Methoden moderner Röntgenphysik I. Coherence based techniques II. Christian Gutt DESY, Hamburg

Transcription

1 Methoden moderner Röntgenphysik I Coherence based techniques II Christian Gutt DESY Hamburg christian.gutt@desy.de 8. January 009

2 Outline Introduction to Coherence Structure determination techniques Oversampling Coherent Diffractive Imaging Fourier transform Holography Correlation Spectroscopy

3 Last lecture

4 Longitudinal coherence Nλ = N + 1 λ λ N N + 1 λ = λ λ λ longitudinal coherence depends on bandwidth

5 Transverse coherence transverse coherence depends on distance and source size

6 1 1 τ + + = t r AV t r AV t r V r1 r Re 1 * τ τ r r A A t r V A t r V A t r I Γ = > + =< Γ 1 * 1 τ τ t r V t r V r r Field amplitude Intensity Mutual Coherence Function MCF 1 1 * 1 t r I t r I t r V t r V r r > + < = τ τ µ Complex degree of coherence Mutual Coherence Function = Visibility µ

7 van Cittert Zernike Theorem + + = S q p ik S q p ik i d d e I d d e I e P η ξ η ξ η ξ η ξ µ η ξ η ξ ψ 0 complex degree of coherence Fourier Transform of the source intensity distribution! = ρ ρ ρ ρ ρ ρθ ρ µ ψ d I d k J I e P i R Y X k R Y q R X p + = = = ψ Axial symmetry z r = θ

8 Speckle Pattern Everything interferes with everything ξ T I q d q~ S q ~ R q q~ R q exp q / q ξ L q π / ξ T

9 Scattering from a Crystal William Henry Bragg William Lawrence Bragg Nobelpreis 1915 Bragg's Law: a r mλ = d sin θ a r 1 r N F q = f q e j e i q Rn crystal j j= 1 n= 1 Unit Cell Structure Factor F uc q i r r q M r r r Lattice Sum

10 Elastic Scattering from a Crystal Differential Scattering Cross Section dσ dω dσ = dω Intrinsic Cross Section Coupling Beam Sample 0 crystal r S q r r S q = F q Properties of the Sample without Beam Phase problem

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12 Solution to the phase problem for periodic objects classical crystallography direct methods using the fact that the density is real and positive anomalous X-ray scattering MAD heavy atoms... + atomic resolution - need for crystals - x-ray damage Structure determination of non-periodic objects a zoo of scanning x-ray techniques scanning transmission x-ray microscope tomography medical imaging... + no need for crystals nm resolution - limited dynamics

13 Coherence based techniques for structure determination Ultrafast femtoseconds imaging techniques for non-periodic objects Coherent diffractive imaging Fourier transform holography Holographic imaging Ptychography and all combinations thereof...

14 Structure Determination from Oversampled Speckle Pattern λ θ iterative algorithm max. resolution λ sin θ J. Miao et al. Nature

15 Phase retrieval and oversampling M ρ x y z L x M x N unknown variables measured quantity F ^ sampled at Bragg peak frequency L Friedel s law L x M x N / independent equations No inversion possible

16 Intensity continuous intensity distribution for non periodic objects Bragg peaks for periodic objects k Idea: sample k finer than Bragg frequency e.g. 3 Number of independent equations = number of unknown variables 3 3 L x M x N / = L x M x N

17 Shannon s theorem in X-ray scattering If a diffraction pattern is sampled at spatial frequencies at least twice that corresponding to the size of the sample the phases can be recovered by means of iterative algorithms. = 1 λd W sampling in reciprocal space W d

18 oversampling parameter σ = speckle size pixel size = λd WP σ = σ =

19 The iterative algorithm due to Gerchberg-Saxton-Fienup

20 The hybrid-input-output HIO algorithm get some a priori knowledge about the support i.e. shape of your object area inside support S x in support x not in support 0 < β HIO < 1 measure of convergence

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24 The Object Claude Monet Seerosenteich II 1899

25 and its reconstruction

26 an unknown object

27 and it s reconstruction

28 Experiment using 8 kev Photonen microns Resolution 30 nm

29 The beamstop problem ccd camera sample intense direct beam

30 Missing Data

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33 First experimental realization at a synchrotron source

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35 First experimental realization at an FEL source H. Chapman et al. Nature Physics

36 pulse #1 pulse # H. Chapman et al. Nature Physics

37 Reconstruction H. Chapman et al. Nature Physics

38 Fourier Transform Holography

39 x 1 x y 1 y Fresnel-Kirchhoff Theory Object o a Reference r z 0 o x r x y i = λz o e i π λ z x + y 0 η Oˆ ξ π i i x + y λ z y = e Rˆ ξ η λz0 e 0 iπξ a ξ = η = x λ z y λ z 0 0 O ˆ ξ η = FT[ o x 1 y 1 ] R ˆ ξ η = FT[ r x 1 y 1 ]

40 y x o y x r y x I + = * * y x o y x r y x o y x r y x o y x r y x I = ξ π ξ π η ξ η ξ η ξ η ξ η ξ η ξ a i a i e O R e O R O R y x I ˆ ˆ ˆ ˆ ˆ ˆ * * ] [ ] [ * y a x o y x r FT y a x o y x r FT + + a ˆ ˆ * η ξ η ξ = O O Convolution with the reference objects limits the resolution

41 Small reference hole

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44 Large reference hole

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47 More than one reference hole

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52 First experimental FTH realization using hard X-rays

53 Experiment with 0.15 nm Photonen

54 1 micron

55 Combination of Holography and Phase Retrieval Resolution 0 nm L.-M. Stadler C. Gutt T. Autenrieth O. Leupold S. Rehbein G. Grübel