Diffraction Imaging with Coherent X-rays

Transcription

1 Diffraction Imaging with Coherent X-rays John Miao Stanford Synchrotron Radiation Laboratory Stanford Linear Accelerator Center

2 The Phase Problem: A Coherence Effect Detector Coherent X-rays Atoms The phase problem is due to the fact that there are no ways to tell where each photon is scattered from. o phase problem for X-ray inelastic scattering. Phase ambiguities: ρ i θ r, ρ r + r + r * i θ e, ρ r e

3 Regular Sampling: Sampling at the Bragg-peak Frequency F k r = 2 / = π i k r ρ r e i k =,, 2, K ρ r : real F k = F * k ρ r : complex # of unknown variables # of independent equations # of unknown variables # of independent equations D /2 D 2 2D 2 2 /2 2D D 3 3 /2 3D Independent equations: intensity points have no crystallographic symmetry. Miao, Sayre & Chapman, J. Opt. Soc. Am. A 5, Miao & Sayre, Acta Cryst. A 56,

4 Oversampling: Sampling at a Spacing Finer than the Bragg-peak Frequency F k r = 2 /2 = π i k r ρ r e ii k =,, 2, K 2 ρ r : real F k = F * k ρ r : complex # of unknown variables # of independent equations # of unknown variables # of independent equations D D 2 2 2D D D D

5 The Oversampling Method g r = ρ r r r 2 iii Eq. 2 F k 2 r = 2 /2 = π i k r g r e k =,, 2, K 2 iv electron density region + no density σ = electron density region region v σ > 2: the phase information exists inside the diffraction intensity!

6 ik e k F / 2 π ρ = = ik e a / 2, π ρ a F = = = = D Case: 2 multiple solutions 2D & 3D Case: Multiple solutions are rare Mathematically, 2D and 3D polynomials usually can not be factoried. Bruck & Sodin, Opt. Commun. 3, Multiple Solutions / / * F F = = = / * F F I = =

7 The Physical Interpretation of the Oversampling Method Real Space F Reciprocal Space Regular sampling Oversampling Better coherence More correlated intensity points Phase information

8 Oversampling and Coherence Oversampling vs. spatial coherence: θ λ 2O a Oversampling vs. temporal coherence: λ λ Oa d O = σ 3 σ for a for a 2 3 D sample D sample Miao et al., Phys. Rev. Lett. 89, a : sample sie d : desired resolution

9 An Iterative Algorithm I ϕ F k = F exp k e ' k II ϕ,, = ' III r ' ρ = FFT - k F IV ρ r = ρ r if r S & ρ ' ' ' r β ρ r if r S or ρ r < ρ r ' V F k = FFT ρ r VI Adopt ϕ k from F k Fienup, Opt. Lett. 3, Miao et al., Phys. Rev. B 67, '

10 Experimental Demonstration of Coherent Imaging a A SEM image b An oversampled diffraction pattern in a logarithmic scale from a. r S γ = ' r S ρ ρ r r c An image reconstructed from b. Miao Charalabous, Kir & Sayre, ature 4, d The convergence of the reconstruction. '

11 Phase Retrieval as a Function of the Oversampling Ratio σ = 5 8 x 8 pixels σ = 4 6 x 6 pixels σ = x 3 pixels σ =.9 x pixels

12 Imaging a anostructure at 7 nm Resolution a A SEM image of a non-crystalline sample made of Au b A coherent diffraction pattern from a the resolution at the edge is 7 nm c An image reconstructed from b

13 Imaging Buried anostructures a A SEM image of a double-layered sample made of i ~2.7 x 2.5 x µm 3 b A coherent diffraction pattern from a the resolution at the edge is 8 nm c An image reconstructed from b

14 3D Imaging of anostructures The reconstructed top pattern The reconstructed bottom pattern An iso-surface rendering of the reconstructed 3D structure

15 Determining the Absolute Electron Density of Disordered Materials at Sub- nm Resolution a A coherent diffraction pattern from a porous silica particle b The reconstructed absolute electron density c The absolute electron density distribution within a x nm 2 area Miao et al., Phys. Rev. B,

16 Imaging E. Coli Bacteria a Light and fluorescence microscopy images of E. Coli labeled with manganese oxide b A Coherent X-ray diffraction pattern from E. Coli c An image reconstructed from b. Miao et al., Proc. atl. Acad. Sci. USA, 23.

17 Coherent X-ray Diffraction of Bone Samples a -74 b -76 c -77 d -78

18 Summary Proposed a theoretical explanation to the oversampling method. Carried out the first to our knowledge definitive experimental demonstration of coherent imaging. The results potentially open a door to near atomic resolution 3D X-ray diffraction microscopy. The methodology can in principle be extended to electrons. Future the potential of imaging large biomoleucles using X-FELs.

19 Acknowledgements B. Johnson, D. Durkin, K. O. Hodgson, SSRL T. Ishikawa, Y. ishino, Y. Kohmura, M. Yabashi, K. Tamasaku, RIKE/SPring-8 J. Kir, D. Sayre, SUY at Stony Brook C. Larabell, UC San Francisco & LBL M. Glimcher, Harvard Medical School M. LeGros, E. Anderson, LBL B. Lai, APS, AL J. Amonette, PL