Andreas Kabel Stanford Linear Accelerator Center

Transcription

1 Numerical Calculation of CSR Effects Andreas Kabel Stanford Linear Accelerator Center 1. Motivation 2. CSR Effects 3. Considerations for Numerical Calculations 4. The Simulation Code TraFiC 4 5. Examples 6. Conclusion Snowmass 2001 Andreas Kabel Slide 1

2 Motivation: Bunch Compression p/p l p/p l - e l E Magnetic Chicane Snowmass 2001 Andreas Kabel Slide 2

3 Retarded Fields On curved trajectories: Fields from the tail of the bunch can interact with the head. purely geometrical effect, independent of energy: forces won t scale down (as space charge) σ D R Overtaking condition: R φ Rϕ σ = 2R sin ϕ 2 L Ot 3 24R 2 σ D L2 Ot 8R Collective effects take place ( Coherent Synchrotron Radiation ) Snowmass 2001 Andreas Kabel Slide 3

4 Forces on a Bend From Liénard-Wiechert Potentials: Source Point 1/qE(x, t) = + β n 1-D R Sampling Point ( ) n β γ 2 D 3 R 2 retarded ( n [(n β) β] ) D 3 R B(x, t) = n E(x, t) retarded Coulomb-like forces and acceleration forces, acting both transversally and longitudinally. Forces due to acceleration: instantaneous transverse force longitudinal force with slow build-up Snowmass 2001 Andreas Kabel Slide 4

5 Talman Force Acceleration gives a centrifugal force ( Talman force ): (From: R. Talman, Phys. Rev. Letters 56, 1429 (1986)) Snowmass 2001 Andreas Kabel Slide 5

6 Talman Force Cancellation But: Considering energy changes effected by E v may lead to a cancellation of the Talman force exact cancellation for coasting beam extended in z direction (E. P. Lee, Particle Accelerators 25, 241 (1990)) partial cancellation for coasting 1-D beam (E. P. Lee, Particle Accelerators 25, 241 (1990)) partial cancellation for finite length relativistic bunch (Y. Derbenev, FERMILAB-Conf-96/125) Snowmass 2001 Andreas Kabel Slide 6

7 Transient Fields Longitudinal Field [V/m] Beamline Position [m] head probe particle # tail Snowmass 2001 Andreas Kabel Slide 7

8 Transient Fields Transverse Field [V/m] Beamline Position [m] head probe particle # tail Snowmass 2001 Andreas Kabel Slide 8

9 Transient Fields Similar behavior for exiting the bend: the fields need some time to decay Snowmass 2001 Andreas Kabel Slide 9

10 Emittance Growth Transverse Emittance growth due to longitudinal and transverse forces: η E and E : Correlation matrix: C(s) = x(s)x (s) = M(s) (C 0 (0) s s + M 1 (s 1 ) F (s 1 )F (s 2 ) M 1 (s2 )ds 1 ds s ) + x(0)f (s 1 ) ds 1 + transp. M (s) 0 = M(s)(C 0 + C F F + C F x + C F x)m (s), For purely longitudinal dependence of F : emittance can be minimized by matching C 0 to C F F. For fat bunches: C F x matters: scan parameter space for minimal emittance or use a descent method. Snowmass 2001 Andreas Kabel Slide 10

11 Emittance Growth "Optics" using 1:2: e-05 9e e-05 8e e-05 7e Figure 1: Emittance growth from numerical simulation for a single bend: R = 2m, L = 1m, q = 1nC, E = 40.7MeV, ε normalized = m Snowmass 2001 Andreas Kabel Slide 11

12 Emittance Growth 8.4e e-05 8e e e e e-05 7e Figure 2: Emittance growth from C F F extrapolation at α = 0, β = 10m Snowmass 2001 Andreas Kabel Slide 12

13 Slice Emittance For FEL operation, beam quality is crucial. The FEL process involves only particles within a certain longitudinal range ( slippage length ). This range may be much smaller than the bunch length: Projection to x,x -space x x l For Phase 1 of TTF-FEL, the slippage length l 10µm l final = 250µm Projective emittance misleading for judging beam quality for FEL operation. (But important for optics considerations) Snowmass 2001 Andreas Kabel Slide 13

14 Slice Emittance Growth Slice Emittance: ɛ(s 0 ) = βγ x 2 c x 2 c xx 2 c where q = 1 q i N s i s 0 < l 2 Emittance growth can take place by means of two mechanisms: (ɛ 2 /2) = Nonlinear forces & uncorrelated energy spread {}}{ Φ ( x,x 2 F (x, x ), x 3,... ) + 1 R ( x 2 δx xx δx ) } {{ } Dispersion mismatch Snowmass 2001 Andreas Kabel Slide 14

15 Nonlinear Transverse Phasespace x [rad] (arbitrary offset) Transversal Phase Space of Slice along Beamline unperturbed motion 1nC, no shielding x [m] (arbitrary offset) Snowmass 2001 Andreas Kabel Slide 15

16 Numerics: Granularity Observer From synchrotron radiation theory: critical frequency ω c R γ 3 giving highly peaked fields of time extension 1 ω c. Required spatial resolution of the bunch: σ R γ3 point particles are needed. Point-to-point code requires C σ2 R 2 γ6 operations/time step. Typical values for γ, σ, R give several years of CPU time (SPARC class) demand for one calculation Use line or sheet particles Snowmass 2001 Andreas Kabel Slide 16

17 Numerics: Macroparticles Higher-dimensional particles are needed for simulations: Granularity Singularities Space charge forces Snowmass 2001 Andreas Kabel Slide 17

18 Shielding Snowmass 2001 Andreas Kabel Slide 18

19 Techniques Used in CSR Calculations Analytic steady-state results [Y. Derbenev] Semi-analytic envelope tracking [P. Emma] Semi-analytic particle tracking [B. Carlsten] Self-consistent macroparticle tracking [R. Li] perturbative/self-consistent tracking, continuous distributions [M. Dohlus, A. K., T. Limberg: TraFiC 4 ] Point particle tracking with onedimensional analytic approximation for the fields [M. Borland: ELEGANT] Snowmass 2001 Andreas Kabel Slide 19

20 The Simulation Code TraFiC 4 What a Simulation Code Should Do: Handle retardation effects correctly use cartesian coordinates Calculate all fields from first principles Don t use linear approximations Consider the full six-dimensional phase space Don t use point particles use continuous charge distributions Use pointlinke probe particles Handle shielding Snowmass 2001 Andreas Kabel Slide 20

21 The Simulation Code TraFiC 4 What TraFiC 4 Does: Handle retardation effects correctly use cartesian coordinates Calculate all fields from first principles Don t use linear approximations Consider the full six-dimensional phase space Don t use point particles use continuous charge distributions Use pointlike probe particles Handle shielding TraFiC 4 = Track particles in the Fields of Continuous Charge distributions in Cartesian Coordinates. (Written by Andreas Kabel, based on the WAKE code by M. Dohlus, A. K., T. Limberg) Snowmass 2001 Andreas Kabel Slide 21

22 The Simulation Code TraFiC 4 In each slice and for each particle, the retarded position of each generating sub-bunch is found. The fields are calculated and are applied to the probe particles. The particles are tracked into the next slice. Probe particle bunch and generating bunch are set up according to the optics of the beamline. The generating bunch is tracked along the beamline. The procedure is repeated up to the exit slice. The phase-space distribution from all slices is written to a file and postprocessed. The beamline is divided into slices. Snowmass 2001 Andreas Kabel Slide 22

23 The Perturbative Kicks Procedure e n t e n+1 n+1 γ n+1 t n γ n Slice #n p n+1 p n Snowmass 2001 Andreas Kabel Slide 23

24 The Perturbative Kicks Procedure 1. The beamline is separated into slices; the field-generating bunch is tracked and its parameters on the slices are stored 2. (Optional: The generating bunch is tracked under its own influence or self-consistently) 3. iteratively track probe particles: To find the retarded position of the generating bunch, a fast binary lookup combined with micro-tracking through one slice is used The resulting forces are applied to the probe particles: γ n+1 = γ n + s F m e n+1 = e n + s F mβ 2 nγ n The particle is tracked into the next slice Snowmass 2001 Andreas Kabel Slide 24

25 The Self-Consistent Tracking Algorithm We can use causality: only slices upstream will contribute Use a leapfrog alorithm: kicks are applied in turns Using two trajectories gives error estimate Self-consistent Tracking Trajectories will deviate; this gives an error estimate t1 t2 t3 t4 t5 Snowmass 2001 Andreas Kabel Slide 25

26 Shielding So far, only free space was considered. Real-world situations impose boundary conditions on the problem. TraFiC 4 handles a limited case of shielding: infinitely extended, perfect conducting horizontal parallel plates. This is done by use of image charges Observation Point As we can t encounter the singularity of the field-generating bunch, one-dimensional charges are used. Benefits of shielding: Power radiated longitudinally (and thus induced energy spread) is reduced. Snowmass 2001 Andreas Kabel Slide 26

27 TraFiC 4 Features 3D multi-particle tracking code calculates fields from first principles (almost) arbitrary beamlines valid for any energy range tons of output: phasespace, correlations,... highly configurable: macro language for beamline setup bunch setup: dimensionality, optics, smearing,... fast/exact calculation control output slice/projective emittance fields at given sampling points only, no tracking it s slow ( hours for production-quality calculation for free-space, days for shielded calculations) huge output files ( bytes) Snowmass 2001 Andreas Kabel Slide 27

28 Multi-Processor Calculations Process 1 Process 2 Field Calculation 1 Synchronization 1 Apply Kicks 1 Field Calculation 2 Snowmass 2001 Andreas Kabel Slide 28

29 Example Applications Example I: Emittance growth cancellation (P. Emma, R. Brinkmann): Place a ±1 transformation between identical sections of the beamline Let CSR do the rest: Kicks will cancel approximately If the field changes due to bunch length changes: scale subsequent sections Example: Proposed DL-1 Transport Line at LCLS Good candidate for emittance growth cancellation: No dispersion in drift space lots of space for quads no bunch length change: constant fields weak fields Snowmass 2001 Andreas Kabel Slide 29

30 Snowmass 2001 Andreas Kabel Slide 30 β (m) New DL1 Double-Bend New Double-Bend DL1 for the LCLS Windows NT 4.0 version 8.23/acc 23/08/ β x β y Dx s (m) δ E/ p 0c = 0. Table name = TWISS Dx (m) Example Applications

31 Dispersion [m] Example Applications 9.8e e e e-07 9e e e Beamline Position [m] Normalized Emittance [m] Emittance Dispersion Snowmass 2001 Andreas Kabel Slide 31

32 Example Applications Example II: Bunch Compression at TTF To reach high current for TESLA Test Facility FEL operation, bunch compression is done in magnetic chicanes. Snowmass 2001 Andreas Kabel Slide 32

33 Example Applications Bunch Compressor II: 140MeV, 1.0nC 18 deflection angle Bunch length: 1000µm 250µm L magn = 0.50m, L Ot 0.40m 0.24m Bunch Compressor III (Phase 2; being designed): 500MeV, 1.0nC 6.5 deflection angle Bunch length: 250µm 50µm L magn = 0.2m or 0.1m, L Ot 0.30m Double chicane Snowmass 2001 Andreas Kabel Slide 33

34 Example Applications Normalized Projective Emittance [m] 1.6e e e-05 1e-05 8e-06 6e-06 4e-06 2e-06 Add l energy spread, unshielded Add l energy spread, 8mm chamber Emittance, unshielded Emittance, 8mm chamber Bend Beamline Position [m] Additional Energy Spread Snowmass 2001 Andreas Kabel Slide 34

35 Example Applications ε N [mm mrad] Transversal Normalized Emittance 250µm, 1nC, no shielding 250µm, 1nC, 8mm shielding 125µm,.5nC, no shielding 125µm,.5nC, 8mm shielding Bending Magnet Beamline Position [m] Snowmass 2001 Andreas Kabel Slide 35

36 Example Applications σ p/p e-05 Transversal Energy Spread 250µm, 1nC, no shielding 250µm, 1nC, 8mm shielding 125µm,.5nC, no shielding 125µm,.5nC, 8mm shielding Bending Magnet Beamline Position [m] Snowmass 2001 Andreas Kabel Slide 36

37 Example Applications Energy Spread in Slice σ δ Position in Beamline [m] Snowmass 2001 Andreas Kabel Slide 37

38 Example Applications 6e-06 5e-06 4e-06 3e-06 2e-06 1e-06 0 ε final [m] β 20 initial [m] ε final 2e e e-06 1e-06 9e α initial [] 0.51 Snowmass 2001 Andreas Kabel Slide 38

39 Example Applications Example III: Bunch Compression at CTF II Exp. 97, q=6 nc γε [mm mrad] Τ bunch,fwhm [ps] γε x (Exp. CTF II) γε y (Exp. CTF II) Τ b (Exp. CTF II) γε x (TraFiC4) γε y (TraFiC4) Τ b (TraFiC4) φ 150 [deg] Snowmass 2001 Andreas Kabel Slide 39

40 Conclusion and Outlook Snowmass 2001 Andreas Kabel Slide 40

41 Conclusion and Outlook Handling beam dynamics on bent trajectories requires special care Conventional simulation codes can t be used TraFiC 4 was written to handle the specific effects of bent trajectories Used as a design tool for the bunch compression sections at TTF Agreement with analytical (steady-state) results Reasonable agreement with experimental data (still sparse data) Inconclusive results for newest CLIC experiments Benchmarking of available code should be done Snowmass 2001 Andreas Kabel Slide 41