Pendulum Equations and Low Gain Regime

Transcription

1 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

2 Interaction with Radiation Field in Helical Undlator Sperimpose a circlar co-propagating plane wave with freqency w and wavenmber k: d d w w E(, t) e E cos k t e E sin k t ee K x y k k t k k t mc ee K dt d e ev E E dt d mc cos( )cos( w ) sin( )sin( w ) cos k k wt mc For now we are leaving the polariation ndefined To change the Energy T, the particle has to move with or against the field lines. So far the wave nmber k has been ndefined. Is there a preferred wave nmber to maximie the energy change? Page

3 The Resonance Condition Energy change is resonant if the phase of the cosine fnction does not change: d ee K cos d d mc dt k k wt k k wt c k k k Becase the normalied velocity is always smaller than nity only the case with the pls-sign can be flfilled: The mins-sign term oscillates as cos(k ) or faster and averages ot to ero The polariation of B- and E- field are opposite The resonance condition is: k 1 K k k k k K Page 3

4 The Resonance Condition The transverse oscillation allows to cople with a co-propagating field The electron moves either with or against the field line, depending on the radiation phase and the injection time of the electron After half ndlator period the radiation field has slipped half wavelength. Both, velocity and field, have changed sign and the direction of energy transfer remains. The net energy change can be accmlated over many period.

5 Ponderomotive Phase Becase the phase in the cosine fnction is considered a slowly changing parameter, we define it as a new variable, the ponderomotive phase: k k wt The resonant energy change becomes: d d eek cos mc The resonance condition links an electron energy for a given wavenmber r 1K Becase we mst allow the case that an electron might not be in resonance, we have energy and resonant energy as two independent parameters For small deviation from the resonant energy the ponderomotive phase changes slowly with d k 1K k k k k k k k d 1 r Z r r Page 5

6 The Pendlm Eqations The particle motion in a helical ndlator with presence of an external field is: d d k r d d ee K cos mc Transform the eqations in a more handy-form: Assme only small deviation from the resonant energy: Use new variables: r k r / d d d eek k sin d mc Classical Pendlm eqations r d dt I d dt I g sin L Page 6

7 Electron Trajectories There are two fix points: Stable:, Unstable:, d d d s sin d For small amplitde arond stable point the motion is a simple oscillation with the freqency s k ee K rmc Sbscript s indicates the similarity to synchrotron oscillation in storage rings Energy (Hamiltonian) of the system is: Bond and nbond motion are split by the separatrix: 1 H s cos 1 cos sep s Page 7

8 Trajectories in Longitdinal Phase Space Energy Gain Energy Loss Unbond Forward Motion Bond Separatrix Backward Motion Page 8

9 Ensemble of Electrons The dynamic of a system with many electrons is best described by a phase space density fnction f. We know the Hamilton fnction. Therefore the density fnction is a constant of motion according to Lioville s theorem: d f (,, ) f f f d We know the expression for and from the Hamilton fnction f f s sin f With the distribtion fnction we can calclate the mean energy of the system: f (,, ) dd Page 9

10 Small Signal Approximation I f f s sin f We assme that the electric field is small and therefore the synchrotron freqency s can be se as a pertrbation parameter: f (,, ) f (,, ) f (,, ) f (,, ) 4 s 1 s The initial beam is niformly distribted in, the first order term incldes the initial energy distribtion g(): f g( ) Fnction g is nspecified so far Sorting powers of s yields recrsive eqation: f fn fn sin n1 Formal 1 st order soltion is: 1 f f1(,, ) cos cos Page 1

11 Small Signal Approximation II The first order term does not show any change in the mean energy: f 1 s f1d d s cos cos d d We have to go to second order. Here we are only interested in the terms which have no explicit dependence on (ero order in Forier series): 1 f f f sin cos cos f sin cos 1 1 f f, f, f 1 cos The term (1-cos) enforces that at = there is no contribtion from this coeficient Page 11

12 Small Signal Approximation III Calclating the mean energy change ( nd order term): cos g 1 cos 4 g s fdd s d s d Integration by parts: 1 cos sin / g d 3 4 s sin s d g d 8 with d Intensity Spectrm of spontaneos ndlator radiation N w k N N w r r r Page 1

13 Small Signal Gain Converting back the gain formla Transfer energy from electron beam to radiation field: mc mc U n mc n n r r e e e k k Express electric field E by energy density: E U Assme no energy spread: g()=d(-<>) 3 3 e ne kk N 3 r mc 3 4 d sin d sin e s k d 4 mc d U n U FEL Gain 3 3 e n sin e kk N d G( ) 3 4 r mc d with N w w r Page 13

14 Madey s Theorems 1 st Theorem: Small signal FEL gain is proportional to derivative of spontaneos ndlator spectrm d sin G d nd Theorem: Connects the indced energy spread to the energy spread 1 Page 14

15 Nmerical Examples Varios initial mean energies Page 15

16 Mean Energy and Energy Spread Optimm detning at arond half the separatrix height Even particles otside of separatrix can transfer energy to field if ndlator is short enogh Energy change is cbic with ndlator length in start p. Mean Energy Energy Spread Page 16

17 A Qantm Level Point of View A very crde argmentation, why the FEL can be considered a laser. The intensity distribtion can be considered as the poplation density of the virtal states E ( wr w) E w r E ( w w) r Spontaneos emission ~ density of state N density In presence of external field the photon can either be absorbed or stimlate the emission of another one. N N ( E w) N ( E w) ph density density dndensity Emission w Absorption de E Assming photon energy mch smaller than electron energy Page 17

18 Planar Undlator The interaction occrs with a radiation field of same polariation as the ndlator E(, t) e E cos k wt The maximm phase change is given by x d e ee K E cos( k)cos( k wt ) d mc mc ee K cos cos k k t k k t mc w w As in the helical case, the cos((k-k )-wt) term cannot be flly in resonance ( <1). However we cannot drop it yet de to the longitdinal oscillation The resonance condition is: k 1 K / 1 1 k k k k K 1 Page 18

19 Planar Undlator d d The longitdinal oscillation has to be taken into accont: ee K w w cos k k sin( ) cos sin( ) k t k k k t mc with k k K 1 K K k K K d d ee K i ik i sin( k ) e e 1 e e mc ee K i ik m imk e e 1 e ( 1) J ( ) m e mc m ee K m e e ( 1) J ( ) J m1( ) e i m mc m ee K m Jm mc m m1 imk ( 1) [ ( ) J ( )]cos( mk ) Page 19

20 Harmonics For the fndamental wavelength on the term for m= is resonant d d ee K m m m1 mc m ( 1) [ J ( ) J ( )]cos( mk ) However if we choose a harmonic of the fndamental with k n =nk: n, m kn k wnt mk nk (m 1) k nwt The resonance condition is then: k k m 1 k n Odd harmonics can be in resonance as well, thogh with a redced copling J m ()-J m+1 () Page

21 Comparison Helical vs Planar Undlator Helical Planar Polariation B- and E- Field Opposite, circlar Same, planar Harmonics None Odd harmonics Resonance Condition 1 K K 1 Copling with E-Field 1 J( ) J1( ) Synchrotron Freqency Low Gain Fnction s d sin G( ) d s J J1 d sin G( ) 4 d Page 1

22 FEL Oscillator Becase the gain can be rather small it needs to be accmlated in an optical cavity: Reflectivity R Reflectivity R 1 Gain G Ot-copled field The bnch repetition rate needs to match the rond trip freqency of the cavity or a harmonic of it. For amplification it reqires: R R 1 G 1 1 n P n P Page

23 Satration-Effects Satration effects occrs once the length of the ndlator matches the period length of the synchrotron oscillation: ee K ee K L 4 L k 16 N s rmc kmc 1 K Normalied Vector Potential of Radiation Field: A r A r 1 K 1 4K N More radiation power can be achieved if the ndlator is shorter. However it makes the FEL oscillator more difficlt to laser becase the small signal gain drops. Page 3

24 Nmerical Simlations Field Growth Growth Rate Limit of Small Signal Gain FEL Oscillator model gets more complicated in reality: Slippage of the radiation field with respect to electron beam Detning of the optical cavity Mode selection in the cavity Page 4

25 Model Limitations The small signal low gain FEL model is limited in its applicability by: Scaling with more nmber of particles the transferred energy can be become larger than the assmption that the electric field is constant. On resonance the beam can be strongly bnched and shold emit coherently. For large field vales the analytical soltion is not valid anymore. In addition the reqirement of being in resonance is diminished. Model mst inclde Maxwell eqation as well High Gain FEL Model Page 5

26 Otlook Contination with High Gain FEL Theory by K.-J. Kim (tomorrow and Monday) More info on FEL Oscillators XFELO (Monday) Page 6