5.16: Diffraction Pattern for Two Concentric Rings

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Create hole positions:

\[ \begin{matrix} A = 32 & R = 1.2 & m = 1 .. 16 & \Theta_m = \frac{2 \pi m}{16} & x_m = R \sin ( \Theta_m) & y_m = R \cos ( \Theta_m) \\ ~ & R = .9 & m = 17 .. A & \Theta_m = \frac{2 \pi m}{16} & x_m = R \sin ( \Theta_m) & y_m = R \cos ( \Theta_m ) \end{matrix} \nonumber \]

Display coordinate-space wave function (mask geometry): \(m = 1 .. A\)

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Fourier transform position wave function into the momentum representation:

\[ \begin{matrix} \Delta = 30 & N = 200 & j = 0 .. N & px_j = - \Delta + \frac{2 \Delta j}{N} & k = 0 .. N & py_k = - \Delta + \frac{2 \Delta k}{N} \end{matrix} \nonumber \]

Hole dimension:

\[ \begin{matrix} d = .1 & \Psi (p_x,~p_y) = \frac{1}{2 \pi d \sqrt{A}} \sum_{m = 1}^A \left( \int_{x_m - \frac{d}{2}}^{x_m + \frac{d}{2}} exp(-i p_x x)dx \int_{y_m - \frac{d}{2}}^{y_m + \frac{d}{2}} exp(-i p_y y) dy \right) \end{matrix} \nonumber \]

Display diffraction pattern:

\[ P_{j,~k} = \left( \left| \Psi (x_j,~Py_k ) \right| \right)^2 \nonumber \]

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