We study experimentally the statistical fluctuations observed in a supercontinuum generated in the normal dispersion regime through cascaded stimulated Raman scattering. Specifically, we show that the statistical distribution of shot-to-shot spectral variations evolves from a quasi-Gaussian in the saturated regime for Stokes orders near the pump to a long-tailed extreme-value distribution for Stokes orders at a large separation from the pump in the unsaturated regime.
©2010 Optical Society of America
It is now well known that supercontinuum (SC) generation in the long pulse regime is very sensitive to the input noise of the pump pulses resulting in poor coherence properties and stability [1 ]. Recently, it was shown that the large shot-to-shot fluctuations inherent to this particular regime originate from noise-seeded modulation instability (MI) that can trigger the generation of pulses with abnormally high intensity . Such pulses are now commonly referred to as optical “rogue” waves and they have been shown to be associated with long-tailed statistical distributions indicating these extreme events occur with an increased probability in contrast to that predicted by Gaussian statistics . These findings have attracted significant interest and stimulated a number of detailed studies reporting similar statistical characteristics in various nonlinear regimes of pulse propagation [3–7].
Extreme-value fluctuations have been found in various types of nonlinear systems where the noise of the initial state can have a dramatic effect on the final state of the system. For instance, in the context of fiber optics, Hammani et al. have demonstrated “rogue-wave-like” statistics in Raman amplification where the noise from a partially incoherent pump wave can lead to significant changes in the properties of the amplified wave . Although the noise-driven nonlinear dynamics associated with “rogue” events have been primarily reported for fiber-based systems, research on extreme-value behavior has also recently expanded in other areas of optics and a range of nonlinear optical phenomena exhibiting similar rogue-like characteristics have been reported. Indeed, extreme-value behavior was observed in Raman amplification on a silicon chip , or during filamentation in air .
In this paper we report on another example of nonlinear pulse propagation where significant shot-to-shot variations occur as a result of nonlinear stimulated Raman scattering (SRS) dynamics leading to a long-tailed, “rogue-wave-like” statistical distribution. Noise transfer mechanisms in SRS leading to non-Gaussian statistics are well known and a number of earlier studies have investigated the statistical variations in the energy of the first Stokes pulse generated by SRS [8–12]. In particular, it was shown both experimentally and theoretically using a simple analytical model [13–14] that fluctuations in the intra-pulse modulation of nanosecond pulses due to longitudinal modes beating can induce significant variations in the energy of the first-order Stokes pulse and that the overall shape of the corresponding statistical distribution was also power-dependent .
Here we focus on the spectral fluctuations observed in the highly nonlinear regime of SRS where a large number of Stokes orders are generated leading to the formation of a broadband supercontinuum spectrum in the UV region where SRS is enhanced. Specifically, we use nanosecond pulses launched into a highly nonlinear fiber to generate a SC whose dynamics are mainly dominated by cascaded SRS. Using a short wavelength pump at 355 nm in combination with a single-mode fiber is one of the simplest and most cost-effective methods to generate a broadband UV-SC source. Therefore, this particular experimental configuration is of significant interest as this type of UV-SC light source can be used in a wide range of applications such as e.g. spectroscopy or microscopy . Significant variations are observed in the SC spectrum and the fully captured single-shot spectra allow us to study in detail the shot-to-shot fluctuations of the SC at the output of the fiber. Our experimental results reveal that “classical pump noise” results in “rogue-wave-like” statistics for wavelengths close to the long wavelength edge of the SC spectrum. Furthermore, we quantify the variations across the entire bandwidth of the SC spectrum and show that the fluctuations are strongly wavelength-dependent, evolving from a quasi-Gaussian towards an L-shape distribution for increased detuning with respect to the pump wavelength.
2. Experimental setup
A broadband UV-SC was generated by launching nanosecond pulses into a 40 m long polarization-maintaining fiber with a 2.3 μm core diameter specially manufactured for low UV attenuation (Nufern PM-S350-HP). The fiber has a pure-silica core and is single-mode for wavelengths above 350 nm. The 8-ns long laser pulses were produced by an electro-optically Q-switched, frequency-tripled Nd:YAG laser (EKSPLA NL202) operating at 355 nm with a 10 Hz repetition rate. The pulse energy was attenuated to the microjoule regime using a variable optical attenuator in order not to exceed the damage threshold of the fiber. Although in our experiments the input polarization was adjusted so as to match one of the principal axes of polarization of the fiber, we have checked that the observations reported here are in fact independent of the input state of polarization. In fact, we have also found experimentally (not shown here) that similar results are observed when a non polarization-maintaining fiber is used with the exception of a much less pronounced individual Raman peaks and smoother spectrum. The pulses were subsequently coupled into the fiber with a microscope objective specially designed for UV light (OFR-LMU-10X-λ) and the fiber input end was mounted on a highly sensitive XYZ-flexure stage for optimal coupling. A silicon photodiode with a 140 ps rise/fall time combined with a 2 GHz oscilloscope was used to monitor the temporal profile of the pulses prior to coupling. The output end of the fiber was connected to the entrance aperture of a spectrograph allowing to capture single-shot spectra through an ICCD camera mounted on the spectrograph exit aperture. The detection setup was connected to a PC allowing for automated measurements. Figure 1(a) shows a schematic of the experimental setup and an example of a generated SC is shown in Fig. 1(b).
The extreme spectral broadening of the pump is mostly caused by cascaded stimulated Raman scattering, which results in distinct spectral peaks separated by c.a. 13 THz. When recording the SC spectra generated by subsequent pulses we observed significant shot-to-shot variations. In order to clarify the origin of these fluctuations we have first characterized the inherent fluctuations in the input pulse characteristics delivered by the Q-switched laser as one can expect these variations to influence the fluctuations of the SC spectra. Specifically, the temporal profile of the pump pulses was observed to be highly modulated due to the beating of multiple longitudinal modes inside the laser cavity. As the number of modes and relative phases are random the modulation varies strongly from pulse-to-pulse as shown in Figs. 2(a)–2(d) where examples of different measured pulse envelopes are plotted. We emphasize that the 2 GHz bandwidth of the detection system is much larger than that used in earlier experiments on the characterization of the first-order Stokes pulse energy fluctuations . On one hand, this allows us to characterize with better accuracy the temporal envelope compared to Ref. 11, where, most probably due to the limited bandwidth of the detection system, a nearly ideal sinusoidal modulation was measured. On the other hand, even with our detection system, it is possible that higher harmonics may still be filtered out and that the ultrafast modulation may thus be not fully resolved.
The variations in the characteristics of the input pump pulses are illustrated in a more quantitative way in Fig. 2(e) which shows a histogram representation of the input pulse peak power corresponding to 9000 input pulses. The peak power distribution is seen to be close to Gaussian but with a small tail and with a relative standard deviation of approximately 23 % of the mean. The tail corresponds to the rare cases where multiple modes are in phase, generating exceptionally high peak power pulses (see Fig. 2(c)). Here we choose to focus on the peak power fluctuations of the input pulse rather than that of the pulse energy as we expect the cascaded SRS process to be more sensitive to the former quantity.
In addition to the “classical pump noise” of the Q-switched laser, possible fluctuations in fiber-coupling efficiency were also quantified as they may impact the overall input noise level. For this purpose, a 10 cm piece of fiber was used to evaluate the shot-to-shot changes in coupling efficiency. A short length was intentionally used in order to prevent any nonlinear process which could alter the result. The coupling efficiency was found to exhibit a near-Gaussian distribution with a mean coupling efficiency of 0.50 and a relative standard deviation of 2.6 %. The shot-to-shot changes in coupling efficiency are therefore a negligible source of additional noise compared to the intrinsic fluctuations of the pulses. Finally, in order to eliminate contribution to the observed spectral fluctuations that would arise from possible multi-mode coupling, we have checked with a selection of spectral filters that all the spectral components generated along the fiber occurred in the fundamental mode.
3. Results and discussion
We recorded two sets of 9000 single-shot spectra at the fiber output corresponding to higher and lower mean energy values of the input pulses, respectively. We focus mainly on the experimental results obtained using the higher mean energy value which was estimated to be 1.0 μJ (coupled energy). The measurement results for the lower energy case of 0.2 μJ are presented briefly at the end of the section as we did not find any major qualitative difference between those two cases. Figure 3(a) superimposes 500 individual spectra recorded at the output of the fiber. The mean spectrum calculated over the whole set of 9000 spectra is also displayed. The spectra show significant broadening towards the longer wavelengths with as many as 6 to 12 Stokes lines generated in the cascaded SRS process. Significantly, the long wavelength edge of the SC spectrum shows large shot-to-shot variations in terms of both spectral location and overall shape. To illustrate further the large discrepancy observed in the recorded single-shot spectra, Figs. 3(b)–3(d) shows selected cases corresponding to narrow, intermediate, and large spectral bandwidth, respectively. A difference as large as 6 Stokes orders between extreme cases can be observed in the output spectra.
To further characterize these variations quantitatively (and for consistency with other related studies [1, 6, 7]), histograms of spectral intensities were calculated at selected wavelengths of 386 nm, 416 nm, 421 nm, and 427 nm corresponding to the 5th, 8th, 9th, and 10th Stokes orders, as indicated by the arrows in Fig. 3(a). The histograms are presented in Figs. 4(a)–4(d). The histograms show a rapid transformation from a bell-like shape distribution into an L-shape distribution for spectral intensities at wavelengths located further from the pump. This type of evolution has been recently observed in a different context where it was attributed to the formation of optical “rogue” waves . Although here there are clearly no optical “rogue” waves in the sense observed in earlier experiments , the measured shot-to-shot fluctuations near the edge of the broadband SC spectrum are the manifestation of extreme-value events. The drastic change in the statistics is more conveniently observed on a log-log scale as shown in the insets in Fig. 4, where the initial bell-shape curve gradually evolves towards a quasi-linear shape, indicative of a power-law dependence.
In addition to calculating histograms at few selected wavelengths, we took advantage of the fact that we could capture entire individual spectra and performed a full wavelength-dependence analysis of the statistics as recently introduced by Kasparian et al. . Following the analysis of Ref. 6, we use the convenient “Pareto-like” metric M(λ) (1) to characterize the skewness of the distribution across the full spectral bandwidth of the SC spetcrum. The metric M at a given wavelength λ 0 is defined as the contribution of 20% of the largest measured intensities to the total sum of all measured intensities and can be calculated from 
where the intensities I i are sorted in a descending order from indices 1 to N. The lowest possible value for M is 0.2 for uniform distribution. M ≈ 0.44 corresponds to a Gaussian distribution, while values exceeding 0.5 indicate long-tailed L-shape statistics.
Figure 5 shows the mean spectrum together with the metric M(λ) calculated for the two measurement ensembles performed at (a) low and (b) high input peak power level, respectively. The emergence of long-tailed L-shape distributions on the long wavelength edge of the spectrum as was observed in Figs. 4(c)–4(d) can be clearly seen to correlate with a rapid increase in the “Pareto-metric” M(λ)in Fig. 5(b). For spectral components above 410 nm corresponding to the 9th Stokes order, the M-value exceeds 0.44 indicating a change in the statistical distribution towards an L-shape, which is consistent with what was observed from the histogram evolution. Figure 5(a) shows that the tendency of the SC to exhibit extreme-value behavior is also present in the lower peak power regime where much less Stokes order are generated on average. Furthermore, we observe that the M-value or skewness of the statistical distribution remains low (i.e. below 0.44) for Stokes orders which are saturated, irrespective of the mean input energy value. Our experimental results are in agreement with the numerical predictions of Ref.  obtained with a pump noise similar to that of our experiments. In fact, we believe that similar results would be obtained independently of the source of noise.
In order to confirm that classical pump noise amplified through cascaded SRS process is responsible for the significant spectral fluctuations, we simulated the nonlinear propagation in our fiber for an ensemble of 1000 Gaussian pulses with a normal peak power distribution having a 23 % standard deviation. The numerical model we use is similar to that introduced in Refs. 13 and 14 and includes spontaneous and stimulated Raman scattering as well as the fiber losses. Although this is clearly a simplified model which neglects the instantaneous nonlinear response and the walk-off effects, it nevertheless reproduces our experimental results qualitatively well. Specifically, it predicts the correct number of Stokes orders generated along the fiber and the calculated M-value increases towards the longer wavelengths for unsaturated Stokes orders similarly to that observed experimentally. The small-scale variations of the measured Pareto metric are, however, not reproduced by the simple model used here. This is because the model does not account for self/cross-phase modulation (as seen from the narrower simulated Raman peaks), only includes spontaneous Raman noise and neglects dispersion effects which transform amplitude noise into phase noise. Yet, the wavelength-dependence of the simulated Pareto metric is well reproduced confirming that classical pump noise alone leads to extreme-value fluctuations through cascaded SRS.
In conclusion, we have experimentally characterized the statistical fluctuations observed in SC generation in the cascaded SRS regime. Quantification of the spectral fluctuations is of high importance for improving the understanding of stability properties of SC generation in this particular regime and the subsequent interpretation of measurements that would employ this type of source. Specifically, we have shown that cascaded SRS can transform classical pump peak power fluctuations into large spectral variations. The statistics of the spectral fluctuations are wavelength-dependent and change from quasi-Gaussian to L-shape form the saturated to unsaturated regime. This constitutes another example of a nonlinear system where initial noise is transformed into extreme-value statistics, complementing other recent studies on extreme-value statistics in nonlinear optical systems.
We thank the Academy of Finland (research grants 130099, and 132279).
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