Postionization regimes of femtosecond laser pulses self-channeling in air

Postionization regimes of femtosecond laser pulses self-channeling in air

Stéphanie Champeaux* and Luc BergéDépartement de Physique Théorique et Appliquée, CEA/DAM Ile de France, B.P. 12, F-91680 Bruyères-le-Châtel, France

sReceived 1 December 2004; published 7 April 2005d

An optical self-guiding of femtosecond filaments in air is identified in a regime where plasma generationceases to support the self-channeling process. Group-velocity dispersion is shown to keep the beam temporallyand spatially localized upon a few meters by taking over the ionization of air molecules, once the pulse peakpower becomes close to the self-focusing threshold. In this regime, the pulse undergoes slow splitting eventsthat maintain a residual self-guiding with light intensities as high as 10 TW/cm2, as soon as the electronplasma density has fallen down below 1015 cm−3.

DOI: 10.1103/PhysRevE.71.046604 PACS numberssd: 42.65.Tg, 42.68.Ay, 42.65.Jx

I. INTRODUCTION

It is well known that the competition between Kerr self-focusing and plasma generation promotes the self-guiding offemtosecond filaments over long distances in airf1–6g.These filaments originate from ultrashort laser pulses, whosehigh input powerPin exceeds the power thresholdPcr abovewhich self-focusing takes placef7,8g. As a result, the atmo-spheric medium behaves as a Kerr lens for the spatial profileof the beam, which shrinks its spatial extent to sizes as smallas 100–200mm. This beam compression forces the opticalintensity to increase rapidly up to 1014 W/cm2, until a tenu-ous plasma with electron densities around a few 1016 cm−3

occurs and maintains the beam in the form of a narrow wave-guide. Along this process, a pulse with near-infrared wave-length self-transforms into a white-light laser pulse. It givesrise to supercontinuum generation and develops a strongconical emissionf9–11g. Because of these remarkable prop-erties, namely, the creation of self-guided filaments throughphotoionization and white-light generation, the propagationof femtosecond light pulses has been the subject of scientificinterest during the past decade, due to its potential applica-tions such as atmospheric remote sensing and lightning dis-charge controlf12–16g.

Although the previous features originally concerned infra-red pulses delivered by Ti:sapphire laser sourcesscentralwavelengthl0=800 nmd, some of them were refound whenmanipulating ultraviolet beamssl0.250 nmd. In Refs.f17–19g, the same processes, except white-light emission,were experimentally or numerically retrieved for UV wave-lengths over comparable distances of propagation. An iden-tical dynamics was, moreover, shown to govern filament for-mation in noble gases at visible wavelengths, as studied inRef. f20g.

From the theoretical point of view, the model equationsdescribing the formation of femtosecond filaments currentlyinclude linear diffraction in the transverse plane, normalgroup-velocity dispersionsGVDd, Kerr self-focusing also re-sponsible for self-phase modulation, plasma generation bymultiphoton transitions and avalanche ionization, together

with multiphoton absorptionsMPAd and plasma lossesf21–25g. Usually, the Kerr response is decomposed into aninstantaneous contribution and a Raman-delayed one for sub-picosecond IR pulses in the atmospheref26g, while plasmageneration is mainly supported by multiphoton ionization.Self-channeling then follows from the complex dynamicsmixing a strong Kerr compression of the beam in space anda severe temporal reshaping caused by the plasma response,that defocuses the most intense time slices in the pulse.Plasma defocusing thus generates multisplit structures. In ad-dition, we have to consider the possibility of promoting pulsesplitting by the action of GVD. Inclusion of self-steepeningterms and higher-order dispersionf27–30g sometimes com-pletes the model. In gaseous media, however, these effects donot change the major dynamics driven by the interplay be-tween self-focusing, plasma defocusing, and GVD, apartfrom a reinforced shock dynamics that asymmetrizes thetemporal profile and blue shifts the pulse spectrumf10g.

Competition between plasma and GVD inherently de-pends on the ratio of the input peak power over critical andon the dispersion characteristic length involving the initialpulse waist over its duration, as established in the pioneeringworks f31,32g. For very small values of the latter parameter,self-focusing triggers ionization at powers even close to criti-cal. This scenario typically applies to air, for what concernsthe onset of the self-guiding stage. Nonetheless, it does notprevent a relevant action of GVD in asymptotic propagationstages when plasma turns off, which will be examined below.

Another important aspect concerns the robustness of thebeam spatial profiles along the propagation axis. Broad cwbeams in focusing media are indeed well known to becomesubject to the modulational instability that breaks up thebeam into small-scale structuresf33g. Generally attributed tothe ambient noise leading to exponentially growing perturba-tions, the resulting structures, also commonly called “fila-ments” or “cells,” are first excited by the initial defects of theinput beam. In a pure Kerr medium, their number can beestimated byPin /Pfil , wherePfil .p2Pcr/4 is the power perunit cell. In a medium with nonlinear saturation, this numbercan noticeably diminish as the saturation competes with theKerr nonlinearityf34g. In the presence of plasma generation,this simple picture of multiple filamentation may no longerhold, because the ionization front defocuses the pulse asym-metrically along the temporal direction. Plasma defocusing*Electronic address: [email protected]

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also creates spatial ringsf21g, which undergo azimuthal in-stability. The number of filaments then change following thepulse time slicesfocused or defocusedd under consideration.Starting with a Gaussian profile, a pulse can develop fewfilaments in the early stage when the rear of the pulse isdefocused, and next decay into several cells when the trail ofthe pulse focuses again in a second propagation stagef35,36g. This complexity yields a turbulent spatiotemporaldynamics, following which high-power pulses may behaveas a turbulent light guide constituted by short-living cellsf37g. These maintain the illusion of a coherent waveguidepropagating over several hundreds of meters. This scenariohas recently been tested through several series of experi-mentsf38,39g. It was observed that the random nucleation ofsmall-scale cells appeared at very high powers inside specificzones located in the vicinity of the beam defects. Thesezones serve as energy reservoirs for further nucleation offilaments. They share broad beams with centimeter waistsinto clusters of filaments or “optical pillars,” which actuallysupport a long-range propagation. For narrower beams withmillimeter waists, a recent workf40g analyzed in detail thecoalescence of two filaments in the presence of plasma gen-eration at moderate peak powerssPin,11Pcrd. It was ob-served that due to the energy exchanges beween the twoparent filaments, several child filaments could be excited lo-cally in time and in space. By decreasing their energy byMPA, they combined into one lobe only, in agreement withthe experimentf41g. The number of child filaments stronglydepends on the initial distance separating their parents. Theshorter this distance, the more child filaments are produced,each of which excites a plasma string and then contributes toincrease the total electron densityf40g.

The present work addresses the point of identifying therole of GVD in the filamentation of millimeter ultrashortpulses in the atmosphere and to discriminate this role fromthat of the Raman-delayed Kerr response. Special emphasisis given to the temporal distortions undergone by the pulsedue to these effects, together with their influence on the mul-tiple filamentation patterns. We demonstrate the ability of thepulse to propagate over meter-range distances as the electronplasma density falls down, so that plasma can no longer bal-ance self-focusing. We thoroughly examine the role of bothGVD and Raman-delayed Kerr effects in sustaining a self-guiding regime with no plasma generation. In this postion-ization regime, GVD arises as a key player: On the one hand,GVD shortens plasma self-channeling; on the other hand, itmaintains the propagation of temporally and spatially local-ized pulses.

The paper is organized as follows: In Sec. II, we recall themodel equations governing femtosecond pulse propagationin air. In Sec. III, we recall a two-scale variational approach,that will provide analytical arguments supporting the mostsignificant aspects revealed by our investigation. In Sec. IV,s3D+1d numerical simulations are performed for initiallyperturbed Gaussian beams with 22 critical powers and 1 mmwaist. The early filamentation stage is analyzed and shown torelax to a single filament. Two distinct self-guiding regimesare identified and discussed, namely, the classical plasmaself-channeling and an optical one involving pulses withweaker intensities. The influences of GVD and Raman-

delayed Kerr effect on both of them are separately discussedin Sec. V. A particular attention is devoted to the relevantmechanism underlying the optical self-guiding regime.

II. MODEL FOR PULSE PROPAGATION IN AIR

The theoretical model for femtosecond pulse propagationin air is based on an extended nonlinear Schrödinger equa-tion governing the slowly varying envelope of the light elec-tric field, coupled to a Drude model for the local plasmagenerated by ionization. The complex scalar envelopeEsx,y,t ,zd of the electric field and the electron densityr ofthe excited plasma evolve asf21–25g

i]zE +1

2k0='

2 E −k9

2]t

2E + k0n2Ss1 − xKduEu2

+xK

tKE

−`

t

e−st−t8d/tKuEst8du2dt8DE − S k0

2rc− i

s

2DrE

+ ibsKd

2uEu2K−2E = 0, s1d

]tr =bsKd

K"v0uEu2K +

s

UiruEu2, r ! rat, s2d

wherez is the propagation distance andt the retarded time inthe frame moving with the group velocity of the pulse alongthe forward direction. The second term in Eq.s1d, where='

2 ;]x2+]y

2, accounts for transverse diffraction and the thirdone for GVD with coefficientk9=]2k/]v2u v0

=0.2 fs2/cm atthe laser wavelengthl0=800 nmsk0=2p /l0=v0/c, wherecis the speed of light in vacuumd. The fourth and fifth termsrepresent the nonlinear self-focusing induced by the Kerr re-sponse of the medium. This includes an instantaneous com-ponent due to the electronic response in the polarization vec-tor and a delayed one in fractionxK=0.5, due to thestimulated molecular Raman scattering with characteristicrelaxation timetK=70 fs. Here, the nonlinear refraction in-dex isn2=3.2310−19 cm2/W and the critical power for self-focusing takes the valuePcr;l0

2/2pn2=3.2 GW. The fol-lowing terms describe defocusing and nonlinear absorptiondue to plasma coupling. The cross section for electron-neutral inverse Bremsstrahlungs in the Drude model is cal-culated from the expressions=sk0/rcdhv0t / f1+sv0td2gj,where t denotes the electron collision time andrc=e0v0

2me/e2 is the critical plasma density, beyond which thelaser pulse no longer propagates. With the present param-eters, one hasrc.1.1131021/ sl0

2 fmmgd=1.7431021 cm−3

ands=5.44310−20 cm2. The last term in Eq.s1d representsenergy losses induced by multiphoton ionizationsMPId,where bsKd=K"v0ratsK is the nonlinear coefficient forK-photon absorptionsMPAd depending on the MPI coeffi-cient sK. K is the minimum number of photons with energy"v0 required for liberating an electron from a medium withionization potentialUi fK=modsUi /"v0d+1g; rat denotes thedensity of neutral molecules at the atmospheric pressure.

As far as plasma formation is concerned, we assume thatphotoionization in airs80% nitrogen, 20% oxygend mainly

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applies to dioxygen molecules, as their ionization potential,Ui =12.1 eV, is lower than nitrogen’ssUi =15.6 eVd. Thedensity of neutrals is then fixed to the valuerat=5.431018 cm−3. For peak intensities reaching1013–1014 W/cm2, MPI and tunnel ionizations should bothcontribute to photoionization. For technical convenience, wechoose to fit the ionization rateWsId provided by the PPTmodel f42g ssee also Refs.f43,44gd by an MPI formulation:WsId<sKIK with sK=2.88310−99 s−1 cm2K /WK. This ex-pression supplies a good approximation ofWsId f23,39g.Avalanche ionization and plasma absorption, although ex-pected to have a weaker influence, account for ionization viacollisions between free electrons and neutrals and they com-plete the model. As in Refs.f24,25g, MPA is self-consistentlyintroduced into Eq.s1d by using the Poynting theorem yield-ing the energy balance between the MPI electron density andthe related optical absorption ofK photons with energy"v0through a constant cross section. Equationss1d and s2d con-tain the principal ingredients for describing the propagationof ultrashort pulses with peak intensities up to 1014 W/cm2

and powers above the GW. Pulse channeling then resultsfrom the competition between the transverse diffraction ofthe beam, normal GVDsk9.0d, self-focusing due to thenonlinear change in the gas refractive index, MPI defocusingand MPA.

In the following, we shall consider collimated inputpulses centered at the wavelengthl0=800 nm, with trans-verse waistw0=1 mm and half width durationtp=100 fs,which enter the Gaussian profile

Esx,y,t,0d = Î2Pin/pw02e−sx2+y2d/w0

2−t2/tp2, s3d

wherePin is the input beam power. We henceforth choose thepower ratioPin /Pcr=22 sI0=2Pin /pw0

2.4.5 TW/cm2d. Thenatural diffractionsRayleighd length for this Gaussian beamis defined byz0;pw0

2/l0=3.93 m. Evolution of the inputs3d, perturbed in space and time by an isotropic 10% ampli-tude random noise, has been investigated numerically by in-tegrating Eqs.s1d and s2d along the propagation axis. Nu-merical simulations were performed with a parallel spectralcode in sx,y,td, running over 128 processors and using anadaptive step size alongz, tuned on the intensity growthf39g.Computations required a numerical box of 51235123768points in thesx,y,td dimensions.

III. PRELIMINARY GUESS FROM A VARIATIONALANALYSIS

Before proceeding with numerical simulations, we wish toargue on the relevant mechanisms participating in the self-channeling process. For this purpose, we find it instructive todetermine the first tendencies in the pulse evolution bymeans of an approximation method. We rescale Eqs.s1d ands2d in dimensionless form, by introducing the substitutionsr →w0r, t→ tpt, z→4z0z, E→ÎPcr/4pw0

2c, and r→ src/2z0k0dr. Assuming a weak contribution from ava-lanche ionization and related losses, the model equations arethen re-expressed as follows:

i]zc + ='2 c − d]t

2c + Rstdc − rc + inucu2K−2c = 0, s4d

]tr . Gucu2K, s5d

Rstd = s1 − xKducu2 + xKtptKE

−`

t

e−tpst−t8d/tKucst8du2dt8, s6d

whered;2z0k9 / tp2 andn;2z0bsKdsPcr/4pw0

2dK−1 denote theGVD and MPA normalized coefficients, respectively. Therescaled MPI coefficient reads as

G ; s2z0k0/rcdsbsKd/K"v0dtpsPcr/4pw02dK.

For simplicity, we also reduce Eq.s6d to an effective in-stantaneous response with suitable weight, influencing thespatial dynamics of the pulse only. To do so, let us considera Gaussian temporal profile scaling asc2,expf−t2/Tszd2g.Figure 1 shows the corresponding curvesRstd for the frac-tion xK=1/2 atz=0 fTs0d=1/Î2g and at later distance, forwhich GVD is expected to linearly stretch the pulse profilefTszd=3Ts0dg. Comparison of the nonlinear refraction indexat z=0 when retainingsxK=0.5d or ignoring sxK=0d thedelayed-Kerr component shows that the latter weakens thenonlinear refraction index and shifts its maximum towardspositive times. At further distances, however,Rstd becomesmore symmetric and approaches the fully instantaneous Kerrresponse in time. From this simple basis, the total Kerr termcan be approximated by an instantaneous contribution havingthe effective refraction indexn̄,1. We estimate this nonlin-ear refraction index at the temporal slice with maximum in-tensity, yieldingn̄.0.8 atz=0.

In order to get analytical insights on the global behaviorof the pulse, we now adopt a two-scale variationnal approachf22,24g, which allows us to describe the radial and temporalmean pulse widthsRszd andTszd along thez axis. We assumea self-similar profile, i.e.,

FIG. 1. Evolution of the functionRstd including both the instan-taneous and delayed Kerr responses in ratioxK=1/2 for aGaussianpulse with temporal extentsTsz=0d=1/Î2 ssolid lined and Tszd=3Ts0d sdashed lined. The dotted and dash-dotted curves refer to thecases whenxK=0, for z=0, and forTszd=3Ts0d, respectively.

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c =ÎJszd

RszdÎTszdfsj',hdexpfiSsj',h,zdg, s7d

S= iRzR

4j'

2 − iTzT

4dh2 − i

Tdzktl2d

h, s8d

where

j' ;Îx2 + y2

Rszd, h ;

t − ktlTszd

. s9d

Here, f=exps−j'2 /2−h2/2d is chosen as a Gaussian test

function with initial centroidkts0dl=0, in accordance withthe input distribution selected in Sec. II. The functionJszdmeasures the power losses caused by MPA, withJs0d=2Î2Pin /Pcr. We insert the ansatzs7d and s8d into somevirial-type equations for the transverse and temporal mean-square radii already derived in Refs.f22,24g, and setRstd. n̄ucu2 for xK=0.5, as justified in Fig. 1. This leads to thedynamical system forRszd andTszd:

1

4Rzz=

1

R3 − n̄JTs0d4R3T

+G

2

ÎpK

sK + 1d2

JK

R2K+1TK−1

−nsK − 1d

4K5/2 RS J

R2TDK−1F 2n

ÎKS J

R2TDK−1

+ sK − 1dTz

T

+ 2sK − 3dRz

RG , s10d

1

4Tzz= dS d

T3 + n̄JTs0d4R2T2D −

nsK − 1d4K5/2 TS J

R2TDK−1

3F 2n

ÎKS J

R2TDK−1

+ 2sK − 1dRz

R+ sK − 5d

Tz

T G ,

s11d

Jz = −2n

K3/2

JK

sR2TdK−1 . s12d

We integrate this variationnal system and compare theevolution alongz of sid the characteristic quantitiesRszd,Tszd, andJszd, sii d the pulse intensity and density maxima atthe beam center, when retaining or omitting the GVD term.These pieces of information are shown in Fig. 2. We observethat GVD is responsible for a significant increase of the tem-poral width Tszd along the propagation direction, as ex-pected. GVD also contributes to shorten the self-guidingrange sustained by plasma generation, as shown by the bot-tom subplots representing the maxima of pulse intensity andplasma density. In fact, from Eq.s7d, it is straightforward toexpress the power in a given time slicePsz,td in the form

Psz,td/Pcr .Jszd

4Tszde−st − ktld2/T2szd. s13d

So, as GVD increases the slope ofTszd, this power de-creases faster down to levels close to critical. FollowingRefs. f31,32g, GVD is then able to inhibit the self-focusing

processsthat triggers ionizationd, before the beam powergoes belowPcr. This property follows from the existence of atypical length scalezNLGVD along which GVD is efficientenough to decrease the pulse power below critical, as in-ferred from Eq.s13d stated att=ktl. Setting zNLGVD ,zc,wherezc denotes the nonlinear focus given by Marburger’sformula f7g, provides the values ofd vs Pin /Pcr, for whichGVD prevents the beam collapsef31,32g.

From Fig. 2, we observe that when GVD is taken intoaccount, the level of power over critical decreases by a factorof T0Jszd /TszdJ0,7.25310−2 over the Rayleigh distance.The maximal effective powern̄322=17.6Pcr conveyed bythe pulse can thus decrease to 1.27Pcr, where GVD begins tobe a key player according to Refs.f31,32g.

In summary, the variational approach predicts that, fromthe Gaussian pulses3d, the self-focusing dynamics increasesthe beam intensity by the maximal factor,17.5. A propaga-tion range of about 5–6 m sustains a GVD broadening intime reaching,5.5 times the initial duration. In plasma re-gime, the waist compression attainsw0/7. This range be-comes enhanced by,1 m, when GVD is discarded. GVD isthus capable of shortening a self-channeled propagation sup-ported by plasma generation. Note that, although they revealan important property, the present results must be consideredwith caution, because of the following reserves:

sid The two-scale variational approach relies on a fixedspatial distribution, which cannot describe the multiple fila-mentation instability and sharp spatial distortions in thebeam profile.

sii d This method, by construction, captures the wholepower of the initial solutions3d. So, it does not permit anyevacuation of radiation to the boundaries.

siii d The variational approach only describes a singlepulse component having a global extent in timeTszd. It can-

FIG. 2. sad,sbd Evolution along thez direction of the mean radialand temporal widthsRszd ssolid lined, Tszd sdashed lined andJszd /Js0d sdashed-dotted lined; scd,sdd on-axissj=h=0d pulse inten-sity sW/cm2d ssolid lined and densityscm−3d sdashed-dotted line;units are indicated on the right-hand side axisd, both computed fromthe variational equationss10d–s12d with GVD sleft columnd andwithout GVD sd=0, right columnd.


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not therefore model any temporal splitting event, as will bemet below.

In spite of these limitations, such approximation methodsprovide global tendencies, which emphasize the importantrole of normal GVD in modifying the propagation range atthe end of the plasma-induced self-channeling regime.

IV. SELF-GUIDING REGIMES

We now comment on direct numerical simulations of thefull s3D+1d model Eqs.s1d ands2d. Results are presented inFig. 3. From top to bottom, Fig. 3sad shows the peak opticalintensity and maximum plasma density integrated from theinitial condition s3d. Two different propagation regimes canbe distinguished:sid a plasma regime corresponding to the

classical self-guiding of femtosecond filaments sustained byplasma formation. The propagation range is then associatedwith two Kerr-focusing–plasma-defocusing sequences.sii dAn optical regime that takes place beyond the Rayleighlength sz0.4 md. When the electron density falls down be-low 1015 cm−3, a remnant pulse survives over half a Rayleighlength, while it can still convey intensities as high as1013 W/cm2. Figures 3sbd–3sed illustrate characteristic fila-mentation patterns computed from the fluence distribution ofthe beam. We detail these regimes in the coming subsections.

A. Self-guided propagation in plasma regime

In the early propagation stage, the beam undergoes Kerrself-focusing achieved at the characteristic nonlinear focaldistance f7g zc.0.367z0/ fsÎn̄Pin /Pcr−0.852d2−0.0219g1/2

=0.43 m, from which an electron plasma is triggered viaoxygen ionization. The arrest of beam collapse here proceedsfrom plasma defocusing. Self-focusing is balanced by MPIaround the threshold intensity Imax<rmax/2rcn2n̄,f2rcn̄n2/ sDtsKratdg1/sK−1d.47.8 TW/cm2 associated withthe electron density levelrmax< tpsbsKd /K"v0dImax

K .4.2431016 cm−3. The beam is then stabilized with a spatial di-ameter decreased toDmin,2ps2k0

2n̄n2Imaxd−1/2.161 mm.This value yields the reduced waists,100 mmd over whichthe filament core is kept localized around one Rayleigh dis-tance. From Fig. 3sad, the beam waist undergoes a compres-sion factor of,7.35, while the peak intensity grows up to 17times its initial value, which reasonably agrees with thevariational model. The filamentation range concerned withhigh intensity values, up toz=5–6 m, also qualitativelyagrees with the variational resultsfsee Fig. 2scdg.

After the early compression of the beam, the pulse breaksup into two filamentssz=1.18 md, which eventually self-attract and coalesce into a single onesz=2.75 md fsee Figs.3sbd–3sedg. It is worth noticing that the number of filamentsarising from the fluence distribution in Fig. 3 does not ex-ceed 2. We should, however, expect aboutn̄Pin /Pfil .7 cellsin optical regime, which do not apparently occur. The expla-nation follows from the temporal dynamics and associatedplasma strings depicted in Fig. 4.

Figure 4sad details the plasma strings issued from the fo-cusing time slices of the pulse. Plasma emission developsalong two distinct sequences. The first one concerns high-plasma levels. The rear of the pulse is defocused while fewfilaments are emitted in the coronal region surrounding thefilament core sz<0.5 md. Then the pulse trail refocuseswhile the Kerr effect dominates again in the outest zones ofthe filament sz<0.78 md. These zones become subject toazimuthal instabilitiesf25,35g, giving rise to 7–8 filamentsfrom the most intense times slices shifted in space from thebeam center and each exciting a burst of free electrons. Thus,locally in space and time, the number of filaments agreeswith the basic expectations from the modulational instabilitytheory. Integrated in time, i.e., from the fluence distributionsF=e−`

+`uEu2dtd, this number is reduced, or at least attenuated,as already observed in Refs.f37,40g.

Figure 4sbd illustrates the corresponding temporal profile.In plasma-driven regime, the filament is not static, but results

FIG. 3. sColor onlined Propagation of 100-fs pulses withPin

=22Pcr in air at l0=800 nm computed from Eqs.s1d and s2d. sadPeak intensityssolid lined and peak electron densitysdashed linedalong z splasma levels are indicated on the right-hand side axisd.sbd–sed Fluence distribution in the transverse planes−1øx,yø1 mmd, plotted at the increasing propagation distancessbd z=0.78 m,scd z=1.18 m,sdd z=2.75 m, andsed z=4.71 m.


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from successive dynamical equilibriums that maintain thebeam. Indeed, as the beam self-focuses, plasma with densityrmax.531016 cm−3 defocuses the pulse trail. The resultingleading peak undergoes MPA damping. Once the intensitydecreases enough, plasma weakens and the back of the pulsecan then refocusf21g. This event can repeat many timesthrough focusing and defocusing cycles, each of which isassociated with one cell in the filamentation patterns lying inthe planesx,y=0,td. Self-guided propagation is then sus-tained by multiple peaks in the temporal profile, which takeover one another. This tricky dynamics creates notches in themaximum peak intensity and electron density plotted in Fig.3sad.

The second sequence is associated with low-plasma levelssz.2 md. Here, the Kerr recompression generates a spikedisplayed in Fig. 4scd. At z=2.75 m, only one peak emerges,while the electron density does not exceed 1016 cm−3. Thissingle-peak structure is maintained up toz=3.2 m, with aninner power decreasing from,70 to 15 GW. Note that thissecond sequence occurs at lower amplitudessboth for theoptical and plasma fieldsd and weaker powers, which mayfurther favor the emergence of a single, long-living structurecomparable with the gas-induced solitary wave discovered inRef. f45g. We recall in this respect that gas-induced solitonsrather occur at low density levelss*1015 cm−3d, for peakpowers close to critical.

B. Optical self-guiding

Beyond the Rayleigh lengthsz.z0.4 md, the electronplasma density falls down below 1015 cm−3 and an optical

regime takes place. In this “postionization” regime, the pulseis still able to survive over half a Rayleigh length, whilekeeping an intensity as high as 1013 W/cm2 fFig. 3sadg. Thepulse relaxes to a lower intensity structure exhibiting aslightly broader radial extent, as illustrated in Fig. 3sed. Thebeam keeps this new shape over 3–4 m, before GVD andtransverse diffraction ultimately spread it out.

Because ionization can no longer counterbalance self-focusing, GVD and the Raman-delayed response may natu-rally be suspected to play a role in sustaining this opticalself-guiding. As plasma turns off atzù3 m, the pulse dif-fracts towards low amplitudes and is then characterized byan increase of both its transverse and temporal global sizes.For z.4 m, a leading component then starts to refocus. Coa-lescence of the neighboring time slices contributes to feedthis single leading peak with about 8–10 TW/cm2 in inten-sity fFig. 5sadg, which propagates over,50 cm. This dynam-ics resembles the fusion of multisplit structures emphasizedin Ref. f28g for nonlinear optical systems undergoing GVDand the delayed-Kerr response, in the absence of plasma for-mation. Afterwards, power is transferred towards the rear ofthe pulse, as shown by Fig. 5sbd. The resulting structure thenslowly undergoes a GVD splitting event over nearly half aRayleigh length, as displayed by Fig. 5scd.

In the following, we examine the respective actions of thegroup-velocity dispersion and of the delayed-Kerr compo-nent in the focusing response, separately.

V. INFLUENCE OF GVD AND DELAYED-KERRRESPONSE

As GVD modifies the pulse temporal shape and since thenonlocal Raman-delayed response is responsible for mixingtemporal components, we focus in this section on their re-spective actions on the optical self-guiding regime. In orderto identify their influence, we compare numerical simula-

FIG. 4. sad Three-dimensionals3Dd plasma stringssmaxt rd ex-cited from the 100-fs, 22Pcr pulse in air. Isodensity levels are ex-pressed in thesx,y,zd space plotted from the minimal levelrmin

=1015 cm−3. Temporal profiles of the optical intensity in thesx,y=0,td plane atsbd z=0.78 m andscd z=2.75 m.

FIG. 5. Intensity of an IR 100-fs pulse in air withPin=22Pcr

along the liney=0, at the increasing distancessad z=4.71 m, sbdz=5.89 m, andscd z=7.46 m in postionization regime.


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tions of the complete systemfFigs. 3–5g with the ones per-formed when omitting either GVD or the delayed-Kerr con-tributions.

A. Dispersionless propagation

Let us discuss numerical simulations performed with thesame parameters as in Fig. 3, but with the GVD term re-moved. Results are shown in Figs. 6 and 7. When comparingfull numerical simulations and those performed withoutGVD, it is clear that no optical self-guiding develops fromz.z0 fFig. 6sadg.

At previous distances, no significant variation is observedin the filamentation patterns. The overall dynamics in thediffraction plane is recovered as shown by the beam fluencesplotted at the same propagation distances as in Fig. 3. Apart

from tiny variations, no major difference occurs along thisfirst propagation range until the Rayleigh distancefFigs.6sbd–6sddg. In contrast, things noticeably change at longerdistancessz.4 md. Here, the self-guiding is still maintainedby plasma generation, that prevents the spatial relaxation ofthe filament corefFig. 6sedg. The main features of the tem-poral dynamics in plasma-driven regimes are retrieved inFig. 7. Pulse distortions induced by plasma defocusing occurat z=0.78 m, before a dominant leading peak remains local-ized at negative timessz=2.75 md. At larger distances, how-ever, sequences of focusing-defocusing events are continu-ously excited at high intensity levels, which constitutes themost important difference with Figs. 4 and 5. As a result, thepulse keeps a multipeaked temporal profile, because plasmageneration continues to be the key saturation process far be-yond the Rayleigh length.

Hence, even if GVD is usually believed to play no sig-nificant role over short propagation distances in airsGVDcharacteristic length is,2tp

2/k9=1 kmd, the propagation dy-namics is drastically affected by dispersive effects upon afew meters only. In the absence of GVD, the propagationrange associated with plasma self-guiding is significantly in-creased and covers over very long distances. This signifiesthat GVD cuts the self-guiding length supported by plasmageneration, in agreement with the variational expectations ofSec. III.

B. Delayed-Kerr component

In the early propagation, the Raman-delayed Kerr effectlowers the maximum level in the nonlinear refraction index.Consequently, omitting this contribution should increase theeffective power ratioPin /Pcr. To check this point, we per-formed a different series of simulations by using the sameparameters as in Figs. 6 and 7, while removing the nonlocalKerr responsesxK=0d. Results are illustrated in Fig. 8. Com-


=22Pcr in air in the absence of GVD.sad Peak intensityssolid linedand peak electron densitysdashed lined along z. sbd–sed Fluencedistributions in the transverse planes−1øx,yø1 mmd plotted atthe increasing propagation distancessbd z=0.78 m,scd z=1.18 m,sdd z=2.75 m, andsed z=4.71 m.

FIG. 7. Temporal profilessx,y=0,td of 100-fs pulses in air withPin=22Pcr and no GVD, at increasing propagation distancessad z=0.78 m,sbd z=2.75 m,scd z=4.71 m, andsdd z=5.89 m.


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pared with Fig. 6, the nonlinear focal point occurs earlieralong the propagation axis. A higher saturation threshold isreached and more filaments are created, although in the endthey coalesce into a single structure. Fromz=2.75 m, nosalient difference with Fig. 6 was observed and we thereforedo not reproduce the fluence patterns at further distances. Inaddition, the Raman-delayed Kerr effect shifts the nonlinearrefractive index towards positive timesssee Sec. IIId. Thisasymmetry results in the emergence of positive time slicescarrying higher power, which then favors refocusing in theback of the pulse. Reversely, neglecting this contributionsmoothes the trailing part of the pulse at early stage in thefilamentation process, which is confirmed by comparingFigs. 8sdd and 8sed, with 7sad and 7sbd.

Concerning the role of this delayed component, the tem-poral evolution of the functionRstd approximated in Sec. III

would indicate that the main differences with a full instanta-neous Kerr response atz=0 tend to be more and more at-tenuated as the pulse propagates, while its temporal meanwidth increases. On this basis, we would not expect the de-layed response to be crucial in sustaining the optical regime.In order to check this statement, we performeds3D+1d nu-merical simulations where GVD is included and the nonlocalresponse in time is again ignored by settingxK=0. To matchthe effective nonlinear refraction index with that of the fullsystem, the input power has also been decreased to 17.6times critical, which simulates a self-guided filament startingfrom the same nonlinear focus as in the complete system. Bydoing so, emphasis is given to the temporal distortions owingto the absence of the delayed response. In this configuration,Fig. 9sad shows a self-guiding dynamics quite similar to Fig.3sad with the same nonlinear focal point, an identical propa-gation range, saturation thresholds close to those observedwith the complete system, and a plasma defocusing also end-ing near the Rayleigh length. As in Figs. 3–5, we recover theoptical postionization regime characterized by the propaga-tion of a residual pulse over half a Rayleigh length, withintensity reaching 1013 W/cm2. Even in the absence of thenonlocal response, we retrieve the relocalization process oftemporal slicesfFigs. 9sbd and 9scdg once the electron densitydrops below 1015 cm−3, leading to pulse components withbroader duration and lower intensity. The only discrepancylies in the persistent localization of the pulse at negative


=22Pcr in the absence of both GVD and delayed KerrsxK=0d. sadPeak intensityssolid line, upper curved and peak electron densitysdashed line, lower curved along thez axis. Fluences in the trans-verse planes−1øx,yø1 mmd, plotted atsbd z=0.78 m andscd z=2.75 m.sdd,sed Temporal intensity profilessTW/cm2d at y=0 forthe same distances.

FIG. 9. Propagation of 100-fs pulses withPin=17.6Pcr in airwhen the nonlocal Kerr response in time is neglectedsxK=0d. sadPeak intensityssolid line, upper curved and peak electron densitysdashed line, lower curved along z. sbd,scd Temporal intensity pro-files sTW/cm2d at y=0 displayed for the distancesz=4.71 and5.89 m.


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times: its power is not transferred towards the rear zone as inFig. 5sbd. This phenomenon is tied to the fact that the non-local character of the delayed-Kerr response and its asymme-try in the nonlinear index are ignored and cannot promoterefocusing of temporal slices at positive times.

VI. CONCLUSION

In summary, we have investigated the filamentation ofmillimeter ultrashort pulses containing several tens of criticalpowers. A two-scale variational approach enabled us to em-phasize the impact of GVD on the beam self-guiding in ion-

ization regimes, at powers becoming close to critical. Thesepower levels are currently accessible, once the pulse energydecreases not only by multiphoton absorption and plasmalosses, but also when the temporal extension of the pulsecomponents broadens, making their own power decreasesignificantly. These expectations were confirmed bys3D+1d-dimensional numerical simulations, which enabledus to examine in detail the respectives roles of GVD and theRaman-delayed response, both from the spatial and temporalpoints of view. From this study, it appears that GVD plays afundamental role in “cutting” the plasma range, while con-tinuing the propagation of pulse components with intensitiesas high as 10 TW/cm2 over a few meters.

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Postionization regimes of femtosecond laser pulses self-channeling in air