A Vlasov-Maxwell Solver to Study Microbunching Instability in the First Bunch Compressor System

Transcription

1 Microbunching in the First Bunch Compressor System / Gabriele Bassi Page 1 A Vlasov-Maxwell Solver to Study Microbunching Instability in the FERMI@ELETTRA First Bunch Compressor System Gabriele Bassi University of Liverpool and the Cockcroft Institute, UK Collaborators James A. Ellison, Klaus Heinemann, Math and Stat, UNM, Albuquerque, USA Robert Warnock, SLAC, Stanford University, Stanford, USA 1. Self Consistent Vlasov-Maxwell Treatment 2. Field Calculation 3. Self Consistent Monte Carlo Method 4. Microbunching Instability Studies Thanks to Office of Naval Research Global for support

2 Microbunching in the First Bunch Compressor System / Gabriele Bassi Page 2 Self Consistent Vlasov-Maxwell Treatment I E = B Y = X h Y R r ( s ) Z E = B Y =

3 Microbunching in the First Bunch Compressor System / Gabriele Bassi Page 3 Self Consistent Vlasov-Maxwell Treatment II Wave equation in lab frame with 2D planar source: ( 2 Z + 2 X + 2 Y 2 u)e = H(Y )S(R, u), E(R, Y = ±g, u) =. where u = ct, E(R, Y, u) = (E Z, E X, B), R = (Z, X). Vlasov equation in beam frame: where f s κ(s)xf z + F z f pz + p x f x + [κ(s)p z + F x ]f px = F z = e E, vēv F x = e Ē β 2[ X (s)e Z + Z (s)e X + vb)], and V = v(t(s) + p x n(s)), E = (E Z, E X ) and B are evaluated at R = R(s) + xn(s) and u = (s z)/ β.

4 Microbunching in the First Bunch Compressor System / Gabriele Bassi Page 4 Field Calculation (Lab Frame) E(R, u) := E(R,, u) = g g H(Y )E(R, Y, u)dy. averaged field computed much more quickly E(R, u) = 1 2π k= ( 1) k (1 δ k 2 ) u kh dv π π dθ S( ˆR, v, k) where ˆR = R + (u v) 2 (kh) 2 (cosθ, sinθ). Issues localization in θ (angular size of the beam) for v u kh and in v delicate calculation (must be done cum grano salis) θ integration: superconvergent trapezoidal rule v integration: adaptive Gauss-Kronrod rule

5 Microbunching in the First Bunch Compressor System / Gabriele Bassi Page 5 Beam to Lab Charge/Current Density Transformation To solve Maxwell equations in lab frame must express lab frame charge/current density in terms of beam frame phase space density To a good approximation lab frame charge/current densities are ρ L (R, Y, u) = H(Y )ρ(r, βu), J L (R, Y, u) = βch(y )[ρ(r, βu)t(βu + z) + τ(r, βu)n(βu + z)], ρ(r, s) = Q dp z dp x f(ζ, s), τ(r, s) = Q dp z dp x p x f(ζ, s), where ζ = (z, p z, x, p x ) Remark: subtlety in the change of independent variable u=ct s Derivation to be published in a forthcoming paper

6 Microbunching in the First Bunch Compressor System / Gabriele Bassi Page 6 Self Consistent Monte Carlo Method Outline and comparison with PIC for Vlasov-Poisson (VP) system fron s to s + s From scattered beam frame points at s smooth/global Lab frame charge/current density via a 2D Fourier method (Charge deposition (+ filtering) in VP PIC). 1D Example: 1D orthogonal series estimator of f(x), x [,1] f J (x) := J θ j φ j (x), θ j = j= 1 φ j (x)f(x)dx, φ (x) = 1,φ j (x) = 2 cos(πjx),j = 1,2,... According to the fact that f(x) is a probability density θ j = E{I {X [,1]} φ j (X)}, therefore a natural estimate is ˆθj := 1 N N I {Xn [,1]}φ j (X n ) n=1 Calculate fields at s from history of Lab Frame charge/current density using our field formula (Solve Poisson Equation in VP PIC) Use fields at s to move the phase space points to s + s (Same in VP PIC)

7 Microbunching in the First Bunch Compressor System / Gabriele Bassi Page 7 Microbuching in FERMI@ELETTRA First Bunch Compressor Microbunching can cause an instability which degrades beam quality This is a major concern for free electron lasers where very bright electron beams are required FERMI@ELETTRA first bunch compressor system proposed as a benchmark for testing codes at the Workshop on the Microbunching Instability I in Trieste.

8 Microbunching in the First Bunch Compressor System / Gabriele Bassi Page 8 FERMI@ELETTRA First Bunch Compressor Parameters Layout first bunch compressor system Table 1: Chicane parameters and beam parameters at first dipole Parameter Symbol Value Unit Energy reference particle E r 233 MeV Peak current I 12 A Bunch charge Q 1 nc Norm. transverse emittance γǫ 1 µm Alpha function α Beta function β 1 m Linear energy chirp u /m Uncorrelated energy spread σ E 2 KeV Momentum compaction R m Radius of curvature ρ 5 m Magnetic length L b.5 m Distance 1st-2nd, 3rd-4th bend L m Distance 2rd-3nd bend L 2 1 m

9 Microbunching in the First Bunch Compressor System / Gabriele Bassi Page 9 FERMI@ELETTRA First Bunch Compressor I ρ(z n,x n,s) z n x n Initial charge density in norm. coordinates for A=.5, λ = 1µm. Init. phase space density = (1 + A cos(2πz/λ))µ(z)ρ c (z, p z )g(x, p x ).

10 Microbunching in the First Bunch Compressor System / Gabriele Bassi Page 1 Gain factor Numeric Analytic gain λ(µm) Gain factor := b(k f, s f )/b(k, ), where b(k, s)= dz exp( ikz)f(z, s) and k f = k /(1 + ur 56 (s f )) for a given initial wavelength λ = 2π/k. Here the compressor factor C = 1/(1 + ur 56 (s f )) = 3.54, s f = 8.29m.

11 Microbunching in the First Bunch Compressor System / Gabriele Bassi Page 11 FERMI@ELETTRA First Bunch Compressor II.1 A= A=.5, λ=1µm -1e mean power path length (m) Mean power

12 Microbunching in the First Bunch Compressor System / Gabriele Bassi Page 12 FERMI@ELETTRA First Bunch Compressor III 9e-5 8e-5 A= A=.5, λ=1µm 7e-5 x-emittance (m-rad) 6e-5 5e-5 4e-5 3e-5 2e-5 1e-5 A=: 1.48 mm-mrad A=.5: λ=1µm: 1.5 mm-mrad path length (m) x-emittance

13 Microbunching in the First Bunch Compressor System / Gabriele Bassi Page 13 FERMI@ELETTRA First Bunch Compressor IV ρ(z n,x n,s) z n x n Charge density in normalized coordinates at s = 8.29m for λ = 2µm.

14 Microbunching in the First Bunch Compressor System / Gabriele Bassi Page 14 FERMI@ELETTRA First Bunch Compressor V ρ(z n,x n =,s) 2.5 s=m s=8.29m z n Section of charge density in norm. coord. at s = 8.29m for λ = 2µm

15 Microbunching in the First Bunch Compressor System / Gabriele Bassi Page 15 FERMI@ELETTRA First Bunch Compressor VI ρ(z n,x n,s) z n x n Charge density in norm. coordinates at s = 8.29m for λ = 1µm

16 Microbunching in the First Bunch Compressor System / Gabriele Bassi Page 16 FERMI@ELETTRA First Bunch Compressor VII ρ(z n,x n =,s) 2.5 s=m s=8.29m Comparison of ρ(z n,, s) at s = 8.29m and s = m for λ = 1µm z n

17 Microbunching in the First Bunch Compressor System / Gabriele Bassi Page 17 FERMI@ELETTRA First Bunch Compressor VIII ρ(z n,x n,s) z n x n Charge density in norm. coordinates at s = 8.29m for λ = 8µm

18 Microbunching in the First Bunch Compressor System / Gabriele Bassi Page 18 FERMI@ELETTRA First Bunch Compressor IX ρ(z n,x n =,s) 2.5 s=m s=8.29m z n Comparison of ρ(z n,, s) at s = 8.29m and s = m for λ = 8µm

19 Microbunching in the First Bunch Compressor System / Gabriele Bassi Page 19 FERMI@ELETTRA First Bunch Compressor X.3 E t z n x n E t in normalized coordinates at s=8.29m for λ = 2µm.

20 Microbunching in the First Bunch Compressor System / Gabriele Bassi Page 2 FERMI@ELETTRA First Bunch Compressor XI E t x n z n Enlargement of E t in norm. coord. at s=8.29m for λ = 2µm.

21 Microbunching in the First Bunch Compressor System / Gabriele Bassi Page 21 FERMI@ELETTRA First Bunch Compressor XII.3 E t x n z n E t in normalized coordinates at s=8.29m for λ = 1µm.

22 Microbunching in the First Bunch Compressor System / Gabriele Bassi Page 22 FERMI@ELETTRA First Bunch Compressor XIII E t x n z n Enlargement of E t in norm. coord. at s=8.29m for λ = 1µm.

23 Microbunching in the First Bunch Compressor System / Gabriele Bassi Page 23 FERMI@ELETTRA First Bunch Compressor XIV.3 E t z n x n E t in norm. coord. at s=8.29m for λ = 8µm.

24 Microbunching in the First Bunch Compressor System / Gabriele Bassi Page 24 FERMI@ELETTRA First Bunch Compressor XV E t x n z n Enlargement of E t in norm. coord. at s=8.29m for λ = 8µm.

25 Microbunching in the First Bunch Compressor System / Gabriele Bassi Page 25 Main Issues and Accomplishments FERMI@ELETTRA microbunching studies at λ 8µm: - Very small effect of µbi on mean power and transverse emittance - Gain factor at short wavelengths indicates weaker µbi than predicted by analytical formula - Simulations done at the HPC at UNM and on NERSC at LBNL, typical runs on NERSC: N procs = 2-7, N particles = , 1-2 hours of CPU time Storage/computational cost very important - Analytical work + state of the art numerical techniques: integration, interpolation, density estimation, quasirandom generator - Parallel computing Delicacy of field calculation, support of charge/phase space density

26 Microbunching in the First Bunch Compressor System / Gabriele Bassi Page 26 Future Work Study wavelengths shorter than λ = 8µm and different amplitudes of the initial modulation Complete studies for benchmark microbunching instability including RF cavities A paper will be submitted shortly to PRSTAB EPAC8 Special Issue