Emittance Calculations based on Maximum Entropy (MENT, MaxEnt,....)

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Emittance Calculations based on Maximum Entropy
(MENT, MaxEnt,....)
rmsEmittance reconstruction based on beam
profiles considering linear beam transformation(s)
Christoph Gabor, ASTeC (c.gabor_at_rl.ac.uk)Acknowle
dgement to Devinder S. Siva, Lecturer Oxford
Universityfor his help with MEMSYS 5 software
The Front End Test Stand Collaboration
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Outline of the presentation
(I.) Introduction Problems with
backward projection with noisy and incomplete
data (II.) Maximum Entropy An image
reconstruction method A very short
motivation about this technique Some
examples about achieved performance (III.)
Outlook Design of a slitpoint emittance
instrument Combination of a slitpoint
principle and movable angle detector applied
to the Front End Test Stand FETS (IV.) Summary
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Experimental Measurements
"Real" data typically suffer from noise and
incompleteness entailing problems for backward
projections
"test object" (object of study)
measured data
measurement instrument
for example spectrometer, telescope Traditiona
l solution
for example Fourier transform, line
integral, profile(s), (motion) blurred
image, image out of focus
for example Density of electrons plasma
density phase space distribution
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Problems with Inversion and more Advanced Approach
Ndata lt Nobject common model fitting, but
difficult to separate out the
features that follow the data and those
which come from the
model itself, always an answer whether assumed
model true/
false. Invent extra data to make Ndata Nobject
(e.g. inverse Fourier transform). Are unmeasured
Fourier components zero (as commonly
assumed)? Ndata lt Nobject should give rise
to a unique solution. Noise on the data often
amplifies errors (unstable,
deconvolution problems). However, very often
more advanced techniques are desirable.
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Probability Theory Bayesian Calculus
Scientific Inference
(Cox 1946, Jaynes 1963, Gull, Skilling MemSys)
"Kangaroo" argument entails a "best" set of
proportions pi on L by maximising the entropy
(Gull Skilling 1984)
No other function will always give the required
uncorrelated form of best proportions.
? proportion must sum to 1 ? positive and
additive distribution ? information theory based
entropy (Shannon entropy)
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The MaxEnt Data Processing
normally distributed error (noise, thermal)
gen. entropy S regularization
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References Literature
G.N.Minerbo, "MENT A maximum entropy algorithm
for reconstructing a source from projection
data", Computer Graphics and Image Processing 10
(1979), p. 4868O.R.Sander, G.N.Minerbo,
R.A.Jameson, D.D.Chamberlin, "Beam tomography in
two and four dimensions", 1979 LINAC--Conference
Montauk, N.Y.U.Rohrer, W.Rohrer, "Introduction
of 2dimensional beam tomography for monitoring
transverse beam emittance at SIN", PSI Ann. Rep.
1982, NL 56C.T.Mottershead (LANL, AT-6)
"Maximum entropy beam diagnostic tomography",
IEEE Trans. on Nucl. Sc., Vol. NS32, No.5,
1985http//pc532.psi.chWeb page of Uli Rohrer,
partly Coauthor of MENTX (previously
used)D.S.Siva, "Data Analysis A Bayesian
Tutorial", Oxford Science Publications (Clarendon
Press Oxford), 1996, 2006MEMSYS 5 software
libraries http//maxent.co.uk by J.Skilling
and S.Gull
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Envelope and Profiles
Use matrix formalism (assumed to be linear!),
invert matrix and use your constraints (beam
profiles) to calculate emittance Backward
projection with MaxEnt to avoid numerical
problems
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Entrance distribution
xx'projection
MAX MIN xy
x 5.7 mm - 6.3 mm
y 7.5 mm - 5.5 mm xx' x
- 5.7 mm - 6.3 mm x'
75 mrad - 73 mrad yy' y
7.5 mm - 5.5 mm y' -
80 mrad 66 mrad
-100.....100mrad
-20....20mm
xyprojection
xx' is "test object" Phase Space Advance
-20.....20mm
-20....20mm
yy'projection
-100.....100mrad
-20.....20mm
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Comparision of MaxEnt Reconstruction
withMeasured (Pepperpot) Entrance Distribution
Pictures use different intensity levels and
pseudo colours. Artefacts at the edges are caused
by starting conditions but sufficient phase space
advance makes it possible to separate remains of
starting point (default model) from reconstructed
object.
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rmsEmittance
Comparison of reconstructed and entrance
distribution in xx'. Numerical errors are bigger
than the shown variances.
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MaxEntVergleich mit verschiedenen Profilen
A 3profiles 25, 90, 150mm B 4profiles 75,
90, 100, 150mm C 8profiles 10....150mm D Entr
ance distribution (all pictures in same scale)
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NonDestructive Emittance Diagnostics for FETS
Front End Test Stand is supposed to demonstrate a
fast chopped beam with ...
? negative ions H- ? high beam current ? high
perveance ? not symmetric beam ? high beam
"quality"
nondestructive photo detachment laser
as a slit scintillator as angle detector ?
slitpoint transfer function (yy', xy) ?
movable detector (MaxEnt xx') ? possible to
rotate coordinate system
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MaxEnt applied to Phase Space Rotated Emittance
First Test with ideal QUAD measured distribution
rotated in phase space F -30...0...30 7
profiles in total
_at_ C.Plostinar M.Clark Gayther
Final design of MEBT (still under discussion) and
emittance diagnostic has to be adjusted that
phase space distribution of missing plane is
convergent (increased phase advance for MaxEnt
computation.
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Summary

• Can benefit of Bayesian probability theory
• Useful to calculate/ reconstruct emittance
  based on profile
• measurements.
• Profiles must have sufficient phase space
  advance to get
• good agreement in rmsemitttance.
• Still some work left ..... e.g. to specify
  phase space advance,
• influence of starting model, stop value a0
• Possible application Photo detachment, i.e.
  slitpoint
• transfer function including movable angle
  detector.
• Due to direct measurement of yy' and xy a
  rotating
• coordinate system gives all missing projections
  for 4dim phase space distribution (xx')a

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Baysian Statistics An Illustration of Bayes' Rule
Bayes theorem tells us how to update our
conclusions when we get more data. Suppose the
conclusion is that the temperature T of an ion
source plasma lies between T and T dT. Noise
is described by normal statistics (Gaussian).
prior
The old data T lies between 12 and 18 eV
The new data We measure T three times,
obtaining 3 different values. We only want one.
likelihood
posterior likelihood x prior
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Simulation about its "transfer function"
Neutralized particles are represented by cut out
of ions at an equivalent position Laser is
assumed at a certain yn
neutralized particles, drifted particles after a
drift of 0.1mm
phase space
beam profile
all photo detachment aspects like cross section
are neglected
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Extension of this concept by a movable detector
Drift of neutrals Variation in x y offers the
chance of determining both x' y'
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MEBT overview output (LINAC)
MEBT is not ready designed. Approximately 10...13
QUADs will be used for beam transportation and
final matching to assumed LINAC or beamdump.
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Pepperpot emittance measurements
Significant advantages considering the slit
extraction and sector magnet, usefull for beam
profiles (along the drift) as well
Pepperpot head

• 4 D phase space sampling
• 3 mm resolution, 60 mm range
• 5 mrad resolution, 100 mrad range
• good agreement with slit slit results

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Slitslit emittance measurements
Measured at the ISIS Penning H- ion source

• 2 x 2 D phase space sampling (uncorrelated)?
• 0.1 mm resolution, 40 mm range (slit length)?
• 2 mrad resolution, 100 mrad range