Exponential-recovery model for free-running SPADs with capacity-induced dead-time imperfections

Abstract

Current count-rate models for single-photon avalanche diodes (SPADs) typically assume an instantaneous recovery of the quantum efficiency following dead-time, leading to a systematic overestimation of the effective detection efficiency for high photon flux. To overcome this limitation, we introduce a generalized analytical count-rate model for free-running SPADs that models the non-instantaneous, exponential recovery of the quantum efficiency following dead-time. Our model, framed within the theory of non-homogeneous Poisson processes, only requires one additional detector parameter -- the exponential-recovery time constant $\tau_\mathrm{r}$. The model accurately predicts detection statistics deep into the saturation regime, outperforming the conventional step-function model by two orders of magnitude in terms of the impinging photon rate. For extremely high photon flux, we further extend the model to capture paralyzation effects. Beyond photon flux estimation, our model simplifies SPAD characterization by enabling the extraction of quantum efficiency $\eta_0$, dead-time $\tau_\mathrm{d}$, and recovery time constant $\tau_\mathrm{r}$ from a single inter-detection interval histogram. This can be achieved with a simple setup, without the need for pulsed lasers or externally gated detectors. We anticipate broad applicability of our model in quantum key distribution (QKD), time-correlated single-photon counting (TCSPC), LIDAR, and related areas. Furthermore, the model is readily adaptable to other types of dead-time-limited detectors. A Python implementation is provided as supplementary material for swift adoption.