Origins of Decoherence in Coherent X-ray Diffraction Experiments

1. Origins of Decoherence in Coherent X-ray Diffraction Experiments I.A. Vartanyants, I.K. Robinson Department of Physics, UIUC, Urbana-Champaign,USA Optics Communications (2003), 222, 29-50.

2. Optical Components on a Typical Beamline Source Double-Crystal Monochromator Sample Be window Slits Slits How coherent properties of the beam are changed passing through all this elements?

3. Laws of Propagation of Mutual Coherence Function Mutual Coherence Function: Γ(P1,P2;τ)=<E(P1,t+ τ)E*(P2,t)>T Mutual Intensity Function: J(P1,P2 )=Γ(P1,P2;0)= <E(P1,t)E*(P2,t)>T P1 R1 Q1 n1 Q2 P2 R2 n2 Σ1 Σ2 Propagation of MIF: Complex Coherence Factor:

4. Propagation of MIF through the Optical Element Be window Sample Source R1 R2 s r u L1 L2 MIF at sample position: Green’s function: (propagator) T(u) – amplitude transmittance function

5. Limits of Coherent and Incoherent Illumination Coherent illumination Incoherent illumination generalized van Cittert-Zernike theorem

6. Transmittance Function Complex transmittance function Be window d(u) Mirror h(u)

7. Propagation of MIF through a random OE • Assumptions: • Optical element (Be window, mirror) considered as a random optical element • Partial coherent incoming beam • in<<eff –transverse coherence length of the incoming beam is smaller then the intensity distribution MIF passing such random OE: autocorrelation function

8. Autocorrelation function for random phase object • Assumptions: • Gaussian statistics • Stationary random process varience normalized phase autocorrelation Autocorrelation function where

9. Propagation of MIF through a random OE Undistorted part: van Cittert-Zernike theorem: Rescattered part:

10. Propagation of MIF through a random OE Source (σsx, σsy) Sample JS(r1,r2) Be window (σwx, σwy) JW(r1,r2) L1 L2

11. Parameters used for calculation of <J( r)> Table II Transverse coherence lengths of the ‘broad’ sand ‘sharp’ wcomponents of MIF <J(r)> on sample position Table I Parameters used for calculation of the CCF (r) x,y-horizontal and vertical source sizes, (eff)-effective source size of the beam on Be window, 2 and -varience and longitudinal correlation length Letters (L) and (G) mean Lorentzian or Gaussian form of CCF W(r).

12. Complex Coherence Factor (r)=1.5 Å, L2=6 m

13. Imaging of small crystals Diffracted intensity (partial coherent illumination): projection of shape function of particle s(r,z) MIF at sample position: Coherent illumination |<(r)>|=1

14. Iterative phase retrieval algorithm FFT sk(x) Ak(q) Real Space Constraints Reciprocal Space Constraints s'k(x) A'k(q) FFT-1 • Real space constraints: • finite support • real, positive Reciprocal space constraint: R.W.Gerchberg & W.O. Saxton, Optic (1972) 35, 237 J.R. Fienup, Appl Opt. (1982). 21, 2758 R.P. Millane & W.J. Stroud, J. Opt. Soc. Am. (1997) A14, 568

15. Imaging of small crystals(partial coherent illumination) MIF <J(r)> on sample position diffracted intensity I(Q) reconstructed crystal images for two different sets of random phases

16. Conclusions • Optical elements in the beamline can change coherence properties of the beam • MIF in the incoming beam (on the sample position) can be multicomponent function due to propagation through different optical elements (windows and mirrors) • Partial coherence of incoming beam can introduce an artifacts in reconstructed images of nanoparticles