Progress on Excel-based Numerical Integration Calculation

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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 - calibrating the model
parameters - sensitivity to the model
parameters - comparison of coded aperture with
pinhole - remaining deficiencies, future
improvements and calibrations
2012-02-02
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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,



(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.
E E0 ? e i 2p r(y) /? T(y)
Pulse height is calculated for 64 pixels of size
25µm, (half diodes) . Paths are separated by
?y0.5 µm at the optic. (Features are 10 µm.
) January 2011 version had many shortcuts
x-ray energy mono-energetic 2.0 keV,
?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.
raw
smoothed 50 n.n.
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History Sum Of Gaussian fit
The beam size0 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 / (2p)½ ?J1,12 AJ / (sJ2 s2)½
e -½ ( x - X0J - X0)2 / (sJ2 s2 )
(2)
The 36 function definition parameters are
calculated, for beam size0, once.
These parameters are ( AJ , sJ , X0J )
J1,12 . In January 2011, there were 4 fitting
parameters C (area), X0 (position),
s (magnified beam size), B (background )
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Performance in January 2011
Results from the 3 optics elements were similar.
2012-02-02
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Attempts to fit December 2011 data.
(3) interference dip - is lower than the fit,
is lower than the background level
uncover several problems

• primary peak is OK,
• but is dominating the fit

(4) left shoulder of the primary peak - is
higher and narrower than the fit
(2) secondary peak - is higher than the fit
(5) background - does not match a flat line -
is from a material property and should be a
fixed contribution
004018, BigD
Shown 3 conditions, 4 turns each
004020, BigD displaced detector
004029, Norm
2012-02-02
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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.
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
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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 expression for the complex index of
refraction
n 1 - d - iß

(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 2p r(y) /? T(y)

(1) becomes
(including the time dependence)
E E0 ? e i (2p / ?)
(-ct r) e - i (2p / ? ) d r e ( 2p
/ ? ) ß r
(4) So far, this is simply an
expression
of the undisturbed wave, a phase shift, and the
transmission.
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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
d re ?2 na / (2p) f1
(5)
iß i re ?2 na / (2p) f2
(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 (2p / ?) (-ct r) e - i 2p
( re ? na 1/(2p) f1 ) r e ( re ? na
f2 ) 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
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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 (2p / ?) (-ct r) e - i 2p ( re ?
na 1/(2p) f1 ) r e ( re ? na ? f2 ) r
(7)
phase shift
transmission
The amplitude transmission is thus
T e ( re ? na f2 ) r
e ( re ? N/AAu ? f2 ) r where
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 Avogadros
number 6.022 x 1023/g
AAu is the atomic weight of gold 197
? is the
density of gold, 19.3 g/cm3
f2 is taken from the plot 31 at 2.4 keV ,
?5.17 x10-8 cm
10 at 2.0 keV ,
?6.20 x10-8 cm

T (2.4 keV) e-1.865
0.155 and T (2.0
keV) 0.49 ( the simple
average is 0.32 )
I express the phase shift as a fraction of 2p.
Again, referring to equation (7)
? - ( re ? na 1/(2p) f1 ) r -
( re ? N/AAu ? 1/(2p) f1 ) r
f1 is taken from the plot 52 at 2.4 keV ,
?5.17 x10-8 cm
50 at 2.0 keV , ?6.20
x10-8 cm ?
(2.4 keV) -0.50
? (2.4 keV) -0.57
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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 2p .
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Input Energy distribution other material in the
beam line
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.
µSi
The energy intensity distribution can be
calculated using the mass absorption
coefficient, µ , I /
I0 e - ? µ ? r
µC
and for I0 the incident power distribution due
to synchrotron radiation.
(Brian Heltsley)
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Input Energy distribution, what I use
The calculated energy intensity distribution is
shown at right. Future calculations should
include this realistic energy intensity
distribution.
(Brian Heltsley)
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.
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The current coded aperture image and Sum Of
Gaussians
(this stuff is for my records)
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 2p ) sigma of SoG match to image
351.48
line is beamsize0
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Fits of the resulting function to beam various
conditions
004018, BigD 18 micron
The background basically follows the shape of
the data, but the background now may be
slightly high and may require wider shape.
Better agreement may be found with
amplitude transmission 0.42 Features in the
very low beam size data mainly fit well,
but the dip in the fit may be too low.
Better agreement may be found with more
low and high energy x-ray contribution,
the calculated distribution.
004020, BigD displaced detector
004029, Norm 13 micron
021442 Coup8 23 micron
2012-02-02
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Image contributions from the 5 energies.
with 0.45 amplitude transmission, at all
energies
with -0.31 x 2p phase , at all energies
1.98, 2.09, 2.3, 2.61, 3.03 keV with average
2.4 keV
In each case, the red line is the
parameterization of the sum.
1.98 keV
3.03 keV
2.61 keV
2.30 keV
2.09 keV
And, the possible contribution from lower and
higher energies
4.0 keV
1.6 keV
2012-02-02
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variation with phase
all with 0.45 amplitude transmission, at all
energies
phase 0
phase -0.1
phase -0.2
phase -0.4
phase -0.3
phase -0.5
phase -0.9
phase -0.8
phase -0.6
phase -0.7
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compare pinhole and CA on coupling-8 scan
subtractor 16 microns
2012-02-02
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summary, future improvements, calibrations
The new 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.) Even
though this is currently using average values,
the transmission and phase of the
resultant calculated image match the expectations
of the averages better than might be
expected. 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.
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