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<!DOCTYPE html PUBLIC "-//w3c//dtd html 4.0 transitional//en">
<html>
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<meta http-equiv="Content-Type" content="text/html;
charset=windows-1252">
<meta name="GENERATOR" content="Mozilla/4.7 [en] (X11; U; OSF1 V4.0
alpha) [Netscape]">
<meta name="Author" content="C. L. Davis">
<title>Light and Optics - Interference from Thin Films - Physics 299</title>
</head>
<body style="color: rgb(0, 0, 0); background-color: rgb(255, 255,
255);" link="#0000ee" alink="#ff0000" vlink="#551a8b">
<center>
<h1><img src="ULPhys1.gif" height="50" align="texttop" width="189">
</h1>
</center>
<center>
<h1>Interference From Thin Films<br>
</h1>
</center>
<center><img src="celticbar.gif" height="22" width="576"> <br>
<br>
<font color="#ff0000"><i>"<span class="bqQuoteLink"></span>Everything
we call real is made of things that cannot be regarded as real<span></span><span
class="bqQuoteLink"></span><span></span>"</i></font><br>
<font color="#ff0000"><i><span class="bqQuoteLink">
<meta http-equiv="content-type" content="text/html;
charset=windows-1252">
</span></i></font> <font color="#ff0000"><i>
<meta http-equiv="content-type" content="text/html;
charset=windows-1252">
</i></font>Niels Bohr<br>
</center>
<img src="netbar.gif" height="40" align="middle" width="100%"> <br>
<ul>
<li>Double slit interference, described on the previous page, is
rarely observed in nature.&nbsp; On the other hand, interference
due to thin films is quite frequently observed - swirling
colours on an oil slick, colours on a soap bubble, the purple
tinge on an expensive camera lens - are all examples of thin
film interference.</li>
</ul>
<div align="center"><img alt="thin films fig1"
src="lo_thinfilm_fig1.jpg" height="218" width="292">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<img alt="thin films fig3" src="lo_thinfilms_fig3.jpg"
height="219" width="291"> &nbsp;&nbsp; &nbsp; <img alt="thin
films fig2" src="lo_thinfilms_fig2.jpg" height="220" width="320"><br>
<br>
</div>
<ul>
<li><img alt="thin films fig4" src="lo_thinfilms_fig4.jpg"
height="257" align="right" width="539">Consider a thin film,
separating two regions of the same refractive index, for example
a soap bubble.&nbsp; Incident light undergoes reflection and
refraction at every interface as shown at right.&nbsp; There
will be multiple "rays" reflected back into the air (only the
first two are shown in the diagram) and multiple "rays"
transmitted through the film (only the first of these is shown).</li>
</ul>
<ul>
<li>The reflected rays will interfere with each other, their phase
difference being determined by their path difference.</li>
<li>Similarly the transmitted rays will interfere with each other.</li>
</ul>
<ul>
<li><img alt="exclamation" src="exclamation-icon.gif" height="30"
width="31"> Note that coherence of the interfering rays is
ensured since there is only one source - the incident ray
coloured blue in the diagram at right.</li>
</ul>
<blockquote>Assuming a monochromatic source and normal incidence, we
can expect that the two reflected rays will interfere
constructively if their path difference is an integer multiple of
wavelengths,<br>
<br>
<div align="center"><img alt="thin films eqn1"
src="lo_thinfilms_eqn1.png" height="30" width="264"><br>
<br>
<div align="left">where d is the thickness of the film.<br>
</div>
</div>
</blockquote>
<div align="center">
<div align="left">
<ul>
<li><big><u><b>First Correction</b></u></big></li>
</ul>
<blockquote>The path difference 2d is measured inside the film,
but the wavelength is the wavelength in air...&nbsp; Therefore
the condition for a maximum should be,<br>
<br>
<div align="center"><img alt="thin films eqn4"
src="lo_thinfilms_eqn4.png" height="34" width="289"><br>
</div>
However, we know that,<br>
<br>
<div align="center"><img alt="thin films eqn2"
src="lo_thinfilms_eqn2.png" height="34" width="354"><br>
<br>
<div align="left">and using the definition of refractive
index,<br>
<br>
<div align="center"><img alt="thin films eqn3"
src="lo_thinfilms_eqn3.png" height="57" width="354"><br>
<br>
<div align="left">This means the condition for
constructive interference can be written,<br>
<br>
<div align="center"><img alt="thin films eqn5"
src="lo_thinfilms_eqn5.png" height="30"
width="275"><br>
<br>
<div align="left">where n is the refractive index of
the film and &#955; is the wavelength of the light in
air.<br>
<br>
</div>
</div>
</div>
</div>
</div>
</div>
</blockquote>
<div align="center">
<div align="left">
<div align="center">
<div align="left">
<div align="center">
<div align="left">
<ul>
<li><big><u><b>Second Correction</b></u></big></li>
</ul>
<blockquote>This result implies that when the film
becomes very thin (d approximately zero), we
should expect constructive interference
(equivalent to m = 0).&nbsp; That is, just before
a soap bubble bursts, as its thickness gets
smaller and smaller, we should expect the swirling
colours to become brighter.&nbsp; Actually the
opposite is observed - bubbles becomes dull just
before they burst - in other words destructive
interference must be taking place.<img
alt="confused" src="confused_smiley.gif"
height="22" width="15"><br>
<br>
The explanation to this apparent conflict is that
the first reflected ray (green in the above
diagram) is <i><b>phase changed by 180</b></i><i><b><sup>0</sup></b></i><i><b>
</b></i>by the act of reflection.&nbsp; <i><b>In
the above diagram, only this ray undergoes
such a phase change.</b></i>&nbsp; In fact the
rule (which can be verified theoretically from
Maxwell's equations) is,<br>
<br>
<div align="center"><big><i><b>"A </b></i><i><b>180</b></i><i><b><sup>0</sup></b></i><i><b>
</b></i><i><b>phase change on reflection
occurs when light incident from a less
dense medium (smaller n) reflects off the
boundary with a more dense (larger n)
medium"<br>
</b></i></big>
<div align="left"><br>
Reflection from more dense to less dense
(large n to small n) causes no phase change
and transmitted rays never undergo phase
changes.<br>
<br>
Incorporating this phase change leads to the
condition for <b>constructive interference</b>,
<br>
<br>
<div align="center"><img alt="thin films eqn6"
src="lo_thinfilms_eqn6.jpg" height="31"
width="345"><br>
<div align="left">and for <b>destructive
interference</b>,<br>
<br>
<div align="center"><img alt="thin films
eqn7" src="lo_thinfilms_eqn7.jpg"
height="31" width="294"><br>
<br>
</div>
</div>
</div>
</div>
</div>
</blockquote>
<ul>
<li><font color="#ff0000"><big><u><b>IMPORTANT !!</b></u></big></font></li>
</ul>
<blockquote><img alt="thin films fig5"
src="lo_thinfilms_fig5.jpg" height="303"
align="right" width="269">The second correction
applies specifically to the air-soap-air
configuration, or more generally to the
circumstance of a film with greater refractive
index than the media on either side.&nbsp; This
correction may not be needed for other thin film
configurations;&nbsp; for example, a film of water
on a glass surface, shown at right.&nbsp; In this
case, since n<sub>air</sub> &lt; n<sub>water</sub>
&lt; n<sub>glass</sub> , there is a <i>180</i><i><sup>0</sup></i><i>
</i>phase change for the reflected rays from both
the air-water and water-glass interface.&nbsp;
That is the condition for <b>constructive
interference</b> reverts to<br>
<br>
<div align="center"><img alt="thin films eqn5"
src="lo_thinfilms_eqn5.png" height="30"
width="275"><br>
</div>
</blockquote>
<blockquote>
<div align="left">
<div>
<div>
<div>
<div>and for <b>destructive interference</b>,<br>
</div>
</div>
</div>
</div>
</div>
</blockquote>
<div align="center">
<blockquote>
<div>
<div>
<div>
<div><img alt="thin films eqn6"
src="lo_thinfilms_eqn6.jpg"
height="31" width="345"><br>
<br>
</div>
</div>
</div>
</div>
</blockquote>
<div>
<div>
<div>
<div>
<div align="left">
<ul>
<li>All of the above analysis assumes
a single wavelength of light.&nbsp;
With incident white light on a thin
film maxima for different
wavelengths will occur at different
film thicknesses.&nbsp; Thus, in
nature, where a thin film will not
typically have a fixed thickness, we
often observe swirling colours, as
the thickness of the film changes.</li>
</ul>
<ul>
<li>Finally, why <i><b>thin</b></i>
films ?&nbsp; A thin film implies
path differences (thicknesses) of no
more than a few wavelengths.&nbsp;
Films thicker than this will not
necessarily exhibit this type of
interference due to re-emission and
scattering of light within the film.<br>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<blockquote>
<div align="center">
<div align="left">
<div align="center">
<div align="left">
<div align="center">
<div align="left"> </div>
</div>
</div>
</div>
</div>
</div>
</blockquote>
</div>
</div>
<blockquote><img src="netbar.gif" height="40" width="100%"><br>
</blockquote>
<center>"<i><font color="#ff0000">
<meta http-equiv="content-type" content="text/html;
charset=windows-1252">
Military intelligence is a contradiction in terms.</font></i><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>
<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>
<p><img src="header-index.gif" height="51" width="92"> </p>
</center>
<p><br>
</p>
</body>
</html>
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