Magali Durand
email: magali.durand@creol.ucf.edu

Khan Lim

Vytautas Jukna

Erik Mckee

Matthieu Baudelet

Aurélien Houard

Martin Richardson

André Mysyrowicz

Arnaud Couairon

Blueshifted continuum peaks from filamentation in the anomalous dispersion regime

Filamentation in the anomalous dispersion regime is experimentally shown to give rise to an extreme blueshifted continuum peak in the visible region even when the filament is formed by near-infrared pulses. Measurements and numerical simulations allow us to identify this peak as an axial component of the conical emission. Its features are characterized and quantitatively reproduced by the effective three-wave mixing model.

The propagation of high-peak-power laser pulses (above the megawatt level for solid materials) through transparent media gives rise to a nonlinear propagation regime called filamentation [START_REF] Couairon | Femtosecond filamentation in transparent media[END_REF][START_REF] Chin | The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges[END_REF]. This regime is characterized by an intense and narrow beam surrounded by a low intensity reservoir. The intense core interacts with matter over distances exceeding the typical diffraction length. The use of filaments has been proposed for a wide range of applications such as THz generation [START_REF] D'amico | Conical Forward THz Emission from Femtosecond-Laser-Beam Filamentation in Air[END_REF], discharge guiding [START_REF] Forestier | Triggering, guiding and deviation of long air spark discharges with femtosecond laser filament[END_REF], atmospheric analysis [START_REF] Kasparian | White-Light filaments for atmospheric analysis[END_REF] and few-cycle pulse generation [START_REF] Hauri | Generation of intense, carrier-envelope phase-locked few cycle laser pulses through filamentation[END_REF]. Pulses with peak power exceeding the megawatt level at near infrared wavelengths (1 -2.5 µm) can be easily generated by the use of an optical parametric amplifier (OPA), and have allowed experimental investigations on filamentation in the anomalous dispersion regime since 2004 [START_REF] Moll | Role of dispersion in multiple-collapse dynamics[END_REF][START_REF] Porras | From X-to O-shaped spatiotemporal spectra of light filaments in water[END_REF][START_REF] Porras | Nonlinear unbalanced O waves: nonsolitary, conical light bullets in nonlinear dissipative media[END_REF][START_REF] Saliminia | Ultra-broad and coherent white light generation in silica glass by focused femtosecond pulses at 1.5 m[END_REF][START_REF] Faccio | Competition between phase-matching and stationarity in Kerr-driven optical pulse filamentation[END_REF][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]. Moll and Gaeta have shown that anomalous group velocity dispersion induces a transfer of energy into the collapse region and a substantial increase of the characteristic distances at which multiple collapses occur [START_REF] Moll | Role of dispersion in multiple-collapse dynamics[END_REF]. Porras et al. [START_REF] Porras | From X-to O-shaped spatiotemporal spectra of light filaments in water[END_REF][START_REF] Porras | Nonlinear unbalanced O waves: nonsolitary, conical light bullets in nonlinear dissipative media[END_REF] have interpreted measurements of filamentation and conical emission in the normal and anomalous dispersion regimes as a manifestation of the nonsolitary stationary solutions to the nonlinear Schrödinger equation. These solutions, called nonlinear X-or O-waves, arise from the self-focusing dynamics in Kerr media with nonlinear losses and normal or anomalous dispersion, respectively. Salimina et al. [START_REF] Saliminia | Ultra-broad and coherent white light generation in silica glass by focused femtosecond pulses at 1.5 m[END_REF], and Faccio et al. [START_REF] Faccio | Competition between phase-matching and stationarity in Kerr-driven optical pulse filamentation[END_REF], have shown the emergence of a broad peak in the visible part of the supercontinuum spectra generated by a 1.5 m filament. Recent reports on supercontinuum emission from filaments at longer wavelengths confirm this observation while showing that filamentation in the anomalous dispersion regime gives rise to a broader continuum of over 3.3 octaves that could ideally self-compress to a single cycle pulse [START_REF] Silva | Multioctave supercontinuum generation from mid-infrared filamentation in a bulk crystal[END_REF][START_REF] Smetanina | Anti-Stokes wing of femtosecond laser filament supercontinuum in fused silica[END_REF][START_REF] Smetanina | Femtosecond laser pulse filamentation under anomalous dispersion in fused silica. Part 1. Numerical investigation[END_REF][START_REF] Smetanina | Femtosecond laser pulse filamentation under anomalous dispersion in fused silica. Part 2. Experiment and physical interpretation[END_REF].

In this paper, we present new measurements of supercontinuum emission by filamentation in the anomalous dispersion region of fused silica. By varying the initial wavelength for filamentation in the near-infrared region, we uncover the physical mechanism at the origin of the conical emission in the visible region.

From the first set of experiments, spectra obtained from filamentation in the anomalous dispersion regime of fused silica are shown in Figure 1. The propagation of 35 fs, 1.9 µm laser pulses generates a broad continuum around the fundamental wavelength, as well as an intense component in the visible close to the third harmonic wavelength. Similar peaks close to the third harmonic were previously reported in measurements of supercontinuum from filaments in a YAG crystal [START_REF] Silva | Multioctave supercontinuum generation from mid-infrared filamentation in a bulk crystal[END_REF]. They were not interpreted as the third harmonic but as the axial component of the angularly resolved spectrum that exhibits a fish-shaped tail formed by scattering of the incident pulse from the induced refractive index change [START_REF] Faccio | Competition between phase-matching and stationarity in Kerr-driven optical pulse filamentation[END_REF][START_REF] Kolesik | Perturbative and non -perturbative aspects of optical filamentation in bulk dielectric media[END_REF]. In Ref. [START_REF] Smetanina | Anti-Stokes wing of femtosecond laser filament supercontinuum in fused silica[END_REF], the pump wavelength was changed from 1.2 to 2.1 m but the central wavelength of the visible peak was not observed to change significantly, ruling out third harmonic generation as a key process responsible for the formation of the visible peak. The authors interpreted the generation of the "Anti-Stokes component" in terms of an interference model [14 -16]. Our measurements reported in this paper show a significant blueshift of the central wavelength of the visible peak when the laser wavelength was redshifted. We interpret the results in terms of the effective-three wave mixing model (ETWM) [START_REF] Kolesik | Interpretation of the spectrally resolved far field of femtosecond pulses propagating in bulk nonlinear dispersive media[END_REF].

In the second set of experiments, the initial filamentation wavelength was varied from 1.2 m to 2.4 m. The laser source was an OPA (Opera F, Coherent) pumped by a Titanium-Sapphire chirped pulse amplified laser system that delivered up to 1.1 mJ, 100 fs pulses at 800 nm with a repetition rate of 1 kHz (Spitfire, Spectra-Physics). The pulses at the output of the OPA had energies up to 100 μJ, at 100 fs. The laser beam was focused with a 50 cm focal length lens inside a 3 cm long fused silica hexagonal rod (Hexagonal Light Pipe, Edmund). The measurements taken by three spectrometers were combined to cover the spectral range from 298 nm to 2571 nm (HR 4000 (298-756 nm), HR 4000 (711-889 nm) and NIRQUEST (853-2571 nm), from Ocean Optics). The spectral response of the spectrometers was calibrated using a NIST-calibrated tungsten lamp. The on axis pulse spectrum was measured via an optical fiber. The energy at each laser wavelength was chosen experimentally to be as close to the critical power of filamentation as possible by visually determining the threshold for supercontinuum generation; this energy was around 3.5 J for central wavelengths from 1.2 m to 1.6 m; then 5 J for 1.7 m, 7J for 1.8 m, 7.2 J for 1.9 m, 7.1 J for 2 m, 7.3 J for 2.1 m, 9.5 J for 2.2 m, 18 J for 2.3 m and 12.5 J for 2.4 m. The resulting spectra are shown in Figure 2. The visible part of the conical emission is clearly shifted to shorter wavelengths when the initial filamenting pulse is varied to longer wavelengths, ruling out third harmonic generation as the mechanism at the origin of the visible emission. In order to interpret the results using the ETWM model, we first note that the nonlinear polarization is generated by well localized intensity peaks of the pulse. In the case of anomalous dispersion, numerical simulations show that, in a range of initial powers above the critical power for selffocusing, the pulse undergoes filamentation but does not split and therefore retains the profile of a single peak with a propagation velocity that can be larger or smaller than the group velocity . Figure 3 shows the result of a numerical simulation with input parameters corresponding to the experimental results shown in Fig. 2, i.e., for a 70 fs, 1.9 m, 7.2 J pulse. (See Refs [12, 19 and 20] for details of the numerical scheme and fused silica material properties). The velocity can be determined by calculating the local slope and from the relation . A subluminal velocity is obtained for positive . The solid black line superimposed in Fig. 3a indicates a quasi-constant over the last centimeter of propagation, corresponding to a velocity . The traveling perturbation of the refractive index acts as a polarization wave with a frequency that scatters the incident pulse of frequency into a scattered wave of frequency and transverse wavenumber . This process is associated with the phase matching condition [START_REF] Kolesik | Interpretation of the spectrally resolved far field of femtosecond pulses propagating in bulk nonlinear dispersive media[END_REF]:

(1) The longitudinal wavenumber of the scattered wave fulfills the condition , where denotes the dispersion relation in fused silica [START_REF] Malitson | Interspecimen Comparison of the Refractive Index of Fused Silica[END_REF]. Introduction of the dispersion relation in the effective three wave-mixing relation (1) allows us to find the loci of the supported new frequencies:

(2) where and . From the simulation shown in Fig. 3a, the far-field pattern of the pulse was reconstructed. Figure 3b presents the angular repartition (, ) of the spectrum. It consists of elliptical structures around the pump wavelength (the so-called O-waves) and a tail in the visible region that represents the conical emission part of the supercontinuum, corresponding to the blueshifted peak measured in the experiments. These features match perfectly with the angular distribution of wavelengths given by Eq. ( 2), depicted by the dashed white curves with the velocity parameter as determined from Fig. 3a and = 0.25mm -1 . The conical emission in the visible region is reproducible independently of the value of the phase mismatch , whereas each ring of the O-shaped structure around the pump is populated for a specific value of , reflecting the interaction length over which the pulse propagates with quasi-constant peak intensity and velocity.

Other than by analyzing the results of numerical simulation, the velocity of the nonlinear polarization can also be determined experimentally by fitting the angularly resolved spectrum with Eq. ( 2). Measurements of (, ) spectra were performed only in our first set of experiments. Figure 4 presents the (, ) spectrum measured for a 35 fs pump pulse at . The solid white curves represents a fit obtained with Eq. ( 2) and the values and Δk = 0.5mm -1 . In spite of the different pulse durations and energies, as well as a different value of , all the features visible in Fig. 4 are in good qualitative agreement with those in the simulation of Fig. 3b. The peak in the visible region represents the axial part of the conical emission which extends to angles up to 8 degrees. The low intensity supercontinuum in the range between 1 and 1.5 μm is more pronounced when the pump energy is increased, in agreement with Fig. 1. The elliptical shape of the spectrum around the fundamental frequency reflects the nature of the nonlinear O-waves generated by spontaneous reshaping of the pump during propagation [START_REF] Porras | Nonlinear unbalanced O waves: nonsolitary, conical light bullets in nonlinear dissipative media[END_REF][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]. We can now check that the interpretation of the spectral peak in the visible region shown in Fig. 2 corresponds to the axial part of the conical emission for any pump wavelength. The ratio of the peak velocity for the nonlinear polarization to the group velocity was assumed independent of wavelength, i.e. the value for α was assumed to be , with varying with the pump wavelength.

The roots of Eq. ( 2), which represent the intersections of the phase matched region of the spectrum with the axis ( ), were evaluated by a numerical procedure. For a pump wavelength in the near-IR region, the dispersion landscape is such that three roots can always be found when the nonlinear polarization wave is subluminal (

). The roots which are closest to the pump belong to an elliptical branch featuring the O-wave, whereas the root at the highest frequency (wavelength λ + ) belongs to the hyperbolic branch representing conical emission in the visible region.

The spectral position λ + of the peak in the visible region was compared with the experimental data, as shown in Figure 5. The black continuous curve corresponds to the wavelength λ + of the visible peak as obtained from the model. From Figure 2, we measured the central wavelength of the visible peak (red circles), and the full width at half maximum (FWHM) of this peak, delimited by (blue squares) and (purple triangles). The black line was generated with the value . The gray area represents new frequencies generated in the visible range, determined by the limits and in the axial phase matching conditions. The vertical line represents the zero group velocity dispersion wavelength. The departure of the analytical estimation from the measurements near the wavelength of zero GVD is attributed to the narrow range of peak velocities used in Eq. ( 2). In fact, pulse splitting becomes more prominent at shorter wavelengths, leading to additional polarization waves propagating at different velocities and modifying the position and width of the visible spectral peak. The quantitative agreement between the model and the measurements in the region of anomalous dispersion is very good, thus leading to the conclusion that the phenomena of conical emission accompanying filamentation can be accurately explained by the effective threewave mixing model, not only in the normal dispersion regime [START_REF] Kolesik | Interpretation of the spectrally resolved far field of femtosecond pulses propagating in bulk nonlinear dispersive media[END_REF][START_REF] Faccio | Generation and control of extreme blueshifted continuum peaks in optical Kerr media[END_REF] but also in the anomalous dispersion regime [START_REF] Faccio | Competition between phase-matching and stationarity in Kerr-driven optical pulse filamentation[END_REF][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]. In particular, the model accurately predicts the presence of a peak in the visible region that is separated from the supercontinuum in the near infrared. The emergence of this peak cannot be attributed to self-phase modulation alone or to third harmonic generation.

Although the formation of the visible spectral peak was observed in Refs 14, 15, 16, its blueshift with increasing pump wavelength has not been reported before. We now comment on the possible reasons for this significantly different result. Our experiment differs from that of Ref. 14 in the range of pump wavelengths used as well as the measurement of the conical emission spectrum. The blue spectral peak narrows as the pump wavelength moves further from the zeros GVD wavelength. The narrowing also makes the movement of the peak towards shorter wavelengths more easily observable, as reflected in Fig. 2 for pump wavelengths between 2 m and 2.4 m. Secondly, in Ref. 14 the entire emission cone from the filament was angularly integrated (focused) into the spectrometer. The collection/integration of the entire conical emission may broaden the blue peak and therefore obscure the frequency shift. Differences in the pump pulse energies used between Ref. 14 and our experiment are minor and should not have had a significant influence on the results.

The wavelength and intensity of the visible spectral peak depend strongly on the filament characteristics, and therefore on experimental parameters such as pulse duration, chirp and focusing geometry. Depending on the dispersion profile of the medium and the initial pulse parameters, the velocity of the pulse intensity peak may vary. This would consequently modify the conical branch of the filament emission and the on-axis emission wavelength in the visible region. A narrow and welldefined visible spectral peak develops if the pulse peak intensity and velocity remains fairly constant over the length of the filament. Conversely, if the velocity of the pulse intensity peak varies significantly during propagation, the visible spectral peak becomes broader as a result of the effective three wave-mixing model applied to the case of a non-uniformly moving material wave. Higher initial peak powers may lead to multiple splitting events and to potentially chaotic multiple collapses [START_REF] Moll | Role of dispersion in multiple-collapse dynamics[END_REF]. In this case, the "polarization wave" is more complicated but each split pulse can still be interpreted as being associated to a well-defined velocity, leading to a phase matching relation for each intensity peak. The spectral content in the visible region is then a result of the superposition of the visible peaks generated by each polarization wave. The numerical simulations performed in Ref. [START_REF] Smetanina | Femtosecond laser pulse filamentation under anomalous dispersion in fused silica. Part 1. Numerical investigation[END_REF] indicate that a well-defined visible peak may still emerge in such a condition.

The observation of a spectral peak in the visible region alone does not allow us to infer any conclusion on the occurrence of pulse splitting during filamentation. Our measurements with increasing pump pulse power reported in Fig. 1 show a sharply defined spectral peak, which we interpret as the result of one dominant polarization wave. This scenario is well represented by the numerical simulations reported in Ref. 12. Pulse splitting is evident in the simulations, but a significantly higher multiphoton absorption compared to Ref. 15 prevented the occurrence of multiple collapses, resulting in an intense subluminal pulse whose intensity remains almost constant over 80 % of the filament length. This results in a visible spectral peak at the same wavelength for the whole range of input pulse powers from 5 to 26 P cr . We therefore emphasize that there is a one-toone correspondence between the central wavelength of the spectral blue peak and the velocity of the material wave. However, there may exist multiple sets of initial pulse parameters, leading to different filamentation dynamics, that can lead to the same final polarization wave with a given velocity.

In this letter, we have presented new measurements of conical emission for filamentation in the regime of anomalous dispersion. We have identified the mechanisms responsible for the generation of a peak in the visible region which is clearly separated from the supercontinuum around the pump wavelength. We have shown that the effective three-wave mixing model and phase-matching arguments give an accurate explanation for this phenomenon.

The authors acknowledge the funding sources allowing this work: the US Army Research Office MURI program # W911NF1110297, the State of Florida and the ONERA.

Figure 1 :

 1 Figure 1: (Color online) Supercontinuum measured at the center of the conical emission after filamentation in 3 cm of fused silica of a 1.9 µm, 35 fs pulse at different input energies. Insert: Photo of the conical emission at 15 µJ.

Figure 2 :

 2 Figure 2: Spectra measured at the center of the conical emission after filamentation in a 4 cm thick fused silica sample. The central wavelength used for filamentation ranged from 1.2 µm to 2.4 µm. The visible part of the spectra up to 1 µm is magnified by a factor of ten for [1.2 -1.5 µm] and by a factor of two for [1.6 -2.4 µm]

Figure 3 :

 3 Figure 3: (Color online) (a) Computed pulse temporal evolution during filamentation in a 3 cm long fused silica sample. The input pulse has central wavelength  0 = 1.9 µm, duration 70 fs, and energy 7.2 µJ. The black line marks the slope for calculating the velocity of the intensity peak. (b) (,) spectrum obtained by numerical simulation . The dashed curves show the fit with Eq. (2).

Figure 4 :

 4 Figure 4: (Color online) Angular distribution of the conical emission, after filamentation of 1.9 µm, 40 fs, 26 µJ pulses in 3 cm of fused silica. This figure has a logarithmic scale. The white curves show the fit with Eq. (2).

Figure 5 :

 5 Figure 5: (Color online) Calculated and measured visible component that is generated during filamentation in a 3 cm thick fused silica sample as a function of the input pulse central wavelength.


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

Yohann Brelet

Guillaume Point

Aurélien Houard

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

Self-seeded lasing in ionized air pumped by 800 nm femtosecond laser pulses

We report on the lasing in air and pure nitrogen gas pumped by a single 800 nm femtosecond laser pulse. Depending on gas pressure, incident laser power and beam convergence, different lasing lines are observed in the forward direction with rapid change of their relative intensities. The lines are attributed to transitions between vibrational and rotational levels of the first negative band of the singly charged nitrogen molecule-ion. We show that self-seeding plays an important role in the observed intensity changes.

Introduction

Lasing in ambient air or its major components (N 2 and O 2 ) when pumped by an intense ultrashort laser pulse has attracted much attention in recent years [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] Shneider | Population inversion of molecular nitrogen in an Ar: N2 mixture by selective resonance-enhanced multiphoton ionization[END_REF][START_REF] Luo | Lasing action in air driven by ultra-fast laser filamentation[END_REF][START_REF] Kocharovsky | Gain-swept superradiance applied to the stand-off detection of trace impurities in the atmosphere[END_REF][START_REF] Yao | High-brightness switchable multiwavelength remote laser in air[END_REF][START_REF] Chu | Multiwavelength amplified harmonic emissions from carbon dioxide pumped by mid-infrared femtosecond laser pulses[END_REF][START_REF] Ni | Harmonic-seeded remote laser emissions in N2-Ar, N2-Xe and N2-Ne mixtures: a comparative study[END_REF][START_REF] Kartashov | Stimulated amplification of UV emission in a femtosecond filament using adaptive control[END_REF][START_REF] Ni | Identification of the physical mechanism of generation of coherent N2 + emissions in air by femtosecond laser excitation[END_REF][START_REF] Yao | Remote creation of coherent emissions in air with two-color ultrafast laser pulses[END_REF]. The generation of coherent radiation with such cavity-less laser sources holds great potential for remote sensing applications. Q. Luo and coworkers were the first to suggest the existence of a backward stimulated emission from filaments formed in air by 800 nm femtosecond pulses, based on an exponential increase of nitrogen fluorescence with filament length [START_REF] Luo | Lasing action in air driven by ultra-fast laser filamentation[END_REF]. Dogariu et al. demonstrated in 2011 that a backward coherent emission at 845 nm can be achieved in air pumped by picosecond UV pulses [START_REF] Dogariu | High-gain backward lasing in air[END_REF]. The stimulated emission was attributed to optical dissociation of O 2 molecules followed by multiphoton excitation of atomic lines of Oxygen. Shortly later, Kartashov et al. reported a strong backward lasing emission at 337 nm and 357 nm in pure Nitrogen gas and its mixture with Argon when driven by powerful mid-infrared lasers ( = 3.9 m or 1.03 m), due to a collision-induced population inversion between the   u C [START_REF] Shneider | Population inversion of molecular nitrogen in an Ar: N2 mixture by selective resonance-enhanced multiphoton ionization[END_REF] and

  g B 3
states of the excited N 2 molecule [START_REF] Kartashov | Free-space nitrogen gas laser driven by a femtosecond filament[END_REF][START_REF] Shneider | Population inversion of molecular nitrogen in an Ar: N2 mixture by selective resonance-enhanced multiphoton ionization[END_REF]. Here, the electronic transitions of nitrogen molecule are the same as in the conventional discharge-pumped nitrogen laser [12]. In 2011, Yao et al. reported a forward stimulated radiation from N 2 molecule-ions pumped by short and intense mid-infrared pulses [START_REF] Yao | High-brightness switchable multiwavelength remote laser in air[END_REF]. The lasing effect corresponded to the

     g u X B 2 2
transition between the excited and ground cation states. It could be triggered by fine tuning of the wavelength of a femtosecond Optical Parametric Amplification (OPA) laser operating in the mid-IR (1.2 m-2.9m). It was found that when the 3 rd or 5 th harmonic of the driving pulse overlapped with specific spectral positions of the above transitions, lasing radiation was initiated by a seeding effect. The authors attributed the lasing effect to the amplification of the weak harmonic pulse in the presence of population inversion [START_REF] Ni | Harmonic-seeded remote laser emissions in N2-Ar, N2-Xe and N2-Ne mixtures: a comparative study[END_REF][START_REF] Ni | Identification of the physical mechanism of generation of coherent N2 + emissions in air by femtosecond laser excitation[END_REF]. The mechanism responsible for population inversion is still a subject of controversy [START_REF] Chu | Multiwavelength amplified harmonic emissions from carbon dioxide pumped by mid-infrared femtosecond laser pulses[END_REF][START_REF] Ni | Harmonic-seeded remote laser emissions in N2-Ar, N2-Xe and N2-Ne mixtures: a comparative study[END_REF][START_REF] Ni | Identification of the physical mechanism of generation of coherent N2 + emissions in air by femtosecond laser excitation[END_REF][START_REF] Yao | Remote creation of coherent emissions in air with two-color ultrafast laser pulses[END_REF]. Later, the same group showed that lasing action of could also be activated with 800 nm femtosecond laser pulses, but required injection of a seed pulse around 400 nm [START_REF] Ni | Identification of the physical mechanism of generation of coherent N2 + emissions in air by femtosecond laser excitation[END_REF][START_REF] Yao | Remote creation of coherent emissions in air with two-color ultrafast laser pulses[END_REF]. This external 400 nm pulse is the counterpart of the 3 rd and 5 th harmonic in the case of mid-IR pumping. Lasing obtained with a pump at 800 nm is particularly interesting because of the wide availability of femtosecond lasers at this wavelength.

In this article, we report on forward lasing effect of in ambient air and pure nitrogen gas pumped by just one pulse at 800 nm, without any external seeding pulse. We attribute the observed lasing lines around 391 nm and 428 nm to a self-seeding effect. Lines around 391 nm and 428 nm have been attributed to

    0 0 ' 2 2          g g X B and     1 0 ' 2 2          g g X B
transitions. Specifically, the 391 nm transition is seeded by the 2 nd harmonic around 400 nm generated by the laser pulses in the plasma. We attribute the 428 nm laser emission dominating at higher gas pressure to the seeding by the super-continuum generated when the intense incident laser pulse undergoes filamentation. It is further found that the gas pressure and incident laser beam convergence play an important role in determining the intensity of the emission lines. We thus confirm the importance of an injection source proposed by Yao et al. to explain the forward stimulated emission between cation states of N 2 and show in addition that several other factors such as gas pressure, nonlinear propagation effects, and the presence of oxygen play a key role for optical gain.

Experimental results and discussion

In the experiment, we employed a commercial Chirped-Pulse-Amplification laser system (Thales, Alpha 100), which provides 45 fs pulses at 800 nm with energy up to 12 mJ per pulse. The laser pulses were focused by a convex focal lens (f = 40 cm, 75cm, or 150 cm) into a gas chamber with variable length. In the experiment, the chamber was first pumped to a pressure as low as 10 -5 mbar and then filled with air or nitrogen gas at different pressures. Visible plasma strings with length of 5 to 30 mm were observed in the gas chamber, with a string length depending on the focal lens, incident pulse energy, and gas pressure. The pulses emerging from the gas chamber after the plasma generation area were first filtered by two color filters (BG 40), which transmit the spectral components shorter than 600 nm but block largely the fundamental pulses and the accompanying super-continuum. The filtered pulses were then focused by a fused silica lens (f = 10 cm) to couple them into a fibre spectrometer (Ocean Optics, HR 4000). A photodiode was also used to measure the radiation intensity. No significant emission was detected in the backward direction in our current experiment, indicating the backward stimulated emission was relative weak comparing to the forward one.

In Fig. 1(a), we present the measured spectrum as a function of gas pressure in air in the case of f = 40 cm. The incident laser pulse energy was 3.5 mJ. A strong narrow-line radiation around 391 nm is observed for a pressure ranging from 20 mbar to 200 mbar. The radiation around 391 nm has been identified previously as the transition between the vibration states v = 0 of second excited state   g B 2 of the nitrogen ions and its ground state [START_REF] Yao | High-brightness switchable multiwavelength remote laser in air[END_REF][START_REF] Ni | Identification of the physical mechanism of generation of coherent N2 + emissions in air by femtosecond laser excitation[END_REF][START_REF] Yao | Remote creation of coherent emissions in air with two-color ultrafast laser pulses[END_REF]. We find that the radiation around 391 nm is actually composed of two separated lines at 390 nm and 391.6 nm (Fig. 1(b)). We attribute the peaks at 390 nm and 391.6 nm to the R-branch and P-branch of the

  g X 2
    0 0 ' 2 2          g g X B
transition, where different levels of quantum rotation of the molecular ions are involved [13, 14]. A higher spectral resolution is needed to reveal the discrete spectral lines corresponding to different rotation states of the transition. The relative strength of the two lines changes with pressure. At very low pressure, the P branch at 391.6 dominates, while at higher pressure p > 150 mbar the R branch becomes comparable or even surpass the P branch. We performed similar measurement in pure nitrogen gas. The results are presented in Fig. 1(c). Here the incident pulse energy was slightly higher, about 4 mJ. At the lowest pressure p = 4 mbar, a weak broad emission peaked at 400 nm is observed, which will be discussed in more detail later. Under a slight increase of pressure, the lasing radiation around 391 nm appears, grows to a maximum for a pressure around 100 mbar and disappears above 200 mbar. Again, the same alternation between the relative importance of the P and R branch is observed with pressure. In contrast to air, the intensity of the lasing line around 428 nm increases significantly from 100 mbar up to p = 750 mbar and then saturates for p > 750 mbar. The line at 428 nm corresponds to the transition between the vibration states v = 0 of second excited state   g B 2 of the nitrogen ions and its v = 1 ground state   g X 2 [START_REF] Yao | High-brightness switchable multiwavelength remote laser in air[END_REF][START_REF] Ni | Identification of the physical mechanism of generation of coherent N2 + emissions in air by femtosecond laser excitation[END_REF][START_REF] Yao | Remote creation of coherent emissions in air with two-color ultrafast laser pulses[END_REF]. A weak radiation peaked around 471 nm, corresponding to the

    2 0 ' 2 2          g g X B
transition [START_REF] Yao | High-brightness switchable multiwavelength remote laser in air[END_REF][START_REF] Ni | Harmonic-seeded remote laser emissions in N2-Ar, N2-Xe and N2-Ne mixtures: a comparative study[END_REF], is observed for pressure larger than 450 mbar. However, it does not grow into a strong emission for the pressure range tested. The evolution of the radiation intensity around 391 nm and 428 nm are presented in Fig. 2(a) and 2(b) as a function of pressure. We have repeated the same experiments in pure N 2 with longer focusing conditions. For focal distance f = 75 cm, the intensity dependence of the two lasing lines as a function of pressure is shown in the Fig. 2(c) and 2(d) for incident pulse energy of 3.5 mJ. The lasing emission around 391 nm appears, dominates, and disappears in the pressure range from 1 mbar to 50 mbar. At the same time, the lasing emission at 428 nm starts to show up at 15 mbar, reaches its maximum around 400 mbar, and then gradually decreases up to 1000 mbar. For f = 150 cm no lasing emission was observed around 391 nm for all pressures tested while a relatively weak radiation around 428 nm was only observed for pressure below 80 mbar. We now come back to the broad emission around 400 nm in the case of p = 4 mbar, presented in Fig. 1(d). We attribute the observed broad radiation at 400 nm to the generation of second harmonic. In previous studies, 2 nd harmonic generation in gas media was supposed to be impossible [START_REF] Ni | Identification of the physical mechanism of generation of coherent N2 + emissions in air by femtosecond laser excitation[END_REF][START_REF] Yao | Remote creation of coherent emissions in air with two-color ultrafast laser pulses[END_REF], due to the isotropy of the medium. However, several groups reported second harmonic by simply focusing a femtosecond laser pulse in gases [15,16]. The mechanism has been attributed to the nonuniform plasma distribution driven by the ponderomotive force or an electric-field-induced third-order mixing [15, 16]. To further check this hypothesis, we filled the chamber with low pressure O 2 , Ar or CO 2 gases. For the three gases, a similar broad emission around 400 nm was observed for pressure below 80 mbar. The harmonic signal appears above an input pulse energy of 2 mJ. It rises with increasing pressure to reach a maximum around 15-30 mbar and then decreases progressively up to 80 mbar. A detailed study on the 2 nd harmonic generation in gases will be published elsewhere.

Also, white-light continuum became visible to human eyes during the experiments for pressures larger than 100 mbar. Above 100 mbar, filamentation occurs under our experimental conditions. Broadband continuum generation accompanying the formation of filaments is a well known phenomenon. The presence of the supercontinuum is only weakly observed in Fig. 1(a) around 490 nm for p > 600 mbar, because the supercontinuum was strongly attenuated by the two BG40 filters. Without filters, we observed that the supercontinuum extends well below 400 nm in the case of p = 1000 mbar both in air and pure N 2 .

We measured the lasing radiation intensity in N 2 as a function of incident laser pulse energy for the two lasing lines. The results are presented in Fig. 3. The chosen nitrogen gas pressures for the two experiments were 100 mbar and 600 mbar for 391 nm and 428 nm, in view of their pressure dependence (Fig. 1(c)). With the current detection system, incident laser pulse energy thresholds around 2 mJ were observed for both lasing lines. This corresponds to the threshold for appearance of the second harmonic. A discussion of this behavior requires proper inclusion of nonlinear propagation effects. Due to nonlinear propagation, there is no simple relation between the incident pulse energy and the local laser intensity in the interaction region where plasma is produced. For instance, at the lowest pressures, the intensity at the focus is fixed by diffraction of the beam, because beam defocusing effects are negligible. The peak incident laser intensity then scales like the incident pulse energy and reaches a maximum value of for a pulse energy of 2 mJ. At such intensities, a large fraction of the molecules are singly ionized. At higher pressures, defocusing of the beam limits the peak intensity. An increase of pulse energy does not change significantly the fraction of ionized molecules but extends the range of ionization (onset of filamentation). In filaments, the peak laser intensity is clamped to a value I ~ 3 × 10 13 W/cm 2 for air or N 2 gas pressure between 200 and 1000 mbar [17].

We now try to explain the evolution of the observed self-seeded lasing effects presented in Fig. 1(c) and compare it with the previous results of other groups. In the previous report of Yao and coworkers, where the incident pulse energy was below the energy threshold (~ 2 mJ) for harmonic generation, population inversion between the   g B 2 state and the   g X 2 state could be achieved [START_REF] Ni | Identification of the physical mechanism of generation of coherent N2 + emissions in air by femtosecond laser excitation[END_REF][START_REF] Yao | Remote creation of coherent emissions in air with two-color ultrafast laser pulses[END_REF]. However, external seeding pulse was required to initiate lasing phenomenon [START_REF] Ni | Identification of the physical mechanism of generation of coherent N2 + emissions in air by femtosecond laser excitation[END_REF][START_REF] Yao | Remote creation of coherent emissions in air with two-color ultrafast laser pulses[END_REF]. In our case, the incident laser energy exceeds the threshold energy for 2 nd harmonic generation. As a result, effective 2 nd harmonic around 400 nm is generated (Fig. 1(d)), which serves as the seeding pulse to start the lasing around 391 nm. With pressure beyond 100 mbar, the 391 nm lasing radiation decreases due to the gradual yield decrease of the 2 nd harmonic. At the same time, the lasing activity at 428 nm takes over, thanks to the effective generation of supercontinuum, which gradually approaches 428 nm with increasing pressure. At still higher pressures, for example, p > 750 mbar in Fig. 1(c) and Fig. 2(b), and p > 400 mbar in Fig. 2(d), the lasing intensity saturates or even decreases. At higher pressure, the supercontinuum seed is expected to be stronger. Therefore, this suggests that the optical gain, i.e. the population inversion between the two involved states, decreases at higher pressure.

Finally, we measured the spatial profile of the lasing radiation as a function of nitrogen gas pressure. Results are presented in Fig. 4 in the case of f = 75 cm. In the experiment, a Charge-Coupled-Digital (CCD) chip was installed closely behind the two BG 40 filters to record the spatial profile of the transmitted lasing emissions. In Figs. 4(a)-4(c), the profiles of the lasing radiation around 391nm are shown for gas pressure of 5 mbar, 15 mbar, and 25 mbar. At the lowest pressure of 5 mbar, a Gaussian radiation pattern is observed (Fig. 4(a)). Upon increase of pressure, an outer ring structure surrounding the central spot appears. The outer ring is better seen in the line-cuts shown in Figs. 4(b) and 4(c). For the radiation peaked at 428 nm, a ring structure is observed at all pressures between 100 mbar and 1000 mbar (Figs. 4(d)-4(f)). The divergence angle of the lasing radiation was measured to be ~ 20 mrad in Fig. 4(e). At this moment, we speculate this ring-shaped profile around 428 nm reflects the spatial distribution of the super-continuum seeding pulse, which is emitted in a conical form during laser pulse filamentation [18,19]. Further experimental and theoretical efforts are under progress for a better understanding of this feature.

Conclusion

In conclusion, we demonstrated that lasing radiation of can be achieved in ambient air or pure nitrogen gas pumped with a single 800 nm femtosecond laser pulse, without a second external seeding pulse. We attribute the initiation of the lasing effect to self-generated seeding pulses, either the 2 nd harmonic or the super-continuum white light. Gas pressure was shown to have a significant role on the lasing intensity and the spatial profile. The discovery of the selfseeded regime of the air plasma lasing pumped by the widely available 800 nm femtosecond pulses simplifies the experimental setup required and may prove useful for applications.

Fig. 1 .

 1 Fig. 1. (a) and (c), Spectrum intensity of the forward lasing radiations in air (a) and nitrogen gas (c) as a function of gas pressure. The focal length was f = 40 cm. The incident laser energy was 3.5 mJ in (a) and slightly higher, about 4 mJ, in (c). (b) Zoomed spectrum around 391 nm for pressure p = 30 mbar in Fig. 1(a). (d), Zoomed spectrum around 400 nm for pressure p = 4 mbar in Fig. 1 (c).

Fig. 2 .

 2 Fig. 2. Intensity of the lasing radiation around 391 nm (left) and 428 nm (right) as a function of nitrogen gas pressure. The focal length was 40 cm in (a) and (b), 75 cm in (c) and (d).

Fig. 3 .

 3 Fig. 3. Intensity dependence of the 391 nm (a) and 428 nm (b) lasing radiation on the incident laser pulse energy. The nitrogen pressure for (a) and (b) are 150 mbar and 600 mbar, respectively.

Fig. 4 .

 4 Fig. 4. (a)-(c), Spatial profile of the 391 nm emission at different nitrogen pressure. From (a) to (c), the pressure are 5 mbar, 15 mbar, and 25 mbar, respectively. (d) -(f), Spatial profile of the 428 nm lasing radiation. The pressures are 100 mbar, 300 mbar, and 800 mbar from (d) to (f). The angle of view of each panel is 60 × 40 mrad. The white lines on the right of each panel present the vertical line-cut of each intensity distribution on the center. As a final remark, we would like to mention that after the submission of our manuscript we became aware of two other independent works reporting self-seeded lasing action in pure nitrogen or air [20, 21]. Wang T-J et al. observed forward lasing at 428 nm in a filament formed in air and attributed it to the filament-induced white light [21]. Chu W. and coworkers observed the 391 nm and 428 nm lasing in air and pure nitrogen gas as well as a donut-shaped lasing emission profile at 428 nm [20].

Acknowledgments

The authors are grateful to Prof. S. L. Chin of Laval University, Prof. Hongbing Jiang of Peking University, Dr. Benjamin Forestier of CILAS, Dr. Jinping Yao of SIOM, Thierry Lefrou of LOA for stimulating discussion and technical help. The project has been partially funded by Contract No. ANR-2010-JCJC-0401-01.


Magali Durand

Aurélien Houard

Bernard Prade

André Mysyrowicz

Anne Durécu

Bernard Moreau

Didier Fleury

Olivier Vasseur

Hartmut Borchert

Karsten Diener

Rudiger Schmitt

Francis Théberge

Marc Chateauneuf
email: marc.chateauneuf@drdc-rddc.gc.ca

Jean-François Daigle

Jacques Dubois

Kilometer range filamentation

Keywords:

We demonstrate for the first time the possibility to generate long plasma channels up to a distance of 1 km, using the terawatt femtosecond T&T laser facility. The plasma density was optimized by adjusting the chirp, the focusing and beam diameter. The interaction of filaments with transparent and opaque targets was studied.

Introduction

The long range propagation of laser beams in the atmosphere and their ability to produce intense illumination on distant target has been intensively studied since the discovery of laser. Recently, the observation with femtosecond lasers of a spectacular phenomenon called filamentation has brought new perspectives in this research area [START_REF] Couairon | Femtosecond filamentation in transparent media[END_REF][START_REF] Chin | The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges[END_REF][START_REF] Kasparian | Physics and Applications of atmospheric nonlinear optics and filamentation[END_REF]. During the propagation of a femtosecond pulse in air, provided the initial peak power exceeds a threshold, a dynamic competition between diffraction and nonlinear optical effects such as Kerr effect and plasma defocusing gives rise to thin and long light strings inside which the intensity is clamped to a value above 10 13 W.cm -2 . This filamentation regime characterized by the formation of a weakly ionized plasma channel in the wake of the pulse is followed by a second propagation stage of weaker intensity in which diffraction is only balanced by the Kerr effect [START_REF] Méchain | Long range self-channeling of Infrared laser pulses in air: a new propagation regime without ionization[END_REF][START_REF] Chen | Evolution and termination of a femtosecond laser filament in air[END_REF]. Numerous potential applications have been proposed for filamentation such as the generation of white-light supercontinuum [START_REF] Théberge | Ultrabroadband conical emission generated from the ultraviolet up to the far-infrared during the optical filamentation in air[END_REF], and its application to teledetection of pollutants [START_REF] Kasparian | White-light filaments for atmospheric analysis[END_REF] or the remote guiding of electric discharges [START_REF] Comtois | Triggering and guiding leader discharges using a plasma channel created by an ultrashort laser pulse[END_REF][START_REF] Tzortzakis | Femtosecond laser-guided electric discharge in air[END_REF].

The longest evidence of filamentation and plasma formation has been observed up to 400 m by Méchain et al. by propagating horizontally a collimated 4 TW laser beam [START_REF] Méchain | Range of plasma filaments created in air by a multi-terawatt femtosecond laser[END_REF]. For the long distance propagation of such beam, atmospheric turbulence can have a significant effect on the formation of filaments. Recent studies have shown that, once formed, filaments are not very sensitive to turbulence, whereas turbulence undergone before the collapse of the beam by Kerr self-focusing can prevent the apparition of filaments [START_REF] Salamé | Propagation of laser filaments through an extended turbulent region[END_REF][START_REF] Houard | Study of the influence of air turbulence on femtosecond filamentation[END_REF].

In this manuscript we will present the details of the experimental setup followed by a characterization of the nonlinear propagation of the TW beam over distances ranging from 150 to 1500 meters. This allowed us to determine the optimal beam parameters to maximize the plasma density at different distances from the laser. The results of the interaction of the long range filaments with different optical windows are then described.

Description of the experimental setup

The laser facility used for these experiments was the portable Terawatt & Terahertz (T&T) laboratory from the Defence Research and Development Canada (DRDC) Valcartier. This 5 TW Titanium-doped Sapphire laser system is embedded in a sea container. The compact laser system sits on a 1.25 m by 2.5 m table in a class 100,000 clean room occupying about half of the container. The remaining part being used as a control room and for data acquisition. A schematic of the interior of the T&T laboratory is shown in Fig. 1(a). The control room is filled with the air conditioning unit, the power supplies and chillers for the laser systems as well as the control rack. The laser operator is able to fully control the laser either from the clean room, the control room or from the exterior of the container via an Ethernet link. The control room also acts as an airlock for the clean room.

The laser pulse energy of the system can reach 200 mJ with pulse duration adjustable from 40 fs up to 15 ps. The pulse repetition rate is 10 Hz, the central laser wavelength is 805 nm and the laser beam diameter at the compressor output is 2.5 cm at full width at half maximum (FWHM).

An all-reflective telescope, composed of a convergent off-axis parabolic mirror with a metallic coating and a divergent dielectric spherical convex mirror mounted on a translation stage, was installed outside the container. The parabolic mirror has a diameter of 15 cm and a focal length of 150 cm. The focal length of the convex mirror is -50 cm which implies a magnification of 3 for a collimated input and output beams. By moving the translation stage it was possible to vary the focusing of the laser beam along its propagating path and therefore, the position of the generated filament. The focal length of the telescope is referred as the geometrical focal length.

These experiments were done at a 2.5 km long firing range located at DRDC Valcartier [see Fig. 1(b)]. The T&T laboratory was installed at one end of the firing range and the laser beam propagated at few meters above the ground. The first 1.5 km of the firing range was flat while the last section was at the base of a mountain. The propagation of high power laser beam into the atmosphere is characterized by numerous linear and nonlinear optical effects that modify the laser beam energy distribution at long distance. Nonlinear optical effect like the Kerr self-focusing or linear optical effect such as the atmospheric turbulence can strongly decrease the achievable laser intensity at long distance.

Reaching high laser intensity at long range in the atmosphere with a compact optical device remains a challenge. Solutions for delivering the high laser intensity at distance involve the use of optical components that pre-compensate for the perturbations induced by the linear propagation effects and modify the laser beam parameters in order to minimize the undesired nonlinear intensity-dependent optical effects.

Because of the self-focusing in air, it is important to adjust the initial intensity of the laser pulses with both the optical compressor and a beam expander in order to optimize the generation of filaments at longer distances. The main concept behind the idea of using a beam expander to induce filaments at long distance is based on the self-focusing property of powerful and ultrashort laser pulses. The self-focusing distance ( ) of a laser pulse, which is approximately the starting point of the filament, is given by 13:

a) b) 2 1/2 2 1/2 0.367 , 0.852 0.0219 f cr ka z P P                            (1)
where P cr is the critical power for self-focusing, k is the wavevector of the laser pulse, a is the beam radius defined at 1/e level of the intensity and P is the initial peak power of the laser pulse. Thus, a laser beam having a larger radius (parameter a is larger) would tend to selffocus at longer distance. Moreover, decreasing the ratio P/P cr (by adjusting the energy per pulse, the pulse duration and/or the laser chirp) would also increase the self-focusing distance ( ).

In the presence of the external focusing from the sending telescope, the new self-focusing distance z f ' becomes [START_REF] Talanov | Focusing of light in cubic media[END_REF]:

' , f eff f f eff zf z z f   (2)
in which f eff denotes the effective focal length of the telescope. Thus, if we adjust the radius of the beam (a) and its peak power (P) in such a way z f  f eff , then and the filament is generated around the geometrical focus of the telescope.

Working with intense and ultrashort laser pulses is different than with a conventional laser beam. The spectrum of the pulse is large (tens of nm) which is subject to chromatic aberrations. More importantly, undesired non-linear effects are much more important in glass than in air and the self-focusing in refractive optics tends to generate "hot spots" in the laser beam profile, which decreases the effective radius of the laser beam and makes the laser beam to self-focus at shorter distances [START_REF] Luo | Effect of beam diameter on the propagation of intense femtosecond laser pulses[END_REF][START_REF] Hosseini | Competition of multiple filaments during the propagation of intense femtosecond laser pulses[END_REF]. That justifies the choice of using an off-axis all reflective telescope instead of an on-axis telescope composed of transparent lenses.

Characterization of the propagation

Optimization of the beam parameters

Our goal was to demonstrate the possibility to achieve kilometer range femtosecond filamentation and to study the interaction of such long range filaments on different optical windows. We studied the long range filamentation at the following distances of observation i the plasma density yields locally was optimized by adjusting the chirp, the magnification and the focal length of the telescope. The plasma density was monitored using an electronic detector sensitive to radiation around 10 GHz placed on the side of the filament. This signal radiated by the plasma permits to evaluate the local plasma density and to determine the best beam parameters to produce a maximum of filaments at this position [START_REF] Tzortzakis | Coherent sub-THz radiation from femtosecond infrared filaments in air[END_REF]. From these measurements we determined that the optimal focusing corresponds to a geometrical focus a few meters after the observation position, while the optimal chirped pulse durations were respectively 1.2 ps, 3.68 ps and 2.97 ps for 150 m, 300 m and 1 km. When comparing the value of negative chirp impressed on the pulse and the value of air dispersion, one can see that in every cases the dispersion of air was negligible compared to the chirp of the pulse. The purpose of chirping the pulse was then not to pre-compensate the dispersion of air but to postpone the filament formation by decreasing the initial peak power.

Length and number of filaments as a function of the distance

For each distances D i , the length and the number of plasma filaments were measured by impact on photographic paper at different positions around the geometrical focus. Results are

eff f f z  
presented in Fig. 2 where the black dots symbolize the distance at which an impact was made revealing the presence of a plasma channel. The grey lines symbolize the presence of light channels, which are millimeter size lower intensity channel emerging from the 100 m plasma filaments [START_REF] Méchain | Long range self-channeling of Infrared laser pulses in air: a new propagation regime without ionization[END_REF][START_REF] Chen | Evolution and termination of a femtosecond laser filament in air[END_REF]. The inset in Fig. 2 shows an example of traces made by a filament and by a light channel on the photographic paper. The light channels allow the energy of the pulse to continue to travel with a minimal diffraction over several hundred of meters but they are not intense enough to ionize air. For each value of D i the length over which ionization occurs was estimated. As shown in Fig. 3 this length increases linearly with the distance of optimization D i and reaches 350 m when the beam is optimized at 1 km. At the same time the averaged number of filaments across the laser beam profile decreases from 17 to ~ 3 beyond 300 m and keeps roughly the same value until 1 km. We note that this number is in direct correlation with the ratio P/P cr where P cr = 5 GW is the critical power for self-focusing in air at 800 nm for pulse duration longer than 200 fs [START_REF] Liu | Direct measurement of the critical power of femtosecond Ti:sapphire laser pulse in air[END_REF]. Indeed, P/P cr was equal to 40 at 150 m and dropped to ~13 for larger distances. This correlation is in good agreement with theoretical predictions [START_REF] Couairon | Femtosecond filamentation in transparent media[END_REF].

Evolution of the laser spectrum at 1 km as a function of the chirp

The spectrum of the laser recorded after propagation shows strong modifications due to nonlinear interactions depending on the intensity of the initial pulse. The inset of Fig. 4 shows two typical results obtained after 1 km propagation corresponding to an initial pulse duration of 0.84 ps (red curve) and 6 ps (black curve) respectively. The comparison of the two curves shows that the pulse corresponding to the shorter duration, which is also the most intense, undergoes a global redshift. The red points in Fig. 4 show the variation of the mean wavelength as a function of the initial pulse duration, when the pulse duration was varied by imposing a negative chirp. We attribute the Stokes shift of the propagated spectrum to the retarded Kerr effect in air [START_REF] Ripoche | Determination of the time dependence of n2 in air[END_REF] and develop a very simplified model in order to explain more quantitatively our measurements. By taking into account only phase effects and neglecting diffraction and ionization, the evolution of the electric field E in the semi-wave approximation reads:

where k is the wavenumber and n(t) represents an intensity dependent index of refraction. By taking the Fourier transform of the electric field E, we obtain the evolution equation of the spectrum F: By multiplying each member of Eq. ( 3) and adding the complex conjugate, we obtain the propagation equation of the modulus of the spectrum:

We characterize the redshift by computing the mean frequency <> of the spectrum defined by

F d F d              (
where n 2 is the instantaneous Kerr effect coefficient and  is the delay. Recall that the variations of the pulse duration are obtained by moving the distance between the compressor gratings. Under this condition the intensity depends on pulse duration in order to keep the energy constant. We model the intensity I(t) by an expression of the form

2 0 0 ( ) exp( / ) t I t I t t t     
which represents the intensity envelope of a chirped pulse of duration t and energy 00 tI   independent of the duration. Here t 0 corresponds to the minimum pulse duration obtained for the compressed pulse.

By using the expressions of n(t) and I(t) the evolution of the mean frequency is given by the following integral :

t t u t u t d kn I t t dtdu dz t                  (9)
The integration gives:

2 2 /2 2 0 0 2 2 1. 2 2 t d kn I t t e erf dz t                                         (10)
In practice, for the values of  and t considered here the second term is negligible so that

2 0 0 / / 2 d dz kn I t t      
. Finally the redshift after a length of propagation z is given by:

2 0 0 0 0 0 , 2 kn I t z t             (11)
showing that the redshift is inversely proportional to the pulse duration t. Figure 4 shows (black curve) the centroid wavelength redshift computed according to Eq. ( 11). This model although very simplified is in good agreement with the measurements (red dots).

Fig. 4. Red shift of the spectrum after 1 km of propagation in air as a function of the initial pulse duration. Measurements (red point) and calculation (continuous black curve) based on Eq. [START_REF] Salamé | Propagation of laser filaments through an extended turbulent region[END_REF]. Insert: spectrum for 6 ps (black curve) and spectrum for 0.84 ps (red curve).

Interaction between long range filaments and solid materials

White-light generation at long distance up to 1.5 km in transparent materials for visible spectrum

We studied at different distances the ability of the filaments to generate conical white light in transparent windows made of Silica, CaF 2 , BK7 and Sapphire, materials which are well suited for supercontinuum generation [START_REF] Ripoche | Determination of the time dependence of n2 in air[END_REF]. For the optimized distances D i between 150 m and 1 km, the presence of high intensity filaments gave rise to significant white light generation. The continuum generated by the filaments in the material was observed on a diffusing screen, and a collecting lens was used to image the whole cone of white light into the fiber of the spectrometer. Measured spectra at 500 m are presented in Fig. 5, where each curve is an average over 10 shots. The black curve is the spectrum of the beam in absence of windows. One can see that the contribution of white light emitted by the solids is not negligible over the white light emitted by the filamentation in air. On average the continuum created in the transparent materials was detected from 400 nm to 1.1 m. As for the number of filaments presented in Fig. 3, we observe a drop in the efficiency of supercontinuum generation in solids after 150 m and then a stable production until 1 km. Only light channels were obtained at the optimized distance of 1.5 km. Those light channels were also powerful enough to induce filamentation in transparent windows and to produce white light emission.

Long range filament effects on IR materials

Different impact tests were performed during this study by changing the distance between the laser source and Germanium samples, the chirp, and the telescope magnification. The measurements have been performed under different weather conditions. The distance between the laser source and the sample was changed from 30 m up to 1300 m. The damage effect on the sample of a single shot and several thousand laser pulses was studied. The weather conditions were measured to investigate the influence of these effects on the filamentation and damaging behaviour. Chirp values varying between 1 ps and 4.3 ps were used to reach a maximum of energy in the filaments for different distances and different weather conditions. Figure 6 presents the damage on Ge samples. The damage caused to Ge remains on the surface. Even with one pulse, the Ge window could be damaged at some points on the surface. The high intensity in the filaments caused this punctual damage behavior [START_REF] Brodeur | Ultrafast white-light continuum generation and self-focusing in transparent condensed media[END_REF]. Displacement of the filaments inside each laser pulse profile caused damages on different position of the Ge irradiated surface. Thus for a series of 600 pulses the whole irradiated surface was damaged. The damage decreases with longer distances and depends also on the weather conditions. Concerning weather conditions we can resume that turbulences and fog reduce the damage impact on the IR-windows, but normal rain does not reduce the damage behavior.

Conclusion

In conclusion, by optimizing the parameters of beam diameter, focalization and chirp we managed to sustain plasma channels formation over 350 m at a target distance of 1 km. It was demonstrated that the effect of chirping the pulse is postponing the filament formation by decreasing the initial peak power and not pre-compensating the dispersion of air. The supercontiuum generated in transparent solids is detected from 400 nm to beyond 1.1 m with different efficiency depending on the material. Above 1 km the filaments transform into intense bright light channels. At 1.5 km, such light channels were observed and were able to induce filamentation in transparent windows with white light supercontiuum generation.

We have shown also that the high intensity filament core is responsible for the damages measured on Germanium samples. Germanium samples could be damaged by means of filaments in the km range. The damage thresholds for Ge samples are much lower with femtosecond laser pulses than for longer laser pulses. Tests have shown that damage effects are possible under free-field propagation conditions. Concerning weather conditions we can resume that turbulences and fog reduce the damage impact on the IR-windows, but normal rain does not reduce the damage behavior.

Fig. 1 .

 1 Fig. 1. (a) Schematic of the interior of the T&T container. The clean room with the laser system is shown on the left-hand side and the control room is at the right-hand side of this schema. (b) Picture of the 2.5 km long firing range at DRDC Valcartier, where the tests were conducted.

D i : D 1 =

 1 150 m, D 2 = 300 m, D 3 = 500 m, D 4 = 1 km and finally D 5 = 1.5 km. For each distance D

Fig. 2 .

 2 Fig. 2. Measurement of position of the plasma channels (black dot) and the position of the light channels (grey large line) for different distances Di. Inset: impacts on a photographic paper showing a plasma channel and a light channel at 700 m for optimized parameters at 1 km.

Fig. 3 .

 3 Fig. 3. Measurement of the length over which ionization occurs (square) and average number (bars) of filaments across the laser profile as a function of the value of Di.

Fig. 5 .

 5 Fig. 5. Generated spectrum in different transparent materials after a propagation of 500 m. The black curve shows the spectrum of the laser pulse before the sample.

Fig. 6 .

 6 Fig. 6. Damage impacts on Ge at different distances (left: 500 m, right: 1000 m).

  By using the property of the inverse Fourier Transform, we obtain after integration over 

	0 d dz  ( ) 0 5 10 15 20 dI k n t Number of filaments dt     now specify n(t) and I(t). The retarded Kerr effect can be modeled by the expression: 500 1000 0 100 200 300 400 * ( ) ( ) C.C. 2 k dE n t E dt I t dt dt          and finally ( ) d dt I t dt dz     , where =  is the circular frequency. Let us (7)  Length of the ionized zone (m) / 2 0 ( ) ( ) , u n n t e I t u du      
												Distance D i (m)
	dE	dz		i	k	t ( n	)	E
															5)
								d	dz 		ik d       	( ) *e n t EF	i	2 	t	dt	  	( ) I t dt		C.C.	(6)

Acknowledgments

We acknowledge the support from a DRDC Applied Research Program and a DRDC Technology Investment Fund and from Délégation Générale pour l'Armement DGA through grant REI 2009 34 0025. We thank also the technical support from Michèle Cardinal, Marcellin Jean and Alain Fernet from DRDC and the participation of Bruno Desruelle from DGA.


A Jarnac

M Durand

Y Liu

B Prade

A Houard

V Tikhonchuk

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

Study of Laser Induced Plasma Grating Dynamics in Gases

Keywords: Non-linear optics, filamentation, plasma grating, ambipolar diffusion, recombination

The relaxation of a plasma grating resulting from the interference of two crossing laser filaments in molecular and atomic gases is studied experimentally. Dissipation of the grating fringes is dominated by ambipolar diffusion in atomic gases and by a combination of ambipolar diffusion and collision-assisted free electron recombination in molecular gases. A theoretical model of the grating evolution is developed and compared to experimental results. Good agreement with simulations allows extracting plasma properties such as electron density, diffusion and recombination coefficients in Ne, Ar, Kr, Xe, N 2 , O 2 , CO 2 and air at atmospheric pressure.

Introduction

Femtosecond laser filamentation has raised much interest in the last decade in various domains of applied physics such as guiding electric discharges [1,2], atmospheric monitoring through LIDAR technique [3], THz generation [4], white-light continuum generation [5] or virtual radiofrequency antenna [6]. Resulting from a dynamic competition between the Kerr effect and plasma defocusing, filamentation appears spontaneously during the propagation of an intense femtosecond laser pulse, provided the initial peak power exceeds a threshold value P cr (a few GW in gases). An intensity reaching 5×10 13 W/cm 2 in air is maintained in the core of the beam over an extended length, producing a long string of plasma in the wake of the pulse [7]. In order to remotely control the filamentation process for long range applications, much attention has been given to intersecting filaments by another femtosecond laser pulse. In the intersection region, interference of the two pulses gives rise to a plasma grating [8][9][10]. Recent studies have highlighted the role of the grating as a way to manipulate femtosecond pulses: energy exchange [10][11][12], interruption of filament [13][14][15] or redirection of a third beam [16,17]. Measurements of the plasma density as well as the relaxation process of the plasma grating are crucial for the understanding of the phenomenon. In a recent article we reported first results concerning the experimental characterization of the plasma grating decay in several molecular and atomic gases [18].

In this manuscript we present the complete theoretical model and the experimental results allowing describing the plasma dynamics in the laser induced grating for all studied gases. We demonstrate that the plasma fringes are washed out by the ambipolar diffusion in atomic gases and by a combination of the ambipolar diffusion and collision-assisted free electron recombination in molecular gases. A theoretical model based on a Bragg diffraction is developed to interpret the experimental results. Data are fitted by the model, allowing extraction of the plasma properties such as the free electron density, diffusion and recombination coefficients in Ne, Ar, Kr, Xe, N 2 , O 2 , CO 2 and air. Some of these parameters were already measured several years ago but for a very low gas pressure and in a fully ionized plasma [19][20][21]. The recombination coefficient for air was also estimated and measured in the case of single filament, showing discrepancy between different values obtained [22][START_REF] Iondi | Recombination[END_REF][START_REF] Tzortzakis | [END_REF]. In the present study we measured these quantities at atmospheric pressure in a weakly ionized gas.

The outline of this paper is the following: first, we study in Sec. 2 the plasma by using temporally resolved in-line diffractometry measurements to determine the initial plasma density and the recombination coefficient of the filament in the crossing area. Due to the laser field interference, this density is higher than the density in single filaments, and needs to be evaluated. We then elaborate in Sec. 3 a theoretical model describing the relaxation of the grating. We finally study the decay of grating using a pump-probe experiment. Best fits between calculations and experimental results yield the initial free electron density, its time evolution as well as the diffusion coefficient in all studied gases.

Study of the plasma relaxation

Experimental set-up

We used in the experiment a chirped pulse amplified (CPA) femtosecond laser system delivering 50 fs pulses at 800 nm with a maximum energy of 15 mJ at a repetition rate of 100 Hz. At the output of the CPA system the laser beam has a Gaussian spatial intensity distribution with a diameter of 14 mm. The output pulse is split into three beams by two sequential beam splitters (10%/90% and 50%/50%). Two pump laser pulses of 1 mJ with a vertical polarization are focused with a 1000 mm focal lens in a cell containing the gas at atmospheric pressure, where they form two filaments. A plasma grating is formed in the intersection region (see details in Refs. [17,18]). Measurements were performed in air, O 2 , N 2 , CO 2 , Ne, Ar, Kr and Xe.

In this first experiment, an unfocused weak probe beam is used to illuminate the plasma grating (Fig. 1). The probe beam is sent transversely through the plasma. Since it propagates perpendicularly to the plane containing the fringes, it is only sensitive to the global phase object it traverses, but not to the plasma density modulation. The probe can be delayed with respect to the plasma formation time to record the temporal evolution of the diffraction pattern caused by the plasma bubble. We used a CMOS camera (uEYE UI-1240SE) placed at the distance H = 38 cm away from the plasma grating. A similar in-line diffractometry method has already been used in Refs. [START_REF] Tzortzakis | [END_REF][25][26] to determine the temporal evolution of the plasma electron density in the case of a single filament produced in air.

Calculation of probe diffraction by a plasma bubble

The plasma electron density evolution was calculated from the diffractometry measurement of the probe beam. The plasma bubble produced at the intersection of the two filaments induces a change of refractive index  on the path of the probe beam. The plasma refractive index reads , where is the plasma frequency,  is the laser angular frequency,  0 is the vacuum permittivity, e is the elementary charge, m e is the electron mass and  e is the electron plasma density. The ratio is small, so the variation of the plasma refractive index is directly proportional to  e :

(

where  c =1.7×10 21 cm -3 is the electron critical density at 800 nm. Based on the crossing geometry of two filamentary beams, we assume a super-Gaussian plasma density profile along x direction and a Gaussian profile along z direction with characteristic lengths L 1 and L 2 . Due to geometric considerations we expect L 2 to be slightly larger than L 1 . Then the change of refractive index in the interaction area can be described as (2) where n 0 is the maximum variation of the index at the center of plasma and a is the super Gaussian order. Then the phase shift due to the probe beam propagation through the plasma reads

(3)

Assuming that the probe beam has a Gaussian spatial profile with a width w 0 ~7 mm, the amplitude of diffracted wave at the distance H from the plasma is expressed as follows:

(4)

The expression above provides a relation between the plasma electron density and the diffraction pattern of the probe intensity . The temporal evolution of the probe diffraction pattern provides information on the plasma relaxation. It is governed by two phenomena: the free carrier diffusion and recombination. In a plasma created by filamentation, the degree of ionization is rather low and the electron recombination is based on two distinct processes: the recombination on parent ions and attachment to neutral atoms. Zhou et al. [27] demonstrated that in gases the first process is faster. The electron diffusion could be important over a distance of a few microns, comparable to the electron mean free path, but it can be neglected on the plasma scale of a few tens of microns. Consequently, the decay of the electron plasma density from the initial value   can be expressed as [START_REF] Tzortzakis | [END_REF] (5). The electron recombination coefficient  can be found from the contrast of the fringes as the ratio , where I 0 is the background intensity and is the difference of the probe beam intensity with and without the plasma.

Experimental results and discussion

Fig. 2 presents an image obtained from the diffraction of the probe beam by the plasma bubble in air. Two images have been recorded: the probe transmitted though the plasma (a) and the background without plasma (b). The pictures shown in Fig. 2(a) and 2(b) were taken when the probe beam was sent  = 1.13 ps after the formation of the plasma grating. After subtraction of the background we obtain the diffraction pattern shown in Fig. 2(c). The x-profile of the fringes extracted from Fig. 2(c) along the y position where the fringes are at their maximum is shown in Fig. 2(d). The contrast is calculated by dividing the profile by the maximum value of the background taken at the same coordinate y. These measurements were performed for different gases as a function of the delay between the probe beam and the formation of the plasma. The contrast extraction procedure was routinely repeated for each delay .

The calculated fringe profile is best fitted to the experimental one (see Fig. 3) through adjustable parameters a, L 1 and L 2 that characterize the shape of the plasma bubble. For all gases it was found a = 2, L 1 = 40 µm and L 2 = 90 µm, except for Xenon for which L 1 = 80 µm. Once the plasma shape is fixed, the maximum contrast is linked to the electron density  e . We considered here only the real part of the refractive index n 0 . By adding a complex part related to absorption one could improve the fit of the central fringe at x = 0.

The model allows retrieving from the maximum fringe contrast the plasma density evolution. The fringe contrasts for each delay  measured experimentally and fitted by Eq. ( 5) are reported in Fig. 4 for all gases studied. The striking result is a significant difference in the electron relaxation times in the molecular and atomic gases. The relaxation time in the diatomic gases is less than 100 ps, which is radically different from the atomic gases, where the electron lifetime exceeds 1 ns. A very slow plasma decay in the case of argon is explained by the fact that the electron recombination proceeds through the long lived Rydberg states [28] with a very low ionization potential. It is possible that in our experiment the intensity of the probe is sufficient to re-ionize the Argon atoms in Rydberg states, which would explain the longevity of the plasma grating in this case.

One can see that measurements can be well fitted using this equation. The inverse dependence of the plasma density with time is indeed confirmed experimentally. The maximum at  = 0 gives the initial density of the plasma  0 and the decay rate gives the recombination coefficient . The values shown in Table 1 are consistent with the data from Refs. [26,29,30].

The electron recombination could occur through several mechanisms in the filament plasma: the direct radiative electron-ion recombination, collision-assisted electron-ion recombination and, for molecules, the dissociative recombination. The direct radiative electron-ion recombination is an intrinsic property of the atom or molecule, which does not depend on the gas temperature or pressure. Its characteristic time is in the range of a few hundred ns, much longer than the decay times measured here. The dissociative recombination process corresponds to a collision of an electron with a molecular ion resulting in two neutral atoms, e + X 2 + → X + X. The corresponding cross-section could be as high as 10 -13 -10 -14 cm 2 at room temperature, but it decreases by 2 -3 orders of magnitude for the electron temperature T e ~ 1 -10 eV [31]. Thus it also leads to a recombination time in the nanosecond scale, which is inconsistent with the observations. The last process is the three body recombination where the electron-ion recombination is assisted by a neutral atom or molecule: e + X + + X → X + X. Assuming the electron density  e of the order of 10 17 cm -3 and the neutral density  at = 2.7×10 19 cm -3 , the collision-assisted recombination time is in the sub-nanosecond range, and it dominates under our experimental conditions.

Several authors have discussed the decay of plasma generated by laser filamentation in air in terms of the collision-assisted recombination [START_REF] Tzortzakis | [END_REF]25]. The coefficient  = K at is proportional to the density of neutrals. The constant K was previously determined at low gas pressures and at the normal temperature 300 K for a few gases: for Helium K ~ 10 -26 -10 -27 cm 6 /s [19], for Oxygen K ~ 10 -30 cm 6 /s [20] and for Neon K ~ 10 -27 cm 6 /s [21]. The laser filament crossing experiment provides the possibility to evaluate K for several gases at higher electron temperatures and at the atmospheric pressure. The results are reported in Table 1. For every gas the initial electron density is of the order of 10 17 -10 18 cm -3 . This value is in agreement with simulations using the PPT (Perelomov, Popov and Terent'ev) model of multiphoton ionization [32], under the assumption of a complete interference between the two crossing laser fields and a filament intensity for each gas taken from ref. [33]. Relying on these measurements we are now in a position to determine the characteristics of the plasma generated from the interference of two laser filaments in different gases. The extracted values of the peak densities and the recombination constants will be used as starting parameters for the analysis of the temporal evolution of the plasma grating.

Air

Study of the plasma grating

Experimental set-up

The experimental set-up is presented in Fig. 5. The plasma grating is generated similarly to the previous experiment. A weaker third probe beam at the frequency 2, this time collinear with one of the filament forming pulse, is diffracted by the grating in the direction of the other filament. In order to satisfy the Bragg condition, the second harmonic probe beam is diffracted by the second order anharmonic term of the grating profile. The diffraction efficiency of the grating is measured as a function of the time delay  between the probe beam and the formation time of the grating. The experiment was performed for three configurations  = 7, 14 and 90°. A photodiode was used to measure the diffracted probe beam. From these measurements, the parameters of the plasma decay, the electron diffusion and the electron recombination, can be retrieved.

Theoretical model

The goal of this part is to provide a model for interpretation of the experimental data concerning the lifetime of the plasma grating. We consider two optical beams crossing at an angle  in the (y, z) plane. The electric field related to two overlapping waves can be expressed as (see scheme in Fig. 6):

The interference pattern is defined by the intensity that presents the periodic modulations along z-axis with the wave vector , where  is the fringe spacing of the grating. For our calculations we assume that amplitudes of both filamentary beams, A 1 and A 2 are equal.

During the filamentation process, a laser pulse of a duration t p leaves in its trail a thin, weakly ionized plasma channel. The peak plasma density can be estimated from the equation (7) where  M is the coefficient of M-photon ionization with the integer M depending on the gas characteristics and the ionization regimemulti-photon or tunnel, according to the Keldysh-PPT model. In the present case, we are interested in the maximum electron density obtained from the constructive interferences of the two intercepting filaments. For each gas studied, the ionization regime was defined according to Refs. [7,[START_REF] Chin | From ultiphoton to Tunnel Ionization[END_REF] and the values of the parameter M are presented in Table 2. According to Eqs. ( 6) and ( 7), the plasma density is proportional by the M-th power of the interference field: (8) with  0 the maximum electron density in the plasma grating. Among the two phenomena that are responsible for the relaxation of the plasma grating, the electron recombination was considered in the previous section. To evaluate the role of the second process, the electron diffusion, we consider the characteristic spatial scales of the problem. For the electron density in plasma ~ 10 18 cm -3 and the electron temperature of a few electron-volts, the electron Debye length  D is about 0.07 m and the electron mean free path  en is approximately 0.15 m. Both of them are smaller than the grating period  that is in the range from 0.6 to 6 µm.

Air
Consequently, the diffusion is not controlled by electrons, but is associated to their parent ion mobility. It is known as the ambipolar diffusion [START_REF] Condon | Handbook of physics[END_REF], whose coefficient reads as , where T i is the ion temperature, is the ion diffusion coefficient, k B is the Boltzmann constant and  i is the ion mobility. Under these assumptions, the temporal evolution of the electron plasma density obeys the partial differential equation:

(9) where the first term in the right hand side describes the electron diffusion along the plasma modulation direction. The initial condition to this equation is given by expression (7). It contains the main grating period  and its harmonics /n with n ≤ M: (10) thus leading to a system of M-coupled differential equations for the electron density harmonics  n , which have been solved numerically.

The lifetime of the plasma grating is measured by the probe beam operating at the second harmonic 2According to the Bragg condition, the diffraction efficiency is proportional to the amplitude of the second harmonic of the grating  2 . The principle is schematized in Fig. 7. After solving the M-coupled differential equations ( 9), we retain only two terms in the electron density while solving the Helmholtz equation for the probe beam. Then, according to Eqs. ( 1) and ( 10), the plasma refraction index reads:

Correspondingly, the field of the probe beam splits into the transmitted  0 and diffracted  r waves:

(12) The amplitudes of these two waves are described by two differential equations: (13) with the boundary conditions corresponding to the incident fundamental wave,  0 (0) = A 0 , and to the absence of the diffracted wave,  R (0) = 0. For a grating size L 2 less than one hundred microns one can neglect the probe wave depletion, and assuming  0 to be constant, solve the equation for the diffracted wave only. Then, one obtains a simple expression for the reflected intensity (14) Thus, I R depends on the second harmonic of the grating amplitude  2 , which follows an evolution ruled by Eq. (9). By measuring and fitting the decay of the reflected intensity we can determine experimentally the values D and . The characteristics of the plasma determined in Sec. 2 served as initial conditions in the simulations. The temporal evolution of the plasma grating and the characteristic electron densities in air are shown in Fig. 8 for relevant parameters. The average electron density decreases with time relatively slowly, approximately by a factor 2 in 40 ps, which is consistent with the result presented for air in Fig. 4. The grating amplitudes corresponding to the main and the second harmonic decrease faster: the main harmonic decreases by a factor 2 in 30 ps while the second harmonic requires less than 20 ps.

Experimental results and discussion

Fig. 9 1 shows the intensity of the probe beam diffracted by the grating as a function of time in molecular and atomic gases for three angles . In all atomic gases, there is a strong dependence of the signal diffracted with the angle . It decreases with increasing crossing angle as experimentally shown in [8,17]. It can be explained by the fringe spacing . A smaller grating period leads to a reduction of the time necessary for a diffusive wash out of the fringe pattern. One can note that the lifetime of the grating is much shorter than the plasma recombination time measured in Fig. 4 for the same gas. We can then conclude that in atomic gases the role of recombination in the grating dynamics is negligible compared to ambipolar diffusion. On the contrary, in molecular gases, the plasma recombination time is comparable to the relaxation time of the grating, so that both the recombination and the diffusion play a role in the grating relaxation. In each gas, the three experimental curves are fitted with one set of parameters ρ 0 ,  and D. The numerical fitting procedure is restrictive, since for a same set of parameters  0 and , the time dependence of the plasma and grating decay must be reproduced. The retrieved parameters ρ 0 ,  and D from this calculation are presented in Table 3. The coefficients D am correspond to the calculated ambipolar diffusion coefficient, with  i from Refs. [START_REF] Condon | Handbook of physics[END_REF][START_REF] Polynkin | [END_REF], for T i = 300 K (room temperature) and T e = 0.5 eV [37]. The diffusion coefficients D obtained from our experiment are in good agreement with calculated values. The value of the electron recombination coefficient of air is consistent with the experiment performed with a single filament [START_REF] Tzortzakis | [END_REF]25]. Values of the diffusion coefficient have not been measured before in this range of pressure and temperature.

Air

Conclusions

We have fully characterized the plasma and the grating formed at the intersection of two crossing filaments in several atomic and molecular gases. Two processes, which are responsible for the plasma relaxation are identified. In atomic gases, the plasma grating lifetime is very sensitive to the filaments crossing angle , but not in molecular ones.

Therefore, the dissipation of the grating is dominated by the ambipolar diffusion in atomic gases and by a combination of the ambipolar diffusion and the collision-assisted free electron recombination in molecular gases. We have elaborated a theoretical model based on the Bragg diffraction that describes the grating relaxation. The numerical results are in good agreement with the experimental observations. By combining the observations with the model we extracted the electron density ρ 0 , the recombination time  and the diffusion coefficient D in all tested gases. Such values have never been determined at the atmospheric pressure and the room temperature. Therefore, we provide a pioneering experimental technique to measure inherent characteristics of a specific type of plasma formed by laser filamentation.

Figure captions

FiguresFig. 1

 1 Figures

  Fig. 1.

Fig. 2 .

 2 Fig. 2.

Fig. 3 .

 3 Fig. 3.

Fig. 4 .

 4 Fig. 4.

Fig 5 .

 5 Fig 5.

Fig. 6 .

 6 Fig. 6.

Fig. 7 .

 7 Fig. 7.

Fig. 8 .

 8 Fig. 8.

Fig. 9 .

 9 Fig. 9.

Figure 1 :

 1 Figure 1: Experimental set-up of the in-line diffractometry measurement.

Figure 2 :

 2 Figure 2: Recorded diffraction patterns for a delay  = 1.13 ps with respect to the plasma formation in air: (a) Image of the strips obtained on the camera after the probe beam has been diffracted by the plasma bubble; (b) Spatial profile of the probe beam without plasma bubble; (c) Images of fringes due to diffraction after the subtraction of the background (b); (d) xprofile of (c) at the coordinate y where the fringes are at their maximum.

Figure 3 :

 3 Figure 3: Theoretical fit of the fringe pattern corresponding to Fig. 2(d).

Figure 4 :

 4 Figure 4: Decay of the experimental (black dots) and fitted (red open circles) diffraction contrast extracted from Fig. 3 as a function of the delay .

Figure 5 :

 5 Figure 5: Experimental set-up for measuring of the plasma grating lifetime

Figure 6 :

 6 Figure 6: Schematic representation of two plane waves of same wavelength  crossing with an angle . The black lines represent the grating formed at the intersection.

Figure 7 :

 7 Figure 7: Schematic of the reflection of the 2 probe beam by the plasma grating.

Figure 8 :

 8 Figure 8: (a) Temporal evolution of one period of the plasma density grating for the following set of parameters:  = 0.8 m,  = 14°, D am = 1 cm 2 /s,  = 5.10 -8 cm 2 /s and  0 = 10 18 cm -3 . (b)Left axistemporal evolution of the average density (circle), the first (square) and the second (solid line) harmonics of the grating. Initially the fringe maxima are 4 times above the average density for M = 5. Right axisreflected intensity of the 2 probe beam on the grating (dash line).

Figure 9 :

 9 Figure 9: Normalized intensity of diffracted probe signal as a function of the delay  (ps) for crossing angles  of 7°, 14° and 90°. Measures are represented by dots. Calculations are represented by solid lines.

Table 1 :

 1 Plasma parameters determined by diffractometry technique. The parameters  0 and  were obtained from best fitting the fringe contrast and its decay in time. The constant K was calculated using the relation  =

		N 2	O 2	CO 2	Ne	Ar	Kr	Xe
	1×10 18  (cm 3 /s) 5×10 -8  0 (cm -3 ) K (cm 6 /s) 2×10 -27	1×10 18 3×10 -8 1×10 -27	1×10 18 9×10 17 4×10 -8 2×10 -7 2×10 -27 7×10 -27	7×10 17 6×10 -9 2×10 -28	3×10 18 2×10 -10 7×10 -30	2×10 18 1×10 -9 4×10 -29	9×10 17 2×10 -9 6×10 -29
	K at .

N 2 O 2 CO 2 Ne Ar Kr Xe

	M	5	7	5	5	5	5	4	4

Table 2 :

 2 Number of photons needed in the ionization process according to the Keldish-PPT model.

Table 3 :

 3

Plasma characteristics determined experimentally. The same set of parameters D,  0 and  is obtained from best fitting the plasma and the grating decay for each gas. The first line presents the calculated value of the diffusion with  i taken from Refs.

[START_REF] Condon | Handbook of physics[END_REF][START_REF] Polynkin | [END_REF]

.

The sharp dips observed in Air, O

and N 2  = 90° are due to spontaneous revivals of the refractive index which diffract the probe pulse before it reaches the grating volume. For clarity, the dips observed at other angles  have been removed.

Acknowledgments

We acknowledge fruitful discussions with Dr B. Forestier, technical help from J. Carbonnel, A. Dos Santos and Dr J. Gautier and financial support from DGA.