592 lines
29 KiB
HTML
592 lines
29 KiB
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<!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 - Double Slit Interference - 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);" alink="#ff0000" link="#0000ee" vlink="#551a8b">
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<center>
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<h1><img src="ULPhys1.gif" align="texttop" height="50" width="189">
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</h1>
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</center>
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<center>
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<h1>Double Slit Interference</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">We only have
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to look at ourselves to see how intelligent life might
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develop into something we wouldn't want to meet.</span></i><span></span><i>"</i></font><br>
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<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> Stephen Hawking<br>
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<br>
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</center>
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<img src="netbar.gif" align="middle" height="40" width="100%"> <br>
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<br>
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<ul>
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<li>We now begin a discussion of wave (physical) optics in which -
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in contrast to geometric optics - we explicitly consider the
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wave nature of light. Remember, in geometric optics light
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traveled in straight lines (light rays). This is a valid
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approximation so long as we do not consider apertures with
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dimensions similar to the wavelength of the light or look too
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closely at the edges of objects.</li>
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</ul>
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<ul>
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<li>So what happens when a wave passes through apertures whose
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size is similar to the wavelength of the wave ? First
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we'll consider the case of double slit interference, in which a
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parallel beam of incident monochromatic (containing a specific
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wavelength) light from the left strikes a screen with two slits,
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S<sub>1</sub> and S<sub>2</sub> , as below.</li>
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</ul>
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<div align="center"><img alt="interference fig1"
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src="lo_interference_fig1.jpg" height="323" width="404"><br>
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<br>
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<img alt="divider" src="divider_ornbarblu.gif" height="64"
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width="100%"><br>
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<br>
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<u><b><big>CONDITIONS for MAXIMA and MINIMA</big></b></u><big><u><b>
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in DOUBLE SLIT INTERFERENCE</b></u></big><br>
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<br>
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<div align="left">
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<ul>
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<li>With L >> d geometric optics predicts that two
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bright spots would be observed on the right hand screen
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immediately opposite S<sub>1</sub> and S<sub>2</sub> .
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The rest of this screen would be in shadow. What is
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actually observed on the right hand screen is an
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"interference pattern" as indicated below,</li>
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</ul>
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<div align="center"><img alt="interference fig2"
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src="lo_interference_fig2.jpg" height="371" width="601">
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<img alt="double slit interference"
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src="lo_interference_fig3_inf.gif" height="362" width="500"><br>
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<div align="left">
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<ul>
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<li>The explanation is that each slit acts as a source of
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spherical waves, which "interfere" as they move from
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left to right as shown above.</li>
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</ul>
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<ul>
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<li>In the diagram at the top of the page, light reaching
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P from S<sub>1</sub> and S<sub>2</sub> will travel
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different distances. Assuming that the light from
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the two sources S<sub>1</sub> and S<sub>2</sub> are
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initially in phase, then due to the path difference S<sub>1</sub>P
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- S<sub>2</sub>P , at P the two waves will be out of
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phase. If the path difference is equal to an
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integral number of wavelengths the waves will interfere
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constructively, leading to a bright spot on the
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screen. Mathematically we can write this <b>condition
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for maximum intensity</b> as,</li>
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</ul>
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<div align="center"><img alt="interference eqn1"
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src="lo_interference_eqn1.jpg" height="30" width="120"><br>
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<blockquote>
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<div align="left">where n can take on integer values, n
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= 0, 1, 2, 3... and we have assumed that θ = θ' or in
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other words the width of each slit is small compared
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to their separation (D >> a above). <br>
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<br>
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Similarly, the <b>condition for minimum intensity </b>at
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P, when the path difference is a multiple of half
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wavelengths, is given by,<br>
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<br>
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<div align="center"><img alt="interference eqn2"
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src="lo_interference_eqn2.jpg" height="34"
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width="176"><br>
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<div align="left">where n can again take on integer
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value, n = 0, 1, 2, 3...<br>
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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="left">
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<div align="center">
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<div align="left">
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<ul>
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<li>When the distance from slits to screen is much
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larger than the distance on the screen, D
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>> y above (or L >> x in first
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diagram), then the angle θ is "small" and we may
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assume tanθ is approximately equal to sinθ which
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is approximately equal to θ (in radians) and we
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may write,</li>
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</ul>
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<div align="center"><img alt="interference eqn3"
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src="lo_interference_eqn3.jpg" height="54"
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width="156"><br>
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<blockquote>
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<div align="left">for the position on the screen
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for maximum intensity.<br>
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<br>
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<img alt="divider" src="divider_ornbarblu.gif"
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height="64" width="100%"><br>
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<br>
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<div align="center"><big><u><b>INTENSITY
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DISTRIBUTION in DOUBLE SLIT
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INTERFERENCE</b></u></big><br>
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<br>
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</div>
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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>The interference pattern shown above was
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first observed for visible light in 1801 by
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<a
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href="http://www-history.mcs.st-and.ac.uk/Biographies/Young_Thomas.html">Thomas
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Young</a> <img alt="Young"
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src="lo_young.jpg" align="middle"
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height="117" width="94">, the experiment
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is still sometimes called Young's slit
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experiment.</li>
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</ul>
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<blockquote><img alt="exclamation"
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src="exclamation-icon.gif" height="30"
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width="31"> In the above description we have
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assumed the incident light is
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monochromatic. If white light
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(containing all the wavelengths in the visible
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spectrum) is used, the maxima for the
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different wavelengths will occur at slightly
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different positions (y) on the screen.
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In this case an interference pattern will only
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be observed if the maximum - minimum
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separation is much larger than the separation
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between the maxima of the extreme wavelengths
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in white light (red and violet) for the same
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"n". <br>
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<br>
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</blockquote>
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<ul>
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<li>In the above description we have shown
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that at certain locations on the screen
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there will be bright spots whereas at other
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locations there will be no light - the
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interference pattern. But exactly how
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does the light intensity vary as a function
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of position on the screen ?</li>
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</ul>
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<blockquote>In the diagram at the top of this
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page the electric field from light originating
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at each of the slits S<sub>1</sub> and S<sub>2</sub>
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can be written,<br>
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<br>
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<div align="center"><img alt="interference eqn
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4" src="lo_interference_eqn4.jpg"
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height="51" width="156"><br>
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<br>
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<div align="left">where each slit has the
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same maximum <b>E </b>field, E<sub>0</sub>
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and φ is the phase difference due to the
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path difference S<sub>1</sub>P - S<sub>2</sub>P.
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<br>
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Therefore the <b>E</b> field at P can be
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written,<br>
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<br>
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<div align="center"><img alt="interference
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eqn6" src="lo_interference_eqn6.jpg"
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height="38" width="660"><br>
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<br>
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<br>
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<div align="left">the product of an
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amplitude and a sinusoidal time
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varying wave. In the case of
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light waves the frequency of the time
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varying part is so large that our eyes
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and most instruments "see" only the
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ampliutde part. Actually, what
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we observe is the <b>intensity</b>,
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which is the <i>square of the
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amplitude</i>. The intensity
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observed at P is then given by,<br>
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<br>
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<div align="center"><img
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alt="interference eqn7"
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src="lo_interference_eqn7.jpg"
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height="42" width="174"><br>
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<br>
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<div align="left">as shown in the
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red shading in the diagram
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above. Note that maxima of
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the above cosine squared function
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occur when φ = 2π n; this leads to
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bright spots on the screen.<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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</blockquote>
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<div align="center">
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<div align="left">
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<ul>
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<li>As we have seen, when the path
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difference is an integer multiple of
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wavelengths, the waves from the two
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sources interefer constructively.
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That is they are in phase, and as we
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have seen above, φ must be an
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integer multiple of 2π,<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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<blockquote>
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<div align="center">
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<div align="left">
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<div align="center"><img alt="interference
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eqn5" src="lo_interference_eqn5.jpg"
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height="130" width="318"><br>
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<br>
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<div align="left">Thus for small values
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of θ (sinθ = θ), φ and θ are
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proportional to each other.<br>
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<br>
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<img alt="divider"
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src="divider_ornbarblu.gif"
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height="64" width="100%"><br>
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<br>
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<div align="center"><big><u><b>COHERENCE</b></u></big><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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</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">
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<ul>
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<li>Throughout the above
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description we have implicitly
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assumed that the light waves
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from the two slits S<sub>1</sub>
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and S<sub>2</sub> are in phase
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with each other. This
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ensures that any phase
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difference in the light from the
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two slits is due entirely to the
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different path lengths the two
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waves travel. If the
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relative phase of the
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sources S<sub>1</sub> and
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S<sub>2</sub> is unknown and
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randomly varying with time then
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the interference pattern will
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also randomly change with a
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frequency similar to the
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frequency of the light.
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Practically this means there
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will be no observed interference
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pattern.</li>
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</ul>
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<ul>
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<li>The difficulty of obtaining
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two sources (S<sub>1</sub> and S<sub>2</sub>)
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emitting waves <i><b>coherently</b></i>
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(in phase) depends on the
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wavelength of the
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electromagnetic wave. <br>
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</li>
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</ul>
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<blockquote>For long wave radiation
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(e.g. radio waves) a single source
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illuminating the two slits results
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in two coherent sources, since
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this type of radiation is
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typically produced in a continuous
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waveform, as shown at left below.<br>
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<br>
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For short wavelength radiation
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(e.g. light) "waves" are typically
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emitted by multiple individual
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atoms in a random incoherent
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manner (no definite phase between
|
||
|
|
these multiple "sources".
|
||
|
|
The radiation is emitted in
|
||
|
|
"packets" rather than as a
|
||
|
|
continuous wave. Thus a two
|
||
|
|
slit configuration, at left below,
|
||
|
|
will not produce an interference
|
||
|
|
pattern. In order to observe
|
||
|
|
an interference pattern with light
|
||
|
|
the configuration at right below
|
||
|
|
must be employed. The single
|
||
|
|
slit to the left of the two slits
|
||
|
|
ensures that light reaching the
|
||
|
|
two slits is from the same part of
|
||
|
|
the source and therefore in phase.<br>
|
||
|
|
<img alt="hot" src="hot.gif"
|
||
|
|
align="middle" height="43"
|
||
|
|
width="79"> Note that a
|
||
|
|
laser beam produces a coherent
|
||
|
|
light source and can be used to
|
||
|
|
create an interference pattern in
|
||
|
|
the left configuration. <br>
|
||
|
|
</blockquote>
|
||
|
|
<div align="center"><img
|
||
|
|
alt="interference fig4"
|
||
|
|
src="lo_interference_fig4.jpg"
|
||
|
|
height="171" width="295">
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
<img
|
||
|
|
alt="interference fig5"
|
||
|
|
src="lo_interference_fig5.jpg"
|
||
|
|
height="211" width="333"><br>
|
||
|
|
<br>
|
||
|
|
<img alt="divider"
|
||
|
|
src="divider_ornbarblu.gif"
|
||
|
|
height="64" width="100%"><br>
|
||
|
|
<br>
|
||
|
|
<big><u><b>QUANTUM LIMIT - DOUBLE
|
||
|
|
SLIT INTERFERENCE</b></u></big><br>
|
||
|
|
<br>
|
||
|
|
<div align="left">
|
||
|
|
<ul>
|
||
|
|
<li><img alt="interference
|
||
|
|
fig2"
|
||
|
|
src="lo_interference_fig2.jpg"
|
||
|
|
align="right" height="371"
|
||
|
|
width="601">Consider
|
||
|
|
double slit
|
||
|
|
interference. In the
|
||
|
|
figure at right there are
|
||
|
|
locations on the screen
|
||
|
|
which have zero light
|
||
|
|
intensity.</li>
|
||
|
|
<li>Now gradually reduce the
|
||
|
|
intensity of the incident
|
||
|
|
light so that instead of a
|
||
|
|
"continuous" wave impacting
|
||
|
|
the slits we have individual
|
||
|
|
"photons" incident.
|
||
|
|
This is the "quantum limit",
|
||
|
|
where we treat a source of
|
||
|
|
light as a source of
|
||
|
|
discrete "wave packets" or
|
||
|
|
photons. with
|
||
|
|
individual photons incident
|
||
|
|
the same interference
|
||
|
|
pattern is observed.<br>
|
||
|
|
</li>
|
||
|
|
<li>But with photons incident
|
||
|
|
one at a time it makes sense
|
||
|
|
to ask, "which slit did the
|
||
|
|
photon pass through ?"</li>
|
||
|
|
<li>Close one of the slits,
|
||
|
|
but continue illumination
|
||
|
|
with individual
|
||
|
|
photons. The intensity
|
||
|
|
pattern observed on the
|
||
|
|
screen changes with only one
|
||
|
|
slit open. But if the
|
||
|
|
photon passes through the
|
||
|
|
lower slit, how does it know
|
||
|
|
whether the upper slit is
|
||
|
|
open or closed ? For
|
||
|
|
the interference pattern to
|
||
|
|
change it <i><b>must know</b></i>.</li>
|
||
|
|
<li>Explanation.... The
|
||
|
|
photon is an extended
|
||
|
|
object, so it never really
|
||
|
|
passes though one
|
||
|
|
slit. Due to its
|
||
|
|
extended nature it can
|
||
|
|
"feel" whether the other
|
||
|
|
slit is open.</li>
|
||
|
|
<li>However, perhaps most
|
||
|
|
intriguing, is the fact that
|
||
|
|
if the two slits are
|
||
|
|
illuminated with a beam of
|
||
|
|
particles, for example
|
||
|
|
electrons, the same
|
||
|
|
interference phenomena is
|
||
|
|
observed. You'd
|
||
|
|
naturally expect electrons
|
||
|
|
to go through one slit or
|
||
|
|
the other, but with both
|
||
|
|
slits open an interference
|
||
|
|
pattern similar to that at
|
||
|
|
left is observed. The
|
||
|
|
electron is displaying "wave
|
||
|
|
properties".</li>
|
||
|
|
<li>This is an example of <i><b>"<a
|
||
|
|
href="http://science.howstuffworks.com/light6.htm">wave-particle duality</a>"</b></i>,
|
||
|
|
an important consequence of
|
||
|
|
the quantum theory of
|
||
|
|
matter.</li>
|
||
|
|
<li>For an accessible
|
||
|
|
description of what's going
|
||
|
|
on in the double slit
|
||
|
|
experiment see what <a
|
||
|
|
href="https://www.youtube.com/watch?v=fwXQjRBLwsQ">Dr.
|
||
|
|
Photon</a> has to say....<br>
|
||
|
|
</li>
|
||
|
|
</ul>
|
||
|
|
<div align="center"><img
|
||
|
|
alt="interference fig6"
|
||
|
|
src="lo_interference_fig6.jpg"
|
||
|
|
height="350" width="500"><br>
|
||
|
|
<br>
|
||
|
|
<br>
|
||
|
|
</div>
|
||
|
|
<div align="center"><img
|
||
|
|
alt="interference fig7"
|
||
|
|
src="lo_interference_fig7.jpg"
|
||
|
|
height="169" width="299">
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
<img
|
||
|
|
alt="interefernce cartoon"
|
||
|
|
src="lo_Interference_GodAsDeveloper.gif"
|
||
|
|
height="247" width="310"><br>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
<blockquote>
|
||
|
|
<div align="center">
|
||
|
|
<div align="left">
|
||
|
|
<div align="center">
|
||
|
|
<div align="left"> </div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</blockquote>
|
||
|
|
<blockquote> </blockquote>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
<ul>
|
||
|
|
</ul>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
<br>
|
||
|
|
<img src="netbar.gif" height="40" width="100%"><br>
|
||
|
|
<center>"<i><font color="#ff0000">One morning I shot an elephant in
|
||
|
|
my pajamas. How he got into my pajamas I'll never know.</font></i><span></span><span
|
||
|
|
style="font-style: italic; color: rgb(255, 0, 0);">"</span><br>
|
||
|
|
<meta http-equiv="content-type" content="text/html;
|
||
|
|
charset=windows-1252">
|
||
|
|
Groucho Marx<br>
|
||
|
|
(in the film <i>Animal Crackers</i>)<br>
|
||
|
|
<br>
|
||
|
|
<img src="celticbar.gif" height="22" width="576"> <br>
|
||
|
|
|
||
|
|
<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>
|
||
|
|
</p>
|
||
|
|
<p><img src="header-index.gif" height="51" width="92"> </p>
|
||
|
|
</center>
|
||
|
|
<p><br>
|
||
|
|
</p>
|
||
|
|
</body>
|
||
|
|
</html>
|