\begin{quotation}
{\it Objective:} Derive the diffraction efficiency of the HEG
or MEG using Phase I ``grating-in grating-out'' measurements made at
fixed (non-scanned, non-continuum) energies.
\end{quotation}

\subsection{System Effective Area}

The system effective area is calculated from the measurements through
the equation:

\begin{equation}
SEA(E_{\rm line},m,{\rm mode}) = {\frac
{\rm focal~plane~counts/s~in~line~order}
{\rm source~flux~in~line} }~~~
[{\frac
{\rm counts/s}{\rm photons/cm^2s}} = {\frac{\rm cm^2{counts}}{photon}}]
\end{equation}

\noindent where the units for the quantities are given as well.  (Although
the terminology ``line order'' suggests grating applications these
considerations apply to HRMA-only measurements as well.) The
numerator is derived from the focal plane data through:

\begin{equation}
{\rm focal~plane~counts/s~in~line~order} = {\rm {\it R}_{fp}(mode) \times
{\it L}_{fp}(aperture,spectrum,{\it QE}_{fp})}
\end{equation}

\noindent where the focal plane rate $R_{\rm fp}({\rm mode})$ is the
observed count rate in the data---generally this count rate will
represent the counts per second within a spatial-spectral ``bump'' in
the data, {\it e.g.}, the counts in the ROI region of the FPC\_X2
spectra.  As such this quantity, and others that follow, {\it has meaning
only in the context of a specific analysis method}.  The variable
``mode'' is used here to indicate the additional effect that this rate
can depend on the focal plane detector mode of operation, {\it e.g.},
continous {\it vs.} timed-exposure modes for ACIS.

The focal-plane line fraction term
$L_{\rm fp}({\rm aperture},{\rm spectrum},QE_{\rm fp})$ is a
conversion factor from the observed count rate to the count rate that
would be due to the line fraction alone.  This factor is required
because the sources are not truly mono-chromatic and the detectors do
not fully separate the line, as indicated in the discussion regarding
Figures~\ref{fig:pha_meg_example}~and~\ref{fig:alk_heg_2c} above.  The
line fraction defined here will generally not depend on the detector
mode, but {\it is} taken in the context of an analysis method, {\it
e.g.}, the line fraction will be different if ROI analysis is
performed or XSPEC\cite{edgar97} fitting is used.

Note that the $SEA$ as defined here is equivalent to the ``total
effective area'' into $2\pi$ and there is an assumption that the
line-fraction correction also includes a small correction for the line
counts that are outside the aperture.  Not discussed here but treated
similarly is the {\it Encircled Effective Area} (often called
``Encircled Energy''), {\it e.g.}, $EEA(E_{\rm line},m,{\rm mode},{\rm
diameter})$ which considers only the counts within the aperture.

The $SEA$ denominator, ``source flux in line'', is given by:

\begin{equation}
{\rm source~flux~in~line} = { \frac{ R_{\rm BND} \times
L_{\rm BND}({\rm spectrum},QE_{\rm BND})}
{QE_{\rm BND}(E_{\rm line}) \times A_{\rm BND@HRMA}}  }
\end{equation}

\noindent where the BND rate $R_{\rm BND}$ and BND line fraction
$L_{\rm BND}({\rm spectrum},QE_{\rm BND})$ are defined analogously to
the focal plane terms above.  The two additional terms in the
denominator provide the key ingredients for the calibration: the BND
quantum efficiency $QE_{\rm BND}(E_{\rm line})$ provides the
conversion from counts to photons for the BND (supporting relative
calibration), and the BND area $A_{\rm BND@HRMA}$ is the equivalent
geometric area of the BND detector at the HRMA aperture and provides
the absolute calibration; for the BNDs near the HRMA ({\it e.g.},
FPC\_HN fully open) this value is 32.55~${\rm cm}^2$.

\subsection{Optic Effective Area}

In the calibration of the optical components, the HRMA and HRMA-Gratings,
it is useful to consider the {\it optic effective area (OEA)} which
represents the ability of the optical system to collect photons and
does not include detector effects:

\begin{equation}
OEA(E_{\rm line},m) = {\frac{\rm focal~plane~photons/s~in~line~order}
{\rm source~flux~in~line}}~~~
[{\frac{\rm photons/s}{\rm photons/cm^2s}} = {\rm cm^2}]
\end{equation}

\noindent where we get the usual ${\rm cm}^2$ units.  The {\it OEA}
may be calculated from the measurements through:

\begin{eqnarray}
\label{equ:oea_first}
OEA(E_{\rm line},m) = & A_{\rm BND@HRMA} &
~~\times ~~{\frac
{ { R_{\rm fp}({\rm mode}) \times
L_{\rm fp}({\rm aperture},{\rm spectrum},QE_{fp})} }
{ R_{\rm BND} \times L_{\rm BND}({\rm spectrum},QE_{\rm BND})}
 } \\
  &   &   
~~\times ~~{\frac
{\rm{ {\it QE}_{BND}({\it E}_{line}) }}
{{\it QE}_{fp}({\it E}_{line},mode)}
}.
\label{equ:oea_second}
\end{eqnarray}

\noindent  When the focal plane detector has a $QE$ identical to
the BND detector the last term is unity and, for non-dispersed
orders ($m=0$ or HRMA-only), the line-fraction corrections are also
close to unity.  In this way the HRMA effective area
can be accurately measured\cite{kellogg97}.


\subsection{Optic Effective Area Results}

Figures~\ref{fig:megheg1}~and~\ref{fig:megheg0} show the results of an
initial analysis of the fixed-energy Phase I optic effective area
measurements of the HRMA-HETG in 1st-order and zero-order.  A crude
region-of-interest analysis ({\it e.g.},
Figure~\ref{fig:pha_meg_example}) was performed and a simplified
version of the optic effective was calculated:

\begin{equation}
OEA(E_{\rm line},m) = A_{BND@HRMA} ~~\times 
~~{\frac
{ R_{\rm fp} }
{ R_{\rm BND}} }
\label{equ:simple_oea}
\end{equation}

The solid curves in these Figures are the combination of the pre-XRCF
mirror model with the pre-XRCF HETG efficiency data; the dotted
curves give an estimate of the error range for the predictions.  The
data points have horizontal lines to indicate the size of the
data statistical errors; the vertical lines indicate an estimate
of the measurement systematic errors.

The 1st-order HRMA-HETG effective area curves can be divided (somewhat
arbitrarily) into 5 regions where different physical mechanisms govern
the effective area of the optical (mirror-grating) system:

\begin{itemize}
\item[ ] {\it below 1 keV}~--~The polyimide membrane of
the gratings is dominating the area changes, with edges due to C,
N, and O.
\item[ ] {\it 1-2 keV}~--~The phase effects of the grating cause an
increasing enhancement of the diffraction efficiency.
\item[ ] {\it 2-2.5 keV}~--~Edge structure from the mirror (Ir) and
grating (Au) dominates, sharply reducing effective area.
\item[ ] {\it 2.5-5.5 keV}~--~Effective area is slowly varying, with some
low-amplitude Ir and Au edge structure.
\item[ ] {\it 5.5-10 keV}~--~The mirror reflectivity and grating efficiency
are decreasing rapidly with energy.
\end{itemize}

\noindent 
Horizontal error bars on the data indicate the corresponding statistical
error of the measurement, which is generally below $\pm 5$~\%.
The vertical error lines indicate the probable systematic errors
estimated for the 1st-order measurements as $+30$~\% and $-10$~\%. 
These data illustrate that 
the absolute effective areas are close to expected values
and they show the large range of energies
sampled at XRCF.

\begin{figure}
\begin{center}
\epsfig{file=megheg1st.eps,height=19cm}
\caption[Preliminary XRCF HRMA-MEG and HRMA-HEG Areas: 1st order]
{Preliminary First-order Optic Effective Area results for the
HRMA-MEG (top) and HRMA-HEG (bottom) at XRCF.  These plots
represent the automated application of the simplified
Equation~\ref{equ:simple_oea}
to 132 fixed-energy data sets from Phase I.  \\
~~~Discrepancies arise from i) the need for ``line-fraction''
corrections (the $L$ functions of Equation~\ref{equ:oea_first} ),
ii) departure of the HRMA area from prediction, and iii)
inaccurate facility setup information ({\it e.g.}, incorrect BND aperture sizes.)}
\label{fig:megheg1}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\epsfig{file=megheg0.eps,height=19cm}
\caption[Preliminary XRCF HRMA-MEG and HRMA-HEG Areas: 0-order]
{Preliminary Zero-order Optic Effective Area results for the
HRMA-MEG (top) and HRMA-HEG (bottom) at XRCF.  These plots
represent the automated analysis
of 88 fixed-energy data sets from Phase I.  Because the zero-order
is not dispersed, the effect of ignoring ``line-fraction'' corrections
is not as severe as in the first-order case.}
\label{fig:megheg0}
\end{center}
\end{figure}

\clearpage
\subsection{Grating Efficiency}

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:

\begin{equation}
G_{\rm config}(E,m) = {\frac{OEA_{\rm grating}(E_{\rm line},m)}
                 {OEA_{\rm HRMA}(E_{\rm line})}}
\label{equ:meas_eff_effic}
\end{equation}

Many of the terms of Equations~\ref{equ:oea_first} and 
\ref{equ:oea_second} will cancel
leaving only:

\begin{equation}
G_{\rm config}(E,m) = {\frac 
{\frac
{ { R^g_{\rm fp}({\rm mode}) \times
L^g_{\rm fp}({\rm aperture},{\rm spectrum},QE_{fp})} }
{ R^g_{\rm BND} } }
{\frac
{ { R^H_{\rm fp}({\rm mode}) \times
L^H_{\rm fp}({\rm aperture},{\rm spectrum},QE_{fp})} }
{ R^H_{\rm BND} } }
 }
\label{equ:meas_eff_effic_gory}
\end{equation}

\noindent Thus, the only corrections required to the efficiency measurements
are for the line fraction terms in the focal plane detector.

\subsection{Grating Efficiency Results}

Using Equation~\ref{equ:meas_eff_effic} the grating effective
efficiencies were derived from the previous optic effective area
data.  The results in Figures~\ref{fig:effmegheg1} and
\ref{fig:effmegheg0} show good agreement with the pre-XRCF efficiency
predictions (solid curves).

\subsection{Effective Area / Efficiency Analysis Software}

The results presented here were automatically processed
by a set of IDL procedures that used as input the
CMDB, TRW ID to run id mapping, raw PHA data files, etc.
The results were kept in a custom rdb database table.  Please see
the following web reference for details:

{\tt http://space.mit.edu/HETG/eae/eae.html}

\begin{quotation}
{\it To-do:} \\
Create s/w to estimate the $L^g_{\rm fp}$ and $L^H_{\rm fp}$ corrections
using the measured grating-HSI source spectra. \\
Work with MST and ASC to incorporate XSPEC fitting results. \\
Repeat the auto analysis here using IDL Gaussian fitting plus a
modeled continuum -- as an XSPEC cross check. \\
Validate all relevant CMDB entries used in the analysis. \\
\end{quotation}


\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~\ref{equ:oea_first} 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~\ref{equ:oea_first} will be nearly the same with and without
the grating in place resulting in a clean measurement.}
\label{fig:effmegheg0}
\end{center}
\end{figure}