1. Introduction

Superconducting nanowire single-photon detectors (SNSPDs) have advantages of wide response spectral range, near unity detection efficiency, low dark counts, and short timing jitter [1]. However, due to several factors, such as the low operation temperature, fabrication challenges, and limited readout resources, scaling up SNSPDs to a large array is difficult. Previous demonstrations of large-scale SNSPD arrays used different readout methods, including RSFQ readout [2,3], superconducting nanowire cryotron readout [4], row-column multiplexing [5], and thermal-coupling readout [6,7]. Several schemes were available for applying multiplexing readout on detectors directly, including frequency multiplexing [8,9], time-of-flight multiplexing [10,11], and orthogonal time-amplitude multiplexing [12].

The above readout methods harness energy-efficient superconducting electronics or superconductivity in superconducting nanowires to bypass scaling difficulties at cryogenic temperatures. However, the current scalability is limited to kilo-pixels [5,12], which is considerably lesser than that in advanced semiconductor photodetector arrays [13,14]. Another problem that remains unaddressed for SNSPD arrays is that most current readout approaches cannot process individual pixels in parallel. There are a few multiplexing methods that can resolve pixels in parallel, however the scalability is current limited to a small number [15,16]. When multiple photons are detected simultaneously or within a duration shorter than the recovery time of the readout, the readout fails and does not give any signal output or ambiguous photon positions. This multiple-photon detection (MPD) problem also occurs in other position-sensing detectors, such as a microchannel plate detector with a delay line readout [17]. For single-photon imaging, the MPD problem limits the SNSPD array to operate only under low photon flux conditions, in which one pixel of the entire array is allowed to fire within one time window. However, when a large array is pursued for a large field of view, the array collects more photons and has a higher probability of MPD. Thus, a tradeoff occurs between the array size and the tolerance of imaging distortion caused by multiple hits. For semiconductor single-photon detector arrays, the pixel-level readout was used to overcome this problem [18–21]. However, the power dissipation of the pixel-level readout is proportional to the size of the array, which would be a critical challenge for superconducting imagers in a cryogenic environment.

In this paper, we investigated imaging distortion caused by the MPD in a superconducting nanowire single-photon imager (SNSPI) using time-of-flight multiplexing readout. First, we constructed a detection model taking into account MPD probability based on experimental results from a discrete 8 × 8 SNSPI, in which neighboring pixels were separated by a sufficiently long delay line for distinguishing their locations with high fidelity. Modeling results showed that, as the array size increased, the maximum single-photon detection (SPD) probability (about 37%) was obtained when the photon flux approached 1 photon/pulse. Even at such a low photon flux, 42% of the detection events were from MPDs, which confirmed the serious impact of MPDs on imaging quality. The discrete geometry allowed the mapping of SPDs and MPDs into signal pixels and idle pixels. We simulated different image distortions at different photon fluxes assuming that an SNSPI scaled up to a 64 × 64 array. Depending on the object, the reconstructed image either was blurred or generated ghost images. Finally, we introduced a correction method by subtracting the MPD image composed of idle pixels from the distorted image. Simulation showed that the image quality increased by a factor of 1.3∼9.3 at a photon flux of µ = 5 photon/pulse, which indicated that a large-scale SNSPI could operate at a relatively higher photon flux without pixel-level readout.

2. Readout ambiguity for multiple-photon detections (MPDs) in an SNSPI

An 8 × 8 SNSPI was fabricated to study the problem caused from MPDs. As shown in Fig. 1(a), the device was designed into a discrete architecture in which adjacent pixels were separated by a long superconducting delay line. Each pixel was designed as a two-element superconducting nanowire avalanche photodetector (2-SNAP), so that the wide delay line can be insensitive to photons but still maintain good impedance matching. Details of the performance of the device are provided in another study [22]. We defined MPDs as events in which two or more photons were detected within a time window that the readout cannot resolve. The time window was limited by the reset time of an SNSPI. The probability of n photons hitting the same pixel was 1/Nⁿ, which could be neglected as N increased to a large number.

 

Fig. 1. (a) SEM images of the 8 × 8 SNSPI. One pixel made of two parallel nanowires was enlarged. (b) Conceptual illustration of the ambiguous readout caused by MPDs. (c-f) MPDs at different incident photon fluxes. The data are plotted in a two-dimensional space governed by pixel index and normalized t_sum/2. Singe photon detections (SPDs) are on top, while the bottom triangle shows MPDs. (d) An enlarged view of (c) showing SPDs and MPDs from pixel 33 to 40. (e) A photon count histogram of pixel 38 over the normalized t_sum/2. (f) Photon count histograms of all detection events, SPD events, and MPD events from pixel 33 to 40.

Download Full Size | PDF

An ambiguous readout of the MPD is illustrated in Fig. 1(b). Due to the series structure, an N-pixel SNSPI can be simplified into a straight line with N nodes. We assumed that the signal transmission time between adjacent pixels was τ, and the time required for transmitting the signal from the outermost pixels to both ends was calibrated to be zero. As shown in the figure, when only one photon was detected at time t_p at the i^th pixel, the pulse arrival time at both ends t₁ and t₂ can be expressed as $\left\{ {\begin{array}{{c}} {{t_1} = {t_p} + ({i - 1} )\cdot \tau }\\ {{t_2} = {t_p} + ({N - i} )\cdot \tau } \end{array}} \right.$. The arrival time and position of the detected photons were obtained by mapping half of the sum time ${t_{\textrm{sum}/2}} = \frac{{{t_2} + {t_1}}}{2} = {{{t_p} + ({N - 1} )\tau}}/{2}$ and half of the difference time ${t_{\textrm{diff/2}}} = \frac{{{t_2} - {t_1}}}{2} = {({N + 1)\tau}}/{2-i} \cdot \tau$, respectively. However, when two photons hit the i^th pixel and the j^th pixel (i < j) as shown in Fig. 1(b), the equations ${t_{\textrm{sum/2}}} = {{{t_p} + (N - 1 - {j + i} )\tau}}/{2}$ and ${t_{\textrm{diff/2}}} = {{(N + 1)\tau/2 - ({i + j} )}} \tau/2$ were not complete for solving i and j. For two-photon detections, when i + j was even, the detections were mapped wrongly on the pixels where the single-photon detections were mapped on. However, when i + j was odd, the two-photon detections were mapped to non-existent pixels, which we referred to as idle pixels. These idle pixels were shifted by half τ from the real pixels.

A femtosecond laser with a repetition rate of 100 kHz at a wavelength of 1,550 nm was used to measure MPDs. The arrival times were acquired by a high-speed digital oscilloscope and then processed offline. The results are shown in Fig. 1(c-f). The distribution of all detections in a two-dimensional space taking t_sum/2 and t_diff/2 as coordinates is shown in Fig. 1(c). For better illustration, these coordinates were normalized and shifted to t_p and i. The clusters of SPDs were on top, which corresponded to a constant arrival time synchronized to the laser. However, as the photon flux increased, the MPDs became more significant, and their uncorrected readings mapped to a triangle profile below the SPD clusters in the plot. An enlarged view is shown in Fig. 1(d). This distribution was used previously to resolve coincidental events by two-photon detections [23]. The SPD events and MPD events could be extracted individually by their t_sum/2 value with the prior information of photon arrival times, as shown in Fig. 1(e). However, in practical imaging applications, photon arrival times are not known beforehand. Therefore, after discarding information from t_sum/2, photon positions were extracted from the 1D histogram of t_diff/2. An example is shown in Fig. 1(f). The photon detection histograms from SPDs and MPDs were separated (Fig. 1(f)). The locations of the real pixels and idle pixels were also marked. It is obvious that the MPDs introduce a background adding on the SPDs. It is worth noting that in order to guarantee a clear separation of the pixel location and an accurate extraction of the MPD events, the length of the delay line between adjacent pixels needs to be enough long. As a role of thumb, the minimum delay is set to the FWHM (full-width-of-half-magnitude) of the differential jitter j_diff, which gives a minimum delay line length L_{delay_min} of $2 \cdot {j_{diff}} \cdot {v_p}$(v_p is microwave signal transmission speed). By taking in the device parameters, L_{delay_min} is 184 µm. For the imager as shown in Fig. 1(a), the length of the delay line was designed to be 300 µm long, which was longer than L_{delay_min} for a better separation of SPD peaks.

3. Modeling multiple-photon detections (MPDs)

The above experimental results confirmed the ambiguous readout of MPDs and showed their effects on the imaging quality. To find the optimum incident photon flux and correction methods, particularly for large arrays, a probability detection model for an SNSPI was introduced. In the model, we assumed a scenario where a pulsed light illuminated on the imager. For an N-pixel SNSPI, the mean photon number on the i^th pixel was λ_i per pulse, and the efficiency of the detector was η. In the following simulations, the value of $\eta $ was set to 1 for an ideal detector. For a single pixel operating independently, the probability distribution of k photons detected by the i^th pixel can be expressed as ${p_i}(k )= \frac{{{{({\eta {\lambda_i}} )}^k} \cdot {e^{ - \eta {\lambda _i}}}}}{{k\textrm{!}}}({k \ge 0} )$ and the probability of having a detection event was ${p_i}({k \ge 1} )= 1 - {p_i}({k = 0} )= 1 - {e^{ - \eta {\lambda _i}}}$. When N pixels composed an array, the conditional probability of only the i^th pixel fired while the other pixels remain silent was shown below:

For calculating the mean probabilities for SPDs and MPDs, irrespective of the imaging objects, we assumed uniform illumination. Then, we simplified Eq. (1) and evaluated the probability for all SPDs in an array as follows:

Here, $\mu = N \cdot \lambda $ represents the average number of photons per pulse integrated over the entire array.

Using Eq. (2), the dependence of the probability of SPDs on incident photon flux and array size was investigated. The probability of SPDs increased to a maximum and then reduced due to the increased MPDs (Fig. 2(a)). By extracting the maximum probability $p_{\textrm{array}}^\textrm{m}$ and the corresponding incident mean photon number µ_m, the dependence of µ_m and $p_{\textrm{array}}^\textrm{m}$ on the array size N was visualized in Fig. 2(b). µ_m and $p_{\textrm{array}}^\textrm{m}$ could also be obtained by calculating the maximum values for Eq. (2). As N increased, µ_m approached 1 and $p_{\textrm{array}}^\textrm{m}$ approached 37%. These values corresponded to the maximum single-photon probability in a coherent light pulse following the Poisson distribution, indicating that $p_{\textrm{array}}^\textrm{m}$ only depended on the incident power for a large array. However, the probability of MPDs also reached 42% at µ_m and kept increasing as the photon flux went higher (Fig. 2(c)). This value was significant and indicated that 42% of the detection events were from MPDs and distorted the reconstructed image. To reduce the portion of MPDs, the incident photon flux needed to be attenuated as well, which certainly limited the dynamic range and imaging speed.

 

Fig. 2. (a) Probabilities of SPDs versus the incident photon intensity µ when N = 2, 4, 8, and 32. The maximum probability of SPDs $p_{\textrm{array}}^\textrm{m}$ and the corresponding incident mean photon number µ_m were extracted and shown in (b). (c) Probabilities of all detections (ADs) and SPDs versus µ. The MPD-to-AD ratio versus µ was also plotted on the right axis.

Download Full Size | PDF

4. Simulations of the image distortion for a 64 × 64 SNSPI

To illustrate how MPDs can distort the image, we used the above model for simulating the imaging process of a 64 × 64 SNSPI for different objects at different photon fluxes. We meandered the nanowire horizontally, and the two ends were at the top-left and bottom-right. In the imaging simulations, we set the total average incident photon flux and calculated the average incident photon flux for each pixel based on the grayscale values of different pixels in the image. Then we calculated the SPD probability and MPD probability for each pixel based on the model. Finally, we combined the two probabilities to obtain the total detection probability of each pixel, thereby generating the distorted image. The mean structural similarity (MSSIM) index of each image was calculated to evaluate the degree of distortion.

The simulated image from a natural scene is shown in Fig. 3(a). It had a bright background, and the object was dark at the center. Therefore, as µ increased, the MPDs were mainly caused by the photons illuminating on the surrounding pixels, and the wrong readout mapped these photon positions to the center of the nanowire. Correspondingly, the intensity of the center pixels was superimposed by the intensity of the MPDs from the background, which caused a bright blurring distortion similar to an overexposure distortion. The details in the middle of the image were lost.

 

Fig. 3. Simulation of distorted images for a 64 × 64 SNSPI for four different scenes (a: cameraman, b: rabbit, c: single-point target, d: three-point target) at varied photon flux.

Download Full Size | PDF

An inverse case is shown in Fig. 3(b), where the object was bright, but the background was dark. Since the object was placed in the center and its shape was almost symmetrical, the MPDs were mainly from the photons illuminating on the object, and the wrong readout mapped these photon positions in the center of the nanowire. As the object was bright and the probability of MPDs mapped to the object was relatively low, the details of the object could still be recognized. However, the dark background near the bright object was brightened, creating an illusion that light was scattered around the object.

The simulated images for point targets are shown in Fig. 3(c) and (d). As µ increased, ghost targets were generated. For a single-point target, as shown in Fig. 3(c), the ghost targets appeared on the horizontal sides of the original target with a lower intensity and a smaller size. The distortion became more disordered for three targets (Fig. 3(d)). The combination of any two or three-photon detections caused more ghost targets. Since target A was between B and C, MPDs from BC blocked the readout of SPDs from A, preventing target A from being distinguished as µ increased. When µ was 3, the probability of MPDs from the bright target B and C was even higher than any SPDs from an individual target. The intensities of the ghost targets were brighter than the real targets, leading to wrong interpretations.

It should be noticeable that the image distortion both depends on the geometry of the nanowire, which determines the spatial mapping relationship from the differential time to pixel locations, and the object, which varied the incident photon flux over pixels. At the maximum probability, where µ = 1, image distortion was already observable, which reduced the MSSIM to 0.8~0.9. The MSSIM decreased as µ increased, and the reduction was greater for more complex objects.

5. Correction method and results

Although the distorted image could be simulated using rigorous mathematics, correcting the distortion reversely was challenging since the origins of a distorted pixel can be mapped to various combinations of multiple pixels. The reverse calculation may use an iterative algorithm by taking the distorted image to estimate the initial photon flux distribution and inputting the errors for refining the result. This algorithm will be computationally expensive and need careful regularization to converge.

Here, we introduced a more straightforward method. As mentioned above, the SNSPI was designed discretely, which provided an idle pixel between adjacent signal pixels. The image reconstructed from the idle pixel was purely based on MPDs, which inspired us for interpolating the probabilities of MPDs superimposed on signal pixels. For an N-pixel SNSPI, the probability of MPDs at the i^th signal pixel can be calculated by summing all possible combinations that any other two pixels fired at locations symmetrical to the i^th pixel, which can be expressed as follows:

The probability of MPDs at the two idle pixels adjacent to the i^th signal pixel can be expressed as follows:

According to Eqs. (3)–(5), the probability of MPDs on a signal pixel was shifted by one pixel compared to the probability of MPDs on the idle pixel next to it. Considering a large array, the difference of one pixel could be smoothed out. Therefore, the probability of MPDs on a signal pixel was interpolated by the idle pixels, which facilitated the separation of the error image from the distorted image. After a direct subtraction, the corrected image was obtained.

The correction performance for several different distorted images is shown in Fig. 4. Both for nature scenes and point targets, the correction method can remove the distortion and ghost points. We calculated the MSSIMs of the corrected ones and compared them to the values of the uncorrected ones (as shown in Fig. 4(e)). Our results indicated that this correction method was valid within a wide dynamic range and gave an improvement by a factor of 1.3~9.3 at µ = 5 (corrected images at µ = 5 are shown in Fig. 4(d)). The improvement was more evident for natural scenes since the distortion was distributed more evenly at the center. For a single-point target, as the pixels of the real target were unlikely to be affected by MPDs generated from itself, subtracting the error image affected the real target slightly so that the correction methods could work at a photon flux as high as µ = 10. However, as the number of targets increased, the correction performance was limited by the photon flux again. This occurred because the probability of SPDs on A was blocked by MPDs on B and C at high photon flux. Thus, the distorted image only contained a few correct detections for target A, thus, invalidating the correction method. It was noticeable that the current correction method removed the distortion but did not compensate for the loss of SPDs on real targets. Thus, although ghost points or background blurring could be removed, the intensity of the objects had a slight variation.

 

Fig. 4. Distortion correction performance for different scenes. (a), (b), and (c) are the distorted images at µ = 2, MPD distortion interpolated from idle pixels, and restored images, respectively. (d) Restored images at µ = 5. (e) The MSSIMs of distorted images (red) and the restored images (green) versus µ.

Download Full Size | PDF

6. Conclusion

To summarize, in this paper, we illustrated the MPD problem in a superconducting nanowire single-photon imager. Readout ambiguity was experimentally demonstrated based on a small 8 × 8 SNSPI, and simulations were conducted on a large 64 × 64 SNSPI. Results showed that MPDs introduced significant distortion in the image, and the distortion depended both on the object and the input photon flux. We showed the advantage of using discrete pixels in an SNSPI, which was good for correcting MPD distortions. Since there was a long delay line inserting between adjacent pixels, the MPDs mapped on idle pixels can be used for interpolating distortions on signal pixels. The correction method efficiently removed background blurring and ghost points, thus, extending the maximum operation photon flux by a factor of 1.3~9.3. The low filling factor of the discrete SNSPI scarifies the detection efficiency. Further improvement of the device efficiency can be achieved by integrating the SNSPI with micro-lens array to improve the total optical absorptance similar to what has been achieved successfully in commercial SPAD arrays [24].

In the simulation, the incident light was modulated in pulses. However, the correction method can also be applied to CW light since photon arrival times (i.e. the depth information of objects) were not included during correcting for simplification. In addition, because the detection efficiency of a superconducting nanowire takes a reset time for recovering, the probability of MPDs in a CW illumination is less than that in a pulsed illumination at the same average power. Thus, a larger dynamic range would be expected. In an active 3D single-photon imaging, photon arrival times provided depth information and affected the distortion differently. The MPDs yielded a wrong interpretation of photon arrival times, based on which MPDs could be distinguished after reverse calculation. Therefore, the depth information could also be useful for optimizing the correction method, although some ambiguity may still exist.