\chapter{Efficiency Physics and Model} \label{chap:effic_phys} The key physics to model the HETGS effective area comes in at the level of the single-grating diffraction efficiency $g_f(E,m)$ appearing in Equation~\ref{equ:G_s}. In order to achieve high diffraction efficiencies, we are operating outside the range of the simple diffraction theory\cite{born80} of opaque bars and instead in a region where the grating bars transmit and phase-shift the X-rays. Guided and assisted by Dale Graessle and his AXAF witness mirror work at NSLS\cite{graessle96}, Tom Markert and Christie Nelson initiated grating synchrotron measurements of diffraction efficiency, reported in Nelson {\it et al.}\cite{nelson94} and Markert {\it et al.}\cite{markert95}. These studies have allowed us to verify the phased non-rectangular model, provided reference standards for laboratory tests, and provided more accurate optical constant data for the materials in our gratings. The state of our synchrotron measurements and modelling efforts was most recently detailed by Flanagan {\it et al.}\cite{flanagan96}. Although this work is ongoing, the fundamental result was a detailed physical model of grating diffraction that applies to individual gratings $g_f(E,m)$ and has been measured at many finely spaced energies. Cross-testing of several gratings in the X-GEF lab and at synchrotron facilities allows us to put constraints on the efficiency models produced from X-GEF tests. \section{Diffraction Theory and Models} \subsection{Scalar Diffraction Theory} The model we use is the simple scalar (Kirchoff) diffraction theory \cite{born80}; rigorous diffraction theory, using the vector properties of the electromagnetic field and appropriate boundary conditions for the grating bars, is not required so long as the wavelength of the light being considered is much less than the period of the grating. Since we are dealing with, at most, wavelengths of ~30 \AA, and the grating period is no less than 2000 \AA (for the HEG gratings) this condition is fulfilled fairly well. We note that the scalar model will begin to err at the lowest energies of interest, and that we will have to consider the more rigorous method (at least at the lower energies) as we continue to refine the model. The general formula for the efficiency of a periodic grating, using the Kirchoff diffraction theory with the Fraunhofer approximation is (\cite{born80,kamer95}): \begin{equation} \eta_{(m)}(k) = 1/p^{2} \times [\int_0^p e^{ik(\nu - 1)z(\xi ) - 2\pi mi\xi /p }d\xi]^{2} \label{eqkaf1} \end{equation} \noindent where $\eta_{(m)}$ is the efficiency in the $mth$ order, $k$ is the wavenumber ( $=2\pi /\lambda$), $\nu$ (= $\nu (k)$) is the complex index of refraction, $p$ is the grating period, and, $z(\xi) $ is the grating bar shape function. The bar shape function $z(\xi)$ may be thought of as the shape of the cross-section of the grating bar for X-rays normally incident on the grating surface. In addition to the parameters indicated in equation~\ref{eqkaf1}, there are other factors which effect the measured diffraction efficiencies. These are the absorptions of the support film (see Figure~\ref{fig:grat_cross_secs}) and the plating base. The grating is built up onto a thin (0.98 $\mu$m for the HEGs, 0.55 $\mu$m for the MEGs) polyimide film which provides mechanical support. In addition, there are very thin metallic films ($\simeq$ 200 \AA\ of gold and 50 \AA\ 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) X-rays. However, their absorption introduces edge structure which has been investigated through testing at synchrotrons (described below). Thus, the parameters for the scalar model (with absorption) of the HETG gratings are: \begin{description} \item $\beta (k)$ and $\delta (k)$, the components of the index of refraction for gold \item $z(\xi)$, the bar shape function (note that the grating period is not an explicit parameter, but only scales the bar shape) \item $t_{au}$, the thickness of the gold plating base \item $t_{cr}$, the thickness of the chromium plating base \item $t_{poly}$, the thickness of the polyimide support film \item $\beta_{polyimide}(k)$ and $\beta_{cr}(k)$, the imaginary parts of the index of refraction, which give the transmission of the support film and the plating base. \end{description} \subsection{Rectangular Models} Consider the grating model without regard to the aborbing support layers. For a rectangular wire profile\cite{schnopper77}, i.e, where \begin{quote} $z(\xi) = h$ for $0 < \xi \leq p-a$, and \end{quote} \begin{quote} $z(\xi) = 0$ for $p-a < \xi \leq p$, \end{quote} equation~\ref{eqkaf1} becomes \begin{equation} \eta_{(m)}(k) = [\frac{sin(\frac{m\pi a}{p})}{m\pi}]^{2} \times [1 + e^{-2k\beta h} - 2e^{-k\beta h}cos(k\delta h)]\;\;\;\;\;\;\; m \neq 0 \end{equation} and \begin{equation} \eta_{(0)}(k) = (a/p)^{2} + (1 - a/p)^{2}e^{-2k\beta h} + 2(a/p)(1 - a/p)e^{-k\beta h}cos(k\delta h)\;\;\;\;\;\;\; m = 0. \end{equation} \noindent Here $a$ is the width of the space between grating lines and $p$ is the period. As before, $\delta$ and $\beta$ are the complex components of the index of refraction $\nu$. Note that $\nu$, $\delta$, and $\beta$ are all functions of the X-ray wavelength, i.e., $\nu (k) = 1 - \delta (k) +i\beta (k)$. If $\beta h$ is very large (i.e., the grating bars are opaque) then there is no wavelength dependence of the diffraction efficiency. The HETG gratings are designed such that the grating bars are partially {\em transparent} at higher energies, and so the phase shift of the X-rays through the bars (the $\delta$ term) actually enhances the efficiency at some energies. Gratings with this property are call {\em phased} transmission gratings. For a quantitative look at the higher efficiency of phased gratings as compared to opaque gratings, equations~\ref{eqkaf2} and \ref{eqkaf3} below show the grating efficiencies (where $\eta$ is now the sum of the grating efficiencies of the $+1$ and $-1$ orders): \begin{equation} \eta_{opaque} = 2 \times \frac{sin^2(\frac{\pi a}{p})}{\pi^2} \label{eqkaf2} \end{equation} \begin{equation} \eta_{phased} = \eta_{opaque} \times [1 + e^{-2kh\beta} - 2e^{-kh\beta}cos(kh\delta)] \label{eqkaf3} \end{equation} \noindent When absorption is ignored (the fully-phased case), equation~\ref{eqkaf3} reduces to the following\cite{schnopper77}: \begin{equation} \eta_{fully-phased} = 8 \times sin^2(\frac{kh\delta}{2}) \times \frac{sin^2(\frac{\pi a}{p})}{\pi^2} \label{eqkaf4} \end{equation} \noindent From equation~\ref{eqkaf4}, it can be seen that at an optimal wavelength ($\lambda = 2h\delta$), the fully-phased grating efficiency is four times the opaque grating efficiency for this simple scalar diffraction theory. (A complete vector treatment of Maxwell's equations gives slightly different results at some energies.) At efficiencies near this one particular wavelength, partial destructive interference of the zeroth grating diffraction order occurs and the first grating diffraction orders are enhanced. The result is a range of wavelengths for which the grating efficiency is maximized. The scalar model fits our data fairly well, even if a simple rectangular grating bar shape is assumed\cite{nelson94}. However, in order to match the data at the 1 per cent level, we find in general that more complex bar shape functions must be invoked. In addition, we have independent evidence, from electron microscope photographs that the bar shapes for the HETG gratings are not simple rectangles. \subsection{Vertex Model} The bar shape function function $z(\xi)$ can be reduced to a finite number of parameters. For example, if a rectangular bar shape is assumed, then $z$ can be computed with two parameters, a bar width and a bar height. A more complex bar shape can be parameterized with arbitrary precision by dividing the bar into small rectangular sections. For our modeling, the grating bar shape is parameterized as piece-wise linear, defined by the location of vertices, {\it e.g.}, Figures~\ref{fig:meg_jfit_shape} and \ref{fig:heg_jfit_shape}. The number of line segments is selectable, but there is little improvement in the quality of the fit after about 4 to 6 segments. \subsection{Tilted Grating Model} Another factor which makes a difference in the computed efficiencies in the various orders is the effect of tilting the grating, {\it i.e.}, changing the orientation of the grating surface normal with respect to the incoming X-rays. Given a grating bar cross-section, and a tilt angle (orientation of the incoming X-rays with respect to the grating plane), one can determine the grating bar shape function $z(\xi)$ from simple geometry. Since tilt and bar shape are complementary, we don't include tilt angle as a model parameter. (It is always possible to find an untilted grating with the same response as a tilted grating of an arbitrary shape. This equivalence only holds true at a given tilt angle, however. If one studies gratings over a range of angles, then one must generally use the actual bar cross-section.) %% \noindent %% \begin{minipage}[b]{.46\linewidth} %% \centering\epsfig{file=trap_assym.ps1,width=\linewidth} %% %%\caption{Assymetry parameter for a trapezoidal grating} %% \label{fig:trap_assym} %% \end{minipage}\hfill %% \begin{minipage}[b]{.46\linewidth} %% \centering\epsfig{file=trap_shape.ps1,width=\linewidth} %% %%\caption{Trapezoidal bar shape assumed for the model at left.} %% \label{fig:trap_shape} %% \end{minipage} \begin{figure}[p] \begin{center} \epsfig{file=trap_assym.ps1,height=4.0in} \caption[Asymetry parameter for a trapezoidal grating]{Asymetry parameter for a trapezoidal grating} \label{fig:trap_assym} \end{center} \end{figure} \begin{figure}[p] \begin{center} \epsfig{file=trap_shape.ps1,height=3.0in} \caption[Trapezoidal bar shape]{Trapezoidal bar shape assumed for the model above.} \end{center} \end{figure} \subsection{Trapezoidal Model} As an example of the complementarity between tilt and bar shape, consider a rectangular cross-section: When viewed at an angle, it has a trapezoidal $z(\xi)$. Moreover, the efficiency vanishes in all orders when the the grating is tilted by an angle $\theta = tan^{-1}(h/p)$, where $h$ is the grating bar height and $p$ is the period. (This relationship is clear because at that angle all of the X-rays incident on the grating see the same path length through the bars, hence there can be no diffraction.) Fischbach {\em et al.} \cite{flanagan88} have reported on the theory and measurements of tilted rectangular gratings. Except for a few bar cross-sections showing great symmetry, the efficiency in the positive and negative orders will be different at all but a few tilt angles. For example, Figure~\ref{fig:trap_assym} shows the efficiency asymmetry in the plus and minus first orders for a trapezoidal grating at $E = 2.290$~keV, the energy of the Mo~L~$\alpha $ line. The asymmetry at a given tilt angle is defined as 100($I_{+1}/I_{-1} - 1$). There is order symmetry at {\em normal} orientation (zero degrees tilt means that the incoming X-rays are normal to the grating plane), but at very few other angles. \clearpage \section{Synchrotron Measurements} \label{sec:sync_meas} \subsection{Synchrotron Measurements Summary} The table below summarizes all the synchrotron tests which have been performed to date on HETG gratings or their prototypes. \input{sync_summary_new.tex} \subsection{Synchrotron Data Analysis Techniques} A complete description of the techniques employed in extracting diffraction efficiencies from synchrotron data is given in Flanagan {\it et al.}\cite{flanagan96}. \subsection{Gold Optical Constants and Edge Structure} When fitting the scalar model to the data we use the simplex technique to minimize the chi-square function. We allow all parameters to vary, except for the optical constants. There are too many optical constants (2 at each energy for gold alone), so fitting them isn't appropriate. On the other hand, the values of the $\beta$ and $\delta$ are not well-enough known to allow us to use the published values uncritically. Our initial tests (January 1994) at the National Synchrotron Light Source (NSLS) at Brookhaven National Laboratory (BNL) indicated a significant disagreement between the Henke values for the gold optical constants\cite{henke}, and those required by our tests (our results were in basic agreement with those obtained earlier by Blake {\em et al.}\cite{blake} from reflection studies of gold mirrors). The most noticeable feature of the disagreement was that the energies of the prominent gold M absorption edges were shifted from the tabulated amounts by as much as 40 eV. In an effort to determine more relevant optical constants, in February, 19994 we measured the transmission of a gold foil with well-determined thickness and density in the energy range 2.03--6 keV at the NSLS using the X8A beam line. The transmission measurements yielded the complex part of the index of refraction ($\beta$) directly. We used the Kramers-Kronig relation (i.e., the dispersion relation linking $\delta$ and $\beta$) to compute the real part of the index of refraction, $\delta$. The optical constants we required outside of this range were obtained by connecting the synchrotron-derived constants with those of Henke (at other energies) in a smooth way. \begin{figure}[h] \begin{center} \epsfig{file=au_optical_constants.ps,height=3.5in,width=5.5in} \caption[Comparison of synchrotron-derived optical constants to Henke tables.] {Comparison of optical constants from transmission data measured in 1994 (solid lines) to those from the Henke tables (dashed lines). The plot on the left shows the values of the real part of the index of refraction, $\delta$, and the plot on the right shows the values of the imaginary part of the index of refraction, $\beta$.} \label{fig:au_oc} \end{center} \end{figure} The optical constants we determined are shown in Figure~\ref{fig:au_oc}, along with the Henke {\it et al.}\cite{henke} values. The constants are displayed near the gold MIII, MIV, and MV edges only (1.5 keV $<$ E $<$ 3 keV). Outside this energy range, the constants we derived were in good agreement with the Henke values. \begin{figure}[h] \begin{center} \epsfig{file=HA2021_p1.ps,height=3.5in,angle=90} \caption[First order synchrotron data of HEG grating HA2021, with best fit model.]{First order synchrotron data of HEG grating HA2021, overlaid with best fit model.} \label{fig:HA2021p1} \epsfig{file=HA2021_resid.ps,height=3.5in,angle=90} \caption[Residuals from the first order fit of grating HA2021.]{Residuals from the first order fit of grating HA2021.} \label{fig:HA2021resid} \end{center} \end{figure} The validity of the phased, non-rectangular model and the effectiveness of updated gold optical constants for energies above 2~keV are demonstrated in Figure~\ref{fig:HA2021p1} , where the synchrotron-measured diffraction efficiencies, sampled over many closely-spaced energies, agree well with the modeled efficiency for a flight-batch HEG grating. The residuals are shown in Figure~\ref{fig:HA2021resid} (only the first order data were used in the fit). The excellent agreement of the model with the data at the gold M edges is seen in better detail in Figures~\ref{fig:MA1047p1} and \ref{fig:MA1047resid} for a flight-batch MEG grating. Note that the residuals at the gold edges are generally within 1\%. There have been subsequent tests of gold transmission in 1996 by Richard Blake at NSLS and by Tom Markert at PTB (at BESSY). Blake's data above 2~keV agree well with the 1994 measurements. Analysis of the PTB gold data (1.9~keV and below) is underway. \begin{figure}[h] \begin{center} \epsfig{file=MA1047_p1.ps,height=3.2in,angle=90} \caption[First order synchrotron data of MEG grating MA1047, with best fit model.]{First order synchrotron data of MEG grating MA1047, overlaid with best fit model.} \label{fig:MA1047p1} \epsfig{file=MA1047_resid.ps,height=3.2in,angle=90} \caption[Residuals from the first order fit of grating MA1047.] {Residuals from the first order fit of grating MA1047. The horizontal lines are at $\pm1$\%.} \label{fig:MA1047resid} \end{center} \end{figure} \clearpage \subsection{Polyimide and Chromium Optical Constants and Edge Structure} \label{sec:poly_synchrotron} 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\cite{flanagan96}. 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 taken from Henke, Gullikson and Davis\cite{henke}. This modeling, however, provided an unacceptable fit at the edges, as shown in Figure~\ref{fig:HAedge}. In this figure are seen oxygen, nitrogen and chromium edges, as well as EXAFs: all are inadequately represented by the model. The residuals are given in Figure~\ref{fig:HAedgeresid}, and exceed 200\% at the C and N edges. Just below 600~eV are seen edge residuals of about 20\% from Cr L. Similar results have been found for the MEG grating MA1047. Taking the model as a whole, the polyimide edges exhibit the worst discrepancies between our physical model and the data overall. In order to refine the optical constants for our polyimide at the C, N and O edges, we used the PTB data for the two flight batch polyimide samples. We began by finding a best-fit thickness for each of the (MEG and HEG) polyimide samples assuming Henke optical constants and fitting over the edge-free energy range 0.6 to 1.6~keV. For each sample, an effective absorption coefficient $\mu$ is obtained assuming $T=e^{-\mu t}$ where {\bf t} is the thickness in microns and T is the transmission through the polyimide foil. The final value for $\mu$ is taken to be the average value of the HEG and MEG $\mu$, between 272~eV and 875~eV, smoothly joined to the Henke values outside this region. In addition, we smoothed the derived $\mu$ in the carbon edge region between 288.8~eV and 300.99~eV because low counting statistics otherwise introduce a jittery structure there. Note that the 1982 Henke constants for carbon were employed as these were found to agree better with our data and have been shown in independent tests by the HRC instrument team as the the better choice. The discrepancy between the average $\mu$ and the $\mu$ extracted from the data provides a first estimate of the error on $\mu$. This technique does not adequately accomodate certain systematic errors. In particular, 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.) The gratings tests at BESSY did not cover the carbon edge, and only one polyimide sample provided an estimate for $\mu$. Richard Blake tested a polyimide sample supplied by Mark Schattenburg (of the HETG team) and at ALS (Advanced Light Source at the Lawrence Berkeley National Laboratory) in 1996. He found a best-fit $\rho$t of 124.25 $\mu$g/$cm^2$. Assuming a density of 1.45 g/$cm^3$, he derived a value for $\mu$/$\rho$ in a similar manner as has been described with regard to the BESSY data. After conversion to comparable units, a comparison of our $\mu$/$\rho$, Henke's and Blakes is given for the nitrogen edge in Figure~\ref{fig:blake_compare}. In general, we found good agreement between the two data sets, except at the carbon edge. This close agreement between data sets generally confirms the detailed edge structure and magnitudes of the absorption coeficients we have derived. The independent ALS measurements will be used to provide a second estimate of errors on the absorption coefficients. Measurements taken on the BNL beamline X8A with the multilayer monochromator are of insufficient resolution to reproduce the details of the edge structure. 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 BESSY) and fit it assuming a fixed thickness of Cr (55 \AA 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. The new Cr and polyimide results have been incorporated into the file optical-constants.981023. Realistic errors of the new optical constants are probed by applying these constants to fit an independent data set. This is provided by BESSY measurements on the two gratings, HA2021 and MA1047, in the case of the polyimide constants, and by HA2021 and first order MA1047 in the case of the Cr constants. (No other high-resolution grating measurements in this energy range are available from other synchrotron facilities.) The data and best-fit model obtained from fitting the first order of HA2021 are shown in Figure~\ref{fig:HAedge_new}. The residuals are given in Figure~\ref{fig:HAedgeresid_new}. In each, we have excluded energies below 600~eV from the fitting process and we have fixed the thickness of the chromium layer to the fabrication measurement. Note that the residuals are reduced by a factor of 3-5 at the C, N and Cr edges relative to the former fitting. The new optical constants file, which improves the edge treatment, will be incorporated into an upcoming revision of grating efficiencies but hasn't been implemented yet. %%%The data and best-fit model for an HEG grating are shown in %%%%Figure~\ref{fig:HAp1BESSY} for the whole energy range tested. %%%%Figure~\ref{fig:HAresidBESSY} shows the residuals over this range, but %%%%excludes the polyimide edges. %%\begin{figure}[H] %%\begin{center} %%\epsfig{file=HA2021_p1_BESSY.ps1,height=3.5in} %%\caption[Data and best-fit model for HEG grating over full energy range] %%{Data and best-fit model for grating HA2021 over full energy range of %%BESSY and Brookhaven tests.} %%\label{fig:HAp1BESSY} %%\end{center} %%\end{figure} %%\begin{figure}[H] %%\begin{center} %%\epsfig{file=HA2021_resid_BESSY.ps1,height=3.5in} %%\caption[Residuals for HEG grating over full energy range] %%{Residuals for grating HA2021 model when fit over full energy range; %%the fit excluded the polyimide edges. Currently, the highest %%residuals are at the polyimide edges, where Henke constants have been %%used.} %%\label{fig:HAresidBESSY} %%\end{center} %%\end{figure} %%\begin{figure}[H] %%\begin{center} %%\epsfig{file=MA1047_p1_BESSY.ps1,height=3.5in} %%\caption[Data and best-fit model of MEG grating over full energy range]{Data and best-fit model of grating MA1047 over full energy range of BESSY and Brookhaven tests.} %%\label{fig:MAp1BESSY} %%\end{center} %%\end{figure} %%\begin{figure}[H] %%\begin{center} %%\epsfig{file=MA1047_resid_BESSY.ps1,height=3.5in} %%\caption[Residuals for MEG grating over full energy range]{Residuals for grating MA1047 over full energy range. The highest residuals are at the polyimide edges, where Henke constants have been used.} %%\label{fig:MAresidBESSY} %%\end{center} %%\end{figure} \begin{figure}[h] \begin{center} \epsfig{file=HA2021edges_better_plot.ps,height=3.5in} \caption[Data and best-fit model of HEG grating at polyimide and Cr edges, with Henke constants used.]{Data and best-fit model of grating HA2021 at polyimide and Cr edges, with Henke constants used in the model.} \label{fig:HAedge} \end{center} \end{figure} \begin{figure}[h] \begin{center} \epsfig{file=HA2021_edge_resid.ps1,height=3.5in} \caption[Residuals for HEG grating at polyimide edges]{Residuals for grating HA2021 at polyimide and Cr edges, where Henke constants have been used.} \label{fig:HAedgeresid} \end{center} \end{figure} \begin{figure}[h] \begin{center} \epsfig{file=blake_compare.ps,height=3.5in} \caption[Comparison of absorption coefficents at nitrogen edge]{A comparison of absorption coefficients at the nitrogen edge shows good agreement between ALS and PTB (BESSY) data sets, and strong departure from the Henke values. There is evidently a small energy offset between the two data sets.} \label{fig:blake_compare} \end{center} \end{figure} \begin{figure}[h] \begin{center} \epsfig{file=HA2021_edge_981023.ps,height=3.5in} \caption[Data and best-fit model of HEG grating at polyimide and Cr edges, with October, 1998 optical constants used.]{Data and best-fit model of grating HA2021 at polyimide and Cr edges, with optical-constants.981023 used in the model.} \label{fig:HAedge_new} \end{center} \end{figure} \begin{figure}[h] \begin{center} \epsfig{file=HA2021_edge_resid_981023.ps,height=3.5in} \caption[Residuals for HEG grating at polyimide edges and Cr edge, with October,1998 optical constnats used.]{Residuals for grating HA2021 at polyimide and Cr edges, where optical-constants.981023 has been used.} \label{fig:HAedgeresid_new} \end{center} \end{figure} %%\begin{figure}[h] %%\begin{center} %%\epsfig{file=MA1047edges.ps1,height=3.5in} %%\caption[Data and best-fit model of MEG grating at polyimide edges]{Data and best-fit model of grating MA1047 at polyimide edges.} %%\label{fig:MAedge} %%\end{center} %%\end{figure} %%\begin{figure}[h] %%\begin{center} %%\epsfig{file=MA1047_edge_resid.ps1,height=3.5in} %%\caption[Residuals for MEG grating at polyimide edges]{Residuals for grating MA1047 at polyimide edges, where Henke constants have been used.} %%\label{fig:MAedgeresid} %%\end{center} %%\end{figure} \clearpage \subsection{Reference Grating Measurements} \label{sec:ref_grats} Although these data provide estimates as to the validity of our grating model, synchrotron tests are impractical to use for more than a few gratings. For the full set of flight gratings, the model parameters are determined from laboratory measurements on each grating element, using a standard (electron impact) type X-ray source and a few X-ray lines in MIT's X-Ray Grating Evaluation Facility (X-GEF). Gratings that have been tested at the synchrotron are used as transfer standards in each X-GEF test, to minimize systematic errors and allow normalization against a known efficiency. (Note, therefore, that the residuals of Figure~\ref{fig:HAresidBESSY} are {\it not} representative of the calibration of the individual gratings constituting the HETG.) Two gratings, HX220 and MX078, were tested early in the program at a synchrotron (Markert {\it et al.}\cite{markert95}). These are used as transfer standards in X-GEF testing; each X-GEF efficiency measurement is performed with an analogous measurement on the reference grating. Based on the assumption that the diffraction efficiency of the reference grating is correctly known at that energy, we obtain the efficiency of the test grating at the measurement energy. From measurements made at 5 or 6 energies in the laboratory, we then estimate the best-fit parameters (bar shape, gold thickness, space-to-period ratio) for the test grating and generate a model for its diffraction efficiency at all appropriate AXAF energies. Figures~\ref{fig:hx220p1} and \ref{fig:hx220resid} illustrate the quality of the model fit to the synchrotron data for the HX220 reference grating. Figures~\ref{fig:mx078p1} and \ref{fig:mx078resid} show the data and fit for the reference grating MX078. These fits used both the first order and zeroth order data. Tilt tests were performed in X-GEF on the two reference gratings. Figure~\ref{fig:mx078_asym} plots the asymmetry of MX078 versus angle at the Mo~L line. MX078 exhibits no substantial asymmetry (confirming results found at Brookhaven). Figure~\ref{fig:hx220_asym} shows the asymmetry parameter found for HX220, the HEG reference grating. For reference, we have overlaid the results found in a tilt test performed at Brookhaven, which shows substantial agreement. The modeling for HX220 is discussed in Markert, {\it et al.}\cite{markert95} %%For a symmetryic grating bar having a symmetric X-ray path function at all angles (such as a rectangle), this plot would be flat with asymmmetry parameter equal to zero. In that case, the top figure would have a single peak angle for both first orders. \begin{figure}[H] \begin{center} \epsfig{file=hx220_p1.ps,height=3.5in} \caption[Data and best-fit model of reference grating HX220]{Data and best-fit model of reference grating HX220.} \label{fig:hx220p1} \end{center} \end{figure} \begin{figure}[H] \begin{center} \epsfig{file=hx220_p1_resid.ps,height=3.5in} \caption[Residuals for reference grating HX220]{Residuals for reference grating HX220.} \label{fig:hx220resid} \end{center} \end{figure} \begin{figure}[H] \begin{center} \epsfig{file=mx078_p1.ps,height=3.5in} \caption[Data and best-fit model of reference grating MX078]{Data and best-fit model of reference grating MX078.} \label{fig:mx078p1} \end{center} \end{figure} \begin{figure}[H] \begin{center} \epsfig{file=mx078_p1_resid.ps,height=3.5in} \caption[Residuals for reference grating MX078]{Residuals for reference grating MX078.} \label{fig:mx078resid} \end{center} \end{figure} \begin{figure}[h] \begin{center} \epsfig{file=mx078_asym.ps,height=3.5in} \caption[Tilt data for reference grating MX078] {The X-GEF tilt test data for MX078 confirms the synchrtron result of a minimal tilt-induced asymetry.} \label{fig:mx078_asym} \end{center} \end{figure} \begin{figure}[h] \begin{center} \epsfig{file=hx220_asym2.ps,height=3.5in} \caption[Tilt data for reference grating HX220] {The X-GEF tilt depencance for HX220 agrees with measurements made in a tilt test at NSLS, overlaid(*). An angular offset if 1.15 degrees has been added to the NSLS measurements.} \label{fig:hx220_asym} \end{center} \end{figure} \clearpage \subsection{Error Estimates for X-GEF Predictions of Grating Efficiency} Four gratings have been cross-tested both at X-GEF (using standard flight testing and analysis prodecures including the reference gratings to normalize their efficiencies) and at the synchrotron. By comparing the X-GEF derived efficiency model with the synchrotron measurements, we have an estimate of the errors implicit in using X-GEF to predict the efficiencies for each grating facet. The residuals between the X-GEF generated model and the synchrotron data are shown in Figure~\ref{fig:xgefsync1} and \ref{fig:xgefsync0}. In effect, this provides an {\it upper limit to the errors of our subassemby efficiency predictions}, because a large strip of the reference grating is in fact illuminated in each X-GEF test, and the diffraction efficiency of this strip is (incorrectly) assumed to have the synchrotron reference efficiency. Only a small portion of the reference grating was actually tested at the synchrotron and it is only this small portion which should actually be assigned the reference efficiency. Therefore, the assignment of this fixed efficiency to a much larger area of the reference grating is incorrect. Tests have been done to cross-normalize the strip efficiency in comparison to the confined area for each grating, but the results have not been included yet. \begin{figure}[h] \begin{center} \epsfig{file=xgef_sync1.ps1,height=3.5in} \caption[Residuals between first order X-GEF predictions and synchroton data]{Residuals between first order X-GEF predictions and synchrotron data. The large residuals at the N and O edges should be reduced when new X-GEF predictions are generated assuming updated (October, 1998) optical constants.} \label{fig:xgefsync1} \end{center} \end{figure} \begin{figure}[h] \begin{center} \epsfig{file=xgef_sync0.ps1,height=3.5in} \caption[Residuals between 0th order X-GEF predictions and synchroton data]{Residuals between 0th order X-GEF predictions and synchrotron data. Edge residuals between 0.5 and 0.6~keV should be reduced when predictions are generated assuming updated (October, 1998) optical constants.} \label{fig:xgefsync0} \end{center} \end{figure} \clearpage %%%\subsection{Edge Structure Data} \clearpage \subsection{Synchrotron Analysis Status} \begin{enumerate} \item The non-rectangular phased model is used in the X-GEF-modeling technique, its errors estimated and it has been documented in the literature. \item Updated gold optical constants are also employed in this modeling. Recent tests on gold confirm our model above 2~keV, but BESSY data below 1.9~keV await analysis before the gold constants at lower energies can be checked. \item Reference gratings have been assigned efficiencies based on early synchrotron tests. These values need to be examined and errors attached, reflecting increased knowledge of the synchrotron beamline systematics obtained in subsequent tests. The biggest considerations are: \begin{itemize} \item Reconsideration of the cross-calibration techniques (high priority) \item Careful examination of the multilayer data (below 2.03keV) which may contain low-energy contamination. (high priority) \item Applying wing corrections and error bars. \end{itemize} \item Reference grating efficiencies are applicable to a small spatial region of each reference grating, but have been assumed for large regions in X-GEF tests. Cross-normalization tests have been performed in X-GEF (in a test procedure labeled BRLONG), so that efficiencies over a large strip can be normalized to the central section, and a proper reference efficiency can be assigned for the large test strip. This needs to be completed. \item The results of the above projects will provide new efficiencies and errors for the reference gratings as used in X-GEF tests. These improved reference efficiencies can then be transferred to the flight grating measured efficiencies. \item New optical constants have been obtained that treat the C, N and O edges of polyimide and strong chromium edges around 577 and 586~eV. Include any appropriate changes from examining gold data and finalize. Attach error bars derived from comparisons among independent measurements. Supply polyimide constants, as required, to ACIS and HRC groups for their filter modeling. \item Repeat model fits for flight gratings incorportaing best optical constants and incorporating revised reference grating efficiencies. \item Finish analysis of NSLS tests of gratings HD2338 and MB1148, to examine ratios of higher orders to first order. Compare these results with apparent systematic behavior of higher orders found with HETG + HRC-I tests at XRCF\cite{Flanagan98}. \item We have looked for evidence of scattering in synchrotron data, and find it to be qualitatively consistent with modeling describing the scattering noted at XRCF\cite{Davis98}. \end{enumerate} \clearpage \section{Scattering Model} \label{sec:scat_theory} \input{scatter_theory}