Motion in an Undulator

Transcription

1 WIR SCHAFFEN WISSEN HEUTE FÜR MORGEN Sven Reiche :: SwissFEL Beam Dnamics Grop :: Pal Scherrer Institte Motion in an Undlator CERN Accelerator School FELs and ERLs

2 On-ais Field of Planar Undlator For planar ndlator with onl one transverse magnetic field component, the field is given on-ais: B Besin( k) with k Using Lorent Force eqation with the assmptions: Relativistic electron eam moves primaril into -direction The energ of the electrons is preserved in the magnetic field. d F ev B mc ecb sin( k) dt The eqation cannot e solved directl ecase the time-dependence of the longitdinal position (t) and velocit (t) is nknown. Page

3 Dominant Motion On-Ais I We assme that the deflection strength per modle is small so that the electron still moves predominantl in the -direction. The transverse motion and the reslting modlation of the longitdinal motion can e regarded as small. ( t) c t e f ( t), ( t) ( e / c) f ( t) The parameter and fnction e and f(t) are ndefined t we assme that the are sfficientl small to treat them as a pertration. Later we will jstif this assmption when the eplicit form of e is known. Becase the motion in a magnetic field does not change the energ the longitdinal and transverse velocit are linked 1 1 Page 3

4 Dominant Motion On-Ais II Integration of leading term in Lorent Force eqation: d d eb mc ecb k k c t dt dt m sin( ) sin( ) eb mck cos( k) The phsical constants and the ndlator parameters are comined into the so-called ndlator parameter eb.93 T cm cos( ) B k mck Two comments: The vale of is tpicall arond nit For a relativistic eam the maimm angle in the orit is ' / / Page 4

5 Dominant Motion On-Ais III Becase total energ is preserved, longitdinal and transverse velocities are linked energ: Using the epression of and the identit cos() =[1+cos()]/, we are getting: cos( k ) 1 / 1 cos( k) 4 The mean longitdinal velocit is given: 1 1 / The integration pertration is well jstified ecase the oscillating term in longitdinal velocit remains small over the entire ndlator length. Page 5

6 Trajector in Planar Undlator Integration ( pertration again) of the velocities in and ield the trajector: ( t) sin( ck( t)) k Co-moving Frame ( t) c t sin( c k t) 8 k Longitdinal wiggle motion has half period length. Cases a figre 8 motion in the co-moving frame. The longitdinal position is effectivel smeared ot Copling to harmonics Redced copling to fndamental trajector Effective position Page 6

7 Harmonics in Planar Undlator Inclding longitdinal oscillation term in transverse oscillation: cos( k) ee e ik i sin( k ) c t 8 ik m e e ( 1) J m( ) e m m ( 1) Jm( ) cos([m 1] k ) m m ( 1) [ Jm( ) Jm 1( )] cos([m 1] k ) m imk Identities of Bessel Fnction iasin im m( ) m e J a e m J ( a) ( 1) J ( a) m m Motion has: Redced amplitde of fndamental oscillation J ( m ) 1for m Occrrence of odd harmonics. On the scale of the ndlator period the harmonics are hardl noticeale (Thogh it ecomes important with respect to a given radiation wavelength) Page 7

8 On-ais Motion in Helical Undlator Helical ndlators has a transverse magnetic field, which rotates along the ndlator ais: B B e sin( k) e cos( k) From the Lorent force we otain: d ecb d ecb sin( k ) cos( k ) dt mc dt mc Integration similar to planar ndlator case: cos( k ) sin( k ) Longitdinal velocit Note that there is no longitdinal oscillation no harmonics are ecited Page 8

9 Comparison Planar and Helical Undlator Eclding higher harmonics in case of planar ndlator Using average position (for convenience) Planar Helical [ J( ) J1( )] cos( k) cos( k ) sin( k ) 1 / 1 cos( k) / 1 1 Page 9

10 Off-Ais Field Components I The simple field dependence B B cannot e sed for the entire esin( k) transverse plane ecase it violates Mawell condition of free space: B We assme a vector potential to derive B: A B Condition I: A Condition II: A Dominant vector component is in : B A cosh( k)cosh( k )cos( k) k Condition I: A ( k k k ) A k k k Meaning of k and k will e eplained later Condition II: B k A A A sinh( k)sinh( k )cos( k) k k Page 1

11 Off-Ais Field Components II With the valid vector potential the field is: It provide focsing of the electron if not injected on-ais: k sinh( k)sinh( k )sin( k) k B B cosh( k)cosh( k )sin( k) k cosh( k)sinh( k )cos( k) k Horiontal Focsing: Effective Field B 1 for [,+] Effective Field B for [-,] Off-Ais: B 1 > B Net kick inwards B B 1 Vertical Focsing: 1 st half-period: F = -ev B nd half-period: F = -e(-v )(-B ) Off-Ais: B ~k Net kick inwards Field Lines Transverse Velocit Page 11

12 Crved Poles Meaning of k and k Note that the magnetic field can also e derived from a scalar potential B cosh sinh sin k k k k Setting the scalar potential to a constant vales defines an eqipotential plane with the dependence for small transverse etensions: k sinh( k)sinh( k )sin( k) k B B cosh( k)cosh( k )sin( k) k cosh( k)sinh( k )cos( k) k c 1 sinh( k ) c1 c 1 k cosh(k ) The parameter k descries the transverse dependence on the pole srface: k > poles are crved inwards k < cosh is replaced with cos poles are crved otwards defocsing k > k < Page 1

13 Helical Undlator In an ideal helical ndlator the focsing is smmetric with: k k k / The simplest vector potential and magnetic field is: ( ) ( ) cos ( ) ( ) sin A I k r I k r k r B A A I kr I kr k k A I( kr) I( kr) cos k ( ) ( ) sin I ( k r)sin k B B I k r I k r k 1 However a helical ndlator field is also otained sperposition of two planar fields. If the smmetr is roken (e.g. APPLE Undlator) the roll-off parameters k and k are different for the two polariation planes. The sm of all coefficient in sqare still have still to e the sqare of k. For APPLE tpe ndlator the reslting net constants can e significantl larger, e.g.: k 5 k, k 6k Page 13

14 The Hamilton Fnction and Electron Motion I The Hamilton fnction is a constant of motion ecase there is no eplicit timedependence in the vector potential (here planar ndlator): 4 H P ea c m c mc The velocities are: B cosh( k)cosh( k )cos( k) k sinh( )sinh( )cos( ) A k k k k k P ea P ea P H, H, H p m p m p m Note that the velocit term proportional A is eactl the fast oscillation term t now with the transverse dependence on the ndlator field: cosh( k )cosh( k )cos( k ) The canonical momentm P descries mostl the slow etatron-oscillation Page 14

15 The Hamilton Fnction and Electron Motion II 4 H P ea c m c mc The transverse momenta are given : 1 F P H P ea m mk eb cosh( k ) cosh( k ) ( k / k ) sinh( k ) sinh( k ) 4 A k k k k k cosh( k)cosh( k )cos( k) k sinh( )sinh( )cos( ) Some terms have een dropped or simplified for the evalation of the slow etatron motion: B p cos 1 p A cos k p A k Page 15

16 Electron Motion for Small Amplitdes The reslting etatron eqations of motion ecome: P e B k sinh( k )cosh( k ) cosh( k ) ( k / k ) sinh( k ) m m k For small amplitdes in : c k Similar calclation for : c k Natral focsing of ndlators Total focsing strength is given ndlator period: k k k Page 16

17 Betatron Motion for Natral Focsing The transport matri for qadrpole focsing is given M 1 cos sin sin cos k c Matching condition is: Note: eam energ and long. velocit T M 1 1 M M Note: twiss parameter 1,, M11 M M1M 1 At aot 1 MeV the matched etatron fnction is arond 1 m At 1 GeV it is 1 m For X-ra FELs there is the need for eternal focsing to redced the electron eam sie Page 17

18 Eternal Focsing If the natral focsing is not sfficient, sperimposed qadrpole fields can provide more focsing. Undlator Modle Qadrpole Eqations of motions for the slow etatron-oscillation are: ( ) ( ), Q Q( ) ( ) Formal soltion of : Special case: Natral focsing onl ( ) I ( ) cos ( ) p I '( ) ( )cos ( ) sin ( ) p () I ( ) cos '( ) I sin Page 18

19 Longitdinal Velocit for Natral Focsing Inclding etatron-motion in longitdinal velocit: cos( ) sin( ) 4 1 / 1 1 k k Averaging ot all fast oscillating terms Betatron motion p p k I 1 / k I 1 sin sin 4 4 (, ) cosh( k )cosh(k ) The -vale shold e evalated at the position of the electron with: I I (, ) 1 k k k cos( ) k cos( ) 1 / 1 k k 4 I I No dependence on the etatron-phase Page 19