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Abstract
The characterization and manipulation of polarization state at single photon level are of great importance in research fields such as quantum information processing and quantum key distribution, where photons are normally delivered using single mode optical fibers. To date, the demonstrated polarimetry measurement techniques based on a superconducting nanowire single photon detector (SNSPD) require the SNSPD to be either highly sensitive or highly insensitive to the photon’s polarization state, therefore placing an unavoidable challenge on the SNSPD’s design and fabrication processes. In this article, we present the development of an alternative polarimetry measurement technique, of which the stringent requirement on the SNSPD’s polarization sensitivity is removed. We validate the proposed technique by a rigorous theoretical analysis and comparisons of the experimental results obtained using a fiber-coupled SNSPD with a polarization extinction ratio of ∼2 to that obtained using other well-established known methods. Based on the full Stokes data measured by the proposed technique, we also demonstrate that at the single photon level (∼ -100 dBm), the polarization state of the photon delivered to the superconducting nanowire facet plane can be controlled at will using a further developed algorithm. Note that other than the fiber-coupled SNSPD, the only component involved is a quarter-wave plate (no external polarizer is necessary), which when aligned well has a paid insertion loss less than 0.5 dB.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Polarization is a fundamental property of light [1]. Owing to the recent development of fabrication techniques [2,3] and computational algorithms [4,5], the discipline of measuring light polarization, i.e., polarimetry, has undergone tremendous changes and has become a highly valuable tool for medical imaging [6], chemical spectroscopy [7], remote sensing [8], quantum information processing [9], and quantum key distribution [10]. One of the key components involved in these polarimetry applications is the photon detector. For cases where the detected optical power is moderate, photodiodes or CCD arrays are normally employed. However, for faint light applications such as quantum information processing and quantum key distribution, these detectors are normally not good options. Candidates that fit into the faint light scenario include avalanched photodiodes or superconducting nanowire single photon detectors (SNSPDs). Owing to advantages such as high system detection efficiency (SDE), fast speed and low timing jitter [11–14], SNSPDs have been considered to be a choice better than the avalanched photodiodes. This is particularly true for the fiber-optics-based quantum key distribution applications that operate in the infrared region, such as 1550 nm.
To date, two research groups have reported polarimetry works that employ SNSPDs [15,16]. Sun et al. demonstrated the first prototype of a polarimeter based on a four-pixel superconducting nanowire array that is intended to be highly sensitive to the polarization state of the received photons [15]. However, owing to the limited polarization extinction ratio (PER) of their fabricated device (only ∼10 instead of ∼100), the accuracy of their reported polarimeter is not satisfying. Moreover, we note that their demonstration was realized in a free-space optic setting and cannot be directly transferred to the case of fiber-optic settings, since in the latter situation, the coupling fiber wrapped on the cooling stage introduces a significant amount of unknown birefringence that is absent and thus not considered in the former free-space case. Recently, Hu et al. also demonstrated a fiber-optics-based full-Stokes polarimetric measurement system that can measure the arbitrary state of polarization of faint light [16]. The key to their success is the use of a specially designed SNSPD that is highly insensitive to the polarization state of the received photons. Since the SNSPD is now polarization insensitive, the unknown birefringent effect of the coupling fiber plays no role in the measurement results and therefore can be neglected. We note that while the demonstrated result is quite pleasing, the fabrication process of their polarization-insensitive SNSPD requires a tremendous amount of exploration and correction of the fractal nanowire layout and lithography exposure dose, and therefore is generally difficult to complete.
In this article, we report the development of a polarizer-free polarimetry measurement technique that employs a fiber-coupled SNSPD. Different from the previously reported work, the polarimetry technique proposed in this work does not require the involved SNSPD to be either highly sensitive (PER ∼100) or highly insensitive (PER ∼1) to the polarization state of the received photon, thereby allowing an SNSPD designed mainly for the optimized system detection efficiency to be used [12]. We first provide a rigorous theoretical treatment that covers the principle of operation and error analysis. We next compare the full Stokes data measured by our technique using an SNSPD with a PER ∼2 with the full Stokes data measured by other well-established polarimetry methods and show that these two sets of experimental data agree well with each other. Finally, we demonstrate the capability of full polarization control of fiber- delivered single photons, by showing that their polarization states can be adjusted to be either parallel or perpendicular to the superconducting nanowire using a further developed algorithm.
2. Method
The operation principle of the proposed polarizer-free full Stokes vector measurement technique is sketched in Fig. 1, which consists of a rotatable quarter-wave plate placed in front of a fiber-coupled SNSPD with a finite polarization extinction ratio (PER). Note that the operation principle of our technique is similar to that of the well-known rotational quarter-wave plate technique [17,18] but with two major differences. Firstly, owing to the birefringent effect of the coupling fiber wrapped on the cooling stage, the polarization state of the incident photon undergoes a non-negligible modification, which is absent in the rotational quarter-wave plate method and needs to be calibrated and subtracted. Secondly, different from the rotational quarter-wave plate method, in this work, since the PER of the SNSPD is finite, an effect of nonideal polarizer is created and needs to be taken into account.
Fig. 1. Operation principle of the proposed polarizer-free full Stokes vector measurement technique that employs a fiber-coupled and polarization-sensitive SNSPD. QP: quarter-wave plate.
In the supplemental document, by following a treatment similar to that of the rotational quarter-wave plate method [17,18], we show that the averaged count rate of the SNSPD reads as follows:
where θ denotes the orientation angle of the slow axis of the quarter-wave plate, andNote that owing to the redundancy of Eq. (2), i.e., solving 4 unknown variables from 5 combined equations, we cast Eq. (2) into four sets of 4 × 4 linear equations. These four sets of equations are as follows:
In the Supplement 1, we show that for a fiber-coupled polarization-sensitive SNSPD, at least one of the four determinants of the above four linear equation sets is nonzero. This indicates that the linear equation set described by Eq. (3) is mathematically solvable, and hence, the full Stokes vector can be obtained. An exception occurs for the case in which the coupling fiber of the SNSPD effectively acts as a quarter-wave plate and its slow axis offsets the superconducting nanowire by 45 degrees. However, we argue that such an exception case occurs rarely in experiments and can always be overcome by placing a compensation waveplate in front of the SNSPD’s fiber input port.
We summarize this section by emphasizing that for a fiber-coupled polarization-sensitive SNSPD, of which the TE and TM detection efficiencies are not identical, it is always possible to measure the full Stokes vector by calibrating the system first, rotating the quarter-wave plate eight times, collecting the corresponding data and postprocessing.
3. Results
We validate the above theoretical analysis by comparing the full Stokes vector measured by our fiber-coupled SNSPD-based technique with the full Stokes vector measured by other photodiode-based well-established methods. Figure 2 describes the measurement setup of our proposed polarimetry technique. It consists of a pulsed light source (NPI: Rainbow 1550) followed by an attenuator A1 as a first-stage attenuator. The output light is then directed to the series of two U-benches (Thorlabs: FBP-A-FC). The first U-bench1 consists of a quarter-wave plate (QP, Thorlabs: FBR-AQ3) and a half-wave plate (HP, Thorlabs: FBR-AH3), which functions as a polarization state generator. The polarization states of incident photons can be modified arbitrarily by rotating these two wave plates. The second U-bench2 consists of a quarter-wave plate, which functions as a polarization state analyzer along with the fiber-coupled SNSPD. After passing through these two U-benches, the light is coupled to a fiber-coupled beam splitter with a splitting ratio of 50:50. One branch of the splitter connects to a power meter (Thorlabs: S155C) and monitors the real-time optical power incident onto the SNSPD, while the other branch is directed toward the second attenuator A2. By setting a proper attenuation value, the light is attenuated to the single-photon level and sent to the fiber-connected SNSPD, which has a measured PER of ∼2 at 1550 nm. The reason to use a pulsed light source instead of a continuous-wave (CW) light source is because the case of CW source does suffer from signal count instability caused by partial reflections. When measuring the full Stokes vector using our technique, we first tuned the attenuator A1 until the display of the power meter was 10 nW (-50 dBm). Then, the light is further attenuated to -100 dBm by setting the attenuation value of attenuator A2 to -50 dB.
Fig. 2. Schematics of the measurement setup using the proposed method. Note that for the classical measurement, the QP is used together with a linear polarizer in U-bench2. PSG: polarization state generator; PSA: polarization state analyzer; QP: quarter-wave plate; HP: half-wave plate; RPM: reference plane of measurement.
As elaborated in the Supplement 1, we first calibrate the fiber-coupled SNSPD system using the four-state method [19] and determine the values of a, b, c, φb and t, p, r, q. Then, for a given input polarization state defined at the reference plane of Fig. 2, we set the axis angle of the quarter-wave plate to be nπ/8 (with n being an integer ranging from 1 to 8) and measure the SNSPD’s corresponding count rate eight times. By plugging these data into Eq. (S4) in the Supplement 1, the values of A, B, C, D, and E are obtained experimentally. Finally, based on the above obtained values of t, p, r, q and A, B, C, D, E, we numerically solve Eq. (3) and obtain four sets of the computed Stokes vectors. Such a process is repeated 18 times, with each time the input polarization state is prepared randomly by setting the axis angles of the waveplates mounted on U-bench1 in a random manner. The measured 18 Stokes vectors are presented in Fig. 3(a). Note that while in general the four computed solutions agree well with each other, deviations between them are identifiable. This indicates that some of the computed Stokes vectors may contain significant errors.
Fig. 3. Results of the polarimetry measurement. (a) Computed full Stokes vectors by solving the four equation sets of Eq. (3). (b) Refined results of (a) and the full Stokes vectors measured by the classical method.
In the Supplement 1, we show that the error of the computed full Stokes vector is proportional to the conditional number of the matrix appearing on the left side of Eq. (3). In general, the smaller the conditional number is, the lower the measurement error. Since the four matrices of Eq. (3) have already been determined experimentally, we numerically evaluate their conditional numbers of the 18 prepared polarization states. The results are summarized in Tab. S1 of the Supplement 1. It can be seen that the conditional number for different matrices of Eq. (3) varies significantly, suggesting that the errors of different computation methods will also vary significantly, in agreement with the fluctuation shown in Fig. 3(a). Based on the calculated matrix conditional number table, for each prepared polarization state, we selected the computation result that has the minimum matrix conditional number. Finally, the refined data of the computed full Stokes vectors with minimized error are summarized in Fig. 3(b).
To validate the Stokes vectors measured by our proposed technique, we measured the full Stokes vectors of the 18 prepared polarization states by the well-established four-step classical method [18]. The setup of the classical method is similar to Fig. 2 except that the quarter-wave plate is used together with a linear polarizer on U-bench2, and we use the photocurrent measured by a fiber-optic power meter as the measurement raw data. It is well-known that for the four-step classical method to work correctly, the photodetector and the photon delivery path directly in front of the photodetector need to be polarization insensitive. To satisfy this requirement, before we take the four-step classical measurement, we measured the polarization dependent detection efficiency of the fiber-optic power meter together with the fiber-optic coupler. We find that the measured polarization dependence is negligible (∼ 0.03%). Using the four-step classical method, i.e., by measuring the intensity of the vertically polarized, horizontally polarized, 45 degree linearly polarized and circularly polarized components of the incident light, the full Stokes vectors of the 18 prepared polarization states are obtained experimentally. The results are plotted in Fig. 3(b), together with the refined results obtained by our proposed method. An excellent agreement between these two categories of measurement results can be identified, indicating a successful experimental validation of our proposed polarimetry measurement technique.
To investigate the sensitivity of the SNSPD-based polarimeter, we measured the Stokes parameters as a function of the incident power with the polarization state fixed. The results are shown in Fig. 4, together with that obtained by the classical method for reference. It can be seen that the deviation remains steady within the power range from -90 dBm to -110 dBm, while it increases after further decreasing the input power. This trend is attributed to the signal-to-noise-ratio (SNR) of the SNSPD [20], which can be estimated by SNR = CR / DCR, where CR is the count rate of incident photons and DCR denotes the dark count rate. In our measurement, the working current of SNSPD is Ib = 7.5 μA, where the dark count rate is ∼100 cps. The SDEs of SNSPD for TE- and TM- polarized photons at Ib = 7.5 μA are 26.9 ± 0.38% and 14.6 ± 0.59%, respectively. In more detail, the measured counts for TE- and TM- polarized photons are ∼210283 and ∼113906, respectively, for incident power of -100 dBm with integration time of 1 s. For input power greater than -110 dBm, the signal counts are over ∼10,000, and the corresponding SNR > 50. However, when the power is further reduced to ∼ -115 dBm, the signal count rate would be ∼ 3000 and the SNR is ∼ 30. It is hence believed that the sudden deviation of the measured Stokes parameter is caused by the significant deterioration of the SNR at the input power of ∼ -115 dBm.
Fig. 4. Stokes parameters as a function of the incident power measured by the proposed method (red solid dots). For reference, the Stokes parameters obtained using the classical method are also shown (black circles).
We summarize this section by pointing out that in our polarimetry measurement setup, there is no need for an external polarizer. The only extra component is a quarter-wave plate, which, when mounted on a fiber bench and optically well-aligned, introduces a paid insertion loss of less than 0.5 dB.
4. Discussion
The polarimetry measurement technique proposed in this article is not only a promising candidate for single photon polarimetry measurement but also benefits several research fields, such as quantum information processing and quantum key distributions. Using quantum key distribution as an example, we note that owing to the large birefringence of the transmission optical fibers (typically ∼100 km), the polarization states of the photons at Bob’s end deviate significantly from that at Alice’s end and undergo slow fluctuations due to environmental perturbations. To the best of our knowledge, the deviations and fluctuations of the photon’s polarization states are normally compensated by manually or electrically scrambling the polarization states until a desired criterion is satisfied [21]. With the help of the proposed polarimetry technique, we show that the polarization state at the facet plane of the superconducting nanowire can be controlled at will using a further developed algorithm. We demonstrate such a capability by showing that the polarization states of the received photon can be aligned to be either parallel or perpendicular to the superconducting nanowire of the fiber-coupled SNSPD.
Referring to Fig. 5(a), we aim to adjust the polarization state of the incident photon defined at the input plane of U-bench2 (RP of Sin) in such a way that when the photons reach the SNSPD facet plane, their polarization states are either parallel (TE) or perpendicular (TM) to the nanowires. Based on the birefringent model of the fiber-coupled SNSPD presented in the Supplement 1, it is straightforward to show that at the input plane of the SNSPD’s fiber input port (RP of Smax,min), the Stokes vectors needed for the maximum (TE) and minimum (TM) SNSPD count rates should read as:
Fig. 5. Demonstration of polarization control at an input power of ∼ -100 dBm. (a) Schematics of the polarization control experiment. (b) Comparison of the detection efficiencies measured by the algorithm-produced TE and TM states and that measured by the four-state polarization algorithm (FSPA) method [19]. PSG: polarization state generator; PSC: polarization state controller. QP: quarter-wave plate; HP: half-wave plate; RP: reference plane.
The system described in Fig. 5(a) can be modeled using the Mueller matrix [1], where we have Smax,min = MQP2·MHP·MQP1·Sin, where MQP1, MHP, and MQP2 are the Mueller matrices of the corresponding waveplates. With some mathematical manipulations, one finds that to align the polarization of the received photon to be either parallel or perpendicular to the superconducting nanowire, the axis angles of the waveplates, i.e., θQP1, θQP2, θHP, need to be:
Note that in Eq. (6), the definitions of S21 and S22 for both cases of parallel and perpendicular alignments are:
Moreover, relating to the definition of S11 and S12, for the case of parallel alignment, we have:
Note that in Eq. (8) and Eq. (9), γ = arctan (c/(a - b)).
Equations (5)–(9) give the axis angles of the waveplates for the polarization alignment of the received photon to be either parallel or perpendicular to the superconducting nanowires. To validate the proposed algorithm, we pick four prepared polarization states (15∼18) presented in Fig. 3 and plug their measured Stokes vectors into Eqs. (5)–(9) to calculate the axis angles needed for their TE and TM alignments. We then carefully adjust the optical power of the four prepared polarization states to reach the single photon level (∼ -100 dBm), set the three waveplates of U-bench2 using the axis angles calculated above, and measure the corresponding detection efficiencies. The measured results are summarized in Fig. 5(b). For comparison, we also measured the maximum and minimum detection efficiencies of the same fiber-coupled SNSPD using the technique described in Ref. [19] and plotted them in Fig. 5(b). These two categories of measured detection efficiencies agree well with each other, suggesting a successful validation of the single photon level polarization-control technique using the algorithm based on Eqs. (5)–(9).
We summarize this section by pointing out that the triple-waveplate setup used for polarization control is compatible with the single-waveplate setup used for polarization measurement, since for the latter case QP1 and HP can be set in a known condition and hence their contribution to the Stokes measurement can be deduced. In other words, one could use the triple-waveplate setup to measure and control the polarization states at the same time without rearranging the waveplate settings. Moreover, although the current demonstration is done manually, it is possible that the measurement and control of the polarization states can be done in an automatic manner by employing rotatable waveplates that are driven by step-motors and controlled by computers. We hence believe that the demonstrated technique could be helpful for several research fields, such as quantum information processing and quantum key distributions.
5. Conclusion
In summary, we have presented the development of a polarizer-free full Stokes vector measurement technique that employs a fiber-coupled SNSPD with a finite PER of ∼2. Using a rigorous theoretical analysis, we show that the proposed technique is capable of measuring the polarization state of the fiber-delivered photons. We also show that by rotating a quarter-wave plate eight times, collecting the data, postprocessing and refining the results, accurate full Stokes vectors can be obtained. As a demonstration of our proposed technique, we show that it is possible to align the polarization state of the incident single photon (∼ -100 dBm) to the TE or TM states of the SNSPD, using algorithm-based operations rather than by polarization state scrambling searches. The insertion-loss cost for completing the proposed polarimetry technique is estimated to be less than 0.5 dB.
Funding
Innovation Program for Quantum Science and Technology (2021ZD0301701, No. 2021ZD0303401); National Natural Science Foundation of China (62071214, 62071218, 62101240, Nos. 12033002); Natural Science Foundation of Jiangsu Province (BK20210177); Key-Area Research and Development Program of Guangdong Province (2020B0303020001); Priority Academic Program Development of Jiangsu Higher Education Institutions; Jiangsu Provincial Key Laboratory of Advanced Manipulating Technique of Electromagnetic Wave.
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.
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