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We demonstrate electronic post-compensation for nonlinear phase fluctuation in a 1000-km 20-Gbit/s optical quadrature phase-shift keying (QPSK) transmission system, where group-velocity dispersion is well managed. The inter-symbol interference (ISI) at the transmitter induces the nonlinear phase fluctuation through self-phase modulation (SPM) of the signal transmitted through a fiber. However, when the optimized phase shift proportional to the intensity fluctuation is given to the complex amplitude of the signal electric field by using a digital coherent receiver, the nonlinear phase fluctuation can be reduced effectively.
©2008 Optical Society of America
1. Introduction
The phase-shift keying (PSK) modulation format offers many advantages for long-haul optical transmission over the conventional on-off keying format [1][2]. However, the performance of optical PSK transmission systems is limited by nonlinear phase fluctuations [3]. Although the intensity waveform of the return-to-zero (RZ) PSK signal is a regular RZ pulse train, the intensity fluctuation may be caused by the inter-symbol interference (ISI) on the modulated signal when a Mach-Zehnder modulator (MZM) is employed for PSK modulation [2]. In addition, amplified spontaneous emission (ASE) from erbium-doped fiber amplifiers (EDFAs) generates the intensity fluctuation of the transmitted RZ PSK signal [4]. Such intensity fluctuations of the transmitted signal are converted to the phase fluctuation through self-phase modulation (SPM) due to Kerr nonlinearity of the fiber for transmission, resulting in bit error-rate (BER) degradation. However, since these nonlinear phase fluctuations are correlated with the received intensity fluctuation, they can partially be compensated for by imposing a reverse phase shift proportional to the intensity fluctuation on the complex amplitude of the electric field of the transmitted signal [5][6][7][8].
In this paper, we examine the origin of the nonlinear phase fluctuation for a 20-Gbit/s optical quadrature PSK (QPSK) signal, which is transmitted through a 1074-km dispersion-managed link, and find that the inter-symbol interference (ISI) at the transmitter is the dominant cause of the nonlinear phase fluctuation. Next, we demonstrate the electronic post-compensation for the nonlinear phase fluctuation using the digital coherent receiver [9]. The optimized phase shift proportional to the intensity fluctuation is given to the complex amplitude of the signal in the electrical domain, leading to effective reduction of the nonlinear phase fluctuation.
2. Compensation for nonlinear phase fluctuation due to inter-symbol interference
The optical signal propagation through a fiber along the z direction is generally described by the nonlinear Schrödinger equation [10] as
In this equation, A(z,T) represents the complex amplitude of the signal electric field on exp(jωcT), where ωc is the angular frequency of the carrier, αp the loss coefficient, β2 the group-velocity dispersion (GVD) parameter, and γ the nonlinearity coefficient. The time T in the moving frame is defined as T=t-z/νg, where νg is the group velocity. At the receiving end, the digital coherent receiver can measure A(L,T), where L denotes the transmission distance. The measured complex amplitude will be distorted due to fiber dispersion and nonlinearity.
Since Eq.(1) is generally valid, its complex-conjugate form shown below is also valid.
Equation (2) describes the signal back-propagating through the fiber where the sign of αp is reversed. The back-propagating signal is the phase-conjugated replica of the forward-propagating signal at every point along the fiber. This fact means that if we virtually let the measured output complex amplitude A(L,T) propagate backwardly through the fiber where the sign of αp is reversed, we can compensate for dispersive and nonlinear effects and restore the input complex amplitude A(0,T) by numerical calculations at the receiver. This principle of operation is the same as that of the midway optical phase conjugation (OPC) scheme [11]. Such calculation is performed by the split-step Fourier method; however, this restoration process may not be practical for the real-time operation of the digital coherent receiver due to the long computation time required when the number of segments for calculations is too large.
In this paper, we focus on a dispersion-managed link [12], where a normal GVD fiber is concatenated with an anomalous GVD fiber in each span so that the local GVD is nonzero but the total GVD is zero. The span loss is compensated for by an EDFA in each span. The increase in the local GVD reduces the nonlinear effect, while it makes the transmission spectral window narrower; therefore, the dispersion map should be optimally designed by compromising these two factors. In a well-designed dispersion-managed link, the relatively large local GVD and the moderate optical power diminish the nonlinear effect, and hence the one-step calculation without segmenting the link is still effective for mitigation of the nonlinear effect. In such a case, Eq.(2) shows that the nonlinear effect is cancelled by using
In this equation, m is the number of spans and (γℓ)eff the effective phase rotation in a span length ℓ defined as
We use t instead of T at the receiver, ignoring the delay time L/νg, and we assume that an EDFA in front of the receiver restores the received power to the power launched on each span.
In a real transmission system, the power fluctuation is induced by the inter-symbol interference (ISI) at the transmitter (z=0) using a MZM. Equation (3) means that the initial ISI generates the nonlinear phase shift while traveling through the link, but that such nonlinear phase shift is cancelled at z=L by knowing the output power fluctuation. Therefore, Eq.(3) is valid as far as the intensity waveform is almost maintained along the link. However, this condition is not always satisfied because the local GVD generates ISI along the path, which in turn interacts with fiber nonlinearity. In such a case, the correlation between the intensity and phase fluctuations is not necessarily perfect, and Eq.(3) should be modified as
In this equation, α shows the correlation degree ranging from 0 to 1, which is dependent on the dispersion map of the link. In the experiment shown in Sec.5, α is used as an adjustable parameter, and the optimized value is confirmed by extensive computer simulations.
3. Compensation for nonlinear phase noise
Another type of nonlinear impairments is the Gordon-Mollenauer effect [3]. In a multi-span transmission system, each amplifier generates amplified spontaneous emission (ASE). Considering an m-span configuration, the intensity fluctuation δI(t) at the receiver is the sum of all amplifiers’ contributions as
where A 0 denotes the constant envelope and ak(t) the in-phase ASE noise from the k-th amplifier counting backwards from the receiver. The amount of the intensity noise is given as its variance as
where . On the other hand, the intensity noise is converted to the phase noise by SPM while the signal propagates through the amplifier chain, and the received phase noise can be written as
Equation (8) yields the phase noise variance
Since δI(t) and δϕ(t)are partially correlated as shown by Eqs.(6) and (8), by applying a reverse phase shift proportional to the received intensity noise, the nonlinear phase noise is partially compensated for. The compensated phase noise is given as
where α is an adjustable parameter. The variance of the compensated phase noise results in
which can be minimized by choosing α=1/2 when m≫1 as
Comparing (Eq.(9)) with (Eq.(12)), we can see that the nonlinear phase noise is reduced by 6 dB by optimal compensation [6][10].
It is evident that such nonlinear phase noise compensation can be performed by the same equation as Eq.(5) where we can fix α at 1/2.
4. Digital coherent receiver
A homodyne phase-diversity receiver used in our experiment is shown in Fig. 1 [9]. The combination of the λ/4 waveplate and the polarization beam splitter (PBS) creates the 90° optical hybrid necessary for phase diversity. The polarization of the local oscillator (LO) is aligned such that it becomes circular after passing through the λ/4 waveplate. On the other hand, the signal is linearly polarized, and the polarization angle is 45° with respect to principle axes of PBS’s. The receiver is composed of free-space optical components packaged in a small metal case. All the input and output ports are fiber-pigtailed.
We represent the complex amplitude of the beat between the signal and LO as A(t)∝A 0 exp[j(θs(t)+θn(t))], where A 0 denotes the signal amplitude, θs the M-ary phase modulation(=2πm/M,m=-M/2+1,⋯,0,⋯,M/2), and θn the phase noise from both of the signal and LO. Balanced photodiodes PD1 and PD2 measure the real and imaginary parts of A(t), respectively. Next, analog-to-digital converters sample outputs from PD1 and PD2 simultaneously at a sample rate larger than 2/T, where T represents the symbol interval. The data are resampled to keep one sample/symbol according to the extracted clock. Thus, we have the digitized complex amplitude A(iT), where i denotes the sample number.
We consider a system where the data composed of M symbols are differentially precoded at the transmitter. To retrieve the phase modulation θs(iT), we must evaluate the carrier phase θn(iT). The procedure to estimate θn(iT) is shown in Fig.2. We take the M -th power of A(iT), because the phase modulation is removed from A(iT)M. Averaging A(iT)M over (2k+1) samples from t=(-k+i)T to (k+1)T constitutes a phase estimate at t=iT as θe(iT)=arg[∑k j=-k A(jT)M]/M. The phase modulation is first determined by subtracting the phase estimate from the measured phase of each symbol. The phase modulation is then discriminated among M symbols. The restored symbols are finally differentially decoded.
Fig. 1. Schematic diagram of the phase diversity homodyne receiver. PBS: polarization beam splitter, HWP: half-wave plate, QWP: quarter-wave plate, HM: half mirror, LO: local oscillator, PD: double-balanced photodiode.
5. Experimental setup for QPSK transmission system
Figure 2 shows the setup for the 20-Gbit/s RZ QPSK transmission experiment. A LiNbO3 IQ MZM [13] was used to form the 20-Gbit/s QPSK signal from precoded two 10-Gsymbol/s data streams with 27-1 pseudo-random binary sequence (PRBS). 10-Gsymbol/s electrical signals were band-limited by 7.5-GHz electrical filters. A pulse carver converted the signal to the return-to-zero (RZ) format with a 50 % duty ratio. The 3-dB spectral width of the distributed feedback (DFB) semiconductor laser used as the transmitter was 150 kHz. The RZ QPSK signal was then transmitted through a 1074-km link in a 26 span configuration. Each span consisted of a 28-km-long large-core SMF and a 12-km-long inverse dispersion fiber (IDF). The GVD value, loss coefficient, and nonlinear coefficient of the SMF were 21 ps/nm/km, 0.2 dB/km, and 0.8/W/km, respectively. Those of the IDF were -50 ps/nm/km, 0.3 dB/km, and 3.0/W/km, respectively. A 0.98-µm forward-pumped erbium-doped fiber amplifier (EDFA) compensated for the span loss of 11dB including the splice loss of 1.8 dB. In the final span, a 74-km-long SMF was used for residual dispersion compensation.
A homodyne phase-diversity receiver described in Sec.4 was used to restore the complex amplitude of the transmitted signal. The local oscillator (LO) was a semiconductor DFB laser, whose 3-dB spectral width was 150 kHz and output power was 10 dBm.
The signal IQ components were sampled and digitized by AD converters at 20 Gsample/s. The stored data were processed offline as follows: Digital signal processing (DSP) circuits performed compensation for the nonlinear phase fluctuation by using Eq.(5) before carrier phase estimation with the averaging span k=10. Symbols were then discriminated among four phase states and the number of bit errors was counted.
6. Experimental results and discussion
Figure 4(a) shows the back-to-back intensity waveform of the RZ QPSK signal measured by a sampling oscilloscope with an 80-GHz bandwidth. The power launched on each span is 0 dBm. We find that ISI is generated by the bandwidth limitation of the transmitter. On the other hand, Fig. 4(b) shows the intensity waveform of the RZ pulse train without QPSK modulation after transmitted through the 1000-km link. The power fluctuation is mainly induced by ASE accumulation in this case. Comparing Fig. 3(a) with Fig. 3(b), we conclude that the initial ISI is more influential in causing nonlinear impairments than the ASE accumulation in the present transmission system.
Fig. 4. Comparison of the intensity waveform between the back-to-back RZ QPSK signal (a) and the RZ pulse train after 1000-km transmission.
Figure 5 shows the measured bit-error rate as a function of the compensation parameter α when the average power at the fiber input in each span is 0 dBm. The minimum BER is obtained by adjusting α≃1/2.
The upper row in Fig. 6 shows constellation maps measured at the uncompensated state (A in Fig. 5), the optimally compensated state (B), the overcompensated state (C), and the under-compensated state (D). The distribution of the phase fluctuation is narrowed owing to the effective compensation in the optimal compensation (B), leading to a significantly improved BER.
To verify the experimental results, we conducted the computer simulation. The initial ISI was created by filtering the complex amplitude of the RZ QPSK signal with an ideal Gaussian filter with a 3-dB bandwidth of 10 GHz. The noise figure of amplifiers was assumed to be 5 dB. Other parameters of fibers for transmission are given in Sec.4. The split-step Fourier method was used to solve the nonlinear Schrödinger equation. The lower row in Fig. 6 shows constellation maps corresponding to the states A, B, C, and D. The agreement between experimental and simulation results is very good. We thus find that the initial ISI generates the nonlinear phase fluctuation, which is partially cancelled out by the post electronic compensation scheme with the digital coherent receiver. The imperfect correlation of α=1/2 between the received intensity fluctuation and the nonlinear phase fluctuation stems from ISI caused along the link in addition to the initial ISI.
Fig. 6. Constellation maps of the transmitted RZ QPSK signal at the uncompensated state (A in Fig. 5), the optimally compensated state (B), the overcompensated state (C), and the under-compensated state (D). The upper row shows the measured results and the lower row shows the simulation results.
7. Conclusion
We have demonstrated electronic post-compensation for nonlinear phase fluctuation of a 20-Gbit/s optical quadrature phase-shift keying (QPSK) signal transmitted through a 1000-km dispersion-managed link. The intensity fluctuation is caused by the inter-symbol interference on the RZ QPSK signal at the transmitter. Such intensity fluctuation is then converted to the phase fluctuation through self-phase modulation (SPM) along the dispersion-managed link, resulting in bit error-rate (BER) degradation. Since the nonlinear phase fluctuation is correlated with the received intensity fluctuation, it can be compensated for by imposing a reverse phase shift proportional to the intensity fluctuation on the complex amplitude of the signal electric field. The optimized phase shift is given to the complex amplitude by using a digital coherent receiver, leading to effective reduction of the nonlinear phase fluctuation.
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