Modelling the Microbunching Instability for Bunch Compressor Systems

Transcription

1 Modelling the Microbunching Instability for Bunch Compressor Systems Gabriele Bassi Page Modelling the Microbunching Instability for Bunch Compressor Systems Gabriele Bassi Department of Physics, University of Liverpool and the Cockcroft Institute, UK Collaborators Jim Ellison, Klaus Heinemann, Dept. of Math and Stats, University of New Mexico, Albuquerque, NM, USA Robert Warnock, SLAC National Accelerator Laboratory, Menlo Park, CA, USA. Motivation. Self Consistent Vlasov-Maxwell Treatment 3. Microbunching Instability Studies for the Bunch Compressor System 4. Discussion of Some Approximation Schemes Workshop on the Microbunching Instability III, 4-6 March, INFN-LNF, Frascati, Italy

2 Modelling the Microbunching Instability for Bunch Compressor Systems Gabriele Bassi Page Self Consistent Vlasov-Maxwell Treatment for Sheet Beam in Beam Frame Sheet Beam model: f(z, X, Y, P Z, P X, P Y ; u ) = δ(y )δ(p Y )f L (Z, X, P Z, P X ; u ). Approximate beam frame equations of motion z = κ(s)x, x = p x, p z = F z(z, x; s) + p z F z (z, x; s), p x = κ(s)p z + F x (z, x; s), where the self forces are F z = q P r c E (R(s, x); s z q ) t(s), F z β r P r c E (R(s, x); s z β r F x = q P r c [E (R(s, x); s z ) n(s) cb Y (R(s, x); s z )]. β r β r where R(s, x) = R r (s) + xn(s) and R r (s) = (Z r (s), X r (s)) T. ) n(s), Associated Vlasov IVP for the evolution of the beam frame phase space density s f B + r r f B + p p f B =, f B (r,p; ) =: f B (r,p). Workshop on the Microbunching Instability III, 4-6 March, INFN-LNF, Frascati, Italy

3 Modelling the Microbunching Instability for Bunch Compressor Systems Gabriele Bassi Page 3 Field Calculation Formula in free space F L (R r (s) + M(s)r, s/β r ) = 4π s/βr u i dv θm θ m dθ S[ R(θ, v;r, s), v], where R(θ, v;r, s) = R r (s) + M(s)r + (s/β r v)e(θ), e(θ) = (cosθ, sin θ) T and M(s) = [t(s),n(s)]. θ integration: superconvergent trapezoidal rule (localization in θ for v s/β r ) v integration: adaptive Gauss-Kronrod rule (non uniform behavior in v) For more details see: G. Bassi, J.A. Ellison, K. Heinemann and R. Warnock, PRSTAB, 874 (9) Workshop on the Microbunching Instability III, 4-6 March, INFN-LNF, Frascati, Italy

4 Modelling the Microbunching Instability for Bunch Compressor Systems Gabriele Bassi Page 4 Microbunching in FERMI@Elettra First Bunch Compressor Microbunching can cause an instability which degrades beam quality, i.e. can cause an increase in energy spread and emittance This is a major concern for free electron lasers where very bright electron beams are required We discuss numerical results for the FERMI@Elettra first bunch compressor system. See G. Bassi, J.A. Ellison, K. Heinemann and R. Warnock, PRSTAB, 874 (9) This system was proposed as a benchmark for testing codes at the first Workshop on Microbunching Instability held in Trieste in 7 Layout first bunch compressor system Workshop on the Microbunching Instability III, 4-6 March, INFN-LNF, Frascati, Italy

5 Modelling the Microbunching Instability for Bunch Compressor Systems Gabriele Bassi Page 5 FERMI@Elettra First Bunch Compressor Parameters Table : Chicane parameters and beam parameters at first dipole Parameter Symbol Value Unit Energy reference particle E r 33 MeV Peak current I A Bunch charge Q nc Norm. transverse emittance γǫ µm Alpha function α Beta function β m Linear energy chirp h -.6 /m Uncorrelated energy spread σ E KeV Momentum compaction R m Radius of curvature ρ 5 m Magnetic length L b.5 m Distance st-nd, 3rd-4th bend L.5 m Distance rd-3nd bend L m Workshop on the Microbunching Instability III, 4-6 March, INFN-LNF, Frascati, Italy

6 Modelling the Microbunching Instability for Bunch Compressor Systems Gabriele Bassi Page 6 Initial D Spatial Density Initial spatial density in grid coordinates for A=.5, λ = µm. Init. phase space density = ( + A cos(πz/λ ))µ(z)ρ c (p z hz)g(x, p x ). Workshop on the Microbunching Instability III, 4-6 March, INFN-LNF, Frascati, Italy

7 Modelling the Microbunching Instability for Bunch Compressor Systems Gabriele Bassi Page 7 D Spatial Density and Longitudinal Force Fz at s = sf,xn,s) z e z e z z λ = µm (top left), λ = µm (top right), λ = µm (bottom left), λ = µm (bottom right) Workshop on the Microbunching Instability III, 4-6 March, INFN-LNF, Frascati, Italy

8 Modelling the Microbunching Instability for Bunch Compressor Systems Gabriele Bassi Page 8 Longitudinal Density s=s f : self fields OFF s= self fields OFF self fields ON.5.5 ρ(z,s) (/mm) ρ(z,s) (/mm) z (mm) z (mm) self fields OFF self fields ON self fields OFF self fields ON.5.5 ρ(z,s) (/mm) ρ(z,s) (/mm) z (mm) z (mm) λ = µm (top left), λ = µm at s = s f (bottom left), λ = µm at s = s f (top right), λ = 8µm at s = s f (bottom right). Workshop on the Microbunching Instability III, 4-6 March, INFN-LNF, Frascati, Italy

9 Modelling the Microbunching Instability for Bunch Compressor Systems Gabriele Bassi Page 9 Longitudinal Density (con t).5.5 ρ(z,s) (/mm) ρ(z,s) (/mm) z (mm) z (mm).5.5 ρ(z,s)(/mm) ρ(z,s) (/mm) z (mm) z (mm) λ = µm at s = s f (top left), λ = 6µm at s = s f (bottom left), λ = µm at s = s f (top right), λ = 4µm at s = s f (bottom right). Workshop on the Microbunching Instability III, 4-6 March, INFN-LNF, Frascati, Italy

10 Modelling the Microbunching Instability for Bunch Compressor Systems Gabriele Bassi Page Gain factor Numeric: ratio of values at k and C f k Analytic: CSR on Numeric: ratio of peak values Analytic: CSR off 3 Gain λ (µm) Gain := ρ(k f, s f )/ ρ(k, ), ρ(k, s) = dz exp( ikz)ρ(z, s) and k f = C(s f )k for λ = π/k. Here C(s f ) = /( + hr 56 (s f )) = 3.54, s f = 8.9m. H. Huang and K. Kim, PRSTAB 5, 744, 993 (); S. Heifets, G. Stupakov and S. Krinsky, PRSTAB 5, 644 (9); G. Bassi, J.A. Ellison, K. Heinemann and R. Warnock, PRSTAB, 874 (9). Workshop on the Microbunching Instability III, 4-6 March, INFN-LNF, Frascati, Italy

11 Modelling the Microbunching Instability for Bunch Compressor Systems Gabriele Bassi Page Spectra Longitudinal Density λ = µm (k = 683 /m), Gain =.649/.48 =.669 λ = 8 µm (k = /m), Gain =.6/.48 =.5. s=8.8 m, A=.5 s= m, A=.5. s=8.8 m, A=.5 s= m, A= ρ(k,s). (683,.48) (74,.649) ρ(k,s). (78534,.48) (7845,.6) k (/m) k (/m) λ = 6 µm (k = 47 /m), Gain =.593/.48 =.387 λ = 4 µm (k = 577, /m), Gain =.9/.48 =.734. s=8.8 m, A=.5 s= m, A=.5. s=8.8 m, A=.5 s= m, A= ρ(k,s). (47,.48) (3735,.593) ρ(k,s). (577,.48) (55685,.9) k (/m) k (/m) Workshop on the Microbunching Instability III, 4-6 March, INFN-LNF, Frascati, Italy

12 Modelling the Microbunching Instability for Bunch Compressor Systems Gabriele Bassi Page Mean Power, Transverse Emittance and D Longitudinal Force F z vs D Wake mean power. -e A= A=.5, λ=µm x-emittance (m-rad) 9e-5 8e-5 7e-5 6e-5 5e-5 4e-5 3e-5 A= A=.5, λ=µm -.7 e-5 A=:.48 mm-mrad path length (m) e-5 A=.5: λ=µm:.5 mm-mrad path length (m) D D at x = Mean power (top left), transverse emittance (top right), D longitudinal force F z (section at x = ) vs. D (steady state CSR wake) at s = s f (bottom left). D steady state CSR wake: W(z, s)= Nr e γ(3r) /3 z λ(z,s). (z z ) /3 z Workshop on the Microbunching Instability III, 4-6 March, INFN-LNF, Frascati, Italy

13 Modelling the Microbunching Instability for Bunch Compressor Systems Gabriele Bassi Page 3 Discussion Microbunching Studies for FERMI@Elettra BC FERMI@Elettra microbunching studies at λ 4µm: - Very small effect of µbi on mean power and transverse emittance - Gain factor at long wavelengths shows breakdown coasting beam assumption - Gain factor at short wavelengths indicates deviations from analytical gain formula Work in progress and future work: - Study wavelengths shorter than λ = 4µm - Study dependence on the amplitude of the initial modulation and on the uncorrelated energy spread - Study initial perturbation with more than one frequency - Study energy modulations Workshop on the Microbunching Instability III, 4-6 March, INFN-LNF, Frascati, Italy

14 Modelling the Microbunching Instability for Bunch Compressor Systems Gabriele Bassi Page 4 Stationary Grid D spatial density without self fields for λ = µm at s =.5sf (top left), s =.75sf (top right), s = sf (bottom left). In the limit σu and σpx stationary grid defined by: z = ( + hr56 (s))z D (s)x x = hd(s)z + x Workshop on the Microbunching Instability III, 4-6 March, INFN-LNF, Frascati, Italy

15 Modelling the Microbunching Instability for Bunch Compressor Systems Gabriele Bassi Page 5 Discussion of F z and F x for A = A= A=, F x = A= A=, F z =F x = mean energy loss -.4 mean energy loss path length (m) path length (m).8685 A= A=, F x =.8685 A= A=, F z =F x = energy spread (%) energy spread (%) path length (m) path length (m) Workshop on the Microbunching Instability III, 4-6 March, INFN-LNF, Frascati, Italy

16 Modelling the Microbunching Instability for Bunch Compressor Systems Gabriele Bassi Page 6 Discussion of F z and F x for A = (con t).4 A= A=, F x =.4 A= A=, F z =F x =.... x-emittance (m-rad) 8e-5 6e-5 4e-5 x-emittance (m-rad) 8e-5 6e-5 4e-5 e-5 e path length (m) path length (m) e-6 A= A=, F x = 9e-6 A= A=, F z =F x = 8e-6.8e-6 x-emittance interaction (m-rad).6e-6.4e-6.e-6 x-emittance interaction (m-rad) 7e-6 6e-6 5e-6 4e-6 3e-6 e-6 e path length (m) e path length (m) Workshop on the Microbunching Instability III, 4-6 March, INFN-LNF, Frascati, Italy

17 Modelling the Microbunching Instability for Bunch Compressor Systems Gabriele Bassi Page 7 Comparison with Elegant for A = A= Elegant: A= 8e-5 A= A=, Elegant mean energy loss x-emittance interaction (m-rad) 6e-5 4e-5 e-5.48 mm-mrad Elegant:.5 mm-mrad path length (m) path length (m) Courtesy of Deepa Angal-Kalinin Preliminary results, planning to calculate gain factor Workshop on the Microbunching Instability III, 4-6 March, INFN-LNF, Frascati, Italy

18 Modelling the Microbunching Instability for Bunch Compressor Systems Gabriele Bassi Page 8 D longitudinal Force F z for A = s=.5m (exit first magnet) s=3m (entrance second magnet) x z x z s=3.5m (exit second magnet) s=4m (mid second drift) x z x z Workshop on the Microbunching Instability III, 4-6 March, INFN-LNF, Frascati, Italy

19 Modelling the Microbunching Instability for Bunch Compressor Systems Gabriele Bassi Page 9 D Longitudinal Force F z for A = (con t) s=4.5m (entrance third magnet) s=5m (exit third magnet) x z x z s=6.5m (mid third drift) s=8m (exit fourth magnet) x z x z Workshop on the Microbunching Instability III, 4-6 March, INFN-LNF, Frascati, Italy

20 Modelling the Microbunching Instability for Bunch Compressor Systems Gabriele Bassi Page Fast Calculation: Kick Particles According to Fields at x = A= A=, APPX e-6 A= A=, APPX.8e-6 mean energy loss x-emittance interaction (m-rad).6e-6.4e-6.e e path length (m) path length (m) Fast calculation: kick particles at s according to fields calculated at (z, ): F z (z, x, s) F z (z,, s). Computational cost for simulations done on Franklin at NERSC: For λ = 5µm, N procs = 6, N particles = 5 8, CPU time: - D calculation: 8 hours - Fast calculation: minutes Workshop on the Microbunching Instability III, 4-6 March, INFN-LNF, Frascati, Italy

21 Modelling the Microbunching Instability for Bunch Compressor Systems Gabriele Bassi Page Fast Calculation: D Longitudinal Force F z for A = at s = 8m s=8m (exit fourth magnet) z F z (z,,s) x Workshop on the Microbunching Instability III, 4-6 March, INFN-LNF, Frascati, Italy

22 Modelling the Microbunching Instability for Bunch Compressor Systems Gabriele Bassi Page Fast Calculation: D Longitudinal Force F z for A = at s = 5m s=5m (exit third magnet) F z (z,,s) z x Workshop on the Microbunching Instability III, 4-6 March, INFN-LNF, Frascati, Italy

23 Modelling the Microbunching Instability for Bunch Compressor Systems Gabriele Bassi Page 3 Fast Calculation: Gain Factor Numeric: D Analytic Numeric: fast calculation Gain λ (µm) Workshop on the Microbunching Instability III, 4-6 March, INFN-LNF, Frascati, Italy

24 Modelling the Microbunching Instability for Bunch Compressor Systems Gabriele Bassi Page 4 Back-up Slides Workshop on the Microbunching Instability III, 4-6 March, INFN-LNF, Frascati, Italy

25 Modelling the Microbunching Instability for Bunch Compressor Systems Gabriele Bassi Page 5 Fields in Terms of Beam Frame Density and Causality Issue To solve the beam frame equations of motion we need (ignoring shielding) F L (R r (s)+xn(s); (s z)/β r )= dr S[R ; (s z)/β r R (R r (s) + xn(s)) ], R 4π R (R r (s) + xn(s)) To compute this we need ρ L [R ; (s z)/β r R (R r (s) + xn(s)) ], as R varies over the support of ρ L in R, given ρ B ( ; s ) for s s. There is a causality issue here, since the calculation of ρ L requires values ρ B for s slightly outside the range s s. This issue can be easily resolved with the following slowing varying approximation f B (r,p; s) f B (r,p; s + ) where is of the order of the bunch size. Workshop on the Microbunching Instability III, 4-6 March, INFN-LNF, Frascati, Italy

26 Modelling the Microbunching Instability for Bunch Compressor Systems Gabriele Bassi Page 6 Density Estimation: Search for Improvement Cloud in cell charge deposition followed by computation of the Fourier coefficients of the truncated Fourier series by a simple quadrature (already implemented). The computational effort is O(N) + O(N z N x J z J x ), where N is the number of simulated particles, N z and N x are the number of grid points in z ans x respectively, and J z and J x the number of Fourier coefficients in z and x respectively. For N z =, N x = 8, J z = 5 and J x = 5, O(N z N x J z J x ) = O( 9 ). Kernel density estimation using standard kernels like bivariate Gaussians or bivariate compact support kernels (e.g. Epanechnikov kernels). The computational effort is O(NÑzÑx), where N is the number of simulated particles and ÑzÑx is the number of grid points inside the circle of radius h (bandwidth) centered at the scattered particle position z, x. For N = 5 8 and Ñz = Ñx = 4, O(NÑzÑx) = O( ). Wavelets-denoising (G. Bassi, B. Terzić, PAC9) Workshop on the Microbunching Instability III, 4-6 March, INFN-LNF, Frascati, Italy

27 Modelling the Microbunching Instability for Bunch Compressor Systems Gabriele Bassi Page 7 Interaction Picture Interaction picture in beam frame (ζ = (z, p z, x, p x )) to isolate CSR dynamics From F z = F x = = ζ = Φ(s )ζ ζ = Φ( s)f, F = (, F z,, F x ). In component form z = R 56 (s)f z D(s)F x, p z = F z, x = (sd (s) D(s))F z sf x, p x = D (s)f z + F x, where D(s) = s κ(τ)dτ and R 56(s) = s D(τ)κ(τ)dτ. Here the principal solution matrix is R 56 (s) D (s) D(s) sd (s) Φ(s ) = D(s) s D (s). Workshop on the Microbunching Instability III, 4-6 March, INFN-LNF, Frascati, Italy

28 Modelling the Microbunching Instability for Bunch Compressor Systems Gabriele Bassi Page 8 Computational Issues Intensive memory requirement and expensive computational cost: - Typical simulations done on the parallel clusters ENCANTO in New Mexico and NERSC at LBNL: N procs = -, N particles = 7-5 8, few hours of CPU time - Memory requirement: for λ = 5µm store 3D array of dimension 5 8 on master processor (to avoid massive communications between slave processors) To reduce storage/computational cost: - Analytical work + state of the art numerical techniques: integration, interpolation, density estimation - Parallel computing Workshop on the Microbunching Instability III, 4-6 March, INFN-LNF, Frascati, Italy

29 Modelling the Microbunching Instability for Bunch Compressor Systems Gabriele Bassi Page 9 Spectra Longitudinal Density II λ = 3 µm (k = 944 /m), Gain =.43/.4=.8 λ = 3 µm (k = 944 /m), Gain =.43/.4=.8 s= m, A=.5 s= m, A= s=8.8 m, A=.5 s=8.8 m, A= ρ(k,s).4 (944,.4) ρ(k,s).4 (7447,.43) k (/m) k (/m) λ = 6 µm (k = 47 /m), Gain =.36/.76 =.349 λ = 6 µm (k = 47 /m), Gain =.36/.76 =.349 s= m, A=.5 s= m, A= s=8.8 m, A=.5 s=8.8 m, A= ρ(k,s).4 (47,.76) ρ(k,s).4 (373,.36) k (/m) k (/m) Workshop on the Microbunching Instability III, 4-6 March, INFN-LNF, Frascati, Italy