## Abstract

We theoretically and numerically explain the power saturation and the additional phase noise brought by the fiber optical parametric amplifier (FOPA). An equation to calculate an approximation to the saturated signal output power is presented. We also propose a scheme for alleviating the phase noise brought by the FOPA at the saturated state. In simulation, by controlling the decisive factor dispersion difference term Δ*k* of the FOPA, amplitude-noise and additional phase noise reduction of quadrature phase shift keying (QPSK) based on the saturated FOPA is studied, which can provide promising performance to deal with PSK signals.

©2012 Optical Society of America

## 1. Introduction

Current optical communications technologies rest upon the principle of repeaterless optical amplification, a passive form of inline signal processing. This is in sharp contrast with earlier systems, where signals were periodically passed through electronic repeaters. A key question is to what extent active inline signal processing, such as all-optical signal regeneration [1], could prove beneficial in future developments, both in terms of system performance and economic return. Fully transparent features in both the time and frequency domain are required for optical signal regenerators because future wavelength-division multiplexed (WDM) networks will utilize ultra-broad bandwidths over 100 nm and data rates will be 160 Gb/s or even higher, and all-optical signal regeneration is an efficient method to extend reach of high-speed optical signal transmission without relying on optical-electric- optical conversion and signal processing in the electric domain. Optical parametric regeneration, which is based on nonlinear optical principles, and also fiber optical parametric amplifiers (FOPA) has been known for years, but only recently has their potential in optical signal processing been recognized, with experiments conducted utilizing them in applications including optical regeneration [2], wavelength conversion and multicast [3], and optical time domain multiplexing (OTDM) [4].

One promising property of the FOPA is its potential to have phase-insensitive, quantum-limited amplification [5–7]. Recent research has applied this property as a signal regenerator, which takes advantage of the instantaneous (femtosecond) saturation properties of FOPA operating within the gain-limited regime. Regeneration in FOPAs works on the principle of ultrafast power-dependent gain saturation, which can be utilized to suppress intensity variation in signals. This suppression can increase the signal-to-noise ratio of signals as well as equalize the optical power level. Matsumoto [8] and Kylemark et al. [9] discussed the noise properties of the saturated FOPAs. Watanabe, et al. have attempted to process phase-shift keying (PSK) and on-off keying (OOK) signals with saturated FOPAs [10]. All the results are focused on the performance of the saturated FOPAs as amplitude limiters. However, in their discussion, it is difficult to get an expression for the output signal power, which is not convenient for real applications.

Another problem is that if we only consider the amplitude limitation properties, the phase fluctuation introduced by the FOPA would be a crucial factor that will degrade the system performance. This phenomenon is illustrated in Fig. 1 . Some researchers have mentioned this problem [8,11], but the origin of this additional phase noise was not explained clearly. In this paper, we discuss the saturation in the FOPA and derive a useful expression for estimating the output signal power. The reason for the additional phase noise in the saturated FOPA is also discussed, and we propose a method to cancel or alleviate this problem.

## 2. Signal output power in a saturated FOPA

For an unsaturated FOPA, the theory and applications can be found in many textbooks and papers [12]. The classical theory is mainly based on these assumptions: the signal power is much less than the pump power; the pump power is unchanged during the amplification process; and the phase-matching condition is always perfect. Under these hypotheses the complex amplitudes of the three waves after propagation in the FOPA can be calculated [9]. However, in the discussion of a saturated FOPA, this theory is no longer suitable. A theoretical analysis without approximation of pump depletion is essential. Usually, the three coupled-mode equations for the degenerate four-wave mixing can be expressed as

*E*

_{0},

*E*

_{1}, and

*E*

_{2}are the electric field complex amplitudes of the pump power, and two sideband amplitudes, respectively. Δ

*k*,

*γ*, and

*z*are the propagation constant mismatch, third order nonlinear parameter, and propagated distance in the fiber, where Δ

*k*=

*k*

_{1}+

*k*

_{2}−2

*k*

_{0}. Since the total power

*P*= |

*E*

_{0}|

^{2}+ |

*E*

_{1}|

^{2}+ |

*E*

_{2}|

^{2}is conserved as we ignore the loss in the fiber, Eq. (1) can be conveniently rewritten in terms of normalized dimensionless amplitude variables. To this end, the normalized pump power

*η*(

*z*)≡|

*E*

_{0}|

^{2}/

*P*and the normalized signal and idler amplitudes

*a*

_{1,2}≡|

*E*

_{1,2}|/

*P*

^{1/2}are introduced. The phase matching condition is defined as

*ϕ*(

*z*) = Δ

*kz*+

*ϕ*

_{1}(

*z*) +

*ϕ*

_{2}(

*z*)−2

*ϕ*

_{0}(

*z*). Kylemark et al. [9] and Cappellini et al. [13] studied the analytical saturation theory for FOPAs. In their discussion, the process of FWM can be expressed by the normalized coupled equations:

*η*,

*a*

_{1}, and

*a*

_{2}are the normalized pump power, and the two normalized sideband amplitudes, respectively.

*ξ*=

*zγP*is the normalized propagation length, and

*κ*= Δ

*k*/

*γP*is the normalized phase-matching term. In Ref [13], the results are expressed in the form of normalized pump power as

In this work, the mathematical notation is the same as in Ref [10]. unless noted otherwise. Function sn stands for the elliptic sine function, which is a periodic function. The parameters *a*, *b*, *c*, and *d*, are the roots of equation d*η*/d*ξ* = 0 (Eq. (2a) in Ref [10].)), which are ordered so that *a > b* > *η*(*z*) ≥ *c > d*. From Eq. (3) it can be known that the pump power at the output is a periodic function of the normalized propagated length *ξ* with periodicity of the sn function. When the total power *P* is unchanged in the fiber, the power will be transferred between the signal/idler and the pump periodically.

However, Eq. (3) does not lend itself to a clear and intelligible understanding of the properties, as the analytical expressions of *a*, *b*, *c*, and *d* are too complicated to calculate the saturated signal output power from this equation. However, in many applications of saturated phase-insensitive FOPA as an amplitude limiter, the results can be simplified, where the input pump power is very large compared with the power of the input signal. In this paper, we choose fiber with *γ* = 12/W/km and dispersion slope d*D*/d*λ* = 0.03 ps/nm^{2}/km. The difference between the pump and the zero-dispersion wave-length is *λ*_{p}−*λ*_{0} = 3 nm. Pump power *P*_{0} is 400 mW. The effect of fiber loss is neglected. Signal frequency is set at the peak position in the unsaturated gain spectra, Δ*ν* = (−2*γP*_{0}/*k*_{2})1/2/(2π), where Δ*ν* and *k*_{2} are the frequency separation between the signal and the pump and the group-velocity dispersion (GVD) coefficient at the pump wavelength, respectively.

When the input signal power is comparable with the input pump power, an additional frequency component is considered as a higher order FWM process takes place. In this case, Eq. (1) yields an inaccurate solution. In this section, however, we proceed with Eq. (1) assuming an input pump with large enough power.

Under such an assumption, we use a numerical method for solving Eq. (1) to show the evolution of the signal power, and the results are shown in Fig. 2 . From Fig. 2 we can see that the signal propagates in the fiber periodically with a period dependent upon the input power. However, at a certain specific position the powers of each signal will have nearly the same value. For example, if we choose a fiber with proper length (1000 m in Fig. 1), then the signals would have nearly the same output power at this position, as shown in Fig. 3 .

In the case of small input signal, the power difference term *α* = *a*_{1}^{2}−*a*_{1}^{2} between the input signal and idler approaches zero, and *a*, *b*, *c*, and *d* can be simplified to

*H*= 4

*ηa*

_{1}

*a*

_{2}cos

*ϕ*−(

*κ*−1)

*η*−3

*η*

^{2}/2 is the Hamiltonian of the degenerate FWM system, which is determined by the initial conditions. Because the pump power only changes in the range of

*bP*and

*cP*, and

*c*corresponds to the saturated length, the saturated output signal power can be written as

*P*is the system power, Δ

*k*is the dispersion difference, and

*γ*is the third-order nonlinear parameter.

## 3. Nonlinear phase shift and phase noise

Much research has been done concerning the satuation regime of the FOPA discussing applications in optical transmission systems to processing a PSK signal. Essential to this application is the characterization of the phase noise in FOPAs. For example, Ref [14]. bases its calculation of the phase noise characteristics of the FOPA on the classical theory of the unsaturated FOPA. This analysis, however, is not valid for the saturated regime, as the phase noise cannot be solely attributed to self-phase modulation (SPM) or cross-phase modulation (XPM), as in the classical theory. In this section, we discuss the phase noise introduced by the saturated FOPA.

We still use the approximation that the input signal power is small enough compared with pump power. The subscripts 1, 2, and 0 stand for signal, idler, and pump, respectively. Generally, based on the discussion concerning Eq. (2), the signal phase in the FOPA can be expressed as a differential equation by

*ϕ*is equal to π/2 by the classical theory, and cos

*ϕ*= 0. If the signal is amplified to a power comparable (tens of mWs) with that of the pump, the last term with cos

*ϕ*cannot be ignored. The FWM affects not only the power of signal, but the phase of it. We can rewrite Eq. (6) thus as

*γPL*, and a combination of SPM and FWM.

The analytical form of *ϕ*_{1} by integrating Eq. (7) is too complicated to be expressed. However, because of the peoriodicity in such a system with perturbation, there is a simple method for us to analysis the phase noise introduced by the power fluctuation from input signals. For example, as shown in Fig. 4(a)
, consider the power evolution of the signals with input power 0.35 mW (red solid line) and input power 0.05 mW (green dashed line) in the fiber mentioned above. If we move the dashed line to the left, it can be found that the two lines coincide with each other except the lengths in the right-red and left-green rectangle parts, which are defined as *L _{phaseG}* and

*L*respectively (shown in Fig. 4(b)). It is obvious that

_{phaseR}*L*=

_{phaseG}*L*. Because the pump power is the same (400 mW) and much larger than the signal (0.05 mW and 0.35 mW), the two FWM processes with the same signal power (the coincided part in Fig. 4(b)) will have nearly same pump power and idler power. During the distance where all the powers (pump, signal, and idler) are the same, the FWM processes will bring the same nonlinear phase shift to the signals. Under such simplification, we can calculate the nonlinear phase difference at the end of the fiber.

_{phaseR}If there is a small power modulation Δ*P*_{1} = Δ*P* in the input signal power, the phase shift difference Δ*ϕ*_{1} can be expressed as

*L*, the energy transfer direction will change at this area. The phase-matching term

_{phaseR}*ϕ*is near zero, so cos

*ϕ*≈1,

*A*

_{2}≈

*A*

_{1}. In the

*L*part, the phase is matched, so cos

_{phaseG}*ϕ*≈0; and compared with

*P*,

*A*

_{1}≈0 and

*A*

_{2}≈0. Using these approximations, the Δ

*ϕ*

_{1}can be rewritten as

In Eq. (9), the *A*_{1} and *A*_{0} can be further simplified as *A*_{1}(*L*) and *A*_{0}(*L*) further, where *L* is the length of the fiber. Here, it is easy to see that the nonlinear phase noise is still composed of a background phase noise, and the noise from SPM and FWM.

## 4. Phase noise alleviation

Usually, the design considerations for FOPAs aims to maximize the signal gain or entirely convert the pump. When designing the regenerator for a PSK signal, we are interested in how to limit the fluctuation of amplitude without introducing additional phase noise. Here, we find a condition for the FOPA under which the phase noise is alleviated in the saturation regime.

We first consider a 1000 m FOPA with nonlinearity 0.012/(W⋅m). For a sufficiently high pump power (400 mW in our simulation), the signal is amplified in a highly nonlinear fiber (HNLF) and the SOP of the signal approaches to that of the pump. If we consider this as a linear FOPA without pump depletion, the phase-matching condition Δ*k* = −2*γP*_{0} ≈−0.096 /m is perfect. The nonlinear phase shift is defined by the difference between the phase of the signal after and before the FOPA (assuming that the input signal phase is zero). Under this parametric gain saturation condition, the FOPA works as a limiter amplifier, suppressing the amplitude noise of the signal. However, as the red dashed line shown in Fig. 5
, the nonlinear phase shift from the FOPA increases with the input power.

From the analysis of Eq. (8) and Fig. 9, it can be found that if we want to alleviate the phase noise from the FOPA, Δ*ϕ*_{1} should approach 0. By the analysis in Section 2 and 3, wihout integration, we try to find the condition under which Δ*ϕ*_{1} = 0.

In fact, Eq. (9) can be rewritten in the form of inequlity as

Substituting Eq. (5) into Eq. (10) and using the energy relation between the signal and the pump, we get the relation

If there is a case that Δ*ϕ*

_{1}= 0, the term 3

*κ*+ 5 should be larger than zero,

*κ*< −5/3. Furthermore, from Eq. (4c), there is a ouput power of the signal only if

*κ*> −4.

The discussion above shows that the alleviating condition relates to the the normalized phase-matching term *κ*. If the phase noise can be alleviated, *κ* should be in a interval that

This told us that, by changing the pump frequency to control Δ*k*, we can find a specific condition that alleviating the phase noise while limiting the amplitude at the same time.

By numerical analysis, we can find the optimized condition. Figure 6
shows resulting transmission relation for the optimized Δ*k* ≈−0.0175 /m.

From Fig. 6, we can see that if the input signal power is around 0.54 mW, both the output signal power and phase shift will be at the saturated state. We can see the relation between the signal-to-noise ratio (SNR) of the output signal and the phase noise of output signal introduced by FOPA from Fig. 7 . The introduced phase noise is calculated by the difference of output and input signal phase fluctuation, only considering the phase noise introduced by the signal power fluctuation at the input. It is clear that the additional phase noise introduced by the FOPA can be reduced by adjusting the dispersion relation.

## 5. Simulation

To test the theoretical analysis and illustrate the effect on the PSK signal more intuitively, we analyze the signal constellation using the system shown in Fig. 8
. The FOPA consists of a 1000 m HNLF, a polarizer, and an optical bandpass filter to select signals at the wavelength *λ*_{S}. The signal and pump were input into the HNLF via an optical coupler with their state of polarization (SOP) properly controlled by automatic polarization controllers. The HNLF has a zero-dispersion wavelength at *λ*_{0} = 1559 nm, dispersion slope d*D*/d*λ* = 0.03ps/nm^{2}/km. and a nonlinear coefficient of *γ* = 12/W/km. The wavelength of the pump is set to *λ*_{P} = *λ*_{0} + 3nm, while a 100 GSymbols/s QPSK signal has wavelength *λ*_{S} = 1550 nm. The signal passes through an optical signal-to-noise ratio (OSNR) controller comprising an optical attenuator and an EDFA. Then, the signal enters the FOPA together with the pump, which is a continuous wave. (We can also use the 100 GHz single polarization pulse train as the pump, which is synchronized with the signal). The output signal of the FOPA was transmitted over a 1nm filter and is detected by the coherent receiver. In this paper, the simulation is based on the fully numerical solution of the nonlinear Schrödinger equation (NLSE). The numerical calculation assumes a non-return-to-zero (NRZ) PSK format for the input signal, and white Gaussian noise is added to the complex amplitudes of both signal and pump.

This scheme of phase alleviating is much more suitable for dealing with higher order *M*-PSK signal, especially for the signal with low SNR. We compare our scheme with the theory in Ref [6], where the energy conversion ratio is near 100%. Figure 9
shows the simulation results of a QPSK signal with Gaussian noise at SNR = 20dB and Fig. 10
is for SNR = 15dB. or each figure, the (a), (b), and (c) parts correspond to the input signal, the output signal without optimization, and the output signal with optimization by changing the pump frequency into 1564.5nm.

The angles in the constellations are the objects of reference for the phase noise. They are equal to the spread of the phase noise of the signal before entering the FOPA. From Fig. 9 and Fig. 10, we can find that when the SNR is at a low level, the amplitude limiting introduces very large phase noise. Furthermore, when the SNR is 15dB, the limiting capability itself no longer performs well. This noise leads to a higher bit errors rate. However, if we use the phase alleviating scheme, not only the amplitude is limited but the phase noise brought by FOPA is compressed.

## 5. Conclusion

The theory of the saturated FOPA is discussed. Based on the theory, we explain the power saturation and the additional phase noise brought by the FOPA, and gave an equation to approximately calculate the saturated signal output power. We proposed a scheme for alleviating the phase noise brought by the FOPA at the saturated state. Amplitude-noise and additional phase noise reduction of BPSK and QPSK based on the saturated FOPA was numerically studied. The dispersion difference term Δ*k* is one decisive factor for the saturated FOPA. The FOPA can be optimized by controlling the dispersion relation between the pump, signal and idler, and this provides the means to deal with PSK signals.

## References and links

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