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Abstract
Superconducting nanowire single-photon detectors (SNSPDs) have attracted remarkable interest for visible and near-infrared single-photon detection due to their outstanding performance. However, conventional SNSPDs are generally used as binary photon-counting detectors. Another important characteristic of light, i.e., polarization, which can provide additional information of the object, has not been resolved using the standalone SNSPD. In this work, we present a first prototype of the polarimeter based on a four-pixel superconducting nanowire array, capable of resolving the polarization state of linearly-polarized light at the single-photon level. The detector array design is based on a division of focal plane configuration in which the orientation of each nanowire division (pixel) is offset by 45°. Each single nanowire pixel operates as a combination of a photon detector and almost linear polarization filter, with an average polarization extinction ratio of ∼10. The total system detection efficiency of the array is ∼1% at a total dark count rate of 680 cps, with a timing jitter of 126 ps, when the detector array is free-space coupled and illuminated with 1550-nm photons. The mean errors of the measured angle of polarization and degree of linear polarization were about -3° and 0.12, respectively. Furthermore, we successfully demonstrated polarization imaging at low-light level using the proposed detector. Our results pave the way for the development of a single-photon sensitive, fast, and large-scale integrated polarization polarimeter or imager. Such detector may find promising application in photon-starved polarization resolving and imaging with high spatial and temporal resolution.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The vast majority of the imaging sensors found in daily life are designed to detect light intensity and wavelength, corresponding to human perception of brightness and color, respectively. Imaging sensors do not generally detect the polarization state of light because of the polarization insensitivity of human's eyes. However, capturing the optical properties of polarized light has proved to be important. It can provide additional information for material classification, target recognition, and biomedical diagnostics [1,2]. Particularly, polarization imaging plays a significant role in environments with severe optical scattering, such as in target contrast enhancement in hazy conditions [3] and underwater imaging [4,5]. Recently, emerging applications, such as long-range remote sensing [6–10], astronomical observations [11,12], quantum-enhanced target detection [13], and fluorescence polarization imaging [14] strongly require detectors capable of resolving and imaging polarization at photon-starved level. However, the conventional polarimeter or image sensors perform poorly in the low-light environments, owing to low signal-to-noise ratio and limited spatial and temporal resolution [9,10,15,16]. For instance, the widely-used CMOS or CCD polarization cameras [5,12,17] lack single-photon sensitivity and temporal resolution, although they can provide a polarimetric integral imaging in a large format through optical or computational reconstruction methods [15]. Meanwhile, the commonly-used single photon detectors [9,10,18], for example, the InGaAs/InP avalanche photodiode (APD) [16] is usually used at near infrared, which exhibits a low system detection efficiency (SDE) of ∼20% at 1550 nm, a long dead time of ∼2 µs (free-running), a large timing jitter of ∼150 ps, and a high likelihood of afterpulsing. Furthermore, most of the single-photon detector are not polarization sensitive and needed additional polarized components to realize polarization resolving and imaging [1,2]. One of the solutions is to place a rotating polarizer before the single-photon detector [11]. However, this approach limits a fast imaging capability as well as the high integration of the image system [1,2]. Another possible solution is to integrate the APD array with a wire-grid polarizer array, however, it will increase the difficulty in sensor design and manufacture (rarely reported).
In recent years, superconducting nanowire single-photon detectors (SNSPDs) have attracted significant attention for their exceptional performance characteristics, including high SDE (> 90%) [19,20], low dark count rate (DCR, < 1 cps) [21], low timing jitter (< 20 ps) [22,23], and high maximum count rate (> 500 M cps) [24]. These detectors have been successfully used for various applications, including long-distance quantum key distribution [25,26], quantum optics [27], fluorescence correlation spectroscopy [28], and laser ranging and imaging [8,29,30]. SNSPDs are single-mode fiber (SMF)-coupled devices and are generally used as binary photon-counting detectors. Their detection mechanism consists in the generation and disappearance of a photon-induced resistive domain across a superconducting nanowire upon photon absorptance [31]. As a result of their meandered nanowire structure, conventional SNSPDs have significantly higher absorption for the transverse electric (TE) mode versus the transverse magnetic (TM) one [32]. The absorption ratio between TE and TM modes is referred to as the polarization extinction ratio (PER) and is used to evaluate the polarization sensitivity of SNSPDs. Numerous previous studies [33–36] have focused on eliminating the polarization sensitivity of SNSPDs. Conversely, SNSPDs’ high polarization sensitivity [37,38] suggests potential application in polarization detection and imaging. For instance, in 2015, Guo et al. fabricated a polarization-sensitive SNSPD that featured a PER of ∼22 and SDE of ∼12% with a low filling factor design [37]. Recently, an emerging multipixel SNSPD array platform [39,40] has provided new opportunities for developing single-photon sensitive, fast, and large-scale integrated polarization polarimeters or imagers. However, to the best of our knowledge, a single-photon polarimeter based on superconducting nanowires has not previously been reported.
To reach this goal, two major challenges must be addressed. First, it is difficult to resolve the polarization state of light using only one SNSPD due to the complexity of the Stokes vector [1,2]. Second, the cross-coupling of optical power between polarization modes caused by the birefringence effect [41] in fibers makes it difficult to calibrate the polarization state of light that actually reaches the detector. Specifically, the polarization state of an SMF-coupled SNSPD varies randomly because the inherent birefringence varies with external perturbations in the fiber. These perturbations can be caused by bending, twisting, or thermal fluctuations during measurement. In this case, one may consider replacing the SMF with a polarization-maintaining fiber (PMF), which can maintain linear or circular polarization along the fibers. However, the use of a PMF prevents the possibility of measuring any other light polarization states for future applications.
In this work, we simulated, fabricated, and characterized a polarization-sensitive four-pixel niobium nitride (NbN) SNSPD array used a division of focal plane (DoFP) configuration [17]. The nanowire was patterned with a width of ∼50 nm and a pitch of ∼200 nm to ensure an average PER of ∼10. The SNSPD array was characterized with a free-space coupling setup that ensured reliable polarization analysis and real-time calibration along the same optical path. By means of Stokes parameter measurements, we successfully demonstrated a working prototype of a single-photon linear polarimeter based on a superconducting nanowire array for the first time. Furthermore, we successfully performed polarization imaging at low-light level. The absence of moving parts and compact pixel size (down to nanoscale) also creates the possibility of developing a high-performance, fast, and highly-integrated polarization detector. Such a detector can be of particular interest for photon-starved polarization resolving and imaging applications with high spatial and temporal resolution.
2. Concept and methods
2.1 DoFP design for SNSPDs
In general, the polarization state of light will be transformed after the light linearly interacts with an optical system (the sample). The transformation process is described by
where Sin and Sout are the light’s input and output Stokes vectors, and M is the sample's Mueller matrix [42]. The Mueller matrix is a 4 × 4 real matrix that characterizes the interaction of the input polarization state with the sample. Using the Mueller matrix, we can obtain all polarization-changing properties.To map polarization state using a superconducting nanowire (or SNSPD), we introduced a method analogous to SNSPD. Since the SNSPD is polarization-sensitive, we treated it as a combination of an analog polarizer (AP) and a polarization-insensitive detector (PID), as shown in Fig. 1(a). The transmission axis of the AP (whose angle is denoted by θ, with respect to an x-axis) is parallel to the orientation of the nanowire. Photons transmitted through the AP were detected by the PID. The most commonly detected photons were polarized in parallel with the nanowire. Based on this hypothesis, we can apply existing theories developed for polarizers to the SNSPD. The theoretical details can be found in Supplement 1.
Fig. 1. Illustration of the operating principle of a superconducting nanowire array-based polarimeter. (a) Schematic of the SNSPD interaction with incoming light. The SNSPD can be viewed as a combination of an analog polarizer (AP) and a polarization-insensitive detector (PID). A polar coordinate system is used, where θ is the transmission axis of the AP with respect to an x-axis. The orientation of the nanowire is parallel to the AP’s transmission axis; (b) A superpixel of the array comprises four nanowire pixels oriented at 0°, 45°, 90°, and 135°, respectively; (c) Cross-sectional schematic of the nanowire on a thermally oxidized Si substrate. Incident light is backside illuminated through the substrate to the nanowires; (d) Simulated optical absorptance ${\mathrm{\eta }_\textrm{A}}$ (${\mathrm{\eta }_{\textrm{A}\parallel }}$ for TE mode, ${\mathrm{\eta }_{\textrm{A} \bot }}$ for TM mode) and PER as a function of nanowire width, for the nanowire with a fixed thickness of 7 nm and a fixed pitch of 200 nm. (e) Simulated optical absorptance ${\mathrm{\eta }_\textrm{A}}$ (${\mathrm{\eta }_{\textrm{A}\parallel }}$ for TE mode, ${\mathrm{\eta }_{\textrm{A} \bot }}$ for TM mode) and PER as a function of wavelength, ranging from 0.4-2.0 µm. The geometrical structure of the NbN nanowire used here is 50-nm-wide, 200-nm-pitch, and 7-nm-thick.
To obtain the optical properties of the input polarized light, according to Stokes’ theorem, three important parameters must be measured: light intensity (I), angle of polarization (AoP), and degree of polarization (DoP). The most common method for defining these quantities is to use the Stokes vector S = (S0, S1, S2, S3)T, where the superscript T is the matrix transpose, and Si (i = 0, 1, 2, 3) is the Stokes parameter. Further, the DoP can be divided into the degree of linear polarization (DoLP) and the degree of circular polarization (DoCP). Since DoFP sensors without a 90° phase retarder do not normally detect circular and elliptical polarization, linearly-polarized light is thus incident to the sample, and the Stokes parameter S3 = 0 (i.e., DoCP = 0). At low light levels, the intensity I is proportional to the number of photons per second received by a single-photon detector, i.e., the detector's photon count rate (PCR). The PCR is defined as PCR = TCR - DCR, where TCR (total count rate) is the total count rate recorded by a photon counter when the photon source (signal) is turned on, and DCR (dark count rate) is the count rate recorded when the photon source is turned off.
To obtain the Stokes parameters Si, we must measure Iθ or PCRθ. Then, we construct the DoFP layout using the SNSPD, as shown in Fig. 1(b). A superpixel is composed of four pixels with nanowires oriented at θ = 0°, 45°, 90°, and 135°, respectively. The corresponding PCR of the nanowire is denoted by PCRθ. If there are plenty of coaxial cables installed in the cryostat, the layout of the DoFP structure can be easily expanded to a large-scale array, e.g., a 16-pixel array.
We can express Si in terms of PCRθ:
Notably, in our study, the incident light is backside illuminated. From the perspective of the incident light, the nanowire orientations at 45° and 135° shown in Fig. 1(b) will be exchanged, while the nanowire orientations of 0° and 90° will remain the same. Using the above relationships, we have
Therefore, according to the definitions of AoP and DoLP [1],
The Stokes parameters are often normalized by the value of S0, such that the differences between Iθ and PCRθ will be eliminated in the process of solving the equations for the DoLP and AoP.
It is known that the PCR of the SNSPD is proportional to its SDE (${\mathrm{\eta }_\textrm{S}}$) when the incident photon rate (Rin) is constant (or calibrated), i.e.,
Generally, ${\mathrm{\eta }_\textrm{S}}$ can be expressed [19,20] as
Assuming ${\mathrm{\eta }_{\textrm{I}\; }} \approx \; \textrm{1}$ for all pixels when they are biased at the saturated plateau of the current-dependent SDE, Eq. (7-1) can be rewritten as
For each run of the SDE measurement, the${\; }{\mathrm{\eta }_\textrm{C}}{\; }$is constant, and the SNSPD obtains a maximum absorption efficiency (${\mathrm{\eta }_{\textrm{A}\parallel }}$) or maximum PCR (PCRmax) when the light polarization is parallel to the nanowire (i.e., TE mode), and a minimum absorption efficiency (${\mathrm{\eta }_{\textrm{A} \bot }}$) or minimum PCR (PCRmin) when the light polarization is perpendicular to the nanowire (i.e., TM mode). Then, we can further define PER in terms of PCR or ${\mathrm{\eta }_\textrm{A}}$ as
2.2 SNSPD finite element simulation
We simulated the optical absorption of the nanowire on a SiO2/Si/SiO2 substrate (268-nm SiO2, 400-µm Si) by varying the nanowire's geometrical parameters. The simulation was performed using commercial finite element software (Comsol Multiphysics, RF module) [37]. Figure 1(c) shows a cross-sectional schematic of the nanowire used in the simulation. To minimize computational time, the thickness of the Si layer in the simulation was set to 2 µm. Figure 1(d) shows the simulated optical absorptance for TE and TM modes, and PER as a function of the nanowire width, with NbN nanowire pitch and thickness fixed at 200 nm and 7 nm, respectively. As expected, ${\mathrm{\eta }_{\textrm{A}\parallel }}$ (TE mode, red circular dots) was higher than$\; {\mathrm{\eta }_{\textrm{A} \bot }}$ (TM mode, blue square dots) for the same nanowire width. The PER continuously increased with reduction in nanowire width. For 100 nm and 50 nm wide nanowires, the PER was 3.6 and 16.8, respectively (See Supplement 1 for a more detailed analysis of the high-PER nanowire geometrical structures). Figure 1(e) shows the simulated optical absorptance ${\mathrm{\eta }_\textrm{A}}$ and PER as a function of wavelength, in a wavelength range of 0.4-2 µm. The oscillations in ${\mathrm{\eta }_\textrm{A}}$ with the increase of wavelength is induced by the 268-nm-SiO2 coating layer and the geometrical structure of the nanowire used (50-nm-wide, 200-nm-pitch, and 7-nm-thick NbN). The fluctuation of the PER is relatively small around the wavelength of 1550 nm.
2.3 Fabrication and experimental setup
A 7 nm-thick NbN thin film was deposited onto a two-inch thermal-oxidized Si substrate by magnetron sputtering, with 268-nm thick SiO2 on both sides of the substrate. The film was then patterned into a nanowire array using electron beam lithography (EBL, JOEL, 100kV accelerating voltage) and reactive ion etched in CF4 plasma. Each pixel of the array comprised a meandering nanowire structure, with nanowire orientations of 0°, 45°, 90°, and 135°, respectively. Figure 2 presents a typical scanning electron microscopy (SEM) image of the array, with a nanowire width of 45 ± 5 nm and a pitch of 200 ± 5 nm for each pixel. The SEM images indicate that the fabricated nanowires featured good uniformity. The total active area of the array was 16 × 16 µm2.
Fig. 2. (a) SEM image of a superpixel of the division of focal plane (DoFP). Four different nanowires are oriented with a 45° offset. Magnified SEM images of each pixel: (b) Pix1; (c) Pix2; (d) Pix3; and (e) Pix4. All four pixels show a uniform width of 45 ± 5 nm and a pitch of ∼200 nm.
We first screened the fabricated devices in a 2K Gifford–McMahon (G–M) cryocooler system, based on the uniformity of the switching currents (ISW) and the saturation of the current-dependent SDE for all pixels in the array. We defined ISW as the highest bias current that the device could sustain before switching to the normal state. The selected device was then mounted in a 1.5-K free-space-coupled cryogenic system (Cryomech Inc., PT410) for further characterization since testing in a free-space optical path can avoid the drawbacks of SMF. Figure 3 shows a schematic diagram of the experimental setup (See Supplement 1 for more details about the setup and measurements). The inset of Fig. 3 shows a photo of the chip package for an SNSPD array chip, which can support up to 16 output signal channels. To demonstrate our device application for polarization imaging, we built a scanning imaging system based on a motorized X-Y scanning stage [29] (see Supplement 1 for details).
Fig. 3. Schematic of the experimental setup of a free-space coupling SNSPD array used for polarization analysis and calibration, and for SDE measurement. Att-1/2: variable optical attenuator; SMF: single-mode fiber; AS: alignment stage; FC: fiber collimator; LP: linear polarizer; HWP: half-wave plate; BPF: bandpass filter. Inset: optical photo of the SNSPD array chip package (front side). The chip size is 5 × 5 mm2. PCB: printed circuit board
3. Results and analysis
3.1 SNSPD performance in the free-space system
Figure 4(a) shows photon-response pulse waveforms for the four pixels, recorded using an oscilloscope. The pulses are nearly identical, with a pulse-decay time of ∼15 ns (1/e criterion). Figure 4(b) shows the typical timing jitter of a single pixel, with a full width at half maximum of 126 ps for the histogram, at a bias current (IB) of 5.2 µA. This jitter can be further reduced by using a cryogenic amplifier [23] or fabricating nanowires with high switching current [22]. Figure 4(c) provides ISW distribution for the four pixels. The ISW values are within the range 6.7 ± 0.4 µA at 1.5 K, confirming good uniformity of the fabricated nanowire array. Here “± 0.4 µA” is the standard deviation of the values deduced from the four pixels. Figure 4(d) shows the SDE of each pixel as a function of IB, when the photons were polarized parallel to the nanowire at a wavelength of 1550 nm. All pixels demonstrated a well-saturated SDE plateau (for IB ≥5.0 µA), indicating ${\mathrm{\eta }_\textrm{I}}{\; } \approx \; \textrm{1}$. The maximum SDEs of the four pixels are distributed in the range 0.1%–0.5%, implying a misalignment between the light spot and the four-pixel array. The misalignment could be caused by the thermal contraction of the lens holder at low temperatures. This contraction also could lead to the lens losing focus, creating a larger than expected light spot.
Fig. 4. (a) Single-shot waveforms of the photon-response pulse for the four SNSPD pixels, recorded at a bias current (IB) of 5.5 µA. (b) Timing jitter of a single pixel with a 50-nm wide nanowire, measured at IB = 5.2 µA. The red line is the Gaussian fit for the experimental data; (c), switching current (ISW) distribution of the four-pixel array; (d) SDE (solid scatters) and DCR (open scatters) of each pixel as a function of IB.
When all the pixels were biased at 5.5 µA (in the saturated SDE region), the total SDE of the array was ∼1%, and the total DCR amounted to 680 cps. Particularly, the DCR of each pixel was ∼170 cps for a given IB. In our experiment, the measured PCRmax for each pixel was of the magnitude ∼1 × 104 cps, which is nearly two orders of magnitude higher than the DCR, guaranteeing high contrast during data acquisition.
To identify the upper limit of this device's SDE, we re-measured it in a 2-K G-M cryocooler system by coupling the device with a gradient-index (GRIN) lens fiber with backside illumination [37]. The average PER measured using fiber coupling was ∼12, which is slightly higher than that measured with free-space coupling (∼10). However, the total SDE measured with fiber coupling was ∼11%, which is about one order of magnitude higher than the free-space coupled measurement. This result confirms that the relatively low SDE in the free-space system is due to low optical coupling efficiency since only the optical coupling differed in these two measurements.
3.2 PCR versus angle of polarization
Figure 5(a) shows the AoP dependence from the PCRθs for all four pixels, measured at IB = 5.5 µA. PCRθs for the four pixels were recorded when the AoP of the incident linearly-polarized light was rotated clockwise from −90° to 90° with the increment of 10°.
Fig. 5. PCR of each pixel as a function of the angle of polarization (AoP). (a) Raw data of the PCRθ; (b) normalized $\textrm{PCR}_\theta ^{\ast }$ ($\textrm{PCR}_\theta ^{\ast }$=$\frac{{\textrm{PC}{\textrm{R}_\theta }}}{{\textrm{PC}{\textrm{R}_{\mathrm{\theta }\textrm{max}}}}}$). The dashed lines are cosine function fits. The $\textrm{PCR}_\theta ^{\ast }$ maximum and minimum values for the four pixels exhibit a ∼45° shift from each other along the AoP axis.
The maximum PCRθ (PCRθmax) in the PCR (AoP) function was significantly different due to the misalignment between the light spot and the pixel array. Thus, we performed calibration on these data, assuming uniform illumination and uniform ${\mathrm{\eta }_{\textrm{A}\parallel }}$ for each pixel. To achieve this goal, the PCRθmax of each pixel should be identified. This requirement can be realized by normalizing the PCRθ of each pixel by its own PCRθmax. Considering Eq. (7-1), we define a normalized PCRθ for each pixel, $\textrm{PCR}_\theta ^{\ast }$, as
Thus, the difference between PCRθmax of the four pixels is eliminated, while the influence in ${\mathrm{\eta }_\textrm{A}}$ due to variations in AoP (i.e., the polarization information of the incident light) is preserved.
Figure 5(b) provides the $\textrm{PCR}_\theta ^{\ast }$ for each pixel as a function of AoP. Due to the limited PER for each pixel, the minimum $\textrm{PCR}_\theta ^{\ast }$ is ∼0.1. However, it exceeds the ideal value of ∼0. Notably, all four curves are offset by ∼45° due to the physical orientations of the nanowires. To perform quantitative analysis, we fitted the experimental data with cosine functions (indicated with dashed lines). A good fit was obtained with the function expressed below:
3.3 Impact of PER on the measurements of Stokes parameters
To study the effect of PER on the accuracy of the polarization state measurements, we extracted the Stokes parameters of the incident light using a different device with a lower average PER value of ∼3 (see Supplement 1). That device was comprised of a 100-nm wide nanowire array with the same pitch and thickness as the 50-nm wide device used previously.
We then performed an experiment to compare these two devices. Figure 6 shows the measured Stokes parameters (S1, and S2), AoP, and DoLP as a function of the sampling sequence number. Reference data obtained from the commercial polarimeter (red dots) were also plotted. As expected, the 50-nm-wide (relatively-high-PER) SNSPD demonstrated better agreement with the reference values than the 100-nm-wide (relatively-low-PER) device. Particularly, the measured values of S1, S2, and DoLP demonstrated significant sensitivity to PER value. We used the measured DoLP value for illustration purposes since it was the most sensitive one to the changes in PER values. The latter can be due to the DoLP [Eq. (4-2)] calculations involved in all PCRθ measurements. Theoretically, DoLP = 1 corresponds to an ideal linearly-polarized state. As shown in Figs. 6(a) and 6(b), the reference DoLP value was ∼1, indicating that on-demand generation and control of the linearly-polarized state was achieved. For a device with a PER of ∼10 or 3, the measured DoLP had a mean value of 0.88 or 0.65, respectively. Correspondingly, the mean error of the measured DoLP was ∼0.12 or 0.35, respectively. Here we provided analysis about the impact of PER on the accuracy of DoLP, and AoP. Firstly, for DoLP, a low PER would lead to an underestimation of its value, owing to the overestimation of the Stokes parameter S0. According to the reported literature [43], it was found that, with simulation, polarimeters with PER (or contrast ratio called in this reference) of 50, 200 and 500 would result in estimates within 4%, 1% and 0.5% of the true level, respectively. Thus, a higher PER value would ensure more accurate DoLP measurements. Secondly, since the determination of the AoP was independent of parameter S0, thus, AoP was quite robust to the variations of PER. Such behavior was also observed in our experiments as shown in Fig. 6(b) and 6(f), where the measured and reference AoP data was almost overlapped in both cases. The 100-nm-wide nanowire array had a mean AoP error of ∼−5.5°, which was slightly larger than that for the 50nm-wide counterpart. However, even for the case with a PER of ∼10, errors appeared in the measured values of S1 and S2. The latter can be due to the limited PER and the simplified model used. These errors could be reduced in the future using a higher PER values or additional calibrations, such as the inverse Mueller matrix approaches [44,45].
Fig. 6. Measured DoLP, AoP, and Stokes parameters S1 and S2 as a function of sampling sequence number, extracted using two devices with different PER values: (a)–(d) 50-nm wide nanowire array with an average PER of ∼10; (e)–(h) 100-nm wide nanowire array with an average PER of ∼3. The reference data in each case are plotted with red dots.
3.4 Demonstration of polarization imaging
To highlight the potential applications of the polarization-sensitive SNSPD array proposed, we conducted a polarization imaging demonstration at low-light level. We used a telescope to receive the photons reflected by the target and reconstructed the images of the target using the measured PCRs of the detector array. The four-pixel array used in the experiments demonstrated an average PER of ∼6–8 and a total SDE of ∼1%. The PCRs of the pixel detector were at an average level of ∼1-4 × 104 cps, i.e., the received photon flux of the entire detector array was ∼1×105 photons/s. The DCR of each pixel was at a level of 100–300 cps, which ensured a high signal-to-noise ratio (∼two orders). The collected data was processed using commercial software (Matlab). Stokes parameters were derived from Eqs. (4-1) and (4-2). For the images, we also used a contrast-enhanced algorithm based on adaptive histogram equalization [46].
Before scanning the target, we adjusted the polarization direction of the incident light so that the PCR of pixel 4 reached its maximum at the polarization angle of ∼90°, i.e., the polarization of the incident light was 90°. The scan covered an area of 80 mm × 70 mm using 80 × 70 pixels, resulting in a pixel-to-pixel spacing of 1 mm in X and Y directions. The dwell time was set to 2 s per pixel to accomplish each scan. Notably, there were some mis-registration errors, due to the finite size of the light spot as well as the step accuracy of the scanning stage. However, since the image was stitched together by pixels, we can smooth the mis-registration errors using linear interpolation between the adjacent pixels. Then we obtained a smooth image as shown in Fig. 7.
Fig. 7. Scanning polarization imaging at low-light level. (a) Optical photo of the target. The scene includes a metal key, a high-reflective tape, and two polarizer with orthogonal polarization directions placed on the tape. The double white arrows mark the directions of polarization of the two polarizers. Reconstructed sample scenes showing the polarization information, recorded using the polarization-sensitive SNSPD array. (b)–(d) Direct-constructed images. (e)–(g) Contrast-enhanced images. (b), (e) Results of reconstruction with intensity; (c), (f) Results of reconstruction with DoLP; (d), (g) Results of reconstruction with AoP. The dark color (blue) represents the low reflection region (low CR region, e.g., PCR at P1∼700 cps, at P2∼2-7 × 103 cps); the bright color (yellow) represents the high reflection region (high PCR regions, PCR at P3∼1 × 104 cps). The notations P1, P2, and P3 are marked in (b).
Figure 7(a) shows the scene of the target, which includes a metal key, a high-reflective tape, and two polarizers [0° (upper) and 90° (lower), respectively]. Figures 7(b)–7(g) provide the scene images reconstructed with different Stokes parameters (intensity, DoLP, and AoP, shown in three columns, respectively) and two different algorithms (direct-constructed and contrast-enhanced, shown in two rows, respectively). The false red (blue) colors shown in the figures represent the high (low) PCR of the regions, corresponding to the high (low) reflectance regions of the scene.
Figures 7(b) and 7(e) show the images reconstructed with the intensity by two algorithms. Figures 7(c) and 7(f) present the DoLP image with two algorithms. The differences between the intensity and DoLP images were clearly observed at the contour of the upper polarizer (see Fig. 7(c), marked by a red arrow) due to a small height difference in the edge thickness of the plastic wrap (black region of the polarizer shown in Fig. 7(a)). Such a difference is more visible in Fig. 7(f), which was reconstructed with the contrast-enhanced algorithm. Moreover, the difference in the intensity of reflected photons from two different polarization directions of the polarizer are visible in Fig. 7(f) and further distinguished in Figs. 7(d) and 7(g) with the AoP mapping (marked with red and yellow dashed circles). However, the AoP images cannot distinguish the details of highly reflective regions, such as the metal key (not visible in the place indicated by the yellow arrow. Thus, AoP mapping mainly reflects the information of the polarization angle while the information of the contour is relatively unclear.
The comparison of common intensity and polarization mappings (DoLP and AoP) confirmed that polarization imaging provides more detailed information about the target. Particularly, it allows to recognize the overall contour of the object, and distinguish polarization-sensitive details (see polarization imaging for a polarized tester in Supplement 1). We also noticed that photon noise appeared during the data collecting process, which showed as discrete PCR peaks in the collected data and can be removed by a post-processing algorithm. The photon noise raised from the strong reflection of the local bumps of the target. The post-processing algorithm mentioned above was referred to an algorithm that can identify and remove the discrete noise data that was significantly higher than the surrounding data, which can be regarded as a denoising process. The post-processing algorithm was done before the reconstruction of the polarization imaging. It is worth noting that the polarization imaging does not require high PER (∼6–8) and a high SDE. For applications such as photon-starved LiDAR and astronomical observations, the requirements for SDE can be further released. Of course, the current method of moving objects has some limitations, such as being time-consuming. This can be improved by building a moving-mirror scanning system or using a large-scale imaging array. In future, by further reducing the pixel size of the detector as well as the time jitter, it is promising to realize a multidimensional sensing and imaging system, which is capable of realizing polarization imaging and three-dimensional imaging at low-light illumination.
4. Discussion
In this study, we have assumed a uniform illumination and uniform ${\mathrm{\eta }_{\textrm{A}\parallel }}$ to calibrate the measured PCRθ, due to inefficient optical coupling. The uniform illumination means that the light illuminated the whole 0°, 45°, 90°, 135° array, where the received photon numbers were the same among the four pixels. However, in the practical experiment, the non-uniform illumination was observed because of the misalignment between the laser spot and the active area of the whole array. Such non-uniform illumination was also indicated by the SDE measurement for each individual pixel, as shown in Fig. 4(d). The assumption (i.e., uniform illumination) has ignored certain information, such as differences in active areas, wire width, and coupling efficiency between the nanowire pixels. PCRθ normalized under this assumption has produced acceptable experimental results indicating that either the differences mentioned above were quite small, or these physical quantities introduced smaller errors compared to others (PER). We speculate that the PER could be a dominant factor in the simplified model. Moreover, this assumption can be relaxed by improving coupling accuracy or using uniform illumination.
The relationship between the mean error of DOLP and PCR also needs to pay attention. In details, the measurement accuracy would depend on the signal-to-noise ratio (SNR) of the detectors. In this study, signal was related to the PCR and noise was related to the DCR. Empirically, a SNR = PCR/DCR of ≥20 would guarantee a relatively high measurement accuracy, which was easy to be obtained for SNSPDs. However, if the intensity of incident light becomes poor, the PCR of detector would be reduced. When the reduced PCR was comparable to the magnitude of the DCR (e.g., ∼680 cps), the measurement accuracy would be significantly degraded, due to the reduction of the SNR. Correspondingly, this phenomenon (reduction of the SNR) reflected in the measurement is that the cosine behavior of the PCR vs. AoP (i.e., Fig. 5) would become blurred and difficult to distinguish. On the other hand, if the accumulating time becomes shorter, the measurement accuracy would also be degraded, due to the influence of the DCR’s randomness. Please find the Ref. [47] for more quantitative analysis.
An SNSPD-based polarimeter offers several advantages. Firstly, it can avoid misalignment issues introduced by the fabrication process of the CCD image sensor integrated with the wire-grid polarizer array [17], owing to the combination of polarization sensitivity and single-photon detection ability. It is worth noting that, although recent advances have shown that great improvements were achieved in the alignment of the wire grating and the underlying detector [48], it still needs an additional overlay process to complete the integration. Besides, the limited signal-to-noise ratio of the CCD sensors would be a bottle neck for the potential applications at the photon-starved regime. Secondly, the multifunctionality of an SNSPD-based polarimeter would greatly improve system integration. This trend is particularly noticeable in the fields of quantum communication and remote sensing, where polarization coding is required [26]. Thirdly, due to the inherent advantages of the SNSPD itself, SNSPD-based polarimeters promise a high SDE and low DCR, as well as low dead time and low timing jitter.
Future development of high performance, scalable SNSPD-based polarimeter or polarization imager presents multiple opportunities and challenges. Firstly, the SDE of the detector in this study was limited by low optical coupling and absorption efficiency. However, promising results have been obtained and indicated that by integrating Al gratings, a single-pixel SNSPD could achieve simulated [49] and experimental [38] efficiencies of ∼95% and ∼48%, respectively. These values were obtained for parallel polarization with a PER of ∼1.5 × 104 and ∼420, respectively. Thus, the development of a high SDE, high-PER, SNSPD-based polarimeter is feasible with proper optical design. Conversely, previous studies of semiconductor DoFP sensors have shown that the high-PER requirement can be relaxed if careful calibration is applied to the Mueller matrix [Eq. (11) in Supplement 1]. Specifically, it has been shown that a PER of ∼3 is sufficient for polarimetry, although PER >10 is preferable for accurate polarization reconstruction [44]. Thus, a DoFP sensor comprising a conventional SNSPD pixel (typical PER ∼3–4 [20]) may also correctly resolve the polarization state, making the fabrication of a high-performance SNSPD-based polarimeter more feasible. Secondly, increasing the scale of the array would not only help to improve coupling efficiency but may also enable the realization of a polarimetric imager. Of course, a large-scale array would require more complex calibration, e.g., elimination of sampling error due to the instantaneous fields of view of neighboring pixels [50]. However, some of these issues have already been addressed in studies of semiconductor DoFP sensors. Thirdly, integrating a phase retarder into the sensor array could enable the detector to resolve circular polarization, facilitating the development of an SNSPD-based full Stokes parameter polarimeter. There are many types of phase retarders, such as, periodic silicon pillar array [51], crossbar antennas array [52] or array of elliptical polarizers [53]. Here, we think that it is more feasible to a layer of periodic silicon pillar array [51], which is stacked on the top of our DoFP nanowire detectors to form dielectric metasurfaces. The silicon pillar array acts as a quarter wave plate (QWP) which can adjust the phase of circularly polarized light by 90° before being detected by nanowires. Thus, the combination of QWP and SNSPD would realize the distinction of circularly polarized light.
The scale expansion of the SNSPD-based polarimeter is limited by the number of cryogenic coaxial cables installed, which would increase the heat load on the refrigerator. This is also one of the challenges encountered in the development of multipixel SNSPD arrays. Luckily, SNSPD array and cryogenic readout technologies have developed rapidly [39,40,54]. For example, the National Institute of Standards and Technology (NIST) has successfully demonstrated a 32 × 32-pixel SNSPD imaging array [39] based on row-column readout architecture, which reduces the number of the readout cable from 322 to 64. NICT has also recently reported a 64-pixel array [40] based on a single-flux-quantum circuit, which reduces the number of readout cables by adopting cryogenic digital multiplexing. Obviously, these techniques can be directly applied to our SNSPD-based polarimeter. To some extent, the scale of 1024 pixels is sufficient to support the requirements of small-scale imaging applications. Furthermore, the reported SNSPD imaging array [39] has demonstrated a maximal counting rate of ∼900 Mcps and timing jitter of 250–400 ps. These performances are better than those of current CCD or CMOS image sensors [5,12]. With the development of readout technology in the future, the scale expansion of the SNSPD-based polarimeter opens up quite promising perspectives.
5. Conclusion
We have designed and fabricated a linear polarimeter based on an array of SNSPDs. Device performance was verified both numerically and experimentally. The pixel of the array comprised a ∼50-nm-wide NbN nanowire, with a 0.25 filling factor and an average PER of ∼10. The four nanowires were oriented with an offset of 45°, forming a division of the focal plane. By characterizing the device in a free-space coupling cryostat, we confirmed successful detection of the polarization state of a linearly-polarized light at 1550 nm, with a total SDE of ∼1% at a low total DCR of ∼680 cps, and a timing jitter of 126 ps. The mean errors of the measured AoP and DoLP were ∼−3° and 0.12, respectively. We also implemented a polarization imaging demonstration using the proposed detectors. Our result indicates that the development of a high-performance, scalable, single-photon sensitive polarimeter and imager based on SNSPDs is feasible. Utilizing the remarkable advantages of SNSPD devices, the proposed SNSPD-based single-photon linear polarimeter has multiple promising applications at the photon-starved regime, such as free-space remote sensing, astronomical observations, and biomedical imaging.
Funding
National Natural Science Foundation of China (61971409); Youth Innovation Promotion Association of the Chinese Academy of Sciences (2019238); Science and Technology Commission of Shanghai Municipality (18511110202, 2019SHZDZX01); National Key Research and Development Program of China (2017YFA0304000).
Disclosures
The authors declare no conflicts of interest.
Data Availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
References
1. J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. 45(22), 5453–5469 (2006). [CrossRef]
2. F. Snik, J. Craven-Jones, M. Escuti, S. Fineschi, D. Harrington, A. De Martino, D. Mawet, J. Riedi, and J. S. Tyo, “An overview of polarimetric sensing techniques and technology with applications to different research fields,” Proc. SPIE 9099, 90990B (2014). [CrossRef]
3. J. Fade, S. Panigrahi, A. Carré, L. Frein, C. Hamel, F. Bretenaker, H. Ramachandran, and M. Alouini, “Long-range polarimetric imaging through fog,” Appl. Opt. 53(18), 3854–3865 (2014). [CrossRef]
4. M. Dubreuil, P. Delrot, I. Leonard, A. Alfalou, C. Brosseau, and A. Dogariu, “Exploring underwater target detection by imaging polarimetry and correlation techniques,” Appl. Opt. 52(5), 997–1005 (2013). [CrossRef]
5. M. Garcia, C. Edmiston, R. Marinov, A. Vail, and V. Gruev, “Bio-inspired color-polarization imager for real-time in situ imaging,” Optica 4(10), 1263–1271 (2017). [CrossRef]
6. J. Qiu, H. Xia, M. Shangguan, X. Dou, M. Li, C. Wang, X. Shang, S. Lin, and J. Liu, “Micro-pulse polarization lidar at 1.5 um using a single superconducting nanowire single-photon detector,” Opt. Lett. 42(21), 4454–4457 (2017). [CrossRef]
7. L. Xue, Z. Li, L. Zhang, D. Zhai, Y. Li, S. Zhang, M. Li, L. Kang, J. Chen, P. Wu, and Y. Xiong, “Satellite laser ranging using superconducting nanowire single-photon detectors at 1064 nm wavelength,” Opt. Lett. 41(16), 3848–3851 (2016). [CrossRef]
8. A. McCarthy, N. J. Krichel, N. R. Gemmell, X. Ren, M. G. Tanner, S. N. Dorenbos, V. Zwiller, R. H. Hadfield, and G. S. Buller, “Kilometer-range, high resolution depth imaging via 1560 nm wavelength single-photon detection,” Opt. Express 21(7), 8904–8915 (2013). [CrossRef]
9. Z. Li, E. Wu, C. Pang, B. Du, Y. Tao, H. Peng, H. Zeng, and G. Wu, “Multi-beam single-photon-counting three-dimensional imaging lidar,” Opt. Express 25(9), 10189–10195 (2017). [CrossRef]
10. P. Yuan, R. Sudharsanan, X. Bai, P. McDonald, E. Labios, B. Morris, and H. Danny, “Single-photon sensitive Geiger-mode LADAR cameras,” Proc. SPIE 8460, 84601A (2012). [CrossRef]
11. G. Kanbach, A. Stefanescu, S. Duscha, M. Mühlegger, F. Schrey, H. Steinle, A. Slowikowska, and H. Spruit, “OPTIMA: A High Time Resolution Optical Photo-Polarimeter,” in High Time Resolution Astrophysics, D. Phelan, O. Ryan, and A. Shearer, eds. (Springer, Netherlands, 2008), pp. 153–169.
12. R. Marinov, N. Cui, M. Garcia, S. B. Powell, and V. Gruev, “A 4-Megapixel Cooled CCD Division of Focal Plane Polarimeter for Celestial Imaging,” IEEE Sens. J. 17(9), 2725–2733 (2017). [CrossRef]
13. M. Malik, O. S. Magaña-Loaiza, and R. W. Boyd, “Quantum-secured imaging,” Appl. Phys. Lett. 101(24), 241103 (2012). [CrossRef]
14. J. A. Levitt, D. R. Matthews, S. M. Ameer-Beg, and K. Suhling, “Fluorescence lifetime and polarization-resolved imaging in cell biology,” Curr. Opin. Biotechnol. 20(1), 28–36 (2009). [CrossRef]
15. K. Usmani, T. O’Connor, X. Shen, P. Marasco, A. Carnicer, D. Dey, and B. Javidi, “Three-dimensional polarimetric integral imaging in photon-starved conditions: performance comparison between visible and long wave infrared imaging,” Opt. Express 28(13), 19281–19294 (2020). [CrossRef]
16. ID Quantique SA, “ID230 Infrared Single-Photon Detector,” (2019), https://www.idquantique.com/quantum-sensing/products/#infrared_photon_counters.
17. V. Gruev, R. Perkins, and T. York, “CCD polarization imaging sensor with aluminum nanowire optical filters,” Opt. Express 18(18), 19087–19094 (2010). [CrossRef]
18. C. Bruschini, H. Homulle, I. M. Antolovic, S. Burri, and E. Charbon, “Single-photon avalanche diode imagers in biophotonics: review and outlook,” Light: Sci. Appl. 8(1), 87 (2019). [CrossRef]
19. F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photonics 7(3), 210–214 (2013). [CrossRef]
20. W. J. Zhang, L. X. You, H. Li, J. Huang, C. L. Lv, L. Zhang, X. Y. Liu, J. J. Wu, Z. Wang, and X. M. Xie, “NbN superconducting nanowire single photon detector with efficiency over 90% at 1550 nm wavelength operational at compact cryocooler temperature,” Sci. China: Phys., Mech. Astron. 60(12), 120314 (2017). [CrossRef]
21. W. J. Zhang, X. Y. Yang, H. Li, L. X. You, C. L. Lv, L. Zhang, C. J. Zhang, X. Y. Liu, Z. Wang, and X. M. Xie, “Fiber-coupled superconducting nanowire single-photon detectors integrated with a bandpass filter on the fiber end-face,” Supercond. Sci. Technol. 31(3), 035012 (2018). [CrossRef]
22. J. J. Wu, L. X. You, S. J. Chen, H. Li, Y. H. He, C. L. Lv, Z. Wang, and X. M. Xie, “Improving the timing jitter of a superconducting nanowire single-photon detection system,” Appl. Opt. 56(8), 2195–2200 (2017). [CrossRef]
23. B. Korzh, Q.-Y. Zhao, J. P. Allmaras, S. Frasca, T. M. Autry, E. A. Bersin, A. D. Beyer, R. M. Briggs, B. Bumble, M. Colangelo, G. M. Crouch, A. E. Dane, T. Gerrits, A. E. Lita, F. Marsili, G. Moody, C. Peña, E. Ramirez, J. D. Rezac, N. Sinclair, M. J. Stevens, A. E. Velasco, V. B. Verma, E. E. Wollman, S. Xie, D. Zhu, P. D. Hale, M. Spiropulu, K. L. Silverman, R. P. Mirin, S. W. Nam, A. G. Kozorezov, M. D. Shaw, and K. K. Berggren, “Demonstration of sub-3 ps temporal resolution with a superconducting nanowire single-photon detector,” Nat. Photonics 14(4), 250–255 (2020). [CrossRef]
24. W. J. Zhang, J. Huang, C. J. Zhang, L. X. You, C. L. Lv, L. Zhang, H. Li, Z. Wang, and X. M. Xie, “A 16-Pixel Interleaved Superconducting Nanowire Single-Photon Detector Array With A Maximum Count Rate Exceeding 1.5 GHz,” IEEE Trans. Appl. Supercond. 29(5), 1–4 (2019). [CrossRef]
25. Y. Liu, Z.-W. Yu, W. Zhang, J.-Y. Guan, J.-P. Chen, C. Zhang, X.-L. Hu, H. Li, C. Jiang, J. Lin, T.-Y. Chen, L. You, Z. Wang, X.-B. Wang, Q. Zhang, and J.-W. Pan, “Experimental Twin-Field Quantum Key Distribution through Sending or Not Sending,” Phys. Rev. Lett. 123(10), 100505 (2019). [CrossRef]
26. Y. Cao, Y.-H. Li, K.-X. Yang, Y.-F. Jiang, S.-L. Li, X.-L. Hu, M. Abulizi, C.-L. Li, W. Zhang, Q.-C. Sun, W.-Y. Liu, X. Jiang, S.-K. Liao, J.-G. Ren, H. Li, L. You, Z. Wang, J. Yin, C.-Y. Lu, X.-B. Wang, Q. Zhang, C.-Z. Peng, and J.-W. Pan, “Long-Distance Free-Space Measurement-Device-Independent Quantum Key Distribution,” Phys. Rev. Lett. 125(26), 260503 (2020). [CrossRef]
27. H.-S. Zhong, Y. Li, W. Li, L.-C. Peng, Z.-E. Su, Y. Hu, Y.-M. He, X. Ding, W. Zhang, H. Li, L. Zhang, Z. Wang, L. You, X.-L. Wang, X. Jiang, L. Li, Y.-A. Chen, N.-L. Liu, C.-Y. Lu, and J.-W. Pan, “12-Photon Entanglement and Scalable Scattershot Boson Sampling with Optimal Entangled-Photon Pairs from Parametric Down-Conversion,” Phys. Rev. Lett. 121(25), 250505 (2018). [CrossRef]
28. T. Yamashita, D. K. Liu, S. Miki, J. Yamamoto, T. Haraguchi, M. Kinjo, Y. Hiraoka, Z. Wang, and H. Terai, “Fluorescence correlation spectroscopy with visible-wavelength superconducting nanowire single-photon detector,” Opt. Express 22(23), 28783–28789 (2014). [CrossRef]
29. H. Zhou, Y. He, L. You, S. Chen, W. Zhang, J. Wu, Z. Wang, and X. Xie, “Few-photon imaging at 2015 nm using a low-timing-jitter superconducting nanowire single-photon detector,” Opt. Express 23(11), 14603–14611 (2015). [CrossRef]
30. G. G. Taylor, D. Morozov, N. R. Gemmell, K. Erotokritou, S. Miki, H. Terai, and R. H. Hadfield, “Photon counting LIDAR at 2.3 um wavelength with superconducting nanowires,” Opt. Express 27(26), 38147–38158 (2019). [CrossRef]
31. A. Engel, J. J. Renema, K. Il’in, and A. Semenov, “Detection mechanism of superconducting nanowire single-photon detectors,” Supercond. Sci. Technol. 28(11), 114003 (2015). [CrossRef]
32. V. Anant, A. J. Kerman, E. A. Dauler, J. K. W. Yang, K. M. Rosfjord, and K. K. Berggren, “Optical properties of superconducting nanowire single-photon detectors,” Opt. Express 16(14), 10750–10761 (2008). [CrossRef]
33. S. N. Dorenbos, E. M. Reiger, N. Akopian, U. Perinetti, V. Zwiller, T. Zijlstra, and T. M. Klapwijk, “Superconducting single photon detectors with minimized polarization dependence,” Appl. Phys. Lett. 93(16), 161102 (2008). [CrossRef]
34. V. B. Verma, F. Marsili, S. Harrington, A. E. Lita, R. P. Mirin, and S. W. Nam, “A three-dimensional, polarization-insensitive superconducting nanowire avalanche photodetector,” Appl. Phys. Lett. 101(25), 251114 (2012). [CrossRef]
35. J. Huang, W. J. Zhang, L. X. You, X. Y. Liu, Q. Guo, Y. Wang, L. Zhang, X. Y. Yang, H. Li, Z. Wang, and X. M. Xie, “Spiral superconducting nanowire single-photon detector with efficiency over 50% at 1550 nm wavelength,” Supercond. Sci. Technol. 30(7), 074004 (2017). [CrossRef]
36. X. Chi, K. Zou, C. Gu, J. Zichi, Y. Cheng, N. Hu, X. Lan, S. Chen, Z. Lin, V. Zwiller, and X. Hu, “Fractal superconducting nanowire single-photon detectors with reduced polarization sensitivity,” Opt. Lett. 43(20), 5017–5050 (2018). [CrossRef]
37. Q. Guo, H. Li, L. You, W. Zhang, L. Zhang, Z. Wang, X. Xie, and M. Qi, “Single photon detector with high polarization sensitivity,” Sci. Rep. 5(1), 9616 (2015). [CrossRef]
38. R. Xu, Y. Li, F. Zheng, G. Zhu, L. Kang, L. Zhang, X. Jia, X. Tu, Q. Zhao, B. Jin, W. Xu, J. Chen, and P. Wu, “Demonstration of a superconducting nanowire single photon detector with an ultrahigh polarization extinction ratio over 400,” Opt. Express 26(4), 3947–3955 (2018). [CrossRef]
39. E. E. Wollman, V. B. Verma, A. E. Lita, W. H. Farr, M. D. Shaw, R. P. Mirin, and S. Woo Nam, “Kilopixel array of superconducting nanowire single-photon detectors,” Opt. Express 27(24), 35279–35289 (2019). [CrossRef]
40. S. Miyajima, M. Yabuno, S. Miki, T. Yamashita, and H. Terai, “High-time-resolved 64-channel single-flux quantum-based address encoder integrated with a multi-pixel superconducting nanowire single-photon detector,” Opt. Express 26(22), 29045–29054 (2018). [CrossRef]
41. S. Rashleigh, “Origins and control of polarization effects in single-mode fibers,” J. Lightwave Technol. 1(2), 312–331 (1983). [CrossRef]
42. R. A. Chipman, “polarimetry,” in Handbook of Optics, 3rd ed., M. Bass, ed. (McGraw-Hill, New York, 1995).
43. D. Vorobiev and Z. Ninkov, “Design, fabrication and characterization of a polarization-sensitive focal plane array,” Proc. SPIE 9403, 94030A (2015). [CrossRef]
44. J. S. Tyo and H. Wei, “Optimizing imaging polarimeters constructed with imperfect optics,” Appl. Opt. 45(22), 5497–5503 (2006). [CrossRef]
45. S. B. Powell and V. Gruev, “Calibration methods for division-of-focal-plane polarimeters,” Opt. Express 21(18), 21039–21055 (2013). [CrossRef]
46. J. B. Zimmerman, S. M. Pizer, E. V. Staab, J. R. Perry, W. McCartney, and B. C. Brenton, “An evaluation of the effectiveness of adaptive histogram equalization for contrast enhancement,” IEEE Trans. Med. Imag. 7(4), 304–312 (1988). [CrossRef]
47. F. Goudail and A. Bénière, “Estimation precision of the degree of linear polarization and of the angle of polarization in the presence of different sources of noise,” Appl. Opt. 49(4), 683–693 (2010). [CrossRef]
48. Sony Semiconductor Solutions Corporation, “SONY IMX250MZR, IMX264MZR, IMX253MZR,” (2021), https://www.sony-semicon.co.jp/e/products/IS/industry/product/polarization.html.
49. F. Zheng, X. Tao, M. Yang, G. Zhu, B. Jin, L. Kang, W. Xu, J. Chen, and P. Wu, “Design of efficient superconducting nanowire single photon detectors with high polarization sensitivity for polarimetric imaging,” J. Opt. Soc. Am. B 33(11), 2256–2264 (2016). [CrossRef]
50. J. S. Tyo, C. F. LaCasse, and B. M. Ratliff, “Total elimination of sampling errors in polarization imagery obtained with integrated microgrid polarimeters,” Opt. Lett. 34(20), 3187–3819 (2009). [CrossRef]
51. A. Basiri, X. Chen, J. Bai, P. Amrollahi, J. Carpenter, Z. Holman, C. Wang, and Y. Yao, “Nature-inspired chiral metasurfaces for circular polarization detection and full-Stokes polarimetric measurements,” Light: Sci. Appl. 8(1), 78 (2019). [CrossRef]
52. J. Bai, C. Wang, X. Chen, A. Basiri, C. Wang, and Y. Yao, “Chip-integrated plasmonic flat optics for mid-infrared full-Stokes polarization detection,” Photonics Res. 7(9), 1051–1060 (2019). [CrossRef]
53. W. L. Hsu, G. Myhre, K. Balakrishnan, N. Brock, M. Ibn-Elhaj, and S. Pau, “Full-Stokes imaging polarimeter using an array of elliptical polarizer,” Opt. Express 22(3), 3063–3074 (2014). [CrossRef]
54. A. N. McCaughan, J. I. Gammell, D. M. Oh, and S. W. Nam, “A high-density customizable microwave vacuum feedthrough for cryogenic applications,” Rev. Sci. Instrum. 91(1), 015114 (2020). [CrossRef]
