ISSN 1054�660X, Laser Physics, 2009, Vol. 19, No. 8, pp. 1776–1792.© Pleiades Publishing, Ltd., 2009.Original Russian Text © Astro, Ltd., 2009.

1776

1 1. INTRODUCTION

The unique and spectacular phenomenon of fem�tosecond laser pulse filamentation in gases and con�densed media [1–4] is possible because of the cubicnonlinearity of the medium, which leads to the spatialand temporal localization of the pulse. The necessarycondition for the pulse localization is the increase ofthe peak power over the critical power for self�focusingPcr in the medium. Self�focusing enhances the localoptical intensity to extraordinarily high values. In thevicinity of a self�focus, the field strength is highenough to initiate a number of nonlinear processes,including multiphoton and tunneling ionization, har�monic generation, stimulated scattering and dielectricbreakdown. All these processes are normally absent atthe incident laser intensity and without self�focusing.

Yablonovitch and Bloembergen [5] showed ava�lanche ionization to be the main mechanism limitingthe intensity growth in the course of self�focusing inwide band�gap insulators, including glasses and trans�parent liquids. The seed electrons initiating the ava�lanche may come from multiphoton or tunneling ion�ization [6, 7]. The pulse durations considered in [5]were of the order of 1 ps and the estimated light field

1 The article is published in the original.

sufficient to stop the collapse was ~107 V/cm givingthe vacuum intensity ~1011 W/cm2. The resulting fila�ment radius in these insulators was found under theassumption that there is one critical power for self�focusing in the filament transverse section. The esti�mated number ranged within 2–20 µm depending onthe specific insulator.

As the experimentally achievable pulse durationsbecame shorter and the pulse peak powers becamehigher, femtosecond light filamentation in gases canbe observed routinely. Visible stable multiple filamentsin a range of hundreds of meter in the atmospheric airwere observed [8] and it raised the question on thelocal characteristics of these uniquely long confinedlight structures. Kasparian et al. [9] provided an esti�mation of the intensity inside a filament generated inair based on the experimental data on the ionization ofoxygen and nitrogen molecules obtained by Talebpouret al. [10]. The critical or clamped intensity wasnumerically calculated from the equation balancingthe contributions from the Kerr and the plasma non�linearity to the refractive index. The electron density

growth was modeled using the fit to the and ion yield curves as functions of the intensity. Thederived intensity was 4.5 × 1013 W/cm2 and it hap�

O2+

N2+

FILAMENTATION

Can We Reach Very High Intensity in Air with Femtosecond PW Laser Pulses?1

O. G. Kosarevaa, *, W. Liub, **, N. A. Panova, J. Bernhardtc, Z. Jid, M. Sharific, R. Lid,Z. Xud, J. Liud, Z. Wangd, J. Jud, X. Lud, Y. Jiangd, Y. Lengd, X. Liangd,

V. P. Kandidova, and S. L. Chinc, ***a International Laser Center, Physics Department, Moscow State University, Moscow, 119991 Russia

b Institute of Modern Optics, Nankai University, Key Laboratory of Opto�electronic Information Science and Technology, Education Ministry of China, Tianjin, 300071 China

c Centre d’Optique, Photonique et Laser (COPL) and Département de physique, de génie physique et d’optique,Universite Laval, Québec, Québec, G1K 7P4 Canada

d State Key Laboratory of High Field Laser Physics, Shanghai Institute of Optic and Fine Mechanics, Chinese Academy of Sciences, Shanghai, 201800 China

*e�mail: kosareva@phys.msu.ru**e�mail: liuweiwei@nankai.edu.cn

***e�mail: See.Leang.Chin@copl.ulaval.caReceived March 11, 2009

Abstract—In the course of femtosecond pulse filamentation in atmospheric density gases, the peak intensityis always limited by optical�field�induced ionization. This intensity clamping phenomenon is universal in allthe cases we studied, namely, single and multiple filament regimes with and without external focusing usingpulses of up to subpetawatt level. Even in the tight focusing cases, the clamped intensity along the propagationdirection does not exceed 30% of the global intensity maximum. The remarkable shot�to�shot stability of theclamped intensity (better than 1% of the maximum value) is revealed both experimentally and numerically ina single filament regime in air.

PACS numbers: 42.65.Jx, 42.65.Re, 32.80.Fb

DOI: 10.1134/S1054660X09150250

LASER PHYSICS Vol. 19 No. 8 2009

CAN WE REACH VERY HIGH INTENSITY IN AIR 1777

pened to be in good agreement with the intensityneeded to create the electron density of 1016 cm–3 inatmospheric air found from independent experiments[11, 12].

Becker et al. [13] studied the molecular nitrogenfluorescence yield from the filament created by 250 fs800 nm pulses in N2 as a function of the laser pulseenergy for different nitrogen gas pressures. It wasfound experimentally that there was a characteristicchange in the slope of the fluorescence yield as a func�tion of the laser pulse energy. The laser pulse energy atwhich this characteristic change occurred increasedlinearly with the inverse gas pressure. Thus, theauthors of [13] concluded that the peak intensityremained the same (clamped) once the pulse peakpower had exceeded the critical power for self�focus�ing in the nitrogen gas. Further research [14] showedsaturation of the maximum positive frequency shift ofthe supercontinuum spectrum in condensed opticalmedia (water, chloroform and glass). This saturationwas attributed to the clamping of the peak intensityinside the localized structures created in condensedmatter be they single or multiple filaments.

The long�range stability of the high peak intensityin the filament is extremely important for applicationssince it makes it possible to enhance and control alarge variety of nonlinear processes. Akozbek et al.[15] as well as Bergé et al. [16] observed and simulatedthe third harmonic generation in the course of fila�mentation of 45 fs 800 nm pulses in air. It was foundthat the generated 3rd harmonic maintains its energyand intensity over distances much longer than thecharacteristic coherence length. The study of The�berge et al. [17] showed a strong decrease in the root�mean�square fluctuation of the 3rd harmonic energyin the filament as compared with the value estimatedbased on the perturbation theory from the initialenergy fluctuations of the pump pulse (807 nm 40 fs)creating the filament.

The dynamic balance between self�focusing andionization in the filament leads to the important appli�cation of filamentation for pulse self�compression.The pulse with an initial duration of 40–50 fs, centralwavelength 800 nm and a few millijoules of energy canbe shortened down to 4–8 fs in argon gas at the densityclose to the atmospheric one [18–20]. The theoreticalinterpretation of the self�compression was given in[21–23]. As the free electron density builds up withincreasing intensity, the ionization�induced nonlinearphase partially compensates the Kerr–induced phaseacquired in the course of self�focusing. The result isthe local flattening of the phase dependence in thetime domain and almost flat phase around the lasercentral frequency λ0 ≈ 800 nm in the spectral domain[24]. Thus, the newly created frequencies due to self�phase modulation are set in phase. The pulse becomestransform�limited at a much shorter duration than atthe laser system output. This is indeed the self�com�

pression in the filament. The peak intensity reached inthe self�compressed pulse is equal to the clampedintensity for at least several centimeters of the filamentproduced in atmospheric density air or argon. Suchlongitudinal extension is sufficient for the confidentregistration of the self�compressed pulses in the exper�iment.

Experimental evidence for creating the filamentpulse compressor at 800 nm inspired the authors of[24] to demonstrate the efficient generation of power�ful and tunable ultrashort laser pulse in the visiblespectrum through four�wave mixing of 800 nm pumpand infrared seed pulses during filamentation in atmo�spheric density argon and air. The central wavelengthof the compressed ~10 fs pulse was tunable from 475 to650 nm depending on the infrared seed wavelength.Due to the intensity clamping during filamentation,the tunable pulse energy’s root�mean�square fluctua�tion was 3 times lower than the expected perturbativelimit.

The detailed investigation of the laser�inducedplasma in the filament was performed in [25]. It wasfound that with increasing initial energy of 45 fs807 nm pulse and a given geometrical focusing dis�tance of the lens, the electron density induced by thispulse in air is independent of the initial pulse energyafter the filament has been created (i.e. the pulse peakpower becomes above the critical power for self�focus�ing in air). The constant electron density as a functionof the initial pulse energy is one more evidence for theexistence of the clamped intensity in air.

Although the intensity clamping process is nowwell understood, the direct measurement of this keyparameter, especially at pulse peak power many timeshigher than the critical power for self�focusing in themedium (Ppeak � Pcr), has not been possible [1–3].This is because any probe that was exposed to the highlaser intensities inside the filaments would bedestroyed.

In this paper we present experiments and simula�tions showing that in atmospheric density gases (air orargon) the peak intensity attained in the filamentshows a remarkable stability, which is independent ofthe initial pulse energy, geometry of the experiment,single or multiple filament regimes. We prove bothexperimentally and numerically that neither externalfocusing of the whole pulse, nor fusion of multiple fil�aments can lead to the peak intensity increase by morethan about 30% of the global maximum in a single fil�ament created by a collimated beam.

Numerical model used to illustrate the intensityclamping in a single and multiple filaments isdescribed in Section 2 and the experimental condi�tions are specified for each case considered in Sec�tions 3–6.

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LASER PHYSICS Vol. 19 No. 8 2009

KOSAREVA et al.

2. NUMERICAL MODEL

Numerical simulations of the pulse propagation inatmospheric pressure air or argon have been per�formed based on the slowly evolving wave approxima�tion, which allows one to consider the propagation ofthe pulses up to single�cycle duration. We have chosenthe approach published in [26] modified with consid�eration of the laser�produced plasma generation [27].The equation for the slowly varying light field ampli�tude A (x, y, z, t) is given by

(1)

The first term on the right�hand side of the Eq. (1)

describes the beam diffraction (here ∆⊥ = +

or in the case of cylindrical symmetry ∆⊥ = +

is transverse Laplacian) and the space�time focusing,the second and the third terms describe material dis�persion, where k0 = ω0n0/c is the wavenumber at thelaser central frequency ω0, n0 ≈ 1 is the refractive indexof air or argon for the case of linear light propagation,k(ω) = ωn(ω)/c, vg = (∂k(ω)/ )–1 ≈ c is the

group velocity, k" = ∂2k(ω)/ ≈ 1.6 × 10–31 s2/cm

in air or ≈2.1 × 10–31 s2/cm in argon is the second orderdispersion coefficient. The fourth, the fifth and thesixth terms grouped in square brackets on the right�hand side of the Eq. (1) describe the contribution ofthe Kerr nonlinearity, self�steepening and the plasmato the pulse transformation. The seventh termdescribes the pulse energy loss due to the ionization.

The Kerr nonlinearity of neutrals in air is definedby anharmonic response of the bound electrons andthe stimulated Raman scattering on rotational transi�tions of molecules:

(2)

where the response function H(t) measured in [28] wasapproximated based on the damped oscillator model

2ik0∂A∂z����� 1

vg

����∂A∂t�����+⎝ ⎠

⎛ ⎞ 1 iω0

����� ∂∂t����–⎝ ⎠

⎛ ⎞ 1–

∆⊥A=

– k0k''∂2A

∂t2

������� 1

inn!( )

�����������∂ n( )k

2

∂ωn�����������

ω0

∂ n( )A

∂tn

���������

n 3=

∞

∑+⎝ ⎠⎜ ⎟⎛ ⎞

+2k0

2

n0

������� 1 iω0

����� ∂∂t����–⎝ ⎠

⎛ ⎞ ∆nk 1 iω0

����� ∂∂t����–⎝ ⎠

⎛ ⎞ 1–

Re ∆np( )+

��+ iIm ∆np( ) A ik0αA.–

∂2

∂x2

������ ∂2

∂y2

������

∂2

∂r2

������ 2r�� ∂

∂r����

∂ω ω ω0=

∂ω ω ω0=

∆nk_air x y z t, , ,( )

= 12��n2air A

2H t t'–( ) A t'( ) 2

t'd

∞–

t

∫+⎝ ⎠⎜ ⎟⎛ ⎞

,

in [29] by the following equation with characteristicresponse time of 77 fs. In the simulations we used n2 ≈10–19 cm2/W, which gives a critical power for self�focusing Pcr [30] equal to ≈10 GW for a 45 fs pulse inagreement with measurements [31]. In argon the Kerrnonlinearity of neutrals was considered as purelyinstantaneous

(3)

where n2Argon ≈ 1.2 × 10–19 cm2/W as extracted from themeasurements made in [32] according to the tech�nique suggested in [31].

In the case of a femtosecond pulse propagation inatmospheric density gases, the effective electron—neutral collision frequency νc is much smaller than thelaser frequency ω0. The imaginary part of the plasmacontribution to the refractive index Im(∆np) can beneglected, while the real part is represented by:

(4)

. (5)

The function α(x, y, z, t) is responsible for the ioniza�tion energy loss, in air:

(6)

or in argon

. (6a)

In the Eqs. (3)–(5) me and e are electron mass and

charge, respectively, = 8 and = mAr = 10 arethe numbers of photons necessary for the multiphotonionization of oxygen O2, nitrogen N2 or argon mole�

cules, respectively. I(x, y, z, t) = is

the light field intensity. The electron densities (x,

y, z, t), (x, y, z, t) are calculated according to therate equations:

(7)

(7a)

∆nk_Argon x y z t, , ,( ) 12��n2Argon A

2,=

Re ∆np x y z t, , ,( )( )ωp

2x y z t, , ,( )

2n0ω02

������������������������,–=

ωp x y z t, , ,( ) 4πe2Ne x y z t, , ,( )/me=

α x y z t, , ,( )�ω0

I�������� m

O2∂Ne

O2

∂t���������� m

N2∂Ne

N2

∂t����������+⎝ ⎠

⎛ ⎞=

α x y z t, , ,( )�ω0

I��������m

Ar∂Ne

∂t�������=

mO2 m

N2

cn0

8���� A x y z t, , ,( ) 2

Ne

O2

Ne

N2

∂Ne

N2 O2 Ar, ,

∂t��������������������

= PN2 O2 Ar, ,

x y z t, , ,( ) Na

N2 O2 Ar, ,Ne

N2 O2 Ar, ,–( ),

Ne Ne

O2 Ne

N2,+=

LASER PHYSICS Vol. 19 No. 8 2009

CAN WE REACH VERY HIGH INTENSITY IN AIR 1779

(7b)

where (x, y, z, t) is the ionization probability

of nitrogen, oxygen or argon molecules and, ,

, are the densities of the respective neutralsbefore the pulse enters the medium. For the laser cen�ter wavelength λ = 800 nm and the peak intensity1013–1014 W/cm2 reached in the filament in air and Ar,the Keldysh parameter γ is in the range 3 < γ < 1, i.e.,in the transition regime from the multiphoton to tun�neling ionization. The model [33], which gives a verygood agreement with the ion yield curves in the tun�neling regime, is not quite appropriate in our case.Thus, after the detailed analysis of the possible ioniza�tion models presented in [34], we have chosen the PPTmodel [35] with fitting parameters derived from theexperimental data [10] to calculate the ionization

probabilities .We assume that the initial pulse is Gaussian with

Gaussian transverse intensity distribution. In the gen�eral case, there can be also natural imperfections onthe beam transverse profile, which can be modeled byadditive Gaussian noise ξ(x, y):

(8)

where τ = t – z/vg is the time in the co�moving coor�dinate system, Rf is the geometric focusing distance, vg

is the group velocity of the pulse, a0 and τp are the ini�tial beam radius and half pulse duration at e–1 intensity

level, δ = is the rate of frequencychange with the local time τ during the pulse. A0 is the

amplitude of a random light field (x, y, z = 0, τ). One

realization of the random field corresponds to a sin�gle laser shot. In the case of cylindrical symmetry ξ(x,y) = 0 and x2 + y2 = r2.

The system of Eqs. (1)–(7) with the initial condi�tions (8) was solved either on an (x, y, z, τ) or a (r, z, τ)grid. The numerical simulations control was per�formed based on the spatial and temporal phasechange. The phase difference between the two neigh�boring grid points in any direction did not exceed π/8radians.

There are several numerical characteristics, whichwe analyze when studying the intensity clamping phe�nomenon. At each propagation position z we search

∂Ne

N2 O2 Ar, ,

∂t��������������������

= PN2 O2 Ar, ,

x y z t, , ,( ) Na

N2 O2 Ar, ,Ne

N2 O2 Ar, ,–( ),

PN2 O2 Ar, ,

Na

N2

Na

O2 NaAr

PN2 O2 Ar, ,

A x y z = 0 τ, , ,( ) A0 –x2

y2

+

2a02

������������� τ2

2τp2

������–⎝⎜⎛

exp=

+ iδτ2

2������ ikx

2y

2+

2Rf

�������������+ ⎠⎞ 1 ξ x y,( )+( ),

τp2– τp

2/τ0

21–( )

1/2–

A

A

for the intensity maximum Imax(z) over both the localtime and the transverse cross section given by (x, y) orr coordinates:

. (9)

The clamped intensity Ic will be defined as the globalmaximum over the whole propagation distance, localtime and transverse coordinates:

. (9a)

In the experiment the theoretically obtained local timecharacteristics can hardly be measured. Therefore, wecalculate the fluence distribution, which is the inten�sity integrated over the pulse at each spatial coordi�nate:

. (10)

The other important characteristics are the localplasma density Ne(x, y, z) calculated at the end of thepulse and its maximum over the cross section calcu�lated at each position z: Ne_max(z). The integratedplasma characteristic is the linear plasma densityDe(z), represented by the local density Ne(x, y, z) inte�grated over the whole transverse section, where theplasma exists:

. (11)

3. INTENSITY CLAMPING IN A SINGLE FILAMENT REGIME

The phenomenon of the intensity clamping in thefilaments occurs when the self�focusing inducedgrowth of the local maximum intensity Imax(z) (seeEq. (9)) is stopped by the free electron generation. InFig. 1a the maximum intensity Imax(z) (solid curve) inair is obtained from the solution of the full set of spa�tio�temporal equations (1)–(7) describing the pulsetransformation in air in the single filament regime[36]. In both the experiment and the simulation theinitial pulse duration was 45 fs, the energy 3.2 mJ, andthe laser central wavelength 800 nm. The actual initialbeam diameter was 2.6 mm FWHM, while in the sim�ulations it was taken equal to 2 mm FWHM so as to fitthe earlier filament formation in the experimentcaused mainly by the slightly convergent wave front. InEq. (8) the noise function ξ(x, y) and the chirp param�eter δ were put to zero, the geometrical focusing dis�tance Rf to infinity. A good quantitative agreementbetween the simulated and experimentally obtainedbeam diameter of the filament core has been observed(Fig. 1b) [36], therefore we can take this case of colli�mated beam propagation resulting in a single filament

Imax z( ) I x y z τ, , ,( )x y τ, ,max=

Ic I x y z τ, , ,( )x y z τ, , ,max Imax z( )( )

zmax= =

J x y z, ,( ) I x y z τ, , ,( ) τd

∞–

∞

∫=

De z( ) Ne x y z, ,( ) xd yd

∞–

∞

∫∞–

∞

∫=

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LASER PHYSICS Vol. 19 No. 8 2009

KOSAREVA et al.

as a reference for the study of the intensity clampingphenomenon. This will be compared later with a mul�titude of other situations; i.e., multiple filamentationwith and without external focusing at very high peakpower (up to nearly sub�petawatt peak power in ourexperiment). We shall first discuss in more detail thisreference case.

In Fig. 1a we show the curve, marked by circles,which would be the maximum intensity Imax(z) in air ifthere were no free electron generation. Note that thesimulated “presence” of pure material dispersionwithout the plasma does not prevent the beam fromcollapsing if the pulse peak power is several times

higher than the critical power for self�focusing in air(Ppeak/Pcr ≈ 7) and the self�focusing distance of thecollimated beam zsf ≈ 235 cm [28] is much shorter thanthe characteristic distance of the pulse dispersivebroadening Ld ≈ 45 m.

The local maxima of the intensity function Imax(z)(Fig. 1a, solid curve) vary from 4.8 to 7.8 × 1013 W/cm2

(or within 30%) along the whole ~6 m extension of theplasma channel including gaps and refocusing(Fig. 1c). Let us consider the first and the most intensepart of the plasma channel extended from z ≈ 236 to380 cm. The corresponding part of the maximumintensity Imax(z) is shown in the inset to Fig. 1a. In

160

120

80

40

0

Max

imu

m in

ten

sity

, T

W/c

m2

3.0

2.5

2.0

1.5

1.0

0.5

0

0.2

0

−0.2

Dia

met

er F

WH

M,

mm

r, c

m

0 250 500 750 1000Propagation position, cm

160

120

80

40

0200 250 300

Propagation distance, cm

A

BC

D

E226 cm

237 cm255 cm

280 cm

310 cm

Max

imu

m in

ten

sity

, T

W/c

m2

(a)

(b)

(c)

(d)

Fig. 1. Propagation characteristics of 45 fs (FWHM), 3.2 mJ, 800 nm pulse in air in a single filament regime. The beam was col�limated with the initial diameter 2.6 mm FWHM in the experiment and 2 mm FWHM in the simulations. (a) The simulated max�imum on�axis intensity obtained from the solution of the full spatio�temporal problem given by the Eqs. (1)–(7) with consider�ation of the plasma generation (solid line) and without the plasma generation (solid line marked by circles). The inset shows mag�nified region from 200 to 310 cm with the positions A, B, C, D, E discussed in the text. (b) Measured from the fluence distributionbeam diameter (FWHM, averaged for horizontal and vertical dimension) (triangles) and calculated from the simulated fluencedistribution (solid line). (c) The simulated plasma distribution Ne(r, z) at the end of the pulse.

LASER PHYSICS Vol. 19 No. 8 2009

CAN WE REACH VERY HIGH INTENSITY IN AIR 1781

order to better understand the intensity variations, wewill follow both the intensity and the refractive indexchange with the local time τ shown in Fig. 2. Figures 2a–2d are shown for the successive distances, indicated bythe solid vertical lines and the letters A, B, C, D, E inthe inset to Fig. 1a. Figures 2b contain the plots for thetwo positions B and C. The left column of each panelFigs. 2a–2d shows the on�axis nonlinear refractiveindex ∆n(r = 0, τ) (upper plot) and the intensity∆I(r = 0, τ) (lower plot), which induces the contribu�tion to the refractive index ∆n(r = 0, τ), given by thesum of the nonlinear Kerr and the plasma contribu�tions:

(12)

The right column of Fig. 2 shows the fluence distribu�tion (Eq. (10)) at the same position as the intensity andthe refractive index in the left column of the currentpanel.

As the peak intensity grows along the propagationdistance through the position A at z ≈ 226 cm and tilljust before the position B z ≈ 237 cm, the positive Kerrnonlinear contribution to the refractive index followsthe intensity according to [37] and the plasma contri�bution is negligible as compared with the Kerr one(Fig. 2a). The cardinal change of the refractive indexbehavior occurs at the position B (z ≈ 237 cm) (Fig. 2b,solid curves of the refractive index and the intensity inthe left column, magnified contributions of the Kerrand the plasma to the refractive index in the inset). Itis the earliest propagation distance at which the non�linear contribution to the refractive index (12) goesbelow zero at τ = 12 fs. Although the accumulation offree electrons necessary to balance the Kerr nonlinear�ity takes place all over the pulse from τ = –30 fs and tillτ = 12 fs, the fastest plasma growth occurs in the vicin�ity of the time moment τ = –4 fs, where the maximumintensity Imax ≈ 6.5 × 1013 W/cm2 is reached: –6 fs < τ <–1 fs (see the curve marked by triangles in the inset toFig. 2b).

The major role of the plasma is to slow down theself�focusing of all the temporal slices located after theslice where the maximum intensity Imax is reached atτ > –4 fs. The consequence of this slowdown is thenear�axis flattening of the transverse intensity slicesI(r, τ). The integral effect of this flattening is well pro�nounced in the fluence distribution in Fig. 2b, wherewe can see the superGaussian beam profile instead ofthe Gaussian one shown in Fig. 2a.

In spite of the self�focusing slowdown due to theplasma, the maximum intensity Imax(z) continues toincrease while the propagation distance changes fromthe position B at z ≈ 237 cm to the position C at z ≈255 cm (inset to Fig. 1a and Fig. 2b, intensity plot,curve marked by filled circles). The intensity maxi�

∆n r = 0 z τ, ,( )

= ∆nk_air r = 0 z τ, ,( )ωp

2r = 0 z τ, ,( )

2n0ω02

����������������������������–

mum shifts towards the front of the pulse and the pulseis compressed to ~6 fs FWHM. Because of the pulseself�compression, there is less time for the electrons tobuild up and to stop the intensity growth. At the posi�tion C (z ≈ 255 cm) the effect of material dispersionbecomes obvious. Indeed, the position C is character�ized by the pulse splitting as seen from the stepwiseaccumulation of energy for the time τ > –10 fs (see thecurve marked by filled circles in the intensity panel ofFig. 2b). In the spatial domain we can see as the super�Gaussian fluence shape transforms into a ring�shapedone, because the plasma accumulated by the end ofthe pulse on the filament axis pushes the radiation outto the periphery of the beam (filled circles in the flu�ence panel of Fig. 2b). Between the positions C at z ≈255 cm and D at z ≈ 280 cm the trailing subpulsedevelops due to both refocusing and material disper�sion. The value of the local maximum in the trailingsubpulse catches up with the maximum value at theleading subpulse by z ≈ 280 cm. Therefore, we canobserve two intensity spikes of equal height (Fig. 2c).

The propagation region within 280 cm < z < 320 cm(inset to Fig. 1a) is characterized by the compressionof the second subpulse and growing of its peak inten�sity. The clamped intensity Ic, defined as the globalintensity maximum over the whole propagation dis�tance 0–1000 cm (Fig. 1a), is attained at the positionE and the distance z = 310 cm (Fig. 2d). This maxi�mum is equal to Ic ≈ 7.8 × 1013 W/cm2 and does notexceed other local maxima by more than 30%. In thespatial domain (Fig. 2d, the fluence panel) the well�developed rings are observed resulting from the refrac�tion of the radiation on the laser—produced plasmaalong the whole propagation range between the posi�tions B and E.

During propagation in the region 320 < z < 380 cmthe degradation of both the leading and the trailingsubpulses due to the plasma and material dispersiontakes place (Fig. 1a). The maximum intensitydecreases dramatically till the appearance of the sec�ond plasma burst between the distances z ≈ 380 to600 cm (see Fig. 1a). The local intensity dynamics inthe second plasma burst develops similarly to what wehave just described for the distances 220–310 cm andis characterized by the energy interchange between thetwo major temporal subpulses.

The general note about the temporal dynamics in asingle filament regime with a collimated beam is thatthe electron density is rather low (Ne < 1016 cm–3) andhas the comparatively narrow transverse section withthe diameter d = 60 µm. Therefore, the absolute valueof the negative contribution to the refractive indexduring the pulse is most of the time essentially smallerthan the positive Kerr–induced contribution. Never�theless, this small contribution is enough to slow downthe self�focusing process and limit the collapse.

The variations of the maximum intensity with thepropagation distance are repeatable from shot to shot

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LASER PHYSICS Vol. 19 No. 8 2009

KOSAREVA et al.

3020100–10–20–30

1.4

0

1.2

1.0

0.8

0.6

0.4

0.2

∆n(r = 0, τ), z = 226 cm, A

Nonlinear contribution to therefractive index, 10–5

0.40.20–0.2–0.4Radius, mm

0.9

0

0.80.70.60.50.40.30.20.1

Fluence (r), z = 226 cm, A

Fluence, J/cm2

3020100–10–20–30

80

10

70

60

50

40

30

20

0

I(r = 0, τ), z = 226 cm, A

Local time τ, fs

Intensity, TW/cm2

(a)

3020100–10–20–30

1.6

0

1.41.21.00.80.60.40.2

–0.2

∆n = 0

z = 237 cm, Bz = 255 cm, C

∆n(r = 0, τ)

Nonlinear contribution to therefractive index, 10–5

3020100–10–20–30

0.2

0.1

0

–0.1

Nonlinear contribution to therefractive index, 10–5

∆nk_air + ∆np∆nk_air∆np

Local time τ, fs

0.40.20–0.2–0.4Radius, mm

1.0

0

0.90.80.70.60.50.40.30.20.1

Fluence, J/cm2

Fluence (r) z = 237 cm, Bz = 255 cm, C

3020100–10–20–30

80

0

70

60

50

40

30

20

10

Intensity, TW/cm2

z = 237 cm, Bz = 255 cm, C

Local time τ, fs

(b)

Local time τ, fs

Fig. 2. Left column: time�dependent nonlinear refractive index and intensity distributions on the beam axis. Right column—flu�ence distribution given by the Eq. (10). Capital letters indicate the positions from the inset to Fig. 1a. The propagation distanceis (a) z = 226 cm, position A; (b) z = 237 and 255 cm, positions B, C; (c) z = 280 cm, position D; (d) z = 310 cm, position E. Theinset to panel (b) shows magnified nonlinear contributions to the refractive index at z = 237 cm. The pulse characteristics are thesame as in Fig. 1. Horizontal lines on the refractive index plots show the level ∆n = 0. Vertical lines show the local time momentsat which ∆n = 0 or the intensity takes the maximum value. Horizontal line at the level ∆n = 0.2 × 10−5 in Fig. 2b, the refractiveindex plot, shows the upper limit of the inset to Fig. 2b in the right column.

I(r = 0, τ)

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CAN WE REACH VERY HIGH INTENSITY IN AIR 1783

3020100–10–20–30

1.6

–0.2

1.41.21.00.80.60.40.2

0

Nonlinear contribution to therefractive index, 10–5

Intensity, TW/cm2

∆n(r = 0, τ) z = 280 cm, D

0.40.20–0.2–0.4Radius, mm

1.0

0

0.8

0.6

0.4

0.2

Fluence, J/cm2

Fluence (r), z = 280 cm, D

3020100–10–20–30Local time τ, fs

80

0

70

60

50

40

30

20

10

I(r = 0, τ) z = 280 cm, D

(c)

30100–10–20–30

1.6

–0.2

1.41.21.00.80.60.40.2

0

Nonlinear contribution to therefractive index, 10–5

Intensity, TW/cm2

∆n(r = 0, τ) z = 310 cm, F

1.0–1.0Radius, mm

0.7

0

Fluence, J/cm2

Fluence (r), z = 310 cm, F

30100–10–20–30Local time τ, fs

80

0

70

60

50

40

30

20

10

I(r = 0, τ) z = 310 cm, F

(d)

20 0.50–0.5

0.1

0.6

0.5

0.4

0.3

0.2

20

Fig. 2. (Contd.)

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LASER PHYSICS Vol. 19 No. 8 2009

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and stable towards reasonable fluctuations of the ini�tial pulse energy [38]. The experiment was performedin the Institute of Modern Optics of Nankai Univer�sity. The 800 nm laser pulse with the duration 45 fs, theenergy 2 mJ and the beam diameter 3 mm FWHM wassent into air through a telescope. The filament startedat z ≈ 97 cm from the telescope. The longitudinal flu�orescence intensity distribution was measured byscanning the photomultiplier tube (PMT) along thepropagation axis (Fig. 3, curve marked by squares). Avariable neutral density filter was added between theinterference filter and the PMT to attenuate theamount of light received by the PMT down to the samelevel at different distances. Then, the root�mean�square fluctuation of the signal was calculated for 200shots at every position. In Fig. 3, the circles show thetrend of the root�mean�square fluctuation of the sig�nal as a function of the propagation distance. Clearly,the data could be divided into three zones as separatedby two dotted lines in Fig. 3. Zone A corresponds tothe region where the propagation distance z < 96 cm.The maximum fluctuation is about 13% at 90 cm. Asthe propagation distance increases, the fluctuationquickly declines to less than 1% at z = 97 cm. After�wards, the fluctuation of the fluorescence signal dis�plays a very high stability until z = 132 cm. We identifythis range as zone B. Throughout zone B, the signalfluctuation is less than 1% and the minimum is evenbelow 0.5%. Note that the energy fluctuation of theinput pump beam is about 1.5%. Since the ionizationrate of oxygen molecules, which are mainly responsi�ble for the plasma generation in air, can be roughlycharacterized by an effective nonlinearity order of 8,one would expect that the fluctuation of the fluores�cence signal should be at least 1.5% × 8 ≈ 12% in theperturbative regime. It is more than one order of mag�

nitude higher than the measured fluctuation withinthe zone B in our experiment. After d = 132 cm, i.e.,in zone C, the stability of fluorescence signal becomesworse rather quickly. It reaches 14% at the distance of140 cm. For further distances, the signal received bythe PMT was not strong enough to make consistentmeasurement.

The shot�to shot fluctuations of the maximumintensity in the filament was studied numerically in asingle filament regime. The simulations were per�formed for 50 fs (FWHM) pulse centered at 800 nm,beam diameter 0.6 (FWHM) and the initial pulseenergy around 800 µJ. We used 20 realization of theinitial regular laser pulse given by the Eq. (8), wherethe noise function ξ(x, y) and the chirp parameter δwere put to zero, the geometrical focusing distance Rf

to the infinity. The difference between the realizationswas in random fluctuations of the overall initial energyintroduced through the fluctuations of the amplitudeA0 only. Therefore, the initial pulse energy was ran�domly distributed in the interval 800 ± 12 µJ or with±1.5% error. In Fig. 4, multiple thin merging lines out�line the normalized on�axis maximum intensity fortwenty different initial pulse energies (left axis, I0

denotes the maximum intensity at z = 0), while a sin�gle thick line indicates “shot�to�shot” root�mean�square fluctuation of the maximum intensity (rightaxis). The most interesting thing about Fig. 4 is merg�ing of twenty maximum intensity curves into almost asingle one as soon as the simulated plasma is producedat z ≈ 50 cm and till the end of the plasma at z ≈ 110 cm.The intensity root�mean�square fluctuation withinthis distance range is always less than 0.5% with theminimum approaching 0.05%. Such extremely stabi�lized laser intensity naturally leads to constant ioniza�tion rate and results in almost invariable free electron

14013012011010090Propagation distance, cm

30000

0

20000

10000Sig

nal

, ar

b. u

nit

s

20

0

15

10

5 RM

S fl

uctu

atio

n,

%

357 nm signal as function of distanceRoot�mean�square fluctuation of signal

A B C

Fig. 3. (from [38]) Experimental data on the stability of fluorescence signal fluctuation in air in a single filament regime. Blacksquares (left label): longitudinal distribution of nitrogen fluorescence signal at 357 nm; red circles (right label): root�mean�squarefluctuation of the measured nitrogen fluorescence signal as a function of propagation distance. Blue dashed line: the measuredroot�mean�square fluctuation by the same detection setup under linear focusing condition (E = 50 µJ/pulse, f = 12.5 cm).

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CAN WE REACH VERY HIGH INTENSITY IN AIR 1785

density. The simulated fluctuation of the free electrondensity within the range from 0.5 to 1.1 m is as low as0.5%, which is in good qualitative agreement with 1%fluorescence signal fluctuation between z = 97 and 132cm in Fig. 3.

Thus, in a single filament regime produced by acollimated beam in air, we managed to observe andconfirm with the numerical simulations the longitudi�nal and shot�to�shot stability of the maximum inten�sity in the filament. Along the propagation directionthe on�axis peak intensity variation may be up to 30%of the clamped intensity Ic (given by the global inten�sity maximum over the whole propagation distance).The shot�to�shot root mean square fluctuation of thesimulated intensity and free electron density at a givenposition in the filament is less than 0.5% of the corre�sponding maximum values. The shot�to�shot rootmean square fluctuation of the measured fluorescencesignal is less than 1% of the signal maximum.

4. INTENSITY CLAMPING WITH EXTERNAL FOCUSING

If the pulse is focused into air externally, it might atfirst sight seem that the peak intensity in the focalregion increases with decreasing geometrical focusingdistance of the lens. Indeed, the intensity increaseproportional to the initial laser pulse energy may beobserved either in a low pressure gas or in an atmo�spheric pressure gas but at low energy, when the maxi�mum intensity in the focal region does not reach theionization threshold intensity. Once we deal with thecase of filamentation, such situation may not be possi�ble, since, at the filament starting position zfil, where

(13)

and Rf is the geometrical focusing distance of the lens,and zsf is the Marburger self�focusing distance, theinfinite intensity growth will be stopped similarly tothe case of the filamentation of the collimated beam.Thus, we can conclude that the clamped intensityconcept is also relevant to the external focusing intoatmospheric density gases.

The information on the clamped intensity reachedwith external focusing into air or argon can be experi�mentally obtained from the data on the electron den�sity in the vicinity of the position z = zfil [25, 39]. In theexperiment [25] a single filament created after focus�ing of 0.2–4.0 mJ, 45 fs, 800 nm pulse into air, wasside�imaged onto an ICCD preceded by a band�passfilter to select the first negative band of the nitrogen

ion . The experimental results [25] show that theplasma diameter increases with decreasing focallength from 380 to 50 cm. The initial beam diameterwas 2.5 mm FWHM and the initial pulse energy1.4 mJ. The systematic simulations to study thedependence of the filamentation dynamics on externalfocusing conditions were performed in 1 atm argon. Inthe simulations the initial 800 nm pulse duration,energy, and the beam diameter were 45 fs (FWHM),3.2 mJ, and 2 mm (FWHM), respectively. The focallength of the lens varied from 100 to 400 cm. The sim�ulations reproduced well the experimentally obtained[25] plasma diameter increase from 58 to 105 µm withdecreasing focal length from 400 to 100 cm (Fig. 5a).Larger diameter values in the simulations (solid curvewith squares in Fig. 5a) are due to the larger energy:

1zfil

���� 1zsf

���� 1Rf

����,+=

N2+

200150100500Propagation distance, m

18

16

14

12

10

8

6

4

2

8

0

7

6

5

4

3

2

1

Nor

mal

ized

max

imum

inte

nsi

ty, I

/I0

RM

S fl

uctu

atio

n, %

Normalized on axis peak intensity, I/I0

RMS fluctuation, %

Fig. 4. (from [38]) The simulated on axis laser peak intensity in air (thin lines, left label) and root�mean�square fluctuation of theintensity (thick blue line, right label) as a function of propagation distance in air. The initial pulse was a regular Gaussian pulsecentered at 800 nm with the duration 50 fs (FWHM), beam diameter 0.6 mm (FWHM) and the initial energy 800 ± 12 µJ. Theinitial energies of 20 realizations were randomly distributed within the interval of ±12 µJ.

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3.2 mJ in the simulations versus 1.4 mJ in the experi�ment.

The increase in the plasma diameter is directlyassociated with the increase in the overall amount ofelectrons at the beginning of the filament (for ourcomparatively long geometrical focusing distance werefer to the first 10 cm of the plasma column). FromFig. 5b we can see that the largest amount of electronsis attained for the focal length of 100 cm and decreasestowards larger focal lengths. At the same time, the

maximum electron density (Fig. 5c, right axis) and theclamped intensity (Fig. 5c, left axis) vary slowly withthe change of the focusing geometry. Note that thevariation of the clamped intensity with the geometricalfocusing distance is less than 7%.

The notable attribute of the intensity clampingphenomenon in the presence of external focusing iswidening of the transverse area, where the clampedintensity is reached. Figure 6 shows the comparison ofthe simulated transverse fluence and plasma distribu�

450400350300250200150100500

110

60

100

90

80

70

50

Pla

sma

diam

eter

(F

WH

M),

µm

(a)SimulationsExperiment Phys. Rev. E 74, 036406 (2006)

450400350300250200150100500

3.0

2.5

2.0

1.5

1.0

0.5

(b)

Am

oun

t of

ele

ctro

ns,

1011

450400350300250200150100500Focal length, cm

200

150

100

50

Cla

mpe

d in

ten

sity

, T

W/c

m2 (c)

8 × 1016

2 × 1016

7 × 1016

6 × 1016

5 × 1016

4 × 1016

3 × 1016

Intensity

Election density

Pea

k el

ecti

on d

ensi

ty,

cm–

3

Fig. 5. Dependence of the FWHM plasma diameter (a); total amount of electrons in the first 10 cm of the filaments (b) peak elec�tron density and peak intensity. The simulation results are presented for 1 atm argon, where the initial pulse duration, energy, andthe beam diameter were 45 fs (FWHM), 3.2 mJ, and 2 mm (FWHM), respectively. The experimental results are taken from [25]with 45 fs (FWHM), 1.4 mJ, 2.5 mm (FWHM) laser pulse focused into air.

LASER PHYSICS Vol. 19 No. 8 2009

CAN WE REACH VERY HIGH INTENSITY IN AIR 1787

tions at the geometrical focusing distance Rf = 400 cmand Rf = 200 cm. At least 1.5 time increase in both thebeam diameter (compare Figs. 6c and 6d) of Fig. 6)and the plasma channel diameters (compare Figs. 6aand 6b) of Fig. 6) is clearly pronounced. The fluencereaches the maximum 2.2 J/cm2 at the tighter focusinginstead of 1.1 J/cm2 at the looser focusing conditions.Besides, the amplitude of surrounding rings is essen�tially larger for Rf = 200 cm than for Rf = 400 cm(compare the fluence distributions in Figs. 6c and 6d).

Thus, the case of single filamentation with externalfocusing differs from the collimated beam filamenta�tion mainly by the total amount of electrons, the widthof the plasma channel and the transverse fluence dis�tribution at the beginning of the filament formation.At the same time, the maximum electron density andthe clamped intensity remain almost independent ofthe geometrical focusing distance (Fig. 5c).

5. INTENSITY CLAMPING IN THE MULTIPLE FILAMENTATION CASE WITH TERAWATT

PEAK POWER

The experiments proving the intensity clampingphenomenon in a multiple filamentation regime wereperformed in Laval University at terawatt powers andin the Shanghai Institute of Optics and Fine Mechan�ics at petawatt powers [39]. In the terawatt experi�ments 800 nm 100 fs negatively chirped pulses at 10 Hzrepetition rate were focused (Rf = 100, 60, and 20 cm)into air or a 1.5 m tube filled with pure argon at 1 atm.The pulse energy varied from 10 to 70 mJ. The fluores�

cence emission from the filament was side imagedonto the entrance slit of a spectrometer. The electrondensity was deduced from the Stark broadening ofeither atomic O triplet (777.4 nm) or Ar 696.54 nmline. The intensity was then derived from the electrondensity using the ionization curve of air or Ar+ mea�sured separately using the setup described in [40, 41].The remarkable stability of the electron density andthe derived clamped intensity (only the latter is shownin Fig. 7) observed in the whole range of the initialpulse energy studied. The noticeable fluctuations ofthe intensity with energy were obtained only for theshortest geometrical focusing distance and were attrib�uted to very high light field gradients near the start offilamentation.

In the course of multiple filamentation in argon,the intensity derived from the electron density mea�surements in air increases as the focal length decreasesto Rf = 20 cm as compared to the free propagation sin�gle filament’s clamped intensity Ic [9] (Fig. 7a).Accordingly, in argon (Fig. 7b), the derived intensityalso increases with decrease in the focal length of thelens (compare curves, marked by triangles, circles andrectangles in Fig. 7b). However, the variation of theclamped intensity averaged over all the energy rangedoes not exceed 30% with decreasing geometricalfocusing distance.

In order to check the amount of the maximumintensity variation in the multiple filament regime wesimulated the propagation of a collimated beam withrandom transverse intensity and phase perturbationsthrough air. The pulse was centered at 800 nm and had

4

2

0.40.2

0

−0.2−0.4

−0.2

00.2

0.4

3.01.5

0.40.2

0−0.2

−0.4−0.2

00.2

0.4

2

1

00.4

0.20

−0.2

−0.4−0.2

00.2

0.4

1.00.5

00.4

0.20

−0.2−0.4

−0.20

0.20.4

y, mmx, mmy, mm

x, mm

Rf = 200 cmRf = 400 cm

Rf = 200 cm Rf = 400 cm

Ele

ctro

n d

ensi

ty,

1016

cm

−3

Flu

ence

, J/

cm2

(a) (b)

(c) (d)

Fig. 6. The simulated transverse plasma distribution at the end of the pulse (a), (b) and transverse fluence distribution (c), (d) fordifferent external focusing conditions in 1 atm argon. The focal length of the lens and the propagation distances for plasma andfluence “registration” are Rf = 200 cm, z = 124 cm (a), (c) or Rf = 400 cm, z = 250 cm (b), (d).

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the energy of 17 mJ, the duration 170 fs (FWHM) andthe beam diameter 2.5 mm (FWHM). The ratioPpeak/Pcr ≈ 20 was sufficient for multiple filament for�mation. The simulations were performed using thesystem of equations (1)–(7) with the initial condition(8). Full (x, y, z, τ) formulation of the propagationproblem was used without the cylindrical symmetryassumption. At each propagation position z we “regis�tered” the transverse coordinates where the local elec�tron density maxima were reached by the end of thepulse. Afterwards we “measured” the intensity at thesetransverse positions. This approach allowed us to fol�low both the maximum intensity in the parent fila�ments born from the initial transverse perturbationsand the intensity in stochastically created child fila�ments. The calculation results are shown in Fig. 8,where at each propagation distance z the local inten�sity maxima corresponding to the plasma peaks are puttogether. For example, at z = 194 cm there are at leastsix local intensity maxima associated with the plasma

channels shown in Fig. 8c. Deviation of any of thelocal intensity maxima from the clamped intensity Ic

shown by the horizontal dashed line in Fig. 8a is lessthan 30% of the value Ic throughout the whole exten�sion of multiple filament zone in the longitudinaldirection. This deviation is in agreement with the vari�ation of the maximum intensity Imax(z) in a single fila�ment regime (Fig. 1a), which is in the range 5.0–7.8 ×1013 W/cm2. Thus, in the multiple filament regime themaximum intensity is also limited by the clampedintensity value similar to the one reached in a singlefilament regime.

Regarding the case of multiple filamentation thereexists a challenging question. If, in spite of theclamped intensity phenomenon, there is any chanceof enhancing the local peak intensity by superimpos�ing several filaments both spatially and temporally. Apartial answer to this question may be given by tryingto merge two filaments produced by the same pulse.Merging was performed by means of the numericalsimulations. A regular 800 nm Gaussian pulse with theduration 100 fs (FWHM) and beam diameter of 2 mm(FWHM) was transmitted through a mask with twocircular apertures with the diameter of 0.85 mm each.The apertures were located symmetrically relatively tothe Gaussian beam maximum with the transversecoordinates (x = 0, y = 0). The distance between theaperture centers was 0.85 mm. The two small beamstransmitted through the mask were totally identical intheir energy, pulse duration, peak intensity. The pulseenergy before the mask was 8.5 mJ and after the mask1.9 mJ. The partial energy transmitted through a singleaperture was 0.85 mJ and the partial ratio Ppeak/Pcr =1.7. After transmission through the mask, the radia�tion was geometrically focused by a lens with focallength Rf = 30 cm. The resulting plasma channels areshown in Fig. 9a. Till the distance z = 24 cm the twofilaments develop independently. A bright plasma spotis then observed at the position of filament merging z ≈24–28 cm. The global maximum or the clampedintensity Ic = 7.5 × 1013 W/cm2 (Eq. 9a) attained at thisposition is just 1.3 times higher than the clampedintensity reached in a single filament formed by thesame circular aperture and the same energy of 0.85 mJ(compare the solid curve and the curve marked bysquares in Fig. 9b). Moreover, the value Ic = 7.5 ×1013 W/cm2 almost coincides with the clamped inten�sities attained in a single (Fig. 1a) and multiple(Fig. 8a) filamentation in air.

The clearly pronounced advantage of the suggestedfilament control tool is the possibility to increase theelectron density at the prescribed position of filamentmerging. Comparing the maximum electron densitiesin Fig. 9c in the case of a single filament (Fig. 9c, solidcurve) and the merged filaments (Fig. 9c, squares),one might see a well�pronounced doubling of the elec�tron density at z ≈ 26 cm. The origin of this electron

70605040302010

10

8

6

4

2

(a)

f = 20 cmI ~ 6.4 × 1013 W/cm2

Ic ~ 4.5 × 1013 W/cm2

I/1013 W/cm2

E, mJ

80706050403020100Laser energy, mJ/pulse

2.0

1.5

1.0

0.5

f = 100 cmf = 60 cmf = 20 cm

Clamped intensity, 1014 W/cm2

(b)

Fig. 7. (from [39]). Maximum intensity reached in thecourse of multiple filamentation in 1 atm air (a) or argon(b). In air (panel a) multiple filaments were produced afterfocusing of 47 fs pulse with the lens, Rf = 20 cm. In argon(panel b) multiple filaments were produced after focusingof 100 fs pulse with several lenses: Rf = 20 cm (triangles),60 cm (circles) and 100 cm (squares). In both cases (a, b),the pulse energy was varied up to 70 mJ.

0

LASER PHYSICS Vol. 19 No. 8 2009

CAN WE REACH VERY HIGH INTENSITY IN AIR 1789

100

75

50

25

0

Inte

nsi

ty,

TW

/cm

2 (a)

0.5

0

−0.5

(b)

y, m

m

z = 194 cm

260240220200180160140Propagation distance, cm

0.50–0.5x, mm

(c)

56 TW/cm2

61 TW/cm2

66 TW/cm2

55 TW/cm2

Fig. 8. Multiple filamentation of 170 fs (FWHM), 17 mJ, 800 nm pulse in air. Beam diameter is 2.5 mm (FWHM). The ratioPpeak/Pcr ≈ 20. (a) Intensity in the local electron density maxima developed by a certain propagation distance. The dashed lineshows the value of the clamped intensity Ic (the global maximum over all the filaments). (b) Plasma channels formed by the endof the pulse and shown in (y, z) projection down to the level 10–4 of atmospheric density. (c) Plasma cross section formed by theend of the pulse at z = 194 cm and indicated by the solid vertical line on panel (b). For each local electron density maximum weindicated the corresponding maximum of the local intensity.

1.0

0.5

0

−0.5

−1.0

y, mm

(a)

80

60

40

20

0

One aperture

Two aperture

Maximum intensity, TW/cm2

(b)

1.5 × 1017

1.0 × 1017

0.5 × 1016

0

One aperture

Two aperture

40302010Propagation distance, cm

(c)Peak electron density, cm−3

Fig. 9. Two�filament merging by means of the mask with two apertures. Pulse duration 100 fs (FWHM), aperture diameter0.85 mm, partial energy transmitted through one aperture 0.85 mJ. (a) Electron density distribution Ne(x = 0, y, z) at the end ofthe pulse along the propagation direction. The black solid contour shows 10–4 level of atmospheric density. Maximum intensity(b) and electron density (c) in the case of the two filament merging (squares) and in the case of a single filament produced by oneaperture with the same diameter of 0.85 mm centered at (x = 0, y = 0) (solid curve).

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LASER PHYSICS Vol. 19 No. 8 2009

KOSAREVA et al.

density increase is in the enlarged duration of thepulse, accompanying filament merging.

6. ESTIMATION OF THE CLAMPED INTENSITY AT SUBPETAWATT PULSE PEAK

POWER

In the sub�petawatt experiment performed at theShanghai institute of Optics and Fine Mechanics [39],laser pulses with a beam diameter of about 150 mmwere focused with a lens (Rf = 9.5 m) into a 25 m longcorridor. After passing through the lens, the pulseswere reflected with two mirrors out of the room of thelaser system. The propagation took place along thecorridor and induced multiple filamentation over adistance of more than 20 m. The backscattered nitro�gen fluorescence signal was collected with a fused sil�ica lens (Rf = 10 cm) and detected by a photomulti�plier tube. It was filtered by placing an UG 11 filter(band pass: 200–400 nm) and an 800 nm high�reflec�tivity mirror before the photomultiplier tube. As com�parison, two laser energies, 40 mJ/60 fs and 4 J/60 fs,were used. The corresponding peak powers were 0.67and 67 TW, respectively. Therefore, the relative inten�sity ratio between the two situations was interpretedfrom the measured fluorescence signal strength. Fig�ure 10 shows the two fluorescence signals induced bythe 4 J/60 fs Fig. 10 (grey line) and 40 mJ/60 fs pulsesFig. 10 (black line).

For the estimate of the intensity ratio reached in 4J and 40 mJ pulses, we assume that the backscatterednitrogen fluorescence signal SBSF is proportional to the

product of the plasma density produced in a single fil�ament and the number of filaments nf in the interac�tion volume “seen” by the collection system:

(14)

where N0 is the number of neutrals before the pulse islaunched into air and the value of η is determined bythe parameters of the collection and detection system.To make the rough estimate possible, we approxi�mated the ionization rate by the power law using theprobability for the oxygen molecule (the 8th order ofthe multiphoton process for λ = 800 nm laser wave�

length). Using the notations , , , ,I4 J, I40 mJ for the fluorescence signal, the number of fil�aments and the clamped intensity obtained in the sub�petawatt and terawatt cases, respectively, we can derivefrom the Eq. (13) that

(15)

In the Eq. (14) we used that the parameters of the col�lection system are the same for both cases of 4 J and40 mJ initial pulse energy. Since from the experimentit is clearly seen by the naked eye that the number of

multiple filaments produced by 4 J pulse is essen�

tially larger than , we can estimate the upperboundary of the intensity ratio as

. (16)

Inserting the peak voltages of the fluorescence signal0.49 and 0.1 V for the 4 J/60 fs (Fig. 10, B') and40 mJ/60 fs (Fig. 10, B) pulses into the Eq. (15), we get

. (17)

Therefore, even though the pulse energy was increasedby about 100 times, the clamped intensity onlyincreased by less than a factor of 1.22, which is withinthe 30% of intensity variation in the multiple filamentsshown in Fig. 8a.

7. CONCLUSIONS

We have shown that the optical�field�induced ion�ization provides us with the intensity limitation in thecourse of femtosecond pulse filamentation in atmo�spheric density gases. The nature of this limitation is inthe interaction of the laser field with a single mole�cule/atom. The type of the molecule/atom predeter�mines the particular laser intensity causing the ioniza�tion probability sufficient to produce enough electronsto stop the intensity growth. However, sufficient gas

SBSF nfηNe∼ nfηN0I8,=

SBSF4 J

SBSF40 mJ

nf4 J

nf40 mJ

SBSF4 J

SBSF40 mJ

�����������nf

4 J

nf40 mJ

����������I4 J

I40 mJ

���������⎝ ⎠⎛ ⎞

8

.∼

nf4 J

nf40 mJ

I4 J

I40 mJ

���������SBSF

4 J

SBSF40 mJ

����������⎝ ⎠⎜ ⎟⎛ ⎞

1/8

≤

I4 J

I40 mJ

��������� 0.490.1

��������⎝ ⎠⎛ ⎞

1/8

1.22≈≤

151050–5Distance, m

0.5

0

0.4

0.3

0.2

0.1

–0.1

AB

A'

Signal, arb. units

40 mJ/60 fs laser4J/60 fs laser

B'

Fig. 10. The backscattered nitrogen fluorescence signaldetected by a photomultiplier tube in the petawatt experi�ment performed at Shanghai institute of Optics and FineMechanics [39]. The peaks A and A' come from the scat�tering from the last mirror reflecting the beam out of theroom to the corridor. The peaks B and B' arise from thebackscattered nitrogen fluorescence irradiated by the fila�ment in the corridor.

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CAN WE REACH VERY HIGH INTENSITY IN AIR 1791

pressure is needed to convert a single molecule/atom–laser field interaction into the propagation effect.There is no intensity clamping phenomenon invacuum.

As soon as the electron density is enough to affectback on the pulse propagation, the pulse peak intensitychanges weakly along the propagation direction. Forall the propagation conditions studied (namely, singlefilament regime with and without external focusing,multiple filamentation with subterawatt to subpeta�watt pulse peak power) the variation of the local inten�sity maxima is within 30% of the highest intensityvalue, reached in the filament. The remarkable shot�to�shot stability of the clamped intensity (better than1% of the maximum value) is revealed both experi�mentally and numerically in a single filament regimein air.

The notable attribute of the intensity clampingphenomenon in the presence of external focusing isthe widening of the transverse area, where the clampedintensity is reached. As a result of this widening theexperimentally measured electron density becomeslarger in the volume “seen” by the registration device.The average clamped intensity deduced from the mea�sured electron density becomes larger. At the sametime, the local maximum intensity remains within thelevel of variation (30%) reached without externalfocusing.

Merging multiple filaments by means of transmis�sion through a mask and subsequent focusing with thepurpose of intensity enhancement through the lightfield interference led to just a 30% enhancement ascompared with a single filament case. The enhance�ment by a factor of two can be obtained in the peakelectron density. The obtained enhancement is due toboth the maximum intensity increase and the pulseduration increase at the position of the filamentmerging.

Finally, using 4 J in a 60 fs, 800 nm laser pulse gen�erated by the petawatt laser system at Shanghai insti�tute of Optics and Fine Mechanics [42], the authors of[39] found out that the maximum intensity obtained ina 4 J pulse is at maximum by a factor 1.22 higher thanthe maximum intensity obtained in a sub terawattregime with 40 mJ, in a 60 fs, 800 nm laser pulse.

Thus, we can conclude that the same gaseousmedium, which allows us to observe nice femtosecondfilamentary structures up to several hundred meterslong, forbids the enhancement of the maximum inten�sity in the channels above 30% of the clamped inten�sity level of a specific gas or air.

ACKNOWLEDGMENTS

O.G. Kosareva, N.A. Panov, V.P. Kandidovacknowledge the support of the Russian Fund forBasic Research, grant N09�02�01200�a. W. Liuacknowledges the support of the 973 Program (grantno. 2007CB310403), National Natural Science Foun�

dation of China (grants no. 10804056), NCET,SRFDP, Fok Ying Tong Education foundation theopen fund of the State Key Laboratory of High FieldLaser Physics (SIOM), and SRF for ROCS, SEM.The experimental work in Laval University was sup�ported in part by Natural Sciences and EngineeringResearch Council, Defence Research and Develop�ment Canada–Valcartier, CIPI, Canada Foundationfor Innovation, FQRNT, and Canada ResearchChairs. The experimental work in the Shanghai Insti�tute of Optics and Fine mechanics was supported byNational Basic Research Program of China (grant no.2006CB806000), Chinese Academy of Sciences, Chi�nese NSFC (grants nos. 10523003 and 10674145) andShanghai Commission of Science and Technology(grant no. 06DZ22015).

REFERENCES

1. S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Théberge,N. Aközbek, A. Becker, V. P. Kandidov, O. G. Kosareva,and H. Schroeder, Can. J. Phys. 83, 863 (2005).

2. A. Couairon and A. Mysyrowicz, Phys. Rep. 441, 47(2007).

3. L. Berg’e, S. Skupin, R. Nuter, J. Kasparian, andJ.�P. Wolf, Rep. Prog. Phys. 70, 1633 (2007).

4. J. Kasparian and J. P. Wolf, Opt. Express 16, 466(2008).

5. E. Yablonovitch and N. Bloembergen, Phys. Rev. Lett.29, 907 (1972).

6. G. S. Voronov and N. B. Delone, Zh. Eksp. Teor. Fiz.50, 78 (1966) [Sov. Phys. JETP 23, 54 (1966)].

7. S. L. Chin, F. Yergeau, and P. Lavigne, J. Phys. B 18,L213 (1985).

8. G. Méchain, C. D’Amico, Y.�B. André, S. Tzortzakis,M. Franco, B. Prade, A. Mysyrowicz, A. Couairon,E. Salmon, and R. Sauerbrey, “Range of Plasma Fila�ments Created in Air by a Multi�Terawatt FemtosecondLaser,” Opt. Comm. 247, 171–180 (2005).

9. J. Kasparian, R. Sauerbrey, and S. L. Chin, Appl. Phys.B 71, 877–879 (2000).

10. A. Talebpour, J. Yang, and S. L. Chin, Opt. Comm.163, 29 (1999).

11. H. Schillinger and R. Sauerbrey, Appl. Phys. B 68, 753(1999).

12. S. Tzortzakis, B. Prade, M. Franco, and A. Mysyrow�icz, Opt. Commun. 181, 123 (2000).

13. A. Becker, N. Aközbek, K. Vijayalakshmi, E. Oral,C. M. Bowden, and S. L. Chin, Appl. Phys. B 73, 287(2001).

14. W. Liu, S. Petit, A. Becker, N. Aközbek, C. M. Bowden,and S. L. Chin, Opt. Commun. 202, 189 (2002).

15. N. Akozbek, A. Iwasaki, A. Becker, M. Scalora,S. L. Chin, and C. M. Bowden, Phys. Rev. Lett. 89,143901 (2002).

16. L. Bergé, S. Skupin, G. Méjean, J. Kasparian, J. Yu,S. Frey, E. Salmon, and J. P. Wolf, Phys. Rev. E 71,016602 (2005).

17. F. Theberge, J. Filion, N. Akozbek, Y. Chen, A. Becker,and S. L. Chin, Appl. Phys. B 87, 207 (2007).

1792

LASER PHYSICS Vol. 19 No. 8 2009

KOSAREVA et al.

18. C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich,A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller,Appl. Phys. B 79, 673 (2004).

19. A. Guandalini, P. Eckle, M. Anscombe, P. Schlup,J. Biegert, and U. Keller, J. Phys. B: At. Mol. Opt.Phys. 39, S257 (2006).

20. G. Stibenz, N. Zhavoronkov, and G. Steinmeyer, Opt.Lett. 31, 274 (2006).

21. A. Couairon, J. Biegert, C. P. Hauri, W. Kornelis,F. W. Helbing, U. Keller, and A. Mysyrowicz, J. Mod.Opt. 53, 75 (2006).

22. S. Skupin, G. Stibenz, L. Bergé, F. Lederer, T. Sokollik,M. Schnürer, N. Zhavoronkov, and G. Steinmeyer,Phys. Rev. E 74, 056604 (2006).

23. O. G. Kosareva, N. A. Panov, D. S. Uryupina,M. V. Kurilova, A. V. Mazhorova, A. B. Savel’ev,R. V. Volkov, V. P. Kandidov, and S. L. Chin, Appl.Phys. B 91, 35 (2008).

24. F. Théberge, N. Akozbek, W. Liu, A. Becker, andS. L. Chin, Phys. Rev. Lett. 97, 023904 (2006).

25. F. Théberge, W. Liu, P. Tr. Simard, A. Becker, andS. L. Chin, Phys. Rev. E 74, 036406 (2006).

26. T. Brabec and F. Krausz, Phys. Rev. Lett. 78, 3282(1997).

27. N. Akozbek, M. Scalora, M. C. Bowden, and S. L. Chin,Opt. Commun. 191, 353 (2001).

28. E. T. J. Nibbering, G. Grillon, M. A. Franco, B. S. Pra�de, and A. Mysyrowicz, J. Opt. Soc. Am. B 14, 650(1997).

29. M. Mlejnek, E. M. Wright, and J. V. Moloney, Opt.Lett. 23, 382 (1998).

30. J. H. Marburger, Prog. Quantum Electron. 4, 35 (1975)31. W. Liu and S. L. Chin, Opt. Express 13, 5750 (2005).32. P. Simard and S. L Chin (in preparation).33. M. V. Ammosov, N. B. Delone, and V. P. Krainov, Zh.

Eksp. Teor. Fiz. 91, 2008 (1986) [Sov. Phys. JETP 64,1191 (1986)].

34. N. B. Delone and V. P. Krainov, Nonlinear Ionization ofAtoms by Laser Radiation (Fizmatlit, Moscow, 2001) [inRussian].

35. A. M. Perelomov, V. S. Popov, and M. V. Terent’ev, Sov.Phys. JETP 23, 924 (1966).

36. Y. Chen, F. Théberge, O. Kosareva, N. Panov,V. P. Kandidov, and S. L. Chin, Opt. Lett. 32, 3477(2007).

37. S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin,Optics of Femtosecond Laser Pulses (Amer. Inst. Phys.,New York, 1992).

38. S. Xu, Y. Zhang, W. Liu, and S. L. Chin (2009, in press).39. J. Bernhardt, Z. Ji, M. Sharifi, W. Liu, R. Li, Z. Xu,

J. Liu, Z. Wang, J. Ju, X. Lu, Y. Jiang, Y. Leng,X. Liang, O. Kosareva, and S. L. Chin, Appl. Phys. B(2009, in press).

40. A. Talebpour, C. Y. Chien, and S. L. Chin, J. Phys. B:At. Mol. Opt. Phys. 29, 5725–5733 (1996).

41. S. F. J. Larochelle, A. Talebpour, and S. L. Chin,J. Phys. B: At. Mol. Opt. Phys. 31, 1215 (1998).

42. X. Y. Liang, Y. X. Leng, C. Wang, C. Li, L. H. Lin,B. Z. Zhao, Y. H. Jiang, X. M. Lu, M. Y. Hu, C. Zhang,H. Lu, D. Yin, Y. Jiang, X. Lu, H. Wei, J. Zhu, R. Li,and Z. Xu, Opt. Express 15, 15335 (2007).