Errors in MeasuringTransverse and Energy

Jitters By Beam Position MonitorsVladimir Balandin

FEL Beam Dynamics Group Meeting, 1 November 2010

BPM 1 BPM 2 BPM 3 BPM n

s = s1 s = s2 s = r s = s3 s = sn

GOLDEN TRAJECTORY

INSTANTANEOUS TRAJECTORY

Problem Statement (I)

RECONSTRUCTION POINT

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Problem Statement (II)

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Standard Least Squares (LS) Solution (I)

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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 whichin the beginning seemed to be static.

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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.

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Beam Dynamical Parametrization of Covariance Matrix of Reconstruction Errors

Compare with previously known representation

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Error Twiss Functions

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Courant-Snyder Invariant as Error Estimator (I)

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Courant-Snyder Invariant as Error Estimator (II)

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Two BPM Case

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XFEL Transverse Feedback System (I)

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XFEL Transverse Feedback System (II)

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XFEL Transverse Feedback System (III)

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Periodic Measurement System (I)

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Periodic Measurement System (II)

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Periodic Measurement System (III)

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Inclusion of the Energy Degree of Freedom

COORDINATE AND MOMENTUMERROR DISPERSIONS

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Three BPM Case

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Three BPMs in Four-Bend Chicane

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Summary

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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.