Fractal superconducting nanowires detect infrared single photons with 84% system detection efficiency, 1.02 polarization sensitivity, and 20.8 ps timing resolution

Because of their near-unity system detection efficiency (SDE) [1, 2, 3, 4, 5, 6, 7], low dark-count rate (DCR) [8], high count rate [9, 10, 11, 12], excellent timing resolution [13, 14], and broad working spectral range [15, 16, 17, 18], superconducting nanowire single-photon detectors (SNSPDs) [19] have been widely used in classical and quantum photonic applications [20], ranging from LiDAR [21], detection of luminescence from singlet oxygen [22], quantum key distribution (QKD) [23], to quantum computing [24, 25]. Indeed, these detectors have become indispensable tools and enabling components in the systems requiring faint-light detection. However, the prevalent design based on meandering nanowires yields polarization-dependent SDE, which could be problematic if information is encoded in polarization states. In particular, when the polarization states of the photons are unknown, time-varying, or random, it may not be possible to rotate the polarization states to maximize SDE of SNSPDs. Therefore, the major merit of these detectors would get severely compromised; and this compromise could be detrimental for many photon-starved applications that stringently require high SDE. For photon-number-resolving detection or coincidence photon counting, the fidelity to resolve $n$ photons or the $n$-fold coincidence count rate scales with SDEⁿ [26, 27], which quickly drops if SDE decreases. In a QKD system using SNSPDs, the polarization-dependent mismatch of SDE makes the system vulnerable for quantum hacking [28]. Furthermore, the subtle trade-offs between SDE and timing resolution [29] make their simultaneous optimization challenging. Recently, 85$\%$ SDE at the wavelength of 915 nm and 7.7 ps device timing jitter [14], and 98% SDE at the wavelength of 1425 nm and 26 ps system timing jitter [6], were demonstrated on meandering SNSPDs, but the SDEs were still polarization-dependent.

To address the issue of polarization dependence of SDE, several approaches have been proposed and demonstrated, including spiral SNSPDs [30, 31], two orthogonal side-by-side meanders [30], double-layer orthogonal meanders [32], SNSPDs involving compensating high-index materials [33, 34], specially designed SNSPDs with low polarization dependence at a certain wavelength [35, 4], and fractal SNSPDs [29, 36, 37]. These demonstrations all have successfully reduced the polarization sensitivity (PS, the ratio of the polarization-maximum SDE, SDE${}_{\textrm{max}}$, over the polarization-minimum SDE, SDE${}_{\textrm{min}}$ [38]) of SNSPDs; however, none of them could simultaneously preserve other major merits, in particular, high SDE and excellent timing resolution. Among these demonstrations, amorphous SNSPDs, made of WSi or MoSi, have exhibited over 80% [32, 35] and even over 90% SDE [4], however, their timing jitter ranges from 76 ps to 465 ps; on the other hand, polycrystalline SNSPDs, made of NbN or NbTiN, have shown better timing resolution, however, so far, the highest SDE demonstrated on polycrystalline SNSPDs with low-PS designs is 60% [29], still significantly lower than the state-of-the-art SDE${}_{\textrm{max}}$ of meandering SNSPDs, which is over 90% demonstrated by several research groups [1, 2, 3, 4, 5, 6, 7]. Therefore, it remains an outstanding challenge how to boost the SDE of SNSPDs with low PS to the level comparable to the SDE${}_{\textrm{max}}$ of their meandering counterparts while simultaneously optimizing the timing resolution.

Although the geometry of the fractal SNSPDs [37, 29] eliminated the global orientation of the nanowire, and therefore, significantly reduced PS, it was also this geometry that brought the major obstacle for further enhancing SDE and timing resolution. The fractal design contains a plethora of U-turns and L-turns that may limit the switching current, $I_{\rm sw}$, due to the current-crowding effect [39], which may further affect SDE and timing resolution. In the past, we demonstrated fractal superconducting nanowire avalanche photodetectors (SNAPs) [29] with 60% SDE, 1.05 PS, and 45 ps timing jitter. However, it is still elusive whether this route, using fractal geometry, is a practical one to combine high SDE, low PS, and low timing jitter.

In this paper, we report on our design and demonstration of a fiber-coupled fractal SNAP, fully packaged in a 0.1-W closed-cycle Gifford-McMahon (GM) cryocooler with the base temperature of 2.0 K, achieving 84$\pm$3$\%$ SDE, 1.02${}^{+0.06}_{-0.02}$ PS at the wavelength of 1575 nm, and 20.8 ps timing jitter. An enabling innovation is that we used an arced fractal geometry [40] for the nanowire to successfully reduce the current-crowding effect and therefore, increased $I_{\rm sw}$ to a level comparable to that in the meandering structure with the same nanowire width, thickness, and fill factor, achieving saturated, near-unity internal quantum efficiency, P${}_{\textrm{r}}$. We integrated the arced fractal nanowire with an optical micro-cavity, supported by dielectric distributed Bragg structures, for enhancing the optical absorptance, A, of the nanowire [3, 41, 7, 5].

Figure 1 (a) presents a schematic of the optical structure of an arced fractal SNAP (AF-SNAP). Six pairs of alternating silicon dioxide (SiO₂) and tantalum pentoxide (Ta₂O₅) layers were deposited on a silicon substrate, functioning as the bottom Bragg reflector; three pairs formed the top reflector; in between a SiO₂ defect layer was sandwiched. The thicknesses of a SiO₂ layer and a Ta₂O₅ layer in the Bragg reflectors are 264 nm and 180 nm, respectively; and the thickness of the SiO₂ defect layer is 529 nm, targeting for the wavelength of 1550 nm with optimal optical absorptance. The red line in Fig. 1 (a) shows the simulated distribution of the light intensity in the dielectric stacks (without the nanowires) at the wavelength of 1550 nm for top illumination; and the NbTiN nanowires were designed to locate in the middle of the defect layer of the optical micro-cavity where the light intensity is the strongest. The thickness of the NbTiN film used in this work was 9 nm. Fig. 1 (b) presents a false-colored scanning-electron micrograph of a fabricated AF-SNAP, before the top Bragg layers were integrated. The photosensitive region of the detector was 10.2 µm by 10.2 µm, and the width of the nanowire was measured to be 40 nm [Fig. 1 (c)]. The photosensitive region of the detector is composed of 64 second-order arced fractal Peano curves [40] that are electrically connected according to the circuitry in Fig. 1 (g). In comparison to the the standard Peano fractal curve [37, 29], the arced Peano fractal curve [40] reduced the current-crowding effect at the turns. We simulated the normalized distribution of the supercurrent density, $\lvert\textbf{{J}}\lvert$, at the proximity of an L-turn and a U-turn [Sec. S1 of Supporting Information (SI)], which are presented in Fig. 1 (d). Fig. 1 (e) further presents the simulated $I_{\textrm{sw}}$, normalized to the critical current of a straight nanowire with the same width and thickness, $I_{\textrm{c}}$, of the meandering, standard fractal, arced fractal nanowires, and the optical absorptance of the arced fractal nanowires, as functions of the fill factor. We used a commercial software COMSOL Multiphysics based on finite-element method for these simulations. At the fill factor of 0.31 used in this work, the simulated optical absorptance for the plane wave at the wavelength of 1550 nm is 96$\%$, and the normalized switching current of the arced fractal nanowire is 0.81. As a comparison, with the same fill factor, 0.31, the normalized switching currents of the meandering nanowire and the standard fractal nanowire, are 0.82 and 0.67, respectively, further evidencing that the current-crowding effect in the arced fractal nanowire is significantly reduced, compared with that in the standard fractal one. Detailed comparison of the distribution of the supercurrent density of these three types of geometry is presented in Sec. S1 of SI. Fig. 1 (f) presents the simulated optical absorptance, A, of the AF-SNAP for the plane wave as functions of wavelength for two orthogonal linear polarization states, denoted as transverse-electric (TE) and transverse-magnetic (TM) states. A peaks at 1550 nm and remains above 50% in the wavelength range from 1490 nm to 1610 nm. The simulation shows that A is completely polarization-independent. To investigate the coupling efficiency, $\upeta_{\rm c}$, we simulated the optical modes of the optical micro-cavity without the nanowires. The simulated mode-field diameter (MFD) at the plane of nanowires coupled with a Corning high-index optical fiber (HIF, HI 1060 FLEX) used in this paper was 6.8 µm, which ensured the $\upeta_{\rm c}$ of 99% assuming perfect alignment. In comparison, the MFD at the plane of nanowires coupled with a Corning SMF-28e+ optical fiber (SMF) was 10.7 µm and the corresponding coupling efficiency is 89%. The detector coupled with HIF is more tolerable to spatial misalignment than the detector coupled with SMF. Detailed simulation regarding optical modes of the cavity is presented in Sec. S2 of SI. Electrically, the detector was composed of sixteen cascaded 2-SNAPs, as we used previously [29], and Fig. 1 (g) presents the equivalent circuit diagram. The chips were etched into the keyhole shape by Bosch process for self-aligned packaging [42]. Detailed fabrication process is presented in Sec. S3 of SI. Fig. 1 (h) shows a photograph of the resulting chip package and the inset presents a photograph of a keyhole-shaped chip. In this package, the detector was self-aligned and directly coupled with a HIF, with a MFD of 6.3$\pm$0.3 µm, which was connected to SMF, with a MFD of 10.4$\pm$0.5 µm, through an in-line mode-field adapter (Sec. S4 of SI).

We used the experimental setup, schematically presented in Sec. S4 of SI, to measure system detection efficiency and the polarization dependence. The base temperature for these measurements was 2.0 K. At this temperature, $I_{\rm sw}=21.67$ µA. A cryogenic, low-noise microwave amplifier was mounted on the 40-K stage and used to amplify the output pulses. We first measured the DCR as a function of the bias current (Sec. S5 of SI). Then, we biased the detector at 21.17 µA, tuned the laser wavelength and found that the system detection efficiency peaked at 1575 nm for this particular detector. The wavelength deviation from the designed wavelength with the maximum optical absorptance is presumably due to deviations of the thicknesses of the deposited dielectric layers and the refractive indices. We then fixed the wavelength at 1575 nm and scanned the polarization states of the input light over the Poincaré sphere, and found the polarization states corresponding to $\rm{SDE}_{max}^{*}$ and $\rm SDE_{min}^{*}$; at these two polarization states, we measured $\rm{SDE}_{max}^{*}$ and $\rm SDE_{min}^{*}$ as the functions of the bias current [Fig. 2 (a)]. To accurately measure the SDE*, we calibrated each optical attenuator at each polarization state and each wavelength for these measurements (Sec. S6 of SI). In the high-bias regime ($I_{\textrm{b}}>20.17\$µA), as shown in Fig. 2 (a) and (b), the $\rm{SDE}^{*}$-$I_{\rm b}$ curves go upward, showing additional false counts other than the dark counts and showing unrealistic $\rm{SDE}^{*}$. Similar observations have previously been reported on meandering SNAPs [43] and also SNSPDs [44]. We re-measured the $\rm{SDE}_{max}$ and $\rm{SDE}_{min}$ using the method based on time-correlated photon counting [44] to exclude the false counts. Note that we use SDE* to refer to the system detection efficiency directly measured with the CW laser [Fig. S5 (a) for the experimental setup], excluding the dark counts; and we use SDE to refer to the system detection efficiency measured by time-correlated photon counting, excluding all false counts [Fig. S5 (b) for the experimental setup]. The values of SDE and SDE* in Fig. 2 (a), (b), (e), and (f) take into account the fiber end-facet reflection that occurred when we used the optical power meter to measure the optical power coming out from the fiber to avoid under-calculating the optical power delivered to the cryogenic AF-SNAP system and, therefore, to avoid over-calculating SDE and SDE* [6] (Sec. S7 of SI). In Fig. 2, the associated error bars present the uncertainties with 68% confidence ($k=1$) [45] of the measurements (Sec. S8 in SI). At the bias current of 19.83 µA, $\rm{SDE}_{max}$ was measured to be 84$\pm$3$\%$, $\rm{SDE}_{min}$ was measured to be 82$\pm$3$\%$, and the resulting PS ($\rm{SDE}_{max}$/$\rm{SDE}_{min}$) was 1.02${}^{+0.06}_{-0.02}$. $\rm{SDE}_{max}^{*}$ and $\rm{SDE}_{min}^{*}$ were measured to be both 85$\pm$2$\%$, and the resulting PS* was 1.00${}^{+0.03}_{-0}$. At this bias current, DCR and false-count rate (FCR) were measured to be $2.1\times 10^{3}$ cps and $2.2\times 10^{3}$ cps, respectively (Sec. S5 of SI). At the bias current of 17.83 µA, $\rm{SDE}_{max}$ and $\rm{SDE}_{min}$ decreased to 80$\pm$3$\%$ and 78$\pm$3$\%$, respectively; $\rm{SDE}_{max}^{*}$ and $\rm{SDE}_{min}^{*}$ decreased to both 81$\pm$2$\%$ [Fig. 2 (b)]; and the measured FCR and DCR at this bias current were 3.7$\times 10^{2}$ cps and 3.0$\times 10^{2}$ cps, respectively. In the low-bias regime, $I_{\textrm{b}}<$15.83 µA, the detector was unstable [46], generating multiple false pulses with low amplitudes after detecting one photon [Sec. S9 of SI], and resulting in the pronounced deviation of SDE* from SDE. PS and PS^∗, as functions of the bias current, were calculated and presented in Fig. 2 (c) and (d). Fig. 2 (e) presents $\rm{SDE}_{max}$ as a function of the wavelength, $\uplambda$, at the bias current of 21.17 µA. The full width at half maxima (FWHM) of the spectrum of $\rm SDE_{max}$ is 110 nm, which is slightly smaller than the FWHM, 120 nm, of the designed spectrum of the optical absorptance [Fig. 1 (f)]. Fig. 2 (f) presents a zoom-in view of the $\rm{SDE}_{max}$ for the wavelengths ranging from 1560 nm to 1595 nm, in which SDE${}_{\textrm{max}}>$75%. We note that here, the measurements of SDE and SDE* were performed at relatively low average input-photon rates going into the cryogenic AF-SNAP system, 1.79$\times 10^{5}$ s^-1 and 1.82$\times 10^{5}$ s^-1, respectively.

As the flux of the input photons increases, SDE would decrease. To characterize the SDE of the AF-SNAP at various input-photon rates, we measured the $\rm{SDE}_{max}$ as a function of the average input-photon rate (Sec. S10 of SI). Note that we used the oscilloscope to measure the $\rm{SDE}_{max}$ in Fig. 3 (a) with the sampling rate of 2.5 Gsample/s and the time span per frame of 1 ms, because the maximum count rate of the input channel of the TAC is 12.5 Mcps, smaller than the photon-count rate we needed in this measurement. As the flux of the incident photon increases, the switching current decreases. Each value of $\rm{SDE}_{max}$ in Fig. 3 (a) was measured at the bias current of 0.99$I_{\textrm{sw}}$. The results show that when the average input-photon rate increases to 5.47$\times 10^{7}$ s^-1, $\rm{SDE}_{max}$ drops to 59%, and the corresponding photon-count rate with FCR excluded was 26 Mcps.

Figure 3 (b) presents the measured timing jitter of the AF-SNAP by using a mode-locked fiber laser with the central wavelength of 1560 nm, a fast photodetector with 3-dB bandwidth of 40 GHz, and a real-time oscilloscope with bandwidth of 4 GHz. The experimental setup is schematically presented in Sec. S11 of SI. Each data point in Fig. 3 (b) is the FWHM of the exponentially modified Gaussian (EMG) fitting [47] to the time-delay histograms [Fig. S12 (b)]. The lowest value of timing jitter was 20.8 ps at 21.67 µA. The time-delay histogram and the EMG fitting is shown in the inset of Fig. 3 (b). Timing jitter monotonically increased with decreasing the bias current in the avalanche regime; for example, at $I_{\textrm{b}}=19.67$ µA, timing jitter increased to 25.6 ps. As for other temporal properties, in the avalanche regime, the exponential fitting to the recovery edge of the output pulse shows a 1/$e$ time constant of 8.68 ns [Sec. S9 of SI].

We estimated the highest possible SDE for the current configuration of our system. The total transmittance of the two types of optical fibers connected through the mode-field adapter was measured to be 98% at ambient temperature; the coupling efficiency, $\upeta_{\textrm{c}}$, between the high-index optical fiber and the photosensitive area was calculated to be 99%, assuming perfect alignment; the optical absorptance, $A$, was simulated to be 96% at the wavelength of 1550 nm; and the internal quantum efficiency, $P_{\rm{r}}$, was assumed to be 100%. The product of these numbers gives an estimation of the highest possible SDE to be 93%. The measured SDE, 84%, is less than this estimation of the highest possible SDE presumably because (1) certain misalignment between the optical mode and the photosensitive area existed in the package and (2) $P_{\rm{r}}$ was less than 100%.

The geometry of arced fractal nanowires can be applied to SNSPDs/SNAPs targeted for other interesting wavelengths by similarly re-designing the optical structures of the devices. In particular, the polarization dependence of SDE becomes more severe at longer wavelengths for meandering SNSPDs, and we think that the geometry presented in this work would be useful for creating SNSPDs working in the mid-infrared with low PS. We simulated and optimized the optical absorptance of meandering [Fig. 4 (a)] and arced fractal SNSPDs [Fig. 4 (b)] at some additional wavelengths, 0.6 µm, 0.9 µm, 1.3 µm, 2 µm, 3 µm, 4 µm, and 5 µm, for TE and TM polarization states. The optical structures are similar to that in Fig. 1 (a) except for that two, rather than three, pairs of alternating top layers maximize the optical absorptance of a meandering SNSPD for TE polarization and except for the modified thicknesses of the dielectric layers for different wavelengths. The simulation took into account the wavelength dependence of the refractive indices of the materials [48, 49, 50], which are listed in Sec. S12 of SI. The calculated absorptance ratios of these two polarizations are presented in Fig. 4 (c). At the longer wavelengths, the absorptance ratio for the meandering SNSPD increases whereas the absorptance ratio for the arced fractal SNSPD remains constantly 1. As the polarization-dependent optical absorptance is the dominant contributor to the PS, and as this work demonstrates that the AF-SNAPs can reach high SDE and high timing resolution at the near infrared, we think that arced fractal SNSPDs/SNAPs should be good device structures for polarization-insensitive single-photon detection in the mid-infrared, as well as the visible, spectral ranges.

In conclusion, we demonstrated a fiber-coupled AF-SNAP with 84$\pm$3$\%$ SDE, 1.02${}^{+0.06}_{-0.02}$ residual PS at the wavelength of 1575 nm, and 20.8-ps timing jitter. The SDE was boosted to the level comparable to the amorphous SNSPDs/SNAPs with low PS [4, 32, 35], but the timing resolution of the NbTiN AF-SNAP exceeded (See Sec. S13 of SI for the comparison; we also note that at their reported SDEs, amorphous SNSPDs/SNAPs listed in Sec. S13 of SI exhibited lower DCR than the FCR and DCR of the NbTiN AF-SNAP that we report in this paper, but amorphous SNSPDs/SNAPs require lower temperatures). These combined properties have not been achieved with any single-photon detectors reported previously and are enabled by our comprehensive device design. In particular, the arced fractal geometry of the nanowires is the key, enabling innovation that reduces the current-crowding effect and increases the switching current to the level comparable to that in the meandering SNSPDs and therefore, enhances both SDE and timing resolution. Since fractal SNSPDs were introduced in 2015 [36], although we kept enhancing their performances [29, 37], it had been elusive whether the fractal designs of the nanowires could be practical device structures; it is this work that gives a positive and unambiguous answer by showing that fractal SNSPDs are practical devices with excellent comprehensive performances, comparable to meandering SNSPDs, on top of which low PS is added. The arced fractal geometry is equally applicable to designing SNSPDs working in other spectral ranges, in particular, mid infrared. This demonstration is a detector coupled with a single-mode optical fiber, but the same geometry can be used for detectors coupled with few- or multi-mode optical fibers and for detecting single photons coming from free space. Additionally, the negligibly small PS of the arced fractal SNSPDs/SNAPs would eliminate the security loophole, due to polarization-dependent mismatch of SDE, in the QKD systems [28]. We believe that this work paves the way for polarization-insensitive single-photon detection with high SDE and high timing resolution.

X. H., Y. M., and K. Z. conceived the project. Y. M., K. Z., N. H., and X. H. designed the devices and performed numerical simulation. S. S., S. G., and V. Z. sputtered NbTiN films. K. Z., N. H., and X. L. fabricated the devices. Y. M. and L. X. performed the measurements. X. H., Y. M., K. Z., and N. H. analyzed the data and wrote the paper. All authors commented and revised the paper. X. H. supervised the project.

The authors declare no conflicts of interest.