\begin{quotation}

{\it Objective:} Imperfections in grating fabrication produce blur in
the dispersion and cross-dispersion directions.  These blurs are
dominated, respectively, by the rms period variation and and the rms
roll (alignment) variation.  The core of the diffracted image is
studied and compared with the core of the no-grating (or zero-order)
image to measure the size of these variations.

\end{quotation}

\subsection{Focus tests}

The focus tests (Focus Check FC and Shutter Focus SF)
were performed by cycling through the quadrants
of the mirror.  When examining the 0th order, the inner shells
were closed for images of the MEG 0th order and opened (while
closing the outer shell pair) for the HEG 0th order measurement.
The displacement of the image centroids gave an estimate of the
detector defocus using a simple thin lens approximation and
the knowledge that the detector was already placed within
0.5 mm of the desired focus position.  The insertion of gratings
was not expected to have an effect on detector focus location,
and this was observed to the measurement accuracy of about 25 microns.
Furthermore, once the fixed offset of about 180 microns was
corrected, an additional defocus measurement gave an answer
consistent with zero defocus error.
All 0th order images
were shaped as bowties or hourglasses, depending on the set of
open shutters, as expected, due to mirror scattering, which
redistributes photons preferentially perpendicular to the scattering
surface.

Higher order images were examined in order to verify the
placement of the gratings and detectors on the Rowland circle.
The detector was offset by the amount appropriate to the Rowland
circle and a shutter focus was performed.

Data from D-HXH-FC-1.010, Figure~\ref{fig:mis_aligned}, were crudely
analyzed to obtain $E/dE~\approx$~978 for a Gaussian fit to
the LRF core.  In the cross dispersion a value of $\gamma = 2$~arc minutes
was determined (including astigmatism contributions.)

\subsection{1D Scans of LRF}

\begin{figure}
\begin{center}
\epsfig{file=mgk_psf1d.eps,height=9cm}
\caption[FPC 1D slit scans: HEG, Mg-K, first and second orders]
{Slit scans with the FPC across Mg-K HEG 1st and 2nd orders.
Gaussian approximations to the cores indicate high resolving
powers are being achieved.
}
\label{fig:mgk_psf1d}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\epsfig{file=mgk_meg_psf1d.eps,height=9cm}
\caption[FPC 1D slit scans: MEG, Mg-K, first and third orders]
{Slit scans with the FPC across Mg-K MEG 1st and 3rd orders.
Gaussian approximations to the cores indicate high resolving
powers are being achieved.
}
\label{fig:mgk_meg_psf1d}
\end{center}
\end{figure}

The direct approach to resolution measurement would be to use an X-ray
source with an intrinsically narrow line at low energy.  The best
candidate emission line for this function was the Mg-K$\alpha$ line at
1.253 keV.  Lower energy K lines have resolvable natural line widths
(e.g. O-K or C-K) and the L lines (of Ti, Fe, and Ni, for example)
were generally much weaker and substantially more complex.  Therefore,
we used an indirect approach, observing the higher energy K lines at
high order.  Using a 200 $\mu$~m by 10 $\mu$~m slit, the HEG+1, HEG+2,
MEG+1, and MEG+3 images were scanned directly (``PSF/1D'' measurement)
to measure the LRF.  Figures~\ref{fig:mgk_psf1d} and
\ref{fig:mgk_meg_psf1d} show the data and a simple fits for the HEG
and MEG orders.

\subsection{Results}

The $E/dE$ measurements made so far are tabulated in
Table~\ref{tab:core_results} and plotted in Figure~\ref{fig:xrcf_res_power}.

\begin{quotation}
{\it To-do:} \\
Create XRCF $E/dE$ predictions ($m=1,2,3$) for comparison
with results. \\
Analyze all data available \\
Remove HRMA effects to determine grating-only contribution
\end{quotation}

\begin{table}[hb]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
Data Set & Grating & Energy & Order & E/dE & dp/p rms limit \\
\hline
FC (Fig.\ref{fig:mis_aligned}) & MEG & 1.486 & 3 & 978. & $<$ 435. ppm \\
1D scan & HEG & 1.254 & 1 & 1049. & $<$ 405. ppm \\
1D scan & HEG & 1.254 & 2 & 1418. & $<$ 300. ppm \\
1D scan & MEG & 1.254 & 1 & 479. & $<$ 888. ppm \\
1D scan & MEG & 1.254 & 3 & 1095. & $<$ 389. ppm \\
\hline
\end{tabular}
\caption[Results of some XRCF LRF core measurements]
{Results of some XRCF LRF core measurements}
\label{tab:core_results}
\end{center}
\end{table}

\begin{figure}[ht]
\begin{center}
\epsfig{file=xrcf_res_power.eps,height=18cm}
\caption[XRCF $E/dE$ Measurements]
{XRCF $E/dE$ Measurements.  Resolving Power for the HRMA-HEG and HRMA-MEG as
measured at XRCF during Phase I.  The $E/dE$ values(*) are from
Table~\ref{tab:core_results} and the high-order points are plotted at 
an energy of $E/m$.\\
Various flight error budget curves are plotted for reference only
(see Figure~\ref{fig:res_power} for their details.)
}
\label{fig:xrcf_res_power}
\end{center}
\end{figure}