5.12: Diffraction Pattern for Pentagonal Finite Point Scatterers

Establish mask geometry:

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

Fourier transform of position wave function (mask geometry) into the momentum representation:

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

Display mask geometry and diffraction pattern: \( \begin{matrix} A = 5 & d = .3 \end{matrix}\)

\[ \begin{matrix} N = 100 & \Delta p = 10 & j = 0 .. N & k = 0 .. N & px_j = - \Delta + \frac{2 \Delta p j}{N} & py_k = - \Delta p + \frac{2 \Delta p k}{N} \end{matrix} \nonumber \]

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

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