\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 only instruments
astronomers can use to look at 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.  In the following lab, we will deal
with the actual telescope instruments and camera devices, 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 an optics kit and worksheet}
\item{{\it Materials:}  optics kit, ray trace worksheet, ruler}
\end{itemize}

\vspace{0.2in}
\begin{flushleft}
{\bf \Large 8.2 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 1
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.}
\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}


\vspace{0.20in}
\begin{flushleft}
{\bf \Large 8.2.1 Reflective Optics: Mirrors}
\end{flushleft}
\vspace{0.20in}

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

~ ~

In small groups, set up the simple optics kits, which contain a
light source, a white metal slit mask, and a flat mirror
that mounts into a protractor.  If not already done, hook the alligator
clips at the ends of the wires of the lamps to the battery to turn on
the light (the lamp simply lays on the desk sideways, while the metal slit
mask sits on top of the lamp).

Place the white metal mask with the slits directly in front of the light 
source.  Line up the slits at the front of the tray with the
arrows on the worksheet (top panel of Figure 2 at the end of this handout).  
Insert the plane mirror upright into the
protractor, and place this in the designated position on the
worksheet (yes, the semicircle in this worksheet is smaller than your 
protractor, but you
get the idea).  Make a dark point at the origin of the protractor and
some dark marks every 10 degrees, so you will be able to measure the
angles and point the light source correctly even after the room lights
are turned off (which they will need to be off for you to see the light rays).
With the light source on (and room lights off), and trace the incident and
reflected light rays (Do your best drawing under the protractor).
Measure the angles of incidence and reflection and write the values
below.  

\vspace{0.7in}

Now experiment using different angles of incidence by rotating the protractor.
Make a small table of angle of incidence vs. angle of reflection.
\textit{(3 points)}

\smallspace

\begin{flushleft}
What do you conclude about how light is reflected from a mirror? \textit{(2 points)}
\end{flushleft}

\smallspace

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{convex} mirror, one which is curved \textit{towards} 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{0.4in}

At the front of the room are two large curved mirrors, and a wooden stand
that holds up to eight laser pointers. 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 pointers in front of the convex mirror. 
{\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 2). \textit{(5 points)}
\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 attached work sheet, labeled ``Concave Mirror'' in Figure 2). \textit{(5 points)}
\vspace{0.2in}

Note that the central six laser beams are arranged in the shape of an arrow. 
What happens to the image of the arrow after reflection? (You may wish to use 
one of the small white plastic screens (or a 3x5 index card) that comes with the optical bench
set to view the image of the arrow)? Draw this result \textit{(5 points)}:
\vspace{2.5in}


The point where the converging of the laser beams happens 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'')? \textit{(3 points)}

\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? \textit{(2 points)}\\


\vspace{0.20in}
\begin{flushleft}
{\bf \Large 8.2.2 Refractive Optics: Lenses}\\
\end{flushleft}
\vspace{0.20in}

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

At your table is an appartus called an ``optical bench''. It is basically
a meter stick with some lens holders and a source of light to allow you to 
mount various bits
of optics and experiment with how lenses work. As with mirrors,
there are two types of lenses: convex and concave. But in the case of lenses,
you have two surfaces, so one side can be convex, and one side can be
concave. Or, you can have both sides being convex, or concave. The latter
types of lenses are called ``double convex'' and ``double concave''. We
will be experimenting with these two types of lenses.\\

Mount the light source on the right hand end of the bench by sliding the
shaft into the hole and clamping it tight with the small screw. Connect the 
light source to the battery using the alligator clips. Take
the double convex lens (the larger of the two lenses) and mount it in the middle of the optical bench 
(tightening or loosing the clamping screw to allow you to slide it into
the mounting hole, and the horizontal screw to hold it in place on the bench). 
At the opposite 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. \\
~ ~\\
The focal length of the lens is:  \hspace{0.7cm} cm\\
~ ~\\
Now replace the double convex lens with the double concave lens. Repeat
the process. Can you find a focus with this lens? What appears to be
happening? \textit{(5 points)}

\smallspace

How does the behavior of lenses compare with the behavior of mirrors?
Draw how light behaves when encountering the two types of lenses using
Figure 3.  Note some similarities and differences between what you have
drawn in Figure 2, and what you drew in Figure 3.  \textit{(5 points)}

\smallspace

\vspace{0.20in}
\begin{flushleft}
{\bf \Large 8.2.3 What can optics do for you?}\\
\end{flushleft}
\vspace{0.20in}

Write down a list of different objects in your everyday experience which
use optics.  \textit{(5 points)}

\smallspace

Consider a camera, binoculars, microscope, and telescope. What is the
function of optics in each of these objects?  \textit{(5 points)}

\smallspace

\subsubsection{Making Images}

One important function of optics is to make images of things. In fact,
your eyes contain small lenses that produce images on our retinas which allow 
us to see! This is also what a camera does (but instead of a retina, you
have film).

To make an image, optics takes the light which comes from an object
and concentrates it all in one place. Considering your experiments with
mirrors and lenses above, which shape of mirror and which shape of lens
makes the light converge?  \textit{(5 points)}

\smallspace

To demonstrate how images are made, look at some object inside the
classroom (such as an overhead light) or outside the window. Hold up a white piece of paper towards the 
object. Note that you
cannot see the object on the paper! Now place the double convex lens in front of
the paper; you should locate it about one focal length (measured above)
in front of the paper. Can you see the image of your object?

\smallspace

\subsubsection{Magnifying/Demagnifying}

Another function of optics is to magnify or demagnify things, that is,
to make them appear larger or smaller. Use your double concave and convex 
lenses to look at nearby objects (for example, the table top, or the lab manual,
etc.) and see how magnification works. Experiment with placing the lens
near to the object you want to look at and then move it farther away.

First try the convex lens. Does it magnify or demagnify? How does
the magnification depend on the distance of the lens from the object?
\textit{(5 points)}

\smallspace

Next try the concave lens. Does it magnify or demagnify? How does the
magnification depend on the distance of the lens from the object?
\textit{(5 points)}

\smallspace

Note that the behavior of magnification can get a bit complicated: it
can change depending on both the location of the lens from the object
and also your eye from the lens. However, the behavior of a lens and
its magnification actually follows a relatively simple mathematical
behavior. We won't go into it here, but ask your instructor if you are
curious about learning more.

\subsubsection{Collecting Light}

Finally we come to the main function of optics in astronomical telescopes,
namely, to collect light. When optics are used to make images, light
coming to different locations is all focussed to a single location.
As a result, if you use a bigger optical element, you can collect
more light into your image, and, as a result, make the image appear
\textit{brighter}. This is the main function of telescopes: to make
faint objects appear brighter, and is the reason why telescope optics
are so big. Astronomical objects are faint, so to collect enough light,
we collect light which is going to fall over a wide area and concentrate
it all in a certain point.

To demonstrate this, use your convex lens to look at the outdoor object
again. Now use a piece of paper to cover up a portion of the lens.
How does the appearance of your image change as different
portions of the lens are covered?  \textit{(5 points)}

\smallspace

\vspace{0.20in}
\begin{flushleft}
{\bf \Large 8.3 Summary}\\
\end{flushleft}
\vspace{0.20in}

({\it 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.
\clearpage

\begin{figure}[h]
\centering
\leavevmode
\epsfxsize=1.05\textwidth
\epsfbox{OPTICS/lensfig2.eps}

\caption{The worksheet needed in section 8.2.1}
\end{figure}

\clearpage
\begin{figure}[h]
\centering
\leavevmode
\epsfxsize=1.05\textwidth
\epsfbox{OPTICS/lensfig1.eps}
\caption{The worksheet needed in section 8.2.2}
\end{figure}
