Universal Iterative Phasing Method for Near-Field and Far ...€¦ · Qun Shen June 15, 2005 8...

A U.S. Department of EnergyOffice of Science LaboratoryOperated by The University of ChicagoOffice of Science

U.S. Department of Energy

Universal Iterative Phasing Method for Universal Iterative Phasing Method for NearNear--Field and FarField and Far--Field Coherent Field Coherent

Diffraction ImagingDiffraction ImagingQunQun ShenShen (APS, ANL)(APS, ANL)XianghuiXianghui Xiao (Cornell)Xiao (Cornell)

⇒⇒ Imaging in different regimesImaging in different regimesnearnear--field phase contrastfield phase contrastinin--line holographyline holographycoherent diffractioncoherent diffraction

⇒⇒ Distorted object approachDistorted object approachFresnelFresnel wave propagation by FFTwave propagation by FFTunified iterative phasingunified iterative phasing

⇒⇒ Recent activities at APSRecent activities at APScoherent imaging coherent imaging beamlinebeamlinephasephase--sensitive topographysensitive topographydose estimates & scaling with resolutiondose estimates & scaling with resolution

⇒⇒ SummarySummary

2Qun Shen June 15, 2005

Different Regimes of XDifferent Regimes of X--ray Imagingray Imaging

absorptionradiograph

phasecontrast

in-line holography

coherent diffraction

x-ray beam

Kagoshima et al. JJAP (1999).

Miao et al. Nature (1999).

Jacobsen (2003).

2a

z >> a2/λz ~ a2/λ

near-fieldFresnel

diffraction

far-fieldFraunhoferdiffraction


Image Reconstruction in Different RegimesImage Reconstruction in Different Regimes

⇒⇒ Absorption regimeAbsorption regimestraightforward, based on intensity attenuationstraightforward, based on intensity attenuation3D 3D tomographictomographic reconstructionreconstruction

⇒⇒ Phase contrast regimePhase contrast regimeedgeedge--enhanced shape recognitionenhanced shape recognitiontransport of intensity equation (TIE)transport of intensity equation (TIE)holotomographicholotomographic method based on method based on TolbotTolbot effecteffect

⇒⇒ InIn--line holographic regimeline holographic regimeholographic reconstructionholographic reconstructiontwintwin--image problemimage problem

⇒⇒ FarFar--field regimefield regimeiterative phasing methoditerative phasing methodFourier transforms in real and reciprocal spaceFourier transforms in real and reciprocal spacerequires requires oversampledoversampled diffraction pattern diffraction pattern

Unified Method ?Unified Method ?


Iterative Method in FarIterative Method in Far--field Diffractionfield Diffraction

Gerchberg & Saxton, Optik 35, 237 (1972)Fienup, Appl. Opt. 21, 2758 (1982)

Elser, JOSA, A20, 40 (2003); Shen et al, JSR 11, 432 (2004)

Question: Can we extend FFTQuestion: Can we extend FFT--based based iterative algorithm to neariterative algorithm to near--field ?field ?


FresnelFresnel Wave Field PropagationWave Field Propagation

Wave-field in the object plane

))),,(),,((exp()0,,(0

∫ ∞−−⋅−= dzzyxizyxikyxu βδ

)),,(exp(0

∫ ∞−−≈ dzzyxik δ (pure phase object)

)0,,()0,,())0,,(exp()0,,( yxibyxayxiAyxu +=−= φ

Fresnel formula for wave propagation

∫∫−

= dxdyr

eyxuiYXFikr

),(),(λ

2/1222 ])()([ yYxXzr −+−+=

Van der Veen & Pfeiffer, J. Phys.: Condens.

Matter 16, 5003 (2004)

( ) ( )∫∫

+−+−−

= dxdyeeyxuR

eiYXFYyXx

ziyx

ziikR

λπ

λπ

λ

222

),(),(

zyYxXz 2])()[( 22 −+−+≈

R = (x2 + y2 + z2 )1/2


Distorted Object ApproachDistorted Object Approach

( )22

),(),(yx

zi

eyxuyxu+−

≡ λπ

( )∫∫

+−−

= dxdyeyxuR

eiYXFYyXx

zikikR

),(),(λ

⇒⇒ Unified wave propagation Unified wave propagation method by Fourier transformmethod by Fourier transform

Xiao & Shen, PRB, in press (July 2005)

Fig.2: 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 = 2mm and (c) z = ∞, using the unified distorted object approach (above) with Nz = 500 zones in (b) and Nz = 0 in (c). Notice that the diffraction pattern changes from noncentrosymmetric in the near-field (b) to centrosymmetric in the far-field (c).

PhasePhase--chirped distorted object:chirped distorted object:

Momentum transfer: (Qx, Qy) = (kX/z, kY/z)

Number of Fresnel zones: Nz = a2/(λz)


Iterative Phasing with Distorted ObjectIterative Phasing with Distorted Object

( )22

),(),(yx

zi

eyxuyxu+−

≡ λπ

Modified u(x,y) G=|G| exp(iφ)

Fourier spaceconstraints

G=|F| exp(iφ)

F

F -1

Real spaceconstraints

New u(x,y)

( )22

),(),(yx

zi

eyxuyxu++

≡ λπ

Start


OversamplingOversampling @ 2x @ 2x NyquistNyquist f = Correct Samplingf = Correct Sampling

LLπ2

=> Sampling at frequency 2π/L in Fourier space is not fine enough to resolve interference fringes!

=> Additional measurements in-between 2π/L are necessary to tell us some interference is going on.

LLQ ππ

=⋅=Δ212D1

max

=> Minimum => Minimum oversamplingoversampling ratio is 2, ratio is 2, regardless whether it is 1D, 2D or 3D.regardless whether it is 1D, 2D or 3D.

LLQ ππ 2

212D2

max =⋅=Δ

3D3

max 212

⋅=ΔL

Q π

X-ray wavelength is λ, object’s half width a, object-image distance z, and oversamplingfactor O, the pixel size of detector

zaN

NOaX z

z ⋅=

⋅≤Δ

λ

2

,2

aOzX⋅

≤Δλ

or


Numerical Simulation ExampleNumerical 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: 2x2.

3

0

-3

1

10-4

10-8

∞

10-10

z = 20cmNz = 1.25

z = 5 cmNz = 5

z = 50cmNz = 0.5

z = 100cmNz = 0.25

z = Nz = 0

ΔX = 0.25 um ΔX = 1 um ΔX = 2.5 um ΔX = 5 um ΔX ∝ z


cm5=z cm20=z cm50=z cm100=z ∞=z

9998.0=R 9954.0=R 9943.0=R 9248.0=R9639.0=R

Phasing ResultsPhasing Results

Twin image?

Xiao & Shen, PRB, in press (July 2005)


Comparisons with FarComparisons with Far--fieldfield

0 1000 2000 3000 40000.0

0.2

0.4

0.6

0.8

1.0

Cor

rela

tion

Iteration Number

z = 20cm z = 50cm z = 100cm z = ∞

• 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.

• 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 4.4x107 photons.

• Maximum intensity in the diffraction patterns are 7.6x105, 6.2x106, 8.8x106 and 1x107 photons, for z = 20cm, 50cm, 100cm, and far-field, respectively.


K.A. Nugent et al, Acta Cryst. A61, 373-381, (2005)

Far-Field

∫ ∫∞

∞−

∞

∞− ⋅⋅

⋅⋅ +⋅⋅⋅= dxdyiiyxuYXU z

YyzXx

Rq ))(2exp()exp()0,,(),( 2

2 2

λλλπ π

)exp()0,,( 22 2

Rqiyxu ⋅⋅ λ

π

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

Distorted Object & Astigmatic DiffractionDistorted Object & Astigmatic Diffraction


Test Experiment at 32Test Experiment at 32--IDID--BB

Dr. Xianghui Xiao (CHESS, Cornell University)

APS APS undulatorundulator AA

horizontalhorizontalslits slits

(~100um)(~100um)

monochromatormonochromatorC(111)C(111)mirrormirror

pinhole pinhole 5um5um

samplesample

lenslens--coupled coupled

CCDCCD8.2 8.2 keVkeV


Specimen Used in ExperimentSpecimen Used in Experiment

Specimen:Specimen: cardiac cardiac myocytesmyocytesBrad Palmer (Univ. Vermont)Brad Palmer (Univ. Vermont)

Differential phase contrast Differential phase contrast image with a configured image with a configured detector (Chris Jacobsen)detector (Chris Jacobsen)

Stefan Vogt (APS)


Preliminary Results on Preliminary Results on MyocytesMyocytes

Z = 910 mmNz = 0.045, ΔX < 27um

Δx = 330nm

Z = 455 mmNz = 0.09, ΔX < 14um

Δx = 160nm

Z = 277 mmNz = 0.15, ΔX < 8um

Δx = 100nm

•• Data obtained June 10Data obtained June 10--12, 200512, 2005•• Multiple images with different exposure times Multiple images with different exposure times to avoid saturation (need stitching)to avoid saturation (need stitching)

•• Images at several z with Images at several z with NNzz = 0.045 = 0.045 –– 0.45 0.45 •• Data processing in progress Data processing in progress ……..


Improving Experimental SetupImproving Experimental Setup

•• Pinhole scattering Pinhole scattering need scatter shieldsneed scatter shields•• Use multiple silicon nitride windowsUse multiple silicon nitride windows•• Need four independent Need four independent xx--yy translationstranslations

McNulty & Lai

•• Need CCD with larger dynamic rangeNeed CCD with larger dynamic range•• Perhaps a hybrid direct/lensPerhaps a hybrid direct/lens--coupled CCDcoupled CCD•• Finer CCD pixel sizeFiner CCD pixel size

•• Optical telescope for sample viewing & centeringOptical telescope for sample viewing & centering


Consideration of making Sector 32 (Com-CAT) a dedicated imaging XOR-Sector:

• 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 high-energy sector.• Potential for future expansion perhaps into a long beam line (~200m) with optimized insertion devices.

Current Status: starting to perform coherent imaging experiments, and to plan for beam line extension.

• Phase imaging / tomography• Diffraction topography• Diffraction enhanced /USAXS imaging• Coherent Fresnel diffraction

Many Benefits:

Imaging Imaging BeamlineBeamline at Sector 32at Sector 32

200m

100m


Phase I: make use of existing hutch and equipment, with upgrades to monochromator & Be windows

Phase II: expansion to ~75m by building a new white-beam capable hutch at 75m and beam transport

Phase imagingDiffraction topographyUSAXS imaging

High-sensitivity phase imagingCoherent Fresnel diffractionProjection microscopy

Ultra-sensitivity phase imagingUltra-plane-wave topographyMedical imaging ?

Phase III: future expansion to ~200m (ID-D) with additional outside funding, and with optimized insertion devices and optics

32-ID-C32-ID-B

42 m 75 m

Proposed Project for a Dedicated Imaging SectorProposed Project for a Dedicated Imaging Sector

Main Research ProgramsMain Research Programs::

•• NearNear--field diffraction and imagingfield diffraction and imaging•• Optics developmentOptics development• UltrafastUltrafast imaging with pink beamimaging with pink beam

Existing Sector 32


Other Applications of Distorted ObjectOther Applications of Distorted Object

Phase imaging sensitivity studyPhase imaging sensitivity study

Coherent diffraction topographyCoherent diffraction topography

R2

1.0E-07

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

0.1 1 10 100 1000 10000 100000

Distance R2 (mm)

Rel

ativ

e C

ontr

ast

0.003

0.007

0.014

0.029

0.143

φ’ (rad/um)

2000

1800

1600

1400

1200

1000

800

600

Inte

nsity

(arb

. uni

ts)

220200180160140120100x (microns)

30 mm 75 mm 100 mm 200 mm 300mm

0.400

0.500

0.600

0.700

0.800

0.900

1.000

1.100

1.200

-4 -2 0 2 4

30

75

100

200

300

x (x2.77 um)

Shen (2005)

Chu et al. (2005)

2D Gaussian phase object Analytical expressions


Dose vs. Resolution & Radiation DamageDose vs. Resolution & Radiation Damage(biomaterials)(biomaterials)

Shen et al. JSR 11, 432 (2004)

• V = (100nm)3

Marchesini, Howells, et al. Optics Express (2003)

CW Sources

23~ λdLI ⋅

24~ λdI

( )

)~

,~,~(.~

1

320

0

−

−Δ

⋅Δ=

λ

λμλ

λρμ

E

tIinConst

EtIDose

Q

L = thickness


SummarySummary

Distorted Object ApproachDistorted Object Approach provides a simple provides a simple universal methoduniversal method for for wave field propagation by fast wave field propagation by fast Fourier transformFourier transform, in both far, in both far--field and field and nearnear--field regimes. field regimes.

Distorted Object ApproachDistorted Object Approach extends the iterative phasing algorithm to extends the iterative phasing algorithm to nearnear--field and provides an field and provides an alternativealternative to to farfar--fieldfield coherent diffraction coherent diffraction imaging and to imaging and to astigmaticastigmatic diffraction with curved beams. It eliminates diffraction with curved beams. It eliminates the the FriedelFriedel enantiomorphenantiomorph phasing ambiguity in the farphasing ambiguity in the far--field.field.

Practical imaging applicationsPractical imaging applications may be in the region where may be in the region where FresnelFresnelnumber number NNzz lies between lies between 0.2 0.2 −− 11, so that requirement on detector pixel , so that requirement on detector pixel size is relaxed but significant size is relaxed but significant FresnelFresnel zone curvaturezone curvature still exists.still exists.

Other applicationsOther applications include design of phase imaging beam line, and include design of phase imaging beam line, and phasephase--sensitive xsensitive x--ray diffraction topography.ray diffraction topography.

APSAPS plans to expand in xplans to expand in x--ray ray imagingimaging, including to plan for a bio, including to plan for a bio--nanoprobenanoprobe beam line, to build a fullbeam line, to build a full--field imaging beam line for coherent field imaging beam line for coherent imaging applications. imaging applications.


AcknowledgmentsAcknowledgments

Financial support:

Advanced Photon Source is supported by the U.S. Department of Energy, Basic Energy Sciences, Office of Energy Research, under Contract No. W-31-109-Eng-38

CHESS is supported by the U.S. National Science Foundation, under Award No. DMR-0225180.

CHESS (Cornell University):CHESS (Cornell University): XianghuiXianghui XiaoXiao

XX--ray Microscopy Group (APS):ray Microscopy Group (APS): Ian McNultyIan McNultyZhonghouZhonghou CaiCaiYong Yong ChuChuStefan VogtStefan Vogt

University of Vermont:University of Vermont: Brad PalmerBrad Palmer

XX--ray Physics Group (APS): ray Physics Group (APS): Peter LeePeter Lee

Detector Group (APS):Detector Group (APS): Tim MaddenTim MaddenJohn Lee John Lee Steve RossSteve Ross

http://www.sc.doe.gov/bes/bes.html

Universal Iterative Phasing Method for Near-Field and Far ...€¦ · Qun Shen June 15, 2005 8...

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Transcript of Universal Iterative Phasing Method for Near-Field and Far ...€¦ · Qun Shen June 15, 2005 8...