Single-photon detectors, which are sensitive enough to register an individual photon’s input, are applied in various leading-edge applications, such as quantum communication [1, 2], quantum computing [3, 4], and quantum sensing [5]. Detectors based on avalanche photodiodes (APD) are widely used in practice, thanks to their compact dimensions and no need for cryogenic operation. However, the detection efficiency (DE), defined as the probability for successful registration of a single-photon input, is limited for APD-based single-photon detectors, especially in the infrared range for optical-fiber communication [6]. Achieving a DE close to unity is essential in quantum cryptography [7], quantum computing [8], and quantum sensing [9]. Superconducting nanowire single-photon detectors (SNSPDs), which use narrow and ultrathin superconducting wires as the photon absorber [10, 11], have recently been proved the best candidate for a unity-DE system with superior performance, such as short dead time and high time resolution over a wide wavelength range [12–16].

The overall system DE of a SNSPD can be modeled as the product of three different efficiencies [11]: η_DE = η_coupling × η_absorption × η_registering. The coupling efficiency η_coupling corresponds to the loss of incident photons before reaching the nanowire absorber, due to absorption, scattering, or reflection. The absorption efficiency η_absorption is the probability that a photon reaching the nanowire is absorbed. The registration efficiency η_registering is the probability that the nanowire generates an output electrical signal after photon absorption. A coupling efficiency close to 99% can be achieved by optimizing the spatial mode coupling between the optical fiber and the nanowire absorber [17, 18]. The registering efficiency relates to the properties, design, and fabrication of the nanowire material used [11, 19–21]. For most silicide and nitride superconductors, a registering efficiency close to unity has been demonstrated [20, 22], which can be confirmed by measuring the saturation of the system’s DE against the bias current. The absorption efficiency can be optimized by avoiding reflection from the surface of the nanowire absorber, and by reflecting the transmitted light back to the nanowire absorber. The most widely used methods are to add an antireflection coating on the front side, and a highly reflective mirror on the back side, of the nanowire material [15, 16, 23].

High-efficiency SNSPDs commonly use an antireflection coating (ARC) having a structure with multiple layers, from double to quadruple, together with a distributed Bragg reflector (DBR) or metal-mirror reflector. The first SNSPD that achieved a DE of higher than 90% used a double-layer ARC [24], and a detector with 98% efficiency was reported with a triple-layer ARC using a high-refractive-index material [15]. However, an increase in the number of layers of ARC does not generally correlate with an increase in DE [25]. The number and design of the ARC layers need to be optimized according to the structure of the individual SNSPD.

In this paper, we describe the design of an antireflection coating (ARC) to optimize the absorption efficiency of a SNSPD. We apply finite-element analysis (FEA) to simulate the optical absorption in various layer structures, both single and double layers, for the superconducting absorber material amorphous molybdenum silicide. A DBR mirror at 1550 nm is fixed to the back side of the nanowire absorber. For a single-layer structure, the simulation results are compared to the experimental realization.

The absorption in the nanowires is numerically simulated using commercial FEA software COMSOL Multiphysics, Wave Optics module (COMSOL Inc., MA, USA). Figure 1(a) shows the construction of the fiber-coupling package, which consists of a fiber ferrule, a mating sleeve, and a SNSPD. We decided on this geometry for the front-side coupling, in which the light from the fiber passes to the nanowire absorber and the DBR mirror fabricated on top of a silicon wafer substrate. Figure 1(b) shows an example of a nanowire absorber, which is designed as a meander pattern to obtain both high registration efficiency and high fill factor under the given fabrication conditions. For the simulation presented in this paper, we apply a design for a 5-nm-thick α-Mo_xSi_1-x absorber with nanowire width of 130 nm and pitch of 200 nm. The area of the nanowire absorber is 14 μm × 14 μm. For the DBR mirror, we apply a 6.5-period (13-layer) reflector, consisting of seven layers of 265-nm-thick SiO₂ and six layers of 180-nm-thick Ta₂O₅.

Figure 1. Schematic diagram of a SNSPD system. (a) Illustration of a fiber-coupling package, (b) SEM image of a nanowire absorber with a meander-pattern design where the superconducting material appears brighter, (c) a vertical cross section of the fiber-coupling package, and (d) enlarged part of the cross section from the fiber end to the nanowire absorber, where the unit cell for simulation is indicated by a red box.

Figure 1(c) shows a vertical cross-sectional view of the construction when the fiber ferrule is assembled with the SNSPD. The yellow area indicates the core of the fiber, which guides the light. The nanowire absorber, indicated as black dots in Fig. 1(c), lies between the green layer of the DBR mirror and the blue ARC layer. Between the fiber end and the top of the ARC layer there is an air gap, which is determined by the mechanical fixture of the mating sleeve. The design principle of this construction is to keep the distance between fiber end and nanowire absorber smaller than 3 μm, so that the beam diameter of the light from the fiber at the absorber is approximately the same as the fiber mode-field diameter of 10.4 μm, at a wavelength of 1550 nm [26].

The simulation is performed for the unit cell shown in Fig. 1(d), by applying a two-dimensional infinite-grating model with in-plane periodic boundary conditions [27]. This can be justified by considering that the light from the fiber is close to a plane wave in the near-field condition, and that the lateral dimension of the nanowire of 130 nm is much smaller than the mode-field diameter.

The thickness of the air gap shown in Fig. 1(d) is neither known nor controllable, because of the large temperature difference between fabrication and operation conditions for the system. We estimate that the realized distance between fiber end and nanowire absorber can be anywhere between 1 and 3 μm [16]. This uncertainty of the air gap is considered in the simulation by averaging the absorptance with respect to variation of the air-gap distance. To demonstrate the averaging method, we perform the simulation without any ARC layer, as shown in Fig. 2(a). The complex refractive indices n + iκ at 1550 nm used in the calculation are taken from literature: n = 1.4682 for fiber core [26], n + iκ = 4.7 + 2.9i for α-Mo_0.75Si_0.25 [28], n = 2.086 for Ta₂O₅ [29], and n = 1.44 for SiO₂ [30]. Since the nanowire’s absorptance depends on polarization, the incident light is set to be in the TE mode (electric field parallel to the nanowire), for the highest absorptance. The simulation results are shown in Fig. 2(b) as a function of the air-gap distance between the fiber end and the nanowire absorber. It shows that the absorptance periodically varies from 63.3% to 90.3% depending on the air-gap distance, with an average value of 76.8%, as indicated with a red dashed line. The uncertainty of the average absorptance is evaluated from the maximum variation of the calculated absorptance as a function of air-gap distance, assuming a rectangular probability distribution. The standard uncertainty of the absorptance from Fig. 2(b) is therefore calculated as ${\mu }_s={\Delta }_{\text{max}}/\sqrt{3}=3.8\%$ with Δ_max of 13.5%. As a result of the simulation, we finally obtain an absorptance of (76.8 ± 12.9)% with a coverage factor of k = 1.65 at the 95% confidence level. We note that all the simulated results for absorptance in the following sections are the average values with respect to the air-gap distance.

Figure 2. Simulation for the case without ARC: (a) unit-cell design, (b) simulation results for absorptance as a function of air-gap distance (square symbols), plotted together with the average value (dashed line).

We first design a single-layer ARC for a SNSPD at a wavelength of 1550 nm. This simulation is important for testing the validity of the simulation method, by comparison to an experimental realization. Figure 3(a) shows the simulation structure. For the value of the refractive index of the ARC layer we used 1.42, the measured value for SiO₂ ARC material, at room temperature and a wavelength 1550 nm.

Figure 3. Simulation for the case with a single-layer ARC. (a) unit-cell design, (b) simulation results for absorptance as a function of normalized optical thickness D_opt (symbols), plotted together with the results without an ARC (dashed line), (c) simulation results for the spatial distribution of the electromagnetic field amplitude at D_opt = 0.25 and an air-gap distance of 770 nm, and (d) simulation results for the spatial distribution of the electromagnetic field amplitude at D_opt = 0.5 and an air-gap distance of 380 nm.

Figure 3(b) shows the simulation results as a function of normalized optical thickness of the ARC layer, which is defined as $D_{\text{opt}}=\raisebox{1ex}{$n\cdot d$}\!\left/ \!\raisebox{-1ex}{$\lambda $}\right.$, where n is the refractive index, d the physical thickness, and λ the wavelength. Each data point in Fig. 3(b) corresponds to the absorptance averaged over the air-gap distance, similar to Fig. 2(b). At a normalized optical thickness of 0.25 and 0.5, we obtain an absorptance of (52.3 ± 12.9)% and (76.7 ± 12.9)%, respectively.

Theoretically, we expect that the maximum absorptance would be achieved at a normalized optical thickness of 0.25 for a single-layer ARC [31]. However, this traditionally prominent role of the ¼λ-thick single-layer antireflection film is not present in this simulation result. Instead, the absorptance shows a minimum at ¼λ thickness. This can be explained based on the spatial distribution of the electromagnetic field amplitude, as shown in Figs. 3(c) and 3(d) for a normalized optical thickness of 0.25 and 0.5 respectively. The air-gap distances in Figs. 3(c) and 3(d) are selected to have the maximum absorptance. Due to formation of optical modes by the DBR and the nanowire with an ultrathin thickness and finite width, the position of the maximum field amplitude in the vertical direction shifts, depending on the optical thickness of the ARC layer. Consequently the maximum absorption is achieved at a normalized optical thickness of 0.5, which corresponds to the case of absence of the ARC. Note that the absorptance value without the ARC layer is indicated in Fig. 3(b) as the red dashed line. In conclusion, the single-layer ARC on a SNSNPD cannot improve the absorption efficiency, compared to the case of no ARC.

In an experiment, we fabricate and tested a SNSPD device with a single-layer ARC. Although the simulation results show that the single-layer ARC does not improve the absorptance, its experimental realization and a quantitative comparison with the simulation results are necessary to validate the simulation method based on the simplest structure. We also note that an ARC also functions to protect the nanowires from physical contact with the fiber end. Our goal is to realize an ARC with a normalized optical thickness of 0.25, which results in maximum deviation from the case without an ARC, as shown in Fig. 3(b).

In fabrication, the DRB mirror is first deposited on a Si substrate by plasma-enhanced chemical vapor deposition (PECVD), followed by the nanowire absorber of 5-nm-thick α-Mo_xSi_1−x thin film, deposited by co-sputtering method. The deposition ratio of Mo and Si is targeted to 75:25, but the exact value is not confirmed. The meander pattern of the nanowire absorber is fabricated by electron-beam lithography to a width of 130 nm and a pitch of 200 nm, as shown in Fig. 1(b). The resulting active area of the absorber is 14 μm × 14 μm. The single-layer ARC on the nanowire is deposited with SiO₂ by low-temperature PECVD, with a normalized optical thickness of 0.25. The obtained normalized optical thickness of the ARC is confirmed to be 0.206, using a dummy substrate fabricated in parallel with the nanowire substrate under the same conditions. Finally, a lollipop-shaped detector is cut from the substrate wafer by deep reactive-ion etching, which is matched to the optical fiber by the self-alignment scheme [17]. The detector is placed and cooled in a cryostat system with a GM cooler and an additional 0.9 K sorption unit.

The realized SNSPD with the single-layer ARC is characterized by measuring its system DE at a wavelength of 1540 nm [24, 32]. Figure 4(a) shows the schematic setup of the SNSPD with optical input and electrical-signal output. For the optical input, we use a fiber-coupled diode laser at 1540 nm with a two-stage programmable variable attenuator. The radiant power from the fiber end that is coupled to the SNSPD is measured at the monitoring port from the variable attenuator by using a power meter HP/Agilent 81532A (Keysight, CA, USA). The splitting ratio between the monitoring port and the output port, as well as the attenuation values of the variable attenuator, are calibrated using the same power meter. The uncertainty of the radiant-power measurement is estimated from the specification of the power meter to be 5%, as an expanded relative uncertainty (k = 2). As the system DE depends on polarization, a manual fiber polarization controller is used to maximize the count-rate signal from the SNSPD for fixed input power.

Figure 4. Experiment for the case with a single-layer ARC. (a) schematic setup of the SNSPD, with optical input and electrical-signal output, for measuring the detection efficiency (DE) at a wavelength of 1540 nm, (b) measured results for DE and dark-count rate (DCR) as a function of the bias current.

At the electrical-signal output, a DC current source is used to operate the SNSPD, while the RF signal from the SNSPD is AC-coupled out via a bias tee to a voltage-follower circuit, followed by a pulse counter. The DE measurement is performed at a radiant-power level of 10 fW at 1540 nm, which produces approximately 10⁵ counts per second. The system DE is determined as the count rate of the SNSPD divided by the photon flux corresponding to the measured radiant flux.

Figure 4(b) shows the results of the system DE measurement as a function of the applied bias current of the SNSPD. In addition, the dark count rate (DCR) is also plotted on a logarithmic scale, to show that the two regimes consist of background DCR and intrinsic DCR. The DE starts to increase at a bias current of 4 μA and saturates at 6 μA. The DCR shows a slow exponential increase for bias currents from 2 to 6.5 μA, which corresponds to a background-limited regime. When the bias current reaches 6.5 μA the DCR shows a rapid increase, indicating that the SNSPD has entered the intrinsic-DCR regime and is in unstable operation [33, 34]. From this result, we determine that the optimal operation condition for the SNSPD is at a bias current of 6 μA, at which the system DE is measured to be 54%, with a DCR of 40 s⁻¹.

From the measured system DE we should determine the absorption efficiency, to compare it to the simulation result. As described in the Introduction, the system DE of a SNSPD can be modeled as a product of coupling, absorption, and registration efficiencies. The coupling efficiency can be regarded as close to unity through the self-alignment scheme [17]. The registration efficiency of close to unity is confirmed by measurement of the saturation of the system DE against the bias current. Therefore, we conclude that the measured system DE is mainly limited by the absorption efficiency. From the simulation in the previous section, we obtain an absorptance of (53.5 ± 13.0)% for a normalized optical thickness of 0.206, which can be compared to the experimental value of 54%. The two values agree within the claimed uncertainty that originates from the uncertain value of the air-gap distance. With this comparison the validity of the simulation method is confirmed, within the evaluated uncertainty.

In the previous sections we showed that a single-layer ARC cannot improve the absorption efficiency, due to the shift of the maximum field amplitude depending on the thickness of the ARC (see Fig. 3). Therefore, an ARC based on a double-layer structure, by alternately stacking a high-refractive-index (H) layer and a low-refractive-index (L) layer, is tested by simulation. The goal of the simulation is to find the structure of the double-layer ARC with the maximum absorptance, by varying the optical thickness of each H and L layer. For the L layer, the refractive index is fixed at 1.42, corresponding to the measured value for SiO₂ in the case of the single-layer ARC. For the H layer, in contrast, various materials such as silicon nitride, titanium oxide, and amorphous silicon are considered. For such materials the refractive index can vary strongly, depending on the deposition conditions [35–37]; thus the simulation is performed with the refractive-index value of the H layer selected from a range of 1.6 to 2.8.

Figure 5(a) shows an ARC structure with a pair of H/L layers. Starting from the substrate, an L layer is deposited as the first layer, and an H layer as the second layer. The absorptance results calculated from the simulation for refractive indices of the H layer of 1.6, 2.0, 2.4, and 2.8 are presented in Figs. 5(b)–5(e) respectively. Each graph shows a two-dimensional plot of absorptance as a function of the normalized optical thickness of the first L layer (x axis) and of the second H layer (y axis). From these results, we see that there exist areas of high absorptance, and that the maximum absorptance is near the point where the values of normalized optical thickness for both layers are close to 0.25. As the refractive index of the H layer increases, the area of high absorptance becomes smaller, but the contrast of the absorptance values increases.

Figure 5. Simulation for the case with a double-layer ARC: (a) unit-cell design; (b)–(e) simulation results for absorptance as a function of D_opt of the H and L layers, for different values of refractive index of each H layer.

We recall that the absorptance data used in the plots of Figs 5(b)–5(e) are the average values with respect to the air-gap distance (see also Fig. 2). Figure 6(a) shows the absorptance as a function of the air-gap distance for different values of the H layer’s refractive index, at the values of normalized optical thickness of the H and L layers where the absorptance has its maximum. It is remarkable that the variation of absorptance with air gap decreases as the refractive index of the H layer increases. Figure 6(b) shows the average absorptance values and their uncertainties, calculated from the results of Figure 6(a). We confirm that the absorptance increases as the H layer’s refractive index increases, while the uncertainty due to the air-gap distance is reduced for high refractive-index values of the H layer. For the double-layer ARC with a pair of H/L layers, a maximum absorption efficiency of 95.7% is achieved, with an uncertainty as low as 1.5%, when the refractive index of the H layer is greater than 2.4.

Figure 6. Simulation results for the case with a multilayered ARC, with the pairs of D_opt for H and L layers showing the maximum absorptance. (a) absorptance as a function of air-gap distance, for different values of the refractive index of the H layer, (b) absorptance at different values of the refractive index of the H layer (square symbols), with uncertainties due to the air-gap distance (error bars).

The simulation results confirm that, compared to the single-layer design, a higher DE and smaller influence of the air-gap distance can be realized with a double-layer ARC with a high contrast of refractive index between the L and H layers, at a design wavelength of 1550 nm. On the other hand, the spectral dependence of the DE is expected to be stronger for the double-layer ARC as the refractive index of the H layer increases [38, 39]. The spectral dependence of absorptance for the double-layer ARC is calculated by changing the simulation wavelength in the neighborhood of the design wavelength of 1550 nm. Figure 7 shows the absorptance as a function of wavelength for different values of the H layer’s refractive index, at the values of normalized optical thickness of the H and L layers where the absorptance has its maximum. Here we confirm that the spectral dependence becomes stronger for the ARC with a higher refractive index of the H layer. Depending on target applications, such a spectral dependence of DE should be also considered for optimizing the design of the ARC. We note that the simulation results of Fig. 7 are obtained for a fixed structure of the 13-layer DRB mirror designed for 1550 nm, which has also a strong influence on the spectral dependence of DE.

Figure 7. Simulation results for the case with a multilayered ARC with the pairs of D_opt for H and L layers showing the maximum absorptance: absorptance as a function of wavelength, for different values of the refractive index of the H layer.

In this paper we optimized the DE of a SNSPD with a fiber coupling, by designing an ARC placed between the fiber end and the nanowire absorber. The design was based on numerical calculations of the absorptance of the nanowire with FEA simulation. We identified that the air-gap distance between the fiber end and the ARC top, which is neither controllable nor measurable in experiment, is the dominant source of uncertainty. By simulation for a case without ARC, we could show a periodic variation of absorptance with air-gap distance and develop a method to evaluate the uncertainty of the absorptance due to the air-gap distance.

To validate the simulation method, the simplest case with a single-layer ARC was designed and realized in experiment. From simulation, the absorptance was found to depend on the optical thickness of the ARC layer, but remained always smaller than in the case without ARC. This could be explained by the spatial distribution of the electromagnetic field’s amplitude. In the experimental realization, at a specific optical thickness of the single-layer ARC the system DE was measured to be 54%, which was in agreement with the simulated value of (53.5 ± 13.0)%, within the evaluated uncertainty.

To design an ARC with a high DE, a double-layer structure consisting of a bottom low-index (L) layer and a top high-index (H) layer was considered. Taking the refractive index of the H layer as a parameter, absorptance was calculated as a function of optical thickness of both H and L layers. As a result, we found that an optimal design with a DE greater than 95% was achievable for a refractive index of the H layer greater than 2.4, with a normalized optical thickness close to 0.25 for both layers. Also, the uncertainty due to the air-gap distance could be reduced to as low as 1.5 % when an H layer with a refractive index greater than 2.4 was used.

In conclusion, we were able find an optimal design for a double-layer ARC for a fiber-coupled high-efficiency SNSPD based on a simulation method, which was validated by comparison to experiment.

This work is supported by the R&D Convergence Program of the National Research Council of Science and Technology (NST) (CAP-15-08-KRISS), the Korea Research Institute of Standards and Science (KRISS) (GP2021-0010-03), and the Engineering Research Center (ERC) supported through the National Research Foundation (NRF) funded by the Ministry of Science and ICT in Korea (2019R1A5A1027055).

We thank Dong-hyung Kim and Yongjai Cho for their help in characterizing the optical thin-film properties.