Coherent Diffraction Imaging (CDI) with X-Rays School on Synchrotron - PowerPoint PPT Presentation

Coherent Diffraction Imaging (CDI) with X-Rays School on Synchrotron and Free-Electron-Laser Methods for Multidisciplinary Applications ICTP Trieste, May 15, 2018 Anders Madsen European X-Ray Free-Electron Laser Facility Hamburg, Germany anders.madsen@xfel.eu

 2 Anders Madsen, European XFEL Outline (50 min) (5-10 min, maybe) (30 min)  Motivation - BREAK -  Sources of Coherent X-rays  X-ray Imaging  The European XFEL project  X-ray Coherence  CDI science at XFELs  All the tricks of CDI  Summary  Image reconstruction  Related methods  Applications

 3 Anders Madsen, European XFEL Coherent scattering. Motivation Slide courtesy of I. Vartaniants (DESY)

 4 Anders Madsen, European XFEL Coherent scattering. Motivation Isolated object Static scattering Coherent Diffraction Imaging (CDI): Dynamic scattering Ensemble of objects Correlation spectroscopy: Temporal: XPCS Spatial: XCCA

 5 Anders Madsen, European XFEL Coherent scattering. Motivation Diffraction imaging of biomolecules with coherent femtosecond X-FEL pulses Very much excitement, now for >15 years: Coulomb explosion of T4 lysozyme R. Neutze et al., Nature 406 , 752 (2000)

 6 Anders Madsen, European XFEL Coherent scattering. Motivation Diffraction imaging of biomolecules with coherent femtosecond X-FEL pulses Simulated coherent scattering image (speckle) of a T4 lysozyme molecule R. Neutze et al., Nature 406 , 752 (2000)

 7 Anders Madsen, European XFEL Different regimes of lensless X-ray imaging Fresnel diffraction Fraunhofer diffraction near-field far-field size: a X-rays L ~a 2 / l L >> a 2 / l L~cm L~m Absorption Phase contrast In-line holography Coherent Diffraction Imaging (CDI) Koch et al. (1998) Zhang (2003) Röntgen (1895)

 8 Anders Madsen, European XFEL Different regimes of lensless X-ray imaging  Absorption regime Easy reconstruction based on attenuation 3D tomographic reconstruction, inverse Radon transformation  Phase contrast regime Edge enhanced contrast Transport-of-intensity (TIE) equation Holotomographic reconstruction (Talbot effect)  In-line holographic regime Holographic reconstuction (detector dependent resolution) Twin image problem  Coherent diffraction imaging Tricky data treatment Resolution like in scattering, i.e. D min = 2 p /Q max ….

 9 Anders Madsen, European XFEL Coherence Quantum mechanics  probability amplitudes (waves) Optics  Young’s double slit experiment, interference X-ray (and neutron) scattering It’s all about probability amplitudes and interference !!! Example: Young’s double slit experiment (Thomas Young, 1801) [wave-character of quantum mechanical particles (photons)] P=| S j F j | 2 Plane, mono- F : probability amplitude chromatic wave F j ~ exp[-i( w t-kl j )] w =ck, k=2 p / l , l j (L,y) Laser beam P(y) ~ cos 2 ( p yd/ l L) D y= l L/d

Anders Madsen, European XFEL Coherence lengths Coherence N l d L (N-1)( l+Dl) p 2p 0 Longitudinal Transverse l l = l 2 /(2 Dl) l t = l L /2 d coherence length coherence length

 11 Anders Madsen, European XFEL The CDI Challenge I(q) Using a lens to form the image Question Speckle pattern does not look like sample. How to determine the sample from I(q) ?

 12 Anders Madsen, European XFEL The phase problem in scattering | q | = 2 p / l Q = q in - q out E(Q) | Q | = 4 p sin( q )/ l q out q in r (R)  r  E ( Q ) ~ ( R ) exp[ i Q R ] d R Reciprocal space E( Q )  FT  r ( R ) Real space = = * 2 I ( Q ) E ( Q ) E ( Q ) | E ( Q ) | But…

 13 Anders Madsen, European XFEL The phase problem in scattering Aim: To find E(Q) from measurements of I(Q) = |E(Q)| 2 But E(Q) is a complex number with both phase and amplitude E(Q) = A exp(i f ) Measurement: I(Q) = |E(Q)| 2 = A 2 No direct access to phase…. Exercise FT of XFEL logo  E(Q) I(Q) = |E(Q)| 2 (simulation of coherent scattering) Construct E(Q): A = sqrt(I(Q)) Take random phases f Inverse FT transform of A exp(i f )

 14 Anders Madsen, European XFEL Phase matters! r (R) |FT{ r (R)}| 2 = |E(Q)| 2 = I(Q) 1 0.9 0.8 ampl 0.7 0.6 A 0.5 FT 0.4 0.3 3 2 phase 1 0 f -1 -2 -3

 15 Anders Madsen, European XFEL Phase matters! r (R)  I(Q) as amplitude 1.4 1.2 1 ampl 0.8 0.6 FT -1 {E(Q)} 0.4 0.2 3 2 1 phase 3 0 2 -1 1 0 -2 -1 -3 random phases -2 -3

 16 Anders Madsen, European XFEL Phase is more important than amplitude Fourier transform my dog  A e i f 50 100 Keep amplitudes A 150 Substitute with another image’s phases f 200 250 Inverse Fourier transform 300 350 400 100 200 300 400 500 600 The phases came from a cat… 50 100 50 100 150 150 200 200 250 250 300 300 350 350 400 100 200 300 400 500 600 400 100 200 300 400 500 600

 17 Anders Madsen, European XFEL How can the phase be determined? r I N 2 pixels N 2 pixels 2N 2 unknowns N 2 equations Question How can we make this solvable? Unique solution?

 18 Anders Madsen, European XFEL N 2 pixels, 2N 2 unknowns Intensity measured in N 2 pixels N 2 equations 1 I (Q) 0.9 0.8 ampl 0.7 0.6 0.5 0.4 0.3 3 2 1 phase 0 -1 -2 -3

 19 Anders Madsen, European XFEL Finite support constraint N I (Q) 1 0.9 0.8 ampl 0.7 0.6 0.5 0.4 0.3 N/sqrt(2) 3 2 1 phase 0 -1 -2 -3

 20 Anders Madsen, European XFEL Finite support constraint N I (Q) 1 0.9 ampl 0.8 0.7 0.6 0.5 0.4 0.3 N/3 3 2 phase 1 0 -1 -2 -3

 21 Anders Madsen, European XFEL Finite support constraint N I (Q) ampl 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 N/6 phase 3 2 1 0 -1 -2 -3 Angular speckle size ~ l /sample size

 22 Anders Madsen, European XFEL Oversampling David Sayre (1952)

 23 Anders Madsen, European XFEL Basic experimental requirements for CDI In 2D: - Need that sample dimension a is at least  2 smaller than beam size D: a < D/  2 (reduce number of unknowns) - Need to measure the speckle pattern with a resolution that is at least  2 finer that speckle size in both dimensions. Therefore, the pixel size D p must fulfil: D p/L < l /(a  2), where L is the sample - detector distance (increase number of equations) - Beam must be coherent over the sample, otherwise the FT relationship does not hold Question How to solve the non-linear system of N 2 equations and M unknowns (M < N 2 ) to find r (R)?

 24 Anders Madsen, European XFEL Iterative Phase Retrieval Algorithm FT Is this the real r ( x ) E(Q) solution? real space constraints reciprocal space constraints = r ' ( x ) E ( Q ) I( Q ) E ' (Q) random phases FT -1 Real space constraints: Reciprocal space constraints finite support ( r = 0 outside sample)  E ( Q ) I( Q ) real r ? positive r ? other?

 25 Anders Madsen, European XFEL Iterative Phase Retrieval Algorithm 4x4 times Computer simulation oversampled I(Q) FT E(Q) positive, real r (R) = | E ' ( Q ) | I ( Q ) shrink wrap support r ’(R ) E’(Q) FT -1 = f E ' ( Q ) I ( Q ) exp[ i ( Q )] random phases f

 26 Anders Madsen, European XFEL Iterative Phase Retrieval Algorithm R. W. Gerchberg and W. O. Saxton, Optik 35, 237 (1972) J.R. Fienup, Appl. Opt. 21 , 2758 (1982) Review: R. Millane et al ., J. Opt. Soc. Am. A14 , 568 (1997) Difference map: V. Elser, J. Opt. Soc. Am. A20 , 40 (2003)

 27 Anders Madsen, European XFEL 1 st Experimental Demonstration with X-rays J. Miao et al , Nature 400 , 342 (1999)

 28 Anders Madsen, European XFEL Bio-CDI with soft X-rays reconstructed CDI from a yeast cell (freeze-dried) image Speckle pattern, l =16.5 Å (ALS) No shrink warp, “hand drawn” support Difference map algorithm Averaging iterates (Elser & Thibault) Resolution ~ 30 nm D. Shapiro et al , PNAS 102 , 15343 (2005)

 29 Anders Madsen, European XFEL Setup for Biological CDI In-line optical microscope Sample preparation plunge-freezing

 30 Anders Madsen, European XFEL Bio-CDI with hard X-rays D. Radiodurans cell 1 m m Data taken at ID10, ESRF Resolution about 30 nm Frozen, hydrated cells Goal: ~10 nm E. Lima et al , PRL 103 , 198102 (2009)

 31 Anders Madsen, European XFEL Ptychography  The requirement that sample < beam is a limitation for many practical purposes  Can we find another constraint so the phase can be determined?  The answer is : Ptychography! Ptych- : (to) fold (from Greek)

 32 Anders Madsen, European XFEL Ptychography Science, 321, 379-382 (2008) SEM absorption Buried zone plate

 33 Anders Madsen, European XFEL Ptychography Proc. Natl. Acad. Sci. USA 107, p. 529-534 (2010) D. Radiodurans Setup at cSAXS beamline, SLS

 34 Anders Madsen, European XFEL Question Do you know another method where the phases are encoded in the image, i.e. easy reconstruction?

 35 Anders Madsen, European XFEL Holography Famous method to encode the phases in the intensity pattern In-line holography (electrons, Gabor 1947); Laser (1960), 1 st optical hologram ~1962 Holographic reconstruction Recording Dennis Gabor Nobel prize 1971 First SR hologram: