E-Jitter Propagation. Tim Maxwell, Franz-Josef Decker, Lanfa Wang, et al. August 15, 2014

Transcription

1 E-Jitter Propagation Tim Maxwell, Franz-Josef Decker, Lanfa Wang, et al. August 15, 2014

2 Jitter Propagation FJD suggested leveraging linac phasing to suppress correlated jitter Lanfa also found this possible in LiTrack optimizations With FJD s input, trying to build analytical model for tracking these correlated time-energy ellipses 2

3 Correlated Elliptical Distributions We know and trust the phase space ellipse Each offset point is a single electron in one bunch 3

4 Correlated Elliptical Distributions But also note the centroid (or jitter) ellipse The same matrix math describes evolution of the timeenergy distribution of the centroids of many shots Each offset point is the centroid of a different bunch* * Not likely same as dist. of 1 bunch 4

5 This talk: Centroid math - This is dry stuff, details and general observations are in slides after the end - Takeaway: Jitter growth/station function of station s jitter ellipse, transformed by the choice of local RF chirp Characterize individual stations Analysis of jitter reduction exp ts - L0B SLED impact (April 22) - -15º Decker phasing - Soft X-ray jitter reduction MDs 5

6 RF covariance parameters Gun Presume initial centroid ellipse cov. is σ gun, known from BSA data - (1) it stands to reason, (2) final results are sensible * Note, E in these figures is the forward energy gain [Amp*cos(φ)] σ t,gun 30 fs σ E,gun kev (0.01%) ρ gun 0.1 6

7 RF covariance parameters L0A (unsleded) For L0A, all expected jit. growth from amp. jit. (0 phase / R 65 ) σ t,l0a 70 fs σ E,L0A 19.5 kev (0.034%) ρ L0A

8 RF covariance parameters L0B SLED took out some (6 kev, ~24%) jit. growth due to phase jit. unsleded SLEDed SLEDed σ t,l0b 76 fs σ E,L0B 21.7 kev (0.031%) ρ L0B 0.71 σ t,l0b 45 fs σ E,L0B 19.0 kev (0.027%) ρ L0B -0.27, typ. 8

9 RF covariance parameters L1S (SLEDed) σ t,l1s 38 fs σ E,L1S 35 kev (0.030%) ρ L1S -0.26, typ. 9

10 RF covariance parameters L1X σ t,l1x 37 fs σ E,L1X 12.5 kev (0.067%) ρ L1X

11 RF covariance parameters L2/L3 L2 / L3 have many dozen klystrons BSA doesn t see correlated RF amp/phase Fast time plots suggest big variation, average per station: σ t = 180 fs (standard deviation 100 fs) σ δ = 0.042% (standard deviation 0.025%) Calculating diff. configs compared to measurements: σ t = 280 fs σ δ = 0.03% ρ = (assume they re like other SLEDed stations) 11

12 RF covariance parameters L2/L3 Discussing with FJD, bad L2/L3 stations treatment so far: A φ These numbers: N stations, similar jitter, all amp. and phase tunable φ Actual: N-S stations + S sub-boosters, similar jitter, only phases changed, amp s kept fixed φ sub Oops. Stations are just Lego blocks now though, can update these numbers later to be as granular as we like. 12

13 Calculate jitter only from RF data For example: L0B SLEDed vs. unsleded Explains most measurable jitter; some missing likely from correlations to current (wakes) or my poor treatment of L2/L3 sub-boosters Ex: Frequently measured DL2 jit. is +1 MeV higher than calculated 13

14 HXR Jitter Reduction HXRSS June 19th Avg. jitter vs. L3 phase, σ δ = 0.032% 0.024% 25% improvement (Goal 0.027%) So how does this work? 14

15 L3 HXR Decker Phasing Jitter ellipse evolution 3) L3 linear transform w/ φ L3 = 0º (same) 4) L3 uncor MeV growth* 1) After L2 (5 GeV, ~3.5 ka) 2) After BC2 R 56 = mm E (MeV) E (MeV) E (MeV) t (ps) E (MeV) t (ps) t (ps) 3) L3 linear transform w/ φ L3 = -14.6º (positive R 65 fights negative BC2 chirp) t (ps) 4) L3 uncor MeV growth* * i.e. Competition: chirp can remove previously corr. E-jitter, but also increases uncor. jitter from L3 E (MeV) E (MeV) From LiTrack, φ L3 = -15º should yield < 1% relative increase of proj. e-spread t (ps) t (ps) 15

16 June 18 th SXR Jitter Reduction Study Constants: 1.5 ka, L3 Energy 4 GeV L2 Klys., BC2 E = 5 GeV, R 56 = mm L2 Klys., BC2 E = 3 GeV, R 56 = mm L2 Klys., BC2 E = 3 GeV, R 56 = mm o Calc. + Meas. 16

17 June 18 th SXR Jitter Reduction Study Yes, L2/L3 sub-booster models wrong, but all else equal, we find the absolute jitter picked up in a linac section has contributions that go as σ E, tot E tot + R 65 1) Reducing L2 energy helps 2) Reducing L2 chirp (incr. R 56 ) helps Constants: 1.5 ka, L3 Energy 4 GeV L2 Klys., BC2 E = 5 GeV, R 56 = mm L2 Klys., BC2 E = 3 GeV, R 56 = mm L2 Klys., BC2 E = 3 GeV, R 56 = mm o Calc. + Meas. For soft x-rays, overshooting beam energy in BC2 should, in general, always hurt final σ E These results repeated on July 9 th 17

18 SXR Jitter Reduction Both days, no clear L3 phase impact (?) L3 chirp (R 65 ) much weaker, per ±1 GeV L3 gain - Need -35º L3 phase, we only went to -15º - Should get to GeV (0.056%): 18

19 Summary For the machine: For > 9 GeV, maintain φ L3 = -15º whenever possible For < 5 GeV, biggest win by reducing BC2 energy May need tweaking in light of diff. sub-booster arrangements For the model: Correct the sub-booster modeling Incorporate more online measurements, esp. L2/L3 Tie in *actual* phase space ellipse (you know, the important one) as additional constraint Approx. wake effects (next order cross-covariances) Much of this can be solved backwards/forwards Fast, full system optimizer 19

20 Math nitty gritty 20

21 Correlated Elliptical Distributions Consider linear theory for a single bunch s time-energy distribution Ellipse is described by longitudinal Twiss (covariance) matrix: 21

22 Correlated Elliptical Distributions We know how linear transformations M operate For some coordinates X / Y (e.g.: z / δ), if then new ellipse σ' around new mean vector µ' : 22

23 Comment on Coordinates For phase space ellipse, usually track (z, δ) For centroid ellipse, tracking absolute ( t, E) proves more convenient Each rf station tends to contribute some known amount of absolute energy jitter Annoying to rescale each contribution to local E 23

24 Comment on Coordinates Do math using covariance matrix Easy to relate to (measurable) rms jitters σ t, σ E and their correlation ρ < 1 Also avoid confusion with Twiss params. - But note: centroid emittance equivalent to the irreducible amount of time-energy jitter - We plan to show how irreducible jitter grows, so emittance not a practically useful, conserved quantity 24

25 Primary Linear Operators For long. coordinates, two transforms of interest RF accelerating structure: - R 65 (sometimes called h) is the rf chirp Erf sinφrf - Shears phase space in E direction, also adds energy offset to mean E ( ) Magnetic chicane: - Shears phase space in time direction - Compresses a bunch in time if initially a positive chirp and R 56 < 0 25

26 Primary Nonlinear Contribution: RF Jitter Linac sections each have an intrinsic phase-amp. jitter ellipse σ rf Seen by the centroid of beam, assuming it adds purely uncorrelated energy jitter per station, it is: Increases irreducible total t-e jitter 2 2 ε = ε + σ 55σ rf,66 All else being minimized, can only control chirp R 65 Point 1: Minimizing rf chirp minimizes irreducible jitter growth from rf phase jitter 26

27 Repeated RF Jitter Growth Combining N accelerating sections in a row, the individual energies (E i ), chirps (R 65,i ) and uncorrelated squared-energy growth terms (σ' 66,i ) sum: 27

28 Repeated RF Jitter Growth If we assume, for example in L2 or L3 with dozens of stations, that: All stations have ~same jitter characteristics Phase jitter per station σ t,i independent of amplitude Absolute σ E,i per station prop. to station s abs. energy gain E i E i = E tot / N and R 65,i = R 65,tot / N E tot and R 65,tot are set by desired total energy gain and chirp Then combined uncorrelated growth is: σ 1 N ( E σ + R ρ σ σ R σ ) 2 E, tot = tot δ, i 2 65, tot i δ, i t, i+ 65, tot t, i 28

29 Repeated RF Jitter Growth Combined uncorrelated growth was: σ 1 N ( E σ + R ρ σ σ R σ ) 2 E, tot = tot δ, i 2 65, tot i δ, i t, i+ 65, tot t, i Point 2: Uncorrelated growth goes as σ E, tot 1/ N, so the more stations the better (until per-station level so low individual stabilities get worse) Point 3: All things equal, σ E, tot Etot, so greater absolute energy gain increases irreducible jitter from that combined section 29