
\documentclass[11pt]{article}

\usepackage[dvips]{graphicx}      %% FOR INCLUDING GRAPHICS
\usepackage{natbib}
\usepackage{epsf}

%% ----------------------------------------------------------------------
%% Add specials here
%% ----------------------------------------------------------------------

\def\ion#1#2{#1\,{\sc #2}\relax}  %% 

%% ----------------------------------------------------------------------
%% Modify: 
%%   Note number, version number and date
%%   Title
%%   Author and affiliation 
%% ----------------------------------------------------------------------

\setlength{\oddsidemargin}{0in}         %  1 inch (0 + 1) from left.
\setlength{\topmargin}{-0.5in}          %  1 inch (1.5 - 0.5) from top.
\setlength{\textwidth}{6.5in}           %  1 inch (8.5 - 1 - 6.5) from right.
\setlength{\textheight}{9in}            %  1 inch (11 - 1 - 9) from bottom.

\addtolength{\parskip}{0.5\baselineskip}

%  The environment centerpage centers text vertically on the page.  This is
%  useful within a titlepage environment.

\newenvironment{centerpage}{\mbox{} \protect\vspace*{\fill}}{
        \protect\vspace*{\fill} \mbox{} \protect\\ \mbox{}}

\newcommand{\lam}{$\lambda$}
\newcommand{\arcsec}{$^{\prime\prime}$}
\begin{document}                
%\bibliographystyle{plain}               % Define bibliography style.

\begin{titlepage}
\hrule
\noindent\textbf{EUV IMAGING SPECTROMETER}\\
\vspace{-0.7\baselineskip}\\
{\noindent\Huge\bf Hinode}
\vspace{2mm}
\hrule
\vspace{3mm}
\centerline{\bf EIS SOFTWARE NOTE No. 3}
\vspace{3mm}
\hrule
\noindent Version 1.0 \hfill 21 July 2011
\vspace{2mm}
\hrule

\begin{centerpage}

\begin{center}
{\Large\bf Spatial Y-offsets between different wavelengths (spectrum
  tilt and CCD offset)}\\
\mbox{}\\
\mbox{}\\
Peter Young\\
George Mason University\\
4400 University Drive\\
Fairfax, VA 22030\\
U.S.A.\\
\mbox{}\\
pyoung9@gmu.edu
\mbox{}\\
\end{center}

\end{centerpage}

\end{titlepage}

%% ----------------------------------------------------------------------
%% The document!
%% ----------------------------------------------------------------------

\section{Overview}

Images obtained in different EIS emission lines  occur at
different Y-positions on the EIS detectors. There are two instrumental
effects that contribute to this:

\begin{enumerate}
\item a tilt of the grating dispersion axis relative to the CCD axes;
  and
\item a spatial offset in Y between the two CCDs of EIS.
\end{enumerate}


The offsets are measured using the EIS spectra themselves, but the
measurements are
made difficult by the fact that solar features appear different in the
various emission lines observed by EIS. This document summarizes
measurements that have been made of the offsets and presents an IDL
routine called \verb|eis_ccd_offset.pro| that is used to give the
offset for any wavelength.

\section{Basic information and notation}

EIS has a single grating that disperses light onto two CCDs, one for
the short wavelength channel (SW; 166--212~\AA) and one for the long
wavelength channel (LW; 246--291~\AA).

Although the grating tilt is the same for both CCDs, the two CCDs
could be tilted relative to each other, meaning that the spectra
observed on the two CCDs could have a different tilt. We therefore
refer to the observed effect (as seen on the CCDs) as ``spectrum
tilt'', with the two CCDs potentially having different spectrum
tilts. Alternative names for this effect are ``spectrum rotation'',
``grating tilt'' and ``slit slant''. Note that the effect is
completely different to the ``slit tilt'' which is described in EIS
Software Note No.~4 and is due to a slight mis-alignment between the
EIS slits and the CCDs.

\section{Definitions}

The spectrum tilt, $T$, is defined as:
%
\begin{equation}
T= { Y_2 -Y_1 \over \lambda_2 - \lambda_1}
\end{equation}
%
where $Y$ is the Y-pixel position of an observed feature on the
detector, and $\lambda$ is the wavelength in which the feature is
observed.  

\ion{He}{ii} \lam256.32 is taken as the reference line and
defined to have a zero Y-offset, and \ion{Fe}{viii} \lam185.21 and
\ion{Si}{vii} \lam275.37 are used to define the offset, $\Delta$, between the two
CCDs.

Therefore the Y-offset for a general wavelength, $\lambda$, is
%
\begin{equation}
Y=\left\{ \begin{array}{l@{\qquad}l}
  T_{\rm SW}(\lambda-185.21) + \Delta & {\rm (SW~channel)}\\
  \\
  T_{\rm LW}(\lambda-256.32) & {\rm (LW~channel)} \end{array} \right.
\end{equation}
%
In reality, the spectrum tilts found for the two CCDs are very similar
and a single tilt value is used for both in the software
implementation (Sect.~\ref{sect.idl}).


\section{Measurements of Y-offsets}

\citet{young09} used solar features observed in \ion{Fe}{viii} and
\ion{Fe}{xii} emission lines to determine the spectrum tilt in the
EIS SW channel. The gradient was found to be $-0.0792$~pixel/\AA. 

\citet{delzanna09} found that the Y-offsets between the endpoints of
both the SW and LW channels were $3.66\pm 0.20$~pixel, after
cross-correlating intensities from lines at different
wavelengths. These offsets convert 
to gradients of $-0.0802$ and $-0.0804$~pixel/\AA\ for the SW and LW
channels, respectively.

Unpublished work of P.R.~Young used images formed from \ion{Si}{vii}
\lam275.37 and \ion{Fe}{viii} \lam185.21 to determine the spatial
offset between the SW and LW channels. An offset of 18.5 Y-pixels
between the two lines was found. An example is shown in
Fig.~\ref{fig.fe8-si7} where some loop footpoints from an observation
from 2009 November 21 23:44~UT are shown. The intensity images from
\lam275.37 and \lam185.21 are usually very similar in appearance, and
the contour plot on the right panel shows that excellent spatial
alignment between the two images is found after a 18.5 pixel
adjustment is made in the Y-direction.

\begin{figure}[h]
\centerline{\epsfxsize=14cm\epsfbox{compare_si7_fe8_offset.eps}}
\caption{The left panel shows an intensity image formed from
  \ion{Si}{vii} \lam275.37, and the right panel an intensity image
  from \ion{Fe}{viii} \lam185.21. A sub-section of the original raster
is shown, highlighting a number of loop footpoints. The raster was
obtained on 2009 November 21 23:44~UT.}
\label{fig.fe8-si7}
\end{figure}


A different approach was applied by \citet{kamio08} who used EIS
observations of the Mercury transit of 2006 November 8. Since
Mercury is unambiguously the same in all wavelengths (it appears as a small
absorption spot against the solar corona), then it can be used to
determine the Y-offsets between different wavelengths. \citet{kamio08}
used slot observations in a number of lines to determine gradients of
$-0.0872$ and $-0.0768$~pixel/\AA\ for the SW and LW channels,
respectively. The work also yielded the offset between the two CCDs
and, expressed as the offset between \lam185.21 and \lam275.37, a
value of 18.5~pixels was found, in excellent agreement with the value
of P.R.~Young.



\section{The IDL routine eis\_ccd\_offset}\label{sect.idl}

This routine is called as
%
\begin{verbatim}
IDL> offset=eis_ccd_offset(wavelength)
\end{verbatim}
%
where \verb|wavelength| is specified in angstroms. The offset is given
in terms of CCD Y pixels, and the reference wavelength is 256.32~\AA,
the position of the \ion{He}{ii} line. 

Giving a wavelength of 195.12~\AA\ yields an offset of 16.2~pixels (as
of 2011 July 21). This means that a solar feature that emits in both
\ion{Fe}{xii} and \ion{He}{ii} will be positioned 16.2~pixels higher, in terms
of CCD Y-coordinates, in \lam195.12 that it is in \ion{He}{ii}.

The first version of \verb|eis_ccd_offset| (circa 2008 August) used the spectrum tilt
gradient of \citet{young09} and assumed that the same gradient applied
to the LW channel. The offset between the two CCDs was taken from the
18.5~pixel offset between \ion{Si}{vii}
\lam275.37 and \ion{Fe}{viii} \lam185.21. Although \citet{delzanna09}
and \citet{kamio08} found slightly different gradients, as described
above, the differences amount to a small fraction of a Y-pixel and so
the original formulation of the Y-offset is retained.

Correctly accounting for the Y-offset between different wavelengths
can be very important for certain EIS analyzes. In particular, the
derivation of densities from lines separated in wavelength, and the
creation of full CCD spectra of compact solar features. Some EIS
software routines automatically correct for the Y-offset. For example,
the routine \verb|eis_mask_spectrum.pro| (see EIS Software Note
No.~16) creates full CCD spectra of user-selected features within a
raster. Also, the routine \verb|eis_density.pro| automatically
corrects intensity maps in different lines when creating a line ratio
map that is then converted to density (see EIS Software Note
No.~15). Both of these routines make use of \verb|eis_ccd_offset.pro|.


\section{Physical offset between the two CCDs}

A spectrum tilt gradient of $-0.0792$~pixel/\AA\ means that, of the
18.5~pixel offset between \ion{Si}{vii}
\lam275.37 and \ion{Fe}{viii} \lam185.21, 7.1 pixels are due to the
grating tilt. The remaining 11.4 pixels are due to a physical offset
in the Y direction between the SW and LW CCDs. The CCD pixel size is
13.5~$\mu$m, therefore the physical offset is 0.15~mm.

\begin{thebibliography}{}

\bibitem[Del Zanna \& Ishikawa(2009)]{delzanna09}
  Del Zanna, G., \& Ishikawa, Y. 2009,
  A\&A, 508, 1517

\bibitem[Kamio \& Hara(2008)]{kamio08}
  Kamio, S. \& Hara, H. 2008,
  Note published on EIS wiki and available at http://msslxr.mssl.ucl.ac.uk:8080/eiswiki/attach/CCDOffset/yoffset.pdf

\bibitem[Young et al.(2009)]{young09}
   Young, P. R., Watanabe, T., Hara, H., \& Mariska, J. T. 2009,
   A\&A, 495, 587



\end{thebibliography}



\end{document}
