252 lines
10 KiB
HTML
252 lines
10 KiB
HTML
<!DOCTYPE html PUBLIC "-//w3c//dtd html 4.0 transitional//en">
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<html>
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<head>
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<meta http-equiv="Content-Type" content="text/html;
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charset=windows-1252">
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<meta name="GENERATOR" content="Mozilla/4.7 [en] (X11; U; OSF1 V4.0
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alpha) [Netscape]">
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<meta name="Author" content="C. L. Davis">
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<title>Light and Optics - Single Slit Diffraction - Physics 299</title>
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</head>
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<body style="color: rgb(0, 0, 0); background-color: rgb(255, 255,
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255);" link="#0000ee" alink="#ff0000" vlink="#551a8b">
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<center>
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<h1><img src="ULPhys1.gif" height="50" align="texttop" width="189">
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</h1>
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</center>
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<center>
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<h1>Single Slit Diffraction<br>
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</h1>
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</center>
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<center><img src="celticbar.gif" height="22" width="576"> <br>
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<br>
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<font color="#ff0000"><i>"<span class="bqQuoteLink"></span></i></font><font
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color="#ff0000"><i>
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<meta http-equiv="content-type" content="text/html;
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charset=windows-1252">
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</i></font><font color="#ff0000"><i>
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<meta http-equiv="content-type" content="text/html;
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charset=windows-1252">
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</i></font><font color="#ff0000"><i>Physics is really nothing
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more than a search for ultimate simplicity, but so far all we
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have is a kind of elegant messiness.” <br>
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</i><font color="#000000">Bill Bryson,</font><i><font
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color="#000000"> <i> A Short History of Nearly Everything </i></font></i></font><br>
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</center>
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<img src="netbar.gif" height="40" align="middle" width="100%">
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<blockquote>
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<div align="center">
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<div align="left">
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<div align="center">
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<div align="left">
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<div align="center">
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<div align="left"> </div>
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</div>
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</div>
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</div>
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</div>
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</div>
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</blockquote>
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<div align="center"> </div>
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<ul>
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<li>Diffraction may be thought of as the "spreading out" of waves
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as they pass through or by an aperture or edge. We now
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investigate this phenomenon more closely for a single slit of
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width "a".</li>
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</ul>
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<ul>
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<li>By considering point sources in pairs across the width of the
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slit we can use the ideas of double slit interference to show
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that minimum intensity is observed at point P on a screen when</li>
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</ul>
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<div align="center"><img alt="eqn1" src="lo_ssdiffraction_eqn1.jpg"
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height="25" width="105"><br>
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<br>
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</div>
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<div align="center"><img alt="fig1" src="lo_ssdiffraction_fig1.gif"
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height="302" width="349">
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<img alt="fig4" src="lo_ssdiffraction_fig4.gif" height="301"
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width="301"><br>
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<div align="left">
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<ul>
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<li><img alt="fig2" src="lo_ssdiffraction_fig2.jpg"
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height="328" align="right" width="245">However, in order
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to obtain the intensity profile as a function of θ we must
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perform a more complex analysis. Considering each
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point in the slit as a point source, for electromagnetic
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waves the electric field at a point P on the screen at right
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can be written,</li>
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</ul>
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<div align="center"><img alt="eqn4"
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src="lo_ssdiffraction_eqn4.jpg" height="38" width="244"><br>
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<div align="left"><br>
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<blockquote>where ω is the angular frequency of the wave and
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Δφ is the phase angle. The relationship between
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phase angle and path difference Δs is given by<br>
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<div align="center"><img alt="eqn2"
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src="lo_ssdiffraction_eqn2.jpg" height="61" width="90"><br>
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<br>
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<div align="left">From the diagram at right the path
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difference Δs, relative to the center, is given by
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ysinθ. In this case<br>
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<div align="center"><img alt="eqn3"
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src="lo_ssdiffraction_eqn3.jpg" height="63"
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width="153"><br>
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<br>
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<div align="left">so that the total electric field
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at a point on the screen is obtained by
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integration over the slit<br>
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<br>
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<div align="center"><img alt="eqn5"
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src="lo_ssdiffraction_eqn5.jpg" height="72"
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width="413"><br>
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<br>
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<div align="left">Performing this integration we
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obtain<br>
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<div align="center"><img alt="eqn6"
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src="lo_ssdiffraction_eqn6.jpg"
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height="59" width="121"><br>
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<div align="left">where <img alt="eqn7"
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src="lo_ssdiffraction_eqn7.jpg"
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height="66" align="top" width="129"><br>
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<br>
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</div>
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</div>
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</div>
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</div>
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</div>
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</div>
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</div>
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</div>
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</blockquote>
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</div>
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</div>
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<ul>
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</ul>
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</div>
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</div>
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<ul>
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<li> The intensity observed is proportional to the square of the
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electric field <br>
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</li>
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</ul>
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<div align="center"><img alt="eqn8" src="lo_ssdiffraction_eqn8.jpg"
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height="83" width="134"><br>
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</div>
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<div align="center"><br>
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<img alt="ssdifffig5" src="lo_ssdiffraction_fig5.jpg" height="334"
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width="583"><br>
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<div align="left"><br>
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<ul>
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<li>Using the intensity function above the minima condition is
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given by sinβ = 0 ( β ≠ 0 ), which means β = nπ where
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n = 1,2,3... Using the definition of β above
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gives the condition for minima</li>
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</ul>
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<div align="center"><img alt="eqn1"
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src="lo_ssdiffraction_eqn1.jpg" height="25" width="105"><br>
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<blockquote>
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<div align="left">as expected.<br>
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<br>
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</div>
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</blockquote>
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<div align="left">
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<ul>
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<li><img alt="exclamation" src="exclamation-icon.gif"
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height="30" width="31"> For n = 0 above, θ = 0, but
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the intensity function leads to the central maximum.</li>
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<li><img alt="exclamation" src="exclamation-icon.gif"
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height="30" width="31"> Note that the width of the
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central maximum - 2λ/a - is double that of secondary
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maxima - λ/a. This is in contrast to the double
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slit interference pattern where all maxima have the same
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width.</li>
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<li><img alt="exclamation" src="exclamation-icon.gif"
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height="30" width="31"> The location of the secondary
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maxima are given <i><b>approximately</b></i> by </li>
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</ul>
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<div align="center"><img alt="eqn9"
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src="lo_ssdiffraction_eqn9.jpg" height="36" width="189"><br>
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<blockquote>
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<div align="left">The exact position of these maxima is
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shifted slightly towards smaller θ.<br>
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</div>
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</blockquote>
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<div align="left">
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<ul>
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<li>In the above analysis we have (implicitly) assumed
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that the source and observation screen are
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infinitely far from the single slit (on opposite
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sides of the slit). This allows us to use the
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plane wave approximation leading to the intensity
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expression above. This is described as <a
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href="http://www.madehow.com/inventorbios/43/Joseph-von-Fraunhofer.html">Fraunhofer</a> <img
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alt="fraunhofer" src="fraunhofer.jpg" height="161"
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align="middle" width="129">diffraction.
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Practically we can approximate a Fraunhofer
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situation by using converging lenses to produce
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parallel rays.</li>
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</ul>
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<ul>
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<li>Without assuming plane waves we must resort to a
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more complex analysis known as <a
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href="http://www.newworldencyclopedia.org/entry/Augustin-Jean_Fresnel"><img
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alt="fresnel" src="fresnel.jpg" height="149"
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align="top" border="0" width="121"></a><a
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href="http://www.rodenburg.org/theory/y1200.html">Fresnel
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diffraction.</a><br>
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</li>
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</ul>
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</div>
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</div>
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<ul>
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</ul>
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</div>
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</div>
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<ul>
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<br>
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<br>
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</ul>
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</div>
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</div>
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<br>
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<img src="netbar.gif" height="40" width="100%"><br>
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<br>
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<center><i><font color="#ff0000">
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<meta http-equiv="content-type" content="text/html;
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charset=windows-1252">
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</font></i><i><font color="#ff0000">
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<meta http-equiv="content-type" content="text/html;
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charset=windows-1252">
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</font></i><i><font color="#ff0000">
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<meta http-equiv="content-type" content="text/html;
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charset=windows-1252">
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</font></i><i><font color="#ff0000">
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<meta http-equiv="content-type" content="text/html;
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charset=windows-1252">
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The Official Unabashed Scientific Dictionary defines a
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transistor as a nun who's had a sex change</font></i><br>
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<br>
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<img src="celticbar.gif" height="22" width="576"> <br>
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<p><i>Dr. C. L. Davis</i><br>
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<i>Physics Department</i><br>
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<i>University of Louisville</i><br>
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<i>email</i>: <a href="mailto:c.l.davis@louisville.edu">c.l.davis@louisville.edu</a>
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<br>
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</p>
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<p><img src="header-index.gif" height="51" width="92"> </p>
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</center>
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<p><br>
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</p>
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</body>
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</html>
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