Nonlinear polarization-mode dispersion

Abstract

258 0 Thursday Morning tively, LB = d b is the beat length), and the terms E and N contain random coefficients driven by the random birefringence fluctuations.'The first term on the right-hand side corresponds physically to the usual linear polarization-mode dispersion while the second term, which we refer to as nonlinear polarization-mode dispersion, is due to incomplete mixing of the nonlinearity on the Poincare sphere. We show here that this term is unimportant for current communication systems but can become important for high-bit-rate time-division multiplexing (TDM) systems approaching 100 Gbitds.In the simplest model of randomly varying birefringence, the birefringence strength is assumed to be constant and the orientation angle 6 assumed to be given by a white noise process. This model has been shown to capture all of the essentials of the more realistic case where both the orientation angle and the birefringence strength vary randomly.' Variations in the orientation angle with distance drive the birefringence fluctuations, and these birefringence fluctuations in turnproduce polarization variations in the electromagnetic field envelope W. A natural way to track the polarization state is by using Stokes' parameters (.$, S, , s,), which satisfy the equations.'where g0 is the white noise process, (gO(z)) = 0 and (@(z)gO(z')) = u$(z-z'), and where (SI, S,, 3, ) = (LO, 0) at z = 0. Note that the fiber correlation length is h+, = 2/u$' It can be shown that the random coefficients present in N satisfy the same equations as the second moments of the Stokes' parameters.'In order to quantitatively determine how the polarization fluctuations couple with the nonlinear terms, it is necessary to work out the statistics of the second moments of the various coefficients 3,. One way to do this is to use averages of the coefficients and their moments, as determined from Eq. (2). Alternatively, one can work with the Fokker-Planck equation for the probability distribution of the polarization state. The field fluctuations produced by the randomly varying polarization are seen from Eq. (1) to be given by integrals of the coefficients present in N, and are therefore proportional to integrals of the second moments of the Stokes parameters.We have calculated the means and variances of the various field fluctuations produced by the random terms in Eq. (1). For example, there is a term that corresponds to J S: dz. Since (S:) = 1/3 (apart from an initial transient) the mean of the integrated term is straightforward to calculate.To calculate Var (JS: dz), we want which for large z is approximately (7) Here R(t -5') = ((S:(t) -:)(St(i') -3) is the autocorrelation function for S: and the average is over all possible states on the Poincare sphere. Note that R(6) is nonzero only for 6 -0. In the limit hfiber < < LB, we have' R(C) = (S:(O) -: ) exp (-24bz/uz) and((Sg(0) -3) = 4/45, so that OFC '97 Technical Digest Similarly, when hfiber > > LB. Var (J S: dz) -2hfiber d135.We thus conclude that nonlinear polarization-mode dispersion become...