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<h2 class="hd hd-2 unit-title">Introduction to Double Slit Interference</h2>
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So far, we have looked at how light interferes with itself upon reflection from multiple boundaries at normal incidence. Now, in this lesson, we examine how light interferes with itself when passing through an aperture. </p><p>
Specifically, we will study the case of <i class="itshape">double-slit</i> interference, where light passes through two slits. </p>
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<h2 class="hd hd-2 unit-title">L37v1: Light Propagating Through two Slits</h2>
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<h3 class="hd hd-2">L37v1: Light propagating through two slits</h3>
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<h2 class="hd hd-2 unit-title">L37Q1: Phase Difference</h2>
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Phase Difference
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Consider a plane wave with wavelength [mathjaxinline]\lambda[/mathjaxinline] and wavenumber [mathjaxinline]k[/mathjaxinline] that is transmitted through two slits with separation [mathjaxinline]d[/mathjaxinline]. The waves transmitted through each slit are coincident at the same location on a screen a distance [mathjaxinline]L[/mathjaxinline] away from the slits, where [mathjaxinline]L \gg d[/mathjaxinline]&#8212;the coincident location can be specified by the angle [mathjaxinline]\theta[/mathjaxinline], which extends from the bisecting line between the slits. The following questions are about the phase difference, [mathjaxinline]\delta[/mathjaxinline], between the two waves that are transmitted. </p>
<p><b class="bfseries">(Part a)</b> Let [mathjaxinline]\Delta r[/mathjaxinline] be the optical path length difference. Which of the following are equivalent expressions for the phase difference [mathjaxinline]\delta[/mathjaxinline]? Select ALL that apply. </p>
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<text>a) [mathjaxinline]\delta =k\Delta r[/mathjaxinline]</text>
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<text>b) [mathjaxinline]\delta =k d\sin {\theta }[/mathjaxinline]</text>
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<text>c) [mathjaxinline]\delta =2\pi k\Delta r[/mathjaxinline]</text>
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<text>d) [mathjaxinline]\delta =2\pi k d\sin {\theta }[/mathjaxinline]</text>
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<text>e) [mathjaxinline]\delta =\dfrac {k}{\Delta r}[/mathjaxinline]</text>
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<text>f) [mathjaxinline]\delta =\dfrac {k}{d\sin {\theta }}[/mathjaxinline]</text>
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<text>g) [mathjaxinline]\delta =\dfrac {2\pi k}{\Delta r}[/mathjaxinline]</text>
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<text>h) [mathjaxinline]\delta =\dfrac {2\pi k}{d\sin {\theta }}[/mathjaxinline]</text>
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<text>i) [mathjaxinline]\delta =\dfrac {\Delta r}{\lambda }[/mathjaxinline]</text>
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<text>j) [mathjaxinline]\delta =\dfrac {2\pi \Delta r}{\lambda }[/mathjaxinline]</text>
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<p><b class="bfseries">(Part b)</b> Which of the following is a condition for constructive interference at the location on the screen where the optical paths converge, where [mathjaxinline]n[/mathjaxinline] is an integer [mathjaxinline]n=0,1,2 \ldots[/mathjaxinline]? </p>
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<text> a) [mathjaxinline]\delta = \dfrac {n \pi }{2}[/mathjaxinline]</text>
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<text> b) [mathjaxinline]\delta = n \pi[/mathjaxinline]</text>
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<text> c) [mathjaxinline]\delta = 2 n \pi[/mathjaxinline]</text>
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<text> d) [mathjaxinline]\delta = (2n-1)\pi[/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">L37Q2: Constructive Interference</h2>
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Constructive Interference - part a
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<p>
Again, consider a plane wave with wavelength [mathjaxinline]\lambda[/mathjaxinline], that is transmitted through two slits with separation [mathjaxinline]d[/mathjaxinline]. The waves transmitted through each slit are coincident at the same location on a screen a distance [mathjaxinline]L[/mathjaxinline] away from the slits, where [mathjaxinline]L \gg d[/mathjaxinline], and the coincident location is specified by the vertical distance [mathjaxinline]y[/mathjaxinline] from the bisecting line between the slits. Consider the small angle limit where [mathjaxinline]y \ll L[/mathjaxinline]. </p>
<p><b class="bfseries">(Part a)</b> Find an expression for the phase difference between the transmitted waves at a position [mathjaxinline]y[/mathjaxinline] on the screen. Express your answer in terms of <code>d</code>, <code>y</code>, <code>L</code>, <code>lambda</code> for [mathjaxinline]\lambda[/mathjaxinline], and any relevant constants like <code>pi</code>. </p>
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<p style="display:inline">[mathjaxinline]\delta =[/mathjaxinline] </p>
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<code>2520</code>
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<code>2/3</code>
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<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
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<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
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<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
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<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
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<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
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<code>e, pi</code>
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enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
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<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
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<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
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<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
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<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
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Constructive Interference - part b
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<p><b class="bfseries">(Part b)</b> Where is the location of the first maximum beyond the central maximum? Express your answer in terms of <code>d</code>, <code>L</code>, <code>lambda</code> for [mathjaxinline]\lambda[/mathjaxinline], and any relevant constants like <code>pi</code>. </p>
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<p style="display:inline">[mathjaxinline]y_{1} =[/mathjaxinline] </p>
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<td class="formulainput">integers</td>
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<code>2520</code>
</td>
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<tr class="formulainput">
<td class="formulainput">fractions</td>
<td class="formulainput">
<code>2/3</code>
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<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
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<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">
<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
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<h2 class="hd hd-2 unit-title">L37Q3: Double Slit Intensity</h2>
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Double Slit Intensity - part a
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We have been studying a plane wave with wavenumber [mathjaxinline]k[/mathjaxinline], that is transmitted through two slits with separation [mathjaxinline]d[/mathjaxinline]. We showed that the intensity of transmitted light depends on the phase difference between the waves transmitted through each slit&#8212;we showed that [mathjaxinline]\langle I \rangle \propto \cos ^{2}(\delta /2)[/mathjaxinline], where [mathjaxinline]\delta = k\left(r_{B} - r_{A}\right)[/mathjaxinline] is the phase difference. The lengths [mathjaxinline]r_{A}[/mathjaxinline] and [mathjaxinline]r_{B}[/mathjaxinline] are the optical path lengths of the transmitted waves. </p>
<p>
Let the two transmitted waves have the following form: </p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\vec{E}_{1}=E_{0}e^{i(\omega t - k r_{A})} \hat{z}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\vec{E}_{2}=E_{0}e^{i(\omega t - k r_{B})} \hat{z}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p><b class="bfseries">(Part a)</b> What EXACTLY is the time-averaged intensity as a function of [mathjaxinline]\delta[/mathjaxinline]? Use the variables <code>c</code>, <code>epsilon_0</code> for [mathjaxinline]\epsilon _{0}[/mathjaxinline], <code>E_0</code> for [mathjaxinline]E_{0}[/mathjaxinline], and<code>delta</code> for [mathjaxinline]\delta[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]\langle I \rangle =[/mathjaxinline] </p>
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<td class="formulainput">integers</td>
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<code>2520</code>
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<td class="formulainput">fractions</td>
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<code>2/3</code>
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<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
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<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
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<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
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<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
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<tr class="formulainput">
<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
</tr>
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<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
</td>
</tr>
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<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">
<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
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</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
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<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
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<tr class="formulainput">
<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</tr>
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<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
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<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
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Double Slit Intensity - part b
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<p><b class="bfseries">(Part b)</b> What EXACTLY is the time-averaged intensity as a function of position [mathjaxinline]y[/mathjaxinline] on a screen a distance [mathjaxinline]L[/mathjaxinline] away from the slits, where [mathjaxinline]y \ll L[/mathjaxinline]? Use the variables <code>y</code>, <code>L</code>, <code>d</code>, <code>c</code>, <code>epsilon_0</code> for [mathjaxinline]\epsilon _{0}[/mathjaxinline], <code>E_0</code> for [mathjaxinline]E_{0}[/mathjaxinline], and <code>lambda</code> for [mathjaxinline]\lambda[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]\langle I \rangle =[/mathjaxinline] </p>
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</tr>
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<td class="formulainput">integers</td>
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<code>2520</code>
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<td class="formulainput">fractions</td>
<td class="formulainput">
<code>2/3</code>
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<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
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<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">
<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
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<h2 class="hd hd-2 unit-title">L37Q4: Varying Double Slit Interference Parameters [WITH SIMULATION]</h2>
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Varying Double Slit Interference Parameters
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Consider the following intensity patterns, which are SOLELY due to double-slit interference. NOTE, we have not considered diffraction phenomena, which we will discuss next week. </p>
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<p><b class="bfseries">(Part i)</b> Assume that both patterns are associated with a configuration where the distance [mathjaxinline]L[/mathjaxinline] and slit separation [mathjaxinline]d[/mathjaxinline] are fixed, but [mathjaxinline]\lambda[/mathjaxinline] is varied. Which pattern corresponds to the shortest wavelength? </p>
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<p><b class="bfseries">(Part ii)</b> Assume that both patterns are associated with a configuration where the distance [mathjaxinline]L[/mathjaxinline] and wavelength [mathjaxinline]\lambda[/mathjaxinline] are fixed, but the slit separation [mathjaxinline]d[/mathjaxinline] is varied. Which pattern corresponds to the smallest slit separation? </p>
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<h2>Run the Interactive Python Visualization that Generated the Plots Above!</h2><p>The widget is run in a Jupyter notebook, accessible through the button below. <b>NOTE: The notebook may take up to 3 mintues to load! Please be patient!</b></p><p><div align="center"><a href="https://mybinder.org/v2/gh/mitx-803/vis/master?filepath=double_slit_interference_1.ipynb" class="btn btn-primary" target="_blank" style="color:#FFFFFF;">ACCESS JUPYTER NOTEBOOK HERE</a></div></p><p><div class="hideshowbox"><h4 onclick="hideshow(this);" style="margin: 0px">How to Run Jupyter Notebooks (expand this section if you need a reminder!)<span class="icon-caret-down toggleimage"/></h4><div class="hideshowcontent"><p><h3>Running Notebooks on an External Server</h3></p><p>To access a simulation, click the "ACCESS JUPYTER NOTEBOOK HERE" button. This will bring you to a loading page, hosted by <i class="itshape">mybinder.org</i> (the loading time is anywhere from 20 seconds to 3 minutes). The Jupyter notebooks are run externally to the course, on a server which runs an instance of Python. There is no need to install Python or related dependencies!</p><div align="center"><iframe src="https://mitx-803.github.io/gifs/python_06.html" width="720" height="590" scrolling="no" frameborder="0"/></div><p><h3>Initializing the Program</h3></p><p>Once loaded, you will see a Jupyter notebook in your browser! You will have to click a button to initialize the program. The button is indicated in the instructions within the notebook, and also shown below.</p><div align="center"><img width="700" src="/assets/courseware/v1/3f6c044fc06f79d82bb2e8a97f7dd11a/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_binder_initialize_button.png"/></div><p/><div align="center"><iframe src="https://mitx-803.github.io/gifs/python_07.html" width="720" height="602" scrolling="no" frameborder="0"/></div><p><h3>Instructions and Source Code</h3></p><p>Each notebook has self-contained instructions on how to use the Python simulation. Additionally, you may toggle the button at the bottom of the notebook to view/augment the source code.</p><div align="center"><iframe src="https://mitx-803.github.io/gifs/python_08.html" width="720" height="608" scrolling="no" frameborder="0"/></div><p><h3>Saving/Running Notebooks Locally</h3></p><p>Finally, you can dowload each notebook to run locally. Additionally, you can visit the git repository to download all notebooks in the course. In order to run notebooks locally, you must install Python and its dependencies. We cannot help with this process, but we encourage you to look at the resources below, if you are interested.</p><div align="center"><iframe src="https://mitx-803.github.io/gifs/python_09.html" width="720" height="609" scrolling="no" frameborder="0"/></div><p><h3>External Links</h3><br/> [mathjaxinline]\bullet[/mathjaxinline] git repository: <a href="https://github.com/mitx-803/vis" target="blank">github.com/mitx-803/vis</a><br/> [mathjaxinline]\bullet[/mathjaxinline] information on Jupyter notebooks: <a href="https://jupyter.org/" target="blank">Jupyter Notebooks</a><br/> [mathjaxinline]\bullet[/mathjaxinline] information on installing Python through Anaconda: <a href="https://www.anaconda.com/distribution/" target="blank">Anaconda</a><br/> [mathjaxinline]\bullet[/mathjaxinline] information on the Binder community: <a href="https://mybinder.readthedocs.io/en/latest/" target="blank">Binder</a><br/></p><p><h3>Dependencies</h3></p><p>
The visualizations run on Python 3. Dependencies for running Python code locally (not through Binder) are stated in the git repository, and include (but are not limited to) the following:
<br/> [mathjaxinline]\bullet[/mathjaxinline] scipy
<br/> [mathjaxinline]\bullet[/mathjaxinline] numpy
<br/> [mathjaxinline]\bullet[/mathjaxinline] ipywidgets
<br/> [mathjaxinline]\bullet[/mathjaxinline] nbinteract
<br/> [mathjaxinline]\bullet[/mathjaxinline] matplotlib
<br/> [mathjaxinline]\bullet[/mathjaxinline] pandas
<br/> [mathjaxinline]\bullet[/mathjaxinline] IPython
<br/> [mathjaxinline]\bullet[/mathjaxinline] ffmpeg
<br/> [mathjaxinline]\bullet[/mathjaxinline] jupyter-contrib-nbextensions
<br/>
</p><p>
You will have to find resources that explain how to install these appropriately for your system, if they are not already installed with your Python package.
</p></div><p class="hideshowbottom" onclick="hideshow(this);" style="margin: 0px"><a href="javascript: {return false;}">Show</a></p></div></p><SCRIPT src="/assets/courseware/v1/631e447105fca1b243137b21b9ed6f90/asset-v1:MITx+8.03x+1T2020+type@asset+block/latex2edx.js" type="text/javascript"/><LINK href="/assets/courseware/v1/daf81af0af57b85a105e0ed27b7873a0/asset-v1:MITx+8.03x+1T2020+type@asset+block/latex2edx.css" rel="stylesheet" type="text/css"/><h2>What You Should See</h2><p>When the notebook is initialized, you will see the following visualization. Follow question prompts within the notebook.</p><div align="center"><img width="800" src="/assets/courseware/v1/fc752381dcc007ec76aec219bca2fcc0/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_double_slit_interference_1.png"/></div><p/>
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