An Endless Reset-Free Polarization Control Method ... - IEEE Xplore

An Endless Reset-Free Polarization Control Method Using Two Wave Plates Volume 2, Number 6, December 2010 Weifeng Rong Laurent Dupont Yixin Wang Yong-Kee Yeo Jian Chen Wu Kuang Varghese Paulose

DOI: 10.1109/JPHOT.2010.2077275 1943-0655/$26.00 ©2010 IEEE

IEEE Photonics Journal

Endless Polarization Control Method

An Endless Reset-Free Polarization Control Method Using Two Wave Plates Weifeng Rong, 1 Laurent Dupont, 2 Yixin Wang, 1 Yong-Kee Yeo, 1 Jian Chen, 1 Wu Kuang, 1 and Varghese Paulose 1 2

1 RF & Optical Department, Institute for Infocomm Research, Singapore 138632 Department of Optics, TELECOM Bretagne, CS 83818, 29238 Brest Cedex 3, France

DOI: 10.1109/JPHOT.2010.2077275 1943-0655/$26.00 2010 IEEE

Manuscript received August 16, 2010; revised September 8, 2010; accepted September 9, 2010. Date of publication September 10, 2010; date of current version September 28, 2010. Corresponding author: W. Rong (e-mail: [email protected]).

Abstract: We present a new, endless polarization control method based on two rotatable wave plates with variable birefringence. We propose two boundary conditions for the parameters of the wave plates so that this method will not have the pole problem. The pole problem is one of the main weaknesses in reset-free polarization controllers. We experimentally demonstrate the control principle of this method and verify its feasibility by using a simple feedback mechanism. Index Terms: Polarization, wave plate, polarization control.

1. Introduction In recent years, there has been a lot of research on high-speed optical networks, such as 100-Gb/s Ethernet and 160-Gb/s optical transmission network [1], [2]. Some techniques require an endless reset-free polarization controller (PC) to implement polarization division multiplexing. In order to realize such a PC, various endless polarization control methods have been proposed [3]–[11]. The main principle, in these cases, is to obtain the desired state of polarization (SOP) by using a set of polarization control devices such as wave plates. This is also an important step leading to polarization dependent loss optimization and polarization mode dispersion (PMD) compensation [12]. An effective PC should endlessly transform an arbitrary polarization state into another arbitrary one with continuous control parameters. Some researchers have proposed a cascade of wave plates [3], [9], [10], while some have proposed simple architectures to reduce the complexity of control, the number of degrees of freedom (DOFs), and the cost of implementation. However, the cascading of many wave plates, to ensure endless reset-free control, is technically difficult. Furthermore, it is also difficult to assemble three rotatable wave plates in free space to obtain a compact device and to focus a light beam into the aperture of the star-like electrodes [15] of the three plates. Other solutions include using a wave plate with birefringence from 0 to 2 [5] and two wave plates with variable birefringence and rotatable optical axis under certain boundary conditions [7]. These two methods have the same Bpole problem[ [8] for endless control, which is a serious weakness for such control methods and may result in discontinuity. The disadvantage in the first method is due to the SOP passing through the pole on the Poincare´ sphere because the rotation axis cannot deviate from the equator. The second method also has the same problem when the resultant rotation axis, which is formed by the two wave plates, passes through the pole [8]. In this paper, we present a method that uses two rotatable wave plates that have variable birefringence to solve the pole problem. We provide experimental results to verify the feasibility of this method.

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Fig. 1. Concatenation of two rotations on the Poincare´ sphere.

2. Control Principle An endless polarization control method that uses two rotatable wave plates with variable birefringence has been proposed in our previous works [7], [8]. Fig. 1 shows us the control principle of this method. A, B, and C are three SOPs represented on the Poincare´ sphere. A is the input SOP, and C is the output SOP. The first wave plate, with azimuth 1 and birefringence 1 , rotates A to B. The second wave plate, with azimuth 2 and birefringence 2 , rotates B to obtain C, which is our target SOP. ~ to By using quaternion algebra [13], we can just use a rotation of angle  about the axis R associate the two rotations of the wave plates. Hence, these two rotations, from A to B and then from B to C, are replaced by a single rotation from A to C directly. We represent the variables of the quaternion as follows:            1 2 1 2 (1) cos ¼ cos cos  sin sin cosð21  22 Þ 2 2 2 2 2 0   1       sin 21 cos 22 cosð21 Þ þ cos 21 sin 22 cosð22 Þ   B C          B C ^ Rsin (2) ¼ B sin 21 cos 22 sinð21 Þ þ cos 21 sin 22 sinð22 Þ C: @ A 2  1   2  sin 2 sin 2 sinð21  22 Þ This is a method that has four DOFs (1 , 2 , 1 , and 2 ), and it is very difficult to control all four variables to control the polarization. In order to reduce the number of DOF and to obtain a simple control method, we first assumed that the resultant rotation is always 180 , i.e.,  ¼ 180 . Then, there are two possibilities to achieve polarization control with just two DOFs: 1) 1 ¼ 2 ¼  and  ¼ 180 ) 2sin2 ð=2Þcos2 ð1  2 Þ ¼ 1, i.e., the two wave plates have the same birefringence. In this method, we have to control the variables  and 1 (or 2 ). 2) 2 ¼ 180 and  ¼ 180 ) 2ð1  2 Þ ¼ 90 , i.e., one of the wave plates is fixed as a halfwave plate. In this method, we have to control the variables 1 and 1 (or 2 ). Under these conditions, the resultant rotation axis may go around the entire sphere, and it may be displaced continuously with different configurations if the variables of the two wave plates are changed continuously. However, there are two singularities: poles, because the transition of the resultant rotation axis by the pole is not continuous. Fig. 2 shows us an ideal case where the rotation axis traverses the pole on the Poincare´ sphere. When the axis is moved from P1 to P2 , a discontinuity of 180 appears in the Stokes space, which corresponds to a physical jump of 90 . This requires the PC to reset. We have previously proposed to define a small exclusion area around the pole, according to the optical system specification, to avoid the pole problem [8]. This has to lead to a detour during the polarization control in relation to optimal PMD compensation and has to be manifested by a non-optimal degree of polarization. On the other hand, this detour decreases the control speed, which introduces another problem in highspeed optical transmission networks.

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Fig. 2. Example of the pole problem on the Poincare´ sphere [(b) is the S1 OS2 plan of (a)].

TABLE 1 Rotation angle and axis depending on 

In order to solve the pole problem, we vary the resultant rotation angle . Based on the second assumption above, we assume two conditions: • 2 ¼ 180 , i.e., one wave plate is fixed as a half-wave plate. •  ¼ 21  22 ¼ 1 =2  90 , i.e., relate the angle between two axes on the sphere to the birefringence of the second wave plate. From the four solutions of , we can obtain the various rotations as shown in Table 1, where 0 1 cosð22 Þ A ~ r ¼ @ sinð2 21Þ : sin 2 ^ where  is the resultant rotation angle, and R ^ is the resultant rotation We define a vector ~  ¼ R, axis according to the quaternion algebra, to represent all rotations on a surface, as shown in Fig. 3(a). When sinð=2Þ is positive, i.e., when S2 9 0, and  varies continuously as per its four solutions, vector ~  changes along the track B   a ! b ! c ! d ! e ! f ! a   ,[ as shown in Fig. 3(b). When sinð=2Þ is negative, i.e., when S2 G 0, we will obtain the same result. Hence, we will only consider the case of sinð=2Þ 9 0 in this paper. We set ð; Þ as the unit spherical coordinates of ~ in which  is the azimuth and  is the latitude, and we have the rotation axis R, 0 1  1  sin B C 2 (3)  ¼ 22 and  ¼ arcsin@qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  A 2 1 1 þ sin 2 where the sign of  is decided in Table 1. We notice that  can be controlled continuously by 2 , and hence, we have to find out a method to continuously vary the parameter . Fig. 4 shows that the parameters of this rotation depend on the variable . We find a discontinuity in  from Bc[ to Bd,[ which matches the discontinuity in Fig. 3(b). In fact, this discontinuity does not disturb the continuous control because we have a null rotation ð ¼ 360 ,  ¼ 0 Þ here. Hence, an endless polarization control is obtained in the given interval by controlling the parameter  continuously.

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Fig. 3. Surface of the vector ~  and its side elevation on the S2 S3 plane (unit: radian). (a) 3-D presentation of Vector ~ . (b) 2-D presentation of Vector ~  on the S2 S3 plane.

From our previous two assumptions [7], [8], the pole problem occurs when the resultant rotation ~ is just on the poles, i.e., the birefringence of the two wave plates are fixed at 90 , and the axis R axes of the wave plates are orthogonal on the Poincare´ sphere ð21  22 ¼ 90 Þ. However, the axes of the wave plates are not orthogonal in the Stokes space under the new boundary condition, because 21  22 ¼ 1 =2  90 . Now, we discuss two cases: • 21  22 ¼ 90 , i.e., the point b or e in Figs. 3 and 4. Obviously, there is no discontinuity induced in this case. • 1 ¼ 2 ¼ 90 , i.e., the point a=f or c=d in Figs. 3 and 4. According to our discussion above, there is no discontinuity induced here too since  ¼ 0 or  ¼ 360 . Therefore, we can finally conclude that this method provides us with a real endless reset-free polarization control without the pole problem.

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Fig. 4. Control parameters depending on  (unit: degree).

Fig. 5. Experimental setup.

3. Feedback Algorithm and Experiment In order to experimentally verify the control method, we use the configuration shown in Fig. 5. A PC, which is a combination of two wave plates which have rotatable axes and variable birefringence, will be tested. The data acquisition and control are implemented in the control algorithm layer. In this system, the feedback signal is obtained from the intensity measured by a photodiode after a polarizer. In the Poincare´ sphere representation of polarization, the transmission through a perfect polarizer is ^ Iout 1 þ s^  p ¼ 2 Iin

(4)

where Iin and Iout are the intensities before and after the polarizer, s^ is the polarization of incident ^ is a unit vector that specifies the pass axis of the polarizer [14]. In this paper, we use light, and p Iu ¼ Iout =Iin as the normalized intensity. The feedback algorithm flowchart is shown in Fig. 6. This control algorithm is divided into two parts: searching and holding. Here, It is defined as the normalized intensity threshold, which is the required intensity and has a value of 0.9 here. This signal is used to control the program alternating between the searching algorithm and holding the axis. The searching process cannot be interrupted until the normalized intensity ðIu Þ exceeds the intensity threshold ðIt Þ. In the searching part, the control algorithm should modify the rotation axis by tuning the variables of the wave plates to find a satisfactory axis, which allows the photodetector to receive a reasonable amount of intensity. In the holding part, the algorithm should maintain all of parameters of the device. During the searching part, we define A ¼ ð; Þ as the vector of the axis parameters and ni as the factor of the increment i ¼ ð; ÞT , and ni ¼ ðcosði=2Þ; sinði=2ÞÞT , i ¼ 0; 1; 2; 3 . . .. The flowchart of the searching algorithm is shown in Fig. 6, where Ii is the measured output intensity ðIout Þ of the ith iteration. When Iiþ1 is more powerful than Ii , then we take Ai þ ni i as Aiþ1 , and if Iiþ1 G Ii , then we take the opposite direction Ai  ni i as Aiþ1 . The searching process should

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Fig. 6. Feedback algorithm flowchart including searching part and holding part.

Fig. 7. Example of the searching process with four iterations.

be implemented until Iu 9 It , and the increment i is dynamic and should be recalculated for each step. In our article [8], we have obtained the error function between the increment of axial evolution  (we choose the same increment here for axial azimuth  and latitude  to simplify the calculation, e.g.,  ¼  ¼ ) and Iu : 

3 pffiffiffiffiffiffiffiffiffiffiffiffiffi 1  Iu : 8

(5)

According to (5), in theory,  is reduced to zero when Iu is equal to 1. However, in a real optical transmission system, it is not possible for Iu to approach 1. Due to this reason, an intensity threshold It should be defined. Fig. 7 shows an example of the searching process with four iterations to understand this method. The points, which are depicted by stars in the figure, are points chosen by

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Fig. 8. Received normalized power and PC parameters versus iterations.

the searching process. We first initialize n0 , 0 , A0 , and I0 . Then, we choose A0 þ n0 0 as A1 and measure the received intensity. In this example, we assume I1 9 I0 so that A0 þ n0 0 is chosen for the next axis. In the next step, we assume I2 9 I1 so that A1 þ n1 1 is chosen for the next step. In the third step, we assume I3 G I2 so that A2  n2 2 is chosen for the next step, and we do a similar thing in the fourth step. This process will be terminated once Iu is greater than It . Finally, experiments are carried out according to the setup as shown in Fig. 5. Matlab and LabVIEW programs are used in the control algorithm layer for data acquisition and controlling the driver of the PC under test. The PC is made of two polymer network-liquid crystal (PN-LC) rotatable wave plates with variable phase shift ð0  Þ. The PN-LC has star-like electrodes. By applying different voltage on each electrode, both axis orientation and birefringence can be controlled by changing the amplitude and the phase of the electric field, respectively. Such a liquid crystal wave plate has a response time of about 0:3 =s [15]. The experimental results are shown in Fig. 8. During the experiment, we change the input SOP by using a mechanical PC. This change can be seen in Fig. 8 by the abrupt drop of Iu . When this happens, the control algorithm immediately enables the searching part, and we observe that the feedback algorithm, under the boundary conditions, quickly finds the optimum axis in fewer than 10 iterations. After the controller finds the satisfactory axis, the algorithm switches back to the holding part and waits for the next change in input SOP. From the parameters curve, we notice that the parameters are controlled continuously to obtain the required optimized intensity. In these experiments, we have used the error function [see (5)]. In order to have a reasonable increment of the wave plates’ parameters during the control, we can suitably modify the parameters of the error function on the basis of the requirements of the system.

4. Conclusion We have presented an endless, reset-free, polarization control method with two wave plates which can completely solve the pole problem. We have also demonstrated a simple and efficient feedback algorithm under two boundary conditions. The experimental results have shown that it is a quick and effective SOP control method, which can be applied to a PC and other devices that are based on PCs, such as the polarization scrambler, the polarization stabilizer, the polarization dependent loss optimizer, and the distributed PMD compensator.

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