The nonlinear power limit of optical links using optical Orthogonal Frequency Division Multiplexing (OFDM) for dispersion compensation can be significantly improved using an optimum combination of nonlinearity precompensation and postcompensation. The compensation is implemented at the transmitter and at the receiver as computationally-efficient power-dependent phase shifts with a single tuning parameter. The system is robust against the exact details of the fiber plant’s dispersion and power levels. Using an optimum combination of pre and post compensation allows a 2-dB increase in launch power for 2000-km standard single-mode fiber (S-SMF) systems and 5-dB when 6 ps/nm/km fibers are used. Using pre or post compensation alone approximately halves these values.
© 2007 Optical Society of America
Electronic Dispersion Compensation (EDC) ,  is of growing interest because it self-adapts to any fiber system, including dynamically-switched optical networks, so reduces engineering and inventory costs. We have previously proposed  and demonstrated  using Orthogonal Frequency Division Multiplexing (OFDM) combined with optical-single-sideband transmission for EDC; however, our simulations showed fiber nonlinearities restricted the optical power launched into each fiber span , limiting transmission to 4000 km of standard single-mode fiber (S-SMF) for 80-km amplifier spacing. A 2-dB increase in launch power could support a 10-km increase in span length at 1550 nm, so any compensation of fiber nonlinearity is useful. Recently, I proposed compensating nonlinearity by modifying the transmitter ; this allowed higher powers and reduced Bit Error Ratios (BERs).
Fiber nonlinearity compensation was first proposed in 1996 using materials with a negative nonlinear coefficient , which are impractical. EDC has allowed this idea to be implemented virtually , , . Ideally, the signals are first propagated through a detailed numerical model of a negative nonlinearity fiber, and then the output of the model is fed into the real system using an optical modulator that is able to generate a time-varying complex optical field. Unfortunately, this requires a detailed a priori knowledge of the system’s dispersion map and optical power map, together with extensive computation. Some simplified approaches to nonlinearity precompensation have been proposed. Goeger  applies a constant optical phase shift to each RZ bit dependent on the two adjacent bits. This allows a 2-dB increase in launch power to 2 dBm for a 700-km S-SMF system. Liu and Fishman  proposed a simplified calculation for the inverse system model, using one step of a split-step fiber model to represent up to 2 fiber spans.
Fiber nonlinearity has also been post-compensated, that is, at the receiver. Unlike precompensation, this can follow rapid variations along the fiber plant, such as power variations. For example, Xia and Rosenkranz  have proposed a nonlinear equalizer stage in a feed-forward/decision-feedback equalizer (FFE/DFE) and simulated a 4.5 dB increase in nonlinear threshold for a 1-dB eye-opening penalty at 43 Gbit/s RZ over 8×80-km spans of S-SMF. For data transmission based on phase-shift-keyed modulation, Xu and Liu  used an optical phase modulator driven by the received optical power to simulate a negative nonlinear coefficient , to give a >3 dB improvement in received signal quality: Ho and Khan  derived a scale factor to optimize the amount of phase modulation in this system. Recently, Shieh et al. have demonstrated nonlinearity post-compensation for OFDM . This gave around 2-dB improvement in signal quality over 1000 km.
This paper proposes applying a combination of nonlinearity pre- and post-compensation to coherent optical OFDM systems , , , using the digital hardware that already exists in OFDM transmitters and receivers. An optimum combination of pre and post compensation allows a 2-dB increase in launch power for standard single-mode fiber over 2000-km, or a 5-dB improvement in signal quality for a given launch power. Low-dispersion fibers offer even better performance in absolute terms after nonlinearity compensation. The pre-compensation and post-compensation are fitted to the system using a single parameter relating applied phase shift to instantaneous signal power, which is not critical. A combination of pre- and post-compensation gives far better results than pre- or post-compensation alone. Furthermore, the nonlinearity compensation at the receiver can be used to adaptively compensate for rapid variations in the nonlinearity of the plant.
2. Optical OFDM system
Figure 1 shows the new OFDM system block diagram. Incoming data of 2N bits is presented to N, 4-QAM modulators which each encode 2 bits. The modulators feed an inverse Fast Fourier Transform (FFT), which creates a complex-valued waveform that is the superposition of the N QAM-modulated subcarriers. Nonlinearity precompensation  is then applied as a time-dependent phase shift, detailed below. The in-phase (real) and quadrature (imaginary) parts of the waveform are then fed in series to two Digital to Analog Converters (DACs), which drive an optical Inphase-Quadrature (IQ) ‘complex’ modulator to create an optical OFDM waveform . The laser’s output is thus optical amplitude and phase modulated, as with other electronic precompensation techniques. The modulator is biased at a null to create a single-sideband optical spectrum with a totally suppressed carrier. At the receiver, inphase and quadrature components of a locally-generated optical carrier are mixed with the optical signal at the receiver to reconstruct the inphase and quadrature electrical signal components. These components are digitized by analog to digital converters for processing in a standard OFDM receiver. The equalizer part of the OFDM receiver after the FFT can compensate electrical phase distortion caused by fiber dispersion . In this work a second equalizer is added before the FFT in the receiver, which imposes a time-dependent phase modulation proportional to the received power waveform, which post-compensates fiber nonlinearity.
OFDM systems are sensitive to nonlinearity because they use very closely-packed channels (tens of MHz), so there is little walk-off due to dispersion that would ordinarily mitigate fiber nonlinearity . Here, a small walk-off allows the transmission path to be approximated as being dispersionless (that is, having zero walk off). Neglecting dispersion greatly simplifies the nonlinear equalization, as the usual split-step algorithm reduces to a single step for the whole transmission span. This single step applies a phase advance, θ(t), linearly proportional to the instantaneous optical power, P(t). The constant of proportionality is defined here using an effective length, Leff, per fiber span, a number of spans, s, a nonlinear coefficient n 2, an effective cross-sectional area of the fiber, Aeff and centre wavelength λ 0:
This time-domain phase modulation can be applied just after the inverse-FFT at the transmitter or just before the FFT at the receiver, or at both points in any proportion. These are relatively simple calculations compared with the FFT and inverse-FFT already required to implement OFDM. Although Eq. (1) is derived assuming no walk off, walk off does reduce the effectiveness of this scheme because the power waveform evolves along the fiber’s length. This means the precompensation is only accurate at the initial stages of the fiber link as it uses the transmitted power in its calculation: the postcompensation is only effective at the final stages of the link as it uses the received power. In the middle of the link, the phase shifts calculated by Eq. (1) may not improve performance because power waveform may not resemble the waveforms at the transmitter or the receiver. Section 3 accounts for walk off using reduced Leff’s in Eq. (1) so the partitioning of the phase shift between transmitter and receiver, and the effect of walk-off, are accounted for.
The data rate was 10 Gbit/s. 512 OFDM carriers each carry 2 bits to give an optical bandwidth of 5 GHz centered around 193.1 THz. The 2000-km link comprised twenty-five 80-km spans, without optical dispersion compensation. The fiber has a loss of 0.2 dB/km, a nonlinearity coefficient of 2.6×10-20 m2/W and an effective cross-section of 80 μm2. The optical amplifiers compensated for the 16-dB fiber loss in each span. The amplifiers were approximated as being noiseless, as noise only limits the signal quality at lower input powers . The output power of each amplifier was controlled to set the input power to each 80-km fiber span. The coherent receiver used a 10-mW local oscillator laser and was also considered to be noiseless. VPIsystems’ VPItransmissionMakerTMWDM V7.0.1 was used to simulate the complete system with an optical bandwidth of 80 GHz and an electrical bandwidth of 40 GHz. The electrical signal quality is defined as shown in Fig. 2(a), where q = μy/σy and Q = 10log10(q 2). The Bit Error Ratio (BER) can be estimated using BER = ½erfc(q/√2); e.g., a q 2 of 13.8× corresponds to a Q of 11.4 dB and gives a BER of 10-4.
3.1. Example Constellations
Figure 2 shows a set of constellation diagrams for the electrical signals just before the QAM demodulator under various conditions. The fiber had a dispersion of 6 ps/nm/km and the input power to each fiber span was -1 dBm. Figure 2(a) compares no nonlinearity compensation with optimum precompensation: the Q has improved from 1.9 dB to 8.7 dB. Figure 2(b) compares no compensation and optimum postcompensation: the Q has improved to 6.4 dB. Thus precompensation  is advantageous over postcompensation . Figure 2(c) shows the result of using an optimum mix of pre and post compensation; the Q has increased to 14.4 dB, far better than pre or post-compensation alone. No erroneous points can be seen in this case.
3.2. Performance against transmitter and receiver effective lengths
Figure 3 shows the Q versus effective length applied as precompensation, for a number of postcompensation effective lengths for standard single-mode fiber (16 ps/nm/km). The data was averaged over 10 runs for each graph point, which is equivalent to 10,240 bits. The launch power into each fiber span was reduced to -2 dBm as -1 dBm gave a maximum Q of only 8.3 dB. The points along the y-axis are for precompensation alone, and show between 8 and 10 km of precompensation is optimum. The dark green line (bullets) is for postcompensation alone, and shows a maximum Q of 7.9 dB at 10 km. Optimum performance occurs with equal amounts of pre and post distortion, giving a Q of 10.9 dB for 8.5 km precompensation plus 8.5 km postcompensation. These effective lengths add to 17 km; more than the optima for pre- or post-compensation alone. This is because the postcompensation is calculated using the received waveform, so better matches the effect of nonlinearity in the last spans of the link than would doubling the precompensation, which only matches the transmitted waveform so can only compensate for the start of the link. Ideally, the compensation should account for the evolving waveform along the whole link: this is what using the split-step Fourier method to predict the evolving waveform would achieve , , , albeit at considerable computational cost.
Simulations for 6 ps/nm/km fiber showed much greater improvements, with Q=18 dB for (10km precomp. and 8 km postcomp.). Because of the lower dispersion, the precompensation or postcompensation could be increased to 14 km for optimum compensation at the transmitter or the receiver.
3.3. Improvement in launch power
The main benefit of nonlinearity compensation is to be able to increase the nonlinear threshold, so systems can be operated at higher powers to allow increased amplifier spacing or longer transmission distances . Figure 4 shows the electrical Q versus the launch power into each fiber span with pre, post and pre- plus post-compensation. The optimum effective lengths were used in each case. For standard single-mode fiber [Fig. 4(a)], the input power can be increased by 1-dB using pre or post compensation alone for a given Q. Using pre- and post-compensation allows the input power to be increased by 2 dB for a given Q. Another way of interpreting the same data is that for a given input power, Q is increased by >5 dB.
Figure 4(b) shows similar plots for 6 ps/nm/km fiber. An optimum combination of pre-and post-compensation allows the launch power to be increased by over 5 dB. For a given launch power, the Q increases by >15 dB at low powers or >10 dB at higher powers. It is clear that the greatest improvements in Q are obtained for the lowest-dispersion fibers. In fact, the nonlinear threshold for low-dispersion fiber is 2-dB higher than for the S-SMF fiber when nonlinear compensation is used, assuming that the low-dispersion fiber is a large-effective area design. Thus, low-dispersion fibers could have an advantage in optical OFDM systems.
This paper shows that, for coherent optical OFDM systems, a combination of nonlinearity pre- and post-compensation gives >2 dB increase in the nonlinear power limit for 16 ps/nm/km fibers, and >5 dB for 6 ps/nm/km fibers. The corresponding increases in Q for a given input power are >5dB and >10 dB. The benefit is approximately double that of pre- or post-compensation alone, because the compensation reflects the shapes of the power waveforms at the transmitter and the receiver, which are different due to dispersion.
A single tuning parameter is used at the receiver and at the transmitter, so that little knowledge is required of the actual fiber plant to achieve a reasonable benefit. The actual value of this parameter need not be accurate: for example, a rough estimate of precompensation could be used, and then the postcompensation could account for dynamic variations in the system. The system would then be robust against variations in input power.
It should be noted that the present results apply to a single-channel system, so that intrachannel nonlinearity is being compensated. However, preliminary simulations show that benefit is also possible in WDM systems as the intrachannel nonlinearity between the closely-spaced OFDM subcarriers dominates over inter-channel nonlinearity across widely-spaced WDM channels.
I should like to thank VPIphotonics (www.vpiphotonics.com) for the use of their simulator, VPItransmissionMakerTMWDM V7.0.1. This work is supported under the Australian Research Council’s Discovery funding scheme (DP 0772937).
References and links
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