Beam Breakup, Feynman and Complex Zeros

Transcription

1 Beam Breakup, Feynman and Complex Zeros Ivan Bazarov brown bag seminar, 16 Jan 04 Ivan Bazarov, BBU, feynman and complex zeros, Brown Bag seminar, 16 January

2 Outline BBU: code-writing and simulation results feynman : diversion # 1 on ERL injector complex zeros: diversion # 2 on similarities between exploding molecules and bunch profile measurements using coherent radiation Ivan Bazarov, BBU, feynman and complex zeros, Brown Bag seminar, 16 January

3 HOM-beam interactions Two basic dangers: Multipass beam breakup Resonant excitation of a higher order mode monopole (m = 0) TM 01 -like y y dipole (m = 1) TM 11 -like y y quadrupole (m = 2) TM 21 -like y y z x z x z x E B E B E B high losses, no kick kick and losses when beam is not centered kick, coupling and losses when beam is not centered Ivan Bazarov, BBU, feynman and complex zeros, Brown Bag seminar, 16 January

4 Wake formalism used R y A resonant mode characterized by,q, ω Q m rˆ a r e c transverse q c longitudinal θ kz θˆ kz x ckm 2Q 2Q Wm = 2 e sin kz W m 2kme cos kz ω W m (z) W m (z) L / 2 L / 2 r F L / 2 L / 2 ds = eqa F ds = eqa m m W m ( z) mr W ( z) r m m m 1 ( rˆ cos mθ ˆsin θ mθ ) cos mθ z L / 2 L / 2 r The above can be ω combined R to say (Panofsky-Wenzel): F ds = F ds loss factor km z L / 2 L / 2 Q R = 4 Q in units of Ω m 2m m m k = ω / c is mode s wave number Ivan Bazarov, BBU, feynman and complex zeros, Brown Bag seminar, 16 January

5 Beam breakup m 12 Transverse mode, m = 1 injected beam V 2 nd pass deflected beam 1, x t ( t) = W ( t t 1 )[ I x (2) e ( t) = m12v c (1) ( t ) x (1) ( t x t r ) ( t ) + I (2) ( t ) x (2) ( t )] dt Longitudinal mode, m = 0 injected beam m 56 2 nd pass modulated beam worst case thresholds: I th,1 I th,0 = = e( R / Q) 2E e( R / Q) 1, λ 2E 0, λ k 1 λ k λ Q Q λ λ 1 R R Ivan Bazarov, BBU, feynman and complex zeros, Brown Bag seminar, 16 January

6 bi - beam instability code Features: allows any ERL topology arbitrary bunch pattern (can setup a cloud to study singe bunch effects) transverse / longitudinal BBU implemented in C++ fast Ivan Bazarov, BBU, feynman and complex zeros, Brown Bag seminar, 16 January

7 Wake arithmetics Wake function due to single bunch (e.g. a {λ} set of dipole HOMs): t τ W ( τ ) = e λ ( R / Q) λ 2 ω 2c λ ω 2Q sinω Electrons in test bunch will get a kick: p e V c e = = λ λ τ Same for test bunch trailing behind a bunch train {t i, q i, d i }: p λ τ e ( τ ) W ( τ ) qed c e =, i W ( τ t c all past bunches, An absolute must: represent the sum above in Horner s form Horner s trick i i e ) q Problem: evaluate polynomial: a n x n + a n 1 x n a 1 x + a 0 Correct answer: ( (a n x + a n 1 ) x + a n 2 ) x + ) x + a 0 i d i Ivan Bazarov, BBU, feynman and complex zeros, Brown Bag seminar, 16 January

8 Basic algorithm Expand beam line into a consecutive list of cavities (pointers) in the same order a bunch sees them (most repeat n_pass times) in its lifetime (from injection to dump); Link pointers to actual HOMs (i.e. cavities); consecutive list of cavities a bunch sees in its lifetime (total number N): actual HOMs or cavities (n N): N 2 N 1 N Start filling beam line with bunch train; Determine which pointer sees a bunch next; hom 1 hom 2 hom n 1 hom n Update wake-field in HOM which the pointer points to; Push the bunch to next pointer, store its coordinates until they are needed by any bunch that will reach this point next; Ivan Bazarov, BBU, feynman and complex zeros, Brown Bag seminar, 16 January

9 Tracking vs. theory (single HOM, one recirc.) Benchmarking: transverse longitudinal BBU BBU bi code threshold current [A] threshold current [ma] tracking old theory old theory corrected theory worst case formula E E E E E E E E E E recirculation HOM frequency time [s][ghz] Ivan Bazarov, BBU, feynman and complex zeros, Brown Bag seminar, 16 January

10 Some features of the longitudinal instability 0.8 I = 22.0 ma 0.6 similar to transverse BBU in its scaling: [(R/Q)Qω] 1, E bad frequencies (n + ¼)ω0, n is an integer does not grow exponentially, but saturates much less of an issue in ERL (time of flight of the lattice is nearly zero) -0.6 arrival time difference [ps] I = 22.1 ma I = 23.0 ma bunch # Ivan Bazarov, BBU, feynman and complex zeros, Brown Bag seminar, 16 January

11 Beam breakup for a 5 GeV ERL 1.E+05 dipole HOM bbu (freq. spread 3 MHz rms) in ERL threshold current [ma] 1.E+04 1.E+03 1.E+02 y = 7.02E+06x -9.44E-01 TESLA 9-cell y = 3.51E+07x -9.44E-01 5 safety factor 1.E+01 1.E+03 1.E+04 1.E+05 1.E+06 (R/Q)Q/f [Ohm/cm^2/GHz] HOM damping spec for dipoles Ivan Bazarov, BBU, feynman and complex zeros, Brown Bag seminar, 16 January

12 HOM frequency spread effect rms = 0 Hz rms = 33 khz rms = 42 khz momentum [ev/c] 4.0E E+06 fixed current 30 ma, 2.0E+06 applying frequency 1.0E+06 spread 0.0E E E E E+06 position [m] momentum [ev/c] 2.5E E E E E E E E E E E E E E E E+04 position [m] momentum [ev/c] 1.0E E E E+02 position [m] -6.00E E E E E- 0.00E E- -2.0E E E E E E E- 06 rms = 46 khz 2.0E+02 rms = 53 khz 2.0E+01 rms = 67 khz 1.5E+01 momentum [ev/c] 1.5E E E E E E E- 0.00E E E E E E+02 momentum [ev/c] 1.5E E E E E E E E E E E+01 momentum [ev/c] 1.0E E E E E E E E E E E+02 position [m] -1.5E E+01 position [m] -1.0E E+01 position [m] Ivan Bazarov, BBU, feynman and complex zeros, Brown Bag seminar, 16 January

13 Stabilizing effect of HOM frequency spread Effectively decreases Q of the mode: Q ~ ω/ ω ½ Limited in its effect by ~ 1/t r (~ MHz) Linearized solution for the simplest case I 1 2c e( R / Q) Qkm ω ω = cot( Ωt 2Q r 12 ) sin( Ωt r, ) ω ω cot( Ω 2Q t r ) ω Periodic in ω = Ω ω with ω = 2π / here ω is HOM frequency r t r Ivan Bazarov, BBU, feynman and complex zeros, Brown Bag seminar, 16 January

14 Threshold vs. frequency spread 300 f = MHz, R/Q = Ω/cm 2, Q = , same seed, uniform threshold current [ma] frequency spread rms [MHz] Ivan Bazarov, BBU, feynman and complex zeros, Brown Bag seminar, 16 January

15 HOM f uncertainty (10 MHz uniform) f = MHz, R/Q = Ω/cm 2, Q = , 1000 seeds, uniform (2.9 MHz rms) # cases average: 204 ma rms: 58 ma threshold [ma] Ivan Bazarov, BBU, feynman and complex zeros, Brown Bag seminar, 16 January

16 Spread in Q s 25 f = MHz, R/Q = Ω/cm 2, uniform spread in Q ( ) 10 4, 200 seeds, uniform frequency spread 2.9 MHz rms (same seed) # cases all Q s all Q s all Q s threshold [ma] Ivan Bazarov, BBU, feynman and complex zeros, Brown Bag seminar, 16 January

17 Outline BBU: code-writing and simulation results feynman : diversion # 1 on ERL injector complex zeros: diversion # 2 on similarities between exploding molecules and bunch profile measurements using coherent radiation Ivan Bazarov, BBU, feynman and complex zeros, Brown Bag seminar, 16 January

18 Injector specs that justify ERL Most often used figure of merit for synchrotron light-sources B I ε ε Storage rings already operate at diffraction limit in vertical plane (for ~ 10 kev photons that corresponds to 0.1 Å-rad rms), also future rings will be designed with yet smaller horizontal emittance 5.11 GeV linac x y e.g. NSLS planned upgrade B 500 ma (Å - rad) 2 Ivan Bazarov, BBU, feynman and complex zeros, Brown Bag seminar, 16 January

19 Space charge in the injector electron bunch ε th = if beam matched to Brillouin eq. flow condition s.c. vs. external focusing leads to reversible emittance oscillations freeze out the s.c. by acceleration when minimum occurs (e.g. Phys. Rev. E, 55, 7565) one can use sliced-beer-can model code HOMDYN to find appr. working points ultimately, a complete simulation with realistic profiles (esp. long.) is required r 2 cath E cath 2 mc Ivan Bazarov, BBU, feynman and complex zeros, Brown Bag seminar, 16 January

20 Bunch dynamics in the injector (77 pc) Ivan Bazarov, BBU, feynman and complex zeros, Brown Bag seminar, 16 January

21 feynman at work on ERL injector results of parameter scan at 80 pc Ivan Bazarov, BBU, feynman and complex zeros, Brown Bag seminar, 16 January

22 Emit. scaling vs. charge, vs. bunch length 0.4 mm q bunch 2.0 mm [mm-mrad] 1 σ z σ z [mm] q [pc] 8 pc 92 pc Ivan Bazarov, BBU, feynman and complex zeros, Brown Bag seminar, 16 January

23 Scaling with charge (6 MeV inj. energy) pc 77 pc 0.6 mm emittance [mm-mrad] diffraction limit for 8 kev 0.8 mm 1.0 mm 1.2 mm charge [pc] Ivan Bazarov, BBU, feynman and complex zeros, Brown Bag seminar, 16 January

24 More optimization results Optimization for emittances in case of transverse uniform, longitudinal gaussian laser profile: mm-mrad for 8 pc/bunch 0.58 mm-mrad for 80 pc/bunch 5.3 mm-mrad for 0.8 nc/bunch final bunch length < 0.9 mm Simulations suggest that thermal emittance is not important for high charge / bunch (~ nc), but is important for low charge bunch (~ pc) Better results if longitudinal laser profile shaping can be employed (easily 2 for uniform profile) Note: results are similar to those of RF guns Ivan Bazarov, BBU, feynman and complex zeros, Brown Bag seminar, 16 January

25 Outline BBU: code-writing and simulation results feynman : diversion # 1 on ERL injector complex zeros: diversion # 2 on similarities between exploding molecules and bunch profile measurements using coherent radiation Ivan Bazarov, BBU, feynman and complex zeros, Brown Bag seminar, 16 January

26 Ivan Bazarov, BBU, feynman and complex zeros, Brown Bag seminar, 16 January Phase problem in crystallography Far field diffraction of many identical copies (Bragg) produces discrete pattern, which is easy to measure but data is undersampled sampling spacing = (unit cell) 1 FT l 1 0,..., 1 0,..., ), ( ), ( ) / / ( 2 = = = = = + m k l k e y x k k F y x l x m y m y k l x k i y x y x π ρ m l m unknowns ρ(x,y) l m/2 Bragg intensities F(k x,k y )

27 How important is phase? amplitude phase IFT FT & amplitude only IFT phase only IFT Ivan Bazarov, BBU, feynman and complex zeros, Brown Bag seminar, 16 January

28 Oversampling (or the right sampling) noncrystalline specimen generate continuous FT, which can be sampled fine enough (equivalent to adding blank support for DFT) to attempt phase retrieval without any prior knowledge of the distribution 2 questions: Can one find phases if given sufficiently sampled amplitudes of FT? Answer: Yes (Fienup, 1978) Is the solution found unique? Answer: It depends It turns out that dimensions 2D and higher generally produce unique solution, while 1D case does not. Ivan Bazarov, BBU, feynman and complex zeros, Brown Bag seminar, 16 January

29 Fienup s algorithm in action Ivan Bazarov, BBU, feynman and complex zeros, Brown Bag seminar, 16 January

30 Grand hope of XFELs 20 40% of all proteins cannot be crystallized If one can collect diffraction data from a single molecule (or tiny microcrystal), then the structure can be solved Radiation damage is a showstopper: most useful particles here are electrons (cryo-em), but so far ~10 Å resolution is a limit If ultraintense, ultrafast X-ray burst is used, one can envision collecting diffraction data before molecule Coulomb-explodes, thus, defeating conventional radiation damage limit Ivan Bazarov, BBU, feynman and complex zeros, Brown Bag seminar, 16 January

31 Bunch profile measurement Same phase problem, only 1D Recall that in forward direction coherent radiation is given by I coh ( ω) = N 2 FF( ω) = ρ ( ω), f ( t)? 2 FF( ω) I 0 ( ω) F( ω) FT{ f ( t)} ρ( ω) e iϕ ( ω) rms value can be inferred from autocorrelation function, i.e. IFT{FF(ω)} profile measurements require knowledge of ϕ(ω) I{ln F(ω)} by either Fienup s algorithm or by applying Kramers-Kronig relation to real part of the same function R{ln F(ω)} = ln ρ(ω) While Kramers-Kronig relation is unique for pair R{F(ω)}, I{F(ω)}, zeros of F( ˆ), ω ˆ ω C in complex plane make phase retrieval dubious Ivan Bazarov, BBU, feynman and complex zeros, Brown Bag seminar, 16 January

32 Simple example F ( ω) = sinc( ω + 1) + sincω + sinc( ω 1) F ( ω) = i sinc( ω + 1) + sincω + i sinc( ω 1) f f ( t) IFT{ F1 ( ω)} = ( t) IFT{ F2 ( ω)} = 1+ 2cost 2π 0, 1+ 2sint 2π 0,, t π t > π, t π t > π F 1( ω) = F2 ( ω) ω Note that 1 ω / i F2 ( ω) = F1 ( ω), while F1 ( i) = 0 1+ ω / i f 1 ( t) f 2 ( t) In general, for zeros ωˆn, multiplication by does not add poles, nor changes amplitude but contributes to phase n 1 ω / ˆ ω 1 ω / ˆ ω n t Ivan Bazarov, BBU, feynman and complex zeros, Brown Bag seminar, 16 January

33 More examples simulated 1 hump bunch profile 2 humps at the end of the injector 3 humps Ivan Bazarov, BBU, feynman and complex zeros, Brown Bag seminar, 16 January

34 The end Ivan Bazarov, BBU, feynman and complex zeros, Brown Bag seminar, 16 January