Expected value and variance of the indirect time-of-flight measurement with dead time afflicted single-photon avalanche diodes

Indirect time-of-flight (TOF) measurement with single-photon avalanche diodes (SPADs) is performed by counting incident photons in several time windows. Since SPADs exhibit dead time not all incident photons can be counted within a given time window. This affects the expected values and, hence, the variance of the distance measurement. For photon detection rates close to the inverse of the dead time, which defines the maximum count rate of a SPAD, the probability of photon detection cannot be assumed constant within the window anymore. In this paper, the effects of the dead time on the photon counts as well as the corresponding variances are analyzed by employing statistical calculations. Based on these a model which can be used to correct systematic error is derived. In addition, the detailed analysis of the variance is useful to estimate the performance of an indirect TOF system in the design phase.