Optical and Quantum Electronics 35: 1367–1379, 2003. 2003 Kluwer Academic Publishers. Printed in the Netherlands.
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Polarization mode dispersion and chromatic dispersion compensation by using a three-stage compensator YUAN ZHENG, XIAOGUANG ZHANG* AND BOJUN YANG Department of Physics, School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China (*author for correspondence: E-mail:
[email protected]) Received 2 August; accepted 28 October 2003 Abstract. A three-stage compensator used for Polarization mode dispersion (PMD) and Chromatic dispersion (CD) compensation is proposed. The compensator is capable of compensating the two components of second-order PMD when no CD exists. Two operating points of compensating second-order PMD have been proposed. Two-stage and three-stage compensators are compared by the outage probability. When CD is introduced, the compensator should retain a quantity of second-order PMD to compensate CD. The outage probability when PMD and CD coexist has been calculated. The results show that a tradeoff must be made in order to compensate CD and PMD at the same time. Key words: chromatic dispersion, chromatic dispersion compensation, optical fiber communication, polarization mode dispersion, polarization mode dispersion compensation
1. Introduction Polarization mode dispersion (PMD) and chromatic dispersion (CD) are the main problems in high-speed optical communication systems. The compensation techniques for PMD (Noe et al. 1999; Rosenfeldt et al. 2001) and CD (Kikuchi et al. 1995; Eggleton et al. 1999) have been discussed separately. Generally, a PMD compensator consists of several units composed of polarization controllers and birefringent fibers. Dispersion-compensation fibers (DCFs) or fiber Bragg gratings (FBGs) can be used as a fixed amount of CD compensators, whereas chirped fiber gratings (Eggleton et al. 1999) or two birefringent fibers connected by a polarization controller (Kikuchi 2001) can be used as adaptive CD compensators. Generally, two kinds of dispersions coexist in the fibers laid early. Schemes have been proposed to compensate them separately, but very few are capable of compensating them at the same time. Here we have proposed a compensator composed of two fixed DGD segments and one variable DGD segment to compensate them together. The first two DGD segments are capable of compensating
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polarization dependent chromatic dispersion (PCD) and the principal states of polarization rotation rate (PSPRR) when compensating PMD and they are also capable of adaptive CD compensation without PMD. The variable segment is used to compensate the DGD induced either by the first two segments when compensating CD or by both the PMD fiber and the first two segments when compensating PMD. It should be emphasized that no extra second-order PMD is introduced in this process.
2. Theory of three-stage compensation scheme In this part, three issues will be discussed, one is the PMD compensation mechanism without CD, and another is CD compensation mechanism without PMD. At the end of this part, we will consider the compensation when PMD and CD coexist.
2.1.
THREE-STAGE COMPENSATOR FOR SECOND-ORDER PMD COMPENSATION
2.1.1. Limitation of PCD on variable first-order PMD compensation To study the impacts of PCD combined with DGD and PSPRR on the bit error rate (BER) of the systems employing variable first-order PMD compensators, we establish a 40 Gb/s NRZ system numerical model similar to that in Zheng et al. (2003) but exclude the effects of ASE noise because BER also varies with ASE noise. By using the approximation given by Forestieri and Vincetti (2001) and averaging the PMD vector orientation, we obtain the numerical results shown in Fig. 1, where each curve is a border of BER ¼ 10)12 for different groups of PCD, PSPRR and a fixed DGD.
Fig. 1. Relationship among DGD, PCD and PSP rotation rate when BER ¼ 1 · 10)12.
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BER > 10)12 if PCD and PSPRR fall into the right area of the curve; otherwise BER < 10)12. If ASE noise is considered, the curve will shift to the left and lose its smoothness, but the trend will remain unchanged. Fig. 1 shows that compensating either PSPRR or PCD before first-order compensation can reduce BER. Additionally, it is shown that the maximum PCD is limited to 180 ps2 at BER ¼ 10)12, even if DGD and PSPRR are very small. From the analyses above, to improve the PMD tolerance of systems, PCD should be compensated.
2.1.2. Analysis of three-stage PMD compensation The output PMD vector of a fiber can be expressed as (Gordon and Kogelnik 2001) ~ s ¼ s^ q, where DGD s ¼ j~ sj and PSP orientation ^q ¼ ~ s=j~ sj. Secondorder PMD vector is given by ~ sx ¼ sx ^q þ s^qx , where PCD sx ¼ ds=dxjx¼0 , q=dxjx¼0 and ^q ? ^qx . Second-order PMD vector of PSP rotation rate ^ qx ¼ d^ the two concatenated fibers 1 and 2 is given by (Gordon and Kogelnik 2001) h i ~ q1 þ s1 ^ q1x þ ðRy2~ s2 Þ ðs1 ^q1 Þ stotx ¼ R2 s1x ^
ð1Þ
where fibers 1 and 2 are regarded as compensated and compensating fibers, respectively. R denotes Mueller matrix and Ry is Hermitian conjugation of R. q1 cannot be compensated in (1) because the multiplication cross term s1x ^ ðRy2~ s2 Þ ðs1 ^ q1 Þ is normal to s1x ^q1 , that is to say, a single compensating fiber is unable to compensate PCD. So we investigate the case of three concatenated fibers 1, 2 and 3. Similarly, fiber 1 is regarded as compensated fiber. The firstand second-order PMD vectors are given by (Gorden and Kogelnik 2001) s1 þ ~ a þ~ bj j~ stot j ¼ j~
ð2Þ
a þ~ bÞ ðs1 ^q1 Þ þ ~ a ~ bj j~ stotx j ¼ js1x ^ q1 þ s1 ^ q1x þ ð~
ð3Þ
b ¼ Ry2~ s2 ¼ ðRy2 Ry3~ where ~ a ¼ Ry2 Ry3~ s3 and ~ s2 . The relation Ry2 ½ðRy3~ s3 Þ ~ s3 Þ s2 Þ is used to obtain (2) and (3), which can be explained as follows: the ðRy2~ effect of Ry2 rotates the vector in Stokes space but keeps its magnitude uns2 can s3 Þ ~ changed. Generally, this rotation is frequency dependent. Ry2 ½ðRy3~ be explained as such a process: two vectors carry out a multiplication cross operation and then the resultant vector is rotated by an angle. This process is equivalent to that in which the two vectors are rotated by the same angle at first and then carry out a multiplication cross operation, as can be expressed s3 Þ ðRy2~ s2 Þ. by the term ðRy2 Ry3~ It is easily understood that ~ a is independent of ~ b because of the independence between fibers 2 and 3. It is shown from (3) that if ~ a and ~ b are chosen
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Coupler
Input PC1
Fixed DGD1
PC2
Fixed DGD1
PC3
Var. DGD
Logic Control Unit
Output
PMD, CD Monitor
Fig. 2. Scheme of the three-stage compensator. PCi: ith polarization controller, Var. DGD: variable DGD.
Fig. 3. Operating point 1 (a) and 2 (b) of three-stage PMD compensators.
properly, s1x ^ a ~ b and at the same time s1 q^1x can q1 can be compensated by ~ be compensated by ð~ a þ~ bÞ ðs1 ^q1 Þ. From the analyses above, we find that the first two birefringent fibers of a PMD compensator are capable of compensating both PSPRR and PCD. The scheme of the three-stage compensator is shown in Fig. 2. 2.1.3. Two operating points when compensating second-order PMD According to the analyses above, we propose two possible operating points distinguished by the relative orientations of ~ a and ~ b which are shown in ~ Fig. 3. Vectors ~ a and b are located on the plane P, which is determined by q1 ^ q1x . ~ c denotes the projection of ~ a þ~ b in the direction of vector ^ q1 and ^ ~ ~ ^ q1x . d denotes the projection of ~ a þ b in the direction of ^q1 in Fig. 3a, q1 ^ but in the direction of ^ q1x in Fig. 3b. For operating point 1, both ~ a ~ b and ~ c ðs1 ^q1 Þ are used to compensate d can be used to compensate firstthe component in the direction of ^q1x , ~ order PMD, but s1x cannot be compensated, which is similar to a two-stage compensator. However, there are some differences between them. A variable DGD is not permitted in the first stage of the two-stage compensator due to the limitations of tracking speed. For a three-stage compensator, the magnitude of ~ a þ~ b varies with the relative angle between ~ a and ~ b, which is equivalent to a variable DGD. However, the DGD of first compensating fiber is fixed. So there exists a tradeoff between the tracking speed and
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Fig. 4. Relationship between ~ a ~ b and ~ c.
compensating performance for the three-stage compensator. In conclusion, operating point 1 is suitable for the cases of larger PSPRR and smaller PCD. Operating point 2 is shown in Fig. 3b, where ~ c ðs1 ^q1 Þ is still used to compensate the components in the direction of ^qx . Different from operating point 1, ~ a ~ b is used to compensate s1x , viz. PCD. So the two components of the second-order PMD can be compensated in this operating point. PCD and PSPRR are uncorrelated (Penninckx and Bruye´re 1998), which requires ~ a ~ b independent of each other. The correlativity of vector ~ c q1 Þ and ~ c ðs1 ^ and ~ a ~ b is shown in Fig. 4, where the DGD values of first two compensating fibers are numerically optimized to 12 ps to obtain the minimum outage probability in 40 Gb/s NRZ systems. When ~ a and ~ b are rotating in the ~ plane P, all the possible values of ~ c and ~ a b fall into the grid area in Fig. 3. When ~ c falls into the region from )17 to +17 ps, ~ c and ~ a ~ b are independent of each other. Although the PSPRR that can be compensated is slightly smaller than that in operating point 1, operating point 2 is still suitable for the case of large PSPRR and large PCD. A three-stage compensator has been proposed to solve the problem that a two-stage compensator cannot compensate PCD of second-order PMD. So we will compare the performance of the two compensators in the following paragraphs. In the numerical model, we create 1000 random fibers where 10 input SOP of 40 Gb/s NRZ pseudo-random sequence (26 ) 1) are used for each fiber realization, resulting in 10,000 BER-samples from which the outage probability is calculated at an input power margin 2 dB.
2.1.4. Numerical results for PMD compensation Firstly, we compare them in a large average DGD condition. Here the average DGD is chosen to be 20 ps. The DGD values of the first stage of the two-stage compensators and the first two stages of three-stage compensators are optimized to be 20, 12 and 12 ps, respectively. Even if BER > 10)12 happens to only one input SOP in one fiber, such a fiber is recorded as a sampling point with error bits, which is marked in Fig. 5a and b, where the big size filled markers in Fig. 5b denote the case that BER > 10)12 happens to more than 5
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Fig. 5. The sampling points with error bits after (a) two-stage and (b) three-stage compensation.
input SOP in one PMD emulator state. It is shown that (1) error bits will happen no matter whether DGD is large or not. There are large amounts of sampling points which scatter in the region of )100 ps2 < PCD
