We present a numerical study of the performance of 40 Gbit/s return-to-zero differential phase-shift keying (RZ-DPSK) transmission with different dispersion maps. The optimum dispersion mapping for RZ-DPSK format are discussed and compared with those for on-off keying (OOK). Two pseudo-linear transmission systems, one using standard single-mode fiber and the other nonzero dispersion-shifted fiber, are investigated, respectively. In principle, the optimum dispersion mapping for RZ-DPSK format is different from that for OOK and depends on intra-channel four-wave mixing and nonlinear phase noise.
©2006 Optical Society of America
Direct-detection return-to-zero differential phase-shift keying(RZ-DPSK) format has been studied extensively for high-speed optical transmission [1, 2]. In transmission systems operating at 40 Gbit/s and above, the intra-channel nonlinear effects, namely, intra-channel cross-phase modulation (IXPM) and intra-channel four-wave mixing (IFWM), are the dominating nonlinearities , which have been investigated mostly focusing on on-off keying (OOK) format [3, 4, 5, 6]. By introducing optimum dispersion mapping, say, by optimizing pre-compensation as well as residual dispersion per span, intra-channel pulse interactions can be suppressed [4, 5]. The spirit of this method is to annihilate the real components of the nonlinear mixing terms . In doing so, pulse amplitude jitter due to the in-phase beating between signal pulses and IFWM products can be inhibited. Moreover, pulse timing jitter due to IXPM will be eliminated up to the first order perturbation. For DPSK systems, which carry the information in optical phase changes between bits, the situation is more complicated. Firstly, as the imaginary part of IFWM distortions directly contaminates the pulse phase, it is more detrimental than amplitude fluctuations . Secondly, nonlinear phase noise, caused by the interaction between amplified spontaneous emission (ASE) and fiber nonlinearities, is another nonlinear limitation to DPSK systems . IXPM itself does not degrade RZ-DPSK signals. However, self-phase modulation (SPM) - and IXPM-induced nonlinear phase noise is harmful . The intra-channel nonlinear impairments in RZ-DPSK systems mainly include IFWM and nonlinear phase noise. To minimize them, one should conduct an optimization towards both the real and the imaginary components of the nonlinear distortions. Therefore, the optimum dispersion mapping for RZ-DPSK is not naturally the same as that for OOK. To our knowledge, there has been no detailed analysis of dispersion map design for DPSK systems thus far. Based on a simple model of ignoring fiber attenuation, Wei et al.  studied IFWM effect in highly dispersive RZ-DPSK systems and showed an advantage of symmetric dispersion management over non-symmetric maps. Coelho et al.  showed that DPSK system with full post-dispersion compensation is better than that with full pre- and hybrid-(half pre- and half post-) compensation. However, both Ref.  and  didn’t optimize pre-compensation and didn’t consider inline residual dispersion and nonlinear phase noise.
In this paper, we present a comprehensive study of dispersion mapping in typical RZ-DPSK systems with 33% duty cycle, by optimizing pre-compensation at given values of residual dispersion per span. At first, we study IFWM-induced optical signal-to-noise ratio (OSNR) penalty in RZ-DPSK systems with different dispersion maps. Then, we consider also nonlinear phase noise – which cannot be studied in terms of OSNR penalty – by estimating the bit error rate (BER) via a semi-stochastic method based on the numerical solution of the nonlinear Schrödinger equation. Since the relative strength of IFWM and nonlinear phase noise changes with local dispersion, we investigate two pseudo-linear systems, one using standard single mode fiber (SSMF, typical dispersion 17 ps/nm/km) and the other non-zero dispersion-shifted fiber (NZDSF, typical dispersion 4.5 ps/nm/km), respectively.
2. IFWM degradation with different dispersion maps
In this paper, we analyze a single 40 Gbit/s RZ-DPSK channel with 33% duty cycle pulse generated by a standard Mach-Zehnder modulator with appropriate clock tone and bias. The fiber link consists of Nspan transmission fibers each of length Lspan dispersion D, attenuation α, nonlinearity γ, and residual dispersion per span Dres. The lumped amplifier is placed at the end of each span to compensate for the span loss. An ideal (linear and without loss) pre-compensation module Dpre is assumed at the input. At the receiver end, residual dispersion is fully compensated. We choose Lspan =90 km, α=0.21 dB/km, and γ=132 /W/km for both the SSMF and the NZDSF systems. The average launch power is set as Pave= -1.2 dBm so that the mean nonlinear phase shift 〈ΨNL〉=1 rad after Nspan=50 spans. Here 〈ΨNL〉=Nspan γPaveLeff. Leff≈l/α is the effective length per span. The nonlinear Schrödinger equation is solved by the split step method with the simulation tool VPItransmissionMakerV6.5 using a De Bruijn sequence of 4096 (212) bits, which contains sufficient bit patterns to capture nonlinear interaction details for system scenarios considered in this paper . The inline amplifier is set noiseless so that the nonlinear degradation mainly results from IFWM. The assumed receiver consists of a preamplifier, followed by a second-order Gaussian optical filter with 3 dB bandwidth of 100 GHz, a subsequent 1-bit delay interferometer, a balanced detector and a fifth-order Bessel electrical low-pass filter with 28 GHz bandwidth. The system performance is specified in terms of the OSNR required to achieve a BER of 10-9, assuming only linear ASE noise added by the preamplifier at the receiver end . The nonlinear transmission will result in OSNR penalty, referencing to the back-to-back measurement, for a resolution bandwidth of 0.1 nm. The BER is optimized with respect to both threshold and sampling time and calculated by using a semi-analytic method based on Karhunen–Loeve series expansions for both signals and noise after the square-law detection .
We investigate different dispersion maps with residual dispersions of Dres =0, 2.5, 5, 10 and 20 ps/nm per span. IFWM impairment in both the SSMF and the NZDSF systems is evaluated by calculating the OSNR penalty against different pre-compensation. The results are summarized in Fig.1. For a resonant map of Dres =0, IFWM products add coherently span by span. Although the penalty curves for the SSMF and the NZDSF are different, both of them have a lump when sweeping over pre-compensation. We believe that this comes from a competition between the real and the imaginary parts of the nonlinear distortions. When inline residual dispersion is introduced, IFWM distortions in each span are different and add to each other with some kind of averaging and canceling. The lump is thus removed. In comparison, Fig.2 shows the nonlinear OSNR penalties of 33% RZ-OOK systems with the same average launch power as above. Note that for OOK format, the penalty is caused by both IFWM and IXPM effects when inline amplifier noise is neglected. Compared with OOK, RZ-DPSK has a much smaller OSNR penalty. The better fiber nonlinearity tolerance in DPSK systems owes to reduced pulse peak power, identical energy in each bit slot and a correlation of nonlinear distortions between adjacent bits . It should be noted that OOK’s OSNR penalty changes greatly for different Dres, while RZ-DPSK has always a small penalty given pre-compensation is optimized. To test this property more generally, we simulate a shorter fiber link of 10×90 km, that allows us observing nonlinear interactions accurately over much larger Dres values without increasing bit sequence length. We choose an average launch power to achieve 〈ΨNL〉=1 rad at the end of the fiber link. The results are shown in Fig. 3, which agrees with Fig. 1. A moderate Dres slightly increases the OSNR penalty by introducing more bit overlapping. However, the penalty does not keep increasing with Dres. For the SSMF, it even decreases again at large Dres. Compared with the resonant map, the penalty variation over other Dres is insignificant (less than 0.1 dB for the SSMF and 0.3 dB for the NZDSF). Consequently, that RZ-DPSK is very robust to inline residual dispersion regarding IFWM.
Note that Killey reported an analytical formula to optimize pre-compensation in OOK systems 
Eq. (1) only considers the suppression of the real components of the nonlinear distortions and is in general not applicable to DPSK formats. It is not surprising that the optimum pre-compensation for RZ-DPSK is totally different from that for OOK in the case of resonant maps. However, when a reasonable residual dispersion is introduced each span, RZ-DPSK requires similar pre-compensation to OOK, but holds a broader optimum region in the case of the SSMF. For the problem of optimum dispersion mapping, the results in Fig. 1 are not sufficient since nonlinear phase noise should also be taken into account. However, we will show in the following that in most systems (with reasonable residual dispersion per span or with transmission fiber of large dispersion) the optimum pre-compensation will be determined by IFWM.
3. Optimum dispersion mapping by considering both IFWM and nonlinear phase noise
Here, we assume the same system configurations as in Fig.1, but set the inline amplifier noise figure as 5 dB, resulting an OSNR of ρs =25.5 (14 dB) at the end of the fiber link. We also neglect the receiver impairment to focus on the dispersion map design alone. The OSNR is defined as ρs=Pave/(Nspan S0 Bd). S0 is the amplifier noise spectral density in a single polarization. Note that in our definition, optical noise bandwidth Bd is set the same as the bit-rate of 40 Gbit/s, and we consider both optical signal and noise in one single polarization. With only linear Gaussian noise, the error probability is ½ exp(-ρs) ≈ 4.2×10-12 . Since the Karhunen–Loeve method is confined to linear noise, in order to include both the fiber nonlinearity and its beating with ASE noise, we adopt a semi-stochastic method developed by Richter et al. , which calculates BER by a statistic analysis of the probability density function (PDF) of the detected samples at the balanced receiver. The moment-generating function (MGF) of the differential electrical signal is given as
Ψ1(s) is the MGF corresponding to the signal and noise after one of the photodiodes (constructive port) and Ψ2(s) the subtracted noise after the other photodiode(destructive port). The parameters N 1, M 1, E, N 2, and M 2 are obtained by fitting the histogram of the detected amplitudes with a noncentral χ2 distribution. Richter has validated the MGF approach against Monte-Carlo simulations and achieved excellent agreement with the PDFs . Therefore, with high accuracy, the BER can be calculated from the resulting MGF by the saddle point approximation technique .
Figure 4 shows the calculated BER as a function of pre-compensation for both the SSMF and the NZDSF systems. To reduce the stochastic randomness, the results have been averaged over ten iterations of BER calculations with different noise random-number seeds. Each calculation is optimized in terms of both threshold and sampling time. Note that in our paper, the OSNR penalty of RZ-DPSK systems results from IFWM, while the BER takes both IFWM and nonlinear phase noise into account. Compared with Fig.1, the BERs for the SSMF systems follow approximately the same trend concerning pre-compensation as their corresponding OSNR penalties. We believe this to be an indication that IFWM dominates over nonlinear phase noise in 40 Gbit/s SSMF systems . For the NZDSF systems, nonlinear phase noise is dominating at Dres=0, and thus the behavior of BER evolution is distinct from that of the OSNR penalty. The inline residual dispersion, however, will lead to rapidly larger IFWM impairments if Dpre is not optimized, resulting in similar optimum pre-compensation between the BER and the OSNR penalty for the NZDSF at Dres=10 and 20 ps/nm. For both the SSMF and the NZDSF, the larger inline residual dispersion always results in better BER performance, since it reduces nonlinear phase noise efficiently. For example, the NZDSF suffers more IFWM degradation (larger OSNR penalties) at Dres=10 and 20 ps/nm, but shows a smaller BER (which considers both IFWM and nonlinear phase noise) as Dres increases. We believe that nonlinear phase noise is suppressed due to the de-correlation introduced by residual dispersion in each span. Additionally, the force of nonlinear phase noise (pulse peak power) becomes smaller as more pulse broadening results from inline residual dispersion.
Table 1 summarizes the optimum pre-compensation for both OOK and RZ-DPSK systems, which is extracted from Fig.2 and Fig.4. For Dres=0, the optimum pre-compensation of RZ-DPSK systems is very different from that of OOK for both the SSMF and the NZDSF. For Dres=10 and 20 ps/nm, the operating point of RZ-DPSK systems with the NZDSF is close to its equivalent of OOK, meaning that Eq.(1) is still valid in this configuration, which is consistent with Chowdhury’s experimental results . For the SSMF, the BER curve fluctuates slightly over a relatively large region around the operation point predicted by Eq. (1), which means a loose pre-compensation design. Therefore, once a sufficiently large residual dispersion per span (e. g. >10 ps/nm) is chosen, the optimum pre-compensation for RZ-DPSK systems can be approximately determined by Eq. (1).
We have discussed the optimum dispersion mapping for RZ-DPSK systems, which is influenced by both nonlinear phase noise and IFWM. Residual dispersion per span will be beneficial to reduce nonlinear phase noise. IFWM can be suppressed by an optimum pre-compensation. In transmission systems with reasonable residual dispersion per span (e.g. >10 ps/nm), optimum pre-compensation will be similar for RZ-DPSK and OOK formats.
The authors acknowledge the financial support of Alexander von Humboldt foundation.
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