1 Introduction

The superconducting nanowire single-photon detector (SNSPD) has been developed in recent years as a new type of single-photon detector. Due to its wide response band, high efficiency, low dark count, high speed, and low time jitter [1,2,3,4], the SNSPD has been widely used in scientific research applications, such as in the fields of quantum information, high-sensitivity optical communication, conditional state preparation, and source characterization [5,6,7,8]. However, the pulse amplitude of traditional SNSPDs is not dependent on the number of photons hitting the nanowire, and thus photon-number resolution cannot be achieved [9]. In addition, the nanowire is modeled as a lumped element inductor in series with a nonlinear dynamic resistor that represents the resistive domain, resulting in the inability of the conventional electrical readout of an SNSPD to obtain information about the resistive domain location [10]. Nevertheless, the spatial resolution of photons is critical to extremely weak-light imaging, including laser radar and molecular fluorescence imaging [11]. Photon-number-resolving detectors are also essential for several applications, such as linear photon quantum computation and near-infrared spectroscopy [12,13,14]. In 2007, the multi-pixel SNSPD was first proposed by Dauler et al. [15]. In 2016, Mattioli et al. [16] demonstrated an array detector that could resolve up to 24 photons; however, while the device adopted a series structure with no delay line between adjacent nanowires, the device could not obtain the spatial information of incident photons. Miki et al. [17] fabricated a 64-pixel SNSPD array in 2014 that could resolve up to 64 photons, although the low nanowire fill factor (~ 35.4%) resulted in a system detection efficiency (SDE) of less than 1%. Recently, Zhang et al. [18] reported a 16-pixel interleaved SNSPD array that could count more than 1.5 GHz photons per second, yet the interleaved structure failed to locate where the photon hit on the device, i.e., the spatial information was missing. Zhu et al. [19] realized the intrinsic photon-number-resolving ability in 2019 using a superconducting tapered nanowire detector; although this is impressive, the detector lost the spatial information of incident photons. Most recently, a kilopixel array of SNSPDs that realized the imaging function was reported by Wollman et al. [20], yet the system detection efficiency (~ 8%) requires improvement. The linear resolution of the photon number primarily depends on the detection efficiency precision and the number of pixels [21]. Although the pixel interval of multi-pixel SNSPDs has been previously explored, the low system quantum efficiency limits the linearity in resolving the photon number. Hence, it is necessary to fabricate a photon-number-resolving SNSPD array that has a high SDE and that can obtain spatial information of incident photons.

To achieve these goals, an SNSPD array that can simultaneously realize photon-number resolution and photon-space resolution is proposed in this paper. The SNSPD array consists of 16 pixels in a 4 × 4 array with a quantum efficiency of 94% and SDE of 46%. The size of a single pixel is 20 × 20 μm, and the total effective photosensitive surface is 80 × 80 μm. This array structure inherits the merits of traditional SNSPDs. Experimental results indicate that the maximum number of resolved photons reaches 16, and the spatial distribution of light can be illustrated by the array device.

2 Device fabrication

The fabrication process of the nanowire has been reported in the authors’ previous work [22]. The refractive index of Si3N4 is 1.86 and its surface flatness is better than 0.5 nm at λ = 1064 nm via optimization of the PECVD process. To reduce the reflectivity of incident light from the back of the device, a Si3N4 layer with a thickness of 135 nm is grown on the back of a silicon wafer. Next, a thin film of niobium nitride (NbN) with a thickness of 6.5 nm and critical temperature of 8.3 K is deposited on the upper resonator by DC magnetron sputtering. Then, an Au electrode with a thickness of approximately 200 nm is fabricated by photolithography. After the lift-off process, the Au electrode is prepared and electron beam exposure is initiated. The substrate is spin-coated with electron beam photoresist (2% PMMA) and is then baked for 3 min on a baking table at a temperature of 100 °C.

After baking, electron beam exposure is started when the substrate temperature decreases to room temperature. During the process of electron beam exposure, the current of the electron beam is set to 0.2 nA, the acceleration voltage is set to 100 kV, and the exposure is set to 1400 μC/cm2. The nanowire pattern is generated by the exposure of the electron beam lithography system, and the development runs at room temperature. Finally, an NbN nanowire is obtained by etching with a reactive ion etching tool. An SEM image of the array device is presented in Fig. 1.

Fig. 1
figure 1

SEM image of the array device. a The location of each pixel is marked from 1 to 16, and the electrode of each pixel is individually extracted. b Enlarged view of pixels No. 3 and No. 4. c Enlarged view of (b), where w1 is the nanowire interval and w2 is the nanowire width. d The light from the multi-mode fiber is focused on the SNSPD chip using a microscope system. The brown square in the middle is the nanowire area, and the brown rectangles surrounding the nanowire area are gold electrodes. The bright circle in the middle is the light spot

Full size image

3 Device measurement and analysis

The resistance of each pixel of the array device was measured at room temperature. The resistance ranged from 3.7 to 4.4 MΩ and the standard deviation was 0.16 MΩ. The device was installed on a Gifford–McMahon refrigerator for cooling, and the I–V curve of each pixel was first measured. The superconducting critical current Ic ranged from 9 to 12.5 μA with a standard deviation of 0.82 μA. The voltage branch of the I–V curve was symmetrical and the hysteresis current Ih was 2 μA. The Ic/Ih value of each pixel ranged from 4.5 to 6.5. The dark count rate (DCR) of each pixel was measured in a dark chamber with the fiber shuttered down and the fiber head carefully wrapped. The results indicate that the DCR of each pixel was less than 1000 cps at 0.9Ic. All the dark count curves had similar shapes; they increased slowly at a low bias current Ib and increased rapidly when Ib was near Ic. The SDE of each pixel was measured when the 1064-nm light (attenuated to single-photon power) illuminated on the device. All the SDE curves also had similar shapes; they increased rapidly at a low bias current and tended to be saturated when Ib was near Ic. At the bias current when the DCR of each pixel was 1000 cps, the total SDE of the device reached 46% at the wavelength of 1064 nm (details in ref [22]). The main reason why the system detection efficiency was lower than the quantum efficiency was the low absorption of TM waves. The polarization of the output light of the conventional multimode fiber is uncontrollable, so the polarization-optimized SDE could not be obtained. This problem could be solved in the future by adopting polarization-maintaining fiber.

An optoelectronic system was built for the array device detector, as shown in Fig. 2. The pulsed laser was used in the experiment (repeat frequency 10 MHz, pulse width 46 ps). The light is divided into two parts by the splitter, one of which is coupled to the detector, and the other of which is input into the oscilloscope as the synchronous trigger signal through a high-speed photoelectric converter. Each pixel detects photons independently, and the signal is read out independently. The pulse passes through a bias-tee (Mini-Circuits ZFBT-4R2G+, 10–4200 MHz) and is then input into a low-noise amplifier (LNA-650, bandwidth 650 MHz). The signals from 16 channels are processed by two methods: one method is independently processing the signals by a counter (SR400); the other method is combining the signals into one signal by the power combiner and then inputting the signal into the oscilloscope (Keysight, Infiniium V, bandwidth 33 GHz, sampling 80 GSa/s). The position information of the photons can be obtained by separately counting each signal through the counter, and the photon number information can be extracted from the combined signals.

Fig. 2
figure 2

Schematic diagram of the measurement system for measuring the photon-number resolution and photon-space resolution. The light is divided into two parts by the splitter; one part is coupled to the detector, and the other is input into the oscilloscope. Each pixel detects photons independently, and the signal is read out independently

Full size image

3.1 Photon-space resolution and counting rate distribution of the device

The spatial distribution of light intensity can be obtained by measuring the photon number arriving at each pixel in unit time through a 300-μm multi-mode fiber. Based on previous work [23], a beam compression system with two aspheric lenses was coupled to the multi-mode fiber to compress the optical beam with a magnification factor of about 1/3. Thus, the image size is about 100 μm in diameter as estimated for the core of the 300-μm multi-mode fiber, which appears similar to a Gaussian spot. The nominal coupling efficiency of 95% can be obtained by the detection area of 80 × 80 μm. In the experiment, the counting rate (CR) of each pixel was used to represent the photon number detected in unit time. Different spatial distributions of the spot could be obtained by changing the position of the focused laser beam for the device. Sixteen output signals were independently connected to the low-noise amplifier and were then, respectively, connected to the counter to separately read the counting rate of each signal. The experimental results are presented in Fig. 3.

Fig. 3
figure 3

Spatial distributions of photons. The sixteen squares represent sixteen distinct pixels in the experiment, and X and Y represent two edges of the device. a, b Multi-mode fiber results. c, d Single-mode fiber results. a, c Light spot slightly off-center of the device. b, d Light spot close to the center of the device

Full size image

Figure 3a and c, respectively, present the detection results using the multi-mode fiber and single-mode fiber with the same spatial distribution of light intensity. The spot is slightly off-center and close to the right of the device. Further, the counting rate of pixels away from the center of the spot declines rapidly, and the coverage of the multi-mode fiber is larger than that of the single-mode fiber. Figure 3b and d, respectively, present the counting rate distributions after adjusting the spot position to the center of the device using the multi-mode fiber and single-mode fiber. The spot position is closer to the center of the device, so the pixel with a higher counting rate is closer to the center area. In addition, the coverage of the multi-mode fiber is larger than that of the single-mode fiber. The measured results are in agreement with the faculae position.

As the intensity of light coupling from the multi-mode fiber is not uniformly distributed within the 16 pixels, the quantum efficiency of each pixel cannot be directly detected. Figure 3b presents the CR distribution of the array device coupled with the multi-mode fiber in the experiment, with a DCR of each pixel of 1000 cps. The system detection efficiency is defined as the ratio of the number of photons measured for all pixels to the total number of incident photons by considering the light coupling loss in this experiment. The light power as measured by a pre-calibrated optical power meter was 13.5 nW. The optical power was attenuated by an attenuator at 50 dB to ensure that the mean photon number per optical pulse was less than 0.1, which ensures that the probability of a single photon is greater than 95%. Accumulating the CR of all the 16 pixels, a total SDE of 46% was obtained at the DCR of 1000 cps. Regardless of other secondary factors, the QE can be calculated as

$${\text{QE}} = \frac{\text{SDE}}{{\alpha_{0} (1 - \alpha_{\text{l}} )\eta_{\upalpha} \eta_{\text{c}} }} ,$$
(1)

where α0 and αl represent the transmittance of light through the silicon substrate and the loss rate of light in the multi-mode fiber, respectively. α0 = 71.1% was measured using a transmission spectrometer and αl = 8.8% was obtained via the measurement of the multi-mode fiber. ηα is the absorption efficiency of photons by nanowires, and ηα = 79% (TE wave at 1064 nm) was achieved by simulation in previous work (details in Ref. [22]). ηc represents the coupling efficiency of the multi-mode fibers, and the value was measured to be greater than 95% in previous work (details in Ref. [23]). Hence, QE = 94.5% was calculated. This indicates a high photoelectric conversion efficiency of the nanowires.

3.2 Photon-number resolution

With the above-described multi-pixel detector, a 16-photon resolution was realized in the experiment. The power combination method was used to realize the photon-number resolution, as shown in Fig. 2. The distribution of photons can be obtained by arranging the height distribution of the output pulses. By adjusting the light power from 0 to 110 pW, histograms of the output signals were obtained, as presented in Fig. 4. Seventeen distinct output levels corresponding to the detections of 0–16 photons were obtained, marked by black numbers in Fig. 4. Figure 5 presents the experimental results extracted from Fig. 4. Figure 5a–c present the photon-number distribution results when the incident light power was 3.4, 28, and 110 pW, respectively. Figure 5d describes the linear relationship between the pulse amplitude and the photon number, which is consistent with the expectation.

Fig. 4
figure 4

Histograms of output signals with different light powers from 0 to 110 pW. Seventeen distinct output levels corresponding to the detections of 0–16 photons were obtained

Full size image
Fig. 5
figure 5

Photon-number distribution of distinct light power. The colored lines in a–c are Gaussian fittings to the experimental data (gray lines). a Photon numbers of 0–4; the full-width at half-maximum (FWHM) of the 1-photon voltage peak is 10.5 mV. b Photon numbers of 5–11. c Photon numbers of 11–16. d Linear relationship between the pulse amplitude and photon number. The red triangles are the mean pulse heights of different voltage peaks extracted from (a–c), and the error bars are also marked

Full size image

The pulse height of a single photon is supposed as H, which takes the attenuation of the power combiner and the gain of the amplifier into account. The power combiner is used to combine the multiple channels into one output; thus, the pulse height of n photons is n × H. Due to the noise influence, all the output voltages will be superimposed by noise. The amount of noise is N0, and the noise will not superimpose linearly with photon number n. To simplify the analysis, the relationship between V and photon number n can be expressed as V = nH + N0.

The process of photon incidence is a random event, and the peak value of the pulse voltage is the superposition of several photons. The Gauss function was used to fit the distribution, and the mean voltage V0 of each peak and the standard deviation of each peak voltage were obtained. The relationship between V0 and the photon number n is shown in Fig. 5d. The linear fitting results represent V = 19.32n + 5.98 mV.

Noise is due to the accumulation of intrinsic noise from each of the n detection events, the variation of the height of output pulses produced by different elements, and the external electronic component noise, such as the noise of amplifiers [9]. The intrinsic noise from each element is related to the inhomogeneities of the nanowires. Suppose the noise of n detection events is Vnn, and the pulse generated by n detection events is Vpn. As discussed previously, Vpn increases linearly with n detection events and scales with n. With a fixed light power, the noise in distinct wires is uncorrelated and completely random. Thus, their total contribution to Vnn is expected to scale as n1/2. When Vnn exceeds Vp1, i.e., when the noise generated by n detection events exceeds the voltage generated by only one photon, the photon-number resolution of this method reaches its limit. To simplify the analysis, the maximum number of photons, n, which can be distinguished by the power-combined method, can be limited as

$$V_{p1} \ge \sqrt n V_{n1} ,$$
(2)

where Vp1 is 25.3 mV according to Fig. 5a, d, and Vn1 is the noise produced by one single channel, i.e., one single element. As there were 16 channels in this experiment, Vn1 could not be detected directly. Fortunately, the full-width at half-maximum (FWHM) of the fitted Gaussian peak of different photon detection events in Fig. 5 represents Vn16 (noise produced by 16 channels). For instance, Vn16 is 10.5 mV when one photon is detected, which is shown in Fig. 5a. The FWHM of all 16 distinct peaks was calculated in Fig. 5a–c, and a mean of 11.98 mV and a standard deviation of 0.86 mV of the FWHM were determined, which indicates a reasonable noise consistency. Therefore, the Vn1 value of one single channel was 11.98/161/2 = 3.00 mV. Thus, according to Eq. (2), the maximum number of photons that could be distinguished in this experiment was estimated to be (25.3/3.00)2 ≈ 71, which demonstrates the potential to distinguish a larger number of photons. Based on the preceding discussion, reducing the noise of the external circuit, improving the homogeneities of the nanowires, and increasing the pulse generated by photon detection events will improve the photon-number resolution.

This detector inherits the merits of traditional SNSPDs without reducing the signal-to-noise ratio (SNR), unlike the series nanowire detector [9] or the uncontrolled switching of the nonfiring wires in parallel nanowire detectors [24, 25]. Additionally, the much higher cost required for this method simultaneously demands a higher cooling power and greater read-out circuit, although it is theoretically scalable. The digitization of the response pulse of photon detection is currently being attempted to reduce its cost and extend it to more pixels over a larger scale.

The fitting result in Fig. 5d is a straight line almost across the origin, i.e., the average photon number of each pulse is proportional to the pulse height. Therefore, the photon number can be deduced by the pulse height, which is also consistent with the theoretical result of the power combination method. The linear relationship between the pulse amplitude and registered photon number is convenient for resolving the photon number.

In Fig. 6, the distribution of registered photons was deduced from Fig. 4. The light power was found to increase by about 10 dB as it increased from 3 to 33 pW, while the mean photon number μ increased from 0.5 to 5.5. The ideal distribution of photon number n in a laser pulse as calculated by the Poisson distribution is also presented in Fig. 6. It is apparent that the distribution measured by this detector is consistent with the theoretical prediction at μ = 0.5 and 1.2. At μ = 3.5 and 5.5, the experimental data are close to the calculated curves. These results indicate that this detector approaches linear resolution.

Fig. 6
figure 6

Photon-number distribution at mean photon numbers μ = 0.5, 1.2, 3.5 and 5.5. The blue squares are the experimental data, and the red circles are the calculation results based on the Poisson distribution

Full size image

While μ approaches 16, this detector should be nonlinear, and the reason for this is complex. Photon-number-resolving detectors are currently mainly used in few-photon applications [19], so the detector could not perform well when detecting a large number of incident photons. When several photons are incident on the same pixel at the same time, the pixel cannot respond to the incident photons during the dead time, i.e., the pixel can respond to only one photon within a detection period, leading to a reduction in the detection efficiency for multiphoton events. This is consistent with Fig. 6. This discrepancy could be reduced in the future by reducing the area and dynamic inductance of a single pixel.

PN, the probability that N photons can be detected by the array device, can be calculated as follows:

$$P_{N} = C_{16}^{N} \prod\limits_{i = 1}^{N} {\eta_{i} } \prod\limits_{j = 1}^{16 - N} {(1 - \eta_{j} )} ,$$
(3)

where \(C_{16}^{N}\) is a combinatorial number, representing the total number of combination of N pixels selected from 16 pixels, and \(C_{16}^{N} = 16!/\left( {N!*\left( {16 - N} \right)!} \right)\). Index i represents the fired pixels and index j represents the unfired pixels. Therefore, ηi is the fired probability of pixel i and (1 − ηj) is the unfired probability of pixel j, where ηi(j) = SDEi(j)(1 − exp(− μi(j))), and μi(j) is the mean photon number incident on the pixel i(j). SDEi(j) is the system detection efficiency of pixel i(j), and μi(j) can be calculated by the Gaussian beam theory based on the known input light power. Hence, \(\prod\nolimits_{i = 1}^{N} {\eta_{i} }\) is the probability of N fired pixels, and \(\prod\nolimits_{j = 1}^{16 - N} {(1 - \eta_{j} )}\) is the probability of (16 − N) unfired pixels.

In this study, it was assumed that the 16 distinct pixels had the same detection efficiency and the input light beam had a uniform electromagnetic field distribution. The assumption performed well for the few-photon events (the mean photon number is 0.5 in Fig. 6). Although adopting the Gaussian beam theory would yield some interesting outcomes, it is difficult to make the SDEs of different pixels exactly the same. A higher-order mode would yield in the 300-μm multi-mode fiber, and the NbN nanowires have a relative high sensitivity for the polarization of the light beam. Therefore, a discrepancy between the measured and calculated photon number distribution occurred when higher numbers of photons were incident on the device. To overcome these problems, polarization-maintaining fibers could be used in future experiments. Accordingly, the experimental results in Fig. 6 show the nonideal distribution of the photon number and present a nonlinear relationship between the mean photon number and input light power. It is expected that the linear resolution of more photons will be possible in the future, resulting in more pixels and greater efficiency.

4 Conclusions

An SNSPD consisting of 16 pixels in a 4 × 4 array was proposed in this paper. The size of a single pixel is 20 × 20 μm, and the total effective photosensitive surface is 80 × 80 μm. This SNSPD works in the free mode and does not require gate control as does the APD system. In addition, the proposed SNSPD adopts the power combination method without an additional readout circuit, and the circuit analysis is simple. A uniform distribution of the counting rate and a maximum quantum efficiency of 94% were achieved, enabling this detector to approach a linear photon-number resolution. Regarding both the photon-number resolution and photon-space resolution, this device is promising and could be widely used in quantum communication, single-photon imaging, astronomical detection, and fluorescence spectrum analysis.