F. V. Hartemann- High-intensity scattering processes of relativistic electrons in vacuum

PHYSICS OF PLASMAS VOLUME 5, NUMBER 5 MAY 1998

High-intensity scattering processes of relativistic electrons in vacuum *

F. V. Hartemann†,a)

Institute for Laser Science and Applications, Lawrence Livermore National Laboratory, Livermore,California 94550 and Department of Applied Science, University of California, Davis, California 95616

~Received 18 November 1997; accepted 11 February 1998!

Recent advances in novel technologies such as chirped pulse amplification and high gradient rfphotoinjectors make it possible to study experimentally the interaction of relativistic electrons withultrahigh intensity photon fields. Femtosecond laser systems operating in the TW–PW range arenow available, as well as synchronized relativistic electron bunches with subpicosecond durationsand THz bandwidths. Ponderomotive scattering can accelerate these electrons with extremely highgradients in a three-dimensional vacuum laser focus. The nonlinear Doppler shift induced byrelativistic radiation pressure in Compton backscattering is shown to yield complex nonlinearspectra which can be modified by using temporal laser pulse shaping techniques. Colliding laserspulses, where ponderomotive acceleration and Compton backscattering are combined, could alsoyield extremely short wavelength photons. Finally, strong radiative corrections are expected whenthe Doppler-upshifted laser wavelength approaches the Compton scale. These are discussed withinthe context of high field classical electrodynamics, a new discipline borne out of the aforementionedinnovations. ©1998 American Institute of Physics.@S1070-664X~98!95905-5#

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I. INTRODUCTION

The physics of laser–electron interactions changesmatically at relativistic intensities, where the transverse mmentum of the charge in the laser wave, as measureelectron units, exceeds unity. Three fundamental vacuprocesses are know to occur in this regime: relativistic pderomotive scattering,1 ultrahigh intensity Comptonbackscattering,2,3 and nonlinear Kapitza–Dirac scattering.4,5

These interactions correspond to the following geometrcollinear propagation, head-on collision, and electron dfraction in a laser standing wave, respectively.

If, in addition, the Doppler-shifted laser wavelength,measured in the instantaneous rest frame of the electroncomes comparable to the classical electron radiusr 0

52.8178310215 m), strong radiative corrections to thelectron dynamics are expected. This is the case for ultrartivistic electron beams, such as the Stanford Linear Accerator Center~SLAC! beam, where the laser field can aproach the Schwinger critical field6 for pair creation.Because the normalized vector potential and the avephoton number of the laser pulse are both Lorentz invariit is possible to observe the scattering event in a highly retivistic frame where the laser light is Doppler-upshiftedextremely short wavelengths, while its intensity remainstrahigh. In this new regime, both nonlinear Doppler shif3

and radiation damping2 dominate the electron dynamicThese effects, as well as three-dimensional~3-D! pondero-motive scattering could play an important role in the physof the g–g collider.

*Paper qThpI2-2 Bull. Am. Phys. Soc.42, 2061~1997!.†Invited speaker.a!Electronic mail: [email protected]

2031070-664X/98/5(5)/2037/11/$15.00

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The physics of the aforementioned radiative correctiois fundamentally related to the electron self-interaction prlem, which is central to the foundations of both classical aquantum electrodynamics~CED and QED!. In the study ofCED at high field strengths, the Dirac–Lorentz equation2,7

describes the covariant dynamics of a point charge, includradiative corrections representing the recoil momentumthe photon field interacting with the particle. This effectassumed to be equivalent to a reaction force connected toself-interaction of the charge with its electromagnetic fieAlthough the quantum electrodynamical nature of telectron–photon interaction must be taken into account fofull description of such phenomena, it is hoped that a laclass of interactions may be appropriately studied withincontext of high field strength CED. In addition, a thorouunderstanding of that topic is required for a comprehensapproach to nonlinear QED. A number of conceptual prolems arise within the classical framework, including electmagnetic mass renormalization, runaway solutions, andacceleration or acausal effects, and must be carefaddressed. In QED, the Dirac equation describes the temral evolution of the wave function of a relativistic spin 1particle. At high field strengths, the Dirac–Coulomb problecan be solved exactly, owing in part to the hidden supersymetry of the problem, but a general treatment of QEDtime-dependent external fields remains to be defined. Inticular, multiphoton~nonlinear! Compton scattering has noyet been fully described in terms of the Dirac equation, athe classical relativistic particle limit remains elusive.8,9 Fi-nally, one might quote Dirac’s comment7 concerning theelectron self-interaction: ‘‘...it seems more reasonable to spose that the electron is too simple a thing for the questionthe laws governing its structure to arise, and thus quan

7 © 1998 American Institute of Physics

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2038 Phys. Plasmas, Vol. 5, No. 5, May 1998 F. V Hartemann

mechanics should not be needed for the solution of theficulty.’’

Coherent synchrotron radiation in a free-electron la~FEL!10 has been extensively studied;11–14 therefore, the fo-cus of this paper will be the interaction of relativistic eletrons with ultrahigh intensity laser pulses, in vacuum. In SII, I review the relativistic dynamics of a single electrosubjected to the classical electromagnetic field of a plwave of arbitrary intensity. The canonical invariants of tsystem are derived, and the spectral properties of the stered light are analyzed. In particular, for circular polariztion, I obtain an exact analytical expression for the full nolinear spectrum. Temporal laser pulse shaping is briediscussed as an experimental method to increase the conbetween the main backscattered line and the nonlinear slites due to the relativistic radiation pressure. Colliding laspulses, where ponderomotive acceleration and Compbackscattering are combined, are also considered andwavelength scaling of this interaction is derived, showingpotential to produce extremely high energy photons. 3-Dfects are presented in Sec. III. To accurately describefocusing and diffraction of the drive laser wave in vacuuthe paraxial propagator approach is used, where the mshell condition~vacuum dispersion relation! is approximatedby a quadratic Taylor expansion in the 4-wave number. Tapproach proves extremely accurate for any realizable lfocus, and yields analytical expressions for the fields. Indition, the gauge condition is satisfied exactly everywhethus yielding a proper treatment of the axial electromagnfield components due to wave front curvature. Sectionfocuses on radiative corrections. The Dirac–Lorentz eqtion is derived using an explicit evaluation of the electrself-interaction, and a classical description of nonlineCompton scattering is proposed. Finally, conclusionsdrawn in Sec. V.

II. LORENTZ–MAXWELL ELECTRODYNAMICS,PLANE WAVE THEORY

A. Canonical invariants

The electron 4-velocity and 4-momentum are definedum5dxm /dt and pm5m0cum , with umum521. Here,t isthe proper time along the dimensionless electron worldxm(t). In the absence of radiative corrections,2,7 the naturallength scale of the problem is the the laser wavelengc/v0 , while time is measured in units of 1/v0 , charge inunits ofe, and mass in units ofm0 . Within these basic unitsany relevant physical quantity can be normalized by simdimensionality considerations: For example, momentumnormalized tom0c, the 4-vector potential is measuredunits of m0c/e, and the 4-wave number is given in unitsv0 /c. The energy–momentum transfer equations are driby the Lorentz force

dum

dt52~]mAn2]nAm!un. ~1!

For plane waves, the 4-vector potential of the laser wavgiven by


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Am~f!5@0,A~f!#, f5kmxm~t!, ~2!

wheref is the relativistically invariant phase of the travelinwave along the electron trajectory. Choosingkm5(1,0,0,1),with the wave propagating in thez direction, we have

df

dt5g2uz5k, ~3!

which defines the light-cone variablek, and the 4-momentum transfer equations read

du'

dt5k

dA'

df~f!, ~4!

duz

dt5

dg

dt5u'–

dA'

df~f!. ~5!

Equation~5! shows thatk is invariant:k5k05g0(12b0);additionally, Eq.~4! is readily integrated to yield the wellknown transverse momentum invariant15

u'~t!5A'~f!, ~6!

and the energy and axial momentum are immediatelytained using the fact that the 4-velocity is a unit 4-vec(g2511u'

2 1uz2):

uz~t!5g0Fb01A'

2 ~f!

2~11b0!G , ~7!

g~t!5g0F11A'

2 ~f!

2~11b0!G . ~8!

The quadratic dependence of the energy and axial momtum on the 4-vector potential, measured in electron undistinguishes the relativistic scattering regime, whereA'

>1. In this regime, the ponderomotive force dominateselectron dynamics, yielding nonlinear slippage and Doppshifts.1,3 Equation~8! also provides a scaling for the maxmum energy in a plane wave:g* /g0.A'

2 , for relativisticelectrons. Finally, the electron position is given byx(f)5(1/k0)*2`

f u(c)dc.

B. Compton scattering, nonlinear spectra

The focus of this section is the spectral characteristicsthe radiation scattered by the accelerated charge. Ascussed by Jackson,15 the distribution of energy radiated peunit solid angle, per unit frequency can be derived by cosidering the instantaneous radiated power, as describethe Larmor formula, and applying Parsival’s theorem to otain

d2N~v,n!

dv dV5

a

4p2 vU E2`

1`

n3~n3b!exp@ i ~vt

2n–x!#dtU2

, ~9!

where v is the frequency measured in units ofv0 , and a51/137.036 is the fine structure constant. The quantityEq. ~9! corresponds to the average radiated photon numb


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2039Phys. Plasmas, Vol. 5, No. 5, May 1998 F. V Hartemann

Using the phase as the independent variable, onehas

d2N~v,n!

dv dV5

a

4p2

v

k02 U E

2`

1`

n3@n3u~f!#exp$ i v@f

1z~f!2n–x~f!#%dfU2

. ~10!

Here, I have used the plane wave invariance ofk5k0 .The most interesting case is the backscattered radia

where most of the power is radiated and where one obtthe maximum relativistic Doppler upshift. In this case,n52 z, z(f)2n–x(f)52z(f), z3@ z3u(f)#52A'(f),and Eq.~10! can now be recast in a manifestly covariaway, to read

d2N~v2 z!

dv dV5

a

4p2 xU E2`

1`

A'~f!

3expH ixFf1E2`

f

A'2 ~c!dcG J dfU2

, ~11!

where I have introduced the normalized Doppler-shifted fquencyx5v(11b0)/(12b0).

The functional dependence of the spectrum is now inpendent ofb0 , which only sets the frequency scale. This fais not surprising, as it results directly from covariance:changing the reference frame in which the scatteringviewed, one can vary the sign ofb0 and go continuouslyfrom the FEL geometry10 to the laser acceleratiogeometry.16–21 For the FEL, the laser frequency is Dopplupshifted, while it is downshifted in the second case. In bcases, the normalized vector potential and the averageton number are conserved as they are both Lorentz invar

C. Circular polarization

Having derived the expression for the nonlinear bascattered light spectrum for arbitrary polarizations and intsities, I will now focus on one important case: circularpolarized plane waves. In this case, the dimensionlesvector potential can be expressed asA'(f)5A0 g(f)3@ x sinf1y cosf#, which implies that the magnitude othe 4-vector potential varies adiabatically as the pulse insity envelope:AmAm5A'

2 (f)5A02 g2(f). A simple physi-

cal model for the pulse envelope is given by a hyperbosecant, namelyg(f)5cosh21(f/h), so that the electron’saxial position can be determined analytically.22 In this case,

E2`

f

A'2 ~c!dc5E

2`

f A02

cosh2S c

h D dc

5A02hF11tanhS f

h D G , ~12!

and the nonlinear backscattered spectrum is now protional to


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xUA0eixA02hE

2`

1` x cosf1 y sin f

coshS f

h D3expH ixhFfh 1A0

2 tanhS f

h D G J dfU2

. ~13!

This Fourier transform can be evaluated analytically by peforming two changes of variable;3 namely, I first sety5ef/h, thenx5(y221)/(y211), with the result that

d2N~v,2 z!

dv dV5

a

8A0

2h2xH UF~m2,1,2iA02xh!

coshFp2 h~x21!GU2

1UF~m1,1,2iA02xh!

coshFp2 h~x11!GU2J . ~14!

Here, F is the degenerate~confluent! hypergeometricfunction,22 andm65 1

2@11 ih(x61)#. The electron dynam-ics are shown in Fig. 1~top!, while the behavior of the non-linear spectral function is illustrated in Fig. 1~bottom! forh55, and different values ofA0 . Within this context, theonset of nonlinear relativistic spectral effects corresponds

FIG. 1. Top: behavior of the normalized axial electron position for circulaand linear polarizations. Bottom: nonlinear spectral function for circularpolarized light and different values ofA0 . In both cases, cosh22 intensityenvelope,h55.


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a situation where the electron phasef and the axial Dopplershift *2`

f A'2 (c)dc become comparable because of the

tense radiation pressure of the drive laser pulse. This isrectly connected with the relativistic mass shift of t‘‘dressed’’ electron within the ultrahigh intensity laser puls

For linear polarization, the nonlinear effects are evstronger because there is an extra modulation of the avelocity at the second harmonic of the laser. As shownFig. 2, chaotic spectra are predicted. It is quite remarkathat an elementary electrodynamical process, such asscattering of coherent light by a single electron describinwell-behaved trajectory can yield such complex relativisnonlinear spectra when the radiation pressure strongly molates the electron’s proper time.

D. Temporal pulse shaping

A very important consequence of the nonlinear Doppeffect described in the previous sections resides in thethat, at ultrahigh intensities, the peak photon number denin each line is approximately constant across the spectru3

This indicates that for ultrashort laser pulses, even in the cof circularly polarized light, the backscattered energy isdistributed over a wide spectral range instead of contributo a single, narrow, Compton backscattered line. This ipotentially serious difficulty for applications, such as t

FIG. 2. Nonlinear Compton backscattered spectra for a linearly polarcosh22 pulse of widthh55 and increasing intensities.


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g–g collider, which require the generation of a single, itense, highly collimated, narrow x-ray org-ray line.

This problem can be alleviated by shaping the tempoenvelope of the pump laser pulse in order to minimizevariation of the nonlinear Doppler shift during the interation. In such a scheme, as illustrated in Fig. 3~top!, the mainpart of the laser pulse is flat, thereby yielding a constant aelectron velocity during most of the interaction. The assoated Doppler shift thus remains nearly constant, resultingthe radiation of a narrow spectral line, as indicated in Fig~bottom!. During the rise and the fall of the laser pulse, trasient lines are radiated, but they are kept to a minimumusing this technique, which is rather analogous to the usa tapered wiggler entrance in a FEL.10

To study the effects of pulse shaping on the Compbackscattered spectra in the nonlinear regime, and to findoptimum temporal profile of the laser pulse envelope, Itroduce variable pulse shapes which are modeled theocally by considering a circularly polarized pulse with a composite envelope, including acosh21 rise and fall, and aconstant flattop :g(f)5cosh21(f/h) for f<0, g(f)51 for0<f<u, and g(f)5cosh21@(f2u)/h# for f>u. The pulseshape is then parametrized by the ratio of the flattop toFWHM ~full width at half-maximum!, which is equal tor5u/@u12h ln(21A3)#. For r50, the pulse is a hyperbolicsecant, and forr51, the pulse is square.

It is easily seen that in the nonlinear Fourier integral@Eq.

d

FIG. 3. Top: hyperbolic secant envelope with a flattop, and axial elecposition. Bottom: nonlinear spectrum forA052, h55, u510.


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~11!#, the contribution of the flattop is simply proportional

A0eixA02hE

0

u

~ x sin f1 y cosf!exp@ ix~11A02!#df,

~15!

where the factor 11A02 corresponds to the relativistic Dop

pler shift due to the slower axial motion of the electrcaused by the constant laser radiation pressure. The intein Eq. ~15! yields asinc spectrum. It is clear that foru @h,the line at the normalized Doppler-shifted frequencyx*51/(11A0

2) dominates the backscattered spectrum,shown in Fig. 3~bottom!. The shorter wavelength lines corespond to a combination of the multiphoton lines resultfrom the nonlinear Doppler shift during the transient partsthe pulse and the oscillations of thesinc.

The beneficial effects of square optical pulses, whichbe generated by holographic filtering at the Fourier planea CPA laser, as demonstrated by Weineret al.,23 can beevaluated quantitatively by considering the evolution ofratio of the energy in the main line to the total backscatteenergy, as a function of the pulse shape. The resultsshown in Fig. 4, clearly validating the pulse shaping aproach at ultrahigh intensities, and demonstrating that sqoptical pulses correspond to the optimum shape for Compbackscattering applications.

E. Colliding laser pulses

In this section, a novel concept to produce extremshort wavelength photons is suggested, in which vacuponderomotive acceleration~VPA! is combined with Comp-ton backscattering@Fig. 5 ~top!#. In this manner, the highenergy acquired by an electron beam within the drive lapulse may be effectively extracted in the form of short walength photons by a colliding probe laser pulse, withoutquiring complex structures to terminate the interactio

FIG. 4. Energy contrast ratio as a function of pulse shape for various insities.


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thereby preserving the robustness and simplicity of VPA,1 ascompared to other laser acceleration schemes. It willshown that for this process, the photon energy scales ashn>4(hc/l0)g0

2A04, wherel0 is the laser wavelength~pump

and probe!, g0 is the initial beam energy, and

A05eE0l0

2pm0c2 5el0

pm0c5/2 A I 0

2e0

is the dimensionless amplitude of the 4-vector potentialsociated with the drive laser wave; as expressed in termthe focused intensityI 0 .

As a numerical example, 1 TeV photons could be pduced with an 8.531020 W/cm2 drive pulse at 800 nm, in-teracting with a 550 MeV beam. These parameters cospond toA0520. By comparison, a FEL using the samwavelength for an electromagnetic wiggler would requover 200 GeV of beam energy. The advantageous enescaling of the proposed interaction results directly fromfact that Compton backscattering occurs when the inielectron energy has been boosted by VPA to its maximvalue, given byg* /g0.A0

2 , as shown in Sec. II A.It is interesting to evaluate the length required by th

acceleration process. For a circularly polarized pulse inacting with ultrarelativistic electrons, the nonlinear slippalength is approximately given byDz>(c/v0)*f0

0 @1

1A02 g2(c)#dc, wheref0 is the injection phase of the elec

tron. For asin4 temporal intensity envelope, which closematches a Gaussian, with a finite durationh5v0Dt, one

n-

FIG. 5. Top: geometry of the colliding pulses interaction process. Bottophoton energy~linear scale! as a function of beam and drive laser energifor a slab geometry.l05800 nm,sy55l0 , 20 fs FWHM.


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findsDz5g02cDt(11 3

8A02). The acceleration length require

to produce 1 GeV photons with a drive laser strength ofand a duration of 20 fs FWHM is estimated at 2.5 m. Testimate represents an upper bound, as 3-D effects reducaverage laser strength along the electron path.

The next step of the theoretical analysis consistsshowing that the drive and probe laser pulses are decoubecause of their very different Doppler shifts. The phasethe counterpropagating probe pulse isw5t1z, and the dif-ferential phase variation between each pulse evolves athe electron trajectory according todw/df5(dw/dt)3(dt/df)5(g1uz)/(g2uz)>@g0(11b0)#2>4g0

2. Thelast two equalities are approximately valid if the radiatipressure of the probe pulse is much smaller than that ofdrive pulse. As a direct consequence, the drive laser fiare ‘‘frozen’’ during the interaction between the acceleraelectron and the probe pulse. It should also be noted tharelationu'5A'1Ap still holds. Here,Ap is the normalizedpotential of the probe laser. In addition, in the case olinearly polarized drive pulse, it is clear that the fields aequal to zero at the maximum electron energy, since theyproportional todAx /df and A'

2 (f)5Ax2(f). Another ad-

vantage of the linear polarization is that the transverse ption of the electron during backscattering is quite smx(f)>(c/v0k0)*2`

f ux(c)dc, which yields x* 5x(f50)>(l0 /4Ap)g0(11b0)A0h exp(2h2/4)>0 for a Gaussianpulse. However, this equation also indicates that the mmum transverse excursion during VPA can be quite larsx>(l0 /p)g0(11b0)A0 . This must be taken into accounwhen defining the dimensions of the slab focus geomewhere the beam cross section isS5sxsy, with sy chosen assmall as possible to minimize the drive pulse energy.

Finally, the angle of the electron trajectory during bacscattering is defined by the ratio of the transverse and amomenta in the drive laser, and is small:

u* 5arctanFux

uz~f50!G>

A0

g0Fb01A02S 11b0

2 D G>

1

g0A0. ~16!

Despite its low value, this angle is critical in obtaining thfull Doppler upshift for the backscattered radiation. To illutrate this point, it is instructive to compare the Doppler ushift along thez axis, (g* 1uz* )/(g* 2uz* )5@(11b0)/(12b0)#(11A0

2)>4g02A0

2, which is seen to scale as the squaof the drive field, to the full Doppler upshift, (g*1u* )/(g* 2u* )5@g* (11b* )#2>4g0

2A04.

The Compton backscattered spectrum is obtained fEq. ~11!, by replacing the initial velocity by the maximumvelocity due to the drive laser:


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-

m

d2N~v,u* !

dv dV5

a

4p2 S 12b*

11b* D vU E2`

1`

Ap~w!

3expH i vS 12b*

11b* D Fw

1E2`

w

Ap2~c!dcG J dwU2

. ~17!

In the case of a Gaussian wave packet, in the linear reg(Ap

2!1), the phase integral can easily be performed to yi

d2N~v,u* !

dv dV>

a

4p2

Ar2h2

4g02A0

4 v

3expF2h2

2 S v

4g02A0

421D 2G , ~18!

where the frequency of the backscattered line has thepected value,v>4g0

2A04. To estimate the total photon yield

Eq. ~18! can be integrated over all frequencies, to obtadN(u* )/dV>a@A(2/p3)#g0

2Ap2A0

4h. The solid angle canthen be approximated byDV.p/g0

2A04 to yield the average

number of photons scattered by a single electron:Ng /Ne

'aA2/pAp2h.

The overall scaling of the process, for a slab geometrysummarized in Fig. 5~bottom!, where the following param-eters are fixed:l05800 nm,sy55l0 , and where the pulseduration is 20 fs FWHM. Using the frequency scaling, trelation between the intensity and the vector potential, athe equation for the transverse excursion, with the constrthat the drive pulse energyW5I 0sxsy, the photon energycan be expressed in terms of the electron beam energyg0 ,and the drive pulse energyW, ashn/e}W4/3g0

2/3, or moreaccurately as

v54S 1

4p2/3

l0

sy

W

PDt D4/3

g02/3, ~19!

where l have introduced the parameterP5e0m02c5/e2

50.6931 GW.Finally, the speculative nature of this proposal should

stressed: A conclusive argument for or against the practfeasibility of this physical process as a useful radiatisource requires detailed studies of some potential probleincluding coherence, electron beam phase space, 3-Ddiffraction effects, and radiation losses in parasitic synchtron radiation channels, all contributing to the final brighness and efficiency of the photon source.

III. THREE-DIMENSIONAL DYNAMICS,PONDEROMOTIVE SCATTERING

Detailed knowledge of the 3-D electromagnetic field dtribution of the focusing laser wave is required to study hiintensity scattering and properly model experimental resuFor example, two ultrahigh intensity relativistic electroscattering experiments are currently underway at SLACCEA. In the first case, nonlinear~multiphoton! Comptonbackscattering is investigated using the SLAC 50 GeV beand a tightly focused TW-class laser;24 at the Commissariat a


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l’Energie Atomique~CEA!, low energy electrons are acceerated by a TW laser.25 In both instances, the 3-D nature othe focused laser pulse is an essential feature of the exment and must be described accurately to interpret the reing data.

A. The paraxial propagator

The 3-D behavior of the laser electromagnetic fiepropagating in vacuum is now described within the contof the paraxial propagator formalism. The wave equatgoverning the 4-vector potential ishAm14p j m5@]n]n#Am

14p j m50, whereI have introduced the 4-gradient operatodefined as]m[]/]xm[(2] t ,“), and the 4-current densitj m[(r,j ). In addition, it is important to note that the 4potential must satisfy the Lorentz gauge condition]mAm

50.In vacuum~no sources!, a general solution to the wav

equation can be constructed as a Fourier superpositiowave packets of the form

Am~xm!51

A2p4 E E E E Am~kn!exp~ iknxn!d4kl ,

~20!

where the 4-wave numberkm satisfies the vacuum dispersiorelation, which is also the mass shell condition for the phofield: \2(kmkm)50.

FIG. 6. Transverse wave number spectrum and comparison betweeparaxial and exact dispersion forf :3.


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When the laser pulse characteristics are defined at fo(z50), the electromagnetic field distribution can be otained in any givenz plane by performing the followingintegral:

Am~x,y,z,t !51

A2p3 E E E Am~k' ,v,z50!exp@ i ~vt

2kxx2kyy2Av22k'2 z!#d2k'dv, ~21!

where I have introduced the focal spectral densAm(k' ,v,z50).

The physics of this solution can be understood aslows: The temporal evolution of each wave packet is dscribed by the frequency spectrum, while the transversefile of the laser wave is expressed as an integral ovecontinuous spectrum of transverse vacuum eigenmodes.dispersion relation indicates how each transverse compoof the wave packet propagates, thus yielding wave front cvature and diffraction.

In vacuum, the gauge condition reduces to“–A50, andcan be satisfied by requiring thatA5“3G. For a linearlypolarized Gaussian-elliptical focus, the generating field,G,takes the form

G~xm!5 yGy~xm!,~22!

Gy~x,y,z50,t !5A0

k0g~ t !expF2S x

w0xD 2

2S y

w0yD 2G ,

wherew0x refers to the beam waist along thex axis andw0y

refers to the beam waist along they axis,A0 is the amplitudeof the vector potential at focus,k0 corresponds to the centra

the

FIG. 7. Transverse and axial vector potential components, in the planpolarization, at three different times. The pulses are six cycles long andf :3.


gr

d

rgcthend

iniarg

ndhaag

aa

fnaodorghae-e

g

one ishe

lin-

ied8,

na-er-n-d

ringcurserPA

tionainse-and


laser wavelength, andg(t) is the temporal variation of thelaser wave, which can be arbitrary. The propagation inteis now approximated by

Gy~x,y,z,t !51

A2p3 E E E Gy~k' ,z50,v !

3expH i Fvt2kxx2kyy2

3S v2k'

2

2k0D zG J d2k'dv, ~23!

where the square root factor has been Taylor expandesecond order aroundv51 and k'50. It is clear that theexact and Taylor expanded axial phase differ only for lavalues of the transverse wave number, where the spedensity is vanishingly small, as illustrated in Fig. 6. Tintegrals overd2k' are readily obtained, as they correspoto complex Gaussians:

Gy~x,y,z,t !5A0

k0g~f!F11S z

z0xD 2G21/4F11S z

z0yD 2G21/4

3expH 2F x

wx~z!G2

2F y

wy~z!G2J

3expF i H 1

2arctanS z

z0xD1

1

2arctanS z

z0yD

2z

z0xF x

wx~z!G2

2z

z0xF y

wy~z!G2J G , ~24!

where wx,y(z)5w0x,yA11(z/z0x,y)2 and z0x,y5 1

2k0w0x,y2

represent the Rayleigh ranges for eachf number.

B. Ponderomotive scattering

The vector potential of the focusing wave is shownFig. 7. The fields are then derived from this vector potentThe algorithm developed to model the dynamics of a chainteracting with the 3-D laser fields employs the secoorder Runge–Kutta method and uses the axial electron pas the integration variable to handle the nonlinear slippand the relativistic Doppler shift.

For a laser focus with extremely largef numbers, excel-lent agreement is found between the numerical resultsthe theoretical analytic expressions obtained for plane wdynamics; for smaller values off , scattering is obtained. Toobtain efficient scattering, the electron must be seededfrom focus, so that by the time it has slipped into the nolinear temporal phase of the pulse, the focus is reachedthe electron interacts with the spatio–temporal maximumthe laser wave. High energy scattering occurs for intermeate values of thef numbers: for low values, the focus is totight and the electron scatters away too early; for very lavalues, we recover the plane wave interaction. The fact tfor higher initial injection energies, efficient scattering rquires larger values of thef number is not surprising, sinc


al

to

eral

l.e-see

ndve

ar-ndfi-

et,

the transverse electron excursion scales like 1/k.1/k0

5g0(11b0). For linear polarization, the largest scatterinenergies are achieved forf x. f y .

To assess the feasibility of laser acceleration basedthis scattering process, the total energy in the laser pulsobtained by integrating the Poynting vector flux over tfocal spot and the pulse duration to obtain

W

m0c2 53p

32A0

2 w0x

l0

w0y

l0

cDt

r 0, ~25!

wherer 0 is the classical electron radius. The case of a cydrical focus (w0x5w0y5w0) is considered, where the pulseFWHM is maintained at 20 fs, and the productA0w0 is keptconstant at 250 mm~constant energy of 20 J!, while A0 isvaried between 1 and 20, and the injection position is varbetween220 mm and focus. The results are shown in Fig.and clearly indicate the existence of an optimum combition of f number and laser intensity for high energy scatting, approximately obtained for a normalized vector potetial of 17.5 and f :23. Here, there is a sharply defineacceptance range in the initial position, and the scatteenergy reaches 0.25 GeV. The acceleration process ocover 3 mm, yielding a gradient of 85 GeV/m. These lasparameters correspond to the next generation of Clasers.26

IV. DIRAC–LORENTZ ELECTRODYNAMICS,RADIATIVE CORRECTIONS

The Dirac–Lorentz equation2,7 describes the covariandynamics of a classical point electron, including the radiatreaction effects due to the electron self-interaction. The msteps of Dirac’s derivation are outlined here. For conciness, I now use electron units, where length, time, mass,charge are measured in units ofr 0 , r 0 /c, m0 , ande, respec-tively. In these units,e051/4p, andm054p. The electron4-current density is

j ms ~xl!52E

2`

1`

um~xl8 !d4~xl2xl8 !dt8, ~26!

and the corresponding self-electromagnetic fieldFmns

5]mAns2]nAm

s satisfies the driven wave equationhAms (xl)

FIG. 8. Scan ofA0 and initial z for a constant pulse energy of 20 J.l0

51 mm, FWHM 20 fs, initial energy 10 MeV,xi5yi50.


ns

d

on

o

-

f-

rd

a

thlti

ioz

isns:

l

or-

tictzthe

aw

onn-

rel toy

mal-thehus

c–

iale

n


54pjms (xl), which can be solved in terms of Green functio

asAms (xl)54p*2`

1`um(xl8)G(xl2xl8)dt8.The self-force can now be evaluated as

Fms 52~]mAn

s2]nAms !un52E

2`

1`

un~xl!@un~xl8 !]m

2um~xl8 !]n#G~xl2xl8 !dt8. ~27!

The advanced and retarded Green functions both depenthe space–time interval s25(x2x8)m(x2x8)m:G6

52d(s2)@17(x02x08)/(ux02x08u)#. The partial derivativescan now be replaced by the operator]m[2(xm2xm8 )]/]s2,and Eq.~27! reads

Fms 522E

2`

1`

un~xl!@un~xl8 !~xm2xm8 !2um~xl8 !~xn2xn8!#

3]G

]s2 dt8. ~28!

At this point, the new variablet95t2t8 is introduced, sothat the range of integration explicitly includes the electr~singular pointt950!. To evaluate the integral in Eq.~28!,one can now use Taylor–McLaurin expansions in powerst9:

xm2xm8 5t9um2 12t92am1 1

6t93dtam1¯, ~29!

um~xl8 !5um~t2t9!5um2t9am1 12t92dtam1¯, ~30!

where the 4-accelerationam5dtum . Using the above expansions, one first finds thats2.2t92, which yields ]G/]s2

.2(1/2t9)(]G/]t9). With this, the expression for the selelectromagnetic force now reads

Fms .E

2`

1` H 2t9

2am1

t92

3 Fdam

dt2um~anan!G J ]G

]t9dt9.

~31!

This equation can be integrated by parts; using the reta~causal! Green function, one finds

Fms 52

1

2amE d~t9!

ut9udt91

2

3 Fdam

dt2um~anan!G . ~32!

The corresponding 4-momentum transfer equation now re

F111

2 E d~t9!

ut9udt9Gam52Fmnun1t0Fdam

dt2um~anan!G ,

~33!

wheret05 23 is the Compton time scale, in the units ofr 0 /c

used here. The divergent integral on the left-hand side ofequation is the infinite electromagnetic mass which muplies the 4-acceleration. Dirac first proposed7 to renormalizethis term away by using the time symmetrical Green functG5 1

2(G22G1); with this one obtains the Dirac–Lorent

equation

am52Fmnun1t0Fdam

dt2um~anan!G . ~34!


on

f

ed

ds

e-

n

One of the conceptual difficulties associated with thequation is the existence of unphysical, runaway solutioContracting Eq.~34! with am, it is easily seen that, in theabsence of an external field it reduces toamam5(t0/2)3(d/dt)(amam), which admits the runaway solution@amam#(t)5@amam#0 exp(2t/t0). To avoid these unphysicasolutions, one must require that the Dirac–Rohrlich~DR!asymptotic condition7,27 be satisfied: limt→6`am(t)50.

Also note that the second radiative correction term cresponds to the radiated 4-momentumHm ; thus I can rewriteEq. ~34! as

am52Fmnun1t0

dam

dt2

dHm

dt. ~35!

In the case of an external electric field deriving from a stapotential, the time-like component of the Dirac–Lorenequation, which describes energy conservation, takessimple form~here,H05W!

dg

dt5u–“w1t0

d2g

dt22dW

dt5

d

dt Fw1t0

dg

dt2WG ,

~36!

and can formally be integrated to yield the conservation l

D~g2w1W!5t0Fdg

dt G2`

1`

, ~37!

which indicates that, provided the DR asymptotic conditilimt→6`@dg/dt#50 is satisfied, the electron potential eergy is converted to kinetic energy and radiation.

Within this context, the small value of the fine structuconstant, which corresponds to the ratio of the classicaquantum electron scale~classical electron radius divided bthe electron Compton wavelength!, guarantees that theacausal effects related to the electromagnetic mass renorization will be smeared by quantum fluctuations beforestrong classical radiative correction regime is reached, tpreventing ‘‘naked acausalites.’’

In the case of nonlinear Compton scattering, the DiraLorentz equation can be given as

am5dum

dt5Lm1t0Fdam

dt2um~anan!G ,

~38!

L'5kE' , Lz5L05u'–E' ,

where we recognize the light-cone variablek5g2uz , andthe laser transverse electric field.t052r 0/3c50.626310223 s is the Compton time scale. Subtracting the axcomponent of Eq.~38! from the temporal component, wobtain an equation governing the evolution ofk,

dk

dt5t0Fd2k

dt22k~anan!G . ~39!

Noting that E'5e(dA' /df), we also obtain an equatiogoverning the evolution of the canonical momentum:


rdar

e

am

ive

edag-t-

the

xbutd

f aticolo-t rflly.as ara-ler

ghm-m-ses,tter-elyor-ser

se inthinew

-

-

g

of

/

. G.

ck


d

dt~u'2A'!5t0Fd2u'

dt2 2u'~anan!G . ~40!

Introducing the small parametere5v0t0 , which measuresthe Doppler-shifted laser wavelength in units ofr 0 , and us-ing f as the independent variable, Eq.~39! now reads

dk

df5eF d2

df2 S k2

2 D2k2S dum

df

dum

df D G . ~41!

Since the right-hand side of Eq.~41! is at least of ordere, wecan replace the terms in the brackets by their zeroth-o~Lorentz dynamics! approximation; in this case, we obtainsimple differential equation for the light-cone variable peturbation

d

df F 1

k~f!G.eA02g2~f!, ~42!

where we recognize the envelope of the circularly polarizlaser pulse. Equation~42! can easily be integrated to yield

1

k~f!5

1

k01eA0

2E2`

f

g2~c!dc. ~43!

This equation describes the electron recoil for beam pareters similar to those of SLAC,24 as illustrated in Fig. 9~top!.It is clear that at sufficient intensities, the relative radiatenergy loss becomes significant.

FIG. 9. Top: fractional final electron energy after nonlinear Compton bascattering. Bottom: Lorentz and Dirac–Lorentz backscattered spectra.


er

-

d

-

Finally, the Dirac–Lorentz equation can be integratbackward in time to avoid runaways due to the electromnetic mass renormalization7,27,28and the Compton backscatered spectrum is obtained by evaluating

d2N~v,2 z!

dv dV5

a

4p2 vU E2`

1` u'~f!

k~f!

3expH i vFf12E2`

f uz~c!

k~c!dcG J U2

.

~44!

For the SLAC beam parameters, and a TW-class laser,result is shown in Fig. 9~bottom!. At this intensity(0.22 TW/mm2), the nonlinear Doppler shift yields complespectra for both Lorentz and Dirac–Lorentz dynamics,the lines are shifted, in similarity with the Lamb shift, andamped at high frequencies.

V. CONCLUSIONS

In this paper, I have reviewed the classical theory onumber of high-intensity scattering processes of relativiselectrons in vacuum, that recent advances in novel techngies such as chirped pulse amplification and high gradienphotoinjectors make possible to investigate experimentaIn particular, ponderomotive scattering has been studiedpossible way to accelerate electrons with extremely high gdients in a 3-D vacuum laser focus. The nonlinear Doppshift induced by relativistic radiation pressure in ultrahiintensity Compton backscattering was shown to yield coplex nonlinear spectra which can be modified by using teporal laser pulse shaping techniques. Colliding lasers pulwhere ponderomotive acceleration and Compton backscaing are combined, also has the potential to yield extremshort wavelength photons. Finally, the strong radiative crections that are expected when the Doppler-upshifted lawavelength approaches the Compton scale, as is the caongoing experiments at SLAC, have been discussed withe context of high field classical electrodynamics, a ndiscipline borne out of the aforementioned innovations.

ACKNOWLEDGMENTS

In collaboration with A. K. Kerman~Massachusetts Institute of Technology!, J. R. Van Meter, University of Cali-fornia, Davis ~UCD!, A. L. Troha ~UCD!, A. Gupta~CCNY!, H. A. Baldis~UCD and Lawrence Livermore Natural Laboratory!, and N. C. Luhmann, Jr.~UCD!. I also wishto personally thank D. T. Santa Maria for very stimulatindiscussions.

This work is supported in part by DepartmentDefense/Air Force Office of Scientific Research~DoD/AFOSR! ~MURI! F49620-95-1-0253, AFOSR~ATRI!F30602-94-2-001, ARO DAAHO4-95-1-0336, and LLNLLDRD DoE W-7405-ENG-48.

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F. V. Hartemann- High-intensity scattering processes of relativistic electrons in vacuum