327 lines
14 KiB
HTML
327 lines
14 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 - Interference from Thin Films - 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>Interference From Thin Films<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>Everything
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we call real is made of things that cannot be regarded as real<span></span><span
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class="bqQuoteLink"></span><span></span>"</i></font><br>
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<font color="#ff0000"><i><span class="bqQuoteLink">
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<meta http-equiv="content-type" content="text/html;
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charset=windows-1252">
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</span></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>Niels Bohr<br>
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</center>
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<img src="netbar.gif" height="40" align="middle" width="100%"> <br>
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<ul>
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<li>Double slit interference, described on the previous page, is
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rarely observed in nature. On the other hand, interference
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due to thin films is quite frequently observed - swirling
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colours on an oil slick, colours on a soap bubble, the purple
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tinge on an expensive camera lens - are all examples of thin
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film interference.</li>
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</ul>
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<div align="center"><img alt="thin films fig1"
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src="lo_thinfilm_fig1.jpg" height="218" width="292">
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<img alt="thin films fig3" src="lo_thinfilms_fig3.jpg"
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height="219" width="291"> <img alt="thin
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films fig2" src="lo_thinfilms_fig2.jpg" height="220" width="320"><br>
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<br>
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</div>
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<ul>
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<li><img alt="thin films fig4" src="lo_thinfilms_fig4.jpg"
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height="257" align="right" width="539">Consider a thin film,
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separating two regions of the same refractive index, for example
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a soap bubble. Incident light undergoes reflection and
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refraction at every interface as shown at right. There
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will be multiple "rays" reflected back into the air (only the
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first two are shown in the diagram) and multiple "rays"
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transmitted through the film (only the first of these is shown).</li>
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</ul>
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<ul>
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<li>The reflected rays will interfere with each other, their phase
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difference being determined by their path difference.</li>
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<li>Similarly the transmitted rays will interfere with each other.</li>
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</ul>
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<ul>
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<li><img alt="exclamation" src="exclamation-icon.gif" height="30"
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width="31"> Note that coherence of the interfering rays is
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ensured since there is only one source - the incident ray
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coloured blue in the diagram at right.</li>
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</ul>
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<blockquote>Assuming a monochromatic source and normal incidence, we
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can expect that the two reflected rays will interfere
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constructively if their path difference is an integer multiple of
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wavelengths,<br>
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<br>
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<div align="center"><img alt="thin films eqn1"
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src="lo_thinfilms_eqn1.png" height="30" width="264"><br>
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<br>
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<div align="left">where d is the thickness of the film.<br>
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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><big><u><b>First Correction</b></u></big></li>
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</ul>
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<blockquote>The path difference 2d is measured inside the film,
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but the wavelength is the wavelength in air... Therefore
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the condition for a maximum should be,<br>
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<br>
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<div align="center"><img alt="thin films eqn4"
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src="lo_thinfilms_eqn4.png" height="34" width="289"><br>
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</div>
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However, we know that,<br>
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<br>
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<div align="center"><img alt="thin films eqn2"
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src="lo_thinfilms_eqn2.png" height="34" width="354"><br>
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<br>
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<div align="left">and using the definition of refractive
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index,<br>
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<br>
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<div align="center"><img alt="thin films eqn3"
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src="lo_thinfilms_eqn3.png" height="57" width="354"><br>
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<br>
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<div align="left">This means the condition for
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constructive interference can be written,<br>
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<br>
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<div align="center"><img alt="thin films eqn5"
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src="lo_thinfilms_eqn5.png" height="30"
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width="275"><br>
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<br>
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<div align="left">where n is the refractive index of
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the film and λ is the wavelength of the light in
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air.<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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<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><big><u><b>Second Correction</b></u></big></li>
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</ul>
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<blockquote>This result implies that when the film
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becomes very thin (d approximately zero), we
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should expect constructive interference
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(equivalent to m = 0). That is, just before
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a soap bubble bursts, as its thickness gets
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smaller and smaller, we should expect the swirling
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colours to become brighter. Actually the
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opposite is observed - bubbles becomes dull just
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before they burst - in other words destructive
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interference must be taking place.<img
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alt="confused" src="confused_smiley.gif"
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height="22" width="15"><br>
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<br>
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The explanation to this apparent conflict is that
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the first reflected ray (green in the above
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diagram) is <i><b>phase changed by 180</b></i><i><b><sup>0</sup></b></i><i><b>
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</b></i>by the act of reflection. <i><b>In
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the above diagram, only this ray undergoes
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such a phase change.</b></i> In fact the
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rule (which can be verified theoretically from
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Maxwell's equations) is,<br>
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<br>
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<div align="center"><big><i><b>"A </b></i><i><b>180</b></i><i><b><sup>0</sup></b></i><i><b>
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</b></i><i><b>phase change on reflection
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occurs when light incident from a less
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dense medium (smaller n) reflects off the
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boundary with a more dense (larger n)
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medium"<br>
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</b></i></big>
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<div align="left"><br>
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Reflection from more dense to less dense
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(large n to small n) causes no phase change
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and transmitted rays never undergo phase
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changes.<br>
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<br>
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Incorporating this phase change leads to the
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condition for <b>constructive interference</b>,
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<br>
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<br>
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<div align="center"><img alt="thin films eqn6"
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src="lo_thinfilms_eqn6.jpg" height="31"
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width="345"><br>
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<div align="left">and for <b>destructive
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interference</b>,<br>
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<br>
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<div align="center"><img alt="thin films
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eqn7" src="lo_thinfilms_eqn7.jpg"
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height="31" width="294"><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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<ul>
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<li><font color="#ff0000"><big><u><b>IMPORTANT !!</b></u></big></font></li>
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</ul>
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<blockquote><img alt="thin films fig5"
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src="lo_thinfilms_fig5.jpg" height="303"
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align="right" width="269">The second correction
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applies specifically to the air-soap-air
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configuration, or more generally to the
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circumstance of a film with greater refractive
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index than the media on either side. This
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correction may not be needed for other thin film
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configurations; for example, a film of water
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on a glass surface, shown at right. In this
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case, since n<sub>air</sub> < n<sub>water</sub>
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< n<sub>glass</sub> , there is a <i>180</i><i><sup>0</sup></i><i>
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</i>phase change for the reflected rays from both
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the air-water and water-glass interface.
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That is the condition for <b>constructive
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interference</b> reverts to<br>
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<br>
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<div align="center"><img alt="thin films eqn5"
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src="lo_thinfilms_eqn5.png" height="30"
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width="275"><br>
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</div>
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</blockquote>
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<blockquote>
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<div align="left">
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<div>
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<div>
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<div>
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<div>and for <b>destructive interference</b>,<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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<blockquote>
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<div>
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<div>
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<div>
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<div><img alt="thin films eqn6"
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src="lo_thinfilms_eqn6.jpg"
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height="31" width="345"><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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</blockquote>
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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 align="left">
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<ul>
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<li>All of the above analysis assumes
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a single wavelength of light.
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With incident white light on a thin
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film maxima for different
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wavelengths will occur at different
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film thicknesses. Thus, in
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nature, where a thin film will not
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typically have a fixed thickness, we
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often observe swirling colours, as
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the thickness of the film changes.</li>
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</ul>
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<ul>
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<li>Finally, why <i><b>thin</b></i>
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films ? A thin film implies
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path differences (thicknesses) of no
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more than a few wavelengths.
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Films thicker than this will not
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necessarily exhibit this type of
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interference due to re-emission and
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scattering of light within the film.<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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</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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</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"> </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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<blockquote><img src="netbar.gif" height="40" width="100%"><br>
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</blockquote>
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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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Military intelligence is a contradiction in terms.</font></i><span
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style="font-style: italic; color: rgb(255, 0, 0);">"</span><br>
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<meta http-equiv="content-type" content="text/html;
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charset=windows-1252">
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Groucho Marx<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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