Pulse Collision Picture of Inter-Channel Nonlinear Interference in Fiber-Optic Communications

Abstract

We model the build-up of inter-channel nonlinear interference noise (NLIN) in wavelength division multiplexed systems by considering the pulse collision dynamics in the time domain. The fundamental interactions can be classified as twopulse, three-pulse, or four-pulse collisions and they can be either complete, or incomplete. Each type of collision is shown to have its unique signature and the overall nature of NLIN is determined by the relative importance of the various classes of pulse collisions in a given WDM system. The pulse-collision picture provides qualitative and quantitative insight into the character of NLIN, offering a simple and intuitive explanation to all of the reported and previously unexplained phenomena. In particular, we show that the most important contributions to NLIN follow from twopulse and four-pulse collisions. While the contribution of twopulse collisions is in the form of phase-noise and polarizationstate-rotation with strong dependence on modulation format, four-pulse collisions generate complex circular noise whose variance is independent of modulation format. In addition, two-pulse collisions are strongest when the collision is complete, whereas four-pulse collisions are strongest when the collision is incomplete. We show that two-pulse collisions dominate the formation of NLIN in short links with lumped amplification, or in links with distributed amplification extending over arbitrary length. In long links using lumped amplification the relative significance of fourpulse collisions increases, emphasizing the circularity of the NLIN while reducing its dependence on modulation format. Index Terms-fiber nonlinearity, NLIN, inter channel interference, XPM, pulse collision, dependence on modulation format, phase noise.1 Other time-domain models for inter-channel effects were considered in [11]- [13]

Modeling the Bit-Error-Rate Performance of Nonlinear Fiber-Optic Systems

“…At the link's end, the COI is detected by a receiver, which is assumed to perfectly compensate for the linear link impairments, so that the received symbols are expressed as |a n + |∆a n , where |∆a n is used instead of |∆a XPM n , in order to simplify notation, as only the XPM contribution is considered analytically in this section. It has been shown in [1,10] that within the first order perturbation analysis,…”

“…In principle, the sums S 1 and S 2 can be evaluated directly [7] based on the knowledge of the X h,k,l coefficients that were derived in [1]. This entails calculating the coefficient for each combination of four pulses that overlap with the n-th pulse [10]. While the computation of a single coefficient involves a simple double integral and can be performed rapidly, the number of pulse interactions can be very high, quickly reaching the point of requiring the computation of thousands of coefficients, and rendering this procedure extremely inefficient.…”

“…The main computational cost is in calculating the sums of Eqs. (10,11), as they describe the full autocorrelation function of the ISI matrices, and are therefore composed of hundreds of coefficients. While the brute-force computation of these sums is impractical, using the method described here they can be evaluated in the order of one minute per WDM channel on a current desktop computer.…”

“…(10,11), h =k X h−n,k−n,l X * h−n ,k−n ,l = h,k X h−n,k−n,l X * h−n ,k−n ,l − k X k−n,k−n,l X * k−n ,k−n ,l = S 2 − S 1 , (29) and using the relations…”

“…Furthermore, the impact of these temporally correlated NLIN contributions may be reduced when using a receiver that employs fast adaptive channel equalization [7][8][9], a fact that should be taken into account when assessing the BER so as not to underestimate system performance. In addition, in certain cases a notable fraction of the NLIN variance can be attributed to phase and polarization rotation noise (PPRN) [10][11][12][13], whose effect on the BER is substantially different from that of AWGN.…”