Single photon detectors are important for a wide range of applications each with their own specific requirements, which makes necessary the precise characterization of detectors. Here, we present a simple and cost-effective methodology of estimating the dark count rate, detection efficiency, and afterpulsing in single photon detectors purely based on their counting statistics. This methodology extends previous work [IEEE J. Quantum Electron., vol. 47, no. 9, pp. 1251-1256, Sep. 2011], [Electron. Lett., vol. 38, no. 23, pp. 1468-1469, Nov. 2002]: 1) giving upper and lower bounds of afterpulsing probability, 2) demonstrating that the simple linear approximation, put forward for the first time in [Electron. Lett. , vol. 38, no. 23, pp. 1468-1469, Nov. 2002], yields an estimate strictly exceeding the upper bound of this probability, and 3) assessing the error when using this estimate. We further discuss the requirements on photon counting statistics for applying the linear approximation to different classes of single photon detectors.

http://vcq.quantum.at/typo3temp/pics/23478694bd.gif

Example of a histogram (drawn in log scale) representing an arbitrary afterpulsing model whose important property is that afterpulses eventually die off after a time T.

Single photon detection at telecom wavelengths has attracted significant research efforts due to its numerous applications in metrology and telecommunications as well as in quantum optics where it is particularly relevant for quantum key distribution (QKD).

Characterization of single photon detectors has become an important task in order to compare and select the right parameters for a specific application. Here we discuss and develop further a method for afterpulsing estimation, which uses a discrete, binned probability density function of the timing distances between the measured events. Based on the theoretical probability density function of time measurement events, as recorded by a perfect detector, which detects photons, generated by a light source at random times and independently one from the other, this method allows separating the imperfection in a very simple way. It even lets detector assessment using only the intrinsic dark counts.

This method is a generalization of a procedure proposed in [1], [2], which is specifically designed for characterizing detectors operating in gated mode with the objective to obtain a robust estimate of the various performance parameters, especially the afterpulsing probability. The advancement presented in this paper extends the applicability to the free-running detection mode and allows using any light generation process if it can be approximated by a Poisson one. Importantly, this includes the intrinsic dark counts of the detector. Our method only requires the time-binned statistical measurement of detection events and is easily realizable in hardware allowing for a quick assessment of single photon counting detectors. Fundamentally, similar to [2] it is based on a linear regression fit of the detection events’ histogram in contrast to an approximation (second order Taylor series expansion) of the afterpulsing waiting probability suggested in [1].

Simultaneously in contrast to [2] a precise mathematical derivation of the waiting probability of detection events is put forward and the waiting probabilities characterizing the different classes of events (source photons, dark counts, afterpulsing) are systematically studied. Moreover we derive bounds for the cumulative afterpulsing probability and use these for estimating the error introduced by the linear regression approach. In any case it should be underlined that unless the exact functional dependence of the afterpulsing probability as a function of time is known, our method (exactly as the approaches of [1], [2]) can only serve to find an upper bound of afterpulsing and thus verify that the detector performs better, i.e., has a lower afterpulsing probability than is determined by the linear regression.

The well-known standard method [3], in contrast can exactly determine the afterpulsing probability as a function of time. The difference of the two approaches lies in the fact that while the standard method requires relatively advanced instrumentation including pulsed sources, the method discussed here does not even require a light source. So it can be used as quick approach to determine an upper bound of afterpulsing.

We have tested our results for different detector classes using simulation tools, and have also done an experimental proof of principle validation using a self-designed and implemented single photon detector (custom-made electronics with a commercial Indium Gallium Arsenide/Indium Phosphide single photon avalanche diode, PGA-400 by Princeton Lightwave, Inc.) that we had at our disposal.

**Publication in "Journal of Lightwave Technology":**

"A Simple and Robust Method for Estimating Afterpulsing in Single Photon Detectors"; G. Humer, M. Peev, C. Schaeff, S. Ramelow, M. Stipcevic, R. Ursin;

Journal of Lightwave Technology, Vol. 33, 3098 - 3107 (2015),

DOI: 10.1109/JLT.2015.2428053