Simulation of Measurement of Transverse Coherence 21 April 2005 Sven Reiche

1
Simulation of Measurement ofTransverse
Coherence21 April 2005Sven Reiche

• Schematic Layout of Experiment
• Numerical Modeling
• Results - Single Wavefront
• Results - Full Radiation Profile

2
Experimental Consideration

• Standard method with slits/holes is unpractical
• Precision of position and width in Angstrom
  regime
• Thick mask to absorbed radiation, not going
  through slits/holes
• Huge background signal from spontaneous
  radiation.
• Small slit/hole separation (d150 ?rad correspond
  to 1 ?rad for angle of observation of the first
  maximum)
• No single-shot measurement of transverse
  coherence.
• Measurement should fit in single experimental
  station.
• Scattering on single crystal with well-known
  properties.

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The Generic Method

• Based on structure analysis with coherent light
  source
• Phase reconstruction possible due to 2D
  oversampling theorem (e.g. used in FROG).
• With structure known the incident field can be
  reconstructed.

Because the process cannot a time-resolving
radiation wave it would reconstruct an equivalent
steady-state field with limited coherence.
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Modeling Problems

• Exposed area 1 mm2
• Crystal grid size 5 Ångstrom
• 3D Crystal generates many Bragg-Peaks with a lot
  of empty space of deflect radiation.
• Strongly deflecting Bragg-Peaks mix longitudinal
  and transverse coherence

Grid plane
Interference plane
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Model Assumption

• 2D Grid only (selects only one Bragg Peak)
• Point-like sources for scattered photons (Huygens
  principle)
• Far-field zone, small angle
• Analytical solution of scattered field As for
  small sub-domain of grid with constant incident
  field Ai

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Testing of Algorithm

• Generic Gauss-mode as incident field with waist
  position on grid (d4 Å, ?1.5 Å)
• Dependence of rms width of Bragg peak on waist
  size of incident field.

Width is inversely proportional to incident beam
size, as expected from this Fourier-like
transformation.
23.77 / w0
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Waist Position

• With waist of Gaussian mode placed on grid the
  curvature of the wavefront is excluded.
• No dependence on waist position, confirming the
  Fourier-like dependence between input and
  scattered field.

With wavelength known, the rms width is a measure
for the mode size at its waist position.
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LCLS - Single Wavefront

• Steady-state run of LCLS with saturation at
  around 100 m (undulator length is 130 m)
• Propagation of the wave front till the
  experimental station (120 m), using Fresnel
  integration of wavefront.

Scattered Field
Incident Field
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Coherence - Single Slice

• RMS size of incident field 120 ?m.
• RMS size of scattered Bragg peak 0.85 ?rad
• Equivalent RMS size of Gaussian mode 27.9 ?m
• However radiation is diffracting from the end of
  the undulator (and even before in the saturation
  regime).

Undulator exit
The RMS size at the undulator exit is 29.3 ?m.
Comparing with an equivalent Gaussian mode the
coherence is 95
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LCLS - Full Pulse

• Time-dependent run of full LCLS profiles
  (start-end simulation) with 15000 slices
• Calculation of diffraction pattern took 3 days on
  a 11 Node Beowulf cluster -gt only one calculation
  done

Scattered Field
Incident Field
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Coherence - Full Pulse

• Comparison to single slice results

Undulator exit
Strong asymmetry in near field profile with
second wide distribution, centered around x100
?m.
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The Time-Dependent Run
profile
spectrum
centroid
Initial centroid in x
20 ?m 50 ?m 62 ?m
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Initial Discussion of Results

• Higher-Mode content
• Pedestal level of radiation profile
• Larger radiation size at grid (115 m downstream)
• Larger opening angle of Bragg Peak
• Background signal in near field distribution at
  undulator exit.
• Possible caused by centroid motion of electron
  slices
• Method of equivalent fundamental Gaussian mode is
  error-prone. A high mode (e.g. Gauss-Hermite Hmn)
  would yield an equivalent waist size of
• Need better method to define transverse coherence
  (reconstruction of incident radiation field)

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Conclusion

• Slits/holes method is unpractical due to its
  extreme dimensions, stringent tolerance on the
  mask, the background signal from the spontaneous
  radiation
• Measurement of transverse coherence is similar to
  structural analysis of 3D objects.
• Reconstruction of incident radiation wavefront
  with recursive algorithm based on 2D oversampling
  theorem.
• Method of equivalent Gaussian only of limited
  value if higher mode content is present.
• Simulation needs to be continued and further
  developped to include the reconstruction
  algorithm.