Estimation of photon number distribution and derivative characteristics of photon-pair sources

The evaluation of a photon-pair source employs metrics like photon-pair generation rate, heralding efficiency, and second-order correlation function, all of which are determined by the photon number distribution of the source. These metrics, however, can be altered due to spectral or spatial filtering and optical losses, leading to changes in the metric characteristics. In this paper, we theoretically describe these changes in the photon number distribution and the effect of noise counts. We also review the previous methods used for estimating these characteristics and the photon number distribution. Moreover, we introduce an improved methodology for estimating the photon number distribution, focusing on photon-pair sources, and discuss the accuracy of the calculated characteristics from the estimated (or reconstructed) photon number distribution through simulations and experiments