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

(1) E = E0 ∑ e i 2π r(y) /λ T(y) 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 2π r(y) /λ T(y) 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: 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. raw smoothed 50% n.n. 2012-02-02

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 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 (4) left shoulder of the primary peak - is higher and narrower than the fit (5) background - does not match a flat line - is from a material property and should be a fixed contribution 2012-02-02 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 2012-02-02 5 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 6

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 (5) iβ = i re λ2 na / (2π) 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 (2π / λ) (-ct + r) e - i 2π ( re λ na 1/(2π) f1 ) r e – ( re λ 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 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 λ na ρ f2 ) r (7) phase shift transmission The amplitude transmission is thus T = e – ( re λ na f2 ) r = e – ( re λ N/AAu ρ 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 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 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 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 8

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 images are ~13 μm beam size fits are forced to beam size=0, for illustration fits are without added background John F current dpp Jan 2012 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 current coded aperture function 1.98, 2.09, 2.3, 2.61, 3.03 keV (average= 2.4 keV ) 1.98 keV 2.09 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 = -0.1 phase = -0.2 phase = -0.3 phase = -0.4 phase = -0.5 phase = -0.8 phase = -0.9 phase = -0.6 phase = -0.7 2012-02-02 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 15