1. Introduction

The fast growing capacity demand in fiber-optic communication systems requires the use of larger optical bandwidth, higher spectral efficiency and advanced modulation formats such as multilevel phase-shift keying (m-PSK) and quadrature amplitude modulation (QAM). The latter, being a promising solution, poses however the problem that the increasing number of states in the constellation diagram makes the signal more sensitive to noise. Hence, efficient signal regeneration is necessary frequently. Especially phase-sensitive amplifiers (PSAs) attracted much attention in the last years for all-optical phase regeneration [1–7]. Several schemes for quadrature or even higher-level PSK have been proposed, e.g. using high-order idlers as pumps for the PSA [5] or phase-conjugated pumps [7] which also fulfill the required phase-matching conditions. However, the performance of these methods has been demonstrated only for pure PSK with one amplitude level.

In this work, we show that a PSA with phase-conjugated pumps can be also used for phase regeneration of so-called star-8QAM signals with two amplitude levels and four phases states. Necessary condition is equalization of amplitudes of the phase-conjugated pumps, which can realized, as we show, using nonlinearity of four-wave-mixing (FWM) process. This leads to identical parametric gain on both amplitude levels and therefore to optimal regeneration conditions. We have also found that the PSA stage can be used in two different operating modes: without format conversion or with phase-rotated amplitude levels, and in both cases efficient phase noise suppression is possible. High phase-regeneration efficiency for both amplitude levels in both regimes has been observed: the initial phase noise with 0.2 rad standard deviation is reduced by a factor of 5.

2. Operation principle of the phase-sensitive amplifier

The operation principle of the dual-stage regenerator [7] is illustrated in Fig. 1. In the first stage, the two phase-conjugated idlers are generated in two phase-insensitive fiber-optic parametric amplifiers (FOPA). Two CW pumps, which are equally frequency up and down shifted to the signal, are injected in highly nonlinear fibers (HNLF 1) where they interact with the star-8QAM signal via a FWM process. Note that the CW pumps have to be phase correlated to provide stable phase-matching in the PSA. The resulting idlers posses the conjugated phase −ϕ in contrast to the signal phase ϕ. At the output of the first stage, the two conjugated idlers are filtered and amplified so as to obtain optimum conditions for the second stage, the PSA. The reference fiber with no nonlinearity is used for the transmission of the signal to the PSA because of strong amplitude distortions of the output signals by the FWM process in HNLFs 1.

 

Fig. 1 Principle scheme of a dual-stage phase-sensitive amplifier with phase-conjugated pumps; HNLF, highly nonlinear fiber.

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In the second stage, in HNLF 2, the amplified idlers from the first stage serve as phase-conjugated pumps for the PSA. The phase relation of the phase-conjugated pumps and the signal allows a regeneration of four signal phase states through a degenerate FWM process with the phase-matching condition

It has been already shown that this operation principle efficiently regenerates quadrature phase-shift keying (QPSK) signals or even higher-level phase-encoded signals [6,7]. However, multiple amplitude levels pose a challenge since the PSA gain is highly dependent on the amplitude of the pumps. At least two options for a 2-amplitude-level PSA have to be taken into account: operating the PSA in small-signal mode or in saturation. In small-signal mode, the gain is the same for all amplitude states and hence the pump amplitude has to be the same for all amplitude states. This option could also work for more than two amplitude states. Operating the PSA in saturation means higher saturation for higher amplitude states and thus the pump power has to be higher for higher power states. This could have the advantage of weak phase-to-amplitude conversion of the high-amplitude states but this option is hardly possible for more than two amplitude states. Therefore, we concentrated on investigating the small-signal mode of the PSA.

In order to handle the two amplitude states we first study the dependence of the phase transfer function on the pump and signal power. This is illustrated in Fig. 2 for different power of the signal and the phase-conjugated pumps before the input of the PSA stage, where the phase distribution at the output of the PSA is shown [for a better understanding: the input distribution is a circle, not a QAM signal, phase states are distributed over all phase values, as in Fig. 2(a)]. For this study we first fixed the power of the phase-conjugated pumps to 14 mW and for the signal to 0.45 mW since it promised efficient phase regeneration (other system parameters can be found in the next section). In the first row [Fig. 2(c) and 2(d)] the phase distribution for different signal power is shown: 1.4 mW and 23 mW. One can clearly see that the phase dependence of the parametric gain and the phase distribution does not change significantly in (c). Increasing the signal power strongly by 17 dB leads to modification of the phase distribution in (d). Strong phase noise which differs a lot from the original phase state is highly attenuated, however weak phase noise is not efficiently suppressed as can be seen at the rounded edges of the four maxima of the phase distribution. Moreover, a slightly asymmetric behavior is observed which can be explained by self phase modulation of the signal. Varying the pump power by several dB while keeping the signal power at 0.45 mW has more influence as can be seen in Fig. 2(e) and 2(f) where the power was changed to 28 mW and 35 mW. This confirms that the signal power influence is negligible compared to the pump power in the small-signal regime. Therefore, different amplitude levels play a major role in the generation of the phase-conjugated pumps and in order to optimize the regeneration ability for multiple amplitude levels, we ideally need to generate conjugated pumps with only one amplitude level in order to conserve the same regeneration efficiency.

 

Fig. 2 Signal constellation diagrams: (a) at the PSA input; (b–f) at the PSA output for different power of the signal and the phase-conjugated pumps before entering the second PSA stage. Signal power / pump power: (b) 0.45 mW / 14 mW, (c) 1.4 mW / 14 mW, (d) 23 mW / 14 mW, (e) 0.45 mW / 28 mW, (f) 0.45 mW / 35 mW.

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One straight forward option to equalize the pump amplitudes is using an additional amplitude modulator between the first and second PSA stages. However, the signal has to be detected in order to extract the amplitude level information for driving the additional amplitude modulator and therefore this method has only little benefit compared to optical-electrical-optical regenerators.

Our proposal for resolving this problem is to operate the first stage in highly nonlinear gain regime, as it is illustrated in Fig. 3(a), adjusting the power of the star-8QAM signal and CW pump so as to obtain the same power levels for both amplitude states. Then the phase constellations of the generated idlers show that the amplitude levels of the conjugated pumps overlap with negligible phase shift. This also means that amplitude noise of the signal is not converted to phase noise in the phase-conjugated pumps. Operating the first stage in the nonlinear gain region is possible because the gain of a parametric amplifier decreases after a critical input power, in our case 28 mW, as it is illustrated in Fig. 3(b). The two amplitude states should be placed around the gain maximum.

 

Fig. 3 (a) Constellation diagrams illustrating the operation principle of the first and second PSA stages using the highly nonlinear gain regime in the first, pump-generation stage; (b) Power transfer function after first, pump-generation stage. The two amplitude values used in the paper are indicated: about 12 mW and 49 mW, which result in the same value for the output power.

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In the second stage, the original signal with two amplitude levels and the conjugated pumps (= idlers, just one amplitude level) interact in the PSA. Since the conjugated pumps exhibit only one amplitude level, the parametric gain and accordingly the regeneration is equally efficient on the low and high signal amplitude levels. As shown in Fig. 1(a), the signal is sent through a reference fiber with no nonlinearity in order to match the bit sequences of the conjugated idlers at the input of stage 2. Because of strong signal amplitude distortions by the FWM process due to the gain nonlinearity (overlap of the amplitude levels), the output signals of HNLFs 1 cannot be used.

3. Regeneration performance

In numerical simulations, we used a 50%-RZ star-8QAM signal with 40 GBaud rate at λ=1.550 μm [9]. The CW pumps are frequency up- and down-shifted by 250 GHz compared to the signal. The HNLFs have the following parameters: zero-dispersion wavelength λ₀ = 1.550 μm, dispersion slope 0.001 ps/nm²/km, nonlinear coefficient 10.5 W⁻¹km⁻¹, and 2500 m length. The spectral filters have 140 GHz bandwidth. In the first stage, operating in regime of nonlinear gain, the conjugated pumps are generated. The peak power of the star-8QAM signal is adjusted to 12 mW / 49 mW for its two amplitude states and to 250 mW for both CW pumps at the input of the first stage.

We found two different regeneration regimes for different conjugated pump and signal powers in the second, PSA stage. In the first operating mode, the average pump power is 3.9 mW and the signal peak power is much lower: 67 μW for the first amplitude level and 270 μW for the second one. The transfer functions for phase and amplitude can be found in Fig. 4 (red data). For both amplitude levels, we find four plateaus in the characteristic phase transfer function which indicates a high phase-regeneration ability for both amplitude levels. Also the characteristic gain curve shows the common behavior with four maxima [Figs. 4(c) and 4(d)]. The PSA gain is 1.4 dB, as is expected according to Eq. (2).

 

Fig. 4 (a,b) Characteristic phase transfer functions; in (b) the output phase of the red data was shifted by π/2 on the y-scale with respect to the black data for better comparison. (c,d) Gain dependence on phase for the first and second amplitude levels; red data belongs to the low-signal-power regime and black data to the high-signal-power operating case. (e–h) Constellation diagrams: (e) Input star-8QAM signal with phase noise; (f) One of the two conjugated pumps, the big cross and circle stand for the mean value of the respective phase state; (g) Regenerated output signal in the low-signal-power regime, (h) Regenerated output signal in the high-signal-power regime, where the low and high-amplitude levels are rotated with respect to each other. Circles and crosses signify two different amplitude levels, different colors stand for different phase states.

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The second, rather unconventional operating regime was found at average pump power 4.3 mW and much higher signal peak power: 8 mW / 32 mW for the two amplitude levels. Here, the FWM process results in attenuation of the regenerated signal, as can be seen in Figs. 4(c) and 4(d), since the optical power is transferred to the low-power conjugated pumps. Inspite of signal attenuation, the condition Eq. (2) for optimal regeneration is satisfied.

The characteristic phase and gain curves are shown in Fig. 4 (black data). For the first amplitude level, the phase and gain curves are similar as in the low-signal-power operating regime except for the maximum and minimum gain values, which represent actually attenuation. The gain difference between the maximum and minimum value is almost the same for both modi. For the second amplitude level, however, the characteristic curves change and the phase transfer function shows broader plateaus with overshoots. This feature for the higher amplitude level is interesting since regeneration in the second amplitude level is more important because this level is stronger affected by nonlinear phase noise. Regarding the gain curve, we find a higher gain difference of 8.6 dB compared to 4 dB in the first operating regime.

To study phase noise suppression we used a star-8QAM signal with added phase noise as shown in Fig. 4(e). Additional phase noise with 0.2 rad standard deviation was imposed on the signal by a phase modulator driven by a Gaussian noise signal. As it can be seen in Fig. 4(f), the two amplitude levels overlap for the conjugated pumps. The figure also shows that both amplitude levels (circles and crosses) are slightly shifted by 0.26 rad respective to each other. This effect is due to self phase modulation in the first stage because of their different original amplitude levels. In Figs. 4(g) and 4(h), the regenerated star-8QAM signals can be found, where (g) corresponds to the low-signal-power regime and (h) to the high-signal-power case. We observe high phase regeneration efficiency for both amplitude levels in both cases: the improvement of the phase Q-factor is about 5 ± 0.4. In the low-signal-power regime, Fig. 4(g), the two amplitude levels are slightly shifted by about 0.13 rad which is due to the phase difference of the conjugated pumps in Fig. 4(f). The phase-to-amplitude noise conversion is higher for the second amplitude level in the high-signal-power case which can present a problem for the discrimination of both amplitude states. However, the four phase states of the high-amplitude level are rotated with respect to the ones of the low-amplitude level. This effect can advantageously be used to separate the different phase and amplitude states. The rotation is dependent on the star-8QAM signal power before entering the second PSA stage and is due to self-phase modulation in the second PSA stage. Hereby, the intensity of the star-8QAM signal has to be adjusted so that the regeneration efficiency is as high as possible for both amplitude states but also the rotation of one amplitude level with respect to the other one should be close to an optimum of π/4 converting the star-8QAM signal into a 8QAM signal with two amplitude states but eight phase states. In the low-signal-power regime [Fig. 4(g)], the phase regeneration is less efficient for high phase noise but the relative phase state of both amplitude levels is kept and the amplitude noise is less pronounced.

Effect of signal amplitude noise can be seen in Figs. 5, where the eye diagrams, obtained by a phase-insensitive receiver as commonly used for on-off-keying, at different points in the regenerator are shown. The amplitude noise on the input signal (a) has standard deviation of 5 % of the state amplitude. The eye diagram of the conjugated pumps in Fig. 5(b) shows clearly the overlap of both amplitude levels. One can also see that the amplitude noise of the pump pulses is about the same or even a bit weaker than that of the signal while there is some noise enhancement of the background between the pulses because of the nonlinearity of the parametric gain in the first, pump-generation stage. As the signal used in the second, PSA stage is provided by a bypass, there is interaction of the input signal with pumps which have some excess noise between the pulses but not on the peaks. This should obviously produce not much noise on the output signal. Excess amplitude noise on the output signal, which is especially strong for the high-signal-power regime [see Fig. 5(c) and 5(d)], is mainly due to phase-to-amplitude noise conversion, common for PSA. Operation of the PSA stage in saturation cannot be used to reduce this noise component because it would lead to a change of the factor m in Eq. (2) and accordingly to inefficient phase noise suppression and, moreover, would change the ratio of the state amplitudes.

 

Fig. 5 (a–d) Eye diagrams provided by a direct detection of (a) input signal, (b) generated pumps (after first stage), (c) signal output in the low-signal-power and (d) high-signal-power regime; (e) dependence of amplitude noise on the phase reduction factor for low-signal-power and high-signal-power regime; inset of (e): output amplitude noise over input amplitude noise for the different power regimes and amplitude levels.

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An interesting issue is how the regenerator performance is affected by amplitude noise. The simulation results presented in Fig. 5(e) show the phase noise reduction factor plotted against amplitude noise for the different power regimes and amplitude levels. A general decrease of the regeneration efficiency, and thus of the phase noise reduction factor, is observed while increasing amplitude noise. Nevertheless, the phase noise reduction remains above 3 even for the amplitude noise of the input signal as strong as standard deviation 16 % of the state amplitude. We also studied the transfer of the amplitude noise in absence of phase noise of the input signal which can be seen in the inset of Fig. 5(e). For the low-signal-power regime the output amplitude noise is about the same as on the input, even a bit less: the input standard deviation of noise of 16 % of the state amplitude, results in about 16 % for the high-amplitude level and 12 % for the low-amplitude level. This result for the low-signal-power regime is quite promising since the amplitude noise is not enhanced. In the high-signal-power regime, the low-amplitude level shows the same noise level at input and output, but the output amplitude noise in the high-amplitude level increases, as expected from Fig. 4(h). In this case, one can use an amplitude regenerator before our phase regeneration scheme or working in the low-signal-power regime.

4. Conclusion

We investigated numerically the performance of a dual-stage phase-sensitive amplifier (PSA) for a star-8QAM signal. Despite the challenging task of multiple amplitude levels, optimal phase noise suppression on both amplitude levels has been achieved due to generating the conjugated pumps by FWM in regime of highly nonlinear gain. We investigated two power regimes and they both possess their advantages with respect to either better phase noise suppression or less phase-to-amplitude noise conversion. The original phase noise with 0.2 rad standard deviation is reduced by a factor of 5. The approach of stripping the amplitude modulation at the pump-generation stage and use then the PSA-stage in linear operation mode can be extended for processing of signals with two amplitude levels and any number of phase states.