NanoBPM Analysis

1
Nano-BPM Analysis

• Sean Walston
• LLNL

2
Review of the Basics

• Fit waveforms from cavity BPMs to
• Define

3
Want x, x, y and y

• First define Position and Tilt signals
• For each BPM
• xsxPx
• xsxTx
• ysyPy
• ysyTy

4
Nano-BPM Apparatus
5
BPM Readout Calibration

• Beam assumed to be a by God! straight line
• Therefore trajectory is a constant from BPM to BPM

Position of beam in BPM
x
Slope12
Slope23
z12
z23
z
6
BPM Position Calibration
Position of beam in BPM
Position of BPM after move
x
Nominal position of BPM
Slope12
Slope23
z12
s1P1
s2P2
z23
s3P3
m2
m1
m3
x2
z
7
Calculating the Position Scales and BPM 2 Offset

• 4 Linearly independent equations from the moves
  m
• m1m2m30
• m10, m2m30
• m20, m1m30
• m30, m1m20
• 4 Unknowns
• 3 Scales, s1, s2, s3
• Offset of BPM 2, x2

8
BPM Tilt Calibration
Orientation of BPM 1 after tilt
Angle of beam in BPM
Nominal orientation of BPM 1
Orientation of BPM 2 after tilt
??
Coordinate axis
?1
Nominal orientation of BPM 2
s1T1
t1
Orientation of BPM 3 after tilt
?2
Coordinate axis
?2
Nominal orientation of BPM 3
s2T2
t2
?3
Coordinate axis
?3
s3T3
t3
9
Calculating the Tilt Scales and BPM 1,3 Angular
Offsets

• 5 Linearly independent equations from the tilts
  t
• t1t2t30
• t10, t2t30
• t20, t1t30
• t30, t1t20
• 5 Unknowns (one ??is indeterminate, say???)
• 3 Scales, s1, s2, s3
• 2 Offsets for BPMs 1 and 3, ?1 and ?3

10
Calculating ??
Orientation of BPM 2 after tilt
x
Nominal orientation of BPM 2
?2
s1P1
s2T2
t2
s3P3
m1
m3
z
11
Calculating the ?1,2,3

• There is more available here
• 4 Linearly independent equations from the moves m
  and t
• m1t2m30
• m10, t2m30
• t20, m1m30
• m30, m1t20

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13 Equations in 13 Unknowns!

• Bringing it all together
• 4 equations from line 1
• 5 equations from lines 2 and 3
• 4 more equations from line 4
• Solve using Singular value decomposition or
  similar technique

13
Very Preliminary Results

• y Position Calibration implemented to date using
  Singular Value Decomposition technique
• Early results promising
• Recent calibration runs appear to have the blobs
  spaced approximately right
• y calibration /- 20 ?m moves shown at right
• But not perfected yet