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TRANSCRIPT
Slide: 1
Coherent ImagingP. Cloetens, ESRF
Propagationbased phasecontrast imaging the forward problem; Fresnel diffraction coherence conditions
The inverse problem phase retrieval methods holotomography
Recent developments Projection microscopy KBbased imaging
Slide: 2
1 mm
From artefacts to 3D phase contrast imaging
Unpolished beryllium window
Rendering of a semisolid
Slide: 3
Phase Contrast vs Absorption
Absorption
Sample Sample
PhaseSimple transmission
• Dream 2: Improve the Sensitivity Absorption contrast too low high spatial resolution
light materialssimilar attenuation
• Dream 1: Zero DoseIncrease the energyAbsorption contrast ↓ replaced by phase contrast
Slide: 4
Transmission through a sample
∆ z
refractive index n = 1 δ + iβ
Transmission functionprojection of the refractive index distribution
uinc(x,y) u0(x,y) = T(x,y). uinc(x,y)z
xy
e ikmedium
z=e
i 2
n z=e
i 2
z. e
−i 2
z. e
−2
z
T(x,y) = A(x,y).eiϕ (x,y)
Phase
Amplitude
x , y =−2 ∫ x , y , z dz
A x , y =e−B x , y =e−
2∫ x , y , z dz
Slide: 5
Phase contrast vs Absorption
Gain of up to a few 1000 !
Hard XraysWater window
BeC
Al
FeAu
Energy (keV)
Slide: 6
inner layerpolystyrenethickness 30 µm
outer layerparylenethickness 15 µm
850 µm
Absorption
D = 0.03 mλ = 0.7 Å
200 µm
Propagation
D = 19 cm
Polymer sphere with two layers
D = 83 cm
Propagation
Phase contrast vs Absorption
Slide: 7
Phase Sensitive Techniques
Phase Retardation
s∆−=∆ δλπϕ∆ s
Deflection⇔
Phase gradients∆ α ~ µrad
=−
2∂
∂ x
At zero distance:Intensity
⇒ all phase information is lost
I 0=∣u0∣2=I inc .exp −∫ dz
Slide: 8
Phase Contrast Imaging methods
Analyzerbased PCI
Gratingbased PCI
Propagationbased PCI
ID17Medical Imaging
BM05/ID19wavefront sensorCh David, F PfeifferT Weitkamp
ID19 / ID22NIEdge enhancementHolotomography
Slide: 9
edge detection versus holography (Fresnel diffraction)
each edge imaged independentlyno access to phase, only to border
access to phase, if recorded at ≠ D’sdefocused imageλ = 0.7 Å
50 µm
D = 0.15 m D = 3.1 m
towardsFraunhoferdiffraction
D≪a D≈a
Slide: 10
Fresnel diffraction
Fresnel Integral (1D)
• Convolution in real spaceMultiplication in reciprocal space
• Small angles paraxial approximation
D
u0(x,y) uD(x,y)
uD x= 1
i D∫u0x 0exp[−i
D x−x 0
2]dx0
Slide: 11
Fresnel diffractionD
u0(x,y) uD(x,y)
For example at λ = 0.5Å (25 keV)a = 1 m ⇒ D = 10 mma = 40 m ⇒ D = 16 m
First Fresnel zone determines the sensed lengthscale Distance to be most sensitive to object with size a Dopt=
a2
2
In principle: complete object contributes to a point of the image In practice: only finite region: first Fresnel zone
radius r F= D
Slide: 12
Fresnel diffraction in reciprocal space
Propagation over distance D along z:Simple phase factor given by: exp(i kz D)
kinc
k
θ = λ f
kinc kinc
k
),(~0 yx ffu
decomposition of wave in its plane wave components
z
each plane wave component is modified in its phase:multiplication with propagator (function of spatial frequency f ):
PD f =exp −iD f 2
Slide: 13
Contrast Transfer Functions
Fourier Transforms of the intensity and phase are linearly related
phase contrast factor
ID(f) = δ Dirac(f) 2 cos(πλ Df 2) . B(f) + 2 sin(πλ Df 2) . ϕ (f)
2
1
0
1
2
0 0.5 1 1.5 2 2.5
amplitudephase
contrastfactor
frequency
valid in case of a slowly varying phase, weak absorptionamplitude contrast factor
Slide: 14
Variable period
2 µm
period 720 nm≈
period 610 nm≈
Contrast depends strongly on period or spatial frequency
period 530 nm≈
Object invisible !
Decreasing linewidthIncreasing spatial frequency
Slide: 15
Coherence
Coherenceof a beam
Temporal Coherencecorrelation in time u(t).u(t+∆ t)
Monochromaticity: ∆ λ /λLongitudinal coherence length: λ 2/ ∆ λ
Spatial Coherencecorrelation in space u(x).u(x+∆ x)
Angular source size: αTransverse coherence length: λ / 2α
∆ x
c.∆ t
Slide: 16
Spatial Coherence Hard Xray / neutron sources are ~ incoherent
Wave is partially coherent when the source is small and far
source object detector
s αblurring= z2.α= z2/z1.s
z1 z2
szl
2.
21
cohλ
αλ ==
Feature size a (1/f)Dopt=
a2
2| µ 12(λ Df)|
lcoh > a / 2
Transverse coherence length
Slide: 17
Temporal Coherence Incoherent sum of intensity for different wavelengths Approximation:
second coherence envelope transfer function ID(f) = δ Dirac(f) + sinc(Df 2/2) . 2 sin(πλ
0Df 2) . ϕ (f)
2
1.5
1
0.5
0
0.5
1
1.5
2
0 2 4 6 8 10
=0
=0.05
2
1.5
1
0.5
0
0.5
1
1.5
2
0 2 4 6 8 10
Slide: 18
Simulation
Direct Space Reciprocal SpaceOBJECT
multiplication wave withtransmission function
convolutionT = eB eiϕ
PROPAGATION
diffraction integral multiplication FT wave withpropagator PD = exp(iπ λ Df2)
INTENSITY
autocorrelation
�
IDcoh x,y( ) = u(x,y) 2
COHERENCE / DETECTORmultiplication FT intensity withdegree of coherencedetector transfer function
| µ12
(λ Df)|R(f)
convolution intensity withprojected sourcePSF detector
Known Object ⇒ Image
Slide: 19
Simulation
λ = 0.52 ÅD = 0.4 m
Hole =dodecahedron seen
along the 2fold axis
2 4 6 µm
Simulations
Porosity in Quasicrystals
Known Object ⇒ Image
Slide: 20
Image Degradation
Stringent requirementson beamline optics:Polished windows,Monochromator surface, …Dust!
Scratches Dust
1 mm
Everything in the beam can act as a phase object
Unpolished Beryllium
no thickness variations (surface)no density fluctuations (volume)
Slide: 21
Diffraction at apertures
aperturegeometrical
shadow
• Fresnel diffraction
z1 z2
edge
λ = 0.7 Å
100 µm
rectangularaperture
λ = 0.6 Å
z1 = 140 mz2 = 6 m
⇒ aperture close to sample
Slide: 22
Edge Detection
Essentially edge enhancementWeak defocusing (and weak contrast!)
Radiography (2D)
Tomography (3D)
Detection of cracksholesreinforcing fibres, particles
absorption term refraction termLaplacian refractive index
aD <<λ
absorption image phase term2D Laplacian phase
I D x , y ≈ I 0 x , y . [1− D2
xyx , y]
o x , y , z ≈ x , y , z − D .xyzx , y , z
Slide: 23
Edge Detection
U. Wegst
Slide: 24
Insect Anatomy
Virtual slices through heads of tiny staphylinid beetles
100 µ m
O. Betz (Univ. Tuebingen), U. Wegst, D. Weide et al, J. Microscopy, 227, 51 (2007)
Slide: 25
Phase Retrieval
How to retrieve the phase and amplitude in the object plane?
Phase in image plane is lost
Image(s) ⇒ Object ???Inverse Problem
Slide: 26
Phase retrieval and holography
ID f = ∣∣2D
* e−iD f 2
f e iD f 2
*− f NonLin.
e iD f 2
ID f ≠0 = * f e i 2D f 2
* − f Backpropagation
FT object FT conjugate object
e iD1
f 2
I 1 f ≠0 = * f e i2D
1f 2
*− f
⋮
e iDN
f 2
IN f ≠0 = * f ei 2D
Nf 2
*− f
Multiple distances to avoid twin image problem
Consider weak object
kinc .kinc
k T x =x ⟨x ⟩=0with
Slide: 27
Phase Retrieval in practice
Flemish School: D. Van Dyck, P. Cloetens, JP GuigayTransfer Function Approach / Focus variation
D1 D2 Dn
Series of images recorded at different distances
Each distance is most sensitive to a specific range of spatial frequencies
Australian School: K. Nugent, T. Gureyev, D. PaganinTIE (transport of intensity) ∂ I
∂ z=−
2∇ I ∇
Slide: 28
Phase Retrieval: multiple distances
• nonabsorbing foam• 4 images recorded• E = 18 keV D
D = 0.21 m D = 0.51 m D = 0.90 mD = 0.03 m
50 µ m
Dvariable
K=∑m∣I m
exp f −I m
calc[ f ]∣2
Least squares minimization of cost functional
Slide: 29
Holotomography
3D distribution of δ or the electrondensityimproved resolutionstraightforward interpretation
processing
2) tomography: repeated for ≈ 1000 angular positions
PS foam
1) phase retrieval with images at different distances
Phase mapD
P.Cloetens et al., Appl. Phys. Lett. 75, 2912 (1999)
Slide: 30
Phase retrieval: transfer function approach
)sin( 2Dfπλ“transfer function”
Noniterative (fast!)
⇒ Optimizationof the choice of distances
1/N∑m sin2 Dm f 2
4 distances
Linear least squaresI m f =2sin Dm f 2
f
f =∑m sin Dm f 2
I m f
∑m2 sin2
Dm f 2
Slide: 31
Phase retrieval: regularization
Remains illposed for low spatial frequencies Requires regularization:
e.g. limit the energy of the solution
K=∑m∣I m
exp f −I m
calc[ f ]∣2∣ f ∣2
Model error Regularization error
f =∑m sin Dm f 2
. I m f
∑m2sin2Dm f 2
⇓
Slide: 32
Phase Tomography of Arabidopsis seeds
3D structure of Arabidopsis seeds in their native state wet sample, no preparation no staining, no fixation, no cutting, no cryocooling
Holotomographic approachContrast proportional to the electron density4 distances, 800 anglesE = 21 keV
P Cloetens, R Mache, M Schlenker, S LerbsMache, PNAS (2006) 103, 14626
Slide: 33
Phase Tomography of Arabidopsis seeds
In situ 3D imaging of a seed of an Arabidopsis plant
wet sample, no preparationRadiograph D = 10 mm Spectrum – Fourier transform
50 µ mContrast factor
2
1
0
1
2
0 0.5 1 1.5 2 2.5
amplitudephase
Slide: 34
Contrast factor
Phase Tomography of Arabidopsis seeds
In situ 3D imaging of a seed of an Arabidopsis plant
wet sample, no preparationRadiograph D = 30 mm Spectrum
50 µ m
Slide: 35
Contrast factor
Phase Tomography of Arabidopsis seeds
In situ 3D imaging of a seed of an Arabidopsis plant
wet sample, no preparationRadiograph D = 60 mm Spectrum
50 µ m
Slide: 36
Contrast factor
Phase Tomography of Arabidopsis seeds
In situ 3D imaging of a seed of an Arabidopsis plant
wet sample, no preparationRadiograph D = 100 mm Spectrum
50 µ m
Slide: 37
Phase Tomography of Arabidopsis seeds
Holotomographic approachFour distancesE = 21 keV
Seed of Arabidopsis
30 µm
Tomographic Slices
CotyledonP Cloetens, R Mache, M Schlenker, S LerbsMache, PNAS (2006) 103, 14626
Slide: 38
Phase Tomography of Arabidopsis seeds
Seed of Arabidopsis
protoderm
organites(protein stocks)
tegumenintercellular spaces
10 µm
Tomographic Slice
5 µm
Slide: 39
Phase Tomography of Arabidopsis seeds
threedimensional network of intercellular air space
gas exchange during germinationand/orrapid water uptake during imbibition
Role?
Seed of Arabidopsis
P Cloetens, R Mache, M Schlenker, S LerbsMache, PNAS (2006) 103, 14626
Tafforeau P., Kaminskyj S.Tafforeau P., Kaminskyj S.
Absorption
D = 6 mm
Oldest terrestrial fossil plant from Rhynie (Scotland)
100 µ m
Phase contrast
D = 100 mm100 µ m
Oldest terrestrial fossil plant from Rhynie (Scotland)
Tafforeau P., Kaminskyj S.Tafforeau P., Kaminskyj S.
Phase contrast
D = 300 mm100 µ m
Oldest terrestrial fossil plant from Rhynie (Scotland)
Tafforeau P., Kaminskyj S.Tafforeau P., Kaminskyj S.
Holotomographicreconstruction
100 µ m
Oldest terrestrial fossil plant from Rhynie (Scotland)
Tafforeau P., Kaminskyj S.Tafforeau P., Kaminskyj S.
Slide: 44
Coherent Diffraction Imaging
I. Robinson, I. Vartaniants, G. Williams, et al., PRL (2001) 87, 195505
Coherent Diffraction Pattern from Au particlemeasured at APS
Far field diffraction pattern (D → ‘∞’)Lensless ImagingNonperiodic objectInterest: high spatial resolution
I. Vartaniants
�
λD >> a
Slide: 45
Coherent Diffraction Imaging
Iterative phase retrieval algorithm
sk(x) Ak(q)
Reciprocal Space Constraints
A'k(q)s'k(x)
Real Space Constraints
FFT
FFT1
Real space constraints:•finite support•real, positive
Reciprocal space constraint:)(I)(A expk qq q
R.W.Gerchberg & W.O. Saxton, Optic (1972) 35, 237J.R. Fienup, Appl Opt. (1982). 21, 2758R.P. Millane & W.J. Stroud, J. Opt. Soc. Am. (1997) A14, 568
I. Vartaniants
Slide: 46
Coherent Diffraction Imaging
3D Reconstruction of Au particle
G. Williams, M.Pfeifer, I. Vartaniants, I. Robinson,PRL (2003) 90, 175501
I. Vartaniants
Slide: 47
The NanoImaging Project
Projection Microscopy PXMStructureDose efficient, fastPhase contrast
Fluorescence mapping XFMMicroanalysisSlowTrace elementsPhase contrast
Slide: 48
Bent graded multilayers in KB geometryBent graded multilayer
KBgeometry
O Hignette, P Cloetens et al, Rev. Sci. Instr. 76, 0637091 (2005)
Good efficiency Large acceptance and NA Large bandwidth / 'achromatic'
Slide: 49
KirkpatrickBaez focusing
Line Focus
0
0.2
0.4
0.6
0.8
1
1.2
200 150 100 50 0 50 100 150 200
Normalized Intensity
x [nm]
FWHM = 45nm
focal line raw data
Line width: 45 nmdeconvolved ~ 40 nm
O. HignetteP. CloetensCh. MoraweW. LudwigP. BernardR. Mokso
E = 24 keV
45 nm
2D Focus76 nm (h) x 84 nm (v)6 109 ph/s in 76 x 84 nm2
1012 ph/s in 80 x 150 nm2
8 107 ph/s/nm2
Energy: 17 keVAcceptance: 0.2 x 0.4 mm2 (0.4 x 1 mm2)∆ E/E ~ 2 102 0
0.2
0.4
0.6
0.8
1
0.6 0.4 0.2 0 0.2 0.4 0.6
Normalised Intensity
Piezo position (‚m)
FWHM = 76 nmaperture: 0.1x0.1 mm2
Slide: 50
Projection Microscopy (PXM)
objectz1 z2 detector
equivalent toD
+
Plane wave illumination Magnification
Limit casesz1 >> z2
D = z2 ; M = 1z1 << z2
D = z1 ; M = z2/z1
Prop. by D Van Dyck, J Spence
M=z1z2
z1D=z1 . z2
z1z2
Slide: 51
Interaction nanoparticles / cells / tissuesCombination PXM SXM
Distance: 200 mm − 10 mm'Pixel size': 540 nm − 27 nmFOV: 1 mm − 50 m
No Xray optics behind the sampleDose efficient
E = 17 keVExp. time = 0.3 s
Cells exposed to Pt nanoarticles, freeze dried
S Bohic, P Cloetens; B. Kysela, The Medical School, Univ. Birmingham
Slide: 52
Projection Microscopy
Distortion CorrectionMagnification CorrectionMutual alignment
20 m
Slide: 53
Projection Microscopy: Phase Retrieval
Relative phase mapComplementary to fluorescence maps
20 mE = 17 keV
Slide: 54
Combination PXM SXM
200 nm/step100 ms / ptfreezedried
K
Zn PtPt nanoparticlesφ : 6 nmtarget the nucleus
50 m
Phase
Health hazard
S Bohic, P Cloetens; B. Kysela, The Medical School, Univ. Birmingham
Slide: 55
Zoom Tomography
R Mokso, P Cloetens, E Maire, W Ludwig, JY Buffière, APL, 2007, 90, 144104
Keep sample preparation straightforwardNot too destructiveSample size >> field of view
Slide: 56
Zoom Tomography
R Mokso, P Cloetens, E Maire, W Ludwig, JY Buffière, APL, 2007, 90, 144104
Inside φ = 1 mm sample → local tomography!E = 20.5 keVXray magnification ~ 80 (voxel size = 90 nm)
Al / Si alloytomographic slice
75 µ m
SiPoreAl5FeSi
Slide: 57
Conclusions
• Coherent Xrays allow to perform, in an easy way,phase (sensitive) imaging through propagation
• The coherence puts stringent requirements on all beamline optics
• Quantitative mapping of the phase (2D) and density (3D) is possible by combining images at different distances with an adapted numerical algorithm
• Nanoprobe Imaging combines in a single setupProjection Microscopy (µstructure / function)
and Fluorescence Imaging (element specificity / trace elements)
Slide: 58
Some references
• E. Hecht, Optics, 3th ed. (AddisonWesley, 1998).• M. Born and E. Wolf, Principle of Optics, 6th ed. (Pergamon Press, Oxford, New York,
1980).• J.W. Goodman, Introduction to Fourier optics, 2nd ed. (McgrawHill, 1988).• D. Paganin, Coherent Xray Optics (Oxford University Press, USA, 2006).
• P. Cloetens, R. Barrett, J. Baruchel, J.P. Guigay and M. Schlenker, J. Phys. D: Appl. Phys. 29, 133 (1996).
• K.A. Nugent, T.E. Gureyev, D.F. Cookson, D. Paganin, Z. Barnea, Phys. Rev. Lett. 77, 2961 (1996).
• P. Cloetens, M. PateyronSalomé, J.Y. Buffière, G.Peix, J. Baruchel, F. Peyrin and M. Schlenker, J.Appl. Phys. 81, 9 (1997).
• P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J.P. Guigay, and M. Schlenker, Appl. Phys. Lett. 75, 2912 (1999).
• S. Zabler, P. Cloetens, J.P. Guigay, J. Baruchel, M. Schlenker, Rev. Sci. Instrum. 76, 073705 (2005).
Slide: 59
Outlook
Beamline for NanoImaging and NanoAnalysisNI NA
1 0.01Energy range
Main goals
Main fields
Length 185 meters 150 metersSpatial Res. 10 – 100 nm 100 nm 1 mE/E (%)
Discrete 10 – 18 – 30 keV Scanning 5 – 100 keVXRF, XRI2D/3D
...XAS, XRD, XRF, XRI2D/3D
insitu experimentsBiology
NanoTechnologyBiology, environmental sciences,geoscience, materials sciences, ...
Medium term: ESRF Upgrade Program
Long high beamline 'NINA''NINA'
Slide: 60
Limits in Xray focusing
Σ
…
sDL = c lNA
J Susini+ Optics imperfections
Slide: 61
Nano XRF of a Siemens star
E = 17.5 keV600 x 600 pixels30 ms / point50 nm step (zap)3 hoursDetector limited
30 m
Slide: 62
Fresnel diffraction of a Siemens star
E = 17.5 keV500 ms exposurez1: 133 → 7 mm 'Pixel' : 250 → 13 nm
Slide: 63
Fresnel diffraction of a Siemens star
E = 17.5 keVD = 7 mm
Simulation