Robert W. Boyd, Svetlana G. Lukishova,Y. R. Shen (Eds.)

Self-focusing: Pastand PresentFundamentals and Prospects

~ Springer

Robert W. BoydUniversity of RochesterRochester, NYboyd@optics.rochester.edu

Svetlana G. LukishovaUniversity of RochesterRochester, NYsluk@lle.rochester.edu

University of CaliforniaBerkeley, CAshenyr@physics.berkeley.edu

ISSN: 0303-4216ISBN: 978-0-387-32147-9DOl 10.10071978-0-387-34727-1

© Springer Science+Business Media, LLC 2009All rights reserved. This work may not be translated or copied in whole or in part without the writtenpermission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connectionwith any form of information storage and retrieval, electronic adaptation, computer software, or bysimilar or dissimilar methodology now known or hereafter developed is forbidden.The use in this publication of trade names, trademarks, service marks, and similar terms, even if theyare not identified as such, is not to be taken as an expression of opinion as to whether or not they aresubject to proprietary rights.

____ 1_

mailto:boyd@optics.rochester.edumailto:sluk@lle.rochester.edumailto:shenyr@physics.berkeley.edu

9. Self-focusing, Conical Emission, and Other Self-action Effectsin Atomic Vapors .Petros Zerom and Robert W. Boyd

10. Periodic Filamentation and Supercontinuum Interference .Xiaohui Ni and R.R. Alfano

II. Reprints of Papers from the past. . . . . . . . . . . . . . . . . . . . . . . . . . . 265

(I) G .A. Askar'yan: Effects of the Gradient of a Strong ElectromagneticBeam on Electrons and Atoms, Sov. Phys. JETP 15, 1088-1090 (1962)-First paper on self-focusing and self-trapping.(2) V.I. Talanov: On Self-focusing of Electromagnetic Waves in Nonlinearmedia,Izv. Vuzov, Radiophysica, 7, 564-565 (1964) - First time translationfrom Russian.(3) M. Hercher: Laser-induced Damage in Transparent Media. Presents thefirst laboratory observation of self-focusing. This paper ispublished here for thefirst .time in its entirety. Previously, only the abstract had been published in J.Opt. Soc. Am., 54, 563 (1964).

Part II Self-focusing in the Present

12. Self-focusing and Filamentation of Femtosecond Pulses in Airand Condensed Matter: Simulations and Experiments .A. Couairon and A. Mysyrowicz

13. Self-organized Propagation of Femtosecond LaserFilamentation in Air 323Jie Zhang, Zuoqiang Hao, Tingting Xi, Xin Lu, Zhe Zhang,Hui Yang, Zhan Jin, Zhaohua Wang, and Zhiyi Wei

14. The Physics of Intense Femtosecond Laser Filamentation . . . . . . . 349See Leang Chin, Weiwei Liu, Olga G. Kosareva, andValerii P. Kandidov

15. Self-focusing and Filamentation of Powerful FemtosecondLaser Pulses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371v.P. Kandidov, A.E. Dormidonov O.G. Kosareva, S.L. Chin, andW. Liu

16. Spatial and Temporal Dynamics of Collapsing UltrashortLaser Pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399Alexander L. Gaeta

17. Some Modern Aspects of Self-focusing Theory. . . . . . . . . . . . . . . . 413Gadi FiNch

18. X-Waves in Self-Focusing of Ultra-Short Pulses. . . . . . . . . . . . . . . 439Claudio Cont·i, Paolo Di Trapani, Stefano Trillo

ond and nanose-135 (2003).'ld laser-written

Chapter 15Self-focusing and Filamentation of PowerfulFemtosecond Laser Pulses

Abstract The physical picture of the phenomenon of filamentation of apowerful femtosecond laser pulse in the bulk of a transparent dielectric isconsidered on the basis of the experimental results at Laval University (Quebec,Canada), and the computer simulations performed at M.V. LomonosovMoscow State University (Russia).

In this chapter the dynamic moving-focus model is discussed. Thequasi-stationary model is offered for the analysis of filament origin. The quasi-stationary estimation of critical power for self-focusing is received with consid-eration of the delayed nonlinear response in gases. Generalization of the knownMarburger formula is given to femtosecond pulses with an elliptic intensitydistribution and to the initially chirped pulses. The influence of turbulenceand scattering particles in the atmosphere on chaotic pulse filamentation isconsidered. Spatial regularization of filament by means of the regular intensityand phase perturbations introduced into the transverse section of the pulse isinvestigated. Chirped pulse propagation is studied in a single filament regime.The scenario of filament competition is considered on the basis of three-dim en-sional nonstationary propagation model.

The phenomenon of filamentation of laser radiation, which has been knownsince the 1960s, has again attracted the attention of scientists. Tightening of apowerful laser pulse in a filament that causes concentration of energy andintensification of nonlinear optical processes is of interest from the points ofview of both fundamental science and its applications.

A.E. Dormidonov (1:Bl)International Laser Center, Physics Department, MV. Lomonosov Moscow StateUniversity, Moscow 119992, Russiae-mail: adorm@list.ru

R.W. Boyd et a!. (eds.), Selj~focusing: Past and Present,Topics in Applied Physics 114, DOl 10.1007/978-0-387-34727-1_15,© Springer Science+Business Media, LLC 2009

mailto:adorm@list.ru

Fig. 15.1 The firstself-focusing experimentwith a 20-MW pulse focusedinto a cell with organicliquids. (a) Side view of onechannel in cyclohexane and(b) of two glowing filamentsin orthoxylol [I].

In 1965, post-graduate and undergraduate M.V. Lomonosov Moscow StateUniversity students N. Pilipetsky and A. Rustamov were the first to demonstratefilament images in the course of laser beam focusing [1]. A glowing filament,which contained about 1% of radiation energy, appeared in the laser pulsewith the power of 20 MW focused into a cell with organic liquids (Fig. 15.1).In some cases two or three filaments appeared. The first report about self-focusing observation was made in [2]. Earlier, the self-focusing effect hadbeen predicted in [3]. In the same paper a waveguide mode of the electromag-netic beam propagation had been suggested. It had been shown that self-focusing was possible in the beams only if their power exceeded the so-called"critical" power for self-focusing in the medium [4]. For the first time the self-focusing in air of focused and collimated beams was registered in [5] and [6],respecti vely.

At high density oflaser radiation power a small-scale self-focusing develops,at which the beam breaks up as a result of modulational instability of an intenselight field in a medium with the Kerr nonlinearity. The theory of spatio-temporal instability of intense light field in a cubic medium was developed in[7] and then was confirmed experimentally in [8]. Small-scale self-focusing was aserious barrier on the way of creation of powerful solid-state lasers [9]. The peakpower in nonlinear foci reached the damage threshold of a material and led tothe destruction of the optical elements [10]. To overcome this obstacle, large-beam-diameter laser systems began to be used in which the peak intensity incascades of amplification was decreased by the scaling of the transverse size of abeam. Now this principle, being transferred on the temporal scale of radiation[11], allows experimenters to create powerful femtosecond laser systems.

The development of powerful femtosecond lasers made it possible to receivethe filamentation of pulses in gaseous media, particularly in air at atmosphericpressure. In the first experiments [12-14] with pulses centered at 0.8 11m,peakpower of 5~50 GW, duration of 150-230 fs, lO-m filaments were observed. In athin filament of about IOOl1m in diameter, 8-10% of the pulse energy waslocalized. The parameters of a filament are stable enough along its length anddo not depend on the initial pulse energy and duration.

Color rings of the conical emission were formed around a filament. It isessential, that the filamentation take place without geometrical focusing of thepulse. The main parameters for the formation of filaments are femtosecondduration and central wavelength of the laser pulse. In the 1970 s-1980 s,

powerfllasers, ,delocalopticalold low

Thepaniedconicalsystem~of the;methocThe ph,tries, aJ

In tlfemtostwell as

ThefOJcan beexceed~appro amediurdefocudynamleads tfilameJa filamself-ch;

Thebasis 0Revie\'ACCOfl

tempo]action

I In air

IThr"" I

••IIIlSCOW State;:monstrateg filament,laser pulse(Fig. 15.1).about self-effect hadlectromag-t that self-e so-calledne the self-[5] and [6],

~develops,an intenseof spatio-veloped inIsing was a. The peakand led toIde, large-atensity inse size ofa.' radiation~ms.to receivemospheric:~m, peak~rved. In a:lergy wasength and

nent. It is;ing of theltosecond) s-1980 s,

powerful laser systems of atmospheric optics were created on the basis of CO2lasers, which generated millisecond pulses. When such pulses propagate in air,delocalization of their power occurs, which is caused by thermal self-influence,optical breakdown, and other nonlinear-optical effects with the power thresh-old lower than that of self-focusing [15].

The phenomenon of femtosecond filamentation of a laser pulse is accom-panied by the formation of plasma channels, generation of white light andconical emission [16-19]. On the basis of these properties of filamentationsystems for broadband laser probing, fluorescent and emission spectroscopyof the atmosphere, and remote control of the high-voltage discharge in air,methods of creation of microoptics devices are now under development [20-24].The phenomenon of femtosecond filamentation in air, liquids and solid dielec-trics, and its possible applications, are discussed in detail in [25].

In the present chapter the phenomenon of filamentation of a high-powerfemtosecond laser pulse constructed on the basis of the moving-focus model, aswell as the quasi-stationary model, are both represented.

15.2 Femtosecond Filamentation of a Laser Pulseand the Moving-Focus Model

15.2.1 The Dynamic Moving-Focus Model

The formation of filament during the propagation of a femtosecond laser pulsecan be explained in! the following way. In a pulse with peak power, whichexceeds the critical power for self-focusing, the intensity increases as itapproaches the nonlinear focus. When the photoionization threshold of themedium is achieved, the laser-produced plasma is generated, in which thedefocusing limits subsequent growth of the intensity in nonlinear focus.' Thedynamic balance between the Kerr self-focusing and the plasma defocusingleads to localization of energy and stabilization of parameters in a lengthyfilament. Nevertheless, the balance of the Kerr and the plasma nonlinearity ina filament does not mean sensu stricto the formation of a waveguide mode andself-channeling of a pulse in the medium.

The physical picture of filamentation can be visually demonstrated on thebasis of the moving-focus model [29]. First, moving foci were registered in [5].Review [30] describes other experimental confirmations of the modeLAccording to the model a laser pulse is represented by a sequence of thintemporal slices, which are transformed in the course of nonlinear-optical inter-action with the medium (Fig. 15.2).

, In air the photoionization threshold IThr = IOl3 + IO'4W jcm-2 [26], in water and methanolIThr"" IOl3Wjcm-2 [27,28].

Fig. 15.2 The moving-focusmodel. Three temporal slicesof the pulse focused atdifferent distances z. Thebold curve indicates thecontinuous succession offoci forming the filament

The moving-focus model explaining beam self-focusing in the medium withinstantaneous Kerr response and the two-photon absorption is modified in [14]and [31] for the analysis of dynamic filamentation of a femtosecond pulse. As aresult, the model of moving foci, which is used for the filamentation, can benaturally called the dynamic model.

The dynamic moving-focus model reflects the influence of one temporal sliceof a pulse on subsequent slices, which originates from nonlinearity of laserplasma accumulated with time. According to the dynamic model [14,31] thefilamentation of a high-power femtosecond laser pulse can be presented asfollows. The temporal slice with the peak power is focused at the minimaldistance, defining the starting point of the filament. With increasing intensityin this slice up to the threshold of multiphoton ionization, the laser plasma isproduced. Defocusing in plasma stops the subsequent growth of intensity.

In other temporal slices of the pulse the power decreases continuously at theirdisplacement from the peak slice. Therefore, slices in the front of the pulse arefocused at the distance, which increases with their removal from the slice with thepeak power. The continuous succession of nonlinear foci, which arises in a pulsefrom the slice with the peak power and then, consequently, in slices of pulse up tothe slice with the critical power for self-focusing, represents a lengthy filament.The electron generation caused by photoionization of a medium in nonlinear fociforms a plasma channel, where defocusing occurs of all subsequent temporalslices of the pulse (Fig. 15.3). Therefore, in the case of femtosecond filamenta-tion, unlike the quasi-stationary self-focusing and filamentation of a long pulse,the back of a femtosecond pulse is exposed to the aberrational defocusing and asa result the intensity ring distribution is formed around the filament [18,32,33].

The ring structure was first observed in [34] in the course of self-focusing ofaruby laser pulse with power up to 100 kW into a cell with CS2. The appearance

········tt···.:.:. '. ...t.I..'·....t>. '-'/'?'/ /,(

Fig. 15.3 The dynamic moving-focus model. Temporal slices at the back of the femtosecondpulse are exposed to the aberrational defocusing in the laser plasma. The tone images ofspatio-temporal intensity distributions are presented

of ringspoint irsaturatisaturati,axis ane

Theregistenunder estructurfilamenformati,simple;shown twith a rwhich i~

In thmodelsenon ojfocusedwas ob~opinionmodel [mode. 1numericIt is shlpacket.explaini

Fig. 15.4filaments(b) Inten,

of rings is explained by interference between a spherical wave diffracting from apoint in the liquid and the untrapped beam. The beam self-focusing withsaturating nonlinearity was numerically considered in [35]. In the course ofsaturation the decreasing of the Kerr-lens focal power takes place at the beamaxis and gives rise to the rings.

The ring formation, surrounding the femtosecond filament in air, wasregistered in [36] by measurements of silicate sample thickness at ablationunder exposure to a focused femtosecond 85-mJ laser pulse. Typical ringstructure, received experimentally and numerically in the course of ",90 mfilamentation of a collimated pulse in air, is given in Fig. 15.4. Physically theformation of intensity rings in the pulse transverse section is explained by asimple model [37], evolving the assumptions stated in [34,35]. In [37] it isshown that rings are the result of superposition of the background light fieldwith a plane wavefront and divergent field of the filament, the wavefront ofwhich is the surface of a cone.

In the journal Optics Letters there was a terminological discussion aboutmodels of moving foci and self-channeling for interpretation of the phenom-enon of femtosecond filamentation. In an experiment [38] with laser pulsesfocused in a cell with gases at various pressures, the filamentation of a pulsewas observed behind the geometrical focus of a lens. The authors are of theopinion that this result is unexplained from the point of view of the moving focimodel [14], and regard it as a confirmation of the formation of a self-guidedmode. The opposite point of view is stated in [39]on the basis of laboratory andnumerical experiments with the femtosecond pulse focused in a cell with water.It is shown, that a light filament does not behave as a self-channeled wavepacket. The authors of [32]used the term "dynamic spatial replenishment" forexplaining long-range filament formation by femtosecond laser pulses.

edium withified in [14]pulse. As aion, can be

11poralsliceity of laser[14,31] the-esented asIe minimal19 intensityr plasma isensity.lslyat theire pulse areicewith thes in a pulsepulse up toy filament.llinear focit temporalfilamenta-long pulse,,ing and as[18,32,33].:using ofappearance

~ 0aD

-1

z-2

-2 -1 0 2 -2 -1 0 2x/aD x/aD

emtoseconde images of

Fig. 15.4 Typical pictures of the interference of ring structures produced by two developedfilaments at z = 87 m in a l4-mJ pulse. (a) Fluence distribution obtained from the experiment.(b) Intensity distribution obtained in the simulations is plotted in the relative units [37,44]

In pulses with peak power of 6-10 times as much as the critical power forself-focusing in the medium, refocusing arises and becomes apparent throughthe formation of fluence maxima along the femtosecond filament. This effect,which first was found out for femtosecond filamentation in [14],is theoreticallyinvestigated in [15,31-33]. Later, the nonmonotonic character of intensitychange in the filament was confirmed in [40]by measurements of a photoemis-sion signal of molecules and ions of nitrogen. The influence of air pressure onthe refocusing of a focused pulse was considered in [41].

Physically, refocusing is explained by recurring self-focusing of pulse tem-poral slices, which were defocused in the laser plasma. An increase in intensityat the beam axis toward the back of the pulse - simultaneously with continuedself-focusing at the pulse front - leads to the fluence growth registered inexperiments. This is schematically explained in Fig. 15.3. Computer experi-ments [31]visually illustrate how divergent rings of aberrational defocusing atthe back of the pulse tighten again to the beam axis (Fig. 15.5).

At the beginning of refocusing the dispersion of the pulse plays an essentialrole. Strong material dispersion of the medium can result in multiple refocusing,at which a filament breaks up into sequence of "hot" spots with high localiza-tion of energy [29].With an increase of group velocity dispersion the number ofhot spots along the filament decreases, and the distance between them reduces.

The dynamic picture of refocusing of a femtosecond pulse has some differentaspects in comparison with the multi-focus model proposed in [42]on the basisof numerical investigation of stationary self-focusing in a medium with two-photon absorption. According to this model, to the first nonlinear focus thepower flows from a circular area on the beam axis. Then this power comes outfrom the "game" due to absorption in a medium and diffraction. To the secondfocus the power flows from the ring enveloping the circular area; to thethird focus the power flows from the following ring; and so on [43].At femto-second filamentation in transparent dielectric energy losses are insignificant,and the collapse of the nonlinear focus stops due to a strong nonlinear

Fig. 15.5 Spatio-temporal evolution of the intensity distribution of the pulse in the filament:z = 27 m (a); 33 m, (b); 40 m (c). The contour interval is 0.25 x 1013W /cm-2, ao = 3.5 mm,Ppeak = SPcr [31]

refracti(Formatitaneous

A narrobackgroreservoiienergy,provide~continu<its para,

Durilverse setWhen tJplasmaparaxialdistancearea, incthe perilengthyit contin

Thedis confirwas cor

Fig. 15.6T = -20f,0.5 and 3

• duration'transverSt

refraction of the pulse in the laser plasma, instead of absorption in the medium.Formation of the maxima of filament fluence at refocusing occurs from simul-taneous self-focusing at different temporal slices of the pulse.power for

lt throughfhis effect,eoreticallyf intensitylhotoemis-ressure on

pulse tem-n intensitycontinued~istered inter experi-'ocusing at

A narrow filament with a high concentration of light field is surrounded by thebackground reservoir of energy that provides its existence [32]. From thisreservoir the sequence of nonlinear foci in the front of the pulse is supplied byenergy, forming a filament. In the pulse trailing part, the energy in the ringsprovides a subsequent refocusing. In the course of filamentation there is acontinuous exchange of energy in the transverse section of the pulse betweenits paraxial area and its periphery [44].

During nonlinear focus formation, power from the periphery of the trans-verse section flows toward the axis in a temporal slice of a pulse (Fig. 15.6).When the intensity of focus reaches the threshold of photoionization, laserplasma is generated. Defocusing in plasma causes a power flow from theparaxial area to the periphery of transverse section of temporal slice. At thedistance z where refocusing arises, the power flows down again to the paraxialarea, increasing intensity on the axis. After refocusing the power again flows tothe periphery of the transverse section of the pulse temporal slice. Thus, alengthy filament exists due to the background reservoir of energy with whichit continuously exchanges.

The determining role of the energy reservoir in formation oflengthy filamentis confirmed visually with laboratory experiments. In [39]a section of filamentwas completely blocked by a small circular screen with diameter 55Jlm.

n essentialefocusing,h localiza-number ofn reduces.e differentn the basiswith two-focus the

com~s out.he second~a; to theAt femto-ignificant,nonlinear 2 \

-----\ 1\,I \>.. /-

/-\\ II ~ ••••••/ __--

3--- ---

Fig. 15.6 Dependences of power in different spatial regions of the pulse temporal sliceT = -20fs on the distance z: in the axial region of radius 0.5 mm (I), in a ring with radii of0.5 and 3.5mm (2), and in a ring with radii on.5 and 8mm (3). Pulse energy W= 10mJ,duration TO = l40fs and ao = 3.5 mm. The insert shows the positions of the regions in a pulsetransverse section [44]

he filament:•= 3.5mm,

Nevertheless, the filament was interrupted only a small distance behind thescreen, but then reappeared again on all its length as in the absence of the screen.

In one experiment [45] thin aluminum foil screen was located on the trace ofthe filament. The intensive field of the filament burned a hole through the foil.The screen with the hole burnt completely blocked the filament behind thescreen. Energy that transferred from the periphery of the transverse section ofthe pulse was cut off by the screen, and in the absence of a background reservoirof energy the filament was eliminated.

intensi:mediUl

For socompot1nk =is definradiatinonlin,consid.followi

15.3 Quasi-Stationary Model of Filament Origination

15.3.1 Initial Stage of Filamentation

The process of filamentation can be separated into two stages: the initial stage,at which the formation of nonlinear focus occurs, and the dynamic stage, atwhich an aberrational defocusing dominates the self-induced laser plasma. Theinitial stage, i.e., the prefilamentation stage, consists of the pulse transforma-tion, with intensity under the threshold of medium photoionization. At thisstage. plasma is not yet generated, and the light field varies only due to the Kerrself-focusing. For pulses with duration of some tens of femtoseconds, thefilamentation in air and, as a rule, in optical glasses begins on distancesconsiderably smaller than the dispersive length, and the influence of materialdispersion on the origin of filament can be neglected [46,47]. Thus, at the initialstage of filamentation the complex amplitude E(x, y, z, t) of the light field intemporal slice of a pulse submits to the equation:

aE a2E a2E 2k22ik-a=-a 2 +-a 2 +-(~ndIEI2)+~n(x,y,z))E,z x y no

wherecoefficTO> >

It i~the ter,

Th(statiorpartic'expres

where ~nk(IE(x,y, z, t)12) is the nonlinear correction to the refractive index dueto the Kerr effect, and ~n(x,y, z) is the random fluctuations of the refractiveindex in the medium. The electric field envelope E(x, y, z = 0, t) at the lasersystem output is usually represented in the form:

{t2} {x2 + y2}E(x,y,z=O,t)=Eoexp --2 exp ---2- .

2TO 2ao

In the central slice of the pulse t = 0 the power achieved the peak valuePpeak = na5/0, where 10 = f,; IEol2 is the peak intensity.

The solution to problem (IS.!) for the central pulse slice t = 0, which con-tains the peak power, allows us to investigate the prefilamentation, i.e., theorigin of filament due to the Kerr self-focusing. The distance ZjU on which the

Th.slice t

intensity in the central pulse slice achieves the threshold of photoionization of amedium is the distance of the filamentation start.

Ice behind thee of the screen.on the trace ofrough the foil.:nt behind theerse section ofound reservoir For solids and liquids the increment D.nk is caused only by the electronic

component, and the nonlinearity can be considered as an instantaneousD.nk = n2/, where n2 is the nonlinearity coefficient of a medium. In gases, D.nkis defined by the electronic nonlinearity and the stimulated Raman scattering ofradiation on rotational transitions of molecules, which leads to delay of thenonlinear response. For air this delay is Tn! ~ 70fs [48], and it is necessary toconsider it for pulses of femtosecond duration. According to [32] in air thefollowing approximation for increment D.nk can be used;

Ie initial stage,lamic stage, at~r plasma. The~e transforma-~ation. At thislue to the Kerrtoseconds, the; on distancesIce of materiallS, at the initiale light field in

"'n,(tl ~ ~n, (IE(t)I' +1H(t ~ t')IE(t'll'dt) ,where H(I) is the response function, and n2 = (1.5 -;- 5.6) x 1O-19cm2/W is thecoefficient of air nonlinearity, which was established for pulses with durationTO> > Tn! [48].

It is possible to introduce the effective coefficient of nonlinearity niSI (I) forthe temporal slice 1of pulse with the known profile £(1) [49]:

The coefficient of nonlinearity received in this way corresponds to the quasi-stationary estimation in approximation of the given profile of the pulse. Inparticular, for the Gaussian pulse the quasi-stationary model leads to followingexpression for the function TJ( 1 = 0) in the central pulse slice:

:tive index due'the refractiveI) at the laser

0, which con-ation, i.e., the! on which the

The quasi-stationary estimation of the critical power JXf:;1 (I) for the temporalslice 1 of femtosecond pulse in air follows from (15.4, 15.6):

The function 7](t) :s 1 and P'g{(t) ~ Per. For example, for a Gaussian pulse(15.1) with duration TO = 27fs the effective coefficient of nonlinearity for thecentral temporal slice equals 4'{ (t = 0) = 0.56n2 [49]. The quasi-stationarycritical power for self-focusing of the central slice of the pulse JXf:~{(t = 0) is1.8 times as mush as the critical power for self-focusing of a long pulse(JXf:;H(t = 0) = 1.8Pcr).

Direct measurements of critical power for self-focusing of Ti:sapphire laserpulses with energy of 0.42 mJ and the pulse duration from 42 to 800 fs wereexecuted in [50]. Critical power for self-focusing was determined by displace-ment of a focal spot of the focused laser pulse, the duration of which changed bymeans of the initial phase modulation. The experimental results received withpositive and negative initial phase modulation are in agreement with an analy-tical estimation.

The concept of critical power for self-focusing was introduced at the analysis ofthe axially symmetric Gaussian beam [3]. With the change of intensity distribu-tion in a beam the value of critical power varies, reaching a minimum for a beamwhose profile coincides with the Townes mode [51]. For beams without axialsymmetry the critical power for self-focusing increases because of significantoutflow of its power at formation of the axisymmetric Townes mode. Forelliptical beams the critical power Per(ajb) depending on the axial ratio ajb ofinitial intensity distribution was determined by means of numerical simulation[52-54]. As the ratio aj b increases the critical power for self-focusing of ellip-tical beam rises several times (Fig. 15.7).

p';" (a/b)/?:" (a/b= 1)

•2.0 ••Fig. 15.7 Dependences of ••the critical power for self-1.5 ••focusing P'f:' on the axial ••••

ratio a/b of the initial elliptic ••transverse section of a ••••collimated Gaussian beam: 1.0 •••circles are numericalsimulation [54]; the solidcurve is the empirical 0.5 a/bexpression [52] I 2 3 4 5 6 7

TheZfli belthat ttgenerathe puthe ditMarbt

wheresectioI'tempo.criticalpulse \fOffiulfilamelPpeak =distrib

Durin£criticalup intequencemediurwhichtempO!of filanintensilrefracti

ThefO!redistriredistriat the

Ltthe analysis oftensity distribu-mum for a beam1S without axial~e of significantIlles mode. Forx:ial ratio a/ b ofrical simulation)cusing of ellip-

The quasi-stationary model allows one to analytically estimate the distancezfil before the start of filamentation. From the computer simulation it followsthat the Marburger formula [55] for the distance of the beam self-focusing isgeneralized on the quasi-stationary model of the initial stage offilamentation ofthe pulse with elliptical intensity distribution. According to this generalizationthe distance before the filament start zfll is calculated with the generalizedMarburger formula [55,56]:

LGaussian pulseLlinearity for thequasi-stationarylse P!::! (t = 0) isof a long pulse

fi:sapphire laser-2 to 800 fs wereoed by displace-hich changed byIts received withIt with an analy-

0.367kabzfll = 1/2 .

{ [(I";:I~:~~a/b))1/2 -0.852] 2 -0.0219}

where a, b are the axes of elliptic distribution of intensity in the transversesection of a pulse, Ppeak is the pulse peak power reached in the central (t = 0)temporal slice, P!::! (t = 0, a/ b) is the quasi-stationary estimation (15.7) of thecritical power for self-focusing of the central temporal slice of the femtosecondpulse with elliptic intensity distribution (vide supra Fig. 15.7). Application offormula (15.8) is confirmed with the computer simulation results of pulsefilamentation in air with duration TO = 27 and 140 fs and peak powerPpeak = 10 - 64GW at a various axial ratio a/b of elliptic intensitydistribution [56).

During the propagation of femtosecond pulses, the power of which exceeds thecritical power for self-focusing, multifilamentation occurs as the pulse breaksup into multiple filaments. The break-up of a pulse is the unavoidable conse-quence of transverse modulational instability of an intense light field in amedium with the cubic nonlinearity. The initial stage of multifilamentation atwhich nonlinear foci are formed is defined by small-scale self-focusing intemporal slices of a pulse. The centers of origins of nonlinear foci and, hence,of filaments in the transverse section of the pulse are chaotic perturbations ofintensity which can be caused by quality of output laser beam, fluctuations of arefractive index, and scattering by particles of a medium.

The formation of nonlinear foci at the initial stage of filamentation causes theredistribution of power in the transverse section ofa pulse. The character of thisredistribution depends on pulse energy and geometry of intensity perturbationat the laser system output. In experiments [57] with the pulses centered at

I053 11m,the peak power 30 GW, and duration 500 fs, it was observed that atsmall distances there were 3-4 maxima of energy density formed in the trans-verse section of a beam.

However, later on not all of these maxima were transformed to filaments. In[58] the complex picture of the formation of filaments in a pulse with nonuni-modal distribution of energy density in the transverse section was investigated.It was noted that at the beginning, due to the modulational instability, the initiallaser beam breaks up into two maxima in transverse section, which then merge,forming one filament.

In experiments [49] in air with collimated pulses with duration of 45 fs(FWHM), filaments were formed independently from large-scale perturbationsof the initial profile [Fig. I5.8(a)]. In the numerical simulation of this experi-ment, the quasi-stationary model of the filament origin for the pulse with anintensity profile coincided with the measured flue nee on the laser system output.Intensity distributions in the transverse section of the central slice of the pulsewere received numerically for the distance, where filamentation began, aresimilar to the flue nee registered in the experiment [Fig. 15.8(b)]. At increasingpulse energy up to 40mJ, a bunch containing many filaments is formed. Thesefilaments are irregularly located in the transverse section of the beam and beginat different distances from the output aperture of the laser [Fig. 15.8(c)]. Asfollows from estimations [5], in such a pulse [Fig. I5.8(a)] the intensity pertur-bations, with transverse section size equal to 0.1 em, have the greatest incrementof growth. Thus, with the pulse energy increasing. filaments arise from small-scale perturbations of intensity of the initial profile, which generally varyirregularly from shot-to-shot.

b) 1."..,(x,y,z=87m) F(x,y,z = 87m)

'.''.'l~;I~tTlAJ-2 -I 0 1 2 -2 -I 0 2

Fig. 15.8 (a) Experimentally measured initial fluence distributions F(x,y,:: = 0).(b) Measured fluence F(x,y,::) and computer intensity distributions Icomp(x,y, z) and fluenceF(c) and intensity Icomp(c) profiles. Pulse energy W = 10mJ, distance z = 87 m. (c) FluenceF(x,y,z) at pulse energy of W= 40mJ [49]

Detaintentwo Jother

whersimupulsebard-[Fig.

filarrAt tI-

TIan IT:[Fig.tionpertlfocwincreat bebati<

Fig.maxizm toaxisy

~d that atthe trans-

ments. In;1 nonuni-estigated.the initialen merge,

Detailed research of the formation of several filaments in a pulse with initialintensity perturbations are performed in [59] on the example of superposition oftwo partial coherent beams whose centers are displaced with respect to eachother in the transverse section plane.

1 of 45 fsurbationslis experi-e with anmoutput.. the pulseegan, arencreasinged. Theseand begin;.8(c)]. Asty pertur-ncrement)m small-'ally vary

E( )) E [(y - d/2)2+X2] [ (y + d/2)2+X2]

x,y = oexp - 2 +Eoexp - 2 '2ao 2ao

where d is the distance between the partial beams of radius ao. From computersimulations on the basis of the quasi-stationary model it follows that for thepulse considered there are two critical powers p~~)and p~;), which defineborders of areas of various modes of filamentation depending on distance d[Fig. l5.9(a)]. The area p).~)< Ppeak < p);) corresponds to the mode of the single-filament formation, the area Ppeak > p);) to the mode of two and more filaments.At the peak power Ppeak < p~~)the filamentation is absent.

The distance to the start offilamentation Zjii changes nonmonotonically withan increase of the peak power at the constant distance between partial beams[Fig. l5.9(b)]. Such a dependence Zjii(Ppeak) is explained by a "power" competi-tion between initial perturbations. For the peak power Ppeak > p~~), the initialperturbations at the beginning of the propagation merge into one, and then self-focusing occurs. In this case, the distance to the filament origin decreases withincreasing Ppeak. For Ppeak :s; p~;), initial perturbations increase independentlyat beginning. Then, the extended competition occurs between enhanced pertur-bations, and the power accumulated in them transfers to the beam axis, where

87m)a) b)

I p~~), pi./;IP~:t 'fl,/ka512 I \6 10 II \8 I1

4 I6 I\

4 \2 ',~=2.5ao

p(l) 2 ,

0cr

00 2 4 6 8 10 dlao 0.4 0.6 0.8 1.0 1.2 PIP~~)

r,y, z = 0).and f1uence(c) Fluence

Fig. 15.9 Dependences of the critical powers r,.~) and r,.;) on the distance d between themaxima of perturbations in the initial intensity distribution (a) and dependences of the distance2fil to the filament start on the peak power PP:R"k (b). P/," is the critical power for self-focusing ofaxisymmetric collimated Gaussian beam; Pc; is the critical power for two-foci formation [59]

one nonlinear focus forms. The redistribution of power in the pulse transversesection increases the distance to the filament starts. For P peak > p~;), the dis-tance z/i1(Ppeak) again monotonically decreases with increasing power.

The stochastic character of the multifilamentation, which is caused by modula-tion instability of the pulse with random perturbations of the initial profile,decreases the effectiveness of the femtosecond laser systems, for example, ofatmospheric lidars because of an unstable back-scattering signal [60]. In recentyears, various methods for spatial regularization of a bunch of filaments inhigh-power pulses have been offered. The opportunity for filament regulariza-tion by means of introducing a circular diaphragm, the mask with apertures,and introducing phase astigmatism in a pulse at the laser output are consideredin [61]. It can be noted that a regular bunch of filament whose centers arelocated on a circle was observed in 1973 [62].

The general method for spatial regularization of multifilament at ion consistsin formation in the pulse transverse section of a regular system of perturbationsof the light field, capable of suppressing the influence of random fluctuations onthe origin and formation of filaments. In this case the modulation instability ofthe light field develops on the preset array of perturbations, creating the centersof origin of regular filaments. For the creation of amplitude perturbations in thepulse transverse section for example a wire mesh can be used, and phaseperturbations can be created by lens arrays.

The method of spatial regularization of multiple filaments by means of amesh imposed in a plane of the pulse transverse section is experimentally andnumerically investigated in [63-66]. For the "stochastic" multifilamentationachieved when only the random mask was located in front of the methanolcell entrance window, filaments were arranged chaotically in the pulse trans-verse section. For the "periodic" filamentation, when only the mesh was in frontof the cell, a regular system of filaments was formed. For "regularization" modeboth the mask and the mesh were placed before the cell entrance window, anordered spatial arrangement of filaments was observed, which is in contrast tothe "stochastic" filamentation.

For computer simulation of spatial regularization of multiple filaments thequasi-stationary model (15.1) was used with ~n(x,y,z) = ° [67]. At the "sto-chastic" filamentation, the light field E(x,y, z = 0, r) for the ith laser shot wasrepresented with the additive perturbations ~;(x,y):

where Cj is the normalizing factor. The random function ~(x,y) obeys thenormal distribution law with zero mean value and variance cl-. The spatialcorrelation of the perturbations is given by the Gaussian function.

Theregion,Reor mtradiusgrowthfield Er

Thein thecompalo-20~

Atsfollow:

Frofilameldistanlulariz2.pertur(Fig. Iinstabion sIT.

Fig. 15periodi~=Operiodparam.

lulse transverseP(2l h d'> er, t e IS-

power.

sed by modula-: initial profile,or example, ofI [60]. In recentof filaments inlent regulariza-with apertures,

c are consideredose centers are

ntation consists,f perturbationsfluctuations on)n instability oflting the centersurbations in thesed, and phase

by means of a~rimentally andltifilamentation,f the methanolthe pulse trans-esh was in frontlrization" modenee window, anis in contrast to

,Ie filaments the;7]. At the "sto-II laser shot was

:X,y) obeys the(12. The spatialIon.

The correlation radius Reor of perturbations was chosen so, that in near-axialregion of a pulse the power P Reo, contained in the perturbation with this radiusReor met the condition: PReo, ~ Pf::/(t = 0). In this case, perturbations with theradius Reor are most "dangerous" because they have the greatest increment ofgrowth [5]. At research of the periodic filamentation the amplitude of the lightfield Eper(x,y, z = 0, t) after a mesh equals:

The mesh transmission factor Tarray(x,y) periodically changed with period din the XY plane. The width h of the opaque part of the mesh was smallcompared with the mesh period d, so that the resulting energy loss was within10-20% of the initial pulse energy calculated before the mesh.

At simulation of the "regularization" mode, the light field Ereg was present asfollow:

From the numerical simulation it was learned [67] that at the stochasticfilamentation the average number of filaments monotonously increases withdistance z, as the "periodic" filaments appear in groups. In the mode of reg-ularization there is a significant suppression of the contribution of chaoticperturbations, and filaments origin on average close to the group formation(Fig. 15.10). For the spatial regularization it is necessary that the modulationinstability on the regular perturbations, which are created by a mesh, developedon smaller distance, than on random fluctuations of the light field. It is

Fig. 15.10 Average number of filaments as a function of the distance. Open circles indicateperiodic filamentation; the curves marked by closed triangles indicate the stochastic mode for,; = 0.0 I; the curves marked by open triangles indicate the regularization mode. The meshperiod d = 0.2ao, the ratio Ppeak/ P'J:t = 370, Punit/ P'f:t = 3.7, PRe",/ P'f.:t = 1.5. Laser pulseparameters: A = 800nm, duration 42fs FWHM, energy 130l-d, and radius ao = 2.4mm [67]

established, that the filament regularization by means of a mesh is most effectiveif a mesh cell contains the power Punit equals (3.1 -;- 3.2)Pgf. Therefore, theoptimum period dapt of regularizing mesh follows the estimation:

Fig. 15picturemultifi40-mJ42 fs (f

In the first experiments on filamentation of femtosecond laser pulse in atmo-sphere [68] a focusing lens with a 30-m focal length was used for positioning offilament in space. Theoretically problems of stochastic filamentation, influ-ences of dispersion at the propagation of femtosecond pulses in atmosphereare discussed in [65].

Pulse'In eXIenergpulse:fiJam,tatiorbothpertu

AtfiJam<Stati~oflO(indepthe pstruClweakbund

In laboratory experiments [12] it was observed that in the plane of observationthe position of lengthy filament wanders from shot-to-shot. In subsequentworks [46,69,70], random displacements of filament at different distancesfrom the laser system output was measured, and statistical investigations ofthe distribution function of these displacements were performed.

The quasi-stationary model is applicable for the stochastic problem aboutrandom displacements of filament in the turbulent atmosphere. In the computersimulation, fluctuations of the refractive index D.n(x,y, z) were represented by achain of 8-correlated phase screens located along the propagation axis [71]. Forthe spatial spectrum of the refraction index Fn(",x, "',v. "'z) the following expres-sion was used [72]:

where "'0 = 2n/ Lo, "'m = 5.92/10, Lo and 10 are the outer and inner scales ofturbulence, respectively, and C~is the structure constant.

The statistical analysis of the results of laboratory experiments and ofnumerical simulations has shown that in the turbulent atmosphere distributionfunction of filament wanderings obeys Rayleigh's law. It is necessary to notethat statistical investigations [49] have shown that for a pulse with randomfluctuation of intensity at the laser system output the distribution function ofdisplacement of the filament does not obey Rayleigh's law.

In lalbeenmedihas ~trans

ismost effectiveTherefore, the

. pulse in atmo-r positioning ofentation, influ-. in atmosphere

~of observationIn subsequent

erent distanceslVestigations ofd.problem about[n the computer'epresented by a)fi axis [71]. For,llowing expres-

riments and ofere distribution'cessary to notee with randomion function of

Fig. 15.11 Typical tonepicture of themultifilamentation of a40-mJ pulse with duration42 fs (FWHM) [60]

Pulses of an atmosphere terawatt-power laser break up into multiple filaments.In experiments [73,74]the phase-modulated pulses with duration of 100-600 fs,energy of 230mJ were used. In experiments [25,60] with spectral-limited 45-fspulses with energy of 50mJ, !he formation of multiple "hot" spots at multi-filamentation have been registered (Fig. 15.11). Such a picture of multifilamen-tation explains that the modulation instability of a powerful pulse developsboth for inhomogeneities of the initial profile of intensity and from irregularperturbations caused by atmospheric turbulence.

At multifilamentation, the quasi-stationary model describes the origin offilaments that are initiated by fluctuations of a refractive index of a medium.Statistical characteristics of a bunch of filaments were determined from a seriesof 100pulses, anyone of which was resolved by problem (15.1) with statisticallyindependent chains of phase screens. The average distance < zfl/(N) > at whichthe pulse can have N "primary" filaments is reduced with an increase in thestructural constant C~ - characterizing the "force" of turbulence - andweakly depends on the internal scale 10 (Fig. 15.12). The average width of abunch of N chaotic filaments increases with the growth of C~.

In laboratory experiments on filamentation of laser pulses in an aerosol it hasbeen established [76,77] that with an increase of the optical thickness of themedium, the number of filaments is reduced. The first natural experiment [47]has shown that filaments can be formed and propagate in a rain. In pulsetransverse section, the formation of a diffraction pattern (typical for scattering

Fig. 15.12 Distance zft! to the filament start in the atmosphere for: C/ = 15 x 10-15 em -2/3,10 = I mm (circles); C/ = 3.0 x 1O-15cm-2/3, 10 = I mm (squares); Cn2 = 3.0 x 1O-15cm-2;;,10 = 5mm (triangles). Pulse power P peak = 1.2 X 1011= Wand ao = 0.82 em [75]. On the tracewith a length of80 m, statistically independent screens with the spectrum (15.14) were located,each 10m long

by spherical particles) was observed. Originating intensity perturbations maybecome self-focusing initiators sites and thus initiators of random filaments.

The influence of a separate particle on the filamentation of a pulse wasinvestigated in the experiment [78] where water droplets with a diameter from30 to 100!-tm were placed on the trace of the filament propagation. It wasrevealed that particles comparable with the filament diameter had an insignif-icant influence on the existence of the filament behind the drop.

"Survivability" of the filament while hitting an aerosol particle is explainedby preservation of the surrounding energy reservoir, which provides its subse-quent existence. The simple model in which the particle is replaced with anabsorbing disk was offered in [79]. On the basis of this model, numericalinvestigations of the propagation of the femtosecond pulse through separateopaque particles displaced from the pulse axis were performed [80]. The modelof absorbing disks for aerosol particles is used in [77] for interpretation oflaboratory experiments on pulse filamentation in clouds of high optical density.The influence of dense aerosol, which is represented by a continuous stronglyabsorbing layer, on the length of filament is numerically considered in [81].

At the same time, water particles of rain, clouds, and fog absorb weakly at800 nm and mainly cause the scattering. The stratified model of pulse propaga-tion in an aero-disperse medium, which describes coherent scattering by parti-cles, diffraction, and Kerr and plasma nonlinearity in air, was offered in [82].The stratified model consists of a chain of screens with particles, at which takesplace coherent pulse scattering leading to the small-scale intensity redistributionin its transverse section. At free sections between aerosol screens, diffractionand nonlinear-optical interaction of a pulse with gas components of air takesplace.

On the basis of the stratified model the pattern of intensity distribution in thetransverse section of a subterawatt pulse that propagated in a drizzling rain wasreceived in [83]. The distribution contains randomly located "hot" spots, and

Fig. 15.13 Dof filamentatconcentratiolof aerosol pamark the uppmultifilament.triangles repnboard of a sinregime; the d;corresponds tvalue of the a,a.~nr

ring formafield on thlthe size of;depend on

From ncles with apulse withlo-l5!-tmstimulate treduced. Lparticles, tJformationtion at the

15.5.4 RbJ

In atmosptransversefluctuatiornot bringspatial reggated (FigstationaryEreg(x,y, zhas the sqtLens focmregulariza 1groups. 0ments at tl

')ution in theling rain was" spots, and

1O-15cm~2(3,

1O-15cm~213,

]. On the traceIwere located,

Fig. 15.13 Different modesof filamentation versusconcentration n and radius rof aerosol particles. Circlesmark the upper board of themultifilamentation regime;triangles represent the upperboard of a single filamentationregime; the dashed curvecorresponds to the constantvalue of the attenuation,r:J.~nr

4000I\,, no filaments\\,

\\,

\ ,si;"!ll~!ilament

is explainedes its subse-ced with an:, numericalIgh separate. The modelpretation ofcical density.,Jus stronglydin [81].'b weakly atIse propaga-ing by parti-ered in [82].which takes:distribution, diffractionof air takes

ring formations such as in [49], which are caused by the diffraction of a lightfield on the droplets. The statistical analysis performed in [84] has shown thatthe size of perturbations with the greatest increment of growth [5] and does notdepend on drops concentration, and is determined only by the beam power.

From numerical simulation it has been established that scattering by parti-cles with a radius of 2-5 Jlm does not practically influence the filamentation of apulse with peak power Ppeak rv lOOPer> whereas large particles with a radius of10-15 Jlm produce in pulses the formation of intensity perturbations thatstimulate the origin of filaments, and the distance before the filaments start isreduced. Depending on the concentration and the size of atmospheric aerosolparticles, the different modes are possible: the formation of many filaments, theformation of a single filament in a pulse, and, at last, the absence of filamenta-tion at the large optical thickness (Fig. 15.13).

Jations mayfilaments.a pulse waslmeter fromtion. It wasan insignif-

15.5.4 Regularization of Filaments in the Atmosphereby a Lens Array

In atmosphere it is reasonable to use the regular phase modulation in thetransverse section of a pulse for suppression of the influence of turbulentfluctuations of a refractive index. Moreover, the phase regularization doesnot bring energy losses, unlike amplitude mask. In [85] the lens array of thespatial regularization of a stochastic bunch of filament is numerically investi-gated (Fig. 15.14). The analysis was performed on the basis of the quasi-stationary models (15.1) for the light field amplitude of the pulseEreg(x,y, z = 0, t), represented in the form of (15.12). Each array elementhas the square form with size d and corresponds to a lens with focal length RI'Lens focusing radius in array Rf = 16.75m, lenses size d/ao = 0.4 -;- 1.0. Uponregularization by means of a homogeneous lens, array filaments are formed bygroups. Optimization of the lens array elements allows formation of all fila-ments at the same distance [86].

390 V.P. Kandidov et al. 15 Self-foc

a) b) where Ry,cm model.

0.5 On thtation (2

0 channelsa filamer

-0.5 and exissupercor

-0.5 0 0.5 x,cm -I -0.5 0 0.5 x,cm

., • •• • • ., • •., • .. ,., •• • • • • •• • • ., 1# ••

•• • •• ••

Fig. 15.14 Intensity distributions in the central slice of a pulse in a turbulent atmosphere afterz = 4.7 m: (a) without lens array, (b) with lens array. The beam radius Go = 1em and the ratioPI Per = 135. The period of the lens array is d = 2.5 mm. The turbulence structure constant isC} = 5 x 1O-l3cm-2I3, the inner scale is 10 = I mm and the outer scale is Lo = 0.16m [85] For fern

inseparadispersi<of a lase

The 1menta titdistancethe dist2outlet c,subsequofa pul

Phys;two factpowerilsign oftcompenmodulapulse \\increas(Tp(8) >

The spatio-temporal scenario of femtosecond pulse filamentation is reproducedcompletely by the dynamic model which covers both the initial and the dynamicstages of the process. The dynamic model includes diffraction, pulse dispersion,non-stationary nonlinear-optical response of medium and induced laser plasma[19,25,33,87,88]:

fJE fJ2E 2k22ik--;:;- = f'::!..lE- kk' ~ + - [(f'::!.nk+ f'::!.np+ f'::!.n)E] - ikaE.

uZ ut no

Equation (15.15) leaves out the self-steepening of the pulse. This effect isinvestigated in [87-90]. The Kerr nonlinearity f'::!.nk(X,y,Z, t) is determined by(15.3). In air the plasma increment of the refractive index f'::!.np(x,y,z,t) isdescribed by the expression:

Atn,tempordistanc[96] onmodelsMarbuPpeak(8

Thelength i

where wp(x,y,z,t) = v4ne2Ne(x,y,z;t)/m is the plasma frequency, Wo is thecentral frequency of pulse spectrum, and m and e are the mass and charge of theelectron. The free-electron density Ne(x,y,z,t) is described by the kineticequation:

fJNe 27it = R(IEI )(No - Ne),

ie. This effect iss determined byL'lnp(x,y,z, t) is

where R(IEI2) is the ionization rates, determined from the accepted processmodel.

On the basis of the dynamic model (15.3,15.17-15.19) of the pulse filamen-tation (2) the spacio-temporal scenarios of formation of filaments and plasmachannels were received in [25,31-33]. The processes of energy exchange betweena filament and the background reservoir, which explains refocusing of the pulseand existence of lengthy robust filament [28,31,32,44], and mechanisms ofsupercontinuum generation and conical emission, are also investigated [17,19].

,t atmosphere afterI cm and the ratio

ructure constant isLo = a.16m [85] For femtosecond pulses the spatial compression of a beam at self-focusing is

inseparably connected with redistribution of power. At self-focusing the normaldispersion ofa medium leads to essential transformation of the temporal profileofa laser pulse, and to its breakdown on some spikes[91-93].

The pulse-phase modulation allows controlling the distance Zftl before fila-mentation starts. The influence of the initial pulse-phase modulation on thedistance before filamentation starts was noted in [94]. It is revealed therein thatthe distance before the beginning of the filament increases at the detuning of theoutlet compressor, independently of a sign of the initial phase modulation. Insubsequent experiments [95] the pre-compensation of group velocity dispersionof a pulse in a medium with a normal dispersion was investigated.

Physically, the chirp influence on the formation of filaments is determined bytwo factors [96]. The first one, quasi-stationary, consists of a reduction of initialpower in temporal slices at pulse-phase modulation and does not depend on thesign of the phase modulation. The second, dynamic, factor, consists of the pre-compensation of group velocity dispersion and depends on the sign of phasemodulation. In a medium with a normal dispersion, recompensation of thepulse with negative chirp occurs, and the power in pulse temporal slicesincreases with distance z. The duration of a phase-modulated pulse isTp(8) > TO, and the peak power of a pulse is Ppeak(8)

1'0 = 70r.II

80604()

20o

300 ISO ISO 300 4!n

o300 200 100 100 200 300 400 500 600 700 't(o),fs

Negative chirp

Fig. 15.15 The distance of a filament start ZfiI calculated from both the quasi-stationary model(.6.) and from the dynamical model (0), as a function of the duration T( 6) of chirp pulse. Theestimate by generalized Marburger formula (15.8) is represented by the dashed curve). Theenergy of the transform-limited pulse is Wo = 60 mJ, its duration is TO = 21 fs, andao = 1.5cm [96]. The inset shows the experimental results [95]

supercontinuum generation [96]. In natural experiments [20] the advantages ofthe phase-modulated pulses for receiving filaments that are kilometers long areevidently shown. Detailed research of the propagation in air of the phase-modulated pulses with energy of 190mJ was performed in [97]. At phasemodulation the pulse duration Tp( 8) varied from 0.2 to 9.6 ps, which corre-sponded to a change of the peak power in the range Ppead8) = (190 --;-4)Per·At duration T(8) = 2Aps, phase modulation was the optimum, and the laserplasma was registered at distances up to 400 m.

The filament number in terawatt and subterawatt pulses increases with distance[25]. The arrangement of the "hot" spots defining the position filament in atransverse section changes. Upon side visualization of filaments in rhodamine Ba merging and birth of new filaments along the propagation axis are visible [98].Changes of number and arrangement of filaments with distance are confirmedby supercontinuum patterns [99]. Its frequency-angular distribution becomesmore complex due to interference of radiation at an increased number offilaments with distance and, hence, of supercontinuum sources.

The basic laws of dynamic multifilamentation are based on the example of asimple model of a pulse with two perturbations at the initial intensity

Fig. 15.16distributicdistances:(z = 0); (bdevelopml"parent" f(z = 7.5 cr"daughter(d) "survi\(z = 20.6(

distribut(15.23) imultifilathe puis-perturb£:the intermaximafilamentaccomp.dynamicthe pulse

Thedscatterinin plasrrreductiodensely,

Self-focIchapterfemtose,namely,enon ofnuum, ,filament

Fig. 15.16 (a) Fluencedistributions at differentdistances z: initial conditions(z = 0); (b) independentdevelopment of two"parent" filaments(z = 7.5 cm); (c) birth of a"daughter" (z = 16.9cm);(d) "survival" of one of them(z = 20.6cm) [100)

....I.-i).fs

;tationary model.chirp pulse. The,hed curve). The

TO = 21 fs, and

distribution [60,100]. The system of equations (15.3), (15.18), (15.19), and(15.23) is numerically investigated for a pulse (15.9). The dynamic scenario ofmultifilamentation is evidently reproduced by fluence distributions F(x,y, z) inthe pulse in water (Fig. 15.16). In the beginning of propagation from eachperturbation so-called "parent" filaments were formed [Fig. 15.16(a),(b)]. Atthe interference of the ring structures from "parent" filaments [37],some localmaxima are formed. These maxima become the center of origin of "daughter"filaments [Fig. 15.16(c),(d)]. The development of "daughter" filaments isaccompanied by energy flow in a pulse transverse section. As a result ofdynamic energy competition between filaments only one filament survives inthe pulse.

The dynamic competition of the filament is the cause of instability of a back-scattering fluorescent signal, which arises at excitation of molecules of nitrogenin plasma channels of competing filaments [100]. It was shown [84] that atreduction of the transverse sizes of a pulse, plasma channels arrange moredensely, and the registered signal of fluorescence increases.

advantages ofleters long areof the phase-97]. At phasewhich corre-(190 -7- 4)Pcr•and the laser

> with distancefilament in a

1rhodamine Bue visible [98].are confirmedltion becomesed number of

Self-focusing is the necessary condition for filament initiation. In the presentchapter we have considered only one aspect of the many-sided phenomenon offemtosecond laser pulse filamentation in the bulk of transparent medium,namely, the formation of a filament that is directly connected to the phenom-enon of self-focusing. The questions of plasma channel formation, superconti-nuum, and conical emission generation influence of various factors on pulsefilamentation - all these issues are subjects of independent reviews. Research

e example of alitial intensity

aspects associated with the filamentation in dielectric solids and applications ofthis research to the solution of microoptics problems are of significant interestas well.

Acknowledgment V.P. Kandidov, A.E. Dormidonov, and O.G. Kosareva thank the supportof the European Research Office of the U.S. Army through contract No. W91INF-05-1-0553,and the Russian Foundation for Basic Research through grants No. 06-02-17508-a andNo. 06-02-08004.

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3. G.A. Askar'yan: Effects of the gradient of a strong electromagnetic beam on electronsand atoms, SOl'. Phys. JETP 15,1088-1090 (1962).

4. R.Y. Chiao, E. Garmire, C.H. Townes: Self-trapping of optical beams, Phys. Rev. Lett.13, 479 (1964).

5. V.V. Korobkin, A.J. Alcock: Self-focusing effects associated with laser-induced air break-down, Phys. Rev. Lett. 21, 1433 (1968).

6. N.G. Basov, P.G. Kryukov, Yu.V. Senatskii et al.: Production of powerful ultrashortlight pulses in a neodymium glass laser, SOl'. Phys. JETP 30,641-645 (1970).

7. V.I. Bespalov, V.I. Talanov: Filamentary structure of light beams in nonlinear liquids,SOl'. Phys. JETP Lett., 3, 307-310 (1966).

8. A.J. Campillo, S.L. Shapiro, B.R. Suydam: Relationship of self-focusing to spatialinstability modes, Appl. Phys. Lett., 24,178-180 (1974).

9. A.N. Zherikhin, Yu.A. Matveets, S.V. Chekalin: Self-focusing limitation of brightness inamplification of ultrashort pulses in neodymium glass and yttrium aluminum garnet,Quantum Electron. 6, 858-860 (1976).

10. I.A. Fleck, Jr., C. Layne: Study of self-focusing damage in a high-power Nd:glass-rodamplifier, Appl. Phys. Letts. 22, 467-469 (1973).

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12. A. Braun, G. Korn, X. Liu et al.: Self-channeling of high-peak-power femtosecond laserpulses in air, Opt. Lett. 20, 73 (1995).

13. E.T.J. Nibbering, P.F. Gurley, G. Grillon et al.: Conical emission from self-guidedfemtosecond pulses in air, Opt. Lett. 21, 62 (1996).

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