SPACE-FREQUENCY MODEL OF ULTRA WIDE-BAND INTERACTIONS IN FREE-ELECTRON LASERS. Yosef Pinhasi Yuri Lurie Asher Yahalom

Transcription

1 מכללת י ה ו דה ו שומרו ן The College of Judea and Samara המ ח לקה ל הנדסת חש מ ל ו א לקטרוניקה Dept. of Electrcal and Electronc Engneerng SPACE-FREQUENCY MODEL OF ULTRA WIDE-BAND INTERACTIONS IN FREE-ELECTRON LASERS Yosef Pnhas Yur Lure Asher Yahalom

2 SPACE-FREQUENCY MODEL Exctaton equatons n the frequency doman Frequency dependent effects (gan, absorpton, dsperson) Consderaton of statstcal features (radaton, gan medum) Ultra wde band nteractons Free-space,wavegude, resonator

3 Schematc llustraton of a pulsed beam free-electron laser

4 WIDE BAND INTERACTIONS Spontaneous emsson and nose Super-radance from ultra short bunches Synchrotron amplfed spontaneous emsson (SASE) Radaton exctaton and buldup n an oscllator Sngle- and Mult- mode operaton (forward and backward, grazng )

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6 MODAL REPRESENTATION OF THE ELECTROMAGNETIC FIELD ~ E [ ] jk z e ε ( x, y) ( ) ( ) + jk z r, f = C z e + C ( z) zq q + q q zq q d dz C ± q 1 N ( ) m jk ( ) ( zqz * z, f = m e J r, f ε x, y)dxdy q ~ q N q [ ~ ( ) ~ ( )] Ε x, y H * x, y = q q zˆ dxdy

7 ENERGY SPECTRUM dw + df ( z) q Cut off 1 = Im q Pr opagatng [ ] ( ) ( ) C z, f C z, f Re{ N } { ( ) ( )} * C z, f C z, f Im{ N } + q + q q q q q +

8 THE DRIVING CURRENT ( ) ( ) ( ) ( ) [ ] δ δ δ = t z z y y x x v e t r J r, ( ) ( ) ( ) ( ) ( ) f u e y y x x v v e f r J z ft j z π δ δ, ~ = r ( ) ( ) ( ) ( ) [ ] = ± ± z z f k z ft j q z q q dz e y x v v N e f z C zq 0 ' ' * ', 1, m r π ε

9 PARTICLE DYNAMICS r dv dz = 1 γ e 1 m v z E [ r, t ( z) ] + v B[ r, t ( z) ] r + r v dγ dz dγ e 1 r = v E r, dz mc v z [ t ( z) ] t ( z) = z 1 t + dz' 0 v ' 0 z ( z )

10 Accelerator Beam energy Beam current Beam pulse duraton Operatonal Parameters E k =1 6 MeV I 0 =1 A T b =0.1 ps Wggler Magnetc nducton B w = kg Perod λ w =5 cm Number of perods N w =0 Wave gude Rectangular Mode TE 01 TE 1, TM 1 TE 41, TM 41 TE 03 15mm 7.5mm Cut-off frequency 0.0 GHz 8.3 GHz 44.7 GHz 60.0 GHz

11 Energy dependence of the dsperson solutons Frequency [GHz] TE 41,TM 41 TE TE 01 TE 1,TM Beam Energy [MeV]

12 SINGLE TRANSVERSE MODE

13 Dsperson solutons for the transverse mode for TE 01 E k = MeV 8 6 E k =.00 MeV f 1 =46.1 GHz f =149.5 GHz f / f co 4 0 wavegude beam grazng E k =1.6 MeV f 0 = 69.6 GHz k z / k w

14 Super-radant emsson from an ultra short E k = MeV bunch wth the sngle mode TE 01 Energy spectrum (a) 0.3 d W / d f [pj/ghz] f [GHz]

15 Super-radant emsson from an ultra short E k = MeV bunch wth the sngle mode TE GHz Temporal wave-packet (b) GHz E x [kv/m] t [ns]

16 GRAZING

17 Super-radant emsson from an ultra short bunch at grazng E k 1.6 MeV Energy spectrum (a) 0.4 d W / d f [pj/ghz] f [GHz]

18 Super-radant emsson from an ultra short bunch at grazng E k 1.6 MeV Temporal wavepacket (b) 1 E x [kv/m] t [ns]

19 MULTI-TRANSVERSE MODES

20 E k.44 MeV ( grazng n the TE 1, TM 1 modes ) 1.5 TE 1 TM 1 0 d W / d f [pj/ghz] E x [kv/m] TE 01 TE f [GHz] t [nsec]

21 E k.6 MeV d W / d f [pj/ghz] TE 1 TM 1 E x [kv/m] 5-5 TE f [GHz] t [nsec]

22 BACKWARD WAVES Two-pont boundary value problem

23 Dsperson Solutons Backward ω ω 6 Wavegude Wave Beam lnes 4 v 1 + z kw π a tan ω λw q k q γ ( α ) = 1 w > 1 k z kw

24 Boundary condtons C+(z) C+(0)=0 0 C - (L)=0 L z C - (z)

25 Operatonal parameters Accelerator Beam energy Wggler Magnetc nducton Perod Number of perods Wave gude Rectangular Fundamental mode E k =50 kev B w = kg λ w =10 mm N w = mm TE 01

26 1.5 C ( + ) ( z = 0) ( ) ( ) C (±) (z=0), arb. unts cut-off C z = 0 0 C (±) (z=l), arb. unts cut-off ( + ) ( z L) C = ( ) ( z L) C = Frequency [GHz]

27 Summary and conclusons Coupled-mode theory, formulated n the frequency doman, enables development of a three-dmensonal model, whch accurately descrbes wde-band nteractons between radaton and electron beam. The approach s appled n a numercal smulaton WB3D!