Techniques for the measurement and analysis of HOM signals for diagnostics

1. Techniques for the measurement and analysis of HOM signals for diagnostics Stephen Molloy European Spallation Source

2. Overview • Well known that HOMs are “bad”, • But can they be “good”? • FLASH experiment • Hardware • Analysis • Results • Future? • Multibunch?

3. Usefulness of HOMs • Mode excitation depends on the beam • Coupling defined by trajectory (for dipole & higher) • Phase of excitation depends on beam arrival time • Each mode carries information • Can this be extracted? • Vacuum/cryo infrastructure already exists • Bandstop filtered coupler extracts power • Replace resistive load with monitoring devices • Very cheap hardware installation!

4. Dipole modes are polarised • Two polarisations, rotated by π/2 • Not necessarily coincident with xy plane • Alignment may depend on cell number • Four degrees of freedom • Phase & amplitude in each polarisation • Horizontal & vertical position • Only 2 trajectory dofs? • What info is carried by the other dofs?

5. “Beam” degrees of freedom Cannot separate these two Analyse bunch with finite length, σz, as two “macro-particles” Head & tail particles excite equal but opposite signals, separated by σz/c. Vector sum is really a time-delayed subtraction – much like a differential. Thus, the “tilt” signal is equivalent to the “offset” signal, but phase rotated by π/2.

6. Our experiment

7. Measurement electronics Mix to 20MHz IF BP filter & amplify LP filter & amplify Input from coupler Digitize at ~100MS/s

8. Raw data Two exponentially decaying sinusoids. Polarisation degeneracy broken, so two frequencies beating against each other. Beat frequency indicates Δf is very small. Calibration tone

9. Analysis • “Standard analysis” • Determine complex amplitude of the signal • Correlate real and imaginary components with position and angle • Matrix transform  rotation & scaling • Problem…

10. Idea? • Data from each pulse is a vector • Actually two vectors (one from each coupler) but they may be concatenated into one. • 1600 points per coupler per acquisition • This could be considered as 1 point in a hugely dimensional space (3200 dimensions!) • A dataset is a “fuzz” plotted in 3200-D! • The data produced by a “pure” move in 1 dof (x, x’, y, y’) is also a vector • New coord system is defined by aligning with each of these • How? • What about the other 3196 dims?

11. X  data matrix (100x3200) Matrix decomposition Singular Value Decomposition: • V contains “time dependent” eigenvectors • Unitary, so (by definition) also orthonormal • New basis for 3200-D data? • X may be represented by U.S in “V space” U & V are unitary S is diagonal. Contains the “singular values”. Each of these is an eigenvalue equation, and U & V are matrices containing the respective eigenvectors.

12. How to use this new vector basis? • For each incoming pulse: • Determine location in 3200-D space • Dot product with V vectors • Correlate this with trajectory info from other devices • SVD calculates and orders V to maximise each subsequent singular value • Only four beam dofs, so we can discard 3196 vectors! • Maybe choose 6 to be safe! • Correlate these with the trajectory

13. Identical to JPEG compression! • Break image into blocks • 8x8 pixels • Dot product each with 2D basis function • Converts 64 pixel image to 64 amplitudes • Compress by discarding information • Remove high frequency blocks

14. Eigenmodes & singular values

15. Results ~0.3 mm Resolution ~4 μm ~0.3 mm

16. 4 μm resolution – is that good? • No! • Compare: • The power in the mode • with • The thermal noise in the electrons • Find the beam position at which these are equal • 130 nm!!! • Problem in the electronics • Signal jitter mixes position & angle • Updates could fix this…

17. What about multi-bunch? • Decay time >> bunch separation • Can bunches be separated? • Subtracting bunch by bunch leads to large errors • Work with single bunch modes • Measure the mode amplitudes in one bunch window • Then again in the second window • No bunch in this region! • Calculate the transformation matrix • Calculate mode amps for each bunch subtracting the previous bunch

18. Multibunch data Multibunch residual is equivalent to a position noise of ~1-2 um. Added in quadrature to 4 um, this increase is negligible.

19. Stability • In real world: • Diagnostics must be stable as well as high res. • Can’t constantly interrupt machine time to calibrate HOM BPMs! • Questions: • Why does the calibration change? • Temperature drifts, cable changes, etc. cause phase rotations and gain changes • Can we prevent it changing? • Solution? • Measure gain/phase of cal. tone. • Remove gain/phase deltas

20. Convert to “pure” modes • Construct pure modes from cal data • Monitor phase/gain of cal tone • Alter incoming data appropriately • Determine amplitude of pure modes

21. Summary • Impossible to completely eradicate HOMs • Dealt with by appropriate accelerator design • This infrastructure can be designed to allow HOMs to be useful • 4D trajectory monitoring • Many machines are dominated by their linac, so such measurements give lots of information • Other measurements possible • Bunch timing wrt accelerating phase • Internal alignment of cryomodules & cavities