2916 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 14, JULY 15, 2009 Intra-Channel Four-Wave Mixing Impairments in Dispersion-Managed Coherent Fiber-Optic Systems Based on Binary Phase-Shift Keying Dong Yang and Shiva Kumar Abstract—In this paper, we mathematically derive the proba- bility density function (PDF) of the received electrical signal for coherent fiber-optic transmission systems. Both amplified sponta- neous emission (ASE) noise and fiber nonlinearity are taken into account. The results show that the PDF of a bit “0” or “1” is asym- metric when intra-channel four-wave mixing (IFWM) is dominant. However, the PDF becomes nearly symmetric when the variance of ASE is much larger than that due to IFWM. The standard devia- tion of the received signal is calculated analytically. It is shown that the variance varies quadratically with launch power. We also inves- tigate the optimum system scheme, including optimum dispersion map and pre-compensation ratio, for the coherent fiber-optic sys- tems based on analytically calculated variance of IFWM. Index Terms—Fiber-optic communication systems, intra- channel four-wave mixing (IFWM), probability density function (PDF). I. INTRODUCTION T HE statistical characteristic of received bits is a funda- mental concern in communication systems. In long-haul fiber-optic communication systems, because of the fiber non- linearity, it is usually hard to obtain the exact analytical ex- pression for the probability density function (PDF) of received bits. The pulsewidth and amplitude of pulse fluctuate due to intra-channel self-phase modulation (ISPM) and also the non- linear interaction of adjacent pulses causes phase modulation of the probe pulse due to intra-channel cross-phase modulation (IXPM), which is translated to timing delay because of disper- sion [1]–[3]. Furthermore, the nonlinear mixing of overlapped pulses generates ghost pulses due to intra-channel four-wave mixing (IFWM), which is one of the dominant penalties for high bit rate fiber-optic systems ( 40 Gb/s) [4]–[13]. Suppose we have three consecutive bits of “1” centered at , and 0. The ghost pulse generated by the nonlinear mixing (IFWM) of pulses at and interferes with the pulse at . This interference leads to the amplitude jitter of the pulse at . In this paper, we consider the impact of ISPM, IXPM, and IFWM in a coherent fiber-optic system based on bi- nary phase-shift keying (BPSK). In many occasions fiber links can be seen as quasi-linear sys- tems, if transmitted power is well controlled such that the non- Manuscript received October 13, 2008; revised February 19, 2009. First pub- lished April 28, 2009; current version published July 22, 2009. The authors are with the Department of Electrical and Computer Engi- neering, McMaster University, Hamilton, ON L8S 4K1, Canada (e-mail: yangd5@mcmaster.ca; kumars.mail.ece.mcmaster.ca). Digital Object Identifier 10.1109/JLT.2009.2019610 linear effect of fiber is not too large. Our work is based on this assumption and the nonlinearity of a fiber is described by the first-order perturbation approximation [7], [13]. In this paper, we derive analytical expressions for PDFs and variances of bit “0” and bit “1” for dispersion-managed coherent fiber-optic sys- tems based on BPSK. Our results show that the probability den- sity functions for both bit “1” and bit “0” are of asymmetric shape when fiber nonlinearity is dominant impairment over am- plified spontaneous emission (ASE) noise. One might expect that the PDFs should be symmetric since bit “0” (amplitude 1) and bit “1” (amplitude 1) occur with equal probability for sys- tems based on BPSK. But the conditional PDF of the received signal given that bit “1” (or bit “0”) is sent, is calculated by fixing the bit “1” (or bit “0”) in the bit slot 0 and other bit slots carry bit “1” or bit “0” with equal probability. The non-degen- erate symmetric four-wave mixing triplet involving the pump pulse in bit slot 0 (which is fixed) is responsible for making the PDFs asymmetric. However, the PDFs become nearly sym- metric as ASE noise increases. Recently, auto correlation func- tions and approximate probability density function of IFWM in- duced phase noise have been obtained in [15] using a different approach. After numerically validating our analytical expres- sion for the variance of IFWM, we used it as a design tool to optimize the various parameters such as pre-, post-, and inline- dispersion compensation and launch power of a dispersion-man- aged coherent fiber-optic transmission system. Our results show that as the average dispersion of the dispersion managed trans- mission fiber becomes large, the optimum dispersion compensa- tion ratio approaches 0.5. This implies that dispersion compen- sation at the transmitter (pre-compensation) should be roughly same as that at the receiver (post-compensation) for all the dis- persion maps analyzed in this paper when the average dispersion of the fiber-optic link (excluding pre- and post-compensation) is large. The ASE noise introduced by an amplifier fluctuates the en- ergy of a pulse. The fiber nonlinearity translates the energy fluc- tuations into phase fluctuations leading to nonlinear phase noise or Gordon–Mollenauer phase noise [16]. In this paper, we ig- nore the nonlinear phase noise and mainly focus on the ampli- tude fluctuations due to IFWM. It is likely that the dispersion map that minimize the IFWM impairments would also mini- mize nonlinear phase noise. However, this would be the subject of a future investigation. In Section II, mathematical derivation of PDF is given, in which first-order perturbation theory is used to solve the non- linear Schrödinger equation (NLS). In Section III, the variance of IFWM is analytically calculated. In Section IV, simulations 0733-8724/$26.00 © 2009 IEEE