\section{LC-1: Diffraction and Interference}\label{lc01_diffint}
\input{toolbar/l01_tb0}

\noindent OBJECTIVES:
\begin{enumerate}
\item[] \latex{\vspace*{-.06in}} 
To observe diffraction and interference and  to 
measure the wavelength, $\lambda$, of laser light.
\end{enumerate}

\noindent APPARATUS: 
\begin{enumerate}
\item[] \latex{\vspace*{-.06in}}  
Optical bench; He-Ne laser; slit of variable width; thickness 
gauge; slide of 4 pair of electroformed double slits; short focal length 
lens; small opaque ball tipped pin; small circular holes in thin 
brass; screen; Pasco Interface with light and rotation sensors, mounting
bracket, light aperature module;
traveling microscope; interference 
demonstrations: optical flats (2), Newton's rings, 18 mm gauge blocks (2), 
interferometer \& Na lamp. \smallskip  \end{enumerate}

\noindent INTRODUCTION: 
\begin{enumerate}
\item[] \latex{\vspace*{-.06in}}  
The He-Ne laser provides plane light waves of wavelength 633 nm. 
The waves in a cross section of the beam are in phase (i.e.~the 
light is ``coherent'').  Any finite plane wave will spread by diffraction.  
The spreading is rapid if the beam is narrow, e.g.~after passing the narrow 
slit in Part I below.  If we illuminate 
two closely spaced narrow slits (narrow relative to the slit to slit spacing) 
by the same laser beam, the two spreading beams will 
overlap and interfere:  Part II.

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\framebox[5.2in]{\parbox[b]{5in}{\smallskip {\em CAUTION:} \noindent 
 The~laser~is~a~very~bright~source.  Do not allow the laser beam  
to enter the eye and do not point the beam at anyone!\smallskip \par }}
 
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<IMG SRC="../../lab/gif/caution.gif" ALIGN=center> <br><br> 
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\end{enumerate}

\noindent{\bf Experiment~I: Single Slit Diffraction}:

\begin{figure}[h] \centering 
\includegraphics[height=2.5in]{figs/lc1-01.eps}
\caption {Schematic of single slit diffraction}
\end{figure} 

\noindent THEORY:
\begin{enumerate}
\item[] \latex{\vspace*{-.06in}}  
The 
angular separation in {\em radians} of the first minimum from the center of the 
pattern is 
\[\theta = \lambda /a \]
\noindent where $a$ is the width of the slit.  (For your derivation you may refer to the
text and  remember that for small $\theta $, sin~$\theta ~\cong ~\theta $).
\end{enumerate}

\noindent SUGGESTED PROCEDURE:
\begin{enumerate}
\item[1.] \latex{\vspace*{-.06in}}  
Mount the laser and variable width slit on the optical bench as shown in Fig.~1.  
Observe the pattern on the supplied white screen and qualitatively explain  in your lab book
that the first minimum occurs as shown in Fig.~1 at $m=1$.

\item[] QUESTIONS:

\begin{enumerate} 
\item[{\em Q1.}]  \latex{\vspace*{-.06in}}Qualitatively, how does the pattern vary as the slit is closed?
\item[{\em Q2.}]  \latex{\vspace*{-.06in}}In ``real'' life when would you begin to be concerned about the appearence 
 of diffraction?  Identify at least one (if not more) example of diffraction in your daily
 life.
\item[{\em Q3.}] \latex{\vspace*{-.06in}}Qualitatively, how does the pattern vary as $\lambda$ decreases (from red to
blue-violet)? \\
 Click on the Single Slit Diffraction link immediately below (web-version) 
 to test your prediction. 
\input{toolbar/l01_j1}
\end{enumerate} 

\item[2.]
To enable more {\em quantitative} observations the PASCO interface module has been configured 
to provide an Intensity vs.\ Linear position of the light sensor plot and table.  Launch the
experiment by CLICKing on the telescope icon below (web-version only).  After the PASCO experiment
window pops up, start the data acquisition by CLICKing once (or twice) on the REC button and move
the combined light and rotary motion sensor in the lateral direction gently and smoothly by hand.  
Pratice starting on one side of the
diffraction pattern and move smoothly towards the other.  
\input{toolbar/l01_tb0}

\item[3.] Configuring and aligning the optical components is very important!  
In particular the laser alignment and apperature settings
are VERY important for getting good results.  \\
There are two aspects for configuring the light sensor: aperature size and detector gain.  Using
a larger aperature lets more light reach the detector.  How does this affect the image profile?
A larger gain increases the sensor output but can peg the output at maximum and increase the
noise.  {\em Adjust and record the setting which give you reasonably good results.}
Ask your instructor for a brief demostration if you are at all uncertain.

\item[4.] Use the thickness gauge to set a slit width, $ a \cong $~0.1 mm. It is important
that your slit is parallel.  What would happen if the slits were not parallel to one another?  
\item[5.] Using the cross-hair feature of the PASCO graph display, measure $\theta $ (from $y/D$) for
$m$=2, 1, 0, -1, -2 plot $m$ vs $\theta$.
 Does this form a straight line?  From the slope and known value of  $a$, calculate $\lambda$.
 \item[6.] Estimate 
uncertainties and then compare with the accepted value of 633 nm.  Are there any obvious sources
of systematic error? 
\end{enumerate}
%\newpage

\noindent {\bf Experiment II: Multiple Slit Interference}
\begin{enumerate}
\item[] \latex{\vspace*{-.06in}}  
\noindent For double slit interference, shown in Fig.~2, the distance $y$ to the m$^{th}$ 
bright fringe from the midpoint 0 is \end{enumerate}

\begin{enumerate}
\item[] 
\[ y = D \tan \theta \cong D \theta \mbox{\hspace*{.3in}and \hspace*{.3in}} \theta \cong m\lambda /d \]
\noindent Hence 
\[ \lambda ~=~yd/mD,\] 
(For more information refer to the text and recall that for small $\theta $, 
$\theta \cong \sin\theta \cong \tan \theta$.)
\end{enumerate}
\begin{figure}[h] \centering 
\includegraphics[height=2.8in]{figs/lc1-02.eps}
\caption {Schematic of double (or multiple) slit interference.}
\end{figure} \vspace*{-.1in}

\noindent SUGGESTED PROCEDURE:
\begin{enumerate}
\item[1.] \latex{\vspace*{-.06in}}  
Mount the slide containing the four electroformed slits in the slit holder 
and illuminate one of the slit pairs with the laser.  Two of the slit pairs 
have a center to center slit spacing of 0.25 mm and the other two have a 
0.50 mm spacing.  The slit pairs with the same spacing differ only in the 
width of each slit:  one has slit widths of 0.04 mm and the other has slit 
widths of 0.08 mm.
\item[2.] Note the difference in pattern and spacing of the resulting
interference fringes on the wall.  Neither slit width is negligible  
compared to the wavelength so both diffraction and interference effects are
apparent in all patterns, and the interference fringes are contained within 
the diffraction envelope.  Observe the interference pattern for all four
electroformed slit pairs.  
\item[3.] Choose one pattern for careful measurements and measure the fringe separations
at a number maxima ($m$= -1, 0, 1, etc.) and 
calculate $\lambda $ using the same general procedure as experiment I. 
The separations marked on the slits are only 
nominal.  If time permits use the  
\hyperref{travelling microscope}{traveling microscope  (Appendix~}{)}{TravellingMicroscope}
 on page~\pageref{TravellingMicroscope}
to measure the slit  separations.


\item[4.] Estimate quantitatively your uncertainties.  Compare with the accepted 
value. 
\end{enumerate}
%\newpage

\noindent {\bf Experiment~III: Fresnel~Bright~Spot and Other Interference
Patterns} \smallskip

\noindent INTRODUCTION: 
\begin{enumerate}
\item[] \latex{\vspace*{-.06in}}  
The Fresnel bright spot (also called Poisson's 
bright spot) is a bright spot in the center of the shadow 
cast by every circular obstacle in 
the path of a plane wave.  The effect is implicit in Fresnel's 
representation of 
a coherent plane wave by half-period zones (e.g.~see Shortley and Williams,
``Elements of Physics'' 5th ed.~p.~737 or, if this is the web-version, see this 
\htmladdnormallink{example}{http://www.physics.ucla.edu/~dauger/fresnel/PoissonAragoStory.html}
\latex{at \\ http://www.physics.ucla.edu/~dauger/fresnel/PoissonAragoStory.html
}), but was first pointed out by Poisson 
(a disbeliever in the wave theory) in an attempt to ridicule Fresnel's wave 
theory.  Arago subsequently showed experimentally that the spot existed.  
You, too, may demonstrate the spot:
\begin{enumerate}
\item[1.] Replace the slit support rod with the rod containing the PASCO Slit Accessory wheel.
Rotate and adjust the assembly to illuminate the circular obstacle in the laser 
beam.  
\item[2.] To obtain a large enough coherent plane wave it may be necessary to diverge the laser beam by 
placing a short 
focal length lens directly in front of the laser. 
\item[3.] Carefully align laser plus lens and obstacle to center 
the shadow along the beam axis. The coherently 
illuminated area (the plane wave) should be several times the 
area of the obstacle.  Proper alignment may help significantly 
in ``cleaning up'' the image.\\
\end{enumerate}
\end{enumerate}
 
\noindent OPTIONAL: 
\begin{enumerate}
\item[] \latex{\vspace*{-.06in}} {\bf Diffraction from a small circular opening}:  
By analogy with the Fresnel bright spot in a circular shadow, you 
might expect a dark spot centered in the image of a circular opening.
In this case the situation is more complex: the on-axis Fresnel bright spot in the 
shadow results from superposition of
a large number of higher Fresnel 
half-period zones whereas the light from a small circular aperture comes 
only from a few low Fresnel half-period zones whose superposition 
on axis may result in either a dark or bright spot: 
dark for an even number of zones, bright for an odd. 

\noindent Test this using the 2 circular aperatures on the PASCO Slit Accessory wheel. 
Illuminate the hole with the diverged laser beam
in the same manner
as described above.
By varying the hole size and/or distance from hole
to screen you may change the small number of half-period 
zones contributing and hence see the central point on axis
as either a bright or a dark spot, e.g.~see Fig.~13 Shortley and Williams, 
p.~740.  Again careful alignment is important to obtain a clean symmetric 
image.

\noindent If time permits you may find it interesting to view the four addition 
two-dimensional diffraction patterns. \\

\noindent DEMONSTRATIONS ON THE DISPLAY TABLE:

\begin{enumerate}
\item[] \latex{\vspace*{-.07in}} A. Interference between optical flats, gauge
blocks, etc.\\
       B. Newton's rings  \\
       C. Michelson's interferometer
\end{enumerate}

\end{enumerate}
