Quantum SASE FEL with laser wiggler. Rodolfo Bonifacio* INFN Sezione di Milano and Laboratorio Nazionale di Frascati (LNF), Italy.

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1 Quntum SASE FEL ith lse iggle Rodolfo Bonifcio* INFN Sezione di Milno nd Lbotoio Nzionle di Fscti (LNF), Itly nd Deptmento de Fisic, Univesidde Fedel Algos, Mceiò, , Bzil Abstct In this lette e specify the physicl pmetes necessy to opete SASE FEL in the quntum egime ith lse iggle We lso sho tht this is moe fesible in the quntum egime thn in the clssicl one Specific exmples e given PACS: 416C; 45Fx Key ods: quntum SASE; FEL; expeimentl pmetes *Coesponding utho Fx: E-mil ddess: odolfobonifcio@tinit Deptmento de Físic, Univesidde Fedel de Algos Cmpus A C Simões, BR 14 km 14, Tbuleio dos Mtins, Mceió-AL, Bzil 1

2 It hs been peviously ecognized tht quntum effects in SASE FEL e detemined by the quntum FEL pmete ρ [1] defined s the usul FEL pmete times the tio beteen the electon enegy nd the photon enegy Quntum effects become elevnt hen ρ < 1 Hoeve the clcultions of Ref [1] e confined to the line egime In Ref [] e hve extended the theoy of Quntum SASE FEL to the non line egime nd e hve shon the phenomenon of quntum puifiction of SASE spectum, ie, the bod supeposition of chotic seies of ndom spikes pedicted by the clssicl theoy shinks to vey no spectum of the emitted dition hen ρ <1 The question is: ht is the expeimentl set up nd the expeimentl pmetes necessy to obseve Quntum SASE in the shot velength egion (hee quntum effects e expected nd moe elevnt)? The fist possibility is the usul configution of GeV cceletos nd vey long mgnetic undultos s in the SLAC LCLS nd DESY TESLA-FEL pojects Hoeve the quntum egime ould equie even longe mgnetic undulto, becuse to ech the quntum egime one needs vey lo vlue of the FEL pmete, hich detemines the gin pe iggle peiod Altentively one cn popose the use of typicl Compton bck sctteed configution: lo enegy electon bem counte-popgting ith espect to n electomgnetic iggle (ve) povided by high poe lse In this note e popose the Compton configution giving the scling ls nd the expessions of ll elevnt physicl quntities s function of ρ, both in the quntum nd the clssicl cse In pticul, e sho tht this expeimentl set up ppes esonble in the quntum SASE egime ρ << 1, hees it is much moe poblemtic in the clssicl egime, due to fct tht the scling ls s function of ρ e much diffeent in the to cses

3 Let us conside lse iggle ith dition popgting in the z diection opposite to n electon bem ith the folloing specifictions: W is the minimum dimete of the lse bem in the focus, σ is the minimum dius of the electon bem, Z is the distnce in hich the dition bem diveges (Ryleigh nge), β* is the nlogous length fo the electon bem, γ is the Loentz fcto, ε n is the nomlized bem emittnce nd is the FEL dition velength length The folloing eltions e ell knon []: (1) Z πw () σ * ε nβ γ In ode to ensue good ovelp nd mtching beteen dition nd electon bem e must impose tht () ε W 1 4σ 1 (4) * β ε Z 1 Condition () is the consevtive [4], but ensues tht the lse intensity is lmost constnt ove the electon bem tnsvese pofile If the iggle pmete is smll condition () cn be elxed The FEL esonnce condition fo n electomgnetic iggle eds: (5) 1 γ + (1 ) We emk tht fo mgnetic iggle the fcto 1/ should be eplced by 1/ Imposing the consistency of (1) - (4) nd using Eq (5) e obtin: (6) (1 + ) ( µ m) ( A) (1 + ) η ( πη )

4 hee (7) η ε ε ε 1 n( mm md ) We emk tht Eq (6) (fomlly independent on the electon enegy) gives diect eltion beteen the dition velength nd the iggle velength in tems of to geometicl pmetes nd ε n (vi the η fcto) nd the iggle pmete The pevious eltions cn be deived using the folloing chin of equtions: Z 1/ * πw 16πε1σ 16πε1εnB πε1εε n Z / γ 1+ Eliminting Z fom the fist nd the lst eqution, e obtin Eqs (6) nd (7) As n exmple if e tke, ε 1 ε ε n 1, 1A, nd <<1, e obtin 1 µm Using Eq (5), Eq (6) cn be itten s (8) 16πη η γ 5 µ ( m) Eqution (8) fix the esonnt enegy only in tems of the pmete η nd of the lse iggle velength Fo exmple fo η 1 nd 1 µ one hs γ 5 Let us emembe the definition of the quntum FEL pmete [1,]: (9) mcγ ρ ρ F F k γρ C hee c h 4 A is the Compton velength Inveting Eq (9) nd using Eq (6) one hs mc (1) ρ F 51 4 η ρ µ ( m) 1+ Using Eq (1), e cn ite the poe gin length s: µ [ m] [ µ m] + +ρ + / 8πρ ρ ηρ (11) Lg[ µ m] ( 1 ρ ) 8 ( 1 )( 1 ) F 4

5 Eqution (11) is not n exct eqution, but is n intepoltion fomul hich gives the coect behviou in the quntum egime ρ <<1 [] nd the clssicl expession [5] in the opposite limit This eqution cn be igoously justified in the symptotic cses ρ vey lge o vey smll We must lso impose tht the electon bem chcteistic length β* is lge thn the gin length, ie, (14) * β L g ε 1 Hence, using Eqs (1) - (4) nd (15), one hs: εεl [ ] + + γ γ η ρ * 4 εβ µ m / n n g (15) σ [ µ m] 166εnε ( 1 ρ )( 1 ) Futhemoe, using Eq () nd (15), it cn be esily shon tht: (16) / K ηε P[ TW] ρ B( T) ( cm) K P[ TW] W µ [ m] ε (1 + ) ( 1+ ρ ) 1/ hee P is the lse poe, B is the ms vlue of the lse mgnetic field nd K 5, if the e- bem hs gussin tnsvese pofile [4], o K 7 fo flt tnsvese pofile ith the sme totl poe nd bem ist Eqution (16) is self consistent eqution fo, ie, 1 + hee: (17) / 41 ε 1+ ρ K P ε η ρ is the iggle pmete hen << 1 Solving the pevious eqution e obtin esily 5

6 (18) F( ) hee (19) F ( ) 1+ > 1 Note tht in the limit 4 << 1,, hees in the opposite limit Futhemoe, the eltive enegy sped is subjected to the limittion () γ 4ρF ρ 1 η γ µ / ( 1+ 16ρ ) F ( ) [ m] ( 1+ 16ρ ) ρ hee Eq (1) nd (19) hs been used Eqution () is NOT n exct expession but it gives n intepoltion fomul hich gives the coect expession in the quntum limit ρ <<1 nd the clssicl expession [5] in the opposite limit This eqution cn be igoously justified in the symptotic cses ρ vey lge o vey smll Substituting γρ F U [5] in Eq (9), e get 1/ / 4/ 16 J B ( S ) ρ 1 J( A/ µ m ) ( µ m) ( A) Using Eq (15) nd the fct tht JI/πσ, the pevious eqution becomes ρ ε ρ F( ) Hence, e obtin: 4 5 K PI 1 εεη n ( + ρ ) 71 1 εεη n (1) I 5 ( 1+ ρ ) K P εf( ) We emk tht in the quntum limit ρ <<1 the cuent is independent on ρ, hees in the opposite limit it inceses s ρ The minimum lse time dution equied is given by: 6

7 Lg () τ( psec) 1 ε Lg[ µ m] ε ( 1 ) F ( ) +ρ / c ηρ hee () Lg ( 1+ ρ ) 8 F( / ) ηρ Futhemoe, e hve (4) F ( ), η η (5) EMeV ( ) 5, (6) (7) γ γ 1 ( 1+ 16ρ ) η ρ F( ) / ( 1+ ρ ) 4 * εβ εε n L n g σ [ µ m] 166 εnε F( ) / γ γ η ρ The units e: in µm, P(TW), in Angstom The othe chcteistic lengths β*, Z, nd W cn be obtined by Eqs (1-4) nd (7) To explicit exmples fo lse iggle t 1 µm nd 1 µm e given in Tble 1, hee ll the othe physicl pmetes e expessed s function of the lse poe, P The vlues in penthesis e fo P1TW In conclusion, Eqs (17)-(7) nd eltions (1)-(4) give the elevnt quntities fo the design of SASE FEL in tems of the quntum FEL pmete ρ, the pump lse velength, the unpetubed iggle pmete, nd of dimensionl pmetes η (Eq (7)) The quntum SASE egime is obtined hen ρ < 1 hees in the opposite limit the clssicl egime ith lse iggle is ecoveed We emk gin tht Eq (1) shos tht in the clssicl limit, hee ρ >>1, the equied cuent inceses s ρ nd is much lge thn in the quntum egime, hee it is independent of ρ This fct mkes moe poblemtic the use of n electomgnetic iggle in the 7

8 clssicl egime thn in the quntum egime If the quntum egime ith the lse iggle is expeimentlly fesible, shot velength coheent FEL cn be tble top object In such cse the technologicl poblem ould go fom high enegy cceletos plus long mgnetic iggles to the idely used high poe lse technology This ok hs been completely suppoted by INFN, Sezione di Milno nd Fscti The utho is gteful to D Mssimo Feio, D Nicol Piovell nd D Luc Sefini fo helpful discussion nd suggestions nd D Luci de Slvo fo continuous ssistnce The utho ould like lso to thnk Pof Segio Betolucci nd Pof Luigi Plumbo fo thei suppot nd inteest [1] R Bonifcio, F Csgnde, Opt Comm 5, 51 (1984), NIM Phys Res A 7, 168 (1985); R Bonifcio, ibid, 4, 165 (1997) nd Opt Comm 15, 16 (1998) [] R Bonifcio, N Piovell, G Robb, (Quntum SASE FEL) to ppe in NIMA [] E T Schlemnn, pg115 nd U Amldi, pg 17 in High Gin, High Poe FEL: Physics nd Appliction to TeV Pticle Accletion, Venn, Itly (1988) [4] L Sefini, Pivte communiction [5] R Bonifcio, CPellegini, LNducci, Opt Comm, 5, 7 (1984) 8

9 (µm) 1 1 ε ε 1 1 ( Angstom) FP ( ) (1) 1 FP ( ) (15) 8 P 9 P F ( P ) ( 1 1 1P ) (1) + + ( P ) (15) PFP ( ) () 9 PFP ( ) (7) I( Amp ) (1) (18) EMeV ( ) 5 5 γ γ FP ( ) (4 1 ) FP ( ) (4 1 ) τ ( psec) 4 56 d( µ m) σ 94 FP ( ) (94) 7 FP ( ) (11) W ( m) µ 18 FP ( ) (18) 5 FP ( ) (7) Z ( µ m) β* ε 1 F ( P) (1 ) 71 F ( P) (161) Tble 1 Exmples of the vious pmetes fo ε 1 e ε n 1, K5, ρ Hee the unique fee pmete is the lse pump poe, P (TW) The numbe in penthesis e fo P 1TW 9

School of Electrical and Computer Engineering, Cornell University. ECE 303: Electromagnetic Fields and Waves. Fall 2007

School of Electrical and Computer Engineering, Cornell University. ECE 303: Electromagnetic Fields and Waves. Fall 2007 School of Electicl nd Compute Engineeing, Conell Univesity ECE 303: Electomgnetic Fields nd Wves Fll 007 Homewok 3 Due on Sep. 14, 007 by 5:00 PM Reding Assignments: i) Review the lectue notes. ii) Relevnt