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Probability Distribution of After Pulsing in Passive-Quenched Single-Photon Avalanche Diodes

The resolution and accuracy of the number of photons detected by a silicon photomultiplier (SiPM) are limited by the excess noise, which consists mainly of after pulsing and optical crosstalk. Since these excess noises are correlated with each other, it is challenging to determine the after pulsing and the optical crosstalk probabilities independently of the output signal of the SiPM. Here we examine the dynamic response of passive-quenched single-photon avalanche diodes to clarify the characteristics of the after pulsing. Optical crosstalk is not present in the output signals of single-photon avalanche diodes; therefore, we can obtain the after-pulsing probability directly from the experimental results. We derive an analytical expression for the probability density function of after-pulse timing and analyze the experimental data based on this expression. We confirm that two types of trap levels with emission lifetimes on the order of 10 and 100 ns are responsible for the after pulsing in our devices. The dependence of the after-pulsing probability on the device recovery time is also well explained by our model.