mode photons generation, and lead to the asymmetric broadening of the SPDC emission cone relative to plane wave, while our analysis shows it will be further intensified when the pump divergence exceeds 1.3°. In this paper, we propose a theoretical model for accurately ascertaining the frequency-angular spectrum of entangled photon pairs produced by arbitrary pumps, then obtain complete spatial distribution and spectral properties of type I and II SPDC photons, and further demonstrate its validity referring to experimental results of other researchers. Our method is available for high-efficient generation of entangled photon pairs through optimizing crystal orientation, pump beam focus, and detection geometry.