1. Introduction

Automatic polarization control systems are important components for coherent optical receivers, polarization division multiplex receivers, distributed PMD compensators, all-optical regenerators, and other optical components or subsystems where polarization matching is critical. In such systems, momentary polarization mismatch during short periods of time may cause data loss and thus must be prevented. For practical applications, an automatic polarization controller should be capable of tracking even the fastest polarization changes (in the order of 10 µs) in the transmission fiber that have been observed in a recent field trial [1].

In order to increase the spectral efficiency of an optical transmission, the combination of differential quadrature phase-shift keying (DQPSK) with polarization division multiplexing (PolDM) to quadruple the channel capacity is an interesting choice. At a rate of 40 Gbaud this corresponds to a per-channel capacity of 160 Gb/s. Demultiplexing of the two polarization channels in a PolDM receiver however remains one of the most difficult challenges in implementing such a transmission system. A fully automatic polarization demultiplexer needs to track polarization fluctuations during the transmission, so fast that the two polarization channels can be demodulated properly. In addition, at a 40 Gbaud line rate (and beyond) polarization mode dispersion (PMD) becomes a significant transmission impairment. An ideal optical PMD compensator needs several differential group delay (DGD) sections where each section is preceded by or contains a polarization controller [2] that can endlessly transform any input state-of-polarization into a principal state-of-polarization (PSP) of the DGD section [3]. In such a compensator, fast polarization controllers are again required so that the PMD compensator can track the birefringence evolution of the transmission fiber.

This paper presents an endless polarization controller which has a very fast response speed and is thus suitable for polarization demultiplexing and PMD compensation. It endlessly tracks one variable polarization while the other is fixed, either the transmitted state of a polarization beamsplitter or a PSP of a subsequent DGD section. Note that certain applications may require polarizations at both ends to be variable, for example a PMD compensator where the connection between the polarization transformer and the DGD section is time-variable in its polarization transfer characteristics or a mid-span polarization controller. Accurate tracking is achieved by using calibrated lithium niobate waveplates as the polarization transformers. The high-speed operation is enabled by implementing the digital controller in a field-programmable gate array (FPGA).

2. Waveplate calibration

An electro-optic polarization waveplate currently is the most promising solution for a broadband, compact, reliable and responsive polarization transformer [4]. In the following section, a method to characterize and calibrate such waveplates is presented. Note that the procedure can also be applied to sections of a distributed PMD compensator [5]. Using the characterization result, it is possible to find the polarization transformation of the device as a function of applied electrode voltages with a very high degree of accuracy.

The electro-optic polarization transformer using LiNbO₃ (lithium niobate) crystals is shown in Fig. 1 (left). It comprises a waveguide along the z propagation axis on an x-cut substrate with three electrodes (V₁, V₂, V₃). If the middle electrode is connected to ground (V ₃=0), the horizontal field component E_y in the region of the waveguide is induced by V ₁-V ₂, while the vertical field component E_x is induced by V ₁+V ₂. After proper normalization, its phase retardation φ and and eigenmode orientation ψ are determined by [4]

The polarization transformation of the device on the Poincaré sphere is shown in Fig. 1 (right). The eigenmodes are given by the rotation axis Ω. The eigenmodes of the device lie in the S ₁ S ₂ plane. The polarization transformer is thus a linear retarder. From Eq. 1 it can be seen that the eigenmodes can be endlessly rotated even with a limited range of V ₁,V ₂. A circular polarization at the input of the device can be transformed into any elliptical polarization provided that V ₁,V ₂ can introduce a retardation in the range of 0…π.

 

Fig. 1. Structure of an x-cut z-propagation lithium niobate retarder (left) and its polarization transformation on the Poincaré sphere (right).

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A model of electro-optic polarization transformer can be used to determine its operation as a function of electrode voltages, as well as to identify the electrode voltages needed to achieve a specific polarization conversion [6]. On the Poincaré sphere, polarization transformations of retarders are represented by rotations. In the following analysis, we will use quaternions to represent the rotation [7]. The retarder model that is presented here is basically the quaternion model as a function of the applied voltages. Generalizing eqn. (1), suppose that

holds with κ_x, κ_y, V _{o, x}, V _{o, y}, V_π as the model parameters, then the quaternion model of the linear retarder can be written as

If the quaternion components in the above equations are denoted as L₀(V₁, V₂), L₁(V₁, V₂), L₂(V₁, V₂), L₃(V₁, V₂), then each of these components defines a parametric surface as a function of V₁, V₂. An ideal linear retarder will have L ₃(V ₁, V ₂)=0. Compared to other polarization transformer models, this quaternion model has the advantage that it can be used to directly identify the eigenmodes and the retardation for specific electrode voltages. In addition, rather than applying elimination procedures systematically (as in [6]) in order to get the model parameters, it is simpler to characterize the retarder [8] so that the quaternion can be obtained for the whole operation range of the electrode voltages.

Figure 2 shows the characterization result of a lithium-niobate retarder. The four contour plots represent the four quaternion components. The electrode voltages are swept over the range (-56 V, 56 V) with a voltage quantization of 3.75 V. As can be seen from the contour plot of L₃(V₁, V₂), the k component of L₃(V₁, V₂), is not completely zero. However, it is very close to zero especially in the vicinity of the centers of the ellipses formed by the contour lines of L₁(V₁, V₂), where the retardation is small. This shows that although the polarization transformer is not a perfect linear retarder, its non-ideal behavior is found to be small.

Using the quaternion model of a linear retarder (eqn. (5)), the necessary voltages which correspond to a specific retardation and eigenmode can be calculated. Graphically, this is also clear from the contour plots of the quaternion components of the linear retarder (Fig. 2). For example, since a contour line in the contour plot for for L ₀ is the locus for constant retardation, voltages that trace along this contour line rotate the eigenmode and thus operate the linear retarder like a rotating fractional waveplate. Finding V₁, V₂ for the whole operating range of phase retardation φ and and eigenmode orientation ψ yields the surface plots as shown in Fig. 3. These calibration data are then tabularized and stored as look-up tables for ultrafast execution in the FPGA. Because the memory capacity is limited, each look-up table has a coarse resolution (granularity). A fast two-dimensional interpolation is carried out to improve the accuracy.

 

Fig. 2. Characterization result of a lithium-niobate retarder.

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Fig. 3. Calibrated retarder voltages for different retardation and eigenmode orientation.

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3. Controller implementation and experimental results

Several endless polarization control algorithm and experiments have been published in the last two decades [4,8–12]. However, fast polarization stabilization with a guaranteed endless operation while tracking worst-case polarization rotations is rarely demonstrated. This would have been important since tracking a complete set of endless polarization changes, including (and most importantly) the worst-case ones, is much harder and slower [13, 14], and thus limits both the speed of polarization control and tracking accuracy. A key to this is that in the search for perfect polarization matching infinitesimal variations in the input (or output) polarization state must never require more than infinitesimal control voltage variations. This problem has been solved for the first time in endless fiber squeezer and electrooptic polarization control systems in [4].

We have realized an endless polarization control system using the linear retarder algorithm with a calibrated electro-optic waveplate as the polarization transformer. It can endlessly transform variable polarization into a fixed circular polarization or vice versa, a capability which is sufficient for most but not for all applications. Algorithm details can be found in [4]. Several waveplates are used because the retardation of just one plate is too small, and for experimental convenience. An ancillary waveplate transforms the fixed circular polarization into that polarization in the output waveguide which becomes an eigenmode of the PBS. The control algorithm is digitally implemented in an FPGA for a very fast execution. The waveplate calibration data stored in the controller allows the electro-optic waveplate to be treated as a near-perfect linear retarder with linear eigenmodes and endlessly adjustable azimuth angle, by compensating for the potential inaccuracies due to the non-ideal behavior and other fabrication imperfections. In similar fashion as in the electrooptic polarization control system of [4] and in line with [13] we guarantee, through the use of several waveplates, that small polarization variations will never require overly large control voltage variations.

The algorithm was designed to track polarization at a speed of about 0.03 rad per full iteration, under all circumstances and also if the fixed polarization deviates slightly from the desired value. The controller acts on the waveplates electrode voltages and uses a gradient descent optimization to maximize (or minimize) the feedback signal. The feedback signal itself is obtained from a photodetector connected to the polarization beam splitter (PBS) placed after the polarization transformer. If the feedback signal is to be maximized, the other output of the PBS contains the remaining cross polarization which is useful to assess the tracking accuracy. However, when the controller is set to minimize feedback signal, maximum photointensity with stabilized polarization is available at the other PBS output. Concurrently with the gradient algorithm it is checked whether the waveplate retardation exceeds π. If this is the case then the algorithm starts turning the waveplate until the gradient algorithm succeeds in reducing the waveplate retardation below π. Waveplate retardation and orientation determine the device voltages by way of the characterization data. Fig. 4 right shows a block diagram of the algorithm.

Figure 4 left shows the full setup of the polarization stabilization experiment. A set of rotating fractional waveplates, quarter- (QWP) and halfwave (HWP) plates, are arranged before the polarization controller. A 1551 nm optical signal from a laser is passed through the waveplates. The bulk-optic HWP in the middle, inserted between two collimators, rotates at an adjustable rate up to 600 Hz. Since the QWPs rotate at unequal rates between −6 and +6 Hz, this effectively randomizes the orientation and the size of polarization trajectories. The maximum speed of polarization changes realized by this arrangement is 15 krad/s, mainly circles with all possible orientations. This stresses a polarization controller much more than limited or forth-and-back polarization changes.

Between the output of the polarization transformer and the PBS there was normal fiber, probably subject to small thermally induced polarization drift. In a commercial version this fiber could easily be replaced by polarization-maintaining fiber which is more stable than our setup probably is. (A suitable version of the polarization transformer is by the way even commercially available.)

The digital controller running on the FPGA can track a polarization mismatch of ≤1 rad essentially within one control iteration (2 µs), fast enough for high practical speed demands [1]. For analysis purposes, the photointensity error, i.e., the residual power falling upon the photodetector, is recorded every 0.4 µs, also during the dithering steps of the gradient algorithm. The performance of the controller was then analyzed in a series of ≥30-minute tracking experiments, one for each of several different HWP rotation rates. The mean polarization change speed is calculated to be π/4 times the maximum one. During the tracking, the total accumulated polarization changes are therefore >67 Mrad.

 

Fig. 4. Endless polarization control experiment setup (left), and block diagram of control algorithm (right).

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Figure 5 shows the complementary distribution (cumulative density) function 1-F(RIE) of the relative intensity error (RIE), i.e., the probability that the intensity becomes worse than the value given on the abscissa. For example, at a HWP rotation rate of 360 Hz (polarization rotations up to 9 krad/s) the RIE is <0.15% during 99% of the time and it never exceeds 0.4%. The photointensity is subject to measurement errors, as shown in the reference measurement without light. Thus the true results are likely to be better.

Figure 6 shows the polarization mismatch (left scale) inferred from the RIE (right scale) as a function of the fastest polarization transformation changes (horizontal scale) for each HWP speed (inside diagram), for F(RIE)=0 (i.e., <10^-8, no worse intensity sample recorded), <10^-5, and <10^-3. The displayed mean polarization error, based on linear RIE averaging, is 0.061 rad. At 15 krad/s a maximum RIE of 0.48% was observed in the cross polarization, corresponding to a-10·log(1-(<0.0048)) dB=<0.021 dB loss for the maximized signal exiting at the other PBS output. The design goal, 0.03 rad/iteration/2 µs/iteration=15 krad/s is obviously reached. This and the lower-speed data with even smaller tracking errors validate the truly endless operation of the controller.

Between characterization and presented experiments there was a delay on the order of hours. But we see no significant degradation in our most recent experiments (which will be reported later) compared to others performed months ago, even though the same set of characterization data is being used. We observe only small discrepancies between characterization data obtained at different times.

The hitherto fastest endless polarization tracking report by other groups was that of Heismann [9]. In Fig. 4 of [9], 10 periods on the Poincaré sphere were tracked in 20 ms with a mean intensity loss of 4% (-14 dB). The system [9] was designed for two endlessly variable polarizations but in that Fig. 4 one of them is fixed and the other rotates only on one particular Poincaré sphere trajectory.

In contrast, we allow endless polarization variations at only one side of the polarization transformer. Control speed is several times higher, and mean and maximum intensity losses are several times lower. Complete Poincaré sphere coverage including worst-case trajectories is assured by the tracking of several rotating waveplates over >67 Mrad.

In practical applications, control speed may be limited by the fact that the optical signal to be controlled is modulated with a broadband digital signal. This modulation may introduce random “noise” in the feedback signal to the controller and, hence, perturb proper control of the output polarization state. Nevertheless we have demonstrated in [15] that endless polarization control of data signals is possible at krad/s speeds.

 

Fig. 5. Complementary distribution function 1-F(RIE) of relative intensity error (RIE) for polarization tracking of different HWP rotation rates.

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Fig. 6. Tracking error for different polarization changes.

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4. Summary

We have presented a fast and truly endless polarization control system with the digital processing realized in an FPGA. The controller is able to stabilize worst-case 15 krad/s polarization changes, which is the fastest endless control speed reported to date, with a maximum polarization mismatch of only 0.14 rad. The performance and endless operation of the controller were verified in long-term polarization tracking measurements. The controller is suitable for polarization demultiplexers, PMD compensators and coherent receivers.