Propagation of the Mutual Intensity Function Through Optical Elements

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Propagation of the Mutual Intensity Function
Through Optical Elements
Ivan Vartaniants HASYLAB, DESY, Hamburg, Germany
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CONTENTS

• Introduction
• Coherent X-ray Diffraction Experiments
• Problems
• Propagation of MIF from Source to Sample
• Problems
• Propagation of MIF Through Optical Elements
• Measurements of Coherence
• Links to Laser Sources
• Conclusions

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Lensless X-ray Microscopy
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Imaging of biomolecules with femtosecond X-ray
pulses
Explosion of T4 lysozyme induced by radiation
damage
R. Neutze, et al., Nature (2000) 406, 752
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Imaging of Biomolecules with Femtosecond X-ray
Pulses
Simulated continuous scattering image of a single
T4 lysozyme molecule under ideal conditions
without sample movement or damage.
R. Neutze, et al., Nature (2000) 406, 752
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Coherent Scattering on Au Crystals

• 4 micron Au nanocrystal
• Coherent X-ray Diffraction
• Measurement at APS sector 33-ID (UNICAT)

I. Robinson, I. Vartanyants, et al., PRL (2001),
87, 195505
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Coherent Diffraction Tomography
G. Williams et al., PRL (2003) 90, 175501
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Definitions
P1
S
P2
L1
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Laws of Propagation of Mutual Intensity Function
R2
Can be further calculated using paraxial
approximation
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Sources
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Van Zittert-Zernike Theorem
Sample
Source
R1
                         
 
s
R2
L
r
MIF from an incoherent source is a FT of
intensity distribution from this source
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Synchrotron source
Intensity distribution (Gaussian)
Single Gaussian function can fully describe
coherence properties of the Gaussian source
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Synchrotron source
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Imaging Experiment (small crystals)
Diffracted intensity (partial coherent
illumination)
projection of shape function of particle s(r,z)
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Imaging of small crystals(partial coherent
illumination)
MIF J (?r) on sample position
diffracted intensity I(Q)
reconstructed crystal images
I. Vartanyants I. Robinson. J.Phys. Condens.
Matter (2001) 13, 10593
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Optics
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Optical Components on a Typical Beamline
Source
Sample
How coherent properties of the beam are modified
passing through all these elements?
 
Start to End Calculations
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Propagation of MIF through an Optical Element
Optical Element
Ein
Et
                         
 
u
Transmitted amplitude
T(u) amplitude transmittance function
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Propagation of MIF through an Optical Element
Optical Element
Sample
Source
R1
                         
 
R2
s
r
u
L1
L2
Coherence properties of the beam in general are
modified passing optical elements in the beamline
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Limits of Coherent and Incoherent Illumination
Coherent illumination
MIF factorizes
Phase object just after incoherent source does
not change coherence
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Propagation of MIF through an Optical Element

• Two ways to proceed
• To take an 'exact' model for an optical
  element
• To use statistical approach

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1-st approach Transmittance Function
Complex transmittance function
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2-nd approach Propagation of MIF through a
random OE

• Assumptions
• Optical element (Be window, mirror) considered as
  a random optical element
• Partial coherent incoming beam
• Transverse coherence length of the incoming beam
  is smaller then the intensity distribution
  ?inltlt?eff

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Autocorrelation function for random phase object

• Assumptions
• Gaussian statistics
• Stationary random process

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Propagation of MIF through a random OE
Undistorted part
Rescattered part
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Propagation of MIF through a random OE
                                       
 
Source (ssx, ssy)
Sample
JS(r1,r2)
Be window (swx, swy)
JW(r1,r2)
L1
L2
I. Vartanyants I. Robinson. Opt.Comm. (2003)
222, 29
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Complex Coherence Factor ? (?r)?1.5 Å, L26 m
?2?0.5 ?1 ?m
?2?0.1 ?1 ?m
?2?0.5 ?0.1 ?m
?2?0.5 ?1 ?m
CCF can not be described by single Gaussian
function
The whole shape of the CCF is important not
only averaged characteristics (first, second,..
moments)
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Imaging of small crystals(partial coherent
illumination)
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Coherence measurements
F. Pfeiffer et al., Phys. Rev. Lett. (2005) (to
be published)
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Coherence measurements
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FEL and Laser Sources
Amplitude of electric field
Only laser in single mode operation will be
completely transverse coherent
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Coherence Properties of FEL
Averaged efficiency lt?gt of SASE FEL versus
undulator length
Degree of transverse coherence ? of the radiation
from SASE FEL
E. Saldin, E. Schneidmiller, M. Yurkov, NIM A
(2003), 507, 106
?1/M
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Coherence Properties of Multimode Laser
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CONCLUSIONS

• For CXD imaging experiments full transverse
  coherent beam is desirable (artifacts in
  reconstruction)
• For partially coherent beams coherence properties
  can be affected by the presence of the optics in
  the beamline
• For FELs coherence properties of the x-ray beam
  can be, in principle, calculated starting from
  the undulator to sample position (Start to End
  calculations)
• Expected coherence properties of the FEL in
  multimode regime will be similar to incoherent
  source

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Thank you for your attention
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Coherence
LL?2/(2? ?)
Longitudinal Coherence Length
Als-Nielsen McMorrow (2001)
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Propagation of MIF through Be window
Yabashi et al., J.Phys. D (2004), to be published
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3D Scan of the Reciprocal Space
G. Williams et al., PRL (2003) 90, 175501
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Coherence and Partial Coherence
R1
P
P1
R2
P2
S
D
L1
L2
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Coherence and Partial Coherence
?12(?)1 coherent scattering ?12(?)
0 incoherent scattering 0lt?12(?)lt1
partial coherent scattering
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Intensity distribution
Solution We have to replace real source with an
effective source with an effective coherence
length (generalized van Zittert-Cernike theorem)
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Intensity distribution
?s
?s
?sltlt?s
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Intensity distribution
MIF at sample position (in the far-field)
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Synchrotron source
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Iterative phase retrieval algorithm
FFT
sk(x)
Ak(q)
Reciprocal Space Constraints
Real Space Constraints
A'k(q)
s'k(x)
FFT-1
Real space constraints
Reciprocal space constraint

• finite support
• real, positive

J.R. Fienup, Appl Opt. (1982). 21, 2758 R.P.
Millane W.J. Stroud, J. Opt. Soc. Am. (1997)
A14, 568
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2-nd approach Propagation of MIF through a
random OE

• Assumptions
• Optical element (Be window, mirror) considered as
  a random optical element
• Partial coherent incoming beam
• Transverse coherence length of the incoming beam
  is smaller then the intensity distribution
  ?inltlt?eff

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Examples