The use of nonlinearity precompensation in direct-detection optical orthogonal frequency division multiplexed links is investigated by simulation. Because of the presence of a strong optical carrier its performance is poorer than for coherent systems: with compensation the signal quality is found to vary almost periodically across the signal band. We propose and explain the operation of two optical, one electrical and one computational method of removing this periodic variation. Optical filtering of one sideband at the receiver is most effective, but a substantial improvement can be obtained by a simple modification to the precompensation algorithm.
© 2008 Optical Society of America
Optical OFDM is receiving much attention – because of its ability to electronically compensate for fiber dispersion in long-haul optical communications systems. However, it requires low optical powers along the link, because inter-sub-carrier Four-Wave Mixing (FWM) between the OFDM subcarriers causes significant degradation . This limits the transmission distance for a given optical amplifier spacing, as the signal cannot be allowed to drop below the noise floor before re-amplification. . Recently, we have shown that the effect of fiber nonlinearity can be partially compensated in a Coherent Optical OFDM system (CO-OFDM) ,  using nonlinearity precompensation , or a combination of precompensation and postcompensation . Shieh, Ma, and Tang  have demonstrated postcompensation experimentally.
Direct-detection optical OFDM (DDO-OFDM) offers a simpler receiver architecture than CO-OFDM, so could offer cost and space savings, albeit with increased Optical-Signal to Noise Ratio (OSNR) requirements . With DDO-OFDM, an optical carrier is transmitted along the fiber with the OFDM subcarriers, so nonlinearity will cause interactions between the carrier and the OFDM sideband, causing the generation of additional optical frequencies. These frequencies could cause some additional nonlinear degradation of the received electrical signal, compared with CO-OFDM. An open question is whether nonlinearity compensation is effective with DDO-OFDM because of these additional mixing products.
In this paper, we demonstrate that nonlinear precompensation is effective for DDO-OFDM systems. However, the nonlinear interactions of the carrier and the OFDM subcarriers have to be considered. Additionally, precompensation causes a periodic variation of signal quality across the signal bandwidth. We propose four methods to mitigate this periodic variation: two using optical filters, one using an electrical filter, and one using a modified calculation of the precompensation waveform. Their performance is compared using numerical simulations.
2. Optical transmitter with precompensation
The principle of operation of optical OFDM is well-known, and the block diagram of a complete system can be found elsewhere . Nonlinear precompensation is discussed in . For the purposes of discussion, the transmitter modulator is divided into two parts, shown in Figure 1; however, the function of the two modulators could be combined into a single “complex” modulator (a cascaded triple Mach-Zehnder Interferometer). For coherent OFDM, the first modulator is biased at its null to create a set of optical subcarriers mimicking the electrical OFDM spectrum carried on the waveform VOFDM(t). For direct-detection systems, the modulator can be biased slightly off its null point to give a carrier : a carrier power equal to the power in the sidebands usually results in optimum performance . The carrier is usually offset from the subcarrier band by a gap equal to the bandwidth of the subcarriers .
The phase modulator implements the nonlinearity precompensation , and is usually driven by to give a phase shift, φ(t), linearly proportional to the instantaneous optical power, P(t) at the input to the first fiber span. This phase shift is intended to compensate for the nonlinear phase shift within the fiber itself. The constant of proportionality is defined here using: effective length per fiber span, Leff; number of spans, s; nonlinear coefficient n 2; effective cross-sectional area of the fiber, Aeff; centre wavelength, λ 0:
The electrical bandwidth of the phase modulator drive will depend on whether there is an optical carrier or not. For example, in a coherent system, the drive will be a result of the superposition of all of the subcarrier fields, and will have a bandwidth equal to the bandwidth of the subcarriers (5 GHz for a 10-Gbit/s 4-QAM system). In a direct-detection system, the drive will include the contribution of the offset optical carrier, so will have strong components in the 5–10 GHz band due to the mixing of the carrier and each subcarrier, and also components from 0 to 5-GHz due to intermixing of the subcarriers with one-another.
The phase modulation (PM) of the optical OFDM signal creates distortion tones that considerably broaden the optical spectrum: Fig. 1 (inset) shows a simulated spectrum (top). These tones can be classified according to their origins, as shown in the lower spectrum of Fig. 1. Distortion A is caused by phase modulation of the subcarriers by the instantaneous power corresponding to the mixing of subcarriers with other subcarriers within the subcarrier band. There are many possible contributions to the instantaneous power, especially for pairs of subcarriers with a small frequency difference, but the maximum modulation frequency is equal to the bandwidth of the subcarrier band. Distortion A is the only component in a coherent system. For direct-detection systems, there are additional tones: Distortion B is caused by phase modulation of the optical carrier by the difference frequencies of the carrier and each subcarrier; Distortion C is caused by the subcarriers being modulated by these same difference frequencies. As shown in Section 5, bandlimiting the electrical drive affects which of the distortion products are produced, hence which nonlinear products are compensated.
3. Comparison of CO-OFDM and DD-OFDM systems without precompensation
The quality, q,  of each subcarrier in coherent and direct detection optical OFDM systems was simulated for a 10-Gbit/s link with lengths up to 4000-km. The Bit Error Ratio (BER) can be estimated using BER=½erfc(q/√2); e.g., where Q(dB)=20log10(q) and q is the mean squared divided by the variance of the constellation points in one Cartesian plane, so a Q of 9.8 dB gives a BER of 10-3 . The link comprised fifty 80-km spans of 2 ps/nm/km fiber, 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 we were interested in the nonlinearity-limited performace . 512 OFDM carriers each carry 2 bits to give an optical bandwidth of 5 GHz centered around 193.1 THz. A total of 128 independent blocks were used to give 65,536 constellation points. A simple photodiode was used in the direct-detection system: for the coherent system a local oscillator laser was used with a double-balanced receiver . VPItransmissionMaker™WDM V7.1 was used for simulation.
Figure 2 plots the signal quality versus subcarrier for direct-detection and coherent systems, both operating with -6 dBm fiber-input power in the sidebands to give similar error rates in the absence of nonlinearity. This is at least 3-dB higher than the power where Q is noise limited. A 17-point (17-subcarriers) moving average was used to smooth across the x-axis. The coherent results are shown as dashed lines. These are symmetrical versus subcarrier index, with the poorest performance in the centre as more FWM products fall on the central subcarriers . In the direct-detection systems, the Q is further degraded because of the nonlinear interaction between the optical carrier and the subcarrier band. This creates an image (Distortion B) of the sideband that falls on the opposite side of the carrier. In short systems and for subcarriers close to the carrier, this image is the conjugate of the subcarrier band, and will actually cancel nonlinearity: a small improvement in Q can be seen for the lowest-frequency carriers in the 400-km case. For longer systems resonance effects can occur, so the lower-frequency subcarriers can become worse than the higher-frequency subcarriers. It is clear that the simple theory  for the signal quality versus subcarrier for coherent systems cannot be used for direct detection systems.
4. Performance of DD-OFDM systems using nonlinearity precompensation
For dispersionless fiber, precompensation can totally compensate for fiber nonlinearity; however, fiber dispersion causes the optical waveform to evolve along the fiber link so that the precompensation waveform of Eq. (1) will only be correct for the first spans in the link. Previous work  showed that this effect means a reduced phase shift should be applied for optimum performance; and this can be thought of as reducing the effective nonlinear length in each span to account for dispersion. The effective nonlinear length is therefore a tuning parameter. Figure 3 shows the effect of precompensation on the received signal quality for a 4000-km system. The optimum results are obtained for effective lengths between 6 and 8 km.
A frequency-dependent ripple occurs when precompensation is used, suggesting that the phase modulation required for compensation is responsible for this ripple. An explanation for the ripple is that the phase modulation, φ(t), is converted to intensity noise along the dispersive fiber, in a similar manner to laser phase noise being converted to intensity noise . This intensity noise is frequency-dependent because PM creates upper and lower sidebands for Distortion B, which undergo opposite frequency-dependent phase shifts relative to the carrier due to dispersion. Upon detection each sideband mixes with the carrier to produce two electrical components that will interfere with one another. Constructive interference produces a large intensity noise, hence a degraded signal quality. Simulation confirmed that the spacing of the ripples depends solely on the dispersion-length product of the system, and this spacing agrees with . Stronger precompensation using strong phase modulation leads to stronger ripples. Thus the optimum effective length will be a compromise between how much nonlinearity is compensated and how much noise is generated by the precompensating phase modulation.
5. Strategies for removing the frequency dependence of Q
If one of the compensation sidebands of Distortion B can be removed, then interference should not occur upon photodetection, removing the ripples in signal quality. Sideband removal could be achieved optically either at the transmitter (which will eliminate the strong ripples due to phase modulation at the transmitter, but not ripples caused by the phase modulation in each span), or at the receiver (which should remove all ripples). The extent of the sidebands can also be limited electrically, by bandlimiting the phase modulator drive.
Figure 4 compares optical (transmitter or receiver) filtering with electrical filtering, no filtering, and no precompensation. The effective length was 8 km. No precompensation produces the direct-detection result of Fig. 2. No filtering reproduces the 8-km precompensation result of Fig. 3, with its associated ripples. Transmitter optical filtering, using a filter removing all tones more than 1 GHz below the carrier, produces a minimum Q improvement of around 3 dB, compared with 2 dB without the filter. The average Q value across the bands is not improved significantly. Filtering at the receiver, using a filter extending 1-GHz below the carrier to 1-GHz above the sideband, produces the best average improvement (>4.5 dB minimum improvement). Electrical bandlimiting of the phase modulation waveform produces a similar result to optical filtering at the transmitter, with the advantage that a narrow-band electrical filter is far easier to implement than a narrowband optical filter.
The bandwidth of the phase modulation can also be reduced by modifying the calculation of the power waveform in Eq. (1). For example, removing the carrier from the calculation means that no components exist above 5 GHz. Additional simulations showed that this is equivalent to a 5-GHz brickwall electrical filter. Obviously this calculation is easy to implement using digital signal processing, and is identical to the calculation used to precompensate coherent systems.
The difference between transmitter filtering (optical or electrical) and receiver filtering is the compensation of the various nonlinear distortion terms. Removing the lower sideband by transmitter filtering reduces the effectiveness of compensation, as the FWM products created within the fiber will not be neutralized by compensation products for frequencies below the carrier. Removal at the receiver is therefore optimal, because the nonlinear phase distortion and precompensation phase modulation will have partially cancelled one-another along the link (in a dispersionless link they will fully cancel). Unfortunately, the optical filters are required to have a sharp cut-off, which is problematic. Receiver filtering is also likely to remove under-compensated nonlinear products, which may explain its better performance.
Electrical filtering of the precompensation phase waveform for frequencies above 5 GHz, removes both the upper and lower phase modulated sidebands at greater than 5 GHz away from the carrier and subcarriers. Unfortunately, Distortion B in the fiber is not precompensated. Histograms of Q (Figure 5) show that electrical filtering will increase the average performance of each subcarrier more than a transmitter optical filter, but both systems have similar worst-case performances. Coding over all subcarriers would obviously be beneficial.
This paper shows that nonlinearity precompensation benefits DDO-OFDM. If the CO-OFDM precompensation scheme ,  is implemented without modification, an image band is transmitted which induces variations in the received signal quality across the subcarrier band. The ripples can be suppressed by optically filtering to remove the image band at the transmitter or preferably the receiver. Since very sharp optical filters are required, this is difficult to achieve. Alternatively, the precompensation input can be electrically band limited to prevent the creation of the image band. This produces similar results to a transmitter optical filter. Alternatively, the phase modulation calculation should exclude the carrier.
We would like to thank VPIphotonics (www.vpiphotonics.com) for the use of their simulator, VPItransmissionMaker™WDM V7.1. This work is supported under the Australian Research Council’s Discovery funding scheme (DP 0772937).
References and links
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