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\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. 2}
\vspace{3mm}
\hrule
\noindent Version 0.8 \hfill 9 March 2010
\vspace{2mm}
\hrule

\begin{centerpage}

\begin{center}
{\Large\bf EIS RADIOMETRIC CALIBRATION}\\
\mbox{}\\
\mbox{}\\
John Mariska\\ 
Code 7673\\ 
Naval Research Laboratory\\
Washington, DC 20375\\
USA\\
\mbox{}\\
\verb+John.Mariska@nrl.navy.mil+\\
\mbox{}\\
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\section{Overview}

This document outlines the factors that went into determining the
prefilight radiometric calibration of EIS and discusses the
sensitivity changes that have taken place since Hinode was
launched.

\section{Preflight Calibration Information}

EIS consists of a multilayer-coated, normal-incidence off-axis
telescope and a normal-incidence, toroidal-grating, stigmatic
spectrometer covering two wavelength bands in the EUV.
Figure~\ref{fig:eis_optical_diagram} shows the basic EIS optical
layout.

\begin{figure}[b!] 
\centering
\includegraphics[width=\textwidth,angle=0]{eis_optical_diagram}
\caption{EIS optical diagram.} 
\label{fig:eis_optical_diagram}
\end{figure}

The performance of each optical element was characterized before
the final instrument was assembled. The following sections
summarize the results of that work. In \S\ref{sect:eff_areas} all
the information is combined to create the EIS effective area
curves.

\subsection{Front Filter and Spectrometer Entrance Filter}

To reject visible light, EIS has two 1500~\AA\ thick Al filters
in the optical path.  The front filters are mounted in four
filter frames and protected by a clamshell that can be evacuated
for launch.  The spectrometer entrance filter is located on a
structure bulkhead just past the slit mechanism.  All the filters
sit on a mesh with an estimated transmission of 0.85. 

All the candidate filters were measured at Brookhaven, and flight
filters and spares were then selected. Figure~\ref{fig:al_trans}
shows the measured filter transmission curves. For comparison,
the figures also show the theoretical data that were used earlier
during instrument development. Note that the measured
transmission curves automatically include the mesh transmission
factor that was carried separately in some earlier calculations
of the EIS effective areas.

The filter transmission data are contained in the files
\texttt{EIS\_FFATrans.002} and \texttt{EIS\_SEFTrans.002} in the
directory \texttt{\$SSW/hinode/eis/response}, where
\texttt{\$SSW} is the path to the beginning of the software in
the SSW system.

\begin{figure}
\centering
\begin{minipage}[b]{0.50\textwidth}
\centering\includegraphics[width=\textwidth,angle=0]{ffa}
\end{minipage}\hfill
\begin{minipage}[b]{0.50\linewidth}
\centering\includegraphics[width=\textwidth,angle=0]{sef}
\end{minipage}
\caption{Measured transmissions for one of the the Al front
  filters (SN 10) and the flight spectrometer entrance filter. 
  The solid lines are the theoretical transmission curves that
  were used for earlier calculations multiplied by a factor of
  0.85 to account for the transmission of the filter mesh.} 
\label{fig:al_trans}
\end{figure}

\subsection{Mirror}

The EIS flight mirror (SN-1) if an off-axis paraboloid with a
clear aperture of 150~mm, a focal length, $f$, of 1938.68~mm, and
an off-axis distance to center of 145.5~mm (Reference is Tinsley
Laboratories sheet accompanying the mirror). 

A multilayer coating was applied to each half of the mirror by
Dave Windt. These were then measured at Brookhaven by John Seely,
who provided a detailed report and reflectance curves as a
function of wavelength for each half of the mirror.
Figure~\ref{fig:mirror_refl} shows the measured reflectivities
for the flight mirror. The reflectances are contained in the
files \texttt{EIS\_MirrorRefl\_A.003} and
\texttt{EIS\_MirrorRefl\_B.003} in the directory
\texttt{\$SSW/hionde/eis/response}, where \texttt{\$SSW} is the
path to the beginning of the SSW software system.

\begin{figure}
\centering
\begin{minipage}[b]{0.50\textwidth}
\centering\includegraphics[width=\textwidth,angle=0]{refl_b}
\end{minipage}\hfill
\begin{minipage}[b]{0.50\textwidth}
\centering\includegraphics[width=\textwidth,angle=0]{refl_a}
\end{minipage}
\caption{Measured reflectivities for the short and long
  wavelength bands on the EIS flight mirror.} 
\label{fig:mirror_refl}
\end{figure}

The EIS mirror mechanism has both coarse movement and fine scan
capability.  For coarse movement---to change the position of the
Sun relative to the other instruments---the mirror is shifted in
the $\pm X$-direction.  The mirror travels $\pm 10$~mm in its
coarse motion.  This motion changes the portions of the mirror
that are obstructed by the support spider in the front filter
assembly.  The narrow segments of the spider are 10.5~mm wide, so
as the mirror travels, one side increases in area while the other
decreases.  Once the travel reaches 5.25~mm, this ceases and one
side loses area and the other remains about constant since the
opposite coating starts to appear from under the spider.  This is
illustrated in the left side of Figure~\ref{fig:mirror_coarse_motion}. 

\begin{figure}
\begin{minipage}[b]{0.50\textwidth}
\centering\includegraphics[width=\textwidth,angle=0]{mirror_shift}
\end{minipage}\hfill
\begin{minipage}[b]{0.50\textwidth}
\centering\includegraphics[width=\textwidth,angle=0]{mirror_area}
\end{minipage}
\caption{Left side: EIS mirror displaced by coarse movement behind the
spider.  Right side: EIS mirror area as a function of the coarse
$X$-position.} 
\label{fig:mirror_coarse_motion}
\end{figure}

A small amount of vignetting due to obstruction by the doghouse
surrounding the CCDs is also found in the long wavelength band as
the mirror moves toward $-X$.  Charlie Brown has calculated all
the vignetting on the telescope side of EIS. 
Table~\ref{tab:telescope_vignetting} and the plot on the right
side of Figure~\ref{fig:mirror_coarse_motion} summarizes these
calculations.  These area changes are not currently included in
any calculations of the EIS effective area.  Instead, the mirror
area of 8835.7 mm$^2$ is reduced by a spider transmission factor
of 0.8.

\begin{table}
\centering
\caption{EIS Mirror Area for Changing $X$-positions}
\label{tab:telescope_vignetting}
\begin{tabular}{rrrrrr}
\hline
$X$ & A1 & A2 & Doghouse & SW Area & LW Area \\
mm & mm$^2$ & mm$^2$ & mm$^2$ & mm$^2$ & mm$^2$ \\
\hline
10 & 5755.4 & 7920.7 & & 7920.7	& 5755.4 \\
9 & 5889.7 & 7914.9 & & 7914.9 & 5889.7 \\
8 & 6024.4 & 7901.4 & & 7901.4 & 6024.4 \\
7 & 6159.6 & 7879.2 & & 7879.2 & 6159.6 \\
6 & 6295.2 & 7850.4 & & 7850.4 & 6295.2 \\
5 & 6431.2 & 7790.0 & & 7790.0 & 6431.2 \\
4 & 6567.5 & 7656.6 & & 7656.6 & 6567.5 \\
3 & 6704.2 & 7522.4 & & 7522.4 & 6704.2 \\
2 & 6841.1 & 7387.5 & & 7387.5 & 6841.1 \\
1 & 6978.3 & 7251.7 & & 7251.7 & 6978.3 \\
0 & 7115.3 & 7115.3 & & 7115.3 & 7115.3 \\
$-1$ & 7251.7 & 6978.3 & $-0.86$ & 6978.3 & 7250.8 \\
$-2$ & 7387.5 & 6841.1 & $-4.32$ & 6841.1 & 7383.2 \\
$-3$ & 7522.4 & 6704.2 & $-10.6$ & 6704.2 & 7511.8 \\
$-4$ & 7656.6 & 6567.5 & $-29.3$ & 6567.5 & 7627.3 \\
$-5$ & 7790.0 & 6431.2 & $-57.3$ & 6431.2 & 7732.7 \\
$-6$ & 7850.4 & 6295.2 & $-96.4$ & 6295.2 & 7754.0 \\
$-7$ & 7879.2 & 6159.6 & $-140.0$ & 6159.6 & 7739.2 \\
$-8$ & 7901.4 & 6024.4 & $-186.6$ & 6024.4 & 7714.8 \\
$-9$ & 7914.9 & 5889.7 & $-241.9$ & 5889.7 & 7673.0 \\
\hline
\end{tabular}
\end{table}

The position of the mirror in its coarse motion is measured with
a resolver that has a range from 0x4195 at the $-X$ limit to
0x1408 at the $+X$ limit with a wraparound at 0xFFFF, i.e.,
0xFFFF, 0, 1 to $-X$. Travel from the $+X$ direction toward the
$-X$ direction is denoted as the forward direction. The resolver
numbers are twos-complement. The relationship between the
resolver measurement and the position of the mirror has been
measured. The relationship between the current position of the
mechanism and the table has not been determined.

\subsection{Slit/Slot}

The EIS slits were fabricated using a Si substrate and a
photolithography process.  Slit widths were measured at the left
center and right side of each slit.  Average values for the
widths in $\mu$m of the flight slits are 9.3, 19.2, 384, and 2507
for the 1, 2, 40, and 266 arcsec slits, respectively. 

The slits are attached to a four vaned paddlewheel located
immediately behind the shutter.  It is possible to move the
slit/slot mechanism in either the forward or reverse direction. 
To avoid damage with time, however, NRL and Perdix recommended
that the mechanism only be moved in the forward direction (vane
through which sunlight passes moves toward the grating).  The
order of the slits on the mechanism is 1~arcsec, 266~arcsec,
2~arcsec, and 40~arcsec.  The slit/slot disk rotates at a rate of
20~ms per step and a $90^\circ$ rotation takes 324 steps.  Thus,
it takes 6.48~s for a $90^\circ$ rotation of the mechanism.

\subsection{Grating}

The EIS flight grating comes from a family of toroidal gratings
fabricated by Zeiss. The flight candidate gratings were ruled
with 4200~l/mm and a multilayer coating was applied to each half
by Dave Windt. The gratings were then measured at Brookhaven.
Details of the various measurements that were carried out on the
grating are contained in a series of documents by John Seely and
others on the NRL team. Figure~\ref{fig:grating_eff} shows the
resulting grating efficiencies (a combination of the multilayer
reflectivities and the groove efficiency) for the current flight
grating (FL-7). The measured efficiencies are contained in the
files \texttt{EIS\_GratingRefl\_A.003} and
\texttt{EIS\_GratingRefl\_B.003} in the directory
\texttt{\$SSW/hinode/eis/response}, where \texttt{\$SSW} is the
path to the beginning of the SSW software system.

\begin{figure}
\begin{center}
\begin{minipage}[b]{0.50\textwidth}
\centering\includegraphics[width=\textwidth,angle=0]{eff_b}
\end{minipage}\hfill
\begin{minipage}[b]{0.50\textwidth}
\centering\includegraphics[width=\textwidth,angle=0]{eff_a}
\end{minipage}
\caption{Measured efficiencies for the short and long
  wavelength bands on the EIS flight grating.} 
\label{fig:grating_eff}
\end{center}
\end{figure}

Due to the tight clearances between the slit/slot assembly and
the light beams from the EIS mirror, it was necessary to accept
some vignetting at the long wavelength end of the long wavelength
band detector by the doghouse enclosure that keeps the slit/slot
assembly light tight. This amount of vignetting has been computed
by Charlie Brown. Figure~\ref{fig:grating_vignetting} shows a
plot of the vignetting factor for the case where the EIS mirror
coarse motion is at the zero position along with a table of
values. The numbers used to generate the figure are contained in
the file \texttt{EIS\_vignetting.001} in the directory
\texttt{\$SSW/hinode/eis/response}, where \texttt{\$SSW} is the
path to the SSW software system. These factors are used in
computing the EIS effective area for the long wavelength band.

% \begin{table}
% \captionstyle{\raggedright}
% \caption{Vignetting Data for Doghouse}
% \label{tab:grating_vignetting}
% \begin{tabular}{lc}
% \hline
% Wave & Vignetting Factor \\
% \hline
% 270.0 & 1.000 \\
% 272.0 & 1.000 \\
% 274.0 & 1.000 \\
% 276.0 & 0.993 \\
% 278.0 & 0.966 \\
% 280.0 & 0.920 \\
% 282.0 & 0.865 \\
% 284.0 & 0.804 \\
% 286.0 & 0.738 \\
% 288.0 & 0.667 \\
% 290.0 & 0.596 \\\
% hline
% \end{tabular}
% \end{table}

\begin{figure}
\begin{minipage}{0.40\textwidth}
\begin{tabular}{lc}
\hline
Wave & Vignetting Factor \\
\hline
270.0 & 1.000 \\
272.0 & 1.000 \\
274.0 & 1.000 \\
276.0 & 0.993 \\
278.0 & 0.966 \\
280.0 & 0.920 \\
282.0 & 0.865 \\
284.0 & 0.804 \\
286.0 & 0.738 \\
288.0 & 0.667 \\
290.0 & 0.596 \\
\hline
\end{tabular}
\end{minipage}\hfill
\begin{minipage}{0.60\textwidth}
\includegraphics[width=\textwidth,angle=0]{vignetting.ps}
\end{minipage}
\caption{Vignetting factor due to doghouse.} 
\label{fig:grating_vignetting}
\end{figure}

\subsection{Camera}

The EIS camera contains two Marconi 42-20 back-illuminated CCDs
with 2048 pixels in the dispersion direction and 1024 pixels in
the spatial direction. In both directions, the pixel size is
13.5~$\mu$m. In the dispersion direction, the CCDs have 50
lead-in pixels at each end of the serial register. Data from the
testing and calibration at RAL has those pixels included. (This
may in the future lead to some confusion about pixel numbering in
the dispersion direction.) The read time for the CCD is 0.8~s.

Engineering models of the CCDs were measured at Brookhaven. For
one CCD (EM443-02-07), the average quantum efficiencies at 190
and 270~\AA, were 0.61 and 0.57, respectively. For the other CCD
(EM443-01-04), the corresponding values were 0.52 and 0.48. After
looking at additional information it has been agreed within the
EIS MSSL team that we will use a constant QE of 0.64.

The EIS camera operates in an integrating mode. The shutter is
opened for a short interval and photon events are summed into the
CCD pixels. Thus the amount of charge read out of the CCD by the
camera provides a measure of the energy that has been integrated
in each pixel and not the actual number of photons counted. Since
EIS is a spectrometer, however, when the 1 or 2~arcsec slit is
used, the photons striking each pixel are essentially
monoenergetic, and the amount of accumulated charge is directly
proportional to the photon flux.

EUV photons striking the CCD produce electron-hole pairs in the
silicon. It takes 3.65~eV to produce an electron-hole pair, so
the number of electrons produced by a photon of wavelength
$\lambda$ (\AA) is given by
%
\begin{equation}
N_{e^-} = \frac{12398.5}{\lambda} \times \frac{1}{3.65}\ . 
\end{equation}
%
Thus, at 195~\AA, one photon produces 17.4 electrons, at 270~\AA,
one photon produces 12.6 electrons, and at 290~\AA---the longest
wavelength covered by the spectrometer---one photon produces 11.7
electrons.  The EIS CCDs have a full-well capacity of about
90,000 electrons. 

The charge accumulated in each pixel is converted by 14-bit
analog-to-digital converters into data numbers (DN) using the
expression
%
\begin{equation}
\mathrm{DN} = N_{e^-}/G + \mathrm{offset}\ ,
\end{equation}
where $G$ is the gain and a digital offset is added.  The value
of $G$ is currently measured to be 6.93~electrons~DN$^{-1}$.
Current values of the offset seem to be between 500 and 550 DN
depending on which CCD and which port the data are read from.

In addition to the photon-shot noise, the EIS camera has
quantization and readout noise. The design criteria for the EIS
camera called for it to have a total signal chain noise
comparable to the photon noise, leading to a requirement for a
total noise level of 10 electrons RMS. For the lowest signal
levels, this puts the camera noise just below the shot noise for
a photon at 290~\AA. Measurements of the flight model camera show
that the noise level is in the range 10.1 to 13.5 electrons. With
the value of $G$ set to 5.65, this corresponds to 1.8 to 2.4~DN.

\section{EIS Effective Areas\label{sect:eff_areas}}

For an ideal optical system with no losses, the number of photons
of wavelength $\lambda$ entering the slit per second in a height
interval corresponding to one pixel in the spatial dimension on
each CCD, $N_\lambda^{\mathrm{slit}}$, is given by
%
\begin{equation}
N_\lambda^{\mathrm{slit}} = \phi_\lambda A \frac{a}{f^2},
\end{equation}
%
where, $\phi$ (photons cm$^{-2}$ s$^{-1}$ sr$^{-1}$) is the
intensity of the solar radiation, $A$ (cm$^2$) is the mirror
area, $a$ (cm$^2$) is the area of the slit corresponding to one
spatial CCD pixel in height times the width of slit, and $f$ is
the focal length of the mirror.  These photons are then imaged
onto the CCD in a manner determined by the magnification in the
spectrograph section of the instrument.  Thus, the number of
photons per second registered on a pixel of each CCD is given by
%
\begin{equation}
N_\lambda = \phi_\lambda A
\frac{a}{f^2}\frac{1}{n_{\mathrm{pix}}},
\end{equation}
%
where the division by $n_{\mathrm{pix}}$ provides the desired
solid angle per detector pixel.  The design goal was to have the
narrow slit match the pixel size.  But the actual slit widths
result in small corrections.  For the narrow slit (nominally 1
arcsec wide), $n_{\mathrm{pix}}$ is 1.067 and 1.087 for detectors
A and B, respectively.  The corresponding values for the 2 arcsec
slit are 2.080 and 2.119.  Note that the combination of factors
results in the same solid angle per detector pixel,
$\omega_{\mathrm{d}}$. 

Of course, EIS is not an ideal optical system.  The actual number
of registered photons will be reduced by additional factors due
to the transmission of the two aluminum filters, the reflectivity
of the mirror, the efficiency of the grating, and the quantum
efficiency of the detectors.  Thus, the basic expression for the
number of photons registered in each detector pixel per second is
%
\begin{equation}
N_\lambda = \phi_\lambda A \omega_{\mathrm{d}}
T_{\mathrm{ffa}}(\lambda) T_{\mathrm{spider}}
R_{\mathrm{m}}(\lambda) E_{\mathrm{g}}(\lambda)
V_{\mathrm{d}}(\lambda) T_{\mathrm{sef}}(\lambda)
E_{\mathrm{d}}(\lambda),
\end{equation}
%
where $T_{\mathrm{ffa}}(\lambda)$ and $T_{\mathrm{sef}}(\lambda)$
are the transmissions of the aluminum front filter assembly and
spectrometer entrance filter, respectively; $T_{\mathrm{spider}}$
is the transmission of the spider that is part of the front
filter assembly; $R_{\mathrm{m}}(\lambda)$ is the reflectivity of
the mirror coatings; $E_{\mathrm{g}}(\lambda)$ is the grating
efficiency, which includes both the groove efficiency and the
reflectivity of the multi-layer coatings,
$V_{\mathrm{d}}(\lambda)$ is is a vignetting factor to account
for the vignetting at the long wavelength end of the long
wavelength detector; and $E_{\mathrm{d}}(\lambda)$ is the
detector quantum efficiency.  Note that the Al filter
transmission factors are a combination of the transmission of the
thin Al material and a factor $T_{\mathrm{mesh}}$, the
transmission of the mesh that supports both aluminum filters. 
This value, estimated to be 0.85, was carried separately in early
calculations of the EIS sensitivity, but is now included in the
measured filter data. 

All the factors apart from the solid angle are often grouped into
an effective area $A_{\mathrm{eff}}(\lambda)$, leading to the
expression
%
\begin{equation}
N_\lambda = \phi_\lambda A_{\mathrm{eff}}(\lambda)
\omega_{\mathrm{d}}. 
\label{eq:photons_per_pixel}
\end{equation}
%
Figure~\ref{fig:eis_eff_areas} shows the effective areas for the
two EIS wavelength bands. 
%
\begin{figure}[tb!] 
\centering
\includegraphics[width=\textwidth,angle=0]{eff_area}
\caption{EIS effective areas.} 
\label{fig:eis_eff_areas}
\end{figure}

\section{EIS Radiometric Calibration at RAL}

\section{EIS Sensitivity Changes on Orbit}

In December 2006, NRL initiated a program of regular observations
of quiet Sun regions near Sun center to monitor EIS sensitivity
changes. The initial studies, SYNOP001 and SYNOP002 took
exposures covering all the wavelengths covered by the CCDs. After
the X-band transmitter failure early in 2008, these were replaced
with studies that only covered selected emission lines
(SYNOP005\_A and SYNOP006).



\end{document}
