High Gain Free Electron Lasers Driven by Flat Electron Beam

LBNL-51615High Gain Free Electron LasersDriven by Flat Electron BeamMing XieLawrence Berkeley National LaboratoryBerkeley,California94720,USAPresented at24th International Free Electron Laser ConferenceArgonne,USA,September9-13,2002To be Published inNuclear Instruments and Methods in Physics Research AThis work was supported by the Director,Office of Science,Office of High Energy and Nuclear Physics,High Energy Physics Division,of the U.S.De-partment of Energy under contract No.DE-AC03-76SF00098..High Gain Free Electron LasersDriven by Flat Electron BeamMing Xie∗Lawrence Berkeley National Laboratory,Berkeley,CA94720,USA AbstractA variational solution of the fundamental eigenmode is presented for high gain free electron lasers driven byflat electron beam having unequal emittances,betatron focusings and beam sizes in two transverse planes.Key words:flat beam,elliptical beam,high gain free electron laserPACS:41.60.Cr1IntroductionThe main objective of this article is to expand our earlier work[1]on the eigenmode solution for high gain FELs into the situation where electron beam distribution is no longer axially symmetrical.Such a beam,generally termed flat beam,may have unequal emittances,betatron focusings,or beam sizes in two transverse planes.Earlier studies on this subject date back more than aWork supported by U.S.DOE under contract No.DE-AC03-76SF00098.∗Corresponding author.Tel.:+1-510-486-5616;fax:+1-510-486-6485Email address:mingxie@(Ming Xie).decade ing a special variational technique[2],the fundamental mode was calculated by Xie[3]for parallel beam having elliptical cross ter, a dispersion equation for the eigenvalue of the fundamental mode was derived using a different technique by Chin et al.[4]for more general cases including unequal emittances and betatron focusings.However,the latter technique was known at birth[4]to introduce a systematic error when approaching the1D limit,thus compromising its accuracy for short wavelength FEL calculations, and furthermore,the dispersion equation was not solved.In this article,we present thefirst generalflat beam solution and specific calculations for the fundamental eigenmode and examine effects offlat beam on high gain FEL performance.The eigenmode equation is formulated forflat beam in section2.The variational solution is derived in section3.Finally, specific calculations for the LCLS case are given in section4.2Eigenmode EquationWe specify the radiationfield by a complex envelope a r(x,z,t)slowly varying with respect to a carrier wave exp(k r z−ωr t),where x≡{x,y},the carrier frequencyωr=ck r=2πc/λr is determined by the resonance condition k r =2γ20k w/(1+a2w),γ0mc2is average electron energy,λw=2π/k w is wiggler period,a w=eB rmsλw/2πmc,and B rms is rms wigglerfield assumed con-stant along wiggler axis.The radiationfield is normalized according to|a r|= eE rms/k r mc2,where E rms(x,z,t)is rms amplitude of the electricfield.Upon introducing Fourier transform byaν=1√2π dθe−iνθ(e iθa r),whereθ=(k r+k w)z−ωr t andν=ω/ωr,the eigenmode of the form aν= a(x)exp(−iµz)can be determined by the mode equation[1,5,6]µ−k w∆ν+12kr∂2∂x2 a(x)=ih16k w L1d∞−∞d2p dη∂F∂η−∞dse−i(µ−ξ)s a(x ),(1)withξ=2k wη−k r2(p2x+k2βx x2+p2y+k2βy y2),x =x cos(kβx s)+p xkβxsin(kβx s),y =y cos(kβy s)+p ykβysin(kβy s),L1d=λw4π 2πhγ30 r e n0λ2w a2w f2B13,whereµis the complex eigenvalue related to our earlier notation[1]byµ= iq/2L1d+k w∆ν,L1d is the1D power gain length,∆ν=ν−1,η=(γ−γ0)/γ0,h=(2/√3)3,fB=1for helical wiggler and f B=J0(χw)−J1(χw)for planarwiggler,χw=a2w/2(1+a2w),r e is the classical radius of electron,n0is the peak electron volume density on the axis,F=F⊥(x,p)F (η)is unperturbed beam distribution function normalized by∞ −∞d2p F⊥(x=0,p)=1,∞−∞dηF (η)=1.Transverse focusing of the beam in wiggler is assumed to have a strength invariant along the axis,thus betatron motion is governed byp x=dxdz,dp xdz=−k2βx x,p y=dydz,dp ydz=−k2βy y,whereβx=1/kβx andβy=1/kβy are constant betafunctions.To perform specific calculations,we use a Gaussian model withF ⊥=12πσx σy k βx k βye−x 2+p 2x /k 2βx2σ2x −y 2+p 2y /k 2βy 2σ2y,F =1√2πσηe −η22σ2η,where σx =√βx εx and σy =βy εy are rms beam sizes matched to the focusingchannel,εx and εy are rms emittances,and σηis relative rms energy ing X ≡{X,Y }≡{x/σx ,y/σy }and τ=s/2L 1d ,Eq.(1)can be expressed in a more compact scaled form2ηdx ∂2∂X 2+2ηdy∂2∂Y 2+¯κa (X )=∞−∞d 2X Π(X ,X )a (X),(2)where ¯κ=κ−ηω,κ=2L 1d µandΠ(X ,X)=−∞τdτhe −Ψ2πΦ,Φ=sin(2√ηdx ηεx τ)sin(2√ηdy ηεy τ),Ψ=iκτ+2η2γτ2+(1+iηεx τ)Ωx 2sin 2(2√ηdx ηεx τ)+(1+iηεy τ)Ωy 2sin 2(2√ηdy ηεy τ),Ωx =X 2+X 2−2XX cos(2√ηdx ηεx τ),Ωy =Y 2+Y 2−2Y Y cos(2√ηdy ηεy τ).There are six scaling parameters in Eq.(2):ηdx =L 1d /2k r σ2x and ηdy =L 1d /2k r σ2y are diffraction parameters;ηεx =2L 1d k r k βx εx and ηεy =2L 1d k r k βy εycharacterize effective spread in longitudinal phase due to emittance and be-tatron focusing,and ηγ=2L 1d k w σηdue to energy spread,respectively;and ηω=2L 1d k w ∆νis a frequency detuning parameter.The 1D power gain length can now be expressed asL 1d=λw4πhI A σx σy k 2w γ3I b a 2w f B13,where I b is beam current and I A =17.05kA is the Alf`v en current.3Variational SolutionNext,we present an approximate solution for the fundamental mode.Accord-ing to the recipe of a special variational technique[2],a variational functional may be constructed from Eq.(2)as∞−∞d2X a(X) 2ηdx∂2∂X2+2ηdy∂2∂Y2+¯κ a(X)=∞−∞d2X d2X a(X)Π(X,X )a(X ).Substituting into the variational functional a trial solution of the forma(X)=exp(−αx X2−αy Y2),whereαx andαy are complex variational parameters to be determined,and applying the variational conditions[2]δκδαx =0,δκδαy=0,to the resulting equation,we obtain three equations from which the eigenvalue κand mode parametersαx andαy can be determined byF1≡¯κ4√αxαy−ηdx2 αxαy−ηdy2αyαx−0−∞τdτhe−f1f2x f2y=0,(3)F2≡−¯κ8αx√αxαy−ηdx4√αxαy+ηdy4αx αyαx+−∞τdτhe−f1f 2x2f2x f2x f2y=0,(4)F3≡−¯κ8αy√αxαy−ηdy4√αxαy+ηdx4αy αxαy+−∞τdτhe−f1f 2y2f2y f2x f2y=0,(5)whereF 2(κ,αx ,αy )=∂F 1(κ,αx ,αy )∂αx,F 3(κ,αx ,αy )=∂F 1(κ,αx ,αy )∂αy,f 1=iκτ+2η2γτ2,f 2x =(1+iηεx τ)2+4αx (1+iηεx τ)+4α2x sin 2(2√ηdx ηεx τ),f 2y =(1+iηεy τ)2+4αy (1+iηεy τ)+4α2y sin 2(2√ηdy ηεy τ),f 2x=∂f 2x ∂αx =4(1+iηεx τ)+8αx sin 2(2√ηdx ηεx τ),f2y=∂f 2y ∂αy=4(1+iηεy τ)+8αy sin 2(2√ηdy ηεy τ).It is noted that in the limit of ηεx =ηεy =ηγ=0,Eqs.(3,4,5)are reduced to the same equations studied earlier [3]for the case of parallel Gaussian beam with elliptical cross section.Given parameter αx and αy ,the mode properties can be determined com-pletely by comparing the mode profilea =exp−αx x 2σ2x −αy y2σ2y,with the usual Gaussian descriptiona =exp−x 2w 2x +ik r x 22R x −y 2w 2y +ik r y 22R y,where w x and w y are mode sizes,R x and R y are radii of phasefront curvature.In particular,we have for mode sizesw x =σx√αxr,w y =σy√αyr,and for far field divergence anglesθx =αxr1+α2xi α2xrλrπσx,θy =αyr1+α2yi α2yrλrπσy.Finally,power gain length of the fundamental mode is related to the eigenvalue by L g =L 1d /κi.5678910L g (m )R εFig.1.L g as a function of R εvaried from 1to 50with βx and βy optimized to minimize L g.20406080100βx a n d βy (m )R εFig.2.Optimized βx and βy as functions of R εvaried from 1to 50.51015202530R σ a n d R wR εFig.3.R σand R w as functions of R εvaried from 1to 50with optimized βx and βy .4LCLS ExamplesGiven the solution of Eqs.(3,4,5),we are now ready to examine effects of flat beam on high gain FEL performance.Let’s first define ratios of emittances,betafunctions,beam sizes and mode sizes respectively byR ε=εx εy ,R β=βx βy ,R σ=σx σy ,R w =w x w y.Consider a scenario in which R εis increased from unity under the constraint that 4D emittance εx εy =ε20is kept constant.At each value of R ε,L g is minimized by varying βx ,βy and frequency detuning.We shall take LCLS nominal design values for all other parameters [7]:λr =1.5˚A ,γ0=28009,I b =3.4kA,γ0ε0=1.5mm-mrad,ση=2×10−4,a planar wiggler with λw =3cm and √2a w =3.7.In this scenario R εis the only free varying parameter.Figure 1shows L g as a function of R εvaried from 1to 50with optimized βx and βy given in Figure 2.At larger value of εx ,focusing has to be relaxed with larger βx to minimize gain reduction due to angular spread.On the otherTable1LCLS ExamplesRε11050γ0εx(mm-mr) 1.5 4.7410.6γ0εy(mm-mr) 1.50.4740.212βx(m)204795βy(m)209.4 6.7σx(µm)3290190σy(µm)32137.1w x(µm)4476120w y(µm)442926θx(µr) 1.6 1.10.83θy(µr) 1.6 2.4 3.1L g(m) 6.0 6.78.3L1d(m) 3.1 3.1 3.3hand,focusing can be enhanced with smallerβy at smaller value ofεy.Figure 3shows Rσand R w as functions of Rεvaried from1to50.Notice the aspect ratio of laser mode is much less than that of electron beam.More details are given in Table1for three cases Rε=1,10,50.5ConclusionsWe have presented an effective solution of the eigenmode for high gain FELs driven by generalflat beam.It is found that the gain length increases with emittance ratio when4D emittance is kept constant,and the rate of increase is rather weak in the LCLS parameter regime if beam focusing is simultaneously optimized.In addition,the aspect ratio of laser mode is much smaller than that of electron beam for larger emittance ratio.Effects offlat beam in other regimes and for other scenarios can be readily evaluated with the solution and methods provided here.References[1]M.Xie,Nucl.Instr.Meth.A445,59and67(2000).[2]M.Xie and D.Deacon,Nucl.Instr.Meth.A250,426(1986).[3]M.Xie,“Theory of Optical Guiding in Free Electron Lasers”,Ph.D.dissertation,Stanford University,(1988).[4]Y.Chin,K.Kim and M.Xie,Phys.Rev.A46,6662(1992).[5]K.Kim,Phys.Rev.Lett.57,1871(1986).[6]L.Yu,S.Krinsky and R.Gluckstern,Phys.Rev.Lett.64,3011(1990).[7]LCLS Design Study Report,SLAC-R-521,(1998).。