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Abstract
We present a time-over-threshold readout technique to count the number of activated pixels from an array of superconducting nanowire single photon detectors (SNSPDs). This technique places no additional heatload on the cryostat, and retains the intrinsic count rate of the time-tagger. We demonstrate proof-of-principle operation with respect to a four-pixel device. Furthermore, we show that, given some permissible error threshold, the number of pixels that can be reliably read out scales linearly with the intrinsic signal-to-noise ratio of the individual pixel response.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Superconducting nanowire single photon detectors (SNSPDs) have become increasingly prevalent across many areas of optical sensing [1]. They are particularly suited to applications which require high efficiency, low noise and low jitter when measuring low numbers of photons [2–4]. These devices do not resolve the number of photons; rather, they fire if at least one photon is incident. Quasi-photon-number-resolution can be obtained by building arrays of these devices [5–13]; counting the number of detectors which fire provides information on the number of photons which are incident. Moreover, placing these devices in spatial arrays means that the incoming light beam may be imaged [14–16]. As the size of these arrays increases, a key challenge is to read out each pixel, whilst maintaining the attractive properties of the constituent detectors. Since SNSPDs require cryogenic temperatures to operate, increasing the number of pixels typically increases the number of readout channels, which in turn increases the heatload of the cryogenic system.
A variety of readout schemes is presented in [17]. A distinction is made between on-chip superconducting signal processing, which may be performed by e.g. single-flux quantum (SFQ) logic (see e.g. [18,19]) and analog on chip multiplexing followed by off-chip processing, such as time [20,21] and frequency [22] signature read-out.
In the context of photon counting, knowing which pixel has detected a photon is not required; instead, measuring the total number of counts is important. This can be achieved by connecting the output of each detector in series, such that the height of the output pulse is linearly proportional to the sum of the number of pixels which fire. This can be read out with a single electrical readout line, however analysis of the pulse requires a multilevel discriminator, which may increase the jitter, or splitting the readout line into several separate time-tagger channels, which quickly increases the hardware requirements.
To address this, in this letter we present a scalable channel readout scheme which can extract the number of pixels firing in a multipixel array. This is achieved by measuring both the rising- and falling edge of the electrical pulse arising from a series connection of the outputs of each pixel, using a suitable time-tagger. The arrival time information is retained in the rising edge. The falling edge is related to the recovery time of the detectors, which increases depending on the number of detectors which fire. Therefore, the time difference between the rising and falling edges, i.e. the pulse duration, is also proportional to the number of activated pixels. This implements a rudimentary time-over-threshold circuit, as has been employed in other contexts [23,24]. As such, information for both the number of pixels and the photons arrival time is preserved and read out on a single channel, in principle independent of the size of the array. This scheme is applicable if the incoming light field is pulsed such that the detector recovery time is shorter than the pulse period from the laser source.
In contrast to single pixel readout, the system jitter in this device in increased for two reasons. Firstly, the series connection of each pixel increases the geometric jitter contribution. In addition, the threshold used to read out the device is not necessarily that which minimizes the jitter (i.e. the maximum slew rate) [4]. However, we avoid introducing any further electrical jitter which may arise from the finite bandwidth of electronic channel dividers.
2. Measurement scheme
To demonstrate this approach, we use a four-element multipixel detector wired in series (Fig. 1(a)) and connect it to the time-tagger. A schematic of the setup is shown in Fig. 1(b). We generate coherent states with a fiber-coupled diode laser (PicoQuant) with a repetition rate of 100 kHz centered around 1550 nm and a pulse duration of 50 ps. This light is attenuated and detected with a four-pixel superconducting nanowire detector (PhotonSpot) which is able to generate electrical pulses with four different amplitudes depending on the number of pixels which detect photons. The electronic response is either measured with a Time Tagger Ultra (Swabian Instruments) that is connected with a single electrical channel, or an oscilloscope which is used as a reference.
Fig. 1. (a) Circuit diagram of a four-pixel SNSPD (marked in red). Each detector is connected in parallel with a resistor $R_{\textrm {p}}$ and connected to a constant current source $I_{\textrm {bias}}$ (via a bias-tee of inductance $L$ and capacitance $C$). The voltage $V_{\textrm {out}}$ across a load resistor $R_{\textrm {L}}$ is measured. All detector areas are connected in series to enable a single-channel read-out. (b) A fiber-coupled diode laser produces light at 1550 nm which is then attenuated and detected with a four-pixel SNSPD at 0.77 K. The electronic response from the detector is either measured with a time-tagger connected to a single electrical channel, or an oscilloscope.
An example of the oscilloscope trace for such a device is shown in Fig. 2(a). If we neglect the risetime of the device, we can model this trace by treating each additional pixel as adding an exponential decay, modulated by some noise of width $\sigma _v$. That is, the detector response function for $n$ pixels firing scales as $n A e^{-t/\tau }$, where $A$ is the amplitude of a single pixel, and $\tau$ is a characteristic decay constant, independent of the number of pixels. The constants $A$, $\tau$ and $\sigma _v$ depend on the electronics of the detector circuitry. To evaluate the number of pixels which fire using a measurement in the time domain, we seek the time taken to decay to a particular threshold $a_0$. This occurs at
Thus we have mapped the number of pixels firing to the time above threshold $a_0$ for this simple device. This equation also demonstrates the scaling of the maximum count rate for this device, namely logarithmically with the number of pixels $n$.Fig. 2. Oscilloscope traces showing the electrical response of the four-pixel detector (a) as an intensity heat-map. The evaluation of the output signal is shown for three different thresholds in subfigures (b-d). Here, the time difference between the first rising slope and the first falling slope at the specific threshold is plotted in a histogram. Subfigures (e-g) shows the comparison between the click probabilities from the oscilloscope reference measurement ($c_k$, based on pulse height), the time-over-threshold value from the oscilloscope ($c'_{k,\textrm {OSC}}$) and the time-over-threshold value from the time-tagger ($c'_k$). See text for further details.
Experimentally, this is achieved by the time-tagger recording timestamps if the incoming electrical signal exceeds (rising edge) and deceeds (falling edge) a defined discrimination level. We investigated 25 different thresholds. Furthermore, in our implementation, an artificial deadtime of 400 ns is set on the time-tagger to avoid extra triggering from electrical noise. The time-difference between a rising edge and a falling edge is calculated for all detection events and used as a measure for the pulse height.
3. Click probability evaluation
Based on the oscilloscope and time-tagger measurement the click probability can be extracted in three different ways. From the oscilloscope traces it is straightforward to analyse the pulse height to determine the click probability $c_k$. Figure 2(a) shows that the response functions for the four detection cases can be well separated at $0.5\;\mu s$. While this method is capable of detecting the click probability with high accuracy, it requires sufficient voltage resolution as well as timing resolution. As many readout schemes involve time-taggers that can only record events at a specific threshold we investigate a readout scheme without any voltage resolution. As an example„ the time difference between the first rising edge and the first falling edge at a specific threshold is shown in Figs. 2(b)–2(d) as a histogram. Counts in a specific area are integrated and ascribed the click probability $c'_{k\textrm {,osc}}$.
The time-over-threshold value can also be recorded directly with a time-tagger, bypassing the oscilloscope entirely, and analysed in a similar way to extract the click probability $c'_k$. The comparison between all three methods can be seen in Figs. 2(e)–2(g), showing good consistency between them. A more detailed error analysis of the time-over-threshold scheme is given in the next section. The form of the measured click distribution is given by the Poissonian photon number distribution of a coherent state convolved with the four-element multiplexing detector response.
4. Error analysis
In order to determine the quality of our approach we compare the time-tagger measurement with the measured oscilloscope traces as shown in Fig. 2(a). Figure 3 shows the comparison between the oscilloscope reference measurement and the time-tagger measurement.
Fig. 3. Comparison between click probabilities from the oscilloscope traces and time-tagger histograms for an ensemble measurement (a). (b)-(d) Start-stop histograms for three example thresholds are shown, including Gaussian fits from which to extract the click probability $c'_k$. See text for further information.
Start-stop histograms from the time-tagger are analysed for 25 different trigger thresholds (three examples for 0.0283 V, 0.0162 V and 0.0119 V are shown in Figs. 3(b)–3(d) respectively) and Gaussian functions are fitted to the histogram (colored-dashed lines in Figs. 3(b)–3(d)). We will label the $k$th Gaussian fit $g_k$. These fits are then integrated to extract the lick probability $c'_k$ for having $k$ simultaneous detection events. In order to compare these probabilities we calculated the error probability as the relative deviation of these two values
which is shown in Fig. 3(a). It can be seen that at low thresholds the error probability is quite high as the Gaussian functions are highly overlapping. For high thresholds on the other hand, the two, three and four-fold events can be distinguished precisely but the threshold is above the maximum voltage for the single click event (compare oscilloscope trace in Fig. 2). An optimal threshold can be found around 0.0162 V where the peaks of the two, three and four event clicks can be identified and the single click event is still fully visible.The presented approach based on Gaussian fits requires ensemble measurements. This is a valid approach, for example, to state characterization where the same state can be investigated repeatedly. For some applications, however, single-shot experiments should be considered, in which the requirements may be more demanding. Here, the task is to estimate directly the number of incident photons for each individual run of the experiment, for example when heralding single photons from parametric down-conversion in the presence of multiphoton events. For this reason we divide the time-tagger histogram into different sections. If one start-stop event has a time duration within one of these time sections, we ascribe this event to this section. The upper and lower limits of these sections are given by the center positions between the Gaussian fit maxima. We will denote these limits with $u_k$ and $l_k$ for the upper and lower limit of the $k$th section respectively.
In contrast to ensemble measurements, two sources of error can appear for the single-shot case, which are known as error of type I and II for this hypothesis test. As an example, we consider the second time section which should identify events where two pixels have detected a photon simultaneously. Firstly, we can miss a two-pixel event if the time duration was too short (and therefore was identified as a one-pixel event) or too long (and therefore was identified as a three-click event). Secondly, we can misidentify a one-,three or four-click event as a two-click event if the start-stop time has the appropriate length. We can use the Gaussian fits $g_k$ from ensemble measurements to estimate the magnitude of these two errors. In general we can write
and with normalization constants $M'_k$ and $N'_k$. If the trigger threshold from the time-tagger is low enough such that all events are detected and all Gaussian fits are well separated, these normalization factors would be equivalent $M'_k = N'_k$. However, physical imperfections causing overlapping Gaussian fits and thresholds above the maximal voltage of an event require careful normalization, as shown in Appendix A.Both errors are shown in Fig. 4. For the single-click event it can be seen that lower thresholds are well-suited to detect these events. The misidentified click probability is low due to the large separation of the single- and two-click events in the histogram. For high thresholds it can be seen that an increasing number of clicks are missed as the threshold surpasses the maximum voltage for this event. At very high thresholds (not shown) the error probabilities for the two-,three- and four-event cases would also increase. Negative values for the missing click factor indicate that the calculated probability is higher than the corresponding normalization constant $M'_k$. This effect is very pronounced for the four-fold event at low thresholds. Here the overlap of three-fold and four-fold events causes an overestimation of the four-fold rate.
Fig. 4. We consider two potential errors for single-shot measurements. (a) Probability of a missing click and (b) probability of a misidentified click. See text for further details. Subfigures (b)-(d) show start-stop histograms for three example thresholds including the time sections (marked with vertical lines).
For the misidentified click probability it is beneficial to choose high thresholds. Especially the four-click event profits from higher thresholds as three- and four-fold events have the highest overlap. Misidentified click probabilities above $>\;0.0257$ V cannot be plotted anymore for the single event case because the normalization diverges (no single clicks can be identified). Again a threshold around 0.0171 V seems to be optimal to reduce both errors simultaneously. Details about the calculations of the error bars can be found in the Appendix B.
5. Scaling
It is prudent to consider the scaling of this readout scheme as the number of individual pixels increases. From Eq. (1), the behaviour of the response function is such that the time gaps between the response of each pixel get smaller the more pixels fire, i.e. the ratio $t\left (n+1\right )/t\left (n\right )$ decreases with increasing $n$. Therefore, for a fixed uncertainty in the time $t$, it becomes harder to distinguish higher numbers of pixels.
In general, the uncertainty in the measurement time $\sigma _t$ is the quadrature sum of the contributions from the detector itself and jitter of the time-tagger, however we neglect the jitter as it is typically much smaller than the noise arising from the detector recovery slope. Using the exponential decay model of the response given by Eq. (1), we therefore seek the noise in the time domain $\sigma _t$ caused by noise in the voltage response $\sigma _v$. This is the product of $\sigma _v$ and the absolute value of the time derivative of the response function evaluated at a time $t=t(n)$, i.e.
To evaluate the scalability of this readout scheme in principle, we seek the largest number of pixels $n_{\textrm {max}}$ that can be successfully read out in this manner, depending on the detector parameters noise $\sigma _v$, pixel height $A$, and decay time $\tau$. Our figure of merit is the misidentified click probability (Eq. (4)) of the $n_{\textrm {max}}-1$ bin, which is the “worst case”, since we consider error contributions from $n_{\textrm {max}}$, as well as $n_{\textrm {max}}-2$. We further assume equal likelihood of each bin $n$ being occupied. A more thorough analysis should consider all error terms from $n_{\textrm {max}}-3\cdots 1$, however we choose to define $n_{\textrm {max}}$ such that the overlaps from $n_{\textrm {max}}-3$ (and therefore all smaller bins) are negligible. Furthermore, one may also wish to take into account the probabilities for the number of pixels firing, which is given by the convolution of the photon statistics of the incident beam and the splitting function of the multipixel detector (see e.g. [25,26]).
To evaluate Eq. (4) explicitly, we begin with the Gaussian distribution functions $g_i$, with $i\in \left (n_{\textrm {max}},n_{\textrm {max}}-1,n_{\textrm {max}}-2\right )$. For a mean $\mu _i=-\tau \ln \frac {a_0}{iA}$ and standard deviation $\sigma =\frac {\sigma _v\tau }{a_0}$, this is given by
If we tolerate errors of up to a given threshold $p_{\textrm {th}}$ when identifying pixels in a single-shot configuration, the maximum number of pixels that can be distinguished, as a function of the signal to noise of the detector $\beta$ is very closely approximated by
Fig. 5. Maximum pixel number as a function of the signal-to-noise ratio parameter $\beta$. Each line indicates the detector quality that must be achieved in order to reach a particular error threshold.
Based on the signal to noise ratio of 10 in our experiment, we were able to distinguish 4 pixels with an error rate around 10%, which is in good agreement with the analysis above.
6. Conclusion
We have presented a proof-of-principle demonstration of a scheme to read out a multipixel detector with a single electrical line, based on the time-over-threshold pulse length of the device when each pixel is connected in series. Accoding to a simple model of the response function of the device and its noise characteristics, we showed that the scaling of the device is depends linearly on the signal-to-noise ratio of readout. Therefore, further improvements to the intrinsic circuitry or amplification, which may increase the signal-to-noise level, are particularly useful at higher pixel numbers.
With this method, we have shown that the total number of fired pixels can be counted, based on the same response from each pixel. However, one could consider combining this technique with a pixel-dependent voltage response, such as shown by Gaggero et al. [27], to enable the extraction of which-pixel information, at least in the few-photon regime before voltage noise dominates errors in the readout.
Appendix
A. Normalization factors
The missing click factor $p_{\textrm {missing click, } k}$ should indicate that events are missed, which may be caused by a threshold level $a_0$ above the detector output voltage $A$. In order to capture this behavior the normalization constant $M'_k$ should be calculated for a fixed threshold $a_0$ that is lower than the corresponding maximal voltage $k\cdot A$ for this $k$-fold event. We suggest to calculate the normalization constant $M'_k$ for a threshold $a_0$ between the voltage level of the $k-1$ and $k^{\textrm {th}}$ level
For our calculations all thresholds for $M'_{k\neq 1}$ were calculated at $a_0 = 2A-\frac {A}{2}$ as the error probability for this threshold is already low enough (compare Fig. 3).The normalization for the misidentified click probability $p_{\textrm {misidentified click, k}}$ is different as we consider a post-selected click probability. We are interested in the misidentified click probability contained within the subset of having $k$ events. The size of this subset, which is equivalent to the desired normalization constant $N'_k$, is given by the total number of events recognized in the $k^{\textrm {th}}$ section
B. Measurement uncertainties
The error bars shown in Fig. 3 and Fig. 4 are based on Gaussian error propagation resulting in
Funding
H2020 Future and Emerging Technologies (820365 (PHoG)); Bundesministerium für Bildung und Forschung (13N14911 (Quantum Futur)).
Disclosures
JT, CS, TJB: (P), VA: Photon Spot (I,E,P), HF: Swabian Instruments GmbH (I,E,P)
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