Polarization variations in installed fibers and their influence on quantum key distribution systems

1. Introduction

Quantum key distribution (QKD) is an advanced technology for sharing secure keys between remote parties. Its security is ensured by the laws of physics rather than unproven mathematical complexity [1, 2]. Quantum mechanics guarantee that any eavesdropping activities will be detected. These unique advantages make QKD a prime secure communication technology. Since Bennett and Brassard proposed the famous BB84 protocol [3], further advances have been made both in theory and experiment [4–11]. As the first commercial application of quantum physics at the single-quantum level [1], QKD has already come out from laboratories to field deployments over telecom networks [12–16]. In most research on QKD, the channel is rarely involved, since it is supposed to be controlled by Eve. But when we deploy QKD systems over the installed fiber channels, the polarizations variations in the channel would severely affect the performances of polarization-sensitive QKD systems, which include the polarization encoded BB84 and the phase encoded BB84 based on asymmetric Mach-Zender interferometer [17], the continuous variable [15, 18] and recent measurement-device-independent [10, 19] (MDI) systems. And so far, most QKD systems are polarization-sensitive, using polarization multiplexing method [15, 17, 18] can obtain relatively high key rate, and indistinguishability of polarization from two senders is primary requirement for MDI systems [10, 19]. For these polarization-sensitive QKD systems to achieve stable and good performance, polarization basis alignment (PBA) modules are essential part [10, 17, 19–26].

Polarization variations depend on both the length and the external environment of the fiber channel. According to the length, the installed fibers can be divided into the metropolitan ones and intercity ones. The installed fibers, can also be classified into buried fibers and aerial fibers on the point of the external environment. In previous experiments, we obtained a large amount of recorded data about polarization variations of different installed fibers, including buried fibers and aerial fibers with different length.

In this paper, we want to establish an analytical methodology to quantitatively evaluate the influence of polarization variations on polarization-sensitive QKD systems, based on the recorded data about polarization variations of installed fibers. We first build a model to evaluate the increased quantum bit error rate (QBER) induced by polarization variations. Then using the increased QBER as the key criteria, we propose two parameters - polarization drift time and required tracking speed - to characterize polarization variations. For different field installed fibers scenarios, we quantitatively evaluate the influence of polarization variations, and provide requirement and suggestions for PBA modules of QKD systems.

2. Influence of polarization variations on QKD systems

We use the typical polarization encoded BB84 protocol as an example, and consider one basis for simplification. Suppose at the initial time t₀, the polarization reference between Alice and Bob is aligned perfectly, and QBER equals to zero. And the increase of QBER is only induced by the polarization variation of the fiber channel. At time t₀, the output state of the fiber is $|{\varphi }_{t_0}〉$, while the state becomes $|{\varphi }_{t_0+t}〉$ after time t rose by polarization variation. Then we have

In our experiment to test polarization variations in fiber channels, one c.w. laser at 1550nm (81960A, Agilent Technologies) was used as the source launched into the fiber under test and a lightwave polarization analyzer (LPA, N7788B, Agilent Technologies) recorded the output state of fiber under test (FUT) with a sampling rate of 2.5 kHz. This rate is high enough to catch any burst changes of polarization. The recorded data of LPA is under the Stokes parameters representation with four parameters (S₀, S₁, S₂, S₃)^T [27], which represent the polarization state on the Poińcare sphere. Since the recorded degree of polarization (defined as $\sqrt{\left(S_1^2+S_2^2+S_3^2\right)}/S_0$) was always close to 1, the three-parameter Stokes vector (S₁, S₂, S₃)^T is sufficient to represent the polarization state.

Suppose the Stokes vector of state $|{\varphi }_{t_0}〉$, ${|{\varphi }_{t_0}〉}_{\perp }$ and $|{\varphi }_{t_0+t}〉$ are ${\overrightarrow{S}}_{t_0}$, ${\overrightarrow{S}}_{t_0}^{\perp }$ and ${\overrightarrow{S}}_{t_0+t}$ respectively. Without loss of generality, the equation $U{\overrightarrow{S}}_{t_0}={\left(1,0,0\right)}^T$ and $U{\overrightarrow{S}}_{t_0}^{\perp }={\left(-1,0,0\right)}^T$ always be satisfied with a unitary transformation U. So $U{\overrightarrow{S}}_{t_0+t}={\left({\left|\alpha \right|}^2-{\left|\beta \right|}^2,2Re(\alpha {\beta }^{*}),2Im(\alpha {\beta }^{*})\right)}^T$ at meanwhile according to the definition of Stokes vector [27], then we have

PBA modules are essential for polarization-sensitive QKD systems to minimize the influence of polarization variations in fiber channels, or in other words, to reduce the increased QBER introduced by polarization variations. And all proposed PBA modules fall into two categories: the interrupted PBA modules start to work when the increased QBER is beyond the threshold value [20–23], and the real-time PBA modules always work to keep the increased QBER below the threshold value [19, 24, 25, 28, 29]. Here we establish two parameters - polarization drift time and required tracking speed - for these two categories, respectively. For QKD systems employing interrupted PBA modules, the polarization drift time represents the uninterrupted operation time of QKD procedure. For QKD systems adopting real-time PBA modules, the required tracking speed reflects the probability of increased QBER below the threshold value.

2.1. Polarization drift time τ

For the interrupted PBA modules, Alice and Bob start to align the polarization basis, if they find the increased QBER is beyond the threshold value e_th, and stop the QKD procedure (see Fig. 1(a)). We can use the following equation to estimate the drift time τ:

 

Fig. 1 The sketch map of two categories of PBA modules. (a) The interrupted PBA modules; (b) The real time PBA modules.

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2.2. Required tracking speed υ

For the real-time PBA modules, Alice and Bob align the polarization basis and distribute quantum keys at the same time (see Fig. 1(b)). For different QKD systems, relative stronger reference light [24, 25, 28], or weak data signals [19, 29] were used as feedback of the PBA modules. As shown in Fig. 1(b), the PBA module keep the increased QBER introduced by polarization variation below a given value e_th. Here we pay more attention to the polarization variation speed, which is defined as

Based on above model, the polarization drift time and the required tracking speed for each fiber are analyzed, also some special phenomena are introduced. Furthermore, requirements and suggestions for PBA modules in different fiber link scenarios are given.

3. Buried fibers

3.1. Buried fibers under test

The sketch map of fields buried fiber is presented in Fig. 2. There are 8 nodes on these fibers which connect three cities: Hefei, Chaohu and Wuhu. These fibers belong to the Anhui provincial telecommunication fiber network of China Mobile Ltd. and can be divided into three parts: (1) The metropolitan links belong to Hefei, which has 5 nodes, one node is in Wan-Tong Post and Telecommunication (WTPT) Co. Ltd., the other 4 nodes are located in campus of University of Science and Technology of China (USTC). Among these nodes, both Library (Lib) and Key Laboratory of Quantum Information (KLQI) are located in east campus of USTC, other two nodes locate in the North campus (NC) and West campus (WC) of USTC respectively. (2) The metropolitan links belong to Wuhu, which have two nodes that are located in the Telecom Room (TR) of China Mobile and Asky Quantum Technology Co. Ltd. (Qasky) respectively. (3) The intercity lines from Hefei to Wuhu, which go through the node lies in the Chaohu Branch (CHB) of China Mobile. Such backbone links should run along highways for easy installation and timely convenient maintenance. And the Chaohu to Wuhu inter-city link, in particular, crosses the Changjiang river along the Wuhu Changjiang Bridge, which combines railway and highway.

 

Fig. 2 Overview of buried fiber networks.

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There are 7 optical fiber links among 8 nodes in the field buried fiber network. All the 7 fiber links are characterized as Table 1, including the length of fibers, the date, the time period and the environment temperature. The weather is sunny during our test period. The length of metropolitan links is below 30 km, and the length is over 60 km for intercity fibers. As the laser travels through the FUT, the Stokes parameters were recorded at the end of the FUT by LPA.

Table 1. Characteristics of all buried fiber links under test.

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3.2. Polarization drift time

For each buried fiber, we first calculated the mean value of polarization drift time with given thresholds of increased QBER introduced by polarization variations. The analysis results are shown in Table 2, with three given threshold equals to 1%, 3% and 5%, respectively. Obviously, if the QKD system could tolerate higher QBER, its uninterrupted duration of key distribution procedure would be longer. And it is an approximate linear relationship between the uninterrupted duration of key distribution procedure and tolerant increased QBER for the same fiber channel.

Table 2. The mean value of polarization drift time 〈τ〉 (in s) of buried fiber over given threshold of e_th of increased QBER introduced by polarization variations.

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In terms of polarization drift time, the polarization variations depend not only on the length of fiber, but also on the external environment of the buried fiber. First, the polarization variations accumulate with fiber length. In metropolitan areas (Hefei and Wuhu cities), the length of fiber is less than 30 km, the polarization drift time is in orders of magnitude slower than 50 s. While, in intercity areas (Hefei-Chaohu and Chaohu-Wuhu), the length of fiber is longer than 60 km, the polarization drift time is in orders of magnitude faster than a few seconds. Second, the external environment significantly affects the polarization variations in the installed fiber. For fibers among campuses of USTC, the polarization drift time of the fiber link Lib-NC is about twenty times longer than the one of the fiber link Lib-WC at e_th = 1%, although the lengths of these two fiber links are similar. For fibers in metropolitan areas, the fiber length of TR-Qasky link is much longer than the length of WTPT-KLQI link, but the polarization drift times of these two links are reverse. And the effect of external environments on polarization variations in the intercity fibers is more dramatic compared with fibers in metropolitan areas. Taking the links Hefei-Chaohu and TR-Qasky for example, the length of Hefei-Chaohu link (85.1 km) is only about four times of the length of TR-Qasky link (23.3 km), while the polarization drift time of Hefei-Chaohu link is more than four hundred times faster than the one of TR-Qasky link. The reason for this difference is that the intercity fiber links are installed along the highway, and the traffics induced the dramatic polarization variations in the fiber.

We also notice that sudden polarization variation happened hundreds of times during our test period in the two intercity fiber links, but such sudden variations rarely occur in the metropolitan links. The sudden polarization variations of the Hefei-Chaohu link is shown in Fig. 3(a). For clarity, only parameter S₁ of the Stokes vector is given, and details of part of one typical sudden polarization variation is shown in the inset of Fig. 3(a), which exhibit the periodicity in the sudden polarization variation duration, there are fifteen peaks in two seconds. The mean value of sudden polarization variation duration is 98.8 s for Hefei-Chaohu link, and 57.1 s for Chaohu-Wuhu link. Then, we calculated the power spectral density (PSD) of sudden polarization variations, the result is shown in Fig. 3(b). The fundamental PSD peaks concentrate at the frequency range from 0.5 Hz to 50 Hz. In terms of duration and fundamental frequency, these sudden polarization variations are not likely induced by the slow change of temperature or fast electromagnetic fields. The intercity fiber links are installed along highways, and the Chaohu-Wuhu links crossed the Changjiang river along the Wuhu Changjiang Bridge. The couplings between traffic-induced vibrations and fiber birefringence must be inevitable, and vibrations with such durations and fundamental frequencies can be produced by automobiles [30, 31] and trains [32, 33], so these sudden polarization variations should come from the mechanical vibrations induced by vehicles.

 

Fig. 3 Sudden polarization variations in inter-city links. (a) The time series of the Stokes parameter S₁ within 3400 s of the CHB-TR link. Inset: 2 seconds time series of S₁ of sudden polarization variations. (b) PSD analysis of sudden polarization variations.

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3.3. Required tracking speed

For PBA modules the statistical results of the required tracking speed υ with given thresholds of increased QBER introduced by polarization variations are shown in Table 3. Here, the tracking speed υ is given with the value of 〈P_sp〉 + 3σ. Obviously, if the QKD system could not tolerate higher QBER, the required tracking speed of PBA module is faster, which is consistent with previous results of the polarization drift time. But the approximate linear relationship no longer exists. Taking the WTPT-KLQI fiber link for example, the requirement on the tracking speed have very small differences for three given values of e_th.

Table 3. The required tracking speed υ (in rad/s) over a given threshold of QBER e_th.

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Compared with the polarization drift time, the required tracking speed has weaker correlation with the length of the fiber, but stronger correlation with the external environment of the buried fiber. The WTPT-KLQI fiber link is shorter than the TR-Qasky link, while the required tracking speed of the WTPT-KLQI link is generally dozens of times that of the TR-Qasky link.

3.4. Suggestions for PBA modules

In terms of polarization drift time and tracking speed, our quantitive analysis results of the polarization variations show different requirements on PBA modules of polarization-sensitive QKD systems. For the interrupted PBA modules, the procedure of polarization basis alignment should be completed in a few seconds in metropolitan areas, and should be completed in several milliseconds in intercity areas to guarantee the time efficiency of the whole system. For the real-time PBA modules, its tracking speed should be no less than a few rads per second in metropolitan areas, and should reach dozens of rads per second in intercity areas, in order to keep relatively low increased QBER induced by polarization variations.

To meet these requirements, the temporal characteristic of polarization controllers which is the key components of PBA modules is important. And, for QKD systems, the insertion loss of the polarization controller should be taken into account, since most PBA modules were placed at Bob’s side. In the metropolitan areas, the polarization controller in a PBA module is suggested to use multiple PZT fiber squeezers, which have the response time in the order of dozens of microseconds [34], can track polarization variations at the speed of close to 10 rad/s [34, 35]. And, these all-fiber squeezers have very low insertion loss. In the intercity areas, the suggested polarization controller is electro-optic wave plates, like LiNbO₃ polarization controllers, whose response time is about 100 ns. The only restriction of this type polarization controller is its relative high insertion loss.

4. Aerial fibers

4.1. Aerial fibers under test

The aerial fiber link locates in the State Grid Corporation of China (SGCC) ultra-high voltage direct current Test Base [36], its satellite image is shown in Fig. 4. The aerial fibers are integrated into one optical fiber composite ground wire (OPGW) parallel (not strictly) to the DC transmission line. The OPGW line can protect the power lines from lightning strike and short circuit currents, and allow data transmission through fibers. This line or single aerial fiber is 1.78 km in length, and a pair of aerial and buried fibers make up a fiber loopback. The section view of OPGW is shown in Fig. 4, and five fibers of the OPGW cable were used in our experiment. Connecting 5 loopbacks in the OPGW, we got a field aerial fiber with the length of 8.9 km.

 

Fig. 4 The sketch map of the polarization variations test of aerial fiber in ultra high voltage (UHV) direct current test yard of China. The rated voltage of the UHV direct current source are ±800 kV, and the tower height is up to 70 m [36]. Laser, 1550nm c.w. laser ; LPA, lightwave polarization analyzer.

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Considering the aerial fiber is exposed in the air, the wind induced vibration will rise the polarization variation more severely [37, 38] than the buried fibers. So we did other simulation experiments on indoor aerial fibers in the Ultra-high Voltage Engineering Mechanics Test Base in Liangxiang of Beijing, for getting more detailed understandings of the unique polarization characteristics and their influence on QKD systems. The polarization variations under two common wind-induced cable vibrations, galloping and vortex shedding, have been tested.

4.2. Polarization variations of the field aerial fiber

The temporal variations of Stokes parameters in the field aerial fiber are shown in Fig. 5. The continuous test lasted about 16 hours, from 4:00 PM the previous day to 8:00 AM the next day. During this period, although the weather got chilly at night, the wind was moderate, and there was no rain and no snow. In short time periods, Stokes parameters vary in a relatively small range but with high frequencies. Compared with the buried fibers, the outdoor aerial fiber would couple more strongly to external environments, including (not exclusively) the wind (or rain) induced vibrations and temperature fluctuations [37, 38]. In Fig. 5, from 6:00 AM to 8:00 AM, when the sun rose, the temperature changed dramatically, and the corresponding state of polarization in the aerial fiber varied sharply.

 

Fig. 5 Temporal variations of Stokes parameters (S₁, S₂, S₃) in the field aerial fiber.

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The polarization drift time and required tracking speed of the field aerial fiber are shown in Fig. 6. Although the length of the aerial part of the field fiber is less than 9 km, the polarization drift time and required tracking speed of the PBA module are equivalent to those of the intercity buried fibers. We also estimated PSD of the polarization variations, and found two types of peaks. One of peaks are numerous continuous ones from 0.3 Hz to 3 Hz, and mainly come from the moderate wind [38]. The other type of peaks are relatively weak and discrete ones at about 50 Hz and its integral multiples, and are are mainly due to Faraday effect of residual AC component of UHV direct current [37, 38].

 

Fig. 6 The polarization drift time 〈τ〉 and the required tracking speed υ on PBA modules for aerial fiber over different threshold e_th of QBER.

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4.3. Wind-induced vibration of aerial fibers

To get more detailed understandings of the unique polarization characteristics and their influence on QKD systems, we did simulation experiment over indoor aerial fibers under two typical wind-induced vibrations: galloping and vortex shedding.

4.3.1. Galloping

Galloping is a common phenomenon for aerial fibers [37–39]. It may occur when the temperature hovers around or below 0 °C, caused by a moderate or strong wind with a speed of 25 to 65 km/h. The frequency of fiber vibration causes typically ranges from merely 0.08 to 3 Hz, which is much lower compared with vortex shedding. But the amplitude is larger, which can reach 300 times cable diameter.

Two OPGW cables with the length of 75 m form the fibers in the simulation experiment. One cable was shaken with the galloping simulation equipment, the other one was static on the floor. Different length fibers were obtained by concatenations of several fibers in these two cables. The temporal variations of Stokes parameters for different length fibers (75 m, 150 m, 300 m, 825 m) are shown in Fig. 7. The frequency and amplitude of the galloping simulation equipment are 1 Hz and 600 mm respectively.

 

Fig. 7 Temporal variations of Stokes parameter (S₁, S₂, S₃) under the galloping induced vibration with different length of aerial fibers. The value (75 m, 150 m, 300 m, 825 m) is the length of aerial part of the test fiber.

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Periodic variations of Stokes parameters are observed in each subfigure of Fig. 7, and their amplitude grow as the length of aerial fiber increases. Similar conclusions can also be found when we estimate the galloping induced vibration on polarization-sensitive QKD systems.

For a given threshold of increased QBER e_th = 3%, we calculated the mean polarization drift time 〈τ〉 and required tracking speed υ on PBA modules for different length of aerial fibers, the results are shown in Fig. 8. When the length of the aerial fiber is 300 m, the galloping-induced polarization drift time is less than 0.05 s, and the required tracking speed on PBA modules is 17.02 rad/s. Considering the accumulation effect with fiber length, the galloping-induced polarization variations are really challenging for polarization-sensitive QKD systems.

 

Fig. 8 The polarization drift time 〈τ〉 and the required tracking speed υ on PBA modules for different length of aerial fibers, the threshold of increased QBER is set at 3%.

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4.3.2. Vortex shedding

Vortex shedding [37–39] is resonant, small-amplitude vibration caused by steady, and low-velocity wind blowing across cables under mechanical tension. It causes optical cable to resonate at frequencies up to 150 Hz and amplitude around 0.01 to 1 expressed in cable diameter depending on the specifics of the installation. For long spans of the round cable installed at high mechanical tensions such as OPGW, this type of vibration is the most common phenomenon.

The fiber we used in the vortex shedding simulation experiment is the same as the one in the previous galloping simulation experiment. Here, the length of the aerial fiber is 300 m, and the length of the fiber on the floor is also 300 m. The frequency and amplitude of the vortex shedding experiment are 40 Hz and 2.55 mm respectively, which corresponds to the wind speed of approximate 12.6 km/h.

Temporal variations of Stocks parameters in the simulation experiment are shown in Fig. 9. The amplitude of variations is relatively small. And in the inset figure, 40 Hz frequency induced by the vortex shedding can be obviously observed. We also calculated the increased QBER introduced by the vortex shedding, and found the increment is less than 1 %. But nevertheless, we should pay more attention on polarization variations induced by the vortex shedding. First, as previously verified, the polarization variations accumulate with length of fiber. Here the length of the aerial fiber under test is only 300 m, the amplitude of the variations would increase hundreds of times in the field aerial fiber scenario. Second, the frequency in the simulation experiment is far from the resonant frequency of the OPGW cable, if it is close to the resonant frequency, the amplitude of the vibration would increase dramatically. Third, the frequency of the polarization variations induced by vortex shedding is relatively high, which would make it rather difficult to align the polarization reference between remote Alice and Bob.

 

Fig. 9 Temporal variations of Stokes parameters (S₁, S₂, S₃) in the vortex shedding simulation experiment with 300 m aerial fiber. Inset: details of S₁ parameter during 0.5 second.

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4.4. Suggestions for PBA modules of QKD systems

Considering the variation accumulation effect with the length of fiber, the interrupted PBA modules are not suitable for polarization-sensitive QKD systems over aerial fibers. For the field aerial fiber with the length of less than 9 km, the uninterrupted time of QKD procedure is about 5.54 s with a given threshold of increased QBER e_th = 3%. And with the same e_th, the uninterrupted time falls to 0.48 s when 300 m aerial fiber was in the galloping situation. Moreover, the frequency of vortex shedding could reach 150 Hz. These aerial conditions leave very little time for the procedure of key distribution, since the switching process between key distribution and polarization basis alignment also needs time.

Real-time PBA modules are recommended for polarization-sensitive QKD systems that are deployed over field aerial fibers. In order to keep relatively low increased QBER induced by polarization variations, the required tracking speed of the real-time PBA modules should reach several hundred rads per second at least. So we suggest that LiNbO₃ polarization controllers would be used in the real-time PBA modules.

5. Conclusions

We establish an analytical methodology to quantitatively evaluate the influence of polarization variations in fiber channels on polarization-sensitive QKD systems. Using the increased QBER introduced by polarization variations as the key criteria, we propose two parameters (polarization drift time and required tracking speed) to characterize polarization variations based on recorded Stocks parameters of LPA, and also provide requirements and suggestions for PBA modules of QKD systems in field-deployed scenarios. Our conclusions are as follows.

Based on the length and external environment, polarization variations of installed buried fibers can be roughly divided into two regions: metropolitan areas, and intercity areas. In metropolitan areas, the length of fiber is less than 30 km, the polarization drift time is in orders of magnitude slower than 50 s, and required tracking speed should reach a few rads per second. The interrupted or real-time PBA modules with multiple PZT fiber squeezers can meet these requirements. In intercity areas, the length of fiber is longer than 60 km, the polarization drift time is in order of magnitude 1 s, and required tracking speed should reach dozens of rads per second. Considering sudden polarization variations, real-time PBA modules with LiNbO₃ polarization controllers are suggestions for polarization-sensitive QKD systems deployed over intercity fiber links.

For installed aerial fibers, the polarization variations couple more strongly to external environments compared with buried fibers. When the length of aerial part of the field fiber is less than 9 km, the polarization drift time and required tracking speed are equivalent to those of the intercity buried fibers. Considering galloping with large amplitude and vortex shedding with high frequency on aerial fibers, real-time PBA modules with LiNbO₃ polarization controllers are recommended for polarization-sensitive QKD system deployed over field aerial fibers.

In addition to test of polarization variations in installed fibers and evaluation their influence on polarization-sensitive QKD system, we also inspected performance of the Faraday-Michelson (FM) phase encoded BB84 QKD system [40] over the installed buried and aerial fibers. In theory, the FM QKD system is insensitive to polarization variations of the channel, and doesn’t need the PBA module. For installed buried fibers, the performance of FM QKD system and polarization variations were tested over the same fiber at different time. Several FM QKD systems were deployed over the wide area network (including metropolitan and intercity areas), and tested for more than 5000 hours (from 21 Dec. 2011 to 19 Jul. 2012). The results are shown in Ref. [16]. Taking the FM QKD system over the Hefei-Chaohu intercity link for an example, the QBER of the signal state was consistent around 1.16% during the whole test period, although the external environment has changed a lot. For aerial fibers, the performance of FM QKD system and polarization variations were tested over different fibers in the same cable at the same time. The same QKD system was employed over the field aerial fiber, and also over the simulation indoor fibers. Taking the FM QKD system over field aerial fiber for an example, the QBER of the signal state kept around 1.39%, even though the polarization state varied sharply from 6:00 AM to 8:00 AM. And in the galloping simulation experiment, the performance of FM QKD system remained almost the same when we started the galloping simulation equipment and then stopped the equipment. Our inspection and test demonstrated that polarization variations of channels have little influence on FM QKD systems.

There is also one notable polarization insensitive QKD scheme - reference frame independent (RFI) QKD [41], which can be implemented without frame alignment. And the RFI QKD scheme has been demonstrated over polarization encoded system [42], phase encoded system [43], even MDI system [10, 44]. These demonstrations show practical that RFI scheme would be another approach to polarization variations in installed fibers.

In a word, our work is believed to be helpful for evaluation channel disturbance on polarization-sensitive quantum communication systems. The results of two proposed parameters provide primary data for the design of PBA modules. Through the analytical methodology is on BB84 system, our results would also be useful for other polarization-sensitive systems.