Precise relationship between registration and arrival counts under dead time

Determine the precise relationship between the distribution of the photon registration count M_r and the distribution of the photon arrival count M_a in an asynchronous single-photon LiDAR system with a nonparalyzable SPAD dead time, including how the mean and variance of M_r relate to those of M_a under varying total energy and signal-to-background ratio.

Background

The paper studies photon statistics in single-photon LiDAR where the arrival process across N laser cycles yields a Poisson-distributed count M_a with parameter NQ and i.i.d. timestamps governed by the arrival flux λ(t). In contrast, the registration process is affected by the SPAD’s nonparalyzable dead time, producing a registration count M_r whose distribution is not Poisson and whose timestamps are dependent.

Using simulations under different total energies and signal-to-background ratios (SBRs), the authors observe that M_r is bell-shaped with reduced mean and variance compared to M_a, but they do not have a closed-form or exact characterization connecting M_r to M_a. They note that changes in energy and SBR do not induce simple scaling behaviors in M_r, underscoring the challenge of accurately modeling the relationship between M_r and M_a.