Selection of Extreme Events Generated in Raman Fiber Amplifiers Through Spectral Offset Filtering

In this work, we discuss the emergence of rare and intense events in a Raman fiber amplifier. The temporal and spectral evolutions of continuous and pulsed signals are investigated, and approaches based on spectral slicing of the broadened spectra are proposed in order to isolate the rarest events. Numerical simulations are confirmed by experimental results.


I. INTRODUCTION
or more than three decades, optical fibers have been recognized as a versatile and highly effective testbed to investigate a very wide range of non-linear concepts, ranging from modulation instability, in scalar [1] or vectorial [2] configurations, generated by coherent [1] or incoherent [3] pumping waves, to the propagation of optical solitons [4] or similaritons [5]. More recently, it has also been demonstrated that the evolution of picosecond pulses or continuous waves in a highly nonlinear optical fiber can be characterized by an "extreme" statistics deviating strongly from usual Gaussian statistics [6][7][8][9][10][11]. An analogy with the hydrodynamics field has then been suggested so that the term optical rogue wave has been proposed to qualify such rare but intense events occurring during supercontinuum generation.
Other recent studies have also outlined than extreme statistics were not restricted to soliton-supporting systems. Indeed, similar highly tailed statistics can be observed as well in other wide-spread fiber-based configurations such as Raman amplifiers at telecommunication wavelengths [12] or optical waveguides [13]. In fact, under the influence of the quasiinstantaneous Raman gain and providing a low group velocity mismatch, fluctuations of a partially incoherent pump can be exponentially transferred on the amplified signal. Similar trends have also been demonstrated in parametric amplifiers Manuscript received April 1, 2009. This work was supported by the Agence Nationale de la Recherche (ANR PERSYST II, SOFICARS and MANUREVA projects: ANR-07-TCOM-014, ANR-07-RIB-013-03 and ANR-08-SYSC-019), by the Conseil Régional de Bourgogne, and was carried out within the framework of the Research Networks GDR Phonomi2, COST action 299 FIDES and PRES UFC-UB.
C. Finot driven by a partially incoherent pump [14]. In such a context, in order to study and explain the appearance dynamic of these rare events, it becomes of primary importance to find a way to control and isolate the rarest and most intense events. In the framework of supercontinuum generation, it has been demonstrated that an adequate spectral filtering by means of a off-centered optical band pass filter (OBPF) could efficiently isolate the optical rogue structures [6,7,10]. Basically, ultrashort pulses arising through the nonlinear dynamic fission/recompression of an initial pumping pulse are indeed strongly affected by the intrapulse Raman response of fused silica which progressively shifts their central frequency towards higher wavelengths. Consequently, recompressed pulses with the highest peak power then leads to the most frequency shifted pulses and can therefore be easily isolated.
In the context of extreme event appearance in Raman amplifiers from a continuous seed, it has been shown that the impact of cross-phase modulation (XPM) induced by a partially incoherent pump on the continuous wave leads to a spectral expansion of the amplified signal. It has then been experimentally suggested that slicing in this broadened spectrum, could allow to efficiently isolate these rare events [12].
In the present article, we further detail how these rare structures can be controlled in a Raman amplifier and how is it possible to discriminate the small-scale fluctuations from the giant structures. We highlight a clear distinction between the behavior affecting a continuous seed and the evolution of an initial ultrashort pulse train. Physical phenomena implied in the rare-event control can indeed significantly vary. Our paper will be thus organized as follows. We will first present the numerical model we developed as well as a simplified model that enables us to achieve some clear physical understanding of the underlying nonlinear physics. Then, we will investigate the amplification of a continuous wave as well as a pulse train in the Raman amplifier. Finally, the numerical conclusions will be compared to experimental results in the last part of our manuscript.

II. NUMERICAL MODEL
During its evolution in a single mode optical fiber, the slowlyvarying envelope of the electrical field ψ(z,T) of an optical wave can be described by the extended nonlinear Schrödinger equation [15] : Selection of extreme events generated in Raman fiber amplifiers through spectral offset filtering C. Finot, K. Hammani, J. Fatome, J. M. Dudley and G. Millot with β 2 and β 3 the chromatic dispersive properties of the optical fiber, γ the Kerr nonlinear coefficient, α the optical losses, R(t) the silica response, which includes both the instantaneous and the delayed Raman response. z and T are the propagation distance and the temporal coordinate in a reference frame moving at the group velocity of the wave.
ψ(0,T) is the initial field made of the temporal superposition of the pump wave ψ P and the signal ψ S which is shifted by Ω = 13 THz with respect to the pump wavelength and therefore leads to the maximum gain response : As no additional frequency is generated by nonlinear frequency mixing, it is possible to isolate the evolution of the pump ψ P from the evolution of the signal ψ S . We can also consider that the Raman gain experienced by the signal is constant on the spectral range under investigation g R = 6.5 x 10 -3 m -1 and that the Raman intrapulse effect and resulting soliton self-frequency shift are not crucial in the normal dispersion regime. Moreover, due to the propagation length under consideration (typically half a kilometer), the consequences of the fiber linear losses are low and can therefore be neglected. Given all those justified assumptions, Eq. (1) can be reduced to the following set of two coupled nonlinear Schrödinger equations [16] : with δ = 1/v gP -1/v gS the walk-off parameter defined as the difference between the group velocities v gS and v gP of the signal and the pump respectively. β 2S and β 2P are the second order dispersion of the signal and pump respectively. Let us note that compared to reference [12], we have here taken into account the effects of second order dispersion β 2 in order to better describe the pulsed signal dynamics. Let us also notice that compared to the widely spread models used for Raman amplification, which rely on a power analysis [16], the model we implement here is based on the evolution of the scalar electrical field. This model will then enable us to describe the dynamics, both in the temporal and spectral domains, which is mandatory to understand our proposed spectral filtering method. A crucial point here, is that the pump involved in our study is a partially incoherent pump ψ P that we modeled in the spectral domain by a Gaussian intensity field having a stochastic spectral phase ϕ(ω) with a uniform distribution between -π and π [17,18] : Ω L is the spectral full-width at half maximum (FWHM) of the pump, Ω L = 17.7 GHz, which corresponds in the temporal domain to intensity fluctuations having a minimum temporal FWHM of 25 ps. We will consider an average pump power of 175 mW, leading to an average gain of 3dB.
Regarding the properties of the initial signals, two kinds of signals will be investigated. First, we will study the evolution of a continuous wave with an average power of 0.5 mW. Then, we will focus on the evolution of a Gaussian pulse train having a 2.5 ps FWHM and an initial pulse energy of 1 pJ.
Our Raman amplifier is based on a 500-metre highly nonlinear fiber, with a second order dispersion 1.3 x 10 -3 ps 2 /m -1 and 2.2 x 10 -3 ps 2 /m -1 at signal and pump wavelengths respectively. The nonlinearity is γ = 10 W -1 .km -1 and the walk-off coefficient is δ = 0.10 ps.m -1 , which leads to an integrated walk-off of 50 ps between the pump and the signal. Our fiber is normally dispersive, which enables us to get rid of modulation instability [1] as well as solitonic recompression [4].

III. NUMERICAL RESULTS FOR A CONTINUOUS SEED
We first investigate the evolution of a continuous signal. We have outlined in a previous work [12] that a continuous signal evolving in a Raman fiber amplifier exhibiting low walk-off between pump and signal will experience an incoherence transfer from the pump to the signal leading to the generation of rare and intense spikes.

A. Simulations based on a simplified model
In order to better understand the physics behind the origin of the formation of those spikes, we propose to consider as a first qualitative approach a simplified model. In this approach, the pump, amplifying the considered continuous seed, is reduced to a single Gaussian pulse with a temporal width of 25 ps corresponding to the shortest temporal width possible in the incoherent pump structure.
We have plotted in Fig. 1(a) the initial pump (a1) and signal (a2) as well as the resulting amplified output signal (a3). As expected, the continuous seed has been amplified and a replica of the pump profile can be observed on the output signal ( Fig.  1(a3) plotted on a semi-logarithmic scale). Figure 1(b) outlines that the amplification undergone by the signal varies exponentially with the pump peak-power, confirming that the process is not affected by any depletion effect. Simply stated, we can expect any fluctuation of the initial pump to be exponentially transferred on the signal. It should be also noted that due to the walk-off value which is superior to the finite temporal width of the pump pulse, the amplification is much lower than the value in absence of walk-off : exp(gR L ψP 2 ) [19].
If we now get concerned with the spectral intensity profile of the amplified signal (see Fig. 1(c)), we can note a significant spectral expansion of the initial continuous signal. Such a broadening is caused by the XPM of the pump on the signal [20]. It is then possible to slice the broadened spectrum by using a frequency offset OBPF in order to remove the central continuous part of the signal [21]. The principle of such a technique has already been successfully demonstrated in the field of wavelength conversion of high bit-rate optical telecommunication signals [22] : the intensity information of the most powerful wave (the pump) is converted on the continuous seed via XPM. (d) Transfer function describing the output signal peak power for various pump powers and various frequency offset filterings.
Peak powers of pump and sliced signal are therefore linked via a function that is usually called transfer function. The shape of this function depends critically on the spectral offset Ω F of the OBPF. In order to illustrate this point, we have used a Gaussian filter with a spectral FWHM of 9 GHz. We make out from Fig. 1(d) that for higher detuning values Ω F , it becomes possible to efficiently reject the low pump powers : the frequency filter behaves as an efficient intensity discriminator which threshold can be continuously tuned according to Ω F .
Let us point out that previous experimental configurations have also taken advantage of Raman amplification [22,23]. But using an external counterpropagating Raman pump enables them to avoid any influence of this additional wave on the expected wavelength conversion process. Our scheme is here more complex as the copropagating Raman pump and the XPM pump are intrinsically a single wave and consequently induce simultaneously correlated intensity and phase changes on the signal.
We can also compare our approach to spectral selection process used in the framework of supercontinuum generation [6,7,10], which essentially relies on the intensity-dependant self-frequency shift undergone by ultrashort soliton pulses. In this context, spectral filtering is only possible at longer wavelength edge and the associated frequency shift is usually of the order of several tens of THz. In our approach, filtering is possible both at shorter and longer wavelengths, but a drawback is that a significant part of the signal structure is lost during the filtering process.

B. Simulations based on an initial incoherent wave
In order to check the validity of the qualitative conclusions obtained relying on our previous simplified model, we have run intensive numerical simulations based on a partially incoherent pump as described by Eq. (3). Figure 2(a1) illustrates the initial pump intensity profile plotted on a 50 ns range. We have also computed the statistical distribution of the peak-powers present in this pump (statistics have required a temporal window of 20 µs). Results plotted in Fig. 2(b1) illustrate that peak powers deviating significantly from the average pump power can exist. It is indeed possible to observe a structure having a peak-power more than 15 times the average power (which justifies a posteriori to study in part A Gaussian pulses with initial peak powers as high as 2.5 W). Compared to previous studies involving the peak-power fluctuations of a nanosecond Q-switched laser Raman pump [24][25][26] or recent experiments carried out in silicon waveguides [13], the partially incoherent pump under investigation presents here much larger deviations. The amplified signal (Fig. 2(a2)) exhibits significant peaks resulting from the instantaneous exponential amplification of the pump initial fluctuations. As expected from our simplified model, we can see from this figure as well as from Fig. 2(b2) that the fluctuation amplitude in the amplified signal is much larger than those present in the initial pump. From the 2 10 5 peaks recorded on a temporal window of 20 µs, several giant peaks have been found to present a peak intensity more than 100 times the average value of the signal. Let us also outline that the probability distribution obtained for the amplified signal ( Fig. 2(b2)) is in clear agreement with a linear function when plotted on a log-log scale, in agreement with the typical shape of extreme events. We can also compare the resulting distribution to the distribution one can expect if we consider the incoherent pump as a succession of isolated 25-ps Gaussian identical pulses (such as the simplified model developed in part IIIA) having a peak power distribution following the results of Fig. 2(b1) and an instantaneous gain as Fig. 1(b). We see that this model is qualitatively correct in retrieving a heavily tailed distribution, but fails in quantitatively predicting the values of the maximum peak powers that can be achieved: our simplified model underestimates the peak power value of the giant structures. This is linked to the fact that in this model, we consider a single isolated pump pulse of fixed temporal width whereas in the initial incoherent wave, due to the walk-off effect, the signal can be subject to amplification by several adjacent substructures, which increases the resulting amplification.
The spectrum of the amplified wave (Fig 2(c)) also exhibits a noticeable reshaping. As discussed above, the initial Dirac spectrum has been significantly broadened under the effect of XPM of the pump on the signal [27,28]. We have finally checked that offset spectral filtering enables us to efficiently isolate the highest peak-power structures. As shown in Fig. 2(d), higher frequency detunings lead to higher intensity discrimination : it is therefore possible to isolate a single rare event generated by the partially incoherent pump.

IV. NUMERICAL RESULTS BASED ON A PULSED SIGNAL
Let us now investigate the evolution of an initial pulsed signal with a temporal width below the coherence time of the pump.

A. Simplified model
As in the previous part, we can first study a simplified approach to obtain some physical insights about the pump to signal interactions. In the context of a pulsed signal, it is necessary to take into account an additional parameter, namely, the initial temporal delay ( noted ∆T ) between the pump pulse and the signal pulse. Indeed, if in the context of a continuous seed, maximum of the amplified pulse and pump were synchronized, in the context of pulsed signal, this is not a priori always the case. The impact of this initial delay ∆T can be seen in Fig. 3(a) where we have plotted the amplified signal for different values of ∆T. From our simulations, we have made out that an initial delay of ∆T = 38 ps leads to the rise of higher peak powers. In other words, if in the continuous wave, the pump amplitude was the single parameter influencing the amplitude of the resulting pulse, in the pulsed case, we will have two parameters that will influence the final state: the amplitude of the initial state and the temporal delay between the pump and signal.
In Fig. 3(b), we have plotted the temporal intensity profiles of the initial and final pump and signal. The temporal intensity profile of the pulse propagating in the normally dispersive amplifier progressively evolves towards a parabolic state [5,29]. Some temporal asymmetry can however be seen due to the temporal walk-off and the depletion effects of the pump during the propagation [30]. The consequences of the strong temporal depletion experienced by the pump (Fig. 3(b)) [31] can also be seen on the gain evolution according to the pump power plotted in Fig. 3(c) which deviates from the usual linear trends. Note that compared to the previous part, we are much more affected by depletion effects here because the initial peak power is much higher than in the continuous case (we have here an initial peak power of 175 mW instead of 0.5 mW). Furthermore, we stress that some negative gain could be observed, which is mainly due to the chromatic dispersion induced temporal broadening of the signal pulse which consequently undergoes a peak power reduction. Fig. 3. (a) Output peak power of the amplified pulse for various initial pump peak powers and various initial delays (b) Temporal intensity profiles of the signal (black) and the pump (grey). Output profiles (solid lines) are compared with initial profiles (dotted lines). Results obtained for the optimum initial temporal delay and a pump peak power of 2.5 W. (c) Evolution of the gain versus pump peak power (gain defined as the ratio between the final and initial pulse peak power) for the optimal initial temporal delay (d) Evolution of the output spectral intensity profile of the pump for various pump powers (results at the optimum initial temporal delay). (e) Transfer function describing the output signal peak power for various pump powers and various values of frequency offset ΩF.
Regarding the optical spectrum, a noticeable broadening can be noticed as well as some spectral asymmetry which is a consequence of the pump-signal walk-off [31,32]. We have numerically checked that the spectral expansion was originating not from the cross phase modulation but rather from the self-phase modulation effect (SPM) [15]. Let us also outline that optical broadened spectra are much larger than the one obtained from a continuous wave, because of a broader initial spectrum and also because the initial peak power being much higher, effects of SPM cannot therefore be neglected.
As in the preceding part, we can use an offset OBPF in order to isolate the structures with the highest peak powers. We have therefore plotted in Fig. 3(e) the transfer function obtained for the optimum delay ∆T for various values of the spectral offset of a Gaussian OBPF having a spectral FWHM of 100 GHz. Similarly to the previous case, increasing the spectral offset Ω F can continuously change the threshold of the transfer function. Let us outline that this method based on SPM in amplifier has already been demonstrated in the telecommunication field with a continuous Raman pumping [33]. It can also be seen as an extension of the well-known signal regeneration technique proposed by Mamyshev [34,35] and the fact that when the offset is increased, the extinction level increases is fully consistent with the previous theoretical analysis [36].

B. Simulations with an initial incoherent wave
In order to confirm our qualitative results, we have performed simulations including an incoherent pump such as described by Eq. (3). We have run more than 12 000 simulations and the resulting shot-to-shot statistics of the peak powers, plotted in Fig. 4(a), black squares, using a log-log scale, exhibits the linear tail characteristic of extreme phenomena.
We have also plotted in Fig. 4(b) the spectrogram of the output amplified signal (i.e. a spectro-temporal representation) where a pronounced event with a behavior deviating strongly from the average behavior can be seen. The presence of this extreme shot confirms that pulses amplified by the highest peak power also exhibit the most broadened spectrum.
By recording the number of pulses after spectral slicing, we obtain the resulting statistics described in Fig. 4(c), which clearly reflects that for a low spectral offset (below 0.4 THz), the proportion of pulses recovered after slicing is constant to 1. A statistical analysis of the peak powers of the filtered pulse train ( Fig. 4(a), grey diamonds) reveals that filtering at the central frequency behaves as a beneficial optical power limiter [37]. Indeed, the peak power fluctuations have been divided by more than 2 and we can also make out a statistical distribution reshaping. For further increase of the spectral offset Ω F , the number of pulses recovered after spectral filtering drops rapidly down to a very low value. We have finally found empirically that the rate of this decrease versus the frequency offset could be well approximated by an exponential decreasing law (a linear function when plotted on a semi logarithmic scale).

A. Experimental set-up
In order to confirm our numerical approach, we have implemented the experimental set-up sketched in Fig. 5(a). The Raman pump is delivered by a fiber Raman laser with an initial average power of 350 mW at an initial wavelength of 1455 nm. As the pump is not polarized, this average pump power of 350 mW corresponds in our scalar numerical model to a value of 175 mW [16]. Such a power experimentally leads to an average experimental gain of 3 dB. With such reduced pump power and gain, we have not observed any spontaneous Raman emission so that we can neglect this effect in the modeling. The autocorrelation function of the pump wave is plotted in Fig. 5(b), grey solid line, and highlights a typical contrast of 1:2 with a coherence time of 25 ps [12,17], with a shape typical of the Gaussian distribution (circles) assumed for our numerical simulations (Eq. 3). The polarized signal is launched into a commercially available highly nonlinear fiber (HNLF) from ofs with parameters described in section 1. We will test the two kinds of signals. The initial continuous wave signal (CW) is delivered around 1550 nm by means of an external cavity laser having a spectral linewidth below a few MHz. Regarding the pulsed signal, it is delivered by a 2.5-ps mode-locked Erbium doped fiber laser running at a repetition rate of 22 MHz. Both signal and Raman pump are coupled into the HNLF by means of a WDM coupler. At the output of the optical fiber, a second WDM coupler enables us to select the amplified signal and remove the residual pump. Previous experiments reported in ref. [12] have demonstrated that the extreme statistics generated by this system is in qualitative agreement with numerical predictions resulting from the model described in section II. In the experimental section of this article, we focus our attention more specifically on the spectral selection process that can be performed by spectral filtering. In the case of the continuous seed, we have used a fiber Bragg grating having a spectral FWHM of 9 GHz. In the case of the pulsed signal, we have inserted a wavelength demultiplexer with eight output channels having each a spectral FWHM of 100 GHz and spaced by 200 GHz.
At the output of the system, the signal is finally analyzed by means of a photodiode.

B. Experimental selection of the extreme events
Experimental results obtained in the case of a continuous seed with an initial power of 0.5 mW are plotted in Fig. 6(a). As the position of the central frequency of the fiber Bragg grating is fixed, the filter-signal frequency detuning Ω F is obtained by simply adjusting the wavelength of our initial CW signal. It is readily apparent from subplots a1 to a3, that higher the signalfilter detuning is, lower is the number of observed spikes. As expected, for largest detuning, only the most intense spikes are conserved. We have also carried out some similar experiments for the 22-MHz picosecond pulsed seed. Initial pulses have a FWHM of 2.5 ps for an average power of -16 dBm (1 pJ energy per pulse). Results are recorded by means of a photodiode and an oscilloscope having a 1-GHz electrical bandwidth so that the experimental observed spikes are mainly correlated to the pulse energy rather than peak-power. Nevertheless, recent works have outlined that statistics of peak-power or energy exhibit similar behavior [10].
If the pulse train is filtered at its corresponding central frequency, we can retrieve after amplification and filtering a pulse train at the initial repetition rate, Fig. 6(b1). However, a high amplitude jitter is clearly visible. Now, if we analyze the output signal at another port of the demultiplexer, corresponding to a higher frequency offset (Figs. b2 and b3), we can observe a dramatic drop in the number of resulting output pulses. The output signal does not present any periodicity, so we have therefore generated a pulse train having a stochastic pulse separation (with a time-spacing corresponding to a multiple of the initial pulse train period). Let us note that compared to the method developed in refs. [38,39], which was also based on the properties of sliced broadened spectra, our resulting pulse train presents here noticeable intensity fluctuations.
Finally, we have experimentally measured the number of output pulses as a function of the initial offset (Fig. 6(c)). In good agreement with results presented in Figs. 6b, for low filter-signal frequency detuning (<200 GHz), the number of detected pulses corresponds to the initial number of pulses.
For higher detunings, we have recorded an exponential drop of the number of intensity spikes after spectral filtering. This result is fully consistent with our numerical study developed in section IV. We can qualitatively see that it is possible to isolate and detect by this technique a pulse having a very low probability of occurrence (around 1 over 1000).

VI. CONCLUSION
In this work, we have proposed a simple model that has enabled us to catch the nonlinear dynamics of the temporal and spectral evolution originating from the rare events with a non Gaussian probability. Based on the evolution of a Gaussian pulse having a temporal width corresponding to the typical coherence time scale of the incoherent wave, it becomes possible to easily understand the physical process stimulating the rogue events. Depending on the nature and power of the initial seed signal (depending if the initial pulse temporal width is below or above the coherence time), there will be a predominating role of the cross or self phase modulations respectively.
Even for a gain as low as 3 dB, resulting fluctuations can be significant, which requires to take into account this effect for the practical design of amplifiers based on quasi-instantaneous gain : large amplitude fluctuations of the partially incoherent pump can therefore lead to large signal distortions and errors under certain conditions. In this context, we believe that our characterization and interpretation of the noise properties in the picosecond temporal domain could be useful to explore further aspects of extreme value fluctuations occurring in optical fiber propagation. Our conclusions could also complement the well-known RIN measurements carried out in the electrical spectral domain to characterize amplifier noise properties [16,40,41]. Our numerical analysis has been confirmed by experimental results.

VII. ACKNOWLEDGMENTS
We would like to thank A. Picozzi and S. Pitois for illuminating discussions.