\section{LC-1: Diffraction and Interference}\label{lc01_diffint}
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\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; Diode laser assembly; two Pasco accessory disks;  short focal length lens; screen; Pasco Interface with light and rotation sensors, mounting bracket, light aperture module \\ 
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 diode laser provides plane light waves of wavelength 670($\pm10$) nm.  The waves in a perpendicular cross section of the beam are in phase (i.e.~the light is said to be ``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. \\ \vspace*{-0.2in}
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{\em CAUTION:} \\
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!\\
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\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}
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\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 \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 why in your lab book that the first {\em minimum} occurs as shown in Fig.~1 at $m=1$. \\ \\
QUESTIONS:

\begin{enumerate} 
\item[{\em Q1.}]  \latex{\vspace*{-.06in}}Qualitatively, how does the pattern vary as the slit width narrows (i.e., rotate the wheel)?
%\item[{\em Q2.}]  \latex{\vspace*{-.06in}}In ``real'' life when would you begin to be concerned about the appearance  of diffraction?  Identify at least one (if not more) example of diffraction in your daily life. (This is not intuitive to novices).
\item[{\em Q2.}] \latex{\vspace*{-.06in}}Qualitatively, how do you expect the pattern vary if the wavelength, $\lambda$, could decrease from red to blue-violet? \\
Click on the Single Slit Diffraction link immediately below (web-version only) to test your prediction. 
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\end{enumerate} 

\item
To enable more {\em quantitative} observations the PASCO interface module has been configured to provide an Intensity (0 to 100\%) 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  on the START icon and move the combined light and rotary motion sensor in the lateral direction {\em gently and smoothly} by hand.  Practice starting on one side of the diffraction pattern and move smoothly towards the other.
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\item Configuring and aligning the optical components is very important!  The laser light should fully illuminate the wheel pattern.  \\
There are two aspects for configuring the light sensor: aperture size and detector gain.  Using a larger aperture lets more light reach the detector (Note: the laser diode intensities can vary from one set up to another).  How does this affect the image profile?  A larger gain increases the sensor output but can more easily saturate the output at maximum and increase the noise.  {\em Adjust and record the gain setting which give you reasonably good results. Ask your instructor for help.} There are four simple steps: 1) CLICK on the SETUP icon 2) DCLICK on the yellow light bulb icon (or Light Sensor) 3) In the Sensor Properties window click on
the ``Calibration" tab and 4) CLICK on Sensitivity pull-down menu. Ask your instructor for a brief demonstration if you are at all uncertain.
\item Turn the PASCO wheel to a single slit width of $ a $=0.1 mm. 
\item Using the cross-hair feature of the PASCO graph display, measure $\theta $ (from $y/D$) for $m$= 1 and -1 plot $m$ vs $\theta$.   From the slope and known value of  $a$, calculate $\lambda$.  Is your value of $\lambda$ comparable with the state value of the diode laser?
%\item Estimate uncertainties and then compare with the accepted value of 633 nm.  Are there any obvious sources of systematic error? (Give me a break, these are 1st and 2nd year undergraduates.)
\end{enumerate}
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\noindent {\bf Experiment II: Two-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, $y=0$, 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 \latex{\vspace*{-.06in}}  Mount the slide containing the four electroformed slits in the slit holder (or rotate the PASCO wheel) and illuminate one of the slit pairs with the diode 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 Note the difference in pattern and spacing of the resulting interference fringes on the screen.  From Experiment I you know that individually these slit exhibit diffraction effects.  Thus you will observe both (i.e., a combination of single and double slit diffraction) interference effects in all patterns.  Observe the interference pattern for four slit pairs.
\item Choose one pattern for careful measurement and measure the separations between a number of adjacent 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 Compare you calculated result with the stated value. 
\end{enumerate}
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\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.] Rotate and adjust the PASCO Slit Accessory wheel 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 the 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 apertures 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}
