Pengji Ding

Sergey Mitryukovskiy

Aurélien Houard

Eduardo Oliva

Arnaud Couairon

André Mysyrowicz

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

Backward Lasing of Air plasma pumped by Circularly polarized femtosecond pulses for the saKe of remote sensing (BLACK)

Keywords:

Recently, S. Mitryukovskiy et al. presented experimental evidence showing that backward Amplified Spontaneous Emission (ASE) at 337 nm can be obtained from plasma filaments in nitrogen gas pumped by circularly polarized 800 nm femtosecond pulses (Opt. Express, 22, 12750 (2014)). Here, we report that a seed pulse injected in the backward direction can be amplified by ~ 200 times inside this plasma amplifier. The amplified 337 nm radiation can be either linearly or circularly polarized, dictated by the seeding pulse, which is distinct from the non-polarized nature of the ASE. We performed comprehensive measurements as to the spatial profile, optical gain dynamics, and seed pulse energy dependence of this amplification process. These measurements allow us to deduce the pulse duration of the ASE and the amplified 337 nm radiation and the corresponding laser intensity inside the plasma amplifier, which indicates that the amplification is largely in the unsaturated regime and further improvement of laser energy is possible. Moreover, we observed an optical gain in plasma created in ambient air, which is of great importance for the future application of this scheme of backward lasing in remote sensing.

Introduction

Stimulated radiation of air plasma pumped by ultrashort intense laser pulses has attracted growing attention in recent 3 years [START_REF] Dogariu | High-gain backward lasing in air[END_REF][START_REF] Luo | Lasing action in air induced by ultra-fast laser filamentation[END_REF][START_REF] Kartashov | Free-space nitrogen gas laser driven by a femtosecond filament[END_REF][START_REF] Hemmer | Standoff spectroscopy via remote generation of a backward-propagating laser beam[END_REF][START_REF] Sprangle | Remotely induced atmospheric lasing[END_REF][START_REF] Peñano | Remote lasing in air by recombination and electron impact excitation of molecular nitrogen[END_REF][START_REF] Mitryukovskiy | Backward stimulated radiation from filaments in nitrogen gas and air pumped by circularly polarized 800 nm femtosecond laser pulses[END_REF][START_REF] Yao | Gain dynamics of a free-space nitrogen laser pumped by circularly polarized femtosecond laser pulses[END_REF][START_REF] Yao | High -brightness switchable multiwavelength remote laser in air[END_REF][START_REF] Yao | Remote creation of coherent emissions in air with two-color ultrafast laser pulses[END_REF][START_REF] Liu | Self-seeded lasing action of air pumped by 800 nm femtosecond laser pulses[END_REF][START_REF] Wang | Self-seeded forward lasing action from a femtosecond Ti: Sapphire laser filament in air[END_REF][START_REF] Kantashov | Remotely pumped stimulated emission at 337 nm in atmospheric nitrogen[END_REF][START_REF] Point | Lasing of ambient air with microjoule pulse energy pumped by a multi-terawatt femtosecond laser[END_REF][START_REF] Li | Signature of superradiance from a nitrogen-gas plasma channel produced by strong-field ionization[END_REF]. Both backward and forward stimulated emission has been observed in experiments. In particular, the backward stimulated emission is very interesting, because it can be potentially employed for remote sensing applications. The employment of a backward stimulated lasing radiation for remote sensing is expected to bring tremendous improvement of measurement precision and sensitivity, because coherent detection methods such as Stimulated Raman Scattering (SRS) can be then used instead of the incoherent detection of laser induced luminescence [START_REF] Malevich | Ultrafast-laser-induced backward stimulated Raman scattering for tracing atmospheric gases[END_REF][START_REF] Kartashov | Mirrorless backward SRS in free-space gas driven by filament-initiated UV laser[END_REF].

Up to now, two different schemes of backward lasing action have been demonstrated. In the first method, a picosecond ultraviolet (UV) pulse (226 nm) was used to excite oxygen molecules in ambient air [START_REF] Dogariu | High-gain backward lasing in air[END_REF]. Population inversion between the 3p 3 P and the 3s 3 S states of oxygen atom was achieved by two photon dissociation of the oxygen molecules followed by resonant excitation of the atomic oxygen fragments. Both backward and forward stimulated emission at 845 nm has been observed in the experiments. However, application of this scheme for backward lasing generation for remote sensing is difficult due to the poor transmission of the UV pump pulse in atmosphere. Another scheme is based on population inversion in neutral nitrogen molecules. Backward stimulated emission from neutral nitrogen molecules inside a laser plasma filament was first suggested in 2003, based on the observed exponential increase of the backward UV emission with the filament length [START_REF] Luo | Lasing action in air induced by ultra-fast laser filamentation[END_REF]. In 2012, D. Kartashov and coworkers focused a mid-infrared femtosecond laser pulse (3.9 m or 1.03 m) inside a high pressure mixture of argon and nitrogen gas. Backward stimulated emissions at 337 nm and 357 nm were observed with an optimal argon gas pressure of 5 bar and nitrogen pressure of 2 bar [START_REF] Kartashov | Free-space nitrogen gas laser driven by a femtosecond filament[END_REF]. The emission at 337 nm and 357 nm have been identified as being due to the transition between the third and second excited triplet states of neutral nitrogen molecules, i.e. C 3 Π u → B 3 Π g . The population inversion mechanism between the C 3 Π u and B 3 Π g states was attributed to the traditional Bennet mechanism, where collisions transfer the excitation energy of argon atoms to molecular nitrogen [START_REF] Kunabenchi | Nitrogen lasers[END_REF]. Unfortunately, this method cannot be applied for remote generation of backward lasing emission because of its requirement of high pressure argon gas (p > 3 bar).

A few months ago, S. Mitrykovskiy et al. showed that a backward Amplified Spontaneous Emission (ASE) at 337 nm can be obtained from laser filaments in nitrogen gas or its mixture with oxygen pumped by circularly polarized 800 nm femtosecond laser pulses [START_REF] Mitryukovskiy | Backward stimulated radiation from filaments in nitrogen gas and air pumped by circularly polarized 800 nm femtosecond laser pulses[END_REF]. We there hence acronymize this scheme of backward lasing as BLACK (Backward Lasing of Air Plasma pumped by Circularly polarized femtosecond pulses for the saKe of remote sensing). Very recently, Yao et al measured the gain dynamics of this free-space laser at 337 nm by sending a weak seed pulse in the forward direction and examining the amplified signal as a function of the delay between the pump and the seed pulse. Their study confirmed the existence of population inversion and revealed a gain lifetime around ~ 20 ps in 1 bar nitrogen gas [START_REF] Yao | Gain dynamics of a free-space nitrogen laser pumped by circularly polarized femtosecond laser pulses[END_REF].

In this paper, we report that an external seed pulse around 337 nm in the backward direction can be amplified by ~ 200 times inside the plasma amplifier. The divergence of the amplified 337 nm emission is found to be significantly reduced compared to that of ASE. Moreover, the amplified lasing radiation inherits the polarization of the seed pulse. These three observations confirm unambiguously our previous assumption of population inversion between the relevant nitrogen molecular states. We further examined the gain dynamics at different gas pressure, the dependence of the amplified lasing signal on seed pulse energy, and the role of the pump laser ellipticity on the lasing action. With the measured gain lifetime, we are able to estimate the pulse duration of the backward ASE and the amplified 337 nm emission. The 337 nm laser intensity inside the plasma amplifier is then calculated, indicating that the current experiments are in the unsaturated amplification regime. As a result, further increase of the backward laser energy is possible by using higher energy pump or seeding pulse. We also observed amplification of seeding pulses in air plasma created by circularly polarized femtosecond pulses, which confirms that population inversion between the C 3 Π u and B 3 Π g states can be achieved in ambient air. This opens up the door for practical application of this BLACK scheme for generation of backward lasing in ambient air. Finally, we discussed the mechanism for population inversion, in analogy with the well-developed collisionally pumped X-ray laser amplifier.

Experimental setup

In our experiment, a commercial Chirped Pulses Amplification (CPA) laser system (Thales Laser, Alpha 100) was used. This system delivers 42 fs laser pulses at a repetition rate of 100 Hz, with maximum pulse energy of 12 mJ. Two experimental configurations have been employed. In the first scheme, the pump IR pulse and the seed pulse counter-propagate. A schematic experimental setup is presented in Fig. 1. The output laser pulse was split into a main pump pulse and a much weaker second pulse by a 1 mm thick 5%/95% beamsplitter. The pump pulse passed through a waveplate and then was focused by an f = 1000 mm convex lens (L 1 ). A dichromatic mirror was used to reflect the focused 800 nm pump pulses into a gas chamber filled with pure nitrogen gas at 1 bar pressure. This dichromatic mirror reflects more than 99% of the 800 nm pump pulse and it is transparent to the backward ultraviolet emission from the laser plasma situated inside the gas chamber. The second weaker 800nm pulse first passed through a mechanical delay line and then through a 1 mm thick type-I BBO crystal in order to generate femtosecond pulses at 400 nm. The 400 nm pulse was linearly polarized in the vertical direction. The obtained 400 nm pulse was further focused by an f = 100 mm convex lens (L 2 ) inside a 20 mm long fused silica sample to broaden its spectrum through intense nonlinear interaction. We selected the spectrum component around 337 nm with an interference bandpass filter, which has a transmission peak at 337 nm and a bandwidth of 10 nm. The resulting pulse centered at 337 nm, referred to as seeding pulse in the following, was focused by another f = 100 mm lens into the gas chamber from the opposite direction of the pump pulses. The separation between the lenses L 2 and L 3 was adjusted to insure that the geometrical focus of the seeding pulse overlapped with the central part the of the long plasma filament in the longitudinal direction. The transverse spatial overlapping between the geometrical focus of the seeding pulse and the center of the plasma filament was carefully assured by translating finely the focal lens (L 3 ) in the transverse plane. The temporal delay between the 800 nm pump pulses and the seeding pulse at 337 nm could be adjusted by the mechanical delay line. For some of our experiments, we installed a  waveplate for 400 nm radiation after the BBO crystal so that a circularly polarized seeding pulse could be obtained after filamentation inside the fused silica sample. The backward emission from the laser plasma filaments was detected by either a spectrometer (Ocean Optics HR 4000), an intensified Charge Coupled Device (iCCD) camera (Princeton Instrument, model: PI-MAX), a calibrated photodiode, or a sensitive laser power meter (model: OPHIR, NOVA, PE9-C). In a second scheme to measure the temporal dynamics of the optical gain of the plasma amplifier, the seed pulse and the pump pulse were arranged to propagate in the same direction. The spatial overlap of the plasma amplifier and the focus zone of the seed pulse was carefully assured. The temporal delay between them was also varied by a mechanical delay line. The analysis of the 337 nm radiation was performed in the downstream of the filament amplifier.

Experimental results and discussion

Manifestation of spectrum narrowing and seed amplification

We first measured the spectra of the backward emissions from the laser plasma without the seed pulse. With pump pulse energy of 8.5 mJ, a 3 cm long filament is formed [START_REF] Mitryukovskiy | Backward stimulated radiation from filaments in nitrogen gas and air pumped by circularly polarized 800 nm femtosecond laser pulses[END_REF]. In Fig. 2, we present the backward emission spectra for circularly and linearly polarized pump pulses. The emission peaks around 315 nm, 337 nm, 357 nm, 380 nm have been well indentified as the transition between the C 3 Π u and B 3 Π g states of the neutral nitrogen molecules with different initial and final vibrational quantum numbers, which has been denoted in Fig. 2. The tremendously increased radiation at 337 nm with circular laser polarization suggests occurrence of stimulated emission, referred to as ASE in our previous work [START_REF] Mitryukovskiy | Backward stimulated radiation from filaments in nitrogen gas and air pumped by circularly polarized 800 nm femtosecond laser pulses[END_REF]. Thanks to a better spectral resolution, a clear spectrum narrowing of the 337 nm radiation in case of lasing is noticeable, which is a common feature of lasing emission compared to spontaneous one. The 337 nm laser bandwidth was measured to be 1.06 nm, which corresponds to the spectral resolution of our spectrometer. Please note that the bandwidth corresponding to the transition between the C 3 Π u and B 3 Π g state is well known to be around ~ 0.1 nm [START_REF] Hariri | Amplified spontaneous emission in N2 lasers: Saturation and bandwidth study[END_REF]. The backward emission spectra in the presence of the seeding pulses are presented in Fig. 3. Considering the intensity of the seed pulse at the 337 nm spectral position, we estimated that the seed pulse is amplified by a factor of ~ 200 times (Fig. 3 (a)). Compared to the backward ASE, the amplified 337 nm laser emission energy was 40 times higher. With an effective amplifier length of 4.5 mm, we estimated the small-signal gain coefficient to be

  eff f l I I g 0 ln  = 11.8 cm -1 .
Here I f , I 0 , l eff are the intensity of the amplified 337 nm emission, the intensity of the seeding pulse, and the effective length of the plasma amplifier. The effective length of the amplifier is different from the geometrical filament length for the backward ASE and will be discussed later. The population inversion density is then estimated as

s g n   
~ 10 15 cm -3 , where  s ~ 10 -14 cm 2 is the stimulated cross-section of the C 3 Π u → B 3 Π g transition [START_REF] Sprangle | Remotely induced atmospheric lasing[END_REF][START_REF] Peñano | Remote lasing in air by recombination and electron impact excitation of molecular nitrogen[END_REF]. In the case of linearly polarized pump pulses (Fig. 3 (b)), no detectable ASE was observed and no amplification of the seed pulse could be observed.

Polarization of the ASE and the amplified 337 nm radiation

We then studied the polarization of the backward ASE and the amplified 337 nm emission. In order to analysis the polarization properties of the lasing radiation, we installed a Glan-Taylor prism before the detecting photodiode. In the experiment, we recorded the intensity of the transmitted 337 nm radiation as a function of the rotation angle of the Glan-Taylor prism. The result for the ASE obtained without seed pulse is presented in Fig. 4 (a), indicating that the ASE is not polarized. For linearly polarized seed pulses in the vertical direction, we observed that the amplified lasing signal is also linearly polarized in the same direction (see Fig. 4 (b)), evidenced by the good agreement between the experimental results and the theoretical fit with Malus' law. The result for circularly polarized seeding pulses is presented in Fig. 4 (c), where a circularly polarized amplified emission is also observed. The maintenance of the pulse polarization during the amplification is in agreement with our hypothesis that that population inversion is present and responsible for the seed pulse amplification.

Spatial profile of the ASE and the amplified 337 nm emission

We measured the spatial profiles of the ASE and the amplified lasing emission with the iCCD camera. In Fig. 5 (a), the spatial profile of the backward ASE is shown. The ASE exhibits a Gaussian distribution with a divergence of 9.2 mrad. In the case of linearly polarized 800 nm pump pulse, no backward emission at 337 nm was observed with the iCCD. We present the spatial profile of the seed pulse in Fig. 5 (b). In the presence of both pump and seed pulse, an extremely intense 337 nm radiation was found, as presented in Fig. 5 (c). This amplified stimulated emission shows a divergence angle of ~ 3.8 mrad, much smaller than that of the ASE and the seed pulse.

Seed pulse energy dependence

In order to determine whether the optical amplification is in the saturated regime, we tuned the energy of the seed pulse with a variable metallic optical density. The result is presented in Fig. 6. For this measurement, the pump laser energy was fixed at 8.5 mJ. The seed pulse energy was less than 1 nJ and we estimated the absolute value with a calibrated photodiode. Due to the large bandwidth of the seeding pulse, only a small portion of the seeding pulse spectrum is amplified (Fig. 3 (a)). We therefore calculated the effective seeding pulse energy by considering the spectrum portion contained in the 1.06 nm bandwidth of the amplified 337 nm radiation. The maximum output amplified radiation at 337 nm was measured to be 1.2 nJ. This output pulse energy corresponds to an energy conversion efficiency of 1.4 × 10 -7 from 8.5 mJ pump pulses. Therefore, the pulse energy of the ASE was deduced to be ~ 0.03 nJ based on the enhancement factor of 40, which gives a conversion efficiency of 3.5 × 10 -9 . In the next section, we will estimate the corresponding laser intensity and show that it is less than the saturation laser intensity of the amplifier. To evaluate that, we measured systematically the 337 nm laser emission intensity by rotating the λ/4 waveplate for the pump pulses. In Fig. 7 (a), the result for the backward ASE is first presented as a function of the rotation angle  of the waveplate, which has been reported in our previous work [START_REF] Mitryukovskiy | Backward stimulated radiation from filaments in nitrogen gas and air pumped by circularly polarized 800 nm femtosecond laser pulses[END_REF]. Intense ASE was observed only with circularly polarized pump pulses and shows dramatic decrease when the ellipticity deviates from 1. In the presence of a constant linearly polarized seeding pulse, a similar dependence on laser ellipticity was observed (Fig. 7 (b)). This confirms that population inversion between the C 3 Π u and B 3 Π g states can be only achieved with circularly polarized pump pulses. The slight asymmetry and the deviation of the peaks from = 135° and = 315° can be due to the fact that the circularly polarized pump pulses reflect on the dielectric dichromatic mirror in this experiment. This mirror has slightly different reflectivity for p-and s-polarized light and thus results in a nonperfect circularly polarized pump pulses after reflection.

Temporal characterization of the backward emission and the gain

In order to obtain temporal information such as pulse duration of the emission, we have tried to measure both the ASE and the amplified 337 nm emission with a fast photodiode (ALPHALAS, UPD-200-SP, rise time: 175 ps, bandwidth: 2GHz) connected to 13 GHz oscilloscope (Agilent, model: Infiniium DSO91304A). We first tested the temporal response of this detection system with the femtosecond pulse, which serves as an impulse excitation. An oscilloscope trace with a width of ~ 500 ps is observed, reflecting the bandwidth limit of the photodiode. We then used this system to examine the duration of the ASE and the amplified 337 nm emission. In both cases, we observed no change of the signal width on the oscilloscope compared to the femtosecond pulse excitation. This suggests that the durations of the backward ASE and the amplified emission are less than 500 ps. To get precise knowledge of the pulse duration, optical methods such as cross-correlation are needed in the future study.

The temporal dynamic of the gain is very important aspect for an optical amplifier. With a seed pulse in the counter-propagation direction, displacement of the optical delay line changes the position of temporal overlapping between the pump and seed pulse. Therefore, we employed a co-propagation configuration to measure the gain dynamics, similar to that reported in ref. [START_REF] Yao | Gain dynamics of a free-space nitrogen laser pumped by circularly polarized femtosecond laser pulses[END_REF]. We presented the results in Fig. 8 for three different nitrogen pressures. The zero temporal delay is defined as the position where noticeable amplification of the seeding pulse is observed. For 1 bar nitrogen gas, a gain built-up time of ~ 4 ps and a decay process of ~ 20 ps are observed. For decreased gas pressure of 600 and 400 mbar, the gain built-up time increases to 7 and 11 ps. This dependence of the gain built-up time on gas pressure supports our hypothesis of collisionally pumped population inversion, which will be further discussed later. We will also show in the discussion section that this relatively short gain lifetime has a fundamental influence on the amplification process and sets limit for the duration of the backward ASE. In the following analysis, we will take the effective lifetime (Full Width at Half Maximum, FWHM) of the gain  g to be ~ 15 ps.

Experiment in ambient air

The results presented above were obtained in pure nitrogen. We have previously found that the presence of oxygen molecules deteriorates the backward ASE [START_REF] Mitryukovskiy | Backward stimulated radiation from filaments in nitrogen gas and air pumped by circularly polarized 800 nm femtosecond laser pulses[END_REF]. In this seeded lasing scheme, we have tried to observe the amplification of the seed pulse propagating in both forward (co-propagation) and backward (counter-propagation) directions. For focal lens of f = 1000 mm, we observed almost no amplification of the seed pulse in ambient air in both copropagation and counter-propagation configuration (Fig. 9 (a) and(b)).

In the work of Yao et al, the authors used an f = 300 mm lens and observed an amplification of ~ 2 times in atmospheric air [START_REF] Yao | Gain dynamics of a free-space nitrogen laser pumped by circularly polarized femtosecond laser pulses[END_REF]. We therefore tested a shorter focal lens of 500 mm in our experiments. In the co-propagation scheme, we observed an amplification of ~ 90 of the seeding pulse in ambient air (Fig. 9 (c)). This observation confirms that optical gain can be achieved in atmosphere with optimized pump conditions, which is very important for the future application of BLACK in atmospheric environment. However, in the counterpropagation configuration, we have observed no amplification. The optical gain of air plasma amplifier is estimated to be   l I I g f 0 ln  = 1.2 cm -1 , with l = 3 cm the length of the plasma amplifier. With such a relatively weak gain of ~ 1.2 cm -1 , the amplification factor for the backward ASE can be approximated as exp(gl eff ) ~ 1.7, with l eff = c g ~ 0.45 cm the effective length of the amplifier for a backward propagating photon. Therefore, no significant amplification of the spontaneous emission photon should be expected in the backward direction with such un-sufficient gain, which agrees with our above observation. Fig. 9. Amplification of the seed pulse in atmospheric air. In (a) and (b), the seed pulse copropagates and counter-propagates with respect to the pump pulses respectively. The focal lens was 1000 mm. In (c), the seed and the pump pulses were in the same direction and the focal lens of f = 500 mm.

In our current study, the optical gain achieved in air is much less than that in pure nitrogen. The detrimental role of oxygen molecules has been attributed to the collision quenching process and the decreased laser intensity inside the filaments due to the lower ionization potential of the oxygen molecules [START_REF] Mitryukovskiy | Backward stimulated radiation from filaments in nitrogen gas and air pumped by circularly polarized 800 nm femtosecond laser pulses[END_REF][START_REF] Yao | Gain dynamics of a free-space nitrogen laser pumped by circularly polarized femtosecond laser pulses[END_REF]. Another possible process is the attachment of electrons to the neutral oxygen molecules [START_REF] Zhao | Femtosecond ultraviolet laser pulse induced lightning discharges in gases[END_REF], which can decrease the electron density.

We noticed that in the work of D. Kartashov and coworkers, the authors reported that the forward 337 nm lasing emission obtained in pure nitrogen were essentially identical to those in ambient air [START_REF] Kantashov | Remotely pumped stimulated emission at 337 nm in atmospheric nitrogen[END_REF]. In this study, they employed linearly polarized picoseconds (2-10ps), 10 J pulses at 1.053 m as pump pulse. The authors pointed out that no forward lasing action at 337 nm can be observed when the pulse duration becomes less than a threshold value of 2 ps. In our current study and that of Yao et al, femtosecond pulses with duration of ~ 40 fs were used. The different roles of oxygen molecules may be related to these distinct experimental parameters. A further study on this point is highly necessary, especially for the sake of future application of this backward 337 nm laser pumped by circularly polarized femtosecond pulses.

Further discussion

Mechanism for population inversion

In our previous work, we have attributed the population inversion to the following inelastic collision process: Π .

The effectiveness of circular laser polarization for population inversion lies in the fact that photoelectrons generated in a circularly polarized laser field are left with a substantial kinetic energy after the passage of the pump laser pulse. In our case of laser intensity around 1.4 × 10 14 W/cm 2 [START_REF] Mitryukovskiy | Backward stimulated radiation from filaments in nitrogen gas and air pumped by circularly polarized 800 nm femtosecond laser pulses[END_REF], most of the electrons obtain kinetic energy around 16 eV, which is sufficiently high to excite the ground state nitrogen molecules to the third triplet state through inelastic collisions [START_REF] Fons | Electron-impact excitation of the second positive band system (C 3 Πu→ B 3 Πg) and the C 3 Πu electronic state of the nitrogen molecule[END_REF].

The measurement of the gain dynamics supports this hypothesis of population inversion. A gain built-up time of ~ 4-11 ps is clear in Fig. 8, which agrees well with the fact that the electron-molecules mean collision time is around 0.3 ps at atmospheric pressure. With decreased nitrogen pressures, the gain built-up time increases gradually due to the reduced collision frequency.

We would like to point out that population inversion by energetic electron collisions achieved with circularly polarized femtosecond pulses has been well developed in the domain of X-ray laser [START_REF] Corkum | OSA Proceedings on Short Wavelength Coherent Radiation: Generation and Applications[END_REF][START_REF] Ph | A high intensity highly coherent soft X-ray femtosecond laser seeded by a high harmonic beam[END_REF][START_REF] Sebban | Saturated Amplification of a Collisionally Pumped Optical-Field-Ionization Soft X-Ray Laser at 41.8 nm[END_REF][START_REF] Almiev | Dynamial descriptin of transient X-ray lasers seeded with high-order harmonic radiation through Maxwell-Bloach numerical simulations[END_REF][START_REF] Oliva | Comparison of nature and forced amplification regimes in plasma-based soft-x-ray lasers seeded by high-order harmonics[END_REF]. As first proposed by Corkum and Burnett in 1988, the hot electron distribution released by a circularly polarized laser field can be suitable to pump the upper level of laser transition by collision excitation [START_REF] Corkum | OSA Proceedings on Short Wavelength Coherent Radiation: Generation and Applications[END_REF]. Since then, numerous experimental demonstrations of X-ray laser in the ASE regime or seed-amplification regime have been reported, with a circularly polarized pump pulse [START_REF] Ph | A high intensity highly coherent soft X-ray femtosecond laser seeded by a high harmonic beam[END_REF][START_REF] Sebban | Saturated Amplification of a Collisionally Pumped Optical-Field-Ionization Soft X-Ray Laser at 41.8 nm[END_REF][START_REF] Almiev | Dynamial descriptin of transient X-ray lasers seeded with high-order harmonic radiation through Maxwell-Bloach numerical simulations[END_REF][START_REF] Oliva | Comparison of nature and forced amplification regimes in plasma-based soft-x-ray lasers seeded by high-order harmonics[END_REF]. For example, with a circularly polarized 30 fs, 1 J pump pulse, Ph. Zeitoun et al have demonstrated a small-signal gain coefficient of 80 cm -1 at 32.8 nm based on the optical transition of Kr 8+ [START_REF] Ph | A high intensity highly coherent soft X-ray femtosecond laser seeded by a high harmonic beam[END_REF].

Pulse duration, laser intensity of the backward ASE

In a swept-gain amplifier such the filamentary plasma here, the pump pulse travels from one end of the amplifier to the other and the population inversion is built up sequentially inside the medium. The pulse duration of the backward ASE is related to three basic factors. The first factor concerns the spectrum bandwidth of the laser emission, which limits the pulse duration by Fourier transformation. The bandwidth of the 337 nm lasing was well known to be of ~ 0.1 nm, which corresponds to a Fourier transform limited pulse duration of ~ 2 ps. The second factor is the gain lifetime. The duration of the radiation emitted from a small segment (

g c z   
) of the amplifier should be less than the gain lifetime, beyond which no amplification is present. In our experiments, the gain lifetime has been measured to be around ~ 15 ps. The third factor is the traveling time of the pump pulse through the amplifier, which is given by c l t   . This corresponds to 100 ps for a 3 cm long plasma filament, which is much longer than the gain lifetime and the Fourier transform limited duration. The real pulse duration of the backward ASE can be estimated as the convolution of the above three characteristic times. Since the traveling time is much larger than the other two, the pulse duration of the backward ASE should be close to the pump traveling time  t ~ 100 ps. With a calibrated maximum pulse energy of 0.03 nJ, the power of the ASE emission can be estimated as p ASE E P   ~ 0.3 W. We have previously measured that the transverse diameter of the plasma amplifier is around 300 m with similar experimental parameters [START_REF] Mitryukovskiy | Backward stimulated radiation from filaments in nitrogen gas and air pumped by circularly polarized 800 nm femtosecond laser pulses[END_REF]. Therefore, the laser intensity of the 337 nm ASE is

  2 2 d E I p ASE ASE   ~ 3.5 × 10 2 W/cm 2 .
The saturation intensity of the C 3 Π u → B 3 Π g transition is given by

  B N e B N e C s sat T h I 2 2 , , 1 2       , where  s , T C , B N e 2 ,  , C N e 2 ,
 are the frequency of the 337 nm emission, the cross-section of stimulated emission, the lifetime of the upper level C 3 Π u state, the temperature-dependent excitation rates to the C 3 Π u and B 3 Π g states by electron collisions, respectively [5 , 6] In the presence of a backward seed pulse, we have observed a strong energy amplification of the seed pulse inside the plasma medium (Fig. 3 and Fig. 6). The pulse duration of our seed pulse is estimated to be around ~ 100 fs, taking into consideration of the dispersion of the 20 mm fused silica sample and other transmission optical elements such as the two lenses (L 2 and L 3 ) and the incident window. Due to the narrow bandwidth (~ 0.1 nm) of the optical gain, such a relatively broadband (~ 10 nm) seed pulse cannot be amplified uniformly in the spectrum domain. The bandwidth of the final output pulse is expected to be close to the bandwidth of the optical gain, i. e. ~ 0.1 nm. In analogy to the plasma amplifier seeded by large bandwidth high order harmonics for intense X-ray laser, we expect the output pulse in our case consists of the amplified seed pulse, a wake field with a complex structure related to Rabi oscillations and coherent decay, and the residual ASE [START_REF] Almiev | Dynamial descriptin of transient X-ray lasers seeded with high-order harmonic radiation through Maxwell-Bloach numerical simulations[END_REF][START_REF] Oliva | Comparison of nature and forced amplification regimes in plasma-based soft-x-ray lasers seeded by high-order harmonics[END_REF]. The temporal scale of the entire output pulse is found to be on the order of the gain lifetime [START_REF] Almiev | Dynamial descriptin of transient X-ray lasers seeded with high-order harmonic radiation through Maxwell-Bloach numerical simulations[END_REF][START_REF] Oliva | Comparison of nature and forced amplification regimes in plasma-based soft-x-ray lasers seeded by high-order harmonics[END_REF]. Therefore, a pulse duration of ~10 ps should be a good approximation for the amplified 337 nm radiation in our experiments. Considering the maximum pulse energy of 1.2 nJ, the peak power is found to be 1.2 × 10 2 W. The corresponding laser intensity inside the amplifier is then estimated to be 1.7 × 10 5 W/cm 2 , which is ~ 1 % of the saturation intensity. A better understanding of this amplification process can be achieved with numerical simulation of this collisionally pumped plasma amplifier, which is under progress at this moment.

Conclusion

In conclusion, we demonstrated that a seed pulse at 337 nm injected into the backward direction of the plasma filaments, which is generated in nitrogen gas pumped by circularly polarized 800 nm femtosecond laser pulses, can be amplified by a factor of ~ 200. The amplified lasing radiation inherits the polarization property of the seed pulse and its divergence angle was found to be around 3.8 mrad, much less than that of the backward ASE. The critical role of pump laser polarization was also observed in the seeded lasing regime, where intense lasing effect was only possible for circularly polarized pump pulses. The amplification phenomenon, the reduced divergence of the seeded lasing radiation, and the critical role of pump laser polarization confirms unambiguously the presence of population inversion between the C 3 Π u and B 3 Π g states of neutral N 2 molecules in the filament plasma. We attributed the population inversion mechanism to the collisions between the ground state neutral nitrogen molecules with energetic electrons, which is produced by the circularly polarized femtosecond laser pulses during ionization. We also point out that this collisionally pumped population inversion scheme has been well developed in the domain of X-ray laser since 1990s.

We measured the temporal dynamics of the optical gain and found a pressure-dependent gain built-up time of 4 ~11 ps, and a gain lifetime of ~ 15 ps. In a swept-gain plasma amplifier, this gain lifetime is one order of magnitude less than the pump traveling time through the ~ 30 mm long plasma column. As a result, the pulse duration of the backward ASE is largely determined by the pump traveling time. With the measured maximum pulse energy of 0.03 nJ, we estimated the 337 nm ASE laser intensity is about 3.5×10 2 W/cm 2 , which is 5 orders of magnitude less than the saturation laser intensity.

In the presence of a backward injected seed pulse, the maximum 337 nm pulse energy reaches 1.2 nJ. The small-signal gain was estimated to be 11.8 cm -1 . With estimated pulse duration of ~ 10 ps, the laser intensity was found to be 1.7 ×10 5 W/cm 2 , which is about 1 % of the saturation intensity.

Finally, we demonstrated that optical gain at 337 nm can be achieved in plasma created in ambient air, by properly optimizing the focusing geometry of the pump pulses. This observation suggests the feasibility of BLACK in atmospheric environment for future applications.

Fig. 1 .

 1 Fig. 1. Schematic experimental setup.

Fig. 2 .

 2 Fig. 2. Backward emission spectrum with circularly (a) and linearly (b) polarized pump pulses. The numbers in the parenthesis denote the vibration quantum number of the initial and final states of the transition.

Fig. 3 .

 3 Fig.3. Spectra of the backward emission with circularly (a) and linearly (b) polarized pump pulsed at 800 nm. The spectra of the seed pulses and those of the backward emission from the pump pulses are also presented for comparison. In (a), the spectrum of the seed pulse is magnified by a factor of 10 for visibility.

Fig. 4 .

 4 Fig. 4. Polarization of the ASE (a), the seeded backward stimulated emission with linearly (b) and circularly (c) polarized seed pulses at 337 nm. The dots present the experimental results and the dashed lines denote the theoretical fitting.

Fig. 5 .

 5 Fig. 5. Spatial profile of the backward ASE (a), the seed pulse (b), and the amplified 337 nm radiation (c). The opening angle of each panel is 12.5 mrad × 10 mrad.

Fig. 6 .

 6 Fig. 6. Energy of the amplified 337 nm emission in the backward direction as a function of the seed pulse energy.

3. 5

 5 Pump laser polarization dependence of the amplified 337 nm laser All the above experimental observations highlight the crucial role of pump laser polarization.

Fig. 7 .

 7 Fig. 7. Dependence of the backward ASE (a) and the seeded backward stimulated radiation (b) as a function of the rotation angle of the quarter-wave plate. The angles φ = 90º × mcorrespond to linearly polarized laser, with m = 0, 1, 2, 3. The angles φ = 45º + 90º × m correspond to circularly polarized laser.

Fig. 8 .

 8 Fig. 8. Measured forward amplified 337 nm lasing signal as a function of the time delay between the pump and the seeding pulse. The pump pulse energy was 7.5 mJ.

  ~ 1.5×107 W/cm 2 . The above estimated laser intensity I ASE is 5 orders of magnitude less than the saturation intensity. 4.3 Pulse duration, laser intensity of the backward amplified 337 nm laser

	. For T e ~ 16 eV, we found B e 2 N ,  ~ 0.7×10 -17 cm -2 and B e 2 N ,  ~ 2×10 -17 cm -2
	[27]. With  s ~ 10 -14 cm 2 and T

C ~ 0.6 ns

[START_REF] Sprangle | Remotely induced atmospheric lasing[END_REF][START_REF] Peñano | Remote lasing in air by recombination and electron impact excitation of molecular nitrogen[END_REF]

, we obtain I sat

Acknowledgments

The authors thank Dr. Aurélie Jullien, Stéphane Sebban, and Philippe Zeitoun of Laboratoire d'Optique Appliquée for important technical support and stimulating discussion. Yi Liu acknowledges stimulated discussions with Dr. Jinping Yao and Prof. Ya Cheng of Shanghai Institute of Optics and Fine Mechanics, Prof. Huailiang Xu of Jilin University, Dr. Daniil Kartashov of Friedrich-Schiller University, Prof. Hongbing Jiang and Chengyin Wu of Peking University.


Guillaume Point

Yohann Brelet

Leonid Arantchouk

Jérôme Carbonnel

Bernard Prade

André Mysyrowicz

Aurélien Houard

Two-color interferometer for the study of laser filamentation triggered electric discharges in air

I. INTRODUCTION

Laser filamentation, that is the self-guided, nondiffracting regime for an ultrashort laser pulse of which peak power is greater than a critical power (about 5 GW at 800 nm), results from the dynamic competition between the self-focusing Kerr effect, on the one hand, and diffraction, multiphoton absorption, and plasma defocusing on the other hand 1 . This phenomenon has many interesting features, among which is the fact that the laser pulse leaves a long underdense (n e ≈ 10 22 m -3 ) plasma column in its trail 2,3 . This column can be used for various applications, such as acting as a waveguide for infrared and microwave radiation [4][5][6] , serving as a gain medium for lasing action 7,8 , generating strong terahertz emission 9,10 , or triggering and guiding electric discharges in atmospheric air [11][12][13] . The latter is very promising due to its potential openings for the laser lightning rod 14,15 , highvoltage, high-current switches 16,17 , or a virtual plasma antenna 18,19 .

The initiation, optimization and maturation of these technologies rely on a good knowledge of the discharge plasma parameters, that is the electron density and, to a lesser extent, the electron temperature. Interferometry had proven to be a very reliable and rigorous electron density diagnostic, able to give absolute density measurements [START_REF] Hutchinson | Principles of Plasma Diagnostics[END_REF] . Therefore, it appears to be an ideal choice as a diagnostic for filamentation-guided electric discharges.

However, a difficulty arises from the nature of studied sparks. Since energy deposition from the discharge in the medium leads to strong hydrodynamic effects [START_REF] Shneider | [END_REF] , and because the dynamics of studied plasmas are slower than the characteristic development time of such effects, two different contributions to the plasma optical index emerge. One of them comes from the free electron population. The other originates from the neutral-bound electrons, because the neutral density along the probe beam and the reference beam are not balanced. A way to a) Electronic mail: guillaume.point@ensta-paristech.fr separate these two contributions is to record the plasma optical index at two different wavelengths, so-called twocolor interferometry [22][23][24][25] . Using this technique, it becomes possible to isolate the free electron density, and to record neutral density as well. Two-color interferometry has also been used to make vibration-insensitive electron density measurements on large experiments, where such effects can become significant [26][27][28] . This application is very similar to ours, consisting in discriminating between two different contributions to the phase shift. In the simpler case where the plasma optical index only depends on the free electron density, dual-wavelength interferometry can also be used to extend the available dynamic range, using a short wavelength to probe high-density areas, and shifting to a longer wavelength for low-density regions 29 .

In this Article, we introduce a two-color interferometer designed for the study of filamentation-triggered electric discharges, but that can however be readily adapted for use on any plasma where bound electron contribution to the refractive index cannot be neglected. Working principles of this kind of plasma diagnostic are described in section II. Interferogram processing from image recording to the extraction of radial density profiles is explained in section III. Finally, experimental results from the study of a centimeter-scale spark discharge triggered by a laser filament are presented in section IV. From these results, we estimate the phase shift resolution of the interferometer to be at most 30 mrad for λ = 532 nm. For a typical spark transverse size of 200 µm, this gives minimum recoverable densities on the order of 4 × 10 22 m -3 for free electrons, and of 10 24 m -3 for neutrals.

II. PLASMA TWO-COLOR INTERFEROMETRY

A. Theoretical aspects

As said earlier, two distinct contributions arise in the plasma optical index: one comes from free electrons, and the other from electrons bound to neutrals and ions. More fundamentally, the polarization density #» P can be written as:

#» P = #» P f + #» P b , (1)
where #» P f corresponds to the free electronic dielectric response, and #» P b to the bound electronic dielectric response. The free electronic polarization density can be found using the Drude model for a cold and non-magnetic plasma:

#» P f = -ϵ 0 ω 2 p ω 2 ( 1 -i 2πνc ω ) #» E, ( 2
)
where ϵ 0 is the vacuum permittivity, ω p = √ n e e 2 /m e ϵ 0 is the plasma frequency (n e , e and m e being respectively the electron density, the electron charge and the electron mass), ν c is the average collision frequency between free electrons and other species in the plasma, and ω denotes the angular frequency of the harmonic exciting laser electric field #» E. In our case, the probe beam is either in the visible range, or in the near-infrared range, that is ω ∼ 2 × 10 15 rad • s -1 . As for the electron collision frequency, if we suppose that our plasma is dominated by electron-neutral collisions because of incomplete ionization, we can estimate this parameter as: [START_REF] Rax | Physique des plasmas[END_REF] . We are thus in the limit ω ≫ 2πν c . Therefore, it is possible to simplify equation ( 2):

ν c ≈ 6 × 10 -15 n n [m -3 ]T 1/2 e [eV] Hz ∼ 1.5 × 10 13 Hz for k B T e ∼ 1 eV
#» P f = -ϵ 0 ω 2 p ω 2 #» E = -ϵ 0 r e λ 2 π n e #» E, (3)
where the classical electron radius r e = e 2 /4πϵ 0 m e c 2 and the wavelength λ = 2πc/ω were introduced.

As for the bound electronic polarization density, it can be evaluated considering bound electrons as harmonic oscillators:

#» P b = e 2 m e ( n i ∑ k1 f k1 ω 2 0,k1 -ω 2 + n n ∑ k2 f k2 ω 2 0,k2 -ω 2 ) #» E, (4)
where f k is the oscillator strength associated with the kth resonance of which characteristic frequency is ω k , n i is the ion density, and n n the neutral density. At this point, the first term of the left-hand side can be neglected for two reasons: first, the studied sparks are only moderately ionized, meaning n n ≫ n i . Second, ionic polarizability, proportional to the left-hand sum in equation ( 4), is less important than neutral polarizability 25 . Consequently, we can write:

#» P b = e 2 m e n n ∑ k f k 1 ω 2 0,k -ω 2 #» E = ϵ 0 r e π n n ∑ k f k λ 2 λ 2 o,k λ 2 -λ 2 0,k #» E.
(

) 5
Since our probe laser has a very weak intensity and because the spark plasma is not magnetized, it can be locally considered as linear and isotropic. Thus we have:

#» P = ϵ 0 (ϵ r -1) #» E, (6)
with the relative dielectric constant ϵ r being linked to the plasma optical index n p following:

ϵ r = n 2 p . (7)
Therefore, the plasma index can be written as:

n p = 1 - r e λ 2 π n e + r e π n n ∑ k f k λ 2 λ 2 o,k λ 2 -λ 2 0,k ≈ 1 - r e λ 2 2π n e + r e 2π n n ∑ k f k λ 2 λ 2 o,k λ 2 -λ 2 0,k , ( 8
)
|n 2 p -1| being on the order of 10 -4 . The last term in this last equation is actually nearly independent of the wavelength in the visible-near infrared range, and is usually written as:

r e 2π n n ∑ k f k λ 2 λ 2 o,k λ 2 -λ 2 0,k = β n 0 n n , (9)
called the Gladstone-Dale relation. The Gladstone-Dale constant β defines the value of the refractive index 1 + β at a reference point characterized by n n = n 0 . Data computed using the Ciddor equation 31 with standard air temperature and pressure condition give β ≈ 2.7 × 10 -4 for a reference neutral density n 0 = 2.47 × 10 25 m -3 . The plasma index is now reduced to:

n p = 1 - r e λ 2 2π n e + β n 0 n n . ( 10
)
When an interferometric measurement of the plasma in air is done, the recorded phase shift of the fringes ∆φ is linked to the plasma refractive index by:

∆φ(λ) = 2π λ ∫ s2 s1 (n p (λ) -1 -β) ds, (11)
s 1 and s 2 defining the length of integration chord. If one records simultaneously the phase shift due to the plasma at two different wavelengths along the same chord, one can then separate the contributions from free and bound electrons in the plasma index as follow:

       ∫ s2 s1 n e (s) ds = λ 1 ∆φ 1 -λ 2 ∆φ 2 r e (λ 2 2 -λ 2 1 ) ∫ s2 s1 (n n (s) -n o ) ds = n o 2πβ λ 2 ∆φ 1 -λ 1 ∆φ 2 λ2 λ1 -λ1 λ2 . ( 12
)

B. Experimental setup

Our interferometer is built in a standard Mach-Zehnder configuration (cf. figure 1). The probe laser (Quanta Ray GCR-290-10 from Spectra Physics) is a Nd:YAG Q-switched, frequency doubled pulsed laser delivering ∼ 100 µJ, 8 ns full width at half maximum (FWHM) light pulses at both 532 and 1064 nm. The probe beam is spatially cleaned and magnified using a beam expander and a 100 µm pinhole, yielding a pseudo-Gaussian collimated beam with a 7.2 mm FWHM at 532 nm and 8.3 mm FWHM at 1064 nm. The delay between the two probe pulses due to the propagation through optical materials is evaluated to a few ps, which is negligible with respect to the duration of laser pulses. The whole interferometer is built on a 60 × 60 cm 2 breadboard mounted on mechanic isolation feet to reduce vibrations, and entirely enclosed to limit the influence of air turbulence on measurements. An ultrashort laser pulse from the Ti:sapphire chirped pulse amplification laser chain ENSTAmobile is focused at f /70 and collapses into a filament in the discharge gap, triggering the studied spark. The resulting discharge plasma is generated in one of the arms of the interferometer, perpendicularly to the probe beam. The probe laser is synchronized with the ENSTAmobile using the latter's internal clock, enabling us to precisely adjust the temporal delay between the laser pulses anywhere between 0 and 100 ms with a ns precision and a 1.5 ns jitter.

Interferograms are recorded by means of two CCD cameras (TaperCamD-UCD12 from DataRay, Inc.) with a 1360 × 1024 pixel matrix and a 10.5 µm pixel size. Probe wavelengths are separated using a dichroic mirror (T1064/R532). Adapted bandpass filters with < 10 nm bandwidth suppress remaining broadband emission from the plasma. The whole detection part is arranged in a 2f /2f fashion using a 75 mm focusing lens, CCD cameras being in the conjugated plane of the plasma plane.

The interferometer is slightly misaligned so that ∼ 10/20 fringes (depending on the wavelength) appear on the detectors in absence of plasma in the probing arm.

III. INTERFEROGRAM PROCESSING

A. Phase recovery algorithm

The first step to perform during the processing of interferograms is to retrieve relative phase planes from images. In our case, this is done using a 1D continuous wavelet transform (CWT) algorithm. This kind of algorithm has been proved to be more reliable than routines based on the fast Fourier transform (FFT), particularly in case of low contrast and/or noisy interferograms 32,33 . We will first describe the CWT, and then how this mathematical transform is used to extract the local phase from interferograms.

The continuous wavelet transform

The CWT is a mathematical integral transform for any square-integrable function f ∈ L 2 (R), much like the Fourier transform. However, where the Fourier transform decomposes f on a complete basis of trigonometric polynomials, the CWT projects f on a base of particular functions, called wavelets, which are well localized both in direct and Fourier space. If we consider a function ψ ∈ L 2 (R), called the mother wavelet, satisfying particular conditions (zero average, normalized energy, and a last condition called admissibility condition [START_REF] Mallat | A wavelet tour of signal processing -The sparse way[END_REF] ), then the wavelet transform of f is defined as:

f CW T (a, b) = ⟨f |ψ a,b ⟩ = 1 √ a ∫ R f (x)ψ ( x -b a ) * dx, (13)
where ψ a,b is the daughter wavelet of parameters a and b, a ∈ R * + is called the scale parameter, b ∈ R is the translation parameter, and * corresponds to the complex conjugation operation. The CWT thus projects a 1D function on a 2D space of which (a, b) forms a suitable basis, called time/frequency space. This is because b is dimensionally similar to x (that is, the time direction) while a defines the daughter wavelet extension in the frequency space, and can be directly linked to the Fourier frequency [START_REF] Meyers | [END_REF] .

The mother wavelet we use is called the Morlet wavelet, which is a trigonometric polynomial modulated by a Gaussian envelope:

∀x ∈ R, ψ M s (x) = 1 4 √ π e -x 2 /2 e isx , ( 14
)
where s ≥ 6 to satisfy the admissibility condition [START_REF] Mallat | A wavelet tour of signal processing -The sparse way[END_REF] . This wavelet is particularly well adapted for the study of sinusoidal signals because of its good localization in both time and frequency space 36 . The power of the CWT over Fourier transform becomes evident when it comes to the study of signals of which frequency varies with time. If we take the example of a linearly chirped cosine function:

f test (x) = cos (2π (5 + 0.075x) x) , ( 15
)
we can compute both the Fourier transform and the CWT using a Morlet wavelet for this function. As seen on figure 2-(a), the spectral power density for f test calculated using a standard FFT algorithm is different from 0 only between 5 and 20 m -1 , which corresponds to the signal frequency f x at x = 0 and x = 100 m, i. e. the space boundaries for the study. Still, no more useful information can be retrieved from this plot. The CWT, however, as a time/frequency analysis tool, gives much more information about the studied signal. Thus, as shown in figure 2-(b), it is possible to retrieve the frequency of this signal for any x value using the curve of maximum magnitude, that is the ridge of the wavelet transform. The ridge is particularly significant in wavelet analysis because along this curve, the phase of f CW T is equal to the phase of f 34 .

The algorithm

The phase recovery algorithm works as follows: each row of the interferogram is selected one after the other, as illustrated in figure 3-(a). For a given line, the CWT is computed by means of a routine based on the one proposed by Torrence and Compo using a Morlet wavelet as mother wavelet 37 . This yields a 2D complex array, which can be split into a magnitude and a phase plane (figure 3-(b) and (c), respectively). The following step is to recover the ridge position on the magnitude plane. Because of noise and low interferogram contrast, the true ridge corresponding to the fringe frequency can be dwarfed by parasite curves on the wavelet magnitude map. A way to overcome this and to enhance the robustness of the algorithm with respect to noise is to add a criterion for the ridge selection other than looking for the curve of maximum modulus. A choice is to discard candidate ridges presenting too pronounced steps along a, since noise tends to generate parasite curves that often shift in frequency. Thus, to recover the correct ridge using this procedure, one must look for the parametric curve (a 0 = g(b)) that minimizes a cost function defined as:

cost(g(b), b) = -A 1 ∫ b -∞ |f CW T (g(b ′ ), b ′ )| 2 db ′ +A 2 ∫ b -∞ ∂g ∂b ′ (b ′ ) 2 db ′ , ( 16
)
where A 1 and A 2 are weighting coefficients 33 . For these parameters, we typically use values such as

A 2 /A 1 ≈ max a,b |f CW T (a, b)| 2 .
Once the ridge position is recovered, it becomes possible to find back the phase of the original signal using the phase of the CWT since on the ridge, both values are equal. Reporting the ridge position on the phase map thus yields directly the phase along the studied line, as illustrated on figure 3-(c). By repeating this operation for every line of the interferogram, we can finally recover the phase for the whole image (figure 3-(d)).

B. Phase unwrapping algorithm

As seen in figure 3-(d), the phase recovery algorithm extracts wrapped phase, that is a phase map presenting 2π discontinuities. This is because the arctangent function used to evaluate the phase of the wavelet transform returns its principal value in the [-π, π[ interval. The following step therefore consists in unwrapping the extracted phase. The most basic algorithm to perform this task simply follows a predefined path, usually one row/column after the other, and looks for phase jumps greater than π between two adjacent pixels, correcting the phase of untreated pixels by adding or subtracting 2π multiples. Of course, this approach is very basic and suffers from several problems: first, noisy phase map and low signal-to-noise ratio can trick the algorithm and result in the appearance of artificial phase discontinuities. Second, any error along the unwrapping path can be propagated and corrupt the whole phase map.

For this reason, we make use of an unwrapping algorithm that follows a non-continuous path defined by decreasing value of a reliability function calculated for every pixel on the phase map. In our case, this reliability function depends on the second difference between adjacent pixels, that is the squared phase difference between a given pixel and its two neighbors along the same direction, corrected, if necessary, from any 2π discontinuity. For example, considering the pixel (i, j), the horizontal second difference for this pixel, H 2 (i, j), would be:

H 2 (i, j) = (u(φ(i, j-1)-φ(i, j))-u(φ(i, j)-φ(i, j+1))) 2 , ( 17
)
where u represents the unwrapping operator. Computing vertical V 2 (i, j) and diagonal D 2 1 (i, j) and D 2 2 (i, j) second differences allows to deduce the reliability function for the pixel (i, j), R(i, j), as:

R(i, j) = (H 2 (i, j) + V 2 (i, j) + D 2 1 (i, j) + D 2 2 (i, j)) -1/2 . ( 18
)
The unwrapping operation works along the path going from the most reliable pixel to the least reliable one, joining processed pixels into groups, which can be unwrapped one with respect to the other as a whole. This algorithm was first proposed and demonstrated by Herráez et al. 38 , both robust and fast. Figure 4 presents the phase map resulting from the unwrapping algorithm applied to the wrapped phase map displayed in figure 3-(d). One can see that no unwrapping error, that is artificial phase jump, was introduced, and that phase noise initially present on the wrapped map is still confined to the same areas. Now that unwrapped phase maps can be reliably extracted from interferograms, it becomes possible to compute phase shift maps.

C. Phase shift extraction

Extracting useful information from interferograms requires the evaluation of the phase shift, that is the removal of the background phase due to the undisturbed spatial carrier. This step is done by recording several blank interferograms (i. e. without plasma) and recovering the corresponding unwrapped phase map, which includes any wavefront distortion induced by optical components in the interferometer. For any interferogram recorded in presence of plasma, a simple algorithm looks for the reference phase plane minimizing the function σ(φ) defined as:

σ(φ) = √ ⟨(φ -φ ref ) 2 ⟩ -(⟨φ -φ ref ⟩) 2 , (19)
i. e. the standard deviation of the difference between the phase plane and the reference candidate phase plane evaluated on the whole image.

Once the best reference candidate is found, it is subtracted from the informative phase plane. Usually, a phase offset persists after this step. To correct this, the average phase outside the plasma-affected region is computed and removed from the whole phase map, giving a globally null phase shift out of the plasma. For instance, the interferogram displayed in figure 3-(a) results in the phase shift map of figure 5 using an appropriate reference phase map. Phase noise can also be evaluated as the phase standard deviation over the undisturbed region. We typically find a limiting value of 30 mrad RMS at 532 nm, which results, for a probed object of size ∼ 200 µm, in a density resolution of 4 × 10 22 m -3 for free electrons and of 10 24 m -3 for neutrals. Phase shift maps can then be used directly to evaluate line-integrated densities following equation system (12).

D. Abel inversion algorithm

Principle

As given by equation (12), only line-integrated densities along the probe beam propagation direction are recovered from interferograms. The very last step therefore consists of recovering true densities. In the general case where no assumption is made about the density function, this inversion problem has no solution. A way to approach a solution is to multiply probing chords in different directions, that is a tomographic reconstruction [START_REF] Hutchinson | Principles of Plasma Diagnostics[END_REF] . However, in the case of cylindrically symmetrical density functions, a rigorous mathematical solution exists. This is particularly important in the field of plasma diagnostic because many laboratory plasmas, such as ours, exhibit this kind of symmetry. Considering the schematic representation for the Mach-Zehnder interferometer displayed in figure 6, the recorded information at the output is:

f (y, z) = ∫ x2 x1 ρ(x, y, z) dx, ( 20
)
where ρ represents either free electron density of relative neutral density. Assuming that ρ is cylindrically symmetrical, we can rewrite this equation:

f (y, z) = ∫ √ x 2 1 +y 2 y rρ(r, z) √ r 2 -y 2 dr+ ∫ √ x 2 2 +y 2 y rρ(r, z) √ r 2 -y 2 dr. ( 21
)
x y beam splitter beam splitter

x 1 x 2 y s n(r) If x 1 and x 2 are large enough with respect to the plasma characteristic dimension, that is if ∀(y, z), ρ(x 1 , y, z) ≈ ρ(x 2 , y, z) ≈ 0 , then:

f (y, z) = 2 ∫ +∞ y rρ(r, z) √ r 2 -y 2 dr, ( 22
)
which is called the Abel transform of the function ρ. At the condition that ∀z, lim r→+∞ rρ(r, z) = 0, we can invert the Abel transform and extract back ρ from f following:

ρ(r, z) = - 1 π ∫ +∞ r ∂f ∂y (y, z) dy √ y 2 -r 2 . ( 23
)
This last step can only be performed if the integral converges, i. e. if the probe spatial extension y s is large enough so that ∀z, f (±y s , z) ≈ 0.

Using this last equation, we are now able to recover true densities n e (r, z) and n n (r, z) from two-color interferograms.

The algorithm

Several methods can be used to numerically implement the inverse Abel transform. Obviously, the most direct approach is to discretize equation (23). However, two major difficulties arise here. The first one lies in the spatial derivative term which intrinsically amplifies data noise. The other one is the pole y = r in the integral.

A way to avoid these problems is to take advantage of the fact that the Abel transform is a component of the FHA cycle, namely is equivalent to a composition of an inverse Fourier transform and of a Hankel transform of order 0. Thus, equation ( 23) can be rewritten as:

ρ(r, z) = 1 2π ∫ +∞ 0 (∫ +∞ -∞ f (y, z)e -iky dy ) kJ 0 (kr) dk, (24)
where J 0 is the 0-order Bessel function of the first kind. In this last equation, the spatial derivative and the pole have disappeared, so it can be used for discretization following:

ρ(r i , z j ) = ∆k∆y 2π N ∑ l=0 k l J 0 (r i k l ) N ∑ m=-N f (y m , z j ) cos(k l y m ), (25)
where it has been taken into account that f is an even function. This kind of algorithm is called Fourier-Hankel, and is particularly fast because of the FFT 39,40 . Another useful feature arising from the Fourier-Hankel algorithm lies in the possibility to implement a low-pass filtering of experimental data during the process. Indeed, the first step of the transform involves a Fourier transform. At this point, one can then convolute the corresponding spectrum with an appropriate filtering window (in our case, a Hann window), before implementing the second step of the transform, that is the inverse Hankel transform. The usefulness of this technique is illustrated in figure 7. Figure 7-(a) gives an example of integrated neutral density profile, while figure 7-(b) shows the corresponding inverted profiles using the Fourier-Hankel algorithm. In the first case, only raw experimental data are used, resulting in few unrealistic oscillations (black solid curve). Using the low-pass filtering step yields a much smoother output profile (red dashed curve). The gap switch consists in two cylindrical coppertungsten alloy electrodes of face diameter 36 mm placed into a Plexiglas construction opened to atmospheric air 17 . The laser beam passes through Ø3 mm holes drilled into the electrodes. The gap spacing between the electrodes is 10 mm corresponding to a self-breakdown level U cr ∼ 30 kV. Working at U 0 = 15 kV thus ensures that no breakdown can occur without external triggering.

-2 22 -1 0 1x10 line integrated n n (m -2 ) -3 -2 -1 0 1 2 3 y (mm) -2.0 25 -1.5 -1.0 -0.5 0.0 0.5 1.0x10 n n -n 0 (m -3 ) 3

IV. EXPERIMENTAL RESULTS

A. Setup for electric discharges

The discharge is triggered with a femtosecond laser pulse (4.5 mJ, 50 fs @ 800 nm) collapsing to form a single filament about 2 cm long. Discharge onset results in a current pulse with a rise time of about 18 ns decaying exponentially with a damping time R b C d ∼ 800 ns. Its maximal amplitude of about 36 A is defined by the resistance R b . The average delay between the laser beam and the discharge onset is 46.0 ns with a jitter of 5.7 ns.

B. Electron density

Examples of radial profiles for electron density are given in figure 9-(a). These profiles are taken midway of the discharge gap and averaged over 20 rows (i. e. 210 µm along z). They are characterized by a central main peak, the amplitude of which is at most 6.8 × 10 23 m -3 and with a FWHM of ∼ 100 µm, flanked by a smaller secondary peak with a comparable width. As time goes on, both peaks decrease in amplitude, with the lateral peak flattening much faster and propagating outwards with an average speed of about 700 m • s -1 , while keeping approximately a constant width. After 2.5 µs, that is almost 3 times the R b C d constant for our circuit, it is still possible to record a peak density around 10 23 m -3 . Past this point, n e drops below our detection limit.

Figure 9-(b) displays the evolution of on-axis electron density and of the discharge current with time. Each data point represents an average taken over five different shots, and the corresponding error bar corresponds to plus/minus one standard deviation. On this figure, it is possible to witness the decreasing trend of n e in time, which loosely follows the exponential decrease of the current. This behavior can be easily explained by the fact that the central density peak does not widen significantly in time, as shown in figure 9-( The occurrence of the electronic side peak propagating outwards is linked to the formation of a shock wave resulting from energy deposition at the center of the channel. Even though this phenomenon is much more flagrant when studying neutral density profiles (as it will be seen in the next section), it appears that a significant portion of free electrons is carried away by this hydrodynamic wave.

The maximum encountered electron density can be compared to an estimated value using the discharge current following:

I = ∫ ∫ S #» j • d #» S ∼ en e (r = 0)µEπR 2 , ( 26
)
where µ = e/m e ν c is the electron mobility, E the electric field amplitude, and 2R the FWHM of the conducting channel. Using E ∼ 15 kV • cm -1 and R ∼ 100 µm, we find:

n e,max (r = 0) ∼ 6 × 10 23 m -3 , (27)
which is indeed in very good agreement with the measured value of 6.8 × 10 23 m -3 .

C. Neutral density

As seen in figure 10-(a), neutral density profiles are characterized by a central, low-density region where n n is typically less than 10 % of the reference neutral density n 0 , surrounded by a high-density ring. With time, the central density hole widens while the high-density peak increases in amplitude at first, and then decreases and enlarges. After a few µs, the hole FWHM stabilizes around 1.5 mm. This behavior is typical of the formation of a blast wave following an important, very localized energy deposition in the medium, with the formation of an outward-propagating shock wave.

Figure 10-(b) represents the time evolution of the shock wave radius, r SW , taken as the radial position where neutral density is maximum. Using this data, we can then estimate the shock wave velocity u SW = dr SW /dt. As seen in the same figure, the velocity remains well above the speed of sound in air in standard temperature and pressure conditions, c s = 343 m • s -1 , up to the maximum probed time, beyond which measurements cannot be reliably done because r SW exceeds the field of view of cameras. Starting at nearly 1 km • s -1 , u SW falls abruptly during the first 500 ns then gently decreases to about 400 m • s -1 after 4 µs.

D. Study of probe beam deflection by the plasma

No interferometric measurement of a nonhomogeneous refractive index can be rigorously made without checking that probe beam deflection due to index gradients is negligible. In our case, this verification is done at times when index gradients are the most pronounced (t ∼ 150 ns) using index profiles measured at 1064 nm, since this effect is more important at longer wavelengths. The first task to perform is to evaluate the profile for deflection angles, α(y) (cf. figure 11-(a)). In the framework of geometrical optics, this profile can be calculated by solving the following equation:

α(y) = π -2y(1 + β) ∫ +∞ r0 dr r √ r 2 n 2 p -y 2 (1 + β) 2 , ( 28
)
where r 0 is the radius solution of the equation r 0 n p (r 0 ) = y(1 + β) 41 . Using a refractive index profile recorded at λ = 1064 nm and t = 150 ns, we find the maximum deviation angle to be less than 0.8 mrad (figure 11-(b)). We can also evaluate the position difference ∆y ′ on the CCD array between the deflected beam and an hypothetical undeviating beam (respectively the red solid beam and the black dashed beam in figure 11-(a)), given by:

∆y ′ = ( ϵ 1 + ϵ 2 + ϵ 1 ϵ 2 f ) tan(α(y)), (29)
where 2f + ϵ 1 corresponds to the distance between the imaging lens and the point where the two beams start to separate, and 2f + ϵ 2 to the distance between the lens and the CCD array. ϵ 1 originates from any misplacement of the imaging lens with respect to the plasma , but also from the spatial extension of the plasma, while ϵ 2 arises solely from the displacement of the CCD array with respect to its intended position.

For ∆y ′ to be as large as one pixel size, the left-hand term in equation ( 29) must be on the order of 1 cm. As the plasma has a typical transverse size of 100 µm, the only significant contribution in ϵ 1 comes from an eventual misplacement of the imaging lens, much like ϵ 2 . As a consequence, even a slight displacement of the imaging lens on the order of a few mm cannot result in a detectable influence of the probe deflection on experimental results. We therefore conclude that probe deflection effects can be completely neglected in our experiment.

V. SUMMARY

In this Article, we presented a sensitive two-color interferometer that can be used as a plasma diagnostic for simultaneous measurements of free electron density and neutral density with a ∼ 10 µm spatial resolution and ∼ 10 ns time resolution. Interferogram treatment starts with a phase extraction algorithm based on 1D continuous wavelet transform using a Morlet wavelet. Recovered phase is unwrapped by means of a non-continuous path phase unwrapping routine, and phase shift is estimated by subtracting adequate phase reference, allowing to recover line-integrated electron and neutral densities. Finally, a Fourier-Hankel algorithm for Abel inversion of integrated profiles is used. After image treatment, we estimate the phase sensitivity of the interferometer to be 30 mrad at the shorter wavelength, giving a resolution of ∼ 4 × 10 22 m -3 for free electron density and 10 24 m -3 for neutral density with a typical size ∼ 200 µm for the studied plasma.

We tested our diagnostic on low-current, cm-scale sparks triggered by means of laser filamentation. We were able to realize space and time resolved measurement of electron and neutral densities and to follow the evolution of corresponding spatial profiles in time until the channel expansion reached the boundary of the field of view of the cameras.

Figure 1 .

 1 Figure 1. Schematic layout of the interferometric setup.

  test )(x,f x )| 2 (a. u.)

Figure 2 .

 2 Figure 2. Comparison between the Fourier transform and the CWT. (a): squared magnitude of the Fourier transform of ftest. (b): squared magnitude of the CWT of ftest.

Figure 3 .

 3 Figure 3. Principle of the phase recovery algorithm. (a): example of recorded interferogram. The test row is indicated by the white dashed line. (b): CWT magnitude for the test line. The ridge is indicated by the black dashed line. (c): CWT phase for the test line. The ridge position is reported here and indicated by the black solid line. (d): phase recovered from the interferogram.

9 Figure 4 .

 94 Figure 4. Unwrapped phase map obtained after running the unwrapping algorithm on the wrapped phase map of figure 3-(d).

Figure 5 .

 5 Figure 5. Phase shift plane evaluated from the interferogram displayed in figure 3-(a).

Figure 6 .

 6 Figure 6. Schematic depiction of the probe arm of the interferometer for the Abel inversion problem.

Figure 7 .

 7 Figure 7. (a): example of line-integrated neutral density profile. (b): corresponding inverted neutral density profiles using raw experimental data (black solid curve) or using a low-pass filter during the inversion process (red dashed curve).

Figure 8 Figure 8 .

 88 Figure 8 gives a schematic overview of the experimental setup. A storage capacitor of nominal value C d = 2 nF charged up to a voltage U 0 = 15 kV DC discharges through a laser triggered spark gap to a ballast resistance of nominal value R b = 380 Ω and a current viewing resistor, or shunt, of 50 mΩ.

Figure 9 .

 9 Figure 9-(b) displays the evolution of on-axis electron density and of the discharge current with time. Each data point represents an average taken over five different shots, and the corresponding error bar corresponds to plus/minus one standard deviation. On this figure, it is possible to witness the decreasing trend of n e in time, which loosely follows the exponential decrease of the current. This behavior can be easily explained by the fact that the central density peak does not widen significantly in time, as shown in figure9-(a). As a consequence, maximum electron density should indeed follow the current time evolution.

Figure 10 .

 10 Figure 10. (a): measured neutral density profiles for various probing times. (b): radial position of the shock wave, rSW (blue circles) and corresponding speed uSW (red squares). The speed of sound cs is given for convenience.

Figure 11 .

 11 Figure 11. (a): schema for probe beam deflection scenario. (b): plasma index at t = 150 ns and λ = 1064 nm (blue solid line) and corresponding deflection angle (red dashed line).

ACKNOWLEDGMENTS

This work was funded by the French Direction Générale de l'Armement (Grant n o 2013.95.0901). We also want to thank Florian Mollica for fruitful discussions.


Sergey Mitryukovskiy

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

Pengji Ding

Aurélien Houard

Arnaud Couairon

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

Plasma Luminescence from Femtosecond Filaments in Air: Evidence for Impact Excitation with Circularly Polarized Light Pulses

Keywords: numbers: 33.50. Dq, 34.80. GS, 33.80. Rv



 u C 3 and   g B 3 of neutral nitrogen molecules (2 nd positive system of N 2 ) and between second excited state   u B 2 and ground state   g X 2 of nitrogen molecular ions (1 st negative system of  2 N ) [3,4]. A detailed understanding of the process leading to this filament luminescence is important for several reasons. Filament luminescence has been used as a tool to characterize the length and width of the plasma column and to extract the electron temperature and laser intensity inside filaments [5][6][7][8]. Due to its dependence on an external electric field, it has found application in the remote measurement of DC electric field or intense Terahertz radiation [9,10]. A good understanding of its excitation process is also central to an interpretation of the recently discovered laser action from either neutral or ionized nitrogen molecules inside filament plasmas [11][12][13][14][15][16][17].

In this letter, we measure and discuss the dependence of the luminescence from both N 2 and  2 N molecules on polarization of the incident femtosecond laser pulse. It reveals new non radiative routes to populate excited neutral and ionic molecular levels through electron collisions. A minimum electron kinetic energy is necessary in order to achieve impact excitation of neutral or ionized molecules. In the range of laser intensities of this study, numerical simulations predict that this threshold kinetic energy is only obtained with circular polarization, in good agreement with our experimental observations. In the experiment, femtosecond laser pulses with pulse energy up to 10 mJ (45 fs, 800 nm) from a commercial laser system(Thales Laser, Alpha 100) were focused by an f = 1000 mm convex lens in ambient air. A quarter-wave plate () was installed before the focal lens to change the laser polarization from linear to circular. Plasma channels with length varying from a few millimeters to several centimeters were created, depending on the incident pulse energy. In order to detect the spontaneous luminescence, a lens of focal distance f = 25 mm was employed to collimate the luminescence emitted in the direction orthogonal to the laser propagation axis and a second lens with f = 100 mm was used to focus the collected emission onto the entrance slit of a monochromator (Jobin-Yvon H-20 UV). The two lenses used were made of fused silica. The luminescence signal from the monochromator was measured by a photomultiplier tube (Hamamatsu, model: H10722). In order to resolve longitudinally the luminescence signal along the filament axis, a rectangular diaphragm (1 mm × 6 mm) was placed 1 mm away from the filament axis. More details about the detection method can be found in ref [17].

We first measured the luminescence spectrum for linearly and circularly polarized laser pulses at two representative incident laser pulse energies E in . Detection was performed at the middle position of the plasma in each case. Figure 1 shows the results for E in = 250 J, just below the threshold for filamentation (Fig. 1(a)) and at 8.3 mJ, where the peak power is 30 times higher than the critical power for filamentation (Fig. 1(b)). The corresponding peak laser intensities were determined by measuring the laser flux and pulse duration transmitted through a circular diaphragms of 80 µm diameter placed in the middle of the plasma string. It yields I = 3 × 10 13 W/cm 2 in case (a) and a value I ≥ 1.4 × 10 14 W/cm 2 for incident pulse energies above 1 mJ, when filaments are formed [17]. No significant difference of laser intensity was observed for linear and circular laser polarization. The luminescence lines visible in Figure 1 belong to two categories, as already mentioned: transitions between excited triplet states of the neutral nitrogen molecules or transitions between different ionic states  2 N of the nitrogen molecule [3]. Their precise identification in terms of rotational levels is given Fig. 1. As can be seen, upon increase of laser intensity, there is a reversal in the relative intensity of luminescence between linear and circular pump laser polarization. At lower intensities, linearly polarized pump light is more efficient. At higher laser intensity, a circularly polarized pump becomes more effective.

The same trend is better observed in the dependence of the 337 nm (representative of luminescence from neutral molecules) and 391 nm signals (due to molecular ion emission) as a function of pump laser ellipticity. The ellipticity of the laser can be changed continuously from linear () to elliptical (0 < ), and circular (by rotating the quarter-wave plate. In Fig. 2, results are presented for 4 different pulse energies as a function of rotation angle of the waveplate. The angles φ = 90º × mcorrespond to linearly polarized laser, with m = 0, 1, 2, 3. The angles φ = 45º + 90º × m correspond to circular laser polarization. All measurements were performed around the center of the plasma. With a laser pulse energy of 250 J, the dominance of linear laser polarization over elliptical and circular is observed for both 337 nm and 391 nm emission lines (Fig. 2(a) and 2(a')). With a pulse energy of 600 J, the intensity of the triplet luminescence line at 337 nm becomes independent of the pump polarization (Fig. 2(b)). Upon a further increase of the pump pulse energy, the signal at 337 nm becomes more intense with a circularly polarized pump (Fig. 2(c) and 2(d)). Concerning the emission of  2 N at 391 nm, its relative intensity increases with circularly polarized laser pulses (see Fig. 2(b') and 2(c')), until almost no dependence on ellipticity is observed for pulses of 10 mJ (Fig. 2(d')).

In Fig. 3, we present the evolution of luminescence lines at 337 nm and 391 nm along the filament axis z for both linear and circular laser polarization, again for three different incident laser pulse energies. Below the threshold for filamentation (Fig. 3(a) and(a')), the length of the plasma corresponds to the Rayleigh distance calculated by assuming a linear laser pulse propagation. As expected for such a case, the plasma luminescence peaks at the geometric focus. Both lines are more intense along the ~ 25 mm plasma string with linearly polarized pump. At higher laser energies (E in = 1.1 mJ), the plasma string moves towards the laser, a signature of filamentation (Fig. 3(b) and 3(b')). The luminescence obtained with circularly polarized light increases and becomes equal or even predominant at still higher energies (E in = 10 mJ), (Fig. 3(c) and3(c')). We verified that other emission lines at 357 nm and 428 nm exhibit a similar behavior.

How should we understand the dependence of the luminescence with pump laser polarization? Populating excited ionic state

     u B N 2 2
during filamentation in air is generally accepted as being due to direct high-field photonionization of inner-valence electrons of the nitrogen molecules [START_REF] Talebpour | Multiphoton Processes[END_REF][START_REF] Becker | [END_REF]. Linear laser polarization is known to be more efficient [START_REF] Talebpour | Multiphoton Processes[END_REF][START_REF] Becker | [END_REF]. Direct high-field photonic excitation of the triplet state

    u C N 3 2
is a spin forbidden process and therefore unlikely. Two indirect excitation processes have been proposed. A first scheme consists in a dissociative recombination through the processes:

2 4 2 2 2 N N N N N       followed by   2 3 2 4 N C N e N u       [20].
Another more recent scenario proposes that collision-assisted intersystem crossing from excited singlet states is the dominant path to produce the triplet state, while the dissociative recombination would be a minor contributor [21]. In any case, both processes should generate a larger signal with a linearly polarized pump. In the case of dissociative recombination, the final density of N 2 )

( 3   u C
molecules depends on the density of  2 N , which is more effectively produced by linearly polarized laser in the laser regime of 10 13 W/cm 2 -10 15 W/cm 2 [START_REF] Talebpour | Multiphoton Processes[END_REF][START_REF] Becker | [END_REF]. With the intersystem crossing mechanism, one expects a similar dependence on laser polarization, because the transition from the fundamental singlet state of N 2 to an intermediate singlet state is more effective with linearly polarized laser pulses. So the question is why circularly polarized laser pulses become more efficient at higher laser intensity?

An important difference between linear and circular laser polarization in gas plasma generation is the kinetic energy of free electrons remaining after the passage of the intense laser field [22,23]. With linearly polarized laser pulses, free electrons are left with low kinetic energy because they experience alternative acceleration and deceleration by the laser field during each optical cycle of the pulse. By contrast, with a circularly polarized laser, electrons are always accelerated away from the molecular ion. As discussed below, free electrons acquire an average energy of ~ 2U p at the end of the laser pulse, where

2 0 0 2 2  I m c e U e p  
is the ponderomotive energy of the electron in the laser field with   , m e , I,   being the vacuum permittivity, the mass of the electron, the intensity and frequency of the laser field. Therefore, a large number of electrons with a kinetic energy around 2U p ≥ 16.7 eV are produced inside a filament.

The distribution of transverse electron kinetic energies can be predicted by semi-analytical laws: integration of Newton's equations for electron motion leads to a transverse momentum , where denotes the vector potential at instant t and denotes its counterpart when the electron is liberated, at rest. After the passage of the pulse, vanishes. The transverse momentum becomes , and the transverse kinetic energy reads

, where indicates that the electron was liberated at instant within the pulse. We can infer the vector potential by integration of by using an analytical form for the electric field with a cosine envelope for -T/2 < t < T/2 For t < -T/2 and t > T/2

where denotes an arbitrary carrier envelope phase. The kinetic energy of an electron, born at time t 0 when the electric field phase

  0 0      t c
, after acceleration by the pulse can therefore found to be .

(2) Here

2 0 2 0 2 4  m E e U p 
is the ponderomotive potential. Therefore, all electrons generated between and with probability , where denotes the electron density calculated from the rate equations and denotes the total electron density generated by the pulse, will have a kinetic energy between and . We retrieve that the maximum kinetic energy of the electrons is 2U p for a circularly polarized pulse () when it is born at the peak of the field envelope (t = 0). A parametric representation of the distribution of kinetic energies is presented as a continuous curve in Fig. 4. In the case of linear laser polarization most electrons are left with energy below 1 eV, as shown in Fig. 4 (a). By contrast, an almost monoenergetic distribution around 16.7 eV is achieved for circular laser polarization (Fig. 4(b)) [24]. We also calculated the electron energy distribution after the passage of the laser pulses by numerical simulations. In the simulations, we assume that electrons are generated by optical field ionization of oxygen and nitrogen molecules, described by a set of coupled rate equations. At the intensity level considered, we also computed second ionization processes so as to accurately determine the fraction of oxygen and nitrogen ions which liberate a second electron and found them negligible. Liberated electrons are assumed at rest and are accelerated by the electromagnetic (Lorentz) force mainly in the polarization plane. The acceleration in the longitudinal (propagation) direction is negligible compared to the transverse acceleration. We calculated the classical motion of a set of electrons under the action of the Lorentz force and then performed statistics by weighting each electron by its probability to be liberated at a given instant during the pulse. The results of these numerical simulations are presented by the bars in Fig. (4) and agree very well with that of the semi-analytical analysis.

Electrons with high kinetic energy can populate excited states of N 2 and  2 N via a collision process. Process populating the triplet molecular state opens up if the electron energy exceeds a rather sharp threshold energy of ~ 11 eV, with a cross section peaking at 14.5 eV with a value of 0.58 Å 2 [25]. In a traditional nitrogen laser pumped by electric discharge, it is actually this inelastic collision that gives rise to population inversion between the

  u C 3 and   g B 3
states [26]. In our case, the energetic electrons produced inside

                                 0 sin cos cos 0 0 0 y x u c t u c t Tc t E E                      0 2 2 0 2 cos 1 1 cos 2 e u U E p kin
4 the filament plasma with circularly polarized laser pulse lead to efficient population build-up in the   u C 3 state, and thus stronger luminescence at 337 nm, as observed in Fig. 2(d) and Fig. 3(c). They experience collisions with neutral air molecules at frequency v c ~ 10 12 Hz for standard temperature and pressure conditions. In the presence of abundant energetic free electrons, electron collision excitation of the excited ionic state

  u B 2 is also possible through the process     e B N e X N u g 2 2 2 1 2        
. The corresponding threshold electron energy is 18.75 eV with a maximum effective collision section of ~ 0.01 Å 2 at [25,27,28]. With circularly polarized laser pulses of intensity I = 1.4 × 10 14 W/cm², energetic electrons just reach the required energy.

We have verified the interpretation of new collision-induced excitation routes by repeating at low air pressure the ellipticity measurements shown in Fig. 2. Low gas pressure effectively suppresses collisions and therefore is also expected to suppress the corresponding excitation routes. Result is shown in Fig. 5 for line at 337 nm at a gas pressure of 10 mbar and a laser pulse energy of 350 µJ. Since the propagation of the pulse is linear at such a low pressure, the intensity at the focus can be readily estimated from diffraction to be 1.45 × 10 14 W/cm², of same order of magnitude as in a filament at normal pressure. As can be seen, the response to ellipticity is the same as that observed at lower intensities (Fig. 2(a)) although the reason is different. At normal pressure, collisions are present, but the electron energy is insufficient to excite the molecules (Fig. 2(a)); at low pressure (Fig. 5), electrons have the required kinetic energy but collisions are suppressed. A similar result is obtained for line at 391nm when the laser pulse energy is varied between 250 and 500 µJ.

In conclusion, we have demonstrated that the luminescence emitted by neutral and singly ionized N 2 molecules inside femtosecond laser filaments in air depends strongly on the polarization state of the incident laser pulses. At lower laser pump energies, the predominance of the luminescence from both species with linearly polarized pump is simply explained by the higher optical field ionization rate that constitutes the prime excitation route. At higher laser intensity, new excitation routes become available, due to the presence of electrons with high kinetic energy left after the laser pulse. A collision-assisted mechanism populates the excited triplet state of N 2 and the excited ionic molecular state of nitrogen directly from the ground state of neutral molecules. These findings are important for the understanding of the stimulated radiation from filaments and may find applications in remote sensing of electric field and THz radiation. We believe that these findings are not just restricted to laser filamentation process and they should intervene in other laser-gas interaction phenomenon, such as laser-induced gas breakdown and its relevant applications.

Figure captions : Fig. 1 . 2 N

 :12 Figure captions: Fig. 1. Luminescence spectrum of the air plasma filaments for linearly and circularly polarized laser pulses of 250 J (a) and 8.3 mJ (b). In (a), the signal obtained with circular laser polarization is multiplied by a factor of 2 for visibility. The measurements in (a) and (b) were performed at z = -2 mm and -45 mm respectively (see Fig. 3(a)). The vibration quantum numbers related to the second positive system of the N 2 and the first negative system of  2 N are denoted.

Fig. 2 .

 2 Fig. 2. Luminescence at 337 nm ((a)-(d)) and 391 nm ((a')-(d')) as a function of the rotation angle of the quarterwave plate. The incident pulse energy was 250 J, 600 J, 2.7 mJ and 10 mJ for (a) and (a'), (b) and (b'), (c) and (c'), (d) and (d'), respectively. All the measurements were performed around the center of the filament. Angle 0º corresponds to linearly polarized light.

Fig. 3 .

 3 Fig. 3. Intensity of the 337 nm ((a)-(c)) and the 391 nm ((a')-(c')) luminescence signal measured along the filaments for linearly and circularly polarized laser pulses. The incident pulse energy was 250 J for (a) and (a'), 1.1 mJ for (b) and (b'), and 10 mJ for (c) and (c').

Fig. 4 .

 4 Fig. 4. Calculated electron energy distribution in the case of linearly (a) and circularly (b) polarized laser pulses. The laser intensity used in the calculation was 1.4 × 10 14 W/cm 2 . The red lines represent analytic results and the bars numerical simulation.

Fig. 5 .

 5 Fig.5. Luminescence at 337 nm as a function of the rotation angle of the quarter-wave plate for air pressure of 10 mbar. The incident pulse energy was 350 J and the calculated laser intensity is 1.45× 10 14 W/cm 2 .

Figure 1 7 Figure 2

 72 Figure 1

Figure

  Figure 3

3

 3 Figure 3

8

 8

Figure 4 Figure 5

 45 Figure 4