POLARIZATION mode dispersion (PMD) is widely regarded

4100 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 24, NO. 11, NOVEMBER 2006 Effect of Nonlinearities on PMD Misha Boroditsky, Senior Member, IEEE, Marianna Bourd, and Moshe Tur, Fellow, IEEE, Fellow, OSA Abstract We show experimentally and theoretically that the magnitude and direction of the polarization mode dispersion PMD) vector of a WDM channel is significantly affected by nonlinear polarization crosstalk from neighboring channels within the PMD correlation bandwidth. We use our model to estimate the effect the nonlinear interactions may have on the PMD-induced penalty. Index Terms Nonlinear optical effects system performance, optical fiber communications, polarimetry, polarization-mode dispersion PMD). I. INTRODUCTION POLARIZATION mode dispersion PMD) is widely regarded as a limiting impairment in high-speed optical communication systems. Since it changes unpredictably in time with temperature and other environmental conditions [1], [2], active mitigation techniques are being developed based on PMD measurement on working channels [3]. Even though PMD is often looked at as a property of a passive fiber, it is in fact affected by anything that changes the birefringence of the optical media. As the network becomes more agile with fast restoration and provisioning, and with the emergence of burst and packet switching, dependence of the PMD vectors on rapidly changing optical power in adjacent channels should be considered. In multichannel long-haul systems, high optical power may cause nonlinear polarization rotation or crosstalk [4] [6]. The question therefore arises to what extent this changes the PMD vector both in direction and magnitude [7]. This paper comprises an experimental and theoretical study of changes in the PMD vector of a given channel that occur when optical power in a close-by channel modifies the local birefringence through the Kerr effect [6]. In the following, we show that in the presence of a strong optical signal the pump signal) in the spectral vicinity of the measured channel, its differential group delay DGD) may vary by 30% or more, and direction of the principal states of polarization PSP) may deviate by up to 20. We show that the process strongly depends on the frequency separation between channels, with a typical frequency bandwidth commensurate with the correlation bandwidth of PMD. Manuscript received April 27, 2006; revised August 5, 2006. M. Boroditsky was with AT&T Laboratories, Middletown, NJ 07748 USA. He is now with Knight Equity Markets, Jersey City, NJ 07310 USA e-mail: mboroditsky@knight.com). M. Bourd and M. Tur are with the School of Electrical Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel e-mail: tur@eng.tau.ac.il). Color versions of all figures are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JLT.2006.884184 II. EXPERIMENTAL SETUP In our measurements, we used a 20-km spool of AllWave fiber with mean DGD of 0.35 ps averaged over the C-band). The experimental setup is shown in Fig. 1. The Probe laser represents the channel, whose PMD properties are under investigation, while the Pump laser stands for a nearby interfering channel. The coherence control of the pump laser was used to suppress Brillouin scattering. Polarization synthesizer built into the HP8509 polarization analyzer controls the probe input state of polarization SOP) for PMD measurements, and another stand-alone polarization synthesizer is used to study the pump effect on PMD as a function of the pump input SOP. After amplification of the pump signal, the two inputs are combined by a coupler and input into the fiber under test. On the receiving side, the probe wavelength was selected by a pumpblocking 0.25-nm tunable filter, and its SOP was measured by a polarization analyzer. All relevant equipment was computer controlled, and the fiber was placed in a temperature-controlled environment to ensure stability. We verified that the output SOP of the probe operated at λ 0 = 1537.5 nm) was not affected by the leakage of pump light at λ p = 1540 1560 nm) through the pump-blocking filter [7]. In all experiments, we used the Jones Matrix Eigenanalysis technique to measure the dependence of the DGD at an arbitrarily chosen wavelength λ 0 = 1537.5 nm as a function of pump input SOP and power. The Jones matrices were measured at λ = λ 0 ± λ/2, where λ =3nm. III. MEASUREMENT RESULTS The DGD at λ = 1537.5 nm measured in the presence of pump, as a function of the pump wavelength for different orientations of the pump, as its launch SOP scans the equator on the Poincaré sphere, is plotted in Fig. 2a). Fig. 2b) illustrates variations of DGD with azimuthal angle of the pump, with a dashed line showing the DGD without pump. As the effect of nonlinearities on the measured value of DGD is expected to become more pronounced as the pump power increases, it is indeed clear from Fig. 2c) that the optically induced PMD deviations grow linearly with the pump power in this 20-km spool. Very similar results were obtained at different wavelengths of the probe and pump. PSPs are also affected by the nonlinear interaction. As can be seen from Fig. 3, as the pump SOP scans an equator of the Poincare sphere, the PSP of a probe precesses, with amplitude increasing with the pump power. To get a better idea of the PMD variation under the nonlinear interaction, we modified the launch conditions of the pump: Instead of tracing a great circle on a Poincaré sphere, pump SOP was sampling 20 icosahedron s face centers which are uniformly distributed on a sphere. Under these pump conditions, 0733-8724/$20.00 2006 IEEE

BORODITSKY et al.: EFFECT OF NONLINEARITIES ON PMD 4101 Fig. 1. Experimental setup. Fig. 3. Example of the PSP evolution with pump SOP scanning the equator of Poincaré sphere for pump power of 5, 8, 11, 14, and 17 dbm, with the inner curve corresponding to the lowest power. The PSP deviations for the highest pump power the outermost curve) are about 20. The probe wavelength is fixed at 1537.5 nm Fig. 2. DGD of a 1537.5-nm channel as a function of a) the pump wavelength at pump power of 17 dbm and b) SOP orientation for wavelength separation of 6, 12, 18, and 24 nm for the pump power of 17 dbm. Dashed line shows DGD of a channel in the absence of a pump. c) Linear evolution of the peak-to-peak PMD deviations for pump at 1542 nm. the probe PMD vector was changing its direction as well as its magnitude, outlining a three-dimensional shape that enclosed the tip of an unperturbed PMD vector not shown). To verify the repeatability of the observed effect at other wavelengths, we performed another set of measurements, where the spectral distance between the pump and the probe was kept constant at 5.5 nm. Both pump and probe were moved across 12 nm in 2-nm steps, and the pump SOP was again switched between 20 SOP states uniformly distributed on Poincaré sphere, and power was switched between 17, 14, 11 dbm, and no power. Again, a linear dependence of DGD variation on the pump power in milliwatts) was observed in all measurements. The DGD variations on the channels for the 17-dBm pump power are shown in Fig. 4a) are dots, and the intrinsic DGD, measured in the absence of the pump, is plotted with a thick line. Clearly, for all channels, in the presence of the pump, DGD fluctuates by up to 20% of its value. The mean angular deviations of the PSP from the unperturbed value are also plotted in Fig. 4b). For the 17-dBm pump, the PSP varies by up to 8, which means that there are, on average, fluctuation of the PMD vector orthogonal to its direction of about 15% of its length. In other words, the PMD vector s changes in length are approximately equal to those in direction. The small under 0.4 ) PSP variation without pump reflects the degree of the experimental setup s stability. IV. THEORY To model the PMD variation induced by nonlinearities, we adopt an approach based on stochastic differential equations. It

4102 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 24, NO. 11, NOVEMBER 2006 Fig. 4. a) DGD variation at the probe wavelength for the constant 5.5-nm spectral spacing between the pump and a signal. b) Mean angular variation of PSP in the same setting. Note that the angular deviation, like DGD deviation, grows with the pump power. allows us to calculate analytically the root-mean-square rms) deviations of the PMD vector averaged over different fiber realizations. Our approach allows us to estimate both self-induced nonlinear PMD [8] and the PMD variations introduced by the strong pump. Further, this approach enables the analysis of other types of birefringence perturbations as well, as described in Appendix A. In the derivation below, we would like to find an rms change in the PMD τ rms caused by an additional small birefringence perturbation β, which may be coming from, e.g., nonlinearities [5], [6], external stress [9], or magnetic field [10], [11]. We denote the PMD vector of an unperturbed fiber as τ 1, and that of perturbed fiber as τ 2, so that τ 2 rms = τ 1 τ 2 ) 2 is the quantity we would like to compute. PMD vectors are functions of the propagation distance z, which can change from 0 to the fiber length L, and signal optical frequency ω S, which changes over the spectral range of interest, corresponding to tens of nanometers in our case. The equations that describe accumulation of the PMD vector along the fiber are driven by position-dependent fiber birefringence β and its frequency derivative β ω = β/ ω [12], which are both assumed to be changing randomly along the fiber τ 1 z = βω S ) + βω ω S ) τ 1 1) τ 2 z = βω S ) + βωs )+ β ω ) τ 2 2) where, as mentioned above, we use β to denote the additional birefringence. Keeping the future analyses of nonlinear PMD in mind, we use an adjacent pump frequency ω P as a reference and expand the birefringence in Taylor s series around this point as βz) = β 0 z)+ω β ω z) so that ω = ω S ω P. It can be shown [13] that by using coordinate transformation Rz), which satisfies d R z) dz = β 0 z) R 3) we can set β 0 =0, as long as we apply the following transformations: β ω z) R β ω z) and βz) R βz). This last transformation has an important implication for our analyses. If the perturbation in birefringence is constant in a stationary coordinate system, for example, linear birefringence coming from external stress, or circular birefringence induced by a magnetic field, it will be randomized in the rotating coordinate system, with a resulting correlation length on the order of the beat length. If, however, the perturbation is caused by an optical nonlinearity, coming from a strong pump light propagating at frequency ω p, its SOP evolves in the stationary coordinate system according to dsω P ) = β dz 0 z) S 4) and stays constant in the rotating frame defined in 3). In the subsection below, we treat the PMD variations caused by the nonlinear birefringence. For completeness, effects of the birefringence perturbation constant in the laboratory references frames are treated in Appendix. When the birefringence is perturbed by the optical pump, so that S P =const, the local perturbation β at location z is approximately β = ks =2π/λ)n 2 P z)/a) S, where λ 1.5 µm is the wavelength of light, A 60 µm 2 is the area of the fiber, and n 2 =2.2 10 20 m 2 /W is the nonlinear coefficient of the glass. In other words, in this paper, we assume that the nonlinear birefringence is simply proportional to the Stokes vector of the pump. This is an oversimplification, since the circular component of the pump does not contribute to the Kerr effect. To compensate for this simplification, we reduce the nonlinear coefficient by one third. Following notation in [14], we introduce the Wiener process with a differential such that dw = β ω dz and dw dw = γdz, with PMD coefficient γ related to the mean-square PMD as γ = τ 2 /L. After we convert the equations above to the Itô form [15], the PMD differentials without and with pump-induced perturbation become, respectively d τ 1 = d W + ωd W τ 1 1 3 γω2 τ 1 dz 5) d τ 2 = d W +ωd W + k Sdz) τ 2 1 3 γω2 τ 2 dz. 6) We are interested in the rms difference between the PMD with and without the pump τ 1 τ 2 ) 2 =2 τ 2 2 τ 1 τ 2.

BORODITSKY et al.: EFFECT OF NONLINEARITIES ON PMD 4103 We used the fact that, as can be seen from 6), an additional rotation does not change the statistics of the PMD, and τ 2 1 = τ 2 2 = γl. That is, while the optical nonlinearities modify the channel-specific PMD, they do not change the mean DGD of the fiber. For the dot product differential we write, following the recipes of stochastic calculus [15], d τ 1 τ 2 )= τ 1 d τ 2 + d τ 1 τ 2 + d τ 1 d τ 2, and, using equations above with identities of vector algebra, we find d dz τ 1 τ 2 = γ 1 3 k S τ 1 τ 2. 7) This leads us to finding the mean of a mixed product d S τ1 τ 2 ) = k τ 1 τ 2 dz 1 ) 3 γω2 S τ1 τ 2 k τ 1 S) S τ 2 ) 8) which, in turn, requires another equation for completeness d τ 1 dz S) S τ 2 ) = 1 3 γ 1 3 γω2 τ 1 τ 2 γω 2 τ 1 S) S τ 2 ). 9) Introducing a three-dimensional vector X = τ1 τ 2, S τ 1 τ 2, τ 1 S) S τ 2 ) ) T, we can rewrite 7) 9) as d X dz = γ 0 + 0 k 0 k 1 1 3 γ 3 γω2 k X 10) 1 3 γω2 0 γω 2 with initial condition X0) = 0, 0, 0) T. Equation 10) above is a linear equation of the form d X/dz = A X + B, which, for k =const, has a solution X = A 1 exp Az) I) B. Then, the relative rms deviation of the PMD vector in the presence of nonlinearly induced birefringence is stdτ)/τ rms = 2 2X1 z)/γz. This quantity is plotted in Fig. 5a), as a function of fiber length and wavelength detuning for the rms PMD of 0.35 ps and pump power of 17 dbm. From the structure of the solution, one can see that the nonlinear interaction has a bandwidth on the order of PMD correlation frequency 1/τ rms. One can also see that for the chosen simulation parameters, the maximum change occurs in the proximity of the pump. That means that the adjacent channels are the main contributors into the PMD variations. For the small pump-probe spacing, the relative variation is expected to average around 22%, effect being four-fold weaker at 20-nm detuning. The power and wavelength separation dependence of the normalized PMD variations are plotted in Fig. 5b) and c). The linear dependence on the pump power is hardly surprising, at least in the first order. Clearly, the relative fluctuations of the PMD cannot grow forever, as nonlinear rotation does not change the frequency statistics of the birefringence. We applied 10) Fig. 5. a) Accumulation of the relative rms difference along the fiber as a function of fiber propagation distance and frequency detuning for pump power 14 dbm. Fiber length is 20 km, and mean PMD is 0.35 ps. b) Relative fluctuation of PMD vector at L =20 km as a function of power and c) detuning between pump and probe. to the 20-km fiber spool we used in our experiments, with 0.2-dB/km attenuation, and assumed a nonlinear coefficient n 2 =2.2 10 20 m 2 /W. In Fig. 6, we plot the wavelength separation dependence obtained from our model together with the wavelength dependence of the PMD fluctuations obtained experimentally by scanning the pump wavelength. We find the qualitative agreement between theory and experiment satisfactory, given that theory assumes data to be averaged over all fiber realizations, which had not been done in the experiment. It is not surprising that the wavelength correlation is, as often happens in PMD-related effects [14], given by the inverse of the mean PMD.

4104 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 24, NO. 11, NOVEMBER 2006 Fig. 6. RMS deviation of nonlinearity-induces DGD, normalized to its rms as a function of wavelength separation for different wavelength separations in a 20-km fiber spool with 0.35-ps mean DGD. Equation 10) can be solved exactly for the case of ω 0, when pump and probe are spectrally very close. In that case, we find τ1 τ 2 ) 2 ω=0 = 4 3 1 ) sin kl τ 2. 11) kl The decaying harmonic term in 11) arises from the fact that in the rotating coordinate systems all primitive random components of the PMD vector precess around the added birefringence vector. For kl 1, which set a criteria for short fiber or low optical power), the equation above further simplifies to τ 1 τ 2 ) 2 ω=0 =2/9kL) 2 τ 2. Another interesting insight into the nonlinearly perturbed PMD can be obtained by conditioning the rms PMD variations on the instantaneous PMD vector. The calculation can be carried out analytically with a Brownian bridge method [16] for ω 0. For the construction of a Brownian bridge, a random process W z) is transformed into W z) = W z) z ) W L) τ0 12) L which guarantees that W L) = τ 0. Then, for zero frequency pump-probe separation ω =0, the perturbed and unperturbed processes are simply given by d τ 1 = dw 13) d τ 2 = dw + ksdz τ 2 14) which can be integrated explicitly to obtain τ1 τ 2 ) 2 τ1 ω=0 = τ 2 sin kl 1 2 kl 1 + τ 2 3 2 3 cos kl +21 k 2 L 2 1 cos kl k 2 L 2 ) ) 15) where τ = S τ 0 denotes the component of the original unperturbed PMD vector orthogonal to the SOP of the perturbation not to the launch SOP of the probe signal). An interesting property of the equation above is that the PMD fluctuations grow both with the instantaneous and mean PMD. In the analyses above, we assumed that the amount of additional birefringence is constant. However, if the pump signal changes along the fiber, it is straightforward to extend the above analyses by introducing a z-dependent coefficient kz) to accommodate the attenuation of the pump, and, if the pump is modulated, the bit-pattern dependence. For the modulated pump signal, besides the bit pattern itself, the coefficient will also depend on the relative propagation speed of the pump and the signal [4], [17]. For a constant dispersion, if the pump is modulated with a signal P t), a bit launched at time t =0 into the probe channel will experience the nonlinear coefficient kz) =2π/λ)n 2 P t z))/a)exp αz) defined by the walk-off in local time t = D λz and attenuation α.inamore general case of an arbitrary dispersion map, the walk-off at position z can be written as t z) = z 0 nz,ω)/ λ) λ/c)dz. To analyze this case, we integrated the 10) numerically for a few cases of interest. We considered two 1000-km routes, one with rms PMD 4 ps and another with rms PMD 15 ps. For each route, two calculations were done. One used continuous wave CW) light as a pump, assuming no attenuation. Another one used a nonreturn-to-zero NRZ) signal at 40 and 10 Gb/s, respectively, modulated with 101010... bit pattern, and took into account the pump power variation coming from attenuation and optical amplification. We also assumed a chromatic dispersion of 17 ps/nm km). However, the launch power of the modulated signal was chosen such that the average power along the fiber equals to that for the CW case. The results of the calculations are plotted in Fig. 7. Fig. 7a) and b) refers to the 4-ps case for CW and NRZ modulation of the pump, respectively, and plots in Fig. 7c) and d) refer to a 1000-km link with mean DGD of 15 ps. It follows from our simulation, that for the relevant range of parameters, the average power matters more than the detail of the power profile and the bit pattern. To take our analyses a step further, we can use the results of this section to estimate the fluctuations of optical signalto-noise ratio OSNR) penalty caused by nonlinear effects. Since the OSNR penalty is related to the values of PMD vector orthogonal to the launch SOP [18], 15) provides an order of magnitude estimate for the fluctuations of the PMD-induced OSNR penalty in the presence of the nonlinear perturbation to the birefringence. In particular, by recalling that OSNR penalty, expressed in decibels, can be written approximately as ε db = A τ S) 2 /4T 2 ), with A 50 for NRZ modulation format, and A 15 for return-to-zero format [19], we can estimate the variance of the OSNR penalty coming from the nonlinearities as varε) = A 2 16T 4 τ 1 S) 2 τ 2 S) 2) 2 τ1 τ 2 ) 2 12εε τ 2 16) where we assume for simplicity that the change of the PMD vector is directionally independent from the PMD vector itself, and that the fluctuations are smaller than the penalty itself. Substituting the values for a 20-km link from Fig. 5, one obtains about 30% rms variations in OSNR penalty. In the case of a perturbation high enough to cause decorrelation of PMD, like at the peak of Fig. 7a) and b), corresponding to a 1000-km link, the fluctuations become comparable to the penalty itself.

BORODITSKY et al.: EFFECT OF NONLINEARITIES ON PMD 4105 Fig. 7. RMS fluctuation of the PMD vector in a) 4-ps 1000-km link for various average optical powers, assuming no loss, and b) realistic power profile with 10-G NRZ modulation at 17 ps/nm km). c) and d) plot the same for a 1000-km link with 15-ps mean DGD. V. C ONCLUSION We have demonstrated theoretically and experimentally that nonlinear polarization crosstalk from neighboring channels significantly changes the magnitude and direction of the PMD vector of a wavelength-division-multiplexing channel while not changing an average PMD. The variation can be on the order of the PMD vector itself, and the strength of the interaction is reduced over the frequency correlation bandwidth. Further, we derived an estimate of nonlinearly induced bit-pattern dependent fluctuations of the PMD-induced OSNR penalty. We have also extended our theory to include the PMD variations from the perturbations, which are static in the laboratory reference frame. APPENDIX For completeness, let us consider the case where the perturbation in birefringence is constant in the stationary reference frame, for example, coming from a magnetic field [10], [11] or stress [20]. Such a perturbation gets randomized in the rotating frame defined by 3). Therefore, in the equations describing the PMD evolution of unperturbed and perturbed fiber in the rotating reference frame d τ 1 = dw + ωdw τ 1 17) d τ 2 = dw +ωdw + db) τ 2 18) where d B denotes a random birefringence process, not a deterministic one as before, and d W is the same Wiener process as before. The properties of these processes are assumed to be such that d W d W = γdz; d B d B = bdz d W d B =0;β γ 19) with the value of b to be determined later. Again, following the stochastic calculus recipe, we have to rewrite differentials in the Itô form as d τ 1 =d W + ωd W τ 1 1 3 ω2 d W 2 τ 1 20) d τ 2 =d W +ωd W + d B) τ 2 1 3 ω2 d W 2 + d B 2 ) τ 1. 21)

4106 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 24, NO. 11, NOVEMBER 2006 As before, the quantity in which we are interested in is the variance of PMD in the presence of perturbation τ 1 τ 2 ) 2 = 2γz 2τ 1 τ 2. For the dot product, we write again d τ 1 τ 2 )= τ 1 d τ 2 + d τ 1 τ 2 + d τ 1 d τ 2 22) and, using the identities of vector algebra, averaging over directions of dw and db, and using properties 19), we arrive at d τ 1 τ 2 = γ 1 ) 3 b τ 1 τ 2 dz 23) which can be integrated with initial condition τ 1 τ 2 z=0 =0 to obtain τ 1 τ 2 = 3γ 1 e bl 3 ) γl bγl2 = τ 2 1 bl ). b 6 6 24) Note that there is no frequency dependence in 24), which, in turn, yields the expression for the average variation of the PMD vector in the presence of the perturbation. τ1 τ 2 ) 2 =2τ 2 2 τ 1 τ 2 bγl2 = τ 2 bl 3 3. 25) In a similar way, we can estimate the SOP variation due to the change in birefringence. If we choose the figure of merit to be the dot product between perturbed and unperturbed SOPs S 1 S 2, its average value will be S 1 S 2 = e bl 3 26) and a typical angular deviation is φ 2 =2bL/3. The intrinsic and optically induced birefringence vectors in the fiber are not actually random processes with delta-function autocorrelation; therefore, we need to adjust the diffusion constants accordingly. If the birefringence is characterized by the correlation length L c, the diffusion constant has to be chosen as γ = β 2 ω L c = τ 2 /L; inasimilarway,wehave to define [21] b = β 2 L P, L p being the correlation length of this perturbation, which is nothing but the beat length of the fiber. This result indicates that by introducing a controlled perturbation into the birefringence, e.g., using magnetic field or stress, one can estimate the beat length of the optical fiber. ACKNOWLEDGMENT The authors would like to thank M. Shtaif and C. Antonelli for fruitful discussions. REFERENCES [1] R. Caponi, B. Riposati, A. Rossaro, and M. Schiano, WDM design issues with highly correlated PMD spectra of buried optical cables, in Proc. OFC, Mar. 2002, pp. 453 455. [2] M. Brodsky, P. Magill, and N. J. Frigo, Evidence for parametric dependence of PMD on temperature in installed 0.05 ps/km 1/2 fiber, presented at the ECOC Conf., pp. 1 2, 2002, Paper 9.3.2. [3] H. Sunnerud, C. Xie, M. Karlsson, R. Samuelsson, and P. A. Andrekson, A comparison between different PMD compensation techniques, J. Lightw. Technol., vol. 20, no. 3, pp. 368 378, Mar. 2002. [4] B. Collings and L. Boivin, Nonlinear polarization evolution induced by cross-phase modulation and its impact on transmission systems, IEEE Photon. Technol. Lett., vol. 12, no. 11, pp. 1582 1584, Nov. 2000. [5] J. H. Lee, K. J. Park, C. H. Kim, and Y. C. Chung, Effects of nonlinear crosstalk in optical PMD compensation, IEEE Photon. Technol. Lett., vol. 14, no. 8, pp. 1082 1084, Aug. 2002. [6] M. R. Phillips, S. L. Woodward, and R. L. Smith, Cross-polarization modulation: Theory and measurement of subcarrier-modulated WDM system, J. Lightw. Technol., vol. 24, no. 11, pp. 4089 4099, Nov. 2006. [7] M. Bourd, M. Tur, and M. Boroditsky, Effect of non-linearities on PMD, presented at the Euro. Conf. Optical Commun., Stockholm, Sweden, 2004, Paper We4.P.028. [8] P. K. A. Wai, W. L. Kath, C. R. Menyuk, and J. W. Zhang, Nonlinear polarization-mode dispersion in optical fibers with randomly varying birefringence, J. Opt. Soc. Amer. B, Opt. Phys., vol. 14, no. 11, pp. 2967 2977, Nov. 1997. [9] R. Ulrich, S. C. Rashleigh, and W. Eickhoff, Bending-induced birefringence in single-mode fibers, Opt. Lett., vol. 5, no. 6, pp. 273 275, Jun. 1980. [10] R. H. Stolen and E. H. Turner, Faraday rotation in highly birefringent optical fibers, Appl. Opt., vol. 19, no. 6, pp. 842 848, Mar. 1980. [11] M. Brodsky, M. Boroditsky, M. D. Feuer, and A. Sirenko, Effect of a weak magnetic field on quantum cryptography links, in Proc. ECOC, Glasgow, U.K., 2005, pp. 19 20. [12] Optical Fiber TelecommunicationsIV-B: Systemsand Impairments, I. P. Kaminow and T. Li, Eds. Amsterdam, The Netherlands: Elsevier, Apr. 15, 2002. [13] Q. Q. Lin and G. P. Agrawal, Effects of polarization-mode dispersion on cross-phase modulation in dispersion-managed wavelength-divisionmultiplexed systems, J. Lightw. Technol., vol. 22, no. 4, pp. 977 987, Apr. 2004. [14] M. Shtaif and A. Mecozzi, Study of the frequency autocorrelation of the differential group delay in fibers with polarization mode dispersion, Opt. Lett., vol. 25, no. 10, pp. 707 709, May 2000. [15] C. W. Gardiner, Handbook of Stochastic Methods, 2nd ed, vol. 13. Berlin, Germany: Springer-Verlag, 1985. [16] M. Shtaif, The Brownian-bridge method for simulating polarization mode dispersion in optical communications systems, IEEE Photon. Technol. Lett., vol. 15, no. 1, pp. 51 53, Jan. 2003. [17] M. Karlsson, Effect of nonlinearities on PMD system degradation, presented at the Optical Fiber Commun. Conf., Anaheim, CA, 2006, Paper OWE4. [18] M. Boroditsky et al., Comparison of system penalties from first- and multiorder polarization-mode dispersion, IEEE Photon. Technol. Lett., vol. 17, no. 8, pp. 1650 1652, Aug. 2005. [19] P. J. Winzer, H. Kogelnik, C. H. Kim, H. Kim, R. M. Jopson, L. E. Nelson, and K. Ramanan, Receiver impact on first-order PMD outage, IEEE Photon. Technol. Lett., vol. 15, no. 10, pp. 1482 1484, Oct. 2003. [20] A. Galtarossa, S. G. Someda, A. Tommasini, B. A. Schrefler, G. Zavarise, and M. Schiano, Stress birefringence in fiber ribbons, in Proc. OFC, Feb. 1997, pp. 267 269. [21] J. P. Gordon, Statistical properties of polarization mode dispersion, in Polarization Mode Dispersion, vol. 1, A. Galtarossa and C. Menyuk, Eds. New York: Springer-Verlag, 2005, pp. 52 59. Misha Boroditsky SM 06) received the M.S. degree in applied physics from St. Petersburg Polytechnic Institute, St. Petersburg, Russia, in 1993 and the Ph.D. degree in physics from the University of California, Los Angeles, in 1999. His Ph.D. dissertation was on the modification of spontaneous emissions in photonic crystals. After graduation, he worked for more than six years with the Optical Systems Research Department, AT&T Laboratories, on various aspects of access architectures, ultrafast optical packet switching, and polarization-mode dispersion. Since May 2006, he has been working in the field of quantitative finance with the Statistical Arbitrage Group, Knight Equity Markets, Jersey City, NJ. He has authored or coauthored more than 50 publications. He is the holder of six patents. Dr. Boroditsky served on the Optical Fiber Communications Technical Committee from 2004 to 2006 and was the Lasers and Electro-Optics Society Meeting Committee Chair for 2005 2006.

BORODITSKY et al.: EFFECT OF NONLINEARITIES ON PMD 4107 Marianna Bourd received the B.Sc. and M.Sc. degrees in physics from Tel-Aviv University, Tel-Aviv, Israel, in 2001 and 2006, respectively. She is currently a Lecturer with the Engineering Academy, Tel-Aviv University. Moshe Tur M 87 SM 94 F 98) received the B.Sc. degree in mathematics and physics from the Hebrew University, Jerusalem, Israel, the M.Sc. degree in applied physics from the Weizmann Institute of Science, Rehovot, Israel, and the Ph.D. degree from Tel-Aviv University, Tel-Aviv, Israel, in 1981. During the academic years 1981 1983, he was a Postdoctoral Fellow and then a Research Associate with the Information System Laboratory and the Edward L. Ginzton Laboratory, Stanford University, Stanford, CA, where he participated in the development of new architectures for single-mode fiber-optic signal processing and investigated the effect of laser phase noise on such processors. He is currently the Gordon Professor of electrical engineering with the School of Electrical Engineering, Department of Indisciplinary Studies, Faculty of Engineering, Tel-Aviv University, where he has established a fiber-optic sensing and communication laboratory. He authored or coauthored more than 250 journal and conference technical papers with emphasis on fiber-optic bit-rate limiters, fiber lasers, fiber-optic sensor arrays, the statistics of phase-induced intensity noise in fiber-optic systems, fiber sensing in smart structures, fiber Bragg gratings, polarization-mode dispersion, microwave photonics, and advanced fiber-optic communication systems. Dr. Tur is a Fellow of the Optical Society of America.