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

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

Problem Statement (II) 3

Standard Least Squares (LS) Solution (I) 4

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

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

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

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

Error Twiss Functions 9

Courant-Snyder Invariant as Error Estimator (I) 10

Courant-Snyder Invariant as Error Estimator (II) 11

Two BPM Case 12

XFEL Transverse Feedback System (I) 13

XFEL Transverse Feedback System (II) 14

XFEL Transverse Feedback System (III) 15

Periodic Measurement System (I) 16

Periodic Measurement System (II) 17

Periodic Measurement System (III) 18

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

Three BPM Case 20

Three BPMs in Four-Bend Chicane 21

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.