d. j. sullivan and b.b. godfrey- em wave electron acceleration

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IEEE Transactions on Nuclear Science, Vol. N S-28, No. 3, June 1981EM WAVE ELECTRON ACCELERATION*

D. J. Sullivan and B. 8. GodfreyMission Research Corporation1400 San Mateo Boulevard, S. E., Suite AAlbuquerque, New Mexico 87108Abstract

Several aspects of the laser electron acceleratorconcept are studied. The minimu m laser intensitynecessary for electron trapping is strongly dependenton frequency. A pulse length equal to the plasmonwavelength (2n wp -II P o uces significantly betteracceleration than one with half that length. Relati-vistic plasma effects enhance acceleration and in-crease the maximum accelerating field. If the laserrisetime or intensity is such that electron trappingdoes not take place, plasma heating by the nonlinearponderomotive force still occurs producing suprathermalelectrons. Application of this mechanism to microwaveshas severe drawbacks.Introduction

The concept of laser electron acceleration ad-vanced by Tajima and Dawsonl has been examined bothnumerically and analytically. The mechanism dependsupon the interaction between an intense electromagneticwave packet and the electrons of an underdense plasma.The nonlinear ponderomotive force associated with thelight waves propagation in the plasma displaces theelectrons. This leads to a charge separation and coin-cident restoring force producing a train of plasmaoscillations. The phase velocity of the wake plasmawave is equal to the group velocity of the EM wave,which is derived from the dispersion relationu* = k*c* + wn2 to ber

I,/k =v =vPP p g = (1 - wp*/u2)%, (11where w is the electron plasma frequency, kp the plas-ma wavePnumber, vp the plasma wave phase velocity, vgthe EM wave group velocity, w the EM wave frequencyand c the speed of light. Because of the mobility ofelectrons, and the fact that large changes in energyfor relativistic electrons translate into small velo-city changes, the electrons are synchronous with thewave front for long periods.

In the wave frame the electrostatic field asso-ciated with the plasma can be viewed as a particle mir-ror with the maximum electron acceleration taking placewhen the electron experiences a momentum change of2ySmc , where m is the electron rest mass, $ is the wavevelocity normalized to c and y is the usual relativis-tic factor given by (1 - 32)-g. Transforming back tothe laboratory frame yields a maximum electron energy0f ymaxmc 2 where ymax = 2 w2/, P*. The critical plasmonelectric field derived from wave-breaking arguments isEz = mcop/e, which implies an accelerating field on theorder of a GeV/cm for a 1018 cmA3 plasma density. Theoptimal pulse length was reported to be half a plasmawavelength or ;I tip-l. This is a severe constraint onthe mechanism, because it implies a 0.056 picosecondlaser pulse for a 1018 cme3 plasma. However, a proposedalternate method1 of using two lasers with a frequencydifference of wp to produce a beat wave has been shownto be feasible*.*This work supported by the Air Force Weapons Labora-tory, Kirtland Air Force Base, New Mexico 87117.

The purpose of this paper is to report on certainimportant aspects of the acceleration mechanism whichwere not addressed in the original paperl. These in-clude the min imu m laser intensity necessary for electrontrapping, relativistic plasma effects, injection of thepulse through the plasma boundary, and effects of tem-perature, plasma gradient, and intensity on the acceler-ation. The role of pulse length was also studied.Finally, microwaves were looked at as a possible driverfor acceleration, since particle energy depends on w/wpand the analysis is valid for any EM radiation.3Minimum Intensity

One can determine a relationship for the min imumwave E-field amplitude, Ezin, from trapping argumentsand the nonlinear ponderomotive force equations. Inthe wave frame trapping requires that the potential belarge enough to stop the electrons movin in the nega-tive direction. Therefore, epave > ymc 5 . Transformingback to the laboratory frame yields that e+lab > mc2 .Using the relation that Ez = kp$lab and vp,= vg, oneobtains for the min imum plasma wave electric field

min mew mewEZ =-A? L->---le vg e (2)This is an interesting result since it appears that theminimum E required for trapping exceeds the wave break-ing limitnoted earlierl.

If one neglects ion motion due to the high frequencynature of the phenomena, the restoring force on theelectrons is due solely to space charge. Thus, the non-linear ponderomotive force per cm2 can be equated tothe energy density gradient of the resulting plasmawave, or

V Ez *I2 m2= nEIT u2 -is (3)

where E is the EM wave electric field.and subgtituting Ez Solving for E.mln from equation (2) yieldsmin

EO =fiF 2 w2 , >fim+ (4)(w -w P2)sThis is valid provided thp laser pulse is shorter thanthe plasma period, ZIT w .P

The min imum laser intensity is readily calculatedfrom equation (4) to be

I min m* c2 w4=- 4v e* w2 - w 2P(5)

This is an absolute minimum, since the analysis assumedan effectively instantaneous risetime. Nevertheless,the strong dependence on laser frequency mus t be noted.For example, a 1.06 urn Nd-Glass laser requires an in-tensit2.4~10 $8 100 times greater than a 10.6 urn CO2 laser orW/cm2 assuming w/o >> 1.P

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An absolute min imum power can also be obtainedassuming E;in = A mew/e and the focal spot size isthe laser wavelength. Then

Pmin 0rr2d*&=T e e

or approximately 21.5 Gigawatts. Since such a tightfocus can only be maintained over a few laser wave-lengths, this is not the best configuration for elec-tron acceleration. Rattler, a more powerful laser beamover a larger area is necessary, which will assure themin imu m intensity over the acceleration saturationlengthl.

SimulationsSimulations were carried out in conjunction withthis analysis using a two-dimensiona l, fully relativis-tic and electromagnetic, particle-in-cell code CCUBE.The code can solve self-consistently for the time de-pendent trajectories of tens of thousands of plasmaparticles over thousands of plasma periods. Allvariables are expressed in dimensionless terms , There-fore, lcngthlisunits of tip. , in units of c/op;.time is measured inand particle velocity is given by

Vi = aiy (1 = 1,2,3), where up is the initial electronplasma frequency.In the laser simulations a plane polarized electro-magnetic wave is launched into a Cartesian geometry.Periodic boundary conditions in the transverse (y)direction make configuration space effectively one-dimensional. In general, the simulation had 1250 cellsin the longitudinal (z) direction modellinp a lengthof 100 C/WI). The cells in the y direction appeareduniform because of periodicity. Each of these macro-cells initially contained 24 particles. The ions weretaken to be an infinitely massive neutralizing back-ground. A vacuum region 10~1 c/othe left hand boundary and the P long was left betweenp asma in order to accu-rately determine the dynamics of laser injection into

the plasma. Different runs were made in which thevalues of u/up, Eo, plasma gradient length, V,z, elec-tron temperature, Te, and pulse length, TI?re canonical simulation has values: lu =Vnz = 0.01 c/up, T, = 0 and p = ~TW l.PWhen the laser pulse encounters the plasma, thenonlinear ponderomotive force resulting from the in-tensity gradient causes the electrons to snow plow.This continues until the force arising from chargeseparation is greater than the ponderomotive force andthe electrons attempt to restore the charge imbalanceby moving in the negative z direction. This motioninitiates a train of large amplitude plasma waves. Theelectrons are trapped by the waves and accelerated. Inseveral code runs the electron density momen tarily ex-ceeded the critical density for the laser pulse andpart of the wave was reflected. The plasma wave trainand resultant electron bunching are shown in Figure 1.The wave steepening evident in the electric field pro-file indicates the nonlinear nature of the mechanismeven at minimum intensity.If the laser intensity exceeds the min imum requiredby a factor of two, the plasma becomes so turbulent thata discernible plasma wave train cannot be established.Coherent acceleration takes place primarily at thepulse front. This is depicted in Figure a). Moreimportant ly, both the particle acceleration and plasmawave accelerating1 field exceed their anticipated maxi-mum s . This is the result of relativistic plasma ef-fects which must be accounted for as seen in Figure 2b).

Tpp 2n w-1P I-4.0 / I0 26.0 60.0 76.0 100.0hJp)

Figure 1. CCUBE diagnostic of the plasma wave electricfield and normalized electron charge densityversus axial distance z.6Oy-- , I -------

w-3WPE,-2.1E,min

II+ta 20 I 7-60wplQl

o ~*~~*~~,, /1I II i 1i

?- 0a"

w-3wp-6.0 E,- 2.1E,mln

T - 60 w p-10.0 I-i ..I--/0 25.0 50.0 75.0 100.0

&/wplFigure 2. particle plots of the longitudinal and trans-verse electron acceleration versus z.

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The laser pulse is coupled to plasma motion in thetransverse direction. In contrast to acceleration inthe longitudinal direction where only a fraction of theelectrons at any given position are trapped and accele-rated, all electrons experience the same transverse ac-celeration for a set value of z and time. It is, there-fore, appropriate to quantify the motion by a plasmarelativistic factor, yp: .An,estimate for Yp can beobtained from the relativistic power equation-& (y-l) mc2 = e E-G

If we assume that the electron velocity approaches thespeed of light, and E has the form E = Eo cos W t then

yP =fi cisinwt F (81gwhere a = E /EmIn. The effect of the relativisticplasma entegs ?nto the wave breaking limit for theplasma electric field and the maximum electron energy,because

rel( )0 = he2 _ 2P yPm yP (9)where o relY is the relativistic plasma frequency. Thecritica electric field is then modified to be

CT-.EZ - Y 3/2 2P e (10)which explains the simulation results where the magni-tude of E, has attained values several times mcwp/e.Similarly the maximum energy becomes

A plot of ymax and energy conversion efficiency, n,versus the laser wave electric field is given in Figure3. The horizontal line indicates Ymax = 2(w/~p)~ or 18.The sloped line is given by equation (11). The symbolsrefer to simulations with various physical parameters.It is clear that laser intensity plays a role in deter-mining the maximum electron energy. The graph alsoshows the results of varying T,, V,,z, and rp from thestandard values. Increasing the electron temperatureto 10 keV and the plasma gradient length by three ordersof magnitude had minim al effect on ymax. In contrast,using a laser pulse length equal to the plasma wave-length, rather than one half that value, significantlyincreases both the maxim um electron energy and theefficiency. Code runs with w = 5 wp produced similarresults verifying the scaling of equation (11).

The simulations disc;fssed thus far were for shortlaser pulses (rp,

d. j. sullivan and b.b. godfrey- em wave electron acceleration

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