Linearity of the Electron Optics

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Linearity of the Electron Optics
Alexey Burov
RR Talks
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Examples of YAG images

• In many cases, YAG images of e-beam had distinct
  triangular shape,
• pointing to a sextupole nonlinearity (Sasha,
  Lionel).
• This nonlinearity has to be seen in differential
  orbit measurements.
• The differential orbit measurements were done by
  Mary
• A5 A6 correctors Oct 29 (not clean in CS) ,
  Nov 6 (OK for CS)
• S2 S5 - Nov 1
• S3 Nov 26 S4 Nov 29.

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Nonlinearity Coefficient

• For every mentioned corrector, 10 equidistant
  kicks were applied, from the highest possible
  negative, to the highest positive. For every
  corrector j, the BPM data were saved in P163
  format.
• To see nonlinearity in these data, I prepared a
  MathCad file, which makes a linear fit for every
  given BPM i , and subtract this fit from the
  data
• For every BPM i, the nonlinearity is
  characterized by
• For every corrector, the measure of nonlinearity
  of its trajectories can be presented as

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Example for CXA05
BXS05, cm
BYS05, cm
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E-Cool Line
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Results

• The numbers are not negligible. Currently, the
  angle is estimated 100µrad on the axis.
• Nonlinearities for below CS2 are small.
• Also nonlinearities for BS2, BS3 were always
  small (not shown here).
• Max nonlinearity are always at XC4, YC8 ( p/2
  of the Larmor phase).
• Hypothesis
• The main source of nonlinearity is located at
  the lens S3

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Accuracy (CYS2 kicks)
Error of CYS2 current (or is it MI perturbation?)
is equivalent to 100-200 microns of the bpm
error. And this case is not unique How can we
avoid this?
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How the error was evolved downstream the line
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How to check the hypothesis

• To check the SS3 hypothesis, a local bump
  CYS2-CYS3-CYS4 can be done.
• Then, for several bump mults, bpm data can be
  taken and compared with the nonlinear parts of
  Nov 1 CYS2 measurements.
• Nov 1 CYS2 CXS2 measurements should be redone,
  due to their high errors for some points.

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Cylindrical Aberrations of Ideal Lens

• Focusing strength of a solenoid can be derived
  as
• Assuming
• In local bumps, this nonlinearity has to be seen
  as a cubic parabola in a plane of the bump, and
  nothing in the other plane, as soon as the lens
  is optically thin.
• For

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Bump nonlinearity

• For a linear local 3-bump around a doublet
• A nonlinear offset in the bpm BS4
  , for

SS3

-
BS4
BS3
CS2
CS3
CS4
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Cx/yS2 Bump Data
CYS2 bump
CXS2 bump
Huge nonlinearity of SS3! 14/2.56 times of the
ideal level
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CS2 bumps show

• Nonlinearities seen in these bumps
• None of them has symmetry of the solenoidal lens
• Y-bump is 3 times more nonlinear as X-bump (as
  in P163, p.5)
• Absolute value of Y-bump nonlinearity is 5 times
  higher the ideal level (also agrees with the P163
  data in p.5).
• Dependences of BS3(CS2) are all linear at the
  noise level.
• Thus, these bump data confirm my hypothesis about
  extremely nonlinear S3 lens

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CS3 bump
CYS3 bump
CXS3 bump
Nonlinearity of SS4 is 5/2.5 2 times of the
ideal level.
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Lens T4

• The same way of measurements with the 3-bump
  around ST4 (Mary, 08Feb21_linearity measurement)
  yields non-linearity as 4 times of the ideal
  level.

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Conclusions

• Bump measurements give the same ratios of
  nonlinearities as P163.
• The hypothesis of extreme SS3 nonlinearity (6
  times more than natural , 3 times more than SS4)
  is confirmed.
• Other measured lenses have following
  nonlinearities
• SS4 2 times of the natural level
• ST4 4 times of the natural level
• It is reasonable to explore the most linear part
  for the beam position in SS3 (-5 mm), and to
  optimize optics, reducing the field in SS3.