Picosecond laser filamentation in air

Item Type Article

Authors Schmitt-Sody, Andreas; Kurz, Heiko G; Bergé, Luc; Skupin,Stefan; Polynkin, Pavel

Citation Picosecond laser filamentation in air 2016, 18 (9):093005 NewJournal of Physics

DOI 10.1088/1367-2630/18/9/093005

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Journal New Journal of Physics

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Picosecond laser filamentation in air

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New J. Phys. 18 (2016) 093005 doi:10.1088/1367-2630/18/9/093005

PAPER

Picosecond laser filamentation in air

Andreas Schmitt-Sody1, HeikoGKurz2, Luc Bergé3, Stefan Skupin4 andPavel Polynkin5

1 Air Force Research Labs, KirtlandAir Force Base, Albuquerque,NewMexico, USA2 Institute ofQuantumOptics, LeibnizUniversityHannover,Welfengarten 1, D-30167Hannover, Germany3 CEA-DAM,DIF, F-91297Arpajon, France4 Univ.Bordeaux—CNRS—CEA,Centre Lasers Intenses et Applications, UMR5107, 33405Talence, France5 College ofOptical Sciences, TheUniversity of Arizona, Tucson, Arizona, USA

E-mail: ppolynkin@optics.arizona.edu

Keywords: laser filamentation, picosecond laser pulses, nonlinear propagation, optical ionization

AbstractThe propagation of intense picosecond laser pulses in air in the presence of strong nonlinear self-action effects and air ionization is investigated experimentally and numerically. Themodel used fornumerical analysis is based on the nonlinear propagator for the opticalfield coupled to the rateequations for the production of various ionic species and plasma temperature. Our results show thatthe phenomenon of plasma-driven intensity clamping, which has been paramount in femtosecondlaserfilamentation, holds for picosecond pulses. Furthermore, the temporal pulse distortions in thepicosecond regime are limited and the pulsefluence is also clamped. In focused propagation geometry,a unique feature of picosecond filamentation is the production of a broad, fully ionized air channel,continuous both longitudinally and transversely, whichmay be instrumental formany applicationsincluding laser-guided electrical breakdown of air, channelingmicrowave beams and air lasing.

1. Introduction

Ionization of gases by powerful opticalfields is one of themost important applications of intense pulsed lasers.Two specific interaction regimes have been particularly well studied. Ionization by nanosecond (ns) laser pulses,frequently termed optical breakdown, utilizes intense opticalfields produced byQ-switched lasers [1] and hasbeen investigated since the invention of laserQ-switching until the early 1980s [2]. In this context, experimentalgeometries with tight beam focusing are typically used. The physics associatedwith the gas response to the laserfield is very complex as it involves numerous photochemical reactions, energy exchange between variousparticles and theirmutual collisions [3, 4]. At the same time, the propagation of the ns laser field itself is usuallytreated in a simple way, e.g., by assuming particular temporal and spatial distributions for the laser intensity inthe interaction zone. The second configuration involves intense femtosecond (fs) laser pulses. Here, atsufficiently high peak power of the pulse, Kerr self-focusing of the beam is counteracted by the defocusing actionof plasma generated on the beam axis, resulting in the self-channeling propagation regime known asfemtosecond laserfilamentation [5, 6]. The temporal and spatial dynamics of the optical field are highlynonlinear [7–11]. However, these complex field dynamics are essentially decoupled from the rich airphotochemistry driven by electron–particle collisions, as the latter happen on the time scales that are longer thanthe laser pulse duration. The disparity of the time scales for the field and plasma dynamics significantly simplifythe analysis of femtosecond filamentation.

Compared to the ns and fs cases discussed above, the propagation of either collimated orweakly focusedintense picosecond (ps) laser pulses in gaseousmedia has been studied to amuch lesser extent. In the picosecondregime, both the highly nonlinear propagation of the laser pulse and the complex response of the gas impactedby electron–molecule and electron–ion collisions occur on comparable time scales. This is a very challengingregime to study not only numerically but experimentally as well, as high energies in themulti-Joule range areneeded to drive significant avalanche ionization of the gas. Fewworks have been published on the subject.Mostof them,with the exception of [12, 13], utilized ps laser pulses with rather low pulse energies (∼mJ) in theUV

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27 July 2016

ACCEPTED FOR PUBLICATION

15August 2016

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[14] or visible [15] spectral domains. The generated plasma densities withmJ-range pulse energies weremuchlower thanwhat can be achievedwith near-infrared Joule-level picosecond laser pulses, as wewill show below.

In [12], near-IR ps laser pulses with tens of Joules of energy per pulsewere used. The focus of that studywason the formation ofmultiple filaments within a collimated laser beam vertically launched into the atmosphereand on the generation of supercontinuum radiation. The peak intensity attained inside the individual filamentswas argued to be similar to that in the fs regime.However, neither numerical simulations nor evaluations of theplasma densities were reported.

In a very recent publication [13],filamentation of laser pulses at 1.03 μmwavelengthwith fixed pulseduration of 1.5 pswas investigated experimentally and numerically. The numericalmodel neglected the effectsof heating of the electron gas by assuming a constant electron-neutral collision time. The reported value of theclamped intensity was 23 TW cm−2, which is very close to the result of ourmeasurement and simulation for theparticular pulse duration of 1.5 ps. Contrary to our results, a significant spectral broadening of the laser pulse,accompanied by pulse self-shortening, was observed. The likely explanation for the differences between ourresults and the results reported in [13]was that a significantly higher input pulse energies were used in ourexperiments, so that a fraction of the pulse energywas sufficient to singly ionize the entire nonlinearlyinteracting volume of air (1–10 J pulse energy in our case against 100 mJ used in [13]).

From the practical angle, studies in this field aremotivated by various atmospheric applications such as airlasing [16], guidance of electrical discharges in air [17], lightning control [18] and channelingmicrowave beams[19]. Plasma produced through optical breakdown of air by tightly focused ns laser pulses is dense but spatiallylocalized. By contrast, plasma channels produced through fs laserfilamentation in air are extended andcontinuous, but they are dilute and short-lived, with plasma densities of 1016–1017 electrons per cm3 [20] andplasma lifetime of less than one nanosecond [21]. Attempts to combine the advantages of the fs and ns excitationregimes, through the use of the so-called igniter-heater scheme, have been shown to produce extended anddense plasma channels in air [22, 23]. However, plasma channels generated via the hybrid fs–ns excitation areusually fragmented into discrete plasma bubbles. All of the above shortcomings severely limit applications. Inthis paper, we report results of experiments and numerical simulations on the propagation of intense, weaklyfocused, ps laser pulses in ambient air.We show that all of the above-mentioned drawbacks of the ns, fs and thehybrid fs–ns regimes can be avoided through the use of energetic ps laser pulses, promoting robust, continuousand dense plasma columns in air.

2. Experimental setup and results

Our experimentsmake use of theComet laser systemwhich is a part of the Jupiter Laser Facility at the LawrenceLivermoreNational Laboratory inCalifornia, USA [24]. Comet is a chirped-pulse amplification chain based onNd-doped glass. It produces 0.5 ps transform-limited pulses withmaximumenergy of 10 J per pulse, at thewavelength of 1.053 μm, in a 90 mmdiameter beam. The laser operates in the single-shot regime (one shot in tenminutes). Laser pulses can be chirped up to 20 ps duration. To avoid the degradation of the 10 mm thick fused-silica exit window of the laser’s vacuum compressor chamber, the peak laser powermust not exceed 1 TW,which ismore than one hundred times the critical power for self-focusing in air. Therefore, for pulses shorterthan 10 ps, the laser energy is reduced accordingly. The laser beam isweakly focused in the ambient air using a5 mm thick, 150 mmdiametermeniscus lenswith 3 m focal length.

Infigure 1we show a single-shot photograph of the plasma channel produced by a 10 J, 10 ps laser pulse. The∼30 cm long plasma string appears to be dense and longitudinally continuous, but the resolution of this imagedoes not guarantee the transverse homogeneity of the plasma. The inset shows the fluorescence produced by thelaser beam in far field, on awhite paper screen. The image is taken through a color-glass filter that blocks theoverwhelming light at the fundamental laser wavelength of 1.053 μm.Darker areas in the image result from thescreening of the trailing edge of the laser pulse by the electron plasma generated on the leading pulse edge. Thelarge dark area in themiddle corresponds to the transversely continuous dense plasma on the beam axis. It issurrounded by several peripheral point-like plasmafilaments. The production of a large-diameter plasmacolumn that is not fragmented intomultiplefilamentsmay be compared to the ‘superfilamentation’phenomenon recently discussed in connectionwith relatively tightly focused fs laser pulses [25]. The differencein the ps case discussed here is the important contribution of avalanche ionization, resulting in plasma densitiessignificantly higher than those achievable in fs superfilaments.

It has been argued that the optical intensity in fs laser filaments in gases is clamped to a value that isindependent of the energy of the laser pulse [26]. The clamping concept, which has been central in the fsfilamentation science, is a consequence of the threshold-like dependence of the ionization yield and theassociated nonlinear absorption on the optical intensity. Once the ionization threshold is reached, nonlinearabsorption and plasma defocusing abruptly stop the further intensity buildup, which can be due to the nonlinear

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New J. Phys. 18 (2016) 093005 A Schmitt-Sody et al

self-focusing or linear focusing by an external focusing optic of both. Although this description is somewhatoversimplified as it does not account for the complex temporal pulse dynamics in fs filaments, the concept ofintensity clampingworks verywell for fs pulses (with the exception of the short temporal spikes thatmaydevelop on the trailing edge of the pulse; those spikes, however, carry only a small fraction of the total energy ofthe laser pulse [27]). The reason is that the relevant ionization rates and thus the generated plasma densitiesdepend very sharply on the pulse intensity (i.e.,∝I11), while the dependence of the plasma density on the pulseduration is only linear. Therefore, the exact shape of the pulse, in particular its duration, are of a lesserimportance than the value of the peak intensity, as long as the pulse remains ultrashort. In the case of ps pulses,where avalanche ionization provides amajor contribution to plasma generation, the validity of the intensityclamping concept is not a priori obvious. Itmay bemore appropriate to talk about clamping of thefluence,which is the integral of the intensity over thewhole pulse and, importantly, provides a direct physical observable.

In the case of ps pulses, the temporal intensity buildup ismuch slower that in the fs case. As our simulationswill show, the temporal and spatial dynamics of energetic ps pulses are decoupled, and the temporal pulse shaperemainsweakly distorted. That conclusion is experimentally supported by themeasurements of spectra of thelaser pulses after passing through thefilamentation zone. Themeasurements show almost no spectralbroadening, i.e. no significant distortions of the pulse shape, up to the highest level of the pulse energy used inthe experiments. If no significant temporal reshaping of the pulse occurs on propagation, the notions of intensityandfluence clamping are equivalent.

In order to evaluate thefluence of the ps laserfield in the interaction zonewe produce single-shot ablation ofthe front surface of a glass slide placed along the beampath close to the position ofmaximumplasmaproduction. This position is determined as the point with the brightest plasma fluorescence on a photographicimage of the plasma string (similar tofigure 1). The ablation patternmarks a circular regionwith a sharpboundary, insidewhich the laserfluence exceeds the threshold value for surface ablation of the glass (seefigure 2).We have separatelymeasured the ablation threshold fluence for the particular type of glass we used(soda-lime glassmicroscope slides), for near-infrared laser pulses with various durations in the range from1 to10 ps.We found that the ablation thresholdfluence steadily growswith pulse duration in that range, from2.5 to5 J cm−2.

Data for the diameter of the ablationmark versus pulse energy, for the case of 10 ps pulse, are shown infigure 2. The corresponding area of themark, which is proportional to the diameter squared, scales linearly withthe pulse energy, as evidenced by the agreement of the data with the dashed regression curve. Thus, the averagefluence computed as the ratio of the pulse energy to the ablation area is approximately constant (or clamped)with respect to the pulse energy, at a given pulse duration.We have conducted thesemeasurements using laserpulses with different durations and found that the average laser fluencewas clamped in all cases. The actual valueof the clamped fluencewas found to be a growing function of the pulse duration, as shown infigure 3(a).

Figure 1.Aphotograph of the time-integrated fluorescence by the plasma column produced in air by a 10 J, 10 ps laser pulse,propagating from right to left and slightly downward. The inset shows fluorescence induced by the laser beamon awhite paper screenplaced in the far field. The dark area in the center of the image results from the screening of the trailing edge of the laser pulse by thedense and transversely uniformplasma channel.

3

New J. Phys. 18 (2016) 093005 A Schmitt-Sody et al

The density of plasma in the ionized channel has been experimentally evaluated using a capacitive plasmaprobe setup. The details of this characterization technique are discussed in [28]. In this particular case, the probehas two 5 cm long electrodes separated by a distance of 3 cm and charged to a voltage of 50 V. The signalproduced by the probe in our experiments isfive to six orders ofmagnitude higher than the signal from the same

Figure 2. (a)Measured (•) and simulated ( )diameters of ablationmarks produced by 10 ps pulses with various energies on a glasssurface placed at the position of peak air ionization. The dashed curves are fits proportional to the square-root of the pulse energy.(b)Photograph of the ablationmark produced by a 10 J, 10 ps pulse. (c)The simulated fluence profile for a 9 J, 10 ps pulse. The5 J cm−2 level corresponding to the ablation threshold fluence for 10 ps pulse duration ismarkedwith a dashed line. The inset in(a) shows the simulated beamdiameter at the ablation threshold level of 5 J cm−2 versus the distance from the geometrical focus for a9 J, 10 ps pulse.

Figure 3. (a)Measured (•) and simulated ( ) average laser fluence at the position of peak air ionization, versus pulse duration. At anygiven pulse duration, the average fluence is clamped, i.e., approximately independent of the pulse energy. Error bars are computedfromdeviations from the regression curves (i.e., see figure 2(a)). (b)–(d)Computedfluence profiles at the positions of peak ionization,for various pulse configurations, showing fluence clamping.

4

New J. Phys. 18 (2016) 093005 A Schmitt-Sody et al

probe applied to a regular femtosecond plasma filamentwith an independently known plasma density between1016 and 1017 cm−3 [20]. Accounting for the difference in the total volumes of the plasma detected by the probein the picosecond case under investigation and that of the known reference femtosecond filament, ourmeasurements indicate a complete or nearly complete single ionization of air by the picosecond pulse in theinteraction zone. The uniformity of plasmafluorescence, inferred from the single-shot photographic images,indicates that the plasma is longitudinally uniform along the ionized channel.

Although the above plasma probes have been applied for plasma characterization in laser sparks andfilaments in gases formany years [1, 29, 30], we point out, following [28], that such a characterization techniqueis at best semi-quantitative. These probes do not detect free electrons directly, as electrons attach to neutraloxygenmolecules in the air on amuch faster time scale thanwhat it would take for the electrons to drift towardsthe probe’s electrodes. Instead, the probe detects negative -O2 complexes and positive oxygen andnitrogenmolecular ions that havemuch longer lifetimes than the lifetime of free electrons. Themeasurements with suchprobes also depend on the electron plasma temperature that, in turn, determines the plasma dynamics.Therefore, thesemeasurements are only good for rough order-of-magnitude evaluations of air-plasma density.

3.Numerical simulations

3.1. The time-dependentmodelFor numerical simulations of our experiments we have devised amodel that combines a unidirectionalnonlinear pulse propagator for the laser field amplitudeU [31]with amaterial responsemodule. Theunidirectional treatment is valid as the estimated peak reflectivity of the plasma is less than 10−4, even assuminga complete single ionization of all airmolecules. Ourmodel accounts formultiphoton, tunnel, and impact singleionization of oxygen and nitrogenmolecules with the initial densities of ( )r = ´2.2 10N

0 192

cm−3 and( )r = ´5.4 10O0 18

2cm−3, respectively. Direct photoionization via Perelomov, Popov andTerent’ev (PPT) theory

[32] is included for oxygenmolecules only, as the ionization potential of oxygen is lower than that of nitrogen.Collisional ionization is included for both neutral species. To that end, we compute the inverse bremsstrahlungheating of free electrons using the electron temperature-dependent collision rates

( ) [ ]( [ ]) ( )n r= ´ - - - -+ +T T7 10 cm eV s , 1e eO6

O3 3 2 1

2 2

( ) ( [ ] [ ])( [ ]) ( )( )n r r= ´ -- - - -+T T2 10 cm cm eV s , 2e eO7

O0 3

O3 1 2 1

2 2 2

( ) [ ]( [ ]) ( )n r= ´ - - - -+ +T T7 10 cm eV s , 3e eN6

N3 3 2 1

2 2

( ) ( [ ] [ ])( [ ]) ( )( )n r r= ´ -- - - -+T T2 10 cm cm eV s 4e eN7

N0 3

N3 1 2 1

2 2 2

in the respective collision cross-sections ( ) s n w= e T m n cX X e e2

0 0 02 [33] (subscriptX indexes different

molecular and ionic species: N2,+N2 , O2 and

+O2 ). Our nonlinear propagatormodel reads as follows:

ˆ ˆ ˆ ∣ ∣

ˆ (∣ ∣ )∣ ∣

( ) ( )( )

w

rr

sr r r

¶¶

= + + *

- - - -

-^

-+

zU

kT U U

cn TR U U

k

nT U U

E W U

UU

i

2i i

i2 2 2

, 5ee

e

0

1 2 02

2

0

02

c

1 O OPPT 2

2 O0

O2 2

2 2

( ) ( ) ( ) ( ) ( )dt tt t

t= + Q+ t-R t t t e t

1

2

1

2sin , 6t1

222

1 22 1

2

(∣ ∣ )( ) ∣ ∣ ( )( )r r rs

r¶¶

= - ++ +

tW U

EU , 7eO O

PPT 2O0

OO

O

22 2 2 2

2

2

∣ ∣ ( )rs

r¶¶

=+

t EU , 8eN

N

N

22

2

2

∣ ∣

∣ ∣ ∣ ∣ ( )

s

s s

¶¶

=

- + - +⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

tT

kU

T

E kU

T

E kU

2

3

2

3

2

3. 9

eB

e

e

B

e

B

2

OO

2

NN

2

2

2

2

2

Here, we use the conventional notation for fundamental constants; w0 is the laser angular frequency, n0 is thecorresponding linear refractive index of air, w=k n c0 0 0 is thewavenumber, rc is the critical plasma density

and r r r= ++ +e O N2 2

is the free-electron density. The operator T̂ accounts for the self-steepening and space–

time focusing effects [34]. Linear dispersion of air is included in equation (5) via the operator ̂ [35]. For thevalue of the nonlinear refractive index of air we use = ´ -n 2.8 102

19 cm2W−1 [36]. For generality, we equally

5

New J. Phys. 18 (2016) 093005 A Schmitt-Sody et al

partition n2 into the instantaneous and delayed contributions according to equation (6)with t = 601 fs andt = 802 fs, although the delayed rotational contribution to nonlinearity is indistinguishable from theinstantaneous nonlinear response for picosecond pulses. EN2

and EO2are the ionization potentials of the neutral

nitrogen and oxygenmolecules, respectively. The compound collision cross-section is defined ass s s s s= + + ++ +e O N O N2 2 2 2

. Electron recombination and attachment to neutralmolecules are neglected asthe time scales for those processes aremuch longer than the pulse duration.

We point out that in a recent publication [37] the value of ´ -3.9 10 19 cm2W−1 was reported for theeffective (instantaneous plus delayed)nonlinear refractive index of air, which is higher than the value that we usein our simulations.We checked that using the two different values of n2 in the simulation of our experimentyields essentially identical results, as the linear focusingwith the lens, under our experimental conditions,produces a significant, if not dominant, contribution to the effective net focusing of the beam.We also checkedthat accounting for the depletion of neutrals in theKerr term and photoionization ofN2 using the correspondingPPT rate in equation (8) does not appreciably change the results of our simulations.

Results of our numerical simulations are shown infigures 2 and 3, together with their experimentalcounterparts. The simulations used here utilize an axially symmetric implementation of themodel discussedabove. In order tomimic the experimental conditions, we useGaussian temporal ( )µ -t texp 2

02 and super-

Gaussian spatial ( )µ -r wexp 404 profiles for the input laserfield. Simulations start 8 cmbefore the position of

the geometrical focus, where intensities do not exceed 10 TW cm−2 and ionization is still negligible. The codepropagates the focused pulse, finds the longitudinal position ofmaximumair ionization and, at that position,computes the diameter of the area insidewhich the fluence exceeds the ablation threshold value (e.g., 5 J cm−2

for the case of 10 ps pulse). Simulation results for the size of ablation agreewith experiments within about 25%.The experimental ablationmarksmay be larger than the computed ones because of (i) the beamdefects that arenot accounted for in themodel; (ii) the uncertainty of the longitudinal placement of the glass sample, estimatedas±0.5 cm. From the simulated beamdiameter at the 5 J cm−2 level versus longitudinal position shown in theinset offigure 2(a), it is clear that the placement uncertaintymay affect themeasurement significantly. Incontrast, the uncertainty of the value of the ablation threshold should not significantly affect the computeddiameter of the ablationmark due to the sharp spatial gradients of thefluence profile.

Infigure 3(a)we show the average fluence, defined here as a ratio of the pulse energy to the area of the beamwithfluence above the corresponding ablation threshold value. Numerical and experimental data agreereasonably well and both show a steady growth of the average fluencewith pulse duration. For both numericaland experimental data, the error bars are computed from the deviations of the data points from the square-rootregression curves, shown infigure 2(a)with the dashed lines for the case of 10 ps pulses.While we report analmost perfect linear dependence of the simulated ablation area on the pulse energy for shorter pulses (notshown), some deviation from linearity is evident for the case of 10 ps pulses. The inspection of the computedfluence profiles (figures 3(b)–(d)) confirmsfluence clamping. Note that the value of the clamped fluence for the1.5 ps input pulse duration is about 40 J cm−2. Assuming that the pulse duration does not significantly changeon propagation, which is supported by simulations andmeasurements of the very small spectral broadening ofthe laser pulse after the filamentation zone, the corresponding value of the clamped intensity is about25 TW cm−2. That number is very close to the peak intensity level of 23 TW cm−2 reported in [13] for 100 mJ,1.5 ps long laser pulses.

Simulations show a complete single ionization of both oxygen and nitrogenmolecules in the psfilament,confirming the results of the semi-quantitativemeasurements using the capacitive plasma probe discussedabove. Computed plasma density exceeds 1019 cm−3 for awide range of pulse energies and durations.Figures 4(a) and (b) show the distributions of plasma produced by 500 fs and 10 ps pulses, eachwith 1 J ofenergy.While the shorter pulse produces plasma near the beam axis only, the 10 ps pulse produces amuchlarger, homogeneous plasma channel with complete single ionization of airmolecules. Electron temperaturesup to 35 eV are achievedwith 1 J pulses in the region of complete ionization, as shown infigure 4(c).With 10 Jpulses, electron temperatures as high as 100 eV are observed. The inspection of the spatio-temporal intensityprofiles (two snapshots are shown infigure 4(d)) reveals that a complete single ionization of air, predominantlythrough collisional ionization, occurs on the leading edge of the pulse, which remains relatively intact withintensity clamped at the level below 100 TW cm−2.

3.2. The time-averagedmodelThe simulations discussed so far utilized an axially symmetric implementation of ourmodel, because fully spaceand time resolved simulations of ps pulses would be prohibitively expensive in terms of computationalresources. Thus, based on the above data we cannot rule out the possible onset ofmultiple filamentation, whichmay break the plasma channel in the transverse plane. To investigate the transverse uniformity of the plasmacolumn,which is evident fromour experimental results, we conduct another set of simulations employing aspatially full 3D ( )x y z, , but time-integrated version of equation (5). To be able to formulate a closed-form,

6

New J. Phys. 18 (2016) 093005 A Schmitt-Sody et al

time-integrated version of equation (5), we resort to the following set of approximations, all of which arejustified, provided that we are only interested in a qualitative conclusion onwhether the transverse beaminstability is strong enough to causemulti-filamentation. First, the possible onset ofmulti-filamentation, shouldmulti-filamentation occur, will happen along the self-focusing stage of propagation, prior to the onset ofsignificant ionization, whichwould tame rather than enhance the instability. Should themulti-filamentationoccur, it will be seeded by the spatial non-uniformities of the initial transverse beamprofile. Accordingly, wesimulate broad realistic beams by approximating their time dependence by aGaussian pulse profile and discardair dispersion. Further, we restrict plasma generation tomultiphoton ionization of oxygenmolecules at the rate

∣ ∣s=W UO 1122

2with the 11-photon cross-section s = ´ -2 1011

4 s−1cm22 TW−11. By doing so, wesystematically underestimate plasma generation, as we neglect bothmulti-photon ionization of nitrogenmolecules and avalanche ionization of both oxygen and nitrogenmolecules. Consequently, we overestimate self-focusing and the corresponding transversemodulation instabilities. As our results will show, even in that casethefilament remains transversely uniform. Therefore the account for avalanche ionization, which could onlymitigate the transverse instability, would not change the qualitative conclusion on the transverse uniformity ofthe plasmafilament.

Following [38], the field envelope is decomposed as ( ) ( )y= ´ -U x y z t t, , exp 202 , where t0 corresponds

to the initial pulse duration. Thenwe find ( ) ∣ ∣ [ ( ) ]r g y» ++ +t t terf 22 1O O22

02 2, where ( )xerf denotes the

usual error function and ( )g p s r=+ t88O 0 11 O0

2 2. After plugging the above expressions into equation (5),

multiplying the entire equationwith the pulse profile ( )µ -t texp 202 , and integrating the propagation equation

over the entire time domain, we arrive at the following extended nonlinear Schrödinger equation for the spatialprofile ( )y x y z, , accounting for self-focusing, plasma defocusing, andmultiphoton absorption:

∣ ∣ ∣ ∣ ∣ ∣ ( )( )

y y a yg

ry y

s ry y

¶¶

= + - -^

+⎛⎝⎜

⎞⎠⎟z k

k nEi

2i

2 2 11. 10

0

20 2

2 O

c

22O 11 O

0

202 2 2

For pulse durationsmuch larger than the dephasing time, tt0 2 (see equation (6)), the contribution of thedelayedRaman response to self-focusing is indistinguishable from the instantaneous Kerr response, andwe finda » 1 2 . An extended version of thismodel has been shown to faithfully reproduce experimental beampatterns inmultifilamentation of fs pulses with peak power significantly above critical [38, 39].

The beamprofile for the laser source used in our experiments has not been characterized because of itsextremely lowfiring rate. In order to numerically investigate the potentialmultifilamentation dynamics for ourps pulses, we instead used two different representative input beam shapes: (i) a noisy fluence profilecharacteristic ofmulti-Terawatt fs lasers [38, 39]; in that case small-scale transverse fluctuations of the beam

Figure 4. (a) Simulated plasma channel produced by a 500 fs, 1 J laser pulse. The longitudinal position of the peak plasma production(themaximumof the linear plasma density ( )ò r r z t r r, , de for t larger than the pulse duration) ismarked by the dashedwhite line.(b) Same for a 10 ps pulse with 1 J energy. (c)Peak ionic densities and electron temperature versus propagation distance for the 10 pspulse shown in (b). (d) Snapshots of the corresponding spatio-temporal intensity profiles. All longitudinal coordinates are shownwithrespect to the linear geometrical focus of the focusing lens.

7

New J. Phys. 18 (2016) 093005 A Schmitt-Sody et al

fluence can efficiently seed the transversemodulation instability, and (ii) a 4th-order super-Gaussian beamprofile perturbed by the addition of 10%of random amplitude noise. Figure 5 shows thefluence distributionsproduced by such input beamprofiles for 10 J, 10 ps pulses. The simulations are started 1 mbefore the positionof the geometrical focus, where intensities are about 0.1 TW cm−2 and nonlinear effects are negligible. Nearfocus,maximum intensities reach tens of TW cm−2. For such intensities, analytical estimates suggest that theinstability growth rate, ( )a g rG = - +k n I I11 20 2 O

11c2, of stationary transversemodulations [40] satisfy G < 0

so that transverse perturbations are not amplified. Thus, a broad plasma channel, once formed, does not havetendency to breakup intomultiple filaments. On the propagation stage towards the nonlinear focus, intensitiesare lower, e. g., I 1TW cm−2, and small-scale beamfluctuations cannot grow on the scale of our experiments.Thus, experimental results, analytical estimates and numerical simulations all show that under our experimentalconditions, no breakup of the beam intomultiplefilaments occurs on propagation and the plasma channelremains transversely uniform.We point out that the role of the external linear focusing in the propagationdynamics of ultraintense ps pulses remains to be quantified; it is likely to contribute to the inhibition ofmulti-filamentation, similar to the case of ‘superfilamentation’mentioned above. Also note that in the simulationusing the noisy input beam (figure 5(a)), several peripheral smallerfilaments emerge outside the plasma region,in agreementwith the experimental observations (see the inset infigure 1).

4. Conclusion

In conclusion, we have investigated the propagation of TW, ps laser pulses in air bymeans of experiments andnumerical simulations. Ourmodel is based on a unidirectional pulse propagator coupled to amaterial responsemodule that accounts formultiphoton, tunnel and impact ionization of airmolecules, as well as heating ofelectron plasma.We report good agreement between experimental and numerical results. Ourmajorfinding isthe possibility to produce a broad, fully ionized, as well as longitudinally and transversely continuous channel,that can be instrumental for various atmospheric applications such as the guidance ofDC electrical breakdownof air, channelingmicrowave beams and air lasing.

Acknowledgments

Thismaterial is based uponwork supported by theUSDefense Threat Reduction Agency under programnumberHDTRA1-14-1-0009 and by theUSAir ForceOffice of Scientific Research under programs number

Figure 5. Fluence profiles computed using the time-integrated propagationmodel equation (10) and (a) a realistic noisy input beamand (b) a 4th order super-Gaussian input fluence profile perturbed by 10%amplitude randomnoise (see text for details). In both cases,the beamprofiles 1 mbefore and at the linear focal plane are shown in the top and bottompanels, respectively.

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New J. Phys. 18 (2016) 093005 A Schmitt-Sody et al

FA9550-12-1-0482 and number FA9550-16-1-0013. The use of the Jupiter Laser Facility was supported by theUSDepartment of Energy, Lawrence LivermoreNational Laboratory, under contract no.DE-AC52-07NA27344.Numerical simulationswere performed atMésocentre deCalcul Intensif Aquitain (MCIA), GrandEquipementNational pour le Calcul Intensif (GENCI, supported under grants no. 2015-056129 and 2016-057594), andThe Partnership for AdvancedComputing in Europe (PRACE, supported under grant no. 2014-112576). LB thanks PatrickCombis for the discussions on ablation thresholds in glasses.

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