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<h2 class="hd hd-2 unit-title">Introduction to Diffraction</h2>
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<p>
In this lesson, we first consider the transmission of light through a single slit, which is called diffraction. Our analysis of this phenomenon is closely related to our analysis of interference. </p><p>
Importantly, diffraction and interference cannot be decoupled from each other! Ultimately, we develop a complete picture of double-slit transmission, which includes both interference and diffraction phenomena. </p>
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<h2 class="hd hd-2 unit-title">L39v1: Diffraction Analysis using 2D Fourier Transform</h2>
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<h3 class="hd hd-2">L39v1: Diffraction Analysis using 2D Fourier Transform</h3>
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<h2 class="hd hd-2 unit-title">L39Q1: Fourier Transform Definitions</h2>
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Fourier Transform Definitions
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Consider the 2D Fourier transform of the aperture characterized by the function [mathjaxinline]f(x,y)[/mathjaxinline]: </p>
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<td class="equation" style="width:80%; border:none">[mathjax]C(k_{x},k_{y})=\dfrac {1}{4\pi ^{2}}\int _{-\infty }^{\infty }dx\, \int _{-\infty }^{\infty }dy\, f(x,y)e^{-i\vec{k}_{(x,y)}\cdot \vec{r}_{(x,y)}}[/mathjax]</td>
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where the subscript [mathjaxinline](x,y)[/mathjaxinline] denotes the [mathjaxinline]xy[/mathjaxinline]-components of the total vector (i.e., [mathjaxinline]\vec{r}_{(x,y)}=\vec{r}(z=0)[/mathjaxinline]). </p>
<p><b class="bfseries">(Part a)</b> What does this Fourier transform represent physically? Select ALL that apply. </p>
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<text>a) The amplitude due to the superposition of many infinitesimal point sources, which emit from the aperture.</text>
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<text>b) A representation of the total field amplitude in k-space.</text>
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<text>c) A representation of the total field amplitude in coordinate space.</text>
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<p><b class="bfseries">(Part b)</b> What is the total vector [mathjaxinline]\vec{r}[/mathjaxinline] for which [mathjaxinline]\vec{r}_{(x,y)}[/mathjaxinline] denotes the [mathjaxinline]xy[/mathjaxinline]-components? </p>
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<text> a) The position vector that points towards the location where the total electric field is measured, relative to a location [mathjaxinline](x,y)[/mathjaxinline] in the aperture.</text>
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<text> b) The position vector that points to the location [mathjaxinline](x,y)[/mathjaxinline] of the infinitesimal point in the aperture where a plane wave "source" is emitted.</text>
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<text> c) An arbitrary direction&#8212;it doesn't matter, it's just a Fourier conjugate variable.</text>
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<p><b class="bfseries">(Part c)</b> What is the total vector [mathjaxinline]\vec{k}[/mathjaxinline]? </p>
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<text> a) The wavevector of the original plane wave that is incident on the aperture.</text>
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<text> b) The wavevector of an infinitesimal source that emits a wave from the aperture in the direction of the observer.</text>
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Aperture with Variable Function
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Consider an aperture defined by the following function, where [mathjaxinline]a[/mathjaxinline] is strictly positive ([mathjaxinline]a&gt;0[/mathjaxinline]): </p>
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<td class="equation" style="width:80%; border:none">[mathjax]f(x)=e^{-|x|/a}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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<p><b class="bfseries">(Part a)</b> Choose the answer(s) that best describe the physical meaning of this aperture. Select ALL that apply. </p>
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<text>a) The aperture forms a slit, which extends in the [mathjaxinline]y[/mathjaxinline] direction.</text>
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<text>b) The aperture forms a slit, which extends in the [mathjaxinline]x[/mathjaxinline] direction.</text>
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<text>c) This is a gradient aperture, whose opacity decreases along the [mathjaxinline]x[/mathjaxinline] direction.</text>
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<text>d) This is a gradient aperture, whose opacity increases along the [mathjaxinline]x[/mathjaxinline] direction.</text>
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Variable Aperture Fourier Transform
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<p><b class="bfseries">(Part b)</b> Calculate the Fourier transform [mathjaxinline]C(k_{x})[/mathjaxinline] corresponding to this aperture function [mathjaxinline]f(x)[/mathjaxinline]. Express your answer in terms of <code>a</code> and <code>k_x</code> for [mathjaxinline]k_ x[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]C(k_{x}) =[/mathjaxinline] </p>
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<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
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<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">integers</td>
<td class="formulainput">
<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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<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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<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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<code>abs, ln, sqrt</code>
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<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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<code>sin, cos, tan, sec, csc, cot</code>
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<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
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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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<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>
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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>
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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>
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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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<h2 class="hd hd-2 unit-title">L39Q3: Characterics of Diffraction Intensity Distribution</h2>
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Characteristics of Diffraction Intensity Distribution - part a
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<p><b class="bfseries">(Part a)</b> In lecture, we noted that [mathjaxinline]\sin (\theta )=\dfrac {x}{r}=\dfrac {k_{x}}{k}[/mathjaxinline]. What assumptions or approximations did we consider here? </p>
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<text> a) [mathjaxinline]D \ll r[/mathjaxinline]</text>
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<text> b) [mathjaxinline]x \ll r[/mathjaxinline]</text>
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<text> c) None</text>
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Characteristics of Diffraction Intensity Distribution - part b
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<p><b class="bfseries">(Part b)</b> We found that the intensity of light from an open slit of width [mathjaxinline]D[/mathjaxinline] was given by: </p>
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<td class="equation" style="width:80%; border:none">[mathjax]I\propto \dfrac {\sin ^2\left(\dfrac {\pi D}{\lambda }\sin \theta \right)}{\left(\dfrac {2\pi \sin \theta }{\lambda }\right)^2}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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<p>
If we define </p>
<table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
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<td class="equation" style="width:80%; border:none">[mathjax]\beta =\dfrac {\pi D \sin \theta }{\lambda }[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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<p>
we can write the intensity as </p>
<table id="a0000000004" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
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<td class="equation" style="width:80%; border:none">[mathjax]I \propto \dfrac {\sin ^2\left(\beta \right)}{\beta ^2}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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(Note that the constants of proportionality are different in these two cases.) </p>
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What are the values of [mathjaxinline]\beta[/mathjaxinline] that will lead to destructive interference in the intensity pattern? Let [mathjaxinline]n=1,2,3\ldots[/mathjaxinline] indicate the first, second, third, etc intensity minima. Express your answer in terms of <code>n</code> and relevant numerical constants. </p>
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<p style="display:inline">[mathjaxinline]\beta =[/mathjaxinline] </p>
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<h2 class="hd hd-2 unit-title">L39Q4: Diffraction Intensity [WITH SIMULATION]</h2>
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Simulation of Diffraction Intensity
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Consider the following intensity patterns, which result from single-slit diffraction. </p>
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<p><b class="bfseries">(Part i)</b> Assume that both patterns are associated with a configuration where the distance to the screen [mathjaxinline]L[/mathjaxinline] and the slit width [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> Now, 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 width [mathjaxinline]D[/mathjaxinline] is varied. Which pattern corresponds to the smallest slit width? </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=diffraction_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/cf2dc977387cc788737a974d02657df3/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_diffraction_1.png"/></div><p/>
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<h2 class="hd hd-2 unit-title">L39Q5: Wavelength of Light from Diffraction</h2>
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Wavelength of Light from Diffraction
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Consider light that travels through a single slit of width [mathjaxinline]50~ \mathrm{\mu m}[/mathjaxinline], and lands on a screen a distance [mathjaxinline]2~ \mathrm{m}[/mathjaxinline] away. The width of the central maximum is measured to be [mathjaxinline]4.4~ \mathrm{cm}[/mathjaxinline]. What is the wavelength of the light, in units of [mathjaxinline]\mathrm{nm}[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]\lambda =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]\mathrm{nm}[/mathjaxinline] </p>
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<th class="formulainput" scope="col">Example Entries</th>
</tr>
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<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">integers</td>
<td class="formulainput">
<code>2520</code>
</td>
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<td class="formulainput">fractions</td>
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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"><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">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
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<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]
</td>
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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]
</td>
</tr>
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<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]
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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>
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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>
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<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>
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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>
<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">L39Q6: Diffraction vs. Interference Pattern</h2>
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Diffraction vs. Interference Pattern - part a
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Light from a coherent monochromatic laser of wavelength [mathjaxinline]\lambda[/mathjaxinline] is incident on double slits separated by a distance [mathjaxinline]d[/mathjaxinline]. Each slit has width [mathjaxinline]a[/mathjaxinline] and the slits are placed a distance [mathjaxinline]L[/mathjaxinline] from a screen, where [mathjaxinline]L \gg d[/mathjaxinline]. </p>
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<p><b class="bfseries">(Part a)</b> How many interference peaks occur within the central diffraction peak? This number must be an integer, so express your answer in terms of the rounded integer value [mathjaxinline]c=\mathrm{nint}(d/a)[/mathjaxinline], where [mathjaxinline]c[/mathjaxinline] is rounded up if the decimal part of [mathjaxinline]d/a[/mathjaxinline] is [mathjaxinline]\geq 0.5[/mathjaxinline] and otherwise is rounded down. </p>
<p>
Note that the outer edge of the central diffraction envelope can cut off part of an interference peak. In some cases, the diffraction minimum will occur close to the location of an interference maximum, resulting in two small peaks on either side. You should NOT count such "partial" peaks&#8212;only include "full" interference peaks. An interference peak is considered "full" if the diffraction minimum occurs at or past the next farther interference minimum. </p>
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<p style="display:inline">[mathjaxinline]N =[/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"><code>3.14</code>, <code>.98</code></td>
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<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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<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]
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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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<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]
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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>
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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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<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>
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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>
</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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Diffraction vs. Interference Pattern - part b
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<p><b class="bfseries">(Part b)</b> The diffraction/interference intensity [mathjaxinline]I(y)[/mathjaxinline] is plotted in the figure below as a function of the distance [mathjaxinline]y[/mathjaxinline] shown in the figure in part (a). </p>
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Based on the information in the figure above, determine the relation between [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]d[/mathjaxinline] </p>
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<text> a) [mathjaxinline]d=9a[/mathjaxinline]</text>
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<text> b) [mathjaxinline]a=9d[/mathjaxinline]</text>
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<text> c) [mathjaxinline]d=4a[/mathjaxinline]</text>
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<text> e) [mathjaxinline]d=5a[/mathjaxinline]</text>
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<text> f) [mathjaxinline]a=5d[/mathjaxinline]</text>
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<text> g) [mathjaxinline]d=8a[/mathjaxinline]</text>
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<text> h) [mathjaxinline]a=8d[/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">L39Q7: Diffraction-Interference pattern [WITH SIMULATION]</h2>
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Simulation of Diffraction Intensity
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Consider the following intensity patterns, which result from two-slit interference. Assume that all patterns are associated with a configuration where the distance [mathjaxinline]L[/mathjaxinline] and wavelength [mathjaxinline]\lambda[/mathjaxinline] are fixed. </p>
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<p><b class="bfseries">(Part i)</b> Which two patterns have the same slit separation [mathjaxinline]d[/mathjaxinline], but different slit width [mathjaxinline]a[/mathjaxinline]? </p>
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<p><b class="bfseries">(Part ii)</b> Which two patterns have the same slit width [mathjaxinline]a[/mathjaxinline], but different slit separation [mathjaxinline]d[/mathjaxinline]? </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=interference_diffraction_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/v1/5fbd1a57e2831cbc2bf9802d0e494df9/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_interference_diffraction_1.png"/></div><p/>
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