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<title>Light and Optics - Double Slit Interference - Physics 299</title>
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<h1>Double Slit Interference</h1>
</center>
<center><img src="celticbar.gif" height="22" width="576"> <br>
<br>
<font color="#ff0000"><i>"<span class="bqQuoteLink">We only have
to look at ourselves to see how intelligent life might
develop into something we wouldn't want to meet.</span></i><span></span><i>"</i></font><br>
<font color="#ff0000"><i>
<meta http-equiv="content-type" content="text/html;
charset=windows-1252">
</i></font> Stephen Hawking<br>
<br>
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<img src="netbar.gif" align="middle" height="40" width="100%"> <br>
<br>
<ul>
<li>We now begin a discussion of wave (physical) optics in which -
in contrast to geometric optics - we explicitly consider the
wave nature of light.&nbsp; Remember, in geometric optics light
traveled in straight lines (light rays).&nbsp; This is a valid
approximation so long as we do not consider apertures with
dimensions similar to the wavelength of the light or look too
closely at the edges of objects.</li>
</ul>
<ul>
<li>So what happens when a wave passes through apertures whose
size is similar to the wavelength of the wave ?&nbsp; First
we'll consider the case of double slit interference, in which a
parallel beam of incident monochromatic (containing a specific
wavelength) light from the left strikes a screen with two slits,
S<sub>1</sub> and S<sub>2</sub> , as below.</li>
</ul>
<div align="center"><img alt="interference fig1"
src="lo_interference_fig1.jpg" height="323" width="404"><br>
<br>
<img alt="divider" src="divider_ornbarblu.gif" height="64"
width="100%"><br>
<br>
<u><b><big>CONDITIONS for MAXIMA and MINIMA</big></b></u><big><u><b>
in DOUBLE SLIT INTERFERENCE</b></u></big><br>
<br>
<div align="left">
<ul>
<li>With L &gt;&gt; d&nbsp; geometric optics predicts that two
bright spots would be observed on the right hand screen
immediately opposite S<sub>1</sub> and S<sub>2</sub> .&nbsp;
The rest of this screen would be in shadow.&nbsp; What is
actually observed on the right hand screen is an
"interference pattern" as indicated below,</li>
</ul>
<div align="center"><img alt="interference fig2"
src="lo_interference_fig2.jpg" height="371" width="601">&nbsp;&nbsp;&nbsp;&nbsp;
<img alt="double slit interference"
src="lo_interference_fig3_inf.gif" height="362" width="500"><br>
<div align="left">
<ul>
<li>The explanation is that each slit acts as a source of
spherical waves, which "interfere" as they move from
left to right as shown above.</li>
</ul>
<ul>
<li>In the diagram at the top of the page, light reaching
P from S<sub>1</sub> and S<sub>2</sub> will travel
different distances.&nbsp; Assuming that the light from
the two sources S<sub>1</sub> and S<sub>2</sub> are
initially in phase, then due to the path difference S<sub>1</sub>P
- S<sub>2</sub>P , at P the two waves will be out of
phase.&nbsp; If the path difference is equal to an
integral number of wavelengths the waves will interfere
constructively, leading to a bright spot on the
screen.&nbsp; Mathematically we can write this <b>condition
for maximum intensity</b> as,</li>
</ul>
<div align="center"><img alt="interference eqn1"
src="lo_interference_eqn1.jpg" height="30" width="120"><br>
<blockquote>
<div align="left">where n can take on integer values, n
= 0, 1, 2, 3... and we have assumed that &#952; = &#952;' or in
other words the width of each slit is small compared
to their separation (D &gt;&gt; a above). <br>
<br>
Similarly, the <b>condition for minimum intensity </b>at
P, when the path difference is a multiple of half
wavelengths, is given by,<br>
<br>
<div align="center"><img alt="interference eqn2"
src="lo_interference_eqn2.jpg" height="34"
width="176"><br>
<div align="left">where n can again take on integer
value, n = 0, 1, 2, 3...<br>
</div>
</div>
</div>
</blockquote>
<div align="left">
<div align="center">
<div align="left">
<ul>
<li>When the distance from slits to screen is much
larger than the distance on the screen, D
&gt;&gt; y above (or L &gt;&gt; x in first
diagram), then the angle &#952; is "small" and we may
assume tan&#952; is approximately equal to sin&#952; which
is approximately equal to &#952; (in radians) and we
may write,</li>
</ul>
<div align="center"><img alt="interference eqn3"
src="lo_interference_eqn3.jpg" height="54"
width="156"><br>
<blockquote>
<div align="left">for the position on the screen
for maximum intensity.<br>
<br>
<img alt="divider" src="divider_ornbarblu.gif"
height="64" width="100%"><br>
<br>
<div align="center"><big><u><b>INTENSITY
DISTRIBUTION in DOUBLE SLIT
INTERFERENCE</b></u></big><br>
<br>
</div>
</div>
</blockquote>
<div align="left">
<ul>
<li>The interference pattern shown above was
first observed for visible light in 1801 by
<a
href="http://www-history.mcs.st-and.ac.uk/Biographies/Young_Thomas.html">Thomas
Young</a> <img alt="Young"
src="lo_young.jpg" align="middle"
height="117" width="94">, the experiment
is still sometimes called Young's slit
experiment.</li>
</ul>
<blockquote><img alt="exclamation"
src="exclamation-icon.gif" height="30"
width="31"> In the above description we have
assumed the incident light is
monochromatic.&nbsp; If white light
(containing all the wavelengths in the visible
spectrum) is used, the maxima for the
different wavelengths will occur at slightly
different positions (y) on the screen.&nbsp;
In this case an interference pattern will only
be observed if the maximum - minimum
separation is much larger than the separation
between the maxima of the extreme wavelengths
in white light (red and violet) for the same
"n". <br>
<br>
</blockquote>
<ul>
<li>In the above description we have shown
that at certain locations on the screen
there will be bright spots whereas at other
locations there will be no light - the
interference pattern.&nbsp; But exactly how
does the light intensity vary as a function
of position on the screen ?</li>
</ul>
<blockquote>In the diagram at the top of this
page the electric field from light originating
at each of the slits S<sub>1</sub> and S<sub>2</sub>&nbsp;
can be written,<br>
<br>
<div align="center"><img alt="interference eqn
4" src="lo_interference_eqn4.jpg"
height="51" width="156"><br>
<br>
<div align="left">where each slit has the
same maximum <b>E </b>field, E<sub>0</sub>
and &#966; is the phase difference due to the
path difference S<sub>1</sub>P - S<sub>2</sub>P.&nbsp;
<br>
Therefore the <b>E</b> field at P can be
written,<br>
<br>
<div align="center"><img alt="interference
eqn6" src="lo_interference_eqn6.jpg"
height="38" width="660"><br>
<br>
<br>
<div align="left">the product of an
amplitude and a sinusoidal time
varying wave.&nbsp; In the case of
light waves the frequency of the time
varying part is so large that our eyes
and most instruments "see" only the
ampliutde part.&nbsp; Actually, what
we observe is the <b>intensity</b>,
which is the <i>square of the
amplitude</i>.&nbsp; The intensity
observed at P is then given by,<br>
<br>
<div align="center"><img
alt="interference eqn7"
src="lo_interference_eqn7.jpg"
height="42" width="174"><br>
<br>
<div align="left">as shown in the
red shading in the diagram
above.&nbsp; Note that maxima of
the above cosine squared function
occur when &#966; = 2&#960; n; this leads to
bright spots on the screen.<br>
<br>
</div>
</div>
</div>
</div>
</div>
</div>
</blockquote>
<div align="center">
<div align="left">
<ul>
<li>As we have seen, when the path
difference is an integer multiple of
wavelengths, the waves from the two
sources interefer constructively.&nbsp;
That is they are in phase, and as we
have seen above,&nbsp; &#966; must be an
integer multiple of 2&#960;,<br>
</li>
</ul>
</div>
</div>
<blockquote>
<div align="center">
<div align="left">
<div align="center"><img alt="interference
eqn5" src="lo_interference_eqn5.jpg"
height="130" width="318"><br>
<br>
<div align="left">Thus for small values
of &#952; (sin&#952; = &#952;), &#966; and &#952; are
proportional to each other.<br>
<br>
<img alt="divider"
src="divider_ornbarblu.gif"
height="64" width="100%"><br>
<br>
<div align="center"><big><u><b>COHERENCE</b></u></big><br>
<br>
</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>Throughout the above
description we have implicitly
assumed that the light waves
from the two slits S<sub>1</sub>
and S<sub>2</sub> are in phase
with each other.&nbsp; This
ensures that any phase
difference in the light from the
two slits is due entirely to the
different path lengths the two
waves travel.&nbsp; If the
relative phase of the
sources&nbsp; S<sub>1</sub> and
S<sub>2</sub> is unknown and
randomly varying with time then
the interference pattern will
also randomly change with a
frequency similar to the
frequency of the light.&nbsp;
Practically this means there
will be no observed interference
pattern.</li>
</ul>
<ul>
<li>The difficulty of obtaining
two sources (S<sub>1</sub> and S<sub>2</sub>)
emitting waves <i><b>coherently</b></i>
(in phase) depends on the
wavelength of the
electromagnetic wave.&nbsp; <br>
</li>
</ul>
<blockquote>For long wave radiation
(e.g. radio waves) a single source
illuminating the two slits results
in two coherent sources, since
this type of radiation is
typically produced in a continuous
waveform, as shown at left below.<br>
<br>
For short wavelength radiation
(e.g. light) "waves" are typically
emitted by multiple individual
atoms in a random incoherent
manner (no definite phase between
these multiple "sources".&nbsp;
The radiation is emitted in
"packets" rather than as a
continuous wave.&nbsp; Thus a two
slit configuration, at left below,
will not produce an interference
pattern.&nbsp; In order to observe
an interference pattern with light
the configuration at right below
must be employed.&nbsp; 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">&nbsp; 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"> &nbsp;
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
&nbsp; &nbsp; &nbsp; &nbsp;
&nbsp;&nbsp;&nbsp; <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.&nbsp; 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.&nbsp;
This is the "quantum limit",
where we treat a source of
light as a source of
discrete "wave packets" or
photons.&nbsp; 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.&nbsp; The intensity
pattern observed on the
screen changes with only one
slit open.&nbsp; But if the
photon passes through the
lower slit, how does it know
whether the upper slit is
open or closed ?&nbsp; For
the interference pattern to
change it <i><b>must know</b></i>.</li>
<li>Explanation....&nbsp; The
photon is an extended
object, so it never really
passes though one
slit.&nbsp; 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.&nbsp; 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.&nbsp; 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">
&nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;&nbsp; &nbsp;
&nbsp; &nbsp; &nbsp; &nbsp;
&nbsp; &nbsp; &nbsp; &nbsp;
&nbsp; &nbsp; &nbsp; &nbsp;
&nbsp; &nbsp; <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>
&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>
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