free- electron lasers I (FEL)

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Willa McLaughlin
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free- electron lasers I (FEL) produc'on of copious radia'on using charged par'cle beam relies on coherence coherence is achieved by insuring the electron bunch length is that desired radia'on

1 free- electron lasers I (FEL) produc'on of copious radia'on using charged par'cle beam relies on coherence coherence is achieved by insuring the electron bunch length is that desired radia'on wavelength Therefore to produce coherent radia'on one needs to form short electron bunches Recent developments make the produc'on of sub- ps bunch possible but lower pulse dura'on is challenging

2 modula6on with a laser coupling of a laser (op'cal wavelength) to an electron beam can lead to an energy modula'on this is similar to the klystron concept How do we couple a laser with a beam? one possible approach is the dielectric- laser accelerator (see Midterm!) 2

3 undulator radia6on planar undulator magnet in the symmetry plane x=0 we have B(z) = B 0 sin(k u z)ŷy undulator magnets are generally magnetosta'c recent development include the use of RF waveguides 3

4 equa6on of mo6on in an undulator dp Lorentz force dt = ev B resul'ng in the equa'on of mo'ons: { ẍ = z = e m B yż e m B yẋ these are coupled equa'ons 4

5 1 st order (linear) solu6on assume that and v z =ż ' v = c v z v x the approximated solu'ons are then { x(t) ' eb 0 m ck 2 u z(t) ' ct sin(k u ct) with ini'al condi'ons, x(0) = 0 ẋ(0) = eb 0 mk u 5

6 1 st order (linear) solu6on II the e- follows a sine- like trajectory x(z) = K k u sin(k u z) where the dimensionless undulator parameters is defined as K = eb 0 u = eb 0 mck u 2 mc ' 0.934B 0[T] u[cm] the radia'on is emi[ed within a 1/ cone centered around the instantaneous direc'on dx dz apple max K 6

7 undulator vs wiggler if K 1radia'on from different axial posi'on overlap this is an undulator photon- science.desy.de if K 1 radia'on divergence is larger than 1/ this is a wiggler 7

8 2 nd order solu6on expand to : explici'ng the 1 st order solu'on for : introducing the average velocity v z O(v 2 x) v z = p v 2 v z (t) = apple 1 v 2 x ' c v z = apple 1 apple 1 1+ K (1 + 2 v 2 x/c 2 ) c 1+ K2 v x ck cos(2! ut) 2 c c 8

9 2 nd order solu6on II the par'cle trajectory is then parameterized as { x(t) = K k u sin! u t z(t) =v z t K k u sin 2! u t 9

10 trajectory in reference ( ) frame In the reference frame of the electron the coordinates are (using the average longitudinal velocity) t 0 = (t z/c) ' t/ x 0 = x = K k u sin! u t z 0 = (z ct) ' K 2 the argument can be wri[en with! 0 =! u 8 k u p 1 K2 /2 sin 2! ut! u t =! 0 t 0 10

11 trajectory in reference frame II trajectory for non- vanishing K follows an 8 pa[ern in the reference frame the e- emits radia'on at the frequency! 0 =! u x 0 /a K =0 K =1 z 0 /a K =5 11

12 proper6es of emiced radia6on In the reference frame the radia'on is going to be of dipolar type with power given by where P = h v 2 i = 1 2 e2 6 " 0 c 3 v 2 2 c 4 ku 2 1+K 2 /2 K 2 (x 0,z 0,t 0 ) (x, z, t) 12

13 wavelength in lab frame apply conserva'on of 4- momentum ~! 0 = [E cpcos ] = ~! l (1 cos ) so that the frequency of the radia'on is given by! l =! u 1 1 cos 13 h[p:// eneduca'onalsoe.htm

/2) u 2 2 cos 1+ K2 2 + 2 2 for a planar undulator, radia- 'on

14 wavelength in lab frame II using Taylor expansion of so the radia'on wavelength in the lab frame is l = 1 =1 2 2 (1 + K2 /2) u 2 2 cos 1+ K for a planar undulator, radia- 'on is linearly polarized and performing a l (m) = 10 = 100 = 1000 u (m) 14

15 radia6on spectrum The electric field in the 'me domain as the e- travel through N u undulator period is: { E l (t) = E oe i! 1t 0 otherwise El(t) if T/2 apple t apple T/2 with Fourier transform sin[(! 1!)T/2] Ẽ l (!) =2E 0! 1! Ẽl(!) 2! 1!! 1 15

16 harmonic frequencies emission at harmonic frequencies occurs as K increases! n = n! 1 heuris'cally can be understood by considering the field detected within a 1/γ angle by a 'me- domain detector 16

17 helical undulators we considered the case of a planar undulator another type of undulator is of helical type linac96.web.cern.ch the magne'c field is B = B 0 [cos(k u z)ˆx +sin(k u z)ŷy] newsline.linearcollider.org 17

18 RF undulators (HW4) a RF waveguide can be used as an undulator where: B u = 1 2 (B? + E? /c) 18

19 op6cal undulator periodic field is provided by a counter- propaga'ng laser pulse enable shorter radia'on wavelengths h[p:// 19

20 undulator as beam- radia6on 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 20

21 Condi6on for sustained energy exchange Light wave as to advance by period of electron trajectory l/2 per half 21

22 Condi6on for net energy exchange 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 radia6on wavelength in an undulator! SLIDE 14!! 22

free- electron lasers II (FEL)

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+ K2 2 + 2 2 angle