1. Introduction

Optical fiber systems are well-known to provide convenient platforms with which to investigate fundamental non-linear phenomena such as modulation instability [1], soliton dynamics [2, 3], and self-similarity [4]. As well as these studies focusing on dynamical aspects of nonlinear propagation, recent work has also considered the associated statistical properties, and there has been particular interest in the observation of statistically-rare high power solitons observed during fiber supercontinuum generation [5, 6]. Significantly, because these experiments were carried out in a regime where modulation instability played a key role in the dynamics, it has been possible to propose an important correspondence with the infamous oceanic rogue waves whose origin has been discussed in terms of similar modulation instabilities that occur in a hydrodynamic context.

The observation of “optical rogue waves” in supercontinuum generation is significant in itself, and has indeed motivated additional work investigating how such instabilities can be harnessed and controlled [6]. But the results are also of much general interest, as such statistically-rare events form part of a general and important class of phenomena that exhibit “extreme-value” statistics, typically characterized by heavy-tailed or L-shaped probability distributions. In this paper, we demonstrate that such extreme-value events are not restricted only to supercontinuum generation in optics, but can also be observed in the nonlinear process of fiber Raman amplification. Specifically, by exploiting Raman amplification with a partially coherent pump, we show that noise transfer from the pump to a coherent signal results in intensity spikes that also exhibit optical “rogue wave-like” extreme value statistics.

Of course, the study of noise in fiber amplification processes has been a subject of much previous research, and important studies have examined noise transfer mechanisms in four-wave mixing [7], Brillouin scattering [8] and Raman amplification [9-12]. Indeed, the presence of non-Gaussian statistics in Raman amplification has been previously noted in the nanosecond regime [10, 11, 13], but connections to extreme-value phenomena were not explicitly examined. The results we describe here extend these previous studies by using high-speed detection techniques to analyze the noise on a picosecond timescale, and by using a semilogarithmic representation of the measured pulse height distributions to more clearly show the heavy-tailed nature of the fluctuation statistics. In addition, numerical modeling based on coupled pump-signal field equations allow us to identify the central physical roles played by dispersive walk-off and cross-phase modulation, effects which were not previously considered in the nanosecond regime. Aside from their fundamental interest, our results will be expected to have important application for improving the physical understanding of noise transfer mechanisms in Raman amplifiers.

2. Experimental set-up

Figure 1(a) shows our experimental set-up, which is all fiber-format and uses only commercially available components at telecommunications wavelengths. We study Raman amplification in two readily-available fibers: a 500 m length of highly nonlinear fiber (HNLF) with dispersion β₂=7×10^-4 ps² m^-1 and nonlinearity γ=10 W^-1.km^-1; and a 3.5 km length of non-zero dispersion shifted fiber (DSF) with β₂=3.2×10^-3 ps² m^-1 and γ=2.4 W^-1 km^-1 at 1550 nm. Note that both fibers exhibit similar integrated Raman gain and cumulative nonlinearity γ L. Both fibers are also normally dispersive so that neither the pump nor the amplified signal experience solitonic or modulation instability effects [1, 2]. This is an important feature of our set up that ensures that the effects stimulating the emergence of rare events does not overlap with the processes involved in supercontinuum generation [5, 6].

The fibers under investigation are codirectionally pumped by an unpolarized Raman fiber laser delivering up to 2 W average power at 1455 nm. The pump delivers a partially coherent quasi-continuous wave (CW) output whose spectral linewidth varies over 20-100 GHz as the pump output power increased from 100-1100 mW [12]. The pump temporal coherence was characterized experimentally via non-collinear intensity autocorrelation, and the solid blue line in Fig. 1(b) shows the results obtained at a power level of 300 mW. Similar results were obtained for higher pump powers. The figure shows the expected 2:1 contrast ratio of a partially incoherent source, and the experimental results are in good agreement with the calculated autocorrelation function (circles) based on a numerical model of the pump assuming chaotic statistical fluctuations with a Gaussian amplitude distribution and a 25 ps coherence time [14]. To illustrate the pump fluctuations explicitly, Fig. 1(c) shows a 1 ns section of the temporal intensity distribution calculated from the numerical model.

 

Fig. 1. (a) Experimental set-up (b) Autocorrelation of the pump at 1455 nm, experimental results (solid blue line) compared to a Gaussian distribution with a coherence time of 25 ps (circles). (c) Corresponding intensity fluctuations normalized according to the median value.

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We have studied the statistical properties of Raman amplification at 1550 nm which is close to the Raman gain peak for a 1455 nm pump. Different aspects of the amplified signal properties were examined using CW and pulsed signal sources. The CW source was an external cavity laser having a MHz spectral linewidth with 0.35 mW average power. The pulsed source was a 22 MHz repetition rate passively-modelocked erbium-doped fiber laser delivering 1 ps pulses (FWHM) and with average power of a few µW.

3. Temporal evolution

3.1 Experimental results

Experiments were first carried to characterize an amplified CW signal for on-off integrated gains in the range of 3-16 dB corresponding to pump powers in the range of 300-900 mW for the HNLF and 350-1700 mW for the DSF. The temporal properties of the amplified signal were studied using both a 1 GHz bandwidth oscilloscope detection system as well as intensity autocorrelation. Figure 2(a) shows a typical single-shot oscilloscope trace spanning 500 ns using the HNLF fiber at 12 dB gain (700 mW pump power). This time series illustrates clearly that the amplified signal exhibits distinct intensity spikes generated without any temporal periodicity. Note that a spectral filter was used to remove possible contributions to the trace arising from spontaneous Raman emission outside the signal bandwidth. The amplified signal characteristics were then examined with higher temporal resolution using intensity autocorrelation as shown in Fig. 2(b). The measured autocorrelation for the HNLF amplifier (black, gray) is compared with that of the pump (blue) and a DSF amplifier (red).

 

Fig. 2. (a). Amplified CW signal in a HNLF amplifier at 12 dB gain. The intensity is normalized relative to the median value. (b). Autocorrelation of pump and amplified signal in HNLF and DSF amplifiers at gains indicated.

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The autocorrelation measurements clearly show the very different characteristics of the pump and the HNLF-amplified signal. Specifically, although the autocorrelation width of the pump and HNLF-amplified signal are comparable, the background level of the signal autocorrelation decreases to near zero, highlighting significant differences between the pump and amplified signal statistics. In addition, the low background level indicates the presence of high contrast pulses in the amplified signal time series, and the similar autocorrelation halfwidths allow us to infer a characteristic temporal pulse width comparable to the 25 ps pump coherence time. In contrast to these measurements of the HNLF-amplified signal, the signal amplified in the DSF yields an autocorrelation signal that decreases only slightly (~10%) over the 120 ps scan range. This indicates that the pump fluctuations transferred to the amplified signal are averaged over a significant timespan due (as we discuss below), to the non-negligible group velocity mismatch between the pump and the signal.

Statistical analysis of the time series in Fig. 2(a) yields qualitative power law behavior of the peak height probability distribution but, given the 25 ps timescale inferred from the autocorrelation measurements, a quantitative analysis of these statistics is not possible because of the limited detection bandwidth of our setup. However, quantitative statistics of the Raman amplified signals can be obtained using a repetitive sampling technique together with the pulsed signal source having a temporal duration well below the coherence time of the Raman pump. In this approach, a high-speed sampling oscilloscope with 50 GHz bandwidth is triggered on the modelocked laser pulse train to measure sampled eye diagrams of the amplified signal. These results are shown in Figs 3(a1)-3(a5), and they clearly illustrate the different behavior of the HNLF and DSF-based amplifiers. Specifically, we can see how the HNLF-based amplifier exhibits noticeably higher fluctuations (even at moderate gains) and we can directly determine the statistical distribution of the output peak-powers as presented in Fig. 3(b). Note that these results are plotted on a semilogarithmic (log-linear) axis. From these measurements we see the different nature of the amplification process in HNLF and DSF directly, and note the much higher probability of seeing high peak power events with HNLF. Quantitatively, the probability distribution for the DSF is well fitted on a log-linear scale by a linear function, indicating a close-to-Gaussian statistics. In contrast, for the HNLF, the tail of the amplified signal statistics is well-fitted with an exponentially decreasing fit, consistent with power law behavior typical of extreme-value processes and previous measurements of the optical rogue waves seen in supercontinuum generation [5, 6].

 

Fig. 3. (a). Eye-diagrams of: (a1) the initial ps pulse train compared with (a2-a5) amplified signals in DSF and HNLF for 3-dB and 12-dB on-off gains (b) Probability of the peak powers for both fibers and different gains. Peak-power values are normalized compared to the median value. Experimental results on a log-scale (lines) are compared with an exponential-decreasing fit (grey and black circles for HNLF amplifier) or with a linear fit (red circles).

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3.2 Numerical simulations

To interpret our results, we have developed numerical simulations based on a standard coupled nonlinear equation model of the Raman amplifier :

where ψ_s and ψ_p are slowly varying signal and fields in comoving frame with the signal. The model includes Kerr nonlinearity through self- and cross- phase modulation of pump and signal waves. The experimental Raman gain of silica is g_r and the optical losses α are also taken into account. The fiber group velocity dispersion is included through the pump-signal walk-off parameter δ. We assume scalar propagation and a factor of 2 correction is thus used to compare the simulations and the experiments that used a non-polarized pump. We have checked that group velocity dispersion acting on pump or signal alone was negligible for the typical temporal durations involved in the experiment. In contrast to previous studies modeling Raman amplification in the nanosecond regime terms of coupled equations for the signal and pump powers [9-11], the use of field amplitude equations here is necessary to capture the effects of cross-phase modulation as an important noise transfer mechanism.

We used this model to simulate the experimental results (with no free parameters) in the case of a CW signal, and the numerical results obtained are shown in Fig. 4. In Fig. 4(a) we show results illustrating the generation and the longitudinal exponential growth of a particular rogue-wave event that occurs for an HNLF amplifier. The time series simulated on a longer time scale [Fig. 4(b)] reproduces qualitatively the fluctuations in the amplified signal seen in the experimental measurements in Fig. 2(a) as well as the characteristics of the experimentally measured autocorrelation function in Fig. 2(b). The calculated peak-power probability distribution is also in good qualitative agreement with the experimental trends, with linear and exponential decreasing evolutions on a log scale for DSF and HNLF, respectively.

Simulations carried out over a wider parameter range allow us to identify the physical origin of the rogue wave statistics as the transfer of intensity noise from the pump to the signal via the exponential Raman gain. Similar extreme nonlinear shaping of the statistical distribution of the pulse energy under the influence of Raman gain has already been mentioned for nanosecond pump pulses delivered by Q-switched lasers [10] or dye lasers [13]. Our study considering optical fluctuations in the picosecond range gives further insight and illustrates that this transfer only occurs efficiently for a weak pump-signal walk-off. This is the case for the HNLF used here with an integrated walk-off (δL=29 ps) which is comparable with the coherence time of the pump. Under these conditions, the temporal fluctuations of the pump are directly translated into signal variations as confirmed experimentally and numerically by the similar autocorrelation widths for the pump and the amplified signals. However, the intensity noise transfer is not linear but exponential, leading to different autocorrelation contrasts.

On the contrary, for the DSF the integrated walk-off is much higher (δL=1.8 ns) so that the pump fluctuations are averaged over a longer timescale. As a consequence the amplifier signal does not exhibit such extreme temporal statistics. Another direct manifestation of the walk-off impact is the influence of the pumping scheme : no rogue wave structure has been experimentally observed in the HNLF when pumping in a counterpropagating scheme where the walk off parameter is nearly twice the speed of light [15].

 

Fig. 4. Numerical results based on the integration of Eq. (1). (a) 3D representation of the longitudinal evolution of the temporal intensity profile of a rogue intensity spike. (b) Amplified continuous signal observed at the amplifier output (c) Autocorrelation signal of the pump and output amplified signal after propagation in HNLF and DSF for gain values as indicated (d) Statistics of the peak powers for DSF and HNLF-based amplifiers (subplots (d1) and (d2) respectively) and for different gains. Results on a logarithmic scale are compared with a linear fit (red dots) or an exponential decreasing fit (grey and black circles).

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4. Spectral evolution and control of the rare events

In order to complement our temporal measurements and to further check the validity of our numerical model, we have also characterized the amplifiers in the (optical) spectral domain. These measurements were carried out for a CW signal. The experimental results are shown in Fig. 5(a1) for both fiber types and for different values of gain as shown. We see that significant spectral broadening of the signal occurs in the case of the HNLF, whereas the broadening is order of magnitude lower in the DSF. This behavior also confirms the very different dynamics of the signal amplification process in these two different fibers. These experimental trends have been studied using our numerical model and these results shown in Fig. 5(a2) are in good agreement with experiment. Importantly, the simulations allow us to clarify that the observed spectral broadening results from the cross phase modulation (XPM) of the pump on the signal [16] which is significantly enhanced by the weak walk-off parameter. To further interpret the spectro-temporal nature of the amplified signal in the HNLF, Fig. 5(b) plots a time-frequency representation of the numerical result, which clearly shows how the rogue wave structure with the highest peak power also exhibits the most broadened spectrum.

 

Fig. 5. (a). Optical spectra of the amplified signal at different gains. Results for the HNLF amplifier are compared with the intial seed and the DSF results. Experimental and simulated results are plotted in (a1) and (a2), respectively. (b). Spectro-temporal plot of a rogue wave-type fluctuation (c). Experimental temporal signal obtained after spectral slicing by a 9 GHz filter and for different spectral offsets of (c1) 0 GHz, (c2) 150 GHz and (c3) 350 GHz.

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The observation that the highest amplitude rogue events are associated with increased spectral broadening suggests that an offset filtering technique can be used to isolate their occurrence within the amplified signal time series. In our experiments, by using a bandpass fiber-Bragg filter having a spectral bandwidth of 9 GHz, we have experimentally recorded the output temporal sequence for different values of signal-filter frequency detuning. On the central part of the spectrum, as can also be seen in Fig. 5(b), both moderate and high peak powers structures are present (Fig. 5(c1)). But (similar to XPM-based regeneration and wavelength conversion techniques [17]) progressively increasing the spectral offset (Fig. 5(c2)) enables the discrimination of the highest peak powers structures from residual noise. This is because the effect of XPM is to lead to preferential spectral broadening for events with higher amplitude and thus spectral filtering can be used to efficiently isolate the rarest and most intense events (Fig. 5(c3)). Although this technique appears at first sight similar to the spectral filtering technique used to isolate the optical rogue waves in supercontinuum generation [5, 6], we stress again that the extreme event generation mechanisms are very different. In fact, a significant practical difference is that in contrast to the detection of red-shifted rogue solitons, the technique for detecting rogue Raman events can use both low and high wavelength filtering.

5. Conclusion

Our results have shown that pump-signal noise transfer in Raman amplification can lead to extreme-value rogue wave-like behavior and power law probability distributions with heavy tails. These results are practically significant as such large amplitude fluctuations could lead to unexpectedly large distortions and errors under certain conditions. A full understanding of the incoherence transfer mechanism is therefore essential for practical amplifier design. In this context, we believe that our characterization and interpretation of the noise properties in the picosecond temporal domain using autocorrelation and eye diagram repetitive sampling measurements can be usefully used to explore further aspects of extreme value fluctuations in fiber propagation, and can also complement the usually used RIN measurements carried out in the electrical spectral domain to characterize amplifier noise properties. Finally, we note that although these results have been carried out in the context of silica fibers, they would be expected to be relevant to any system where Raman amplification plays an important role. Indeed, during the final stages of production review of this manuscript, we became aware of a preprint reporting related studies of noise characteristics in silicon Raman amplifiers [18].