Initialization of Beam Simulations using Particle Distributions Reconstructed from Experimental Data*

1
Initialization of Beam Simulations using Particle
Distributions Reconstructed from Experimental
Data

• Alex Friedman and David P. Grote, LLNL
• Frank M. Bieniosek, Christine M. Celata and
  Lionel R. Prost, LBNL
• Presentation BP1.110
• Annual Meeting of the APS-DPPNovember 10-15,
  2002
• Orlando, FL

Work performed under the auspices of the U.S.
Department of Energy by the University of
California, Lawrence Livermore and Lawrence
Berkeley National Laboratories under Contract
Nos. W-7405-Eng-48 and DE-AC03-76SF00098
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Abstract

• In present-day experiments exploring the behavior
  of intense particle beams for Heavy Ion Fusion, a
  variety of diagnostics yield information on the
  beam particle distribution function. Good
  shot-to-shot reproducibility enables the
  acquisition of detailed 2D projections of the 4D
  transverse phase space using moving slits and
  Faraday cups
• In trials using simulated beam data, we have
  observed a significant improvement in simulation
  fidelity when runs are initiated using
  tomographically synthesized (as opposed to
  simple-model) distributions based on slit scan
  data
• We are beginning to employ such synthesis
  techniques to launch simulations with initial
  conditions developed from diagnostics at an
  upstream station on the High Current Experiment
  at LBNL
• See also Thursday afternoon poster RP1.065 by C.
  M. Celata, et. al., and other posters in that
  session

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Introduction problem and methods
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Simulations can be carried out at three levels

• ideal beams straightforward
• realistic beams harder
• use ensembles of integrated runs for planned
  experiments
• the real machine difficult!
• need detailed machine characterization,
  diagnostics
• integrated simulations tend to drift from
  actual experiments
• ? iterate, varying misalignments, voltage
  errors, etc.
• ? couple with synthesis approach

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Slit-scan diagnostics yield time-dependent data
Two-Slit Emittance Scan Used to measure the beam
emittance in direction perpendicular to the slits
slit 2 selects for transverse momentum slope
x? ? px/pz
slit 1 selects for position x
detector
Crossed-Slit Intensity Map Used to examine the
distribution of beam current density A
scintillator may yield similar data, while a
kapton foil yields a time-integrated intensity map
The second paddle typically has a shallow,
reverse-biased faraday cup attached to measure
the beam signal
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Statement of the mathematical problem

• A longitudinally-cold steady flow can be
  described by the particle distribution
  ƒ(x,y,x?,y?) in the 4D phase space, where x?
  px/pz and z is the axial coordinate
• In reality, beams have a small spread in pz and
  transverse motion isnt independent of pz
  nonetheless, this is a useful approximation
• Experimental diagnostics typically yield 2D
  projections of ƒ(x,y,x?,y?), e.g., ƒ(x,x?) ?
  ƒ(x,y,x?,y?) dy dy?
• We want to launch simulations using an initial
  beam state derived from experimentally measured
  data a goal is to reproduce the observed beam
  behavior in detail
• So we are trying to solve the problem
• Given measured ƒ(x,x?) and ƒ(y,y?) projections
  of ƒ(x,y,x?,y?), and perhaps ƒ(x,y) as well,
  synthesize a reasonable ƒ(x,y,x?,y?)
• This problem is underdetermined

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The problem is underdetermined as stated to make
it well posed, we add constraints

• The available data is insufficient to fully
  specify the 4D distribution
• For ƒ(x,y,x?,y?) on a 4D grid with, e.g., 20
  points along each Cartesian axis, need 204
  1.6?105 points
• The measured (x,x?) and (y,y?) projections each
  consist of 400 data points, so we have a total of
  800 data points, far fewer than is demanded by a
  complete description
• We may also know the intensity ƒ(x,y) then we
  have 1200 points
• We usually dont know the other Cartesian
  projections (x,y?), (y,x?), or (x?,y?) but
  even if we knew them, plus a few other
  projections at angles to the axes, it would not
  suffice
• We impose additional constraints to complete the
  specification
• We use a Monte-Carlo approach related to maximum
  entropy
• This is augmented (optionally) by imposing a
  sampling region with rounded corners

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This 2-plane method reproduces, in the limit of
many particles, the measured (x,x?) and (y,y?)
data

• Assign target counts N(x,x?) and N(y,y?) the
  numbers of particles to be loaded into each bin
  proportional to the measured ƒ in each bin
• These bin counts should sum to the desired total
  number of particles
• Repeat the following until all particles have
  been set
• Generate a random point in a 4-box, and accept it
  only if it
• (a) falls in a pair of bins whose counts are
  nonzero, and
• (b) falls within a user-specified sampling
  region
• If it is accepted, decrement the counts in the
  corresponding bins
• The sampling region is the intersection of
• four 4-cylinders, the extrusions of circles in
  (x,y), (x?,y?), (x,y?), (y,x?) (nominally of
  radius 1 relative to the data extent, but
  adjustable)
• a 4D hyperellipsoid (nominally of radius v2, but
  adjustable)
• Method can paint itself into a corner (fail to
  load all particles desired)
• One 3-plane method proceeds similarly, with
  addition of counts N(x,y)

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An attractive 3-plane version uses bin counts
N(x,y) only in the spatial plane

• Compute bin counts N(x,y) ? ƒ(x,y) mapped onto
  working grid
• Correct probabilities to account for rounded 4D
  sampling region
• Compute V(x,x?) number of (y,y?) grid points in
  sampling region
• Compute ƒ(x,x?) ƒ(x,x?) Ny Ny? HV(x,x?) /
  V(x,x?)10-99
• i.e., divide the acceptance probabilities in
  (x,x?) by the area of the slice of the sampling
  region in the (y,y?) plane at that (x,x?)
• Similarly for ƒ(y,y?)
• Until done, generate random points in the
  four-space accept each if
• It falls inside the sampling region
• ƒ(x,x?)/Max(ƒ(x,x?)) gt Random(0,1) and
  similarly for (y,y?)
• The remaining bin count N(x,y) gt 0
• If a point is accepted, decrement the
  corresponding N(x,y)
• This algorithm avoids the painting into a
  corner problem

These algorithms, and all other processing, were
implemented in the very high level scripting
(interpreted) language Yorick, using vector
constructs for efficiency (very few
element-at-a-time loops are used)
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Tests using simulated data
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Tests of synthesis at the exit of the HCX ESQ
injector are based on a WARP3d run (so we know
the 4D distribution)
(frame from a WARP movie by J-L. Vay)
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Matching section compresses beam significantly
before it enters the HCX transport line
(frame from a WARP movie by J-L. Vay)
(frame from a WARP movie by J-L. Vay)
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Self-consistent simulations of HCX use WARP3d in
injector, and WARPxy in matching section and
transport line
x? ? px/pz -0.002 0 .002
-0.01 0 .01
-0.01 0 .01
End of injector
End of matching section
50 lattice periods (100 quads) past matching
section
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Reconstructions of self-consistent simulated ƒ at
injector exit
self-consistent simulation (E. Henestroza)
2-plane synthesis using (x,x?) (y,y?) data
3-plane synthesis using (x,x?), (y,y?), (x,y)
data
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Comparison of HCX simulations shows benefit of
initiating the run with a good approximation to
ƒ(x,y,px,py)
2-plane synthesis using (x,x?) and (y,y?)
3-plane synthesis adding (x,y) data
Integrated simulation, starting at source
Normalized x emittance (?-mm-mr)
semi-Gaussian

• HCX simulations initialized using synthesized 4D
  particle distributions based on ESQ-exit
  slit-scan data (simulated, here) are far better
  than those using an ideal distribution
• An (x,y) scan, in addition to (x,x?) and (y,y?),
  is important

z (m)
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Synthesized ƒ leads to WARPxy simulations with
higher fidelity than RMS-equivalent semi-Gaussian
(y,y? shown)
2-plane synthesis using (x,x?) and (y,y?)
3-plane synthesis adding (x,y) data
Integrated simulation, starting at source
Normalized y emittance (?-mm-mr)
semi-Gaussian
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Initial synthesis and simulation using
experimental data
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High Current Experiment (HCX) layout
start - simulation - end
Detail of diagnostics at transport line entrance
(QD1)
upstream slits
downstream slits would collide, cant be used
simultaneously crossed-slit map uses upstream
x slit and downstream y slit
kapton
y
x y
x
beam
9.1 cm
(x slits are vertical, move horizontally from
shot to shot y slits are horizontal )
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Quite a few manipulations are needed to put the
experimental data in a usable form

• Preliminaries
• Read in raw data and transform to HCX Global
  Coordinates
• Plot beam moments vs. time, and select a
  beam-frame slice for analysis / synthesis (WARPxy
  follows a single slice per run)
• Apply threshold to experimental ƒs (to remove
  noise bias)
• Map emittance scan data to working grid (finer
  than the raw data grid)
• Compute and then remove ?x?, ?x??, ?y?, ?y??
• Compute and then remove net beam envelope
  convergence in each plane, proportional to
  ?xx?? and ?yy??
• For each data column at constant x (which may
  start at its own unique x?), fit a cubic spline
  and compute ƒ on working-grid x? nodes
• Then use a spline fit along x, for each row in
  x?, to get a 2D ƒ on working grid
• Zero-out nodes outside the raw-data domain
  (spline fits can do weird things)
• Normalize the data to range from 0 to 1, yielding
  ƒ(x,x?) and ƒ(y,y?)
• Map crossed-slit data to working grid
• Use the net convergence from the emittance scan
  (for each time slice) to rescale the y values,
  since the synthesis is done at a plane well
  upstream of the y slit
• Threshold and normalize
• Map kapton data from its native very-fine grid to
  working grid, for comparison

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HCX data at exit of matching section (station
QD1) at one time-slice is the preferred input
to WARPxy simulations
Data at a single time slice

• emittance scan data
• (after removal of net envelope convergence /
  divergence)

raw crossed-slit (distorted because y slit
is far downstream)
Dots denote raw data locations
remapped crossed-slit
Time-integrated spatial intensity is smeared
kapton
remapped crossed-slit
Crossed-slit plots at time slice clipped to 85
of peak
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First 3-plane synthesis using HCX data at
entrance to transport line was based on
emittance scan and kapton data
x,x slit data
n(x)
y,y slit data
x
x,y as synthesized
x,y kapton data
The kapton data is time integrated since Ibeam
varied with time, that image is smeared and
results were marginal
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3-plane syntheses are now based on time-resolved
emittance scanner and crossed-slit data, mapped
to a common station
Projections of synthesized distribution agree
well with input data here, ƒ(x,x?)
effects of round-edge constraint
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This early D-end data shows the corresponding
time-slice of the beam, at the exit of the
10-quad HCX transport line
(data has been recentered beam was sent off-axis
by misaligned quadrupole lenses)
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Initial simulations are only in coarse agreement
with early data at D-end of HCX
remapped crossed-slit
emittance scan y,y?
Spatial hollowing is a common feature
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Discussion and plans
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Future Plans

• We plan to track down the sources of discrepancy
  between HCX data and simulation output, in search
  of quantitative agreement
• In addition to this inverse approach, we are
  also working the forward problem via integrated
  simulations starting from the source
• To achieve agreement with experimental data at
  downstream stations, it may be necessary to vary
  (within the error bars) voltages, alignment,
  and/or source inhomogeneities.
• While the forward approach is more consistent,
  it is laborious, and we anticipate that much of
  our day-to-day work will be done using the
  mid-stream initialization procedures described
  here
• Recently, slits have been combined with
  scintillator-based imaging methods, and the use
  of multi-aperture hole plates is under study
• These methods yield 3D (projectional) and 4D
  (full) phase space information, but with
  relatively coarse sampling.
• We plan to develop the capability to make use of
  such data to initialize simulations, and in
  general, to fold any extra information, e.g.,
  localized data at high resolution, into the
  synthesis

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Extras
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Integrated WARP runs, compared w/ data
Simulation Pg. 108 data
current 555 mA 175 mA
energy 1.79 MeV 1.045 MeV
beta 9.9 ? 10-3 7.6 ? 10-3
emittance (RMS) 1.2 ? 10-5
emittance (edge) 4.8 ? 10-5
norm emit (edge) 5 - 12 ? 10-7 3.6 ? 10-7
norm emit (edge) scaled by square root of current ratio 6.5 ? 10-7
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WARPxy simulations initiated using a synthesis
based on data at QD1 views of simulation phase
space at D-end
phase space x,x? view

• In this run, the simulated distribution is only
  in coarse agreement with early HCX data
• This work is quite new (Nov. 2002) we plan to
  track down the source of the discrepancy
• Ongoing experiments are yielding well-centered
  beams with improved temporal flat-tops and
  better diagnostic resolution

spatial intensity x,y
phase space y,y? view

• Spatial hollowing is a common feature

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HCX transverse phase space data at exit of
matching section (station QD1) is input to
WARPxy PIC simulations

• Emittance scan data
• (after removal of net envelope convergence /
  divergence)

Raw crossed-slit map
At time slice
At time slice
At time slice
Dots denote raw data locations
Kapton foil damage pattern
Crossed-slit data remapped to z(kapton)
At time slice
Time integrated
Time integrated
Crossed-slit plots at time slice clipped to 85
of peak