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% Author: D. Huenemoerder
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\Header{L1.5 Tool Spec: HESF Mask /}{\Revision}

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      \vspace*{0.1in}
      \huge\bf AXAF Science Center}

      \leavevmode{\scalebox{0.17}{\includegraphics{asc_logo.eps}}}
      \vspace*{-0.14in}
      \hrule
      \vspace*{0.1in}
      
      {\LARGE\bf Grating Tool Specifications}

      \vspace*{0.1in}
      {\LARGE\bf Level 1.5:}

      \vspace*{0.1in}
      {\LARGE\bf  LETG/HESF Region Mask Definitions}

      {\LARGE\bf  \&}
      \vspace*{0.1in}
      
      {\LARGE\bf Wavelength Determination}
      
      \vspace*{0.1in}

      {\Large\bf David Huenemoerder}

      \vspace*{0.1in}

      ({\tt http://space.mit.edu/ASC/docs/TBD})

      \vspace*{0.1in}

      \Revision
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%      Submitted: & \rule{3.25in}{0.01in} & \rule{0.55in}{0.01in} \\
%      & David Huenemoerder & Date   \\
%      & Grating Scientist, ASC Science Data Systems & \\[0.25in]
%      Concurred: & \rule{3.5in}{0.01in} & \rule{0.75in}{0.01in} \\
%      & Janet De Ponte & Date   \\
%      & ASC Data Systems Group Leader, L1.5 Pipelines & \\[0.25in]
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\section{Introduction}

This memo specifies how region masks can be defined for filtering data
taken with the LETG in conjunction with the High Energy Suppression
Filter (HESF, also known as the Drake Flat).  The regions are used to
assign identifiers to photons according to which part of the spectrum
they fell, as specified in the Grating Level 1.5 ICD\footnote{
  ``Grating Data Products: Level 1.5 to ASC Archive Interface Control
  Document'', {\tt http://space.mit.edu/ASC/docs/ICD\_L1.5.ps.gz}}.
%
Subsequent processing uses the region information in assigning
wavelengths to photons.  This transformation will also be outlined in
this document.

\section{Region Masks}

The HESF is comprised of two facets which reflect the incident beam
and divide it into two segments.  Since there is a hole in the middle
of the HESF, each of the two reflected spectra are discontinous.  In
addition, there is a non-reflected portion which falls in the same
location as a non-HESF observation.
\begin{figure}[htb]
  \leavevmode{\scalebox{0.85}{\includegraphics{../Ps/HESF_regions.eps}}}
  \caption[LETG/HESF Field View]{\it The predicted regions defining the
    masks are shown by boxes, in which the predicted central locus of
    photons are shown by a solid line, and the photons positions from
    a {\sc Marx} simulation are shown by discrete points.  This view
    is in detector focal-plane coordinates ($y,z$) whose origin is the
    optical axis. No aspect dither motion was applied in this
    simulation.}\label{fg:fieldview}
  \end{figure}
Figure~\ref{fg:fieldview} shows the HRC-S
field with regions outlined by boxes, the predicted central loci of
photons, and results of a {\sc Marx} simulation.
  
The locations of the spectrum for any aim-point and source position
are determined by the HESF geometry and the HRC-S geometry. The tilts
of the reflected spectra are caused by the HRC-S outer micro-channel
plates' (MCP) tilts.  Since the optical axis is not on the detector
centerline, and since the tilts of the $+y$ and $-y$ sides are not
identical, the regions are not exactly symmetric.

The memo by Drake\footnote{``Locations of the Reflected Spectra from
  the HRC-S High Energy Suppresion Filter'', Jeremy Drake, AXAF
  Science Center Memorandum, November 2, 1997} derives the spectral
locations from the HESF and HRC-S geometry.  I have used his equations
4 and 5\footnote{There is an error in Equation 5: a factor of
  $\tan2\theta_u$ should multiply the quantity in $[\ ]$.} to derive a
single linear expression of the form
  \begin{equation}
    \label{eq:zeqABy}
    z = A_i + B_i y
  \end{equation}
  which defines the focal-plane $z$ position of the reflected spectrum
  given the focal-plane $y$-coordinate. It is actually a piecewise
  linear equation, since some of the coefficient terms have several
  discrete values, depending on the facet and on $y$.
  Keeping Drake's notation, the coefficients become:
  \begin{equation}
    \label{eq:Acoef}
    A_i = \tan2\theta_i \left[ h_u 
        \left(1- \frac{\tan\theta_l}{\tan\theta_i}\right)
      + \left(\frac{z_p - z_f}{\tan\theta_i}\right) 
      + h_l(0) + y_0(y)\tan\delta(y)
      \right] 
      + z_s,
  \end{equation}
and
  \begin{equation}
    \label{eq:Bcoef}
    B_i = -\tan2\theta_i\,\tan\delta(y).
  \end{equation}
Here, $\theta_i$ is to be replaced with $\theta_l$ or $\theta_u$ for
the lower or upper facets, respectively.  New terms I have introduced
are:  
\begin{description}
\item[$h_l(0)$,] which is Drake's $h_l(y)$, evaluated at $y=0$;
\item[$y_0(y)$,] which is the $y$-coordinate at which the MCP
  boundaries occur, and at which a tilt begins.  This term has two
  values, one for each MCP boundary;
\item[$\delta(y)$,] which defines the tilt of the outer MCPs.  This
  takes on one of three values, depending upon which MCP the value of
  $y$ implies.
\item[$z_s$,] which is the zero-order source $z$-centroid.  This
  changes the coordinate system from a zero-order referenced system to
  an optical-axis referenced system.  Hence, the optical axis is by
  definition at $(y,z)=(0,0)$.
\end{description}
%

$A$ and $B$ completely specify the locations of the spectrum and
are sufficient to define a region mask around the spectrum.  $A$,
however, depends on observation-specific parameters: the SIM
$z$-position, and the source $z$ location, which both specify $z_p$.
Hence, we could separate $A$ into the ``calibration'' portion
($A_{ci}$) and the ``observational'' portion ($A_{oi}$) such that 
\begin{equation}
  \label{eq:Asum}
  A_i = A_{ci} + A_{oi}
\end{equation}
 as follows:
\begin{equation}
  \label{eq:Aci}
    A_{ci} = \tan2\theta_i \left[ h_u 
        \left(1- \frac{\tan\theta_l}{\tan\theta_i}\right)
      - \left(\frac{z_f}{\tan\theta_i}\right) 
      + h_l(0) + y_0(y)\tan\delta(y)
      \right] ,  
\end{equation}
\begin{equation}
  \label{eq:Aoi}
    A_{oi} = z_p \frac{\tan2\theta_i}{\tan\theta_i} + z_s
\end{equation}
To a good approximation, the ratio if tangents is equal to
$2(1+\theta_i^2)$. So in terms of the {\sc Sim} $z$-offset, $z_0$, and
zero-order centroid, $z_s$, we can write
\begin{equation}
  \label{eq:Aoiapprox}
  A_{oi} \approx 2(1+\theta_i^2)z_0 - (1+2\theta_i^2)z_s.
\end{equation}
{\em Note that this is the {\bf only} parameter of the reflected
  spectrum location which depends upon observational quantities!}  The
other terms are ``just'' calibration.


%
\clearpage
All the terms, their definitions, and default values are given in the
following table (see the Drake memo for defining diagrams).
\medskip

\begin{tabular}{|lp{1.5in}p{3in}|}\hline
\bf Term & \bf Nominal Value & \bf Description \\[1mm] \hline 
$z_p$ 
& 6.25 mm 
& The distance in the $z$-axis between the center of
  the HRC-S and the observation mean aim-point (and location of the
  zeroth order).
\\[1mm] \hline
%
$z_f$
& 5.35 mm
& The distance in the $z$-axis between the center of the HRC-S and the
lower edge of the filter.\footnotemark{}
\\[1mm] \hline
%
$\theta_l$
& $4.5^\circ$
& the angle in the $x-z$ plane between the optical axis ($x$) and the
plane of the upper facet.
\\[1mm] \hline
%
$\theta_u$
& $7.0^\circ$
& The angle in the $x-z$ plane between the optical axis ($x$) and the
plane of the lower facet.
\\[1mm] \hline
%
$h_u$
& 22.3 mm
& The $x$ axis distance between the lower edge of the upper facet
(nearest the HRMA) and the lower edge of the lower facet.
\\[1mm] \hline
%
$h_l(0)$
& 26.3 mm
& The $x$ axis distance (or ``height'') of the lower edge of the lower
facet (furthest from the HRMA) above the MCP surface, at $y=0$.
\\[1mm] \hline
%
$y_0(y)$
& $-46$ mm, $y \le -46$\newline
   54 mm, if $y > 54$ \newline
   0, otherwise (irrelevant)
& The ``boundaries'' of MCP segments, relative to optical axis (which
is different from the array centerline).  These mark the locations at
which the MCP tilts begin and $h_l(y) \ne h_l(0)$.
\\[1mm] \hline
%
$\delta(y)$
& $-1.23^\circ$, $y \le -46$ mm\newline
  $ 0.0^\circ$, $ -46 < y \le 54$ mm\newline
  $1.38^\circ$, $ y > 54$
& The tilt angle of the MCP (which which puts the $y$ into
  $h_l(y)$). The negative is a convention to get the proper signs on
  the negative-$y$ side.  Note the $y$-limits should be the same as
  the values of $y_0(y)$.
\\[1mm] \hline
$y_{gm}$
& $-36.7$ mm
& The HESF central gap limit on the minus-$y$ side.
\\[1mm] \hline
$y_{gp}$
& 28.7 mm
& The HESF central gap limit on the plus-$y$ side.
\\[1mm] \hline
$z_0$
&6.25  mm
& The $z$-distance from the optical axis (by definition at $z=0$) to
the HRC-S centerline.  This is determined by the SIM $z$-offset.
\\[1mm] \hline
$z_s$
& 0.0 mm
& The $z$-coordinate of the zero-order centroid.  This is nominally at
the on-axis position, but may deviate from it.  Given dither, a mean
value is used.
\\[1mm] \hline
\end{tabular}
 \footnotetext{If there is mis-alignment of the
  HESF with the $y$-axis, then $z_f$ becomes a funtion of $y$:
  $z_f(y)=z_{f0} + z_{f1}y$.  If $z_{f1}$ is non-zero, $A_i$ will keep
  the same form, but $B_i$ will have an additional term.}

\medskip
\noindent The three observationally variable $z$'s are related via
\begin{equation}
  \label{eq:zzzs}
  z_p = z_0 - z_s.
\end{equation}
For the tabulated nominal values, the coefficients become:

\begin{center}
\begin{tabular}{rllllllll}\label{pg:Atbl}
$y$ range [mm] & $A_l$ &  $B_l$ &  $A_u$ &  $B_u$ & 
$A_{cl}$ & $A_{ol}$ & $A_{cu}$ & $A_{ou}$ \\ \hline
%
$y\le -46.0$
& 6.133
& 0.00340
& 10.627
& 0.00535
& -6.445 
& 12.558 
& -2.064 
& 12.671
\\
%
$-46.0 < y \le -36.7$
& 5.977
& 0.000
& 10.381
& 0.000
& -6.601 
& 12.558 
& -2.310 
& 12.671
\\
%
$-36.7 < y \le +28.7$
& 0
& 0
& 0
& 0
& 0
& 0
& 0
& 0
\\
%
$+28.7 < y \le +54.0$
& 5.977
& 0.000
& 10.381
& 0.000
& -6.601 
& 12.558 
& -2.310 
& 12.671
\\
%
$y>+54.0$
& 6.183
& -0.00382
& 10.705
& -0.0060
&
-6.395 &
12.558 &
-1.986 &
12.671
\\ \hline
%
\end{tabular}
\end{center}


\section{Making a Mask}

The previous sections described the locations of the spectrum in terms
of focal-plane coordinates.  Filtering is done, however, more easily
in sky coordinates since the aspect correction restores the
zero-order image.  The purpose of the mask is to enclose photons of
interest. Hence, the boundaries to not need to undergo rigorous
coordinate transformations.  Hence, we can simply rotate, translate, and
scale the detector-plane coordinates to approximate sky coordinates.

There are also some simplifying factors for specifying the HESF
regions. First, the zero-order will be near the on-axis point, so we
do not need to consider far off-axis cases.  Second, you will notice
that there is no zero-order $y$-dependence in the reflected spectral
positions --- it is only the $z$-offset which determines where the
reflected spectra lie.

The procedure for defining the regions is as follows:
\begin{enumerate}
\item Determine the mid-points between the gap inner edge and the array
  edge, on both the plus and minus sides.  These won't change -- they
  depend only on the calibration.  These are:
  $$ y_{0p} = \frac{1}{2}\left( \max(y) + y_{gp} \right);$$
  $$ y_{0m} = \frac{1}{2}\left( \min(y) + y_{gm} \right).$$
\item Calculate the values of $z$ at these values of $y$ for the upper
  and lower facets:
  $$ z_{up} = A_u + B_u y_{0p},$$
  $$ z_{um} = A_u + B_u y_{0m},$$
  $$ z_{lp} = A_l + B_l y_{0p},$$
  $$ z_{lm} = A_l + B_l y_{0m}.$$
\item   Define the centers of rectangular regions with these coordinates:
  $$(y_{0p}, z_{up}),$$
  $$(y_{0p}, z_{lp}),$$
  $$(y_{0m}, z_{um}),$$
  $$(y_{0m}, z_{lm}).$$
\item Assign a width to the rectangular region.  Widths are specified
  via error-budget calculation as in non-HESF Level 1.5 processing
  (incorporating PSF vs off-axis angle lookup and defocus,
  astigmatism, and an arbitrary multiplication factor); typical widths
  are about $\pm 0.2$ mm.
\item Assign a length to the rectangular region.  The length need not
  be precise.  It should extend in the $y$-direction from the central
  HESF gap to the detector edge, or about 
  $$L_p \approx \max(y) + y_{gp};$$
  $$L_m \approx -(\min(y) + y_{gm}).$$
\item Determine the rotation of each rectangle.  These do not depend
  upon any observational parameters, but only on calibration
  parameters.  The rotations are given by the $B_i$ values:
  $$ \rho_u = \arctan(B_u);$$ 
  $$ \rho_l = \arctan(B_l).$$
  (Remember that each of these has two values, due to the asymmetric
  MCP tilt angles, $\delta(y)$).
\item Transform the centers, widths, lengths, and rotations to the sky
  frame.
\end{enumerate}

The predicted positions for the nominal parameters are are shown in
Figure~\ref{fg:fieldview} for the entire field, and in
Figure~\ref{fg:zoomview} with expanded scales.

\begin{figure}[htb]
  \leavevmode{\scalebox{0.85}{\includegraphics{../Ps/HESF_regions_zoom.eps}}}
  \caption[LETG/HESF Zoomed View]{\it The predicted regions defining the
    masks are shown by boxes, in which the predicted central locus of
    photons are shown by a solid line, and the photons positions from
    a {\sc Marx} simulation are shown by discrete points.  This view
    is in detector focal-plane coordinates ($y,z$) whose origin is the
    optical axis. No aspect dither motion was applied in this
    simulation.}\label{fg:zoomview}
  \end{figure}

\section{Diffraction Coordinate Specification}

For non-HESF observations, diffraction coordinates are computed by
transforming the 3D location of each event to the diffraction angles
relative to the grating node parallel ($r$) and perpendicular ($d$) to
the dispersion.  The zero-order centroid is the origin for $r$, and
the line through the zero-order centroid perpendicular to the grating
bar direction is the reference for $d$.

With the HESF, the $y$-coordinate is unchanged, since the HESF diverts
the photons perpendicularly to the dispersion
direction.\footnote{\ldots if the HESF alignment is parallel to the
  $y$-axis.  The magnitude of the misalignment affect is TBD.}  Hence,
the wavelength can be calculated as if the photon were in the
un-reflected region.

The cross-diffraction angle must be referenced to the line,
$z=A_i+B_iy$.  The difference in cross-dispersion angle between any
two photons will be the same whether they were reflected by the HESF,
or not, but for an inversion in position.  Hence, we can ``fool'' the
current coordinate transformations by transforming from the deviation
from the reflected line as if it were the deviation from the
un-reflected line:
\begin{equation}
  \label{eq:tgd}
  z^\prime = (A_i+B_iy) - z + z_s
\end{equation}
Here, the photon must first be tested for inclusion in region $i$
(which is $u$ or $l$ for upper-facet or lower-facet reflection,
respectively). Then it is transformed from $(y,z^\prime)$ to $(r,d)$ 
(the $x$-position is also the same as if it were un-reflected, to
within mis-alignment affects).


\clearpage
\appendix
\section{Examples {\it vs.} Source \& SIM Positions}

The following tables and figures are for different source off-axis
positions or for different {\sc Sim} $z$-offsets from the nominal
parameters.  There is currently a difference between the {\sc Marx}
coordinate system and that in the Drake memo.  The former has the HESF
at positive $z$ if the optical axis is at the center of the HRC-S, and
thus requires a translation toward $-z$ to enable the HESF. The Drake
memo shows the opposite case.  To reconcile the two, I have negated
the {\sc Marx} $z$-coordinates (the {\tt zpos} vector) before plotting
on the predicted mask positions.
The relationship between the {\sc Marx} parameters and HESF parameters
defined here is as follows:
\begin{description}
\item[$z_0$:] $-\mathrm{DetOffsetZ}$
\item[$z_s$:] $(\mathrm{SourceElevation}/60)\,(\pi/180)\, X_\mathrm{HRMA}$
\item[$y_s$:] $(-\mathrm{SourceAzimuth}/60)\,(\pi/180)\, X_\mathrm{HRMA}$.
\end{description}
Here, $X_\mathrm{HRMA}$ is the imaging focal-length of the HRMA (about
10065.5 mm).


The following table lists the observational component to the
$A$-coefficient for lower and upper facet reflections.  (Units are in
mm. The other coefficients can be found in the previous table on
page~\pageref{pg:Atbl}; they don't change w/ source or {\sc Sim}
location.)
\begin{center}
\begin{tabular}{|cccc|}\hline
$z_s$ & $z_o$ & $A_{ol}$ & $A_{ou}$ \\ \hline
0.000 & 6.240&  12.558 &12.671\\
1.464 & 6.240&  11.076 &11.162\\
%2.928 & 6.240&   9.593 & 9.653\\
0.000 & 6.240&  12.558 &12.671\\
0.000 & 5.471&  11.010 &11.109\\
-0.878& 5.471&  11.900 &12.015\\
0.878 & 6.250&  11.689 &11.786\\
0.000 & 6.250&  12.578 &12.691\\ \hline
\end{tabular}
\end{center}

The following figures show the predicted spectrum and mask locations
as well as a {\sc Marx} simulation for the same cases.  Each figure is
labeled with the source and {\sc Sim} positions.

\begin{centering}

\leavevmode{\scalebox{0.85}{\includegraphics{../Ps/HESF_regions_0.eps}}}
\leavevmode{\scalebox{0.85}{\includegraphics{../Ps/HESF_regions_0_zoom.eps}}}

\leavevmode{\scalebox{0.85}{\includegraphics{../Ps/HESF_regions_1.eps}}}
\leavevmode{\scalebox{0.85}{\includegraphics{../Ps/HESF_regions_1_zoom.eps}}}

%\leavevmode{\scalebox{0.85}{\includegraphics{../Ps/HESF_regions_2.eps}}}
%\leavevmode{\scalebox{0.85}{\includegraphics{../Ps/HESF_regions_2_zoom.eps}}}

\leavevmode{\scalebox{0.85}{\includegraphics{../Ps/HESF_regions_3.eps}}}
\leavevmode{\scalebox{0.85}{\includegraphics{../Ps/HESF_regions_3_zoom.eps}}}

\leavevmode{\scalebox{0.85}{\includegraphics{../Ps/HESF_regions_4.eps}}}
\leavevmode{\scalebox{0.85}{\includegraphics{../Ps/HESF_regions_4_zoom.eps}}}

\leavevmode{\scalebox{0.85}{\includegraphics{../Ps/HESF_regions_5.eps}}}
\leavevmode{\scalebox{0.85}{\includegraphics{../Ps/HESF_regions_5_zoom.eps}}}

\leavevmode{\scalebox{0.85}{\includegraphics{../Ps/HESF_regions_6.eps}}}
\leavevmode{\scalebox{0.85}{\includegraphics{../Ps/HESF_regions_6_zoom.eps}}}

\leavevmode{\scalebox{0.85}{\includegraphics{../Ps/HESF_regions_7.eps}}}
\leavevmode{\scalebox{0.85}{\includegraphics{../Ps/HESF_regions_7_zoom.eps}}}

\end{centering}
\end{document}
 
