% file pre_intro.tex
\chapter{Introduction}
\label{chap:intro}

This document details the ground calibration of the HETG, including
the basic models, subassembly (``laboratory'') data, XRCF
measurements, and the resulting calibration products.
This section provides an overview of the HETG and its calibration.

Part I of this document presents the basic operating principles
and models of the HETG (Sections~\ref{chap:lrf_phys} and \ref{chap:effic_phys})
and the laboratory measurements made on the component facets of
the flight HETG (section~\ref{chap:lab}.)

Part II consists of an introduction to the XRCF calibration activities
(Section~\ref{chap:xrcf_intro}) followed by XRCF analyses broken down
into a set of projects (Sections~\ref{sec:periods_angles}
through \ref{sec:area_hrc}).  The status of
these analysis efforts is given in Table~\ref{tab:status}.

Lastly, in Part III the complete set of ground calibration results
are sythesized to produce ``Calibration Products'' for use in
flight analysis software.  Methods of accessing the calibration
data are described in Section~\ref{chap:cal_data}.  The current
best estimates of the parameters describing the HETG and HETGS
are available in Section~\ref{chap:cal_prods}.

\section{The HETG on AXAF}
\label{sec:hetg_on_axaf}

The High Energy Transmission Grating (HETG) is one of four Scientific
Instruments (SIs) that will operate with the High Resolution Mirror
Assembly (HRMA) in NASA's Advanced X-ray Astrophysics Facility (AXAF).
Two of the AXAF SIs are imaging detectors at the HRMA focal plane: the
AXAF CCD Imaging Spectrometer (ACIS) and the High Resolution Camera
(HRC).  These detectors each consist of two sub-imagers: an ``-I''
imager (ACIS-I, HRC-I) of large area and square aspect ratio for
imaging applications and an ``-S'' imager (ACIS-S, HRC-S)
with a more rectangular aspect ratio designed for grating
readout.

Either of the two grating SIs, HETG or
LETG\cite{brinkman97,predehl97}, can be inserted into the optical path
just behind the HRMA and, through diffraction, deflect the converging
rays by angles roughly proportional to their wavelength,
Figure~\ref{fig:hetgs_diagram}.  By combining the HRMA's high angular
resolution (of order 1 arc second) and the grating's large diffraction
angles (as high as 100 arc seconds/\AA), the HRMA-grating-detector
systems are capable of spectral resolving powers up to $E/dE \approx
1000$ in the AXAF energy band.

\begin{figure}[ht]
\begin{center}
\epsfig{file=hetgs_diagram.ps,height=6.5cm}
\caption[Schematic of the HETGS Configuration]
{Schematic of the HETGS Configuration.  The HETG can be inserted
into the optical path behind the AXAF mirror (HRMA) to intercept and diffract
the converging X-rays.  The X-rays are diffracted by an angle $\beta$ 
given in Equation~\ref{equ:dispersion} and are detected by the
six chips of the ACIS-S CCD array.  The HEG and MEG gratings
have their dispersion axes oriented $\pm5$~degrees to the ACIS-S
long axis.}
\label{fig:hetgs_diagram}
\end{center}
\end{figure}


\section{The HETG Itself}
\label{sec:hetg_itself}

Details of the HETG have been presented previously in SPIE
conferences\cite{canizares85,markert94}.  The HETG is a faceted
Rowland torus design\cite{beuermann78} that has been engineered to
place each facet at its prescribed location and orientation on a
Rowland torus with Rowland Diameter of 8634 mm.  Specifically, in
Figure~\ref{fig:hetg_drawing} the precision, lightweight HETG Element
Support Structure\cite{pakmcguirk94} (HESS) is populated with 336
individual grating facets.  This design combined with the period and
period variation properties of the facets allows high resolving
powers.

The 336 grating facets of the flight HETG were fabricated in-house at
M.I.T.  in the Space Microstructures Laboratory (SML) using
state-of-the-art holographic lithography
techniques\cite{schattenburg94} to create a fine period, high aspect
ratio grating structures supported on a thin polyimide membranes,
Figure~\ref{fig:grat_cross_secs}.  Each facet was then tested in-house
for period and diffraction efficiency characteristics\cite{dewey94}.

\begin{figure}
\begin{center}
\epsfig{file=grat_cross_secs.ps,height=6.5cm}
\caption[Cross-section schematic of the HETG gratings]
{Cross-section schematic of the HETG gratings.  The fine-period gold
grating bars are supported on a polyimide membrane.  A uniform
``platingbase'' layer provides adhesion of the bars to the polyimide.
}
\label{fig:grat_cross_secs}
\end{center}
\end{figure}

The four rings of facets on the HETG are designed to intercept and
diffract X-rays from the corresponding four HRMA mirror shells.
Because most of the high energy area of the HRMA is due to the inner
two shells, very fine period (2000~\AA) ``High Energy Grating''
(HEG) facets are used here.  The outer two shells have ``Medium Energy
Grating'' facets (``MEG'', 4000~\AA~period) that are efficiency-optimized
below 2 keV.  These two grating sets have their dispersion axes offset
by 10 degrees from each other so that their spectra are spatially
separated on the detector.  In this way the HETG enables
high-resolution spectroscopy ($E/dE > 100$) in the 0.4~keV to 9~keV
band.

\begin{figure}
\begin{center}
\epsfig{file=HETGdiag.eps,height=16cm}
\caption[HETG drawing with facet IDs]
{The HETG.  A lightweight ($\approx$ 10 kg) aluminum structure
supports 336 individual grating facets in a Rowland geometry.  This
view of the HETG is from the HRMA side and the facet-location
identification scheme is indicated.  The HETG, as installed for ground
calibration at XRCF, was rotated 180 degrees about the optical axis
from this view.}
\label{fig:hetg_drawing}
\end{center}
\end{figure}


\section{HETGS Calibration Definition and Goals}
\label{sec:hetgs_cal}

The complete flight instrument that will be utilized and must be
calibrated is the combination of the HRMA, HETG, and the ACIS-S---the
{\it HETG Spectrometer (HETGS)}.
The HETGS calibration requirements generally fall into two categories:
calibration of the {\it Effective Area} and calibration of the {\it
Line Response Function (LRF)}.  Scientific goals set calibration
requirements on the HETGS in these areas which are then flowed-down to
requirements on the individual HEG and MEG elements themselves.

\subsection{Effective Area and Efficiency Definitions}

The HETGS response to an on-axis monochromatic point source consists
of images in the various {\it diffracted orders} produced at locations
given by the grating equation\cite{born80}:

\begin{equation}
sin(\beta) = {\frac{m\lambda}{p}}
\label{equ:dispersion}
\end{equation}

\noindent where $m$ is the order of diffraction (an integer 0, $\pm 1, \pm 2,$
....), $p$ is the grating period and $\beta\ $is the dispersion
angle.  Knowledge of the dispersion axis and the grating-to-detector
distance (the Rowland spacing) allows a conversion of the angle
$\beta$ to a physical location on the detector.

These diffracted images are, for the most part, spatially localized
and so it is possible and convenient to express the system response
to a monochromatic source with incident flux of $1~{\rm photon / cm^2 s}$
as the sum of weighted spatial functions:

\begin{equation}
{\rm HETGS~response}(E,\dots ) = \sum_m SEA(E,m,\dots )
\times PSF(E,m,\dots )
\label{equ:hetgs_response}
\end{equation}

\noindent where $SEA(E,m,\dots )$ is the {\it system effective area},
described further below, and $PSF(E,m,\dots )$ is a unit-normalized
2-dimensional point spread function which describes the spatial distribution of the
$m$-th order detected events, see Section~\ref{sec:lrf_defn} below.
The dots ($\dots$) in the arguments indicate that there are
other dependancies, {\it e.g.}, off-axis angles, defocus, 
detector modes, etc.

The system effective area for the HETGS, in ${\rm cm}^2
{\rm counts}/{\rm photon}$, can be calculated from the
following terms:

\begin{equation}
SEA(E,m,\dots ) = \sum_{s=1,3,4,6} A_s(E,\dots ) ~G_s(E,m)
~QE(E,\dots )
\label{equ:sea}
\end{equation}

\noindent Here the sum is over the
HRMA mirror shells, $A_s$ is the {\it optic effective area} for HRMA shell
$s$ in ${\rm cm}^2$, and $QE(E,\dots )$ is
the detector quantum efficiency including effects of detector spatial
uniformity ({\it e.g.}, BI/FI chips, gaps) and detection mode
({\it e.g.}, event grading) in counts per photon.

$G_s(E,m)$ is the average diffraction efficiency for the
gratings on shell $s$ and is calculated from the individual facet
efficiencies $g_f(E,m)$:

\begin{equation}
G_s(E,m) = ~\nu_s~ {\frac{1}{N_s}} \sum_{f \in \{ s \} } ~g_f(E,m)
\label{equ:G_s}
\end{equation}

\noindent where $N_s$ is the number of facets on shell $s$
and $\nu_s$ is a unitless shell-by-shell vignetting factor
to account for the inter-grating gaps.  Thus, the physics of
$g_f(E,m)$ governs the HETG contribution to the HETGS effective area.

Finally, it is useful to define the grating ``effective efficiency''
for combinations of more than a single HRMA shell:

\begin{equation}
G_{\rm config}(E,m) = {
       \frac{ \sum_{s \in \{ {\rm config} \} } A_s(E) ~G_s(E,m) }
                  { \sum_{s \in \{ {\rm config} \} } A_s(E) }}
\label{equ:eff_effic}
\end{equation}

\noindent where ``config'' may usefully be the HEG (shells 4,6),
 MEG (shells 1,3), or HETG (shells 1,3,4,6).
In practice, it is the effective efficiencies of these configurations
that are measured and used.

\begin{figure}
\begin{center}
\epsfig{file=mis_aligned.eps,height=14cm}
\caption[PSF and LRF Example]
{PSF and LFR example.  
The event distribution (top left) may be described by a 2-D point spread
 function PSF, $\rho (y,z)$ (top right).  The
resulting 1-D line response function $l(y')$ (lower plot) is shown with a
Gaussian plus quadratic fit to its core. \\
\indent ~~~~~~The events are from an XRCF
HSI image of the 3rd-order MEG Al-K line; the HSI is centered at
${\rm HSI}_Y = $~54.76~mm, ${\rm HSI}_Z = $~4.496~mm.  Visible in addition to the
main peak is the ``satellite line'' at ${\rm HSI}_Y = $~-400.  PSF outliers at
${\rm HSI}_Z = $~+400 and -100 represent mis-aligned grating facets.
}
\label{fig:mis_aligned}
\end{center}
\end{figure}

\subsection{Effective Area Calibration Goals}

Calibration of the effective area of the HETGS is driven by the desire
to extract information about the physical parameters of an emitting
region (e.g., temperature, ionization age, elemental abundances) from
the observed intensity of spectral lines through plasma
diagnostics\cite{canizares90}.  Analysis of the sensitivity of
scientific conclusions on the calibration accuracy has led to the
requirements that the HETGS effective area be known with an absolute
accuracy of order 10\% (1$\sigma$) at all energies, and that the
relative effective area at two different energies be known to of order
3\% (1$\sigma$).  Note that the effective area needs to be known at
least on an energy grid comparable to the coarser of the astrophysical
spectrum feature scale and the instrumental response variation scale.

These requirements for HETGS calibration have implications for the
calibration of the HRMA and focal plane detectors as well as the
HEG/MEG elements.  Ideally the effective area and/or efficiency
of all components would be know accurately enough so that the composite
system would be calibrated to the 3\% (1$\sigma$) level on an
energy scale of $\Delta E \approx  E/1000$, especially around
instrumental edges.

For our laboratory calibration of the HEG/MEG elements we have set a
goal of 1\% (1$\sigma$).  This allocates most of the error to the HRMA
and focal plane detector calibrations where the calibration process is
inherently more difficult (i.e., measuring an effective area or an
absolute detection efficiency as opposed to a transmission
efficiency.)

Though not as central as the (first-order) diffraction efficiency
measurements, it is important that the zero-order and
higher orders be calibrated as well.

\subsection{LRF Definition}
\label{sec:lrf_defn}

As described above the focal plane image that results from a
monochromatic near-axis point source can be considered as a set of
images, one for each grating-order.  The distribution of each of these
images can be described by a {\it 2-dimensional Point Spread Function
(PSF)} which gives the normalized density of detected photons in the
detector plane:

\begin{equation}
  \rho (y,z) \Leftarrow PSF(E,m,\dots)
\label{equ:rhoyz}
\end{equation}

\noindent where $y,z$ are the detector coordinates, $E,m$ are the 
energy and grating-order, and ``$\dots$'' is again a placeholder for
other dependancies, {\it e.g.}, the telescope defocus,
location of the source with respect to the optical axis, etc.
As an example, an X-ray event
plot and the corresponding $\rho (y,z)$ PSF is shown
in the upper plots of Figure~\ref{fig:mis_aligned} (these are data
from XRCF and are used here for illustration purposes; they are discussed
in Section~\ref{sec:lrf_core}.)

Because the spectroscopic information of a grating dispersive
instrument is along the dispersion direction, it is useful to define
the {\it one-dimensional Line Response Function (LRF)} to be the PSF
integrated over the cross-dispersion direction:

\begin{equation}
  l(y') = \int dz' \rho (y',z')
\label{equ:lrf}
\end{equation}

\noindent where the $y',z'$ indicate axes aligned with the 
dispersion and cross-dispersion directions in the detector plane.
The lower plot in Figure~\ref{fig:mis_aligned} gives the LRF
corresponding to the example PSF.
Because of the image properties of the mirror, it is generally
useful to (conceptually) break the LRF into at least two regions:
a core or inner LRF and the wings or outer portion of the LRF.

The {\it Resolving Power}  of the spectrometer at energy
$E$ is defined as 

\begin{equation}
 R(E,m)  ~=~ E/dE ~=~ y'_{\rm centroid}/dy'_{\rm FWHM}
\label{equ:ede}
\end{equation}

where $dE$ is the full-width at half-maximum
(FWHM) of the LRF.  Typically the LRF core can be well fit by a Gaussian
profile and in this case $dE \approx 2.35\sigma$ approximates the FWHM.

\subsection{LRF Calibration Goals}

The goal of HETGS LRF calibration is to produce LRF models which are
accurate (have low $\chi^2$) when fitting a line containing of order
1000 counts.  Additionaly, it is important that any ``wings'' or
``ghosts'' in the dispersed spectrum be identified and quantified.
Specifically, the contribution of wings at all scales should be known
or limited to 1\% of the peak line flux in a resolution element.

\section{Ground Calibration Overview}

The HETG ground calibration program consists of 4 main activities:
\begin{itemize}
\item Creating and validating models for the HETGS LRF performance,
Section~\ref{chap:lrf_phys}.
\item Synchrotron testing of sample and reference gratings to 
understand and validate the diffraction model $g_f(E,m)$,
Section~\ref{chap:effic_phys}.
\item Laboratory testing and assembly of all flight gratings,
Section~\ref{chap:lab}.
\item XRCF tests of the flight hardware, Part II,
Sections~\ref{chap:xrcf_intro}-\ref{chap:xrcf_eae}.
\end{itemize}

\clearpage
\section{HETG Documentation}

  It is assumed that the reader is familiar with AXAF, the HETG, and
XRCF calibration and has access to the documents below.

\begin{table}[hb]
\caption{HETG Documentation}
\label{tab:hetg_docs}
\begin{center}
\begin{tabular}{ccc}
\hline
\multicolumn{3}{l}{\it Printed documents:} \\
AXAF Observatory Guide & ASC/User Support & October 1997  \\
AXAF Proposers Guide & ASC/User Support & October 1997  \\
 \mx  Manual & ASC/SDS & October 1997  \\
HETG Ground Calibration Report & HETG and ASC & October 1997 \\
HETGS Flight Performance and Calibration Report & ASC/Cal & TBD 1997 \\
\hline
\multicolumn{3}{l}{\it Electronic documents:} \\
{\tt http://asc.harvard.edu} & ASC & dynamic \\
{\tt http://space.mit.edu/HETG/xrcf.html} & HETG & dynamic \\
\hline
\end{tabular}
\end{center}
\end{table}