Spontaneous and Stimulated Radiative emission of Modulated Free- Electron Quantum wavepackets - Semiclassical Analysis

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1 Sponanous and Simulad Radiaiv mission of Modulad Fr- Elcron Quanum wavpacks - Smiclassical Analysis Yiming Pan, Avraham Govr Dparmn of Elcrical Enginring Physical Elcronics, Tl Aviv Univrsiy, Rama Aviv 69978, ISRAEL Absrac Hr w prsn a smiclassical analysis of sponanous and simulad radiaiv mission from unmodulad and opically-modulad lcron uanum wavpacks. W show ha h radiaiv mission/absorpion and corrsponding dclraion/acclraion of h wavpacks dpnd on h conrollabl hisory-dpndn wavpack si. Th characrisics of h radiaiv inracion whn h wavpack si (duraion) is shor rlaiv o h radiaion wavlngh, ar clos o h prdicions of h classical poinparicl modling. On h ohr hand, in h long-sid wavpack limi, h inracion is uanum-mchanical, and i diminishs xponnially a high fruncy. W xmplify hs ffcs hrough h schm of Smih-Purcll radiaion, and dmonsra ha if h wavpack is opically-modulad and priodically-bunchd, i xhibis fini radiaiv mission a harmonics of h modulaion fruncy byond h limi of high-fruncy cuoff. Bsids, h radiaion analysis is furhr xndd o h cass of suprradian mission from a bam of phas-corrlad modulad lcron wavpacks. Th faurs of h wavpack-dpndn radiaiv mission xplain h classical-o-uanum hory ransiion, and indica a way for masuring h uanum lcron wavpack si. This suggss a nw dircion for xploring ligh-mar inracion. 1

2 Inroducion Acclrad fr lcrons mi lcromagnic radiaion whn subjcd o an xrnal forc (.g. synchroron radiaion [1], Undulaor radiaion [], Compon scaring [3]). Radiaion can also b mid by currns ha ar inducd by fr lcrons in polariabl srucurs and marials, such as in Chrnkov radiaion [4], ransiion radiaion [5], Smih-Purcll radiaion [6]. Som of hs schms wr dmonsrad o opra as cohrn simulad radiaiv mission sourcs, such as Fr Elcron Lasrs (FEL) [7-9], as wll as acclraing (simulad absorpion) dvics, such as Dilcric Lasr Acclraor (DLA) and Invrs Smih-Purcll ffc [1-1]. Th simulad radiaiv mission of an nsmbl of lcrons (an lcron bam) is cohrn (o h xn of cohrnc of h inpu radiaion wav bing amplifid). Th sponanous mission of an lcron bam in any of hs radiaion schms is incohrn, unlss h paricls ar mad o mi in phas wih ach ohr. This can b don by pr-bunching h bam. In his cas, h radiaiv mission is proporional o N - h numbr of lcrons suard (whil h mission of a randomly disribud lcron bam is proporional o N). This cohrn sponanous radiaion procss is analogous o Dick s suprradianc of aomic dipols [13]. I has bn xndd in h classical limi o h cas of bunchd lcron bams, mploying a gnral formulaion ha is applicabl o h wid variy of h aformniond fr lcron radiaion schms. [14]. Mos of h fr lcron radiaion schms of mission or acclraion opra in h classical horical rgim of lcrodynamics, whr h lcrons can b considrd poin-paricls and h radiaion fild is dscribd by Maxwll uaions (no fild uaniaion). Howvr a variy of fr lcron radiaion schms [15,16], and paricularly FEL [.g. Rfs: 17,18,19] hav bn analyd in h framwork of a uanum modl in which h lcron is dscribd in h inhrnly uanum limi - givn as a plan-wav uanum wav funcion h opposi limi of h poin-paricl classical prsnaion. Quanum dscripion of h lcron wavfuncion is also usd in anohr rcnly dvlopd rsarch fild of lcron inracion wih radiaion: Phoo-Inducd Nar-Fild Elcron Microscopy (PINEM) [,1] In his schm a singl lcron uanum wavfuncion inracs wih h nar-fild of a nanomric srucur illuminad

3 by a cohrn lasr bam. Of spcial rlvanc for h prsn discussion is a rcn PINEM-kind xprimn of Fis al [], in which i was dmonsrad ha opical fruncy modulaion of h nrgy and dnsiy xpcaion valus of a singl lcron wavpack ar possibl in his mhod. All hs horical modls and xprimns in h classical and uanum limis of h lcron dscripion rais inrs in h horical undrsanding of h ransiion of h lcron-radiaion inracion procss from h uanum o h classical limi. This is also rlad o dpr undrsanding of fundamnal physics usions, such as h pariclwav dualiy naur of h lcron, [3] and h inrpraion and masurabiliy of h lcron uanum wavpack. Th wavpack rgim of lcron inracion wih radiaion is no wll foundd in hory. Rcn xprimnal sudy of sponanous Compon scaring by h xpanding wavpack of a singl lcron, rvald no dpndnc on h wavpack si and hisory, as also was prdicd by a horical QED analysis of his problm [4-6]. W assr, hough, ha his conclusion dos no carry ovr o h cas of simulad inracion (mission/absorpion or acclraion/dclraion). W hav shown in an arlir publicaion [7] ha h classical phas-dpndn acclraion/dclraion of a singl lcron in h poin-paricl limi is valid in a crain opraing rang also in h uanum-wavpack rgim. Th momnum ransfr from h fild o h wavpack (acclraion) is smallr han in h poin-paricl limi, and i diminishs in h inhrn uanum limi, whr h wavpack si T of h inracing radiaion wav xcds h opical radiaion priod 1 (1) Thus, masurmns of h lcrons nrgy spcrum afr inracion wih radiaion wavs a diffrn fruncis would nabl drminaion of h hisory-dpndn wavpack si. In h smi-classical analysis of lcron wavpack inracion wih radiaion [7] h wavpack-dpndn nrgy (momnum) acclraion/dclraion of h lcron was 3

4 calculad using firs ordr analysis and xac numrical soluion of Schrodingr uaion. Evidnly, such a chang in h lcron wavpack nrgy mus involv also corrsponding chang in h nrgy of h inracing radiaion wav. In h prsn aricl, w xamin his nrgy xchang procss on h sid of h radiaion fild, again using a smi-classical formulaion, in which h lcron currn dnsiy is rprsnd by h xpcaion valu of h wavpack probabiliy dnsiy disribuion, and h radiaion fild is classical, and modld in rms of a modal xpansion formulaion of classical Maxwll uaions. Th following analysis rsuls in full agrmn wih h arlir Schrodingr smi-classical analysis and prsns h sam disincion bwn h uanum, classical and wavpack inracion rgims. Furhr, hr w also prsn for h firs im xprssions for h sponanous and simulad mission from a modulad lcron wavpack and from an nsmbl of unmodulad or modulad lcron wavpacks. In h following scions, w prsn a daild smi-classical hory for wavpackdpndn radiaion of singl lcron and lcron bams. In scion Modling and Mhods, w driv h probabiliy dnsiy currn of an unmodulad and modulad lcron uanum wavfuncion from is Schrodingr Euaion soluion. W hn us hs currn xprssions o find h spcral opical paramrs of h mid radiaion basd on a gnral mod-xpansion formulaion soluion of Maxwll uaions. Th rsuls and discussions ar xhibid in h following scions Rsuls and Discussions, which ar dividd ino four cass of sponanous and simulad missions: In subscions I.A and I. C rspcivly w analy h cass of unmodulad and modulad singl lcron wavpack; in subscions IIE-IIG w analy h cass of suprradian and simulad-suprradian radiaion mission by a mulipl-paricl lcron bam of unmodulad wavpacks and modulad wavpacks. In subscion I.B w driv a classical Einsin rlaion bwn sponanous mission and simulad mission/absorpion of an lcron wavpack. In subscion I.D w prsn a daild xampl of wavpack radiaiv mission/inracion in a Smih-Purcll radiaion schm. Finally, in h las scion Conclusions and Oulook w summari all nw rsuls, and propos an xprimnal sup for sing h dpndnc of wavpack radiaiv mission/inracion on h wavpack characrisics. 4

5 Figur 1: Exprimnal sup of wavpack-dpndn sponanous/simulad Smih- Purcll radiaion mission/absorpion and corrsponding dclraion/acclraion of uanum lcron wavpack. Modling and Mhods Hr, w prsn a smi-classical analysis of sponanous, suprradian and simulad suprradian mission by modulad and unmodulad lcron uanum wavpacks and muli-paricl bams. A proposd schm for masuring sponanous and simulad radiaion mission and lcron nrgy spcrum of an lcron wavpack is shown in Fig. 1. This inracion schm, basd on h Smih-Purcll radiaion ffc was usd in [7] o calcula h wavpack-dpndn lcron nrgy spcrum du o radiaiv inracion wih an inpu radiaion fild, injcd ino h inracion rgion abov h graing, in conrolld phas corrlaion wih h incoming lcron wavpack. Th wavpack si dpnds on h drif im from h cahod o h graing. Hr w includ also opical ligh dcion for masuring h sponanous and simulad mission from h lcron wavpack. Fig. shows schmaically an laboraion of h firs schm, including an nrgy modulaion rgion whr h lcron wavpack ravrss h nar fild rgion of a ip illuminad by a lasr ip [], and gs nrgy-modulad a h fruncy b of h "modulaing radiaion wav". Th nrgy modulaion urns ino dnsiy modulaion of h wavpack nvlop wihin h drif lngh L D ' L d L c. Thn, in h inracion rgion 5

6 <<L G abov h graing, i inracs wih h nar-fild of an inpu radiaion wav a a fruncy nar h fruncy of h "modulaing wav" or is harmonic fruncy - l. Undr h forc fild of h "inpu wav", h modulad lcron wavpack b xprincs acclraion/dclraion, and xhibis a corrsponding simulad radiaionmission/absorpion, dpnding on h phas diffrnc bwn h inpu wav and h modulaing wav. According o classical lcrodynamics analysis, h lcron wavpack can mi sponanously radiaion, also whn h inpu wav is urnd off, a harmonics of h modulaion fruncy, byond h fruncy cu-off condiion (1) of an unmodulad wavpack. Figur : Exprimnal sup of wavpack-dpndn Simulad-Suprradian Smih- Purcll radiaion mission/absorpion and dclraion/acclraion wih a harmonic of a dnsiy-bunchd uanum lcron wavpack. Th wavpack is nrgy modulad a a ip by muli-phoon mission/absorpion procss, and urns o b dnsiy modulad afr a drif lngh L D. Smi-classical drivaion of h probabiliy dnsiy currn of an lcron uanum wavpack W modl h lcron wavfuncion afr is mission from h cahod by a Gaussian wavpack in momnum spac 1, dp p i( p Ep)/ () 6

7 p 1/4 p p 4 p p xp (3) To valua h voluion of h wavfuncion () in im and spac w us h Taylor xpansion of h rlaivisic lcron disprsion rlaion E c m c p E v p p p p * p p m (4) whr E p p mv, m mc, mar h cnr nrgy and momnum and h * 3 ffciv longiudinal mass of h wavpack, and 1 1 v c. Subsiuion of (4) and (3) in () rsuls in [15,7] i pe p v, xp 14 1 i 4 1 i (5) wih * m and h iniial wavpack "wais p. Th xpcaion valu of h fr drifing lcron currn dnsiy can b wrin in rms of h xpcaion valu of h lcron probabiliy dnsiy * * Jr, v * r, m (6) In our 1-D modl, h axial currn is whr J, v, v f f v (7) r 1 v f v xp 1 (8) 7

8 and h ransvrs xpansion of h ransvrs profil funcion f r is nglcd. Now ransform h coordinas fram, so ha h origin is a h nranc o h inracion rgion, and h lcron wavpack arrivs hr a im afr drif im D, and furhr assum ha h lcron wavpack dimnsions hardly chang along h inracion lngh:, or corrspondingly L D D, hn w can wri r J, f f v 1 f L d L D LD 1 v Ld (9) Furhr laboraion is ruird for dscribing h wavpack voluion a h modulaion rgion in h cas shown in Fig. and h subsun drif hrafr (s Appndix A). In his cas, h lcron wavpack undrgos a muli phoon mission/absorpion procss in h shor nar-fild modulaion rgion, and is uanum wavfuncion gs modulad in momnum spac a harmonics of h phoon rcoil momnum p b is h fruncy of h modulaing lasr. Afr h modulaion poin whr b / v, whr 1/4 p p n p p p J xp n g (1) n 4 p L M g ( / ) df( ) is h avragd xchangd phoon numbr gaind from b h nar-fild wih F () h slow-varying spaial disribuion of h nar fild of h ip illuminad by h lasr. Th modulaion ampliud of h n-h ordr muliphoon procss is characrid by h Bssl funcion and h Gaussian nvlop of momnum widh, shifd rlaiv o cnral momnum o p n p. p As in h cas of h unmodulad wavpack (.5), w obain h spaial voluion of h wavpack in ral spac away from h ip ( L ) by subsiuing (1) in (): c 8

9 v n p, J g xp 4 1 i i ( p p )/ n i n p b v v n (11) 1/4 1 i n Th voluion of h modulad wavpack along h drif scion is bs prsnd by a Wignr disribuion wih rspc o h rlaiv posiion v and h shifd momnum p' p p ha is dfind as 1 * i W, p', d p' / p' / xp i Ep' / Ep' / (1) Fig. 3 shows h Wignr disribuion W, p', ' D afr opimal (Maximum bunching) drif im D' L D' v, whr LD ' LD L c is h drif lngh from h modulaion poin o h graing (s Fig. ). W also show h projcd dnsiy disribuions in boh momnum and spaial spacs in Fig. 4. Th shown disribuion paramrs ar.7, g 11.4, in corrspondnc o Fis's xprimn. [] Th Wignr funcion in phas-spac dmonsras h urning of momnum modulaion ino h igh dnsiy micro-bunching a an simad opimal drif im [36] ' D,max 1 Tb p m p (13) whr T b b is h bunching priod, and h maximal ffciv momnum gain p g pdpnds on h xchangd phoon numbr a h modulaion poin. No m ha h momnum spcrum dos no vary wih drif propagaion, hus w canno rval h volvd micro-bunchd srucur of h modulad wavpack by masuring is momnum spcrum alon. Finally, h probabilisic xpcaion of h currn dnsiy of h modulad wavpack (.6) afr drif lngh L D ', is found o b priodically modulad in im and spac as shown in Fig. 3, which displays igh and narrow microbunching a h maximal bunching drif im (.5), wih widh 75as (s in h b 9

10 insr) []. Classically, w can wri hn h currn dnsiy disribuion in h inracion rgion, dfind in h coordinas rang L,, v v J f r f f (14) mod G whr is h modulaion rfrnc im a h nranc o h inracion rgion =. Assuming again ha h wavpack dimnsions hardly chang along h inracion rgion, and h funcion 1 f wavpack nvlop (9) wih v, Sinc fmod 1/ L L D D is h unmodulad is priodic wih priodt b, i can b xpandd as a Fourir sris in rms of h harmonics of h bunching fruncy mod whr l dnos h l-h ordr harmonic. l il b (15) l f B Subsiuing (11) in (14) w driv in Appndix A h cofficin B l of h Fourir sris xpansion afr a drif im L v away from h modulaion ip ' ' D D ' n ' ( n l) D D ' Bl Jn g Jnl g xp xp i( n l) lbd il n 4 1i 4 1 D i D (16) whr * / m v, and w kp h dpndnc on h iniial phas, which is imporan for h subsun xnsion of h analysis o h muli-paricl cas, whr all paricl wavpacks ar modulad. 1

11 Figur 3: Formaion of aoscond igh lcron dnsiy bunching. (a) Th Wignr disribuion afr opimal drif lngh ' L D,max. (b) Th modulad momnum (nrgy) disribuion of lcron wavpack afr inracion wih h nar-fild on a ip. (c) Th dnsiy micro-bunching of a singl-lcron wavpack. Formulaion of h spcral opical paramrs - Radiaion mod xpansion W now urn o calcula h radiaion mission by h currn of a bam of lcron uanum wavpacks. W bas our analysis on a gnral radiaion-mod xciaion formulaion for bunchd bam suprradianc [14]. Th radiaion fild xcid by a gnral fini of currn Jr, is xpandd in h fruncy domain in rms of a s of orhogonal dircional ransvrs mods, E r H r ha ar h ransvrsly confind homognous soluion of h lcromagnic wav uaions of fr spac or a sourc-lss guiding srucur 11

12 whr C, Er,, Hr, C, E r, H r (17), h slowly growing fild ampliud along h propagaion dircion () of a radiaion mod a spcral fruncy is drivd from Maxwll uaions [14]. Th incrmn of h fild ampliud of mod is 1 C C C, d r P J r E r (18) whr ou in * P ˆ E r H r d r is h normaliaion powr of mod. Th spcral radiaiv nrgy mission pr mod, drivd from Winr-Khincin horm (s Appndix B), is givn (for ω>) by dw ou C d P (19) Subsiuing (18) in (19), h mid spcral radiaiv nrgy pr mod can b wrin in rms of hr pars dw dw dw dw d d d d in SP/SR STSR () wih dw C d P in dw d in J r r * 3 P SP/SR STSR C, E d r dw 4 in* 1 in* * 3 RC C R C, E d r d P J r r (1) whr h firs rm is h spcral nrgy of h inpu radiaion wav; h scond rm corrsponds o radiaion mission indpndn of h inpu wav random sponanous 1

13 (SP) or cohrn (suprradian) (SR); h hird rm is simulad-suprradianc (ST-SR) corrsponding o simulad inracion (mission/absorpion) of h inpu radiaion wav in C and h spcral componn of h currn. A daild drivaion of h radiaion mod xpansion formulaion is givn in Appndix B. Rsuls and Discussion I. Radiaion of a Singl lcron wavpack A. Unmodulad uanum wavpack Firs w considr h simpl cas of h probabilisic currn of a singl lcronwavpack (. 6) in h Fourir ransform fruncy domain J r, d J r, () i W us s. (9) for h currn of h unmodulad lcron wavpack, and hn, afr Fourir ransformaion (using / / F / ), subsiu i ino (18) C M ( ) 4P / i (3) whr w dfind hr an ovrlap ingral paramr (analogous o "marix lmn" in spaial spac) E M f r r d r (4) * i (/v) 3 Subsiuing (3) in (1), w g h xprssions for h sponanous and simulad mission of a singl wavpack dw M d 8P,SP dw R C in* M i / d,st (5) 13

14 and w s righ away ha boh sponanous and simula mission xprssions ar wavpack-si-dpndn in h smi-classical rgim, and vanish in h uanum wavpack limi ωσ >>1 (E. 1). W furhr rduc hs xprssions in h cas whr h axial componn of h radiaion mod is a ravling wav of wavnumbr E E, Thn h axial ingraion in (.4) can b carrid ou i r r (6) il/ M M L sinc L / E (7) whr w dfind a normalid cofficin dscribing h ransvrs ovrlap bwn h fild of h radiaion mod and h lcron wavpack Hr E E r wavpack profil. Th paramr 1 E M f r r d r E, whr r is h ransvrs coordina of h cnr of h ( ) (8) v is h lcron/radiaion-wav synchronism (duning) paramr. In hs rms, h sponanous mission and simulad mission ar givn by dw E L M sinc L / W sinc L / d 8 dw d P,SP in* i i L/ / EL R C M sinc L /,ST (9) EL W M 9A 8P 14

15 No ha if h ransvrs wavpack funcion is narrow rlaiv o h ransvrs variaion of h fild, hn M 1. Also no ha for a long inracion lngh L, fficin inracion can ak plac only if (.6) is a slow-wav radiaion fild componn (.g. in Crnkov radiaion or Smih-Purcll radiaion), whr a synchronism condiion v v can b sablishd, so ha. Wid fruncy band mission can ph ak plac also in a shor inracion lngh L 1wihou saisfying a synchronim condiion (.g. in ransiion radiaion). In h limi 1, E. 9 rducs o h classical xprssion for sponanous mission of a singl poin paricl [14]. B. "Einsin Rlaions" and spcral corrspondnc of nrgy xchang consrvaion in lcron inracion wih radiaion Now l us concnra on h simulad mission rm. Assum ha h inracing fild componn of h incidn wav is a singl fruncy harmonic wav (.g. a lasr bam fild) in E E cos (3) In rms of h coninuous spcral formulaion (. 17) and spcral normaliaion of (.18) for, his corrsponds o (s Appndix C): C i E (31) in Thn from ingraion of. 1 ovr ω, h incrmnal simulad-mission radiaion nrgy from a singl lcron wavpack is W E L M cos L sin c L (3),ST whr ( ). This radiaiv nrgy gain/loss is in compl agrmn wih h nrgy loss/gain of a singl lcron uanum wavpack as calculad smi-classically by h soluion of Schrodingr uaion in [7]. I is also consisn wih h classical poin-paricl limi [14] whn 1. This shows ha consrvaion of nrgy 15

16 xchang bwn a cohrn radiaion fild (lasr) and an lcron wavpack, conaind and inracing nirly wihin h spaial volum of a singl radiaion mod, is mainaind wihin h minimal spcral phas-spac volum rprsning h cohrn singl radiaion mod: RAD W W,ST Anohr imporan nw rsul is a univrsal rlaion bwn simulad mission radiaiv nrgy gain a fruncy and sponanous mission spcral radian nrgy ino h sam cohrn phas spac volum (singl radiaion mod) a h sam fruncy. A maximum mission (synchronous) inracion condiion and maximum dclraion phas of h lcron wavpack rlaiv o h wav, his rlaion is GAIN,ST 8E dw W ( ),ST,max E / P d SP,max (33) This univrsal rlaion is only valid in h classical poin-paricl limi and in h uanum o classical ransiion rang of h wavpack 1. In h opposi, inhrn uanum wavpack limi, 1 (.1), boh simulad and sponanous mission xprssions vanish. No ha his smi-classical "Einsin rlaion" bwn classical sponanous mission and simulad mission is diffrn from h classical limi rlaion bwn simulad mission and uanum sponanous mission driv in [15] in a QED modl. Of cours, h smi-classical analysis of an lcron wavpack canno produc h uanum sponanous mission. This aspc is addrssd in a companion aricl basd on QED formulaion [8]. I is insruciv o obsrv ha h proporionaliy cofficin in. 33 can b rlad o Pirc's known "inracion impdanc" paramr K E P [9]. No ha in Pirc ordr o us h rlaion (33) in pracic,.g. for h cas of Smih-Purcll radiaion, on mus solv firs analyically or numrically h classical lcromagnic problm of h 16

17 ampliud of h inracing axial fild componn E rlaiv o h normaliaion powr of h nir mod - P [37]. C. Modulad uanum wavpack Scondly, w considr h cas of a modulad lcron wavpack. In h cas of a modulad wavpack, using (s.14-15,) on gs l / i l MB l b C b b M B 4P 4P i/v l i l (34) whr i/v l i l l / i l b b l b B B. Figur 4 shows h harmonics of h currn bunching ampliud facor B as funcion of fruncy (a) and drif im (b) for opically-modulad lcron wavpack, whr B l was valuad from. 16., D Figur 4: (a) Th currn bunching facor B as funcion of fruncy for opicallymodulad lcron wavpack a h opimal drif im ' D,max. (b) Th dpndnc of h l h -ordr harmonic bunching facor B l on h drif im ' D. Th opimal drif im ' D,max is markd by h vrical dashd-lin. Using. 35 in. 1, h xprssions for sponanous mission by a singl lcron modulad wavpack is: 17

18 dw dw, l d d l,spmod,spmod (35) whr dw d, l,spmod l b W Bl sin c L / (36) whr in (35) w assumd ha h ovrlaps bwn h spcral lins of h harmonics l ar ngligibl, as shown in Figur 4.a. Th incrmnal simulad nrgy mission/absorpion of a modulad lcron wavpack in h cas of a cohrn incidn radiaion fild (.3-31) is whr W W, l,stmod l,stmod, l l lb lb l b / W E L M B cos L / sinc L /,STMOD (37) whr v. A sriking diffrnc bwn hs xprssions and h corrsponding cass for h unmodulad uanum wavpack (9, 3) is ha h modulad wavpack can mi/absorb radiaion a fruncis byond h uanum cuoff condiion (1), which occur a all harmonic fruncis l b of h wavpack modulaion of significan componn ampliud B l. D. Smih-Purcll Radiaion (SPR) - An Exampl A vivid prsnaion of radiaion mission xincion and rvival ffcs of a modulad uanum lcron wavpack is prsnd hr for h cas of h Smih-Purcll radiaion xprimn as shown in Figur 1&. Th mods of h SPR graing srucur ar Flou mods 18

19 E r r whr k /, G is h graing priod and G G i mkg m (38) m 1 c k cos (39) c Th angl is h ig-ag angl of mod in a wavguid srucur. W us in (. 6) mk G whr m is h m-h ordr spac harmonic of h Flou mod, and apply all h xprssions for sponanous and simulad mission o ach of h spac harmonics wih a duning paramr (nglcing h inrfrnc bwn h spac harmonics) m mk G (4) v Th sponanous mission xprssion (. 9) for an unmodulad wavpack and Es for a modulad wavpack, can b modifid o includ inracion wih any mk v spac-harmonics m, which can b synchronous wih h lcron G dw dw W sinc ml / d,m,m m d m,sp,sp dw l dw W B sinc ml / d, m,m l l,m d l,m,spmod,spmod l b (41) W E L,m G,m M,m 8P Figur 5a displays h SPR spcrum in rms of wavlngh c / and angl in h classical limi 1, whr h wavpack appars as a poin-paricl. Emission lins appar for arbirary angls in h rang a fruncis or wavlnghs corrsponding o h synchronism condiionm 19

20 λ m m 1 G 1 mck G cos, or λm Θ β cos Θ (4) in agrmn wih h classical SPR mission rlaion. [6] Figur 5: Th wavpack limi of Smih-Purcll spcrum. a Th classical SP spcrum for poin-lik paricls. b Th ffc of shor wavlngh cu-off of a Gaussian wavpack. d Th sam for a modulad Gaussian wavpack, displaying apparanc of non-classical suprradian-sp harmonic radiaion ligh-spos a h shor wavlngh par. c Th modld bunching facors of poin-lik, un-modulad and modulad currns of lcron wavpack as a funcion of rlaiv wavlngh. / G In Fig.5b, w show h xpcd SPR sponanous mission spcrum of an unmodulad lcron wavpack in a s-up shown in Fig. 1. Th plo shows ha a low fruncis 1, or long wavlnghs c, h low-pass filring ffc of h fini si

21 wavpack xincion paramr of.41a (Fig. 5c, blu curv) kps h long wavlngh par of h classical SPR spcrum of fig.5a unaffcd, and cus off h shorr wavlngh and highr harmonic scions whn 1 1, namly, m1 m1 G (43) G 1 1 This cas is displayd in Fig. 5b&c for h paramrs G G G G.,.7, N L 9. Hr h firs ordr SPR harmonic is parly cuoff, h scond ordr harmonic is barly obsrvabl and highr harmonics ar xinc. Howvr, a mor dramaic chang in h spcrum aks plac whn h wavpack is modulad (h "modulaing lasr" in Fig. is urnd ON ). In his cas h wavpack bunching facor l b l b in. 41 prmis rsonan mission only a harmonics l and his harmonics-spcrum is cu-off only a much highr fruncis by h filring ffc of h narrow micro-bunchs l 1, or b l 1 Tb (44) Figur 5c displays h (classical) SPR sponanous mission spcrum (.43) in rms of, for h sam paramrs and wavpack si G as in h un-modulad wavpack cas (Fig.5b). I is sn ha h bunching facor xhibis rsonanc a harmonics of h bunching fruncis b in h fruncy rang ha was cu-off bfor modulaion. This shows up in Fig.6 as hr rsonan spos, rviving h 1 s, nd and 3 rd ordr SP spac harmonics a disinc mission angls. Inspcion of E.43 rvals ha rsonan mission spos will appar, wihin h spcral rang abov h fruncy cu-off of h unmodulad bam radiaion, 1 c a h fruncis and mission angls in which h narrow band

22 filring funcion ( l b) and m sinc L / ovrlap. Th cnrs of hs spos ar a posiions l, lm, whr lm is h soluion of h uaion and Θ m G l Θ (45) b m lm is givn by E. 4. Th spcral widh of h spo dpnds on which filring funcion is narrowr. Th spcral widh of h synchroniaion funcion is m Θ 1 (46) mn G and of h bunchd wavpack bunching facor 1 (47) l ln b whr N T is h numbr of micro-bunchs in h modulad wavpack. b b Figur 6 displays h mission spcrum for h wo opposi cass N G N and b N N as funcions of and. G b Th sam configuraion of Smih-Purcll xprimn can b usd for masurmn of simulad mission/absorpion and corrsponding dclraion/acclraion wih h inracion inpu lasr bam urnd ON. In his cas, h incrmnal xchangd nrgy from h lcron wavpack o h radiaion wav and vic vrsa is givn by a modifid vrsion of Es. 3&37 (for h unmodulad and modulad cass rspcivly) G,m m G m G W W E L M cos L / sinc L,ST m,st,m m (48) /

23 W W, l,m,stmod m, l,stmod l l E L M B cos L / sin c L G,m l m b b m G m, l )94( l / b Evidnly a spcral diagram similar o Fig.5-6, wih cuoff ffcs and r-mrging spos (in cas of a modulad bam), would b masurd in h incrmnal nrgis of h radiaion wav and h lcrons whn h incidn lasr bam is scannd ovr wavlnghs and incidn angl. Figur 6: Byond-cuoff Smih-Purcll radiaion (SPR) spcrum of a modulad uanum lcron wavpack. (a) SPR for N N. (b) SPR for N N. G b I is argud ha his schm and h characrisic spcral map can b usd for masuring h lcron uanum wavpack si 3 G b a h nranc o h graing. W no, howvr, ha h smi-classical calculaion of sponanous mission from a singl lcron uanum wavpack may hav limid validiy, as discussd in h companion papr [8] basd on QED formulaion, and is masurmn may b jopardid by uanum sponanous mission nois, no inclusiv in a smi-classical formulaion. On h ohr hand, h validiy of using smi-classical formulaion for simulad radiaiv inracion is wll foundd. W hrfor assr ha such a simulad inracion xprimn can b a way for masuring h uanum wavpack si wih a SP xprimn as shown in Figur 1-. Th availabl conrol ovr h inpu radiaion fild

24 innsiy E can hlp o ovrcom xpcd uanum and ohr nois facors in h xprimnal masurmn. II. Radiaion of muli-lcron wavpack bams Byond h singl lcron radiaion cass, w now considr h radiaion of a muliparicl wavpacks bam ( N 1 ). Radiaion masurmns wih singl lcron wavpack would b challnging xprimns. To g significan wavpack-dpndn masurmn, rpad xprimns mus b prformd wih carful pr-slcion filring, o assur similariy (or idniy) of h wavpacks in succssiv masurmn xprimns [38]. W now considr h cas whr w masur a onc a of lcron wavpacks ha may b corrlad a nranc o h radiaiv inracion rgion (s Fig. 7). Now, assum ha h -bam is composd of lcron-wavpacks whos currn dnsiy disribuion is givn by J r, J r, N j1 j. Consunly, h sponanous/suprradian mission and simulad-suprradian mission of h bam (. 1) ar rspcivly N dw P Cj d j1 SP/SR dw 4 N in* P R C Cj d STSR j1 (5) whr w dfin as in (18) and h currn J j, 1 C J r, E rd r (51) * 3 j j 4P r of h j lcron-wavpack is givn by i J j r, d J j r, 4

25 E. Quanum lcron wavpacks bam W go back o s.1-5 and considr h coopraiv mission of a of N lcrons. Assuming all wavpacks ar idnical (xcp for arrival im j ), h avragd spcral nrgy of h is N dw i dw j d SP/SR j1,sp N dw i dw j d STSR j1,st d d (5) whr h singl lcron sponanous and simulad spcral nrgis ar givn in (9). I is vidn ha whn h lcron bam is longr han h radiaion opical priod T, and oj ar random, all h phasor rms in (5B) and all h mixd rms in (5A) inrfr dsrucivly. On gs hn no avrag simulad-mission (and no avrag acclraion) of h random lcron bam, bu hr is a rsulan "classical sponanous mission" (Sho-nois radiaion) of h bam, originaing from h diagonal rms in h produc in E. 5A. Using (9A) dw dw N NW sinc L / d d SP,SP (53) Namly, h sponanous mission from N paricl is N ims h spcral mission from a singl lcron. In considraion of h high fruncy cu-off of h singl lcron sponanous mission E. 1, on concluds ha sponanous sho-nois radiaion of an lcron bam is no whi nois, bu diminishs in h uanum limi 1 [3]. In h opposi limi of a shor lcron bam rlaiv o h opical priod T /, all phasor rms in (5) wihin h, sum-up wih h sam phas and sinc ncssarily, h xponnial dcay facors in (9) ar uniy, and h lcrons radia as poin paricls wihou any dpndnc on. This is acually h 5

26 classical cas of suprradianc analyd in [14], whr h collciv mission of h lcron dpnds only on h paricls arrival im disribuion in h cofficins of. 5, b 1 N N i j (53A) j1 On can rplac h summaion ovr j by ingraion ovr h mporal disribuion of h paricls in h bam f, Appnd. D). For a Gaussian mporal disribuion, whr f ( j)d j 1 [3,14] (s i j / i, j, j b f ( ) d (54) and b (55) Using hs xprssions and E. 9 wih in (5) dw dw N N Wsinc L / d d SR STSR,SP,ST dw dw in* iil/ N NEL RC M sincl / d d / (56) Which is h sam as h singl lcron xprssions (9) wih rspciv N and N facors, and rplacd by. Thus, w obaind in his limi h supr-radianc and simulad-supr-radianc xprssions of classical poin-paricls 1 [14]. Th classical poin paricl limi of simulad suprradianc (. 56B) is nohing bu h acclraion formula in convnional paricl acclraors. Th classical suprradianc formula (. 56A) is of inrs primarily for TH radiaion gnraion dvics, sinc 1 aainabl shor lcron bam duraions ( ) ar in h ordr of 1 s 6 [39]. I is of lss inrs in h prsn conx, sinc i has no uanum wavpack dpndnc.

27 F. Modulad-wavpacks lcron bam: Suprradianc Considr a cas whr all lcron wavpacks in an lcron bam ar modulad in phas corrlaion by h sam cohrn lasr bam. In his cas, h modulad currns of h wavpacks in h (. 14) ar also corrlad, and hir radiaion missions in h inracion rgion ar phas corrlad as wll (s Fig. 7). Assum ha h xpcaion valu of h lcron wavpacks probabiliy dnsiy of an nsmbl of N lcrons is modulad cohrnly a fruncy b by a lasr bam, as shown in Fig. ; h singl lcron wavpack dnsiy funcion in (14) is for lcron =j: whr fmod r J, f f v f v (57) j n j mod is h priodic modulaion funcion (15), composd of many harmonics of b. Imporan o no ha h modulaion phas b (drmind by h modulaing lasr phas). Hr is common o all h wavpacks fn is h wavpack dnsiy nvlop givn by h Gaussian (.8). Th incrmnal spcral ampliud C du o inracion wih h nir -bam is hn found by sing =j, and summing up C j (E. 34) ovr all paricls N N C C M ( ) B b / i b i b j E j l l l l j1 4P l j1 (58) W now dfin h bam bunching facor of l h -ordr harmonic fruncy ovr h nir b () l N 1 l N j1 b i j (59) 7

28 and for N 1 rplac h summaion wih ingraion, ovr h mporal disribuion funcion. For a Gaussian disribuion Appnd. D): 1 f j 1/ j (s b () l l b / (6) () l lb (61) b Subsiuion of s. 7,34 in (5) rsuls in dw dw, l d d l SRMOD SRMOD (6) dw d l, l SRMOD b N W Bl sinc ( L / ) (63) for all harmonics l. Th rm l= ha corrsponds o sponanous mission from a random unmodulad bam, bu wih a facor B, and jus as in. 53, i dcays as and cus-off for 1. All ohr harmonics can radia byond his cuoff (as in Fig 5 in h Smih-Purcll xampl) wih a narrow bandwidh filring-facor l b similar o h singl lcron modulad wavpack cas (E. 36), bu wih h lcronbam duraion rplacing h uanum wavpack si paramr. No ha vn hough h modulad wavpack bam radias suprradianly byond h uanum cu-off condiion 1, i sill has a high harmonic cu-off (E. 44) ha dpnds on h ighnss of h bunching. This inrsing nw rsul suggss ha all lcrons in h mi in phas suprradianly vn if h lcrons nr h inracion rgion sparsly and randomly, s fig.7. Bcaus all wavpacks ar modulad (by h sam lasr) a h sam 8

29 fruncy and phas, hy coopraivly radia consrucivly a all harmonics l b wih cohrnc im ual o h lcron bam duraion and corrsponding spcral linwidh 1/. In a modulad wavpack Smih-Purcll xprimn as shown in Fig. w xpc o g a spcrum similar o h on shown in Fig. 6 bu h radiad nrgy would b nhancd by a facor of N, and h spcral linwidh would b narrowr by a facor. Anohr inrsing obsrvaion is ha h dpndnc on h wavpack si disappard aloghr in (61). In fac, h mission is h sam as h suprradian mission of a bunchd poin-paricl bam [14]. Hr is anohr xprssion of h wavparicl dualiy naur in sponanous mission no disincion bwn h spcral suprradian mission of a bunchd poin-paricls bam and a of phas corrlad modulad wavpacks, vn if h lcron bam is nuous and h wav packs ar spars and random (w do no rul ou h possibiliy ha h phoon saisics may b diffrn). I is also inrsing o poin ou ha h configuraion of bunching h lcron wavpacks by a lasr and masuring hir suprradian mission a h lasr modulaion fruncy shown in Fig. is rminiscn of h Schwar-Hora xprimn [31] and is inrpraion by Marcus [3] as cohrn coopraiv opical ransiion radiaion. According o Marcus h signal o nois calculaion, basd on his modl and h rpord xprimnal paramrs, is blow h masurabl lvl in h paramrs of h xprimn [3], and unforunaly hr is no indpndn xprimnal confirmaion of his ffc. W suggs ha h Smih-Purcll radiaion schm of Fig. will b a mor fficin radiaion schm for obsrving suprradian mission from a lasr modulad lcron wavpacks bam. 9

30 Figur 7: A schmaic diagram of suprradian cohrn mission from a of phascorrlad dnsiy-modulad lcron uanum wavpacks nring h inracion rgion a random. Modulad-wavpack lcron bam: Simulad Suprradianc In simulad inracion (acclraion/dclraion) of a of modulad wavpacks w sum up h incrmnal nrgy conribuions W, l,j (. 37) of all modulad lcron wavpacks j,stmod N lb /, l l l b j l b STMOD j1 W E L M B sinc L / cos L / (63A) Rplacing summaion wih ingraion ovr h Gaussian saisical disribuion of h lcrons in h (s Appnd. D), on obains lb /, b STMOD W l NE L M Blsinc L / cos L / l (64) Rsonan simulad suprradian mission/absorpion (dclraion/acclraion) aks plac a synchronim if h inracion lasr (s Fig. ) is und o on of h bam bunching harmonic fruncis a propr dclraion/acclraion l b 3

31 phas rlaiv o h bunching -,. Th lasr fruncy ( ) rsonan duning rang is drmind by h duraion of h lcron bam - 1. Conclusions and Oulook Dos h hisory-dpndn dimnsion of a fr-lcron uanum wavpack hav physical ffc in is inracion wih ligh? Can i b masurd? Answring his usion has bn h main hrus of his aricl. Basd on h prsn analysis of h smi-classical (Maxwll uaions) modl for h radiaion mission and h corrsponding arlir analysis of h uanum-mchanical modl for h lcron wavpack dynamics in h sam s-up [7], our answr is affirmaiv. In his aricl, w sudid h sponanous and simulad radiaion procss of a singl fr lcron wavpack, as wll as h suprradianc procsss in an nsmbl (bam) of lcron wavpacks. This analysis was carrid ou in h framwork of a smiclassical modl, in which h fr lcron charg dnsiy is rprsnd by h xpcaion valu of h probabiliy dnsiy of wavfuncion, and h radiaion fild is akn o b h classical fild soluion of Maxwll uaions (solvd in h framwork of a mod xpansion modl). This work is complmnary and fully consisn wih our arlir analysis of simulad inracion (acclraion/dclraion) of a singl lcron uanum-wavpack, basd on soluion of Schrodingr uaion for h lcron inracing wih a cohrn classical (lasr) fild [7]. Basd on h complmnariy and consisncy of h wo indpndn formulaions, w mad h following obsrvaions, as lisd in Tabl 1: A. Th singl lcron sponanous mission and simulad mission/absorpion procsss saisfy a wavpack si-dpndn cu-off fruncy condiion 1/ (. 1) (row in Tabl 1). Howvr, if h wavpack is dnsiymodulad, hs radiaiv procsss can sill ak plac byond h cuoff condiion around harmonics of h wavpack modulaion fruncy (row in Tabl 1). 31

32 B. Th consisncy of h smi-classical formulas, drivd indpndnly for wavpack-dpndn radiaiv mission/absorpion and corrsponding lcron dclraion/acclraion, rvals a "phas-spac corrspondnc" consrvaion of nrgy rlaion bwn h fr lcron and h radiaion fild in h corrsponding phas-spac volum of h radiaion mod ha ovrlaps h lcron wavpack spac and rajcory. C. W hav rval a gnralid "Einsin rlaion" bwn h spcral sponanous radiaion mid by a fr lcron wavpack ino a radiaion mod and is simulad inracion nrgy xchang wih h fild of inpu radiaion launchd ino h sam mod. This rlaion can b a usful mhod for prdicing h gain of a variy of fr lcron radiaion schms and dvics [15], or h acclraion ra of various lasr-acclraion schms (lik DLA [1]), basd on masurmn of sponanous mission in h sam sup configuraions. D. Our analysis of h radiaiv inracion of a fr lcron, rprsnd by a uanum wavpack, rvals h ransiion from h uanum inracion rgim (.g. as in PINEM) o h classical poin-paricl rgim (.g. as in DLA). This ransiion givs physical maning o h uanum wavpack funcion, and suggss how is si and characrisics can b masurd. Such wavpack-dpndn masurmn can b basd on obsrving h characrisic long wavlngh cuoff ffc for (L D) in a simulad radiaiv inracion Smih-Purcll xprimn in Fig. 1, and vn mor disincly, by masuring h harmonic spcrum signaur (Figs. 5,6) of a modulad wavpack xprimn (Fig. ). I is mphasid ha hs suggsd xprimnal schms masur h hisory-dpndn si of h wavpack D, and no is fundamnal (cohrnc lngh) si p. Ohr schms for masuring h uanum wavpack characrisics ha hav bn considrd arlir, such as Compon-scaring by an lcron wavpack [4-6] or wavpack slf-inrfrnc [33,34], canno provid such informaion. 3

33 I is nod ha smi-classical analysis of sponanous mission is no consisn wih convnional QED hory. Th smiclassical analysis producs rsuls consisn wih h classical poin-paricl limi hory of sho-nois sponanous mission and suprradian mission from an lcron bam [14], bu i canno produc h uanum sponanous mission xprssions ha ar drivd in h uanum limi of an lcron plan-wav funcion [15,4-6]. Howvr, h smiclassical xprssions of simulad inracion of fr uanum lcron wavfuncion ar fully consisn wih QED [8]. Masuring simulad inracion of singl lcrons is fasibl wih rcn significan advanc in conrolld gnraion, manipulaion and modulaion in ral spac and im of singl lcron uanum wavpacks [35-36, ]. Sinc nihr lcron nrgy spcrum, nor radiaion mission spcrum of a singl lcron ar possibl, an xprimn of masuring h wavpack dimnsions ruirs mulipl singl lcron xprimns undr h sam condiions, including prslcion of h lcron wavpacks in spac and im domains bfor nring h inracion rgion, in condiions similar o wak masurmns [38]. Finally, w hav also analyd h cas of sponanous and simulad suprradianc from an nsmbl (muli-paricl bam) of modulad lcron wavpacks, which ar phas-corrlad whn modulad by h sam lasr (rows 3, 4 in Tabl 1). Qui inrsingly, a bam of phas-corrlad modulad lcron wavfuncions radias suprradianly lik a classically bunchd poin-paricls bam, vn if h modulad wavpacks ar injcd a random and sparsly rlaiv o h opical priod. Unforunaly, in his cas, h rsulan radiaion spcrum dos no rval anymor h individual uanum propris of h lcron wavpacks. 33

34 Gallry: wavpackradiaions Singl unmodulad wavpack Singl modulad wavpack Mulipl unmodulad lcron bam Mulipl modulad lcron bam Sponanous mission (.9) B l l b (.36) N (.53) N B (.63) Suprradianc Null Null N (.56) lb N Bl, l (.63) Simulad mission/ Simulad suprradianc / (.9) B l l b / (.37) / N (.56) N B l l b / (.64) Tabl 1: Summary of h radiaiv inracion fruncy scaling of fr uanum lcron wavpacks. Acknowlgmns W acknowldg A. Fridman and P. Kling for usful discussions and commns. Th work was suppord in pars by DIP (Grman-Israli Projc Coopraion) and by h PBC program of h Isral council of highr ducaion. Corrspondanc and russ for marials should b addrssd o A. G.(govr@ng.au.ac.il). 34

35 Appndix A Elcron uanum wavpack modulaion by nar-fild muliphoon mission/absorpion W analy h muliphoon mission/absorpion procss ha aks plac whn a singl lcron uanum wavpack ravrss hrough h nar fild of a nanomric srucur lik a ip ha is illuminad by an IR lasr, as shown in Fig.. For simpliciy, w assum hr ha for shor nough inracion disancs (or a ip, s fig.) h diffracion and disprsion procsss ar small nough o assum ha h ransvrs dimnsion and h longiudinal dimnsion c say consan hroughou h nar-fild rgion, and saisfy h uncrainy condiion. Following Fis al.[], w modl h lcron wavpack nrgy modulaion by solving h rlaivisically modifid Schrodingr uaion[15,7] wih h opical nar-fild prurbd Hamilonian p F() p A= * m m mb H H psinb (65) whr m m is h ffciv mass, b is h opical fruncy of h modulaing * 3 lasr bam o modula h wavfuncion, and F () is h slow-varying spaial disribuion of h nar-fild. Assuming ha F () may b considrd consan for all rlvan momnum componns of h wavpack, h soluion of h Schrodingr uaion i H is xprssd by Flou xpansion, cn ( ) n i( pep )/ in b (66) Using h Raman-Nah approximaion * p m Ep n b Schrodingr uaion as h sandard Bssl funcion rcurrnc rlaion, w can wri h (67) cn cn 1cn1 35

36 which has h gnral soluion c n F() p Jn() and m b, whr J n is Bssl funcion of ordr n. Thus, h wavfuncion in h inracion rgim is givn by n F p Jn m b () i( pep)/ inb (68) Thn h longiudinal wavpack afr inracion is givn by, J xp p 1/4 dp F() p p p i( pep)/ inb n 4 n m b p (69) Now l us mak a simpl approximaion for h nrgy gain in h inracion rgim of lngh L I, and dfin a paramr g L I m b L /v I F( ) p F( ) d b whr g is h avragd phoon numbr gain from h nar-fild. Thus, h wavfuncion afr passing hrough h inracion rgim can b xprssd as, p xp Jn g 1/4 dp p p i( pep )/ in b 4 n p (7) To simplify h xprssion, w xpand h nrgy disprsion o scond ordr: E c m c p E v p p p p * p p m (71) * 3 whr Ep mc, p mv, m m ar h ffciv nrgy, momnum, and mass rspcivly wih h Lorn facor 1 1 v c. For shor inracion lngh (h nar fild of a ip) h scond ordr uadraic rm in h nrgy-momnum disprsion xpansion (71) is nglcd. Afr subsiuion in (7) wih rplacing 36

37 p p n b v for ach rm in h summaion of h modulad momnum disribuion (h ingrand of E. 7) is: 1/4 p p n p p p Jn g xp n 4 p (7) whr p v. For fr-spac drif afr h modulaion, w hav o xpand h nrgy-momnum disprsion rlaion xpansion (71) o scond ordr, and hn g:, dp p i( p ' ' )/ p ip p v * m dp ' p' i( p p )/ i( pep )/ 1/4 p, Jn g Fn n (73) wi p' p p, and h ingral F, n is: ' ' ip p p ' n p v * m 4 p Fn, xp dp ' 1 n p v * 1 i v xp m n p n p xp * i * 4 p m i m * p m whr w us h ingraion formula x a b xp ibx xp, dx a a (74) a 4 Finally, afr prforming h momnum ingral, on obains 37

38 n p v, J g xp 1 i ( p p )/ * np np i v * 1/4 m m n n 4 1 i i i ( p )/ p * nb * v n i cb vn cb v J 1/4 xp n g 4 1 i i n 1 (75) Th nvlops of h diffrn harmonics dvlop in spac similarly o h unmodulad fr drifing wavpack (E. 5 in h x). Hr p is h iniial wavpack widh, and * m is h drif chirping facor, * c * mc is an ffciv Compon wavlngh, and v /c. Dnsiy micro-bunching in fr spac drif In uanum mchanics, h currn dnsiy opraor is givn by J * * * mi (76) Considring h longiudinal currn solv h rms J along h -dircion (E. 6 in h main x), w i( p ' ' )/ p ip p v ' * m p' ip ip,, ' dp i( p )/ ' ' p ip p v * * * ip ip ' m,, dp ' p' (77) Hr w assum p p ', and hn w only considr h conribuion of h firs rms. Thus, w obain 38

39 , v, J (78) wih v. Thn h dnsiy probabiliy is givn by: np mp * * 1, xp m m Jn g Jm g ( ) nm, ( n m) p ( n m) p xpi * m i i (79) whr h spaial wavpack si of h sprading wavpack nvlop is 1. No ha in h limi of no modulaion ( g in E. 79) rducs o h limi of E. 5 in h x: a Gaussian wavpack, xhibiing xpansion and phas chirp as i drifs wih im. Is dnsiy probabiliy afr drif is: v 1, xp ( ) ( ) (8) Also no ha h harmonic dnsiy modulaion ha is implid by E. 15 is a dirc consunc of h nonlinariy of h nrgy disprsion rlaion (hird rm in E. 71). * In is absnc ( m 1) h nrgy modulaion dos no convr ino dnsiy bunching: i( p p )/ 1/4 4 i( p p )/ 1/4 4 v n p, xp xp v Jn g i n v xp xpi g sin v (17) 39

40 and h dnsiy disribuion is hn indpndn of drif, similarly o E. 8, wihou dnsiy modulaion or xpansion, vn hough h wavpack rmains nrgy (phas) modulad. Finally, w xplain hr ha h choic mad in h main x analysis ha h wavpack is mid a is longiudinal wais a = (E. 5 in h main x), or uivalnly wih symmrical momnum disribuion wihou chirp (E. 3 in h main x), dos no limi h gnraliy of our analysis. If h iniial wavpack bfor modulaion is mid from h lcron sourc chirpd, or acuirs chirp du o ranspor o h modulaion poin, or by a conrolld procss by sraking chniu [35-36], hn h chirp acuird du o h disprsiv ranspor afr modulaion will combin wih his prior chirping. Th wo ffcs may add oghr o nhanc h wavpack widning, or wih ngaiv chirping lad o comprssion of h wavpack. Insad of an unchirpd momnum disribuion wavpack (E. 3 in h x) on would sars in his cas wih a complx Gaussian wavpack: p 1/4 1 p xp ic p p 4 p (81) whr C is dfind as a prior chirp facor and 1 1 () i4c. Thn h Fourir p p ransformaion o ral spac (E. in h main x) rsuls in a modifid complx Gaussian wavpack (E. 5 in h x) wih complx wavpack si a im =: () p() (8) and hn si sprading in im is givn by 4

41 i ( ) () m 1 i C m i * * C i 4 4 * C C m (83) If h prior chirp facor C is slcd such ha C 4 c 4 * C m 1 (84) hn on obains ( ) () 1 c i 4 * C m (85) Wih his subsiuion, w could considr h prior chirp ffc for dnsiy bunching, as w did in h.1 in h conx. Appndix B Spcral nrgy of radiaion mod gnral formulaion In his scion, w prsn h formalism mployd hroughou his aricl for analying h xciaion of lcromagnic filds by currn sourcs disribud along a wavguid, (channl, or wigglr). Th cross-corrlaion funcion of h im dpndn lcric Er, componn Hr, and magnic componn is givn by R, d E r, H r, ˆ dxdy EM (86) According o h Winr-Khinchin horm [J. Goodman, saisical opics, p.73-79, Wily, ], h spcral dnsiy funcion of h lcromagnic signal nrgy SEM, is h Fourir ransform of h cross-corrlaion funcion, which is obaind as 41

42 ,, i S d R EM EM (87) Subsiuing h xprssion (86) ino h spcral dnsiy funcion (87) rsuls in S, d d r, r, ˆ EM E H dxdy i der, Hr, dˆ dxdy i Er, Hr, d ˆ dxdy, H, E r r ˆ dxdy *, H, E r r ˆ dxdy i (88) * In h las sp, w us h rlaion Hr, H r, bcaus Hr, is ral. Finally, h oal nrgy carrid by h lcromagnic fild is calculad by ingraing h spcral dnsiy S, EM ovr h nir fruncy domain, rsuling in 1 1 * W ( ) S, d r, r, ˆ dxdy d EM E H 1 * * r, r, r, r, ˆ dxdy d E H E H 1 * r, r, ˆ dxdy d E H (89) Dfining w idnify dw ( ) d as h spcral nrgy disribuion of h lcromagnic fild ( ), dw ( ) 1 * r, r, ˆ dxdy d E H Hr w dfind nw xprssions o spara h ngaiv and posiiv fruncis (9) 4

43 E r,, i Er, Er, d * E r,, H H i r d H, H r, * r,, r,, (91) and h invrs Fourir ransforms of h filds ar 1 i Er,, E r d 1 i Hr,, H r d And according o h Parsval horm, h oal nrgy is xprssd as 1 W ( ) S, d P, d R, EM EM (9) whr P, E r, H r, ˆ dxdy is h insananous powr. Th radiaion fild ha is xcid by a gnral currn Jr (, ) is xpandd in h fruncy domain in rms of a s of orhogonal dircional ransvrs mods E,H r r ha ar h ransvrsly confind homognous soluion of h lcromagnic wav uaions of fr spac or a sourc-lss guiding srucur Er,,Hr, C E r,h r For calculaing axial flow of radiaiv nrgy, only ransvrs componns of h filds nd o b akn ino accoun. Using h modal xpansion formalism, w rprsn h filds in rms of a compl s of forward and backward propagaing ransvrs mods propagaing in h -dircion): 43

44 r C r ik C r ik E, (, )E (, )xp (, )E (, )xp r C r ik C r ik H, (, )H (, )xp (, )H (, )xp (93) whr C, k,e,h ar h slow-varying ampliud, h wav numbr, and h lcric and magnic fild ransvrs profil funcions of h lcromagnic mod, rspcivly. Th spcral radiaiv nrgy mission pr mod from dw () dw is givn d d by (.13 in h conx) whr P E r H r C dw C, C, P (94) d 1 ˆ dxdy h normaliaion powr of mod, and, is h slowly growing fild ampliud of h radiaion mod a spcral fruncy along is propagaion dircion (). Mod propagas in h invrd -dircion. If h lcromagnic wav is known o propaga only in h +-dircion C,, hus Appndix C dw d C, P (95) Th formulaion in his papr is basd on spcral (Fourir ransform) analysis of fini im signals. A shor drivaion is ncssary in ordr o mach h spcral formulaion o h singl fruncy formulaion ha is gnrally usd in conncion o simulad inracion wih a cohrn lasr bam such as E E cos (96) in 44

45 This should b machd o h spcral prsnaion of h axial lcric fild in im domain i in E E d C (97) in in In ordr o hav a fini im radiaion wav w runca h fild (.96) ino a im window T win / T win / long nough, such ha h inracion of h lcron wavpack inracion aks plac nirly wihin his im duraion T win. Thus, T E E cos d T sinc sinc T win / i E win win in win (98) T win / Sinc w considr only posiiv fruncis, w g C ET in win win sinc. In h limi win E.31 in h main x Appndix D T T C E - in, his can b wrin as For a larg numbr of paricls N 1, boh long and shor lcron bam s can b prsnd in a gnral way. Dfin h paricls bam bunching facor T b 1 N N i j (99) j1 On can rplac h summaion ovr j by ingraion ovr h mporal disribuion of h paricls in h bam f, Gaussian mporal disribuion, whr f ( j)d j 1 [3,14]. For a 1 f j 1/ j : i j / i, b f ( j, ) d j (1) 45

46 b N N N i ' j i i k j N j1 N k1 N N jk ' ik 1 N(N 1) f ( ) d f ( ) d N N N N i j ' i j ' j, j j, j (11) Th drivaion also works for h bam bunching facor of l-h ordr harmonic () b l dfind in.59 in h mainx. Wih his formulaion w can wri oghr xplicily h sponanous/suprradian spcral nrgy of h as dw 1 1 W N sinc ( L / ) 1 d N N SP/SR W sinc ( L / ) N N N 1 (1) Sinc usually, E. 1 rproducs h classical paricl bam xprssions [14] for suprradian cohrn radiaion N in h fruncy rang 1 (s. 56A), and incohrn sponanous sho-nois radiaion N - band-limid by h uanum wavpack condiion 1 (. 53). In h calculaion of h simulad-suprradianc of a of corrlad modulad wavpacks w rplac h avraging ovr paricls in E. 63A by avraging ovr h dnsiy disribuion funcion N b / i( L/ i b ) b j W E L M B sinc L / l R l STMOD l j1, l l lb E LN M B sinc L / cos( L / l ) l b )31( whr in h avraging w us. 1 wih subsiud wih l b. / 46

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