s.l. chin et al- filamentation of femtosecond laser pulses in turbulent air

8/3/2019 S.L. Chin et al- Filamentation of femtosecond laser pulses in turbulent air

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DOI: 10.1007/s003400100738

Appl. Phys. B 74, 67–76 (2002)

Lasers and Optics

Applied Physics B

s.l.chin1

a. talebpour1

j. yang1

s. petit

1

v.p. kandidov2

o.g.kosareva2,✉

m.p. tamarov2

Filamentation of femtosecond laser pulses inturbulent air1 Center for Optics, Photonics and Lasers, Department of Physics, Laval University, Quebec, G1K 7P4,

Canada2 International Laser Center, Physics Department, Moscow State University, 119 899, Moscow, Russia

Received: 23 May 2001/Revised version: 26 September 2001

Published online: 29 November 2001 • © Springer-Verlag 2001

ABSTRACT Formation and wandering of filaments in air arestudied both experimentally and numerically. Filament-centerdeflections are collected from 1100 shots of 190-fs and 800-nmpulses in the plane perpendicular to the propagation direction.To calculate the filament wandering in air we have developeda model of powerful femtosecond laser pulse filamentationin the Kolmogorov atmospheric turbulence and employed theMonte Carlo method to model the propagation of several hun-dred laser pulses. Statistical processing of experimental andnumerical data shows that filament-center displacements in thetransverse plane obey the Rayleigh-distribution law. Parametersof the Rayleigh distribution obtained for numerical and experi-mental data are close to each other.

PACS 42.68.Bz; 42.65.Jx; 02.70.Uu

1 Introduction

First experiments on long-range filamentation of powerful femtosecond laser pulses in air were performedin the middle-1990s in several laboratories [1–3]. In theseexperiments 150–230 fs and 5–50GW laser pulses fromTi:sapphire laser amplification systems operating at775–800 nm produced light filaments with the length of tensof meters. The remarkable feature of these filaments is thatalong most of the propagation distance more than 10% of thepulse energy is localized in the near-axis area with the diam-eter of about 100 microns.

From the point of view of nonlinear optics the filamen-tation phenomenon is a small-scale transient self-focusing of laser radiation in air [4]. Self-focusing originates from theKerr effect, which provides the increase in the nonlinear re-fractive index with the increase in the light-field intensity. If the pulse peak power is not larger than approximately eightthreshold powers for self-focusing in air, only one filamentis created [3]. With increasing peak power two or more fila-ments can be observed [5]. For an initially focused beam thefilamentation starts right beyond the lens focal point [6]. Fil-

✉ Fax: +7-095/939-3113, E-mail: [email protected]

amentation of focused subpicosecond laser pulses with cen-tral wavelength 248 nm was experimentally and numericallystudied [7]. For the laser pulse used in this experiment 50% of the input energy was lost in the nonlinear focus and 40% was

containedin the divergent beam surrounding the filament. Thefilament itself contained 10% of the input laser energy.

Filamentation with various temporal lengths and variouslaser wavelengths is discussed in [8]. Experiments were per-formed using two laser systems: a Nd:YLF laser system with525-fs pulses and a wavelength of 1053 nm and a Ti:sapphirelaser system with 60-fs pulses and a wavelength of 795 nm.The length of the filament produced by 60-fs pulses with peak power of the order of 300 GW was more than 200 m. At thesame time thelength of the filament producedby 525-fs pulseswith peak power of the order of 40GW was approximately50 m.Inthecaseof 525-fs pulses anda peak powerlarger than30GW several (3–4) filaments were observed at the distanceof filament formation.

Small-scale self-focusing of laser pulses in condensedme-dia was intensively discussed in the 1960s and 1970s in thecontext of laser thermonuclear fusion studies (see e.g. [9–12]). Observation of self-focusing in air has become avail-able with the advent of laser systems that produce powerfulfemtosecond and picosecond laser pulses. For longer pulseduration self-focusing in air is suppressed by the nonlineareffects with lower threshold powers: heat defocusing and op-tical breakdown [12]. For femtosecond and subpicosecondpulses these effects do not develop due to a long responsetime. Filamentation of laser pulses in atmospheric air can beobserved only if two conditions are simultaneously satisfied:ultrashort duration and high power of the radiation.

The mechanisms that stop self-focusing in gases and con-densed media are different. In optical glass two-photon ab-sorption stops the growth of intensity in the nonlinear fo-cus [13]. In gases self-focusing is stopped by defocusing of laser radiation in the plasma arising from photoionization of molecules in the nonlinear focus. As a result, the maximumlight intensity in the filament does not exceed 1014 W/cm2 forinfrared pulses at 770–800 nm [2–4] and 1012 W/cm2 for ul-traviolet pulses at 248 nm [7].

Nonlinear interaction of laser pulses withneutralmoleculesof air and self-produced laser plasma results in strong spatio-temporal gradients of light-field intensity and phase. The



68 Applied Physics B – Lasers and Optics

intensity peak produced by Kerr self-focusing is followed bya dynamic ring structure arising from strong nonlinear refrac-tion in the plasma. The rings surrounding the filament con-verge and diverge alternately. As a result, the energy containedin the filament changes nonmonotonically with propagationdistance. This effect was called the refocusing phenomenon.It was studied in [3, 4].

Filamentation is accompanied by the conical emission –wide-band coherent radiation in the visible range. This ra-diation propagates forward in the narrow cone surroundingthe filament [2, 14]. Experiments on self-focusing in water,methanoland CCl4 [5, 15] demonstrated thatthe conical emis-sion in the range 500–800 nm has the same length of co-herence as the input laser pulse with central wavelength of 800 nm.

Nowadays wide-band radiation is regarded as a prospec-tive coherent white-light source for sounding the atmosphere.The first experiments with femtosecond terawatt lidar wereperformed in Jena [16, 17]. Laser pulses with 100-fs duration,2-TW peak power and 790-nm central wavelength were sentto a vertical atmospheric path. The back-scattered signal was

registered from the height of 12km. The absorption spectraof oxygen in the range 758–772 nm and water vapor in therange 826–836 nm were obtained at the height of 600–800 m.In recent experiments [18] with terawatt laser pulses, infraredspectral broadening up to 4.5 microns was registered. Thisopens a new possibility for the application of laser-inducedsupercontinuum to the registration of pollutants in the atmo-sphere, because the absorption lines of most pollutants are inthe range 3–3.7 microns.

Theoretical studies of long-distance propagation in air arebased on the nonlinear Schrödinger equation for the com-plex amplitude of the electric field. The propagation equationincludes diffraction, group-velocity dispersion, nonlinear re-fraction and absorption in the neutral gas and the plasma.The propagation equation is solved together with the equa-tions describing transient nonlinear responses of the air andthe plasma produced in the course of photoionization of aircomponents in a strong laser field. Usually numerical simu-lations are applied to the solution of this three-dimensionalnonstationary problem [3, 4, 19, 20]. In the case of the single-filament regime, a cylindrical geometry of the experimentalsetup is used to reduce one spatial coordinate in the trans-verse direction and to save computational resources. For theinterpretation of the simulation results several models havebeen applied: the self-channeling model [2, 6], the moving-focus model modified by the contribution of the self-producedlaser plasma [3, 4] and the dynamic spatial replenishment

model [20]. Numerical simulations allowed us to recapturethe physical picture of the filamentation and accompanyingeffects of conical emission [4, 14] and refocusing [3, 20].In [21] variational analysis has been applied to the study of the filamentation phenomenon. The equations for the beamwidth and the curvature radius of the laser-beam wavefrontwere derived and solved under the assumption of a Gaussiandistribution of spatial intensity in the transverse direction.

Filamentation is a stochastic process. This fact was firstmentioned in [3] where random deflections of the filamentcenter from the propagation axis were observed in the trans-verse plane perpendicular to the propagation direction. La

Fontaine et al. [8] reported that not every perturbation in thebeam profile formed earlier in the propagation was developedinto a filament. In the femtosecond lidar experiments [16, 17]the pulse peak power was several hundred times larger thanthe thresholdpower for self-focusing in air. Therefore a bunchof filaments was created. In the conditions of natural atmo-spheric turbulence the formation of multiple filaments wasrandom along the propagation distance. In order to regularizetheprocess of the filament formation Woste et al.[16, 17] useda focusing lens with a 30-m focal length. In the focal point of the lens a long light filament was formed from which white-light conical emission was generated.

Theoretically, formation of filaments is the result of spatialinstability of the light field in the strong optical nonlinear-ity. In the single-filament regime in a regular (nonstochastic)medium the instability is located at the point with the highestintensity, i.e. on thebeam axis. However,in the real conditionsthe place of the filament formation depends on the transverseintensity fluctuations caused by limited spatial coherence of the laser radiation and natural perturbations of the refractiveindex in atmospheric air.

The effect of atmospheric turbulence on filamentation of powerful femtosecond laser pulses in air was numericallystudied [22]. Here the stochastic model of phase screens wasemployed to model refractive-index fluctuations in air. Kan-didov et al. [22] showed that filaments were created in theplaces of random focusing of the radiation. Since in the con-ditions of self-focusing the light field is unstable relative tothe local intensity fluctuations, the distance between the lasersystem output and the place of the filament formation is,on average, shorter in turbulence. The simulated picture of the filament wandering in the transverse plane is in qualita-tive agreement with the one obtained in the experiment. TheMonte Carlo method was used to find the change of the trans-verse root-mean-square deviation of the filament from thepropagation axis with distance. Numerical simulations of fil-amentation of a femtosecond pulse with peak power manytimes exceeding the threshold power for self-focusing in airwere performed [23]. Dynamics of multiple filaments pro-duced by a pulse with initially regularly positioned intensityperturbations was studied.

In this paper we present experimental andtheoretical stud-ies of filament wanderingcaused by natural fluctuations of therefractive index in air. On the basis of statistical processing of the filament parameters obtained in the experiment and in thesimulations, we develop a theory of filament formation in theconditions of natural atmospheric turbulence.

2 Experiment

Our laser system consists of a Ti:sapphire os-cillator followed by a regenerative and two multiple-passTi:sapphire amplifiers. The pulse duration is 190 fs (FWHM)and the central wavelength is λ = 800 nm. The laser pulsewith 9.6-mm (at e−2 intensity level) beam diameter and en-ergy of 4.6 mJ propagates in the air of the laboratory building.The beam starts to self-focus at a distance of 30 m from theoutput of the compressor. Pulse peak power is 21GW, i.e.3.7 times larger than the threshold power for self-focusing inair, Pcr, the value of which was estimated as 6.1 GW [3].



CHIN et al. Filamentation of femtosecond laser pulses in turbulent air 69

To characterize the spatial profile of the filament its en-ergy is attenuated by consecutive reflections from the surfaceof two prisms and, after further attenuation by neutral-densityfilters, the beam profileis capturedby a CCDcamera with a di-mension of 6×6 mm2 (120×120 pixels). The output of theCCDcamera is a two-dimensional matrix F ( x, y), the elementof which at each point is proportional to the time-integratedvalue of the intensity of the filament at that point. A sampleoutput of the fluence distribution in a single shot is shownin Fig. 1. A clearly pronounced maximum of the fluence dis-tribution corresponds to the filament-center position at thispropagation distance. From shot to shot the filament-centerposition changes randomly in the transverse plane X OY .

A good fit to the fluencedistribution is a Gaussian functionof the form:

F ( x, y)= A√ w x√ w y

exp

−2

x− X c

w x

2

+

y−Y c

w y

(1)

where X c, Y c are the coordinates of the filament center and

w x , w y are the filament radii in x and y directions, respec-tively. To calculate the values of these parameters we proceedas follows. First, we integrate the function F ( x, y) over the

area y = ±60 pixels to find a function f ( x)=+∞ −∞

F ( x, y)d y.

At the left-hand and right-hand sides of this area the valuesof F ( x, y) are negligibly small. After calculating f ( x) wefitit

to a Gaussian function of the form A√ w xπ/2

exp

−2

x− X cw x

2

,

where A is a constant, and obtain parameters X c and w x . Re-peating this procedure, we integrate F ( x, y) in the x directionand then we find Y c and w y. The diameter of the randomlydisplaced filament, d , is calculated as

d =w2 x +w2

y . (2)

In Fig. 2 we present the dependence of the diameter of thelaser beam d on the propagation distance, z, that has beenmeasured from the output of the compressor. Each point cor-responds to a 20-shot average. The average value of d and its

X (pixels)

Y

(pixels)

FIGURE 1 Fluence distribution in a single pulse measured by a CCD cam-era in the plane perpendicular to the propagation direction

propagation distance (m)

Diameterofthefilamentd(pixel)

FIGURE 2 The filament diameter d as a function of the propagation dis-tance. Each point corresponds to a 20-shot average

standard deviation were calculated according to the formulasof mathematical statistics [24]. At a distance of 30 m (Fig. 2)the filament diameter rapidly decreases. This indicates the

start of the filamentation process. After the filament is created,the pulse energy remains localized in the narrow near-axisarea with a diameter of about 1.5 mm. This localization of en-ergy takes place at least up to 105 m along the propagationdirection. The large standard deviation of the diameter d ata distance of 30 m demonstrates that the filament formation isan irregular process. For example, for the data shown in Fig. 2approximately one-third of all the pulses are far from beingfocused since the beam diameter is too large – around 6 mm.

For statistical analysis of the filament parameters we col-lected 1100 shots at z = 30 m and another 1100 shots at z = 105 m. For each shot the position of the filament center

X, mm

X, mm X, mm

X, mm

Y,mm

Y,mm

a

b

FIGURE 3 The filament-center positions in the transverse plane at dis-tances z = 30 m and z = 105 m from the output of the compressor:a experiment, 1100 laser shots; b simulations, 200 laser shots




( X c, Y c) and the diameter of the filament at 1/e2 of fluence,d , were determined. In Fig. 3a we demonstrate the ensembleof the filament-center positions at these two distances. Theaverage values of filament displacements in x and y direc-tions are zero. The density of the filament-center positions isaxially symmetric. From these observations two conclusionscan be derived. First, there is no correlation between the fil-ament displacement in x and y directions. Second, randomdisplacements of the filament are statistically isotropic in thetransverse plane X OY . Thus, statistical characteristics of thefilament wandering depend only on the displacement of thefilament center Rc from the origin of the coordinate systemlocated on the CCD camera. The distance Rc is defined as

Rc =

X 2c +Y 2c . (3)

This result means that random factors that lead to filamentwandering are statistically isotropic in the plane perpendicu-lar to the propagation direction. Among these factors are per-turbations of the intensity and the phase caused by limitedspatial coherence of the laser radiation and by refractive-index

fluctuations in the propagation path.In Sect. 3 we develop a model of filamentation of power-

ful femtosecond laser pulses propagating in the conditions of refractive-index fluctuations in turbulent air.

3 Filamentation model

A powerful femtosecond laser pulse propagates inthe conditions of diffraction, group-velocity dispersion, Kerrself-focusing, plasma production and refractive-index fluc-tuations in atmospheric turbulence. The propagation equa-tion for the slowly varying amplitude of the electric field E ( x, y, z, t ) is given by

2ik 0

∂

∂ z+ 1

vg

∂

∂t

E =∆⊥ E − k 0

∂2k

∂ω2

∂2 E

∂t 2

+2k 02n2| E |2 E + 2k 0

2npl( I ) E (4)

+2k 02n( x, y, z) E − ik 0α E ,

where ñ( x, y, z) describes random fluctuations of the re-fractive index in turbulence and k 0 is a wavenumber. Theionization-energy loss is described by the coefficient α. Thescattering and absorption of the radiation in the atmosphericaerosol are not taken into account.

Nonlinear change of the refractive index in a strong laserfield is defined by the Kerr effect and the response of the

laser-induced plasma by npl( I ), where I is the laser intensity.Kerr nonlinearity leads to self-focusing, while plasma causesdefocusing of the radiation. The plasma contribution to therefractive index is proportional to the electron density. Freeelectrons are produced due to multiphoton or tunneling ion-ization of the air components, mainly from the molecules of oxygen and nitrogen. Thus, the dependence npl( I ) character-izes the process of plasma production from the oxygen andnitrogenmolecules of airin the course of multiphoton andtun-neling ionization.

According to recent studies [2, 3, 14, 19, 20] the filamen-tation of a powerful femtosecond laser pulse consists of the

following processes. The pulse with peak power larger thanthe threshold power for self-focusing in air undergoes self-contraction during the propagation. In the nonlinear focuslocated in the central or leading part of the pulse the inten-sity reaches 1013 –1014 W/cm2 and the ionization probabilityrapidly increases. The electron density accumulates duringthe pulse. The growth of the electron density is described bykinetic equations where the ionization rate is calculated ac-cording to models [25–27]. For a 800-nm laser wavelengththe ionization rate is roughly proportional to the 8th powerof the laser intensity. Therefore, the plasma contribution ishighly nonlinear. The laser-produced plasma leads to thestrong defocusing of the radiation and limits the intensitygrowth caused by self-focusing. At the trailing edge of thepulse (beyond the nonlinear focus) a complex dynamic struc-ture of aberrational rings surrounding the filament center iscreated.

Filamentation in the conditions of atmospheric turbulenceshould be described in terms of a 3D + time stochastic equa-tion with strong nonlinearity. Numerical simulation of thisequation is not available with our computational facilities.

However, the physical mechanism of filamentation allows usto build up a simpler model of formation and wandering of fil-aments in a randomly inhomogeneous medium. This modelallows us to consider the initial stage of filamentation. By thisinitial stage we mean the nonlinear growth of the intensityup to the ionization threshold and formation of the nonlinearfocus. At this stage of investigation we do not consider com-plex dynamics of the light field in the pulse along the wholepropagation path. Below we present the justification of thisassumption.

According to the moving-focus model [3, 4] themost pow-erful slice of the pulse focuses the closest from the laser sys-tem output and defines the start of the filament. The slicesat the front of the pulse contain less power and focus furtheralong the propagation direction. The succession of the nonlin-ear foci formed by slices from central and leading parts of thepulse form the filament. Slices at the trailing part of the pulsedefocus in the plasma.

In the physical mechanism of filamentation it is essentialthat the ionization starts only after the intensity reaches theionization threshold, which is about three orders of magnitudelarger than the peak intensity of the input pulse. In the condi-tions of our experiment in air the threshold intensity for theionization is estimated as 3×1013 W/cm2 while the peak in-tensity in the input pulse is 3×1010 W/cm2. Therefore, thelaser-produced plasma does not affect the initial stage of thefilament formation.

The dispersion length for a 190-fs and 800-nm laserpulse propagating in air is of the order of 1 km. Therefore,the group-velocity dispersion becomes important only aftera 20–30fs contracted pulse [28] is created in the leading partof the pulse due to self-steepening. This peak is created inthe nonlinear focus. Earlier in the propagation, group-velocitydispersion does not affect the filament formation.

Stochastic simulations of powerful femtosecond laserpulse propagationin the turbulent atmosphere along the wholepropagation path will require taking into account the effectof the ionization and group-velocity dispersion on the spatio-temporal transformation of the pulse in the filament.




FIGURE 4 Schematic picture of filament wandering in atmospheric turbulence. Sturb j−1, Sturb j , Sturb j+1 are phase fluctuations on the screens that imitateatmospheric turbulence, ∆ zturb is the distance between the phase screens, 1, 2 are temporal slices of the pulse and ˜ znf1, ˜ znf2are the positions of the nonlinearfoci created by the slices 1, 2

At the initial stage of filamentation only Kerr self-focusingin randomly inhomogeneous media defines the position of thefilament formation in a 3D space. Schematically this processis shown in Fig. 4. Before the filament is formed, refractive-

index fluctuations in air cause phase perturbations in the cen-tral slice of the pulse. Random small-scale perturbations nearthe maximum intensity location can initiate the formation of the nonlinear focus. The position of this focus is the place of the filament formation in the transverse plane. The place of the filament formation in the transverse plane changes ran-domly from shot to shot. Phase fluctuations, the size of whichis larger than the transverse size of the laser pulse, are re-sponsible for the wavefront tilt of the slice. The joint effectof large-scale and small-scale refractive-index fluctuationscauses a random deviation of the nonlinear focus and the fil-ament center from the propagation axis.

Each slice in the leading edge of the pulse covers itsown distance in randomly inhomogeneous air until it isself-focused. Therefore, random distortions of the wavefront(mainly tilts and focusing) formed in the course of this initialstage of propagation are different in each slice. As a result,the position of the filament center changes randomly withincreasing propagation distance.

According to our considerations the initial stage of the fil-ament formation can be described by the following equation:

2ik 0

∂

∂ z+ 1

vg

∂

∂t

E =∆⊥ E +2k 0

2n2| E |2 E

+2k 02n( x, y, z) E (5)

The model described by (5) does not take into account iner-

tial response of the Kerr nonlinearity [19, 20], since the lat-ter does not influence the location of the nonlinear focus inspace. The coefficient n2 is chosen to be 1, 55×1019 cm2/W.The corresponding critical power for self-focusing is Pcr =6.1×1019 W [3].

Statistical characteristics of a three-dimensional field of refractive-index fluctuations ñ( x, y, z) are given by the modelof atmospheric turbulence. For subpicosecond laser pulsesthe field ñ( x, y, z) may be regarded as stationary. Statisticalisotropy of the filament-center wandering (Fig. 3a) allows usto assume thatthe experiment was performed in the conditionsof ‘the developed turbulence’, which obeys the Kolmogorov

11/3 law [29]. To describe a wide range of refractive-indexfluctuations we use the von Karman spectrum:

Φn(κ)= 0, 033C n2(κ2+κ0

2)−11/6exp{−κ 2/κm2} (6)

where C n2 is a structure constant of atmospheric turbulence,

index n shows thatthe valuesΦn and C n2 referto the refractive

index n. Parameters κ0 and κm in (6) are given by:

κ0 = 2π/ L0 and κm = 5.92/l0 (7)

where L0 and l0 are outer and inner scales of turbulence,respectively. The direct measurement of atmospheric turbu-lence parameters C n

2, L0 and l0 has not been performed inthe framework of our experiment. Therefore, to find the quan-titative values of these parameters we will use the estimatesfrom [30]. If the height of the atmospheric path is H = 1–2 m

then the outer scale of turbulence, calculated from the expres-

sion L0 = 0.4 H, is L0 ≈ 1 m. The inner scale of turbulence isl0 = 1 mm. The structure constantC n

2, which shows the inten-sity of atmospheric fluctuations, is a free parameter. We varieditsvalue in therange C n

2 = 5×10−13−1.5×10−14 cm−2/3 inorder to fit the experimental results.

The initial distribution of the electric field E ( x, y, z = 0, t )

is close to Gaussian in space and time [2, 3]. For coherent ra-diation the field E ( x, y, z = 0, t ) is given by

E ( x, y, z = 0, t )= E 0exp

− x2+ y2

2a20

exp

− t 2

2τ 02

(8)

where a0 is the beam radius and 2τ 0 is the pulse duration. The

peak power P0 in the pulse exceeds the critical power for self-focusing Pcr:

P0 > Pcr, P0 = πa02 I 0, I 0 =

cn

8π| E 0|2 , (9)

where c is thespeedof light. In this formulationthe problemof filamentation in the turbulent atmosphere is reduced to thecal-culation of the nonlinear focus position in ( x, y, z) space forthe successive infinitely thin slices of the pulse. The nonlinearfocus position of the central slice of the pulse is the beginningof the filament, and the nonlinear focus position of the slicewith P ≈ Pcr is the end of the filament. By the nonlinear focus




position of any slice of the pulse we mean the point in ( x, y, z)

space at which the intensity reaches the ionization threshold.The solution to (5) for a random realization of refractive-

index fluctuations ñ( x, y, z) corresponds to the experimentalmeasurements performed for one laser shot. A set of solu-tions to (5) obtained for statistically independent realizationsof refractive-index fluctuations ñ( x, y, z) forms an ensemblethat we used to calculate statistical characteristics of the fila-ment in turbulent air.

4 Numerical simulation technique

Numericalsimulations of (5) and (8) are performedon the basis of the phase-screen model [31]. The calculationsare performed on an ( x, y, z) grid.

In the simulations a randomly inhomogeneous mediumis represented by a chain of phase screens located along thepropagation axis. This chain made up of a finite number of scattering screens reproduces adequately the properties of a continuous medium, provided that the distance between thescreens∆ z is small compared with the characteristic scales of

the field variation along the propagation coordinate z. Thesescales include the length of nonlinearity Lnl, the diffractionlength Ld and the length of turbulence L turb [31]:

∆ z min{ L ln, Ld, L turb} . (10)

The length of nonlinearity is defined as a distance alongwhich the maximum phase growth due to self-focusingϕnl =n2k 0 Lnlmax{| E |2} does not exceed 1 rad. Then

Lnl = (n2k 0max{| E 2|})−1 . (11)

The length of turbulence L turb is defined as a distancealong which the mean-square deviation of the phase due to

refractive-index fluctuations does not exceed 1 rad. For thevon Karman model of atmospheric turbulence the length of turbulence is given by

L turb =

2, 4π2k 02 0, 033C 2n

2π

L0

−5/3−1

, (12)

The diffraction length Ld is given by the expression Ld =k 0a( z)2. The value a( z) coincides with the initial beam radiusa0 at z = 0. Through the value a( z) for z > 0 we denote thespatial scale of the nonlinear focal region in the beam crosssection.

In the course of Kerr self-focusing the intensity increases

sharply in the region where a nonlinear focus is formed. Sim-ultaneously, the lengths Lnl and Ld decrease. In order to sat-isfy the inequality (10) we decreased the interval between thephase screens as the plane of the nonlinear focus formationwas approached. Since the simulation of phase screens thatadequately reproduce atmospheric turbulence requires a lotof calculations we used two systems of phase screens. Thefirst system imitates random phase fluctuations Sturb( x, y, z)

caused by refractive-index perturbations in the atmosphere. Inthis system of phase screens the ‘turbulent’ phase screens areplaced equidistantly along the propagation direction with theinterval ∆ zturb. The interval ∆ zturb is selected from both the

inequality (10) and the condition governing the applicabilityof the δ-correlated phase screens for the turbulent atmospheregiven by

L0 ≤∆ zturb min{ L turb, Ld} . (13)

Here L0 is the outer scale of atmospheric turbulence definedearlier.

The second system of phase screens reproduces the non-linear phase growth ϕnl( x, y, z) arising due to self-focusing.The distance ∆ znl between ‘nonlinear’ phase screens de-creases with increasing intensity in accordance with the con-ditions (10), (11). In the initial stage of propagation, wherethe nonlinear phase growth is small and ∆ znl >∆ zturb, ‘non-linear’ phase screens are located in the same plane as the‘turbulent’ phase screens.

Between both ‘nonlinear’ and ‘turbulent’ phase screensthe light field undergoes only linear diffraction.

In order to obtain a path of the nonlinear focus from onelaser shot, we formed the chain of phase screens to simulatethe atmospheric turbulence over the entire length of the fila-

ment. Then we considered the self-focusing of the successionof pulse slices passing through the same chain of the ‘turbu-lent’ phase screens. The position of the nonlinear focus fora certain slice was defined as a point in ( x, y, z) space, wherethe slice intensity reaches the ionization threshold. A set of such points calculated for all slices of the pulse, from thecentral slice at t = 0 to the slices in the leading front of thepulse, creates the trajectory of the filament for one laser shot.Statistical processing of the ensemble of such trajectories cal-culated for statistically independent chains of phase screensallows us to find the average distance from the output of thecompressor to the beginning of the filament and the varianceof the transverse deviation of the filament in the turbulentatmosphere.

For the simulation of the ‘turbulent’ phase screens weemploy the modified method of subharmonics [32–34]. Thismethod increases essentially the range of spatial scales of

random phase fluctuations reproduced on the grid. As demon-strated in [35], the modified method of subharmonics withfour iterations of phase-screen generation makes it possibleto obtain a random field of phase fluctuations (6), the outerscale L0 of which is two orders of magnitude larger than thetransverse size of the grid in the plane X OY .

In the simulations of filamentation in turbulence we useda square grid (512×512) with the step h = 0.08 mm in thetransverse section. This grid reproduces adequately phasefluctuations with the spatial scale ranging from L0

=1 m

to l0 = 1 mm. The distance between the ‘turbulent’ phasescreens was ∆ zturb = 1 m. The distance between the ‘non-linear’ phase screens was decreased to ∆ znl = 1 cm as weapproached the nonlinear focus. The diffraction length was Ld( z = 0) ≈ 100 m for the input beam and Ld( z = z f ) =2.5 cm for the beam near the start of the filament.

5 Numerical simulation results

Spatial distributions of the light field obtained fromthe numerical solution of the stochastic equation (5) demon-strate a random process of the nonlinear focus formation in




FIGURE 5 Spatial intensitydistribution in the slice with8.5-GW power (P = 1.4Pcr)at several distances from theoutput of the compressor. Thestructure constant of atmo-

spheric turbulence is C n2

=1.5×10−14 cm−2/3, the outerscale L0 = 1 m and the innerscale l0 = 1 mm. Intensity dis-tribution at each distance isnormalized to the maximumintensity I max at this distance.At z = 20 m I max = 1.4 I 0,at z = 40 m I max = 2.5 I 0, at

z = 60 m I max = 3.6 I 0, at z =80 m I max = 7.3 I 0, at z =90 m I max = 16.2 I 0 and at

z = 100 m I max = 554 I 0. Here I 0 = 3×1010 W/cm2 is themaximum intensity at z = 0

the course of laser pulse filamentation in the turbulent atmo-sphere. Figure 5 shows the intensity distribution in the slicewith P = 8.5 GW = 1.4Pcr at several distances z from theoutput of the compressor. In air the ‘turbulent’ lenses inducedistortions in the initially smooth beam profile. The strongestrandom focusing in the paraxial region becomes the nucleusof a nonlinear focus. At the beginning of propagation the in-tensity growth in the vicinity of the nucleus for the nonlinearfocus is relatively slow. Near the distance of the nonlinear fo-cus formation the intensity growth is very fast. One can seethat in this random shot the nonlinear focus is displaced fromthe center of the input beam. The intensity in the nonlinear fo-cusis much higher than the maximum values of intensity in all

FIGURE 6 Trajectories of the filament center in the plane perpendicular tothe propagation direction. The solid curve tracks the filament-center positionsfor the pulse. The random field n( x, y, z) is the same that was used to cal-culate the spatial intensity distribution shown in Fig. 5. The dashed curve isobtained for another realization of the random field n( x, y, z)

the other slices of the pulse that do not focus at this distance z.Therefore, the positionof the nonlinear focus nearly coincideswith the maximum of the fluence at the same distance z. Asa result, the filament center, definedas theposition of the max-imum of the fluence, is randomly displaced in the transversesection of the pulse.

In the simulations we can follow the displacements of thefilament center for a single pulse. For this purpose we cal-culated the nonlinear focus position for a succession of thepulse slices starting from the central slice towards the slicesin the leading front of the pulse. The chain of the ‘turbulent’phase screens was the same for each slice. Figure 6 showstwo trajectories of the nonlinear foci (and, consequently, thefilament center) in the transverse plane X OY . The trajecto-ries are obtained for two different realizations of the randomfield ñ( x, y, z) that describes refractive-index fluctuations inthe atmosphere. Two random realizations of the fieldñ( x, y, z)

correspond to the propagation of two pulses in the labora-tory experiment. Figure 6 demonstrates that for the pulseswith 21-GW peak power the filament starts at z = 26–28 m.This is in agreement with experimental data shown in Fig. 2.With increasing propagation distance the transverse deflec-tion of the filament-center position from the position of theinput beam center increases. See, for example, the trajectory(solid curve) in Fig. 6 that is obtained for the same random re-alization of refractive-index fluctuations as a set of intensitydistributions in Fig. 5. At a distance z

=100 m the nonlinear

focus position is shifted by 1.3 mm along the x-axis and by0.5 mm along the y-axis (compare Figs. 5 and 6). The sec-ond trajectory in Fig. 6 (dashed curve) is obtained for anotherrealization of the random field ñ( x, y, z). At z = 100 m the fil-ament center is shifted down and to the left from the inputbeam center.

6 Statistical processingof experimental and

numerical simulation results

For statistical analysis of simulation results wehave numerically solved equation (5) with 200 statisticallyindependent chains of random phase screens. This corres-




FIGURE 7 The normalized bar charts for the filament-center displacements R at distances z = 30 m and z = 105 m from the output of the compressor:solid curve corresponds to the experiment (1100 shots) and dashed curve tothe simulations (200 shots)

ponds to 200 laser shots in the experiment. The filament-center positions in simulations were measured at z = 30 m

and z = 105m as was done in the experiment. The value of the structure constant C n

2 was a fitting parameter in the sim-ulations. The filament-center positions were calculated forseveral series of 200 shots and for each series we used itsown C n

2 constant in order to find the best fit to the observeddata.

Figure 3b shows the simulated filament-center positionsfor C n

2 = 1, 5×10−14 cm−2/3. There is an obvious qualita-tive agreement between the simulated and experimentally ob-tained filament-center positions (compare Figs. 3a and 3b).

For the quantitative analysis of the results we have plottedbar charts for the samples of experimental { Rlab} and simu-lated { Rcomp} values for the filament-center displacements.The sample size in the experiment is N lab = 1100 laser shotsand in the simulations N comp = 200 laser shots. Figure 7shows the normalized bar charts Plab( Rm) and Pcomp( Rm),where m is the number of the interval. The area of each barchart is equal to unity. The sample size N comp in the simula-tions is much less than the sample size N lab in the experiment.Therefore, in orderto obtain a valid estimate of the probabilityPcomp( Rm), the width of the intervals in the bar chart plottedfrom the simulated data was chosen to be larger than the widthof the intervals in the bar chart plotted from the experimen-

tal data. The empirical distribution functions of the filament-center displacements corresponding to these bar charts are

Propagation distance Experiment, Numerical simulations, 200 shots

1100 laser shots C n2 = 1, 5×10−14 cm−2/3 C n

2 = 3×10−14 cm−2/3

30 m 0.466±0.033 mm 0.315±0.022 mm 0.510±0.028 mm105 m 1.338±0.033 mm 1.592±0.069 mm 2.080±0.172 mm

TABLE 1 The absolute values of w= 1 M

M m=1

wm and σ w =

1 M −1

M m=1

(wm −w)2

1/2

obtained from statistical processing of experimental and numerical

data. The values σ w and w are shown in the form w±σ w

given by:

Ψ lab/comp( Rm)=m

i=1

Plab/comp( Ri) (14)

The distance of the filament center Rc =

X c 2+Y c2 fromthe origin of the coordinate system X OY depends on the dis-

tances X c and Y c, which are statistically independent. Dis-placements X c and Y c are due to uncorrelated fluctuations of the refractive index in air on the path of the pulse propaga-tion and, consequently, they obey the normal-distribution law.Therefore, it is reasonable to assume that the filament-centerdisplacements Rc obey the Rayleigh-distribution law:

Ψ( Rc)= 1− exp

−

Rc

w

2, (15)

where w is the parameter that characterizes the mean value of the filament-center displacements. The normalized probabil-ity function for the Rayleigh distribution has the form:

P( Rc)=2 Rc

w2exp

−

Rc

w

2

(16)

Experimentally obtained and simulated bar charts Plab( Rm)

and Pcomp( Rm) correspond to the Rayleigh probability func-tion(16) with the parameterswlab andwcomp, respectively. Theparameters wlab and wcomp are calculated as follows. For aninterval m parameter wm can be expressed using (14), (15):

wm = Rm√ − ln(1−Ψ( Rm))

, m = 1, 2, . . . M , (17)

where M is the number of intervals in the bar chart.

The mean value of w and its root-mean-square deviationare given by:

w= 1

M

M m=1

wm σ w =

1

M −1

M m=1

(wm −w)2

1/2

. (18)

The absolute values of w and σ w obtained from statisticalprocessing of experimental and numerical data are shown inTable 1.

The comparison of the values of w and σ w shows that thestructure constant of turbulence C n

2 = 1, 5×10−14 cm−2/3

better corresponds to the experimental conditions than thestructure constant C n

2

=3

×10−14 cm−2/3. However, for z

=30 m the parameter wcomp calculated with C n 2 =1, 5×10−14 cm−2/3 is less than wlab. At the same time for




FIGURE 8 The distribution function of thefilament-center displacements Ψ( R) in theprobability scale of the Rayleigh law with pa-rameter wlab: solid curve corresponds to theexperiment (1100 shots) and dashed curve tothe simulations (200 shots)

z = 105 m the value of the parameterwcomp is larger thanwlab.The reasonfor this discrepancy maybe thefinite spatial coher-ence of the laser pulse that has not been taken into account inthe simulations. Random fluctuations in the laser beam maybe responsible for essential deflections of the filament centerat the initial stage of propagation, where the contribution of

turbulence to the nonlinear focus formation is not very large.To analyze the applicability of the Rayleigh law to the

description of random displacements of the filament center,let us plot the distribution function Ψ( Rm) in the Rayleighprobability scale. If we define the probability scale from theRayleigh distribution with the parameter wlab then the ana-lytical dependence for the distribution function (15) will beplotted as a straight line with an angle of 45 degrees betweenthis straight line and each of the coordinate axes (Fig. 8). Thevalues of the empirical distribution function Ψ lab( Rm) calcu-lated for m intervals are shown by dots. The analytical depen-dence of the Rayleigh-distribution function on the parameterwlab fits the distribution function obtained from the experi-mental data very well. In this probability scale the straight linecalculated for thedistributionfunctionΨ comp ( Rm)with the pa-rameterwcomp makes an angle with the analytical dependenceΨ( R) with the parameter wlab. This is the consequence of thediscrepancy between the Rayleigh parameters wcomp and wlab

obtained from the simulations and from the experiment.The distribution functions Ψ lab( Rm) and Ψ comp( Rm) plot-

ted in the Rayleigh probability scale fit the straight lines. Thisproves the validity of our assumption that the filament-centerdisplacements obey the Rayleigh-distribution law.

7 Conclusions

We presented the results of an experimental and

theoretical study of filament wandering in the propagation of powerful femtosecond laser pulses in a randomly inhomoge-neous medium.

We have experimentally obtained theensemble of filament-center displacements in the plane perpendicular to the propa-gation direction. The observed picture of filament displace-ments obtained for 1100 laser shots proves the isotropiccharacter of random perturbations that cause these displace-ments. For the perturbations associated with refractive-indexfluctuations in air the isotropic character of the perturba-tions corresponds to the Kolmogorov atmospheric turbulence.Statistical processing of experimental data shows that the

distribution function of filament-center displacements obeysthe Rayleigh law. The parameter of the Rayleigh distribu-tion characterizes the average value of displacements andincreases with filament length up to 1.5 mm at z = 100m.

We have developed a model of filament formation andwandering in the course of propagation of a powerful fem-

tosecond laser pulse through atmospheric turbulence. In thismodel the filament wandering is associated with refractive-index fluctuations in air. These fluctuations disturb the wave-front of the laser radiation and cause random formation anddisplacements of the filament. For numerical simulations of a femtosecond pulse self-focusing in atmospheric turbulencewe have developed the phase-screen model thatdescribes bothturbulent and nonlinear perturbations of the wavefront of thelaser radiation. Thedevelopedmodelallows us to simulate theinitial stage of the powerful femtosecond laser pulse propaga-tion in atmospheric turbulence.

Using the Monte Carlo method and the Kolmogorovmodel of atmospheric turbulence we have found that thesimu-lated filament displacements as well as experimentally ob-tained filament displacements obey the Rayleigh-distributionlaw. The parameter of the Rayleigh distribution calculatedin the simulations is close to the one obtained from the ex-perimental data. The discrepancy can be associated with thelimited spatial coherence and random angular deflections of the laser radiation. The effect of these factors on the filamentwandering in the atmosphere demands further study.

Further study of stochastic filamentation, including mul-tifilamentation and white-light generation, demands a moreadvanced model in order to perform the simulations of thepulse transformation along the whole propagation path in theturbulent atmosphere. Thismodel should takeinto account theeffect of the ionization and group-velocity dispersion on the

spatio-temporal evolution of the pulse in the filament.

ACKNOWLEDGEMENTS The authors thank K.Yu. Andrianov

for statistical processing of simulation results. This work was supported by

the Russian Fund of Fundamental Research (Grant No. 00-02-17 497).

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