Methoden moderner Röntgenphysik I Coherence based techniques II Christian Gutt DESY Hamburg christian.gutt@desy.de 8. January 009
Outline 18.1. 008 Introduction to Coherence 8.01. 009 Structure determination techniques Oversampling Coherent Diffractive Imaging Fourier transform Holography 15.01.009 Correlation Spectroscopy
Last lecture
Longitudinal coherence Nλ = N + 1 λ λ N N + 1 λ = λ λ λ longitudinal coherence depends on bandwidth
Transverse coherence transverse coherence depends on distance and source size
1 1 τ + + = t r AV t r AV t r V r1 r Re 1 * 1 1 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
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! = 0 0 0 0 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
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
Scattering from a Crystal William Henry Bragg 186-194 William Lawrence Bragg 1890-1971 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
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
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 -+ 0-30 nm resolution - limited dynamics
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...
Structure Determination from Oversampled Speckle Pattern 50 100 λ θ 150 00 50 300 50 100 150 00 50 300 iterative algorithm max. resolution λ sin θ J. Miao et al. Nature 400 34 1999
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
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
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
oversampling parameter σ = speckle size pixel size = λd WP 4 6 4 8 5 0 5 σ = 1 5 4 5 6 6 6 6 8 7 0 7 7 4 7 6 1 500 1 550 1 600 1 650 1 700 1 750 σ = 770 1 800 1 850 155 0 160 0 165 0 170 0 1 750 1800
The iterative algorithm due to Gerchberg-Saxton-Fienup
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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The Object Claude Monet Seerosenteich II 1899
and its reconstruction... 6 0 60 8 0 80 1 0 0 1 00 1 0 1 0 1 4 0 1 40 1 6 0 1 60 1 8 0 1 80 0 0 00 0 0 4 0 6 0 8 0 1 0 0 1 0 1 40 1 60 1 8 0 0 0 0 4 0 40 6 0 8 0 1 0 0 1 0 1 4 0 1 6 0 18 0 0 0 0 40 1 0 0 1 0-1 50 1 0-1 00 1 0-3 1 50 1 0-4 1 0-5 00 1 0-6 50 1 0-7 0 0 0 0 4 0 0 0 6 0 0 0 8 0 00 1 00 0 0 1 0 0 0 1 4 0 0 0 3 00 5 0 1 0 0 1 5 0 0 0 5 0 30 0
an unknown object 1 00 00 3 00 4 00 5 00 6 00 7 00 1 0 0 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 00 1 0 0 1 0-1 1 0-0 5 00 1 00 0 1 5 0 0 00 0 50 0
and it s reconstruction 50 300 350 400 450 100 150 00 50 300 350 400 450 500 550 600
Experiment using 8 kev Photonen microns Resolution 30 nm
The beamstop problem ccd camera sample intense direct beam
Missing Data 1 50 100 150 00 50 300 50 100 150 00 50 300
First experimental realization at a synchrotron source
First experimental realization at an FEL source H. Chapman et al. Nature Physics 839 006
pulse #1 pulse # H. Chapman et al. Nature Physics 839 006
Reconstruction H. Chapman et al. Nature Physics 839 006
Fourier Transform Holography
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 ]
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
Small reference hole 100 150 00 50 300 100 150 00 50 300
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Large reference hole 50 100 150 00 50 300 350 400 450 500 50 100 150 00 50 300 350 400 450 500
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More than one reference hole 50 100 150 00 50 300 350 400 450 500 50 100 150 00 50 300 350 400 450 500
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First experimental FTH realization using hard X-rays
Experiment with 0.15 nm Photonen
1 micron
Combination of Holography and Phase Retrieval Resolution 0 nm L.-M. Stadler C. Gutt T. Autenrieth O. Leupold S. Rehbein G. Grübel