1. Design of a New Coded Aperture
Dan Peterson,

2. We are doing this because:
we decided we may need more optics chips for the remainder of the CesrTA program,
the FZP has not been developed and will not be used,
the standard Coded Aperture provides insufficient beam size resolution
at “normal” beam current.
Careful design is necessary; the chip costs 15 k$, and has day delivery.
Design of a Coded Aperture requires an understanding of the x-ray energy spectrum;
the image is diffractive.
I will
describe the calculation of the energy spectrum,
describe a figure-of-merit for evaluating Coded Aperture designs
describe several attempts at improved designs.

3. This design study uses the data-driven calculation of the energy spectrum.
Data was taken with the PINHOLE optic, with
3 beam energies: 1.8, 2.085, 2.3 GeV
4 filter conditions: none, 4 μm Diamond, μm Aluminum, μm Molybdenum
The energy spectrum is determined by the relative intensities in these conditions.
We find that day-to-day PH calibrations
are not repeatable at the precision required for this study.
Thus we look at ratios only within an energy.
C-line and D-line, at GeV, are averaged.
Thus, there are 9 data points.
December 2012 data

4. The model uses the Jackson formula for the energy distribution
in the synchrotron plane:
δI(ω)/δΩ|θ=0 = 3/(2π) e2/c γ2 ω/ωc exp( -2ω/ωc ) (Jackson 14.88)
In “fitting” this model to the (limited) April 2012 data,
the detector absorption and response could be
approximated by a power function , E *
Model, detector response = E1.15
Relative intensities for the December 2012 data force a reevaluation of the approximation of the detector absorption and response.
The RMS of the deviations,
(locking the relative intensity for 2.085:Diamond to the model)
is 26%.
This quantity was 15% for the April 2012 data.)
({model}-[data})/{data}
* A term of E1 has been absorbed into the normalization of the numerical integration , fixing it to be independent of energy.

5. The new fitted detector absorption and response if derived from a fit to the data.
A vector in the space of the 10 energy bin amplitudes is determined by the errors and the present energy distribution for each data point.
There is only one iteration, with the E1.15 response used for the input energy spectra. A second iteration could improve the result.
However, the RMS is 8% ;
the maximum deviation is 14% ,
which is as good as the consistency of the data.
The function described by the red line, modulating the Jackson formula, is my approximation of the energy spectrum.
({model}-[data})/{data}

6. The current Coded Aperture (CA) works well at 2.085 GeV beam energy.
I would like a way to quantify the quality of optics elements for different conditions
I use a figure of merit (FoM) for the xBSM measurements based on the modeled image.
∑pixel [ h(σ+Δσ) – h(σ) ]2 / h(σ) ( simplified for transparency; the actual formula is symmetric )
where h(σ) is the signal height of a pixel for the given beam size, σ .
So, this is a χ2 ,
measuring the change in the spectrum
with an error due to statistics only.
The FoM has units of signal height;
an increase in FoM is equivalent to increased
beam current.
It is plotted for constant Δσ /σ , and
measures the ability to resolve the beam size
for constant current.

7. threshold for resolving the beam size = ~ 0.6 ,
Some observations:
PINHOLE: The NO Filter is better than Diamond, above 20 μm beam size.
This is expected; NO filter has 2x the intensity.
Above the subtractor the FoM for NO filter should approach 2x that for Diamond.
It is surprising that the diamond is not better at low beam size.
Coded Aperture:
The current Coded Aperture (CA) performs better than the PH below ~13μm beam size, as expected.
Our experience is,
with typical beam current ~ 0.5ma:
below a beam size of ~10μm, the PH is no longer useful,
the CA performance is useful up to a beam size of ~ μm.
I estimate that the
threshold for resolving the beam size = ~ 0.6 ,
for a “typical current”, based on the experience described above .

8. At 1.8 GeV, we do not have a usable optic.
The point of the study is to design the best optic for
beam energy 1.8 GeV
for the range of beam size, μm.

9. First attempts to design a new CA had mixed results.
Images made at 7 μm beam size.
“White lines” at 15 and 20 μm beam size.
Brian suggested 2 widely spaced pairs of slits. Total opening 120 μm.
John used a “pseudo random” pattern
with 20μm minimum feature size.
Total opening 180 μm.
dpp tried a design with 2 symmetric sets of slits. Total opening 170 μm.
dpp tried a “grating, with 62 μm features.
(To operate as a real grating would require smaller features.)
Total opening 186 μm.

10. All show improvement over the standard CA with 0
All show improvement over the standard CA with 0.7 μm gold, above 12μm beam size.
The grating with 62μm features is the first to show real improvement at 20μm beam size.
But, none of these designs are above threshold at 10μm beam size.
Also, at the time of the meeting, I showed that less gold is better.
But we require at least 0.7 μm gold

11. Jim A. had the idea to start with a pseudo Fresnel Zone Plate (with 3 transmitting zones).
The “FZP” has only one significant feature
and total opening of 120 μm.
I kept the feature of a large central opening, but departed from the FZP characteristic of smoothly decreasing feature size moving away from the center.
Total opening is increased to 190 μm.
The performance is significantly improved
and for the first time,
there is visible improvement at 10 μm.

12. The tuning as of 2012-11-19 was somewhat random.
I have developed my program to directly plot the FoM in real time
in response to input changes to the design.
When feature sizes are changed in the spread sheet, the graph responds in ~5 seconds.
This allows rapid tuning of the pattern at the 1 μm level.
I have done further tuning, and now have 2 variations to consider:
a 5 slit design, with wide center slit, μm total opening ;
a 6 slit design, with the center slit split, 188 μm total opening .

13. For reference, not one of the 2 designs,
this is the
“alt3” design

14. “alt3”
“alt4” design
5 slits

15. the center slit is split and all slits are re optimized
“alt6” design
6 slits,
the center slit is split
and all slits are re optimized
“alt4”
“alt7” is a minor change,
moving the outer pairs out by 1 μm.

17. I would choose “alt 6”, with the 10 μm center slit.
There is a significant improvement at 10 μm beam size
at the cost of a minor loss at 30 μm beam size.
In use in IBS, the beam current will be less at the smaller beam size,
which justifies the tradeoff.
At 1.8 GeV , the new Coded Aperture will provide resolution similar
to that of the standard CA at GeV.
At GeV, the new Coded Aperture will provide resolution better than the PINHOLE,
for beam size up to 30 μm .