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<h2 class="hd hd-2 unit-title">Introduction to Diffraction Phenomena</h2>
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Finally, in this lesson we consider other aspects of diffraction phenomena—namely, resolution. We will see that there is a limit at which two objects can be distinguished from each other spatially—this is the "diffraction limit." </p>
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<h2 class="hd hd-2 unit-title">L40Q1: Understanding Diffraction Limit</h2>
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Understanding Diffraction Limit - part a
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Diffraction through a circular aperture (i.e., pinhole) creates what is known as an "Airy pattern," shown below. </p>
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Image source: https://commons.wikimedia.org/wiki/File:Airy-pattern.svg </p>
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Attribution: Sakurambo at English Wikipedia / Public domain </p>
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The angle at which the first minimum occurs is determined by the following relation: </p>
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<td class="equation" style="width:80%; border:none">[mathjax]\sin {\theta } \approx 1.22 \dfrac {\lambda }{D}[/mathjax]</td>
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where [mathjaxinline]D[/mathjaxinline] is the diameter of the aperture. </p>
<p><b class="bfseries">(Part a)</b> Compare this to the condition for the first diffraction minimum for a single slit. Suppose that the slit width is the same as the diameter of the pinhole. How does the width of the central maximum of the slit compare to the diameter of the central maximum of the Airy pattern for the pinhole? </p>
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Understanding Diffraction Limit - part b
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<p><b class="bfseries">(Part b)</b> Consider diffraction through a single slit with width [mathjaxinline]D^{\prime }[/mathjaxinline]. What should the width of the single slit be in order to produce a central band with the same width as the diameter of the central spot in an Airy pattern with pinhole diameter [mathjaxinline]D[/mathjaxinline]. Express your answer in terms of <code>D</code>. </p>
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<p style="display:inline">[mathjaxinline]D^{\prime } =[/mathjaxinline] </p>
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Understanding Diffraction Limit - part c
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<p><b class="bfseries">(Part c)</b> Now consider a square with sides of length [mathjaxinline]a[/mathjaxinline] that has the same area as a circle with diameter [mathjaxinline]D[/mathjaxinline]. What is the length [mathjaxinline]a[/mathjaxinline] in terms of [mathjaxinline]D[/mathjaxinline]? Express your answer using <code>D</code>. How does your answer compare with [mathjaxinline]D^{\prime }[/mathjaxinline]? </p>
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<p style="display:inline">[mathjaxinline]a =[/mathjaxinline] </p>
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<h2 class="hd hd-2 unit-title">L40Q2: Telescope Resolution</h2>
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Resolution of Hubble Space Telescope
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<p>
The "resolution" of a telescope is defined as the closest angular separation that two objects can have and still be distinguished from each other. Since one is talking about two fuzzy Airy patterns getting closer and closer together, this criterion is not that well defined. The "standard" specification for resolution is the angular radius of the central maximum, the separation at which the maximum of one Airy pattern coincides with the first minimum of the other. </p>
<p>
In astronomy, the apparent "size" of an object and its features is described in units of arcseconds (or fractions thereof), where one arcsecond is [mathjaxinline]1/3600[/mathjaxinline] degrees, and one arcminute is [mathjaxinline]1/60[/mathjaxinline] degrees. </p>
<p>
For instance, the apparent diameter of the full moon is approximately [mathjaxinline]30[/mathjaxinline] arcminutes (or half a degree). The star with the largest apparent size (excluding the Sun) is R Doradus, which has an apparent size of [mathjaxinline]0.057[/mathjaxinline] arcseconds. </p>
<p><b class="bfseries">(Part a)</b> The resolution of the Hubble telescope is [mathjaxinline]0.05[/mathjaxinline] arcseconds for blue visible light ([mathjaxinline]\lambda \sim 476\, \mathrm{nm}[/mathjaxinline]). What is the diameter of the Hubble telescope? </p>
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<p style="display:inline">[mathjaxinline]D =[/mathjaxinline] </p>
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<code>2/3</code>
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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>
</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>
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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>
<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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Note that even with its excellent resolution, the Hubble telescope cannot measure the size of stars outside the Solar System. Instead, these sizes are measured using correlations between multiple detectors (a variation of the interference effect) in the so-called HBT technique, named for Robert Hanbury Brown and Richard Twiss who first used this procedure in 1956. <p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/></p>
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<h3 class="hd hd-3 problem-header" id="lect_26_08b-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_26_08b-problem-progress" tabindex="-1">
Resolution of Arecibo Radio Telescope
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<p><b class="bfseries">(Part b)</b> The Arecibo telescope is capable of resolving features the size of [mathjaxinline]3.5[/mathjaxinline] arcminutes in the radio spectrum&#8212;it has a diameter of [mathjaxinline]305[/mathjaxinline] meters. </p>
<p>
What would be the resolution of a radio telescope with a diameter equal to the diameter of the Earth? Consider observations at [mathjaxinline]\lambda \sim 21\, \mathrm{cm}[/mathjaxinline]. Is this better than the resolution of the Hubble telescope for visible light? Enter your answer in units of arcsec. </p>
<p>
Note that [mathjaxinline]21\, \mathrm{cm}[/mathjaxinline] is the wavelength of radiation that is emitted by a fundamental transition in neutral hydrogen (called the hyperfine transition). Hydrogen is the most abundant element in the universe, and therefore observations at this wavelength reveal many important structures at different distance scales. </p>
<p>
<p style="display:inline">[mathjaxinline]\Delta \theta =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]\mathrm{arcsec}[/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>
</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>
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<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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<h3 class="hd hd-2">L40v4: Single Slit Diffraction vs. Two Slit Interference [DEMO]</h3>
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<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_L40v4">Downloads and transcripts</h3>
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<h4 class="hd hd-5">Transcripts</h4>
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