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<meta name="Author" content="C. L. Davis">
<title>Light and Optics - Single Slit Diffraction - Physics 299</title>
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<h1><img src="ULPhys1.gif" height="50" align="texttop" width="189">
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<h1>Single Slit Diffraction<br>
</h1>
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<center><img src="celticbar.gif" height="22" width="576"> <br>
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<font color="#ff0000"><i>"<span class="bqQuoteLink"></span></i></font><font
color="#ff0000"><i>
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</i></font><font color="#ff0000"><i>
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</i></font><font color="#ff0000"><i>Physics is really nothing
more than a search for ultimate simplicity, but so far all we
have is a kind of elegant messiness.<2E> <br>
</i><font color="#000000">Bill Bryson,</font><i><font
color="#000000"> <i> A Short History of Nearly Everything </i></font></i></font><br>
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<img src="netbar.gif" height="40" align="middle" width="100%">
<blockquote>
<div align="center">
<div align="left">
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<div align="center">
<div align="left"> </div>
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<div align="center"> </div>
<ul>
<li>Diffraction may be thought of as the "spreading out" of waves
as they pass through or by an aperture or edge.&nbsp; We now
investigate this phenomenon more closely for a single slit of
width "a".</li>
</ul>
<ul>
<li>By considering point sources in pairs across the width of the
slit we can use the ideas of double slit interference to show
that minimum intensity is observed at point P on a screen when</li>
</ul>
<div align="center"><img alt="eqn1" src="lo_ssdiffraction_eqn1.jpg"
height="25" width="105"><br>
<br>
</div>
<div align="center"><img alt="fig1" src="lo_ssdiffraction_fig1.gif"
height="302" width="349">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<img alt="fig4" src="lo_ssdiffraction_fig4.gif" height="301"
width="301"><br>
<div align="left">
<ul>
<li><img alt="fig2" src="lo_ssdiffraction_fig2.jpg"
height="328" align="right" width="245">However, in order
to obtain the intensity profile as a function of &#952; we must
perform a more complex analysis.&nbsp; Considering each
point in the slit as a point source, for electromagnetic
waves the electric field at a point P on the screen at right
can be written,</li>
</ul>
<div align="center"><img alt="eqn4"
src="lo_ssdiffraction_eqn4.jpg" height="38" width="244"><br>
<div align="left"><br>
<blockquote>where &#969; is the angular frequency of the wave and
&#916;&#966; is the phase angle.&nbsp; The relationship between
phase angle and path difference &#916;s is given by<br>
<div align="center"><img alt="eqn2"
src="lo_ssdiffraction_eqn2.jpg" height="61" width="90"><br>
<br>
<div align="left">From the diagram at right the path
difference &#916;s, relative to the center, is given by
ysin&#952;.&nbsp; In this case<br>
<div align="center"><img alt="eqn3"
src="lo_ssdiffraction_eqn3.jpg" height="63"
width="153"><br>
<br>
<div align="left">so that the total electric field
at a point on the screen is obtained by
integration over the slit<br>
<br>
<div align="center"><img alt="eqn5"
src="lo_ssdiffraction_eqn5.jpg" height="72"
width="413"><br>
<br>
<div align="left">Performing this integration we
obtain<br>
<div align="center"><img alt="eqn6"
src="lo_ssdiffraction_eqn6.jpg"
height="59" width="121"><br>
<div align="left">where <img alt="eqn7"
src="lo_ssdiffraction_eqn7.jpg"
height="66" align="top" width="129"><br>
<br>
</div>
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</blockquote>
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<ul>
</ul>
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<ul>
<li> The intensity observed is proportional to the square of the
electric field <br>
</li>
</ul>
<div align="center"><img alt="eqn8" src="lo_ssdiffraction_eqn8.jpg"
height="83" width="134"><br>
</div>
<div align="center"><br>
<img alt="ssdifffig5" src="lo_ssdiffraction_fig5.jpg" height="334"
width="583"><br>
<div align="left"><br>
<ul>
<li>Using the intensity function above the minima condition is
given by sin&#946; = 0&nbsp; ( &#946; &#8800; 0 ), which means &#946; = n&#960; where
n = 1,2,3...&nbsp;&nbsp; Using the definition of &#946; above
gives the condition for minima</li>
</ul>
<div align="center"><img alt="eqn1"
src="lo_ssdiffraction_eqn1.jpg" height="25" width="105"><br>
<blockquote>
<div align="left">as expected.<br>
<br>
</div>
</blockquote>
<div align="left">
<ul>
<li><img alt="exclamation" src="exclamation-icon.gif"
height="30" width="31"> For n = 0 above, &#952; = 0, but
the intensity function leads to the central maximum.</li>
<li><img alt="exclamation" src="exclamation-icon.gif"
height="30" width="31"> Note that the width of the
central maximum - 2&#955;/a - is double that of secondary
maxima - &#955;/a.&nbsp; This is in contrast to the double
slit interference pattern where all maxima have the same
width.</li>
<li><img alt="exclamation" src="exclamation-icon.gif"
height="30" width="31"> The location of the secondary
maxima are given <i><b>approximately</b></i> by&nbsp;</li>
</ul>
<div align="center"><img alt="eqn9"
src="lo_ssdiffraction_eqn9.jpg" height="36" width="189"><br>
<blockquote>
<div align="left">The exact position of these maxima is
shifted slightly towards smaller &#952;.<br>
</div>
</blockquote>
<div align="left">
<ul>
<li>In the above analysis we have (implicitly) assumed
that the source and observation screen are
infinitely far from the single slit (on opposite
sides of the slit).&nbsp; This allows us to use the
plane wave approximation leading to the intensity
expression above.&nbsp; This is described as <a
href="http://www.madehow.com/inventorbios/43/Joseph-von-Fraunhofer.html">Fraunhofer</a>&nbsp;<img
alt="fraunhofer" src="fraunhofer.jpg" height="161"
align="middle" width="129">diffraction.&nbsp;
Practically we can approximate a Fraunhofer
situation by using converging lenses to produce
parallel rays.</li>
</ul>
<ul>
<li>Without assuming plane waves we must resort to a
more complex analysis known as <a
href="http://www.newworldencyclopedia.org/entry/Augustin-Jean_Fresnel"><img
alt="fresnel" src="fresnel.jpg" height="149"
align="top" border="0" width="121"></a><a
href="http://www.rodenburg.org/theory/y1200.html">Fresnel
diffraction.</a><br>
</li>
</ul>
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<br>
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<br>
<img src="netbar.gif" height="40" width="100%"><br>
<br>
<center><i><font color="#ff0000">&nbsp;
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The Official Unabashed Scientific Dictionary defines a
transistor as a nun who's had a sex change</font></i><br>
<br>
<img src="celticbar.gif" height="22" width="576"> <br>
&nbsp;
<p><i>Dr. C. L. Davis</i><br>
<i>Physics Department</i><br>
<i>University of Louisville</i><br>
<i>email</i>: <a href="mailto:c.l.davis@louisville.edu">c.l.davis@louisville.edu</a>
<br>
&nbsp; </p>
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