In this paper, we use numerical simulations to show that the symbol rate has a significant effect on the nonlinearity-limited performance of coherent optical communication systems. We consider the case where orthogonal subcarriers are used to maximize the spectral efficiency. Symbol rates from 0.78125 Gbaud to 100 Gbaud and links of up to 3200 km, without inline dispersion compensation, were simulated. The results show that the optimal symbol rates for the 800-km link and 3200-km link were 6.25-Gbaud and 3.125-Gbaud respectively. The optimal baud rate decreases as the length of the link is increased. After 3200 km, the performance of the 100-Gbaud system was worst in the nonlinearity-limited regime producing a received Q 2.4-dB lower than the 3.125-Gband system. The variation in the nonlinearity-limited performance is explained by using Cross-Phase-Modulation (XPM) theory and by considering the RF spectra of the intensity fluctuations of the signal along the link. The findings of the paper suggest that the maximum capacity of nonlinear dispersive optical links can only be achieved by using multiple subcarriers carrying a few Gbaud each, and not by high symbol rate systems.
©2011 Optical Society of America
Coherent detection with digital equalization enables compensation of most optical effects, such as chromatic dispersion (CD), polarization mode dispersion (PMD) and fiber nonlinearity . However, compensating for fiber nonlinearity is very computationally intensive as it requires split-step methods . Because the number of computations increases with signal bandwidth, inter-channel fiber nonlinearity compensation was not used in , where the ultimate capacity of an optical link was estimated. This has lead to the proposal of a Nonlinear Shannon’s Limit (NSL) . However, these works have not explored the effect of channel granularity has on fiber nonlinearity.
In 1996, Forghieri showed that the performance of on-off keying (OOK) systems, carrying data at any given rate, would be affected by the channel granularity in a nonlinear channel . If this is also true for coherently detected systems, then transmitting information on a single high baud-rate carrier on each wavelength  may be sub-optimal. An alternative is to transmit information on multiple orthogonal subcarriers, where the symbol rate is equal to the subcarrier spacing, which will give the same spectral efficiency for any given constellation. Coherent optical OFDM (CO-OFDM) systems use digital equalization to avoid linear crosstalk between orthogonally spaced subcarriers . Electrically generated CO-OFDM typically uses long symbols with hundreds of closely-spaced subcarriers [1, 7]. In contrast, no-guard-interval (No-GI) CO-OFDM typically uses fewer optically generated subcarriers with a higher symbol rate . Linear crosstalk can also be avoided optically by phase matching the optical carriers as in coherent WDM  or by tight optical filtering as in Nyquist WDM . Demonstrations have shown that a large number of orthogonal subcarriers can be used to form a continuous spectrum capable of terabit/s transmission and beyond . These technologies allow the granularity of the subcarriers to be decoupled from the spectral efficiency and the total data throughput.
In this paper, we investigate the effect that subcarrier granularity has on the nonlinearity-limited performance of coherent optical communication systems without inline dispersion compensation. We simulated a super-channel with multiple orthogonal subcarriers. We find that the symbol rate causes nonlinearity limited performance to vary by over 2 dB for all simulated distances. The optimal symbol rate is 6.25 Gbaud for an 800-km standard single mode fiber (SMF) system: the optimal symbol rate decreases to 3.125 Gbaud for a 3200-km system. This shows that the optimal symbol rate decreases with transmission distance. This phenomena can be explained with Cross-Phase-Modulation (XPM) theory  and by observing the intensity fluctuations along the link. These findings suggest the performance of transmission systems can be improved by optimizing the symbol rate of the subcarriers for each link and granularity should be considered when estimating the capacity of optical links.
2. Simulation setup
Figure 1 shows the simulation setup. Each subcarrier was modulated with QPSK using separate Complex Mach-Zehnder Modulators (C-MZM) which generates an optical spectrum similar to that shown in Fig. 1a. Each optical subcarrier was passed through a rectangular optical filter with a passband equal to the symbol rate (a Nyquist filter); this reduced the bandwidth of the signal as shown in Fig. 1b. This extreme truncation in the frequency-domain causes each time-domain signal to be sinc-like in shape (near Nyquist pulse). Symbol rates from 0.78125-Gbaud to 100-Gbaud were simulated, where subcarrier spacing is equal to the symbol rate. An ideal optical multiplexer combined the orthogonally-spaced subcarriers to form a continuous 400-GHz wide optical super-channel carrying 800 Gb/s on a single polarization. Figure 1c shows the spectrum of the super-channel for a 50-Gbaud system. The ideal multiplexer and Nyquist filters prevented any linear crosstalk. This is similar to an ideal Nyquist WDM system  except that our optical filter does not suppress the center of each subcarrier in order to make the optical spectrum flat. Signals were limited to a single polarization.
We consider links without inline dispersion compensation because CD can be compensated digitally in coherently-detected systems. The fiber spans were 80-km each. Links of 800 km, 1600 km and 3200 km were considered. The attenuation of the SMF spans was compensated with EDFAs with noise figures of 5 dB. PMD was not simulated.
A separate coherent receiver was used for each subcarrier . Another Nyquist filter was used to remove neighboring subcarriers. Figure 1d shows an example optical spectrum after the receiver filter. The local oscillator frequency was identical to that of the transmit laser. Ideal zero-linewidth lasers were simulated. Analog-to-digital converters (ADC) sampling at two samples/symbol were used for all systems. In the digital signal processor, the bulk of the CD was compensated with a frequency domain equalizer . Then a 12-tap fractionally-spaced time-domain equalizer (FS-TDE) was used to compensate for residual CD and to perform the required downsampling . The least-mean-squares algorithm (LMS) was used to estimate the channel response.
A total of 218, or 262144, symbols were simulated. This equates to 65536 symbols per subcarrier for the 100-Gbaud system and 512 symbols per subcarrier for the 0.78125-Gbaud system. The Q was calculated from the spread in the received constellation assuming a Gaussian distribution in each Cartesian coordinate; the Q values for subcarriers within the central 200 GHz were averaged.
3. Simulation results
Figure 2 shows the received Q against the launch power for 1600-km and 3200-km systems at four different symbol rates. At low powers, the systems are limited by amplified spontaneous emission (ASE). In this region, the Q is almost identical for all systems. The 1.5625-Gbuad system was slightly poorer because of the long impulse response of the filters causing degradation on a large number of symbols. This effect was even greater for the 0.78125-Gbaud system, shown in Fig. 3 . At high powers, the systems are limited by fiber nonlinearity. The spread of the Q is over 2 dB for launch powers of 2 dBm and above. This spread shows that the nonlinearity-limited performance is dependent on the symbol rate of the subcarriers.
Figure 3 plots the received Q against subcarrier carrier symbol rate after 800 km, 1600 km and 3200 km at a launch power of 2 dBm, which is in the nonlinearity-limited regime of Fig. 2. At 800 km, the optimal symbol rate is 6.25 Gbaud; at 1600 km, the optimal symbol rate reduces to between 3.125 and 6.25 Gbaud; at 3200 km, the optimal symbol rate is 3.125 Gbaud. These results show the optimal symbol rate decreases as the transmission distance is increased. These optimal symbol rates are consistent with an independent study conducted in , where the optimal subcarrier spacing of a 1000-km eight-wavelength 107-Gb/s DFT spread OFDM system was found to be around 3.6 GHz. These results show the variation in Q across with symbol rate is significant for all transmission distances: choosing the correct baud rate will improve system performance.
The nonlinear Kerr effect causes intensity fluctuations to induce phase modulation. However, simply minimizing the Peak-to-Average-Power-Ratio (PAPR) at the transmitter will not achieve optimal performance in a dispersive link [13, 14]. This can be explained by using XPM theory  and by considering the RF spectra of the intensity fluctuations of the signal along the link. Whilst simulations of this paper are limited to sinc-like symbols, this theory applies to all coherent systems regardless of the pulse shape because the pulse spreading and distortion caused by CD dominates over the variations in pulse shape for different systems.
XPM theory states the efficiency of the nonlinearity-induced phase modulation is reduced for high-frequency intensity fluctuations. The efficiency of XPM for regular fiber spans without inline dispersion compensation is given by :12], ω is the frequency of the intensity fluctuation in rad/s and N is the number of spans in the link.
Firstly, the XPM efficiency for high frequency-intensity fluctuations is low because CD decorrelates the nonlinear products generated at different points along the fiber. This is because the fields of the nonlinearity products generated along the link add vectorially so the net phase modulation is reduced by this decorrelation : Low-frequency intensity fluctuations induce strong net phase modulation because the nonlinearity products remain phase-aligned (correlated). Therefore, it is only the low-frequency intensity fluctuations that are important. Secondly, the nonlinearity products are generated throughout the entire link, most strongly just after the optical amplifiers where the optical power is greatest. Because CD causes the signal waveform to evolve along the link, intensity fluctuations should be minimized throughout the link . Putting these observations together, we should focus on reducing the low-frequency components of the intensity fluctuations just after each amplifier.
To explore this idea, the frequency content of the intensity fluctuations for different baud rates were investigated by simulation, at different points along the link. Figure 4 shows the RF power into a 1-A/W photodiode into a 1-ohm load. Figure 4a shows the intensity fluctuations of all four systems at the transmitter. These were lowest at DC and increased with frequency at different rates (with respect to frequency). The lowest baud-rate transmitters had the most RF power at low frequencies. This is expected because QPSK is a constant-modulus modulation format so should suppress intensity fluctuations at frequencies below the symbol rate.
After 80 km of fiber, the intensity spectra have different forms as shown in Fig. 4b. CD has caused the low-frequency intensity fluctuations of the 100-Gbaud system to be as strong as in the 1.5625-Gbaud system, even at frequencies below the baud rate. The intensity fluctuations of the 25-Gbaud system have also increased to similar levels as the 6.25-Gbaud system. CD causes the symbols to broaden, which means the low-frequencies of one symbol will overlap with the high-frequencies of a following symbol; overlapped symbols beat together to produce strong intensity fluctuations. For any amount of CD, the number of overlapped symbols increases quadratically with the symbol rate. For example, 400 km of S-SMF will cause 512 adjacent symbols to overlap in a 100-Gbaud system whereas only two adjacent symbols will overlap in a 6.25-Gbaud system. Therefore, the intensity fluctuations will increase most rapidly in systems with high symbol rates. This means that although the 100-Gbaud system will generate the smallest nonlinearity products in the first span, it will generate the largest nonlinearity products of the four systems shown in Fig. 4 in all subsequent spans. This explains why its performance at high powers was the poorest for all lengths, as shown in Fig. 2.
Figure 4c shows that for a transmission distance of 400 km, the intensity fluctuations of the 25-Gbaud system are stronger than the 1.5625-Gbaud system. Also, the 6.25-Gbaud system now has the weakest low-frequency intensity fluctuations. This also explains why the 6.25-Gbaud system has the best performance in 800-km and 1600-km links. After 1600 km, the intensity fluctuations of the 6.25-Gbaud system have increased to a similar level to those of the 1.5625-Gbaud system as shown in Fig. 4d. Therefore, the 6.25-Gbaud and 1.5625-Gbaud systems have a similar performance for 3200 km as shown in Fig. 2 and Fig. 3.
A split-step nonlinearity compensator could be used to compensate for all signal-signal induced fiber nonlinearity if the entire optical signal is encapsulated in a single digital signal . In this case, ASE-signal Four-Wave-Mixing (FWM) will determine the nonlinear limit . Also, we would expect the nonlinear limit to be independent of the granularity. However, if the split-step method operates on sub-bands of the optical signal with a certain bandwidth , then only the fiber nonlinearities within each sub-band can be compensated. From the results of Fig. 4, we expect that an optimized granularity will produce less nonlinear mixing between each sub-band. Therefore, we expect granularity to have an effect on systems using nonlinearity compensation on sub-bands.
In this paper, we have shown that the subcarrier symbol rate has a significant effect on the nonlinearity limited performance of coherent optical systems. The results suggest that the optimal symbol rate decreases for longer links. This result can be explained by considering the spectra of the intensity-fluctuations together with the XPM efficiency. An important conclusion is that the symbol rate should be considered when investigating the ultimate capacity of a long-haul optical link: simply transmitting at the maximum symbol rate allowed by the electronics will not be optimum. That is, it may be possible to increase the capacity of a 100-Gbaud system  by decreasing the symbol rate and so increasing its granularity.
This work is supported under the Australian Research Council’s Discovery funding scheme (DP1096782).
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