1. Introduction

Propagation of high energy femtosecond laser beams in the atmosphere has drawn a considerable attention both for the study of fundamental characteristics of filaments [1–4] and for the development of many applications like atmospheric remote sensing [5] or triggering and guiding of electric discharges in the prospect of lightning control [6–9]. Femtosecond intense laser pulses can propagate in the atmosphere over several Rayleigh lengths as self-guided filaments. The basic mechanism of self-guiding consists of a dynamic balance between the nonlinear Kerr effect in air, which self-focuses the beam until ionization of oxygen molecules by multiphoton absorption, and the defocusing effect of plasma, which prevents optical collapse and sustains long-distance propagation. Besides this basic image, many nonlinear physical processes accompany filamentation, e.g. spectral broadening by self-phase modulation [10, 11], self-compression [12, 13], generation of THz radiation [14] or third-harmonic generation [15–17]. Moreover, when the pulse power largely exceeds the critical power (P_crit ~ 3 GW in air at 800 nm), multiple filamentation occurs [18], with a number of filaments proportional to the power [19].

A visually spectacular effect associated with filamentation is conical emission [20, 21]. It generates a wealth of coloured rings around the beam in the forward direction, whose peak wavelength decreases from infrared to ultraviolet with increasing distance from the propagation axis. The mechanism at the root of conical emission is still debated. It may imply Cerenkov radiation [20, 22], self-phase modulation (SPM) [21], four-wave mixing (FWM) [23, 24] or X-Waves modelling [25–27]. The first experimental measurements of conical emission in air have been performed in the 500–700 nm range [20, 21]. More recently, Méjean et al. estimated a divergence of the UV conical emission, although this estimation was based on indirect observation [28, 29]. In addition, the first high-resolution angle-wavelength spectrum of air filaments and the link with X-Waves was recently introduced [27].

In this Letter, we extend the measurement of conical emission to the ultraviolet, down to 225 nm. Moreover, we show that conical emission is not modified by multiple filamentation, as had been suggested based on indirect observations [30].

2. Experimental setup

Figure 1 shows the experimental setup. Intense femtosecond pulses at λ₀ = 800 nm (FWHM 23 nm) were generated by the Teramobile facility [18] whose laser system consists of a Ti:Sa oscillator followed by a chirped pulse amplification (CPA) chain. The compressed pulses, of 100 fs pulse duration at a 10 Hz repetition rate, had energies of 305 mJ. The corresponding peak power is 3 TW, i.e. 1000 P_crit, which yields a few tens of filaments. The amplified femtosecond beam, with an initial diameter of 15 cm, was focused by a sending telescope (f = +42 m), leading to a filamentation onset at z_f1 = 30 m and a filamentation length of 22 m until z_f2 = 52 m. Downstream of the filaments, at z = 82.5 m, 86 m and 106 m respectively, we measured the forward-emitted spectrum as a function of the transverse distance from the beam center, hence of the emission angle θ over ± 8 mrad. At each angle, the spectrum was recorded between 225 and 1080 nm with 0.4 nm resolution by a computer-interfaced spectrometer (OceanOptics HR2000+), whose input optical fiber was oriented toward the incoming laser source and swept across the beam profile.

 

Fig. 1. Experimental setup. A fiber collector is swept across the beam downstream of the filamenting region and the collected conical emission is analyzed on a spectrometer.

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To correctly record the spectra in spite of their high dynamics (8 orders of magnitude), we employed different neutral filters with high optical density (up to OD 4.0) and varied the integration time from 1 s to 65 s (10 to 650 laser shots). For most of the positions, we acquired at least two spectra: one with higher optical density and shorter integration time, to avoid saturation when measuring the intense 800 nm peak, and another with lower densities and longer integration times, for resolving very weak conical emission features at remote wavelengths. Data were smoothed by running averages over 10 adjacent points, i.e. over 4 nm.

3. Results and discussion

The data recorded at distances z = 82.5 m, 86 m and 106 m yield consistent angles if considering that the origin of the conical emission lies at z₀ = 44.5 m, which lies equally spaced between z_f1 and z_f2, i.e. in the middle of the filamenting region. This suggests that conical emission is generated all along the filamenting region, a picture consistent with the dynamical replenishment model, in which relatively short filaments are generated randomly all along the filamenting region [31] and independently emit conical emission.

 

Fig. 2. Main graph: Normalized light intensity spectra as a function of angle θ with respect to the axis of propagation. The spectral dependence of conical emission angle is clearly visible (see also Fig. 3). Back panel: integrated energy of the conical emission as a function of wavelength.

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Figure 2 displays the forward-emitted spectra over the 225–710 nm region as a function of θ. Each spectrum is independently normalized to unity. The energy of the conical emission peak is also displayed as a function of wavelength on the rear panel of Fig. 2. It is obtained by integrating the intensity of the un-normalized spectra over the wavelength as well as spatially over the ring (i.e., over 2π on the azimuthal angle). The peak of conical emission shifts from 250 nm up to the fundamental wavelength, and broadens from 20 to 140 nm (FWHM inferred from Lorentzian fits) for decreasing values of θ. Note that no conical emission was observed in the infrared between 800 and 1080 nm.

Figure 3 displays the wavelength dependence of the conical emission, as extracted from Fig. 2. Our data agree well with previous ones obtained in the case of single filamentation by Nibbering et al. [20] and Kosareva et al. [21] between 500 and 700 nm. In particular, the conical emission angle decreases regularly with increasing wavelength. Note that our measurements are a convolution of the emission angles from individual filaments and of the divergence of the filament bundle. Due to the overall beam refocusing near the focus [30], this divergence is smaller than initial geometrical beam divergence of 1.8 mrad (half angle), but could not be quantitatively determined in our experiment. This error shall be below Δθ = 1 mrad, in line with observations over longer distances [30]. Hence, our data show that the indirect measurements of Méjean et al. [28, 29] underestimated the conical emission angles, especially in the 300 – 450 nm range, by approximately 2 mrad. The origin of this underestimation could lie in an improper modelling of their Lidar geometrical compression.

 

Fig. 3. Conical emission angle versus central wavelength.

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We compared our data with the respective angular dispersion of conical emission predicted by FWM [24], X-Waves stationary solutions [27] and by the Cerenkov emission interpretation [20]. In the case of FWM, phase matching imposes the following angular dispersion of conical emission:

where k(ω) = n(ω)ω/c (ω = 2πc/λ being the frequency and n(ω) the refractive index of air, given by [32]), Ω = ω − ω ₀ is the difference between the generated frequency and the input pump frequency and k ₀ ⁿ = ∂ⁿk(ω)/∂ωⁿ|_ω0 [24]. On the other hand, according to the X-Waves solution theory [27], the angle θ is given by:

where v_g = 1/(k′₀ − α) is the filament group velocity and α is a free fit parameter which accounts for the input conditions [33], which we set to zero in our curve. This is equivalent to considering v_g = 1/ k′₀ ≡ v_c, where v_c is the laser pulse group velocity at the carrier frequency. In the case of Cerenkov emission [20], the same dispersion relation writes:

Note that by introducing the series expansion cos(θ) = (1 − θ ² + …)^1/2 in Eq. (3) we obtain the same approximate expression as in Eq. (2) for the particular case v_g = c/n ₀ ≡ v_ϕ In other words, the Cerenkov relation can be considered as a sub-case of the more general X-Waves relation when the filament group velocity v_g equals the carrier frequency phase velocity v_ϕ. While the three models fit the previous data adequately, our extension of the data to the UV partially lifts the ambiguity. Our data show that the measured CE spectra are well reproduced by using the Cerenkov expression, i.e. by putting v_g = v_ϕ in the X Waves relation [34]. On the other hand, the FWM predicts a slope significantly different from that of the experimental data, especially under 450 nm. Our measurements therefore suggest that the latter process alone is not sufficient to fit data. In addition, when points are shifted downward to correct for the beam divergence (Δθ ≤ 1 mrad), X-Waves dispersion relation of Eq. (2) always fits the data precisely by using a value of α ranging from 0 to 2·10^-14 s/m.

 

Fig. 4. Conversion efficiency from the fundamental wavelength into white light. Arrows point coherent anti-Stokes Raman scattering (CARS) emission and water vapor absorption.

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By integrating the intensity spectra over the full θ angle distribution i.e. from -8 to 8 mrad, and normalizing by the total beam energy, we calculated for the first time the absolute conversion efficiency from the fundamental wavelength of the incident laser into the white-light continuum. Fig. 4 displays the conversion efficiency rate per spectral unit, η(λ), expressed in nm^-1, which describes the amount of the incident energy at 800 nm which is converted into an infinitesimal wavelength range centered at λ (resolution 4 nm). The spectrum after propagation is centered at 806 nm, slightly red-shifted as compared with the incident carrier wavelength at 800 nm. In addition, it features both the water vapor absorption line at 940 nm (3 v polyad vibrational states of H₂O) [35] and the coherent anti-Stokes Raman scattering (CARS) emission of N₂ at 678 nm, corresponding to a Raman shift of 2329 cm^-1 from the 806 nm central wavelength [36]. The observation of these spectral features illustrates the capability of the supercontinuum to provide a suitable light source for both absorption and Raman spectroscopy in the atmosphere.

4. Conclusion

As a conclusion, we have characterized the conical emission from multiple filamentation down to 225 nm in the ultraviolet. The data prolong well the previously existing ones, while we observed no conical emission between 800 and 1080 nm. Our data exclude the phase-matching of FWM as the only mechanism driving conical emission in the UV, and favor X-Waves modeling and Cerenkov emission. They are also useful for applications which require a good characterization of the emission geometry, such as white-light Lidar.