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Modeling the Chandra High Energy Transmission Gratings
below 2 keV
K.A. Flanagan a , T.H. Markert a , J.E. Davis a , M.L. Schattenburg a , R.L. Blake b ,
F. Scholze c , P. Bulicke c , R. Fliegauf c , S. Kraft c , G. Ulm c , E.M. Gullikson d
a Center for Space Research
Massachusetts Institute of Technology, Cambridge, MA, 02139
b RDS, P.O. Box 6880, Santa Fe, NM, 87502
c Physikalisch­Technische Bundesanstalt
Abbestr. 2­12, 10587 Berlin, Germany
d Center for X­Ray Optics
Lawrence Berkeley National Laboratory, Berkeley, CA 94720
ABSTRACT
The High Energy Transmission Grating Spectrometer of the Chandra X­Ray Observatory is a high spectral resolution
instrument utilizing gold X­ray transmission gratings. The gratings have been subjected to a rigorous program of
calibration, including testing at synchrotron facilities for the purpose of refining and testing the grating model. Here
we conclude our investigation of the optical constants of gold, extending it below 2 keV to complete the coverage over
the Chandra energy range. We investigate the carbon, nitrogen, oxygen and chromium edge structures introduced
by the grating support membrane. Finally, we summarize the state of the grating model, identifying those energy
regions where the residuals are most significant and suggesting where the model might be improved.
Keywords: X­ray, X­ray astronomy, X­ray spectroscopy, transmission gratings, diffraction, synchrotron, scattering
factor, index of refraction
1. INTRODUCTION
The High Energy Transmission Grating Spectrometer (HETGS) of the Chandra X­Ray Observatory is a high spectral
resolution instrument including 336 gold X­ray transmission gratings. The instrument is described in Markert et
al.. 1 The gratings have been subjected to a rigorous program of calibration testing with a goal of modeling the
first order efficiency to 1% outside of absorption edges. The gratings are of two types: High Energy Gratings (HEG)
and Medium Energy Gratings (MEG), which consist of gold bars (of 2000 š A and 4000 š A periods, respectively) atop
polyimide support membranes. Details of fabrication are given in Schattenburg et al. 2
Calibration of the Chandra High Energy Transmission Gratings has involved several distinct phases: (1) Sub­
assembly calibration, including laboratory testing of each flight grating facet against transfer standard gratings 3;4 ;
(2) synchrotron facility testing of selected gratings and sample foils and filters 5;6;7;8 ; (3) testing of the assembled
instrument at the Marshall Space Flight Center, including end­to­end testing with the flight optics and detec­
tors 9;10;11;12;13;14;15 ; and (4) in­flight calibration. 16
The synchrotron radiation measurements serve several purposes. Transmission measurements of polyimide, plat­
ing base and gold foil samples allow the optical constants and edge structures of these materials to be determined.
Absolute efficiency measurements of a few gratings serve to validate and constrain our model, and provide estimates
of its intrinsic errors. Measurements of individual gratings HX220 and MX078 have enabled their use as transfer
standards in laboratory tests. Finally, a comparison of synchrotron measured efficiencies of a few gratings with their
predicted efficiencies based on laboratory measurements allows us to assess the limitations of our subassembly tests.
Further author information: (Send correspondence to K.A.F.)
K.A.F.: E­mail: kaf@space.mit.edu

The present paper addresses the first two of these purposes. Here we complete our revision of the optical constants
of gold and our characterization of the edge structure of the materials in the support membrane. The revised
grating model, which incorporates this new information, is then compared against detailed synchrotron radiation
measurements on two flight gratings. Since these measurements provide efficiencies at hundreds of energies, we have
a clear picture of the overall fidelity of the model and a practical estimate of its limitations. These tests, and their
results, are described below.
1.1. Components of the Grating Model
We view the ''grating model'' from a calibration point of view: Our ability to fit measured grating efficiencies with
a model will depend on our input data, as well as on our understanding of the grating physics. As we shall see in
Section 4, the largest remaining discrepancies between the measurements and the model are at the absorption edges.
Our measurements of the optical constants at these edges has greatly improved our model, yet this input data set is
still the limiting factor overall. Nevertheless, some areas invite investigation at the conceptual level as well.
The model we use for the diffraction efficiency is based on the simple scalar (Kirchhoff) diffraction theory. 17 A
discussion of some aspects of the model is given in Markert et al.. 7 Apart from diffraction by the grating bars, other
factors affect the measured efficiency of the gratings and are included in the model. These include the absorptions
of the film and the plating base. The grating is built up onto a thin (0.98 ¯m for the HEGs, 0.55 ¯m for the MEGs)
polyimide film which provides mechanical support. In addition, there are very thin metallic films (' 200 š A of gold
and 50 š A of chromium) which are used for the electroplating process. These films are essentially uniform over the
grating and serve only to absorb (and not diffract) X­rays. However, their absorption introduces edge structure
which will be discussed in detail in this paper.
A full specification of our grating model includes the grating bar shape, the thicknesses of the plating base
and support polyimide support film, as well as the energy­dependent optical constants for the gold, polyimide and
chromium. The geometric components are regarded as parameters to be deduced from a fitting procedure. Hence,
the model consists of the following:
fi au (k) and ffi au (k), the components of the index of refraction for gold
z(¸), the bar shape function
t au , the thickness of the gold plating base
t cr , the thickness of the chromium plating base
t poly , the thickness of the polyimide support film
fi polyimide (k) and fi cr (k), the imaginary parts of the index of refraction, which give the transmissions of the support
film and the plating base, and
A, the amplitude factor.
The amplitude factor accounts for grating imperfections which affect the measured diffraction efficiencies. One
example of a factor which can affect measured diffraction efficiency is grating scatter. 15 The observations of scatter
are compatible with fluctuations in grating bar geometry. Other factors may also be considered, such as minute
pinholes or small regions which are non­diffracting. Since each of these affects the relative measurements of zero and
higher order efficiencies, we use a multiplicative factor (the amplitude factor) to account for this in modeling the
efficiency of each order.
The optical constants are not fitted as free parameters: There are too many (2 at each energy for gold alone), so
fitting them isn't appropriate. Any errors in these values will be manifested in the fit, and will significantly impact our
ability to accurately model the first order efficiencies. Therefore, we have employed synchrotron transmission tests
to provide independent information about the optical constants of the grating and plating base materials (fi au (k),
ffi au (k), fi polyimide (k) and fi cr (k)). Prior measurements are described in Section 1.2. Revisions to the gold constants
below 2 keV are described in Section 2; revisions to the constants for polyimide and chromium are described in
Section 3.
In practice, when fitting a model to an individual grating, we allow all parameters to vary except for the optical
constants. The bar shape function may be thought of as the cross­section of the grating bar for X­rays normally
incident on the grating surface. It is taken to be piece­wise linear, nominally with five vertices. There are two free
parameters to fix the positions of each vertex. The plating base components and amplitude factor each provide
another free parameter, totaling 14 when a model with five vertices is employed.

This model has been compared against detailed synchrotron radiation measurements on two flight quality gratings,
HA2021 and MA1047. This gives us a practical measure of the validity and limitations of the grating model. These
tests, and their results, are described in Section 4.
1.2. Summary of Synchrotron Radiation Tests
Synchrotron radiation tests for the High Energy Transmission Gratings have been performed at four facilities over
a timeframe of several years. The tests are summarized in Table 1. Our earliest modeling efforts were based upon
a rectangular grating bar model and employed scattering factors (f1, f2) published by Henke et al.. 18 (The real
and imaginary parts of the index of refraction, ffi and fi, are obtained from the scattering factors.) However, early
tests (January 1994) at the National Synchrotron Light Source (NSLS) at Brookhaven National Laboratory (BNL)
indicated significant disagreement with the Henke values for the gold optical constants. The most noticeable feature
was that the energies of the gold M absorption edges were shifted from the tabulated amounts by as much as 40 eV
(a result obtained earlier by Blake et al. 19 from reflection studies of gold mirrors.) In an effort to determine more
relevant optical constants, the transmission of a gold foil was measured over the range 2.03--6.04 keV, and the values
of fi and ffi were revised. 6 (The Henke tables were modified in 1996 to reflect these results.)
Subsequent tests on gratings explored bar shape, tilt and asymmetry, 7 and tests at the radiometry laboratory
of the Physikalish­Technische Bundesanstalt (PTB) below 2 keV identified the need to accurately model the edge
structures of the polyimide support membrane to improve the overall fit. 8 The analysis of the tests performed on
gold and polyimide membranes at PTB in October 1995 has now been completed and is detailed in Sections 2 and 3.
As a consequence of this analysis, our model now includes revised gold optical constants over the full energy range
appropriate to HETG, and detailed structure for absorption edges of C, N, O and Cr. In addition, cross­checks of
the revised gold constants (above 2 keV) and polyimide were performed (in August and November, 1996) and have
confirmed our revisions. For reference, the revision date for these changes to our modeling is May 10, 1999.
Table 1. Summary of HETG synchrotron radiation tests
Date Facility Sample Energy Comments Ref.
July '93 NSLS HX101 2.03 ­ 4 preliminary tests Nelson 5
Nov '93 Daresbury HX101 8.442 period variations (UL) Nelson 5
Jan. '94 NSLS HX101 0.7 ­ 6 first tests Nelson 6
Feb. '94 NSLS 1Au 2.03 ­ 6.04 gold optical constants Nelson 6 ; also y
Jun. '94 NSLS HA04 2.03 ­ 6 Nelson 6
'' '' HX101 2 ­ 3.5 Nelson 6
'' '' MA12 0.7 ­ 5 \Sigma1 order assymetry Nelson 6
Feb. '95 NSLS HX220 0.5 ­ 6.4 X­GEF reference grating Markert 7
'' '' MX078 0.5 ­ 3.5 X­GEF reference grating Markert 7
May '95 NSLS HX220 1.05 ­ 1.95 X­GEF reference grating Markert 7
`` `` poly 0.4 ­ 1.83 poly transmission this paper \Lambda
Oct. '95 NSLS HA2021 2.03 ­ 6.5 flight lot 2, dense 0&1 orders Flanagan 8
'' '' MA1047 2.1 ­ 5.0 flight lot 3, dense 0&1 orders Flanagan 8
Oct. '95 PTB HA2021 0.4 ­ 1.9 flight lot 2 Flanagan 8
'' '' MA1047 0.4 ­ 1.5 flight lot 3 Flanagan 8
'' '' HA2049 0.2 ­ 1.5 polyimide sample, flight lot 2 this paper \Lambda
'' '' MA1066 0.2 ­ 1.5 polyimide sample, flight lot 3 this paper \Lambda
`` `` HX507 0.05 ­ 1.9 gold optical constants this paper y
Mar. '96 NSLS HD2338 2.0 ­ 6.4 flight lot 4, \Sigma1 and higher orders in prep
'' '' MB1148 2.1 ­ 4.9 flight lot 9, \Sigma1 and higher orders in prep
Aug. '96 ALS poly 0.06 ­ .940 polyimide transmission, R. Blake this paper \Lambda
Nov. '96 NSLS 1Au 2.01 ­ 7.0 gold optical constants, R. Blake this paper y

2. GOLD OPTICAL CONSTANTS
The optical constants for gold have been revised according to the results of three synchrotron tests: two tests
(February, 1994 and November, 1996) examined the range above 2 keV, and one (October, 1995) probed energies
below 2 keV.
2.1. Gold below 2 keV
In order to investigate the optical constants of gold below 2 keV, transmission tests were made on a free­standing
gold sample, HX507, at the radiometry laboratory of the PTB at BESSY. Details of test procedures are given in
Flanagan, et al.. 8 Information about the facility may be found in Scholze, et al. 20 and Ulm and Wende. 21 The
gold foil sample contained a residual amount of Cr adhesion layer, and this needed special treatment in the analysis.
We assumed Cr optical constants and edge structure as determined according to Section 3.2, and found a best­fit
thickness for Cr of 38.7 š A by fitting near the Cr edge features. We then fixed the Cr thickness to this value and fitted
over 0.2 to 0.5 keV to obtain a best­fit thickness of 1,075.53 š A for Au. Finally, from the measured transmission,
we divided out the contribution due to Cr. Although this left an artifact around 580 eV and did not remove any
contribution at the Cr LIII edge (near 696 eV), these effects were comparatively small. The resultant transmission
was thus attributed to pure gold of thickness 1,075.53 š A and density 19.3g/cm 3 . This yielded the scattering factor
f2 directly.
The scattering factor for gold has been independently measured at PTB on a different gold sample by Henneken, et
al.. 22 These results agree well with our measurements. This is illustrated in Figure 1, where the scattering factor f2
is plotted (from which fi is directly obtained) for Henneken's data, for our measurements, and for the Henke values
(as updated in 1996). Clearly, the new gold measurements represent a significant difference from earlier values in
the range 0.5--1.5 keV. The close agreement between the HETG and Henneken results 22 is illustrated in Figure 2,
where the HETG f2 curve is overlaid with Henneken's data and its associated error bars. The HETG results are in
y Section 2
\Lambda Section 3
0 0.5 1 1.5
0
10
20
30
energy (keV)
solid line: 1999 f2's based on 1995 PTB data
dotted line: Henneken's independent measurement set
dashed line: Henke constants revised in 1996
Figure 1. Gold scattering factor f2 obtained from inde­
pendent transmission measurements by HETG team and
Henneken et al., 22 compared with Henke values (updated
in 1996). The optical constant fi is derived directly from
f2 at each energy.
Figure 2. Gold scattering factor f2 obtained by HETG
team overlaid with measured values and error bars from
Henneken et al. 22

1.5 2 2.5 3
0
0.0002
0.0004
0.0006
0.0008
0.001
energy (keV)
solid line: 1999 constants based on 1996 NSLS data
dotted line: 1996 constants based on 1994 NSLS data
dashed line: Henke 1992
Figure 3. Gold optical constants were revised in May,
1999 for grating modeling. The real part of the index of
refraction, ffi, is shown here as determined from NSLS
1994 and 1996 measurements, along with Henke 1992
values. The two NSLS results are virtually indistinguish­
able.
1.5 2 2.5 3 3.5
0
0.0001
0.0002
0.0003
energy (keV)
solid line: 1999 constants based on 1996 NSLS data
dotted line: 1996 constants based on 1994 NSLS data
dashed line: Henke 1992
Figure 4. The imaginary part of the index of refrac­
tion, fi, as determined from NSLS 1994 and 1996 mea­
surements contrasts sharply with Henke 1992 values.
agreement with Henneken's to within the published error bars, except between .13 and .26 keV where the HETG
value of f2 is slightly higher (within about 2 oe).
Our data were inadequate below 96 eV, so at this energy we have merged our f2's with those of Henneken's. A
complete file of f2 was obtained by joining the measured set from PTB below 2 keV with a newly measured set from
NSLS above 2 keV. From this, we generated corresponding f1's for a complete table of scattering factors. All of these
revisions are incorporated in the improved grating model.
The impact of the changed gold optical constants on modeling the grating efficiency is small to moderate. Use of
the new constants will result in modeled efficiencies that change by five percent or less except at the Au NIII edge
near 0.55 keV, where the change is about 7%. There are larger variations at energies below 0.11 keV, but HETG
is not intended for use at energies below 0.4 keV for the medium energy gratings (or 0.9 keV for the high energy
gratings).
2.2. Gold above 2 keV
The gold sample (1Au) which was measured at NSLS in 1994 was remeasured above 2 keV at NSLS in November, 1996
by Richard Blake and Tony Burek. The assumed thickness was 11,304 š A and the assumed density was 19.32 g/cm 3 .
In this test, the experimental procedure was improved by continuous beam monitoring and normalizations taken
adjacent in time to the transmission measurements. The edges were sampled in 0.5 to 1 eV step sizes. The 1996
data agree well with the 1994 measurements, and have been incorporated into revised optical constants (May 10,
1999). A comparison of the two data sets against Henke 18 values is given in Figures 3 and 4. The two data sets are
virtually indistinguishable in the figures. The good agreement between the two measurements of sample 1Au serves
to confirm our revisions, and allows a means of assessing some of the errors associated with these measurements.
The largest fractional differences in the two NSLS measurements of fi are seen at the gold M edges (in the energy
range 2.2 to 3.5 keV), but even there agreement between the two data sets is within 2% (or 4% at the MV edge
around 2.2 keV). This corresponds to a 2--3% error in first order efficiency at the gold M edges. Since these two tests
were performed on the same beamline, other systematic effects may not be accounted for.

380 400 420 440
0
energy (eV)
solid is KAF's raw BESSY data
dashed is Henke's values
points are Blake's ALS data
Figure 5. Comparison of polyimide absorption coeffi­
cients of the nitrogen edge region at PTB and ALS with
Henke values (which represent the HETG model prior to
the 1999 revision). The Henke values do not reproduce
the complicated edge structure of the HETG polyimide.
380 400 420 440
0
energy (eV)
solid is KAF's raw BESSY data
points are Blake's ALS data
Added 1.3 eV to Blake's data
Figure 6. Comparison of PTB nitrogen edge data with
ALS data, after an energy shift of 1.3 eV has been applied
to the ALS data to accomodate a presumed beamline
energy offset. Note that the structure and amplitude
of the edge region is confirmed by the two independent
beamline measurements.
3. POLYIMIDE AND CHROMIUM EDGE STRUCTURE
3.1. C, N and O Edges
Measurements of the gold optical constants have enabled the detailed gold edge structure to be well represented. A
similar approach has been taken toward modeling the C, N and O edges of the polyimide. We have tested samples
of polyimide from MEG and HEG flight batches at the radiometry laboratory of the PTB at BESSY. 8 These data
show that there is considerable edge structure at the C, N and O edges in our polyimide. Our approach is to model
the polyimide assuming the chemical formula (C 22 H 10 O 4 N 2 ) and nominal density (1.45 g/cm 3 ) for the polyimide
formulation we use (Dupont 2610). In prior modeling, the optical constants for the polyimide support film and the
chromium plating base have been based on scattering factors for the constituent atoms taken from Henke, Gullikson
and Davis. 18 This modeling, however, provided an unacceptable fit at the edges, with residuals up to 200% at the
C and N edges in fitting polyimide transmission data. Just below 600 eV are seen edge residuals on the order of
20% from Cr L in fitting HEG grating data. Similar results have been found for the MEG grating MA1047. Taking
the model as a whole, the polyimide edges have exhibited the worst discrepancies between our model and the data
overall. As discussed below, synchrotron testing has allowed improvements in our modeling of these edges, although
they remain the largest contributors to the errors of the model.
3.1.1. Nitrogen edge
In order to refine the optical constants for our polyimide at the C, N and O edges, we used the PTB data for two
flight batch samples of polyimide, HA2049 and MA1066. We began by finding a best­fit thickness for each of the
(MEG and HEG) polyimide samples assuming Henke optical constants and fitting over the edge­free energy range
0.6 to 1.6 keV. For each sample, an effective absorption coefficient ¯ was obtained assuming T = e \Gamma¯t where t is
the thickness in microns and T is the transmission through the polyimide membrane. The final value for ¯ was
taken to be the average value of the HEG and MEG ¯, between 272 eV and 875 eV, smoothly joined to the Henke
values outside this region. In addition, we smoothed the derived ¯ in the carbon edge region between 288.8 eV and

520 530 540 550 560
0
energy (eV)
solid is KAF's raw BESSY data
points are Blake's ALS data
Added 1.0 eV to Blake's data
Figure 7. Comparison of PTB oxygen edge data with
ALS data, after an energy shift of 1.0 eV has been applied
to the ALS data. Note the substantial agreement of the
two sets of absorption coefficients, confirming the double
structure of the oxygen edge region.
260 280 300 320 340
0
energy (eV)
solid is KAF's raw BESSY data
dashed is Henke's values
points are Blake's ALS data
Figure 8. Comparison of PTB carbon edge data with
ALS data and Henke values.
300.99 eV because of the jittery structure there. (Although this structure might be real, the low counting statistics
and limitations of the experiment discourage reliance on it.) Note that the 1982 Henke constants for carbon were
employed in our initial fitting as these were found to agree better with our data and have been shown in independent
tests (M. Zombeck, private communication) as the the better choice.
A different polyimide sample, manufactured with the same formulation, was tested at the Advanced Light Source
(ALS) at the Lawrence Berkeley National Laboratory in 1996. We found a best­fit aet of 124.25 ¯g/cm 2 . Assuming
a density of ae= 1.45 g/cm 3 , we derived a value for ¯/ae in a similar manner as has been described with regard to
the PTB data. After conversion to comparable units, a comparison of ¯/ae from PTB, from ALS and from Henke 18
is given for the nitrogen edge in Figure 5. As seen in the figure, the Henke values represent the nitrogen edge by a
single simple discontinuity, in sharp contast to the measured structure. The PTB and ALS measurements appear to
be compatible if one accounts for an apparent slight energy shift (which we attribute to an energy offset in the ALS
beamline.) This is shown in Figure 6, where a relative shift of 1.3 eV has been included. The two different beamlines
independently trace virtually the same structure and amplitude in this region, giving confidence in the result.
3.1.2. Oxygen edge
Analogous plots of the absorption coefficient in the oxygen edge region are given in Figure 7, where a 1.0 eV energy
offset attributed to the ALS beamline has been removed. Note that the Henke representation, which corresponds to
our former modeling, cleanly misses the sharp double structure. This structure is accomodated by our updated optical
constants. The close agreement generally confirms the detailed edge structure and magnitudes of the absorption
coeficients we have derived.
3.1.3. Carbon edge
In general, there was good agreement between the two polyimide data sets (at BESSY and ALS), except at the
carbon edge (see Figure 8). Synchrotron beamlines have notorious difficulty with measurements near the carbon
edge. (Carbon buildup on the monochromator absorbs much of the incident flux, heightening the relative percentage
of contaminant energies, and giving low overall counting statistics.) As discussed above, we take our carbon edge

Figure 9. First order synchrotron data of flight batch
HEG grating HA2021, overlaid with best fit model.
These data come from two different synchrotrons (PTB
and NSLS) to cover the full energy span. The data sets
join at 2 keV.
Figure 10. Residuals from the first order fit of grating
HA2021 shown in Figure 9. The largest residuals, at
the N and O edges of polyimide, have been truncated.
(The region containing the N and O edges is detailed in
Figures 11 and 12.)
structure from the PTB measurements, but (arbitrarily) smoothing the data between 288.8 eV and 300.99 eV. We
do not have the reassuring agreement between the ALS and PTB measurements for this region as we did for the
nitrogen and oxygen edges, and model residuals remain high at the carbon edge. However, this region falls below
the minimum HETG energy of 400 eV and the true edge structure does not matter for our modeling purposes.
3.2. Chromium edge
In order to accomodate the Cr edge structure below 600 eV, we employed a different approach since we do not
have transmission tests of a Cr filter of known thickness. We took zero order grating data from MA1047 (measured
at PTB) and fit it assuming a fixed thickness of Cr (55 š A from fabrication measurements). We assumed that the
absorption features seen at 577 eV and 586 eV could be modeled as a perturbation on the absorption coefficient as
derived from the Henke constants, and thereby obtained a modified absorption coefficient. This allows us to obtain
a transmission for any thickness of Cr. (It is unnecessary to extract new values of f1, since the chromium absorbs
but does not diffract.)
The updated grating model is evaluated in the next section. By refining our treatment of C, N, O and Cr edges,
we have reduced the residuals by a factor of 2--3 relative to the former treatment.
4. ACCURACY OF THE GRATING MODEL
4.1. Overview
The accuracy of the phased, non­rectangular model and the effectiveness of updated optical constants for gold,
polyimide and chromium can be assessed by examining how well the model fits the measured efficiencies of a well­
tested grating. There are two flight­batch gratings that have been tested at synchrotrons over most of the applicable
energy range. These gratings are MA1047 and HA2021, which were tested at in October, 1995 at PTB below 2 keV,
and at NSLS above 2 keV. (Although other gratings have been through synchrotron testing, the experiment was
limited to energies above 2 keV for these other gratings.) As discussed below, the grating model shows excellent

Figure 11. First order synchrotron data of HEG grat­
ing HA2021 overlaid with best fit model, in the polyimide
and plating base edge region. The contrast between the
current model (solid line) and the 1994 Henke model
(dashed line) illustrates the remarkable level of improve­
ment provided by the polyimide and gold transmission
tests at the synchrotrons.
Figure 12. Residuals from the first order fit of grating
HA2021, in the polyimide and plating base edge region.
Although the residuals at the N and O edges are high,
they are nevertheless significantly improved by the use of
the new optical constants.
agreement (at the level of a few percent) with synchrotron measurements of first order efficiencies, except at a limited
set of energies. In particular, modeling the edges remains the largest contributor to the residuals, despite significant
advances in this area. Future work on the modeling is not expected to improve the edge residuals. The second
obvious energy range where the model inadequately represents the data is in the vicinity of the first order efficiency
peak (or zeroth order efficiency trough). Future work on the modeling may result in improvements in this energy
range.
4.2. First order fit to HA2021
The agreement between the model and the data is demonstrated in Figure 9, where first order diffraction efficiencies
for flight­batch grating HA2021 have been measured at many closely­spaced energies. The residuals are shown in
Figure 10, where have defined the residuals to be the fractional discrepancy between the modeled efficiency and the
data (i.e. (model \Gamma data)=data) without consideration of the error bars on the data. To obtain the best­fit model,
only the first­order data for HEG grating HA2021 were used in the fit. (No other orders were used, nor did we impose
the constraint that the +1 and ­1 orders were equal.) The largest residuals are generally due to the polyimide and
plating base edges, and have been truncated in Figure 10. Detailed views of the model and the residuals in this
energy region are given in Figures 11 and 12. The excellent agreement of the model with the data at the gold M
edges is seen in detail in Figures 13 and 14.
By far the largest residuals (tens of percent) between the model and the data occur at the polyimide edges (N and
O). (The testing range did not include the carbon edge.) Given the steep changes in response expected there and the
large systematic errors found in measuring the polyimide optical constants at independent synchrotron beamlines,

Figure 13. First order synchrotron data of HEG grating
HA2021 overlaid with best fit model, in the gold M edge
region.
Figure 14. Residuals from the first order fit of grat­
ing HA2021, in the gold M edge region. Note that the
residuals are small, a few percent at most.
these residuals are perhaps not too surprising and are restricted to a relatively small region of the energy range.
(Furthermore, the residuals as we have defined them do not include any impact of the error bars in evaluating the
significance.) A close examination of Figure 11, in fact, shows that the model is actually quite impressive in its
treatment of the complicated edge structures, despite the formal residuals.
Figure 15 summarizes the model to the first order data of HA2021. The model to which the data are fitted is
a five­vertex polygon bar shape function with three absorbing layers: polyimide, chromium, and gold plating base.
For HA2021, the nominal fabrication thicknesses are 0.0200 ¯m for the Au plating base, 0.97 ¯m for polyimide, and
0.005 ¯m for Cr. The fitted values are displayed in Figure 15, and are in the ballpark of the expected values. Also
shown is the amplitude factor: it close to 1.0, as expected.
As indicated in Figure 15, the reduced chisquare of the fit shown in Figure 9 is about 3. Part of this is attributable
to error bars that are too small. If the statistical errors from the synchrotron tests are increased in order to more
realistically reflect systematic errors, then the reduced chisquare drops to 2.1, but not much lower. (The edge regions
continue to be significant contributors.) Thus, despite improvements gained with the recent synchrotron tests, there
still remain some discrepancies between the model and the data, mostly attributable to limitations in our input data,
fi and ffi over the edge regions.
Table 2 shows the improvement that polyimide and gold transmission tests have provided in understanding the
optical constants at these edges. The improvement in the optical constants at the edges has decreased the relative
residuals (improvements of a factor of 2 or 3 are typical). Note that in Table 2, most of the edges fall outside the
useable energy range of the high energy grating (above 0.9 keV). Thus, from the standpoint of Chandra calibration,
the largest applicable edge residual is 26% (i.e., the MEG grating at the oxygen edge.) Future improvements
in modeling are not expected to improve the edge residuals, since these are assumed due to systematics in the
synchrotron testing, location of the edge energy, variations in polyimide, and other factors outside our ability to
address. (Further improvements in our treatment of the edges must await improved optical constants in these
regions.) We can, however, further examine the model at the conceptual level, as discussed below.
A perusal of Figure 9 shows two regions where the model fit does not agree well (systematically) with the data.
One region is about 6 keV. This is probably due to inadequate separation of the first order from the zeroth order in
the synchrotron test, and is therefore a fault of the test rather than a failure of the model. The second region is the

0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
distance in units of one grating period
BEST FIT to HA2021 FIRST ORDER
Gold plating thickness: 0.012859
Polyimide thickness: 1.007301
Cr thickness: 0.003694
Amplitude Factor: 0.923437
Reduced chisq: 3.14
Figure 15. Summary of the best­fit model to the first
order data of HA2021. The model includes a five­point
vertex bar shape, three plating base thicknesses, and an
amplitude factor.
Figure 16. Zero order synchrotron data of flight batch
HEG grating HA2021, overlaid with the best fit model.
The region around the ``trough'' is highlighted to illus­
trate where conceptual improvements may be made in
the grating model.
Table 2. Typical edge residuals (percent)
Asterisk marks energies outside the applicable range for the gratings.
Model Year Grating N O Cr MV
1999 MA1047 13 26 4.3 1.6
1996 MA1047 42 66 13 2.7
1999 HA2021 70* 43* 12* 3.3
1996 HA2021 93* 114* 28* 5.6
low­energy side of the efficiency peak (about 1.5 to 2 keV). This region is also poorly fit for zeroth order, as shown
in Figure 16. The reasons for the poor fit over the resonance peak (and conversely in the resonance trough of the
zeroth order) are not known, although several possibilities may be considered.
The region near 2 keV corresponds to the energies in which the xray undergoes nearly a 180 degree phase shift
after traversing the grating bar. This phase shift results in a near cancellation of the emerging xray wavefront in
zeroth order and an enhanced first order efficiency. Hence this region is extremely sensitive to the detailed bar shapes
and any attempt to model the shapes as a single shape may ultimately fall short. In fact, we believe that the unusual
bar shape depicted in Figure 15 is indicative of this fact as its distorted shape will give rise to a complex pattern of
phase shifts. Similarly modeling the grating efficiency as a linear combination of efficiencies from different bar shapes
may also not provide an improved fit because this technique does not account for interference effects. In other words,
this energy regime may be impossible to model at the desired level of accuracy using a model based upon diffraction

Figure 17. First order synchrotron data of flight batch
MEG grating MA1047, overlaid with the best fit model.
These data come from two different synchrotrons (PTB
and NSLS) to cover the full energy span. The data sets
join at 2 keV. Only first order data have been included
in the fit.
Figure 18. Residuals from the first order fit of grating
MA1047 shown in Figure 18. Residuals have not been
truncated.
from a periodic structure, or a superposition of periodic structures.
Other possible explanations may also be considered:
ffl The peak of the efficiency curve (2 keV) is also the energy where the data from the different synchrotrons meet.
The actual location tested on grating HA2021 may be slightly different in the two tests, so that in effect two
different gratings are being inappropriately represented by a single model. This should be easy to verify by
individually modeling the two energy regions.
ffl The PTB beamline is known to have stray light contamination above 1500 eV, and this may affect the quality
of the data being fitted in the peak.
ffl Since macroscopic areas of the grating are illuminated in the synchroton tests, it may be appropriate to assume
more than one grating thickness. This might be expected to broaden the efficiency peak overall. Simple models
with two thicknesses have not been found to significantly improve the model fit in this region, however.
ffl It may be necessary to abandon the scalar theory altogether in favor of the much more complex vector theory
that includes the effects of polarization.
ffl The amplitude factor should probably vary with energy.
Thus, it is possible that improvements in the grating model at the conceptual level (i.e., vector model, multiple
thicknesses, etc.) may improve the fit around the first order efficiency peak.
4.3. First order fit to MA1047
A fit to the first order efficiency of MEG flight batch grating MA1047 is shown in Figure 17. The residuals are shown
in Figure 18, and have similar characteristic regions to those described for HA2021. In the case of the MEG, the
residuals at the polyimide edges are less than for the HEG grating, as expected since the absorbing layer of polyimide
is about half that of the HEG grating. Typical results for MA1047 are also given in Table 2.

5. SUMMARY
The grating model shows excellent agreement (at the level of a few percent) with synchrotron measurements of first
order efficiencies, except at a limited set of energies. In particular, the edges remain the largest contributor to the
residuals, although our modeling of the edge structures has improved dramatically. Our measurements at PTB and
NSLS have gone far to improve the optical constants fi and ffi which are inputs to our model. Future efforts will be
directed at conceptual improvements in the model; in particular, the resonance peak around 2 keV represents one
area that invites further investigation.
ACKNOWLEDGMENTS
We thank the HETG and CXC groups for helpful suggestions, in particular C. Canizares, D. Davis, D. Dewey,
J. Houck, H. Marshall, S. Taylor and T.T. Fang. We are grateful to T. Burek for assistance in gold transmission
tests. This work was prepared under NASA contract NAS8­38249.
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