
\section{Introduction}

Unlike other scientists, astronomers are far away from the objects
they want to examine.  Therefore astronomers learn everything about an
object by studying the light it emits.  Since objects of astronomical
interest are far away, they appear very dim and small to us.  Thus
astronomers must depend upon telescopes to gather more information.
Lenses and mirrors are used in telescopes which are the instruments
astronomers use to observe celestial objects.  Therefore it is
important for us to have a basic understanding of optics in order to
optimize telescopes and interpret the information we receive from
them.

The basic idea of optics is that mirrors or lenses can be used to
change the direction which light travels.  Mirrors change the
direction of light by \textit{reflecting} the light, while lenses redirect
light by \textit{refracting}, or bending the light.  

The theory of optics is an important part of astronomy, but it is also
very useful in other fields.  Biologists use microscopes with multiple
lenses to see very small objects.  People in the telecommunications
field use fiber optic cables to carry information at the speed of
light.  Many people benefit from optics by having their vision
corrected with eyeglasses or contact lenses.  

This lab will teach you some of the basic principles of optics which
will allow you to be able to predict what mirrors and lenses will do to
the light which is incident on them.  
At the observatory you use real telescopes, so the basic
skills you learn in this lab will help you understand telescopes better.\\

\begin{itemize}
\item{{\it Goals:}  to discuss the properties of mirrors and lenses, and 
demonstrate them using optics; build a telescope}
\item{{\it Materials:}  optical bench, ray trace worksheet, meterstick}
\end{itemize}

\vspace{0.2in}
\begin{flushleft}
\section{Discussion}
\end{flushleft}
\vspace{0.20in}

The behavior of light depends on how it strikes the surface of an
object.  All angles are measured with respect to the {\bf normal} direction.
The normal direction is defined as a line which is perpendicular to
the surface of the object.  The angle between the normal direction and
the surface of the object is $90^{\circ}$.  Some important definitions
are given below.  Pay special attention to the pictures in Figure 
\ref{fig:normals}
since they relate to the reflective (mirrors) and refractive (lenses)
optics which will be discussed in this lab.
        
\begin{figure}[h]
\centering
\leavevmode
\epsfxsize=1.05\textwidth
\epsfbox{OPTICS/Optics_fig1.eps}
\caption{The definition of the ``normal'' direction {\bf n}, and other angles
found in optics.}
\label{fig:normals}
\end{figure}
 
\begin{itemize}
\item{{\bf n} = line which is always perpendicular to the surface; 
also called the {\it normal}}
\item{{\bf $\theta$$_I$} = angle of {\it incidence}; the angle between the {\it incoming} 
light ray and the normal to the surface}
\item{{\bf $\theta$$_R$} = angle of {\it reflection}; the angle between the 
{\it outgoing} light ray and the normal to the surface}
\item{{\bf $\alpha$$_R$} = angle of {\it refraction}; the angle between the 
{\it transmitted} light ray and the normal direction}
\end{itemize}


\section{Reflective Optics: Mirrors}

How do mirrors work? Let's experiment by reflecting light off of a simple
flat mirror.

As part of the equipment for this lab you have been given a device that
has a large wooden protractor mounted in a stand that also has a flat
mirror. Along with this set-up comes a ``Laser Straight'' laser alignment tool.
Inside the Laser Straight is a small laser. There is a small black switch
which turns the laser on and off. Keep it off, except when performing the
following exercise (always be careful around lasers--they can damage your
eyes if you stare into them!).

With this set-up, we can explore how light is reflected off of a flat mirror.
Turn on the Laser Straight, place it on the wooden part of the apparatus
outside the edge of the protractor so that the laser beam crosses across the
protractor scale and intercepts the mirror. Align the laser at some angle on 
the protractor, making sure the laser beam passes through the {\it vertex} of 
the protractor. Note how the ``incident'' laser beam is reflected. Make a 
sketch of what you observe in the space below.

\vspace{2.5in}

Now experiment using different angles of incidence by rotating the Laser
Straight around the edge of the protractor, always insuring the laser hits
the mirror exactly at the vertex of the protractor. Note that an angle
of incidence of 90$^{\rm o}$ corresponds to the ``normal'' defined above
(see Fig. 9.1.a).
Fill in the following table of angle of incidence vs. angle of reflection.
({\bf 3 pts})
\clearpage

\begin{table}
\caption{Data Table}
\begin{center}
\begin{tabular}{|c|c|}
\hline
Angle of Incidence & Angle of Reflection \\
\hline
20$^{\rm o}$ & \\
\hline
30$^{\rm o}$ & \\
\hline
45$^{\rm o}$ & \\
\hline
60$^{\rm o}$ & \\
\hline
75$^{\rm o}$ & \\
\hline
90$^{\rm o}$ & \\
\hline
\end{tabular}
\end{center}
\end{table}

What do you conclude about how light is reflected from a mirror? ({\bf 2 pts})

~ \\
~ \\

The law governing the behavior of light when it strikes a mirror is
known as the {\bf Law of Reflection}:

\begin{center}
angle of incidence = angle of reflection \\
$\theta$$_I$ = $\theta$$_R$  
\end{center}

OK, now what happens if you make the mirror curved? First let's consider
a \textit{concave} mirror, one which is curved \textit{away} from the
light source.  Try to think about the curved mirror as being made up of
lots of small sections of flat mirrors, and make a prediction for what
you will see if you put a curved mirror in the light path. You might
try to make a drawing in the space below:\\

\smallspace
\vspace{0.6in}

At the front of the classroom is in fact just such a device: A curved wooden
base to which are glued a large number of flat mirrors, along with a metal stand
that has three lasers mounted in it, and the ``disco5000'' smoke machine.
Have your TA turn on the lasers, align them onto the multi-mirror apparatus,
and spew some smoke! Was your prediction correct?\\
\vspace{0.3in}

Also at the front of the room are two large curved mirrors. There are two 
types of curved mirrors, ``convex'' and ``concave''. In a convex mirror, the 
mirror is
curved outwards, in a concave mirror, the mirror is curved inwards (``caved''
in). Light that is reflected from these two types of mirrors behaves in
different ways. In this section of the lab, you will investigate how
light behaves when encountering a curved mirror.

\vspace{0.1in}
 
1) Have your TA place the laser apparatus in front of the convex mirror, and
spew some more smoke. 
{\bf BE CAREFUL NOT TO LET THE LASER LIGHT HIT YOUR EYE.} What happens
to the laser beams when they are reflected off of the convex mirror? Make a
drawing of how the light is reflected (using the attached work sheet, the
diagram labeled ``Convex Mirror'' in Figure \ref{fig:opticws1}. ({\bf 5 pts})
\vspace{0.2in}

2) Now have your TA replace the convex mirror with the concave mirror. Now
what happens to the laser beams? Draw a diagram of what happens (using
the same worksheet, in the space labeled ``Concave Mirror''). ({\bf 5 pts})
\vspace{0.2in}

Note that there are three laser beams. Using a piece of paper, your hand,
or some other small opaque item, block out the top laser beam on the stand.
Which of the reflected beams disappeared?  What happens to the images of the
laser beams upon reflection? Draw this result ({\bf 5 pts}):
\vspace{3.0in}


The point where the converging laser beams cross is 
called the ``focus''. From these experiments, we can draw the conclusion that 
\textit{concave mirrors focus light, convex mirrors diverge light.} Both
of the mirrors are 61 cm in diameter. Using a meter stick, how far
from the mirror is the convergent point of the reflected light (``where
is the best focus achieved'')? ({\bf 3 pts})

\smallspace

This distance is called the ``focal length''. For concave mirrors the
focal length is one half of the ``radius of curvature'' of the mirror.
If you could imagine a spherical mirror, cut the sphere in half. Now you
have a hemispherical mirror. The radius of the hemisphere is the same as the 
radius of the sphere. Now, imagine cutting a small cap off of the 
hemisphere--now you have a concave mirror, but it is a piece of a sphere that 
has the same radius as before!\\

What is the radius of curvature of the big concave mirror? ({\bf 1 pt})\\
\vspace{0.20in}

Ok, with the lasers off, look into the concave mirror, is your face larger
or smaller? Does a concave mirror appear to magnify, or demagnify your
image. How about the convex mirror, does it appear to magnify, or demagnify?
({\bf 1 pt}):
\vspace{0.20in}

\section{Refractive Optics: Lenses}

OK, how about lenses? Do they work in a similar way?

For this section of the lab, we will be using an ``optical bench''
that has a light source on one end, and a projection (imaging) screen on the 
other end. To start with, there will be three lenses attached mounted on 
the optical bench. Loosen the (horizontal) thumbscrews and remove the three 
lenses from the optical bench. Two of the lenses have the same diameter, and 
one lens is larger. Holding one of the lenses by the steel shaft, examine 
whether this lens can be used as a ``magnifying glass'', that is when you look 
through it, do objects appear bigger, or smaller? You will find that two of the 
lenses are ``positive'' lenses in that they magnify objects, and one is a 
``negative'' lens that acts to ``de-magnify'' objects. Note how easy it is
to decide which lenses are positive and which one is the negative lens.

Now we are going to attempt to measure the ``focal lengths'' of these lenses.
First, remount the smaller positive lens back on the optical bench. Turn on
the light by simply connecting the light source to the battery or transformer 
using the alligator clips (be careful not to let the alligator clips touch
each other or else the transformer will be damaged).  Take
the smaller {\it positive} lens move it to the middle of the optical bench 
(tightening or loosing the vertical clamping screw to allow you to slide it 
back and forth). At the one end of the optical bench mount the white plastic 
viewing screen. It is best to mount this at a convenient measurement spot--let's 
choose to align the plastic screen so that it is right at the 10 cm position on
the meter stick. Now slowly move the lens closer to the screen. As you
do so, you should see a circle of light that decreases in size until you
reach ``focus'' (for this to work, however, your light source and lens
have to be at the same {\it height} above the meter stick!). Measure the 
distance between the lens and the plastic screen. Write down this number,
we will call it ``$a$''.\\
~ ~\\
The distance ``$a$'' =  \hspace{1.7cm} cm ({\bf 1 pt})\\
~ ~\\
Now measure the distance between the lens and the front end of the light source. 
Write down this number, we will call it ``$b$'': \\
~ ~ \\
The distance ``$b$'' =  \hspace{1.7cm} cm ({\bf 1 pt})\\
~ ~ \\
To determine the focal length of a lens (``F''), there is a formula called ``the
lens maker's formula'':
~ ~\\
\beq
\frac{1}{F} = \frac{1}{a} + \frac{1}{b}
\eeq
~ ~ \\
Calculate the focal length of the small positive lens ({\bf 2 pts}): F =  \hspace{1.7cm} cm\\
~ ~\\
Now replace the positive lens with the small {\it negative} lens. Repeat
the process. Can you find a focus with this lens? What appears to be
happening? ({\bf 4 pts})

\vspace{4.0cm}
\clearpage
How does the behavior of these two lenses compare with the behavior of mirrors?
Draw how light behaves when encountering the two types of lenses using
Figure \ref{fig:opticws2}.  Note some similarities and differences between what you have
drawn in Fig. \ref{fig:opticws1}, and what you drew in Fig. 
\ref{fig:opticws2} and write them in the space below.  ({\bf 5 pts})

\vspace{6.0cm}

Ok, now let's go back and mount the larger lens on the optical bench. This
lens has a very long focal length. Remove the light source from the optical
bench. Now mount the big lens exactly 80 cm from the white screen. Holding
the light source ``out in space'', move it back and forth until you can get the
best focus (Note that this
focus will not be a point, but will be a focused image of the filament in
the light bulb and show-up as a small, bright line segment.
This is a much higher power lens, so the image is not squished down
like occured with the smaller positive lens). Using the wooden meter stick, 
have your
lab partners measure the distance between the light source and the lens.
This is hard to do, but you should get a number that is close to 80 cm.
Assuming that $a$ = $b$ = 80 cm, use the lens maker's formula to calculate F:\\
~ ~ \\
The focal length of the large lens is F = \hspace{1.7cm} cm ({\bf 2 pts})\\

\subsubsection{Making a Telescope}

As you have learned in class, Galileo is given credit as the first person to 
point a 
telescope at objects in the night sky. You are now going to make a telescope
just like that used by Galileo. Remove the white screen (and light source)
from the optical bench and mount (and lock) the large positive lens at the 
10 cm mark on the yardstick scale. Now mount the small {\it negative} lens about 
40 cm away from the big lens. Looking at the furthest ``eyechart'' mounted in 
the lab room (maybe go to the back of the room if you are up front--you want
to be as far from the eyechart as possible),
focus the telescope by moving the little lens backwards or forwards.
Once you achieve focus, let your lab partners look through the telescope too.
Given that everyone's eyes are different, they may need to re-focus the little
lens. Write down the distance ``N'' between the two lenses: \\
~ ~ \\
The distance between the two lenses is N = \hspace{1.7 cm} cm ({\bf 2 pts})\\
~ ~\\
Describe what you see when you look through the telescope: What does the image 
look like? Is it distorted? Are there
strange colors? What is the smallest set of letters you can read? Is the image
right side up? Any other interesting observations? ({\bf 5 pts}):

\vspace{6.5cm}

This is exactly the kind of telescope that Galileo used. Shortly after Galileo's
observations became famous, Johannes Kepler built his own telescopes, and
described how they worked. Kepler suggested that you could make a better
telescope using two {\it positive} lenses. Let's do that. Remove the small
negative lens and replace it with the small positive lens. Like before, focus
your telescope on the eyechart, and let everyone in your group do the same.
Write down the distance ``P'' between the two lenses after achieving best
focus:\\
~ ~ \\
The distance between the two lenses is P = \hspace{1.7 cm} cm ({\bf 2 pts})\\
~ ~\\
Describe what you see: What does the image look like? Is it distorted? Are there
strange colors? What is the smallest set of letters you can read? Is the image
right side up? Any other interesting observations? ({\bf 5 pts}):

\vspace{6.5cm}

Compare the two telescopes. Which is better? What makes it better? Note
that Kepler's version of the telescope did not become popular until many years 
later. Why do you think that is? ({\bf 5 pts}):
\vspace{4.0cm}

\subsubsection{The Magnifying and Light Collecting Power of a Telescope}

Telescopes do two important things: they collect light, and magnify objects. 
Astronomical objects are very far away, and thus you must magnify the objects
to actually see any detail. Telescopes also collect light, allowing you
to see fainter objects than can be seen by your eye. It is easy to envision
this latter function as two different size buckets sitting out in the rain. The 
bigger
diameter bucket will collect more water than the smaller bucket. In fact,
the amount of water collected goes as the area of the top of the bucket. If 
we have circular buckets, than given that the area of a circle is 
$\pi$R$^{\rm 2}$, a bucket that is twice the radius, has four times the area, 
and thus collects four times the rain. The same relationship is at work for your
eye and a telescope. The radius of a typical human pupil is 4 mm, while the
big lens you have been using has a radius of 20 mm. Thus, the telescopes
that you built collect 25 times as much light as your eyes.

Determining the magnification of a telescope is also very simple: \\
~ ~ \\
\beq
M = \frac{F}{f}
\eeq

Where ``M'' is the magnification, ``F'' is the focal length of the ``objective''
lens (the bigger of the two lenses), and ``f'' is the focal length of the
``eyepiece'' (the smaller of the two lenses). You have calculated both ``F'' and ``f'' in the preceeding for the two positive lenses, and thus can calculate
the {\it magnification of the ``Kepler'' telescope}:\\
~ ~ \\
The magnification of the Kepler telescope is M = \hspace{1.5cm} times. ({\bf 1 pt})\\

Ok, how about the magnification of the Galileo telescope? The magnification
for the Galileo telescope is calculated the same way:\\
 \beq M = \frac{F}{f} \eeq 

But remember, we could not measure a focal length (f) for the negative lens. How
can this be done? With specialized optical equipment it is rather easy
to measure the focal length of a negative lens. But since we do not have
that equipment, we have to use another technique. In the following two 
figures we show a ``ray diagram'' for both the Kepler and Galileo telescopes.\\

\begin{figure}[h]
\centering
\leavevmode
\epsfxsize=0.75\textwidth
\epsfbox{OPTICS/Negative_Eye_Lens.eps}
\caption{The ray diagram for Galileo's telescope.}
\label{fig:negtelly}
\end{figure}
\begin{figure}[h]
\centering
\leavevmode
\epsfxsize=0.75\textwidth
\epsfbox{OPTICS/Positive_Eye_Lens.eps}
\caption{The ray diagram for Kepler's telescope.}
\label{fig:postelly}
\end{figure}

Earlier, we had you make various measurements of the lenses, and measure
separations of the lenses in both telescopes once they were focused. If you 
look at Figure \ref{fig:negtelly} and Figure \ref{fig:postelly}, you will see 
that there is a large ``F''. This is
the focal length of the large, positive lens (the ``objective''). In Kepler's
telescope, when it is focused, you see that the separation between the two
lenses is the sum of the focal lengths of the two lenses. We called this 
distance ``P'', above. You should confirm that the ``P'' you measured above is
in fact equal (or fairly close) to the sum of the focal lengths of the two 
positive lenses: P = f + F (where little ``f'' is the focal length of the
smaller positive lens).

Ok, now look at Figure \ref{fig:negtelly}. Note that when this telescope
is focused, the separation between the two lenses in the Galileo telescope
is N = F $-$ f (where F and f have the same definition as before). 

Find ``f'' for the Galileo telescope that you built, and determine the
magnification of this telescope ({\bf 3 pts}):

\vspace{5cm}

Compare the magnification of your Galileo telescope to that you calculated
for the Kepler telescope ({\bf 2 pts}):

\vspace{3 cm}

What do you think of the quality of images that these simple telescopes
produce? Note how hard it is to point these telescopes. It was hard work
for Galileo, and the observers that followed him, to unravel what they 
were seeing with these telescopes. You should also know that the lenses you
have used in this class, even though they are not very expensive, are far 
superior to those that could be made in the 17$^{\rm th}$ century. Thus,
the simple telescopes you have constructed today are much better than what
Galileo used!
\clearpage
\section{Summary (35 points)}
 Please summarize the important concepts of this
lab.

\begin{itemize}
\item{Describe the properties of the different types of lenses $and$
mirrors discussed in this lab}
\item{What are some of the differences between mirrors and lenses?}
\item{Why is the study of optics important in astronomy?}  
\end{itemize}

\noindent
Use complete sentences, and proofread your lab before handing it in.

\section{Possible Quiz Questions}

\begin{flushleft}
1) What is a ``normal''?\\
2) What is a concave mirror?\\
3) What is a convex lens?\\
4) Why do astronomers need to use telescopes?\\
\end{flushleft}

\section{Extra Credit (ask your TA for permission before attempting, 5 points
)}

Astronomers constantly are striving for larger and larger optics so that
they can collect more light, and see fainter objects. Galileo's first 
telescope had a simple lens that was 1" in diameter. The largest telescopes
on Earth are the Keck 10 m telescopes (10 m = 400 inches!). Just about
all telescopes use mirrors. The reason is that lenses have to be supported
from their edges, while mirrors can be supported from behind. But,
eventually, a single mirror gets too big to construct. For this extra
credit exercise look up what kind of mirrors the 8 m Gemini telescopes
have (at http://www.gemini.edu) versus the mirror system used by the
Keck telescopes (http://keckobservatory.org/about/the\_observatory). Try to find
out how they were made using links from those sites. Write-up
a description of the mirrors used in these two telescopes. How
do you think the next generation of 30 or 100 m telescopes will be built, like
Gemini, or Keck? Why? 

\begin{figure}[h]
\centering
\leavevmode
\epsfxsize=1.05\textwidth
\epsfbox{OPTICS/lensfig2.eps}
\caption{The worksheet needed in section 9.2}
\label{fig:opticws1}
\end{figure}

\begin{figure}[h]
\centering
\leavevmode
\epsfxsize=1.05\textwidth
\epsfbox{OPTICS/lensfig1.eps}
\caption{The worksheet needed in section 9.3. The positive lenses used
in this lab are ``double convex'' lenses, while the negative lens is a
``double concave'' lens.}
\label{fig:opticws2}
\end{figure}

