Numerical Modeling of Collective Effects in Free Electron Laser

Transcription

1 Numerical Modeling of Collective Effects in Free Electron Laser Mathematical Model and Numerical Algorithms Igor Zagorodnov Deutsches Elektronen Synchrotron, Hamburg, Germany ICAP 1, Rostock 1. August 1

2 Overview motivation FEL physics in 1D mathematical model in 3D numerical methods quiet start and shot noise time dependent simulations problems and challenges Igor Zagorodnov ICAP 1. August 1 Seite

3 Motivation Free Electron Laser accelerator undulator laser light very short and tunable wavelength extreme short pulses with very high energy John Madey, Appl. Phys. 4, 196 (1971) Igor Zagorodnov ICAP 1. August 1 Seite 3

4 Motivation FLASH ( Free Electron LASer in Hamburg) gun accelerator undulator laboratory Igor Zagorodnov ICAP 1. August 1 Seite 4

5 Motivation FLASH ( Free Electron LASer in Hamburg) undulator 7m Igor Zagorodnov ICAP 1. August 1 Seite 5

6 Motivation Numerical methods undulator (~ 1 m) S. Reiche, FEL simulations: history, status and outlook, FEL 1, Malmö, 1 1 orders of magnitude bunch (~ 1 µm) slice (~.1 nm) approximations are neccecary λ Wellenlänge FEL numerical algorithms Igor Zagorodnov ICAP 1. August 1 Seite 6

7 FEL Physics in 1D Motion of one electron in an undulator θ u = K γ z λ u = π k u Field on the axis K = - undulator parameter m e ck u γ - electron energy By = B sin( ku z ) electron trajectory eb Igor Zagorodnov ICAP 1. August 1 Seite 7

8 FEL Physics in 1D Energy exchange in FEL v < c an electron is slower than the EM field z v E x EM field ~ plane wave E z t E kz t x (, ) = cos( ω ) Igor Zagorodnov ICAP 1. August 1 Seite 8

9 FEL Physics in 1D Energy exchange in FEL v < c an electron is slower than the EM field z Igor Zagorodnov ICAP 1. August 1 Seite 9

10 FEL Physics in 1D Energy exchange in FEL v < c an electron is slower than the EM field z The electron has to be slower exactly by one wave length. λ u 1 K λ = + γ Igor Zagorodnov ICAP 1. August 1 Seite 1

11 FEL Physics in 1D Longitudinal equations of motion (1D model) d dt ψ = d η = dt k c η u ek[ JJ ] E m e cγ r cosψ ψ = ( k + ku ) z ωt, [ JJ ] ~ 1 γ γ η = r γ r 4π π π 4π ψ s Igor Zagorodnov ICAP 1. August 1 Seite 11

12 FEL Physics in 1D Low gain FEL γ γ r I [ka] z = m E x () = 3MV/m r beam =.1 mm λ u = 7mm ψ K =1. λ = 6 nm I = 1.6 ka peak ψ Igor Zagorodnov ICAP 1. August 1 Seite 1

13 FEL Physics in 1D Low gain FEL γ γ r.5 z = 15m I [ka] ψ ψ Igor Zagorodnov ICAP 1. August 1 Seite 13

14 FEL Physics in 1D Low gain FEL γ γ r.5 microbunching I [ka] ψ ψ Igor Zagorodnov ICAP 1. August 1 Seite 14

15 FEL Physics in 1D Microbunching i j = R (% j ) = j + R( % j ( z) e ψ ) z z 1 s λ K.N. Ricci ant T.I Smith, PR- STAB 3, 381 () experimental evidence of microbunching in Stanford Igor Zagorodnov ICAP 1. August 1 Seite 15

16 FEL Physics in 1D High gain FEL P rad ~ N el P rad ~ N el experimental data (FLASH) E[µJ] z[ m ] the amplification is very high λ[nm] W. Ackermann et al, Nature Photonics 1, 336 (7) Igor Zagorodnov ICAP 1. August 1 Seite 16

17 FEL Physics in 1D Field equation The EM field variation has to be taken into account 1 j 1 µ E = + ρ c t t ε ( z t ) E( z, t) = R E % (, ) slowly-varying envelope approximation ( ) E% x( z, t) Ex( z) e i kz ω = % t E% x ( z) << k E% x ( z) 1 K + ik + E% = ikµ c % j z c t γ 1 µ ck E z j 4γ % x ( ) = % 1 Igor Zagorodnov ICAP 1. August 1 Seite 17

18 FEL Physics in 1D Longitudinal equations of motion d n ku n, n 1,,... N dz ψ = η = d ek[ JJ ] iψ η ( n n = R E % xe ) dz m c Field equation µ ck d E % x( z ) = % j 1 dz e γ r 4γ Igor Zagorodnov ICAP 1. August 1 Seite 18

19 FEL Physics in 1D High gain FEL P[GW] 4 z = m E () =.1MV/m x γ γ r z[m] I [ka] ψ ψ Igor Zagorodnov ICAP 1. August 1 Seite 19

20 FEL Physics in 1D High gain FEL P[GW] 4 z = m γ γ r z[m] I [ka] ψ ψ Igor Zagorodnov ICAP 1. August 1 Seite

21 FEL Physics in 1D High gain FEL P[GW] 4 z = 1m γ γ r z[m] I [ka] ψ ψ Igor Zagorodnov ICAP 1. August 1 Seite 1

22 FEL Physics in 1D High gain FEL P[GW] 1 saturation linear regime z[m] Igor Zagorodnov ICAP 1. August 1 Seite

23 Mathematical Model in 3D 3D Effekte FEL codes transverse beam dynamics space charge effects beam properties (emittance, energy spread etc.) EM field properties undulator properties Igor Zagorodnov ICAP 1. August 1 Seite 3

24 Numerical Methods 3D FEL codes used at DESY Equations of motion EM field Boundary conditions FAST (M.Yurkov) Integral representation Genesis 1.3 (S.Reiche) Runge-Kutta finite-difference, alternating direction ALICE (I.Zagorodnov) leap-frog finite difference, Neumann free space Dirichlet free space with PML the codes are parallelized a cluster with 36 processors is used 7:5:16 Igor Zagorodnov ICAP 1. August 1 Seite 4

25 Mathematical Model in 3D Equations of motion ( ( ) ).5 H ( r, p, t) = c m c + p + ea eϕ change of independent variable 1+ p + a + ( p, a ) H( r, ct; p, pt ; z) = γ 1 + a z γ wiggler-period-averaging (helical undulator) T.M. Tran, J.S.Wurtele, Review of free-electronlaser (FEL) simulation. techniques, Phys. Reports 195 (199) 1.5 p = p mc a = A e mc 1+ p + a H( r, ct; p, pt ; z) = γ 1 + a γ a = K + ask sin( ψ + ϕs) + as.5 z a s Ee mcω Igor Zagorodnov ICAP 1. August 1 Seite 5

26 Mathematical Model in 3D Equations of motion longitudinal dψ 1+ p + K = kw dz γ dγ ek e = R( Ee % ) dz m c mc e iψ n γ r transverse EM field longitudinal space charge E z transverse (slow) d r d z = p γ β d px K kx eg = + x dz γβ z z mc d py K ky eg = y dz γβ z mc Igor Zagorodnov ICAP 1. August 1 Seite 6

27 Mathematical Model in 3D Field equations transverse field (slowly-varying envelope approximation) r E(, t) = E% r (, t)exp i kz t + c. c. ( ω ) ( ) 1 K + ik + E% = ikµ c % j z c t γ 1 longitudinal space charge (on λ scale) E = E + ie x y n ( k + k ) in( k + k ) = w ( n) w ( n) Ez ρ γ z εγ z ψ = ( n) in Ez ( r, ) Ez ( r, z) e ψ Igor Zagorodnov ICAP 1. August 1 Seite 7

28 Mathematical Model in 3D Normalized Equations equations of motion dψ ˆ B ( C ˆ η xˆ yˆ ) dzˆ d ˆ η = uˆ cos( ψ + ϕ ) ˆ s Ez dzˆ = + + ˆ = ( ˆ ) x + field equations 1 N jk x k gˆ xˆ ( ˆ ) y yˆ = k gˆ yˆ [ π ψ ψ ψ ψ ] ˆ ˆ () ˆ z = Ez Λ p sgn( i ) ( i ) N jk i= 1 E 1 ˆ d u ˆ ˆ z ˆ a ˆ z ˆ + ib dzˆ (1) ( r, ) = ( r, ) Zagorodnov I., Dohlus M., Numerical FEL studies with a new code ALICE, FEL9, Liverpool, 9. Igor Zagorodnov ICAP 1. August 1 Seite 8

29 Numerical Methods Equations of motion leap-frog scheme for the longitudinal equations ψ j+ ψ.5 j.5 ˆ B = ˆ η ( ˆ ˆ ) j + C j x j + y j j = zˆ s s ˆ η ˆ ˆ ˆ j + 1 η j u j u j ϕ j+ 1 + ϕ j = cos ψ ˆ j Ez, j+.5, z matrix formalism for the transverse equations 1: N z xˆ ˆ j+ 1 x j M x xˆ = ˆ j+ 1 x j yˆ yˆ. j + 1 j y ˆ = M y ˆ j + 1 y j Igor Zagorodnov ICAP 1. August 1 Seite 9

30 Numerical Methods Field equation Perfectly Matched Layer 1 1 F. Collino, Journal of Computational Physics 131, 164 (1997) ( m) im uˆ( r, z, ) u ( r, z) e ϕ r u a r, m d + m = azimuthal expansion ϕ = ib r r r dz ( ) (1)( m) Igor Zagorodnov ICAP 1. August 1 Seite 3

31 Numerical Methods Perfectly Matched Layer (PML) Im r%,, σ >, ( r), PML, Re r% i r% = r + d r ( ) B σ ξ ξ σ ( r), r r, =, >, r. > - a deformed contour in the complex plane 1 1 ib r% r% r% r% dz m d ( m) (1)( m) r% u a + = r,. Igor Zagorodnov ICAP 1. August 1 Seite 31

32 Numerical Methods Neumann implicit scheme with PML n n n + 1 = n q q + 1 q q q q 1 q c u b u a u f a The matrix, of the system has only three diagonals q and we can use the sweep method = z, 1 r% j.5 % % % % %, 4 ib r ( r r )( r r ) j j +.5 j.5 j j 1 b = (1 a c ) q q q z m ib r% j c q = z 1 r% j ib r% ( r% r% )( r% r% ) j j +.5 j.5 j + 1 j complex numbers! Igor Zagorodnov ICAP 1. August 1 Seite 3

33 Numerical Methods PML performance.8.6 Field power [a.u.] D max r ˆ = 3.3 ˆr ẑ ˆPML PML max r ˆ = D ˆr max = 1 ˆr Igor Zagorodnov ICAP 1. August 1 Seite 33

34 Quiet Start and Shot Noise Quiet start in FEL amplifier P[ GW] 4 3 Genesis v1. (N=6e4) Genesis v1. (N=3e4) 19% Alice (N=6e4) Alice (N=3e4) The difference in saturation length is 7 %. The difference in power gain is 19 %. z[m] Igor Zagorodnov ICAP 1. August 1 Seite 34

35 Quiet Start and Shot Noise Quiet Start? n n ALICE Genesis v y / σ y y / σ y What is the reason? clustering Igor Zagorodnov ICAP 1. August 1 Seite 35

36 Quiet Start and Shot Noise Quiet Start? Uniform William H. Press et al, Numerical Recipes in Fortran. The Art of Scientific Computing, 199 Normal The polar form of Box-Muller algorithm (in Genesis, ASTRA) maps the quiet uniform distribution in a clustered normal distribution. Igor Zagorodnov ICAP 1. August 1 Seite 36

37 Quiet Start and Shot Noise Quiet Start? ALICE Genesis v1. x [a.u.] y [a.u] y [a.u] inverse error function Box-Mueller algorithm transformation Igor Zagorodnov ICAP 1. August 1 Seite 37

38 Quiet Start and Shot Noise Properties of the normal distribution % δ = Genesis v1. ALICE σ 4 4 σ σ 4 1% N δ x, % δ y, % Igor Zagorodnov ICAP 1. August 1 Seite 38

39 Quiet Start and Shot Noise New Genesis v. vs. ALICE P [ GW] 3.5 Genesis v. (N=6e4) Alice (N=6e4) z [m] Igor Zagorodnov ICAP 1. August 1 Seite 39

40 Quiet Start and Shot Noise Convergence P [ GW] at saturation Genesis v. ALICE Genesis v1. N δ = σ % σ 4 4 σ Genesis v. ALICE 1% Genesis v1. N Genesis 1.: Genesis.: ALICE: Hammersley and Box-Mueller Hammersley and the inverse error function Sobol and the inverse error function Igor Zagorodnov ICAP 1. August 1 Seite 4

41 Quiet Start and Shot Noise Shot noise algorithm b b ( n) ( n) b Ne 1 = e N = e m= 1 ( n) 1 = π 4N N e e inψ m beamlets and variation in position (W.Fawley) variations in charges (B.McNeil et al.) N << N e x ψ beamlets - a set of M particles with the same position N b beamlets with M particles each. x = ( x, y, p, p, γ ) x y ( j) k b = ; k = 1: N, j = 1: M b Igor Zagorodnov ICAP 1. August 1 Seite 41

42 Quiet Start and Shot Noise Shot noise algorithm Small position variations m= M δθ = a cos( mθ ) + b sin( mθ ) j m j m j m= 1 a = b =, m M?! m, rms m, rms Nbm W.M. Fawley, Phys. Rev. STAB 5 () 771 A more careful analysis for the highest harmonic m=m yields 1 am, rms = bm, rms = N M b j = 1: M Igor Zagorodnov ICAP 1. August 1 Seite 4

43 Time Dependent Simulations Slice parameters are extracted from Gun-to-Undulator simulations γ γ ε ε β β x y x ' y ' α α I x y x y x y ε [ µ m] x slice emittance Q = 1 nc I [ka].5 current Q = 1 nc Q = pc s s σ s 1.5 Q = pc -5 5 s [µm] Igor Zagorodnov ICAP 1. August 1 Seite 43

44 Time Dependent Simulations Field equation t = t + z c 1 K + ik + E% ( z, t) = ikµ c % j1 ( z, t) z c t γ d z z + E% z, t + = F z, t + ik dz c c t j ( z n +.5) - the time when slice j reaches position z n+.5 zn z n + z.5 n z.5 z E% zn+ 1, t j ( zn+.5 ) + E% zn, t j ( zn.5) d + c c E% ( zn+.5, t j ( zn+.5 )) = + O() dz z Igor Zagorodnov ICAP 1. August 1 Seite 44

45 Time Dependent Simulations zn z n + z.5 n z t j ( zn+.5 ) = t j.5 ( zn ) c j j 1 j j 1 E% n+ 1 + E% n E% n+ 1 E% n + = ik z (, ( )) E% E% z t z j = n n j +.5 n.5 z t j ( zn+.5) + = t j+.5( zn+ 1) c field of the current slice j field of the previous slice j-1 F( z, t ( z )) n+,5 j n+,5 Igor Zagorodnov ICAP 1. August 1 Seite 45

46 Time Dependent Simulations Parallel algorithm Slice for slice" we start from the last slice and track the particles of this slice through the undulator; the radiated EM field is saved. then we track the next slice in the radiation field of the previous slices Igor Zagorodnov ICAP 1. August 1 Seite 46

47 Time Dependent Simulations SASE in the European XFEL W [mj] P [GW] ALICE Genesis ALICE Genesis z [m] I.Zagorodnov, Ultra-short low charge operation at FLASH and the European XFEL, FEL 1, Malmö, 1 s [µm] Igor Zagorodnov ICAP 1. August 1 Seite 47

48 Problems and Challenges Comparison with SPARC FEL experiment L. Giannesi et al, PR STAB 14, 671(11) Medusa is a code based on non-averaged equations Igor Zagorodnov ICAP 1. August 1 Seite 48

49 Problems and Challenges Limitations of the considered FEL model fast slalom motion is not modeled only forward propagating field is considered macroparticles are locked in the slice with periodic boundary conditions only low order harmonics are simulated correctly macroscopic space charge, bunch shape changes are not modeled narrow frequency bandwidth near the resonance frequency, narrow energy spread Igor Zagorodnov ICAP 1. August 1 Seite 49

50 Problems and Challenges Table-Top-FEL M.Fuchs et al, Nature Physics 5, 86(9) strong space charges, large energy spread fast bunch shape change charge or macroparticles redistribution macroscopic space charge equation non-averaged modell Igor Zagorodnov ICAP 1. August 1 Seite 5

51 Summary FEL codes based on averaged equations are checked by experiments new FEL schemes (Table Top FEL, Echo-Enabled High- Harmonic Generation etc.) require to consider more physics FEL codes based on non-averaged FEL equations, with macroscopic space charge are under development H.P. Freund, S.G. Biedron, S.V. Milton, IEEE J. Quantum Electron. 36 () 75. (Medusa code) C.K.W. Nam, P. Aitken, and B.W.J. McNeil, Unaveraged threedimensional modelling of the FEL, FEL 8, Gyeongju, Korea, 8 Igor Zagorodnov ICAP 1. August 1 Seite 51