1. Introduction

Rogue or extreme events (frequently termed rogue waves) are rare high-amplitude events, observed in various physical systems, where nonlinear wave interactions take place, see [1] for a review. The occurrence probability of rogue events is described by so-called extreme-value or heavy-tailed (L-shaped) statistical distribution, which suggests that events with very high amplitude (as compared to the mean value) occur much more frequently than could be expected from normal-value (Gaussian) statistics. A considerable progress in understanding the rogue waves in hydrodynamics and optics has been achieved thanks to close analogy between the underlying physical mechanisms that initiate rogue wave formation [2, 3]. Optical rogue waves [4] are extensively studied in one-dimensional optical systems, i.e. photonic crystal fibers, where nonlinear light propagation gives rise to complex temporal behavior and spectral superbroadening. Here optical rogue waves manifest themselves as high amplitude red-shifted soliton pulses [4] and blue-shifted dispersive waves [5] that emerge from noise-seeded modulation instability, with soliton collisions identified as the main mechanism responsible for rogue wave formation [6]. The rogue-wave-like behavior in optical fibres was observed with different pulsewidths, ranging from continuous wave to picosecond and femtosecond pulses [7–9], and under a variety of operating conditions [10–12].

More recently, rogue wave-like behavior was observed in bulk media, in the femtosecond filamentation regime in gasses (air) [13] and liquids (water) [14] and predicted to occur in even more complex regimes of propagation, such as multiple filamentation [15]. In contrast to optical fibres, where nonlinear dynamics takes place only in one (temporal) dimension, in the filamentation regime, the nonlinear wave interaction occurs in full three-dimensional space. Optical rogue waves in bulk media are therefore not linked to soliton dynamics and formation of particular soliton pulses with extreme characteristics; here, due to space-time coupling, they emerge from modulational instability-mediated pulse splitting and energy redistribution in space and time, taking a form of the X waves [14], which are weakly localized nondiffractive and nondispersive wave packets spontaneously occurring in media with normal group velocity dispersion [16, 17]. Many aspects of the white-light continuum (WLC) generation in thin (typically, of few-millimeter thickness) bulk media share a close similarity with the filamentation phenomena, since the threshold for white-light continuum generation almost coincides with the threshold for self-focusing and filamentation. In this paper we present a statistical study of the WLC generation in a sapphire plate, which reveals that in the certain input-pulse energy interval the blue-shifted spectral components of the WLC exhibit large shot-to-shot intensity variations that obey extreme-value statistical distribution.

2. Results and Discussion

The white-light continuum was generated by tightly focusing 130 fs, 800 nm central wavelength pulses from an amplified Ti:sapphire laser system (Spitfire PRO, Newport-Spectra Physics) into sapphire, a wide-bandgap material, which is widely used for WLC generation in the visible and near infrared [18]. We used a typical scheme for WLC generation: a collimated laser beam with 1.6 mm FWHM diameter was focused by an f = +50 mm lens onto the input face of 3-mm-thick sapphire plate. The input-pulse energy was varied in 0.1–1 μJ range by a half-wave plate and thin-film polarizer. A small fraction of the laser pulse energy was directed to a pyroelectric detector (Pulsar-2, Ophir Optronics) for energy monitoring. The WLC radiation was then re-collimated by an f = +30 mm lens and after passing several neutrally scattering layers of thin teflon-film, was recorded by a fiber spectrometer with thermo-electrically cooled 16-bit detector (QE65000, Ocean Optics), which has a spectral detection range from 200 to 950 nm. The laser repetition rate was set to 10 Hz in order to synchronize the laser pulse with the energy meter and spectrometer that were synchronously operated in a single-shot regime.

Under these experimental settings, the spectral broadening was detected at the input-pulse energy E _in = 0.3 μJ, that corresponds to ∼ 1.3 critical power of self-focusing, P _cr, which is 1.8 MW for sapphire. With increasing the input-pulse energy, an asymmetric, strongly blue-shifted spectral broadening was observed, that is a typical feature of the WLC generation in bulk wide-bandgap media with normal group velocity dispersion [19, 20] and which is resulted by highly asymmetric pulse splitting [21]. Figure 1(a) illustrates a series of consecutive 2000 WLC spectra, recorded at E _in = 0.44 μJ. In this condition, shot-to-shot variations of the spectral intensity in the plateau (550–700 nm) and in the cut-off (490–550 nm) regions revealed markedly different magnitudes of excursions from the average values, in particular, the spectral intensity of the blue edge varied considerably from shot to shot, as highlighted by plotting the blue-shifted portion of WLC spectra in Fig. 1(b). In order to quantify the observed differences, in Figs. 1(d) and 1(e) we plot the statistical distributions of the WLC spectral intensity at 500 nm and 650 nm, respectively; whose spectral width is 1.5 nm, corresponding to a single pixel resolution of the spectrometer. The spectral intensity distribution at 650 nm has a symmetric, nearly Gaussian shape, that yielded a standard deviation of 8%. In contrast, the spectral intensity at 500 nm exhibited a distinct long-tailed statistical distribution, characteristic to an extreme-value process. It is worth mentioning that the rogue events become stronger as the wavelength shift from the pump increases, and such feature seems to be general [10, 13]. Figure 1(c) presents a corresponding histogram of the input-pulse energy fluctuations, which was well fitted by a Gaussian distribution. A standard deviation of 0.8% attested fairly stable laser operation.

 

Fig. 1 (a) 2000 individual WLC spectra (gray curves) generated at E _in = 0.44 μJ. The average spectrum is shown by the black curve. (b) Expanded view of the blue edge of WLC. Statistical distributions of (c) the input-pulse energy fluctuations, (d) and (e) the spectral intensity of the WLC at 500 nm and 650 nm, respectively.

Download Full Size | PDF

It is important to note that the L-shaped statistical distribution of the WLC spectral intensity at 500 nm is not a sole result of the laser pulse energy shot-to-shot fluctuations, but rather originates from shot-to-shot input-pulse intensity modulation due to presence of the noise (intensity and phase noise), which is rather difficult to characterize experimentally. To verify this, in Fig. 2 we plot the statistical distribution of the spectral intensity at 500 nm, which corresponds only to a single input-energy bin (±0.1% from an average value) and despite fewer laser shots (∼ 200) included in the statistics, retains a similar L-shaped statistical distribution as in Fig. 1(d).

 

Fig. 2 Statistical distributions of (a) the input-pulse energy, selected in ±0.1% energy range from the average value, (b) the resulting spectral intensity at 500 nm.

Download Full Size | PDF

For the input-pulse energies above 0.55 μJ, no further spectral broadening was observed, the blue edge of the WLC spectrum was limited to the cut-off wavelength of 410 nm, as a result of the combined effect of intensity clamping [19] and chromatic dispersion of the medium [20]. However, in this case the statistical distributions of the spectral intensity are dramatically modified at all wavelengths. This is illustrated in Fig. 3 by plotting a series of 2000 WLC spectra and resulting statistics of the WLC spectral intensity at 425, 500 and 650 nm, as recorded at E _in = 0.94 μJ. Specifically, the statistical distributions of the WLC spectral intensity acquire a nearly symmetric shapes across the entire wavelength range, see Figs. 3(c)–3(e), thus indicating absence of the extreme events. Moreover, shot-to-shot fluctuations of the spectral intensity in the WLC plateau region were markedly suppressed, for example, the standard deviation of the spectral intensity at 650 nm was measured as low as ∼ 3%. Note that throughout the measurements, the input-pulse energy fluctuations were essentially the same as illustrated in Fig. 1(c).

 

Fig. 3 (a) and (b) the same as in Fig. 1, recorded at E _in = 0.94 μJ. Statistical distributions of the spectral intensity at (c) 425 nm, (d) 500 nm and (e) 650 nm.

Download Full Size | PDF

From more detailed statistical investigations carried out by varying the input-pulse energy from 0.2 to 1 μJ, we found that distinct L-shaped statistical distributions were characteristic just to spectral components in the blue edge of the WLC spectrum. Moreover, the L-shaped statistics were observed only within a narrow input-energy interval between 0.40 and 0.46 μJ, when the blue edge of the WLC was confined in the wavelength interval of 480–590 nm. In order to quantitatively explain this finding, further we examined the spectral broadening dynamics versus the input-pulse energy. We measured the average (over 2000 laser shots) wavelength of the WLC blue edge, which was detected at a fixed intensity level of 10^−3.5 I _max, where I _max is the peak spectral intensity around 800 nm. The corresponding numerical simulations were performed using the numerical model, which solves the extended nonlinear Schrödinger equation, and which accounts for diffraction, dispersion, the instantaneous Kerr effect, self-steepening, 5-photon absorption, and the effect of free electron plasma, as described in detail in [14]. The noise was not included in the simulations, thus yielding fairly average values of the wavelength and peak intensity. The experimental and numerical data are compared in Fig. 4(a).

 

Fig. 4 (a) Dynamics of the spectral broadening expressed as the wavelength of the blue edge. Dashed curve depicts the peak intensity inside the sapphire sample. (b) Shape parameter of the Weibull fit function vs input-pulse energy. Shaded area marks the energy (intensity) range where extreme-value statistics is detected. (c)–(e) Numerically simulated on-axis intensity profiles. Dashed curve shows the input-pulse. See text for details.

Download Full Size | PDF

The experimentally measured non-Gaussian statistical distributions of the WLC spectral intensity at the blue edge were quantified by using a three-parameter Weibull fit function [6]:

Summarizing the results illustrated in Figs. 4(a) and 4(b), we conclude that the dynamics of the spectral broadening undergoes three distinct stages, that are linked to the temporal dynamics of the pulse and resulting peak intensity inside the nonlinear medium, as shown by a dashed curve in Fig. 4(a), and where the blue-shifted spectral components of the WLC exhibit different statistical distributions. The initial WLC generation stage (E _in=0.28–0.40 μJ) is mainly associated with self-phase modulation-induced almost symmetric spectral broadening, where no extreme-value statistics were detected. In the time domain, the pulse profile becomes gradually reshaped [Fig. 4(c)], however no pulse splitting occurs yet. The second stage (E _in=0.40–0.46 μJ) is related to an abrupt increase of the peak intensity, which clearly indicates the pulse splitting event. The spectral broadening is governed by highly asymmetric pulse splitting, pulse front steepening and generation of the shock-front at the trailing edge of the pulse, as shown in Fig. 4(d), which is consistent with WLC generation scenario [21]. On the other hand, the spectral broadening might be described in terms of four-wave mixing and occurrence of two split X waves in full three-dimensional space [17], whose mutual interaction provides suitable phase matching condition for generation of the blue-shifted spectral components. In this stage, the build-up of the intense trailing pulse (which may be called transient stage of WLC generation) strongly depends on the input-pulse intensity. In fact, this suggests that any small intensity variation (intensity noise) in the input beam has a large impact on the resulting peak intensity of the trailing pulse, thus giving rise to the observed rogue-wave-like statistical distribution. In the final stage of the WLC generation (E _in > 0.46 μJ), the spectral broadening markedly slows down and finally stops at E _in = 0.55 μJ as the intensity of the trailing pulse is clamped at ∼ 41 TW/cm² due to multiphoton (5-photon, in our case) absorption. The corresponding intensity profile is shown in Fig. 4(e). Here the dynamical picture settles and no extreme events occur. This in fact suggests that the intensity clamping acts as the mechanism, which suppresses occurrence of the extreme events. The extreme events therefore occur only within a narrow input-pulse energy (intensity) range, which is defined by the pulse splitting and intensity clamping.

3. Conclusion

In conclusion, we have demonstrated that the rogue-wave-like statistics of the spectral intensity of the WLC generated in a wide-bandgap solid (sapphire) are observed only in the transient stage of the WLC generation, and are associated with pulse splitting and build-up of intense trailing pulse, that strongly depend on the input-pulse intensity. The intensity clamping was found to be a mechanism which suppresses the occurrence of the extreme events. These findings also explain why the extreme events in WLC generation in bulk medium are observed only in a narrow input-energy (intensity) interval.