## Abstract

We numerically investigate the 2R-regeneration technique utilizing self-phase modulation and off-center filtering. Our numerical simulations take into account the incoherent nature of noise through its spectral representation. This approach allows to evaluate a Q-factor improvement of 2 dB for this regenerator. Furthermore, our study points out the role of both the input and the output filter of this regenerator. We show that the input filter must be suitably chosen in order to obtain the best Q-factor improvement. The output filter must also be suitably chosen in order to preserve the modulation format.

©2006 Optical Society of America

## 1. Introduction

All-optical signal regeneration technique utilizing self-phase modulation in fiber and subsequent filtering was first proposed by Mamyshev [1]. This 2R-regenerator design is very attractive because of its simplicity : it only uses a fiber and an optical bandpass filter located at the output. Moreover, this technique being based on the Kerr effect, it does not suffer of any time response limitation. Since its first demonstration, this kind of regenerator has been extensively studied, both experimentally [2, 3, 4, 5] and theoretically [6, 7, 8, 9, 10].

In some experimental studies, a noise source is superimposed on the data stream to simulate degradation of the signal by accumulation of amplified spontaneous emission (ASE) from optical amplifiers. Experimental Q-factor improvements of 1.5 dB have been reported with this regenerator [3, 4]. In these experimental studies, an input filter is usually present [2, 3, 4] but its role is not clearly identified.

From a theoretical point of view, the most commonly used theoretical approach to study the Mamyshev regenerator consists of solving the nonlinear Schrödinger equation for an input pulse [6, 7, 8, 9]. Noise is generally taken into account through variations of the input pulse peak power. This method leads to a nonlinear transfer function for the regenerator. This function is known to reduce noise but does not take into account the incoherent nature of ASE noise. In reference [7], Matsumoto uses a more realistic approach of including ASE noise, but the paper is focused on performance comparison between several kind of 3R regenerators rather than on the physical understanding of the Mamyshev regenerator in the presence of incoherent noise.

Interesting work has been recently proposed by Rochette *et al*. both theoretically and experimentally [5, 10] to explain the bit-error-rate improvement observed with this kind of regenerators. The authors argue that the regenerator needs to have different transfer functions for the signal and the noise to make BER improvement possible. However, in these papers the role of the filters is not investigated.

The aim of our paper is to propose an accurate simulation of the Mamyshev regeneration technique. We demonstrate, for the first time in the best of our knowledge, that a maximum Q-factor improvement of about 2 dB is possible only if an input filter is present and suitably chosen. Our conclusion is that the Mamyshev regenerator can discriminate noise from signal only if ASE noise has been previously eliminated from the spectral region where the signal is transposed.

In section 2, we present the approach which consists of taking into account the incoherent nature of ASE noise through its spectral representation. In section 3 we briefly present the principle of this self-phase-modulation-based 2R-regeneration technique. Section 4 presents the results of the simulations leading to the evaluation of the Q-factor improvement in the presence of the regenerator while section 5 is focused on the role of optical filters in this regeneration technique.

## 2. Relation between Q-factor and white noise

In this section we give the relation between the Q-factor of a degraded data stream and its corresponding optical signal-to-noise ratio (OSNR). Our calculation technique is based on the spectral representation of white noise. We use the schematic diagram shown in Fig. 1 to model the acquisition process of an optical data stream degraded by the accumulation of amplified spontaneous emission (ASE) of erbium-doped fiber amplifiers (EDFA).

The data encoder gives a 40 Gb/s random sequence of 1024 Fourier-limited gaussian pulses at 1550 nm with a full width at half maximum (FWHM) of 6.25 ps. The extinction ratio *ER* of the signal is defined as the ratio (in dB) between the peak power of the pulses and the background level.

Let *E _{S}*(

*t*) be the function representing the magnitude of the electrical field of the optical signal in the time domain. The spectral representation

*Ẽ*(

_{S}*v*) is numerically calculated from

*E*(

_{S}*t*) by using the discrete Fourier transform (DFT):

where *N* is the number of points dividing the time window *T*.

The spectral representation *Ẽ _{N}*(

*v*) of ASE noise is assumed to be similar to white gaussian noise [11] and can be written, in the frequency domain, as :

where *A* is the magnitude of white noise. We take *A* constant over a frequency range *B* centered around the carrier frequency *v*
_{0} (see Fig. 2(a)). Quantity *ϕ* is the random phase uniformly distributed between –*π* and *π* (see Fig. 2(b)). This simple formulation allows us to take into account the incoherent nature of the spontaneous emission from optical amplifiers.

The representation *E _{N}*(

*t*) of the noise in the time domain can be obtained numerically by using the inverse discrete Fourier transform (IDFT) of the spectrum. Figures 2(c) and 2(d) represent respectively the real part

*r*(

_{N}*t*) and the imaginary part

*i*(

_{N}*t*) of

*E*(

_{N}*t*). By virtue of central limit theorem [12], both

*r*(

_{N}*t*) and

*i*(

_{N}*t*) can be approximated by gaussian random variables. Figures 2(e) and 2(f) represent the probability density function of

*r*(

_{N}*t*) and

*i*(

_{N}*t*) assuming a gaussian law with a zero mean and a variance

*σ*

^{2}=

*A*

^{2}

*B*/2

*T*. Let us mention that this method to generate gaussian noise is similar to the method which consists to decompose the electric field in Fourier series [13].

In the frequency domain, the signal arriving at the optical bandpass filter input is written as :

Before detection, we evaluate the OSNR, defined as the ratio (in dB) between the signal power and the noise power for a given spectral range.

The detection stage generally includes a bandpass optical filter to reduce noise. We use a flat-top filter represented by a fifth-order supergaussian transfer function *F*
_{1}(*v*). The central frequency of the filter is the carrier frequency of the signal and its 3 dB-bandwidth is 150 GHz. After the filter, the signal is written as:

The photodetector converts the optical intensity, proportional to |*E*
_{2}(*t*)|^{2}, into a voltage *v*
_{1}(*t*):

where *R* is the photodetector responsivity. In the following, *R* will be set to unity and we will neglect shot noise and thermal noise of the detector. The IDFT is used to obtain *E*
_{2}(*t*) from *E*̃_{2}(*v*).

The finite bandwidth of the electronics is taken into account through a fifth-order Bessel lowpass filter of transfer function *H*(*v*) whose cutoff frequency is 32 GHz. After the filter, the voltage, in the frequency domain, becomes:

where *v*̃_{1}(*v*) is the DFT of *v*
_{1}(*t*). To represent the eye-diagram of the signal we calculate the IDFT, *v*
_{2}(*t*), of *v*̃_{2}(*v*).

An example of an eye diagram is given in Fig. 3(a) for ER= 15 dB and OSNR=20 dB/0.1nm. As expected, noise induces variations in the level of detected pulses. The corresponding intensity histogram is presented in Fig. 3(b). From this histogram we calculate the unregenerated Q-factor, *Q*
_{1} defined by:

where *V*
_{1}, *V*
_{0}, *σ*
_{1} and *σ*
_{0} are shown explicitly in Fig. 3(b).

By varying the white noise magnitude, *A*, and the extinction ratio, ER, of the pulse train we can plot the evolution of the Q-factor as a function of the OSNR for different ER. The results are shown in Fig. 4.

## 3. Self-phase-modulation-based 2R-regeneration technique

The schematic diagram of the regenerator is presented in Fig. 5 [1]. The principle of the Mamy-shev regenerator is usually explained as following. Due to the effect of self-phase modulation (SPM), the spectral bandwidth of the pulses broadens during propagation along the fiber of length *L*. The spectral broadening is proportional to the input peak power. The frequency of the filter is shifted of *δv _{f}* with respect to the input carrier frequency. Low peak power pulses do not induce enough SPM and are rejected by the filter. When the pulse peak power is high enough, a part of the SPM-broadened spectrum passes through the filter. This results in a nonlinear transfer function that can reduce noise in the signal. The output filter has usually a gaussian shape with the same spectral bandwidth as the input spectrum. Thus, the output pulsewidth is the same as the input pulsewidth but with a shift in wavelength from λ

_{0}to λ

_{f}.

To study this regenerator, a single-pulse approach, which consists of studying the nonlinear propagation and filtering of a single pulse and plotting the ouptut intensity as a function of the input intensity, is generally used [1]. In Fig. 5, *E*̃_{2}(*v*) is obtained from *E*̃(*v*) by numerically solving the nonlinear Schrödinger equation (NLSE) [14]. :

where *E* is the slowly-varying envelope of the electric field in the time domain, *z* the propagation distance, *t* the time (in a frame of reference moving with the pulse at the group velocity *v _{g}*),

*β*

_{2}the second-order dispersion parameter,

*α*the fiber loss and

*γ*the nonlinear coefficient. Filtering of

*E*̃

_{2}(

*v*) is achieved by multiplying it by the transfer function

*F*

_{2}(

*v*) of the filter to give the filtered signal

*E*̃

_{3}(

*v*):

We performed the simulation, using this method, for a 6.25 ps Fourier-limited gaussian pulse at 1550 nm. The NLSE is solved using the fourth-order predictor-corrector split-step Fourier method [15] for the following parameters : *β*
_{2} = +0.89 ps^{2}/km, *α* = 1.4 × 10^{-4} m^{-1} (0.6 dB/km), *γ* = 8.4 W^{-1}m^{-1}, *L* = 2.5 km, *δv _{f}* = 280 GHz. Figure 6 represents the output peak power of the filtered pulse as a function of its corresponding input peak power.

The nonlinear shape of this function allows for the reduction of both the level of undesirable low-level pulses and the fluctuations of high-level pulses. The peak power of high-level pulses must correspond to the flat-top area of the curve (around 1.5 W in our case).

However, even if the single-pulse approach allows the nonlinear behavior of the regenerator be described, we want to point out that it does not properly describe regeneration of an input signal degraded by EDFA noise. Indeed, in the single-pulse approach noise is taken into account through variations of the pulse peak power. In reality, the wave propagating in the fiber is the superposition of the coherent data stream and the incoherent ASE noise. Then, particular attention must be paid to the description of noise in such regeneration technique. Furthermore, the single-pulse approach cannot investigate the important role played by an input filter.

## 4. Q-factor improvement using the white noise approach

In this section we use the approach already presented in section 2 to describe more accurately the performance of the regenerator. The nonlinear fiber and the filter used in section 3 are inserted in front of the detector (Fig. 7). The detection process does not need any optical filter since the regenerator already includes one (filter # 2). However the previously used flat-top optical filter (filter # 1) is not removed but is now inserted in front of the regenerator. The role of this filter is to eliminate noise in the bandpass of the second filter. As we will show later, the performance of the regenerator depends now of the characteristics of this filter.

The post-regeneration Q-factor *Q*
_{2} can now be evaluated and compared with the Q-factor *Q*
_{1} calculated previously. The Q-factor improvement *Q _{IM}* is defined as the ratio (in dB) of

*Q*

_{2}to

*Q*

_{1}. An improvement of Q-factor occurs if

*Q*> 0. The Q-factor improvement of the signal has been calculated for different values of the peak power of high-level pulses by keeping the ER and the OSNR constant. The result is shown in Fig. 8 for ER = 20 dB and OSNR = 27 dB/0.1 nm.

_{IM}Q-factor improvement occurs between 1.6 W and 2.1 W of peak power. The maximum value of *Q _{IM}* is about 2 dB and is obtained for 2 W of peak power. This value of

*Q*is comparable to the values of 1.5 dB previously reported in experimental studies [3, 4]. In Fig. 8, we have also plotted the transfer function of the regenerator using the single pulse approach (dashed line). The two approaches differ for the evaluation of the working power of high-level pulses : around 1.5 W for the single-pulse approach and around 2 W for the white noise approach. This difference can be partly attributed to the finite extinction ratio of the pulse train used in the white-noise approach.

_{IM}In Fig. 8, we have also plotted *Q _{IM}* when the input optical bandpass filter is removed. In this condition we observe no improvement of the Q-factor. This means that, to properly regenerate a signal degraded by EDFA noise, the Mamyshev regenerator needs to include an input filter. In the best of our knowledge, no demonstration of this point has been proposed previously, even if, in experimental studies [2, 3, 4], an input filter is systematically used.

## 5. Role of the filters

In the previous section we have shown that the Mamyshev regenerator improves the quality of a degraded data stream only if an input filter is included in front of the regenerator. We are now able to propose a new meaning of the Mamyshev technique. Figure 9 shows the magnitude of the spectrum of both signal and noise at different points of the regenerator : (a) before the input filter, (b) after the input filter, (c) after the nonlinear fiber and (d) after the output filter. In Fig. 9(a), spectra of both signal and noise are represented separately. The noise spectrum covers the whole frequency domain while the signal spectrum is located around the carrier frequency. After the input filter, the spectra of both signal and noise can still be represented separately since filtering is a linear process [Fig. 9(b)]. The role of the input filter is to suppress noise over the bandpass of the output filter while preserving the signal spectrum. After nonlinear propagation in the fiber, signal and noise are mixed together because of nonlinear coupling and the spectrum is broadened by SPM [Fig. 9(c)] . However, noise is less affected by SPM because of its relatively low instantaneous power. Due to SPM, higher-order harmonics components appear on both side of the initial spectrum. Figure 9(d) then represents the filtered output signal. This spectrum has the characteristics of the input spectrum but is frequency-shifted. Noise is reduced but not totally eliminated by this technique because of phase-amplitude coupling introduced during the nonlinear propagation in the fiber.

To study the influence of the input filter, we have plotted the Q-factor improvement *Q _{IM}* as a function of the input peak power for different values of the input filter bandwidth. The results are shown in Fig. 10. Maximum Q-factor improvements occur for filter bandwidths around 150 GHz. For higher filter bandwidths, the noise power under the output filter increases and degrades the output Q-factor. For lower filter bandwidths, Q-factor improvement requires higher peak powers because the filter attenuates the input power. When the bandwidth is too small, no Q-factor improvement occurs because the filter degrades the signal too much. Our study shows that an appropriate choice of the input filter bandwidth allows to optimize the regenerator efficiency. This argument could be important for designing Mamyshev regenerators in the context of wavelength division multiplexing systems where optical demultiplexers play the role of input filters [2].

Figure 9(d) also demonstrates the role of the output filter. By choosing the frequency shift *δv _{f}* of the output filter to be a multiple of the repetition rate, the spectrum shape and then the modulation format are preserved. When this is so, the carrier frequency component is centered in the spectrum. This is the case of our study where

*δv*was chosen to be 7 times (280 GHz) the repetition rate. In comparison, we have also represented, in the inset of Fig. 9(d), the output spectrum for a frequency shift corresponding to an odd half multiple of the repetition rate (260 GHz). In this case, the carrier frequency component is suppressed. However, Q-factor improvement still occurs with a maximum value of

_{f}*Q*equal to 1.9 dB.

_{IM}## 6. Conclusion

We have numerically investigated the 2R-regeneration technique utilizing self-phase modulation and filtering. We have proposed a slightly different interpretation of this technique by using the spectral representation of signal and noise. Indeed, the basic principle of the Mamyshev regeneration technique is to transpose more signal than noise to a spectral region where ASE noise has been previously eliminated. Q-factor improvement of 2 dB has been numerically evaluated. This value is in good agreement with the experimental values recently reported. This approach has allowed us to point out the role of the input filter which has to be suitably chosen. We have also demonstrated that the frequency shift of the output filter must be a multiple of the repetition rate to preserve the modulation format.

## Acknowledgments

Authors thank Art O’Hare (School of Physics, Dublin Institute of Technology, Dublin, Ireland) for his comments on the manuscript. This work is supported by the Conseil Régional de Bretagne (convention No.1193).

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