Estimates of the Maximum Lasing Frequency and of the Gain in FEL with Parabolic Potential. K.S. Badikyan *, D.K. Hovhannisyan.

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1 Etimat of th Maximum Laing Frquncy and of th Gain in FEL with Parabolic Potntial K.S. Badikyan *, D.K. Hovhanniyan National Univrity of Architctur and Contruction, Yrvan, Armnia * badikyan.kar@gmail.com Abtract Th ronanc frquncy of th ytm i found and th linar gain i drivd for th odd harmonic of thi frquncy. Avraging of th gain i carrid out ovr th initial ditribution of lctron in a tranvr cro ction of th bam. Etimat ar obtaind of th imum laing frquncy and of th gain at thi frquncy 1. Introduction Th quantum thory of amplification in rlativitic trophotron i dvlopd in [1]. Th quation for th amplitud of tranition probabiliti ar found. Th potntial through i ud in trophotron [-] in contrat to uual FEL [1-45] whr magntic undulator ar ud. Th claical thory of amplification in RS i dvlopd in [46]. W hall analyz th dpndnc of th ronanc frquncy on th initial tranvr coordinat of an lctron r( x) W hall how that thi dpndnc rult in a trong inhomognou broadning and ovrlap of th gain profil at diffrnt harmonic of th fundamntal ronanc frquncy. W hall aum th condition in dtrmining th gain avragd ovr x, i.., ovr th ditribution of lctron in a tranvr cro ction of th bam. W hall how how th avrag gain rtain it ronanc tructur, but ronanc amplification includ contribution of only a mall fraction of th bam lctron that ntr th ytm clo to th axi (mall valu of x ). W hall conclud with numrical timat of th imum attainabl laing frquncy and of th corrponding gain. A gomtry of th fild imilar to that conidrd in th prnt tudy appli alo to th ca of miion of radiation a a rult of channling of lctron and poitron in crytal

2 [17,,3]. In contrat to th convntional channling, th intraction of lctron with an xtrnal macrocopic fild (which can b dcribd a macrocopic channling) i much implr bcau thr ar no uch problm a th aborption of lctromagntic wav in a crytal, poibl limit on th channling lngth, tc. In th prnt articl th mittd nrgy i calculatd in RS uing quantum mchanical approach. Th avraging of gain i givn and it imal valu i found.. Ronanc frquncy and output nrgy Equation (8) of [1] giv (1) / (1 l / ) ( 1), (1) () r th ronanc frquncy of th ytm r xprd in trm of th numbr of an ocillatory lvl l filld mot ffctivly at th initial momnt in tim t =. Th valu of l can b xprd in trm of th initial paramtr of th lctron bam. A corrct dtrmination of l rquir th knowldg of th initial tranvr wav function of an lctron in th form of a packt localizd at om point x. Expanding thi function in trm of ( x) of Eq. () of [1], w find that th quar of th modul of th cofficint in th l xpanion hav a narrow imum at whr a( x ) x / 1/ l l p /ε x ε / ε / x ε a /, () i th claical amplitud of tranvr ocillation of an lctron in a trophotron with an initial coordinat x and th angl of ntry into th fild. Subtitution of l from Eq. () into Eq. (1) giv r λ / [1 (γ a) / ]. (3) Thi xprion for th ronanc frquncy ha bn obtaind arlir in th thory of pontanou miion during channling of particl in a crytal. If γa 1, thn th frquncy r of Eq. (3) i idntical with th rult obtaind in Rf. []: r r γ. W hall now calculat th nrgy mittd by an lctron at a frquncy ω in th dirction of th OZ axi and govrnd by th xprion na. (4) n n

3 linar in According to Rf. [3], th implt mthod of finding in an approximation which i which giv E involv olution of th ytm (9) of [1] by prturbation thory in trm of int, whr 3 ( E ) t ω 1 ( a) d in u ( a) F, 64ε γ 4 du u ( anht) t 1 ( a ) u ω ( 1), 4 γ ( 1) F ( z); F ( z) J ( z) J ( z), int int 1 ( / anh) ( 1) 1/ 1/ l /, anh ( / ) / ( 1) 3 l / 4. (5) (6) Th quantity rprnt th nrgy mittd by an lctron at a frquncy r ( 1) r. For fixd valu of α and x th width of th gain profil aociatd with th finit intraction tim btwn an lctron and th ytm ar / r Noc, whr N t/ i th numbr of tranvr ocillation of an lctron in th tranit tim t acro oc th ytm. Th paration btwn nighboring gain profil i r, i.., in th ca of a ingl lctron th lin do not ovrlap if N t/ > >1. Howvr, in viw of th dpndnc r ( x ) of Eq. (3), diffrnt lctron (corrponding to diffrnt valu of x hav diffrnt ronanc frqunci amplification lin. oc r and, conquntly, diffrnt poition of th ronanc Th nrgy mittd by a bam at a frquncy ω can b found by umming Δ and avraging th total mittd nrgy ovr (7) with rpct to x. Thi procdur and th corrponding condition ar dcribd in th nxt ction, whr th avrag gain i found. Howvr, w hall firt conidr th problm of th imilarity and diffrnc btwn fr-lctron lar with an undulator and a parabolic trough (trophotron). It i known that a fr-lctron lar with a planar linarly polarizd undulator can amplify a wav travling along th axi at a frquncy ω clo to th odd harmonic of th fundamntal ronanc frquncy. Th formula (3), (5), and (6) govrning th nrgy mittd by a ingl

4 lctron can b rducd to th form imilar to th corrponding formula for a fr-lctron lar with an undulator [3-5] if w introduc a paramtr K γ a / c, which tr rplac th uual paramtr for an undulator K B mc / und ( B and ar th undulator intnity and priod). Th major diffrnc btwn th two ytm ar obrvd whn w conidr a bam a a whol rathr than on lctron. If K und i indpndnt of th initial condition, thn in th ytm undr dicuion w hav Ktr Ktr ( x, ). For thi raon th avraging ovr x i not a trivial a in th ca of an undulator. Th rlativ fficincy of timulatd miion of th ( + 1 )th harmonic by a ingl lctron i govrnd by th factor F of Eq. (6) in Eq. (5). If γa 1, thn γ and r th factor i z[ ( 1) r ] (1/ 4)( 1)(γ a) ζ. Sinc th argumnt of th Bl function in Eq. (6) i mall compard with th indx, it follow that amplification of high harmonic i thn impoibl ( fall rapidly on incra in ). Effctiv amplification at highr harmonic rquir that th paramtr K tr hould b larg: K > > 1, which w hall hncforth aum to b corrct. If K 1, thn th ronanc tr tr frquncy r of Eq. (3) diffr conidrably from γ ( r γ ). Howvr, th laing frqunci ( 1) nd not b mall compard with r r bcau th numbr i fairly larg. Th poibility of incraing i govrnd, a uual, by th condition of a modratly trong fall of th avrag gain on incra in. 3. Avraging ovr th ditribution of lctron in a bam. Gain W hall conidr a pcific ditribution of lctron in a bam along th initial tranvr coordinat dcribd by th Gauian function 1 x f( x ) xp, d d (8) whr d d / ( ln ) ( d i th diamtr of th lctron bam atifying th condition d d ). W hall firt timat th cal of inhomognou broadning of th lin du to th cattr of x and aum that th condition d / 1 i atifid.

5 Th chang in r of Eq. (3) on variation of x from to d / r r d i qual to (1/ 16) d. Th corrponding hift of th miion lin rprnting th (+l)th 1 harmonic i ( 1) r ( 1) d r. 16 Thi hift xcd th homognou lin width / ditribution (8) atifi th inquality If thn r r Noc, if th paramtr of th d x 16 / Noc. (9) d x1 3 /, xcd alo diipation btwn th nighboring miion lin r and th lin ovrlap du to inhomognou broadning. Clarly, if th condition d x i obyd, an inhomognou broadning play no ignificant rol and avraging ovr x do not altr or, conquntly, th gain. Thrfor, th gain found in Rf. [5] i corrct if Ktr 1 and d x. For ufficintly larg valu of Ω and t, ncary to nur accptabl valu of th laing frquncy ω and of th gain, th paramtr x i vry mall o that th condition (9) i alway atifid by ral bam. W hall now carry out avraging of th total output nrgy of Eq. (7) intgrating trm by trm thi um with rpct to x allowing for th ditribution function (8). In ach trm of th um (7) th main contribution to th intgral with rpct to x i mad by mall (~ x ) rgion nar th point () x uch that u ( x ) : ( ) ( ) 4 8 min ( x ) ( 1) ( ), ω γ ω (1) whr min γ (11) Th argumnt of th Bl function zx ( ) at th point x x ( ) ( ) 1 z( x x ) z. 4γ (1)

6 In viw of th nd to conidr th poibility of amplification of highr harmonic [ 3 ( ) ], whn z and z, 1, w hall xpr th Bl function in trm of th MacDonald function [47]; thn, th rlvant factor i whr F K ( z ) ( z ) π 3 3 3πγ min 3/ /3 K /3 3 3γ., (13) (14) Th rang of valu of that mak th main contribution to th avrag total nrgy ( ), mittd at a givn frquncy ω i found from th condition min. At a fixd frquncy ω th trm in Eq. (7) with th numbr bcom th dominant on aftr avraging with rpct min to x. Thi man that th ronanc radiation appar mainly bcau of lctron which ntr a trophotron nar th axi ( x ) and ar charactrizd by th coordinat cattr x. Such ( ) radiation i mittd at frqunci ( x ) ( 1) ( x ) (8 ) /. r r Avraging of all th othr trm with giv ri to a nonronanc background, min th intnity of which i l than th ronanc valu bcau of th factor N 1. 3/ oc Th final rult of avraging th um (7) with rpct to x can b rprntd in th form whr 5/3 ( ) 1/ r K 5/ 4 /3 4 π d εγ ( E ) t ( x ) F r ( ), (15) F r ( ) ( ) in r ( x ) 8 d d (16) d t ( ) r ( x ) 8 r ( x ) i th form factor of th ronanc curv of width ~ ( r x ) and t t th valu of th imum i F ( r ).5. xprion 8 Finally, th gain xprincd by an xtrnal wav of frquncy ω i givn by th t 1/ 1/ 5/3 1/ N Nr L 3/ 7/ 1/ /3 E π dγ 8 (ln ) G K Fr ( ), (17)

7 whr N i th dnity of lctron in a bam; r mc i th claical radiu of an lctron; / d / a( x ) d / i a factor which allow for th ovrlap of th lctron bam and th wav bing amplifid. Th dpndnc of th gain G of Eq. (17) on th frquncy ω xhibit a tp fall at, whn th argumnt of th MacDonald function bcom larg. Thrfor, of Eq. (14) i th imum frquncy in a fr-lctron lar of th trophotron typ and right up to thi frquncy th gain can b ignificant. 4. Concluion Th gain of an xtrnal wav in th trophotron i givn by a uprpoition of contribution from amplification at diffrnt (odd) harmonic of th main ronanc frquncy r( x). Th main ronanc frquncy i hown to dpnd on th initial condition of th lctron, and in particular on it initial tranvral coordinat x. Thi dpndnc r( x) i hown to giv ri to a vry trong inhomognou broadning of th pctral lin. Th broadning can bcom larg nough for th pctral lin to ovrlap with ach othr. Th gain i avragd ovr x. In th avragd gain th ronanc pak ar hown to b much highr than th nonronant background. A phyical natur of th ronanc rmaining aftr avraging i dicud. Th imum achivabl avragd gain and frquncy of th trophotron FEL ar timatd. Auming that, can writ Eq. (3) and (17) in th form d, (.5), ( K /3 (1).5, and [ Fr ( )].5, w (18) γ dg, G 7.61 N r L / d γ.1 1 J r L / d γ (19) 3 5/ 1/ 7/ 6 5/ 5/ 7/ whr g i th gradint of th fild in a parabolic potntial trough on th axi of th ytm OZ (xprd in gau pr cntimtr in th ca of magntic quadrupol ln); J i th total imum currnt in th bam (in ampr). It i intrting to not that th gain G of Eq. (19) at th imum frquncy i indpndnt of g. Adopting in th timat th valu d 1cm, J 1A, γ = 1 A, L = m, and g = 1 kg/cm, w find that c ( 36 ), G 1%.. ()

8 Th timat how that in th ca of th abov paramtr of th bam and othr componnt of th ytm, it i in principl poibl to contruct a trophotron fr-lctron lar oprating in th infrard rang. W hall now not th ingulariti which ditinguih a trophotron fr-lctron lar from a convntional undulator lar. 1. Th undulator paramtr K tr dpnd only on th initial condition and not on th rlativitic factor γ; it can b numrically larg. In th abov xampl, w hav Ktr ( x ) 7.7. Th high valu of K tr how that th imum rial numbr of th harmonic in which mor or l ffctiv amplification i till poibl i high: x K x. 3 / r ( ).3[(1/ ) tr ( ] 168 Th gain profil xtnd from r ( x ) to r ( x ) and it conit of a larg numbr of quiditant narrow gain lin.. Th frquncy of a wav amplifid in a trophotron dpnd trongly on th angl at which an lctron ntr a ytm. For a fixd valu of, w hav 1/, whra at, w find that. Hnc, it follow that a trophotron provid an opportunity for continuou tuning of th miion frquncy bcau of a chang in th ntry angl. Clarly, thr i a margin for incraing th gain. In particular, prliminary haping of an lctron bam o that th angl of ntry of lctron into a trophotron dpnd on th initial tranvr coordinat x i a promiing approach. In thi n th optimal ditribution i that for which th quantity x rmain contant ovr th whol bam diamtr. Thi can b achivd by, for xampl, u of a focuing dvic bad on lctron-optical ln. In a ytm of thi kind thr i no trong inhomognou broadning bcau of th ditribution of lctron ovr x, o that th gain of Eq. (19) incra by a factor i.., by th two ordr of magnitud. ( d / ) x, A quantum-mchanical dcription of th motion of an lctron in claical fild i not in conflict with th fact that th final rult obtaind hr do not contain th Planck contant. It i natural to xpct th main rult of th prnt tudy apply alo in th claical approach.

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