\begin{quotation} {\it Objective:} Derive the diffraction efficiency of the HEG and MEG using Phase I ``grating-in grating-out'' measurements made at fixed (non-scanned, non-continuum) energies. {\it Publication(s):} Dewey {\it et al.}~\cite{dewey97,dewey98} \end{quotation} \subsection{Introduction and Overview} In Phase 1 of AXAF testing at the X-Ray Calibration Facility (XRCF), calibrated flow proportional counters (FPCs) and solid-state detectors (SSDs) were used both in the focal plane and as beam-normalization detectors. This use of similar detectors in the beam and focal plane combined with detailed fitting of their pulse-height spectra allowed accurate measurements of the HRMA absolute effective area with minimum influence of source and detector effects. This paper describes the application of these detectors and fitting techniques to the analysis of effective area and efficiency measurements of the AXAF transmission gratings, the High Energy Transmission Grating (HETG) and the Low Energy Transmission Grating (LETG). Because of the high dispersion of these gratings the analysis must be refined. Key additional ingredients are the inclusion of detailed X-ray source models of the K and L lines based on companion High-Speed Imager (HSI) microchannel-plate data and corrections to the data based on high-fidelity ray-trace simulations. The XRCF-measured efficiency values that result from these analyses have systematic errors estimated in the 10--20~\% range. Within these errors the measurements agree with the pre-XRCF laboratory-based efficiency models of the AXAF grating diffraction efficiencies. In this paper we report on progress towards the detailed analysis of the Phase 1 measurements which were designed to measure the grating effective area and efficiency using the FPC and SSD in the focal plane. First the general concepts of effective area and efficiency are reviewed; next the general XRCF Phase 1 measurement configuration is described including summary properties of the gratings and examples of the acquired data and their simplistic analysis. As will be demonstrated, analysis improvements require detailed knowledge of the source spectral composition which we derive from contemporary Phase 1 HRMA-grating-HSI observations of the source. Using the resulting spectra, accurate ray-traces of the system allow us to better understand the FPC/SSD data. Finally, analysis making use of the detailed source spectra and simulations is being carried out to derive accurate grating efficiency values and assess measurement errors. \subsection{Measurements and Example Data} In this paper we focus on the Phase 1 measurements that used the conventional electron impact X-ray source (EIPS) and that had either the FPC or SSD as the focal plane detector. There are of order 160 grating measurements of this kind and over 50 no-grating measurements of direct relevance. The measurement process involved positioning the detector aperture $D$ at one or more locations in the focal plane and acquiring simultaneous pulse-height spectra from the focal plane detector and the BNDs. We illustrate our analysis methods with example data sets taken with the Fe-L and Ti-K source lines, Table~\ref{tab:example_measurements}. Note that non-grating measurements were made as well (Grating = NONE) to allow a direct measurement of efficiency by dividing the grating-in by the grating-out effective areas. Examples of the pulse-height spectra obtained for some of these Ti-K and Fe-L measurements are presented and described in Figures~\ref{fig:TiK_phas}~and~\ref{fig:FeL_phas}. \begin{table} [h] %>>>> here, top, bottom, page_of_floats \caption{ XRCF Phase 1 Grating Effective Area Measurements at Ti-K and Fe-L. These measurements are used to illustrate the analysis. A two-mean-free-path source filter was used with these measurements to reduce above-line continuum. For the Ti-K measurements a $D=2.0$~mm aperture was used; for the Fe-L measurements $D=1.0$~mm. The BND detector FPC\_HN was fully open for all measurements here except for '9.004 where it was closed to a nominal 36 mm diameter. } \label{tab:example_measurements} \begin{center} \begin{tabular}{|c|c|c|c|c|c|} \hline %---------------------- \rule{0pt}{2.5ex} TRW-ID & Source & Shells & Grating & Orders & date/runid/iteration \\[0.2ex] \hline %---------------------- \hline %---------------------- E-IXF-3D-9.001 & Ti-K,TiKx2 & 1,3,4,6 & NONE & -- & 970203/111140i0 \\ E-HXF-3D-10.001 & '' & 4,6 & HEG & 1,2,3,0,-1,-2,-3 & 970203/111126i0--'i6 \\ E-HXF-3D-10.002 & '' & 1,3 & MEG & 1,2,3,0,-1,-2,-3 & 970203/111123i0--'i6 \\ E-LXF-3D-9.004 & '' & 1,3,4,6 & LEG & 1,-1 & 970203/111144i0,'i1 \\ \hline %---------------------- \hline %---------------------- D-IXF-3D-11.003 & Fe-L,Fex2 & 1,3 & NONE & -- & 970110/108163i0 \\ D-IXF-3D-11.004 & '' & 4,6 & NONE & -- & 970110/108164i0 \\ D-HXF-3D-11.020 & '' & 4,6 & HEG & 1,0,-1 & 970110/108161i0--'i2 \\ D-HXF-3D-11.019 & '' & 1,3 & MEG & 1,0,-1 & 970110/108162i0--'i2 \\ D-LXF-3D-11.018 & '' & 1,3,4,6 & LEG & 1,0,-1 & 970110/108165i0--'i2 \\ \hline %---------------------- \end{tabular} \end{center} \end{table} \begin{figure}[t] \psfig{file=TiK_phas.ps,height=11.cm} \caption[Ti-K FPC Spectra] {Examples of FPC Spectra for Ti-K Tests. The main features at this high energy, 4.51~keV, are the main photo-peak around channel 170 and the Ar escape peak around channel 60. {\it Top}: the spectrum seen by the BND FPC\_HN directly viewing the source, continuum well-above the line energy is visible (channels 280 and up). {\it Middle}: the spectrum from FPC\_X2 in the focal plane after the HRMA (no grating): the high-energy cutoff of the HRMA is visible in the continuum above channel 370. {\it Bottom}: the spectrum at the HEG $m=+1$ diffraction order -- the detector here is seeing an essentially monochromatic input of the Ti-K$\alpha$ line. To better show the differences in continuum, the histograms here have been smoothed and a dashed reference line at a relative rate of $10^{-3}$ has been included. } \label{fig:TiK_phas} \end{figure} \begin{figure}[t] \psfig{file=FeL_phas.ps,height=7.cm} \caption[Fe-L FPC Spectra] { Examples of FPC Spectra for Fe-L Tests. The histogram-style curve shows the spectrum seen at the LEG $m=+1$ order: this is the monochromatic response to the 0.705 keV Fe-L$\alpha$ line with some 2.1~keV $m=3$ continuum counts around channel 220. The smoothed solid line shows the FPC\_X2 spectrum at the HRMA focus without a grating (BND spectra are comparable): the ``Fe-L bump'' is visible peaking at channel 65 and continuum above the filter edge appears above channel 120. The extension of the bump to lower energies is due to the presence of unresolved lines, see Figure~\ref{fig:FeL_spectra}. } \label{fig:FeL_phas} \end{figure} \subsection{Problems with Effective Area Analysis} A simple calculation of the measured optic effective area, Equation~(\ref{equ:oea_defn}), from the pulse-height spectra would be: \begin{equation} OEA_{D}(E_{\rm line},m) ~~= ~~ {\frac {R_{\rm fp}} {QE_{\rm fp}(E_{\rm line})} } ~\Bigl/~{ {\frac {R_{\rm BND}} {A_{\rm BND@HRMA}~\times~QE_{\rm BND}(E_{\rm line}) } } } \label{equ:oea_simple} \end{equation} \noindent The subscript $D$ indicates the measured optic effective area is into a finite focal plane aperture. The detector rates $R$ are the counts per second in the pulse height ``bumps'' and the BND effective area $A_{\rm BND@HRMA}$ is the equivalent geometric area of the BND detector at the HRMA aperture and, together with the quantum efficiencies $QE$, provides the absolute calibration. There is poor agreement between this simple analysis and our expectations especially for the dispersed $|m|=1$ effective areas which are shown in Figure~\ref{fig:simple_first_orders} and often appear lower than predicted. As the next sections will show, the source spectra often consist of multiple lines which are not resolved by the FPC detectors but are spatially separated in the focal plane by the grating diffraction. Thus, the BND is measuring several lines while the focal plane detector because of its finite aperture may be seeing only one or a fraction of the source lines. By measuring the source spectra and simulating the measurements we can implement improved analysis techniques. % IDL> SET_PLOT, 'PS' % IDL> device, /portrait,font_size = 12, XSIZE=18.0, YSIZE=6.0, $ % IDL> YOFFSET=0.0, XOFFSET=0.0 % IDL> !p.multi = [0,3,1] % IDL> nohirefs = where(STRPOS(eae.source,'HIREF') EQ -1) % IDL> eae_plots, eae(nohirefs), 6 % IDL> eae_plots, eae(nohirefs), 2 % IDL> eae_plots, eae(nohirefs), 3 % IDL> device, /close \begin{figure}[bh] \psfig{file=simple_first_orders.ps,height=5.6cm} \caption[Simple first-order effective areas] { The problem with the standard effective area analysis: these plots show the grating first-order effective areas, calculated from simply extracted, uncorrected count rates, compared with model predictions\cite{schulz98}; these first-order effective areas are often measured low due to the presence of multiple and contaminating lines in the source spectrum. } \label{fig:simple_first_orders} \end{figure} \subsection{Ray-trace Simulation of Phase 1 Measurements} It would be straight forward to use the grating equation and parameters of the measurement to decide which region of the detailed source spectrum will fall in the detector aperture $D$. While this approach works, it has difficulty including a variety of effects such as: broad lines that may overfill the aperture $D$, HRMA and grating PSF effects which blur the events, higher grating diffraction orders of higher-energy source photons, and the LETG support structure diffraction pattern. In the end it is best to accurately model the complete source-HRMA-grating-detector system to completely understand what the detector sees. The ``Model of AXAF Response to X-rays'' simulation package\cite{marx97} (MARX, 2.04) has been used as the engine to produce these simulations. Custom IDL code was produced to create modified MARX parameters for a given XRCF measurement, see Section~ref{sec:marx}. Simulated focal plane images for the the diffracted Ti-K and Fe-L lines are shown in Figure~\ref{fig:TiKFeL_sim}. These simulations can be used to calculate several quantities relevant to the efficiency measurements being analyzed here: \begin{itemize} \item The encircled energy correction value $EE_{\rm corr}$ is the ratio of the number of all line events to the number of line events that fall in the aperture. \item The pulse-height distribution of events that are within the detector aperture can be formed and used to estimate what fraction of the measured count rate is due to a given line or feature in the sectrum. \item The simulations can also be used to assess the sensitivity of either of the above values to variations in the aperture placement in the focal plane. \end{itemize} \noindent The use of these simulation-derived values is described in the context of our efficiency analysis in the next section. % The feature fraction plot is too detailed... % IDL> print, where(eae.trw_id EQ 'D-LXF-3D-11.018') % IDL> pre_print_sqr % IDL> feat_fraction, eae(84),/PS_P % IDL> device,/close ; cp idl.ps to ... % \psfig{file=D-LXF-3D-11.018_ff.ps,height=10.0cm} % plot the X-Y images of the Ti-K HEG+1 and Fe-L LEG+1 orders % to show aperture effects and encircled energy correction esp % for LEG % IDL> SET_PLOT, 'PS' % IDL> device, /portrait,font_size = 12, XSIZE=18.0, YSIZE=8.0, $ % IDL> YOFFSET=0.0, XOFFSET=0.0, /COLOR % IDL> !p.multi = [0,2,1] % IDL> feat_frac_image, eae(284),/PS_P, MAXE=10000 % IDL> feat_frac_image, eae(84),/PS_P % IDL> device,/close ; cp idl.ps to ... \begin{figure} \psfig{file=tikfel_images.ps,height=7.5cm} \caption[Ti-K and Fe-L simulated first-order images] { Ti-K and Fe-L simulated first-order images. In order to better undestand the FPC/SSD aperture measurements, the AXAF ray-trace package MARX was adapted to simulate XRCF measurements. These figures show the spatial distribution of events relative to the detector aperture used in the measurement. {\it Left}: The FPC aperture, 2~mm diameter circle, is centered on the Ti-K$\alpha$ line; the Ti-K$\beta$ line, at aperture edge, is partially included in the aperture and hence in the resulting pulse-height spectrum. The EE correction value of 1.009 indicates that very little of the Ti-K$\alpha$ line flux is outside of the aperture. {\it Right}: Simulation of this Fe-L$\alpha$ measurement clearly shows the LETG coarse-support diffraction pattern extending in the cross-dispersion direction, $Z_{\rm det}$. The bright continuum on the low-energy (right) side of the line is cut off by the Fe-L filter just above the line. The continuum streak to the high-energy side (left) of the Fe-L$\alpha$ peak is actually 2.1~keV continuum that has been diffracted into the LEG $+3$ order. } \label{fig:TiKFeL_sim} \end{figure} \subsection{Analysis Formalism} Now that we've seen the source and aperture effects, it is clear that the ``failure'' of the analysis specified by Equation~(\ref{equ:oea_simple}) is due primarilly to inaccurate or contaminated values for the measured rates and the effects of the aperture. In addition, the best estimate of the effective flux at the HRMA entrance requires a detailed beam uniformity analysis and consideration of data from all of the BND detectors. These effects are included in the revised equations: \begin{equation} OEA_{2\pi}(E_{\rm line},m) = EE_{\rm corr} \times OEA_{D}(E_{\rm line},m) = EE_{\rm corr} \times {\frac {{\cal R}_{\rm fp}} {QE_{\rm fp}(E_{\rm line})} } ~\Bigl/~{F_{\rm effective}(E_{\rm line})} \label{equ:oea_refined} \end{equation} \begin{equation} F_{\rm effective}(E_{\rm line}) = ~\Bigl\langle~{ {\frac {BU_{\rm corr}~{\cal R}_{\rm BND}} {A_{\rm BND@HRMA} \times QE_{\rm BND}(E_{\rm line}) } } }~\Bigr\rangle_{\rm BNDs} \label{equ:oea_flux} \end{equation} \noindent where the ${\cal R}$'s are the rate in a specific line or narrow energy region, an explicit encircled energy correction factor, $EE_{\rm corr}$, converts from the count rate measured in an aperture to the total focal plane rate, and the incident flux is now an average over the properly weighted ($BU_{\rm corr}$) BND measurements taking into account beam uniformity variations\cite{patnaude98,swartz98}. The grating effective diffraction efficiency, Equation~(\ref{equ:eff_effic}), is measured as the ratio of the optic effective area with the grating in place to the optic effective area without the grating (HRMA-only): \begin{equation} G^{\rm meas}_{\rm config}(E_{\rm line},m) = {\frac{OEA^{\rm g}_{2\pi}(E_{\rm line},m)} {OEA^{\rm H}_{2\pi}(E_{\rm line})}} \label{equ:meas_eff_effic} \end{equation} \noindent where the $g$ superscript refers to grating-in measurement and the $H$ superscript is a HRMA-only measurement. \subsection{Count Rate Corrections} In general pulse-height analysis is performed to provide measured focal plane and BND rates, $R$, in some, possibly broad, energy range. A ``line-fraction-correction'' $L$, can be defined for each pulse-height spectrum to convert this measured rate to an estimate of the rate in a line or narrow energy region: ${\cal R} = L \times R $. The value of $L$ can be numerically determined from the simulation of a measurement as the ratio ${\cal R_{\rm sim}}/R_{\rm sim}$; this requires precise definitions of $R_{\rm sim}$ and ${\cal R_{\rm sim}}$, however. To determine $R_{\rm sim}$ from a simulation, the analysis method itself must be faithfully modeled. In order to determine ${\cal R_{\rm sim}}$, that is the count rate in the line, the ``line'' must be defined. Because some lines are in fact naturally broad and continuum is present in the modelled spectra, it is convenient instead to talk of a spectral ``feature''; here a useful and reasonable definition of the feature is ``all photons in the range $E_{\rm line} \pm E_{\rm line}/100$'' -- thus continuum under the line is part of the feature as well. Substituting ${\cal R} = L \times R $ into Equation~(\ref{equ:meas_eff_effic}), many of the terms will (exactly or very nearly) cancel leaving: \begin{equation} G_{\rm config}(E_{\rm line},m) = {\frac {EE^g_{\rm corr}~L^g_{\rm fp}} {EE^H_{\rm corr}~L^H_{\rm fp}} }~~\times~~ {\frac {R^g_{\rm fp} \bigl/ (R^g_{\rm BND}/A^g_{\rm BND@HRMA} )} {R^H_{\rm fp} \bigl/ (R^H_{\rm BND}/A^H_{\rm BND@HRMA} )} } \label{equ:EEL_eff_effic_meas} \end{equation} \noindent Thus, the simple count-rate based efficiency, the ratio on the right, is corrected for after-the-fact by simulation-derived parameters. This ``feature-fraction'' correction (the product $EE_{\rm corr}~L_{\rm fp}$) can be calculated and applied to different pulse-height analysis methods. The next sections describe our specific analysis efforts that implement this approach to more accurately calculate the measured grating efficiency. In all cases the $EE_{\rm corr}$ factor is derived from the ray-traces and post-applied. Two of the analysis methods (ROI and ``counts-in-bump'') have non-trivial $L$ values, the third method (``counts-in-line'') includes the source spectral information ``upstream'' of the pulse-height analysis and so has $L=1$. \subsection{Region-of-interest (ROI) Analysis} \label{sec:roi_rate} The most simple pulse-height analysis method determines a count rate based on the total number of counts in a region of interest, for example the counts in channels 20 through 120 in the Fe-L spectra of Figure~\ref{fig:FeL_phas} may be summed. This has the advantage of simplicity and computational speed and robustness. Likewise the quantity $R_{\rm sim}$ can be generated with high accuracy for a coarse region of interest by counting all events in an energy range. The very robustness of the ROI analysis points to its main defect: because the data are not evaluated with respect to any model there is no check that the assumed model is realistic (except perhaps when the ROI limits are viewed on the pulse-height histogram), errors in lines present, detector operation, continuum levels, etc. can all cause erroneous results without an indication of a ``failure'' of the assumptions. For our purposes the ROI analysis with the corrections applied serves as an initial robust result and a sanity check and guide to the more complex and detailed fitting analyses, below. The results of the corrected ROI analysis are shown by the ``x''s in Figure~\ref{fig:jdkp_effics_dkff}. % Statistical errors are %shown by the small horizontal lines on each measurement; systematic %errors including the likely errors on the values of $EE_{\rm corr}$ %and $L_{\rm fp}$ are indicated by the vertical error lines. % IDL> SET_PLOT, 'PS' % IDL> device, /portrait,font_size = 12, XSIZE=18.0, YSIZE=20.0, $ % IDL> YOFFSET=0.0, XOFFSET=0.0 % IDL> !p.multi = [0,2,3] % IDL> nohirefs = where(STRPOS(eae.source,'HIREF') EQ -1) % IDL> eae_plots, eae(nohirefs), 17 % IDL> eae_plots, eae(nohirefs), 16 % IDL> eae_plots, eae(nohirefs), 14 % IDL> eae_plots, eae(nohirefs), 12 % IDL> eae_plots, eae(nohirefs), 15 % IDL> eae_plots, eae(nohirefs), 13 % IDL> device, /close %\begin{figure} %\psfig{file=ff_effics.ps,height=20.0cm} %\caption[Feature-fraction corrected ROI efficiencies] %{ %Feature-fraction corrected ROI efficiencies. %} %\label{fig:ff_effics} %\end{figure} \begin{figure} %\psfig{file=acq111123d2i0.pgplot.ps,height=15.0cm} \psfig{file=tik_fpc_5_uf.ps} \caption[JMKMOD fit to Ti-K BND Spectrum] { JMKMOD fit to Ti-K BND. Three of the components of the JMKMOD spectral model are highlighted here: the main K$\alpha$ plus K$\beta$ photo-peak, the Ar escape peak, and the broadband continuum generated by the source and given structure by the Ti source filter. } \label{fig:tik_jmkmod} \end{figure} \subsection{JMKMOD ``counts-in-bump'' Analysis} \label{sec:bump_rate} An improvement on the simple ROI analysis is the detailed pulse-height spectral fitting provided by the JMKMOD software\cite{edgar97,tsiang97}. This software is an add-on package to the XSPEC\cite{xspec} x-ray spectral fitting package and was created to model the XRCF FPC and SSD detectors. Figure~\ref{fig:tik_jmkmod} shows the application of the JMKMOD model to a Ti-K BND spectrum. Because these detectors cannot resolve the K$\alpha$ and K$\beta$ peaks, the fitting process is not able to reliably determine accurate count rates for the separate lines -- rather the combined count rate in the K$\alpha$ plus K$\beta$ ``bump'' is determined. Two main advantages of this fitting technique over a simple ROI analysis are: i) the measured data are fit by a model and therefore data quality and measurement assumptions are tested and ii) the continuum level in the spectrum is measured rather than depending on a modeled continuum. The measured rate $R$ is similar to the ROI rate but does not include a continuum contribution; this requires that slightly different values for the line fraction corrections $L$ be applied in Equation~(\ref{equ:EEL_eff_effic_meas}). As a starting point, however, the efficiency results from this ``counts-in-bump'' analysis were corrected with the same correction as the ROI results and are shown in Figure~\ref{fig:jdkp_effics_dkff} by the triangular symbols. As in the ROI case, the correction has the greatest effect on the efficiency of the L-lines of Mo, Ag, and Sn; its effect on the Ti-K HEG first-order efficiency is an increase of 9.3~\%\~. \begin{figure} \psfig{file=fel_fpc_x2_uf.ps} \caption[JMKMOD fit to Fe-L Spectrum] { JMKMOD fit to Fe-L FPC\_X2. In this ``counts-in-line'' analysis the three known source lines have had their relative intensities fixed based on the HSI-derived relative fluxes; the continuum component is allowed an independent normalization in the fit. The fit intensity of the Fe-L$\alpha$ line is the rate in the line, that is $L=1$ and ${\cal R} = R$. A good fit to the data indicates an agreement of the source model with the data to an accuracy allowed by the detector resolution and count statistics. } \label{fig:fel_jmkmod} \end{figure} \subsection{JMKMOD ``counts-in-line'' Analysis} \label{sec:line_rate} Rather than applying the $L$ line-fraction corrections to the rates in a region or bump, it is possible to fit the pulse-height spectra using as input a modeled source spectrum. Lines and features which are resolved in the spectrum can be fit with independent intensities; those that are poorly resolved must have their relative intensities fixed. This technique has the potential advantage over the ``counts-in-bump'' analysis in that the assumed model is directly tested against each pulse-height data set. An example of a JMKMOD ``counts-in-line'' fit to a non-grating focal plane spectrum is shown in Figure~\ref{fig:fel_jmkmod}. All three lines in the Fe-L spectrum, have had their relative intensities fixed based on the HSI counts spectrum, Figure~\ref{fig:FeL_hsi_dispersion}, and modeled HSI, grating, HRMA, and FPC properties. The fit intensity of the Fe-L$\alpha$ line is then directly the line rate, ${\cal R}$, {\it i.e.}, $L=1$.. The only correction required to this analysis is the ratio $EE^g_{\rm corr}/EE^H_{\rm corr}$ which corrects for finite aperture effects; for LEG measurements this correction can be as much as a $\approx$~10~\% increase. For the dispersed orders of the broad low-energy lines (Be-K and B-K) this ratio must be calculated using an accurate source spectral line shape and source filter transmission curve. The efficiency results of this ``counts-in-line'' analysis are shown by the square sysmbols in Figure~\ref{fig:jdkp_effics_dkff}. % IDL> SET_PLOT, 'PS' % IDL> device, /portrait,font_size = 12, XSIZE=18.0, YSIZE=20.0, $ % IDL> YOFFSET=0.0, XOFFSET=0.0 % IDL> jdkp_first_orders % IDL> device,/close %\begin{figure} %\psfig{file=jdkp_results.ps,height=20.0cm} %\caption[JMKMOD-Analysis Efficiencies] %{ %JMKMOD-Analysis Efficiencies. %} %\label{fig:jdkp_effics} %\end{figure} % IDL> SET_PLOT, 'PS' % IDL> device, /portrait,font_size = 12, XSIZE=18.0, YSIZE=20.0, $ % IDL> YOFFSET=0.0, XOFFSET=0.0 % IDL> jdkp_first_orders, /FF ; /FF to correct DK's data % IDL> device,/close \begin{figure} \psfig{file=jdkp_results_dkff.ps,height=20.0cm} \caption[Efficiency Analysis Efficiencies] { Efficiency Analysis Results. Measured efficiencies for the gratings in zero and first orders are shown here compared with the pre-XRCF model predictions, based on facet-by-facet laboratory measurements. The plotting symbols indicate the pulse-height analysis methods: ``x''s are from region of interest rates (Section~\ref{sec:roi_rate}), triangles are from ``counts-in-bump'' rates (Section~\ref{sec:bump_rate}), and the squares are based on ``counts-in-line'' rates (Section~\ref{sec:line_rate}). Statistical errors on the measurements are generally less than 3~\%\~; understanding and reducing the systematic errors, here estimated to be of order 10--20~\%\~, is an ongoing effort. } \label{fig:jdkp_effics_dkff} \end{figure} \begin{figure} \begin{center} \epsfig{file=effmegheg1st.eps,height=19cm} \caption[Preliminary XRCF MEG and HEG Efficiencies: 1st order] {Preliminary XRCF MEG and HEG Efficiencies: 1st order. Because of the ``ratio-of-ratios'' nature of the efficiency measurement, there is good agreement with the laboratory predictions except for the L-lines where the ``line-fraction'' corrections are large; these $L$ function corrections of Equation were not included in this simple, first-pass analysis. \\ The solid curve is the model prediction, the dotted curves give an estimate of the error range for the prediction. The data points have horizontal lines to indicate the size of the statistical errors; the vertical lines indicate an estimate of the measurement systematic errors. } \label{fig:effmegheg1} \end{center} \end{figure} \begin{figure} \begin{center} \epsfig{file=effmegheg0.eps,height=19cm} \caption[Preliminary XRCF MEG and HEG Efficiencies: 0 order] {Preliminary XRCF MEG and HEG Efficiencies: 0 order. Because the zero-order is non-dispersed the $L$ functions of Equation will be nearly the same with and without the grating in place resulting in a clean measurement.} \label{fig:effmegheg0} \end{center} \end{figure} \subsection{Discussion} The agreement of the XRCF-measured efficiencies with the laboratory-measurement-based efficiency predictions shown in Figure~\ref{fig:jdkp_effics_dkff} is a wonderful confirmation of the grating diffraction models and facet-by-facet laboratory measurements: taken on their own, these XRCF Phase 1 EIPS measurements do in general verify the laboratory-based efficiency models at the 10--20~\% level in most energy regions. These analyses are nearly complete, what remains is to study and assign systematic errors to these measurements, {\it e.g.}, by obtaining error estimates for the HSI-derived source spectra and folding them through the analysis methods. % From jdrake@localhost.localdomain Thu Jul 16 13:13:29 1998 % Dan: I think this should fit in reasonably after the % first paragraph in the discussion (paper version c. last night). % It sort of breaks the second paragraph you have though; any feel % free to do whatever...! Other potentially large systematic errors yet to be accounted for in the analyses are thought to originate in the FPC detectors in ways that are difficult to account for in the detector models. Apparently subtle effects, such as a bowing outward of the FPC windows due to internal gas pressure and obscuration and reflection of focussed focal plane light by the supporting wire mesh, are calculated to affect HRMA measurements at levels below a percent\cite{kellogg97}. However, the former of these effects can be more important at the lowest energies where the detectors and windows are more optically thick: the window bowing alters the effective thickness and location of the window seen by incoming photons, and low energy photons penetrate to smaller depths in the detector. One of the next major challenges is to understand these detector effects and to account for them in the refined spectral modelling process ($\S6$) in order to determine more accurate fluxes at the telescope aperture and focal plane. In the case of the LETG, which is designed to operate at wavelengths as long as 170~\AA\ (0.07 keV), the lowest energies available for efficiency and effective area measurements---provided by the B (183~eV) and Be (108~eV) EIPS---pose special difficulties because of the very low QE of the XRCF detectors in this regime. Here, detector background and counting statistics can be significant sources of error; during the Be tests for example, the HRMA BND count rates were generally too low to be of any quantitative use for beam monitoring and normalization, and the ``Building 500'' FPC detector provides the only means of determining the beam flux. In the near future, these analysis techniques will be applied to the Phase 1 measurements made with a monochromator as the source in place of the EIPS source. The spectral analysis may in general be cleaner (no closely spaced L-lines for example), however the monochromators do have substantial beam uniformity variations\cite{swartz98} that must be dealt with. The ultimate target of all our analyses is to produce grating efficiency models that are of sufficient accuracy that they are not a dominant source of uncertainty in the flight HRMA+grating+detector optical system. When all known sources of systematic error have been included in the modelling processes and flux determinations, it is hoped that the final uncertainties in the individual grating efficiency measurements will be as low as 2-3~\%\~. At this level, the XRCF measurements will be capable of providing quite stringent tests of the grating efficiency models. % However, this in turn could present a dilemma: it %will be difficult based on the limited nature of the XRCF tests to %determine what component of the models might be the cause of any %discrepencies. We also point to the Au optical constants that are %critical for determining the grating efficiencies at energies where %the grating bars begin to become transparent to X-rays. The %uncertainties in the currently available data (ref?: Schattenburg? %Hencke?) have not been rigorously assessed but are likely to be larger %than 2-3~\%\~. % Taken on their own, the XRCF Phase 1 EIPS measurements analyzed here %do in general verify the laboratory-based efficiency models at the %10--20~\% level in most energy regions. By combining these results %with other analyses, {\it e.g.} of Phase 1 monochromator data and %Phase 2 effective areas\cite{schulz98,marshall98,flanagan98}, we will %be at a point to detect, confirm, and reconcile discrepencies with our %models. \begin{quotation} {\it To-do:} \\ \begin{itemize} \item create ffs\_dkr.rdb to have accurate corrections for dk's ratio values. \item Pete to analyze D-LXF-3D-12.003a (in place of '12.003) \item Pete: ``too much continuum in XSPEC fits to Fe-L,O-K, Si-K, Ti-L contnorm in 109286, 109287? \item CMDB mods for Be-K D-LXF-3D-12.003a : w/filter and 10 mm aperture and B-K 22.043 (10 mm aperture), re-MARX simulate, new ff corrections \item Improve Si-K line source model, what is 2.03 keV line in Si-K spectra? I think the Zr filter lets this line appear in FPC measurements...: Brad suggests it is W line! \item dd modify his Fe-K Mn filter to include the little lead in it...(is this the one? \item re-do HSI spectral analysis using new HRMA EA curves: check Mo line for any HRMA-induced ``lines''? \item Two-step source line intensity process: use HSI spectra to decide what lines are present and then use SSD spectra to set relative normalizations where possible, e.g., Ti-K Ka and Kb. \item include support grid effect in MARX simulations \item resimulate (eae\_sim) with new released MARX version \item include DeltaZ aperture errors in feature\_fraction error estimate \item Make page or two of HSI source spectra plots... \item Make a no-grating BND simulation too... \item Double check the HRMA 1,3 - 4,6 area fractions used to convert shell measurements to other shell combinations \item Pete's Be and B analysis: broad line over fills aperture. \item Make sure released rdb files have N and S 's in column def.s \item Create simulated .pha files from MARX to check analysis techniques. \item Analyze the Phase 1 monochromator data - easier and harder! \end{itemize} \end{quotation}