Dispersion Free Steering ( BBA) for the SASE Undulators of the XFEL (Work in Progress!)

Transcription

1 XFEL BD meeting / M.Vogt / Dispersion Free Steering for the SASE Undulators of the XFEL 1 Dispersion Free Steering ( BBA) for the SASE Undulators of the XFEL (Work in Progress!) Mathias Vogt (DESY MPY) lots of input from: W.Decking, P.Castro, B.Faatz,... Why BBA What the Heck is Dispersion Free Steering? The Model Preliminary Results

2 XFEL BD meeting / M.Vogt / Dispersion Free Steering for the SASE Undulators of the XFEL 2 Orbit Requirements for the SASE Process Resonant interaction of charged particle and undulator radiation Particle orbit and radiation cone ( 1/γ) must overlap Beam orbit excursion in undulator rms beam envelope longitudinal scale gain length BAD ORBIT : Strong orbit fluctuations overlap only over short ranges radiation length weak (or no) SASE signal GOOD ORBIT : Flat orbit overlap only over most of undulator > radiation length potentially: saturation

3 XFEL BD meeting / M.Vogt / Dispersion Free Steering for the SASE Undulators of the XFEL 3 β (m) SA1LNE SASE-1 and half of T4 XFEL Undulator SASE-1 6. Unix version 8.51/16 19/2/ β x β y s (m) δ E/ p c =. Table name = TWISS Target for orbit : rms (over 2m) < 3µm BBA will be tricky (mainly due to large unknown offsets) Option 1 : try Dispersion Free Steering (dispersion measurement only needs a difference orbit!) misaligned quads perturbed orbit initial quad misalignment & BPM offsets 3µm (?) beam based alignment (=BBA) necessary in SASE-1 : high resolution cavity BPMs: res 1µm 3µm correctors quad movers in T4 : most likely only: res 2µm 5µm

4 XFEL BD meeting / M.Vogt / Dispersion Free Steering for the SASE Undulators of the XFEL 4 (What the heck is) Dispersion Free Steering Orbit (= dipole) kicks create (spurious) dispersion + given N perturbations (=correctors) {Ki}1 i N and M BPMs + yields M measured orbits {Xi}1 i M + and M measured dispersions {Di}1 i M + measured X offset + statistical fluctuations + measured D statistical fluctuations only M_bpm M_bpm N_kick N_kick N_kick N_kick X K D = * ORM DRM causality in beam line : each upper right 2M conditions for N corrector settings overdetermined system : w/o errors conditions linearly dependent w/ errors least squares solution SVD

5 XFEL BD meeting / M.Vogt / Dispersion Free Steering for the SASE Undulators of the XFEL 5 Introduce weight w ( orbit-only, 1 dispersion-only) ( ) ( ) (1 w) X (1 w)o wd = wd or shorthand: Ξ(w) = A(w) K Ξ R 2M := real orbit/dispersion, A R 2N M := combined orbit dispersion response matrix i-th Measurement: add systematic (const C) and statistical ( S i ) errors ξ i (w) = A(w) K i + C + S i and iterate corrected dipole kicks Φ i with error i K i = K i 1 Φ i i Dispersion Free Steering (2) K How to compute Φ i? assuming NO orbit/dispersion from upstream SASE-1! iff C S i i i (& assuming A is completely known) ξ Ξ = A K is fully redundant, i.e. A R M 2N with K Φ := A Ξ The pseudo inverse A can be computed using a Singular Value Decomposition (SVD) In fact SVD + τ regularization allow some control over correcting the highly correlated (= potentially real ) orbit/dispn. components rather than the weakly correlated (= contaminated) components...

6 XFEL BD meeting / M.Vogt / Dispersion Free Steering for the SASE Undulators of the XFEL 6 SVD + for DispFree Steering A = U diag({σk}) V T N_kick N_kick N_kick M_bpm M_bpm N_kick N_kick N_kick N_kick M_bpm M_bpm N_kick ORM DRM T V * * diag = A = U U R 2M N, U T U = 1 N N U T Ξ := orthogonal orbit/dispn mode V O(N) V T K := orth. knob for mode {σk}1 k N, σk : singular values knob strengths for non degenerate phase advances A has full rank σk> k A := V diag({σ 1 k }) U T if system is underdetermined solution of Ξ = A K is K K part + kern(a) SVD gives minimal solution : A Ξ 2 = min if system is overdetermined solution only in the least square sense SVD yields solution with minimal residue : Ξ A (A Ξ) 2 = min

7 XFEL BD meeting / M.Vogt / Dispersion Free Steering for the SASE Undulators of the XFEL 7 τ regularization for DispFree Steering What if some σ i =??? just redefine A := V diag({(σ k > ) 1,...}) U T yields least square solution! MORE GENERAL : condition of A : cond(a) := max i{σ i } min i,σi >{σ i } large cond means that solutions K of linear system A K = Ξ strongly depend on small variations ( errors!) of Ξ to improve (=decrease) condition : set σ j, σ j < τ with some regularization parameter τ... and redefine A (τ) := V diag({(σ k > τ) 1,...}) U T for Dipersion Free Steering : use only highly correlated orbit/dispn modes!!! & ignore stronlgy contaminated orbit/dispn modes!!! correct orbit/dispn with: Φ i = A (τ) ξ i 1

8 XFEL BD meeting / M.Vogt / Dispersion Free Steering for the SASE Undulators of the XFEL 8 Model of BBA for SASE-1 β (m) SA1LNE 6. Unix version 8.51/16 19/2/ β x β y s (m) δ E/ p c =. Table name = TWISS no orbit/dispn from upstream SASE-1 PERT: 33 misaligned quads in SASE-1 CORR: 33 quad movers in SASE-1 51 BPMs : 33 in SASE in T4 upstream dispersive section ORM & DRM w.r.t. quadmisalignment mad-8 ( lmad ) all errors ( K, C, S i, i ) : independent Gaussian RV initial rms quad misalignment : 3µm rms BPM offset : 2µm rms BPM statistical error in SASE-1 : 1µm rms BPM statistical error in T4 : 5µm rms mover error : 1µm means : as a starting point take BPMs in T4 as good as in SASE-1 and no mover errors correction method (A) : global, variable gain, weight, τ : Φ i = ga (w, τ) ξ i 1 correction method (B) : local (l to m), variable gain, weight, const τ = : l,m Φ i = ga (w, ) ξi 1 l,m

9 XFEL BD meeting / M.Vogt / Dispersion Free Steering for the SASE Undulators of the XFEL 9 Simulation Parameters (1-st try) initial quad misalignment : -rms : 3µm systematic offsets : C -rms : 2µm ; C -rms : difference orbit! X D resolution : S i -rms = S i -rms : 1µm only 3% dp/p acceptance X D multi shot average to reduce S D i -rms mover errors : i =, i > Correction Sequence v1a : step w I s.v. max g Correction Sequence v3a : step w I s.v. max g : all singular values!

10 XFEL BD meeting / M.Vogt / Dispersion Free Steering for the SASE Undulators of the XFEL 1 Singular Values for chosen w / v1a Singular Values (CorrSeq v1a) 1 1 w=. w=.8 w=.95 w=1. 1 σ i <= i <= M bpm

11 XFEL BD meeting / M.Vogt / Dispersion Free Steering for the SASE Undulators of the XFEL 11 Finding the Right I s.v. max for Each Step / v1a orbit estimated rms orbit /µm estimated rms-orbit vs. max number of used S.V. / CorrSeq v1a / seed x pre step 1 pre step 2 pre step 3 pre step 4 post step <= I s.v. max <= M bpm

12 XFEL BD meeting / M.Vogt / Dispersion Free Steering for the SASE Undulators of the XFEL 12 Finding the Right I s.v. max for Each Step / v1a dispn estimated rms dispersion /µm estimated rms-dispn vs. max number of used S.V. / CorrSeq v1a / seed x pre step 1 pre step 2 pre step 3 pre step 4 post step <= I s.v. max <= M bpm

13 XFEL BD meeting / M.Vogt / Dispersion Free Steering for the SASE Undulators of the XFEL 13 Singular Values for chosen w / v3a Singular Values (CorrSeq v3a) 1 1 w=. w=.95 w=.99 w= σ i <= i <= M bpm

14 XFEL BD meeting / M.Vogt / Dispersion Free Steering for the SASE Undulators of the XFEL 14 Finding the Right I s.v. max for Each Step / v3a orbit estimated rms orbit /µm estimated rms-orbit vs. max number of used S.V. / CorrSeq v3a / seed x pre step 1 pre step 2 pre step 3 pre step 4 post step <= I s.v. max <= M bpm

15 XFEL BD meeting / M.Vogt / Dispersion Free Steering for the SASE Undulators of the XFEL 15 Finding the Right I s.v. max for Each Step / v3a dispn estimated rms dispersion /µm estimated rms-dispn vs. max number of used S.V. / CorrSeq v3a / seed x pre step 1 pre step 2 pre step 3 pre step 4 post step <= I s.v. max <= M bpm

16 XFEL BD meeting / M.Vogt / Dispersion Free Steering for the SASE Undulators of the XFEL 16 Initial Orbits / all seeds CorrSeq : v1a, v3a initial orbit /µm seed a seed b seed c seed d seed e seed f seed g seed h seed i seed j seed k <= i <= M bpm

17 XFEL BD meeting / M.Vogt / Dispersion Free Steering for the SASE Undulators of the XFEL 17 Result of Correction Sequence v1a CorrSeq : v1a final orbit /µm 5-5 seed a seed b seed c seed d seed e seed f seed g seed h seed i seed j seed k <= i <= M bpm

18 XFEL BD meeting / M.Vogt / Dispersion Free Steering for the SASE Undulators of the XFEL 18 Result of Correction Sequence v1a (BEST) CorrSeq : v1a 6 4 seed b seed e seed g seed h seed k 2 final orbit /µm <= i <= M bpm

19 XFEL BD meeting / M.Vogt / Dispersion Free Steering for the SASE Undulators of the XFEL 19 Result of Correction Sequence v3a CorrSeq : v3a final orbit /µm seed a seed b seed c seed d seed e seed f seed g seed h seed i seed j seed k <= i <= M bpm

20 XFEL BD meeting / M.Vogt / Dispersion Free Steering for the SASE Undulators of the XFEL 2 Result of Correction Sequence v3a (BEST) CorrSeq : v3a 6 4 seed b seed c seed g seed h seed j seed k final orbit /µm <= i <= M bpm

21 XFEL BD meeting / M.Vogt / Dispersion Free Steering for the SASE Undulators of the XFEL 21 Parameters : initial quad misal. : -rms : 3µm systematic offsets : C -rms : 3µm X but C -rms : difference orbit! D resolution : S SASE1 i -rms : 1µm X S T4 i -rms : 2µm cheaper BPMs X S SASE1 i -rms : 2µm only 3% dp/p D S T4 i -rms : 4µm acceptance & BPMs D mover errors : i -rms : 1µm, i > A More Realistic Example... CorrSeq v1 (1-st attempt = yesterday!!) : step range w I s.v. max g : all singular values!

22 XFEL BD meeting / M.Vogt / Dispersion Free Steering for the SASE Undulators of the XFEL 22 All BPMs in T4 : 2 worse Resolution (1-st attempt = yesterday!!) CorrSeq : v1 / seed x orbit /µm step step 1 step 2 step 3 step 4 step 5 step 6 step 7 step 8 step 9 step 1 step <= i <= M bpm

23 XFEL BD meeting / M.Vogt / Dispersion Free Steering for the SASE Undulators of the XFEL 23 All BPMs in T4 : 2 worse Resolution (1-st attempt = yesterday!!) CorrSeq : v1 / seed x orbit /µm step 2 step 3 step 4 step 5 step 6 step 7 step 8 step 9 step 1 step <= i <= M bpm

24 XFEL BD meeting / M.Vogt / Dispersion Free Steering for the SASE Undulators of the XFEL 24 All BPMs in T4 : 2 worse Resolution (1-st attempt = yesterday!!) CorrSeq : v1 / seed x 1 step 8 step 9 step 1 step 11 5 orbit /µm <= i <= M bpm

25 XFEL BD meeting / M.Vogt / Dispersion Free Steering for the SASE Undulators of the XFEL 25 A Slightly More Expensive Example... Parameters : initial quad misal. : -rms : 3µm systematic offsets : C -rms : 3µm X but C -rms : difference orbit! D resolution : S SASE1 + 1 st 5 in T4 i -rms : 1µm X S T4 (rest) i -rms : 2µm X S SASE1 + 1 st 5 in T4 i -rms : 2µm D S T4 (rest) i -rms : 4µm D mover errors : i = 1, i > CorrSeq v2 (2-nd attempt = today!!) : step w I s.v. max g

26 XFEL BD meeting / M.Vogt / Dispersion Free Steering for the SASE Undulators of the XFEL 26 All but 1-st 5 BPMs in T4 : 2 worse Resolution (2-nd attempt = today!!) CorrSeq : v2 / seed x orbit /µm step step 1 step 2 step 3 step 4 step 5 step 6 step 7 step 8 step 9 step 1 step <= i <= M bpm

27 XFEL BD meeting / M.Vogt / Dispersion Free Steering for the SASE Undulators of the XFEL 27 All but 1-st 5 BPMs in T4 : 2 worse Resolution (2-nd attempt = today!!) CorrSeq : v2 / seed x orbit /µm step 2 step 3 step 4 step 5 step 6 step 7 step 8 step 9 step 1 step <= i <= M bpm

28 XFEL BD meeting / M.Vogt / Dispersion Free Steering for the SASE Undulators of the XFEL 28 All but 1-st 5 BPMs in T4 : 2 worse Resolution (2-nd attempt = today!!) CorrSeq : v2 / seed x 15 1 step 5 step 6 step 7 step 8 step 9 step 1 step 11 5 orbit /µm <= i <= M bpm

29 XFEL BD meeting / M.Vogt / Dispersion Free Steering for the SASE Undulators of the XFEL 29 All but 1-st 5 BPMs in T4 : 2 worse Resolution (2-nd attempt = today!!) CorrSeq : v2 / seed x 15 1 step 8 step 9 step 1 step 11 5 orbit /µm <= i <= M bpm

30 XFEL BD meeting / M.Vogt / Dispersion Free Steering for the SASE Undulators of the XFEL 3 TODO : Larger parameter space to be scanned (including varying of BPM distribution) y plane!!!! & SASE-2,-3,... Include deviations of actual (=unknown) ODRM from design ODRM (=known) Include x/y coupling Implement also uniform RVs, etc Implement drifts (time domain correlations) Include non linear dispersion into application of the kicks Work in progress!! SUMMARY : Even with state of the art diagnostics : orbit constrains for SASE very tight! In particular : initial misalignment and BPM offsets are tough Strategy : dispersion free steering with variable weighting between orbit and dispersion, variable τ ( strongly vs. weakly correlated modes) and variable gain. Result so far : with realistic tolerances and reduced BPM resolution upstream of the undulators the constraints seem extremely hard to meet!