## Abstract

We present a thorough investigation aimed at the optimization of a phase-sensitive optical parametric amplifier capable of simultaneous phase and amplitude regeneration. The regeneration potential, quantified in terms of the phase-sensitive extinction ratio, has been carefully assessed by a scalar model involving high-order waves associated with high-order four-wave mixing processes, going beyond the usual three-wave approach. Additionally, this model permits to unveil the physics involved in the high-order waves assisted regeneration. This permits a multi-dimensional and comprehensive optimization that fully exploits the underlying regenerative capability and expedites the design of a transparent regenerator, showing the potential to act as basic building block in future all-optical processing. We also compare different strategies when such regenerators are configured in concatenation. The approach can be readily applied to virtually any similar applications for different all-optical processing functionalities.

© 2017 Optical Society of America

## 1. Introduction

In order to meet explosive traffic requirements, advanced modulation formats with complex constellations have been extensively employed [1] to scale up the transmission capacity. In such communication systems, all-optical processing, which outperforms electronic approaches in terms of traffic cost, bandwidth, power-consumption, and flexibility, is highly desired in addition to the ceaseless pursuit of low noise repeating. To this end, phase-sensitive optical parametric amplifier (PSA), with its intrinsic ultra-fast response and quantum-limited amplification [2, 3], has become a promising candidate towards all-optical functionalities [3–6]. Particularly, the unique (phase and amplitude) squeezing property has led to the realization of all-optical regeneration and quantization of complex phase-encoded signals and the mitigation of nonlinear phase impairments [6, 7].

For a well-designed binary regenerator, a transparent step-like nonlinear phase transfer function [8, 9] is required. This is indeed expected in an ideal PSA thanks to the inverse gains experienced by orthogonal quadratures: the in-phase field components are amplified with maximum gain while the quadrature components undergo maximum de-amplification. However, such phase-sensitive gain leads to an increase of amplitude fluctuations, which can nevertheless be alleviated by operating the amplifier in the saturated regime [9]. It has been inferred that maximizing the ratio of the maximum and minimum phase-sensitive gains, the so-called phase-sensitive extinction ratio (PSER), is a fundamental approach to improve the processing efficiency [8–11]. Introducing large nonlinearities can directly lead to a high PSER. However, practical implementation of such solution requires large nonlinearities and/or large pump powers that are not always available. An alternative strategy consists in exploiting the extra waves stemming from high-order four-wave mixing (FWM) processes. Indeed, large PSER can be attained as a consequence of large phase-sensitive gain asymmetry (PSGA), defined as the product of the maximum and minimum gains, especially the resulting large de-amplification owing to the high-order processes. This has led to efficient phase regeneration and decomposition [10, 11] at relatively low nonlinear phase shifts (NPS). Regeneration and quantization can also be achieved by coherent superposition of weighted signals and idlers in sequentially concatenated stages [12–15]. More recently, polarization-assisted vector PSA, taking advantage of polarization effects in FWM, has been proposed for regeneration and phase quantization [16, 17]. This method mitigates the NPS requirement at the expense of diminished net gain and additional complexity [8]. To study the squeezing property and in turn optimize the regenerator, phase-sensitive transfer functions under some particular phase matching conditions have been discussed both in highly nonlinear fiber (HNLF) [9, 18] and semiconductor optical amplifiers (SOA) [19], based on the conventional 3-wave model [2, 3] and time-dependent rate equations, respectively. The influence of NPS to PSER has been initially introduced and explored using a 7-wave model [20]. As a promising candidate for future all-optical processing, the limit of such a fundamental PSA setup in the context of regeneration is of particular importance. To date, a thorough analysis and optimization for the basic quadrature operations is still imperative and remains to be done.

In order to fully exploit the regenerative potential of a single PSA based signal processor, we exploit the scalar 7-wave model [20, 21]. Using this precise model, we study quadrature squeezing in terms of nonlinear phase-sensitive transfer characteristics and trajectories in a wide spectrum range with varied phase matching conditions in both small signal and saturation regimes. This enables, for the first time, multi-dimensional optimization of quadrature operations from the regeneration point of view. In addition, different schemes to concatenate two amplifier sections are also discussed to optimize their operation. Based on our investigations, the quadrature squeezing potential acting as a basic building block of the regenerative and processing operations is thoroughly exploited. The proposed approach can be considered a general guiding tool for a variety of application scenarios.

## 2. Theoretical model

In the simplest 3-wave scalar optical parametric process, one considers two pumps denoted *P _{1}*,

*P*and a degenerate signal and idler, denoted

_{2}*S*(see the left picture in Fig. 1). These waves are supposed to be co-polarized along the entire nonlinear medium, the effect of random walk-off and differential group delay [22] is neglected. Governing by the phase matching condition, extra high-order waves will be naturally generated according to the energy conservation. To take extra high-order four-wave mixing processes into account, we follow a more general 7-wave model [21] operating in continuous-wave regime involving four additional 1st order pumps and signals, as shown in the right panel of Fig. 1. Then the powers, complex slow-varying amplitudes, and phases are noted

_{0}*P*,

_{j}*A*and

_{j}*φ*, respectively, at distinct angular frequencies

_{j}*ω*and wavelengths

_{j}*λ*where

_{j}*j = 0 ~*6. As depicted in Fig. 1,

*A*,

_{3}*A*and

_{4}*A*,

_{5}*A*are referred to as 1st order signals and pumps, respectively, while Δ

_{6}*λ*is the pump-pump wavelength separation. The wavelength offset

_{PPS}*δλ*=

_{OFS}*λ*-

_{0}*λ*corresponds to the deviation of the signal wavelength

_{ZDW}*λ*

_{0}with respect to the zero dispersion wavelength

*λ*of the medium, whose dispersion is supposed to evolve linearly with wavelength.

_{ZDW}As a consequence of the energy conservation, all the waves are symmetrically located in frequency with respect to the signal located at frequency *ω _{0}*. The effective phase mismatch ${\kappa}_{mnkl}$ governing the corresponding FWM process involving waves

*m, n, k,*and

*l*reads

*β*represents the linear phase mismatch and is calculated by expanding the propagation constant

_{mnkl}*β*up to 4th order in detuning with respect to

*ω*. The second term on the right hand side of Eq. (1) holds for the contributions from nonlinear phase mismatch owing to the interacting waves. The relative phase of the input waves, which determines the principle gain axes of the regenerator, is defined as

_{0}*φ*The field evolution of the waves co-propagating along the nonlinear medium is governed by a set of seven complex coupled nonlinear ordinary differential equations involving 22 four-wave mixing terms. This theoretical model exhibits sufficient accuracy with a manageable time and calculation complexity as shown in details in [21], which allows to explore the potential possibilities without scaling up to models containing more waves. The evolution of each wave can be obtained by numerically solving the set of equations. Along the whole paper, a typical commercial fiber (OFS standard HNLF, the same as in [21] for the sake of consistency) with length

_{r}= 2φ_{0}- φ_{1}- φ_{2}.*L =*1011 m,

*λ*1547.5 nm, nonlinear coefficient

_{ZDW}=*γ =*11.3 W

^{−1}.km

^{−1}, dispersion slope

*D*0.017 ps/(nm

_{λ}=^{2}.km), and attenuation coefficient

*α*= 0.9 dB/km is considered as the nonlinear media. The stimulated Brillouin scattering threshold of this HNLF is measured to be about 14 dBm, and can be technically mitigated by pump phase modulation.

## 3. Numerical results

#### 3.1 Gain properties, PSER, and PSGA

In order to investigate the potential regenerative capabilities, let us first study the relation, or more precisely the dependence, between PSER and PSGA. The maximum and minimum gain profiles for varying values of *δλ _{OFS}* and Δ

*λ*have been preliminarily calculated [21], based on which PSER and PSGA estimated by the 7-wave model can be derived, as depicted in Figs. 2(a) and 2(b), respectively. These results were obtained with the same 23 dBm total pump power.

_{PPS}According to the standard 3-wave model, the predicted PSER barely depends on the PSGA since the maximum and minimum gains are almost symmetric with respect to the unitary gain. The PSGA is then almost around 0 dB, and the PSER is about twice the maximum gain. By contrast, according to the 7-wave model, the maximum and minimum gain profiles are drastically modified owing to the high-order FWM processes. Thus the PSER and PSGA exhibit completely disparate behaviors. Though in most cases, the PSER evolution agrees with the PSGA one to some extent, the gain profiles are no longer symmetric, leading to distorted and abruptly varying PSER and PSGA maps as functions of *δλ _{OFS}* and Δ

*λ*, as can be seen in Fig. 2.

_{PPS}In the configuration where *δλ _{OFS}* = 0, where all the waves are centered at

*λ*, it is worth noticing that for 10 nm

_{ZDW}*≤*Δ

*λ*20 nm, the PSER exhibits an anomalously low value around 10 dB while the PSGA shows a positive value larger than 20 dB. This implies that the minimum gain is positive (in dB) in this region. This can be understood by the fact that owing to the multiple interactions of all high-order processes, phase-insensitive processes dominate over phase-sensitive ones, resulting in non-negative gain of the output signals in this region. Similar behaviors are also found for small Δ

_{PPS}≤*λ*in the normal dispersion (

_{PPS}*δλ*≤ 0) regime, corresponding to the positive PSGA area (see the yellow part in the left part of Fig. 2(b)). This parameter range thus exhibits no promising operation point. In the rest part of the normal dispersion regime, where the PSGA profile is almost flat, implying more symmetric gain profiles, and it is therefore not surprising to find only a moderate PSER value of about 20 dB.

_{OFS}The situation appears to be quite different in the anomalous dispersion regime (*δλ _{OFS}* > 0). As indicated in Fig. 2, when Δ

*λ*is relatively small, various nontrivial zigzag-shaped negative PSGA dips are located in this regime, exhibiting a strong overlap with regions of large PSER. The minimum PSGA reaches almost −50 dB, revealing the large deamplification that leads to a high PSER reaching up to about 60 dB. This configuration is particularly promising and will be carefully investigated in the following sections.

_{PPS}Besides, in both normal and anomalous dispersion regimes, one can observe two diagonal stripes of PSGA dips and peaks, respectively, expanding towards relatively large absolute values of *δλ _{OFS}* and Δ

*λ*. However, these dips and peaks do not coincide with large values of the PSER except for some inconspicuous periodical ripples. This is due to the fact that in large

_{PPS}*δλ*and Δ

_{OFS}*λ*regime, as a result of the strong dispersion, the large phase mismatch hinders the emergence of high-order FWM processes, eliminating the influence of high-order waves. For instance, when

_{PPS}*δλ*≈0, all spurious processes vanish when Δ

_{OFS}*λ*gets large, leaving only the fundamental 3-wave FWM leading process. In this case, PSGA sums up to nearly zero because of the almost symmetric gain profiles, giving rise to a moderate PSER.

_{PPS}To summarize, the PSER and PSGA are not always positively related, and a large PSGA is a necessary but not sufficient condition to obtain a large PSER.

The substantial PSER predicted by 7-wave model allows improving the steepness of the phase transfer function, thus permitting excellent squeezing efficiency. On top of this, the amplitude fluctuation associated with the phase-to-amplitude conversion becomes prominent and will be studied in the following section.

#### 3.2 Optimization of the nonlinear phase shift

Starting from the wavelength allocation giving rise to a large PSER as predicted in Fig. 2, we calculate the evolution of the amplifier performance as a function of the NPS, defined as the product of *γ*, the total input pump power *P* = *P*_{1} *+ P*_{2} *(with P*_{1} *= P*_{2}*)*, and *L*. The results are reproduced in Fig. 3(a) for the 3-wave (dashed lines) and the 7-wave (full lines) models with *δλ _{OFS}* = 3.0 nm and Δ

*λ*= 4.5 nm.

_{PPS}When the NPS increases, the maximum gain (*G _{max}*) and minimum (negative) gain (

*G*) calculated by 3-wave model evolve almost symmetrically with respect to the unitary gain (0 dB). As a consequence, the PSGA remains close to 0 dB, and the PSER grows almost independently from the PSGA. This is consistent with the results regarding the wavelength allocation

_{min}*δλ*and Δ

_{OFS}*λ*discussed in the last section for the 3-wave model. Now, in the 7-wave model,

_{PPS}*G*is slightly smaller than in the 3-wave model. This phenomenon was expected since the high-order waves consume a significant part of the pump power that becomes unavailable for amplification of the signal. On the contrary, the most surprising difference between the two models comes from the minimum gain

_{max}*G*, which exhibits two abrupt dips for particular NPS values. These dips are also visible on the PSGA at the same NPS values. This is reflected on the PSER as well. By conducting an analysis similar to [21], we can identify the FWM mechanisms whose total phase mismatch is close to zero, and thus play a significant role in the energy transfer between the 7 waves. In the vicinity of the gain dips, this analysis shows that the processes that extract power from

_{min}*A*are much more efficient than those that amplify

_{0}*A*. Taking advantage of the first PSER peak, excellent performances can be expected with moderate total pump power (

_{0}*P =*18.9 dBm), for which one still remains in the small NPS region.

As mentioned above, the associated phase-to-amplitude conversion of the amplifier can induce severe amplitude noise, which is detrimental to the quality of regeneration. This problem should be carefully addressed to achieve regeneration without sacrificing amplitude noise. Analogous to the phase-to-phase transfer characteristics, a step-like phase-to-amplitude response is therefore highly desirable. To this end, the instantaneous saturation property, which exploits the power-dependent gain saturation, due to which the output signal amplitude is distorted, is conventionally investigated to cope with the accompanying intensity variations. However, cares should also be taken as operation in saturation regime will in turn change the PSER performance. For this reason, the influence of the degree-of-saturation (DoS), i.e. the ratio of the incident signal power to the total input pump power, should be investigated. This is performed in Fig. 3(b) for the optimized wavelength allocation and pump power that we found above.

Starting from the small signal region where DoS *≤* −30 dB, Fig. 3(b) shows that both *G _{max}* and

*G*reduce with saturation. However,

_{min}*G*decreases faster than

_{min}*G*, leading to a drop of the PSGA and a small increase in the PSER. With a further increase of the DoS towards the full saturation regime, the absolute values of

_{max}*G*and

_{max}*G*both diminish as well as the PSER as a result. The fluctuations in gain properties after DoS

_{min}*≥*0 dB are quite similar to the behavior observed in conventional optical parametric amplifiers attributed to the intense interaction between comparably strong waves. Nonetheless, the particular demand on signal power makes this fully saturated region impractical. It appears that operation in the gain-limited saturation brings about limited improvement in terms of PSER. However, it could be particularly beneficial for the mitigation of the associated amplitude variations as will be discussed in the following sections regarding the overall optimization.

#### 3.3 Phase-sensitive transfer characteristics

We have just seen that the DoS directly modifies the phase-to-amplitude response and in turn the PSER as well. Consequently, in order to provide a visual picture of the trade-off between phase squeezing and amplitude/power fluctuation management, we plot three different phase transfer characteristics of the amplifier in Fig. 4 for both 3- and 7-wave models. We represent in false colors the phase-to-gain, phase-to-power, and phase-to-phase transfer functions for the DoS varying from −40 to 5 dB and a total pump power equals to *P* = 18.9 dBm. Other parameters are chosen based on the preceding optimization results.

First, the phase-to-gain transfer characteristics from 3- and 7-wave models are shown in Figs. 4(a) and 4(b), respectively. Compared to the quite uniform evolution in small signal region in the 3-wave model, the 7-wave transfer function reveals a much deeper minimum gain, which confirms our preceding discussions. Considering the effect of saturation, the gain distortion is obviously more intense in the 7-wave model than in the 3-wave one. When the DoS lies between −3 dB and −20 dB, both maximum and minimum gain plateaus become much flatter than in 3-wave model. Assisted by the high-order waves, one can observe strong signal depletion when the signal power just becomes comparable with the pump power in the full saturation region, permitting potential all-optical processing opportunities.

The estimation in signal gain is also illustrated by the phase-to-power transfer properties shown in Figs. 4(c) and 4(d). One can also observe the fluctuations along the maximum and minimum plateaus, which will influence the amplitude behavior of the squeezing effect. Both transfer characteristics indicate the possibility to mitigate the associated amplitude noise. However, as discussed from Fig. 3(b), gain limitation using saturation in turn degrades the PSER and thus affects the regeneration capability. Considering now the phase-to-phase response plotted in Figs. 4(e) and 4(f), in the small signal regime, the 7-wave model offers a high detail precision for the transition between two squeezed quadratures. As the squeezing effect is directly connected with PSER, the substantially higher PSER estimated by the 7-wave model indeed leads to better squeezing of both quadratures. This much higher PSER offers some extra margin before the phase squeezing is affected by gain saturation, eventually allowing better suppression capability for amplitude variations. In the 3-wave model, the phase squeezing is compromised for saturation regimes corresponding to −10 dB ≤ DoS ≤ −3 dB. The steepness of the transitions between quadratures is impaired and, more seriously, the phase plateau is no longer flat but fluctuates around 0 rad input signal phase. This is not the case according to the 7-wave model, as the impact due to the saturation does not degrade the sharp transition that much. Only negligible ripples in phase plateaus are then observed.

#### 3.4 Complex plane trajectory

The output signal trajectories in complex plane from linear to saturation regime provide intuitive observations for the simultaneous squeezing effects on both phase and amplitude, as illustrated in Fig. 5. The input signal phase is varying along the unit circle and the output trajectories are normalized with respect to the input signal. They clearly exhibit the gain degradation due to the high-order waves in the 7-wave model. Except for the distortion on the trajectories, it should be noted that in both 3- and 7-wave models the maximum gain axis rotates with saturation towards the same direction. This shift of the principle gain axis is more significant with stronger DoS.

In small signal linear regime, instead of amplified circles for conventional phase-insensitive amplifier, squeezed elliptical output trajectories are explicitly observed. Phase squeezing along the gain axes occurs simultaneously with a large associated amplitude variation in both models. Notably, in this small signal regime, the elliptical trajectory predicted by the 7-wave model is more closed than that in 3-wave model, indicating a more efficient squeezing effect in 7-wave model. This is attributed to the higher PSER with the assistance of the high-order waves.

With the increase of the saturation, the principal gain axis further rotates towards the horizontal axis. This is more clearly observed started from DoS = −10 dB, possibly due to the relative large signal power that gives rise to intense interaction between waves. When DoS = −5 dB, we can observe some moderate amplitude squeezing occurring together with the phase squeezing. This is especially striking in the 7-wave model where some small spirals are buckling around the two terminal vertex of the trajectory. It is well confirmed that, with adequate saturation, it could be possible to achieve desired phase squeezing while alleviating the amplitude noise.

## 4. Processing capability

#### 4.1 Regeneration of BPSK signal

Using the proposed optimization approach, the regenerative capability is firstly numerically evaluated in the case of BPSK modulation. The BPSK signal is contaminated by additive white Gaussian noise with 20 dB signal-to-noise ratio. The root-mean-square error vector magnitude (EVM_{rms}, the ratio of the amplitude of the error vector to the root mean square amplitude of the reference) is used as the performance metric and is assessed with respect to the varied DoS as presented in Fig. 6. It can be easily inferred that the 7-wave model predicts better EVM improvement. The most efficient regeneration configuration in terms of EVM_{rms} is achieved at about −7.5 dB DoS, which agrees well with the predictions of sections 3.3 and 3.4. It is worth noticing that this DoS does not correspond to the optimized PSER (see Fig. 3(b)). However, taking into consideration the phase-to-amplitude noise transfer, the total quadrature noises, including phase and amplitude portions, become well balanced, thus leading to this promising value of the EVM_{rms} around such DoS.

The BPSK constellations at input and output with −30 dB, −7.5 dB, and 0 dB DoS are reproduced in Fig. 7, respectively. Unlike at DoS = −20 dB where the amplitude squeezing is almost negligible, both phase and amplitude squeezing are observed at −7.5 dB as shown in Fig. 7(c). The constellation is then distorted in a similar manner as inferred in Fig. 5. In contrast, further increase of the DoS only brings intense energy interchange amongst waves, leading to highly distorted squeezing in both quadratures that is of no application interest.

#### 4.2 Field decomposition of QPSK signal

All-optical decomposition and regeneration of both in-phase and quadrature components are indispensable functionalities for future all-optical networks that require transparent modulation format conversion as well as some other signal processing capabilities. To this end, by precisely managing the gain axis, i.e. the relative phase between the signal and pumps for example, it is feasible to obtain the orthogonal quadratures individually. In this scenario, we take the decomposition of a QPSK signal as an example test bench.

By controlling the phases of the incident pumps, the gain axis of the regenerative PSA can be aligned to either in-phase (I) or quadrature (Q) for decomposition. In Fig. 8, the EVM_{rms} of both I and Q components are numerically estimated by the 3- and 7-wave models in the cases of the decomposition of each component, respectively. The overall tendency of the EVM_{rms} is quite similar to that in BPSK case. Though the EVM_{rms} evolution with respect to the DoS in both 3-wave and 7-wave models exhibits similar trend, the values from the two models are different. In the small signal regime, although the squeezing is more efficient in 7-wave model thanks to the high-order waves, the EVM performances in the two models are quite close to each other. However, while entering the saturation regime, the divergence increases more and more. The optimum EVM_{rms} for I and Q quadratures occurs at about −6.5 dB DoS, suggesting that some degree of saturation is helpful. Meanwhile, the 7-wave model predicts about 1.5 dB further improvement of EVM_{rms}. Similar to the BPSK case, distortion of the constellations (see Fig. 9) is observed in fully gain-saturated regime, which in turn deteriorates the EVM performance.

It is worth noting that in the saturation regime, the occurrence of the distinct distortions on the squeezed constellations is mainly attributed to the fact that the transfer characteristics have been severely modified in the saturation regime as implied by Fig. 4 and Fig. 5. This eventually leads to the non-orthogonal relation between the maximum and minimum gain axes, resulting in the differences in the constellations.

## 5. Concatenation

According to the discussions above, the simultaneous regeneration in both amplitude and phase in a saturated PSA has been investigated and systematically optimized. One question remains: whether the simple concatenation of two such regenerative PSAs can possibly outperform the separated regeneration in two successive stages. It is intuitive to first consider cascading the same PSA, extending to twice the length of the nonlinear medium, hence doubling the NPS. The resulting gain properties are shown below in Fig. 10.

It appears that, compared at the same amount of NPS, both strategies exhibit quite similar tendencies. However, more precisely, for two stages of HNLF, G_{max} is slightly lower as well as the G_{min}, while similar PSGA and PSER are achieved, but with a slightly higher NPS requirement. This is simply due to fact that in the cascaded case, the higher NPS comes at the price of stronger losses attributed to the longer HNLF.

Since the generation of the high-order waves aids at improving the regeneration capability, we evaluate different strategies with different combinations of waves at the input of the second stage, as illustrated in Fig. 11. The final outputs are assessed with respect to either the initial input or the output of the first stage.

In the former case, no other strategy exhibits a superior performance compared with the straightforward concatenation (the first case “Two stages” with black solid line in Fig. 11(a)) or just directly exciting all the seven waves into the second stage (the last case “1s2p2hs2hp” with dark brown dashed line in Figs. 11(b)). In another general application scenario, considering regeneration at relay points or intermediate sections or relay stages, the output is assessed against the input of the second stage as presented in Fig. 11(c) and 11(d). The most promising strategy is still the overall excitation of all the waves for the second stage. In both scenarios, the more waves from the first stage are used, the better obtainable efficiency in terms of PSER. It is important to note that, even if compared with the tactics in which the dual pumps are regenerated before sending to the second stage thus achieving higher NPS, the direct excitation of as many as possible waves from the first stage still offers the most competitive prospect. Such phenomena could be probably interpreted as according to the phase matching conditions determined by the nonlinear medium and the wavelength allocation, the high-order processes will inevitably take place, therefore leading to the emergence of high-order waves stemming from those processes with certain phase relation in between. This way, no matter what extra waves would be excited, all the waves will unavoidably grow and propagate along the medium with an expectable phase relation, and share the limited total power. It should be also noted that, the regeneration of some designated waves, usually the pumps, could increase the NPS in the second stage, however, at the expense of modifying the phase relations amongst the waves. As these kinds of regeneration does not change or impair the overall tendency of the evolution of all the waves, the growth of the waves has to be steered and reconstituted according to the phase matching condition and eventually again comply with their specified trends. Such that, to arrive at the same PSER peak, the cases with regenerated pumps demands a bit larger PSER. And the peak value is slightly higher due to the stronger NPS induced. Thereby, in a word, the most promising and potential concatenation strategy lies in the straightforward excitation of all the waves with their well-established phase relation.

## 6. Conclusion

In this work, a systematic optimization approach has been proposed for a regenerative PSA based on the more precise 7-wave model. On top of this, the performance of regenerative dual-pump degenerate PSA has been thoroughly investigated and analyzed for simultaneous phase and amplitude regeneration, an important building block of future all-optical processing. To bring the untapped regenerative potentialities into full play, we perform numerical estimations to exploit fundamental regenerative PSA through precise optimization comprising the wavelength allocation, nonlinear phase shift, and degree-of-saturation. Compared with the conventional 3-wave model, the predicted high PSER provides a relaxed margin for the underlying gain-saturation operation. Therefore, it permits to alleviate the accompanying amplitude fluctuation in phase squeezing with moderate power requirement. Such an optimized configuration facilitates the realization of simultaneous regeneration without compromising the performance. It also exhibits great potential to relieve the rigorous power restriction allowed for the input of a HNLF due to the stimulated Brillouin scattering from application point of view. With the proposed approach, the phase and amplitude noise reduction for regeneration of either BPSK or QPSK has been numerically studied.

Moreover, the interpretations and discussions regarding different concatenation strategies unveil the fact that, governed essentially by the phase matching conditions, one could hardly find a more promising solution than the straight excitation of all the relevant waves involved for cascaded identical PSA stages. Such comprehensive and versatile optimization approach could fully exploit the potential regenerative capability of PSA, providing a powerful tool for the application oriented optimization in future all-optical processing networks.

## Funding

French Agence Nationale de la Recherche (ANR) (ANR-12-BS03-001-01); China Scholarship Council (201406230161).

## Acknowledgments

W. Xie is partially supported by Chinese Government Scholarship (CSC) (201406230161).

## References and links

**1. **R. J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity Limits of Optical Fiber Networks,” J. Lightwave Technol. **28**(4), 662–701 (2010). [CrossRef]

**2. **G. P. Agrawal, *Nonlinear Fiber Optics*, 4th ed. (Academic, 2007).

**3. **Z. Tong and S. Radic, “Low-noise optical amplification and signal processing in parametric devices,” Adv. Opt. Photonics **5**(3), 318 (2013). [CrossRef]

**4. **R. Slavík, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Anderson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrom, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics **4**(10), 690–695 (2010). [CrossRef]

**5. **J. Parra-Cetina, A. Kumpera, M. Karlsson, and P. A. Andrekson, “Phase-sensitive fiber-based parametric all-optical switch,” Opt. Express **23**(26), 33426–33436 (2015). [CrossRef] [PubMed]

**6. **M. Karlsson, “Transmission systems with low noise phase-sensitive parametric amplifiers,” J. Lightwave Technol. **34**(5), 1411–1423 (2016). [CrossRef]

**7. **S. L. Olsson, M. Karlsson, and P. A. Andrekson, “Nonlinear phase noise mitigation in phase-sensitive amplified transmission systems,” Opt. Express **23**(9), 11724–11740 (2015). [CrossRef] [PubMed]

**8. **F. Parmigiani, G. Hesketh, R. Slavík, P. Horak, P. Petropoulos, and D. J. Richardson, “Polarization-assisted phase-sensitive processor,” J. Lightwave Technol. **33**(6), 1166–1174 (2015). [CrossRef]

**9. **C. Lundström, B. Corcoran, M. Karlsson, and P. A. Andrekson, “Phase and amplitude characteristics of a phase-sensitive amplifier operating in gain saturation,” Opt. Express **20**(19), 21400–21412 (2012). [CrossRef] [PubMed]

**10. **M. Gao, T. Kurosu, T. Inoue, and S. Namiki, “Efficient phase regeneration of DPSK signal by sideband-assisted dual-pump phase-sensitive amplifier,” Electron. Lett. **49**(2), 140–141 (2013). [CrossRef]

**11. **M. Gao, T. Kurosu, T. Inoue, and S. Namiki, ”Low-penalty phase de-multiplexing of QPSK signal by dual-pump phase sensitive amplifiers.” *39th European Conference and Exhibition on Optical Communication*, OSA Technical Digest (CD) (Optical Society of America, 2013), We.3.A.5 (2013).

**12. **J. Kakande, R. Slavík, F. Parmigiani, A. Bogris, D. Syvridis, L. Grüner-Nielsen, R. Phelan, P. Petropoulos, and D. J. Richardson, “Multilevel quantization of optical phase in a novel coherent parametric mixer architecture,” Nat. Photonics **5**(12), 748–752 (2011). [CrossRef]

**13. **T. Kurosu, M. Gao, K. Solis-Trapala, and S. Namiki, “Phase regeneration of phase encoded signals by hybrid optical phase squeezer,” Opt. Express **22**(10), 12177–12188 (2014). [CrossRef] [PubMed]

**14. **R. P. Webb, M. Power, and R. J. Manning, “Phase-sensitive frequency conversion of quadrature modulated signals,” Opt. Express **21**(10), 12713–12727 (2013). [CrossRef] [PubMed]

**15. **K. R. H. Bottrill, F. Parmigiani, L. Jones, G. Hesketh, D. J. Richardson, and P. Petropoulos, “Phase and amplitude regeneration through sequential PSA and FWM saturation in HNLF,” in Proceedings of the European Conference on Optical Communication (IEEE, 2015), p. We.3.6.3. [CrossRef]

**16. **F. Parmigiani, K. R. H. Bottrill, R. Slavík, D. J. Richardson, and P. Petropoulos, “Multi-channel phase regenerator based on polarization-assisted phase-sensitive amplification,” IEEE Photonics Technol. Lett. **28**(8), 845–848 (2016). [CrossRef]

**17. **F. Parmigiani, G. D. Hesketh, R. Slavík, P. Horak, P. Petropoulos, and D. J. Richardson, “Optical phase quantizer based on phase sensitive four wave mixing at low nonlinear phase shift,” IEEE Photonics Technol. Lett. **26**(21), 2146–2149 (2014). [CrossRef]

**18. **C. Lundström, Z. Tong, M. Karlsson, and P. A. Andrekson, “Phase-to-phase and phase-to-amplitude transfer characteristics of a nondegenerate-idler phase-sensitive amplifier,” Opt. Lett. **36**(22), 4356–4358 (2011). [CrossRef] [PubMed]

**19. **K. Saito and H. Uenohara, “Analytical investigation of operating conditions for simultaneous intensity and phase noise suppression using phase sensitive semiconductor optical amplifiers,” *Optical Fiber Communication Conference*, OSA Technical Digest (CD) 2013 (Opitcal Society of America, 2013), JTh2A.31 (2013). [CrossRef]

**20. **M. Gao, T. Inoue, T. Kurosu, and S. Namiki, “Evolution of the gain extinction ratio in dual-pump phase sensitive amplification,” Opt. Lett. **37**(9), 1439–1441 (2012). [CrossRef] [PubMed]

**21. **W. Xie, I. Fsaifes, T. Labidi, and F. Bretenaker, “Investigation of degenerate dual-pump phase sensitive amplifier using multi-wave model,” Opt. Express **23**(25), 31896–31907 (2015). [CrossRef] [PubMed]

**22. **M. E. Marhic, A. A. Rieznik, G. Kalogerakis, C. Braimiotis, H. L. Fragnito, and L. G. Kazovsky, “Accurate numerical simulation of short fiber optical parametric amplifiers,” Opt. Express **16**(6), 3610–3622 (2008). [CrossRef] [PubMed]