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5.16: Diffraction Pattern for Two Concentric Rings

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    150541

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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): \(m = 1 .. A\)

    Screen Shot 2019-05-09 at 1.26.47 PM.png

    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:

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

    Screen Shot 2019-05-09 at 1.31.40 PM.png

    This page titled 5.16: Diffraction Pattern for Two Concentric Rings was last modified on Wed, 11 Jan 2023 07:23:21 GMT and is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Frank Rioux via source content that was edited to the style and standards of the LibreTexts platform.