\section{L-4: The optics of the eye and resolving power}\label{l04_eyeres}

\noindent OBJECTIVES: 
\begin{enumerate}
\item[] \latex{\vspace*{-.06in}}
To study eye optics and how resolving 
power depends on aperture.
\end{enumerate}

\noindent APPARATUS:
\begin{enumerate}
\item[] \latex{\vspace*{-.06in}}PART I:  Eye model, (fill with distilled water to simulate the aqueous \& vitreous humors); light source, lens set, 
6`` plastic ruler. 
\item[] \latex{\vspace*{-.06in}}PART II: (2) Optical benches, Na lamps, transparent mm slides, vernier calipers, L-2 telescopes; (1) 15 meter tape.

\begin{figure}[h] \centering 
\includegraphics[height=2.2in]{figs/l4-01.eps}
\caption{Various schematics for the eye}
\end{figure} 

\item[] \latex{\vspace*{-.06in}}
You may fix
the moveable retina (cylindrical) in one of three positions:
\begin{enumerate}
\item[] \latex{\vspace*{-.1in}}R for the normal eye, 
\item[] \latex{\vspace*{-.1in}}R$_{h}$ for the hypermetropic (farsighted) eye, 
\item[] \latex{\vspace*{-.1in}}R$_{m}$ for the myopic (nearsighted) eye. 
\end{enumerate}

 (An eye retina is actually 
spherical, but for small images the results are similar.)\\

\noindent The eye model has a {\bf fixed cornea C} and racks for other lenses.
\\

\noindent A lamp box serves as source for most of the experiment.\\

\noindent Set of lenses: (Strength in diopters = 1/f where f is in meters.)\\

\begin{tabular}{|lr@{.}lr@{.}l|} \hline
&\multicolumn{2}{c}{\bf Diopters}\hspace*{.5in} & \multicolumn{2}{r|}{\bf Focal~lengths~in~air}\\
1. Converging spherical &  + 7 & &0 &14 m\\
2. Converging spherical &  +20 & &  0&05 m\\
3. Converging spherical &  + 2&00 & 0&50 m\\
4. Diverging spherical & - 1&75 & -0&57 m\\
5. Diverging cylindrical & - 5&50 & -0&182 m\\
6. Converging cylindrical & + 1&75 &  0&57 m\\
7. Diaphragm with small hole & ~& ~& ~&~ \\\hline 
\end{tabular}
\end{enumerate}


\newpage
\begin{center}
PART I - EXPERIMENTS WITH THE EYE MODEL
\end{center}
\begin{enumerate}
\item \latex{\vspace*{-.07in}}{ACCOMMODATION}:
With the retina in the normal position, point the eye model at a window or 
other bright object 4 to 5 meters away.  Insert the +7 diopter lens in 
the water at the inside mount farthest from the cornea.  An 
image of the bright object should be in focus on the retina.

\noindent Next use as object the lamp box and place it about 30 cm from the cornea. 
The image is blurred until one replaces the +7 diopter lens by the +20 
diopter lens.  This change illustrates accommodation.  
Eye muscles make the lens thicker for close vision.
(Under relaxed conditions the ligaments supporting 
the lens are in radial tension and hence thin the lens.  The ciliary 
muscle contracts against these ligaments allowing the lens to 
thicken for viewing near objects.  See Fig.~1.)
\item \latex{\vspace*{-.07in}}{NEAR~and~FARSIGHTEDNESS} (MYOPIA and HYPERMETROPIA):
With lamp box at 30~cm and the 20 diopter lens at L, place the retina 
in the position R$_{m}$.  Decide which lens placed in slot S$_1$ or S$_2$ will 
bring the image into focus. Try it.

\noindent Make the eye {\em farsighted} by moving the retina to R$_{h}$, and pick the 
proper lens to place in front of the eye to bring the image into focus.  

\noindent Focus again on a window 4 to 5 meters away when the +7 diopter lens is in the 
{\em normal} position L and the retina in position R.  

\noindent Make the eye {\em nearsighted} by moving the retina to R$_{m}$.  Decide which 
lens will bring the image into focus. Try it.

\noindent Remove the correcting lens; the image again blurs.  At S$_2$ 
place the diaphragm with hole and note image improvement.  
With sufficient light
try a smaller hole (e.g.~a hole thru masking tape 
on the diaphragm).  Explain the effect of the reduced aperture.
\item \latex{\vspace*{-.07in}}{ASTIGMATISM}:
With the +20 diopter lens at L in the eye and the retina in the normal 
position, adjust the distance to the lamp box to get a sharp image on the 
retina.  Then produce astigmatism by placing
the strong diverging cylindrical lens in mount G$_1$.  (In the human eye 
astigmatism generally results from
a cornea with different curvatures in 
the horizontal and vertical directions.)  
Turn this lens and note the effect on the image. 

\html{ \begin{tabular}{p{3.5in}l} }
\latex{\parbox{0.4\linewidth}}
{Correct this defect by placing the converging 
cylindrical lens in front of the cornea and turning it to the position for 
which the image becomes sharp.  Note the directions of the axes of the two 
cylindrical lenses.  ({\em They are not necessarily aligned with the tab}.)
}
\html{ & }
\latex{\parbox{0.50\linewidth}}
{\hspace*{.55in}
\latex{\includegraphics[height=1.2in]{figs/l4-02.eps}
\begin{center} {Figure~2:} Cylindrical lens. \end{center}}
\html{\begin{center} {\bf Figure~2:} Cylindrical lens.
\end{center}}
\html{\includegraphics[height=1.8in]{figs/l4-02.eps}}
}
\html{ \cr \end{tabular}  }


\item \latex{\vspace*{-.12in}}{COMPOUND~DEFECTS}:
Combine tests 2 and 3, i.e.~make a nearsighted astigmatic eye. 
Correct by using both a cylindrical and spherical lens in front of the 
cornea.  How is this double correction managed in actual spectacles?
\item \latex{\vspace*{-.07in}}{CATARACTS~and~LENS REMOVAL}:
A cataract results from a cloudy lens.  A cataract operation involves 
removal of the lens. An implanted plastic lens or a strong convergent lens 
in front of the eye restores vision.
Remove 
the eye lens (L) of the model and place the +7 diopter lens in 
front of the cornea.  A clear image reappears for nearby objects.

\item \latex{\vspace*{-.2in}}{Action~of~a~SIMPLE~MAGNIFIER}:
With the lamp box at 35~cm, the +20 diopter lens and the retina in the 
normal position, measure the size of the image.  Now place the +7 diopter 
lens in front of the cornea and move the lamp box toward the eye until the 
image becomes clear.  Record the increased size of the image.
With the lamp box at 35~cm, the 20 diopter lens and the retina in the 
normal position, note the size of the image.  Now make the eye nearsighted 
and bring the lamp box toward the eye until the image becomes clear.  Note 
the increased size of the image in the nearsighted eye.\latex{\vspace*{-.1in}}
\end{enumerate} 
\noindent NOTE: For a real eye and lens the index of refraction, $\mu $, of the lens varies 
from 1.42 at its nucleus to 1.36 at its edges, and these values are not so 
different from those of the adjoining aqueous and vitreous humor 
($\mu $~=~1.330 and $\mu $~=~1.337).  Hence the real eye lens is very weak and is 
primarily for fine focus adjustment.  Most of the refraction in the eye 
actually occurs at the cornea where $\mu $ changes from 1.0 (in air) 
to 1.37 in the cornea.\latex{\vspace*{-.05in}}
\begin{center}
\hrule \latex{\vspace*{-.12in}}
\end{center}
\begin{center}
PART II  - RESOLVING POWER \\
{\em Before doing this part, read in your text about resolving power}.\latex{\vspace*{-.08in}}
\end{center}

\noindent {SUGGESTED EXPERIMENTS}:
\begin{enumerate}
\item \latex{\vspace*{-.08in}}Use as a source a slide with mm divisions and illuminated by a sodium 
discharge lamp.  The wavelength of the sodium yellow light is 589.3~nm.

\item \latex{\vspace*{-.08in}}Back away from the slide until the mm lines become indistinct.
Then advance far enough so that they are just clearly visible; this 
procedure insures that eye defects or the density of rods and 
cones do not (for this measurement) limit the resolving power.  

\noindent Observing now with only one eye, place a vernier 
caliper\footnote{In all the following be sure you are not looking through the hole at one
end of the jaws.  It is advisable to cover the hole with masking tape.}
in front of the eye pupil 
with the caliper jaws parallel to the mm marks and open about 3 mm.  
Slowly close the jaws until the mm marks disappear.  Note the 
separation of the jaws, $w$, and measure the distance, $D$, to the slide from 
the jaws. 

\noindent Compare your measurements of the angular separation $\alpha $ of the mm marks at 
your observing distance,
\[\alpha = \frac{1~\mbox{mm}}{D~\left( \mbox{in~mm}\right) },\] 
\noindent with the expression $\theta $~=~$\lambda/w$ where $\theta $ 
is the angular limit of resolution, (Rayleigh's criterion).
\item \latex{\vspace*{-.1in}}Mount a telescope so its objective is at the same distance from the slide. 
Repeat the experiment with the caliper in front of the 
objective.\footnote{Be sure no light enters the telescope 
except through the slit jaws.  (You 
may need masking tape on the caliper jaws to prevent this from happening.)}
How do the results compare to your measurement without the telescope?
\item \latex{\vspace*{-.1in}}Move the telescope $\geq 4$ m from the slide and repeat the 
experiment.
\item \latex{\vspace*{-.1in}}Turn the vernier jaws perpendicular to the mm lines and repeat 
experiment \#1.  Comment 
on the result.
\end{enumerate}
\noindent {QUESTIONS}:
\begin{enumerate}
\item \latex{\vspace*{-.08in}}Would an eye without any defects of vision have better 
resolving power at dusk or in bright sunlight?  Explain.
\item \latex{\vspace*{-.1in}}Discuss how a contact lens functions.  Would a ``soft'' contact 
lens correct well for astigmatism?
\end{enumerate}
