Universal Iterative Phasing Method for Near-Field and Far-Field Field Coherent Diffraction Imaging

Transcription

1 Universal Iterative Phasing Method for Near-Field and Far-Field Field Coherent Diffraction Imaging Qun Shen (APS, ANL) Xianghui Xiao (Cornell) Imaging in different regimes near-field field phase contrast in-line line holography coherent diffraction Distorted object approach Fresnel wave propagation by FFT unified iterative phasing Recent activities at APS coherent imaging beamline phase-sensitive sensitive topography dose estimates & scaling with resolution Summary Office of Science U.S. Department of Energy A U.S. Department of Energy Office of Science Laboratory Operated by The University of Chicago

2 Different Regimes of X-ray X Imaging a near-field Fresnel diffraction far-field Fraunhofer diffraction x-ray beam z ~ a /λ z >> a /λ absorption radiograph phase contrast in-line holography coherent diffraction Kagoshima et al. JJAP (1999). Jacobsen (003). Miao et al. Nature (1999).

3 Image Reconstruction in Different Regimes Absorption regime straightforward, based on intensity attenuation 3D tomographic reconstruction Phase contrast regime edge-enhanced enhanced shape recognition transport of intensity equation (TIE) holotomographic method based on Tolbot effect In-line holographic regime holographic reconstruction twin-image image problem Unified Method? Far-field regime iterative phasing method Fourier transforms in real and reciprocal space requires oversampled diffraction pattern 3

4 Iterative Method in Far-field Diffraction Gerchberg & Saxton, Optik 35, 37 (197) Fienup, Appl. Opt. 1, 758 (198) Question: Can we extend FFT-based iterative algorithm to near-field? Elser, JOSA, A0, 40 (003); Shen et al, JSR 11, 43 (004) 4

5 Fresnel Wave Field Propagation Fresnel formula for wave propagation F( X, Y ) ikr i e = u( x, y) dxdy λ r r = [ z + ( X x) + ( Y y ) ] 1/ z + [( X x) + ( Y y) ] z Wave-field in the object plane F( X, Y ) ikr iπ iπ i e ( x + y ) ( Xx+ Yy) λz λz = u( x, y) e e dxdy λr R = (x + y + z ) 1/ 0 u( x, y,0) = exp( ik ( δ ( x, y, z) iβ ( x, y, z)) dz) u ( x, y,0) = Aexp( iφ( x, y,0)) = a( x, y,0) + ib( x, y,0) 0 exp( ik δ ( x, y, z) dz) (pure phase object) Van der Veen & Pfeiffer, J. Phys.: Condens. Matter 16, 5003 (004) 5

6 Distorted Object Approach Phase-chirped distorted object: u ( x, y) u( x, y) e iπ λz ( x + y ) ikr ik i e ( Xx+ Yy) z F( X, Y ) = u ( x, y) e dxdy λr Unified wave propagation method by Fourier transform Momentum transfer: (Q x, Q y ) = (kx/z, ky/z) Number of Fresnel zones: N z = a /(λz) Xiao & Shen, PRB, in press (July 005) Fig.: Simulated diffraction amplitudes F(X,Y), of an amplitude object (a) of 10μm x 10μm, with λ = 1 Å x-rays, at image-to-object distance (b) z = mm and (c) z =, using the unified distorted object approach (above) with N z = 500 zones in (b) and N z = 0 in (c). Notice that the diffraction pattern changes from noncentrosymmetric in the near-field (b) to centrosymmetric in the far-field (c). 6

7 Iterative Phasing with Distorted Object u ( x, y) u( x, y) e iπ λz ( x + y ) F Start Modified u(x,y) G= G exp(iφ) Real space constraints Fourier space constraints New u(x,y) G= F exp(iφ) u( x, y) u ( x, y) e iπ + λz ( x + y ) F -1 7

8 x Nyquist f = Correct Sampling L π L => Sampling at frequency π/l in Fourier space is not fine enough to resolve interference fringes! => Additional measurements inbetween π/l are necessary to tell us some interference is going on. => Minimum oversampling ratio is, regardless whether it is 1D, D or 3D. ΔQ ΔQ Δ Q 1D max = π L 1 π = L π 1 π L L D max = = 3D max = π L 3 1 X-ray wavelength is λ, object s half width a, object-image distance z, and oversampling factor O, the pixel size of detector a a ΔX, N z = O N λ z or ΔX z λ z O a 8

9 Numerical Simulation Example Material: Carbon; Object size: 10x10 micron; Maximum thickness ~10μm; X-ray: 1Å; Maximum phase difference ~1.87rad; Absorption contrast~0.1%; Oversampling factor: x. 3 0 z = 5 cm N z = 5 z = 0cm N z = 1.5 z = 50cm N z = 0.5 z = 100cm N z = 0.5 z = N z = ΔX = 0.5 um ΔX = 1 um ΔX =.5 um ΔX = 5 um ΔX z

10 Phasing Results Xiao & Shen, PRB, in press (July 005) Twin image? z = 5cm z = 0cm z = 50cm z =100cm z = R = R = R = R = R =

11 Comparisons with Far-field Correlation coefficient between reconstructed phase map and the original phase map as a function of number of iterations in the iterative phase retrieval using the distorted object approach. Correlation Iteration Number z = 0cm z = 50cm z = 100cm z = Statistical Poisson noises are included in all diffraction patterns in these simulations. All these diffraction patterns are assumed to have the same total integrated intensity of 4.4x10 7 photons. Maximum intensity in the diffraction patterns are 7.6x10 5, 6.x10 6, 8.8x10 6 and 1x10 7 photons, for z = 0cm, 50cm, 100cm, and far-field, respectively. 11

12 K.A. Nugent et al, Acta Cryst. A61, , (005) U Distorted Object & Astigmatic Diffraction π q ( X, Y ) u( x, y,0) exp( i λ R x X + λ z y Y λ z = ) exp( iπ ( )) dxdy π u( x, y,0) exp( i λ q R ) Far-Field Astigmatic diffraction (curved beam) method: create parabolic or spherical wave front with K-B mirror, FZP lens Distorted object method: move detector a little bit closer to the sample comparing with conventional far-field imaging 1

13 Test Experiment at 3-ID ID-B lens- coupled CCD pinhole 5um mirror monochromator C(111) APS undulator A sample 8. kev horizontal slits (~100um) Dr. Xianghui Xiao (CHESS, Cornell University) 13

14 Specimen Used in Experiment Stefan Vogt (APS) Differential phase contrast image with a configured detector (Chris Jacobsen) Specimen: cardiac myocytes Brad Palmer (Univ. Vermont) 14

15 Preliminary Results on Myocytes Data obtained June 10-1, 1, 005 Multiple images with different exposure times to avoid saturation (need stitching) Z = 77 mm N z = 0.15, ΔX < 8um Δx = 100nm Z = 910 mm N z = 0.045, ΔX < 7um Δx = 330nm Z = 455 mm N z = 0.09, ΔX < 14um Δx = 160nm Images at several z with N z = Data processing in progress. 15

16 Improving Experimental Setup Pinhole scattering need scatter shields Use multiple silicon nitride windows Need four independent x-y translations McNulty & Lai Need CCD with larger dynamic range Perhaps a hybrid direct/lens-coupled coupled CCD Finer CCD pixel size Optical telescope for sample viewing & centering 16

17 Imaging Beamline at Sector 3 100m 00m Consideration of making Sector 3 (Com-CAT) a dedicated imaging XOR-Sector: Phase imaging / tomography Diffraction topography Diffraction enhanced /USAXS imaging Coherent Fresnel diffraction Many Benefits: Provides immediate home for the imaging group to satisfy users demand, to expand user base, and to test new application & ideas. Frees up 1-ID so Sector 1 can proceed to become a dedicated highenergy sector. Potential for future expansion perhaps into a long beam line (~00m) with optimized insertion devices. Current Status: starting to perform coherent imaging experiments, and to plan for beam line extension. 17

18 Proposed Project for a Dedicated Imaging Sector Phase I: make use of existing hutch and equipment, with upgrades to monochromator & Be windows Phase imaging Diffraction topography USAXS imaging Phase II: expansion to ~75m by building a new white-beam capable hutch at 75m and beam transport High-sensitivity phase imaging Coherent Fresnel diffraction Projection microscopy Existing Sector 3 3-ID-B 3-ID-C Main Research Programs: Near-field diffraction and imaging Optics development Ultrafast imaging with pink beam 4 m 75 m Phase III: future expansion to ~00m (ID-D) with additional outside funding, and with optimized insertion devices and optics Ultra-sensitivity phase imaging Ultra-plane-wave topography Medical imaging? 18

19 Other Applications of Distorted Object D Gaussian phase object Analytical expressions 000 Coherent diffraction topography Phase imaging sensitivity study 1.0E E-01 R φ (rad/um) Intensity (arb. units) Chu et al. (005) x (microns) mm 75 mm 100 mm 00 mm 300mm 00 0 Relative Contrast 1.0E E E E E-06 Shen (005) 1.0E Distance R (mm) x (x.77 um) 19

20 Dose vs. Resolution & Radiation Damage (biomaterials) V = (100nm) 3 Dose = ( Q I 1 E ~ λ ) ( I Δt) ~ Const. in λ 0 Δt 0 ~ λ μ E ρ 3, μ ~ λ, CW Sources Shen et al. JSR 11, 43 (004) I 3 ~ L d λ L = thickness Marchesini, Howells, et al. Optics Express (003) I 4 ~ d λ 0

21 Summary Distorted Object Approach provides a simple universal method for wave field propagation by fast Fourier transform,, in both far-field field and near-field regimes. Distorted Object Approach extends the iterative phasing algorithm to near-field and provides an alternative to far-field field coherent diffraction imaging and to astigmatic diffraction with curved beams. It eliminates the Friedel enantiomorph phasing ambiguity in the far-field. field. Practical imaging applications may be in the region where Fresnel number N z lies between 0. 1,, so that requirement on detector pixel size is relaxed but significant Fresnel zone curvature still exists. Other applications include design of phase imaging beam line, and phase-sensitive sensitive x-ray x diffraction topography. APS plans to expand in x-ray x imaging,, including to plan for a bio- nanoprobe beam line, to build a full-field field imaging beam line for coherent imaging applications. 1

22 Acknowledgments CHESS (Cornell University): X-ray Microscopy Group (APS): University of Vermont: X-ray Physics Group (APS): Xianghui Xiao Ian McNulty Zhonghou Cai Yong Chu Stefan Vogt Brad Palmer Peter Lee Detector Group (APS): Financial support: Tim Madden John Lee Steve Ross Advanced Photon Source is supported by the U.S. Department of Energy, Basic Energy Sciences, Office of Energy Research, under Contract No. W Eng-38 CHESS is supported by the U.S. National Science Foundation, under Award No. DMR