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Construction of a Talbot Interferometer forphase-contrast imaging
Miriam Robinson
Mentors: Darren Dale and Robin Baur
August 13, 2010
Miriam Robinson (Lewis & Clark) August 13, 2010 1 / 17
What is the Talbot interferometer?
The Talbot interferometer is a X-ray imaging system that is sensitive to slightvariations in the density of matter.
More sensitive than common absorption imaging
Gives info on phase shift, absorption and scattering. All from one scan!
Miriam Robinson (Lewis & Clark) August 13, 2010 2 / 17
How does it work?
Monochromatic light passed through a silicon phase grating formsperiodic self-image downstream at distances zT
Beam
Diffraction grating
(B. Goodman, CC Attribution Share-Alike)
Miriam Robinson (Lewis & Clark) August 13, 2010 3 / 17
How does it work?
Placing an object in the beam path changes the phase and thus pattern
(Engelhardt et al., 2008)
Miriam Robinson (Lewis & Clark) August 13, 2010 4 / 17
How does it work?
Gold coated analyzer grating period matched to fringes magnifies patternthrough the Moire effect
(Momose et al., 2003) (S. Rasouli, 2007)
Miriam Robinson (Lewis & Clark) August 13, 2010 5 / 17
How does it work?
Step analyzer grating through one period to change pattern
Intensity in each pixel oscillates sinusoidally
Using a reference scan we can detect the changes due to a sample
Use DFT or fit to functionI(xg) = a0 + a1 sin(2πxg/p2 + ϕ1) + a2 sin(πxg/p2 + ϕ2)
Miriam Robinson (Lewis & Clark) August 13, 2010 6 / 17
What I’ve Done
Characterizing the Nova600 microfocus X-ray source
− Resolution - what’s the smallest thing we can see?− Sensitivity - how different or thick do the materials have be?
(Oxford Instruments Nova600 microfocus X-ray source)
Image analysis
− Run data through DFT or curve fitter to produce images
Miriam Robinson (Lewis & Clark) August 13, 2010 7 / 17
Resolution
Limiting factors:Fresnel diffraction: Fresnel number & 1
− F =a2
λLf
Phase grating period: ∼ 4µm
− Effective feature size: a′ =L
Lsa
Source size: 20µm
− Fringe visibility is limited by source size: V = e−(1.887Σd∗m/Lp2)2
* *
Miriam Robinson (Lewis & Clark) August 13, 2010 8 / 17
Resolution Results
Calculated F and magnification for range of phase grating and specimendistances ⇒ max V for range of feature sizes
For a required visibility of 3%, minimum detectable feature is ∼ 2.3µm
For a required visibility of 20%, minimum detectable feature is ∼ 2.8µm
Figure: The maximum visibilities for a range of feature sizes.
Miriam Robinson (Lewis & Clark) August 13, 2010 9 / 17
Phase Sensitivity
We want to know how sensitive the interferometer will be to phase differences.This is determined by the amount of noise in our data.Sources of uncertainty:
Counting statistics
− σp =√Np
− More incident photons ⇒ smaller uncertainty, so if you wait long enoughyou can decrease σp as much as you like
Dark current
− Counts recorded even without incident photons− Nd ∼ 0.3 e−/sec ⇒ negligible
Miriam Robinson (Lewis & Clark) August 13, 2010 10 / 17
Monte Carlo Method
Create a perfect sinusoid a0 + a1 sin(2πx+ ϕ) and sample N equallyspaced points (the number of images in a phase-stepping scan)
Add Gaussian noise with standard deviation√Np to each point
Apply the DFT to the noisy points
Recover estimated a0, a1 and ϕ
Repeat many times.
Repeat for curve fitter
The standard deviations of all the extracted ϕDFT and ϕcurve give areasonable estimate of the sensitivity.
Miriam Robinson (Lewis & Clark) August 13, 2010 11 / 17
Results
Since increasing N constrains the sinusoid further, uncertainty improvesas N increases
Curve fitter has less uncertainty than the DFT
4 6 8 10 12 14 16 18 20N (samples per phase-stepping scan)
0.0015
0.0020
0.0025
0.0030
0.0035
0.0040
0.0045
∆ϕ
(rad
ian
s)
��∆ϕDFT
��∆ϕcurve
Miriam Robinson (Lewis & Clark) August 13, 2010 12 / 17
Analysis
Phase shift caused by real part of index of refraction n = 1− δ + iβSuppose we are imaging a bug (δB) in polished amber (δA)
z
x
(beam axis)
TB
A
B
A
B
Top viewSide view
(through dashed line)
x
Miriam Robinson (Lewis & Clark) August 13, 2010 13 / 17
Analysis
Then, through convolution of Gaussian source and decrement profile, the
decrement difference is δ ∼ δA +δB − δA
Σp
z
x
(beam axis)
TB
A
B
Top viewSide view
(through dashed line)
x
A
B
x0
Miriam Robinson (Lewis & Clark) August 13, 2010 14 / 17
Analysis
Then recorded phase shift is ϕ = 2πmp22λΣp
(δB − δA) · TB
Rearrange this to get: (δB − δA) · TB =ϕmin
2π
2λΣp
mp2We can use this to find a minimum decrement difference, given athickness, or a minimum thickness, given a decrement difference
Miriam Robinson (Lewis & Clark) August 13, 2010 15 / 17
Image Analysis - Hymenoptera
Below: Phase-contrastimage
Above: Absorptionimage
Above: Dark field image
Miriam Robinson (Lewis & Clark) August 13, 2010 16 / 17
Acknowledgments
Darren Dale
Robin Baur
CLASSE and NSF
Miriam Robinson (Lewis & Clark) August 13, 2010 17 / 17
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