1. Imaging of Quantum Array Structures with Coherent and Partially Coherent Diffraction
I.A. Vartanyants, I.K. Robinson
Department of Physics, UIUC, Urbana-Champaign,USA
J. Synch. Rad. (2003), submitted

2. X-ray scattering on nanostructures GISAXS measurements
-0.03
Qy (Å-1)
0.03
High-resolution AFM images of PbSe dots on
PbEuTe showing the pyramidal island shape
-0.03
0.03
0.0
Qx (Å-1)
J.Stangl et al. Material Science & Engineering, C19 (2002) 349

3. Imaging of Quantum Dots with Coherent Beams
Source
Sample
CCD
Electron density of periodical array of QD’s
S(r) – shape of coherently illuminated area
sz(r) –projection of shape of one island s(r,z)
Scattered amplitude
Diffracted intensity
cross terms

4. 2D array of QD’s and it’s diffraction pattern
Diffraction pattern of 2D array
Image of individual
island
Diffraction pattern of
individual island

5. Reconstructed image of 2D array of QD’s
Support used for reconstruction
Diffraction intensity of reconstructed image
Reconstructed image with
superposition of twin images
Reconstructed image

6. Imaging of QD’s with partially coherent beams
Intensity distribution with partially coherent illumination
p(r) - electron density distribution,
Jin(r) - mutual intensity function
lcoh – transverse
coherence length
Complex coherence factor
Intensity distribution
Diffracted intensity from 2D array of QD’s:
cross terms

7. Tests of Partial Coherence Effects in Imaging
2D array used for tests of
partial coherence effects
Diffraction pattern of 2D array
Shape of individual island
Diffraction pattern of individual island
A. Szöke Acta Cryst. (2001) A57, 586

8. Diffraction intensity from QD’s with partial coherent illumination
lcoh= 124 px
lcoh= 64 px
lcoh= 31 px
Complex Coherence Factor
in(r)
Diffraction intensity
distribution

9. Equatorial cross section of the diffraction patterns

10. 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

11. Reconstructed images with reduced coherence
of the whole array
Reconstructed image
of central island

12. Ge islands on prepatterned Si substrates
AFM micrographs of 1D (a) and 2D ordered Ge
islands (b) with corresponding line scans
Zhenyang Zhong, A. Halilovic, F. Schäffler, G. Bauer
Institut für Halbleiter-und Festkörperphysik, Johannes Kepler Universität Linz

13. Diffraction pattern around Ge (202) peak
1.95 Qz
1.89 Qx
2.025
1.965
Experiment on ID34 APS (June 2003)

14. Diffraction pattern from Ge islands
GISAXS measurements i c f Qy Qy
i> c
i< c

15. Conclusions
Diffraction pattern from 2D array of QD’s can be inverted to give the correct shape and orientation of individual island
For successful reconstruction experimental diffraction pattern has to be measured up to Q~2/a (a - average size of an island)
Test calculations for very small transverse coherence lengths lcoh of the incoming beam were made
Correct shape of the individual island can be obtained when lcoh~a