Free electron lasers

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1 Preparation of the concerned sectors for edcational and R&D activities related to the Hngarian ELI project Free electron lasers Lectre 1.: Introdction, overview and working principle János Hebling Zoltán Tibai TÁMOP C-1/1/KONV projekt 1

2 Otline Introdction Prodction and description of relatvistic electron beams Electron trajectory in magnetic ndlator Power and anglar distribtion of the radiation of accelerating reletivistic electron Wavelength, power and spectrm of ndlator radiation Importance and development of microbnching Gain spectrm of small gain FELs. References to books and web pages TÁMOP C-1/1/KONV projekt

3 What is a free electron laser (FEL) FEL is a large-scale device which can generate intensive laser beam by ndlator radiation of free electrons having relativistic velocity FEL s working ranges can be anywhere on the spectral range from the microwave to the hard X-ray TÁMOP C-1/1/KONV projekt 3

4 Free Electron Lasers (FEL) FLASH SLAC sorce: 4 sorce:

5 Free Electron Lasers (FEL) SACLA FERMI sorce: 5 sorce:

1.C-1/1/KONV-01-0005 projekt 6 Conventional (bond-electron) laser Radiation is generated by stimlated emission Main components: 1.

6 Comparison of conventional laser and free electron laser P. Schmüser, et al.: Ultraviolet and Soft X-Ray Free-Electron Lasers, Springer, 008 TÁMOP C-1/1/KONV projekt 6 Conventional (bond-electron) laser Radiation is generated by stimlated emission Main components: 1., Active (gain) medim., Resonator (cavity) for feedback 3., Pmping Free electron laser (FEL) Radiation is generated by ndlator radiation of microbnched electrons Main components: 1., Relativistic electrons in ndlator., Resonator for feedback 3., Prodction of high energy electrons in LINAC or synchrotron

7 Prodction of relativistic electron beam Synchrotron LINAC Energy recovery LINAC Velocity: v Relative velocity: b = v/c Relativistic factor: Relativistic energy: W r 1 1 b W r mc Electron rest energy: mc MeV TÁMOP C-1/1/KONV projekt 7

8 Energy recovery LINAC Prodction of relativistic electron beam TÁMOP C-1/1/KONV projekt 8

9 Prodction of relativistic electron beam Wavelength of ndlator/fel radiation is (see later): where B0 is the ndlator period, and K eb K r 1 0 mc is the ndlator parameter. is the amplitde of magnetic indctance, changing harmonically with the distance along the axis of the ndlator (1) sometimes is called to resonance condition. Strong dependence of r (see Eq. 1) makes possible FEL on very different range. For example: = 30 mm, W r = MeV ( = 4), K = 1 r = 1.5 mm microwave 6 GeV ( = 1.x10 4 ), K = 1 r = 1.6 A, X-ray (1) W r = MeV electron bnch directly from electron gns W r = 6 GeV electron bnch only from very complex few hndred m long LINAC TÁMOP C-1/1/KONV projekt 9

10 Magnetic ndlator Inside the ndlator the magnetic indctance along the z axis is: B B sin( k ) y 0 z k The acceleration of an electron arriving into the gap of the ndlator in the z direction is: e x Bz y 0 z m e m Bx () TÁMOP C-1/1/KONV projekt 10

11 Electron trajectory If dx dt dz dt ( the soltion of eq. system () with first order approximation is: eb0 x t) sin( kvt) y( t) 0 mvk Used initial conditions: x( 0) 0 z( t) vt (3) eb0 x ( t) y( 0) 0 y( 0) 0 mvk According to (3), the trajectory of the electrons inside the ndlator is: K x( z) sin( kz) bk eb eb0 where 0 K B0[ T ] [ ] mck mc cm Meaning of K: the direction of the electrons deviates from the average (z) direction by a maximm amont of Q max =K/. is the ndlator par. TÁMOP C-1/1/KONV projekt 11

12 1 TÁMOP C-1/1/KONV projekt Electron trajectory If K is not mch smaller than 1 v z v, instead x x z c v v v b z k K c z v z cos ) ( c K c v z b The main velocity is: In second order approximation, and introdcing ) sin( ) ( t k K t x ) sin( 8 ) ( t k K t v t z z z k v (5B) According to Eq 5 the electrons oscillate transversally with, and longitdinally with anglar freqency. The ratio of the transversal acceleration to the longitdinal one is: (6) K a a l t (4) (5A)

13 Power radiated by accelerating charge Charge moving with constant velocity do not radiate Charge moving with changing velocity radiates electromagnetic wave 1., v << c power of the emitted radiation is: P e 6 c 6 e., v c power of the emitted radiation is: P β ββ F A: acceleration is parallel with the velocity: b 0 3 v 6 0c 6 e P β 6 c 0 (Larmor) (7A) d b b cm bb b bcm b 1 d( mv) d( mb ) c cm dt dt db F cm 3 P e 3 6 0c m F (8A) (Liénard) (7) TÁMOP C-1/1/KONV projekt 13

14 Power radiated by accelerating charge B: acceleration is perpendiclar to the velocity: P e 6 b W 0 = constant F bcm (7) (7B) 3 6 0c 6 0c m F P e F (8B) Comparing (8A) with (8B): A force with a given magnitde creates times more radiation if it is instead of parallel- perpendiclar to the velocity. Radiation means a loss in particle accelerators. In a synchrotron, where the particles moves on a circlar path the energy loss per rond is: Here r is the radis of the circlar path. E 3 4 e b 3 r 0 Althogh E decreasing with increasing r, for large E is large even for g=100 m and the radiation energy loss limits the achievable electron energy. In longitdinal accelerators (LINACs) the radiation loss is mch smaller ( does not appear in (8A)). TÁMOP C-1/1/KONV projekt 14

15 Power radiated by accelerating charge Comparing (8A) with (8B): Accelerators in which the electrons has acceleration in the direction perpendiclar to its velocity are more efficient radiation sorces. Synchrotron radiation sorces consists of straight and crved paths, approximating together a circle. The crved path of the electrons are cased by bending magnets. The electrons emits radiation when moving on the crved paths. The emission will be intense for short radis of crvatre and large of electrons and large electron crrent. Other advantage of large is that increasing the solid angle of emission is decreasing. A modern synchrotron radiation sorce is the Advanced Photon Sorce (APS) in the Argonne National Laboratory ( sitated close to Chicago. Here W r = 6 GeV. In order to prodce narrowband radiation ndlators are sed instead of bending magnets. For the sed different ndlators the ndlator period is on the = mm range, and the radiated wavelength is on the r = 1 3 A range. TÁMOP C-1/1/KONV projekt 15

Anglar distribtion of the radiation A: acceleration is parallel to the velocity: dpl d e a sin L 3 16 0c 1 b cos 5 d sindd K = 1, = 100 is the elemental solid-angle The angle is measred from the

16 Anglar distribtion of the radiation A: acceleration is parallel to the velocity: dpl d e a sin L c 1 b cos 5 d sindd K = 1, = 100 is the elemental solid-angle The angle is measred from the direction of the velocity of the electron The azimthal angle f is measred from the direction of the acceleration of the electron In this case there is no radiation in the forward direction not sefl for FEL D. J. Griffiths: Introdction to Electrodynamics, Prentice-Hall, New Jersey TÁMOP C-1/1/KONV projekt 16

Anglar distribtion of the radiation B: acceleration is perpendiclar to the velocity: dpt d e a 1 b cos 1 b T 3 16 0c 1 b cos 5 sin cos K

17 Anglar distribtion of the radiation B: acceleration is perpendiclar to the velocity: dpt d e a 1 b cos 1 b T c 1 b cos 5 sin cos K = 1, = 100 / 0.8 FWHM D. J. Griffiths: Introdction to Electrodynamics, Prentice-Hall, New Jersey TÁMOP C-1/1/KONV projekt 17

18 18 TÁMOP C-1/1/KONV projekt Wavelength of ndlator radiation The oscillation freqency of the electron moving with velocity v z z v f The radiation emitted by this electron shold have a wavelength z r v c f c ' Since the electron approaches the point P with a velocity of z v, the wavelength of the radiation approaching point P is: b cos 1 cos cos ' z z r r v v c Approximating the cos fnction with the first term of its series and applying Eq. 4 we get: 1 K r (9)

19 Power radiated by an electron in the ndlator From (7B) P e 6 c 4 v T 3 0 K From (5A) v ( t) sin( t) k From these we get that dring traveling thogh the ndlator an electron radiate with P e c k K 1 1 K average power on the wavelength given by (9). 0 (10) Becase of the longitdinal oscillation, and of the deviation of the transversal oscillation from the harmonic one, the electron radiate at higher freqencies (harmonics), too. The total radiated power is: P e c k K 1 total (11) 0 TÁMOP C-1/1/KONV projekt 19

20 Spectrm of the ndlator radiation The intensity spectrm is given by the sqare of the Forier transform of temporal shape of the radiation. Since lathe latter is a trncated harmonic fnction with r anglar freqency, the intensity spectrm is given by: sin I( ) where N r. (1) r According to (1) the spectrm become narrower with the increase of the nmber of the ndlator periods N. The half-width of the spectrm is inversely proportional to N. TÁMOP C-1/1/KONV projekt 0

21 Radiation by an electron vs. electron bnch A: Energy radiated by one electron in an ndlator with N = 100. For K = 1, = 3 mm, and = 00 comes from (10): P = 4.0x10-11 W. From this with N = 100 comes E = 4.0x10-19 J. B: Energy radiated by an electron bnch having 1 nc charge (6.5x10 9 electrons) in case of spatially randomly distribted electrons (incoherent radiation): E = 6.5x10 9 x 4.0x10-19 J =.5x10-9 J =.5 nj C: Energy radiated by an electron bnch having 1 nc charge (6.5x10 9 electrons) in case of perfectly microbnched electrons (coherent radiation): E = 6.5x10 9 x 6.5x10 9 x 4.0x10-19 J = 15.6 J!!! D: Plse energy of real word FEL s (microbnching is not perfect): few mj. TÁMOP C-1/1/KONV projekt 1

22 Microbnching Microbnching is cased by interaction of the electrons with the magnetic field of the ndlator and the electromagnetic field of the FEL. Electric field of FEL: E ( z, t) E k z t x 0 cos The rate of change of the electron energy cased by this interaction is given by: dwr ck v F evx ( t) Ex ( t) e cos 0 cos r r 0 dt ecke ecke r r 0 r r 0 r r 0 k ze k z t 0 cosk k z t cosk k z t cos cos 0 If (1) is flfilled constant. In this case, since ( z) ( z) k z it is averaged ot on one ndlator period, so its term can be omitted. According to (13) the rate of the change of the electron energy constant, bt depends on its position inside the bnch. There are places with a distance of r / from each other, where dw r /dt has maximm, bt opposite sign. (13) TÁMOP C-1/1/KONV projekt

23 Microbnching The periodic spatial dependence of energy change reslts periodic spatial dependence of electron velocity and this reslts microbnching. Electrons have random phase, Radiation is incoherent Electrons are micronbnched Radiation is coherent TÁMOP C-1/1/KONV projekt 3

24 Microbnching The periodic spatial dependence of energy change reslts periodic spatial dependence of electron velocity and this reslts microbnching. TÁMOP C-1/1/KONV projekt 4

25 Developing of laser working in FELs The random distribtion of electrons emits radiation. This radiation interacts with the electrons starting a microbnching. In FELs working on the microwave, infrared, visible or near UV spectral range, -where the FELs consist resonator-, the development of the bnching/coherent radiation is a relatively long process, on the range of microseconds. The gain spectrm of this small-gain FEL s is proportional to the derivative of the spontaneos spectrm: 3 e K N ne d sin G( ) 3 4 0mc d The gain has a maximm at lower then the resonant freqency. The reason: electrons loss energy dring radiation. The average electron energy smaller then the original. TÁMOP C-1/1/KONV projekt 5

Elleame, Wigglers, Undlators and Their applications,

26 References [1] J. A. Clarke, The science and technology of ndlators and wigglers, Oxford University Press, 009 [] H. Onki and P. Elleame, Wigglers, Undlators and Their applications, Taylor & Francis, 004 [3] P. Lchini and H. Motz, Undlators and Free-Electron, Oxford University Press, 1990 TÁMOP C-1/1/KONV projekt 6

27 Web pages if important FEL s TÁMOP C-1/1/KONV projekt 7

28 Controlling qestions 1. How bilds-p a FEL?. How large is the rest energy of an electron? Define the relativistic factor! 3. Which qantities determined the wavelength of FEL? Describe the formla of the resonance condition! On what spectral range can work a FEL? 4. What is an ndlator? Define the ndlator parameter! 5. Describe in first order the electron trajectory in the ndlator! What is the meaning of the K ndlator parameter? 6. For the same accelerating force, how is related the radiated power for transversal acceleration to radiated power for longitdinal acceleration? 7. What is the main difference between the anglar distribtion of radiation generated by longitdinal and transversal acceleration, respectively? 8. Why are generated harmonics beside the fndamental freqency? 9. Why is necessary the microbnching? How develops it? 10. Describe the freqency dependence of the small signal gain in FELs! How connected it to the spectral shape of the spontaneos ndlator radiation? TÁMOP C-1/1/KONV projekt 8

Pendulum Equations and Low Gain Regime

Pendulum Equations and Low Gain Regime WIR SCHAFFEN WISSEN HEUTE FÜR MORGEN Sven Reiche :: SwissFEL Beam Dynamics Grop :: Pal Scherrer Institte Pendlm Eqations and Low Gain Regime CERN Accelerator School FELs and ERLs Interaction with Radiation