Self-Focusing and Guiding of Short Laser Pulses in Ionizing Gases and Plasmas.pdf

IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 33, NO. 11, NOVEMBER 1997 1879

Self-Focusing and Guiding of Short LaserPulses in Ionizing Gases and Plasmas

Eric Esarey, Phillip Sprangle,Fellow, IEEE,Jonathan Krall,Member, IEEE,and Antonio Ting

(Invited Paper)

Abstract— Several features of intense, short-pulse (1 ps)laser propagation in gases undergoing ionization and in plasmasare reviewed, discussed, and analyzed. The wave equations forlaser pulse propagation in a gas undergoing ionization and ina plasma are derived. The source-dependent expansion methodis discussed, which is a general method for solving the parax-ial wave equation with nonlinear source terms. In gases, thepropagation of high-power (near the critical power) laser pulsesis considered including the effects of diffraction, nonlinear self-focusing, ionization, and plasma generation. Self-guided solutionsand the stability of these solutions are discussed. In plasmas,optical guiding by relativistic effects, ponderomotive effects, andpreformed density channels is considered. The self-consistentplasma response is discussed, including plasma wave effects andinstabilities such as self-modulation. Recent experiments on theguiding of laser pulses in gases and in plasmas are brieflysummarized.

Index Terms—Acceleration, gases, laser beam focusing, op-tical propagation, photoionization, plasmas, relativistic effects,transient propagation, waveguides.

I. INTRODUCTION

T HE propagation of intense laser pulses in gases and plas-mas is relevant to a wide range of applications, includ-

ing laser-driven accelerators [1]–[16], laser-plasma channel-ing [17]–[43], harmonic generation [44]–[54], supercontinuumgeneration [55]–[57], X-ray lasers [58]–[63], and laser-fusionschemes [64]–[67]. For many of these applications, it is highlydesirable that the laser pulse propagate extended distances(many Rayleigh lengths) at high intensity. In the absence of anoptical guiding mechanism, the propagation distance is limitedto approximately a Rayleigh (diffraction) length ,where is the laser wavelength and is the laser spot sizeat focus. High intensities require a tight focus and, hence, arelatively short Rayleigh length, e.g., 0.7 mm for15 m and 1 m. In conventional optics, laser pulses canbe guided in optical fibers [68]. In addition, at sufficientlyhigh laser power , where is the critical power fornonlinear self-focusing, the propagation distance is stronglyaffected by nonlinear self-focusing [69]–[73] as well as othernonlinear effects (e.g., ionization and instabilities). Analogousprocesses can occur in gases and plasmas.

Manuscript received July 2, 1996; revised June 23, 1997. This work wassupported by the Office of Naval Research and the U.S. Department of Energy.

The authors are with the Beam Physics Branch, Plasma Physics Division,Naval Research Laboratory, Washington, DC 20375-5346 USA.

Publisher Item Identifier S 0018-9197(97)07831-7.

In this paper, several features of intense, short-pulse (1ps) laser propagation in gases undergoing ionization and inplasmas are reviewed, discussed, and analyzed. The waveequation is discussed, including the nonlinear response ofthe medium (ionizing gas or plasma) to the intense laserpulse. The source-dependent expansion (SDE) method [74],[75] is discussed, which is a general method for solvingthe paraxial wave equation with nonlinear source terms. TheSDE is used to derive an envelope equation which gov-erns the evolution of the laser spot size, the solutions ofwhich describe laser diffraction, self-focusing, and guiding.In gases, the propagation of high-power (near the criticalpower for self-focusing) laser pulses is considered includingthe effects of tunneling ionization [76]–[79] and ionization-induced refraction [80]–[88]. Self-guiding solutions are dis-cussed which result from a proper balancing of diffraction,nonlinear self-focusing in the neutral gas, and defocusing dueto plasma generation [17]–[19], [89], [90]. Self-guided pulsesare subject to an ionization-modulation instability which limitsthe propagation distance [90]. In plasmas, optical guidingby relativistic effects, ponderomotive effects, and preformeddensity channels is considered [91]–[167]. The self-consistentplasma response is discussed, including plasma wave effectsand instabilities such as self-modulation [108], [109], [114],[148]–[159]. Several recent optical guiding experiments arealso briefly discussed [17]–[43].

In conventional optics, the diffraction of laser pulses canbe prevented either by using an optical fiber or by relyingon nonlinear self-focusing [68]–[73]. Laser propagation isoften characterized by the index of refractionof the medium, where is the axial laser wavenumber and

is the laser frequency. Refractive guiding canoccur when the the transverse (radial) profile of the index ofrefraction is peaked along the propagation axis (theaxis), i.e.,

. When this occurs, the phase velocityis smaller on axis than it is off axis and the phase frontsof the optical field can become curved such that the opticalfield focuses toward the axis. Propagation can be affected byboth the linear and nonlinear components of the index ofrefraction , where and is the laser intensity.An optical fiber is created by tailoring the transverse profileof the linear component of the index of refraction such that

[68]. For example, a parabolic variation in thelinear refractive index of the form

can guide a Gaussian laser profile of spot size

0018–9197/97$10.00 1997 IEEE

1880 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 33, NO. 11, NOVEMBER 1997

Fig. 1. Schematic of a laser beam propagating in a gas undergoing ionization. WhenP > PN , the beam self-focuses and the intensity increases causingionization. Plasma is generated along the beam axis, tending to defocus the beam. Self-guiding results by balancing nonlinear self-focusing with plasmadefocusing. Here,� = z � vt, where v ' c is the group velocity.

provided that , where

(1)

is the critical fiber depth. Alternatively, nonlinear effects canlead to self-focusing when , as is typical the when thelaser frequency is below the resonant frequencies of themedium. Note that if the laser intensity is peaked along theaxis, , then for and nonlinearself-focusing may occur. For a laser pulse with a Gaussianradial profile, nonlinear self-focusing occurs when ,where

(2)

is the critical power [69]–[73]. For example, 2.8 GWfor a 1- m pulse propagating in air at 1 atm.

In a neutral gas, pulse propagation is affected by diffraction,refraction, nonlinear self-focusing, ionization, and plasma de-focusing [17]–[19], [76]–[90]. In an unionized gas, the indexof refraction is given by , where 1 andboth the linear and nonlinear contributions aretypically positive, much less than unity, and proportional to theneutral gas density . The basic propagation dynamics can beillustrated by considering a laser pulse with a power slightlyabove the critical power for nonlinear self-focusing (seeFig. 1). Since , the pulse self-focuses and the peakintensity increases resulting in ionization and the generationof a plasma along the axis. In the region of the plasma, therefractive index is modified

(3)

where is the plasma frequency andis the electron plasma density. The local decrease in the

refractive index due to the plasma tends to defocus theoptical beam (ionization-induced refraction) [80]–[88]. As thepulse diffracts, the intensity decreases and ionization ceases.

Nonlinear self-focusing will again cause the beam to focuswhich causes ionization and plasma defocusing. In such away, periodic multiple ionization sparks can be formed alongthe axis [17]. On the other hand, if diffraction, self-focusingdue to , and defocusing due to plasma generation areproperly balanced, a self-guided optical beam can be formedand propagated over extended distances, i.e., many vacuumRayleigh lengths [17]–[19], [89], [90]. The propagation ofsuch a self-guided beam, however, is limited by the ionization-modulation instability as well as other nonideal effects (e.g.,collisional and ionization losses) [90].

In a fully ionized plasma, optical guiding is affected by den-sity channels, relativistic self-focusing, ponderomotive forces,and plasma wave generation [91]–[167]. In a plasma, the indexof refraction including nonlinear effects is given by

(4)

where is the electron plasma frequencyevaluated at the ambient plasma density is the localelectron plasma density, is the relativistic factor associatedwith the plasma electrons, and is assumed. Theradial profile of the refractive index can be affectedeither through the plasma density or the relativisticfactor . This paper is concerned with the interaction ofultrashort ( 1 ps) laser pulses with underdense plasmas. Onthis ultrashort time scale, the motion of the plasma ionsis typically insignificant. Plasma ion motion, however, issignificant in the interaction of long (1 ns) laser pulses inplasmas. For long pulses, self-focusing can result from themotion of the plasma ions (i.e., the formation of a densitychannel) in response to ponderomotive [91], [168]–[174] orthermal [175]–[186] forces induced by the laser fields.

An important parameter in the discussion of intense laser-plasma interactions is the laser strength parameter, defined

ESAREY et al.: SHORT LASER PULSES IN IONIZING GASES AND PLASMAS 1881

as the peak amplitude of the normalized vector potential ofthe laser field, . The laser strength parameter isrelated to the peak intensity and power by

, which gives

[ m] [W/cm ] (5)

and [GW] , where a linearly polarized laserfield with a Gaussian radial profile is assumed. Furthermore,the peak laser electric field amplitude is given by

, i.e., [TV/m] [ m]. Physically,is the normalized transverse “quiver” momentum of a

plasma electron in the laser field, as indicated by conservationof transverse canonical momentum in the one-dimensional(1-D) limit . When 1, the electron quivermotion is highly relativistic and the laser-plasma interactionis highly nonlinear. Highly relativistic electron motion (1) requires laser intensities 10 W/cm for wavelengthsof 1 m. Such intensities are routinely produced bycompact solid-state laser systems based on the technique ofchirped-pulse amplification [187]–[189].

The primary contribution to the relativistic factor of aplasma electron in a laser field is the quiver motion,

, where . For a laser intensity peakedon axis, , the relativistic quiver motion resultsin and the possibility of guiding. This isthe principle underlying relativistic self-focusing [91]–[128]which can occur when the laser power exceeds a critical power[91]–[96] with or

[GW] (6)

where is the plasma wavelength. For example,1.9 TW for a 1- m laser propagating in a plasma of

density 10 cm . Relativistic self-focusing, however,is ineffective in guiding short laser pulses, , where

is the laser pulse length [108], [109], [114]. Furthermore,long laser pulses, , which are affected by relativisticfocusing, are subject to instabilities such as self-modulation[114], [115], [148]–[159], which can limit the propagationdistance.

Alternatively, a preformed plasma density channel can guidea short laser pulse [36]–[43], [113]–[115], [129]–[147]. Forexample, a parabolic variation in the plasma density of theform can guide a Gaussian laser pulsewith spot size provided that the channel depth satisfies

, where

(7)

is the critical channel depth [113]–[115] andis the classical electron radius. This can be written as

[cm ] 1.13 10 [ m] or .Short laser pulses have been guided in plasma channels createdby an axicon focus in a gas chamber [36]–[41], by a slowcapillary discharge [42], [43], or by an intense pump laserpulse in a gas jet [34]. In addition to a preformed plasmachannel, the local plasma density can have perturbationsdue to the ponderomotive force of the laser pulse. This canlead to ponderomotive self-channeling [96], [104], [112]–[125]

plasma wave guiding [106]–[110], and self-modulation of longlaser pulses [114], [115], [148]–[159].

In this paper, the mathematical model for describing laserpulse evolution is based on the paraxial wave equation

(8)

where is the laser field,is the slowly varying amplitude, is the laserfrequency, and is assumed. Here, is generallya nonlinear function of and is determined by the dynamicresponse of the medium. Solutions to (8) can be found in termsof the independent variables and by using theSDE method [74], [75], as discussed in Section II. An equationdescribing the evolution of the laser spot size can bederived via the SDE method by assuming that the transverseprofile of the laser field remains approximately Gaussian andof the form , where andare independent of and . The evolution of the spot size

is given by the envelope equation

(9)

where the angular brackets represent the intensity-weightedradial average, i.e., is anaverage of over the radial intensity profile

of the laser pulse. The first term on the rightdescribes vacuum diffraction and the second term describes therefractive properties of the medium. In vacuum,and , where is the minimum spotsize at the focal point and is the Rayleighlength. Clearly, refractive guiding requires .

The remainder of this paper is organized as follows. SectionII presents a heuristic derivation of the envelope equation forthe laser spot size as well as the critical channel depth andthe critical power necessary for guiding. A formal derivationof the envelope equation for the laser spot size, (9), usingthe SDE method is also presented in Section II. Section IIIdiscusses the wave equation for a gas undergoing ionization.Nonlinear atomic effects are retained to third order in thelaser field. Plasma generation is described self-consistentlyby tunneling ionization. Section IV discusses propagation ina gas undergoing ionization based on dynamic solutions tothe envelope equation. Self-guided solutions are presented andthe ionization-modulation instability, which affects self-guidedpulses, is discussed. The wave equation for a fully ionizedplasma is discussed in Section V. Equations are presentedwhich describe the dynamical response of the plasma to alaser pulse in the linear regime and in the nonlinearquasi-static regime. Section VI discusses propagation in aplasma. Included are discussions of relativistic self-focusing,tailored pulse propagation, guiding in straight and curvedplasma channels, ponderomotive self-channeling, plasma ionmotion, plasma wave guiding, and self-modulation of guidedlaser pulses. Recent experiments on optical guiding in ionizinggases and in plasmas are briefly summarized in Section VII.This paper concludes with a discussion in Section VIII.


II. L ASER PULSE EVOLUTION

The starting point for describing laser pulse evolution inan ionizing gas or a plasma is the wave equation, which canwritten in the form

(10)

where is the transverse electric field of the laser andis the laser frequency, ,

and is the axial propagation direction. Here, isthe effective index of refraction that, in general, is a nonlinearfunction of and is determined by the dynamical response ofthe medium. Alternatively, (10) can be written in terms of thetransverse vector potential of the laser , as is convenientwhen discussing a plasma. In the following, it will be assumedthat is: 1) close to unity, as is the case in a gas or underdenseplasma, and 2) real, i.e., dissipative effects such as collisionsand absorption will be neglected.

To proceed, the paraxial approximation will be made,which assumes that the laser field can be written as

with the complex amplitude slowlyvarying in and compared to , i.e.,and . In terms of the independent variables

and , the paraxial wave equation describing theevolution of is given by

(11)

Formal solutions to (11) with a general nonlinear can befound using the SDE method [74], [75], as discussed in SectionII-B. Before discussing the SDE method, however, a heuristicanalysis of the paraxial wave equation is presented that allowsfor a straightforward discussion of several features of laserpulse self-focusing and guiding.

A. Heuristic Theory

Approximate forms for the the envelope equations describ-ing laser pulse evolution can be found by the following simplebut nonrigorous method. A solution to the paraxial waveequation, (11), will be sought of the form

(12)

where and are real constants, is the spot size and isreal, is inversely proportional to the wavefront curvatureand is real, and is the wavenumber shift and is real.This form for conserves power, i.e.,is independent of . The index of refraction is assumed to beof the form

(13)

where is the depth of the optical fiber,is the nonlinear index, is the laserintensity, and , and are constants. Equations(12) and (13) are then inserted into (11), the wave operator onthe left side of (11) is carried out, and the intensity profile isexpanded for , i.e., . Envelopeequations for , and follow by equating the

constant terms (independent of) and the terms proportionalto . This indicates that

, where ,and

(14)

or, in terms of the normalized spot size ,

(15)

where is the critical fiber depth,is the laser power, is the heuristic

critical power, and is the Rayleigh length. Thefirst term on the right of (15) represents vacuum diffraction,the second term is the focusing effects of the parabolic fiberprofile, and the third term is the self-focusing arising from theintensity-dependent nonlinear index of refraction. Note that invacuum, , where is the focal spot sizeat .

When compared to more accurate calculations, such as theSDE method discussed in the following section, (14) is of thecorrect form, however, the third term on the right is off bya numerical factor. Specifically, the heuristic critical power

is smaller than the actual nonlinear focusing power[69]–[73] by approximately a factor of four, i.e.,

. This difference is due to the fact that a Gaussianradial profile is not an exact solution to the paraxial waveequation with an intensity-dependent nonlinear indexand the expansion overestimates the effects ofnonlinear self-focusing. The general condition for nonlinearself-focusing can be written as

(16)

where is the nonlinear component of the refractive indexevaluated along the optical axis and it is assumedthat , and . Fora gas, and . For a plasma,the nonlinear component of the refractive index due to rel-ativistic effects is andthe critical power for relativistic self-focusing [91]–[96] is

GW.In the low power limit , a parabolic radial

variation in the linear refractive indexcan guide a laser pulse with provided .The expression for given by the heuristic theory is theexact result, since a Gaussian radial profile is an exact solutionto the paraxial wave equation with a quadratic radial variationin the linear refractive index. The general condition for guidinga laser pulse with in a parabolic fiber is

(17)

assuming a linear index of the formwith , and . For a plasma,

, and a density channel of the formcan guide a laser pulse with provided

, where and


[113]–[115]. Methods for creating such a plasma densitychannel are discussed in Section VII.

By analogy to plasma channel guiding, a radial variation inthe neutral gas density profile can guide a laser pulse. Recallthat for a gas, is typically positive and proportionalto the neutral gas density . In particular, if the gas densityis maximum along the axis and of the form

, then a Gaussian laser pulse with spot sizecan beguided provided that the channel depth satisfieswith

(18)

where is typically independent of the gas density. For standard air at 1 atm, 4.6 10

cm or . Creating such a radialprofile in the gas density can be problematic in practice.

The above heuristic theory assumed that the gas andplasma densities are nonevolving. In general, however, theself-consistent interaction with the laser pulse will lead toa nonlinear density evolution. For example, an intense laserpulse can cause ionization of a gas and plasma generation.Initially, plasma generation will be localized along theaxis where the laser intensity is maximum. This radiallylocalized plasma generation causes a local decrease in thelinear refractive index, . Localizedplasma generation implies and,hence, enhanced diffraction (ionization-induced refraction)[80]–[90]. Alternatively, in a fully ionized plasma, theponderomotive force of the laser pulse can cause significantperturbations in the plasma density. For example, theponderomotive force can expel plasma electrons radiallycreating a density channel (ponderomotive self-channeling)[96], [104], [112]–[125]. Furthermore, a laser pulse can exciteplasma waves (wakefields) which can result in the diffractiveerosion of laser pulses [108], [109], [114], plasma waveguiding [106]–[110], and envelope self-modulation [114],[115], [148]–[159]. These effects can be analyzed using theSDE method.

B. SDE Method

The SDE method [74], [75], [90], [99], [142], [151] is ageneral method for solving the paraxial wave equation withnonlinear source terms, e.g., (11). In the SDE method, thelaser field is expanded in a complete set of source-dependentorthogonal Laguerre–Gaussian functions. These functions areimplicitly functions of the propagation distance throughthe laser field parameters (spot size, wavefront curvature,amplitude and phase). The laser field can be described byfour coupled first-order differential equations for the fieldparameters in the variable. Although the SDE method iscapable of describing an arbitrary laser field composed of anarbitrary number of source-dependent modes, the analyticaland numerical results presented in the following sectionsassume that the laser field is adequately described by a singlesource-dependent Laguerre–Gaussian mode. This assumptionis not valid when the laser power greatly exceeds the nonlinearfocusing power, since the laser beam is expected to filament

into higher order modes. As discussed below, however, asingle source-dependent Laguerre–Gaussian mode is itself asuperposition of many vacuum Laguerre–Gaussian modes.

In the following, it is assumed that the laser field amplitudeis axisymmetric, . In general, can be writtenin terms of a complete set of Laguerre–Gaussian functions,i.e., source-dependent modes,

(19)

where is the complex amplitude,is real and denotes the spot size,

is real, is the radius of curvature associated withthe wave front, and is a Laguerre polynomial, e.g.,

and .To proceed with the SDE analysis, (19) is substituted into

(11), the differential operations are performed, and both sidesare multiplied by and integrated over

from 0 to (the details are given in [75]). The resultingequation for is

(20)where

(21a)

(21b)

(21c)

The dot denotes the operator and the asterisk denotesthe complex conjugate.

Equations (20) and (21a)–(21c) describe the evolution ofthe various source-dependent modes. However, this set isunderdetermined since there are more unknowns than equa-tions. An additional constraint, i.e., a specification of thefunction , is necessary to solve (20). The individualsource-dependent modes in (19) are functions of the spot size

, wavefront radius of curvature , and theamplitude and phase . Since is also a function of

and , the evolution of the source-dependent modeis governed by the particular choice for the function. Forexample, recovers the conventional vacuum modes. Ingeneral, however, expansion in terms of the vacuum modes

requires many modes to accurately describe a guidedlaser beam over distances of many Rayleigh lengths. A moreappropriate choice for will depend on the particular problemunder consideration.

In the following, it is assumed that the dynamics of thelaser beam can be adequately described by the behavior of asingle source-dependent mode, in particular, the fundamentalGaussian mode. Note, however, that a higher orderradially polarized, axisymmetric beam, which has applicationsto the inverse Cherenkov accelerator [190]–[193], is consid-ered in [90]. Furthermore, nonaxisymmetric high-order modes


are considered in a SDE analysis of the laser-hose instability[142].

To derive analytic expressions for the envelope equationsdescribing the evolution of the SDE mode, it is assumedthat the coupling to, as well as the amplitude of, the higherorder SDE modes are small. In fact, an optimal choice forcan be determined from (20) by requiring that the higher orderSDE modes are small. Assuming for

1, it is clear from (20) (with ) that the optimalchoice for is

(22)

With this choice for , (20) (with ) yields

(23)

Equations (22) and (23) completely determine the evolu-tion of the fundamental Gaussian source-dependent mode.Substituting (21a)–(21c) into (22)–(23) and setting

, where and are real, gives [75]

(24a)

(24b)

(24c)

(24d)

where and , i.e.,

(25a)

(25b)

The Gaussian ansatz holds best for radiation beams thatare near their matched beam solutions, i.e., optically guidedbeams. For beams far from these equilibrium solutions,or for highly nonlinear regimes, the beam can develophigher order modes. For example, numerical solutions to theSchroedinger equation with a saturable nonlinearity indicatethat the radial beam profile can form rings in the regionafter the first focus [174]. Similarly, numerical simulationsof ionization-induced refraction indicate the formation ofrings, i.e., horseshoe-shaped intensity contours, in the regionafter the focus [82]–[84]. Fluid simulations of self-modulationin the highly nonlinear regime indicate that the laser profileoscillates between a sharply peaked profile and a hollow profile[151]. Similarly, particle-in-cell simulations of beat-waveexcitation [100] and of highly Raman-unstable pulses [162]indicate the formation of higher order modes. Furthermore,for powers much greater than the critical power for nonlinearself-focusing, the beam can undergo a transverse break-upvia the filamentation instability [170]–[173], [194]–[197]. Inprinciple, these effects can be accounted for in the SDEmethod by including higher order modes.

1) Envelope Equations:Equation (24a) indicates that thetotal laser power is a constant (independent of

) at each position, which is consistent with theparaxial approximation and the assumption ofreal. Hence,

, where and are independent of . In

terms of the lowest order SDE mode, the laser field is givenby the real part of

(26)

The time-averaged intensity and power associated with (26)are, respectively, and

, where is the peak intensityalong the axis . The laser spot size obeys theenvelope equation

(27)

where

(28)

represents an average of over the radial intensity profileof the laser pulse. In addition, and

the wavenumber shift obeys

(29)

The specific behavior of the laser envelope depends on theform of . The nonlinear form of for an ionizing gas anda plasma are discussed in Sections III and V, respectively. Itshould be noted that the envelope equation for the spot size,(27), can also be derived using a variational approach [198].

2) Vacuum Solution:For propagation in vacuum ,the solution of (24a)–(24d) yields the conventional vacuummode [71], [72]: is the spot size, isthe minimum spot size at the focal pointis the Rayleigh length, is thewavefront radius of curvature, is the amplitude,and is the phase factor.

III. W AVE EQUATION FOR AN IONIZING GAS

The dynamics of a laser pulse propagating in a gas under-going ionization is governed by the wave equation for theelectric field

(30)

where is the polarization field associated with the gas(bound electrons) and is the plasma current density as-sociated with the ionized gas (free electrons). In obtaining(30), a small source term proportional to the gradient of theplasma density was neglected. In the following, the nonlinearpolarization field of the gas is included to third order in thelaser field whereas the plasma current is included to firstorder [90]. Ionization is considered in the high field limit andis modeled by the tunneling ionization rate [76]–[79]. Theattenuation of the laser field due to collisional and ionizationlosses is neglected.


A. Linear and Nonlinear Polarization

The polarization field can arise from a number of processes:electronic polarization, molecular orientation, electrostriction,saturated absorption, and thermal effects [69]–[73]. On the fasttime scale ( 1 ps) of interest, the electronic polarization is thedominant contribution to the nonlinear refractive index and isdue to the laser field modifying the atomic electronic distribu-tion. In the simple Lorentz model [69]–[73] of the atom, theelectrons are assumed to consist of a charge distribution oscil-lating in an effective potential. Nonlinearities in the effectivepotential result in a field-dependent refractive index. In thefollowing description of the polarization field, only isotropicmatter having ensemble-averaged inversion symmetry, i.e.,centrosymmetric ensemble-averaged effective potentials, willbe considered. This includes all gases.

The electric polarization field is defined by ,where is the electronic charge, is the density of atomsor molecules, and is the displacement of the electronicdistribution from equilibrium due to the laser field. Thepolarization field in the classical single resonant frequencymodel is given by [69]–[73]

(31)

where is the characteristic resonant frequency of theelectronic distribution, is a constant associated with thenonlinear (nonparabolic) nature of the effective potential,is a normalizing polarization field amplitude, and is adamping term. Equation (31) is an accurate description of thepolarization field when the laser frequencyis far from anyresonant frequency (typically in the ultraviolet regime).

In the limit where 1) dispersive effects are weak (farfrom resonance, ); 2) damping effects can be ne-glected ; and 3) the nonlinear term in (31) issmall , the polarization field can beapproximated by

(32)

where is the linear susceptibility,is the third-

order susceptibility of the neutral gas,is the linear refractive index of the neutral gas (will be assumed in the following), is thenonlinear component of the refractive index,is the laserintensity, and has been assumed. For example, forstandard air at 1 m, [cm ] and

[cm /W] [cm ] ( 2.7 10 cmat 1 atm).

The validity of (32), i.e., the validity of the expansion ofthe polarization field in powers of the laser field, requires that

, which in terms of is

[ m] [cm /W] (33)

As an example, for nitrogen at 1 atm ( 3 10 and10 cm /W) with 1 m, the normalized laser

field amplitude is limited to 0.05, which corresponds to

an intensity 3 10 W/cm . In this limit, saturation ofthe nonlinear susceptibility due to higher order effects, e.g.,

, is neglected.

B. Plasma Generation

The ionization of the background gas by the laser pulseresults in the generation of plasma electrons. The plasmacurrent density is given by , where and arethe plasma density and fluid velocity, respectively. To lowestorder in , the continuity and fluid velocity equations are

(34a)

(34b)

where is the plasma source term proportional to theionization rate, the force and thermal effects areneglected in (34b), and the electrons are assumed to be createdwith zero velocity when ionized. Combining (34a), (34b), andkeeping terms to lowest order in, the plasma current densityis given by [199]–[201]

(35)

where is the electron plasma fre-quency evaluated at the initial gas density . The evolutionof the plasma density depends on the photo-ionization rate.In obtaining (35), nonlinear and collisional effects have beenneglected. Nonlinear plasma effects are smaller than nonlinearneutral gas effects by approximately the ratio of the nonlinearfocusing power in a gas to the critical power for relativisticfocusing which is negligibly small.

C. Photo-Ionization Model

Ionization can occur by electron collisional processes or bythe intense laser fields directly, i.e., photo-ionization [76]–[79].At high intensities and for laser pulses short compared to a col-lision time, photo-ionization is the dominant process. Photo-ionization can take place by either tunneling or multiphotonprocesses. These regimes are characterized by the Keldyshparameter , where is the ionization energyand is the electron oscillationenergy. Here, the electric field is assumed to be linearlypolarized and given by , whereis the slowly varying amplitude and . The lowfield limit corresponds to the multiphoton ionizationregime, whereas the high field limit corresponds tothe tunneling ionization regime.

The evolution of plasma density in (34a) is given by

(36)

where is the initial neutral gas density, is theionization rate, and the convection term is neglected.For a linearly polarized laser field, the ionization rate in thetunneling limit is given by [76]–[79]

(37)


where 4.1 10 s is the characteristicatomic frequency, is the fine structure constant,

5.3 10 cm is the Bohr radius, isthe ionization energy, 13.6 eV is the ionization energyof hydrogen, and 5.2 GV/cm is the atomicfield of hydrogen. Equation (36) assumes that the gas is atmost singly ionized.

The solution of (36), for low levels of ionization, ,is

(38)

where is defined in the region (corresponds to the front of the laser pulse). The weakly ionizedlimit is sufficient to describe self-guiding of laser beams [90].The assumption places limits of the laser intensityand pulse duration. In addition to (33), the laser intensity mustsatisfy , where is the ionization intensity thresholdof the gas. Furthermore, the laser pulse duration must besufficiently short so as to avoid high levels of ionization andadditional ionization by electron collisional processes.

D. Effective Refractive Index

The propagation of the laser pulse is described by (30),(32), and (35) together with the tunneling ionization model.The source term for the wave equation is given by

(39)

where is the plasma frequency associated withthe initial neutral gas density. An expression for the effectiveindex of refraction can be found by equating the sourceterm, (39), with . This gives

(40)

The second term on the right is the nonlinear effects of thegas and the third term is the nonlinear effects of the plasmaelectrons via tunneling ionization. Neglected in (40) is thesmall term that is approximately constant (independentof ) and, hence, does not affect the focusing properties ofthe laser pulse.

IV. PROPAGATION IN AN IONIZING GAS

The propagation of laser pulses in gases is affected bya combination of diffraction, nonlinear self-focusing, andionization-induced refraction. The refractive index of a neutralgas is typically of the form with .Refractive guiding requires , which is the casefor an intensity profile peaked on axis . Thenonlinear index implies that the laser pulse will self-focuswhen the laser power exceeds the nonlinear focusing power

. Plasma generation occurs via tunnelingionization, wherein the ionization rate depends on the locallaser intensity. For , the pulse self-focuses andthe peak intensity increases which results in ionization. The

generation of plasma leads to a decrease in the local refractiveindex

(41)

where the last term on the right is proportional to the localplasma density, . The plasma density profilewill be peaked along the axis where the laser intensity ismaximum, hence, and the effect of the plasmawill be to defocus the laser pulse [80]–[88]. If diffraction, self-focusing due to , and defocusing due to plasma generationare properly balanced, a self-guided laser pulse can be formedand propagated over extended distances, i.e., many vacuumRayleigh lengths [17]–[19], [89], [90]. Such a self-guidedpulse, however, is subject to an ionization-modulation insta-bility that limits the propagation distance [90], as is discussedin the following.

A. Envelope Equation for Ionizing Gas

The propagation of a laser pulse in a gas undergoingionization is described by the envelope equation for the spotsize , (27), along with the nonlinear refractive index(40). This assumes that the radial profile of the laser pulseis a fundamental Gaussian (a higher order Gaussian profileis treated in [90]). To analyze the pulse propagation, thesource function must be evaluated.Substituting (40) into (25b) and integrating overgives

(42)

where

(43)

represents a filling factor that accounts for the radial gradientsin the plasma density profile. Ionization is maximum wherethe laser field amplitude is maximum, i.e., at . Sincethe tunneling ionization rate depends exponentiallyon the field amplitude, the radial profile of the plasma densitywill be highly peaked about the axis . Equation (43)can be simplified by expanding the integrand about ,which gives

(44)

with

(45a)

(45b)

The quantities , and are functionsof and , whereas the power is only a functionof as implied by (24a).

Using (42)–(45), the envelope equation (27) for Gaussianlaser pulse is [90]

(46)

where is constant, is the Rayleigh lengthassociated with the spot size is the normalizedspot size, is the total power, and

is the nonlinear focusing power for a laser pulse


with a fundamental Gaussian radial profile [69]–[73] (for thehigher order radially polarized laser field considered in [90],the nonlinear focusing power is four times larger). The termson the right-hand side of (46) denote, respectively, vacuumdiffraction, nonlinear focusing in the gas, and defocusing dueto plasma generation.

In the absence of ionization , the envelope equation(46) has the solution

(47)

where at is assumed. For thelaser beam diffracts with an effective Rayleigh length givenby . For , diffractive spreadingbalances nonlinear focusing and a matched self-guided beamcan, in principle, be obtained. However, small changes awayfrom will result in loss of equilibrium. Forthe laser beam self-focuses. In the absence of ionization thebeam focuses down to zero spot size with a focal length givenby . However, as the beam focusesthe intensity on axis increases, resulting in ionization andplasma defocusing, as is described by (46).

The propagation dynamics including the effects of ioniza-tion can be studied by numerically solving [90] the envelopeequation (46). The laser pulse is assumed to be linearlypolarized with a Gaussian radial profile and an initialaxial profile given by for ,where is the initial peak electric field,

3.0 10 W/cm is the initial peak intensity, and60 m is the pulse length. With wavelength 1 m andinitial spot size 80 m, the peak power is 3.0 GWand the diffraction length is 2.0 cm. The laser pulsepropagates in air at 1 atm: neutral gas density 2.710 cm , nonlinear index [202] 5.6 10 cm /W,normalized ionization potential 1.07, and nonlinearfocusing power 2.8 GW ( 1.1).

The simulation begins with the laser pulse at focusin the neutral gas. With the initial value of

the filling factor computed via (44), the envelope equation(46) is integrated in the simulation variables and

. Fig. 2(a) and (b) show the initial laser intensityand plasma density versus with the direction of

propagation toward the right. Plots ofand versus radius atthe pulse center ( 30 m) are shown in Fig. 2(c) for thiscase. Note that the nonlinear nature of the ionization processcauses the plasma density gradient versus bothand to beconsiderably sharper than the intensity gradient.

The evolution of the laser pulse is shown in Fig. 3(a)–(d),where the spot size (dashed line), intensity on axis (solidline), and plasma density on axis (dotted line) are plottedversus at (a) 0, (b) 6, (c) 8, and (d) 10 cm. Initially, thespot size is constant along the laser pulse, as shown in Fig.3(a). Because , the center of the pulse is focusedwhile the front and back portions diffract, as seen in Fig. 3(b).At 25 m, where and , diffractionbalances nonlinear focusing and the spot size remains constantat . Behind this point, focusing increases the laserintensity, producing a corresponding increase in the ioniza-

(a)

(b)

(c)

Fig. 2. Surface plots of (a) laser pulse intensityI and (b) plasma densitynpplotted versus(r; �) at z = 0 for air at 1 atm withIp0 = 3 � 1013 W/cm2

(propagation is toward the right). (c) The intensityI (solid line) and plasmadensitynp (dashed line) versusr at the pulse center (� = �30 �m).

tion rate. Because ionization is a highly nonlinear process,the steepness of the plasma density gradient also increases.Increased ionization and increased plasma density gradientsare shown in Fig. 3(b)–(d). Increased ionization causes thelatter portion of the laser pulse to diffract, as can be seen inFig. 3(c) and (d). The rapid change in the plasma density at thesteepening ionization front results in a correspondingly rapidchange in the defocusing of the laser pulse. This results in anincreasingly narrow intensity spike at the ionization front. Thegeneral behavior depicted in Fig. 3(a)–(d) occurs even whenthe power exceeds the nonlinear focusing threshold bya factor of two [90].


(a)

(b)

(c)

(d)

Fig. 3. Spot sizers (dashed line), intensityI (solid line), and plasma densitynp (dotted line) plotted versus� on axis at (a)z = 0, (b) 6, (c) 8, and (d) 10cm. The initial peak intensity isIp0 = 3.0� 1013 W/cm2, the initial spotsize isrs0 = 80 �m, and the peak power isP0 = 3.0 GW' 1:1PNG. Thedirection of propagation is toward the right.

Fig. 4. Equilibrium profiles of powerP (solid lines) and plasma densityon axis np (dashed lines) plotted versus� for three different values ofintensity Ip: I1 = 5 � 1013W/cm2, I2 = 6 � 1013 W/cm2, and I3 =

7 � 1013 W/cm2. Here,Es = (8�Ip=c)1=2 is constant versus� such that

rs = (2P=�Ip)1=2.

B. Self-Guided Pulse Profiles

In the presence of ionization, self-guided solutions to (46)can be obtained [90]. The condition for a self-guided beam,i.e., , is

(48)

This implies , where

(49)

is a function of only . The solution gives the self-guidedpulse power as a function of

(50)

Equation (48) or (50) describes a family of equilibrium solu-tions, i.e., there are various equilibrium profiles ,and which satisfy these equations. For example, ifan equilibrium is chosen such that is constant alongthe pulse, then is constant and (50) implies

and the spot size profile is given by. Behind the beam front, , the laser beam

power and plasma density increase such that the nonlinearself-focusing term and the plasma defocusing term remainbalanced.

Examples of matched beam equilibria are shown in Figs.4 and 5. Both cases consider a linearly polarized 1-

m laser pulse with a Gaussian radial profile propagating inair at 1 atm ( 2.7 10 cm 5.6 10cm /W, 1.07, and 2.8 GW). Fig. 4 showslaser power profiles (solid lines) and plasma density profiles(dashed lines) plotted versusalong the axis for equilibriawith constant profiles. Equilibria are shown for threedifferent values of the peak laser intensity: 5 10W/cm , 6 10 W/cm , and 7 10 W/cm .Here, is constant along the length ofthe pulse, such that the variation in power correspondsto a variation in spot size, . Note thatthe constant profile produces a constant ionization rateand a linear rise in . Also, the power profiles areexponential functions as given in (50).


Fig. 5. Equilibrium profiles of powerP (solid lines) and plasma densitynp on axis (dashed lines) plotted versus� for three different values ofleading-edge(� = 0) intensityIp(0): I1 = 5.0� 1013 W/cm2, andI2 = 5.1� 1013 W/cm2, andI3 = 5.2� 1013 W/cm2. Here,rs is constant versus� such thatI(�) = 2P (�)=�r2

s0.

Fig. 5 shows laser power and plasma density profiles forequilibria with constant profiles. In this case,matched profiles are determined numerically from (48)for three different values of the leading-edge intensity:

5 10 W/cm , 5.1 10 W/cm , and5.2 10 W/cm . In this case, the variation in powercorresponds to a variation in intensity , such that

increases with along the length of the pulse. As a result,the ionization rate increases as a function of. Increasedionization (defocusing) requires increased power (focusing)to compensate, further increasing the ionization in a highlynonlinear manner. As a result, the constantequilibriumprofiles can be very sensitive to the value ofas indicatedby Fig. 5.

C. Ionization-Modulation Instability

The self-guided laser pulse equilibria described above areinherently unstable, i.e., the pulse will undergo an ionization-modulation (IM) instability [90]. The IM instability is due tovarying degrees of ionization along the pulse and results inthe modulation of the pulse envelope and the disruption of theback of the pulse. To examine the stability of the self-guidedequilibrium, the envelope equation, (46), is expanded about theequilibrium solution. The perturbations andare such that and denote the perturbedspot size and laser field amplitude respectively. Furthermore,since the laser power within the paraxial approximation isnonevolving (independent of), the perturbations andare related by . Expansion of the envelopeequation, (46), yields

(51)

where has been assumed (typically the case).For the an equilibrium with a nearly constant spot size,

, the asymptotic (large ) behavior of canbe found using standard methods [90]. The growth of theperturbation is given by , where

(52)

is the number of -folds and is the equilibrium plasmadensity profile along the axis as given by

. If the equilibrium is nearly constant in , theplasma density profile is given by andthe number of -folds is , where

. The IM instability grows as afunction of the distance behind the head of the laser pulse,

, and the propagation distance.The dependence of on indicates that the number

of -folds at the back of the pulse is greater than nearthe front. The IM instability disrupts the back of the pulse,and the disruption point propagates toward the front. Thedisruption point can be defined as the point on the pulsewhere the initial perturbation is increased by , where

is the number of -folds necessary for disruption.This point moves toward the front of the pulse with relativevelocity , which is evaluatedat . For the case where the plasma densityprofile is linear, i.e., , the disruption velocity inthe pulse frame is .

To gain some understanding of the IM instability, considerincreasing the spot size of an initially matched laser pulse, i.e.,

. In this case, the intensity and ionization rateare reduced resulting in less plasma generation and enhancedfocusing of the pulse. The focusing laser pulse overshootsits equilibrium value such that some distance behindthe pulse front. When , the intensity, ionization rate,and plasma density increase, causing the pulse to defocus andovershoot its equilibrium value. This focusing and defocusingresults in the IM instability. The modulation amplitude andperiod are functions of the distance back from the head of thelaser pulse, , and the propagation distance,, as indicatedby (52).

An example of the IM instability for a Gaussian laser pulseis shown in Fig. 6, obtained by numerical solution of theenvelope equation, (46). Here, the propagation in air of aconstant equilibrium is considered with 3.0

10 W/cm , 78 m, and 1.9 cm. In thiscase, there is very little initial ionization and the growth ofthe instability is extremely slow with GWalong the length of the laser pulse. The evolution of the laserpulse is shown in Fig. 6(a)–(d), where the spot size(solidline) and plasma density on axis (dashed line) are plottedversus at (a) 0, (b) 400, (c) 500, and (d) 600 cm. InFig. 6, the direction of propagation is toward the right.

The simulation begins, Fig. 6(a), with the laser pulse at focusin the neutral gas. In Fig. 6(a), the spot size

is constant along the pulse and increases linearly sinceis approximately constant. At later times, Fig. 6(b)–(d),

oscillations in cause oscillations in the ionization rate suchthat each region where has decreased corresponds to anincrease in ionization. Eventually, there is a large enoughincrease in the plasma density so that the latter portion of thelaser pulse is defocused, i.e., the guiding is disrupted. Whenthe laser pulse is sufficiently defocused, the ionization rate fallsand . Thus, an “ionization front” develops whichpropagates forward in the pulse frame. Fig. 6 indicates that thedisruption velocity is in good agreement with the theoreticalvalue .


(a)

(b)

(c)

(d)

Fig. 6. Spot sizers (solid line) and plasma densitynp on axis (dashed line)plotted versus� at (a)z = 0, (b) 400, (c) 500, and (d) 600 cm, for an initiallymatched laser pulse withIp = 3.0 � 1013 W/cm2 and rs(�) ' 78 �mpropagating in air (propagation is toward the right).

(a)

(b)

Fig. 7. Perturbed radiusln(j�rj) (solid line) and number ofe-folds Ne

(dashed line) plotted (a) versus� at fixedz = 550 cm and (b) versusz = �at fixed� = �49 �m. Here,�r = (rs � rs0)=rs0 is determined from theintegration of the envelope equation (46) whileNe is given by (52).

The growth of the instability of Fig. 6 is plotted versusat fixed 550 cm in Fig. 7(a), where , from thenumerical integration of the envelope equation, is comparedto the number of -folds from (52). Here,

. Similarly, versus at fixed 49 mis shown in Fig. 7(b). For both plots, excellent agreement isobserved between the slope of and the peaks of the

curve. As expected, agreement tends to break downfor , where the growth is not yet asymptotic, and for

, where the growth is nonlinear.

V. WAVE EQUATION FOR APLASMA

For a fully ionized plasma, it is convenient to represent theelectric and magnetic fields by the scaler and vector

potentials, and , and touse Coulomb gauge, . In terms of the normalizedpotentials and , the wave equationand Poisson’s equation are given by, respectively,

(53)

(54)

where is the normalized electron fluid velocity,is the plasma electron density, is the initial density profile(prior to the passage of the laser pulse), with

corresponding to the direction of propagation (theaxis), and . Here and in the


following, it is assumed that the ions remain stationary, whichis typically the case for short-pulse lasers (1 ps) propagatingin underdense plasma . Furthermore, collisionsand thermal effects are neglected, since the collision time istypically much greater than the laser pulse length; and sincethe thermal velocity is typically much less than the quivervelocity of an electron in the laser field.

The first term on the right of (53) is the contribution due tothe plasma current. In the cold fluid limit, , wherethe plasma density and velocity satisfy the continuity andmomentum equations, which are given by, respectively,

(55)

(56)

where is the normalized electron fluidmomentum and is therelativistic factor. Defining , where is oftenreferred to as the quiver momentum, (56) implies [114]–[117]

(57)

which is exact under the assumption that the quantityis initially (prior to the passage of the laser pulse) zero. Thefirst term on the right of (57) is the space charge force and thesecond term represents the generalized ponderomotive force

.It is also convenient to introduce the independent variables

and , where is a constant equalto the linear group velocity of the laser pulse in a plasma,

. Initially, the front ofthe laser pulse is assumed to be at and the pulse bodyextends into the region (the plasma is unperturbed inthe region ). In terms of the coordinates, the waveequation is given by [53], [108], [109], [114], [115]

(58)

where . On the right side of (58),the term has been neglected, since the fast part ofthe electrostatic potential, , is typically smallcompared to relevant terms contributing to the fast part of theplasma current. Typically, the third and fourth terms on the leftside of (58) can be neglected. In addition, if the approximation

is made to the second term of the left side of(58), assuming and , then the resultingequation is referred to as the paraxial wave equation.

A. Linear 3-D Regime

In the linear regime, , it is straightforward tocalculate the plasma fluid response for an initially uniformplasma, [97], [98], [148]–[151], [203]. This is doneby introducing a perturbation expansion for the various fluidquantities of the form , where . Thezeroth order describes the equilibrium plasma (in the absenceof the laser pulse), , and . Thefirst-order response is simply the electron quiver motion in the

laser field, and . To secondorder, expansion of the momentum equation yields

(59)

The first term on the right is the space charge force and thesecond term is the ponderomotive force .Combining (59) with the second-order continuity and Pois-son’s equation yields [97], [98]

(60a)

(60b)

(60c)

along with and .Letting , where and

, the wave equation in the paraxial limit isgiven by

(61)

where terms to third order in have been retained on theright side of (61). The term arises from the second-ordercontribution to the relativistic factor, . Thesecond-order density response is given by (60b), i.e.,

(62)

Equations (61) and (62) describe completely the 3-D laserplasma interaction in the linear regime within the paraxialapproximation. Furthermore, the effective index of refraction isdefined by setting the right side of (61) equal to ,which gives

(63)

B. Nonlinear 1-D Regime

A useful approximation in the study of short-pulse laser-plasma interactions is the quasi-static approximation (QSA)[108]–[110], [114], [115], [127]. In the QSA, the plasma fluidequations are written in terms of the independent variables

and , where is the velocity of the driver(e.g., laser pulse). The QSA assumes that in the time it takesthe laser pulse to transit a plasma electron, the laser pulsedoes not significantly evolve. In other words, , where

is the laser pulse duration and is the laser pulseevolution time, which is typically on the order of a Rayleigh(diffraction) time . Thus, the plasma electrons experiencea static (independent of) laser field. In the QSA, thederivatives are neglected in the plasma fluid equations whichdetermine the plasma response to the laser pulse. Thederivatives, however, are retained in the wave equation whichdescribes the evolution of the laser pulse. The QSA allowsthe laser-plasma interaction to be calculated in an iterativefashion. For a fixed , the plasma response to the laser fieldis determined as a function of by solving the QSA fluid


equations. Using this fluid response, the wave equation is thensolved to update the laser pulse in.

The plasma response in the nonlinear 1-D regime can beexamined within the QSA by neglecting the derivativesin the fluid equations (54)–(57). In effect, the QSA fluidresponse assumes that the drive beam is nonevolving, i.e., thedrive beam is a function of only the coordinate . The1-D limit applies to broad laser pulses, , where isthe laser spot size. Setting ,and in (54)–(57) imply [53], [108], [109]

(64a)

(64b)

(64c)

where . Equations (64a)–(64c) can be interpreted asconservation of transverse canonical momentum, conservationof particles, and conservation of energy in theframe, respectively. Furthermore, implies in1-D. Equations (64a)–(64c) along with

can be solved to specify the fluid quantities in termsof the field quantities and [53], i.e.,

(65a)

(65b)

(65c)

where and .Using the above expressions for the fluid quantities, expres-

sions for the normalized transverse plasma current, ,and the normalized charge perturbation, , canbe found and inserted into the wave equation and Poisson’sequation, (58) and (54). This results in two coupled nonlinearequations for the potentials and [53], [204]–[206]

(66)

(67)

The has been retained in the wave operator to correctlyaccount for the effects of diffraction. Equations (66) and (67)completely describe the nonlinear laser-plasma response in 1-D within the QSA, i.e., . For example, (66) and (67)have been used to analyze relativistic harmonic generation [53]or, for the case where the laser pulse lengthis approximatelyequal to the plasma wavelength , laser wakefieldgeneration [204]–[206].

In the limit , (65)–(67) reduce to [108]–[110], [207],[208]

(68a)

(68b)

(68c)

(69)

(70)

The paraxial approximation is equivalent to neglecting theterm and replacing with in (69).

Furthermore, the effective index of refraction is definedby setting the right-hand side of (69) equal to ,which gives [53]

(71)

where is determined from (67) or, in the limit , (70).

C. Nonlinear 3-D Regime

The nonlinear fluid model can be generalized to include 3-D effects within the QSA with and asthe independent variables [114], [115], [126]. The 3-D modeldeveloped in [114], which takes advantage of the separationbetween the fast and slow time and space scales, isvalid when and , where is the laserpulse length and is the plasma wavelength.Assuming a linearly polarized laser field with a transversecomponent of the form , thewave equation describing the evolution of the slowly varyingamplitude is given by

(72)

where , and the subscripts anddenote the fast and slow components, respectively. The

small term has been neglected in the wave operator,however, the term is retained so as to correctlydescribe variations in the laser pulse group velocity (in theparaxial approximation ).

The fluid response is described by the quasi-static forms of(54)–(57). All quantities are expanded in slow and fast terms,

, where and. The resulting equations are retained to first order in the

parameters and . For example,the quasi-static form of the momentum equation (57) can bewritten in general as

(73)

The first term on the right of (73) is the effective nonlinearponderomotive force and the second is thespace charge force. The fast transverse component of (73)yields , whereas the component of (73) yieldsthe quasi-static constant of the motion .This allows the slow component of the relativistic factor tobe written as

(74)

where . Furthermore, the slow component ofthe wave equation and Poisson’s equation, (53) and (54), are


given by

(75)

(76)

respectively, where . The slow part of the conti-nuity equation (55) follows from the divergence of (75).

In the axisymmetric limit, (73)–(76), along with ,can be combined to yield a single equation [114], [115], [126]of the form . This equation can bewritten as

(77)

where is given by (74),and are given by

(78a)

(78b)

Substituting this expression for into (72) gives [114], [115],[126]

(79)

The wave equation (79) together with , (77),completely describe the 2-D axisymmetric, quasi-static laser-plasma interaction. In the 1-D (broad pulse) limit, ,(77)–(79) reduce to the set of coupled equations describingthe nonlinear 1-D model given by (69) and (70). In thelong-pulse limit, , whichneglects the generation of plasma waves, , and

.Note that the refractive index is solely a function of

through , i.e., . In terms of

(80)

where is determined from the quasi-static fluid equations(77) and (78). Also, the wake potential is related to the axialelectric field induced in the plasma by ,where and [cm ]V/cm is the cold, nonrelativistic wavebreaking field.

VI. OPTICAL GUIDING IN PLASMAS

The optical guiding mechanisms discussed below are basedon the principle of refractive guiding. Refractive guidingbecomes possible when the radial profile of the index ofrefraction, , exhibits a maximum on axis, i.e.,

. Since implies that the phasevelocity along the propagation axis is less than it is off-axis.This causes the laser phase fronts to curve such that the beamfocuses toward the axis.

The index of refraction for a small amplitude electromag-netic wave propagating in a plasma of uniform density ,in the 1-D limit, is given by . For

large amplitude waves, however, variations in the electrondensity and mass will occur, i.e., . Hence,the general expression for the index of refraction for a largeamplitude electromagnetic wave in a plasma is given by [108],[109], [114]

(81)

assuming . The profile can be modifiedby the relativistic factor or the radial density profile

. The leading order motion of the electrons in the laserfield is the quiver motion and, hence,

. A laser intensity profile peaked on axisleads to and the possibility of

guiding (i.e., relativistic self-focusing). The density profilecan have contributions from a preformed density channel

or a plasma wave ,where . A radial density profile whichhas a minimum on axis (i.e., a channel) implies .In the limits , and , therefractive index is

(82)

In the above expression, the term is responsible forrelativistic optical guiding [91]–[128], the term isresponsible for preformed density channel guiding [36]–[43],[113]–[115], [129]–[147], and the term is responsiblefor self-channeling [96], [104], [112]–[125], plasma waveguiding [106]–[110], and self-modulation of long laser pulses[114], [115], [148]–[159].

A. Relativistic Optical Guiding

The self-focusing of laser beams by relativistic effects wasfirst considered by Litvak [91] and Maxet al. [92]. In thestandard theory of relativistic optical guiding [91]–[95], onlythe effects of the transverse quiver motion of the electronsare included in the expression for , i.e., and

, where and circular polarizationis assumed. Inclusion of the self-consistent density response,however, indicates that relativistic self-focusing is ineffectivein preventing the diffraction of short laser pulses[108], [109], [114]. Before discussing the effects of the densityresponse, the standard theory of relativistic self-focusing willbe discussed in the mildly and in the highly relativistic limits.

1) Mildly Relativistic Limit: In the mildly relativistic limit, the refractive index is given by

(83)

where the density response has been neglected .Refractive guiding requires , which is the casefor a laser intensity profile peaked on axis, .The paraxial wave equation with a refractive index given by(83) has the form of a Schroedinger equation with a third-order nonlinearity, as is the case in nonlinear optics where

. Hence, self-focusing will occur when the laserpower exceeds a critical power [91]–[95].


By applying the SDE method to the paraxial wave equation,as discussed in Section II, an equation for the laser spot size

can be derived, (27). Using the index of refractiongiven by (83) with a Gaussian intensity profile of the form

, the laser spot size evolvesaccording to [95], [99]

(84)

where is the normalized spot size, is theminimum spot size in vacuum, and is thevacuum Rayleigh length. The first term on the right of(84) represents vacuum diffraction, whereas the second termrepresents relativistic self-focusing. Here,for circular polarization (for linear polarization, ).The critical power for relativistic self-focusing is

, where , or in practicalunits [95]

[GW] (85)

The solution to (84) with at is

(86)

which indicates that the spot size diffracts for , remainsguided for , and focuses for . Infact, (84) predicts “catastrophic” focusing. This is due to theapproximation in the limit.In actuality, higher order nonlinearities will prevent the laserfrom focusing indefinitely [95].

Equation (84) describes the basic theory of relativistic self-focusing of a single laser beam in a plasma in the limit

. Furthermore, it is possible for a high-powerrelativistically self-focused beam to guide a second low-powerbeam that copropagates with the first beam but has adifferent wavelength [99]. This is the case since the low-powerbeam will experience the relativistic modifications to the indexof refraction profile created by the high-power beam.

2) Highly Relativistic Limit: In the highly relativisticlimit, , an equation for the laser spot size can be derived[95] by retaining the full relativistic factorin the index of refraction

(87)

As before, however, the self-consistent density response isneglected in the standard theory of relativistic self-focusing,i.e., is assumed. By applying the SDE to the paraxialwave equation with given by (87), the normalized spot size

obeys the equation [95], [99]

(88)

where

(89)

with . The first term on the right of (89)represents vacuum diffraction, whereas the remaining terms

represent the relativistic self-focusing effects of the plasma. Inthe limit ( ), (88) reduces to (84).

Equation (88) describes the position of a particlemoving in an effective potential [95]. The shapeof the potential depends only on the parameter .For , the potential has a minimum (at

) and bounded oscillatory solutions for arepossible. This corresponds to a guided laser beam with aspot size that oscillates about its matched beam value.As examples, the effective potential is plotted in Fig.8(a) for and in Fig. 8(b) for

. The shape of the potential atsmall clearly indicates that catastrophic self-focusing willnot occur. The specific trajectory for the spot sizedepends on the initial conditions, i.e.,and (the initialconvergence/divergence angle) at . In particular, if theinitial is too large, then the laser beam will not beguided, even if . Instead, for example, the beam willfirst focus to a smaller spot size than it would in vacuum andthen diffract rapidly out of the effective potential [95].

3) Self-Consistent Theory:The previous discussion of rel-ativistic guiding neglected the electron density response inthe expression for the index of refraction. The effects of theself-consistent density response can be included by using thequasistatic theory discussed in Section V-B. In particular, itcan be shown that relativistic optical guiding is ineffectivein preventing the diffraction of sufficiently short pulses,

[108]–[110], [114]. This is due to the fact that theindex of refraction becomes modified by the laser pulse onthe plasma frequency time scale, not the laser frequency timescale. Typically, relativistic guiding only effects the body oflong pulses, .

In the 1-D limit, , nonlinear quasi-static theory[108]–[110] indicates that the self-consistent electron fluidresponse satisfies , hence

(90)

where is the normalized electrostatic potential which satisfiesthe nonlinear Poisson equation (70), assuming .For long laser pulses with sufficiently smooth envelopes,

can be neglected in (70) (whichneglects the generation of plasma waves) and

. Hence, in the long-pulse limit , the index ofrefraction has the form given by (87) and the standard theoryof relativistic focusing discussed in the previous section canbe applied to the body of long pulses. Although long pulsescan be guided by relativistic effects, they can also be unstableto self-modulation and laser-hose instabilities [114], [141],[142], [148]–[157], which are discussed in more detail in thesubsequent sections.

The fact that short pulses diffract even whencan be most easily shown in the mildly relativistic

limit . Here, and satisfies (60a)

(91)

Again, in the long-pulse limitimplies and the index of refraction is given by


(a)

(b)

Fig. 8. The effective potentialV (X) as a function ofX = rs=a0r0for (a)P=Pc = 1.2 and (b)P=Pc =5. The well minimum occurs atXf = 1.8 in(a) andXf = 0.4 in (b).

(83), i.e., standard relativistic guiding applies. This, however,neglects the generation of plasma waves, ,which can lead to the self-modulation of long pulses. For shortpulses , the term can be neglected on the left of (91).For example, a short pulse with a constant intensity profile

induces a space charge potential which is given byand the refractive index becomes

(92)

Fig. 9. Laser spot sizers versus propagation distancec� for (a) vacuumdiffraction, (b) an ultrashort pulse withL = �p=4, and (c) a short pulsewith L = �p. Here,P = Pc, a0 = 0.9, and�p = 0.03 cm. Guiding ofthe L = �p pulse in a preformed, parabolic plasma density channel with�n = 1=�rer2s is shown by (d).

as opposed to (83). This indicates that the effective criticalpower for a short pulse [108]–[110] is

, since for a short pulse. In particular,becomes infinite at the leading edge of the pulse . Hence,the leading portion of a laser pulse will diffractivelyerode even when .

Simulations [114], based on the 2-D axisymmetric fluidmodel discussed in Section V-C, confirm the inability ofrelativistic guiding to prevent the diffraction of short laserpulses. The results are shown in Fig. 9 for the parameters0.03 cm ( 1.2 10 cm ), (Gaussian radialprofile), 1 m ( 28 cm) and . The initialaxial laser profile is given by for

, where 0.9 for the above parameters.Simulations are performed for two laser pulse lengths,( 1 ps) and ( 0.25 ps). The spot size at thepulse center versus propagation distanceis shown in Fig.9 for (a) the vacuum diffraction case, (b) the pulse,and (c) the pulse. The pulse diffracts almostas if in vacuum. The pulse experiences a small amountof initial guiding before diffracting. A preformed parabolicplasma density channel, however, is effective in guiding the

pulse, as shown in Fig. 9(d), where the channel depthis given by 1.3 10 cm and the densityon axis is 1.2 10 cm .

B. Tailored Pulse Propagation

A laser pulse with an appropriately tailored profile canpropagate many Rayleigh lengths without dramatic distortion[114], [115], [126]. Consider a long laser pulse, ,in which the spot size is tapered from a large value at thefront to a small value at the back, so that the laser power,

, is constant throughout the pulse and equal to. The leading portion of the pulse will diffract as

if in vacuum, as discussed in the previous section. However,since is large at the front of the pulse, the local Rayleighlength is also large. Hence, the locally large spot size allowsthe pulse front to propagate a long distance, whereas the bodyof the pulse will be relativistically guided. Also, since


(a)

(b)

Fig. 10. Surface plot of the normalized laser intensity,ja2f j, at (a) � = 0and (b)c� = 10ZR for a tailored-pulse LWFA.

increases slowly throughout the pulse, detrimental wakefieldeffects (e.g., self-modulation) are reduced.

The effectiveness of pulse tailoring has been observedin simulations [114], [115], [126]. The results of such asimulation [115], based on the 2-D axisymmetric fluid modeldiscussed in Section V-C, are shown in Figs. 10–12. The initialnormalized intensity profile, , is shown in Fig. 10(a). Thelocal spot size at the front of the pulse is large,

, and tapers down over to (aGaussian radial profile assumed throughout). The initial axiallaser envelope is given by for

such that at each slice, i.e.,. Also, 1 m and 30 m

( 1.2 10 cm , initially a uniform plasma), such that16 TW. This gives a peak value of 0.9 at

the back of the pulse where , which corresponds to aRayleigh length of 0.28 cm. The pulse intensity thenterminates over a distance of . The pulse energyis approximately 3 J and the pulse length is approximately

60 m (200 fs). Because , a largeamplitude wakefield will be excited behind the pulse.

Fig. 10(b) shows the normalized intensity profile afterpropagating 2.8 cm. The pulse is somewhatdistorted, but largely intact. The evolution of the pulse spotsize at the position of peak intensity verses propagationdistance is shown in Fig. 11(b), indicating that guiding hasbeen achieved over the simulation region. The axialelectric field of the wake on axis after is shown inFig. 12. The evolution of a continuous electron beam with aninitial normalized emittance 1.0 mmmrad, RMS radius

5 m, and energy 2 MeV was simulated using the

Fig. 11. Laser spot sizers at the position of peak intensity versus propaga-tion distancec� for (a) a channel-guided LWFA, (b) a tailored-pulse LWFA,(c) vacuum diffraction, and (d) a self-modulated LWFA.

Fig. 12. Axial electric fieldEz on axis atc� = 10ZR for the tailored-pulseLWFA of Fig. 10.

self-consistent wakefields. After 2.8 cm, approximately60% of the beam electrons were trapped and accelerated. Thepeak energy of the beam electrons experienced an averagegradient of 27 GeV/m (750 MeV in 2.8 cm).

C. Preformed Plasma Density Channels

The concept of using a plasma density channel to guidea laser beam dates back to early studies of laser fusion[129]–[132]. Density channels in plasmas have been createdby a number of methods. An intense laser pulse propagatingin a plasma can create a channel through a combination ofponderomotive and thermal effects. The creation of a densitychannel through the hydrodynamic expansion of the radialplasma profile was observed in the early 1970’s in long-pulse(150 ns) CO laser experiments [131]. The length of such achannel, however, is limited to the propagation distance of thelaser pulse which creates the channel, and the utility of usingsuch a channel to guide a laser pulse many Rayleigh lengthsis limited. For high-power short laser pulses, the propagationlength and, hence, the channel length can be increased byrelativistic self-guiding, as has been observed in recent pump-probe experiments [34]. Alternatively, a long focal region canbe created with an axicon lens, and this method has been usedsuccessfully to create extended plasma channels in which laser


pulses have been guided for many Rayleigh lengths [36]–[41].Another approach for creating density channels is to use slowcapillary discharges. Capillary discharges have been used tocreate straight or curved plasma channels, in which laser pulseshave been guided for many Rayleigh lengths [42], [43]. Theseexperiments are discussed in more detail in Section VII.

1) Straight Plasma Channels:A preformed plasma den-sity channel can guide short, intense laser pulses [36]–[43],[80], [113]–[115], [129]–[147]. Consider a parabolic densitychannel of the form , where

is the channel depth. For a low-power ,low-intensity laser pulse, the index of refraction isgiven approximately by

(93)

Analysis of the paraxial wave equation with an index ofrefraction of this form indicates that the spot size of aGaussian laser beam withevolves according to [106], [151]

(94)

where and . The first term on theright represents the effects of vacuum diffraction and thesecond term represents the focusing effects of the channel. Thisindicates that a Gaussian beam will be guided at the matchedbeam spot size provided that the channel depth isequal to the critical channel depth given by [106], [113], [114]

(95)

or [cm ] [ m], where isthe classical electron radius.

The general solution to (94) for the initial conditionsand is [151]

(96)

where and is the injectedspot size. A matched beam requires , e.g.,

and . If the beam is not matchedwithin the channel, the spot size oscillates betweenand with an average value

. The oscillation period within thechannel is . The laserbeam will remain confined within the channel provided thatthe maximum radius of the channel is sufficiently large,i.e., .

To facilitate comparison to experiments [43], it is convenientto express these results in terms of the maximum radius (outeredge) of the channel and the maximum density of thechannel at the outer edge , i.e., a channel ofthe form for , where

, and are determined experimentally. In this notation,(94) can be rewritten as

(97)

Fig. 13. Surface plot of the plasma electron densityn=n0 at c� = 20ZRfor a channel-guided LWFA. The initial density profile is parabolic with adepth given by�n = 1=�rer

2

s .

where the matched beam spot size is given by

(98)

The solution to (97) for the initial conditionsand is given by (96) with

and , where . If ,the spot size oscillates between and withan average value and an oscillationperiod . The laser beam will remainconfined within the channel provided . For ,confinement requires . For , confinementrequires , which implies . Forthe matched case , confinement requires ,which implies

(99)

To illustrate the effectiveness of optical guiding using pre-formed density channels, the results of three simulations willbe presented, all based on the 2-D axisymmetric fluid modeldiscussed in Section V-C. The first simulation [115] is of achannel-guided laser wakefield accelerator (LWFA) with anultrashort , high-intensity laser pulse,the results of which are shown in Figs. 11, 13, and 14.In this example, the initial axial laser profile is given by

for , with 0.72and 120 m (400 fs). Also, 1 m and 60 m(Gaussian radial profile), which implies 1.1 cm and

40 TW. The density on axis is chosen such that( 7.8 10 cm ) and a parabolic profile is assumedwith 3.2 10 cm .

Fig. 11(a) shows the evolution of the laser spot size versuspropagation distance, . The laser pulse remains guidedby the density channel, the laser spot size exhibiting smalloscillations about its initial value over the full 23cm simulation length. After , the pulse profileshows very little distortion from its initial profile. A surfaceplot of the electron density profile at is shownin Fig. 13. The initial unperturbed parabolic profile can be


Fig. 14. Axial electric fieldEz on axis atc� = 20ZR for the channel-guidedLWFA of Fig. 13.

seen at the right , and the distortion of the channel bythe laser pulse, including the excitation of a large amplitudewakefield along the axis, is evident in the region . In thisexample, nearly all the electrons have been expelled from thevicinity of the laser pulse. The radial variation in the channeldensity causes a radial variation in the plasma wavelength anda curvature of the plasma wavefronts. A slight axial dampingof the plasma wave also occurs, as evident in Fig. 14, whereinthe axial electric field is plotted versus along the axisat . The effects of the wakefields on a continuouselectron beam with an initial normalized emittancemm mrad, RMS radius m, and energyMeV was also simulated. After 20 cm, approximately70% of the beam electrons were trapped and accelerated. Thepeak electron energy increases nearly linearly with propagationdistance with an average acceleration gradient of 5.25 GeV/m(1 GeV in 20 cm).

The second simulation [209] is of a channel-guided LWFAin the self-modulated regime with , the results of whichare shown in Figs. 15–18. Here, the initial laser parameters are

100 fs 30 m 0.3 TW (30 mJ),1 m, 10 m ( 0.3 mm), and 2 10

W/cm ( 0.38); and the channel parameters are 810 cm ( 12 m), , and 40 m.Also, the on-axis density is slightly tapered such that it risesto over a length of 1 cm. The pulse remains guided after

0.75 cm of propagation, but a large modulation(in with period ) in the intensity (Fig. 15) and power(Fig. 16) has developed due to self-modulation and forwardRaman instabilities, as discussed in Section VI-G. The densityprofile at 0.75 cm is shown in Fig. 17 along with thecorresponding on-axis electric field (Fig. 18) which clearlyshows a well-defined wakefield of amplitude 50 GV/m.

The third simulation [43] is an example of mismatched laserpropagation in a channel, which extends from 0.5 cm1.5 cm with 5 10 cm , and150 m (parameters near those of the experiment in [43]).Here, a 0.8- m, 100-fs, 30-GW (3 mJ), 1.6 diffraction-limited laser pulse is focused on the channel entrance withspot size 15 m. Due to the low laser power, the pulsedoes not become self-modulated. Fig. 19 shows that the laser

Fig. 15. Surface plot of the normalized laser intensitya2 after propagat-ing c� = 0.75 cm (24ZR) for a channel-guided self-modulated LWFA(propagation is to the right).

Fig. 16. Laser powerP versus� at c� = 0.75 cm (24ZR) for a chan-nel-guided self-modulated LWFA.

Fig. 17. Surface plot of the electron densityn=n0 at c� = 0.75 cm(24ZR)for a channel-guided self-modulated LWFA.

spot size oscillates about its matched value of 28 m,emerging from the 1-cm-long channel with a radius of 45mand a divergence angle of 14 mrad, in approximate agreementwith the experiment of [43].

The above discussion concerned essentially parabolic chan-nel profiles. Other channel profiles, however, may offer differ-ent advantages. Durfeeet al. [37], [38] discuss the formation


Fig. 18. Axial electric fieldEz of the wake atc� = 0.75 cm(24ZR) fora channel-guided self-modulated LWFA.

Fig. 19. Laser spot sizerL versus propagation distancez for vacuum (dashedline) and in a channel (solid line) located at 0.5 cm< z < 1.5 cm for alow-powerP=Pc � 1 mismatched pulse.

of “leaky” channels, in which the channel is approximatelyparabolic out to some radius, after which the density falls offto zero. Higher order transverse modes may not be guided bysuch a channel, and Antonsen and Mora [143] have describedhow leaky channels can stabilize certain instabilities, suchas small angle forward Raman scattering [149], [155], [156]self-modulation [114], [148], [151], and laser-hosing [141],[142]. Hollow channels (e.g., a square channel with densityzero on axis out to the channel radius) may have somebeneficial properties with regard to acceleration [133], [134],[136]–[139]. Within the channel, where the plasma density isessentially zero, the transverse profile of the axial wakefieldis uniform, thus providing uniform acceleration of an injectedbeam [137]. The wakefield in such a channel, however, maybe damped due to resonant absorption in the channel walls[139].

2) Curved Plasma Channels:A curved density channelcan be used to guide a laser pulse through a bend providedthat the radius of curvature is sufficiently large [43]. Laserpropagation in a curved plasma channel can be examinedanalytically in the low-laser-intensity limit [43]. In this limit,nonlinear (ponderomotive) effects are neglected and the chan-nel is assumed to be unaffected by the laser pulse. In a straightchannel, the slowly varying laser field amplitudeobeys theparaxial wave equation

(100)

where the linear index of refraction for a plasma channel isgiven by . Consider a channel which iscurved in the ( ) plane with a constant radius of curvature

, where is the distance along the curved channel axisand , and are defined with respectto the channel axis. It can be shown that the paraxial waveoperator in the curved coordinate system becomes

where andhigher order terms (smaller by at least ) have beenneglected. Hence, the laser field envelope,, obeys a paraxialwave equation of the form of (100) with an effective index ofrefraction given by

(101)

where the term represents the effects of curvature.Consider a density channel with radius of curvature

and a parabolic transverse profile of the form, where , and a laser field amplitude of the

form

(102)

where is the laser centroid, is the laser spot size,and is the matched spot size. The solution to the paraxialwave equation (100), with given by (101), indicates thatthe laser spot size evolves according to (94), as is thecase for a straight channel. The laser centroid , however,evolves according to

(103)

where and is the critical channel depthgiven by (95). Equation (103) indicates that the laser centroid

oscillates in about an equilibrium offset value

(104)

with an oscillation period . For amatched spot size and , the equilibriumoffset value is given by . This offset must beless than the maximum channel radius or the laser will belost from the channel. This sets a minimum acceptable radiusof curvature

(105)

As an example, consider parameters near those of a curvedchannel experiment [43] with a 0.8- m laser pulse in achannel with 150 m and 4 10cm . Equations (98) and (105) give a matched spot sizeof 27 m ( 0.31 cm) and a minimum radius ofcurvature of 6.4 cm.

D. Ponderomotive Self-Channeling

The radial ponderomotive force of a long laser pulsepropagating in an initially uniform plasma can expel elec-

trons from the axis, thus creating a density channel (i.e., self-channeling or electron cavitation) [96], [104], [112]–[125].This can enhance the effects of relativistic self-focusing.


Consider a long axially uniform laser pulsepropagating in an initially uniform plasma. The steady-stateradial force balances indicates that the space charge forceis equal to the ponderomotive force, i.e., ,where (circular polarization). This implies adensity perturbation via Poisson’s equation, ,given by [96], [104], [112], [114], [118]

(106)

assuming . The corresponding index of refractionis given by

(107)

This can also be derived from 3-D nonlinear plasma theoryvia (80). In the long-pulse limitand , which gives (107). Neglected in(107) is the generation of plasma waves, which can lead tothe self-modulation of long pulses.

In the limit , a Gaussian laser profilecreates a density profile

. Along the axis, the depth of theponderomotive channel is given by

(108)

where is given by (95). Analysis of the paraxial waveequation with a density perturbation given by

indicates that the normalized spot size of aGaussian laser pulse evolves according to [112]

(109)

where is given by (108) and is assumed.Hence, in the limit , the ponderomotive channeldepth required to guide a laser pulse is .Clearly, when , the ponderomotive self-channel alonewill not guide the laser pulse. Furthermore,implies and .Hence, implies , which againindicates that the ponderomotive channel alone will not guidethe laser pulse. For laser powers approaching the critical power

, guiding is achieved predominantly by relativisticself-focusing. Ponderomotive self-channeling can enhance thiseffect, but does not dramatically alter the power threshold forguiding. More detailed studies [96], [118] which included theeffects of relativistic self-focusing and ponderomotive self-channeling conclude that the threshold power for guiding is

GW.

E. Plasma Ion Motion

The above discussion of ponderomotive self-channelingneglected plasma ion motion. The ions were assumed to forma stationary, uniform background, and an electron densitychannel was formed by equating the radial ponderomotiveforce of the laser with the radial space charge force dueto the electron density perturbation. However, for intense,

tightly focused laser pulses of moderate pulse duration, where is the electron (ion) plasma

frequency, the effects of ion motion can be significant. This isthe case for laser wakefield accelerator experiments [15], [34]in the high-density self-modulated regime ( 1 ps,10 cm ). In particular, a large plasma density channel, inwhich both the electrons and ions are expelled radially, can beset up in the region behind the laser pulse [15], [34], [146],[147].

The ponderomotive force associated with an intense laserpulse leads to the creation of a plasma density channel asfollows. The laser pulses exerts a ponderomotive force onthe plasma electrons and expels them radially, as is the casein ponderomotive self-channeling. This sets up a large spacecharge force which subsequently drags the ions outward fromthe axis. After the passage of the intense laser pulse, the ionscontinue to drift radially at approximately the ion acousticspeed , thus creating a plasma channel,where is the number of plasma electrons per ion,is the electron (ion) temperature, and is the electron(ion) mass.

The plasma electron motion is described by the radial forcebalance

(110)

where is the space charge force,is the ponderomotive force, is the electron pressure,and is the electron density. This expression neglects thegeneration of plasma waves and assumes that the axial lengthof the laser pulse is large compared to the laser spot size andthe plasma wavelength. In the linear regime, the ion motionis described by the continuity equation

, and the momentum equation ,where and are the perturbed and ambient ion densities,

is the radial ion velocity, and is assumed.Combining the electron radial force balance, the ion continuityequation, and the ion momentum equation yields [15], [34],[146], [147]

(111)

assuming , and an isothermal equationof state.

Assuming a nonevolving laser intensity profile, which is a function of only the variables and

, (111) can be solved using the 2-D Green’sfunction for the wave equation [34], [147],

(112)

where , , and the integrand is onlynonzero in the region . Equation (112) hasbeen evaluated for a nonevolving laser pulse of the form

, with forand otherwise. The evolution of the density channel is


Fig. 20. Density channel created behind a shortL = 120-�m laser pulse witha0 = 0.25, r0 = 10 �m, Te = 100 eV,Z = 2, mi=m = 7300, and�s = 2.3 � 10�4. The laser pulse is located at 0–120�m (very near the left edge).

shown in Fig. 20 for the parameters 0.25, 10 m,120 m, 100 eV, 2, 7300, and

2.3 10 . Analysis indicates that the density channel reachesa maximum depth ofafter a distance of , where are constantswhich depend on the shape of the laser pulse profile (for theexample in Fig. 20, 0.46 and 0.78). For

, the channel depth increases roughly linearly,, and for , the channel depth

decreases roughly as , whereare constants ( 0.77 and 0.40 in Fig. 20).

If the intense laser pulse is of sufficiently high power,, it can be guided through the plasma over a distance

of many Rayleigh lengths by a combination of relativisticself-focusing and ponderomotive self-channeling. A large den-sity channel can be formed behind the pulse as describedabove. Such a channel is capable of guiding a second low-power probe pulse, injected some distance behind the intensepump pulse. This effect has been observed in pump-probeexperiments in the self-modulated LWFA regime [15], [34].

F. Plasma Wave-Guiding

An ultrashort laser pulse can be guided bya plasma wave wakefield, provided that the laser pulse isproperly phased within the wakefield and the wakefield ampli-tude is sufficiently large [106]–[110]. The effective index of

refraction for a low-power , low-intensitylaser pulse propagating in a plasma wave is given by

(113)

where is the density oscillation of the plasma wave, whichis assumed to be unaffected by the low-intensity laser pulse.Consider a plasma wave of the form ,where and . In regions where ,the plasma wave acts as a local density channel and enhancesfocusing; in regions where , the plasma waveenhances diffraction.

The evolution of a “test” laser pulse in an externallygenerated plasma wave can be analyzed using the paraxialwave equation. A nonevolving plasma wave of the form

with will beassumed, i.e., the plasma wave has a Gaussian radial profilewith a radius and a phase velocity . It can beshown that the spot size of a Gaussian laser pulse evolvesaccording to [106]

(114)

where is the critical channel depth,, and and have been assumed.

Consider an ultrashort pulse centered about


Fig. 21. Schematic of the focusing effects of an externally generated plasmawave on an initially uniform low-intensity laser beam. This illustrates thetendency for the laser to form beamlets which are optically guided in theregions of minimum density.

such that . Equation (114) indicates that thispulse will be guided by the plasma wave provided [106]

(115)

which gives for . Notice that a testlaser pulse experiences maximum focusing at the minimumof (i.e., ). In addition, it can be shown that ashort laser pulse can be frequency upshifted by a plasma wavewakefield provided that it resides in the phase region where

. In particular, maximum frequency upshiftingoccurs at the maximum of (i.e., for theabove example). In general, for a sinusoidal plasma wave, atest laser pulse will experience both enhanced focusing andfrequency upshifting over a phase region ofthe plasma wave. Furthermore, (114) describes how a plasmawave can lead to the modulation of a long ( ) laserpulse [106], [107], as illustrated schematically in Fig. 21.

In addition to a plasma wave acting as a local density chan-nel and providing periodic regions of enhanced focusing anddiffraction as described above, a plasma wave can enhance theself-focusing of long laser pulses by several othermethods. For example, the electric field profile of theplasma wave can provide an additional radial ponderomotiveforce via [210]. In addition, the oscillatory motion ofthe plasma electrons in the plasma wave can contribute tothe relativistic factor [100]. Furthermore, the plasma wavecan lead to the generation of higher order Stokes and anti-Stokes light waves (i.e., energy cascading) which can affectself-focusing [101], [102]. These effects have been observedin experiments [210] and simulations [100]–[102] of two-frequency laser-plasma interactions, in which the plasma waveis resonantly driven by the laser beat wave.

G. Self-Modulation of Guided Laser Pulses

A long laser pulse that is guided by relativistic self-focusing or by a preformed plasma channel, i.e.,

, is subject to severe self-modulation [114], [115],[148]–[159]. Specifically, a plasma wave, excited initially bythe ponderomotive force associated with the finite rise of thelaser pulse, can strongly affect the focusing properties of thepulse body [106], [107]. The process of self-modulation canbe understood by considering a long, , opticallyguided, , laser pulse on which afinite wakefield exists. Associated with the wake is a den-sity modulation of the form , whichmodifies the plasma refractive index, as indicated by (82)and (113). This density oscillation acts on the laser pulseas an axially periodic density channel. In regions of a localdensity channel, i.e., where , the radiation focuses.In regions where , diffraction is enhanced.This causes the laser pulse envelope to become modulatedat . The axial ponderomotive force associated with themodulated laser intensity subsequently enhances the growthof the plasma wave, and the process proceeds in an unstableand highly nonlinear manner. The end result can be a fullyself-modulated laser pulse, composed of a series of laser“beamlets” of length , which can remain opticallyguided over many Rayleigh lengths. Associated with theperiodic beamlet structure are large amplitude wakefieldswhich can trap and accelerate a trailing electron beam [115],[150]. This process forms the basis of the self-modulatedLWFA.

An equation for the evolution of the laser spot size, (27), canbe derived by analyzing the paraxial wave equation with theSDE method, as discussed in Section II. An index of refractionof the form given by (82) will be assumed, which includes aparabolic density channel , as well as thedensity perturbation due to the self-consistent plasma responsein the linear regime, i.e., ,as discussed in Section V-A. In the limits and

, the spot size evolves according to [151]

(116)

where and . The second, third andfourth terms on the left in (116) represent the effects ofvacuum diffraction, relativistic focusing, and channel focusing,respectively, whereas the term on the right side representsthe nonlinear coupling of the laser envelope to the plasmawave. Equation (116) correctly describes well-known laserpulse evolution, such as the inability of relativistic guidingto prevent the diffraction of short pulses [108]–[110],[114].

The evolution of a long, axially uniform laser beam can beexamined in the limit where the effect of the plasma wave isneglected, i.e., the nonlinear coupling term on the right sideof (116) is set equal to zero. Neglecting the coupling term, the


solution to (116) for the initial conditionsand is [151]

(117)

where and is the injected spotsize. For , the spot size oscillates betweenand with an oscillation period

. A matched beam withrequires , where [122], [151]

(118)

i.e., the effective critical power for guiding is reducedby a finite density channel (assuming ). Noticethat for and , (117) reduces to

. This indicates that beamwill initially focus for or diffract for withan effective Rayleigh length of .

The effect of the plasma wave on the spot size evolution isdescribed by the right-hand side of (116). The initial effect ofthe plasma wave can be estimated by approximating

within the integral in (116), i.e., initially the spot sizeis uniform throughout the pulse. Equation (116) takes on theform

(119a)

where is the initial density perturbation given by

(119b)

The rise associated with the front of the pulse gives a nonzerovalue of which generates a finite amplitude densitywake. Throughout the body of a long flat-top pulse, this densitywake has the form . In particular, for aflat-top pulse with a fast rise, , (119b) gives

and the third term on the rightof (119a) can be written as .Hence, at the phase regions where , focusingrequires (for , the initial wakevanishes and focusing requires ). Clearly, the effectof the initial density wake in (119a) is to produce -periodic regions of enhanced focusing and diffraction [106],[107]. This causes the laser intensity to become modulatedat , which subsequently enhances the density wake at latertimes. This is the basis of the self-modulation instability.

The growth of the self-modulation instability in a long, optically guided laser pulse can be

analyzed by perturbing (116) about the matched-beam spotsize , where . Asymptoticgrowth rates can be obtained in various regimes using standard

methods. The number of-folds , where , inthe various regimes is given by [142], [151]

Long pulse regime:

(120a)

Intermediate regime:

(120b)

Short pulse regime:

(120c)

Here, ,and . The numberof -folds is a function of the normalized power ,the normalized distance behind the pulse head , andthe normalized propagation distance . Self-modulationconsists of a 2-D axisymmetric “sausaging” of the laserpulse envelope. In 3-D, a related instablity, the laser-hoseinstability [141], [142], can occur which causes a “kinking”of the laser pulse envelope, i.e., the laser pulse centroidcan become unstable to a periodic transverse displacement.Analysis [142] indicates that the number of-folds for thelaser-hose instability is given by (120a)–(120c) with

.An alternative mechansim that can lead to laser pulse mod-

ulation is forward Raman scattering (FRS) [143], [155]–[157].In the 1-D limit, FRS produces modulation via an axialtransport of laser energy. This is in contrast to the envelopeself-modulation instability described above, which is an inher-ently 2-D process wherein modulation results from a radialtransport of laser energy. Some insight can be gained bynoting that the ratio of for self-modulation in the long-pulse regime, (120a), to that of 1-D FRS in the four-wavenonresonant regime [151] scales as . This sup-ports the assertion that self-modulation dominates in the 2-Dlimit, whereas FRS dominates in the 1-D limit, roughlyspeaking, when . These two growth rates,however, occur in different spatial-temporal regimes, hence,comparison of the growth of self-modulation and FRS is morecomplicated. It should also be noted that the growth rate ofsmall-angle FRS in the four-wave nonresonant regime [155],[156], in which the scattered light propagates at an angle of

with respect to the axis, is similar to that of (120a)when .

The self-modulation instability is illustrated by numericallysolving [151] the envelope equation (116) for a long laserpulse (a long rise, , followed by a long flat-top region, ) with and

. Fig. 22 shows the evolution of(a) the normalized spot size (initially, ) and (b)the normalized intensity on axis , plottedversus after a propagation distance of .Note that the front portion of the pulse with is


(a)

(b)

Fig. 22. Numerical solution of (116) showing self-modulation: (a) thenormalized spot sizeR = rs=r0 and (b) the normalized envelope profilea2

versus� at c� = 1:4ZR. Initially, the spot size is uniform and the intensityhas a slow rise followed by a long flat-top region(� � �5�p) with P = Pc.

diffracting. Clearly, the modulation is growing as a functionof both the propagation distance and the distance behind thepulse front. The growth rates obtained numerically are ingood agreement with the theoretical values given by (120).Additional simulations [151] of a flat-top pulse with a fastrise indicate that strong self-modulation occurswhen , in agreement with theory. Formodulation is reduced, since the pulse envelope is everywhere(at all ) diffracting. This is illustrated in Fig. 23, where thenormalized modulation amplitude is plotted versus

, where is the -averaged value of , as obtainedfrom (116) with , and 0.25, 0.5, and 0.75.

In the highly nonlinear regime, self-modulation of a laserpulse can be simulated using the nonlinear quasi-static fluidmodel discussed in Section V-C. In this model, the effectsof FRS are included by retaining the term in thewave operator. The following example [115], [150] is relevantto a LWFA in the self-modulated regime, in which the axialelectric field of the wake is used to accelerate an electronbeam. The simulation begins at 0 as the 1 mGaussian laser pulse enters the plasma, initially convergingsuch that in vacuum it would focus to a minimum spot sizeof 31 m ( 3 mm) at with0.7. The initial pulse length is 90 m (300 fs) with

Fig. 23. Numerical solution of (116) showing the normalized modulationamplitude�R=hRi versusc� for a flat-top pulse withLrise = 0:1�p andP=Pc = 0.25, 0.5, and 0.75.

Fig. 24. Ambient plasma densitynp=n0 (solid line) and spot sizers=�p(dashed line) versus propagation distancec� for a self-modulated LWFAwith n0 = 2.8 � 1018 cm�3.The laser is initially converging such thatthe minimum spot size in vacuum is reached atc� = 3ZR. Here,rs is thespot size of the leading beamlet and is defined to be the radius enclosing86.5% of the laser power.

a peak power of 10 TW and a total energy of 1.5 J.The plasma density is initially increasing, reaching a flat-topdensity of 2.8 10 cm at , as shownin Fig. 24. This gives 20 m ( ) and6.7 TW . Fig. 25 shows the laser intensity at (a)

and (b) . The axial electric field andthe plasma density response on axis at are shownin Fig. 26(a) and (b), respectively. The laser pulse has becomemodulated (three peaks are observable, separated by) andthe plasma wave is highly nonlinear. In addition, relativisticeffects have focused the laser to a much higher intensity thanwould occur in vacuum. The evolution of the laser spot sizeis shown in Fig. 24, indicating that the pulse has focused toa smaller spot size and remains guided over . Highlynonlinear wakefields persist over the distance the laser pulseis guided, and a maximum accelerating field of130 GV/m isobtained at . Additional simulations show that testelectrons, injected into the wakefield with an initial energy of3 MeV, are accelerated to an energy of 430 MeV at

1.8 cm. Beyond this point, the electron slip out ofphase with the wakefield and become decelerated.


(a)

(b)

Fig. 25. Normalized laser intensityjaf j2 for a self-modulated LWFA at (a)c� = 2ZR and (b)c� = 3:2ZR (propagation is toward the right).

Several experiments have observed the self-trapping and ac-celeration of electrons in the self-modulated LWFA [10]–[16].In addition, the plasma wave generated in the self-modulatedregime has been observed using coherent Thomson scatteringwith a frequency-doubled probe laser pulse [158], [159].Optical guiding has also been observed in several of theseexperiments, as discussed in the following section.

VII. OPTICAL GUIDING EXPERIMENTS

Several short-pulse (1 ps) experiments have observedsome form of optical guiding in underdense partiallyand fully ionized plasmas [17]–[43]. In the following, someaspects of these experiments are briefly discussed.

A. Self-Guiding in Ionizing Gases

Ionization-induced refraction, whereby the generation ofplasma can cause the laser pulse to diffract more quickly thanit would in vacuum, has been observed in several experiments[81]–[84], [87], [88]. This can limit the ability to obtain a tightfocus within the ionizing gas, i.e., the laser intensity remainsless than that achieved in vacuum. Likewise, the spot sizeremains greater than the vacuum focal spot and the propagationdistance is limited to , where is theminimum spot size obtained within the ionizing gas.

Liu and Umstadter [17] observed the formation of multi-focifor short pulses propagating in a gas at powers well below thecritical power for relativistic self-focusing in a plasma. Theseexperiments were performed in a chamber containing Hat0.6 atm with a 1- m, 400-fs, 0.2-TW laser pulse withan incident intensity of 10 W/cm . Light emitted from theplasma was diagnosed at 90with respect to the propagationaxis. For these parameters, . In the neutralgas, the pulse self-focused due to nonlinear effects associated

(a)

(b)

Fig. 26. (a) Axial electric fieldEz and (b) plasma electron density on axisn=n0 versus� plotted atc� = 3:2ZR for a self-modulated LWFA.

with the refractive index. As the pulse focused, the intensityexceeded the ionization threshold, thus forming a plasma.Within the plasma, the pulse diffracted. In such a way multipleionization “sparks” were observed with a period of a few mmalong the propagation axis.

A balance between self-focusing in the gas and diffractiondue to plasma formation [17]–[19], [89], [90] has also beenproposed as an explanation for the observations of Braunetal. [18] on the propagation of a 0.8- m, 200-fs laserpulse in air at 1 atm. The laser pulse was injected with alarge spot size (1 cm) and a peak power several times thenonlinear focusing power, 10 GW. After 10 mfrom the injection point, a narrow filament was formed whichcontained a significant portion of the initial laser power. Thenarrow filament extended over 20 m and had a radius of 40

m, as measured by imaging the spatial profile off a diffuserat grazing incidence. Partial ionization was observed along thepropagation axis and the intensity in the filament was estimatedto be near W/cm . This apparently stable mode ofpropagation may result from a balance between self-focusingin the gas and plasma defocusing, as discussed in SectionIV. However, the observed threshold for self-focusing issubstantially higher than the theoretical value and a significantportion of the laser power is observed to reside in a large halosurrounding the central filament. This large-radius (1 cm)


halo can strongly affect the propagation dynamics, e.g., haloenergy can be radially transported along the focal length tocontinually support the filament. Hence, these experimentalconditions are far from the theoretical conditions [90] for self-guiding of a laser pulse with a Gaussian radial profile, asdiscussed in Section IV-B.

Experiments similar to those of Braunet al. [18] were car-ried out by Nibberinget al. [19] using 0.8- m, 150-fs laserpulses with a energy per pulse30 mJ. In these experiments,a laser pulse was injected into an 80-m-long corridor (air at1 atm) with a initial beam diameter of 30 mm. Typically,nonlinear self-focusing in the air caused the formation of oneor more narrow filaments over a “turbulent” region of20m. For an initial pulse energy of 4–8 mJ, a single filamentemerges and propagates a distance up to 40 m. Propagationwas diagnosed by analyzing the light reflected from glassplates at grazing incidence, measuring the conical emissionof light at near forward angles (Cherenkov radiation), andusing single-shot autocorrelation measurements. The measuredenergy in the filament was 0.7–1 mJ, with a spatial size of80–100 m and a duration of 150–180 fs, which implies anintensity of 10 W/cm . The degree of ionization withinthe filament was estimated to be 10 . When theresidual beam surrounding the filament was blocked by anaperture of 1-mm diameter, the filament was observed torapidly diverge. Hence, the self-guided filament is supported,to some degree, by energy transfer from the halo surroundingthe filament.

Sullivan et al. [20] report the propagation of 0.8- m,120-fs laser pulses with 2.3 TW in helium, nitrogen,and argon. In vacuum, the focal spot size was 4 m,which corresponds to a maximum intensity of 9 10W/cm . Propagation was diagnosed by imaging the Thomsonsidescattered laser light. In nitrogen and argon, no guiding wasobserved even when and ionization-induced refractionwas believed to dominate in these cases. When an aperturewas used to constrict the diameter of the laser beam beforethe focusing optic, which reduced the power to 0.5 TW andproduced a vacuum focal radius of 5 m, the pulse wasobserved to undergo vacuum-like diffraction in helium at 30torr, but at 100 torr propagation was extended over .Since 1.8 TW and 8.1 TW for 100 torr ofhelium, the 0.5-TW pulse is not affected by self-focusing.The extended propagation observed is most likely due to thediffractive effects of the aperture, which produced a higherorder mode (non-Gaussian) at focus. If the intensity profilecontained a minimum along the axis, the plasma generatedby ionization could be peaked off axis (in effect, a plasmachannel), which may provide some focusing.

Borisovet al. [21], [22] studied the propagation of a 270-fs,0.46-TW, KrF laser (248 nm) pulse with an incident intensityof 7 10 W/cm through a chamber containing gas (Kror N ) at a pressure of 1–5 atm. The diagnostic consistedof imaging the diffracted laser light at an angle of 15withrespect to the propagation axis. At 5 atm of Kr, the laser lightwas observed to form a filament of length 3–4 mm .A filament radius on the order of 1m was inferred from thediffraction angle of the light at the exit of the gas. Furthermore,

the emission of x-rays (0.5 keV) from the plasma wasobserved to dramatically increase when a extended filamentwas formed. Since the laser power was somewhat above therelativistic guiding threshold, the long propagation distancewas attributed to a combination of relativistic guiding andponderomotive self-channeling. At these intensities, however,Kr and N are not fully ionized and atomic effects, e.g.,nonlinear self-focusing in a gas, are likely to be important.

Kando et al. [23] report preliminary results on the prop-agation of a 2-TW, 100-fs laser pulse through a gas-filled(helium or nitrogen) chamber. Transverse imaging of theplasma emission indicated that the laser pulse propagated over2 cm through 90 torr of He, which is much greater than thevacuum Rayleigh length, even though 5 TW. Alsoobserved were the emission of low energy (10 keV) electronjets in the transverse direction (observed in Nat 50 torr).In addition, preliminary results indicate that some fraction ofa 17-MeV electron beam, which was injected along with thelaser pulse into He at 20 torr, was accelerated to a maximumenergy of 80 40 MeV.

B. Self-Guiding in Plasmas

Experiments on relativistic guiding and ponderomotive self-channeling have also been performed for laser pulses prop-agating in gas-filled chambers, pulsed gas jets, or plasmasgenerated by exploding foils [24]–[35]. Monotet al. [24]–[27]and Chironet al. [28] report the propagation of a 1-m, 15-TW, 400-fs laser pulse through a pulsed hydrogen gas jet( 10 cm ). In vacuum, the focal spot radius was

15 m ( 700 m) giving a peak intensity near4 10 W/cm . Propagation was studied by measuring theThomson sidescattered laser light at an angle of 90withrespect to the propagation axis. For , thelaser pulse was observed to propagate through the entire 3.5-mm length of the gas jet. For , multiplepeaks where observed in the Thomson signal which suggestthe formation of multiple foci. Gibbonet al. [26], [27] andChiron et al. [28] point out the difficulties in interpretinglaser self-guiding based on Thomson scattered light as thesole diagnostic, i.e., the Thomson signal is very sensitive tofocal location, nonlinear effects such as ionization and electroncavitation, and background emission via bremsstrahlung.

Young and Bolton [31] studied the propagation of a1- m, 600-fs laser pulse with an energy of 0.5–2 J through aplasma of relatively high density, , where

1.1 10 cm is the critical density at which .The plasma, formed by exploding a CHfoil with a long pulselaser, inherently contains large axial density gradients. At apulse energy of 500 mJ ( 830 GW) and for densitiesof ( 85 GW),imaging of the sidescattered laser light indicated that the pulsepropagated through the entire length (1 mm ) of theplasma. Interferograms, obtained with a 0.35- m, 600-fslaser pulse synchronized to the 1-m pulse to within 50 ps,also indicated the formation of a density channel through thelength of the plasma. Channel formation was also supportedby a single-shot frequency-resolved optical gating (FROG)


diagnostic, which provided spectrums of the transmitted lightversus time with 10-fs resolution. The FROG signal showeda spectral redshift on the first half of the pulse, indicative ofa decreasing plasma density (channel formation), and a blue-shift on the late time signal, indicative of a increasing plasmadensity (channel collapse). At higher densities or higher pulseenergies, the pulse does not propagate through the entireplasma and a “spraying” of the laser light into larger anglesis observed, presumed to be due to transverse break-up of thepulse by relativistic filamentation.

Borghesiet al. [32] performed experiments on the propa-gation of a 1- m, 1-ps, 10-TW laser pulse in an axiallynonuniform plasma produced by exploding a plastic film witha 400-ps laser pulse which preceded the 1-ps pulse. The 1-ps laser pulse had an incident irradiance of 710 W/cmand a spot size of 10 m ( 300 m). A fewpicosecond, 0.5-m laser pulse, which was delayed from the1- m pulse by 5–30 ps, was used to probe the plasma densityprofile. Both Schlieren photography, which measures densitygradients, and interferometry, which measures relative density,indicated that a density channel was formed, initially at adensity of , which extended into the plasma about 130

m . Over this length, the ambient plasma densityincreased roughly linearly by a factor of two. Note that at

0.4 TW and . Self-emission ofsecond harmonic radiation from the plasma implied a channeldiameter of 5 m.

Wagneret al. [33] studied the propagation of 1- m,400-fs laser pulses with energies3 J in a 1-mm-diameterhelium gas jet 3.6 10 and the subsequent effectof guiding on the accelerate electron beam. In vacuum, themaximum intensity was 5 10 W/cm and the focal radiuswas 8.5 m ( 220 m). Propagation was diagnosedvia imaging the Thomson sidescattered laser light. For lowpowers, , the propagation length was short .The propagation length increased as increased, and for

, the laser pulse propagated through the entire gas jet. Electrons accelerated from the background plasma

were also diagnosed: total number, by use of a Faraday cup;the electron beam spatial profile, by use of a fluorescentscreen imaged by a CCD camera; and the electron energydistribution in the range 0.8–2.3 MeV, by use of a bendingmagnet. The accelerated electron beam was found to have alow divergence angle of 10, and over 1 nC of charge perlaser shot [14]. Moreover, acceleration occured only when

, which is consistent with acceleration from wakefieldsgenerated by self-modulation, as discussed in Section VI-G.For , the electron energy distribution variedas , where is the relativistic factor of the electron.For (when self-guiding was observed), theenergy distribution discretely changed to , i.e.,the ratio of the number of electrons generated at 2 MeVto those at 1 MeV increased. This abrupt change in thedistribution was correlated with and consistent with the laserpulse guiding through the entire gas jet. Furthermore, thedivergence angle of the accelerated electrons was observedto decrease from 10at to 5 at 7.5. Thisis consistent with the electrons gaining energy preferentially in

the axial direction due to the increased propagation distance.In additional experiments [14], [159], the wakefield amplitudewas measured by use of the forward scattered satellites ofthe pump, and time-resolved by use of coherent Thomsonscattering in a pump-probe configuration. Self-modulation wasobserved to occur at a laser power threshold of [14],which is consistent with theory [151]. Plasma wave amplitudesin the range 0.1–0.4 ( 200 GV/m) were directlymeasured, with lifetimes observed to be on the order of 1.5 ps( 50 plasma periods) [159].

Krushelnick et al. [34] and Ting et al. [15] studied theguiding of a 1- m, 400-fs, 2-TW laser pulse through ahydrogen gas jet of density 10cm and length 2.5 mm. Invacuum, the pulse had focal spot radius of 4.5 m (60 m) and a peak intensity of 8 10 W/cm . By imagingthe Thomson sidescattered laser light, the pulse was observedto propagate through the entire 2.5-mm gas jet when1 TW 2 TW. Since 1.7 TW, the onset of self-focusing at may be due to the enhanced guidingeffects of a self-modulated laser wakefield, as discussed inSection VI-G. The existence of a self-modulated laser wake-field was supported by the observation of up to five anti-Stokeslines in the near-forward laser spectrum [15]. No broadeningof the anti-Stokes lines was observed [16]. A 0.5-m, 500-fs, 10-mJ probe pulse, time delayed with respect to the 1-mpump pulse, was used to measure the wakefields via coherentThomson scattering [158]. Large amplitude wakefields with

0.1 were observed with a lifetime 5 ps. Alsoobserved was the production of high-energy electrons fromthe plasma, with energies as high as 30 MeV, the detection ofwhich was correlated to appearance of anti-Stokes lines in thenear-forward pump laser spectrum.

Clayton et al. [35] performed experiments on the propa-gation of a 1- m, 1.2-ps, 20-TW laser pulse through a

4-mm-diameter helium gas jet of density 1.710 cm( 1 TW). In vacuum, the optic provided a focusedintensity of 10 W/cm , a focal spot size (FWHM) of12–15 m, and an effective Rayleigh length of 350 m.The major diagnostic consisted of a 1-cm-diameter, 0.5-

m, 20-ps, 5-mJ probe pulse which intersected the plasmaat 106 with respect to the pump propagation direction. Theprobe pulse was used for: 1) spatially resolved measurementsof the frequency of the plasma waves excited within thechannel via collective Thomson scattering and 2) measure-ments of the refractive index gradients within the plasma viaSchlieren photography. In addition, the spectrally resolvedThomson sidescattered pump laser light was imagined in twoseparate orthogonal planes. These diagnostics indicated thepump pulse was guided through the entire3.8-mmlength of the gas jet plume, typically, within two narrowchannels. Plasma waves of amplitude 0.1–0.6 weredetected within the channels. Spatially local variations in thebandwidth of the observed plasma frequency suggest that rela-tivistic/ponderomotive refractive index changes (which wouldimpose an intensity-dependent decrease in the observedshift during the 1.2-ps pump pulse) were responsible for theobserved laser guiding. Large plasma wave amplitudes atthe far end of the gas jet suggest a channeled intensity of


10 W/cm . The generation of plasma waves was correlatedwith the detection of accelerated electrons [12], [13], [35]with energies in the range 30–100 MeV obtained using asingle-shot, six-channel electron spectrometer, as well as withthe observation of multiple, broadened anti-Stokes sidebandswithin the pump laser spectrum.

C. Guiding in Plasma Channels

In a series of experiments, Milchberget al. [36]–[40]demonstrated the guiding of laser pulses (with intensities

5 10 W/cm ) by preformed density channels. In theseexperiments, a 1-m, 100-ps pump laser pulse with0.7 J waspassed through an axicon lens to create a long focus in a gaschamber (e.g., 200 torr of argon, 7 10 cm ). Thepump pulse ionized and heated the gas, creating a long plasmachannel (1–3 cm) by hydrodynamic expansion with channelradii on the order of 20–40m. After an appropriate delay (1ns), a second probe pulse (either a time-delayed fraction of thepump pulse or a separate dye laser pulse) was propagated alongthe axis of the channel. Diagnostics consisted of imaging theThomson sidescattered laser light from the plasma and imagingthe transverse profile of the laser light exiting the channel. A

565-nm, 500-fs pulse was channeled 3 cm in a 20-torrN 0/200 torr Ar mixture after a 2-ns delay. At the channelexit, the pulse energy was 4 mJ (75% transmission), witha channeled spot radius of 11 m, which correspondsto a channeled intensity in the mid-10W/cm range anda channeled length of (where with

11 m). Single-mode, multimode, and “leaky” modepropagation of the channeled pulse were observed. Additionalexperiments [41] have measured the time- and space-resolveddensity profiles of a 1-cm-long channel by interferometry witha 0.5- m, 70-ps probe pulse synchronized to the pump pulseand passed through a delay line of1 to 11 ns.

Ehrlich et al. [43] demonstrated the guiding of intense (10W/cm ), 0.8- m, 100-fs laser pulses in straight andcurved plasma channels (1 cm long) formed by slow capillarydischarges. The laser pulse was focused at the capillaryentrance with a focal spot of 15 m ( 880 m)and a peak intensity of 10 W/cm at a pulse energy of4 mJ. X-ray absorption measurements [42] of the capillarydischarge channels indicate a roughly parabolic density profilewith , where typically 410 cm with a channel radius of 150 m. For theseparameters, the theoretical matched laser spotsize is 28 m and the minimumradius of curvature allowable for guiding is 6.4 cm(see Section VI-C2). Diagnostics consisted of measuring theinput and transmitted laser energy through the capillary, andimaging the transverse profile of the transmitted laser light atvarious locations between 0 and 1 cm from the capillary exit.For straight channels, 75% of the laser light was transmittedthrough the 1-cm channel in a well-confined mode with spotradius at the exit of 30–50 m. Note that in terms of thevacuum focal spot 15 m and vacuum Rayleigh length

, the 1-cm propagation length corresponds to. In terms of the theoretical matched-beam radius within

the channel 28 m, 1 cm , where. For a radius of curvature smaller than the theoretical

minimum, 4.2 cm , only 2% of the laser energywas transmitted; whereas for 10 cm , the energytransmission was as high as 85%.

Krushelnicket al. [34] and Tinget al. [15] demonstrated theguiding of a 0.5- m, 500-fs, 10-mJ, 5 10 W/cm probelaser pulse in a plasma channel created by a 1-m, 10 W/cmpump laser pulse, which was self-guided through a hydrogengas jet of density 10 cm and length 2.5 mm. Diagnosticsincluded imaging the Thomson sidescattered laser light at 0.5and 1 m and imaging the transverse profile of the probe pulseat the exit of the gas jet. In vacuum, the 400-fs, 1-m pumppulse had focal spot radius of 4.5 m ( 60 m)and a peak intensity of 8 10 W/cm for 2 TW. Thepump pulse was observed to self-guide through the entire 2.5-mm gas jet when 1 TW 2 TW( 1.7 TW).Behind the pump pulse, a density channel was presumed toform due to plasma ion motion, as discussed in Section VI-E.The plasma channel expands at approximately the ion acousticspeed and theory predicts that the channel reaches a maximumdepth 30 ps after the pump pulse for an electron temperature of100 eV. The frequency-doubled probe pulse with wasobserved to undergo vacuum-like diffraction for small delays( few picoseconds) behind the pump pulse. For larger delays(10–40 ps), the probe pulse was observed to propagate throughthe entire gas jet, which is consistent with the formation of aplasma channel on a time scale of tens of picoseconds. Imagingof the probe pulse at the exit of the gas jet indicated a well-defined multimode structure with 75% of the initial pulsepower contained within a central lobe of radius5 m. Thisindicates that the 0.5-m probe pulse was channeled 2.5 mm

at an intensity near 5 10 W/cm .

VIII. D ISCUSSION

This paper has reviewed some of the basic aspects ofself-focusing, guiding, and stability of short laser pulses ingases undergoing ionization and in plasmas. The analysis hasbeen limited, for the most part, to solutions of the paraxialwave equation using the source-dependent expansion (SDE)method, as discussed in Section II. The propagation of thelaser pulse has been characterized by the behavior of the localspot size of the laser envelope. Assuming that thelaser field could be adequately approximated by the lowestorder Gaussian mode, i.e., , theSDE method was used to derive an envelope equation for thelaser spot size, i.e., (27). The source term in the envelopeequation is a function of the index of refraction profile ofthe medium, which in general is a nonlinear function of laserfield and depends upon the dynamical response of the medium.The Gaussian ansatz used in deriving the envelope equationholds best for laser beams near their self-guided solutions.For beams far from their self-guided solutions, e.g., for laserpowers greatly exceeding the critical power for self-focusing,the propagation dynamics can be dominated by the formationof high-order transverse modes, as is the case in laser beamfilamentation. In principle, these effects can be accounted for


by including high-order modes in the SDE method. The SDEmethod has been developed to analyze the paraxial waveequation, hence, group velocity dispersion effects have beenneglected. To account for longitudinal dispersive effects, high-order derivatives must be retained in the wave operator.

The propagation of short laser pulses in gases undergo-ing ionization has been analyzed including the effects ofdiffraction, nonlinear self-focusing in the neutral gas, plasmageneration, and ionization-induced refraction, as discussed inSections III and IV. The polarization current associated withthe neutral gas was retained to third order in the laser field,i.e., inclusion of the linear and nonlinear refractive indices,

. The plasma current was retained to firstorder in the laser field. Plasma formation was determined bytunneling ionization, hence, the plasma density is a highlynonlinear function of the laser field. The SDE was used toderive an envelope equation for the spot size, (46), which isvalid when and for small amounts of plasmaformation, . Self-guided laser pulse profiles wereanalyzed, which result from a proper balancing of diffraction,nonlinear self-focusing in the gas, and defocusing due toplasma formation. These self-guided profiles are subject to anionization-modulation instability which leads to a disruptionof the pulse profile and limits the self-guided propagationdistance.

Short laser pulse propagation in fully ionized, underdenseplasmas is affected by a variety of phenom-

ena, including relativistic self-focusing, ponderomotive self-channeling, plasma wave generation, and preformed densitychannels, as discussed in Sections V and VI. Relativistic self-guiding, which occurs when , only affects the body ofa long laser pulse. Here, the critical power for self-focusing is GW and is a function of only thelaser wavelength and the plasma density. The leading portionof the pulse , however, will diffractively erode due tothe self-consistent response of the plasma density to the laserfield. The self-focusing of a long pulse can be enhanced bythe ponderomotive blowout of the plasma electrons from theaxis, i.e., electron cavitation. In addition, the body of long,relativistically-guided pulse is subject to a self-modulationinstability, which leads to an axial break-up of the pulse profilewith period and is due to the generation of a plasma wavewakefield. Preformed plasma density channels are effectivein the guiding of intense laser pulses when ,where is the critical depth for a parabolicdensity channel and is independent of the laser wavelength.For long pulses, , relativistic effects can reduce thiscriterion, i.e., . In addition, if the pulseis sufficiently short, , the detrimental effects of variousinstabilities (e.g., self-modulation) may be reduced.

The analysis of laser propagation in plasmas presented inthis paper has been limited, for the most part, to a fluiddescription of the laser-plasma interaction. Fluid models aregenerally incapable of describing highly nonlinear phenomenasuch as wavebreaking, particle trapping, and particle cross-ing effects that can occur in complete electron cavitation.Furthermore, additional approximations frequently made insimplifying the wave operator can neglect the effects of

competing instabilities such as Raman backscattering [149].However, much recent progress has been made in the de-velopment of particle simulations of intense laser-plasmainteractions [160]–[167]. For example, the “quasi-static” parti-cle code developed by Antonsen and Mora [166], [167] makesuse of time-averaged particle trajectories and is capable ofstudying electron cavitation and particle crossing effects. Moriet al. [11], [100], [138], [157], [160]–[162] have performedextensive full-scale particle-in-cell (PIC) simulations in 2-Dto model underdense plasma experiments and have beensuccessful in the prediction and interpretation of many ofthe observed phenomena. Since PIC simulations must resolvethe particle orbits on the time scale of the laser frequency,they have been limited to relatively small values of

and, for the most part, to 2-D. Pukhovet al. [165]have done PIC simulations in 3-D, but in a very limitedparameter regime (e.g., ). PIC simulations havethe advantage of showing the numerous competing effects ofvarious instabilities, wavebreaking, particle trapping, electronacceleration, pump depletion, self-focusing and cavitation, theformation of highly non-Gaussian intensity profiles, and theself-generation of high magnetic fields. In 1-D and in mildlynonlinear 2-D regimes, good agreement with theory has beendemonstrated.

This paper has also briefly discussed several experimentsefforts [17]–[43] that have observed evidence of self-focusingand guiding of short laser pulses in ionizing gases and inplasmas. In many respects, however, these experiments canbe considered preliminary, due to the limited diagnosticsemployed and the limited comparison to detailed simulations.For example, the self-guided propagation through air overmany meters as observed by Braunet al. [18] and by Nibberinget al. [19] is not fully understood. In particular, the role of theouter halo region in the propagation dynamics has not beendetermined. Alternatively, the relative importance of pondero-motive self-channeling (cavitation), relativistic self-focusing,and plasma wave wakefield effects has not been measured inintense-laser plasma experiments. Although density channelshave been successfully created by a variety of methods (axiconfoci [36]–[41], capillary discharges [42], [43], or self-guidedhigh-power laser pulses [34]) and subsequently used to guidemoderate intensity (10 W/cm ) laser pulses, the guiding ofultrahigh-intensity ( 10 W/cm ) laser pulses in preformeddensity channels has not yet been demonstrated. At highintensity, various instabilities and the evolution of the channelunder the influence of the laser pulse are likely to be important.Many more detailed experiments on self-focusing and guidingare necessary in order to assess the viability of the variousapplications that require high laser intensity over extendeddistances, such as electron acceleration, harmonic generation,supercontinuum generation, X-ray lasers, and advanced laserfusion schemes.

ACKNOWLEDGMENT

The authors acknowledge useful conversations with C.Clayton, T. Katsouleas, W. P. Leemans, H. Milchberg, W.B. Mori, and D. Umstadter.


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Eric Esarey was born in Wyandotte, MI, onSeptember 12, 1960. He received the B.S. degreein nuclear engineering from the University ofMichigan, Ann Arbor, in 1981 and the Ph.D. degreein plasma physics from the Massachusetts Instituteof Technology, Cambridge, in 1986.

From 1986 to 1988, he was employed byBerkeley Scholars and Research Associates, Inc.Since 1988, he has been a Research Physicist inthe Beam Physics Branch of the Plasma PhysicsDivision at the U.S. Naval Research Laboratory,

Washington, DC. He has published over 60 technical articles and letters inrefereed scientific journals. His research has concentrated on the areas ofadvanced accelerators, advanced radiation sources, intense laser interactionswith beams and plasmas, nonlinear optics, and free electron lasers.

Dr. Esarey is a fellow of the American Physical Society.

Phillip Sprangle (M’90–SM’92–F’97), photograph and biography not avail-able at the time of publication.

Jonathan Krall (M’95), photograph and biography not available at the timeof publication.

Antonio Tang, photograph and biography not available at the time ofpublication.

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Self-Focusing and Guiding of Short Laser Pulses in Ionizing Gases and Plasmas.pdf