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VP14 Circular Aperture Diffraction
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<div id="doc" class="markdown-body container-fluid comment-enabled" data-hard-breaks="true" style="position: relative;"><h1 id="VP14-Circular-Aperture-Diffraction"><a class="anchor hidden-xs" href="#VP14-Circular-Aperture-Diffraction" title="VP14-Circular-Aperture-Diffraction"><span class="octicon octicon-link"></span></a>VP14 Circular Aperture Diffraction</h1><p><img src="pic/VP14/1.png" alt="1"></p><ul>
<li><a href="pdf/VP14.pdf" target="_blank" rel="noopener">官方PDF檔</a></li>
<li><a href="VP14_mistakes.html" target="_blank">同學常見錯誤</a></li>
</ul>
<h3 id="同學們的影片">同學們的影片</h3>
<ul><li><a href="">這部份當然是還沒有啦</a></li></ul>
<h2 id="作業繳交格式"><a class="anchor hidden-xs" href="#作業繳交格式" title="作業繳交格式"><span class="octicon octicon-link"></span></a>作業繳交格式</h2><p>請上傳一個 zip 檔（壓縮檔，請注意副檔名要是 <code>zip</code>）到 CEIBA，zip 檔內需要包含一個<strong>名稱是自己學號的資料夾</strong>，裡面包含一個名為 <code>must.py</code> 的 py 檔。而若本次作業有拍攝說明影片，請<strong>將影片連結寫在</strong> <code>video.txt</code> <strong>裡面，並一併放入學號資料夾中</strong>。</p><p>範例：（zip 檔名可以隨便取，CEIBA 系統會自己把檔名改掉）</p><pre><code>the_zip_file.zip
└── r07222060
├── must.py
└── video.txt
</code></pre><blockquote>
<h4 id="助教註"><a class="anchor hidden-xs" href="#助教註" title="助教註"><span class="octicon octicon-link"></span></a>助教註:</h4>
<p>如果你是用 JuPyter notebook 寫的，請記得將你的程式用 <code>.py</code> 格式下載下來，不要給我 <code>.ipynb</code> 或甚至 <code>.html</code>。</p>
</blockquote><h2 id="Deadline"><a class="anchor hidden-xs" href="#Deadline" title="Deadline"><span class="octicon octicon-link"></span></a>Deadline</h2><p><code>2020/06/11 THU 21:00</code></p><h2 id="Homework"><a class="anchor hidden-xs" href="#Homework" title="Homework"><span class="octicon octicon-link"></span></a>Homework</h2><p>This homework is to find the diffraction pattern of a circular aperture far from the screen.</p><p>Assume there is a circular aperture of diameter <code>d = 100 μm</code> and there is a spherical screen at <code>R = 1 m</code> away. The light source is of wavelength <code>𝜆 = 500 nm</code>. To obtain the diffraction pattern, you can assume there are many point light sources at the lattice points, sitting at the cross points of the <code>N = 100</code> vertical lines and <code>N = 100</code> horizontal lines, each separated by <code>s = d/N = 1 μm</code> apart. If the lattice point’s position is within the circular aperture, then it is allowed to radiate light. Then on the screen you add all the electric field from the light sources of the grid points and you will be able to obtain the diffraction intensity pattern by squaring the electric field. (The detailed derivation is listed below.)</p><p>In addition to (1) generating the intensity plot of the diffraction pattern, also (2) find and print the radius of the first dark ring and check whether the <strong>Rayleigh criterion</strong> <code>θ = 1.22 𝜆 / d</code> is satisfied.</p><p>Also notice that, to calculate intensity is to do the square of the amplitude. However, the ‘real’ diffraction pattern is really faint to be observed on the computer screen. Therefore, we also do a ‘false’ intensity image of the diffraction pattern by taking the absolute value of the amplitude as the intensity. Of course, when you want to calculate the radius of the first dark ring, you should use the ‘real’ intensity image.</p><pre><code class="python hljs"><span class="hljs-keyword">from</span> vpython <span class="hljs-keyword">import</span> *
<span class="hljs-keyword">from</span> numpy <span class="hljs-keyword">import</span> *
N = <span class="hljs-number">100</span>
R, lamda = <span class="hljs-number">1.0</span>, <span class="hljs-number">500E-9</span>
d = <span class="hljs-number">100E-6</span>
dx, dy = d/N, d/N
scene1 = canvas(align=<span class="hljs-string">'left'</span>, height=<span class="hljs-number">600</span>, width=<span class="hljs-number">600</span>, center=vector(N*dx/<span class="hljs-number">2</span>, N*dy/<span class="hljs-number">2</span>, <span class="hljs-number">0</span>))
scene2 = canvas(align=<span class="hljs-string">'right'</span>, x=<span class="hljs-number">600</span>, height=<span class="hljs-number">600</span>, width=<span class="hljs-number">600</span>, center=vector(N*dx/<span class="hljs-number">2</span>, N*dy/<span class="hljs-number">2</span>, <span class="hljs-number">0</span>))
scene1.lights, scene2.lights = [], []
scene1.ambient, scene2.ambient = color.gray(<span class="hljs-number">0.99</span>), color.gray(<span class="hljs-number">0.99</span>)
side = linspace(<span class="hljs-number">-0.01</span>*pi, <span class="hljs-number">0.01</span>*pi, N)
x,y = meshgrid(side,side)
E_field = cos(<span class="hljs-number">10000</span>*((x<span class="hljs-number">-0.005</span>)**<span class="hljs-number">2</span> + (y- <span class="hljs-number">0.002</span>)**<span class="hljs-number">2</span> )) <span class="hljs-comment"># change this to calculate the electric field of diffraction of the aperture</span>
Inte = abs(E_field) ** <span class="hljs-number">2</span>
maxI = amax(Inte)
<span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> range(N):
<span class="hljs-keyword">for</span> j <span class="hljs-keyword">in</span> range(N):
box(canvas=scene1, pos=vector(i*dx, j*dy, <span class="hljs-number">0</span>), length=dx, height=dy, width=dx,
color=vector(Inte[i,j]/maxI, Inte[i,j]/maxI, Inte[i,j]/maxI))
Inte = abs(E_field)
maxI = amax(Inte)
<span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> range(N):
<span class="hljs-keyword">for</span> j <span class="hljs-keyword">in</span> range(N):
box(canvas=scene2, pos=vector(i*dx, j*dy, <span class="hljs-number">0</span>), length=dx, height=dy, width = dx,
color=vector(Inte[i,j]/maxI, Inte[i,j]/maxI, Inte[i,j]/maxI))
</code></pre>
<p><img src="pic/VP14/aperture.png" alt="aperture" width="650"></p>
<p><br>
The amplitude of the electric field at position (x, y) on the screen gets its contribution from all point source sitting inside the aperture:</p>
<p><img src="pic/VP14/eq1.png" alt="eq1" width="700"></p>
<p><br>
In this problem, <img src="https://latex.codecogs.com/gif.latex?\dpi{150}&space;\iint\frac{1}{R}\sin(k_{x}X+k_{y}Y)dXdY=0" title="\iint\frac{1}{R}\sin(k_{x}X+k_{y}Y)dXdY=0" height="40"> due to the symmetry of the integration on the ‘circular’ aperture. Therefore,</p>
<p><img src="pic/VP14/eq2.png" alt="eq2" width="535"></p>
<p><br>
is the amplitude of the electric field on the spherical screen, which is the one you should calculate by summation over all the grid point on the aperture.</p><h2 id="Grading-Policy"><a class="anchor hidden-xs" href="#Grading-Policy" title="Grading-Policy"><span class="octicon octicon-link"></span></a>Grading Policy</h2><ol>
<li>Correctly generated the diffraction pattern (40%)</li>
<li>Correctly calculated the theoretical value (angle or radius) by Rayleigh criterion (20%)</li>
<li>The simulated angle or radius is close to the theoretical value (40%)</li>
</ol></div>
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