free- electron lasers II (FEL)

Transcription

1 free- electron lasers II (FEL) undulator radia+on in the lab frame has the wavelength undulator period l = u 2 2 beam Lorentz factor K = eb 0 = eb 0 u mck u 2 mc ' 0.934B 0[T] u[cm] 1+ K angle of observa+on! l Simula'on by I. Vi'

2 undulator as beam- radia8on coupler Consider an undulator with an electron co- propaga+ng with an e.m. wave Energy exchange occurs if de dt = TEM wave can exchange energy because of non- vanishing v x ee.v Pending some proper phasing condi+on the energy exchange can be sustained 2

3 Condi8on for sustained energy exchange: qualita8ve treatment Light wave as to advance by period of electron trajectory l/2 per half 3

4 Condi8on for net energy exchange: qualita8ve treatment II the difference in +me of fligh between the electron and light for a length u/2 is 1 1 t = u/2 v z c we need c t = l /2 the resul+ng rela+onship between undulator period and radia+on wavelength is l = u 2 2 (1 + K2 /2) same as radia8on wavelength in an undulator (see SLIDE 1)!! 4

5 Quan8ta8ve analysis of energy exchange energy exchange between an electron and an e.m. field within an undulator is dw dt = ev x (t)e x (t) = ee 0 ck cos(ku z) cos(k l z! l t + 0 ) where ecke 0 2 cos ecke 0 2 cos (k l + k u )z! l t + 0 (k l k u )z! l t + 0 ponderomo+ve phase note that z = z(t) 5

6 ponderomo8ve phase Considering requires only, net energy exchange = const. approximate z(t) = v z t (ignore the sine term) so that the condi+on on the pond. phase is imposing v z = (t) =(k l + k u ) v z t d /dt =0 apple K2 2 l = c c u and explici+ng leads to 2 2 (1 + K2 /2)! t t + 0 = const 6

7 dw/dt 2 nd term in. the 2 nd term cannot be kept constant as (t) =(k l k u ) v z t! l t 0 = const. would imply no+ng that the condi+on behaves as k l < 0 (z) = (z) (z) = const. / cos 2k u z - > light direc+on is and given the 2 nd term 2k u z 2 oscilla+on per undulator period and cancels ẑz 7

8 energy exchange neglect 2 nd term so that dw dt = ecke 0 2 cos we introduce the internal bunch coordinate = + /2 ' + /2 k l + k u 2 posi+on of en e- inside the undulator is z(t) = v z t + (t) l 0 = /2 8

9 interpreta8on of ini8al ponderomo8ve phase at t=0: so that 9

10 FEL pendulum equa8ons from the radia+on wavelength we find the resonant energy (Lorentz factor) we introduce the frac+onal energy devia+on r r 10

11 FEL pendulum equa8on take the +me deriva+ve of the pond. phase from the resonant energy defini+on (previous slide) we have and obtain 11

12 FEL pendulum equa8on therefore d dt =2k uc and d dt = ee 0K 2mc 2 r cos 2 nd equa+on is obtained from the energy gain equa+on dw/dt the 2 above equa+on are the FEL pendulum equa+ons they can be combined into ( ) with 2 ee 0Kk u m 2 r 12

13 phase space (over on wave period) mo+on in longitudinal phase space of 15 e- subjected to an E- field within the undulator for net energy transfer from e- to field (e.g. e- energy loss) we need > r = r > r 13

14 E- field gain An FEL gain (during one pass in the undulator) can be defined as and is derived to be where 14

15 E- field gain e- with >0 enhance the intensity of the E- field while e- with <0 reduce it. G 15

16 laser vs. FEL configura8ons conven+onal solid- state or gaseous laser FEL oscillator FEL amplifier 16

17 seeding of the FEL process if the e- bunch has a high peak current the spectral noise within the bunch can have a significant contribu+on at the resonant undulator wavelength this radia+on can seed the FEL process and result in high gain (within a single pass) This is called the self- amplified s+mulated emission (SASE) regime 17

18 SASE FEL field- beam interac+on within a SASE leads to microbunching at the resonant wavelength In the SASE regime the gain theory outline in the previous slide does not hold as E changes signifi- cantly with +me adapted from PRL 88, (2002) 18

19 SASE FEL Longitudinal phase- space evolu+on du- ring the SASE FEL process at different axial posi+ons along the undulator 19

20 FEL landscape nowdays x h`p://pubs.rsc.org/en/content/ar+clehtml/ 2014/fd/c4fd00156g 20

21 high- average- power FELs no medium so free- electron can be used to produce high- average- power radia+on 10kW of average power produced using an energy recovering linac at JLab G. R. Neil,* C. L. Bohn, S. V. Benson, G. Biallas, D. Douglas, H. F. Dylla, R. Evans, J. Fugitt, A. Grippo, J. Gubeli, R. Hill, K. Jordan, R. Li, L. Merminga, P. Piot, J. Preble, M. Shinn, T. Siggins, R. Walker, and B. Yunn Thomas Jefferson National Accelerator Facility, Newport News, Virginia (Received 3 September 1999) Sustained Kilowatt Lasing in a Free-Electron Laser with Same-Cell Energy Recovery VOLUME 84, NUMBER 4 P H Y S I C A L R E V I E W L E T T E R S 24 JANUARY

22 probing ultra- short phenomena using short (fs) X- ray pulses X- ray FELs 22