1. Introduction

Optical OFDM is a potential candidate for 100 Gbps⁺ systems using polarization multiplexing [1, 2]. However, inter-sub-carrier Four-Wave Mixing (FWM) between the OFDM subcarriers causes significant degradation at high launch powers [3]. Recently, we showed by simulation that this effect of fiber nonlinearity can be partially compensated in a Coherent Optical OFDM system (CO-OFDM) [4, 5] using nonlinearity precompensation [6], or a combination of precompensation and postcompensation [7]. Shieh, Ma and Tang have demonstrated postcompensation experimentally [8]. We have also shown that predistortion can be applied to direct-detection optical OFDM systems (DDO-OFDM) [9].

We have previously provided an analytical performance bound for the signal quality in CO-OFDM systems affected by fiber nonlinearity for moderate dispersion (D)×length (L) products [10]. However, practical demonstrations of CO-OFDM have operated with signal qualities in excess of our theory [1, 2]. More recently, Nazarathy et al. have explained this improvement in signal quality using a phase-array model for FWM efficiency in multiple-span systems [11]. This predicts resonances in the FWM efficiency due to the phases of the FWM contributions from each span. These resonances are similar to those observed by Schadt [12] and predicted by Ellis et al. [13], Chiang et al. [14] and Inoue et al. [15] for early optically-amplified multiple-span optical systems.

In this paper we show using simulations that the performance of nonlinearity precompensation can be improved by adding an electrical band limiting filter to the predistortion signal path. While introducing the theory behind this result, we confirm that the theory presented by Schadt [12], Ellis [13], Chiang [14] and Inoue [15, 16] is sufficient to predict the uncompensated signal quality per channel in CO-OFDM systems operating in the nonlinear limit.

2. Review of FWM theory

Inoue [15] and Ellis [13] proposed that the FWM efficiency, η, is dependent on the frequency separation between the three frequencies producing the fourth. For a link with uniform amplifier spacings and uniform frequency separation between the subcarriers, η is given by:

where: α is attenuation in Nepers/m, l ₀ is the fiber’s length and Δβ, for equally-spaced pump frequencies, is

where: i, j, k are integers denoting the locations of the pumps (optical subcarriers in OFDM) on a frequency grid discretised to Δf, c is the velocity of light, λ is the mean wavelength. Note that Δβ is proportional to the product of the two frequency differences between the three responsible subcarriers. Similar arguments were proposed by Schadt [12] and Chiang [14] (where a similar formula is presented) to explain nonlinear interactions between different WDM channels, due to cross-phase modulation (XPM); however, these arguments are equally valid for interactions between the subcarriers within an OFDM subcarrier band.

To help explain the efficiency factor, Fig. 1 plots the loci of the accumulated FWM field along a single span of S-SMF for 2 input pumps (subcarriers); that is where i=j (degenerate FWM). For each position along the fiber, the accumulated FWM field is the vector sum of the FWM fields produced in infinitesimal segments from the fiber input to that position. The segments close to the input of the fiber (at the graph’s origin) produce strong contributions to the FWM, as the FWM is proportional to the pump-power cubed. This cubed-power effect is partially mitigated by the loss that the FWM will incur as it travels to the summation point. The contributions of infinitesimal segments more than 30-km away from the fiber input becomes negligible because of the attenuation of the pump powers and the cube-power relationship. For narrow frequency separations of the pumps (–), the contribution of each infinitesimal segment will be in phase, leading to strong accumulated FWM. For wider frequency separations the contributions will evolve in relative phase (–), leading to spiraling towards a steady-state FWM field at >30 km.

 

Fig. 1. Loci of nonlinear fields for a single span. Six pump-frequency separations shown.

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In a multi-span system, optical amplifiers between each span restore the powers of the pumps. Figure 2 shows the loci of the accumulated FWM for three consecutive spans. Because the phase of the accumulated FWM at the output of an earlier span becomes the phase at the input of the next span, the FWM contributions of subsequent spans can constructively (3C), or destructively interfere (3B).

 

Fig. 2. Loci of the nonlinear fields for three spans.

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Figure 3 evaluates the product of Eq. (1) and Eq. (2) versus the geometric mean of the separations of three input tones for a 10×80 km span S-SMF system. The FWM efficiency is unity for small frequency separations, but reduces significantly at higher separations, apart from some narrow resonant situations such as the case shown in Fig. 2(C), where the FWM of each span adds, but dispersion still reduces the contribution of each span. It is obvious that if a significant proportion of the FWM products are generated from frequency separations greater than the first null in Fig. 3, then the analytical bound for received signal quality in [10] will be pessimistic since FWM efficiency is low. This also suggests that the compensation of FWM produced from widely spaced subcarriers should include this efficiency factor (as we will demonstrate in Section 4); otherwise the precompensation will produce distortion.

In a long link with multiple spans, destructive interference between the outputs of each span reduces FWM efficiency more than the walk off within each span. Thus systems with inline dispersion compensation fiber (DCF) will have minimal phase shift between each span and hence perform significantly worse than SMF links without DCF [17] or using lumped dispersion compensation [18]. Increasing the residual dispersion in each span is similar to increasing the dispersion of NZ-DSF in that it increases the phase shift between spans, thus improving the nonlinear performance [17].

 

Fig. 3. FWM efficiency versus geometric mean of the frequency separation of three pumps for an 800-km S-SMF system.

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3. Validation of theory

To validate that FWM theory is sufficient for predicting nonlinear degradation in optical OFDM systems, we compared the predicted signal qualities from our MATLAB model (similar to that in [10] but each FWM product is multiplied by the efficiency corresponding to the Δβ of the three tones that generated that product) to simulations using the nonlinear split step models of VPItransmissionMaker^™WDM. We used a subcarrier power density of -6 dBm/5 GHz for all bit rates, a subcarrier spacing of 10 MHz, 10×80-km spans, no Amplified Spontaneous Emission (ASE) noise, a nonlinear coefficient, γ, of 1.3/W/km and 4-QAM modulation of the subcarriers meaning that the bit rate is double the signal bandwidth.

 

Fig. 4. Simulated Q [dB] against signal bandwidth with a spectral density of -6 dBm/5 GHz for three fiber types. Split-step method (points) versus MATLAB evaluation (lines).

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Figure 4 plots the signal quality, Q, from our MATLAB model (lines) and VPI simulation (symbols) against subcarrier bandwidth: the advantage of high dispersion is apparent for high signal bandwidths. For optical OFDM systems, each doubling in signal bandwidth will require a doubling of signal power to maintain a given electrical signal to noise ratio. For low-dispersion systems at low data rates, our earlier analytic theory predicted that doubling power will reduce signal quality by 6 dB in the nonlinear-limited region [10], equivalent to the dispersionless case in Fig. 4. This penalty is greatly reduced by walk off for higher bandwidth systems; becoming 2 dB/doubling of subcarrier bandwidth for large walk offs. This explains why extremely high bandwidths can be supported using OFDM on closely-spaced WDM channels [1, 2], particularly if low-dispersion fibers are avoided [3]. The MATLAB predictions are in very good agreement with split-step simulations for reasonable Q-values, showing that the earlier analytic model is easily extended to include dispersion. The slight discrepancy at low Q’s is possibly a result of the MATLAB model not including the depletion of the pumps (subcarrier powers) in the FWM process.

4. Electrically filtered nonlinear precompensation

We have previously shown that nonlinearity precompensation works well for low dispersion fibers [6, 9], because the applied phase modulation (in proportion to the transmitted optical power) is a good prediction of the accumulated nonlinear phase shift for a significant proportion of the link. However, for high-dispersion fibers and long links, nonlinear precompensation is less effective. Unfortunately, precompensation itself induces a penalty as its phase modulation is converted to amplitude noise [9], by fiber dispersion [19]. Thus, the precompensation should be reduced below the value for zero-dispersion fibers to maximize signal quality, Q [7]. For S-SMF, precompensation only provides a small improvement in Q.

We propose an alternative way of reducing the distortion caused by precompensation; by bandlimiting the precompensation signal before applying the phase shift. This means that only the FWM products generated by close ‘pump’ subcarriers will be compensated for. To illustrate the effectiveness of bandlimiting, a precompensator with filter was added to the system presented in Section 3. The effective length for precompensation [6] was set to 20 km/span (zero dispersion fiber has an effective length of 21 km/span). A trapezoidal filter was used with a stopband attenuation of 40 dB and a transition bandwidth of half the passband’s width (Fig. 5). Two fibers were simulated: S-SMF (16 ps/nm/km) and NZ-DSF (2 ps/nm/km).

 

Fig. 5. Signal quality, Q, versus 0-dB electrical filter bandwidth of the precompensator.

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Figure 5 shows that systems with low signal bandwidth (B)×dispersion (D) products, such as 10 Gbps systems using NZ-DSF, improve with wider precompensation bandwidths. Note that 10 Gbps systems have an optical bandwidth of only 5 GHz, so filter bandwidths above 5 GHz (‘A’) have no effect. For higher bit rate systems, the signal quality will initially improve with filter bandwidth, however, high bandwidths reduce Q due to phase-to-intensity conversion [9]. For NZ-DSF systems, filtering is only beneficial above 30 Gbps. For S-SMF, filtering becomes beneficial above 20 Gbps; improvements of more than 3 dB can be achieved with a compensation bandwidth of 1.4 GHz. The optimal bandwidth reduces to 0.8 GHz (‘B’) for a 60 Gbps system using S-SMF. Thus only FWM from close subcarriers is compensated for.

5. Comparison of precompensation systems

Figure 6 summarizes the improvement in Q over our analytical bound [10] due to precompensation and dispersion for S-SMF. The optimal effective length is used for unfiltered precompensation (–). For filtered precompensation (–), the optimal result from Fig. 5 is used. Precompensation works best for low bandwidths where walk off is low. Filtering aids precompensation at bandwidths above 10 GHz: producing a total benefit of 7 dB for a 5-GHz channel, 3 dB at 10 GHz and 2 dB at 15 GHz.

For high signal bandwidths, the reduced nonlinear efficiency (Fig.3) provides a significant improvement in Q (–). This benefit unfortunately limits the effectiveness of compensation, even if an optimal filter is used. Further simulations using NZ-DSF showed similar trends though the curves were shifted rightwards to around 3 times the bandwidth of the S-SMF results. Thus NZ-DSF has its minimum delta-Q at a signal bandwidth of 30 GHz, so it is less suitable for 60 Gbps/pol systems compared to S-SMF.

 

Fig. 6. Increase in Q over the analytical bound in [10] versus signal bandwidth due to (filtered and unfiltered) precompensation and dispersion.

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6. Conclusions

This paper demonstrates that analytical bound in [10] in combination with established FWM theories are sufficient in predicting the performance of all CO-OFDM systems. A model was produced using MATLAB where each FWM product was multiplied by the efficiency term before the summation to find the total FWM power falling on a particular subcarrier.

This phenomenon allows us to improve nonlinearity precompensation by using an electrical lowpass filter to restrict the bandwidth of the compensation signal. This will allow precompensation to deliver a benefit for higher data rate channels. A filtered precompensator provides a 1.5-dB increase in Q for an 800 km, 60 Gbps/polarization S-SMF system.