davisnotes/lo_dsdiffraction.html

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<title>Light and Optics - Double Slit Diffraction/Interference -
Physics 299</title>
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<h1>Double Slit Diffraction/Interference<br>
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<font color="#ff0000"><i>"<span class="bqQuoteLink"></span></i></font><font
color="#ff0000"><i> </i></font><font color="#ff0000"><i>
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All of physics is either impossible or trivial. It is
impossible until you understand it, and then it becomes
trivial.
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<span></span><span class="bqQuoteLink"></span><span></span>"</i></font><br>
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Ernest Rutherford<br>
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<li> <big><b>Intensity pattern for single slit diffraction:</b></big></li>
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<div align="center"><img alt="eqn1" src="lo_ssdiffraction_eqn8.jpg"
height="83" align="middle" width="134">&nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;&nbsp; where&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp; <img
alt="eqn2" src="lo_ssdiffraction_eqn7.jpg" height="66"
align="middle" width="129"><br>
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<li><big><b>Intensity pattern for double slit interference:</b></big></li>
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<div align="center"><img alt="eqn1" src="lo_dsdiffraction_eqn1.jpg"
height="37" align="middle" width="118">&nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp; where&nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;&nbsp; <img alt="eqn4"
src="lo_dsdiffraction_eqn4.jpg" height="59" align="middle"
width="119"><br>
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<div align="left">Note that &#966;' = &#966;/2, where &#966; is defined in the
double slit interference analysis.<br>
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<li>Remember that in the double slit interference analysis we
(implicitly) assumed that the two slits were "point" wave
sources.&nbsp; Of course in reality this is impossible, no
sources are true "point" sources.&nbsp; An accurate
description of double slit interference must include the
effect of the finite width of each slit - that is, the
diffraction phenomena from each slit.&nbsp; A detailed
mathematical analysis leads to an intensity pattern at point
P given by,</li>
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<div align="center"><img alt="eqn5"
src="lo_dsdiffraction_eqn5.jpg" height="63" width="175"><br>
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<div align="left">Qualitatively, the interference pattern is
modulated by the diffraction pattern, as indicated below.<br>
<div align="center"><img alt="fig1"
src="lo_dsdiffraction_fig1.gif" height="339"
width="393"><br>
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<div align="left"><img alt="exclamatiom"
src="exclamation-icon.gif" height="30" width="31"> Note
that the slit width "a" must be smaller than the slit
separation "d" (center to center).<br>
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<div align="left"><img alt="exclamation"
src="exclamation-icon.gif" height="30" width="31"> The
number of interference maximum underneath the central
maximum of the diffraction envelope depends on the
relative values of a and d.<br>
<br>
<img alt="exclamation" src="exclamation-icon.gif"
height="30" width="31"> Diffraction minima occur at &#952; =
&#955;/a , 2&#955;/a , 3&#955;/a...&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; [ asin&#952;
= n&#955; ]<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
Interference minima occur at&nbsp; &#952; = &#955;/2d , 3&#955;/2d ,
5&#955;/2d...&nbsp;&nbsp;&nbsp;&nbsp; [ dsin&#952; = (n+1/2)&#955; ]<br>
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<li>In the examples below the first intensity pattern
has&nbsp; d = 3a, whereas the second pattern has d =
(15/2)a<br>
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<div align="left"><br>
<div align="center"><img alt="fig2"
src="lo_dsdiffraction_fig2.jpg" height="231"
width="489">&nbsp; <img alt="fig3"
src="lo_dsdiffraction_fig3.jpg" height="204"
width="340"><br>
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<center><i><font color="#ff0000">&nbsp;
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</font></i><i><font color="#ff0000">A year after almost failing
her high school physics class, a girl told her older brother,
<br>
"You know, my physics teacher <i><b>was</b></i> right about
the optical Doppler effect. You see those cars. The lights of
the ones approaching us are white, but the lights of the ones
moving away from us are red." </font></i><br>
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<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>
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