% Thesis: Chapter 2.
\chapter{The WHAM Spectrometer: Theory and Design}
%\footnotetext{Mention collaborators here.}
\clearpage
\section{Introduction}
The Wisconsin H-Alpha Mapper (WHAM) is a unique new instrument for the
detection of faint emission lines from diffuse ionized gas in the disk
and halo of the Galaxy. WHAM consists of a 15 cm aperture,
dual-etalon Fabry-Perot spectrometer coupled to a 0.6 m siderostat.
This provides a one degree diameter beam on the sky and produces a 12
km s$^{-1}$ resolution spectrum over a 200 km s$^{-1}$ interval that
can be selected at any wavelength within the range 4800 $\AA$ to 7200
$\AA$. In this chapter I present the basic theory behind Fabry-Perot
systems and describe how the WHAM spectrometer was designed to exploit
the properties of the Fabry-Perot for the purpose of high-resolution
spectroscopy of emission lines from diffuse ionized gas. I include a
description of our method of employing Fabry-Perots, which involves
the use of dual-etalons in series, pressure tuning, and imaging of the
ring pattern passed by the etalons onto a CCD detector.
\section{Introduction to Fabry-Perot Spectroscopy}
In this section the basic theory behind Fabry-Perot systems is
described. I first derive the wavelength dependent transmission
function of the Fabry-Perot and then discuss some of the important
spectroscopic properties of this device.
\subsection{The Transmission Function of a Fabry-Perot}
The heart of a Fabry-Perot system consists of two accurately parallel
reflecting surfaces with some spacing, $l$, between them. This is
often accomplished by depositing a reflective coating on one side of
two disks of fused silica, and then separating the reflecting surfaces
by spacers of precisely identical length. Such an arrangement is
referred to as a Fabry-Perot `etalon'. Imagine that this system is
illuminated from one side as in Figure~\ref{fig2-fabry}. The basic
idea is that multiple reflections occur between the reflective
surfaces and the interference between these reflections gives rise to
the unique wavelength dependent reflection and transmission properties
of this device.
A brief digression is in order here to derive the expression for the
the phase lag, $\phi$, due to the optical path difference between two
consecutive transmitted rays. Figure~\ref{fig2-pathdiff} shows an
etalon system with an index of refraction $n_{g}$ in the gap, immersed
in a medium with index n, and illuminated from the left. Also
depicted is a ray representing the portion of the incoming wave that
is transmitted directly (labeled 1), and one that represents the
portion of the incoming ray that is transmitted after two internal
reflections (labeled 2). Further internal reflections result in an
identical optical path difference between consecutive transmitted
rays.
Let us derive the phase lag between rays 1 and 2 in
Figure~\ref{fig2-pathdiff}. In general,
\begin{equation}
\label{eq2-phidef}
\phi = \frac{2 \pi}{\lambda} \Delta L_{opt},
\end{equation}
where $\Delta L_{opt}$ is the optical path difference between the two rays.
From Figure~\ref{fig2-pathdiff} it is apparent that
\[ \Delta L_{opt} = n_{g}(AB + BC) - n(AD). \]
But,
\[ (AB + BC) = \frac{2 l}{\cos \theta_{t}} \]
and,
\[ AD = (2 l \tan \theta_{t}) (\sin \theta_{i}). \]
Hence,
\[
\Delta L_{opt} = n_{g} \frac{2 l}{\cos \theta_{t}} -
n (2 l \tan \theta_{t}) (\sin \theta_{i}).
\]
Using Snell's law, $n \sin \theta_{i} = n_{g} \sin \theta_{t}$, we have
\begin{eqnarray*}
\Delta L_{opt} &=& 2 l n_{g}\left( \frac{1}{\cos \theta_{t}} -
\frac{\sin^{2} \theta_{t}}{\cos \theta_{t}} \right) \\
&=& 2 n_{g} l \cos \theta_{t}.
\end{eqnarray*}
Then, using Equation~\ref{eq2-phidef}, we have
\begin{equation}
\label{eq2-phi}
\phi = \frac{4 \pi n_{g} l}{\lambda} \cos \theta_{t}.
\end{equation}
Let us now derive the transmission function, $I_{t}$, for the simple
configuration drawn in Figure~\ref{fig2-fabry}, where we have two
surfaces separated by a gap of length $l$ and index of refraction
n$_{g}$ immersed in a material of index n. We begin by calculating
the amplitude transmission function, $a_{t}$. A plane wave of
amplitude $a_{\rm o} \exp[i(\omega \tau + kx + \alpha)]$ is incident
at an angle $\theta_{i}$ to the surface normal, where $\alpha$ is the
phase advancement necessary to make the phase of the directly
transmitted ray equal to zero at the plane where the transmitted
intensity will be evaluated (represented by the dotted line in
Figure~\ref{fig2-fabry}). Let r be the amplitude fraction that is
reflected at the surface, and t be the amplitude fraction that is
transmitted. Note that $\tau$ is used here to represent the time to
avoid confusion with the transmission coefficient, t.
The amplitude of the wave that is transmitted directly is $a_{\rm o}
t^2 \exp(i \omega \tau)$. After two internal reflections a wave of
amplitude $a_{\rm o} t^2 r^2 \exp(i(\omega \tau - \phi))$ is also
transmitted. Similarly, there is a wave that suffers four internal
reflections with amplitude $a_{\rm o} t^2 r^4 \exp\{i(\omega \tau - 2
\phi)\}$ that is transmitted. In fact, there are an infinite number
of successively weaker reflections that are transmitted, and it is the
sum of these interfering waves that determine the amplitude of the
transmitted wave. To wit, if we call the transmitted wave $a_{t}$,
\begin{eqnarray*}
a_{t} &=& a_{\rm o} t^2 \exp(i \omega \tau) +
a_{\rm o} t^2 r^2 \exp[i(\omega \tau - \phi)] +
a_{\rm o} t^2 r^4 \exp[i(\omega \tau - 2 \phi)] + \cdots \\
&=& a_{\rm o} t^2 \exp(i \omega \tau) [1 + r^2 \exp(-i \phi) +
r^4 \exp(-2i \phi) + \cdots].
\end{eqnarray*}
Let $x \equiv r^2 \exp(-i \phi)$. Then,
\[ a_{t} = a_{\rm o} t^2 \exp(i \omega \tau) \sum_{j=0}^{\infty} x^{j}. \]
But,
\[ \sum_{j=0}^{\infty} x^{j} = \frac{1}{1-x}. \]
Hence,
\[ a_{t} = \frac{a_{\rm o} t^2 \exp(i \omega \tau)}{1 - r^2 \exp(-i \phi)}. \]
To get the transmitted intensity, $I_{t}$, we must multiply
$a_{t}$ by its complex conjugate,
\begin{eqnarray*}
I_{t} &=& a_{t} a_{t}^* \\
&=& \frac{a_{\rm o}^{2} t^4}{1 - r^{2}\{\exp(i \phi)
+ \exp(-i \phi)\} + r^{4}} \\
&=& \frac{a_{\rm o}^{2} t^4}{1 - 2 r^{2} \cos \phi
+ r^{4}}.
\end{eqnarray*}
Therefore, with the identity $\cos \phi = (1 - 2 \sin^{2}\frac{\phi}{2})$,
we have
\begin{eqnarray*}
I_{t} &=& \frac{a_{\rm o} t^4}{1 - 2 r^2 + 4 r^{2} \sin^{2}\frac{\phi}{2} +
r^{4}} \\
&=& \frac{a_{\rm o} t^4}{(1 - r^2)^2 +
4 r^{2} \sin^{2}\frac{\phi}{2}} \\
&=& \frac{a_{\rm o} t^4}{(1 - r^2)^2}
\frac{1}{(1 + \frac{4 r^2}{(1-r^2)^2} \sin^{2}\frac{\phi}{2})}.
\end{eqnarray*}
Define $R \equiv r^2$, $T \equiv t^2$, and $A$ such that $T + R +
A = 1$. These are called the reflectance, transmittance, and
absorbtance, respectively, of the reflecting surface. Then,
\begin{equation}
\label{eq2-it1}
\frac{I_{t}}{I_{\rm o}} = \frac{T^{2}}{(1-R)^2} \mathcal{A}(\phi),
\end{equation}
where $I_{\rm o}$ is the incoming intensity and where we have made use
of the definition of the Airy function, $\mathcal{A}(\phi)$,
\begin{equation}
\label{eq2-airy}
\mathcal{A(\phi)} \equiv \mathit{\frac{1}
{1 + \frac{4 R}{(1-R)^2} \sin^{2}\frac{\phi}{2}}}.
\end{equation}
This may also be written
\begin{equation}
\label{eq2-it2}
\frac{I_{t}}{I_{\rm o}} =
\left( 1 - \frac{A}{(1-R)} \right)^2 \mathcal{A}(\phi).
\end{equation}
In the special case where $A = 0$ and thus $T = 1 - R$, we have
\begin{equation}
\label{eq2-it3}
\frac{I_{t}}{I_{\rm o}} = \mathcal{A}(\phi).
\end{equation}
A very similar analysis can be performed to derive the wavelength
dependence of the \emph{reflected} light. The main difference is that the
first reflection is $\pi$ out of phase with all the subsequent
reflections because it is an external reflection. The result, in the
case of no absorption, is
\begin{equation}
\label{eq2-ir}
\frac{I_{r}}{I_{\rm o}} = \frac{\frac{4R}{(1-R)^2} \sin^{2}\frac{\phi}{2}}
{1 + \frac{4 R}{(1-R)^2} \sin^{2}\frac{\phi}{2}}.
\end{equation}
As I will show below, it is desirable to make R fairly high for good
resolution and contrast, and Equation~\ref{eq2-it2} shows that good
peak transmission therefore requires A to be quite low. It wasn't
possible to make a reflecting surface with both high R and low A until
the development of techniques for fabricating dielectric multi-layer
coatings (circa 1950). This explains why it is that even though the
Fabry-Perot concept is quite old (one was constructed by Charles Fabry
and Alfred Perot in the late 1800s), they have not been used
extensively until fairly recently.
\subsection{Some Important Spectroscopic Properties of the Fabry-Perot}
Figure~\ref{fig2-airy} shows a graph of the wavelength dependent
transmission function for an ideal Fabry-Perot system with R = 0.90,
and Figure~\ref{fig2-airy2} shows the reflection function.
The peaks in the transmission function occur when $\sin^{2} (\phi /
2)$ is at a minimum, which is at $\phi = 0, 2\pi, 4\pi,
\ldots$ Thus, the criterion for maximum transmission is
\begin{equation}
\label{eq2-maxcrit1}
2 n_{g} l \cos \theta_{t} = m \lambda, {\rm \ where \ } m = 0,
1, 2, \ldots
\end{equation}
It is often convenient to express things in terms of the reciprocal
wavelength, $\sigma = 1 / \lambda$. Our last equation then becomes
\begin{equation}
\label{eq2-maxcrit2}
2 n_{g} l \cos \theta_{t} \sigma = m {\rm \ where \ } m = 0,
1, 2, \ldots
\end{equation}
As is apparent in Figure~\ref{fig2-airy} and in
Equation~\ref{eq2-maxcrit1}, the Fabry-Perot transmission is a periodic
function, and for a given $n_{g}$, $l$, and $\cos \theta_{t}$, there
will be multiple orders passed. Different orders are labeled by the
index m and correspond to different numbers of wavelengths fitting
into the path difference. Since the Fabry-Perot spectroscopist will
be counting photons in one way or another, and cannot distinguish
between photons from different orders, it is usually necessary to
block all but a single order. An important parameter in a Fabry-Perot
is the distance to the next order, which is called the free spectral
range (it is the spectral range that is free of confusion from other
orders) and is denoted $Q$ (see Figure~\ref{fig2-airy}). A subscript
is used to indicate which parameter is being considered, so for
instance, Q$_{\phi}$ is the amount by which $\phi$ needs to change to
reach the next order. From Equation~\ref{eq2-airy} it is clear that
Q$_{\phi}$ = 2$\pi$. It is also interesting to know
$Q_{\sigma}$. Consider two adjacent orders:
\begin{eqnarray*}
2 n_{g} l \cos \theta_{t} \sigma_{1} &=& m_{1}, \hspace{0.12in}{\rm and} \\
2 n_{g} l \cos \theta_{t} \sigma_{2} &=& m_{2}.
\end{eqnarray*}
Subtracting these two equations gives
\[ 2 n_{g} l \cos \theta_{t} (\sigma_{1} - \sigma_{2}) = m_{1} - m_{2}. \]
But, we assumed the orders were adjacent implying $m_{1} - m_{2} = 1$, and
hence
\[ \sigma_{1} - \sigma_{2} = \frac{1}{ 2 n_{g} l \cos \theta_{t}}. \]
Therefore, on axis, where $\cos \theta_{t} \simeq 1$, we have
\begin{equation}
\label{eq2-fsr}
Q_{\sigma} = \frac{1}{2 n_{g} l}.
\end{equation}
As I will discuss below, one of the main reasons to employ two etalons
in series is to extend the free spectral range of the system.
The spectral resolution achievable with a Fabry-Perot is simply
related to the width of the transmission peaks shown in
Figure~\ref{fig2-airy}. Let $(\Delta \phi)_{FWHM}$ denote the full
width at half maximum in $\phi$ of the transmission function. In the
ideal Fabry-Perot, the half maximum occurs where
\[ \frac{1}{1 + \frac{4 R}{(1-R)^2} \sin^2 \frac{\phi_{1/2}}{2}} = 1/2, \]
where $\phi_{1/2}$ is the value of $\phi$ at which the transmission function
is 0.5 times the peak value. Using $\sin (\phi_{1/2}/2)
\simeq \phi_{1/2} / 2$, we have
\[ \frac{4R}{(1 - R)^2} \frac{\phi_{1/2}^2}{4} = 1, \]
and thus
\[ \phi_{1/2} = \frac{(1-R)}{\sqrt{R}}. \]
But, the full width at half maximum is just twice $\phi_{1/2}$, so
\begin{equation}
\label{eq2-deltaphi}
(\Delta \phi)_{FWHM} = \frac{2(1-R)}{\sqrt{R}}.
\end{equation}
A number that is commonly used to characterize a Fabry-Perot system is
the number of spectral elements in a free spectral range. This is
referred to as the reflective finesse (or often just the finesse),
$\mathcal{F}$, where
\begin{equation}
\mathcal{F} = \mathit{Q_{\phi} / (\Delta \phi)_{FWHM}}.
\end{equation}
Using $Q_{\phi} = 2 \pi$ and Equation~\ref{eq2-deltaphi}, we have
\begin{equation}
\mathcal{F} = \mathit{\frac{\pi \sqrt{R}}{(1-R)}}.
\end{equation}
Figure~\ref{fig2-finesse} shows a graph of the finesse versus R.
Another important property of the Fabry-Perot (or for any spectrometer
for that matter) is the resolving power, $\mathcal{R}$, defined by
\begin{equation}
\label{fig2-Rdef}
\mathcal{R} \mathit{\equiv \frac{\lambda}{(\Delta \lambda)_{min}}},
\end{equation}
where $(\Delta \lambda)_{min}$ is the smallest resolvable wavelength
difference. There are various specific ways of defining $(\Delta
\lambda)_{min}$, but for convenience we will take it to be the full
width at half maximum (FWHM) of the transmission function. We call
the resolving power for the ideal Fabry-Perot the `theoretical
resolving power', $\mathcal{R}_{\rm o}$. The resolution of a real
Fabry-Perot system will be less than $\mathcal{R}_{\rm o}$ due to
effects such as non-flatness of the plates (see
Sections~\ref{sec2-resolution} and \ref{sec2-CCD}). From
Equation~\ref{eq2-phi} it is clear that
\begin{equation}
\frac{\phi}{\Delta \phi} = \frac{\sigma}{\Delta \sigma} =
\frac{\lambda}{\Delta \lambda}.
\end{equation}
Using Equation~\ref{eq2-phi} and Equation~\ref{eq2-deltaphi}, we have
\begin{equation}
\label{eq2-respow}
\mathcal{R}_{\rm o} = \mathit{\frac{\phi}{\Delta \phi} =
\frac{\pi \sqrt{R}}{(1-R)} 2 n_{g} l \cos \theta_{t} \sigma},
\end{equation}
or, more compactly,
\begin{equation}
\label{eq2-respow2}
\mathcal{R}_{\rm o} = \mathcal{F} \mathit{m},
\end{equation}
where m is the order number.
Another important spectroscopic property is the contrast, $C$,
the ratio of the maximum transmission to the minimum
transmission of the spectrometer,
\begin{equation}
\label{eq2-contrastdef}
C = \frac{I_{t}(max)}{I_{t}(min)}.
\end{equation}
I will derive the contrast for the case of a single etalon with no
absorption. The transmission maxima occur where the $\sin^2(\phi /
2)$ term in Equation~\ref{eq2-it3} is zero,
\begin{equation}
I_{t}(max) = I_{\rm o}.
\end{equation}
This is an amazing thing: one can shine light onto two 90\% reflecting
surfaces and obtain nearly 100\% transmission (but only for certain
wavelengths). The minimum occurs where the $\sin^2(\phi / 2)$ term is
unity,
\begin{equation}
I_{t}(min) = \frac{I_{\rm o}}{1 + \frac{4R}{(1-R)^2}}.
\end{equation}
And hence, the contrast for the ideal Fabry-Perot with no absorption
is
\begin{equation}
\label{eq2-contrast}
C = 1 + \frac{4R}{(1-R)^2} = \frac{(1+R)^2}{(1-R)^2},
\end{equation}
or, in terms of the finesse,
\begin{equation}
\label{eq2-contrast2}
C = 1 + \frac{4 \mathcal{F}^{\mathit 2}}{\mathit{\pi^2}}.
\end{equation}
In a real Fabry-Perot system, the contrast will be somewhat degraded
due to scattered light. The contrast is important to us for two
reasons: 1) It determines how much continuum light gets through off
line and thus affects the background level and hence the noise, and 2)
it determines the strength of the ghost peaks in the transmission
function which are passed by one etalon but rejected by the other (see
below).
\subsection{The Throughput Advantage of Fabry-Perots Over Grating
Spectrometers for Diffuse Sources of Emission}
\label{sec2-etendue}
The main advantage of using a Fabry-Perot spectrometer instead of a
diffraction grating based spectrometer is that the Fabry-Perot is
capable of a much higher throughput at high spectral resolution than
is a grating system. This advantage arises from the ability of the
Fabry-Perot system to pass a large solid angle, $\Omega$, of light
into a small spectral element size, $d\lambda$. Consider the basic
relations governing the two types of spectrometers:
Fabry Perot:
$\lambda \propto \cos \theta$
Diffraction Grating:
$\lambda \propto \sin \theta$ \\
The crux of the issue lies in the difference between the behavior of
the $\cos \theta$ function and the $\sin \theta$ function for small
$\theta$. For a given wavelength interval, d$\lambda$, the magnitude
of the opening angle d$\theta$ is given by:
Fabry-Perot:
$d\theta \propto \frac{1}{\sin \theta}$ $d\lambda$
Grating:
$d\theta \propto \frac{1}{\cos \theta}$ $d\lambda$. \\
Hence, for a given $d\lambda$ a much larger $d\theta$ is allowed in the
case of the Fabry-Perot.
This throughput advantage is only realized if the source of emission
is extended and thus able to fill the large $d\theta$ allowed by the
Fabry-Perot. For a star, the solid angle passed by the system is
usually limited by the source and not the spectrometer, and thus the
diffraction grating is no worse off than the Fabry-Perot. But for
diffuse emission from the ISM, the solid angle advantage of the
Fabry-Perot can be enormous; gains of more than 100 times over a
diffraction grating operated at the same spectral resolution are not
atypical.
\section{Using Fabry-Perots for High-Resolution Interstellar \\
Spectroscopy}
The ultimate goal of the spectroscopist is to measure
the intensity of a source as a function of wavelength. In this
section I will examine how we have exploited the properties of the
Fabry-Perot described above to construct a spectrometer optimized for
high-resolution spectroscopy of faint optical emission lines from the
diffuse interstellar medium.
\subsection{Choosing and Setting the Resolution}
\label{sec2-resolution}
The desired resolution of a spectrometer is determined by the
characteristics of the source under study and the objectives of the
study. The WHAM spectrometer is designed to study emission lines
originating in the interstellar medium. The natural width of these
lines is very small, and the profile shape arises from the motions of
the gas atoms, both thermal motions and bulk motions, through the
Doppler effect. For this reason we often consider the abscissa of our
spectra a velocity scale instead of a wavelength scale. Emission
lines from the warm ionized component of the interstellar medium
typically have widths of 15 -- 50 km s$^{-1}$, and the bulk motions of
the gas are usually within $\pm$~100 km s$^{-1}$ of the earth's
velocity. A resolution of 12 km s$^{-1}$ is adequate to separate the
interstellar emission from any terrestrial emission and thus probe the
kinematics of the emitting gas, as well as to probe line-broadening
mechanisms, both thermal and non-thermal, through determinations of
the line widths. We set our theoretical resolving power to achieve a
velocity resolution of 8 km s$^{-1}$ in order to allow for other
instrumental broadening effects described below. The wavelength
change due to the Doppler shift is governed by the formula
\begin{equation}
\label{eq2-doppler}
\frac{\Delta \lambda}{\lambda} = \frac{v}{c},
\end{equation}
where v is the velocity of the emitter with respect to the observer,
and c is the velocity of light. Hence, our required velocity
resolution of 8 km s$^{-1}$ leads to a resolving power,
$\mathcal{R}_{\rm o}$ = 37500.
From Equation~\ref{eq2-respow}, we see that the two ways of adjusting
the resolution of the Fabry-Perot etalon are through the reflectivity
of the etalon surfaces, $R$, and through the gap spacing between the
plates, $l$. Since increasing $l$ also decreases the free spectral
range, it would seem preferable to increase R to reach high
resolution. However, the tolerance for errors in the flatness of the
plates, and for nonparallelness of the plates decreases with
increasing R. Therefore, the design strategy for reaching high
resolution is to acquire the flattest plates that can be afforded (or
that can be manufactured), set R as high as possible without letting
the resolution be degraded by the non-flatness of the plates, and then
set $l$ to achieve the desired resolving power. What non-flatness can
be tolerated for a given R? For on-axis rays, a change in the gap
spacing, $l$, by $\lambda / 2$ moves the transmitted wavelength to the
next order, that is, $Q_{l}$ = $1 / (2 \lambda)$. Hence, changing $l$
by $Q_{l} / \mathcal{F}$ moves the transmission peak by a half-width.
Therefore, the tolerance for plate defects and non-parallelness in the
plates is
\begin{equation}
(\Delta l)_{\rm defects} < \frac{\lambda}{2 \mathcal{F}}.
\end{equation}
The plates used in the WHAM spectrometer are flat to around $\lambda /
150$, implying that $\mathcal{F}$ should be less than 75, implying $R
<$ 0.96. The surfaces of the WHAM etalons are coated such that R =
0.90, hence $\mathcal{F}$ = 29.8 and thus a plate defect of $\lambda$
/ 60 will lead to a degradation of the resolution in our system.
Hence, with our plate flatness we do not expect such a degradation.
With the finesse $\mathcal{F}$ = 29.8, and a desired resolving power
$\mathcal{R}_{\rm o}$ = 37500, Equation~\ref{eq2-respow2} determines
the order, $m$ = 1258. Hence, near \halpha, $l$ =
(6563$\AA$)(1258.5)/2 = 0.041. In fact, the gap in the high-resolution
etalon of the WHAM spectrometer is $l$ = 0.0471 cm, and the theoretical
resolving power is $\mathcal{R}$ $\simeq$ 43000.
\subsection{The Dual-etalon Concept}
\label{sec2-dual_etalon}
The primary purpose of placing two Fabry-Perot etalons in series is to
increase the free spectral range of the system. One can also increase
the free spectral range by decreasing the gap spacing $l$, but that
simultaneously degrades the spectral resolution. There are two other
added advantages in that the product of the second Airy function with
the first both reduces the wide, Lorentzian-like wings of the
primary Airy function and greatly increases the contrast of the
system.
For the WHAM spectrometer the primary, higher resolution etalon is
combined with a lower resolution etalon. The transmission peaks of
the two etalons can then be lined up for the desired wavelength (see
Section~\ref{sec2-tuning} below), and the different free spectral
ranges of the two etalons are such that only after a large number of
free spectral ranges are the peaks lined up again.
Figure~\ref{fig2-dual_etalon} shows the transmission function of each
of the WHAM etalons along with the combined transmission function,
which is the product of the individual transmission functions. The
high resolution etalon has a gap spacing of 0.0471 cm, and the lower
resolution etalon has a spacing of 0.0201cm, leading to
$Q_{\sigma}$(high) = 10.62 cm$^{-1}$ and $Q_{\sigma}$(low) = 24.88
cm$^{-1}$. It can be seen in Figure~\ref{fig2-dual_etalon}c that the
combined transmission function, which is shown here with the two
etalons tuned to \halpha, has a single strong transmission peak along
with some very weak 'ghost' peaks that appear where only one etalon
has a transmission peak. These ghost peaks in the transmission
function are down by of order the contrast, which for R = 0.90 is C
$\simeq$ 360. Thus we expect the ghost peaks to be of order 1/3
percent. This is confirmed in Figure~\ref{fig2-dual_etalon}d where
the combined transmission function is plotted on a log scale. Where
neither etalon has a transmission peak, the dual-etalon transmission
function goes to a very low level, and the contrast of the combined
system is of order the product of the contrasts of the individual
systems.
It is still necessary to use an interference filter as a
pre-monochrometer since the transmission peaks of the two etalons will
eventually line up again, but the filter can have a much broader pass
band than would be necessary for a single etalon system. The filter
is also useful in suppressing the parasitic light, that is, continuum
light leaking through away from the transmission peak due to the
finite contrast. A typical transmission function for one of the WHAM
interference filters (20 $\AA$ FWHM) is shown as a dashed line in
Figure~\ref{fig2-dual_etalon}a.
An important point to note about the dual-etalon transmission function
relates to the $\cos \theta_{t}$ dependence in
Equation~\ref{eq2-maxcrit1}. The transmission functions plotted in
Figure~\ref{fig2-dual_etalon} correspond to on-axis rays, where $\cos
\theta_{t} = 1$. For rays at an angle $\theta_{t}$ with respect to
the optical axis, the entire pattern of the transmission peaks is
shifted to shorter wavelengths. Fortuitously for Fabry-Perots, the
angular dependence of the transmitted wavelength is independent of the
resolution of the etalon, and thus the two etalons maintain the same
relationship for off-axis rays.
The Airy function is not a Gaussian: it has significant wings,
allowing wavelengths fairly far from the central peak to leak through
at a low level. Figure~\ref{fig2-dual_etalon_zoom} shows how the
second, lower resolution etalon serves to suppress these wings.
Figure~\ref{fig2-dual_etalon_zoom}a shows the transmission function of
the high resolution etalon alone and allows a detailed look at the
shape of the Airy function, and Figure~\ref{fig2-dual_etalon_zoom}b shows
the low resolution transmission function.
Figure~\ref{fig2-dual_etalon_zoom}c shows the product transmission
function as a solid line and shows the profile of the high resolution
etalon alone overlayed as a dashed line in order to demonstrate the
wing suppression provided by dual etalons.
\subsection{Pressure Tuning}
\label{sec2-tuning}
In order to successfully implement a system with two Fabry-Perot
etalons in series, it is necessary to be able to align transmission
peaks of the two etalons at a desired wavelength. This is called
\emph{tuning} the system. In our case, this is accomplished by making
slight changes to the optical gap, $n l$, in each etalon through the
dependence of the index of refraction $n$ of the gas in the gap, on
pressure, $p$. Each etalon is immersed in a separate chamber,
allowing the pressures and thus the optical gaps to be adjusted
independently. By finding the appropriate pressures for the two
chambers one can then line up a transmission peak of each etalon with
the desired wavelength. Note that it is not always the transmission
peak of the same order that is used.
I will now calculate the amount by which the index of refraction, $n$,
must be changed in order to move a transmission peak by one free spectral
range, that is, I will calculate $Q_{n}$. Consider on-axis rays for
two adjacent orders,
\begin{eqnarray}
m_{1} \lambda &=& 2n_{1}l \\
m_{2} \lambda &=& 2n_{2}l.
\end{eqnarray}
Subtracting these two equations gives,
\begin{equation}
(m_{1} - m_{2}) \lambda = 2(n_{1} - n_{2}) l.
\end{equation}
But, we assumed adjacent orders, and hence $m_{1} - m_{2} = 1$. Therefore,
\begin{equation}
n_{1} - n_{2} = \frac{\lambda}{2 l}.
\end{equation}
Hence,
\begin{equation}
Q_{n} = \frac{\lambda}{2l}.
\end{equation}
The worst case, that is, the circumstances under which the largest
change in $n$ will be required, is for the low resolution etalon and
the longest observed wavelength. The gap spacing of the low
resolution etalon is $l = 0.0201$ cm and the reddest light that the
WHAM spectrometer can study effectively is $\lambda$ = 7200 $\AA$.
Hence for the worst case, $(\Delta n) = 1.63 \times 10^{-3}$. We thus
must change the chamber pressures by enough to achieve this change in
the index of refraction.
The increase in index of refraction above the vacuum value ($n_{vac}$
= 1.0) for a gas scales approximately linearly with the number density
of atoms, and thus for an ideal gas scales approximately linearly with
pressure. Hence we can write,
\begin{equation}
\label{eq2-index_p}
n_{p} = 1 + \beta p,
\end{equation}
where $n_{p}$ is the optical index of the gas at pressure p measured
in atmospheres, and $\beta$ is the proportionality constant,
characteristic of the gas. For air, $\beta \simeq 3 \times 10^{-4}$.
The gas we use in our chambers is sulfur hexafluoride (SF$_{6}$). The
reason for this is that SF$_{6}$ has an unusually high value of
$\beta$, $\beta_{SF6} = 7.2 \times 10^{-4}$, and it is also inert and
environmentally safe (as opposed to freon, e.g.). Our chambers were
designed to withstand 3 atmospheres of pressure, and hence we can
change $n$ by 2.2 $\times 10^{-3}$, safely larger than the value
needed for the worst case calculated above.
The relationship between the change in index of refraction of a real
gas with the pressure isn't precisely linear as
Equation~\ref{eq2-index_p} would suggest. This is due to attractive
inter-molecular forces that cause the pressure to rise more slowly
than the number density of atoms. This difference is parameterized
using the the Van der Waals' a and b coefficients in the Van der
Waals' equation of state:
\begin{equation}
(p + \frac{N^2 a}{V^2})(V - Nb) = NRT,
\end{equation}
where the values of a and b for sulfur hexafluoride are
\begin{eqnarray}
a_{SF6} &=& 0.7857\ {\rm Pa}\,{\rm m}^{6} {\rm mol}^{-2} \\
b_{SF6} &=& 8.786 \times 10^{-5} {\rm m}^{3} {\rm mol}^{-1},
\end{eqnarray}
and N represents the number of molecules expressed in units of
Avogadro's number. The relationship between the pressure and the
number of moles of sulfur hexafluoride in a 1 m$^{3}$ volume is
plotted as the solid curve in Figure~\ref{fig2-vanderwaala}. The
dotted line shows the relationship for an ideal gas. In
Figure~\ref{fig2-vanderwaalb}, the fractional difference between the
real (SF$_{6}$) and ideal gas cases versus the number of moles N is
plotted. One can see from this figure that at the extreme end of the
range in N, which corresponds to 3 atmospheres of pressure, there is
about a 3\% difference. In fact, it is ($\epsilon$-1) that scales
linearly with the number density of molecules, where $\epsilon$ is the
dielectric permittivity of the gas. The index of refraction, $n$, is
proportional to the square root of $\epsilon$, so we expect the
deviation from linearity in $n$ to be about 1.5\% at the upper limit
of our pressure range.
\subsection{Scanning versus Imaging}
\label{sec2-scan_vs_image}
If confined to using a single element detector (commonly a
photo-multiplier tube), a natural way to measure a spectrum is to use
an aperture to pass only rays close to the axis ($\cos \theta_{t}
\simeq 1$) and then scan either $n_{g}$, or $l$ to move the
transmission peak through a range of wavelengths and thereby measure
the spectrum of the source. The aperture must be small enough so that
the $\cos \theta_{t}$ dependence does not degrade the resolution of
the system, and this requirement severely limits the amount of light
per time that can be admitted. In scanning $l$, one moves one of the
Fabry-Perot plates with respect to the other while maintaining the
very accurate parallelism that is required. This is usually
accomplished by using piezo-electric spacers wherein the variation in
the length of the spacer is proportional to the applied voltage. One
can then scan a range of $l$ by ramping the applied voltage.
Alternatively, if the gap is filled with a gas of some kind, one can
then scan $n_{g}$ by changing the pressure of the gas. This is
accomplished by enclosing the entire etalon in a sealed chamber with
windows, and then varying the pressure in the chamber. Both of these
methods suffer the disadvantage of only accepting light from one
spectral element at a time.
The advent of imaging detectors such as charge-coupled devices (CCDs)
has allowed a more efficient use of the Fabry-Perot for spectroscopy.
Here the variation with $\theta_{t}$ is used to allow simultaneous
measurement of multiple spectral elements. This `multiplexing
advantage', or more properly `multichannel advantage', leads to a gain
in efficiency equal to the number of spectral elements in the spectral
range. Another advantage of this method derives from the fact that
CCD detectors have much larger quantum efficiencies ($\simeq 4\times$)
than photo-multiplier tubes (PMTs), although there can be a disadvantage
due to noise associated with reading out the number of electrons in
each pixel of the CCD. This method has been demonstrated to give
large gains in efficiency in laboratory as well as field tests (see
below). In the next section I will describe our ring imaging method
in some detail.
\subsection{Fabry-Perot Spectroscopy via Ring Imaging}
\label{sec2-ringim}
The theory and application of ring imaging spectroscopy is discussed
in detail by Coakley et al 1996 and Nossal et al 1996. A brief
summary is presented here.
Consider the basic condition for constructive interference in a
Fabry-Perot system,
\begin{equation}
m \lambda = 2 n_{g} l \cos \theta_{t}.
\end{equation}
For a given order, $m$, and for on-axis rays, $\cos \theta_{t}$ = 1,
the wavelength $\lambda_{\rm o} = 2 n_{g} l / m$ is passed by the system.
Progressively shorter wavelengths are passed by the system as we increase
the angle of the rays with respect to the optical axis,
\begin{equation}
\label{eq2-thetadep}
\lambda_{\theta} = \lambda_{\rm o} \cos \theta_{t}.
\end{equation}
Since there is no dependence on the orientation in a plane perpendicular to
the optical axis, a monochromatic light source will produce a symmetric
ring pattern if a lens is used to image the angles passed by
the Fabry-Perot into positions (see Figure~\ref{fig2-ring_im}).
Using Equation~\ref{eq2-doppler} and Equation~\ref{eq2-thetadep}, we can
derive
\begin{equation}
\Delta \lambda = (\lambda_{\rm o} - \lambda) =
\lambda_{\rm o}(1 - \cos \theta_{t}).
\end{equation}
And thus, converting to a radial velocity scale,
\begin{equation}
\label{eq2-vel_theta}
-\frac{v}{c} = \frac{\Delta \lambda}{\lambda} = (1 - \cos \theta_{t}),
\end{equation}
where v is the velocity with respect to the on-axis ($\theta_{t}$ = 0)
velocity. As discussed above, we desire our spectrum to cover a range
of 200 km s$^{-1}$. Plugging this into Equation~\ref{eq2-vel_theta}
and solving for $\theta_{t}$ yields $\theta_{t}$(max) = 2.09$^{\rm
o}$. This is a very important number that dictates many elements of
the WHAM optical system, as will be seen.
Since the maximum angle passed is $\simeq 2^{\rm o}$, we may safely
approximate $\cos \theta_{t} \simeq 1 - \theta_{t}^{2} / 2$.
Hence,
\begin{equation}
-\frac{v}{c} = \frac{\theta_{t}^2}{2}.
\end{equation}
We wish to image angles less than $\theta_{t}$(max) onto a 1 cm$^2$
area of the CCD chip. The chip we ultimately acquired for the WHAM
system is 2.4 cm on a side, but we didn't know what size chip we would
be getting when the optical system was designed. This also allows us
to swap in other CCD cameras, already in our possession, which have a
1 cm$^2$ format, if we need to. Opting for an image that is 2.4 cm on
a side also generates problems caused by the need for very large
binning factors (and the resulting increase in noise) as well as
problems related to the fact that the generated electrons would be
spread over a larger area and the charge transfer efficiency is lower
for lower charge levels. For our 1 cm$^2$ ring image, we can
calculate the effective focal length of the ring imaging lenses,
f$_{eff}$,
\begin{eqnarray}
0.5\hspace{0.06in} {\rm cm} &=& {\rm f}_{eff} \tan \theta_{t}(max) \\
{\rm f}_{eff} &=& 13.7 \hspace{0.06in}\rm{cm}.
\end{eqnarray}
Note that the aperture defined by the etalons has a diameter of 15cm
(the largest available size) and thus this f$_{eff}$ implies an
f-number $\simeq$ 0.9. The actual imaging of the rings is done with
more than one lens, and it was a non-trivial problem to design the
optical system to meet the imaging specifications required for this
application (see below).
I will now derive the relationship between velocity and position on the
CCD chip. Let r be the polar radius vector measured from the center of
the ring pattern on the chip (see Figure~\ref{fig2-ring_im}). Then,
\begin{equation}
r = {\rm f}_{eff} \tan \theta_{t} \simeq {\rm f}_{eff} \theta_{t}
\end{equation}
implies
\begin{equation}
\frac{v}{c} = \frac{r^2}{2 f^2}.
\end{equation}
One important consequence of this relationship is that equal spectral
intervals, $\delta \lambda$, cover equal areas on the CCD chip. Let
$dA$ represent an infinitesimal element of annular area
\begin{equation}
dA = 2 \pi r\ dr.
\end{equation}
Since,
\begin{equation}
r\ dr = \frac{{\rm f}_{eff}\ dv}{c},
\end{equation}
Then,
\begin{equation}
\label{eq2-area}
dA = \frac{2 \pi {\rm f}_{eff}^2}{c}\ dv.
\end{equation}
Equation~\ref{eq2-area} is independent of r, and thus the annuli corresponding
to equal spectral intervals get progressively narrower as one moves
away from the center of the pattern in such a way as to make the
annular area constant. This leads to the peculiar optical
specification that the tolerance for image aberrations is
significantly more stringent at the edge of the field than at the middle.
The area on the chip of an annulus corresponding to an 8 km s$^{-1}$
velocity interval is, $dA$ = 3.15 $\times$ 10$^{-2}$ cm$^{2}$. At the
edge of the field, this must equal $2 \pi r\ dr$ with r = 0.5 cm, which
leads to a spatial width for the outermost spectral element
$(dr)_{edge}$ = 100 $\mu m$. This dimension defines the required image
quality and determines the degree to which on-chip binning of the CCD
can be employed (see next section).
\subsection{CCD Detection of the Ring Pattern}
\label{sec2-CCD}
To measure a spectrum, we use the chamber pressure to adjust the
optical gaps, such that the red edge of the desired wavelength range
is passed for rays on the optical axis. An exposure is then started
and electrons accumulate in the CCD pixels from the absorbed photons
which are distributed radially across the chip according to their
wavelength. After the exposure is finished, the number of electrons
in each pixel is read out and stored on the computer disk. The
spectrum is then reconstructed in software by averaging the counts in
each annular bin in order to form a one-dimensional plot of intensity
versus wavelength.
There is a limit to the accuracy with which the number of electrons in
a pixel can be determined. The error in the determination of the
number of electrons is termed the \emph{read noise} of the CCD system
and is a fixed number, independent of the number of electrons in the
pixel. For this reason, it would be advantageous to reduce the number
of pixels that need to be read within each equal area annulus. This
is possible through on-chip binning of the pixels. It is possible to
combine pixels on the CCD chip to form super-pixels of arbitrary size.
Then the total number of electrons falling within a super-pixel are
read as one with nearly the same noise as in the reading of a single
pixel (we have discovered that the read noise is slightly higher for
binned pixels, an effect which I discuss in the next chapter). The
optimal super-pixel size is the largest size that will not
significantly degrade the ring imaging resolution. This means that
the super-pixel cannot be larger than the width of the outermost ring,
which was calculated above to be $(dr)_{max}$ = 100 $\mu m$. The size
of a pixel on our CCD chip is 24 $\mu m$ and hence 4$\times$4 binning
is optimal. This leads to 16$\times$ fewer reads and thus, in a
regime where the read noise is dominant, a factor of four reduction in
the noise. We actually do suffer a decrease in spectral resolution
near the edge of the chip due to the fact that the pixel width is a
non-negligible fraction of the ring width, but this liability is more
than outweighed by the reduction in read noise afforded by using this
large super-pixel.
\section{The WHAM Optical design}
The requirements of the WHAM optical system are to accept light from a
1$^{\rm o}$ beam on the sky, use this light to illuminate the 15 cm
etalons, with light filling angles corresponding to a 200 km s$^{-1}$
velocity interval, and then to image the resulting ring pattern onto a
1 cm$^{2}$ area of the CCD chip. We also need to pass the light
through a narrow-band interference filter along the way in order to
eliminated unwanted orders that are passed by the two Fabry-Perots.
Further, we need to have the option of imaging the sky onto the CCD
chip for verifying our look direction and also to allow the
possibility for velocity-resolved imaging work at $\sim$ 1$\arcmin$
resolution. This section describes the optical design we have
developed to accomplish these goals. Note that the conceptual design
for the optical system was done by Ron Reynolds and the detailed
optical design was worked out by John Harlander of St.\ Cloud
University using the Code V ray tracing program.
\subsection{Overview of the Optical System}
A schematic drawing of the overall WHAM system hardware is shown in
Figure~\ref{fig2-wham_system}. The two main parts of the system are
the siderostat and the WHAM spectrometer. On the left is depicted the
all-sky siderostat which contains two folding mirrors and a 0.6 m
diameter, 8.6 m focal length lens. On the right is the WHAM
spectrometer, which along with the computer systems and various control
systems is housed in a small trailer.
Light from a chosen place on the sky enters the siderostat and is
guided by the siderostat folding mirrors into the spectrometer. After
passing through the 0.6 m telescope lens the light travels along a
tube into the WHAM trailer. The optical path crosses the trailer near
the ceiling, then reflects off a folding mirror that sends it down
through the two Fabry-Perot etalons chambers. Next, the light passes
through the ring imaging lens just after the etalons, reflects off
another folding mirror, and an image of the ring pattern is then
formed at the location of the interference filter. From there, the
ring pattern is reimaged by a sophisticated 4 element camera lens
onto the CCD chip, where the image is recorded. When an image of the
sky is desired, three additional lenses are inserted between the last
folding mirror and the filter which have the effect of imaging the sky
onto the interference filter. Since the interference filter plane is
imaged onto the CCD chip, this images the sky onto the detector.
\subsection{The Siderostat}
The purpose of the WHAM siderostat is to steer light from an arbitrary
point in the observable sky along the optical axis of the
spectrometer. The siderostat consists of two 90\arcdeg\ folding
mirrors that are mounted on perpendicular rotatable axes. The
siderostat structure also houses the 0.6 m telescope lens, located
between the last mirror and the light tube.
The two movable axes form an altitude-azimuth mounting system, but
tipped over such that the zenith of this ``alt-az'' system is pointing
toward the northern horizon. Our tipped system needs only 180\arcdeg\
rotations for both axes to cover the entire sky, avoids the need for
more than 360\arcdeg rotations of a conventional alt-az, and doesn't
have a zenith exclusion zone problem. The problem still exists, but
for us is located at the northern and southern horizon points. It is
never advisable to observe near the northern horizon point since an
object there is always better observable at another time, and having a
tracking problem at the southern horizon point is clearly preferable
to having one at the zenith.
The siderostat hardware was developed at Physical Sciences Laboratory
and the control software was developed as a joint project between
PSL and the WHAM team.
\subsection{The Pre-etalon Optics}
The desire for a 200 km~s$^{-1}$ spectral interval requires a
2.09$^{\rm o}$ half angle for the cone of rays passed through the
etalons (Section~\ref{sec2-ringim}). Our desire for a 1$^{\rm o}$ beam
on the sky requires that we map the 2.09$^{\rm o}$ half angle cone
onto a 0.5$^{\rm o}$ half angle cone. This is the purpose of the
pre-etalon optics.
The basic idea is to add an additional lens of appropriate size and
focal length in front of the etalons (see
Figure~\ref{fig2-optics_pre}). This `telescope' lens images the sky
onto a plane between the etalons, which has the added benefit of
separating the spatial and spectral information. In many Fabry-Perot
systems there are no pre-etalon optics and thus the detector
simultaneously records the Fabry-Perot ring pattern and an image of
the sky. Each annular ring on the detector corresponds to a
particular spectral interval, as before, but in these systems the
amount of light reaching that ring on the detector is now determined
by the corresponding spatial ring on the sky. Hence, an ambiguity
exists, and a bright spot on the detector may indicate either a
spectral feature, a spatial feature, or some combination of the two.
With the WHAM system, the light from each point on the sky is spread
uniformly over the CCD detector, and no such ambiguity exists.
The conservation of etendue (area $\times$ solid angle product through
the optical system) requires the telescope lens to be 15 cm $\times$
2.09 / 0.5 $\simeq$ 63 cm in diameter. In order to illuminate the
full area of the etalons, the focal length of the telescope lens must
satisfy $\tan 0.5^{\rm o}$ = 15 cm / f$_{L1}$. This implies f$_{L1}
\simeq$ 860 cm. Without the use of an additional lens in front of the
etalons, the cone of rays illuminating outer parts of the etalons
would be tilted on average with respect to the optical axis. Since
only light within 2.09$^{\rm o}$ of the optical axis is passed by the
rest of the optical system, the net effect is to selectively
underilluminate the blue side of the spectrum. For this reason, we
add an additional lens that is placed just ahead of the etalons with
nearly the same focal length as the telescope lens which results in
parallel (telecentric) illumination of the etalons.
\subsection{The Post-etalon Optics}
The Fabry-Perot etalons pass different wavelengths of light at
different angles. Hence after passing through the etalons, the
spectrum of the light is encoded in angles according to
Equation~\ref{eq2-thetadep}. The primary purpose of the post-etalon
optics is to map these angles into positions so that the spectrum can
be recorded. The simplest method is to place a lens just after the
etalons and then place an imaging detector in the focal plane of the
lens. However, we also need a narrow-band interference filter in our
system to eliminate unwanted orders passed by the Fabry-Perot.
Because of the unacceptably high cost of good quality 15 cm diameter
filters, we have opted to image the ring pattern at an intermediate
location in order to allow the use of 7.5 cm interference filters.
Our post-etalon optical design is schematically illustrated in
Figure~\ref{fig2-optics_post}.
The lens just after the etalons, L3, serves to image the ring pattern
onto the filter. The 7.5 cm filter size along with the 2.09$^{\rm o}$
maximum angle determine the focal length, $\tan 2.09^{\rm o}$ = 1.5
$\times$ 2.54 cm / f$_{L3}$ implies f$_{L3} \simeq$ 104 cm. This lens
in combination with L2 also serves as a field lens, imaging the
entrance pupil (L1) onto the interference filter aperture. The next
lens in the optical path is L4 which serves the purpose of
parallelizing the ray cones for passage through the interference
filter. The lens L5 along with L4 also acts as a field lens to image
the etalons, and hence the sky, onto the final set of lenses L6 - L9,
which we collectively refer to as the camera lens. The four element
camera lens focuses the ring image onto a 1 cm$^{2}$ region of the CCD
chip, where the spectrum is recorded. The conservation of etendue
requires that the cone of rays incident upon a particular point on the
chip has a half-angle of 2.09 $\times$ 15 cm / 1 cm $\simeq$ 30$^{\rm
o}$. Hence the camera lens must have an f/\# $\simeq$ 0.9. Due to
chromatic aberration in the system, the camera lens is mounted on a
movable track, and its precise distance from the chip is adjustable
with a micrometer dial, whose setting is different for each emission
line. For example, 1.6 mm of travel is required to move between
H$\alpha$ (6563\AA) and H$\beta$ (4861\AA) (see
Section~\ref{sec3-focus}). The lens diagrams shown in
Figures~\ref{fig2-optics_pre} and \ref{fig2-optics_post} are
schematic. In order to demonstrate the true complexity of the ring
imaging system, I have included Figure~\ref{fig2-cam_lens}, which shows
a scale drawing of the actual camera lens system. Three sets of rays
are traced through the lenses to the CCD chip, corresponding to
on-axis rays, rays tilted by 1.48\arcdeg\ off-axis at the etalons, and
rays tilted by 2.09\arcdeg\ off-axis at the etalons (the blue edge of
our bandpass).
\subsection{The Sky Imaging Lenses}
In order to image the sky on the CCD chip instead of the spectral ring
pattern, lenses can be inserted into the beam between L3 and L4. Three
lenses are used, and they are mounted on a motor driven movable table
that slides into and out of the beam. These lenses take the sky image
that is between the etalons and reimage it onto the filter IF, which as
before is then imaged onto the CCD. An additional feature of our
arrangement of these lenses is that the spectrum is imaged on an
intermediate plane where we have placed an adjustable iris. Hence,
when we are imaging the sky, we can adjust the wavelength interval that
is passed from a minimum value equal to the instrumental resolution of
12 km s$^{-1}$ up to a maximum value equal to the full spectral range,
which is 200 km s$^{-1}$.
\clearpage
\newpage
\begin{references}
Coakley, M. M., Roesler, F. L., Reynolds, R. J., \& Nossal, S. 1996,
Applied Optics, Vol. 35, No. 33, 6479
Nossal, S., Roesler, F. L., Coakley, M. M., \& Reynolds, R. J. 1997,
J. Geophys. Res., in press
\end{references}
% begin figs
\begin{figure}
\plotone{pathdiff.eps}
\caption{The optical path difference between consecutive rays.}
\label{fig2-pathdiff}
\end{figure}
\begin{figure}
\plotone{fabry.eps}
\caption{A Simple Fabry-Perot with multiple reflections between the
plates.}
\label{fig2-fabry}
\end{figure}
\begin{figure}[t]
\centerline{
\epsfysize = 3.0in
\epsffile{airy.eps}
}
\caption{I$_{t}$ / I$_{\rm o}$ versus $\phi$ for an ideal Fabry-Perot
with R = 0.9.}
\label{fig2-airy}
\end{figure}
\begin{figure}[b]
\centerline{
\epsfysize = 3.0in
\epsffile{airy2.eps} }
\caption{I$_{r}$ / I$_{\rm o}$ versus $\phi$ for an ideal Fabry-Perot
with R = 0.9.}
\label{fig2-airy2}
\end{figure}
\begin{figure}
\plotone{finesse.eps}
\caption{Reflective finesse vs. R.}
\label{fig2-finesse}
\end{figure}
\begin{figure}
\plotone{airysigma.eps}
\caption{The theoretical transmission functions of the WHAM etalons:
a) solid line: the high-resolution etalon, dashed line:
interference filter, b) the low-resolution etalon, c) the product
of the etalon transmission functions, d) same as c) but on a log scale.}
\label{fig2-dual_etalon}
\end{figure}
\begin{figure}[p]
\centerline{
\epsfysize = 6.5in
\epsffile{airysigma_zoom.eps}
}
\caption{The theoretical transmission functions of the WHAM etalons:
a) the high-resolution etalon, b) the low-resolution etalon, c)
solid line: the product of the two, dashed: the high-resolution
etalon alone.}
\label{fig2-dual_etalon_zoom}
\end{figure}
\begin{figure}[t]
\centerline{
\epsfysize = 3.25in
\epsffile{vw_n_p.eps}
}
\caption{Sulfur hexafluoride compared to ideal gas, solid line: pressure
vs number of moles of SF$_{6}$ gas in a 1 m$^{3}$ volume, dotted line:
same for an ideal gas.}
\label{fig2-vanderwaala}
\end{figure}
\begin{figure}[t]
\centerline{
\epsfysize = 3.25in
\epsffile{vw_n_pdiff.eps}
}
\caption{Sulfur hexafluoride compared to ideal gas, fractional difference
between SF$_{6}$ and an ideal gas vs. number of moles in a 1
m$^{3}$ volume.}
\label{fig2-vanderwaalb}
\end{figure}
\begin{figure}
\plotone{ringim.eps}
\caption{Fabry-Perot ring imaging.}
\label{fig2-ring_im}
\end{figure}
\begin{figure}[p]
\centerline{
\epsfxsize = 6.0in
\epsffile{wham_ov.eps}
}
\caption{The WHAM system.}
\label{fig2-wham_system}
\end{figure}
\begin{figure}[p]
\plotone{opt_pre.eps}
\caption{The WHAM pre-etalon optics (schematic).}
\label{fig2-optics_pre}
\end{figure}
\begin{figure}[p]
\plotone{opt_post.eps}
\caption{The WHAM post-etalon optics (schematic).}
\label{fig2-optics_post}
\end{figure}
\begin{figure}[p]
\centerline{
\epsfysize = 7.5in
\epsffile{whamlens_fixed.eps}
}
\caption{The WHAM camera lens.}
\label{fig2-cam_lens}
\end{figure}