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PLASMA WAVE WIGGLER S FOR FREE ELECTRON LASERS

C. Joshi,T. Katsouleas, J. M. Dawson, Y. T. Yan, and F. F. ChenUniversity of California, Los Angeles, CA 90024

ABSTRACTWe explore the possibility of using relativistic plasma density

waves as wigglers for producing free electron laser radiation. Two possi-ble wave and beam geometries are explored. In the first, the wiggler is apurely electric wiggler with frequency ma (plasma frequency) but(approximately) zero wavenumber k,. If an electron beam is injectedparallel to a wide plasma wave wavefront, it is wiggled transversely withan apparent wiggler wavelength h, = 2ncJwr,. In the second scheme, theelectron beam is injected down the axis between two narrow plasmawaves propagating opposite to the beam. The electrons are wiggled bythe radial fields of the waves and are guided (focused) by the wavesponderomotive force. With either scheme, effective wigglerwavelengths of order 50-100 urn may be obtained in a plasma of density10*7(cm-3) , thereby permitting generation of short wavelength radiationwith modest energy beams. The effective wiggler strengthsa, = eA/mc2 = 0.5 can be extremely large. We discuss the excitationmethods for such wigglers and examine these with PIC compute r simula-tions.

I. INTRODUCIIONThere is a great deal of interest in developing short wavelength

wigglers for free electron lasers and synchrotrons. Unfortunately, thehigh magnetic field strengths needed for such wigglers are not attainablewith present technology for conventional static magnetic wigglers withwavelengths less than 1 cm. Operation at wavelengths shorter than this,therefore, would require a large increase in the number of wigglerperiods and thus result in a very small extraction efficiency as well asrequire a very small electron beam energy spread and emittance In thispaper, we explore the suitability of two possible electric wiggler schemeswith effective wiggler wavelengths in the 100 pm range. In our proposalthese purely electric wigglers are relativistic plasma density waves. Suchwigglers, because of their extremely short wavelengths, allow the use ofa rather modest energy electron beam for FEL action in the visible orVUV range.

In the first plasma wiggler scheme, the wiggler consists of a purelyelectric field oscillating perpendicular to the electron beam with a fre-quency e+, but with no spatial dependence (kpx = 0) (see Fig. 1). Theamplification of radiation by such a field in vacuum has been modelledrecently by Y. T. Yan and J. M. Dawson, and the use of the plasmawave was proposed in Ref. 2.

tEp - Ep sinw,,t 2,

y 1 1 1 1 lyl$rz+@~~x-r

Fig. 1. Geometry of plasma wiggler FEL scheme 1 @l$.In the second plasma wiggler scheme, the wiggler consists of two

narrow plasma waves propagating anti-parallel to the electron beam (seeFig. 2). The two waves are chosen to be x out of phase so that theirradial fields add and wiggle the electron beam transversely as it passesbetween the waves.

plasma wavedensjty compressions

Fig. 2. Geometry of plasma wiggler FEL scheme 2 ($, / &,)Using a plasma wave as a purely electric wiggler is attractive for

two reasons: F irst, the effective wiggler wavelength (=2xc/0,, typicallyof order 100 ttm) is shorter than that available with conventional magnetwigglers and second, the effective wiggler strength can be extremelylarge (equivalent to 1 megagauss wiggler fields at 100 pm wavelength).However, when a plasma med ium is used for exciting the purely electricwiggler, the FEL electron beam can be strongly influenced by it. Thelimitations to FEL gain imposed by the plasma medium were discussedin Ref. 2. In this paper we discuss how plasma density waves suitablefor FEL actions can be excited and present 2-D particle-in-cell compute rsimulations of wave excitation and test particle trajectories of the FELbeam.

II. FEL MECHANISM IN A PLASMA WIGGLERThe equivalence between magne tic, electromagnetic and the two

purely electric (plasma) wigglers schemes can most easily be seen hornthe point of view of the electrons. Although the four cases appear quitedifferent in the laboratory frame, in the electron frame all appear to beelectromagnetic waves. Here, w, and CI+, epresent the frequencies of theelectromagnetic and electrostatic (plasma) waves respectively, and b, k,and k, (=odc) represent the wavenumbers of the electromagne tic, mag-netic and plasma wave wigglers respectively in the laboratory frame. Inscheme 1 , the purely electric wiggler has zero k parallel to the beamvelocity (vi,) whereas the purely magnet ic wiggler has a zero 0. In theelectron frame the momentum four-vectors are

Ii21 LbYl [ikj.-mmagnti,cII7urelymagnetic0I 1w, Scheme 1purely eledric

Scheme 2

(1)

For p = 1, the transformed quantities satisfy o = kc for all thecases except their wavelengths are different; k = &,./y (magnetic ), &J2y(electromagnetic), 2xc/mr,7 (purely electric, $ , 12) and Kc/O,y (purelyelectric, $ 1 1 3). The choice of plasma density determines the effeC-tive wiggler wavelength in the plasma schemes:

h, = 27tcfFc/wp 3X106/~ tvb I;,IL = ltc/wp T;b I I 3,

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Thus for plasma densities in the range 1Or3 o lOI8 cm-, h, ranges from 1cmto 15pm.

Each electromagnetic wave has a wiggler strength a,. = eE / mwc= eA/mc2 equal to its corresponding value, eBti2xmc* (magnetic),eE,Jmw,c (electromagnetic) and eEr/mwpc (purely electric or plasma) inthe lab frame. The effective maximum undulator strength parameter a,of the plasma wiggler can be easily estimated from 1-D Poissons equa-tion: V.E = -4rcen1, where nl is the perturbed electron density associatedwith the plasma wave. The maximum density rarefaction occurs whenn, = n,, known as the cold p lasma wavebreaking limit. In this limit,I k&p,mnx1 = 4xen, or eE/mr+ = 1 for wdksc = 1. Thus the maximumvalue of E, attainable in the plasma corresponds to a, = 1, independent ofthe plasma density. The effective max imum magnet strength of a plasmawave is

Befr(gauss) = F a, 5 F 3X10-a a

It can be seen that effective magnetic field strengths of order 1 mega-gauss are possible in a plasma of density 10 cm-3 .

In the plasma wigglers, as in the other two cases, the noise thatseeds the lasing process comes from Compton scattering, with the radi-ated frequency in the exact forward direction o, = ywp (%lj;) or0, = 4+wp &, l 1 3). Once, there is spontaneous em ission at r&, there isa resonant ponderomotive force on the radiating electrons. This force isr = -ev, x B, ic, where vw is the wiggle velocity and B, is the magneticfield of the spontaneous or stimulated radiation.

The first plasma wiggler Scheme has been described in some detailin Ref. 2. Here we turn to scheme 2. For the geometry of our secondplasma wiggler scheme we propose two narrow plasma waves withpotentials of the form

$1 = $+ sin(w,t + kpZ) e-YVw* and4 = ~,sin(Opt+kpZ+k)e~y-A~U.

(2)These represent two waves of width w travelling in the -a, directionseparated (radially) by an amoun t A. If the width of the waves (w) ismade comparable ot kp =c/or, it is easy to see by differentiating thepotentials with respect to y or z that the radial field E, = -&$/a~ will becomparable in magnitude to the longitudinal field E, = -&$/azg. The max-imum wiggler field (Er) between the two waves is approximatelyl.f$k,o,,) or 1.6 times the max imum longitudinal field of each waveseparately and is optimized for A = &v. The phase factor n in 02 assuresthat the wiggler fields add rather than cancel on axis (y = A/2). The long-itudinal field does not play much of a role in this geometry since it isparallel to $,. Furthermore, it vanishes on axis.

At this point we comment on the need for two waves, since onewave would appear to suffice to wiggle the electron beam. The reasonthat one wave does not make a good wiggler is that the electrons wouldbe blown radially outward by the waves so-called ponderomotive force.This force arises from the transverse gradient of the time averaged longi-tudinal field ( i.e., F = (+*2/m@&

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Two-dimensional simulations of the wake-field excitation of aplasma wave wiggler have been carried outs. In Fig. 3 contour plots ofconstant E, and Ey are shown for a 10 MeV driving beam moving in thez-direction. The driving beam was gaussian in z of length xc/w, andsquare in y of width 2&e,,. Note that the radial field (E ,) vanishes on axisas expected (e.g., by differentiating one of Eqs. (3)). Fig. 4 illustrates thereal space trajectories of counterstreaming test particles (7 = 1.5) in thesingle driving beam wake-fields of Fig. 3. The ponderomotive blowoutdescribed in section IIB is clearly visible.

J;. -I-. I. .~ -1100@WCFig. 4. Ponderomotive blowout of test particles moving

right to left in the fields of Fig. 3.Figure 5 illustrates the double plasma wave wiggler solution to theponderomotive problem (scheme 2). Fig . 5a shows location of the two

driving beams and Fig. 5b and 5c show slice plots of the resultingwiggler field (Ey) and longitudinal field (E,). Note that E, is maximumdown the axis of symmetry and that -aYE? provides a focussing pondero-motive force. Fig. 6 shows the phase space and real space trajectories oftest particles in the fields of Fig. 5. Particles in the central region areclearly focussed to the axis and wiggled.

02ZE1 0mco,

-.02

06 r-- ---T--

; I,/c. __- ; I / I-W :y-7,/ -1

Detailed calculations and simulations of the plasma wiggler FELwhich take into account the variation in aw across the beam cross section,edge effects, beam energy spread, emittance and plasma wave harmonicsare being carried out presently.IV. PROSPECTS

In this paper we have described two concepts for using plasmawaves as wigglers, which simultaneously offer the prospect of a smalleffective wiggler wavelength and a high wiggler strength. We havedescribed how such wigglers may be produced and modelled the trajec-tories of test particles in the wigglers with compute r simulations. Thereis a considerable on-going effort on generating coherent relativisticplasma wave wavefronts for high energy accelerator applications Anatural extension of this work will lead to characterization of thesewaves for wiggler application. In this paper the effect of the plasmamedium on the FEL process has been neglected. It was shown in Refs. 2and 11 that plasma instabilities provide a limit on the effective length ofthe FEL beam and that plasma wakefields may induce unacceptableenergy spread on the beam unless beam shaping or precursor beams areemployed. Extremely high quality electron beams are required for FELaction in the visible or VUV region using any kind of wigglerlo, includ inga plasma wave wiggler. Finally, although lasing may not be possible ateven shorter wavelengths than this, the strong a, and h, of a plasmawiggler may enable significant spontaneous emission (say of y rays) to beobtained by using a modest energy electron beam (such as that from acompact GeV storage ring).ACKNOWLEDGMENTS

The idea of using a wide plasma wave as an undulator wasindependently proposed by J. Slater of Spectra Technology, Inc. Wehave benefited from some valuable d iscussions with him. We wish tothank W. Mori for some 2D simulations.

This work was supported by DOE contract DE-AS03-ER 40120 ,NSF grant ECS 83-10972, ONR contract N00014-86-K-0584 and LosAlamos National Laboratory.REFERENCES

Fig. 5. Double plasma wave wiggler.g ---;: T- -,-i...

F- .-~

-*~ ~~ -/_~

--:I -L--:L_~ _- .

J!!g Lq,i2 _.. .I -1!-Fi :--

oki [ j 10.0 100WCFig. 6. Phase space and real space traJectories

in the double plasma wave wigglers. 11.

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