A first order perturbation theory is used to develop analytical expressions for the power spectral density (PSD) of the nonlinear distortions due to intra-channel four-wave mixing (IFWM). For non-Gaussian pulses, the PSD can not be calculated analytically. However, using the stationary phase approximations, we found that convolutions become simple multiplications and a simple analytical expression for the PSD of the nonlinear distortion is found. The PSD of the nonlinear distortion is combined with the amplified spontaneous emission (ASE) PSD to obtain the total variance and bit error ratio (BER). The analytically estimated BER is found to be in good agreement with numerical simulations.
© 2012 Optical Society of America
In a highly dispersive fiber, a signal pulse broadens significantly and thereby, it interacts nonlinearly with a large number of pulses in its neighborhood. This nonlinear interaction leads to ghost or echo pulses which is known as intra-channel four-wave mixing (IFWM) [1–3]. The propagation impairments due to IFWM in direct detection systems are analyzed in Refs. [4, 5]. Recently, the modeling of nonlinear distortion in coherent fiber optic systems has drawn significant interest [6–12]. In Ref. , propagation impairment due to four wave mixing (FWM) in coherent orthogonal frequency-division multiplexing (OFDM) systems is studied and symmetries are used in the conventional FWM model. In Ref. , an analytical expression for the probability density function (PDF) of the IFWM impairments is derived for coherent fiber optic systems based on phase-shift keying (PSK). In Ref. , an analytic expression for the nonlinear threshold is found by assuming that signal pulses in each symbol slot are delta functions. In Ref. , an analytical expression for the power spectral density (PSD) of the nonlinear interference in a WDM system is developed. To evaluate the PSD, it is necessary to carry out a triple numerical integration and the computational cost scales as M3 where M is the number of samples in frequency domain. In this paper we have developed an analytical expression for the PSD of intrachannel nonlinear distortion. With analytic simplifications, we found that the computational cost scales as ∼ N2M/8 where N is the total number of significant neighboring signal pulses. Typically, N is smaller than M leading to significant reduction in computational time. However, the direct comparison between these two approaches is not appropriate since Ref.  primarily focused on interchannel impairments whereas this paper deals with intrachannel impairments only. In Ref. , a general first order perturbation theory of a multichannel optical transmission system is developed and stationary phase approximation is done to evaluate the cross-phase modulation fluctuations. In Ref. , it is proposed to apply a large predispersion to an optical signal before the fiber transmission and stationary phase approximation is employed to approximate the solution of nonlinear Schrodinger equation (NLSE) in the limit of very strong initial predispersion. In this paper, the stationary phase approach is used to approximate the Fourier transform of the echo pulse in a single channel so that the computational cost of the PSD calculations can be reduced. The stationary phase approximation translates convolutions into simple multiplications leading to a simple closed form expression for the spectrum of the echo pulse. As a result, the spectrum of the echo pulse is found to be proportional to the product of the signal pulse spectra shifted by the amounts proportional to the temporal positions of the signal pulses. Finally, the PSD of the nonlinear distortion is added to the PSD of the amplified spontaneous emission (ASE) and the integration of the PSD over the receiver bandwidth leads to the total variance which is used to calculate the bit error ratio (BER). In this paper, we consider only the case of single polarization. It is straightforward to extend the approach for the case of two polarizations.
This paper is organized as follows. Analytical expressions for the PSD of the nonlinear distortions are derived in section 2. Stationary phase approximation to calculate the spectrum of the echo pulse due to IFWM is also discussed in section 2. In section 3, the analytical expressions for the variance of the nonlinear distortion and BER are validated using numerical simulations. Finally, in section 4, the contributions of this work are summarized.
2. Mathematical derivation of power spectral density
Let the fiber input be7] 13], the field envelope can be expanded as Eq. (6), we find Eq. (7) is solved to yield, Eq. (5) with the initial condition given by Eq. (1) is Eq. (8), we have Eq. (11) in Eq. (14), we find Eq. (15) is assumed to be from −N/2 to N/2. Substituting Eq. (15) into Eq. (10), we find Eq. (17), Eq. (19) can be written as
The PSD can be divided into two groups. They are (i) non-degenerate intra-channel four-wave mixing (ND-IFWM), and (ii) degenerate intra-channel four-wave mixing (D-IFWM). For constant intensity modulation such as QPSK, it can be shown that self-phase modulation (SPM) and intrachannel cross-phase modulation (IXPM) produce only a deterministic phase shift, which can be removed by the electrical equalizer. So, in this paper, we ignore SPM and IXPM.
Let us first consider the case l ≠ m ≠ n and l′ ≠ m′ ≠ n′. For QPSK signals, we haveEqs. (21)–(23), Eq. (20) becomes, Equation (24) can be rewritten as
The signal pulses located at lTs, mTs, and nTs generate a echo pulse at qTs = (l + m − n)Ts [1–5]. Therefore, qth term on the right-hand side (RHS) of Eq. (25) represents the nonlinear distortion on the qth symbol interval. Due to symmetry, the ensemble average of the nonlinear distortion should be the same on each symbol interval. In the other words, each term on the RHS of Eq. (25) should be equal, which yields
Next, let us consider the case, l = m ≠ n and l′ = m′ ≠ n′. In this caseEqs. (28) and (29) in Eq. (20), PSD due to D-IFWM is
2.3. Correlation between D-IFWM and ND-IFWM
Consider the case l = m ≠ n and l′ = m′ ≠ n′. NowEq. (20), we have ρNL(f) = 0. In other words, there is no correlation between D-IFWM and ND-IFWM.
2.4. Total PSD
Total PSD is given byEqs. (26) and (31), respectively. For Gaussian pulses, X̃l,m,n(f) can be calculated as (see Appendix A)
2.5. Stationary phase approximation
For non-Gaussian pulse shapes, the convolution in Eq. (16) cannot be evaluated analytically. Due to the rapidly varying phase of p̃(f, z) in Eq. (13), stationary phase approximation can be employed to approximate X̃l,m,n(f). Stationary phase method is a standard technique for evaluating the integrals of the form Eq. (42), Eq. (41) may be approximated as Eq. (16), it has double convolutions which can be analytically integrated using the stationary phase approximation when the dispersion is sufficiently large. Now Eq. (16) becomes (see Appendix B) Eq. (16) is hard to evaluate numerically unless the pulse shape is Gaussian. But the stationary phase approximation translates the convolutions into simple multiplications as shown in Eq. (45), which can be easily computed. When the Nyquist pulse such as sinc pulse is used, Eq. (45) can be further simplified. A sinc pulse has a rectangular spectrum, Eq. (46), Eq. (45) can be approximated as
Xl,m,l+m is invariant under the exchange of l and m.Eq. (16)) even without the stationary phase approximation.
Since we have assumed that p(t) is real and symmetric, it follows that p̃(f) is symmetric and from Eq. (45), it is easy to see that there is a mirror symmetry,
Using the Wiener-Khinchin theorem, the variance is obtained asEq. (35), Eq. (53) may be written as Eq. (57) may be written as
2.7. Computational cost
Figure 1 shows the classification of intrachannel impairments for N = 6 and computational cost associated with SPM, IXPM and IFWM. When Property 1 and Property 2 are not used, the computational cost per frequency calculations are as shown in Table 1.
When Property 1 is used, the computational cost per frequency for ND-IFWM is N(N −1)/2. If both Property 1 and Property 2 are used, the cost per frequency for ND-IFWM and D-IFWM are N2/4 and N/2, respectively, and total computational cost per frequency (ND-IFWM + D-IFWM) is N2/4+N/2. If there are M samples in the frequency domain, total computational cost is (N2/4+N/2)M. In addition, if |l +m| > N/2, |n| > N/2, and from Eq. (26), it follows that the signal pulse centered at nTs with |n| > N/2 does not contribute significantly for the formation of the echo pulse at t = 0 and hence, such a triplet may be ignored. With this approximation, total computational cost scales as ∼ N2M/8 for large N. Validation of the stationary phase approximation is carried out in section 3.1.
3. Results and discussions
We carried out the numerical simulations of the fiber optic sytem using the split-step Fourier method in order to test the validity of our analytical model. The following parameters are used: fiber loss α = 0.2 dB/km, fiber nonlinear coefficient γ = 1.1 (W.km)−1, symbol rate = 25 Gbaud, and modulation = QPSK. Gaussian pulses with full width at half maximum (FWHM) of TFWHM = 20 psec are launched to the fiber to obtain Figs. 2, 3 and 6. Amplifiers spacing is 80 km. Multi-span fiber-optic system is simulated here. The dispersion is uncompensated in each span. At the receiver, the transmission fiber dispersion is fully compensated either optically or electrically. Laser phase noise, polarization effects, and the coherent receiver imperfections are ignored since the primary focus of this paper is to validate our analytical model for the fiber nonlinear impairments. For numerical simulation, a pseudo-random bit sequence (PRBS) of length 215 − 1 is used for the calculation of the PSD as well as BER. A Gaussian filter with a full bandwidth of 100 GHz is used as the receiver filter. The significant number of neighbors, N = 20. Equation (45) provides a guideline for choosing the frequency resolution. The minimum frequency shift of the pulse spectrum is πTs/δ. If the frequency resolution Δf is larger than πTs/δ, errors occur in the computation of X̃l,m,l+m(f, z). For a 20-span system and for |β2| = 21 ps2/km, Δf = πTs/δ = 0.189 GHz. With the 100 GHz bandwidth of the receiver filter, number of frequency samples, M = 527. Since N is much smaller than M, in this example the computational cost savings would be ∼ O ((527/20)2).
Figures 2(a) and 2(b) show the analytical and numerical variances as a function of the launch peak power for a 5-spans and 20-spans systems, respectively. To calculate the PSD numerically, we proceed as follows. The SPM and IXPM introduce a constant phase shift which is removed by multiplying the received signal by exp(iθ) where θ is found adaptively. We used adaptive least mean square (LMS) equalizer to compensate the phase shift. We assumed the following parameters for the LMS algorithm: Number of filter taps = 10, number of training sequence = 210, number of samples/symbol = 2, and step size = 0.1. The numerical PSD due to the nonlinear distortion is computed by subtracting the optical field envelope at the transmitter from that at the receiver (after dispersion compensation and the phase shift removal) and then taking the Fourier transform of the difference. To account for the bit-pattern variations, numerical simulations were performed 20 times with different bit patterns and the average PSD is computed. Over a range of powers that is of practical interest for QPSK-based system (−6 dBm to 0 dBm), the discrepancy between the analytical model and the numerical model is less than 4% and 12% in 5-span (Fig. 2(a)) and 20-span (Fig. 2(b)) systems, respectively. Figures 3(a) and 3(b) show the analytical and numerical variances versus the accumulated dispersion for 5-spans and 20-spans systems, respectively. As can be seen, there is a good agreement between numerical simulations and analytical results for a 5-span system. For a 20-span system, there is a small discrepancy at large launch powers which is probably due to the truncation of the field up to the first order (see Eq. (4)). For the 5-span system, when the dispersion is small, the variance of the nonlinear distortion is quite small. However, it grows quickly and beyond 7 ps/nm.km, it decays slowly. For the 20-span system, the variance decreases slowly with the transmission fiber dispersion.
3.1. Stationary phase approximation with raised-cosine pulse
The raised-cosine spectrum is commonly used in communication because of its compact spectrum. In this case, p̃(f) is of the form ,Eq. (45)), total PSD and the variance is calculated. The following parameters are used for raised-cosine pulse. Roll-off factor a = 1 and symbol time interval Ts = 40 psec are assumed. Figures 4(a) and 4(b) show the variance as a function of the launch peak power for a 5-span and 20-span systems, respectively. In the range of −6 dBm to 0 dBm, the discrepancy between the analytical model and the numerical model is less than 7% in Figs. 4(a) and 4(b). Figures 5(a) and 5(b) show the dependence of the variance on the fiber dispersion for a 5-span and 20-span systems, respectively. For a 5-span system, when the dispersion is very low, we see that the stationary phase approximation becomes inaccurate. This inaccuracy is due to the fact that the phase (∝ β2 f2) does not vary rapidly at low dispersions. However, practical fiber-optic systems use fibers with moderate to large dispersions to suppress nonlinear effects and therefore, stationary phase approximation leads to reasonably accurate results for dispersion parameter range that is of practical interest.
3.2. BER calculation
In this section, analytical BER is compared with numerical BER. For the analytical BER, total noise variance is calculated first and then SNR is obtained. Ignoring the interplay between the amplifier noise and nonlinearity (so called Gordon-Mollenauer noise), total PSD is given byEq. (59), total noise variance σtot is calculated and the probability of error Pe for QPSK is given by  15] and SNR is the signal to noise ratio,
Figure 6 shows the analytical and numerical BER versus the launch power. We assumed the following parameters: nsp = 10 dB, and number of spans = 20. We chose a relatively large noise figure intentionally so as to reduce the computational time of Monte Carlo simulations at large launch power. We found that the maximum discrepancy between the analytical and numerical Q-factor is less than 0.6 dBQ20 which is attributed to non-Gaussian distribution of the IFWM pdf . In Eq. (61), it is assumed that the noise is Gaussian distributed. However, in Ref. , it is shown that the pdf of intrachannel impairments are asymmetric and non-Gaussian. Ref.  has modeled the pdf of the nonlinear interference as Gaussian distribution. When the perturbation includes the Gaussian-distributed ASE and the non-Gaussian-distributed IFWM, accurate evaluation of the BER may be carried out using the Gram-Charlier technique , which would be the subject of a future investigation.
We have developed analytical expressions for the PSD of the nonlinear distortions due to IFWM. Combining this PSD with that of the ASE, BER is estimated analytically which is found to be in good agreement with numerical simulations. For non-Gaussian pulse shapes, the spectrum of the the echo pulse can not be calculated analytically. When the phase is varying rapidly as in the case of a high dispersion transmission fiber, stationary phase approximation can be employed to calculate the spectrum of the echo pulse and hence, the PSD can be calculated analytically. The stationary phase approximation translates convolutions into simple multiplications leading to a simple analytical expression for the spectrum of the echo pulse. These analytical expressions significantly reduce the computational time to estimate the BER.
5. Appendix A: Gaussian pulse case
For a Gaussian pulse shape, we haveEquation (16) can be rewritten as Eq. (68) in Eq. (76), we find Equation (77) may be rewritten as Eq. (79) in Eq. (78), we find Eq. (80) in Eq. (75) and noting that Eq. (75) becomes
6. Appendix B: Stationary phase approximation
From Eq. (76), we haveEq. (93) comes when Eq. (95) in Eq. (92), we find Eq. (93) becomes Eqs. (95) and (98) in Eq. (97), we find Eq. (99) in Eq. (90), we find Eq. (100) in Eq. (75) and simplifying, we obtain Eq. (102). Differentiating Eq. (104) with respect to f2 and setting the result to zero yields Eq. (102) can be simplified as
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