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Columbia
Astrophysics
Laboratory
XMM­RGS
Doc. : RGS­COL­CAL­96002
Page : 1
Auth. : Frits Paerels
Date : June 5, 1996
Substrate Flatness Statistics for the First 100 Substrates
Document title: Substrate Flatness Statistics for the First 100 Substrates
Document version: 1
Distribution:
SRON: J.W. den Herder
A. Brinkman
H. Aarts
Columbia: S. Kahn
F. Paerels
A. Rasmussen
LLNL: T. Decker
S. Pratuch
PSI:
MSSL: G. Branduardi­Raymont
ESTEC: C. Erd
P. Videler
Introduction
We analyze the statistics of the flatnesses of the first set of almost 100 substrates (97,
to be precise), and evaluate the impact of the observed distribution of flatnesses on
the predicted resolution of the RGS. We find that on average the substrates are flat­
ter than was initially budgeted for, with the result that the resolution as currently
predicted from known sources of image broadening will slightly improve over previous
estimates.
Statistics of the Flatnesses of the First 100 Substrates
The Zygo interferograms for the first 97 substrates were analyzed in the following
way. The phase maps are converted into slope error maps, by calculating the slope
along the dispersion direction from nearest­neighbor differences (i.e. pixel­by­pixel).
The distribution of these slopes is computed, and a Gaussian distribution is fitted to
it. From the sigma of this best­fitting Gaussian, the HEW of the slope angle distri­
bution is calculated by HEW = 1.35oe. This procedure is identical to the one initially

Columbia
Astrophysics
Laboratory
XMM­RGS
Doc. : RGS­COL­CAL­96002
Page : 2
Auth. : Frits Paerels
Date : June 5, 1996
used to characterize the flatness of the EOBB substrates, from which the current flat­
ness criterion was derived. Note that closer inspection of the images may reveal that
the rms (oe) width of the slope angle distribution may overestimate the broadening
contributed by a particular substrate (cases where a few outlying points dominate the
rms width of the distribution). In that sense the following analysis is a conservative
estimate.
Figure 1 shows the frequency distribution of the non­flatness parameters, or 'bows',
binned in 0.4 ¯rad bins, for the first 97 substrates. Superimposed on the measured
distribution is a Gaussian of the same mean and variance as the measured distribu­
tion (i.e. the Gaussian is not derived from a formal fit). The observed mean bow is
6.8 \Sigma 1.6 ¯rad (1.4 \Sigma0:3 arcsec). Note that these numbers refer to the average HEW
bow of a substrate, and to the standard deviation (oe) of this distribution of HEW bows.
Analyzing the data in two subgroups ('early' and 'late' substrates) shows no signifi­
cant evolution in these flatness values.
Figure 1. Distribution of the bows of the first 97 substrates, binned in 0.4 ¯rad bins.
Superimposed is a Gaussian of the same mean and variance as the sample.

Columbia
Astrophysics
Laboratory
XMM­RGS
Doc. : RGS­COL­CAL­96002
Page : 3
Auth. : Frits Paerels
Date : June 5, 1996
Impact on the Resolution of the RGS
The original RGS resolution budget simulations (cf. ISVR, p. 61) were based on as­
suming a distortion of the grating substrates amounting to 0.7 waves peak­to­valley
from the center to the edge of the substrates (along the dispersion direction; ISVR,
p. 23). This non­flatness produces an image size of 130 ¯m HEW at blaze. The im­
age size due to all perturbations except the telescope is 217 ¯m. The total image size,
with the standard 20 arcsec HPD mirror performance, is 302 ¯.
All simulations after the Panter EOBB test were done based on an analysis of the
set of eight EOBB gratings. From that analysis a mean bow (HEW of slope distri­
bution) of 10.5 \Sigma1:4¯rad (2.2 \Sigma 0.3 arcsec) was derived. In the most recent update
of the simulations (after the last slight loosening of the tolerances on the accuracy of
the boss positions, February 1996), the width of the spectral image at the blaze wave­
length is 287 ¯m (assuming a 20 arcsec telescope beam). The non­flatness by itself
produces a size of 106 ¯m; all perturbations other than the telescope together pro­
duce 181 ¯m.
I used the 'equivalent Gaussian' distribution plotted in Figure 1, to calculate the ef­
fect of the flatter substrates. The combined effect of all perturbations other than the
telescope now gives an image size of 166 ¯m (as opposed to 181 ¯m). After adding
the effect of a 20 arcsec telescope beam, the width is 280 ¯m, a very slight improve­
ment over the previous 287 ¯m. For reference, 280 ¯m at 15 š A corresponds to \Delta– =
0:034 š A, or a resolving power of 447.
Acceptance Criteria for Future Substrates
The above simulations indicate that even though the bow is the single largest contri­
bution to the resolution budget after the telescope blur, a decrease in the average bow
from 2.2 arcsec to 1.4 arcsec (or 10.5 to 6.8 ¯rad) improves the resolution only by a
few percent. This of course also implies that the final resolution will not be affected
by accepting grating substrates much worse than the present average. In particular,
there is no reason to reject substrates with worse than 9 ¯rad bow. The largest bow
encountered among the first 97 is 11.8 ¯rad. This suggests that a sensible acceptance
criterion is 12 ¯rad. The long­term evolution of the distribution of substrate bows
will be monitored periodically to ensure the average bow is not creeping up above 10
¯rad.

Columbia
Astrophysics
Laboratory
XMM­RGS
Doc. : RGS­COL­CAL­96002
Page : 4
Auth. : Frits Paerels
Date : June 5, 1996
Current Resolution Budget
For reference, the current resolution budget, as well as the ISVR budget, is listed in
Table 1. The last column of the table indicates how the quantities listed in the 'Cur­
rent' columns are to be interpreted.
Table 1: Error Budget, Status June 1996.
ISVR current
error type value type value
Tx 2 ffi's 0.3 mm square \Sigma0.4 mm FW (1)
Ty 2 ffi's 1.0 mm square \Sigma1.0 mm FW
Tz 2 ffi's 30 \Theta 10 \Gamma3 mm --- --- ---
Rx 2 ffi's 5:0 \Theta 10 \Gamma5 rad --- --- ---
Ry 2 ffi's 5:0 \Theta 10 \Gamma6 rad --- --- ---
Ry 2 ffi's 1:0 \Theta 10 \Gamma2 rad square \Sigma1:0 \Theta 10 \Gamma2 rad FW
bosses --- --- square \Sigma1:2 \Theta 10 \Gamma3 mm FW
D.MB (2) --- --- square \Sigma1:5 \Theta 10 \Gamma3 mm FW
S.MB (3) --- --- square 0:5 \Gamma 2:5 \Theta 10 \Gamma3 mm FW
rails --- --- square \Sigma0:15 \Theta 10 \Gamma3 mm FW
bow(z) (4) 1 ffi 7– square \Sigma 20 arcsec FW (1)
bow(y) (4) 1 ffi 0.7– Gaussian 1:4 \Sigma 0:3 arcsec HEW (5)
twist 1 ffi 0:7 \Theta 10 \Gamma3 mm --- --- ---
twist(RGA) --- --- square 0.0 rad oe
\Deltad square \Sigma3:0 \Theta 10 \Gamma7 mm square \Sigma3:0 \Theta 10 \Gamma7 mm FW
Ry(RGA) Gaussian 4:85 \Theta 10 \Gamma6 rad Gaussian 4:85 \Theta 10 \Gamma6 rad oe
Tz(R) Gaussian 50 \Theta 10 \Gamma3 mm Gaussian 50 \Theta 10 \Gamma3 mm oe
(1) FW: Full Width, i.e. a flat distribution between --a and +a
(2) Differential embedment, (3) Systematic embedment
(4) Bow(z) is the bow around the z­axis (cross­dispersion)
(5) see text for explanation