\section{L-6: Polarization}\label{l08_pol}

\noindent {OBJECTIVE}:  To study polarization and double refraction of light.

\noindent {APPARATUS}:  
\begin{enumerate}
\item[] \latex{\vspace*{-.07in}} 
Polarization kit plus a polariscope; 
Brewster angle assembly 
(laser + mount on M-2 force table, slotted pin \& black piece of plastic).
\end{enumerate}

\noindent {SUGGESTED~EXPERIMENTS}:  
\begin{enumerate}
\item \latex{\vspace*{-.05in}}{\em Polarization by ABSORPTION}: Observe a light source (desk lamp) thru
each of two polaroids, and then thru both together.  Rotate one of the two 
about the line of sight. Explain what occurs.  

\noindent Orient the two polaroids of the 
{\em polariscope} so no light is transmitted.  Then 
introduce a third polaroid 
between the first two and rotate it about the line of sight.  
Explain what happens.\\ HINT:  there will be an amplitude component 
$E \cos \theta $ in the direction of the transmission axis.

\item \latex{\vspace*{-.05in}}
\html{\begin{tabular}{p{5.5in}l}}
\latex{\parbox[t][1.2in][s]{0.74\linewidth}}
{{\em Polarization by REFLECTION from a dielectric}:  Observe thru 
the small polaroid (the one with the rim index marks)
the light reflected from the black 
plastic (see figure at right). 
Rotate the polaroid about the line of sight.  
Note position of 
the index marks (i.e.~the E vector direction) 
on polaroid when the transmitted reflected light is a minimum.\\}
\html{ & }
\latex{\parbox[b][0.in][b]{0.22\linewidth}}
{\hspace*{.0in} \latex{\vspace*{-1.in}}
\latex{\includegraphics[height=1.0in]{figs/l8-00.eps}}
\html{\includegraphics[height=0.8in]{figs/l8-00.eps}}
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\begin{figure}[h] \latex{\vspace*{-.1in}}\centering 
\latex{\includegraphics[height=1.8in]{figs/l8-01.eps}}
\html{\includegraphics[height=2.2in]{figs/l8-01.eps}}
\caption{Reflection and transmision of light at Brewster's angle.}
\latex{\vspace*{-.1in}}
\end{figure} 


\noindent Now find an incident angle $\theta_i$ (and reflection angle 
$\theta_r$) 
for which this 
minimum is zero.  Then $\theta_i$ is {\em Brewster's angle},~$\theta_B$, and 
$\tan \theta_B = n_2/n_1$.  Estimate $\theta_B$ roughly.  What is the refractive index 
$n_2$ of the plastic?
\begin{enumerate} 
\item[] {\em Alternatively} for greater accuracy use a laser plus polaroid to prepare a 
beam polarized in the plane containing the incident ray and the normal to the 
reflecting surface.  {\em The reflected beam then vanishes at Brewster's 
angle}.  Mount laser plus polaroid on the M-2 force table and point it 
toward a dielectric (plastic or microscope slide) in the slot of the 
central pin.  
Rotate the sample until the reflected beam (on the wall) disappears.  A 
thread from the center pin to this point on the wall will help locate the 
angles on the M-2 table.
\end{enumerate}
\noindent Observe any polarization of light reflected from 
painted or varnished surfaces, floor tile, {\em clean metal} (e.g.~aluminum 
foil) etc.
\item {\em Polarization by SCATTERING}:  Try to detect the polarization 
of skylight: {\bf one needs a blue sky}!  Explain (use a diagram) in
what direction relative 
to sunlight should one look to see this polarization best.
\item {\em Polarization by a DOUBLY REFRACTING CRYSTAL}:  Uniaxial
doubly refracting 
crystals (e.g.~calcite) are highly anisotropic and, as such,
contain an axis of symmetry (the optic axis) for 
which the velocity of light depends on whether the $E$-field vector 
(polarization plane) is $\perp $ to this axis (the ordinary ``$o$'' ray) or 
parallel to the optic axis (the extraordinary ``$e$'' ray).  In the principal 
plane (i.e.~a plane containing the optic axis but $\perp $ to the 
surface) the new wave front 
of the ordinary ray involves the usual spherically expanding wavelets,
but to construct the front for the extraordinary ray requires 
ellipsoidally expanding wavelets where the two axes of the 
ellipse are proportional to the velocities of the parallel and 
$\perp $ vibrations to the optic axis.  

\noindent Calcite (CaCO$_3$) has a large velocity differences 
between the $o$ and $e$ rays, and its optic axis is the direction at equal 
angles with the crystal edges at the obtuse corners.

\noindent Observe through 
the calcite crystal a dot on a piece of paper.  
Rotate the crystal about the line of sight.  Describe the behavior of the 
two images of the dot.  Introduce a polaroid between your eye and the 
crystal. Note how the appearance of the two images changes as you rotate the 
polaroid about the line of sight.  Explain.
\end{enumerate}
\begin{description}
\item {NOTE: For the remaining experiments use the POLARISCOPE.}  The {\em polaroid next to 
the light is the ``POLARIZER,'' the other is the ``ANALYZER.''}  Place the 
samples to be studied on top of the polarizer.
\end{description}

\begin{enumerate}
\item[5.] {\em CIRCULAR POLARIZATION}: 
A thin doubly refracting crystal cut so that 
the optic axis is parallel to a surface constitutes a retardation plate.  
The difference in velocity of 
the $o$ and $e$ rays then gives a phase difference between the emerging 
orthogonally polarized rays.  If the phase difference is $90^\circ $, 
the retardation is $\lambda /4$:  hence a ``{\em quarter wave plate}''. 

\begin{figure}[h] \latex{\vspace*{-.1in}}\centering 
\includegraphics[height=1.8in]{figs/l8-02.eps}
\caption{Obtaining circularly polarized light from unpolarized light}
\latex{\vspace*{-.1in}}
\end{figure} 

\noindent Set the analyzer of the 
polariscope to transmit no light.  
Then place the $\lambda /4$ plate between the polarizer and analyzer and rotate the 
$\lambda /4$ plate so that extinction again results.  You will find two such 
extinction positions $90^\circ $ apart.  Why? \\  HINT:  
When the optic axis is 
aligned with the polarization direction of the incoming plane polarized 
wave, will there be any ordinary ray amplitude?  Then when you turn the 
plate (and thus the optic axis) thru  $90^\circ $, will there be any extraordinary 
ray amplitude.

\noindent Rotate the $\lambda /4$ plate to a position 
halfway between the two extinction positions; thus the optic axis will make 
a $45^\circ $ angle to the incident plane of polarization.  Explain why the 
plate now looks bright.  Leaving the $\lambda/4$ plate in this position, rotate the 
analyzer through $2\pi $ radians.  Why does the brightness stay approximately 
constant$^{\ast}$? \\
HINT: Recall the conditions for a circular Lissajous figure (see E-8 Part 
B, Fig.~3).

\begin{enumerate}
\item[$^{\ast}$]  If it doesn't, repeat the preceding operations.  You may not 
have set the $\lambda/4$ plate close enough to the halfway or $45^\circ $ position. 
Also remember that a $\lambda /4$ plate is exactly $\lambda /4$ only for a single 
$\lambda $. 
\end{enumerate}
\item[6.] {TWO $\lambda /4$ PLATES}:  Place a $\lambda /4$ plate on the 
polarizer so as to produce circularly 
polarized light.  Place a second $\lambda /4$ plate on top of the first and 
similarly oriented.  Observe and explain what happens when you now rotate 
the analyzer.  Turn the top $\lambda /4$ through $90^\circ$ about the line of sight.
Observe and explain what happens now when you rotate the analzyer.
\item[7.] {COLOR~EFFECTS}:  If the plate is thick enough for retardations of several 
(but not too many) wavelengths, {\em color~effects} may result.  For 
a constant refractive index:
\[\lambda _{\mbox{\small red}} \; \sim \; 660 \; \mbox{nm} \; \sim \;
\frac{3}{2}~(\lambda _{\mbox{\small blue}} 
\;\sim \; 440 \; \mbox{nm}) .\]
\noindent Thus a $\lambda /2$ plate for red is a $3\lambda /4$ plate for blue.  
[In general, a 
n($\lambda /2$) plate for red $\sim 3n (\lambda /4$) plate for blue where n is an odd 
integer.]  Hence for white light such a plate at $45^\circ$ between parallel 
polaroids would completely extinguish the red but pass the circularly 
polarized blue light.  The resultant color is complementary to 
that removed.  The 
intensity (hue and saturation) will of course vary with the relative 
orientation of polarizer, the retardation plate, and the analyzer.  As n 
becomes large the range of $\lambda $ extinguished narrows. Hence that 
passed is more nearly white.
\begin{enumerate}
\item[a)] Place thin mica on the polarizer (with analyzer crossed).  Try various 
mica orientations.  Explain.
\item[b)] Repeat a) for the mounted specimen of cellophane tape.
\end{enumerate}
\item[8.] {PHOTO-ELASTIC~EFFECTS:  Strained isotropic 
materials become doubly-refracting}.

\noindent With the polariscope set for extinction:
\begin{enumerate}
\item[a)] Insert and flex the U-shaped lexan (a polycarbonate resin) sample.  
\item[b)] Insert a microscope slide vertically.  View the long edge 
and stress the slide by {\em gently} flexing it.  Note result. (Glass 
blowers use polarized light to test for residual glass strains). 
\end{enumerate}
\item[9.]  {\em LCD APPLICATIONS}: Watches and calculators often use {LIQUID CRYSTAL 
DISPLAYS (LCD)} in which {\em nematic} liquids (thread like molecules arranged 
nearly parallel to each other) can become optically active, 
rotate the plane of polarization and thus permit light to pass thru crossed 
polaroids. A reflector after the analyzer returns the light through the 
cell.

\begin{figure}[h] \latex{\vspace*{-.1in}}\centering 
\includegraphics[height=2.5in]{figs/l8-03.eps}
\caption{Construction of a liquid crystal display}
\latex{\vspace*{-.1in}}
\end{figure} 

\noindent {\em To produce the optical rotation}: two glass plates, which have 
been rubbed in 
one direction to produce invisible scratches, attach aligned {\em nematic} 
molecules.  If a few micron thick {\em nematic} liquid separates the plates, and 
if one plate is rotated relative to the other, then the helical arrangement 
of the {\em nematic} liquid produces a rotation of the plane of polarization.

\noindent {To extinquish the light} one applies a voltage between the transparent 
(but conducting)
tin-oxide coated glass surfaces. A sufficient voltage gradient will 
align the dipole moments of the 
molecules and thus destroy the optical rotatory power.  
Voltages applied to segments 
of the seven segment pattern (see figure) permit display of any numeral 0 to 

\noindent The power consumption of such an LCD is almost negligible. 
\end{enumerate}
\noindent {OPTIONAL~QUESTIONS}:
\begin{enumerate}
\item How experimentally can one tell whether a light beam is 
unpolarized?  Plane polarized?  Circularly polarized?
\item How could you tell whether an object is a gray plastic, a polarizing 
sheet, a $\lambda /4$ plate or a $\lambda /2$ plate?
\item How could you change right circularly polarized light to left 
circularly polarized light?
\item Why does the flexed lexan show colors but not the microscope 
slide?
\end{enumerate}
