## Abstract

It is theoretically and experimentally shown that phase-preserving amplitude regeneration by an all-optical amplitude limiter using saturation of four-wave mixing in a nonlinear fiber can enhance DPSK transmission performance. The limiter suppresses amplitude fluctuations of the signal, by which the nonlinear phase noise caused by self-phase modulation of the transmission fiber is reduced. A 10-Gbit/s short-pulse DPSK transmission experiment shows that the limiter inserted either after a transmitter or inside a recirculating transmission loop can enhance the performance. Theoretical expressions for the linear and nonlinear phase noise are derived, with which the influence of imperfections of the limiter is examined.

© 2007 Optical Society of America

## 1. Introduction

Phase modulation as a method for information encoding in optical fiber communications has attracted intense interest in recent years. Advantages of using phase-shift keying (PSK) modulation formats include higher receiver sensitivity over conventional on-off keying systems and suitability for multi-level signaling such as quadrature PSK (QPSK) or differential QPSK (DQPSK) formats [1]. Maximum transmission distance of the PSK signals is mainly determined by the phase noise imposed on the signal. In long-distance systems, nonlinear phase noise, which is caused by the translation of amplitude noise to phase noise through nonlinearity of the transmission fiber, contributes significantly [2]. Impairment by the nonlinear phase noise will become severer for higher-speed systems that require higher signal peak power.

The nonlinear phase noise can be reduced or compensated for by several means. Because the self-phase-modulation (SPM) induced phase shift is positively correlated to the peak power of the signal pulses at the receiver, it is possible to compensate for the nonlinear phase shift by giving each pulse a negative phase shift whose magnitude is proportional to the peak power of the pulse before detection [3–6]. This can be realized by inserting an optical element having effective negative nonlinearity [3] or a phase modulator driven by the received signal intensity [4–6] in front of the receiver. A phase conjugator followed by an additional nonlinear fiber offers the same effect [7]. The correlation between the received power and the nonlinear phase shift can be exploited also in the electric domain after detection for the phase noise compensation using electric signal processing [8]. The optical compensation can be distributed over the transmission system, which realizes better nonlinear phase noise suppression [9, 10].

Another approach to reduce the nonlinear phase noise is to suppress the amplitude noise that is the origin of the nonlinear phase noise. This can be achieved either passively by inserting narrow-band filters in soliton systems [11] or actively by inserting all-optical limiters whose response time is smaller than a symbol period [12–14]. We have shown that an optical limiter utilizing saturation of four-wave mixing (FWM) in a nonlinear fiber actually suppresses the amplitude noise leading to the reduction of SPM-induced nonlinear phase noise [15]. In this paper, we present a theoretical and experimental study of the reduction of nonlinear phase noise by an optical limiter based on the saturation of FWM. Phase noise in a multi-span system where the limiters are inserted is calculated with several imperfections of the limiter taken into consideration. A 10-Gbit/s short-pulse DPSK transmission experiment is performed and it is shown that the FWM-based limiter can enhance the transmission performance when the nonlinear phase noise is a dominant source of performance degradation.

## 2. Reduction of nonlinear phase noise by all-optical amplitude limiter

Figure 1 shows an optically-amplified transmission system consisting of M amplifier spans. In such a system, a major contribution of the phase noise comes from the amplified spontaneous emission (ASE) from the inline amplifiers. The quadrature component of the ASE noise relative to the signal gives direct phase fluctuations whose variance accumulates proportionally to the number of amplification stages. The in-phase noise component, on the other hand, does not produce phase noise but amplitude noise at the amplifier. The amplitude noise is translated to phase noise after propagation over the transmission fiber, which is more or less nonlinear, through the effect of SPM of the fiber [2]. The nonlinearity-induced phase noise (nonlinear phase noise) dominates over the direct phase noise (linear phase noise) when transmission distance and/or the signal power in the fiber are large. The noise generated in the source also contributes to the linear and nonlinear phase noise at the receiver. The variance of the phase noise is given by

where N_{s}, B, P_{sig}, γ, L_{eff}, and N_{a} are the power spectrum density of the source noise, noise bandwidth, peak signal power launched into the transmission fiber, nonlinear coefficient and effective span length of the transmission fiber, and spectrum density of ASE from each inline amplifier, respectively. N_{s} is related to the source OSNR (noise bandwidth of 0.1nm) as N_{s}=sP_{sig}/(12.5GHz-OSNR) , where s is the duty ratio of the signal (averaged signal power is given by P_{ave}=sP_{sig}) while N_{a} is given by h*v*n_{sp}(G-1), where h*v*, n_{sp}, and G are the photon energy, spontaneous emission factor, and gain of the inline amplifier compensating for the span loss, respectively. The first and second two terms in (1) are contributions from source noise and ASE from the inline amplifiers, respectively.

When an optical limiter that perfectly suppresses the amplitude noise is inserted after the transmitter (point A in Fig. 1), the nonlinear phase noise induced by the source noise, that is, the second term in (1), is eliminated and (1) becomes

where the ASE contribution from an additional amplifier with gain G_{r} located in front of the limiter is added as the last term. Such an amplifier is usually needed to boost the signal power to the saturation level of the limiter. N_{r} is given by h*v*n_{sp}(G_{r}-1). The nonlinear phase noise originating from the inline amplifier noise, the third term in (2), is further eliminated when the optical limiters are inserted every span at point B in Fig. 1. The phase noise then becomes

The variance of the phase noise (3) grows at most linearly as the number of spans M is increased and is inversely proportional to the signal power P_{sig}, indicating the effectiveness of the amplitude limiter in long-distance systems. The actual limiter, however, cannot perfectly eliminate amplitude noise, and the limiter itself generates additional phase noise. Influence of these imperfections of the actual limiter will be discussed in the next section.

## 3. Amplitude limiter using saturation of FWM in a fiber

FWM in fibers has a response time as short as a few femtoseconds leading to its wide applications in ultra-high speed nonlinear signal processing. Saturation of FWM interaction caused by pump depletion and/or excitation of higher-order FWM components is also ultrafast. The FWM saturation can be used as an amplitude limiter that suppresses bit-to-bit amplitude fluctuations of signal pulses [16]. Since the saturation of FWM can take place at relatively small signal power, the input power required for amplitude limitation is smaller compared with other types of 2R (re-amplification and re-shaping) regenerators based entirely on SPM [12]. This has been previously shown both by numerical simulation and an experiment [12, 17]. Smaller required input power leads to smaller additional phase noise generated by the limiter itself. In this paper we use the FWM-based amplitude limiter for the reduction of nonlinear phase noise. It is noted that the additional phase noise introduced by the limiter itself may be further suppressed by the use of limiters that have flatter output phase variation in response to the input power variation as proposed in [14].

A schematic of the FWM amplitude limiter is shown in Fig. 2. It consists of an EDFA, a continuous-wave pump source, a highly nonlinear fiber (HNLF), and an optical bandpass filter (OBPF) for the extraction of the signal wavelength component. Although not shown in Fig. 2, a polarization controller is inserted after the EDFA for the alignment of pump and signal polarizations in the experiment. The structure of the limiter is the same as that of a single-pump parametric amplifier. Because large parametric gain is not needed for the application to an amplitude limiter, a relatively small pump power of a few tens of milliwatts is sufficient. Fig. 3 shows an example of saturation behavior of the limiter obtained by numerical simulation for continuous-wave signals. The HNLF has zero-dispersion wavelength, dispersion slope, nonlinearity, loss, and length of λ_{0}=1556nm, dD/dλ=0.026ps/nm^{2}/km, γ=12/W/km, α=0.78dB/km, and L=1500m, respectively. Pump wavelength and power are 1561nm and 20mW, respectively. For this pump power the output power saturates at input powers ~ 50mW. The saturation input power becomes minimum for frequency separation between signal and pump 400~500GHz. It is noted that the unsaturated parametric gain takes a peak value approximately at a signal-pump separation Δ*v*
_{0}=(-2γP_{pump}/β_{2})^{1/2}/(2*π*) ≅ 270GHz, where β_{2}=d^{2}β/dω^{2} at the pump wavelength. The fact that the signal gain saturates at lower signal powers at signal-power separation a little larger than Δ*v*
_{0} has been pointed out in [18]. For the purpose of signal limiter, larger frequency separation between signal and pump (~ 600GHz) will be preferred because it gives better equalization of output power after saturation as shown by the dash-dotted curve in Fig. 3.

The expressions for the phase noise in the previous section assume an ideal amplitude limiter. Actual all-optical amplitude limiters have several imperfections. One is the extra phase fluctuation that the limiter itself produces. Since all-optical amplitude limiters inevitably use nonlinear medium, phase fluctuations are usually induced in the process of amplitude equalization. In the case of the limiter using FWM in a fiber, signals having different input powers suffer different phase shift due to SPM in the nonlinear fiber although the signal power is equalized at the output of the limiter. The induced phase shift δϕ is proportional to the power fluctuation of the input signal δP to be suppressed by the limiter. The proportionality coefficient k is defined by the relation δϕ = kδP/P_{sat}, where P_{sat} is the saturation input power. For the limiter whose saturation behavior is shown in Fig. 3, k≅0.8 rad is derived by the numerical simulation for a saturation power P_{sat}=50mW. The value of the proportionality coefficient k=0.8 at P_{sat}=50mW is almost equal to the value P_{sat}γ[1-exp(-αL)]/α obtained under a condition that the power fluctuation δP persists along the fiber. When relative power fluctuation of 10% around P_{sat}=50mW, for example, is to be suppressed by the limiter, a phase shift of 4.6 degrees is induced by the amplitude limitation for the value of k=0.8 rad. It is noted that k is approximately proportional to the input signal power. This indicates that small P_{sat} is favorable for the suppression of the limiter-induced nonlinear phase noise.

Another imperfection of the limiter is the residual amplitude fluctuation at the output of the limiter. This can be quantified by a residual power fluctuation ratio r , where the input signal power with fluctuation P_{sat}+δP leads to the output signal power P_{out}(1+rδP/P_{sat}). r=0 corresponds to perfect amplitude limitation while r=1 corresponds to no amplitude limitation. The coefficient r is rather a qualitative parameter accounting for the residual amplitude noise of the limiter. Actual causes of the residual amplitude fluctuation include the curvature of the transfer function P_{out}(P_{in}) shown in Fig. 3 at the operation point where dP_{out}/dP_{in}=0. Although the curvature of the transfer function should be taken into account for the accurate estimation of the probability density of the phase fluctuations, such a second-order effect is difficult to be included in the simple estimation of the magnitude of phase noise. We, therefore, include the influence of the residual power fluctuation assuming that it is proportional to the input power fluctuation.

Inclusion of these imperfections modifies the variance of the phase noise (2) where the limiter is inserted at the output of the transmitter to

The variance of the phase noise (3) with the limiter inserted every span is similarly modified to

## 4. Phase noise variance with and without using amplitude limiters

Here we present examples of phase noise variance for the system shown in Fig. 1 to show the effectiveness of the amplitude limiter and the influence of its imperfections. Fig. 4 shows standard deviation of phase noise versus signal power launched into the transmission fiber. The loss and nonlinearity of the transmission fiber are α=0.3dB/km and γ=3.5/W/km, respectively. The span length is 40km and the total loss per span is assumed to be 22dB, which are close to the values at the recirculating loop transmission experiment reported in the next section. The number of spans is 5, or the transmission distance is 200km. The noise figure of all the EDFAs in the system is 6dB (n_{sp}=2) and noise bandwidth B=2nm. Source OSNR(0.1nm noise bandwidth) is 24.5dB. The horizontal axis is the average power assuming duty ratio of 6.8%, which is also relevant to the experiment using 10Gbit/s 6.8ps pulses. Saturation input power to the HNLF in the FWM-based amplitude limiter P_{sat} is assumed to be 50mW (corresponding average power is 3.4mW) and the coefficient of generation of phase fluctuation is k=0.8 rad. Suppression of amplitude fluctuation is assumed to be perfect with r=0. Solid, dashed, and dash-dotted curves in Fig. 4 correspond to the cases without using limiters, with a limiter inserted at the output of the transmitter, and with limiters inserted every amplifier span, respectively. When no limiters are used, the phase noise becomes minimum at P_{ave}=0.17mW corresponding to SPM-induced phase shift to the signal ϕ_{SPM}=P_{sig}γL_{eff}M=0.59 rad. This is somewhat smaller than the optimal value predicted in [2]. This is mainly because of the inclusion of the effect of source noise in our calculation. When amplitude limiters are inserted in the system, the phase noise is greatly reduced especially at large signal power where nonlinear phase noise contribution is significant. It is noted that the phase noise is steadily decreasing with the increase of the signal power when the limiter is inserted every span. When perfect amplitude regenerators are inserted every span, signals propagate through the transmission fiber with no amplitude fluctuations and, therefore, nonlinear phase noise disappears. Because the remaining linear phase noise is smaller for larger signal power, the total phase noise is steadily decreasing with the increase of the signal power.

Figure 4 assumes perfect suppression of the amplitude fluctuation by the limiter. Fig. 5 shows the amount of phase noise versus average signal power when an imperfect limiter (r≠0) is inserted after the transmitter. As the residual power fluctuation ratio r increases, the effectiveness of the limiter is gradually lost. The influence of the residual amplitude noise is
even severer when the limiters are inserted every amplifier span as shown in Fig. 6. In this case with r greater than ~ 0.5, the phase noise is larger than that without limiters for P_{sig}<0.7mW. This severe degradation comes from enhanced nonlinear phase noise induced by the HNLF in the limiter. The unsuppressed amplitude noise after a limiter induces large nonlinear phase noise in the HNLFs in the succeeding limiters in the system. Fig. 6 shows that the amplitude limitation should be as perfect as possible especially when multiple limiters are inserted in the system.

## 5. Experiment

Figure 7 shows the setup of a DPSK transmission experiment. 10GHz short pulses (1.5ps) at 1558nm are generated by a mode-locked semiconductor laser diode and their phase is modulated by a 256-bit pseudo-random programmed bit pattern. After addition of ASE noise, spectrum of the signal is narrowed by an optical bandpass filter (OBPF) with resultant pulse width of 6.8ps. Then the signal is launched into a recirculating fiber loop. The transmission fiber is a densely-dispersion managed (DDM) fiber consisting of alternating normal- and anomalous-dispersion (≅±3ps/nm/km) non-zero dispersion-shifted fiber sections. Length of each fiber section is 2km and the total length is 40km. The DDM fiber was originally designed and fabricated for DDM soliton transmission at 80Gbit/s [19]. In such a fiber, dispersive pulse broadening is limited, which enhances the effect of SPM and consequently nonlinear phase noise. In the recirculating fiber loop, a manually controlled polarization controller and a polarizer are inserted. They stabilize the signal polarization in the loop and reject ASE noise whose polarization is orthogonal to that of the signal. The amplitude limiter based on saturation of FWM is inserted in the system either at the entrance of recirculating loop (point A) or inside the recirculating loop (point B). Effect of the amplitude limitation is observed by measuring the bit error rate (BER) with turning on and off the pump power in the limiter.

Figure 8 shows averaged output signal power versus input signal power to the HNLF when unmodulated 10GHz pulses (6.8ps) are launched into the amplitude limiter. Optical signal to noise ratio (OSNR) is 23dB with noise bandwidth 0.1nm. The HNLF has the same dispersion, nonlinearity, loss, and length as those used in the simulation in Section 3. Pump wavelength and power are 1561nm and 15mW, respectively. Q factor defined as μ/σ is also plotted where μ and σ are the mean value and standard deviation of the peak voltage of the detected electrical pulses after a lowpass filter with 3dB cutoff frequency 7.5GHz. Results with pump power on and off are compared in Fig. 8. When the pump is turned on, the output signal power shows saturation and the Q factor increases from 10 to 15 as the input power is increased.

Figure 9 shows measured BER after transmission over 200km (M=5) when the limiter is inserted after the transmitter. OSNR of the input signal is 21.5dB including noise in both polarizations. We find that the BER is remarkably lowered by the amplitude limitation for large signal power. This is qualitatively consistent with the calculation of phase noise as shown in Fig. 4. The BER degrades, however, at averaged signal power larger than ~ 1mW even when pump is on. This is considered due to the residual unsuppressed amplitude noise after the limiter, see Fig. 5. The imperfectness of the amplitude-noise suppression is indicated by the finite, relatively low, Q value even at its maximum shown in Fig. 8, although some additional noise is added after the limiter in the Q-value measurement. It is also noted that quantitative difference exists between the behaviors of the BER and phase noise shown in Fig. 9 and Fig. 5, respectively. The difference between the BERs with pump on and off at high signal power is decreasing in Fig. 9 while the difference of the phase noise with r<1 and r=1, corresponding to the cases with pump on and off, respectively, is increasing in Fig. 5. Quantitative discussion of the performance with and without amplitude limiters needs detailed BER analysis including non-Gaussian phase statistics and influence of amplitude noise. Such an analysis will be a subject of future study.

Figure 10 shows BER also after transmission of 200km when the limiter is inserted inside the recirculating loop. OSNR of the input signal is increased to 25.7dB in this experiment. Error-free transmission was not obtained even when the pump is on when the transmitter OSNR is lower at 21.5dB. It is considered that this is again because the residual amplitude noise after the limiter induce large nonlinear phase shift in the HNLF at each circulation. However, the range of usable signal power is extended when the limiter is inserted every amplifier span. The behavior can be qualitatively explained by the phase noise behavior shown in Fig. 6 with non-zero r.

## 6. Conclusion

In this paper we studied both theoretically and experimentally the reduction of nonlinear phase noise by an all-optical amplitude limiter using saturation of FWM in a fiber. Recirculating-loop transmission experiment of 10Gbit/s short-pulse DPSK signals showed that the limiter, which is inserted either at the output of transmitter or in the recirculating loop, can improve the performance at large signal powers where the nonlinear phase noise is significant.

In the present experiment, our main purpose was to show the effectiveness of the FWM-based amplitude limiter by comparing BERs with pump power to the limiter on and off. Significant performance improvement, however, could not be obtained by the limiter if we compare the BERs with the limiter inserted in the system to that with the limiter removed from the system. This is especially true when the limiter was inserted in the recirculating loop. We attribute this to imperfectness of the limiter. If the amplitude stabilization is not perfect, residual amplitude noise induces large nonlinear phase noise in the HNLF in succeeding limiters.

Theoretical expressions for the phase noise with the imperfect limiter inserted are derived, with which the influence of imperfections can be examined. It is shown that the residual amplitude noise after the limiter degrades severely the effectiveness of the limiter in suppressing the nonlinear phase noise. The residual power fluctuation should be as small as possible especially when multiple limiters are inserted in the system.

Finally it is noted that the single-pump FWM-based limiter is polarization sensitive. For the limiter to be used in real systems as inline devices, this issue should be solved, for example, by the use of orthogonally-polarized dual-pump scheme [20 ].

## Acknowledgments

This work is supported by the Japan Society for the Promotion of Science (JSPS) Grant-in-aid for Scientific Research on Priority Areas.

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