On the Statistics of Intra-Channel Four-Wave Mixing (IFWM)

1. On the Statistics of Intra-Channel Four-Wave Mixing (IFWM) in Phase-Modulated Optical Communication Systems Alan Pak Tao Lau, Sahand Rabbani and Joseph M. Kahn Department of Electrical Engineering, Stanford University Probability Distribution of IFWM-induced Phase Noise Motivations Correlation Properties of IFWM • IFWM-induced phase noise and amplitude noise on bit slot 0 • Coherent/differentially coherent transmission systems are needed to achieve higher spectral efficiency. • Interaction of chromatic dispersion and Kerr nonlinearity results in intra-channel four-wave mixing (IFWM), a form of nonlinear intersymbol interference. • Better understanding of IFWM statistics in phase-modulated systems is essential to characterize and improve system performance in PSK/DPSK systems. • Correlation function of IFWM-induced phase noise • Insight: terms in are pairwise independent. For example, • are independent of each other • Consequence of modulo addition of phase of • Terms in are not jointly independent, but treating them all as independent random variables allows us to obtain an approximation for the probability distribution of Intra-Channel Four-Wave Mixing (IFWM) in Phase-Modulated Systems • It can be analytically shown that and are uncorrelated except for BPSK modulation format (in contrast with nonlinear phase noise). • Scatter plots of and • Originally known as ‘ghost pulse’ in bit ‘0’ for On-Off Keying (OOK) systems • Causes phase as well as amplitude noise in phase-modulated systems • Consider pulse train Pulse shape QPSK DQPSK • Nonlinear Schrödinger equation (NLSE) Phase modulated info. • Tail probability of IFWM-induced phase noise and receiver phase noise Nonlinear perturbation • First order perturbation theory: QPSK 8-PSK BPSK Linear solution to NLSE Signal Processing Techniques to Combat IFWM-induced Phase Noise • Previously observed that IFWM-induced phase noise across neighboring symbols are correlated in phase-modulated systems • Optimal linear phase noise predictor based on R(k) QPSK DQPSK • Analytical approximations correspond very well with the true distribution and tail probability References Ho, PTL vol. 17, no. 4, 2005 Wei et al., Optics Letters, Vol. 28, no. 23, 2003 • X. Wei and X. Liu, ``Analysis of intrachannel four-wave mixing in differential phase-shift keying transmission with large dispersion," Optics Letters, vol. 28, no. 23, pp. 2300-2302, Dec. 2003. • K.P. Ho, ``Error probability of DPSK signals with intrachannel four-wave mixing in Highly Dispersive Transmission Systems," IEEE Photon. Tech. Letters, vol. 17, no. 4, pp. 789-791, Apr. 2005. Problem: • No analytical knowledge of the correlation structure of IFWM-induced phase noise nor its probability distribution • Need correlation structure to potentially minimize IFWM • Need probability distribution to characterize system performance • 1.8 dB improvement when dominates • 0.8-1.2 dB in presence of ASE phase noise