1. Introduction

Filamentation is a self-guided propagation regime typical of ultrashort, high-power laser pulses [1–4]. Beyond a critical power (3 GW in air), the Kerr effect balances diffraction. The beam then self-focuses until higher-order nonlinear defocusing effects like ionization or the saturation of the Kerr effect [5] come into play. The resulting dynamical balance gives rise to self-guided light structures known as filaments. Filaments can extend over atmospheric scale distances [6,7], opening the way to various applications from remote sensing [8] to lightning control [9–11], THz generation [12–16], fog clearing [17], or the triggering of condensation in sub-saturated atmospheres [18,19]. Filaments show very characteristic and even spectacular features, including long-distance propagation, bright white light emission due to both plasma emission on the side and spectral broadening in the forward direction, and noise associated to the shockwave [20–22]. Together, these features give rise to the intuitive notion of a qualitatively distinct propagation regime. However, defining a clear threshold for the filamentation regime turns out to be difficult [23], especially in focused beams where linear propagation can be sufficient to reach the ionization threshold [24], and even produce a denser plasma than Kerr self-focusing does [25]. Similarly, along the propagation axis, defining the onset and end of filaments, and therefore their length, requires somewhat arbitrary choices.

Here, we statistically investigate the transition to filamentation. Shot-to-shot fluctuations of the spectral amplitude of a high-power laser beam, whether filamenting or not, are known to display characteristic probability distribution functions (PDF): A regular probability distribution is observed in the center of the spectrum, while on its edge the distribution is long-tailed (optical rogue wave [26]) [27]. We extend this analysis to the sub-filamenting regime and to the transition to filamentation, in order to characterize how the statistics evolves when the incident power progressively reaches the critical power for filamentation. The evolution of spectral intensity PDFs turns out to display typical and contrasted signatures on the edges and in the center of the spectrum, respectively, suggesting that transition to filamentation cannot be characterized as a whole. By doing so, we suggest that the ambiguities in the definition of the boundaries of the filamentation regime are intrinsic to the physical process rather than instrumental or conceptual limitations, shedding a new light on the long-standing lack of explicit definition of this boundary.

2. Experimental setup

The experimental setup (Fig. 1) relied on a Coherent Astrella laser delivering ultrashort pulses at 1 kHz repetition rate. A half-wave plate installed in a motorized rotating stage ensuring $\pm 1^{\circ}$ precision, followed by a linear polarizer, allowed to adjust the pulse energy with an accuracy of $\sim 10$ $\mu$J. The angle-to-energy calibration was performed before each experimental run with a powermeter (Coherent PM10). Due to nonlinear chirp that could not be compensated by the compressor, the Fourier-limited 40 fs duration of the pulse was stretched to $55$ fs (as measured with PulseCheck, APE Berlin), reaching close to the filamentation threshold in air within the tuning range of the energy. The beam was slightly focused from an initial diameter of 11 mm at $1/e^2$ by an $f = 2$ m BK7 plano-convex lens located approximately at 15 cm after the polarizer. We checked that the dispersion of the transmissive optics had a negligible influence on the pulse duration. The beam subsequently propagated through a $13.5$ cm-long turbulent region generated by a heat gun (Steinel HL1502 S) transversally blowing air at $565^{\circ }$C. $15$–$27.5$ cm downstream of the lens, the beam self-focused and filamentation started if the incident peak power was sufficient. The filamentation region was shielded against air displacements in the room by a PVC tube of $18$ cm diameter so as to improve its stability. After the end of the filamenting region, the beam was collimated by an $f = 1$ m lens and focused by a BK7 lens ($f = 20$ cm) onto the entrance slit of an OceanOptics USB2000 spectrometer providing $\lesssim$0.5 nm spectral resolution over the spectral range of the measurements. A neutral density (ND) filter prevented the spectrometer from saturating. The spectrum was measured and recorded independently for each laser shot. For each incident pulse energy, $5000$ individual spectra were recorded.

 

Fig. 1. Experimental setup.

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We characterized the shot-to-shot fluctuations by recording the PDF of the spectral intensity as a 100-bin histogram, using identical bins in all conditions to facilitate comparison. In the characterization of the PDF, we use among others Hogg’s unbiased kurtosis estimator [28], defined as

3. Results and discussion

Figure 2(a) displays the typical evolution of the measured spectrum, for incident energies ranging from $0.017$ to $0.173$ mJ/pulse. As is typical for self-phase modulation (SPM) [27,29], the spectral broadening is characterized by an oscillatory plateau. Higher incident pulse energies correspond to a broader as well as a wider plateau.

 

Fig. 2. Evolution of the spectrum for increasing peak energy. (a) Average spectra; (b)-(f) Intensity fluctuations for incident pulse energies of 0.017, 0.054, 0.102, 0.130, and 0.173 mJ, respectively. In each panel, the solid thick line is the average intensity, the dark shaded area marks range between the 20$^{th}$ and the 80$^{th}$ percentiles and the light shaded area displays the range between the 5$^{th}$ and the 95$^{th}$ percentiles.

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The turbulence imposed to the beam propagation path, together with the pre-filamentation fluctuations of the laser itself in terms of both energy and initial beam profile, as well as the nonlinear transformation of the spectrum performed by the spectral broadening in the high-intensity region of propagation result in strong shot-to-shot fluctuations of the output spectrum, as shown by the shaded areas of Fig. 2(b)–(f). The dark (respectively lighter) shaded area encompasses the 20$^{th}$ – 80$^{th}$(resp. 5$^{th}$ – 95$^{th}$) percentile range. Shot-to-shot fluctuations of the spectrum appear essentially vertical (i.e., in intensity) in the plateau, and essentially horizontal (i.e., in spectral range) in the case of its edges.

The shot-to-shot fluctuations have an order of magnitude comparable to the mean spectral intensity all over the spectrum. In particular, above the threshold for appreciable spectral broadening, fluctuations reach a factor of 4, ranging from the non-filamenting to the filamenting regimes even at average incident pulse energies up to the critical power (173 $\mathrm{\mu}$J, $P$ = 3.15 GW $\sim P_\textrm {cr}$) where filamentation visually seems to be already well established. Such behavior randomly alternating filamenting (spectrally broadened) and non-filamenting pulses is consistent with the previously observed effect of turbulence on filamentation [30].

Figure 3 displays the evolution of the PDF of the spectral intensity as a function of the incident energy, for several wavelengths. The behaviors are contrasted. On the plateau (805 nm, panel e), the mode of the PDF continuously shifts towards more intense values when the incident pulse energy is increased. This shift becomes faster and faster when moving away from the center of the spectrum and approaching the side of the plateau (See 785 nm, panel d). A similar behavior is observed far away from the beam center (760 nm, panel a): The PDF initially peaks close to zero intensity and continuously rises when the incident pulse energy is sufficient for the broadening to reach the considered wavelength.

 

Fig. 3. Evolution of the histogram of the spectral intensity at (a) 760 nm (continuous shift of the distribution mode), (b) 765 nm (bimodal transition), (c) 773 nm (intermediate regime), (d) 785 nm and (e) 805 nm (smooth offset of the peak), and (f) 860 nm (bimodal transition). Each histogram is normalized to a total of 1. Insets display the position of the wavelength in the spectrum (see also Fig. 2(a)).

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Over a spectral range of $\sim$10 nm on each side of the spectrum, corresponding to both edges of the plateau, the transition however occurs in a qualitatively different way (765 nm, panel (b) and 773 nm, panel (c), as well as 860 nm, panel f). The low-intensity mode progressively vanishes when the incident pulse energy is increased, and a high-intensity mode simultaneously emerges. For incident energies within the transition, the PDF of the spectral intensity is therefore bimodal.

Figure 4(a) summarises the qualitative behaviors (negligible broadening, fully deployed broadening, continuous transition, and bimodal transition) depending on the incident pulse energy and wavelength. Skewness (Fig. 4(b)) and Hogg’s unbiased kurtosis estimator $H_g$ [28] (Fig. 4(c)) have low values for bimodal transitions regions and high values for continuous transitions. This can be understood by considering the corresponding evolutions on Fig. 3. In the course of continuous transitions, the PDF tail rises on the high-spectral intensity side, resulting in a heavier tail and increased asymmetry. In contrast, during bimodal transitions, the secondary mode of the PDF appears and grows at the expense of the PDF tail. As a result, the central region of the PDF broadens, so that the skewness and Hogg estimator decreases. Further statistical moment like the kurtosis and coefficient of variation (i.e., the ratio between variance and mean) display qualitatively similar behaviors, as illustrated in Fig. 5.

 

Fig. 4. (a) "Phase diagram" displaying the different regimes as a function of wavelength and incident pulse energy, from non-broadened to fully-broadened, with continuous and bimodal transition modes in between. (b), (c) Borders of the same diagram overlaid on (b) the skewness and (c) the Hogg’s unbiased kurtosis estimator [28].

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Fig. 5. "Phase diagram" displaying the different regimes as a function of wavelength and incident pulse energy, from non-broadened to fully-broadened, with continuous and bimodal transition modes in between overlaid on (a) the kurtosis and (b) the the coefficient of variation (variance/mean).

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Bimodal transitions occurs on the edge of the plateau, where the shot-to-shot fluctuations of the spectrum mostly consist in horizontal variations of the cliff delimiting the plateau. At these wavelengths, the sharp rise of the spectrum on its edge (large $\left |\frac {1}{I}\frac {\textrm {d}I}{\textrm {d}t}\right |$), surrounded by two relatively flat regions (the plateau and the low-intensity spectrum side, respectively), implies that the spectrum mainly features two ranges of attainable values, at low spectral intensity on the tail and at high intensity on the plateau, respectively, resulting in the observed bimodal distribution. The intensity fluctuations of the plateau itself are of second order here. This interpretation also explains why the bimodal behavior is observed on a narrow spectral range close to 840 nm (see Fig. 4(a)), where the plateau oscillates more. In contrast, on the plateau, the fluctuations are mostly vertical. They consist in intensity fluctuations, i.e., a rise of frequencies pre-existing in the incident pulse, so that the transition occurs continuously as the plateau progressively rises.

These two qualitatively different transition regimes coexist within the same spectrum, at frequencies only some nanometers apart and coupled by the nonlinearity of the pulse propagation, and in the course of the same self-phase modulation process. The discontinuous, bimodal transition to filamentation on the edges of the plateau would point to a system where filamenting and non-filamenting regimes are qualitatively distinct and co-exist over some range of input power. In contrast, the smooth, continuous transition on the plateau in the center of the spectrum prevents from defining a threshold for the filamenting regime. Note that the latter spectral range bears most of the pulse energy.

We argue that this dual behavior within the same spectrum during the transition to filamentation intrinsically implies ambiguities in the definition of the edges of a filamenting regime that would be qualitatively different from an essentially linear extended focus [24].

4. Conclusion

In summary, we characterized the evolution of intensity fluctuations across the spectrum of an ultrashort beam propagating in air, for a wide range of powers covering below and up to the critical power for filamentation. While on the edges of the spectrum the transition to filamentation is discontinuous and displays a bimodal distribution of spectral intensities around the critical power, it is continuous around the fundamental incident laser wavelength. This dual behavior provides an explanation for the lack of unambiguous definition of the boundary of the filamentation regime in experiments or numerical simulations in spite of the intuitive understanding that filamentation qualitatively differs from a more linear propagation regime.