energy spread

8/6/2019 Energy Spread

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OPTKS (OU4lLiSlC.4l IONS

III (;H-CAIN SMAI.I.-SIGNAl. MODKS OF I-HE: I;REkZ-I~I.E


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Here I is the currcnl. H is rhe RMS magnetic ticld ofthe wiggler, ltic cons1imI rj is one for a tlCtiCill wiglcrand a well-known diffcrcncc of Bessel functions for alinear wigglclY h, is the Iascr wavclcng~ti, X,, is ihcwiggtcr wavelcngtti. and M is related 10 the clcctronmass WIhbM=rrrll t(rBX,li?nrrrc)*ll*. (2)For large ri diffraction is unimportrnl, and onc-dinicn-sional theory gives the corrccI gain. For small ci thegain is less than that predicted by one-dirriensionaltheory.

A mode I:, with longitudinal dependence cxp(@)has the transverse depcndenccI:;(r) = .I(,( x+7) for I < u. 0)I:;(r) = hH,$gr/u) for r > II. (4 )where Jo is the Bcsscl function of order zero, I-/,, isthe Hanket funcrion of the first kind of order Lero. his a constant, and x and Q are complex numbers. Thecomplex number ~3 ics in I h e lirst quadrant and is rc-latcd to fi according ro 0 = Xp*i$, where k, = ?r/X,.In order that I;, and its firs1 derivative be continuousar r = ~2,wc require rtiat@/iI (Q)J(I(X) =x.~l(W()(o). (5)In addilion. x and 4 arc related by the cquar ionx2 = 0 . ($/ti* . p) 2n (0)where P is a dimensionless cncrgy detuning paramclcl.

For rhe cast of a monoencrgelic clcctron beam. ccnicasurcs tlic detuning of the electron energy 1. fromrcsonancc according 10I:= Afc~(X&~,)I.* 1 + Xyh,jl/(4W$ I. (7)This equation definesp. (Note thar P is related 10 lhcdctuning fi used in rcfs. [3pI according ro P = G.li.)WC assume rhat the second term in square hrackels ismuch less than one. For a lorcntsian distribution 01elcclron cnergics of I tic formf(jJ)=(tln(i?ll +(p /$,X2) I: (8,which is cenlerctl at ;U and has width U: WC mercl!:need 14.0.7 I I O make rhe reptacemcnlI-c+p ii. (1)in cq. (6).I2

The high-gain modes oi lhc FEL. are oblainctl h)numcricatly finding values of x and 0 which are simul-t,ancous solutions of cqs. (5) and (6). This is done bha Ycwton-Raphson procedure. The gain per unit Icnglhis given b>fi -: 2 Kc($) = I;.k,u;. (10)wticrrri = I n 1 ( d 2 ) h i 2 . ( 1 1 )

T h e physical meaning of the scaled gain i can heseen by noting that g and k vary proportionally whena is changed and all other physical paramelcrs archeld consiam. For each mode ri is a function of t,hct Mr i l l l l C t C r S l i , . a n d L ' . I t is typically 011 the order 01unity or less. and is only slightly sensitive to small (fac-t o i 0 1 ' Iwo) changes in ir when ri is small. If wc letI.,, - ifi = k,u: lx ;I characteristic gain length of IhcI-I: t., cYi*( IO) shows that wc can interpret uC =(Gk,) as the diffraction width associated wirhI I& gain lcngt h.

For large i and modes of low order, diffraction cf-facts are unimportant and x takes on values near suc-ccssivc Lcros ofJo. tr is a good approximation then fo~qlcct x2 in cq. (6). which becomes a cubic equationli)r o*. This approximation is equivalent to treatingt,hc FLIL by a one-dimensional theory. lhc low-orderIIWJCS all have ncnrly the same gain. There is at IIWSIone root of the cubic for which I$ has a positive ima-

.Ic OI I. 1P .5 ,*:dk.01.5:: 5 3 - 5 &ii


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\ ollllllr .54. IlllllllWr 2 OPIKS ~oMHI:XI~~~TIoss

@nary pa rt. Suc h a root. as shown by eq . (I 1). leadsIO am plification. The ga in obta ined hy solving the CU-hit equation is shown in fig. I. This,figure showscurves of 2li g a s a function of li2$i for variousVdlJeS ol li 3L!. Peak ga in is obta ined for ii = 3 I2 ,:.II i= 0 . the pea k ga in isi = 32/ 2ri2! and iilllpliihtiori is po ssihlc for jIi < 3(2ii) 2 . II 1 .> OF hen arn-plificiltil~rl is po ssihlc fo r ji > (rib) . Clea rly dc-c rcilS rlg u irlCrC;lSeS IhC gain arid thC gain ba riclwid~h.as well IS rllikkirlg thC gain ICSSsensitive to energySpWild. He shall see that these ClfCcts pe rsist when riis rCl~uc Cd 10 SrTal l Vd lJeS. hlJt that quarllitiltivc~!,~hcy ilrC grcutly Inotlilietl by diffrac tion. WC tind thatdiffrac tion is unimpo rtant for ri .a 20, is riiodc rarclyimportant for li = I. illld is a d0rui~Ulrlt CffCCl forsrllall ri. Snc e the ;iChiCvea blC urrent I is 1101 nc lepen-1ler11 in p rac tice of the clcc tron-bea m rad ius, o11C Ciul-riot say iii ge neral whc rher it is bc ttel to operalc withI i lVW Or SllUll U.z

WC prc scm results in this pa pe r only for the lowest-order irodc. which is the mode for which x approach-es I~IC firs1 de ro of.l,, as i + z:. This rnod c has theIargCst ga in. The ga irl is rllaximal above rcsonancc , butis nc arlq iriaxirrlul on rc sona rIcC f L = 0.

Fig. 3 Show s IIIC gain #iIS ;1 firnclion Of p for Vi l r -iOUS VilllJCS Of I, WhCll li= 2. / \ S CxpCClCl~. CnCrgyspread tlCc rC;lsCs hC ga irl and thC ga in ba ndwid th.ThC dc llJrlirlg giViilg rllilXirlllJl?l ga irl do ts rlol dcpclldap prcc inbly OII 1.: While the bardw idth for C = 0 cx -

b.C.

:.I \,u :!

i;I : (1)! ,I,.;.

.A / --\...., u - 0; . ,,,,,.,,,,....-.,,,,,,.,,.. .,,,,,,,,,,.,,,,,,,,,,,,,.,.Ili .

,, ;, ,: - _. _ __ .- _ -i l; . .

tend s well beyond the cutoflal J(4) 2LJ * I .I9 pre-dicted by one-d irncnsional theory, the band width forli = 0.5 d oes riot cxlcrtd in the nega tive i direction asfar as lhc c utoff al I prediclcd by one-dirriensionaltheory. Moreo ver, there is also a cutoffal posilivc ji.

Fig. 3a shows thC gain as l function Of deturiing i l li = 0 . 0 4 Tar 15= 0 autl for 1; = 0.025. Fig. 3b showsthe pea k near resonance on 811 expand ed sca le. Incornparisori with fig. 2. we see that the peak piri forU = 0 has incrcascd by ahm ~t ;1 f:lCtor oi three. Also,the gain is reduc ed by a lesser pcrcc ntag for a givenam ount o fencrgy spread. Conside ring that ci has bee nrcduml by a factor ol SO, both of these cha ngesarcquite sr~~all. The rnosf dramatic chgc in reduc ing rifrom 2 to 0.04 is the large increase n the gain ba nd-wid th. This la rge ba ndw idth sugc sts that it co 11lt1 bc


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difficult to ac hievesin@frequenc y ope ration in situ-ations where diffrac tion is a dom inant c ffcc t. Asshown in rels. [3.4J. positive dc tuning inc reases thewidth of the laser mod e and the d ivergenc eol thewave fronts. while nega tive de tuning c onc entrates thelaser mode within rhc electron hc am and gives wc ll-collimated wavefronts.

Ifi i s smi l~ l, i l i s s t rd igh t fonwr d to derive a nana lytic c xprcssion Ibr the gain on resonance, whc rcit is nearly ma xima l. In this cast one can neglec t thefirst term O2 on the righl side of cq_ 6) to g etx = (g/ G? + L> 1. (12)It turns out that Q2 is mo stly ima gina ry and x is mosl-Iy real. Also. 0 is very sma ll, and x is fairly sma ll.Using sma ll-argument app roximations For the func-tions in q. (5). and also using cq. (I I). wc deriveln(2jti) = (Ink)/ 2 + q t ?(a t G)? - 114. (13)whcrc yl, = 0.577 is the Euler-Maschc roni c onstant.Lc l. ( 13) rctluc es for 1.= 0 to a rcsuh dcrivcd in rcfs.[3.41.

Fig. 4 shows c urves of the m axim al g ain g a s a furls-tion of ri Ib r scve nl va lues of L?. Thc sc curves wereob tained using NJ. (13) ior sma ll ri, one -d imensionalIheory Ibr large r. and cxa cl numerica l c alculation inthe intermediate regime.

For an arbillary norm&cd transverse c lcc trondistribution U(T). WC expe c t exac t calc ulation of thehigh-gain mod es to bc somc what harder than Ior thesharp-edgeddislribu lion. since the field ES(f) is not

I lg. 4. Ua simul Sc;llCllpin % i\ ShOHn l!i il Iunclion 0l Wllellclcctron-h*lm rrbcliuc for sewn11 values ol lllc cncrp)~~wld i:

known ana lytica lly lsee cq . (3)j inside the elec tronbea m. I lowevc r, the limiting casesof large and sma llbea m radius are easy to trea t. II the beam dc nsily atits peak is nearly uniform over a width muc h grea terthan uC, one- dimensional theory should give the c or-rect gain, provided that the ratio ol current to lasermod e area s replac ed by the pea k current density.This assunlcs, however, tha t transverse excursions olthe electrons due to c mittanc c and betatron oscilla-tions arc muc h less then+ lfc i Q I, it ca n be shownthal c q. (13) is va lid for an arb itrary distribution u(r).providctl that u is defined by the condition

32ir2 $ dr f u(r) In(alr) j dr r u(f) = I .0 0

Let us consitlcr a deiinitc physica l example. Wechoose I = IO A, B = 0.24 T. X,, = 3.2 c m, rj = 1, andA, = 10 /AIll. kille CC]. (1 ), WC GlkUk itC! th at UL = 0.843mm . Then, using cq . (7), we see that the dev iation 15/zof the mean clec i.ron energy from the resonant energyI:(i isSE,X(, = (2.85 x IO )j i. (14)Also. using cq . (I 0). the ga in isg = i$2.34!m). (IS)The results in fig. 2 ap ply ilri = 2 and u = 1.61 mm .In lhis case he maximum ga in for L = 0 is # = 0.47and R = I .052!lni. The rcsulls in lip. J ap ply ifli = 0.04and N = 33.7 / ,HII. In the lultc r USC the maximum gainIbr ( . = 0 isi = I .33 and g = ?.%!m. Fig. 3 indic ates1hill appreciablega in is po ssible c vc rl wheu c%;:/ ~~,c -comus olo~dcr unity. -rheprcscf i l theoretical formula-tion is villid. howc vc r, only whc rl SE/ / .,j is sma ll. I I isunfortunate rhat IIIC iurge increase in g ain bandwidthill small li is accompaniedby such a sma ll inc rease ntolcrancc 10 energy spread.

This rcsc;lrc ll was supp orted in part by the 1i.S.Ollicc 01 Naval Rcscurch.

ReferencesI I I ..\.\I. Koraltrwnko antI I .L. Lldin. Ihkl. Akrcl. Nauk.

249 (lY79)843 jSo\. Ihg. Ihkl. 24 (lY79) 9861.(2 I V.\. hwr ;rntl \.I. Hilshrcin. Ihkl. Mad. Nouk. 250\ IYXO) 1304 J Sw. Ihys. Ihbkl. 25 rlYH(lb I 121.


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volulllc 54. l l l l l l l t ,cr 2 OYIICS COMMI:SICATO~S I5 U;I~ I98513 J (;. I. Uoorch.Opric% OII~III . 52 (19X4) 46.II! (;. I. Vloorc. The high-gain rqtiiuc ol llic trcc -c lcc lron

law. Proc. Inlcrn. Horkshop on (ohc rcnt ;Ind co ltcc tivctmpcrricr I I I I t i c iriltm~1iori ol rcki i v i s l i c c lcc i ro f i c antlr:lCrl 0l l l i lpnr l iC r i l t l iar ioi i . ( 01110. Ilrl). IYX4. to IN!txihtirlictl in hucle nr lnsrruirienls imc l WI hod s ii i Pti! s-ia (Sccl. A)

I.51 I .l. !ktlilrl~lltillll. ..\.\I. Smlm ar id J.S. u ur~dc . OpriGllpuitlinp iii il Ircwlc ~iron IilWX. ibid.16I J. (;al-llil~l;lrloc tlc. (;. I . Voo rc ;IWJ MO. Scull>. in:I w-c lCc Iroii txiic rdtion 0I cotic rCnI rirtliiilion. A. (:.:I.

thu. S.I Jam b\ an tI 11.0. SCIIII~, hoc . SPII: 4.and C. Idlc grini (Aiiicric ;in Inslilutc 01 Ptiysitr. NcuYork. 1984) 1 161.

18 I J.B. .5lurph). C. Pellc grm i ;md H. Ilo llil;l~~o. Op lia( 0l111l1. 13 (1985) 197.

19 J TJ. Or~cc liowski. WC. Muc tms. I.:\ . Icnko. I). Prowits.I.). Rop ers. ( 5 Chidis. K. Ihttxh . 11.11. lr)pkins. H.W.Kurnninp . AC. Paul. :\ .%I. Scsslc r, (;.I). So vc r. .l.TTam be . ILM. Yaumwto and J.S. Wurtelc . Il~cstulus01 rtlc Lawrence Ihkelq l.abo rr~ory antI IIN ImrrcnwI.iwriiiorc Xil l ioni l l tnborat ory I :rcc- l : lcctr ol i I i laCr.Vol . c i r c t l in rd. 14 I, p. 65I IO I .I.U J. Uad~y. Iillk at St:lnlortl F;e\n Rings Nork\lloli.Julx. 19X3.

II I I J.51.J. Udr:~. (OIICCtlIUill Sy\lLl l l dc sipi ()I Xl:\llI.5. vo l. ~ilc tl iii rd. 15 I. 1). 12.I I2 I ILI.. SCWIU~IIL J.C. (d tlsrc in. J.S. I riiscr i 11 l t1 ILK.Coop er. A liiiwl-Jriwn XC\ I rsc-IIccIrori law.ibid . r?. I90

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