\documentclass[journal]{rmaa}


\title{Star Image Shape Transformer for Astronomical\\ Slit Spectroscopy}

\author{O. Cardona, A. Cornejo-Rodr\'iguez, and P.~C. Garc\' ia-Flores\affil{Instituto Nacional de Astrof\' isica, \' Optica y Electr\' onica, Puebla, Mexico}}


\shortauthor{Cardona, Cornejo-Rodr\'iguez, \& Garc\'ia-Flores}

\shorttitle{Image Shape Transformer}


\listofauthors{O.~Cardona, A.~Cornejo-Rodr\'iguez, \& P.~C.~Garc\'ia-Flores}

\indexauthor{Cardona, O.}
\indexauthor{Cornejo+Rodr\'iguez, A.}
\indexauthor{Garc\'ia-Flores, P.~C.}


\fulladdresses{
\item Octavio Cardona, Alejandro Cornejo-Rodr\'iguez and Perla C. Garc\'ia-Flores: Instituto Nacional de Astrof\'isica, \'Optica y Electr\'onica,  
Apdo. Postal 51 y 216, 72000 Puebla, Pue., Mexico (ocardona@inaoep.mx).}


\SetYear{2010}
\SetVolume{46}
\ReceivedDate{2010 June 15}
\AcceptedDate{2010 August 15}


\abstract{We present a new star image shape transformer for astronomical slit
spectroscopy that consists of a plano-convex lens attached to a
plane-parallel rectangular glass plate on one of the small sides, with
the aim of improving the performance of spectrometric systems. The
system transforms the shape of the objects into a rectangular form with
the purpose of filling with light the entrance slit aperture in
spectroscopic systems. The plate is set sidewise with the lens as the
entrance aperture and the exit aperture of the plate is the illuminated
slit seen by the spectrometric system. The aim of this transformer is also 
to maintain the beam characteristics, the intrinsic resolution of the 
spectrometers and to improve the throughput of the spectrometer and detector systems.\smallskip}

\resumen{Presentamos un nuevo transformador de la forma de las im\' agenes estelares para espectroscop\' ia astron\' omica de rendija que 
consiste de una lente plano convexa pegada a una placa rectangular plana paralela de vidrio en uno de sus lados menores con el fin de mejorar 
el desempe\~ no de los sistemas espectrom\' etricos. El sistema cambia la forma de los objetos a la forma rectangular con el prop\'osito de 
llenar con luz la abertura de entrada de la rendija en los sistemas espectrosc\' opicos. La placa se pone de lado con la lente como la abertura 
de entrada y la abertura de salida es la rendija iluminada vista por el sistema espectrom\' etrico. El objetivo de este transformador es 
tambi\'en el de mantener las caracter\' isticas del haz de luz, la resoluci\' on intr\' inseca del espectr\' ometro y de mejorar la eficiencia de los 
sistemas espectrom\' etricos y detectores.}


\addkeyword{instrumentation: miscellaneous}
\addkeyword{stars: imaging}
\addkeyword{techniques: imaging spectroscopy}
\addkeyword{techniques: spectroscopic}



\begin{document}

\maketitle


\section{Introduction}

The main purpose of spectroscopy is to collect the maximum possible
radiant power from the given source, and to disperse and detect it
appropriately and efficiently. The slit spectrometers used in stellar 
astronomy consist of an entrance slit, a collimator, a dispersing element,
a camera, and a detector. The camera is used to demagnify and focus the 
slit on the detector. The image produced by the optical system
or telescope on the entrance slit of the spectrometers is in general
not a sharp point and its size is larger than the optimum slit width.
This is a common problem in astronomical spectroscopy where the 
stellar image is distorted by the earth atmosphere, the seeing, producing 
in general greater images than the required slit width. For that reason the 
light lost is great and is necessary to address this problem especially if one 
observes faint objects. For instance, if the width of the slit of the spectrometer 
is opened to accept, say, one second of arc in the given telescope and the 
seeing is of the order 2.5 seconds of arc, one is loosing fifty percent of the 
stellar light. Therefore, it is essential to find the way to introduce all the star 
light into the spectrometer. This can be accomplished by changing the shape 
of the image while trying to maintain the beam properties. 
This is realized converting the star circular image shape into a rectangular 
one, with the required size to maintain the resolution and to improve the throughput 
of the spectrometric systems. 

The intrinsic resolution of spectrometers and
the intensity transmitted by the slit are interconnected. Assuming
perfect illumination and that the light from the source fills or
overfills the spectrometer solid angle, then the resolving power,
the intensity at the peak of a spectral line, and the slit width are
interrelated. The maximum of the product of resolution and intensity is
of the order of the wavelength of the incoming radiation times the
aperture ratio of the spectrometer. This product becomes the optimum
slit width of the spectrometer \citep{jac54, dav70, tho74} and is given by
\vspace{-0.15cm}
\begin{equation}
w\ =\ \lambda \frac{f_{s}}{D_{s}}\,,
\end{equation}
\noindent where ${w}$ is the slit width, ${\lambda}$ is the wavelength of the incoming
radiation, ${f}_{s}$ and ${D}_{s}$ are the focal length and diameter of the
collimator of the spectrometer. 

To make full use of a spectroscopic
instrument, the dispersion at the receptor must be matched to the
resolution. The match between the size of the detector elements
(pixels) and the projected slit width on the detector, based on the
Nyquist criterion for discrete sampling, is that the slit width should
cover two pixel, required to recover the resolution of the images, that
are just resolved according to the Rayleigh criterion \citep{sch00}. 

There have been several solutions for changing the form of the
circular stellar images into rectangular ones using image slicers
\citep{bow38, ric68}, optical image transformers \citep{ben51}, fiber 
optics systems \citep{fur88,avi08}; and some other systems for beam shaping 
\citep{osh85}. In this work we propose a new system, different from the ones
mentioned before, for changing the form of the stellar images to fit
the entrance slit of the spectrometric systems without changing the
$F$ number of the exit beam of light in any direction with respect to the $F$
number of the entrance beam from the telescope. 

Of course, when one uses active and adaptive optics instruments in the telescope
it is not necessary to have the instruments mentioned before for
changing the form of the image \citep{len98, har98}. But these 
systems are expensive and complicated, and also, one cannot use
them all the time because of the lack of useful guiding stars. The system
presented here can be used in any observational situation and in any
telescope-spectrometer system. 

In what follows, \S~2 introduces the historical background for transforming the shape 
of stars in spectrometric systems that sets the basics of the development of the transformer. 
The conception of the transformer is introduced in \S~3. In \S~4 the design of the transformer 
is described. Finally in \S~5 some conclusions are presented. 


\section{Historical Background}

In the image slicers, the image is cut into slices, which are then
arranged end-to-end to fit the entrance slit of the spectrometer
system. Basically there are two types of slicers, the one designed by
Bowen (1938) and the other by Richardson (1968), but
there exist also several other types (Strong \& Stauffer 1964; Simmons et al. 1982; Diego 1993; Suto \& Takami 1997). 
The Bowen slicer, properly used, can result in a large advantage, especially 
for telescopes with large focal ratios. One of the difficulties with an 
image slicer is the minute scale, because the field to be sliced is very 
small in diameter. Benesch \& Strong (1951) describe some 
of the problems of a Bowen slicer in an f/10 beam. The Bowen slicer 
is composed of a stack of glass plates, aluminized in some 
predesignated regions. The light is reflected by ninety degrees to align 
end-to-end the slices on the slit. However they are not in the same 
plane: they are arranged in steps like in a staircase. This is the principal 
reason for limiting the focal-ratio to values greater than 30 \citep{hun74}.

A modification of the Bowen slicer was introduced by \citep{wal72}. The 
Bowen-Walraven image slicer in which the mirrors were replaced 
by total internal reflections inside a thin parallel plate where the seeing disk 
is sliced by a tilted edge of a prism in contact with the plate. In practice, the 
non-uniform illumination of the spectrograph collimator led to substantial 
problems and they were not extensively used. 

A modified version of the Bowen-Walraven slicer that overcomes most of the problems of this type of 
slicers is the Confocal Image Slicer, which delivers a large number of slices, all 
in focus \citep{die94}. These new developments are very useful for 
high-resolution spectrography. These system are more complicated and difficult 
to construct in comparison with the transformer reported in this article. 

The Richardson slicer resembles a confocal cavity composed of two concave
mirrors one in front of the other. Each concave mirror is divided into
two symmetric parts producing a slit between the parts. The two parts
of each concave mirror are slightly tilted in opposite directions
around the perpendicular to the slit. One of the concave mirrors is
rotated by ninety degrees around the optical axis. In this way, the slits
of each concave mirror are aligned perpendicular to each other in order
to reflect the entrance image in the four inclined mirrors several
times and exits through one of the slits forming a long slit with the
desired size. This slicer is difficult to construct and the multiple
reflections diminish the amount of light in the system. In this type
of slicer all the slices are in focus. 

To avoid some of the problems of
the Bowen slicer in a fast system Benesch \& Strong (1951) 
introduced an alternative, the optical image transformer, a redirector of 
rays. The slices are diverted to the sides by small plane mirrors, except 
the central one, and reflected back by concave mirrors to a central small
plane mirror above or below the central slice, and are deflected through
the slit. The concave mirrors refocus each slice back on the slit. In
this way, there is no limitation on the focal-number of the system.
The problem of many reflections and the alignment of tiny mirrors make it
difficult to implement this system. 

A scheme proposed by Fastie (1967) is appropriate for segmented mirror telescopes composed of 
several independent mirrors. The individual images may be aligned along 
the slit, and the pencils superposed on the grating by tiny individual
prisms. 

To solve the image-slicer problem Kapany (1958)
suggested that a fiber optic system would help and gave a detailed 
exposition of that possibility but no description of a working unit was 
given. One can use a fiber bundle arranged in a round profile at the focal 
plane of a telescope and then ordered linearly in the exit of the bundle to
become the entrance slit of the spectrometer. The size of the circular
entrance aperture can be made to cover several seconds of arc to match
the distorted star image produced by the seeing in the telescope. The
exit of the fiber bundle is composed of individual fibers to form the
slit with the required dimensions to be projected onto two pixels
\citep{fur88}. This fiber bundle is very efficient and the light loses are only 
due to the light lost in the inter bundle holes and in the cladding of the 
individual fibers, and also by the absorption of light in the long fiber bundles. 
The individual fibers in this linear array should have a diameter that covers 
two pixel sizes when demagnified by the optics, as mentioned before. For a 
spectrometer where the magnification is $m=-1$, like in an
Ebert-Fastie spectrometer, this imposes a very restrictive design
parameter. 

All the systems mentioned above increase the signal to
noise ratio of the spectrometric system because the length of the slit
is defined by the number of slices or fibers in the entrance slit of
the spectrometer, thus covering more pixels in the perpendicular
direction to the dispersion of the detector.  

O'Shea (1985) has described several techniques to change
the light distribution in one direction using cylindrical lens
telescopes and the prism beam compressors that can be used in
spectroscopy. The problem with these systems is that the beam becomes
anamorphic, that is, they produce a beam with different divergences in two
orthogonal directions. The system reported in this work overcomes most
of the problems mentioned before and is introduced in the following
section.


%%%%% FIGURA 1

\begin{figure}[!t]
\vspace{0.1cm}
\centering
\includegraphics[clip,width=0.65\columnwidth]{ocardonafig1.eps}
\caption{The flow of light from the star to the detector, 
passing through the shape transformer.}
\vspace{0.3cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%% FIGURA 2

\begin{figure}[!t]
\centering
\includegraphics[clip,width=0.99\columnwidth]{ocardonafig2.eps}
\caption{Plate shape transformer showing the round entrance star image on 
the front face and the slit formed on the back face of the plate.}
\vspace{0.3cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Conception of the System}

One of the objectives of the design of the shape transformer is to
produce a slit with a width ${w}$ and length ${l}$ having the
aspect ratio ${r} = l/w$ equal to at least six or any other desirable number.
This is with the purpose of producing an image on the detector that, when
projected, covers two pixels in the dispersion direction and at least
twelve in the perpendicular one, that at the same time obeys the
sampling theorem of Shannon and improves the signal to noise ratio,
when one sums up all the pixels in the direction perpendicular to the
dispersion direction. In order to take into account the imperfections, the intrinsic
noise and different responses of the pixels of the detector (CCD), and the statistics
of the arriving photons, it is necessary to obtain all the parameters that describe the 
distribution as well as the mean and the standard deviation. A Gaussian of a data set is 
recovered when one has more than ten values of the function, by the law of large 
numbers in statistics \citep{bev02}.


%%%%% FIGURA 3

\begin{figure}[!t]
\centering
\includegraphics[clip,width=0.99\columnwidth]{ocardonafig3.eps}
\caption{Real glass plate showing the dimensions and the width $w$ of the
plate in the laboratory.}
\vspace{0.3cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


The location of the shape transformer in the observational set up is
shown in Figure~1, which displays the flow of light from the star to the detector 
passing through the telescope, the shape transformer, the demagnifying optics 
that forms the slit, the spectrometer, and the detector.

The simplest image shape transformer is a thin glass plate seen edgewise, 
which can be used to change the form of a circular image into a more or less 
rectangular one, shown in Figures~2 and 3 for the design and for the glass 
plate in the laboratory. The width ${w}$ of the plate is fixed to accept the 
entire image that comes directly from the telescope, or from an optical fiber bundle. 
The light beam, after bouncing back and forth inside the plate with total internal
reflections, exits on the opposite side of the plate forming an illuminated 
strip of length ${l}$. The form of the image is flat in the upper and lower
sides of the plate and round at the sides and that becomes the slit
of the spectrometer. The plate does not change the $F$ number of the
entrance beam in any direction, but it scrambles the light rays inside it. This 
system can be analyzed using the unfolding ray
trace technique for plates (Cardona, Cornejo-Rodr\'iguez, \& Flores-Garc\'ia 2010). In order to fulfill the condition 
imposed by the aspect ratio given above, the plate must be large in the 
direction of propagation, depending on the entrance aperture ratio. This 
distance depends in the $F$ number of the entrance beam of the star 
light produced by the telescope. The size of the image produced by a
telescope is given by the plate scale in seconds of arc per millimeter as
\begin{equation}
s\ =\ \frac{206.265}{f(m)}\ (''{\rm mm}^{-1})\,,
\end{equation}
and the F number or focal ratio is defined as
\begin{equation}
F =\ \frac{f}{D}\,, 
\end{equation}
where ${f}$ is the focal length, and ${D}$ the diameter of
the telescope in meters. 

Equation~(1) defines the width of the slit of
the spectrometer when the average seeing disc of the star is given. 
The width in general should be larger because in order to funnel all 
the star light into the slit when the seeing disc of the star is larger
than, say 2.5 seconds of arc. Then, for example, a telescope
of 2.1 m diameter with focal ratio of 12, would produce, using 
equations~(1) and (2), a star diameter of about $300~\mu$m, for the 
seeing mentioned above. Thus, the width of the plate should have this 
dimensions and must be demagnified by around 7.5 times by the optics 
for a detector with square $20~\mu$m pixels. The plate would be long 
and it would lose much light due to the distance traveled by the rays inside 
the plate. For this demagnification it is necessary to employ fast cameras 
that, in general, are expensive and difficult to construct. 

The same problems exist in a normal slit spectrometer for a slit opened to $300~\mu$m 
in order to gather all the star light. For an existing spectrometer-telescope 
system the demagnification should not be great, because the image produced 
by the existing optics of the spectrometer is already used to reduce the size 
of the image produced by the slit, in general opened to accept star images of 
one second of arc, that becomes the entrance slit of the optics of the spectrometer. 
But the main problem is that the $F$ number is changed and is difficult 
to convert back that number to the one necessary for the system. This
plate can also be used with an optical fiber bundle that for the case
considered here should have a diameter of $300~\mu$m and should be
connected directly to the plate. This plate does not fulfill the necessary requirements
due to the problems mentioned above.


%%%%%% FIGURA 4

\begin{figure}[!t]
\centering
\includegraphics[clip,width=0.98\columnwidth]{ocardonafig4.eps}
\caption{Design of the shape transformer showing the star image from the telescope, 
the magnifying lens, the plate, and the exit slit.}
\vspace{0.3cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


The system proposed in this work is similar to the one described before
but has a small plano-convex lens attached to one of the smaller sides 
of the plate to overcome the problems of the single plate mentioned above.
The design is shown in Figure~4. This system permits decreasing the width 
and the length of the plate compared to the case of the single plate. 

Figure~5 shows a meridional ray trace for the system using a commercial package. The transformer is put
in front of the image produced by the telescope or by an optical fiber bundle at a given 
distance of the lens to magnify the image to the required size and therefore 
the aspect ratio. The lens is used to magnify the image to give the desired 
aspect ratio of the slit with respect to the width of the plate, as mentioned 
before; also, due to the bending of the rays by the lens the plate can be 
reduced in width depending on the characteristics of the lens and size of 
the object. The rays inside the plate are reflected between the upper and lower 
faces of the plate by total internal reflections; therefore, no light is lost (Figure~5)
and the entrance beam $F$ number is not changed in the plate, only its form. 
The light absorption in the plate is due only to the distance traveled inside of 
material and in this case is small. 


%%%%% FIGURA 5

\begin{figure}[!t]
\centering
\includegraphics[clip,width=0.99\columnwidth]{ocardonafig5.eps}
\caption{Meridional ray trace of the transformer showing the waist of the beam and the 
internal reflections.}
\vspace{0.3cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


When a wide rectangular glass plate set sidewise is used as a waveguide it behaves 
as an optical fiber in one dimension when the input end is one of the small 
sides and the light is transported inside the plate. In a fiber, from theoretical grounds, 
the focal ratio degradation (FRD) is proportional to the square root of the length of 
the fiber \citep{glo72, car94, pop10}. 
From the experimental side, the FRD slightly increases with the length of the fiber 
\citep{mur08}. The output $f$-ratio for an input $f$-ratio of less than 5 shows little 
to no appreciable FRD (Murphy et al. 2008; Crause, Bershady, \& Buckley 2008; Ramsey 1988). The 
experiments with fiber immersion show that the end effects diminish drastically the FRD 
for all the input $f$-ratios \citep{mur08}. Fiber immersion involves contacting the 
input end of the fiber to an anti-reflective cover plate. The  fiber-glass interface is made 
with an index matching gel.


%%%%%% FIGURA 6

\begin{figure}[!t]
\centering
\includegraphics[clip,width=0.99\columnwidth]{ocardonafig6.eps}
\caption{Star shape transformer in the laboratory, composed of a one-millimeter diameter lens and a glass plate.}
\vspace{0.5cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


In a plate there are no problems of macro and micro bending that produce part of the FRD 
in a fiber because it is a rigid piece of glass. When the length of the plate is small compared 
with the fibers considered in the experiments mentioned above, one expects little or no 
appreciable FRD. In the shape transformer, as the lens is attached to the plate it diminishes 
the end effects of the entrance surface of the plate. The lens magnifies the image of the object 
producing a beam faster than the beam coming from the object. The beams that enter the plate 
are fast and in most of the applications have an $f$-ratio of less than 5. Also, far from the 
plate the output beam that leaves the plate is symmetrical around the optical axis, showing that 
FRD does not occur in the plate, as can be seen in the experimental results of \citet{flo06}. 
Therefore, the shape transformer proposed preserves the $f$-ratio, as required.

If one desires a projected slit width on the 
detector of two pixels one has to demagnify the image of the illuminated side 
of  the transformer maintaining the entrance $F$ number of the original beam. 
To accomplish this it is necessary to magnify the image with the lens by the same 
amount as that of the necessary demagnification. Therefore, we fulfill all the requirements 
on the projected slit width, the aspect ratio of the slit, and the beam characteristics. 
The $F$ number of the entrance beam to the plate is not changed inside it, 
and therefore the beam characteristics are not changed in the system, by the Lagrange 
optical invariant, since the magnification produced by the first lens is equal to the 
demagnification produced by the demagnifying optics. As was the case for the single plate, 
this system also scrambles the light rays inside the plate. Therefore this shape 
transformer is very efficient for astronomical spectroscopy with telescopes of any size.


%%%%% FIGURA 7

\begin{figure}[!t]
\centering
\includegraphics[width=0.97\columnwidth]{ocardonafig7.eps}
\caption{Ray trace of the system for a round entrance image that produces 
the illuminated strip on the back side of the plate.}
\vspace{0.3cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Transformer Design}

We present the design of the shape transformer for some special cases in order to 
show the main ideas and peculiarities of the system. We have designed the transformer 
for a spectrometer for a $f=8$ beam and a star image of 300~${\mu}$m diameter. 

The transformer is composed of a commercial plano-convex lens one millimeter in diameter 
and an effective focal length of 0.6 millimeters, with an index of refraction of 1.85 and 
a radius of curvature of the convex side of 0.51 millimeters. The plate has dimensions of 
five millimeters by five millimeters and a width of 0.160 millimeters, and has an index of
refraction of 1.458. The lens is cemented on one of the thin sides of the plate as shown 
in Figure~5. Then, for a detector with 13.3~${\mu}$m square pixel size, one has to demagnify 
the illuminated image on the side opposite to the lens on the plate (the slit of the spectrometer) 
by six times. For a detector with 20 ${\mu}$m pixel size one has to demagnify the image size 
by four times to fit two pixels on the detector. To maintain the beam parameters for the 
spectrometric system one has to magnify the image with the entrance lens by six and four 
times respectively for the cases given above. In the first case the image is formed on the 
rear face of the transformer, in the second, inside the plate. In this second case the beam 
of light continues traveling to the rear face of the plate forming a longer illuminated
slit. For the two cases mentioned the slit length on the back of the plate will be of order 
of 1.8 millimeters and it will cover 22 pixels in the direction perpendicular to the dispersion 
of the spectrometer, for a five millimeter plate. For the second case one can use a plate of 
3.38 millimeters, and therefore the length of the illuminated strip will be 1.2 millimeters 
in length, giving 15 pixels in the direction perpendicular to the dispersion. The demagnifying 
optics should be focused near the back side of the plate illuminated by the beam of light, in 
order not to show the imperfections of the rear face of the plate. 


%%%%% FIGURA 8

\begin{figure}[!t]
\centering
\includegraphics[clip,width=0.98\columnwidth]{ocardonafig8.eps}
\caption{The laboratory round entrance image (left) used to produce the 
illuminated strip (right) seen with a microscope.}
\vspace{0.2cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


The number of pixels in the direction perpendicular to the dispersion is small, but larger than 
ten to have a good average, as was mentioned above, and to increase the signal to noise ratio. 
In the fiber bundle of 34 fibers covering 68 pixels of the Ebert-Fastie spectrometer in the 
direction perpendicular to the dispersion, the signal to noise ratio is large \citep{fur88}. 

In Figure~6 we show one of the implementations of the transformer in the laboratory \citep{flo06}. 
Figure~7 shows a ray trace with a commercial ray trace package for a round image as the input that 
produces a strip (slit) in the output. Figure~8 shows the illuminated side of the glass plate, seen 
in the laboratory with a microscope, produced by an illuminated circular image. Figure~9 shows 
the transformer with two three-mm diameter commercial lenses with focal lengths of two mm, for the 
demagnifying optics. This simple demagnifying optics does not absorb much light, resulting in a 
very efficient system.  

Of course, the transformer can be changed in 
its dimensions and type of lens used, to be adjusted to any telescope and application, 
not only for astronomy. The transformer is simple and easy to construct and to implement 
in a laboratory, and for any telescope-spectrometer system. From the observational 
side, the first lens has a field of view larger than the star images, permitting the entrance 
of light from the surrounding sky into the spectrometer, which can be used for calibration
purposes. Of course, when the light from the telescope is piped through an optical fiber 
bundle one encounters, as is usual, the problem of sky light subtraction.
The scrambling of the light rays inside the plate makes the illumination of the slit of the 
transformer independent of slight changes in the position of the stars around the optical axis. 
Therefore, no problems of guiding the stars arise at the telescope. 


%%%%%% FIGURA 9

\begin{figure}[!t]
\centering
\includegraphics[clip,width=\columnwidth]{ocardonafigura9.eps}
\caption{Design of the shape transformer with the auxiliary demagnifying optics composed
of two 3-mm diameter lenses.}
\vspace{0.2cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5


\section{Conclusions}

We have designed a simple image shape transformer for astronomical slit 
spectroscopy that changes the circular images of stars into a rectangular 
form by using a lens and a glass plate set edgewise. The exit or back side of 
the plate becomes the slit for the spectrometer with the required aspect ratio 
for the width and length to comply with the sampling theorem and, at the same 
time, increases the efficiency of the spectrometric systems and the signal to 
noise ratio, with low light losses. The system reported here overcomes most of 
the problems of other systems known. This transformer is easy 
to construct and implement for any telescope and spectrometer system.

\vspace{0.1cm}

\acknowledgements

The authors acknowledge the technical support of Mar\'ia de los \'Angeles L\'opez Castillo.


\vspace{0.15cm}

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