A Houard

Y Liu

A Mysyrowicz
email: andre.mysyrowicz@ensta-paristech.fr

Recent developments in femtosecond filamentation

We review recent developments in the field of femtosecond laser filamentation.

Introduction

We review recent developments in the field of femtosecond filamentation. In a first part, we describe the physical effects contributing to induce filamentation during the propagation of intense short laser pulses through transparent media. The importance of group velocity dispersion is illustrated in the case of fused silica. In the normal dispersion regime, the pulse evolution during propagation is complex and depends critically on input parameters such as the beam convergence, pulse duration, laser intensity. By contrast, a simple reproducible behavior is obtained in the anomalous dispersion region. The filamentary pulse is then closely parented to a lossy spatio-temporal soliton.

In a second part, we discuss the interaction between two air filaments in the normal dispersion regime. Interference between the fields of the crossing pulses leads to the formation of a plasma grating. We show how one can characterize the plasma density, its decay and the mechanism for this decay. Both ambipolar diffusion and plasma recombination play a role in molecules, whereas the decay is dominated by ambipolar diffusion in atoms. We show how energy can be exchanged between two crossing filaments due to scattering by a moving plasma grating. We also discuss the interaction between two counter-propagating filaments. The two filaments act as a retro-reflecting mirror: a third probe pulse of same frequency, incident at an arbitrary angle on this mirror, is retro-reflected with a spectacular improvement in beam profile quality and contrast ratio (ratio between precursor light and the pulse peak intensity).

In a third part, we discuss two applications making use of filaments in air. The first concerns the generation of THz pulses. A laser pulse undergoing filamentation emits a radially polarized THz pulse along a forward oriented cone. We discuss the origin of this effect and describe methods to enhance the THz emission by several orders of magnitude. This includes the application of an electric field, either static or optic, or the constructive THz emission from several filaments. The second application is the triggering of guided electric discharges by filaments and their use as RF antennas.

Filamentation in the normal/anomalous dispersion regime

Ultrashort, intense laser pulses are now commonly available, thanks to the development of laseramplifiers systems based on Ti: Sapphire as the active medium and chirped pulsed amplification (CPA) technique. Pulses at 800 nm emitted by such lasers have typically a duration between 10 and 100 fs. A strongly non linear propagation regime sets in when such pulses, even with a modest energy of a few millijoules, are launched through atmosphere. This nonlinear propagation is complex and involves a subtle interplay between many effects. No analytical or semi-analytical treatment can be used to describe it. One must resort to a numerical treatment. This is usually done by solving an extended non linear Schrödinger equation for the evolution of the pulse envelope treated as a scalar quantity. It allows a proper description in the paraxial approximation for rapid temporal changes, down to the single optical cycle limit, but fails to describe fast spatial changes, on a scale comparable to the wavelength. We refer to references [START_REF] Couairon | Femtosecond filamentation in transparent media[END_REF][START_REF] Couairon | Practitioner's guide to laser pulse propagation models and simulation The European[END_REF] for a description of the numerical code and its limits. Figure 1 shows the result of a simulation for the propagation in air of an initially collimated pulse of Gaussian profile (pulse duration: 100 fs, pulse energy: 5 mJ, beam diameter: 40 mm, wavelength: 800 nm). The first effect coming into play is the optical Kerr effect. It induces a change of the medium refractive index according to the law: [START_REF] Couairon | Femtosecond filamentation in transparent media[END_REF] This leads to beam self focusing, because the index is larger around the beam center, where the intensity is highest, than on the wings and therefore bends the pulse wavefront. The effect is cumulative upon propagation because the focusing reinforces the change of n which in turn increases the intensity. This self focusing overcomes the defocusing effect of diffraction and leads to a collapse of the beam upon itself, if the initial pulse peak power exceeds a critical value P cr [START_REF] Marburger | Self Focusing: Theory Progress in Quantum Electronics[END_REF][4][START_REF] Polynkin | Critical power for self-focusing in the case of ultrashort laser pulses[END_REF] where .

(2)

In the example given, the initial region dominated by the optical Kerr effect extend over a distance of 6 meters at which point the beam diameter decreases abruptly and the pulse intensity increases correspondingly.

When the collapsing pulse intensity reaches a value above 10 13 W/cm², multi-photon ionization of air molecules sets in. The creation of free electrons leads to a defocusing effect. Indeed, the refractive index decreases according to the law: ,

where is the density of free electrons and

2 2 0 0 e me c    
the critical plasma density above which the plasma becomes opaque (m e and e denote the electron mass and charge). The plasma of highest density is created at the center of the beam and acts therefore as a defocusing lens. The combined dynamic interplay between beam self focusing and defocusing leads to the formation of a contracted pulse with an intense ionizing core that maintains a radius of ~ 50 µm over large distances. The filament core is surrounded by a reservoir of laser energy that sustains the core by supplying an inflow of energy. Figure 1(b) illustrates the action of the plasma. It surges and acts to prevent collapse when the pulse intensity reaches ~ 10 15 W/cm². It first gives rise to a continuous meter long plasma string, followed by a quasi-periodic cycles of focusing and defocusing. At larger distances, plasma surges only intermittently. In the regions between and beyond these occasional surges of ionization, the nonlinear propagation is dominated by a dynamic competition between self focusing and diffraction (bright channels formation). It should be stressed that the behavior shown in figure 1 is not generic but depends on initial beam conditions (beam size and convergence, pulse energy, pulse duration, etc). With multi TW lasers, continuous plasma strings have been observed at a distance of 1 km with the plasma string extending over more than 50 m [START_REF] Durand | Kilometer range filamentation Opt[END_REF]. Bright channels persist over several km. [START_REF] Mechain | Long range self-channeling of infrared laser pulses in air: a new propagation regime without ionisation[END_REF][8] There is a profound restructuring of the pulse time profile during filamentation. Figure 2 shows the calculated evolution of an initial converging pulse with initial Gaussian profile upon propagation in Argon. A complex pulse splitting pattern takes place, which evolves during propagation and leads to the fortuitous emergence of isolated compressed pulses at certain distances. Pulses of initial duration 35 fs have been shown to compress down to 5 fs during filamentation in Argon [START_REF] Zaïr | Spatio-temporal characterization of few-cycle pulses obtained by filamentation[END_REF]. It should be stressed again that the location and duration of such shortest isolated pulses depends on initial conditions. A spectacular effect of filamentation is beam profile self-cleaning. It occurs in the region dominated by the Kerr effect, during the catastrophic collapse. The filament core acquires a Townes mode profile that is retained in the bright channels emerging after the end of the ionization region [START_REF] Moll | Self-Similar Optical Wave Collapse: Observation of the Townes Profile[END_REF][START_REF] Prade | Spatial mode cleaning by femtosecond filamentation[END_REF].

We now address the influence of group velocity dispersion on the filamentation process. One can readily understand why it plays an important role. Consider the evolution of the pulse spectrum in the self-focusing region. Because of self-phase modulation [START_REF] Agrawal | Nonlinear fiber optics[END_REF], instantaneous frequencies are created that shift to the red (blue) the ascending (descending) part of the pulse, according to the following formula:

20 0 ( , ) () n I r t tz t c t          . ( 3
)
In the normal dispersion region, red frequencies move faster than blue frequencies and therefore there is a detrimental pulse stretching during propagation. In the anomalous dispersion region, by contrast, red frequencies have a slower velocity than blue components. The generated frequencies are swept back to the main pulse. As a consequence, one expects lower threshold for filamentation and more stable filamentary pulses in the region of negative group velocity dispersion (GVD). Experimental verification of the role of the dispersion regime would be difficult to achieve in gases, because of the lack of extended regions of negative group velocity. Transparent solids such as fused silica offer a compact system where it can be experimentally verified [START_REF] Durand | Self guided propagation of ultrashort laser pulses in the anomalous dispersion region of transparent solids: a new regime of filamentation[END_REF]. Indeed, there is a wide frequency range of anomalous dispersion starting around 0.97 µm and extending beyond 3.85 µm in SiO 2 . The critical power in SiO 2 is a few MW instead of GW for gases. Figure 3 shows a comparison between the calculated filamentation in both dispersion regions. With positive GVD, the behavior is complex, showing complex pulse breaking, analog to what happens in air. With negative GVD, a single compressed pulse is formed after 0.5 cm of propagation. The slightly subluminal pulse then conserves its diameter and duration over the remaining length of the 3 cm thick sample. Experiments confirm these simulations. Figure 4 shows a side image of the blue luminescence emitted by the sample at high intensity I >10 13 W/cm². The 20 µm wide filament is formed after 0.5 cm and is maintained over the rest of the sample. Figure 5 compares the beam profile emerging from the sample to the profile measured at an equivalent distance in the absence of sample. Pulse self cleaning, a signature of filamentation is manifest, as well as beam contraction. Measurement of pulse duration of the pulse exiting the sample is shown in figure 6. In agreement with the numerical simulations, the pulse contracts in time and keeps the contracted value over the sample length. From the numerical calculations, one deduces that the filament core has a duration of 8 fs. Together with its surrounding reservoir, the pulse duration becomes 25 fs, in agreement with the measurement where the central core and its reservoir are not resolved. Thus, the filamentation process in the anomalous region leads to the formation of a pulse that is almost invariant with propagation and is therefore closely parented to a spatiotemporal soliton. It must be underlined however that the pulse is lossy since the pulse consumes energy to ionize the medium. Also, part of the light energy is dissipated in the form of a conical emission. [START_REF] Durand | Conical emission and extreme blueshifted continuum peaks from filamentation in the regime of anomalous dispersion in fused silica[END_REF]

Interaction between two filaments

Interaction of filaments has been observed to give rise to many interesting phenomenon such as energy exchange between filaments, enhanced third harmonic [START_REF] Suntsov | Filamentation produced third harmonic in air via plasma enhanced third-order susceptibility[END_REF][START_REF] Yang | Noncollinear interaction of femtosecond filaments with enhanced third harmonic generation in air[END_REF][START_REF] Liu | Efficient Generation of Third Harmonic Radiation in Air Filaments: a Revisit Optics[END_REF], plasma grating formation [START_REF] Suntsov | Femtosecond laser induced plasma diffraction gratings in air as photonic devices for high intensity laser applications[END_REF][START_REF] Yang | Plasma waveguide array induced by filament interaction[END_REF][START_REF] Liu | Energy exchange between femtosecond laser filaments in air[END_REF], molecular lensing effect [START_REF] Varma | Trapping and Destruction of Long-Range High-Intensity Optical Filaments by Molecular Quantum Wakes in Air[END_REF], coherent control of filament fusion or repulsion [START_REF] Xi | Interaction of Light Filaments Generated by Femtosecond Laser Pulses in Air[END_REF][START_REF] Shim | Controlled interactions of femtosecond light filaments in air[END_REF], etc. In the following we discuss several aspects of the interaction between two filaments that enable to extract properties of the plasma created during filamentation, to exchange energy between crossing filaments and to improve spectacularly a third pulse probe pulse of same wavelength.

Plasma grating 3.1.1. Temporal dynamic of a plasma grating.

Field interference of two crossing filamentary laser pulses give rise to laser intensity spatial modulation in the overlapping area. As a result, a strong modulation of the plasma density is created due to the highly nonlinear nature of optical ionization. This modulated plasma structure, i.e. plasma grating, is formed instantaneously, on the time scale of the pulse duration  p . On the other hand, its subsequent decay is long and can last for several hundreds of picoseconds in atomic gases. It is important to analyze this decay in view of possible applications. Recently, we measured the decay process of the plasma grating in different molecular and atomic gases. In the experiment, we employed a probe pulse at 400 nm co-propagating with one of the pump pulse. This 400 nm probe pulse is expected to be diffracted into the propagation direction of the other pump pulse in the presence of the non-sinusoidal plasma grating. Therefore, the measured 400 nm signal as a function of the temporal delay between the probe pulse and the two synchronized pump pulses allows characterizing the temporal evolution of the plasma grating. More details of the experimental setup can be found in ref [START_REF] Durand | Dynamics of plasma gratings in atomic and molecular gases[END_REF] It was found that in molecular gases like N 2 and O 2 the plasma grating decays rapidly, with a lifetime about 10-20 ps (Figures 7(a) and 7(b)). However, in atomic gases such as Ar and Xe the plasma grating persists for more than 300 ps (Figures 7(c) and 7(d)). The decay of a plasma grating process is expected to be due to free electron recombination and ambipolar diffusion. To obtain insight on the respective importance of the two effects, we varied the crossing angle of the two pump pulses. The resulting variation of grating period leads to a change of the ambipolar diffusion while the free electron recombination remains the same. From such experiments, we conclude that the grating evolution is ruled by ambipolar diffusion in atomic gases and by a combination of ambipolar diffusion and collision-assisted free electron recombination in molecular gases. From the systematic measurements, electron diffusion and recombination coefficients are extracted for Ne, Ar, Kr, Xe, N 2 , O 2 , CO 2 and air at 1 bar [START_REF] Jarnac | Study of Laser Induced Plasma Gratings Dynamics in Gases Optics[END_REF].

Energy exchange between filaments in the presence of travelling plasma grating.

The first experimental observation energy exchange between two filaments was reported in 2009 by Bernstein et al. [START_REF] Bernstein | Two-Beam Coupling between Filament-Forming Beams in Air[END_REF]. They demonstrated an energy exchange ratio of 7% between two filament-forming pulses with slight frequency difference. In their work, a travelling refractive index grating is formed due to the Raman effect of air molecules. As a result, the classical two-beam coupling (TBC) scheme applies [START_REF] Bernstein | Two-Beam Coupling between Filament-Forming Beams in Air[END_REF][START_REF] Boyd | Nonlinear Optics[END_REF] and laser energy transfers from the high frequency pulse to the low frequency pulse. This provides a control over filaments propagation by modifying their characteristics in flight.

Recently, we observed a new regime of energy exchange where laser energy transfers from the low frequency pulse to the high frequency one. This new type of energy exchange was also found in monatomic noble gases such as Ar and Xe, excluding a traditional TBC mechanism based on a retarded optical Kerr nonlinearity. We attribute this new energy exchange to the effect of a moving plasma grating formed at the crossing point of the two filamentary pulses. Due to its plasma nature, this refractive grating can sustain very high laser intensity. With tightly focused laser pulses, we have demonstrated energy exchange efficiency of 50% in air. The result is presented in figure 8 [START_REF] Liu | Energy exchange between femtosecond laser filaments in air[END_REF].

Figure 8. Energy exchange between two filaments crossed with

an angle of 20º.

Plasma grating in motion.

To detect the movement of a plasma grating, we employed an experimental setup similar to that in 3.1.1. except that there was a difference in the wavelength of the two pump pulses [START_REF] Durand | Experimental observation of a traveling plasma grating formed by two crossing filaments in gases[END_REF]. We spectrally analysed the diffracted 400 nm pulse with a spectrometer. The movement of the plasma grating imparts a frequency shift on the probe pulse owing to the Doppler effect. Figure 9 presents the spectrum gravity of the reflected 400 nm signal pulse as a function of the time delay (frequency difference) between the two pump pulses with linear chirp. The central frequency of the signal pulse shifts proportional to the frequency difference between the pump pulses, i.e., to the velocity imparted to the grating. A calculation of the measured frequency shift based on the classical Doppler effect agrees well with this result [START_REF] Durand | Experimental observation of a traveling plasma grating formed by two crossing filaments in gases[END_REF].

Figure 9. Spectrum center of gravity of the signal pulse as a function of the time delay between the two pump pulses.

Filaments mirror

A special geometry is obtained when two filaments meet head on [START_REF] Jarnac | Spatiotemporal cleaning of a femtosecond laser pulse through interaction with contra-propagating filaments in air to be[END_REF]. The encountering filaments form a mirror to a third probe pulse of same frequency impinging at an arbitrary angle. The mirror has special properties because the reflection occurs in the retro-direction. In addition, the reflected beam profile and pulse contrast ratio is dramatically improved. Figure 10 shows the beam fluence profile of the probe pulse before and after reflection. One can understand this spectacular property by considering the plasma grating formed by either pump with the probe pulse. As shown in section 3.1.1, the other pump will diffract into the direction of the incoming probe. Therefore, the reflected pulse is replaced by a fraction of the pump pulses with self-cleaned beam profile. The profile fits indeed a Townes mode profile, as expected if the retro-signal corresponds to a diffraction from the filaments. In addition, the small size (4500 µm 3 ) acts as a pulse cleaner removing distortions in the phase profile of the incoming probe pulse. Figure 11 shows the reflection of the filament mirror as a function of pump laser intensity for a probe incident at 90° from the filaments. Before the onset of ionization, the reflection is very small, because no plasma grating is present. There is a very steep increase of the reflection once the threshold for pump beam filamentation is reached, due to the high-order dependence of the ionization rate with intensity, followed by a near saturation, with a reflection efficiency of 20-30%, reflecting the intensity clamping inside a filament [START_REF] Couairon | Femtosecond filamentation in transparent media[END_REF]. At an incident angle of 30°, the reflection increases to 40%.

This leads to the possibility of temporal pulse cleaning by removing light precursors. Indeed, if the chronology of the pump and probe pulses is adjusted such that they all meet at the same time, any precursors present in the pump and probe pulses will not be reflected because no plasma mirror is formed at the time of their encounter, due to the corresponding lower intensity. The time profiles of the incident and retro-reflected pulse have been measured with a third order -2 cross-correlator. No precursor was detected in the reflected pulse within the dynamic range of the detector, as shown in figure 12.

Applications of femtosecond laser filaments

Intrigued by the spectacular effects accompanying filamentation process, researchers have demonstrated plenty of scientific applications of the filaments such as triggering and guiding of lightning [START_REF] Zhao | Use of self-trapped filaments in air to trigger lightning in Ultrafast Phenomena[END_REF], Light Detection and Ranging Technique (LIDAR) [START_REF] Kasparian | White-light filaments for atmospheric analysis[END_REF], few-cycle light pulse generation [START_REF] Hauri | Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation[END_REF], virtual antenna [START_REF] Brelet | Radiofrequency plasma antenna generated by femtosecond laser filaments in air[END_REF], micromachining in transparent solids [START_REF] Yamada | In situ observation of photoinduced refractive-index changes in filaments formed in glasses by femtosecond laser pulses[END_REF], Terahertz (THz) pulses generation [START_REF] D'amico | Conical forward THz emission from femtosecond laser filamentation in air[END_REF], control of aerodynamic flows [START_REF] Dufour | Numerical modelling of supersonic flow actuated by laser-induced plasma[END_REF], water condensation [START_REF] Rohwetter | Laser-induced water condensation in air[END_REF] and snow formation [START_REF] Ju | Laser-filamentation-induced condensation and snow formation in a cloud chamber[END_REF]. Some applications such as few-cycle pulse generation are already quite matured, while others THz generation have just been discovered and require further investigation. In the following two sub-sections, we will detail two applications: THz wave generation by filaments and control of electric discharges with filaments.

THz generation by one filament, two filaments

In 2007, D'Amico et al. reported a forward THz pulsed radiation stemming from a femtosecond laser filament in air [START_REF] D'amico | Conical forward THz emission from femtosecond laser filamentation in air[END_REF]. They observed that the THz is radiated in a hollow-cone structure in the laser propagation direction, with a radial polarization (see figure 13). The authors developed a Transition-Cherenkov model which reproduced the above observed features [START_REF] Amico | Forward THz radiation emission by femtosecond filamentation in gases: theory and experiment[END_REF]. In this model, the heavily damped longitudinal plasma oscillation produced in the wake of the propagating pulse is at the origin of the THz pulsed emission. Terahertz (THz) radiation produced by ultrashort laser pulses in air attracted much attention in recent years, since it has some unique properties. The THz source can be positioned at the proximity of a remote target by displacing the onset of filamentation. This solves the long-standing problem of the poor propagation of THz through atmosphere due to strong absorption by water vapor. In contrast to other THz generation techniques with short laser pulses, there is no obvious limit to the laser power that can be used, since there is no material in the path of the pulse that can be damaged.

However, the energy conversion efficiency from laser to THz radiation by a single filament is low, on the order of 10 -9 [START_REF] Houard | Calorimetric detection of conical THz radiation from femtosecond laser filaments in air[END_REF]. Therefore, it is desirable to improve the conversion efficiency and control the physical properties of this Terahertz radiation. Several techniques have been developed along this line.

In a first scheme, an external electric field is applied on the filament either longitudinal or transverse. A THz enhancement by 3 orders of magnitude is obtained with an applied field E ext ~ 10 kV/cm (figure 13). At the same time, the radiation pattern and polarization property of this amplified emission keep the same as that of the pure transition-Cherenkov THz radiation [START_REF] Liu | Amplification of Transition-Cherenkov Terahertz Radiation of Femtosecond Filament in Air[END_REF]. This amplified THz pulse is of particular significance since a radially polarized THz wave has been demonstrated to be compatible with the propagating mode of a metal wire THz waveguide [START_REF] Wang | Metallic wire for Terahertz wave guiding[END_REF]. Although, this method of enhancement suffers from the fact that the electric field must be applied on the plasma region of the filament, not an easy task for stand-off THz applications. For most THz applications, a THz beam with maximum intensity on the axis is preferable. This requires the presence of a transverse current in the filament and an orthogonal electric field should be able to fulfill this task. In the experiment, we applied an external electric field perpendicularly to the filament axis. With the increase of the external electric field strength, the asymmetrical radiation pattern becomes more and more symmetrical (figure 14). As a result, a THz beam with above desired feature was obtained when E ext exceeds 3 kV/cm. More important, the THz intensity was enhanced by 3 orders of magnitude with respect to the transition-Cherenkov THz with E ext = 9 kV/cm. Concerning the THz polarization, we found it is linear polarized and the polarization plane is dictated simply by the direction of the external field. In addition, we demonstrated that the polarity of the THz can be easily controlled by the direction of the electric field [START_REF] Houard | Strong enhancement of Terahertz radiation from laser filaments in air by a static electric field[END_REF]. Another approach is to use an oscillating transverse field, by combining a short IR pulse at 800 nm with its harmonic generated in a thin BBO crystal. After the first demonstration of this two-color method [START_REF] Cook | Intense terahertz pulses by four-wave rectification in air[END_REF], it has been extensively studied with a millimeter scale long filament (L ~ THz) [START_REF] Xie | Coherent control of THz wave generation in ambient air[END_REF][START_REF] Dai | Coherent polarization control of Terahertz waves generated from two-color laser induced gas plasma[END_REF][START_REF] Thomson | Broadband THz emission from gas plasma induced by femtosecond optical pulses: From fundamental to applications[END_REF][START_REF] Houard | Polarization analysis of terahertz radiation generated by four-wave mixing in air[END_REF][START_REF] Kim | Terahertz emission from ultrafast ionizing air in symmetry-broken laser fileds[END_REF][START_REF] Kim | Coherent control of terahertz supercontinuum generation in ultrafast laser-gas interactions[END_REF]. It was found that the optical biased at second harmonic frequency can lead to a THz amplitude enhancement by two orders of magnitude compared to the air plasma driven by just 800 nm femtosecond laser pulses [START_REF] Xie | Coherent control of THz wave generation in ambient air[END_REF][START_REF] Thomson | Broadband THz emission from gas plasma induced by femtosecond optical pulses: From fundamental to applications[END_REF]. The relative phase between the fundamental laser field and its second harmonic was found to be critical for the yield and polarity of the THz radiation [START_REF] Xie | Coherent control of THz wave generation in ambient air[END_REF][START_REF] Liu | Maker fringes in the Terahertz radiation produced by a 2-color laser filed in air[END_REF]. Recently, this method was also extended to long air plasma (L >> THz) obtained in the filamentation regime [START_REF] Wang | High energy THz generation from meter-long two-color filaments in air Las[END_REF][START_REF] Daigle | Remote THz generation from two-color filamentation: long distance dependence[END_REF]. For such long filaments, it was confirmed that the two-color scheme is more efficient than just a pulses at 800 nm. But, the temporal separation of the 800 nm and 400 nm pulses during their propagation inside the long filaments due to group delay dispersion of air sets a limit of interaction length (~ 2 m) and results in a strong THz yield decrease for standoff THz generation [START_REF] Daigle | Remote THz generation from two-color filamentation: long distance dependence[END_REF].

One advantage of the filaments-based THz source lies in the fact that it is laser damage-free due to its plasma nature, and one can expect high energy THz radiation with powerful incident femtosecond laser pulses. However, it is well known that multiple filamentation forms when the incident laser power well exceeds the critical power for self-focusing in air P cr (~ 5 GW [START_REF] Couairon | Femtosecond filamentation in transparent media[END_REF]). The underlying mechanism of multiple filamentation is the imperfection of the laser beam spatial profile as well as the turbulence in air during propagation. As a result, the process of multiple filamentation changes from shot-to-shot, which hinders the usage of multiple filaments as a stable THz source. Is there possible solution?

Inspired by the phased antenna array in the RF domain, we recently proposed and demonstrated experimentally the coherent synthesis of THz radiations of an array of organized filaments [START_REF] Mitryukovskiy | Coherent synthesis of THz fields from filament antenna array[END_REF]. We find that the THz intensity scales up with N 2 provided that proper filament separation and laser pulse time delays are chosen. Moreover, the THz radiation pattern can be controlled, which is a useful property for many applications. For example, in figure 15 we presented the THz radiation patterns obtained from two filaments separated by a distance of 2.4 mm for increasing time delay between the two pulses. Upon increase of the delay  p , the radiation pattern becomes asymmetric. With p = 3.7 ps, the radiation is totally directed along one lobe. This is particular interesting for applications where irradiation of targets with intense laser pulse should be avoided. Our simulations based on the Cherenkov-Transition model reproduced well our experimental observations. We further extrapolated the simulation to a higher number of filaments. We have calculated the THz radiation distribution for N = 4, 6, 8, 9, 12, 16 filaments organized in a square grid. With a properly optimized delay, the total THz energy can be channeled towards a preferential direction. With two filaments and zero delay, a butterfly wing pattern appears in the XOZ plane (figure 16 (a)). A single leaf radiation pattern is obtained with  p = 3.5 ps (figure 16(b)). For a larger number of filaments, similar leaf-shape radiation pattern appear with properly chosen pulse delays, as presented for 16 filaments in figure 16(c), where its peak intensity is found to be ~ 250 times that of a filament.

Control of electric discharges with laser filaments

Plasma strings produced by femtosecond laser filaments proved to be particularly useful for remote manipulation of high voltage discharges. The physical mechanism is the following [START_REF] Tzortzakis | Time-evolution of the plasma channel at the trail of a self-guided IR femtosecond laser pulse in air[END_REF][START_REF] Bodrov | Plasma filament investigation by transverse optical interferometry and terahertz scattering[END_REF]. In the presence of a static electric field, free electrons formed during filamentation release their kinetic energy in the gas by Joule heating, leading to the appearance of a depressurized channel at the center of the filament path. The resulting low density column offers a privileged path for discharge [START_REF] Vidal | Modeling the triggering of streamers in Air by ultrashort laser pulses[END_REF][START_REF] Tzortzakis | Femtosecond Laser-guided Electric Discharge in Air[END_REF]. In parallel, a fraction of the free electrons does not recombine on the parent ions but become attached to neutral oxygen molecules. Such loosely bound electrons can be easily released by current heating, leading to a decrease of the leader inception voltage [START_REF] Comtois | Triggering and Guiding of an Upward Positive Leader From a Ground Rod With an Ultrashort Laser Pulse-II: Modeling IEEE Trans[END_REF]. As a consequence, a filament decreases the breakdown voltage in a gas by more than 30% and guides the discharge over the perfectly straight path defined by the laser.

Femtosecond filament can trigger and guide megavolts discharges over several meters [START_REF] Pepin | Triggering and guiding high-voltage large-scale leader discharges with subjoule ultrashort laser pulses[END_REF][START_REF] Rodriguez | Triggering and guiding megavolt discharges by use of laser-induced ionized filaments[END_REF][START_REF] Fuji | Leader effects on femtosecond-laser-triggered discharges[END_REF], carry high DC currents with reduced losses [START_REF] Houard | High current permanent discharges in air induced by femtosecond laser filamentation App[END_REF] or deviate arcs from their natural path [START_REF] Forestier | Triggering, guiding and deviation of long air spark discharges with femtosecond laser filament[END_REF]. These properties are of great interest for applications such as the laser lightning rod [START_REF] Zhao | Use of self-trapped filaments in air to trigger lightning in Ultrafast Phenomena[END_REF], virtual plasma antennas for radiofrequency transmission, plasma aerodynamic control or high voltage switch.

High current spark gap.

A previous experiment had shown that plasma filaments were able to initiate high current DC discharge with reduced losses [START_REF] Houard | High current permanent discharges in air induced by femtosecond laser filamentation App[END_REF]. To analyse further this property we studied the triggering of atmospheric kA spark gap by laser filamentation [START_REF] Arantchouk | A simple high-voltage high current spark gap with subnanosecond jitter triggered by femtosecond laser filamentation[END_REF]. The beam (300 mJ, 700 fs at a central wavelength λ 0 = 800 nm) was focused in air with a 5 m focal length lens leading to a quasi-homogeneous plasma column with an effective length of 2 m and a diameter of ~ 3 mm. The experimental spark gap was installed in the middle of the plasma column. Figure 17 gives a schematic overview of the experimental setup. Electrodes with a central hole optimized the contact between the laser created plasma column and the high voltage. The high voltage (up to U 0 ≤ 30 kV) was applied to the spark gap set in parallel with a 50  resistance by activating a conventional switch T. It created on the spark gap electrodes a voltage pulse with rise time (10% -90%) ~ 50 ns and exponential decay  = RC ≈ 14 µs. In this case the laser triggered discharges yielded a current maximal amplitude reaching ~12.5 kA with a 30 kV charge.

An important parameter of the spark gap is the jitter, which is usually defined as the standard deviation of the switching time delay. Figure 18 presents the measured jitter for different gap lengths D as a function of U norm , the breakdown voltage normalized to the self-breakdown voltage threshold in static regime. One can see that it is possible to have a jitter of 0.2-0.4 ns in the region 0.7 < U norm < 1.2 with a 10 mm gap. The measured jitter is close to the time resolution of the oscilloscope (1 GHz and 10 GSa). Triggering via filamentation has several advantages. Since filament generated plasma column can be created at a distance of several hundreds of meters from the laser, remote triggering can be achieved. Because filaments have a near constant longitudinal electron density over several meters, they connect the two electrodes quasi-instantaneously, circumventing the complex process required in the case of an initial localized plasma, and reducing thereby the jitter. Thus, this spark gap combines the advantages of large inter-electrode air gaps capable of switching high voltages and the low jitter of narrow spark gaps or SF 6 filled cells. The long and uniform plasma channels formed by the filament also offer the possibility to trigger simultaneously several spark gaps with an excellent synchronization.

Tesla coil discharges.

Experiment on triggering and guiding of large scale (0.5 -3 m) discharges by use of laser-induced ionized filament have been performed so far with unipolar pulsed or DC high voltage sources, where the electric field direction remains constant, typically with MARX generators [START_REF] Comtois | Triggering and Guiding of an Upward Positive Leader From a Ground Rod With an Ultrashort Laser Pulse-II: Modeling IEEE Trans[END_REF]. Recently demonstration of large discharge triggering and guiding by femtosecond laser filaments using a Tesla coil generator has been reported by our group and by Henrikson et al. [START_REF] Brelet | Tesla coil discharges guided by femtosecond laser filaments in air[END_REF][START_REF] Henriksson | Laser guiding of Tesla coil high voltage discharges[END_REF].

The Tesla coil generator is basically a voltage elevator transformer, where the coupling of two resonant R L C circuits allows obtaining voltage bursts of voltage oscillations. Our system delivers voltage pulses oscillating at 100 kHz with peak amplitude ~ 350 kV. The particularity of our Tesla transformer is that it can be synchronized with an external TTL signal with a temporal jitter below 20 nanoseconds. The laser beam is focused in air producing a continuous plasma column about 2 m long, touching tangentially the two spherical electrodes connected to the Tesla output. The laser beam comes from the left.

The discharges can be triggered and guided with 100 % success; however successful operation requires a precise timing of the trigger with respect to the laser arrival time. Optimum repeatability is achieved when the laser initiates the discharge during a maximum of the oscillatory voltage at the output of the Tesla. No electric discharge is produced by the Tesla generator in the absence of laser when the electrode gap is larger than 32 cm. Guided discharges activated by the filament are obtained up to an inter-electrode distance of 200 cm corresponding to a decrease of 80 % of the breakdown voltage. With optimal laser parameters, reproducible guided discharges of 1 meter at a repetition rate of 10 Hz were demonstrated.

Measurement of voltage and discharge current allows an estimation of the plasma column resistance around 1 kΩ. This resistance appears to increase linearly with the gap length. The influence of the laser input energy on the triggering and guiding effect is also studied showing that the multifilamentation regime significantly increases the probability to trigger long discharges.

Virtual RF antenna.

An application of long laser guided electric discharge is the virtual plasma antenna [START_REF] Brelet | Radiofrequency plasma antenna generated by femtosecond laser filaments in air[END_REF]. Plasma antennas, where plasma replaces metal as the conducting element, have long been known and used [START_REF] Borg | Plasmas as antennas: Theory, experiment and applications[END_REF]. However, most designs use low-pressure plasmas confined inside solid dielectric vessels. We experimentally demonstrated a functional plasma antenna in air, which brings about many advantages like tunability in a large frequency range (100 MHz -1 GHz), stealth when de-activated and quick reconfiguration capabilities.

In this experiment we used the filament induced high-voltage electric discharge generated by the compact Tesla coil as a plasma antenna at atmospheric pressure. The guided discharge had a length of about 100 cm. Radio-frequency (RF) power was injected in the plasma by means of an inductive coupler in the form of a hollow metallic cylindrical cavity, fed by a 35 W solid-state RF amplification chain. Radio emission was then detected using a remote patch antenna with a 100 MHz -1 GHz bandwidth (see setup in figure 20). As shown in the example given in figure 21, when the coupler is excited at 990 MHz and no plasma is present, there is no signal observed at this frequency (black curve), whereas a clear emission peak appears when the laser guided discharge is generated (red curve). RF energy coupling in the plasma has consequently been achieved, which resulted in the plasma column behaving as an emitting antenna. The emission level from a copper rod with a length similar to that of the plasma is approximately equal to four times the plasma signal strength, demonstrating the applicability of this technology in real situations.

Conclusion

In summary, the physics underlying femtosecond laser filamentation has been introduced. Several examples of interaction between filaments have been described. Finally we have discussed applications of femtosecond filamentation to the generation of THz pulses in air and to the initiation and control of high voltage discharges.

Figure 1 .

 1 Figure 1. (a) Beam width of a collimated femtosecond laser pulses during filamentation propagation in air, obtained from numerical simulation. (b) The corresponding plasma density as a function of propagation distance.

Figure 2 .

 2 Figure 2. Calculated evolution of an initial converging femtosecond pulse with initial Gaussian profile upon propagation in Argon.

Figure 3 .

 3 Figure 3. Numerical calculation of the pulse evolution during filamentation in silica for a femtosecond pulse with a central wavelength at 1.6 µm (a) and at 800 nm (b).

Figure 4 .

 4 Figure 4. Side photography of the blue fluorescence of SiO2 induced by a self-guided laser pulse at 1.9 µm with P = 30 Pcr. The laser pulse is focused on a 70 µm spot (FWHM) at the left surface of the 3 cm thick sample.

Figure 5 .

 5 Figure 5. Beam profile of the laser pulse at 1.9 µm (a) with filamentation and (b) without filamentation in the fused silica sample. Intensity scale is normalized.

Figure 6 .

 6 Figure 6. Measurement of the 1.9 µm pulse temporal shape after propagation in the sample performed with a WIZZLER.

Figure 7 .

 7 Figure 7. Normalized intensity of diffracted probe signal as a function of the delay  (ps) for crossing angles  of 7°, 14° and 90° in different gases (Air, CO 2 , Ar and Kr). Measurements are represented by dots. Calculations are represented by solid lines.

Figure 10 .

 10 Figure 10. Fluence profile of a femtosecond pulse before (a) and after (b) reflection on the filament mirror, as recorded in a single shot with a CCD camera.

Figure 11 .

 11 Figure 11. Reflectivity as a function of the pump pulses energy. Probe pulse has energy of 160 and 340 µJ and is incident at an angle of 90°. The three pulses are coincident in time.

Figure 12 .

 12 Figure 12. Temporal contrast of the probe (in black) and of the reflected beam (red) measured with a third order cross correlator.

Figure 13 .

 13 Figure 13. Amplification of the transition-Cherenkov THz emission. The pure transition-Cherenkov THz (E ext = 0 kV/cm) is multiplied by a factor of 200 for visibility.

Figure 14 .

 14 Figure 14. THz Radiation pattern from a laser filament in the presence of increasing transverse electric field. In the top (bottom) row, the external electric field is directed downward (upward).

Figure 15 .

 15 Figure 15. Evolution of emission diagrams of the THz radiation from two neighbouring filaments as a function of the temporal delay between the two laser pulses. The two filaments are separated by d = 2.4 mm. Top row: experimental results. Bottom row: calculations.

Figure 16 .

 16 Figure 16. Calculated THz radiation diagram from two parallel filaments separated by 2.4 mm without time delay (a) and with a time delay  p = 3.5 ps (b). (c) Calculated THz radiation diagram from 16 filaments with proper time delays.

Figure 17 .

 17 Figure 17. Schematic view of the experimental set-up with a pulsed voltage applied to the axial gap switch. The laser beam passes through the axial holes in the electrodes.

Figure 18 .

 18 Figure 18. Jitter of the switch as a function of normalized voltage for gaps of 6, 8 and 10 mm.

Figure 19 .

 19 Figure 19. Photography of a Tesla guided discharge of 1.7 m.The laser beam comes from the left.

Figure 20 .

 20 Figure 20. Experimental setup used for RF coupling in the plasma.

Figure 21 .

 21 Figure 21. RF signal strength with (red) or without (black) the plasma column when the coupler is excited at 990 MHz. The insert corresponds to a zoom of the highlighted region around 990 MHz.

Acknowledgments

Authors wish to acknowledge contributions from A. Couairon for the numerical studies and from M. Durand, Y. Brelet, B. Prade, A. Jarnac, J. Carbonnel, S. Mitryukovskiy, Y. B. André, G. Point, B. Forestier and L. Arantchouk for experimental measurements. This work has been partially supported by a grant from the French DGA (grant EPAT n° 200795091), by ANR (grant n° ANR-2010-JCJC-0401-01) and by LASERLAB-EUROPE (grant agreement n° 284464, EC's Seventh Framework Programme).


Sergey Mitryukovskiy

Yi Liu
email: yi.liu@ensta-paristech.fr

Pengji Ding

Aurélien Houard

André Mysyrowicz
email: andre.mysyrowicz@ensta-paristech.fr

Backward stimulated radiation from filaments in Nitrogen gas and air pumped by circularly polarized 800 nm femtosecond laser pulses

Keywords:

We report on strong backward stimulated emission at 337 nm in Nitrogen gas pumped by circularly polarized femtosecond laser pulses at 800 nm. A distinct dependence of the backward UV spectrum on pump laser polarization and intensity is observed, pointing to the occurrence of backward amplified spontaneous emission inside filaments. We attribute the population inversion to inelastic collision between the free electrons produced by the pump laser and neutral N 2 molecules. The addition of Oxygen molecules is detrimental for the gain, reducing it to near threshold at atmospheric concentration.

Introduction

Ambient air pumped by intense ultrashort laser pulses can give birth to stimulated radiation in both the forward and backward directions [START_REF] Dogariu | High-gain backward lasing in air[END_REF][START_REF] Kartashov | Free-space Nitrogen gas laser driven by a femtosecond filament[END_REF][START_REF] Hemmer | Standoff spectroscopy via remote generation of a backward-propagating laser beam[END_REF][START_REF] Kocharovsky | Gain-swept superradiance applied to the stand-off detection of trace impurities in the atmosphere[END_REF][START_REF] Luo | Lasing action in air driven by ultra-fast laser filamentation[END_REF][START_REF] Sprangle | Remotely induced atmospheric lasing[END_REF][START_REF] Sprangle | Remote lasing in air by recombination and electron impact excitation of molecule nitrogen[END_REF][START_REF] Owada | Amplified spontaneous emission and rotational and vibrational state distributions in state of N2 in femtosecond laser induced filaments in air[END_REF][START_REF] Yao | High-brightness switchable multiwavelength remote laser in air[END_REF][START_REF] Yao | Remote creation of coherent emissions in air with two-color ultrafast laser pulses[END_REF][START_REF] Zhang | Abnormal dependence of strong-field-ionization-induced nitrogen lasing on polarization ellipticity of the driving field[END_REF][START_REF] Liu | Self-seeded lasing action of air pumped by 800 nm femtosecond laser pulses[END_REF][START_REF] Chu | Self-induced white-light seeding laser in a femtosecond laser filament[END_REF][START_REF] Wang | Self-seeded forward lasing action from a femtosecond Ti:Sapphire laser filament in air[END_REF][START_REF] Point | Lasing of ambient air with microjoule pulse energy pumped by a multi terawatt IP femtosecond laser[END_REF]. The backward stimulated radiation (propagating in the direction opposite to the pump laser) is of particular interest for remote sensing, and has therefore attracted considerable attention [START_REF] Dogariu | High-gain backward lasing in air[END_REF][START_REF] Kartashov | Free-space Nitrogen gas laser driven by a femtosecond filament[END_REF][START_REF] Hemmer | Standoff spectroscopy via remote generation of a backward-propagating laser beam[END_REF][START_REF] Kocharovsky | Gain-swept superradiance applied to the stand-off detection of trace impurities in the atmosphere[END_REF][START_REF] Luo | Lasing action in air driven by ultra-fast laser filamentation[END_REF][START_REF] Sprangle | Remotely induced atmospheric lasing[END_REF][START_REF] Sprangle | Remote lasing in air by recombination and electron impact excitation of molecule nitrogen[END_REF][START_REF] Owada | Amplified spontaneous emission and rotational and vibrational state distributions in state of N2 in femtosecond laser induced filaments in air[END_REF]. The advantage of the backward stimulated radiation for remote sensing lies in the fact that it can bring information about pollutants towards the ground observer with a well defined directionality. This has to be compared to the omni-directionality of the fluorescence or scattered optical signal from the same pollutants. In addition, if the information is carried to the ground observer via coherent Stokes-(SRS) or anti-stokes Raman scattering (CARS), the signal is proportional to N², as compared to the N dependence of the fluorescence or spontaneous scattering process. Here N is the number of pollutant molecules.

Up to now, two different schemes for backward stimulated radiation in air have been demonstrated experimentally, based on population inversion either of O atoms or N 2 molecules. In the first scheme, inversion in the population of Oxygen atoms was obtained with an intense UV pulses at 266 nm. The UV pulse served two functions, first to photo-dissociate Oxygen molecules, then to pump the Oxygen atoms to the state via two-photon excitation [START_REF] Dogariu | High-gain backward lasing in air[END_REF]. Strong stimulated emission at 845 nm, corresponding to the transition, was observed in both the backward and forward direction. A serious limitation of this scheme is the significant absorption of 226 nm light by atmosphere, preventing its use for remote pollutant sensing.

The second scheme deals with population inversion in neutral N 2 molecules. About 10 years ago, Q. Luo and coworkers announced the existence of a backward amplified spontaneous emission (ASE) from femtosecond laser filaments in air with linearly polarized 800 nm laser pulses. The evidence was based on the observation of a weak exponential increase of the backward UV luminescence at 357 nm upon filament length [START_REF] Luo | Lasing action in air driven by ultra-fast laser filamentation[END_REF]. The UV emission corresponded to a transition between the ( = 0) and the ( =1) triplet states of the N 2 molecules, where  is the vibrational quantum number. Similar results have been confirmed recently by S. Owada and coworkers [START_REF] Owada | Amplified spontaneous emission and rotational and vibrational state distributions in state of N2 in femtosecond laser induced filaments in air[END_REF]. In 2012 D. Kartashov et al. reported a strong backward ASE lasing from N 2 molecules. A high power mid-infrared (3.9 m or 1.06 m) femtosecond pulses was used to induce a filament plasma column in a high pressure mixture of Nitrogen and Argon gas. Microjoule amounts of backward ASE from the plasma column were observed with a well-defined spatial profile [START_REF] Kartashov | Free-space Nitrogen gas laser driven by a femtosecond filament[END_REF]. In this experiment, the population inversion mechanism was the traditional Bennet scheme, where collisions transfer the excitation energy of Argon atoms to molecular Nitrogen. Unfortunately, this scheme cannot be employed for remote sensing application in atmospheric air, because of its requirement of high pressure Argon gas (p > 3 bar).

In this paper, we demonstrate that an intense backward stimulated radiation from filaments in Nitrogen gas can be achieved with circularly polarized femtosecond laser pulses at 800 nm. Filaments offer a favorable geometry for ASE because of the high aspect ratio of the plasma strings at the origin of the emission. Existence of the stimulated radiation is confirmed by the distinct dependence of the backward UV spectrum on the incident laser polarization and intensity. We attribute the population inversion between the triplet and states to inelastic collision between the electrons liberated by the pump laser and surrounding neutral ground state N 2 molecules. The dependence of the lasing effect on the incident laser pulse polarization is explained by the fact that a circularly polarized laser produces more energetic electrons than a linearly polarized one. We find that the presence of Oxygen molecules results in a significant quenching of the lasing action. With our present experimental parameters, significant gain occurs up to an Oxygen concentration of ~ 12%.

Experimental setup

In our experiments, femtosecond laser pulses with duration of 50 fs were focused by a convex lens of 1000 mm in a gas chamber filled with 1 bar of pure Nitrogen gas or a mixture of Nitrogen and Oxygen. A broadband dielectric beam splitter was used to steer the incident pulses into the gas chamber, while transmitting the backward UV emission from the gas plasma to the detector. A quarter-wave plate was inserted before the incident windows of the gas chamber to change the laser polarization between linear and circular. The backward emission was focused by a f = 100 mm fused silica lens to the slit of a monochromator (Jobin-Yvon H-20 UV, grating: 1200 g/mm) combined with a photomultiplier tube (PMT). Typically, an average over 500 laser shots was performed for each individual measurement. The transverse fluorescence from the plasma channel was also measured. In this case a vertical slit of 1 mm was installed close to the center of the filament in order to limit the investigated plasma volume. The fluorescence transmitted through the slit was first collimated by a f = 2.5 mm fused silica lens and then focused by another f = 100 mm fused silica lens to the incident slit of the same monochromator and PMT (see Fig. 1).

Experimental results

Spontaneous UV emission from filaments in Nitrogen gas

We first examine the spontaneous UV fluorescence emitted by filaments in a pure Nitrogen gas. The fluorescence spectra recorded in the transverse detection geometry are shown in Fig. 2(a) for linearly and circularly polarized pump laser pulses. The emission peaks centered at 315, 337, 357, 380, 405 nm have been previously identified as due to transitions between various vibronic levels of the triplet and states of the neutral N 2 molecule, i.e. the second positive band of the N 2 molecules [START_REF] Kartashov | Free-space Nitrogen gas laser driven by a femtosecond filament[END_REF]. For all these five spectral lines, the signals are ~ 2 times stronger with circularly polarized laser than with linear laser polarization. A possible explanation will be discussed below. Before discussing the backward emission at 337 nm with circularly polarized laser, we first examine in more details the results obtained with linear laser polarization in the backward direction, a geometry where ASE was reported previously [START_REF] Luo | Lasing action in air driven by ultra-fast laser filamentation[END_REF][START_REF] Owada | Amplified spontaneous emission and rotational and vibrational state distributions in state of N2 in femtosecond laser induced filaments in air[END_REF]. In Fig. 3(a), we present the measured backward 337 nm signal as a function of the incident laser pulse energy up to 9.3mJ. We also measured the length and the width of the plasma channel (defined at 1/e 2 level of the plasma luminescence intensity) as a function of laser pulse energy (see Fig. 3(b)). With an increase of the laser energy, the filament length increases linearly, in agreement with a recent observation [START_REF] Owada | Amplified spontaneous emission and rotational and vibrational state distributions in state of N2 in femtosecond laser induced filaments in air[END_REF].

In Fig. 4, we plot the 337 nm signal as a function of the filament length, as it was previously done for identification of ASE [START_REF] Luo | Lasing action in air driven by ultra-fast laser filamentation[END_REF][START_REF] Owada | Amplified spontaneous emission and rotational and vibrational state distributions in state of N2 in femtosecond laser induced filaments in air[END_REF]. A weak nonlinear dependence is observed, which is tentatively fitted with an exponential law. We note that a more satisfying procedure consists in plotting the signal as a function of plasma volume, in view of the fact that the filament size increases slightly with pump laser energy (see Fig. 3(b)). A plot of the backward signal as a function of plasma volume is best fitted by a linear dependence, as shown in Fig. 4. Therefore, we come to the conclusion that stimulated radiation at 337 nm is not achieved with linearly polarized femtosecond pulses in our experiments. We also performed measurements at 357 nm and observed a similar dependence.

Stimulated UV emission from filaments in Nitrogen gas

We now concentrate on the backward emission at 337 nm obtained with circularly polarized femtosecond laser pulses. It corresponds to the (0-0) vibronic transition of the second positive band system of N 2 molecule. In Fig. 2(b), the spectra of the backward UV emission are shown for circular and linear polarization of the laser. The emission intensity at 337 nm is now about 10 times larger with circular pump laser polarization than with linear polarization. Others lines at 315, 357, 380, and 405 nm increase by a factor of ~1.5, an increase similar to that of the spontaneous fluorescence presented in Fig. 2(a). The remarkable behavior of the 337 nm signal suggests that backward stimulated emission is initiated for this particular line with circularly polarized laser pulses.

In Fig. 5, the backward emission intensity at 337 nm is plotted as a function of incident laser energy. The 337 nm signal displays a superlinear dependence on incident laser energy.

To get further insight into the distinct pump polarization dependence presented in Fig. 5, we measured the backward radiation intensity at 337 nm as a function of the incident laser ellipticity. In Fig. 6, the measured signals are presented as a function of the rotation angle of the quarter-wave plate for different incident laser energies. The angles φ = 90º × mcorrespond to linearly polarized laser, with m = 0, 1, 2, 3. The angles φ = 45º + 90º × m correspond to circularly polarized laser. For a low pulse energy of 300 J, linearly polarized pulses generates a UV radiation with an intensity twice that obtained with circular polarization (Fig. 6(a)). For an increased incident energy of 660 J, an octagon-shaped dependence is observed (Fig. 6(b)), indicating the onset of a new mechanism for the emission at 337 nm. In the case of E in = 9.3 mJ, the signal obtained with circular laser polarization totally dominates and decreases rapidly when the laser polarization deviates slightly from circular. This critical dependence of the emission signal at 337 nm with laser polarization reinforces the hypothesis that backward stimulated emission at 337 nm occurs inside the filament plasma.

Discussion of results

We now discuss possible mechanisms for population inversion between the and states of the neutral N 2 molecule, at the origin of the backward stimulated radiation. First, it is worth reminding that a direct population transition between the ground singlet state and the excited triplet state is forbidden in the electric dipole approximation. One widely discussed mechanism to populate state inside filaments in air is the following reaction [START_REF] Xu | The mechanism of Nitrogen fluorescence inside a femtosecond laser filament in air[END_REF]:

. With this mechanism, one expects that a linearly polarized laser produces a stronger fluorescence signal for two reasons. First, it is known that the ionization rate of atoms and molecules is higher for linear polarization both in the multiple photon ionization and tunneling ionization regime [START_REF] Kolesik | Polarization dynamic of femtosecond pulses propagating in air[END_REF]. As a result, the densities of electrons and positive ions are higher in the case of linear laser polarization and should result in a higher density of . Second, the nonlinear refractive index of ambient air n 2 is higher for linear polarized laser [START_REF] Kolesik | Polarization dynamic of femtosecond pulses propagating in air[END_REF]. This results in slightly higher laser intensity inside filaments, which should again lead to higher densities of electrons and ions. We therefore assume that this collision-assisted recombination process of the electron on the parent ion is responsible for the fluorescence of filaments at relatively low laser energy, such as that of E in = 300 J in Fig. 6(a), but not to the stimulated emission observed at higher pump powers.

Another mechanism for the transition from the ground state to the state is the electron-molecule inelastic collision: . This is actually the main reaction responsible for population inversion in a traditional N 2 laser, where the electrons are accelerated to obtain sufficient energy by the discharge electric field [START_REF] Kunabenchi | Nitrogen lasers[END_REF]. The cross section of the above reaction is sensitive to the kinetic energy of the incident electron. It is nearly zero for electron energy below the threshold kinetic energy E th ~ 10 eV, exhibits a resonant peak around 14.1 eV, and then decreases progressively for higher energy electrons [START_REF] Fons | Electron-impact excitation of the second positive band system and the electronic state of the Nitrogen molecule[END_REF]. For electrons born in the intense laser field inside filaments, the distribution of kinetic energies depends strongly upon pump laser polarization. With linear polarization, free electrons are accelerated back and forth along the laser polarization direction at each optical cycle, so that at the end of the pulse most of them have a low kinetic energy, with a weak tail distribution extending to a few eV. For circularly polarized light, where electrons are always accelerated away from the ion core, the final electron distribution is nearly mono-kinetic with a peak at twice the ponderomotive energy. Calculations performed for a filament show that the peak of this distribution is between 5 and 7 eV assuming a laser intensity of 5×10 13 W/cm² and should scale up linearly with laser intensity [START_REF] Zhou | Measurement and control of plasma oscillations in femtosecond filaments[END_REF]. The widely quoted clamped intensity value in air filaments of 5×10 13 W/cm² is therefore insufficient by a factor 2 to be an effective excitation mechanism. On the other hand, it is now well established that intensity clamping is not a rigorous concept. Intensity spikes exceeding the clamped value by order of magnitude have been measured in filaments [START_REF] Schulz | Tracking spectral shapes and temporal dynamics along a femtosecond filament[END_REF].

In our current experiment, we estimated the laser intensity inside the filament by inserting an Al foil around the middle of the filament in the case of E in = 9.3 mJ, in ambient air. After several laser shots, a micrometric hole was drilled by the filament itself [START_REF] Mitryukovskiy | A simple method to estimate the laser intensity in air filaments[END_REF]. After 300 laser shots the transverse size of the hole became stable and its diameter (d) was measured to be 170 m. At the same time, the transmitted laser energy E t was measured to be about 6.6 mJ. The average laser intensity can be estimated as:

1.45× 10 14 W/cm 2 , where  p is the initial laser pulse duration of 50 fs. This indicates that the electron energy required for impact excitation of neutral N 2 molecules is well reached in filaments. More detailed measurements of the intensity inside filaments will be published elsewhere [START_REF] Mitryukovskiy | A simple method to estimate the laser intensity in air filaments[END_REF]. For linearly polarized laser field, this mechanism is turned off because of the low electron kinetic energy, even with a laser intensity above the clamped value. As a conclusion, we believe that the inversion of population inside the triplet manifold of N 2 is due to impact excitation by electron collisions. The threshold character for the inelastic collision cross section with respect to electron energy is reflected in the sharp dependence of the lasing emission with ellipticity [START_REF] Fons | Electron-impact excitation of the second positive band system and the electronic state of the Nitrogen molecule[END_REF].

Backward emission from filaments in ambient air

For applications of this backward stimulated radiation source for remote sensing, operation in ambient air is required. We therefore performed measurements in atmospheric air. We first examined the backward UV emission from filaments formed in ambient air driven by both circularly and linearly polarized laser pulses. Emission spectra similar to those of Fig. 2(b) were observed, except that the signal at 337 nm increased only by a factor 2 for circular laser polarization. In Fig. 7, the measured backward 337 nm signal is shown as a function of the rotation angle of the quarter-wave plate. To further assess the influence of Oxygen gas for the lasing action, we measured the backward 337 nm signal in different mixtures of pure Nitrogen and Oxygen. We present in Fig. 8 the measured 337 nm signal for both circularly and linearly polarized laser as a function of the percentage of Oxygen. A slow decrease of the signal is observed upon increasing Oxygen concentration up to 10 %. Beyond 10 % Oxygen, the signal shows a rapid decrease, indicating the termination of significant lasing action. For Oxygen concentration more than 13%, the signal obtained with circular laser polarization becomes ~ 2 times larger than that of linear laser polarization, similar to the transverse fluorescence presented in Fig. 2(a).

The detrimental influence of Oxygen molecule for the conventional discharge-pumped Nitrogen laser is well documented [START_REF] Kunabenchi | Nitrogen lasers[END_REF]. The underlying physical mechanism is attributed to the collision reaction , which efficiently reduces the population density in the upper state of the lasing emission. To obtain a significant backward stimulated emission in atmospheric air, a higher population inversion density (or a higher gain) is required to overcome the quenching effect of the Oxygen molecules. A possible approach to achieve this is to use a pump laser at longer wavelengths, because the kinetic energy of the produced electrons increases like I², where I is the laser intensity and  its wavelength.

Conclusion

In summary, we have shown that a strong stimulated radiation at 337 nm can be achieved in the backward direction from filament plasma in a Nitrogen gas pumped by a circularly polarized laser pulse at 800 nm. This stimulated radiation shows distinct dependence on laser pulse energy, compared to that obtained with linear polarized laser pulses. In a mixture of Nitrogen gas and Oxygen, the presence of Oxygen molecules suppresses the lasing action to a large extend. As to the mechanism responsible for population inversion, attribute it to inelastic collisions between electrons liberated by the pump laser and neutral Nitrogen molecules, a process which is more efficient with circularly polarized laser pulses. We believe that this simple scheme for backward stimulated emission from Nitrogen gas pumped by the widely available 800 nm femtosecond laser pulse is a significant step towards applications for remote sensing.

Fig. 1 .

 1 Fig. 1. Schematic experimental setup. The incident femtosecond pulse is focused by an f = 1000 mm lens. The dichromatic beam splitter (BS) steers the 800 nm pulses inside the gas cell filled with Nitrogen gas or its mixture with air. The backward radiation from the plasma is collected by a f = 100 mm fused silica lens into the slit of a monochromator. The signal is measured with a photomultiplier tube (PMT).

Fig. 2 .

 2 Fig. 2. Spectra of (a) transverse fluorescence and (b) backward UV emission for circular and linear laser polarization. The incident laser pulse energy is 9.3 mJ.

Fig. 3 .

 3 Fig. 3. (a) Backward 337 nm signal as a function of incident laser pulse energy. (b) Measured length and width of plasma as a function of pump laser energy. In both cases the pump laser is linearly polarized.

Fig. 4 .

 4 Fig. 4. Backward 337 nm signal as a function of plasma length (black squares, lower scale) and plasma volume (red circles, upper scale). The curves are best fitted with an exponential or linear law respectively.

Fig. 5 .

 5 Fig. 5. Measured backward signal at 337 nm as a function of incident laser pulse energy, for both circular and linear laser polarization.

Fig. 6 .

 6 Fig. 6. Backward emission signal at 337 nm as a function of the rotation angle of the quarterwave plate. The incident laser energy for (a), (b), (c) are 0.3 mJ, 0.6 mJ, and 9.3 mJ, respectively. Angle 0º corresponds to linearly polarized light.

Fig. 7 .

 7 Fig. 7. Dependence of the backward 337 nm signal obtained from filaments in air as a function of the rotation angle of the quarter-wave plate. The incident laser energy is 9.3 mJ and the focal length is f = 1000 mm.

Fig. 8 .

 8 Fig. 8. Measured backward 337 nm signal from filaments in air as a function of Oxygen gas concentration, for both circular and linear polarized laser pulses. The incident pulse energy is 9.3 mJ.

Acknowledgments

We gratefully acknowledge useful discussions with Paul Corkum of Ottawa University.


Guillaume Point
email: guillaume.point@ensta-paristech.fr

Yohann Brelet

Aurélien Houard

Vytautas Jukna

Carles Milián

Jérôme Carbonnel

Yi Liu

Arnaud Couairon

André Mysyrowicz
email: andre.mysyrowicz@ensta-paristech.fr

Superfilamentation in air

Keywords: numbers: 42.65.Jx, 42.65.Sf, 52.70.Kz

The interaction between a large number of laser filaments brought together using weak external focusing leads to the emergence of few filamentary structures reminiscent of standard filaments, but carrying a higher intensity. The resulting plasma is measured to be one order of magnitude denser than for short-scale filaments. This new propagation regime is dubbed superfilamentation. Numerical simulations of a nonlinear envelope equation provide good agreement with experiments.

Introduction.-Much attention has been given to the study of propagation of short intense laser pulses through transparent media. When the peak power of such pulses exceeds a particular critical power P cr (P cr ≡ 3.72λo 8πnon2 ∼ 5 GW in air at λ o = 800 nm with n o and n 2 respectively the linear and Kerr optical indexes), filamentation sets in. In this case, the Kerr effect prevails and leads to beam self-focusing. The resulting collapse is stopped by the combination of multiphoton absorption by nitrogen and oxygen molecules in air and the defocusing effect induced by the resulting plasma. A dynamic competition between these opposite trends is then established, and the beam is able to maintain a very high intensity over long distances without significant defocusing [1]. When P > P cr , total beam collapse occurs, with the generation of a single filament. For P ≫ P cr , the laser pulse unavoidably breaks down in many filaments. This short-scale filamentation [2] is initiated by intensity fluctuations in the beam profile that are amplified by modulational instability, leading to local self-focusing of sub-beams instead of a total beam collapse [3][4][5].

So far most studies have focused on the characterization of single filaments (P P cr ) or the interaction between pairs of filaments. Such binary interaction leads to the fusion, repulsion, energy redistribution or spiral motion of the filaments, depending on their relative optical phase [6][7][8]. There is also a large cumulative nonlinear phase shift acquired during these processes so that any initial phase correlation between the filaments is lost [9].

In this Letter, we experimentally characterize a dense filament bundle which can no longer be described by binary interactions. This regime is obtained by weakly focusing a terawatt femtosecond laser pulse in air. In the focal region, a reorganization of the interacting filament bundle takes place with the emergence of a few filamentary structures reminiscent of standard filaments, but carrying a significantly higher intensity and persisting over tens of centimeters. The resulting plasma is measured to be one order of magnitude denser than for shortscale filaments. Simulations solving a non-linear envelope equation [10] reproduce well these features. This effect is measured to be stable in time with a high shot-to-shot reproducibility. The occurrence of such stable events contrasts with other extreme and unpredictable optical phenomena arising from nonlinear wave-matter interactions, so called rogue waves, which have been recently reported [11,12]. This may have impact on the recently discovered laser action in air induced by filaments, where the plasma strings form the active medium [13][14][15], but also on the generation of long-lived virtual optical waveguides in the atmosphere [16].

Experimental results.-The experimentally studied multifilament bundle was generated by focusing a 200 mJ, 30 mm full width at half maximum (FWHM) pulse at 800 nm from the Ti:Sapphire ENSTAmobile laser chain using a 5 m fused silica converging lens (numerical aperture NA = 3 × 10 -3 ). The pulse duration was adjusted to 175 fs in order to avoid white light emission inside the lens. This corresponds to a peak power of about 230 P cr . The experiments took place in an airconditioned room with standard temperature and pressure conditions. In order to record the formation of filaments during the propagation of the laser pulse, we used a method based on the single-shot darkening of photographic plates [17]. Once the propagating laser pulse reaches the intensity required for high-field ionization of air, characteristic circular burns become apparent on the photographic plates (cf. figure 1-(a)), signaling the presence of plasma strings. We recorded such imprints every 2 cm along the beam propagation direction (taken as the z axis, the origin being at the position of the lens), allowing the 3D-reconstruction of the beam undergoing multifilamentation. A longitudinal projection of this reconstruction in the (yz, x = 0) plane is given in figure 1-(b).

The first filaments, displayed in black in the figure, appear around z = 2.5 m. They multiply and converge under the focusing action of the lens, eventually merging together around z = 4 m. After the geometric focus (z = 5 m), they start disappearing with only few of them left at z = 6.5 m. However, this method of detection cannot resolve any structure around the geometrical focus because the photographic paper is completely saturated.

To address this problem, we investigated the luminescence emitted in this compressed plasma region. Side pictures of the plasma luminescence in the focal region were taken using a standard camera (Nikon D50 fitted with a AF-5 Micro NIKKOR 60 mm objective). As seen in figure 2-(a), these images reveal a long and thin intense string embedded in a millimeter-wide fluorescent channel. This string maintains a constant diameter over at least several centimeters. To study more quantitatively these intense strings, a time-gated intensified CCD camera (PI-MAX from Princeton Instruments fitted with a 4× microscope objective and a BG39 colored filter) was used to take single shot, transverse pictures of the multifilament bundle in the focal zone. The spatial resolution of the pictures is 5 µm and the field of view is about 1 mm. The transverse FWHM of these inner plasma structures is the same as that of the single filaments recorded at z = 4.4 m (cf. figure 2-(b)).

We can go further and use a spectroscopic analysis to characterize the plasma strings in the focal region. The corresponding luminescence spectrum was recorded between 330 and 800 nm using a monochromator (model H-20 UV, Jobin Yvon) and a photomultiplier (model 56 UVP). We averaged all the data over more than 200 laser shots per point. The measured spectrum is similar to the one observed in the case of a single filament, consisting mainly of molecular nitrogen lines, with almost all the signal lying between 330 and 450 nm [18,19]. No nitrogen or oxygen atomic emission line was detected. It has been shown that, for a single filament, the luminescence signal is directly proportional to the total number of free electrons in the plasma [19]. We can therefore use the luminescence signal as a diagnostic to evaluate the electron number in the focal region. Evaluating the plasma transverse cross-sectional area S(z) along z therefore allows us to estimate the average electron density n e in the plasma following this relation:

n e (z) ∝ L(z)/S(z), (1)
where L(z) is the average recorded plasma luminescence. This is done by correlating the darkening of the previously used photographic plates to the occurrence of ionization [17]. We can define three different regions from this last graph:

• the first one, up to z = 4.5 m, is characterized by an almost constant electron density, which can be attributed to the intensity clamping inside shortscale filaments [20,21]. At the end of this region, filaments are still discernible on the photographic plates, with about 45 being recorded.

• the second zone sees the quick rise of n e , reaching a maximum value around z = 4.9 m. This position lies within 5% of the full-beam collapse distance given by the Marburger formula in the case of a Gaussian pulse [1]. The maximum plasma density culminates at about 18 times the plateau level of the previous region.

• once z = 5 m is reached, the L/S ratio quickly falls, marking the end of the dense plasma region. This results in a 50 cm long region where the mean plasma density and intensity are significantly higher than the mean values encountered in short-scale filamentation. This length corresponds to about 6 linear Rayleigh lengths, which we measured using the same laser at very low intensity.

This estimation, however, only deals with an average value of the electron density in the plasma. While this method gives insight about the relative evolution of the plasma along z, it lacks a reference with which we could estimate the absolute value of the plasma density. On the other hand, the electron density at the core of standard filaments is well known to be on the order of 10 22 m -3 [22]. To compare the peak values of the electron density between the dense plasma strings and the short-scale filaments, iCCD pictures displayed in figure 2-(b) can be used. Figure 3 shows that filaments seen at z = 4.4 m are still standard clamped filaments, the averaged electron density being at the plateau level characteristic of short-scale filaments. Thus, we have n e,peak (z = 4.4 m) ∼ 10 22 m -3 . Because the dense strings seen around z = 4.9 m have the same FWHM as the short-scale filaments, the ratio of their electron density is directly given by that of their corresponding luminescence levels. This gives a maximum electron density in the focal zone lying between 6 and 11 times the clamping density, that is one order of magnitude higher than this plateau. Thus, the peak density encountered along z is on the order of 10 23 m -3 .

Simulations.-We model the time-harmonic propagation of the electric field envelope E(x, y, z) in air (n 0 ≈ 1) by means of a unidirectional beam propagation equation accounting for diffraction, optical Kerr effect, multiphoton absorption, plasma absorption and defocusing (see, e. g., [10] for details):

∂E ∂z = ic ∂ 2 x + ∂ 2 y 2ω 0 E + i ω 0 c n 2 |E| 2 E - β 8 2 |E| 14 + σn e 2 [1 + iω 0 τ c ] E. (2)
Here ω 0 = 2πc/λ 0 is the carrier frequency, n 2 = 2 × 10 -19 cm 2 • W -1 , β 8 = 8 × 10 -98 cm 13 • W -7 , σ = 5.6 × 10 -20 cm 2 , and τ c ≈ 350 fs. Equation ( 2) was solved numerically by means of a split-step technique [START_REF]Nonlinear Fiber Optics[END_REF]. A first step is performed in Fourier (k x , k y ) space to account only for diffraction. A second step, performed in direct (x, y) space, accounts for amplitude and phase changes due to nonlinear terms. Because of the absence of temporal dynamics in our modeling, the presence of plasma is accounted for by mapping the electric field intensity to the electron-plasma density generated by optical field ionization and avalanche at the temporal center of a 175 fs (pre-chirped) Gaussian pulse with peak intensity I. Simulations are initialized with noise levels of 10% in intensity and 0.2% in phase, mimicking experimental input beam irregularities, triggering modulational instability and the subsequent multifilamentation. The spatial resolution was 6.2 µm in the x and y directions, and 2.5 mm in the z direction, which is sufficient to resolve short scale filamentary structures in a reasonable computational time.

The simulation results are presented in figure 4. Figure 4-(a) shows the same intensity side-view as in figure 1-(b). A logarithmic color scale was chosen to mimic the response of the photographic paper. Even without taking a saturation effect into account, simulation and experiment show good agreement with respect to the beam shape, waist at the focus and the observation of only few filaments past z = 6 m.

Figures 4-(b) and (c) give respectively the simulated maximum intensity and the maximum electron density encountered in the transverse xy plane at a given z position. Both values are very low before the onset of the first filaments, which appear around z = 2.5 m. These short-scale filaments present the typical intensity and plasma density clamping values reported in the literature (I ∼ 4.5 × 10 13 W • cm -2 , n e ∼ 2.5 × 10 22 m -3 ) [20][21][22]. From z = 3.5 m, peak intensity and electron density start to rise slowly, eventually reaching maximum values of I = 6.2 × 10 13 W • cm -2 and n e = 4 × 10 23 m -3 at z = 4.85 m. These results correspond respectively to 1.4 and 16 times the clamping values. The simulated data is thus in excellent agreement with the experiment, both qualitatively and quantitatively, and confirm the existence of a meter-long zone where filaments result in significantly denser plasma channels. Numerical computations also enable us to estimate the peak intensity encountered along z, which would otherwise be extremely difficult to measure.

Discussion.-Two main features characterize the studied multifilament bundle: first, we observed a meter-sized region along the beam propagation, shortly before the geometrical focus of the lens, where the plasma density generated by the pulse was significantly higher than the usual clamping value. Secondly, we showed that the same region has a strongly inhomogeneous structure with a few strings, about the size of a standard filament, carry-ing much of the electron density. These inner structures, which are formed over a meter-long distance, maintain a uniform density and radius over at least several centimeters. We call them superfilaments because of the properties they share with standard filaments.

All these features are well reproduced by simulations. The code solves a nonlinear envelope equation, which is a widely used tool to describe known filamentation regimes [10]. Numerical computations show that intensity rises moderately above the clamping value for standard filaments in the same area. To date, a similar "overclamping" behavior has only been observed in very localized regions either in strong focusing conditions [START_REF] Théberge | [END_REF]25] or during refocusing cycles of the beam [26,27].

The sustaining of filaments with a significant higher plasma density compared to short-scale filaments can be explained by two facts. First, energy losses due to multiphoton absorption are insufficient to balance the energy input due to the focusing action of the lens. As a consequence, intensity and plasma density rise. Second, the defocusing effect of the plasma in the focal region is reduced. Indeed this effect is directly linked to the local electron density gradient. Bringing a large number of filaments very close then smooths the transverse plasma density profile, and therefore results in a weaker defocusing.

The occurrence of superfilaments thus appears strongly correlated to the external focusing action induced by the lens, as evidenced by their sudden disappearance after the geometric focus. However, the moderate beam numerical aperture is also a very important factor leading to this propagation regime. If it is too low, as it is in the case of a collimated beam, then only short-scale filaments are observed, and excess energy is carried by the energy bath [28]. On the other hand, if it is too high (NA > 0.1), a very high intensity can be reached, resulting in the complete ionization of air [29]. However, the length of the dense area is limited in this case to a millimeter-scale.

Superfilaments show great promise for the generation of a strong lasing effect from femtosecond filaments, in which the gain medium consists of the filamentation plasma. Up to now, such emission has been reported by several research groups, both in the foward [15,[30][31][32] and backward direction [13,14]. Although the underlying mechanisms for population inversion have not been fully elucidated yet, the emission characteristics appear strongly correlated to the value of plasma parameters. The description of a new filamentation regime leading to the formation of dense plasma structures is therefore of the utmost importance. This could pave the way for the development of a strong, coherent, remotely generated ultraviolet light source for long-range probing.

Conclusion.-To conclude, we proved that the global interaction of a dense multifilament bundle leads to the emergence of a new filamentation regime. This regime is characterized by the emergence of a few plasma strings over a meter-long region. These strings have a width comparable to that of short-scale filaments, while being one order of magnitude denser than them on average, and are dubbed superfilaments.

Figure 1 .

 1 Figure 1. (Color online) Experimental investigation of the structure of the multifilament bundle. (a): impacts on photographic paper in the transverse xy plane for z = 3.5, and 4.5 m. (b): reconstructed side view (yz, x = 0 plane) of the intensity map using these impacts.

Figure 2 .

 2 Figure 2. (Color online) Luminescence of the plasma channels. (a): side-view picture of the filament bundle in the focal zone. (b): side-view iCCD images of the plasma luminescence centered at z = 4.40 m and corresponding z-integrated luminescence profile. (c): other iCCD picture centered at z = 4.90 m and corresponding z-integrated luminescence profile.

Figure 3

 3 Figure 3 shows the evolution of the plasma luminescence along z by measuring the cumulative surface of the three most important emission lines of neutral molecular nitrogen, respectively at 3371, 3577 and 3805 Å (C 3 Π u -B 3 Π g band). It is worth noting the low shotto-shot fluctuation of the plasma luminescence, with a standard deviation of less than 10% over more than 200 shots. It indicates that the plasma strings are highly reproducible.Evaluating the plasma transverse cross-sectional area S(z) along z therefore allows us to estimate the average electron density n e in the plasma following this relation:

Figure 3 .

 3 Figure 3. (Color online) Evolution of the plasma luminescence (red solid line and circles), the plasma surface (blue solid line) and the average electron density ne (black dashed line and squares) with z.

Figure 4 .

 4 Figure 4. (Color online) Simulated evolution of the multifilament bundle. (a): side view (yz plane, x = 0) of the intensity map. (b): evolution of the maximum intensity in the xy plane with z. (c): evolution of the maximum electron density in the xy plane with z.

Acknowledgments.-The authors would like to thank Dr. Benjamin Forestier for fruitful discussions.