Progress on Excel-based Numerical Integration Calculation
of the Coded Aperture Image
to include mask partial-transmission and phase shift.
Dan Peterson
- reminder of the calculation, including known deficiencies
-issues that are apparent in the December 2011 data
- amplitude transmission and phase shift
- calibrating the model parameters
- sensitivity to the model parameters
- comparison of coded aperture with pinhole
- remaining deficiencies, future improvements and calibrations
2012-02-02
1
History: calculated image
Recall, the image calculation and fitting technique were developed in January 2011, and
uses an Excel spreadsheet to perform the numerical integration of the amplitudes,
E = E0 ∑ e i 2π
r(y) /λ T(y)
(1)
where y is the vertical position at the plane of the optics element,
r(y) is the path length from the source, through the optic, to the detector pixel,
T(y) is the transmission of the optics element as a function of vertical position.
Pulse height is calculated for 64 pixels of size 25μm, (half diodes) .
Paths are separated by Δy=0.5 μm at the optic. (Features are 10 μm. )
January 2011 version had many shortcuts:
raw
x-ray energy: mono-energetic 2.0 keV,
λ=6.2 x10-10 m, 6.2 x10-4 microns.
transmission of the gold coded aperture mask is 0
the semitransparent mask contributes to the diffraction pattern
The integration produces the point source image at right.
2012-02-02
smoothed
50% n.n.
2
History: Sum Of Gaussian fit
The beam size=0 image is parameterized as a Sum of Gaussians, SoG.
The parameterization allows a simple convolution with the broadening due to beam size.
The fitting function is then:
F = B + C / (2π)½ ∑J=1,12 AJ / (σJ2 + σ2)½ e
-½ ( x - X0J - X0)2 / (σJ2 + σ2 )
(2)
The 36 function definition parameters are calculated, for beam size=0, once.
These parameters are
( AJ , σJ , X0J ) J=1,12 .
In January 2011, there were 4 fitting parameters:
C (area), X0 (position), σ (magnified beam size), B (background )
2012-02-02
3
Performance of the original, January 2011, function
December 2010 data:
results were similar from the 3 optics elements: PH, CA, FZP
December 2011 data:
several problems were uncovered
(1) primary peak is OK,
but is dominating the fit
004018, BigD
004029, Norm
(3) interference dip
- is lower than the fit,
lower than the background
(2) secondary peak
- is higher than the fit
004020, BigD
displaced detector
(5) background
- does not match a flat line
- is from a material property and
should be a fixed contribution
2012-02-02
(4) left shoulder of the primary peak
- is higher and narrower than the fit
4
Calibrating the image function
I take an empirical, data driven , approach to calibrating the image function.
Continue using Excel, which provides rapid feedback.
The limitation of Excel is that the energy spectrum is limited to 5 discrete energies.
calibration variables:
For the current calibration, these variables are tuned without much theoretical input.
(1) the energy spectrum ( within the limitation of 5 discrete energies )
(2) semi-transmission of the coded aperture mask
(3) phase shift in passing through the gold mask
There are still shortcuts:
The transmission and phase shift are
( as yet ) energy-independent.
004029, Norm, turn 1
The goal of the calibration is to match
major features of the observed image
(1) primary peak
(2) secondary peak
(3) interference dip
(4) left shoulder of the primary peak
(5) background
(6) side peak at channel 7
(7) side peak at channel 31
2012-02-02
5
transmission, phase shift, and the complex index of refraction
ref. X-ray Data Booklet, LBNL/PUB-490
The expression for the complex index of refraction
n = 1 - δ - iβ
(3)
describes the phase velocity of the wave and absorption due to atomic interactions.
The phase velocity is greater than c, speed of light in a vacuum.
The wave equation (1) stated in slide 2
E = E0 ∑ e i 2π r(y) /λ T(y)
(1)
becomes (including the time dependence)
E = E0 ∑ e i (2π / λ) (-ct + r) e - i (2π / λ ) δ r e – ( 2π / λ ) β r
(4)
So far, this is simply an expression
of the undisturbed wave, a phase shift, and the transmission.
2012-02-02
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transmission, phase shift, and the complex index of refraction
ref. X-ray Data Booklet, LBNL/PUB-490
The transmission and phase shift terms are calculated
from atomic scattering of the x-rays and expressed as
δ = re λ2 na / (2π) f1
iβ = i re λ2 na / (2π) f2
(5)
(6)
where re is the electron radius,
na is the atomic number density
and f1 and f2 are unit-less form factors,
which are provided in the reference in graphs.
Substituting the definitions, (5) and (6) into equation (4)
E = E0 ∑ e i (2π / λ) (-ct + r)
e - i 2π ( re λ na 1/(2π) f1 ) r
e – ( re
λ na f 2 ) r
(7)
We can calculate the image shape with this equation,
knowing the energy spectrum and material thicknesses (windows and coded aperture) .
2012-02-02
7
calculation of transmission and phase shift:
what is reasonable for the specification gold thickness
Amplitude transmission and phase shift can be calculated from equation (7):
E = E0 ∑ e i (2π / λ) (-ct + r) e - i 2π ( re λ na 1/(2π) f1 )
r
e – ( re
phase shift
λ na ρ f 2 ) r
(7)
transmission
The amplitude transmission is thus
T
where
= e – ( re
λ na f 2 ) r
= e – ( re
λ N/AAu ρ f2 ) r
r is the thickness of the gold: specification: 0.7 x 10-4 cm
re is the electron radius: 2.818 x 10-13 cm
N is Avogadro’s number 6.022 x 10+23/g
AAu is the atomic weight of gold: 197
ρ is the density of gold, 19.3 g/cm3
f2 is taken from the plot: 31 at 2.4 keV , λ=5.17 x10-8 cm
10 at 2.0 keV , λ=6.20 x10-8 cm
and
T (2.4 keV) = e-1.865 = 0.155
T (2.0 keV)=
= 0.49 ( the simple average is 0.32 )
I express the phase shift as a fraction of 2π. Again, referring to equation (7):
θ = - ( re λ na 1/(2π) f1 ) r
= - ( re λ N/AAu ρ 1/(2π) f1 ) r
f1 is taken from the plot: 52 at 2.4 keV , λ=5.17 x10-8 cm
50 at 2.0 keV , λ=6.20 x10-8 cm
θ (2.4 keV) = -0.50
θ (2.4 keV) = -0.57
2012-02-02
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Calculation of transmission and phase shift:
what is reasonable based on the observed background
(the gold thickness issue)
The specification gold thickness, 0.7 micron, leads to an
average transmission that does not account for the large background
under the coded aperture image.
The data indicates a lower gold thickness, ~0.5 micron,
approximating with an energy-independent transmission and phase.
(As described earlier, the current calibration
uses average values for the transmission and the phase.)
for illustration,
this actually corresponds to
~0.8 micron gold thickness.
The plots at the right, show the transmission
and phase shift for 0.5 micron gold thickness.
Loosely averaging
{1.7 to 4.0 keV}
leads to an expected
average amplitude transmission of ~0.43
and
average phase shift of ~ -0.35 x 2π .
2012-02-02
9
Input Energy distribution, what I use
Other material in the beam line are :
2.5 microns of Silicon in the Coded Aperture substrate,
~6 ??? microns of Carbon in the diamond window,
and 0.16 microns Si3N4 in the diode passivation layer, which I ignore.
The energy intensity distribution can be calculated from
the synchrotron radiation spectrum
and the mass absorption coefficient, μ : I / I0 = e - ∑ μ ρ r
(Brian Heltsley)
Future image calculations should include
this realistic energy intensity distribution.
But, for now…
The numerical integration includes 5 discrete energy values (lower right).
In the red distribution, the 5 delta functions are spread
into equal-area blocks centered on the discrete energies.
This is the energy distribution used in the current calculation,
along with energy-independent gold amplitude transmission
and energy-independent phase shift.
2012-02-02
10
The current coded aperture image and Sum Of Gaussians
photon energy distribution,
1.98, 2.09, 2.30., 2.61., 3.03 equal weight
amplitude transmission of mask = 0.450
phase shift = -0.31 ( x 2π )
sigma of SoG fit to image = 351.48
dpp
Jan 2012
images are ~13 μm beam size
John F
current
fits are forced to beam size=0,
for illustration
fits are without added background
2012-02-02
11
Image contributions from the 5 energies.
In each case,
the red line is the parameterization of the sum.
with 0.45 amplitude transmission, at all energies
with -0.31 x 2π phase , at all energies
1.98 keV
2.09 keV
current coded aperture function
1.98, 2.09, 2.3, 2.61, 3.03 keV
(average= 2.4 keV )
2.30 keV
2.61 keV
3.03 keV
And, the possible contribution from lower and higher energies
1.6 keV
4.0 keV
2012-02-02
12
variation with phase
In each case,
the red line is the parameterization of the current function
all with 0.45 amplitude transmission, at all energies
phase = 0
phase = -0.5
phase = -0.1
phase = -0.6
phase = -0.2
phase = -0.7
2012-02-02
phase = -0.3
phase = -0.8
phase = -0.4
phase = -0.9
13
compare pinhole and CA on coupling-8 scan
The Coupling 8 scans are used to compare
pinhole and coded aperture results.
Top:
pinhole and coded aperture (2 fits) vs. Coupling 8.
Bottom:
coded aperture beam size vs. pinhole
pinhole subtractor = 16 microns
2012-02-02
14
summary, future improvements, calibrations
The new image calibration results in a coded aperture function has only 3 free parameters, no floating background.
It provides stable fits up to 55 micron beam size.
The main features have sizes of about 8 micron (referred to the source) ,
which should provide stable beam size measurements to about 6 microns.
(The pixel size will be a significant effect below 6 microns.)
The new image calibration leads to average values for transmission (0.45) and phase (-0.31 x 2π),
which match expectations of the averages (0.43)
and (-0.35 x 2π).
After further comparison, the current fit could be improved to have
less background (~0.42)
and less extreme interference dip (which can result from a harder input energy spectrum.
We plan to input a detailed energy spectrum seen by the coded aperture,
add energy-dependent transmission and phase shift,
geared to a variable input gold thickness.
Various experiment can be done to understand the transmission and phase:
take measurements with various filters, and beam energies
measure the details of the image from the “box” on the optics chip
to provide uncomplicated measurement of the amplitude transmission and phase
take offset images (moving the detector) to efficiently place the major peaks and sufficient background.
Daily monitoring the image must be performed to test for changes in the transmission.
2012-02-02
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