1. Transverse and Energy Jitters By Beam Position Monitors
Errors in Measuring
Transverse and Energy Jitters
By Beam Position Monitors
Vladimir Balandin
FEL Beam Dynamics Group Meeting, 1 November 2010

2. Problem Statement (I) 2 BPM 1 BPM 2 BPM 3 BPM n s = s1 s = s2 s = r
s = sn
GOLDEN TRAJECTORY
INSTANTANEOUS TRAJECTORY
Problem Statement (I)
RECONSTRUCTION
POINT 2

3. Problem Statement (II)3

4. Standard Least Squares (LS) Solution (I)4

5. Standard Least Squares (LS) Solution (II)
Even so for the case of transversely uncoupled motion our minimization problem can
be solved “by hand”, the direct usage of obtained analytical solution as a tool for designing of a “good measurement system” does not look to be fairly straightforward. The better understanding of the nature of the problem is still desirable. 5

6. Better Understanding of LS Solution
A step in this direction was made in the following papers
where dynamics was introduced into this problem which
in the beginning seemed to be static. 6

7. Moving Reconstruction Point
In accelerator physics a beam is characterized by its emittances, energy spread, dispersions, betatron functions and etc. All these values immediately become the properties of the BPM measurement system. In this way one can compare two BPM systems comparing their error emittances and error energy spreads, or, for a given measurement system, one can achieve needed balance between coordinate and momentum reconstruction errors by matching the error betatron functions in the point of interest to the desired values. 7

8. Compare with previously known representation
Beam Dynamical Parametrization of Covariance Matrix of Reconstruction Errors
Compare with previously known representation 8

9. Error Twiss Functions 9

10. Courant-Snyder Invariant as Error Estimator (I)10

11. Courant-Snyder Invariant as Error Estimator (II)11

12. Two BPM Case 12

13. XFEL Transverse Feedback System (I)13

14. XFEL Transverse Feedback System (II)14

15. XFEL Transverse Feedback System (III)15

16. Periodic Measurement System (I)16

17. Periodic Measurement System (II)17

18. Periodic Measurement System (III)18

19. COORDINATE AND MOMENTUM
Inclusion of the Energy Degree of Freedom
COORDINATE AND MOMENTUM
ERROR DISPERSIONS 19

20. Three BPM Case 20

21. Three BPMs in Four-Bend Chicane21

22. Summary 22
It was shown that properties of BPM measurement system can be described in the usual accelerator physics notations of emittance, energy spread and Twiss parameters.