Analytical Form of Frequency Dependence of DGD in Concatenated Single-Mode Fiber Systems

Transcription

1 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 21, NO. 10, OCTOBER Analytical Form of Frequency Dependence of DGD in Concatenated Single-Mode Fiber Systems M. Yoshida-Dierolf and V. Dierolf Abstract An analytical form for the lightwave angular frequency ( ) dependence of the squared value of the differential group delay in concatenated single-mode optical fiber systems is introduced. We show that the dependence can be described by a combination of cosine functions of rotation angles and their sums and differences of the fiber pieces, which are located between the two end pieces of the concatenated system. Using the analytical expression, the dependence of real fiber systems can be simulated, providing information about the birefringence configuration of the system. Comparison with experimental data taken on real systems yields good qualitative agreement even if only few concatenations are considered. Index Terms Concatenation effect, differential group delay (DGD), fiber birefringence, polarization mode dispersion (PMD). I. INTRODUCTION AS optical fiber communication systems require higher bit rates and cover longer distances, higher order polarization mode dispersion (PMD) becomes an important limiting factor for the bandwidth. To handle the problem, characterization of the fiber system in terms of its dependence on frequency ( )of propagating lightwave is essential. In recent years, there have been several approaches to elucidate the problem. Higher order PMD expansion theories [1] [3] provide a Jones matrix form, which contains second and even higher order PMD. These theories are based on the model constructed by a single piece of fiber characterized by an dependent rotation angle and an eigenvector, which rotates on the Poincare sphere as changes. For this reason, this approach is useful to describe the higher order PMD behavior in a small range of, but not appropriate for describing the typical oscillating dependence of the differential group delay (DGD) observed for a relatively wide range of in real concatenated long fiber systems [4] [6]. In order to deal with the frequency dependence in such systems, Karlsson and Brentel [7] proposed the autocorrelation function of the PMD vector, and Marks et al. [8] used the function to distinguish different configurations of PMD emulators. The autocorrelation function is an useful tool to estimate the system bandwidth, which limits the bit rate. As Karlsson mentioned in his paper [7], this function characterizes the system as an averaging over an ensemble of fibers of the same type, or as an average over long times for a specific fiber that drifts. This statement is true for the other statistical approaches [6], [9] [14] as well. Manuscript received March 19, The authors are with the Physics Department, Lehigh University, Bethlehem PA USA ( myd2@lehigh.edu). Digital Object Identifier /JLT Long concatenated fiber systems thus far have been treated as a random system, and statistical methods to deal with the features are more often discussed. However, the phenomenon, which engineers are facing and struggling to suppress, is not representative of the system statistics, but one realized case of the system at a time. Each realized case exhibits its own dependence, and it drifts as the configuration drifts. As will be seen in this paper, the dependence of the DGD contains some information of the system configuration; therefore analyzing the dependence is very important to investigate the configuration present during the dependence measurement. Gisin and Pellaux [15] calculated the dependence of the DGD for concatenated fiber systems directly from the product of Jones matrices using random variables. However, the relation between the features of the dependence and the system variables was, unfortunately, not analyzed. The purpose of this paper is to elucidate this point, and as a result to provide a detailed form of the dependence of the DGD in terms of the physical variables of the system. To describe the procedure, we obtain, in the following section, a general equation for the squared value of the DGD of a concatenated system with pieces of fiber, using the recursion relation of the PMD vector [7], [16]. First we explain that the PMD vector of a single piece of fiber does not depend on, which simplifies the form of the squared DGD. In Section III, we apply the general equation to the case of three pieces of fiber and obtain an expression for the dependence, which contains the rotation angles and eigenpolarization directions of the individual fiber pieces in the system. In Section IV, we develop the expression for the case of pieces and show some results of the calculations for the case of five fiber pieces. At the end of the section, a general form containing cosine functions of rotation angles and their sums and differences will be derived. In our treatment, we use the rules of the central and surface angle relations of a spherical surface triangle. In the conclusion, we propose possible applications of our method. Some details of the calculations are shown in two Appendixes. II. DEPENDENCE OF DGD The output PMD vector for a concatenated system with pieces of fiber is given by the following recursion relation [7], [16]: is the output PMD vector of the th fiber piece and is its 3 3 rotation matrix (reduced Muller matrix). Through (1) /03$ IEEE

2 2218 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 21, NO. 10, OCTOBER 2003 the relation, can be expressed in terms of only and as is used. The same expression as (2) is provided by Gordon and Kogelnik [16]. We shall use this expression now to obtain the dependence of the DGD. As a first step, we show that the PMD vector of a single piece of fiber does not depend on. This fact has been used [6], [15] without a precise explanation. An output PMD vector of a general birefringent element is given by [17] (2) Each individual fiber piece in the concatenated system described in (2) is supposed to be taken as a uniform birefringent element, and can be neglected, as commonly done in optical fiber systems [15], [19]. Therefore, the th fiber piece in the system has an independent PMD vector written as is the length of the fiber piece. Note that this vector meets the following relation: Because and are related as (8) (9) (10) is the Stokes vector of the eigenpolarization of the element and is the rotation angle on the Poincare sphere. The first term describes the contribution from the dependence of the rotation angle. The second and the third terms describe that of the eigenpolarization. The latter two terms vanish in a uniform birefringent system because the direction of the eigenpolarization is determined by the system anisotropy, which is in general independent of. Therefore, only the first term remains in (3), indicating that the output PMD vector of the uniform birefringent element has the same direction as the eigenpolarization of the element. When we introduce as the PMD vector of a uniform element, we have the following relation: is the rotation matrix of the uniform element and is the input PMD vector. designates the transposed matrix of. According to (4), we do not distinguish between the input and the output PMD vectors in the uniform element. The rotation angle corresponds to the birefringence strength, which is given as is the speed of light and is the length of the element. is the difference in refractive indexes given by,,, and are the refractive indexes for slow, fast, right circular, and left circular polarization components [18]. Using these variables, the vector in this simple system becomes the prime represents the derivative in respect to. When, the first term can be neglected and the DGD becomes independent. (3) (4) (5) (6) (7) and are rotational matrices, we can take (11) Using (9), the squared value of the DGD can be calculated from (2) as (12) (13) representing the contribution from successive pieces of fiber starting with the th fiber piece. Equation (12) shows that the squared DGD of the whole system contains the terms of squared DGD of individual fiber pieces, and of the contributions from successive two, three, four, pieces of fiber. Because has no dependence as mentioned before, the dependence in the DGD comes only from the rotation angle in the rotation matrices, i.e., only the terms from more than three successive pieces of fiber can cause the dependence. Therefore, (13) carries the dependence of the DGD and has to produce the typical oscillating dependent curve. The second and third terms in (12) are cancelled out when the ensemble average for random concatenation is taken, because the angle dependence is smeared out with the uniform distribution of in the directions on the Poincare sphere. Under this condition we obtain the following result: (14) Therefore, the ensemble average of the squared DGD should have no dependence, as pointed out by Gisin [20]. If we include the dependence in (see (7)), (14) exhibits rather smooth dependence [21], which is a quite different feature from the oscillating structure caused by the concatenation.

3 YOSHIDA-DIEROLF AND DIEROLF: FREQUENCY DEPENDENCE OF DGD IN FIBER SYSTEMS 2219 between and is the same as the angle between and, the angle relation gives (18) This equation exhibits the direct dependence in the squared DGD in (15). Heffner [23] and Hakki [24] experimentally showed that the DGD of a concatenated system with three pieces of polarization maintaining fiber exhibits sinusoidal dependence to the light wavelength. The period in wavelength, which gives 2 deviation in the rotation angle, is given as (19) Fig. 1. Relationship among the Stokes vectors s, s, s, s, the central angles,,, and the surface angle ' on the Poincare sphere. S, S, S, S represent the end points of the vectors s, s, s, s. S, S, S are located on the equator. The counterclockwise direction about the vector s is taken as positive for the rotation angle '. III. CASE OF THREE PIECES OF FIBER To obtain the explicit form for the dependence of the DGD, we start with the case of. In this case, the DGD (12) has a simple form as (15) If all the matrix elements of are plugged into (15) and some transformations are employed, the form of in Gisin and Pellaux [15] can be derived. For simplicity, we suppose that the three individual pieces of fiber have only rectangular birefringence with slow axis directions,, and. The angles between and, and are taken as and, which correspond to central angles in the Poincare sphere. The counterclockwise direction looking from the north pole is taken as a positive direction. The rotation angle of the matrix is given by (16) This angle corresponds to a surface angle on the sphere. The relationship among the angles and the vectors is shown in Fig. 1. The vector is moved to the vector by the rotation ; then we have (17) is the central angle between and. The value can be obtained using the relation among the angles of a spherical surface triangle [22]. Because the central angle The values in the plots of Heffner [23] and Hakki [24] are 40 and 30 nm. The DGD of the second piece is 0.19 and 0.26 ps, respectively, and both measurements are carried out around the 1520-nm range. Using these parameters in (19) gives good agreement, confirming that this expression describes the dependence for three concatenated fiber pieces very well. IV. CASE OF PIECES OF FIBER In this section, we extend the argument of the previous section to a concatenated system with pieces of fiber. We obtain the term of the contribution from successive pieces of fiber starting with the th fiber piece. For convenience in numbering and notation, we use the second expression of (13) but take as If we take as an example, then we have (20) (21) As before, we suppose that each piece of fiber has only rectangular birefringence, so the vectors are aligned on the equator. We take the central angle between and as, and the rotation angle of as. The vector that results from the rotation of the vector is denoted as and the central angle between and as. Using this angle, (21) can be written as (22) The surface angle between and corresponds to the rotation angle. The surface angle between and is taken as. The relations among the vectors and the angles are shown in Fig. 2. Using the relations among the angles of a spherical surface triangle [22], we obtain the following recursion relations for the angle : (23)

4 2220 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 21, NO. 10, OCTOBER 2003 Fig. 3. Calculated! dependence of the squared DGD for a concatenated system with five pieces of fiber. The DGD of the fiber pieces is , , , , ps. The directions of the slow axes on the Poincare sphere are 0.8, 142, 173.2, 31, 255.8, respectively. The constructing components of the DGD curve D(1; 3), D(2; 4), D(3; 5), D(1; 4), D(2; 5), and D(1; 5) are also shown. Fig. 2. Relationship among the Stokes vectors s, s, s, s, s s, the central angles,,, and the surface angles ', ',,, on the Poincare sphere. S, S, S, S, S, S represent the end points of the corresponding vectors. S, S, S are located on the equator. The counterclockwise direction about the corresponding vectors is taken as positive for the surface angles. (24) (25) The derivation of the relations is shown in Appendix A. Through these recursion relations, in (22) can be written in terms of the angles, which are determined by the directions of the slow axes, and, which are the rotation angles of the individual fiber pieces. The procedure to obtain the terms of is shown in Appendix B. Here we give as an example the result for (26) On this basis, we can now calculate all the terms in (12). A typical result for is shown in Fig. 3, along with the constructing components of,,,,, and. The detailed parameters used in the calculation are given in the figure caption. Although the feature of the dependence of the curve is very sensitive to the changes in the parameters, it is clear that the calculated curve is very similar to the experimental results observed in the real optical fiber systems [4] [6]. To observe the evolution of the DGD induced by the change in the system, we follow the dependence of the DGD while gradually varying the variables of the system as shown in Fig. 4 (see the figure caption for details). The result is shown as a contour plot in Fig. 5. Some features in the figure exhibit a tendency similar to the contour plot of the DGD observed by Karlsson et al. [5] in installed fiber systems. Investigating the recursion relations (23) (25) or the example (26), we find that the general term has either a cosine function or a product of cosine and sine functions of the rotation angles of the fiber pieces, which are located between the two end

5 YOSHIDA-DIEROLF AND DIEROLF: FREQUENCY DEPENDENCE OF DGD IN FIBER SYSTEMS 2221 Fig. 4. Assumed change in the fiber system. The slow axis directions are changed 10 alternatively for the first, third, and fifth fiber pieces. The connecting points are moved in 0.25% of each fiber piece as shown in the figure. independent terms in (13). is the amplitude of the following cosine functions. As the dependence is the concern, we use this simple expression for the amplitudes. When necessary, the amplitudes can be written using the DGD values and relative angles between the slow axes of the individual fiber pieces. For our example with, (29) yields (30) Fig. 5. Calculated contour plot of DGD versus light frequency and variable change in the system. Fifty data sets of the! dependence are calculated to make the plot. The curve in the Fig. 3 corresponds to the fifth data set from the bottom. fiber pieces of the succession. More interestingly, each term is an even function of, because it contains an even number of factors of. Therefore, each term can be written as an addition of cosine functions of sums and differences of the contributing angles. For instance, the fifth and sixth terms in (26) can be written as (27) and also the tenth, eleventh, twelfth, and thirteenth terms can be written as (28) are constants determined by the coefficients of the original terms. Therefore we can put all the contributions from in (12) together to obtain the form for the squared DGD as Equation (29) seems reasonable in the sense that the dependence of the concatenated system is given by the addition of dependence of individual fiber pieces located between the two end pieces of the system and the coupling terms among the contributing fiber pieces (i.e., sums and differences). Therefore, the configuration of the concatenation should be derived from the observed dependence of the DGD. In order to check this point, Fourier analysis was employed to the calculated DGD curve in Fig. 3. The result yields 13 components that have the frequencies corresponding to the DGD values and their sums and differences of the second, third, and fourth fiber pieces in the concatenation. However, a method to assign the main components that directly correspond to the rotation angles is not yet established. It should be closely related to the procedure to derive the relative angles between the fiber pieces from the amplitudes of each components. These points are interesting challenges for the future. Finally, we briefly mention the effect from the circular birefringence in the individual fiber piece. Thus far, we supposed that each individual fiber piece has only rectangular birefringence, so that all the eigenpolarization vectors are aligned on the equator. When we allow that each fiber piece has both rectangular and circular birefringence, the eigenpolarization is no longer aligned on the equator, and takes an arbitrary position on the sphere. As shown in Fig. 6, we take the surface angle counterclockwise from to as positive and the central angle from to as positive. Then the recursion relations (23) (25) receive a minor change of substituting with. This change results in phase shifts in the cosine functions in (29). (29) represents the cosine functions of sums and differences of the angles. sums up the cosine functions of all the possible sums and differences of the angles. sums up all the different numbers and the possible combinations of fiber pieces between the two end pieces of the system. is a constant term containing the contributions from the first two summations in (12) and other V. CONCLUSION The main contribution of this paper is (29), which shows that the dependence of the squared DGD is expressed as a summation of all the cosine functions of the rotation angles and their sums and differences. The contributing rotation angles are of the fiber pieces, which are located between the two end pieces of the concatenated system. The amplitudes of the functions depend on the relative angles among eigenpolarization directions in a rather complicated but tractable way. However, it can be at

6 2222 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 21, NO. 10, OCTOBER 2003 into the drift in the physical variables of the system. Then detailed information such as the drifting variables, the ranges of the drift, the speed of the drift, etc., could be obtained and compared to the environmental parameters, change of which may cause the drift. Such investigations will help to understand the mechanism of the drift and will contribute to subdue the higher order PMD effects. Fig. 6. Relationship among the Stokes vectors s, s, s, s, s, s, the central angles,,, and the surface angles ', ',,,,, on the Poincare sphere. S, S, S, S, S, S represent the end points of the corresponding vectors. The direction from s to s in the central angle is taken as positive. least concluded from (22) that the contribution from the succession that has two end pieces with large DGD is relatively strong. The model we used in this paper is constructed by concatenation of waveplates corresponding to fiber pieces, supposing that each piece has uniform birefringence. This model would be reasonable as a rough description of real fiber systems, although the configuration of the birefringence should have gradual change along the length of the system. According to (29), the dependence of the DGD contains the information about the system configuration. Therefore, analyzing the dependence is very important. If an available data set of the dependence covers enough range of, the Fourier analysis could be employed and the result would give the physical variables of the elements constructing the system. Of course, the analysis of the dependence of the DGD in the real fiber system would not be so simple. The number of the concatenation would be large, so the combination of the rotation angles would become tremendously complicated. However, according to the long-term measurement of PMD in installed fiber systems reported by Karlsson et al. [5], the dependence of the DGD appears not much more complicated than the curve in Fig. 3. This fact suggests that the birefringence configuration of some real fiber systems might not be so complicated, and be possible to be modeled by a concatenation with a relatively small number of fiber pieces. More interestingly, the time average for 36 days in their measurement [5] does not totally smear out the dependence of the DGD, but a certain structure remains. This remaining structure exhibits the typical multiple oscillation caused by the concatenation. The fact that such a structure remains even after the time average of 36 days indicates that there should exist a birefringence configuration, which moves very slowly. If the Fourier analysis could be employed for an adequate data set of DGD evolution, the drift in the DGD could be converted APPENDIX A The relations among the Stokes vectors,,,,,, the central angles,,, and the surface angles,,,, on the Poincare sphere are shown in Fig. 2. In the spherical surface triangle constructed with the end points of the vectors,,, it is rather easy to derive the first relation (23) using the cosine formula for the central angle [22]. Similarly, the cosine and sine formula for the surface angles give (31) (32) (33) (34) Plugging (32) (34) into (31), we get (35) as shown at the bottom of the page. Equations (24) and (25) can be obtained using (33) and (35). When we take (23) (25) become APPENDIX B (36) (37) (35)

7 YOSHIDA-DIEROLF AND DIEROLF: FREQUENCY DEPENDENCE OF DGD IN FIBER SYSTEMS 2223 (38) (39) Because and, the recursion relation ends with, and the rest gives the following factors: Using (36) and (40), the terms (40) can be written as a summation of (41) for all possible combinations of ( ). Note that the first parameter in the first factor must be zero, so that can take only zero or one. For five successive fiber pieces ( ), 13 terms appear. As an example, we show a term which corresponds to the sixth term in (26). (42) [4] R. M. Jopson, L. E. Nelson, and H. Kogelnik, Measurement of second-order polarization-mode dispersion vectors in optical fibers, IEEE Photon. Technol. Lett., vol. 11, pp , Sept [5] M. Karlsson, J. Brentel, and A. Andrekson, Long-term measurement of PMD and polarization drift in installed fibers, J. Lightwave Technol., vol. 18, pp , July [6] E. Ibragimov, Statistical correlation between first and second-order PMD, J. Lightwave Technol., vol. 20, pp , Apr [7] M. Karlsson and J. Brentel, Autocorrelation function of the polarization-mode dispersion vector, Opt. Lett., vol. 24, no. 14, pp , [8] B. S. Marks, I. T. Lima Jr, and C. R. Menyuk, Autocorrelation function for polarization mode dispersion emulators with rotators, Opt. Lett., vol. 27, no. 13, pp , [9] G. J. Foschini and C. D. Poole, Statistical theory of polarization dispersion in single mode fibers, J. Lightwave Technol., vol. 9, pp , Nov [10] G. J. Foschini, R. M. Jopson, L. E. Nelson, and H. Kogelnik, The statistics of PMD-induced chromatic fiber dispersion, J. Lightwave Technol., vol. 17, pp , Sept [11] A. Djupsjobacka, On differential group-delay statistics for polarization-mode dispersion emulators, J. Lightwave Technol., vol. 19, pp , Feb [12] G. J. Foschini, L. E. Nelson, R. M. Jopson, and H. Kogelnik, Statistics of second-order PMD depolarization, J. Lightwave Technol., vol. 19, pp , Dec [13] A. Bononi and A. Vannucci, Statistics of the Jones matrix of fibers affected by polarization mode dispersion, Opt. Lett., vol. 26, no. 10, pp , [14], Statistical characterization of the Jones matrix of long fibers affected by polarization mode dispersion (PMD), J. Lightwave Technol., vol. 20, pp , May [15] N. Gisin and J. P. Pellaux, Polarization mode dispersion: Time versus frequency domains, Opt. Commun., vol. 89, pp , [16] J. P. Gordon and H. Kogelnik, PMD fundamentals: Polarization mode dispersion in optical fibers, Proc. Nat. Acad. Sci. USA, vol. 97, pp , [17] M. Karlsson, Polarization mode dispersion-induced pulse broadening in optical fibers, Opt. Lett., vol. 23, no. 9, pp , [18] R. C. Jones, A new calculus for the treatment of optical systems. VII: Properties of the N-matrices, J. Opt. Soc. Amer., vol. 38, pp , [19] S. C. Rashleigh and R. Ulrich, Polarization mode dispersion in single-mode fibers, Opt. Lett., vol. 3, no. 2, pp , [20] N. Gisin, Solutions of the dynamical equation for polarization dispersion, Opt. Commun., vol. 86, pp , [21] D. A. Flavin, R. MacBride, and J. D. C. Jones, Dispersion of birefringence and differential group delay in polarization-maintaining fiber, Opt. Lett., vol. 27, no. 12, pp , [22] Handbuch der Mathematik. Koeln: Buch und Zeit Verlagsgesellschaft mbh, [23] B. L. Heffner, Accurate, automated measurement of differential group delay dispersion and principal state variation using Jones matrix eigenanalysis, IEEE Photon. Technol. Lett., vol. 5, pp , July [24] B. W. Hakki, Polarization mode dispersion in a singlemode fiber, J. Lightwave Technol., vol. 14, pp , Oct REFERENCES [1] D. Penninckx and V. Morenas, Jones matrix of plarization mode dispersion, Opt. Lett., vol. 24, no. 13, pp , [2] H. Kogelnik, L. E. Nelson, J. P. Gordon, and R. M. Jopson, Jones matrix for second-order polarization mode dispersion, Opt. Lett., vol. 25, no. 1, pp , [3] E. Forestieri and L. Vincetti, Exact evaluation of the Jones matrix of a fiber in the presence of polarization mode dispersion of any order, J. Lightwave Technol., vol. 19, pp , Dec M. Yoshida-Dierolf, photograph and biography not available at the time of publication. V. Dierolf, photograph and biography not available at the time of publication.