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5.13: Pentagram Diffraction Pattern

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    150538

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    Establish mask geometry:

    \[ \begin{matrix} R= 2 & m = 1 .. 5 & \Theta_m = \frac{2 \pi m}{5} & x_m = R \sin ( \Theta_m ) & y_m = R \cos ( \Theta_m) \\ R = 0.75 & m = 6 .. 10 & \Theta_m = \frac{2 \pi (m - 0.5)}{5} & x_m = R \sin ( \Theta_m) & y_m = R \cos ( \Theta_m) \end{matrix} \nonumber \]

    Fourier transform of position wave function (mask geometry) into the momentum representation: \( m = 1 .. 10\)

    \[ \begin{matrix} d = .15 & \Phi (p_x,~p_y) = \frac{1}{2 \pi d \sqrt{10}} \left[ \sum_{m = 1} \left( \int_{x_m - \frac{d}{2}}^{ x_m + \frac{d}{2}} exp(-i p_x x) dx \int_{y_m - \frac{d}{2}}^{y_m + \frac{d}{2}} exp (-i p_y y) dy \right) \right] \end{matrix} \nonumber \]

    Display mask geometry and diffraction pattern:

    \[ \begin{matrix} N = 100 & \Delta p = 20 & j = 0 .. N & k = 0 .. N & px_j = - \Delta + \frac{2 \Delta p j}{N} & py_k = - \Delta p + \frac{2 \Delta p k}{N} \end{matrix} \nonumber \]

    \[ \text{Diffraction pattern}_{j,~k} = \left( \left| \Phi ( px_j,~py_k ) \right| \right)^2 \nonumber \]

    Screen Shot 2019-05-09 at 12.30.40 PM.png

    This page titled 5.13: Pentagram Diffraction Pattern 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; a detailed edit history is available upon request.