1. Introduction

Measurements of faint states of light are at the forefront of modern science, with applications ranging from astrophysics to molecular biology, and from quantum information science to fundamental tests on the nature of reality. The breadth of applications makes single-photon-detector technology an area of intense and sustained interest [1]. There are a variety of technologies used for single-photon detection, particularly: photomultipliers, semiconductors, and superconductors, and within each of these systems there are multiple device designs. While they differ radically in their nature and operation, in all cases their response to and recovery from a detection event is a complex and temporally evolving process. This makes the state of the detector at any given time dependent on the device’s prior history, as with, for example, a detector with a dead time followed by a re-arming or reset process, and afterpulsing. For many types of measurements, and particularly high-precision measurements, a detector’s complex history dependence can lead to systematic errors that must be accounted for in analysis, and this sort of accounting requires a comprehensive knowledge of the detection system. For a typical non-photon-number-resolving detector (or click detector) accurate characterization includes not only the detection efficiency (DE), jitter, and noise properties when the device is fully armed and ready, but also the device’s dead time, its behavior during reset, and afterpulsing, as well as the history dependence of these properties.

There are a few methods that are used to identify and characterize properties of single-photon detectors [2] so that they can be used for quantitative radiometric measurements. Particularly, a calibrated continuous wave (CW) or pulsed light source can be used to approximately determine DE. The use of primary standard sources, such as a blackbody [3] or a synchrotron source [4], is preferred for traceable radiometry because their output depends on independently verifiable parameters of the physical system. In addition, there is another primary calibration method based on quantum sources that produce photons in pairs [5], so that a detection of one photon of the pair indicates the presence of another; a so-called correlated-photon-pair calibration method [6–9]. However, the systematic uncertainty of most accurate radiometric measurements with single-photon counters (including primary calibration measurements) is by far inferior to that with classical detectors. This is because additional sets of measurements are required to characterize properties other than DE [9–15], significantly inflating the experimental overhead of precision radiometric measurements with photon counters. To this end, it is important to offer a uniform way to characterize single-photon detectors as accurately as possible regardless of the differences in their underlying technology. At the same time the method must be simple in implementation.

Here we introduce a method that enables the complete and simultaneous characterization of all the parameters of a non-photon-number resolving (non-PNR) detectors relevant to high-precision measurements (except jitter), including the history dependence of these various parameters, with a simple and straightforward measurement. This method is based on using an attenuated calibrated CW laser beam as an input to the detector and a nearly deadtime-free time-tagging system that monitors the detector output. The critical requirement is that the time-tagging system’s inactive time, the time when it is not capable of detecting the next output from the detector (if any), is shorter than the detector’s recovery time. The relative simplicity of the physical implementation of this technique is supplemented with advanced statistical analysis. In addition to raw detection counting, we perform the calculation of an auto-correlation of second and third orders on the collected data. We expose single-photon avalanche diodes (SPADs) to a series of input conditions, from no input light to relatively high photon fluxes from an attenuated laser, and study correlation functions. We show, for the first time, that second-order detector models that are commonly used to characterize a detector have limited accuracy and lead to systematic errors, which is particularly important for precision measurements and substantial at high count rates. We attribute the observed third-order effects in our devices under test to heretofore under-appreciated transient properties of single-photon avalanche detectors. As detection systems with ever higher speed and performance are developed, the ability to identify and characterize such transient properties will become more important. The method we present here provides that, along with a straightforward means to characterize most of the other properties of a detector.

2. Models of a photon-counting detector

2.1. A generalized second-order model

It is known that a typical non-PNR detector’s characterization consists of finding its detection efficiency, dark count rate, recovery time, and afterpulsing [1]. DE definitions vary, but the most typical definition is the probability that a fully armed detector fires in response to a single-photon input. To first order, DE is a proportionality coefficient between the rate of incoming photons and the detection rate and the dark count rate is the rate of detections when no light is present at the input. These two parameters can be found from just the total number of single counts, a first-order measurement. Such a measurement gives the apparent rate of detected photons per exposure time, r_measured. However, this approach leads to significant systematic errors. The recovery time, or the time immediately after a detection during which the detector is incapable of producing an electronic output, and afterpulsing, excess dark counts at the end of recovery time are effects that are correlated with a prior detection event. Therefore, they can be assessed through a second-order autocorrelation measurement, C⁽²⁾(Δt). Historically, a device characterization that relies on just one prior detection has been deemed entirely sufficient. One could formalize any such second-order detector model in the most general way as follows:

The features of this histogram can be used to identify the detector’s behavior under a second-order model. Specifically for SPADs, some of these features, particularly the perceived afterpulse probability, strongly depend on the probability of a recent prior event, i.e. they are strongly average-countrate dependent [9, 12]. This particular behavior can still be explained within a second-order model of the detector by introducing a two-stage recovery process, i.e. with a dead time and reset time (t_recovery = t_dead + t_reset). This is a typical situation in actively quenched solid-state single-photon detectors. During t_dead the detector is truly dead, i.e. it produces no avalanches. Photoelectronic detections that occur during the reset time are delayed and can contribute to a sharp afterpulsing peak. Other contributions to such a peak are true afterpulses that occur during the detector’s reset time and, usually to a lesser extend, primary dark counts. The exponentially decaying tail is typically due to true afterpulses. They occur because on rare occasions carriers get trapped for a significant time and may cause a delayed avalanche. Under a second-order detector model, the two-stage recovery can be characterized by subtracting the true afterpulse profile from histograms at a range of detection rates, and checking the dependence of any observed excess afterpulse peak Ξ on the count rate. Note that the sum is computed for delays in the interval (t_recovery, 2t_recovery). Using this range of delays guarantees that the afterpulsing correction is only applied once, thus eliminating systematic errors from higher-order effects that will be discussed later on. We write:

2.2. A higher-order model and its significance

A second-order model has been quite successful in describing single-photon detectors. However, its implicit assumption that the detector’s behavior may only depend on the time passed since a previous detection has not been directly verified. This verification is particularly important, because a second-order model has difficulties with extrapolation to high count rates. To verify a second-order model, we employ a third-order autocorrelation.

Models based on fourth- or higher- order correlation functions can in principle be similarly developed. To evaluate these correlation functions, the same experimental dataset f(t) is used. A fourth-order correlation function can be used to validate a third-order model of a detector in an exact analogy with our use of a third-order correlation measurement to validate a generalized second-order model.

3. Measurements

3.1. Setup

A schematic of the experiment is shown in Fig. 1. An attenuated laser source produces a weak CW coherent state (i.e. a state with Poisson photon statistics) at 852 nm. Times of photoelectronic detections are recorded with a custom FPGA-based circuit. This system time tags events in 100 ps time bins and its recovery time is <1.6 ns, [16]. Such a short FPGA recovery time satisfies the chief requirement of this characterization technique for a broad range of single-photon detectors. We obtain and analyze experimental data collected with two commercially available SPADs. As expected, experimentally acquired correlation functions provide extremely rich information about the device under test. Owning to the simplicity of this method, characterizing a device at a range of input photon fluxes is a straightforward exercise. Measurements at very low input flux are useful to characterize true afterpulsing, whereas measurements at higher count rates provide a better characterization of recovery time, and in our case reveal countrate-dependent effects.

 

Fig. 1 Schematic of the characterization setup.

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We illustrate our method by characterizing two silicon SPADs: one with a thick and one a thin junctions, [1]. These detectors are commercially available and employ different logic circuits. Throughout the text, we will refer to these detectors as detector A (thick junction SPAD) and detector B (thin junction SPAD).

3.2. Second-order correlation measurements

As we established before, to characterize a detector, a source can be set to a low count rate, and then to a relatively high count rate. The former is important to measure purely electronic (or primary) afterpulsing properties of a detector. The higher-count-rate measurement increases the probability of closely-spaced detection events, and thus is useful in determining recovery-related effects. As seen in Fig. 2, an afterpulsing profile is a function with a complex shape that can contain a sharp afterpulsing peak and a slow exponential tail, a structure similar to that discussed in [15]. If two such features of the afterpulsing behavior are identifiable, as is the case for detector A, it suggests that this SPAD has a two-stage recovery process, t_recovery = t_dead + t_reset. At significantly high input photon fluxes other effects become evident. This behavior illustrates a limitation of a second-order detector model and may lead to a systematic uncertainty in radiometric measurements.

 

Fig. 2 Typical second-order measurements on detectors A (a through c) and B (d through f) at an average count-rate of ≈500000 s⁻¹. (a, d) High count rate start-multistop histogram, C⁽²⁾. Main features are the recovery time, afterpulsing peak. (b, e) Start-single stop histograms of the same experiment as above. Histograms of first, second, third and fourth detection are shown and marked by color. (c, f) Afterpulsing profile of a detector, obtained under low-average count-rate conditions, with photon counts and dark counts subtracted, therefore the effect of higher-order detections not associated with the laser source is negligible.

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Detector B demonstrates a significantly different behavior. Its afterpulse only contains an exponentially decaying tail, seen in Fig. 2(f) (cf. [15]). The high-mean-count-rate histogram has a decaying tail, similar to detector A, indicating that the excess counts seen in this peak are primarily pure electronic (“true”) afterpulses. We see that the second-order theory can accurately describe the behavior of this detector without appealing to a two-stage recovery time. However, even at this moderately high count rate, there already are signs of the excess saturation seen in Fig. 2(d) histogram. Particularly, the histogram in Fig. 2(d) falls slightly below a steady-state asymptote at delays near 140 ns to 160 ns (approximately equal to twice the recovery time). This saturation, unaccounted for by the model, becomes even more significant with higher count rates, as can be more clearly seen in Fig. 3(b). Behaviors such as this can cause a systematic uncertainty in high-accuracy measurements.

 

Fig. 3 Afterpulsing seen as second-order correlation features for a range of input powers for detector A (a) and B (b).

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The shortcomings of a second-order model can be seen in Fig. 2(b) and Fig. 2(e). First, we observe that the shape of the first afterpulse peak is significantly different from that of the second one for both detectors. This observation does not automatically invalidate the model, since this could be due to random electronic jitter in detection times causing the second and further afterpulse peak features to be less sharp. It could be a plausible explanation to some extent in case of detector A, but the evidence seen in this figure is insufficient. In case of detector B, the minimal perceived recovery time between the first and third detections is longer than twice the recovery time between the two subsequesnt detections. This feature points at a significant violation of the assumption of a second-order model and will be investigated further.

Next, we turn to the dependence of the afterpulsing peak on count rate, presented in Fig. 3. Both detectors exhibit a significant countrate dependence of their afterpulsing features. Detector A has a tall peak during the first 2 ns to 3 ns after the recovery. We also see that the recovery time of detector A slightly depends on the count rate, i.e. for higher input photon fluxes the afterpulsing peak is delayed by ≈ 0.5 ns. This dependence of the afterpulse peak position on the count rate cannot be accounted for within a second-order model. Detector B presents a strong suppression of counts at high average count rates, best seen at ≈ 2t_recovery delay. A long exponential tail of the afterpulsing feature does not depend on countrate for either detector A or B.

To determine if the two-stage recovery mechanism applies, we use a second-order model of a detector with a one-stage recovery time (i.e. we assume that recovery time equals dead time, found as the maximal delay time on a second-order histogram during which there is no detector output), see Eq. (2). To use Eq. (1), we compute the photon detection rate corrected for deadtime r_corrected and use electronic afterpulsing found from data presented in Fig. 2(c) and Fig. 2(f) according to Eq. (2). The results of this calculation are plotted in Fig. 4 as a function of corrected count rate.

 

Fig. 4 The two-step detector recovery calculations. A plot of observed afterpulse counts at different input powers with true electronic afterpulses subtracted. Detector A exhibits a rise in apparent afterpulsing whereas detector B exhibits their suppression. Dots: measured values. Solid lines: linear fits based on low count rates. Dashed lines: extrapolation of linear fits. Deviation of the measured data from these extrapolations shows saturation effects at higher count rates.

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In case of detector A, this method works well. We see that for low count rates this function follows a linear dependence closely, and then saturates at rates above 10⁶ s⁻¹. Using a linear regression, we find a correction to an inactive time. Our method gives t_recovery =53.5(5) ns; t_reset =6.2(5) ns. Therefore, t_dead = t_recovery −t_reset =47.3(5) ns. The uncertainty is due to variations of recovery time with the count rate, seen in Fig. 3(a).

We now apply the same formalism to analyze the detector B. Detector B’s t_recovery =70.6(1) ns. It produces no sharp afterpulsing feature in Fig. 3(b) and a negative slope in Fig. 4, pointing to a different effect. It is highly likely that some of the time the detector B goes into an additional recovery cycle that is approximately as long as the first recovery cycle. This assumption is supported by the shape of the saturation region (it is uniform between the delay times from t_recovery to 2t_recovery), as seen in Fig. 3(b). A model of two-stage recovery developed eariler, Eq. (2), can be modified based on this assumption:

Findings of the second-order model for both detectors are compiled into a table 1. In this demonstration we use a set of pre-calibrated filters to extract the DE. Note that the accuracy of DE characterization is limited by filter calibration accuracy, due to laser instability and power meter accuracy, and not by the method described here. We list calibration results for detector B with and without the assumption introduced above. We see that this assumption leads to a significant increase of the afterpulsing probability. Note the difference between the apparent and inferred afterpulsing probabilities with and without the double-recovery cycle assumption made for detector B, Eq. (6). This is because the degree of detection suppression seen in Fig. 3(b) for high input powers is very significant.

Table 1. Summary of SPAD parameters found under a second-order detector model, see text.

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3.3. Third-order correlation measurements

To better understand rate dependence effects, we evaluated a third-order autocorrelation function C⁽³⁾, as in Eq. (3), and $C_{\text{theor}}^{(3)}$, as given by Eq. (4), from a single experimental set of photoelectronic detection times. The higher-order autocorrelation histogram is presented in Fig. 5(a) and Fig. 5(d). We compare this histogram to $C_{\text{theor}}^{(3)}$ constructed from C⁽²⁾. The difference between the observed third-order autocorrelation function C⁽³⁾ and the expected third-order autocorrelation function under the second-order approximation $C_{\text{theor}}^{(3)}$ is best represented as a logarithm of the ratio of the two, see Eq. (5), Fig. 5(b) and 5(e). We see that R ≈ 0 for a wide range of delays (Δt⁽¹²⁾, Δt⁽¹³⁾) for both detectors, indicating that the underlying assumption of the second-order model holds true most of the time. This is not surprising because the second-order model is successful in describing SPADs to within experimental uncertainties considered so far. There are, however, significant deviations from the model at time delays that correspond to second afterpulses (i.e. afterpulses of afterpulses). As it is expected from the previous discussion, these deviations are very different for detectors A and B.

 

Fig. 5 Third-order measurements on detectors A (a through c) and B (d through f). (a, d) Raw third-order start-multistop histogram, C⁽³⁾. Surface cuts (marked with horizontal lines) represent the second afterpulsing peak, and are plotted separately. (b, e) Test of a second-order detector model. Difference between expected and actual histograms expressed as a logarithm of the ratio of the measured and expected histograms, see Eq. (5). The second-order model is correct except for times immediately following the inactive time, indicating that more than one prior detection affects the detector’s behavior. (c, f) Structure of second afterpulses, i.e. afterpulses that occur immediately after an earlier afterpulse, and a transition to the steady-state. A significant change in the second afterpulse shape with respect to that of the typical afterpulse is seen for detector A. A significant decrease in the afterpusing events is seen for detector B.

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For detector A, we observe that if a first avalanche occurred within the afterpulse peak (just after the reset time) then there is a non-trivial effect. At certain delays, the probability of an afterpulse in the vicinity of the previous afterpulse is reduced by as much as nearly an order of magnitude; at other delays it is increased by nearly an order of magnitude. Fig. 5(c) shows the series of second-afterpulse profiles for different delays between the first and the second events. This heretofore undiscovered effect can be ascribed to the reset time in this actively quenched SPAD: when an avalanche occurs during the reset time, the active quenching circuitry is not activated in its usual manner, but rather is activated at the end of the reset time (as with the output signal, producing the ’spike’ in afterpulses observed in Fig. 1). As a result, the avalanche current through the SPAD can persist for longer than usual, populating a larger number of charge traps within the device than in the case of a normally quenched avalanche. The systematic uncertainty due to a varying afterpulsing probability is count-rate independent for true electronic afterpulses and can be metrologically significant (i.e. contributing to systematic errors on the level of ≈0.1%) in some devices. The systematic uncertainty due to valid photon detections during t_reset grows with the count-rate and can become metrologically significant when count-rate reaches 10⁶. This effect can only be seen by considering higher-order correlations in the device’s output. The dependence of the afterpulse probability on the total charge that flows through the SPAD during the avalanche is a very important property. It can be used to develop new detectors with better characteristics, such as shorter inactive times, [17].

For detector B, we also observe a significant third-order effect, also in a vicinity of the second afterpulse. The comparison of the second-order and third-order detector models, Fig. 5(e), shows a significant difference between the actual detector’s output and the output predicted by a second-order model. Here we see that the second recovery time is longer than usual if it immediately follows a prior recovery process, Fig. 5(d). This result compliments the histogram seen in Fig. 2(e), where it is observed that the recovery time and the shape of a third detection differ significantly from those of the second detection. Here we see that the second recovery time strongly depends on the time difference between the preceding two counts; for preceding counts that are separated by just one recovery time (71 ns) the extra recovery delay could reach 25 ns, this second recovery time is more than 33 % longer than that of a single isolated detection event! This extra delay vanishes when the the time difference between the preceding two counts reaches ≈ 85ns. The systematic uncertainty due to recovery time variations is count-rate dependent. This uncertainty can become metrologically significant at rather modest count-rates (above 10⁵ s⁻¹). It is also evident that the actual afterpulsing rate is slightly higher for second afterpulses when the delay time between the two preceding counts is within the range of 85 ns to 100 ns, probably due to a process similar to that described for detector A.

The above analysis demonstrates that while a generalized second-order model describes actively quenched SPADs reasonably well, there are higher-order effects that lead to systematic errors that are significant for high-accuracy measurements. Also, this result validates summing just over one recovery time in Eq. (1), because the observed transient behavior around 2t_recovery, ignored in any second-order model, would otherwise contribute multiple times.

4. Conclusion

In conclusion, our method provides a straightforward characterization of the key properties of detectors and it is maximally device-independent. We applied this method to SPADs from different manufacturers, and have established drastically different, and previously unidentified behavior during transient times. We accurately characterized a complete set of SPAD parameters, including t_dead, t_reset and the afterpulse probabilities. A nonlinear behavior of the SPAD beyond simple saturation is identified. A second-order single-photon detector model, commonly used for detector characterization is generalized. Third-order correlation analysis is offered for the first time to validate the second-order model and to identify and explain some of those nonlinear effects.

The use of third-order correlations opens a way of creating third-order detector models, when needed. Once a third-order model is suspected to cause a detectable systematic uncertainty, a fourth-order correlation measurement will help in its experimental validation, analogous to the validation method of a generalized second-order detector model described in this article.