5.11: Diffraction Pattern for Pentagonal 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:

\[ \Phi (p_x,~p_y) = \frac{1}{ 2 \pi \sqrt{A}} \sum_{m = 1}^{A} ( exp(-ip_xx_m) exp (-i p_y y_m)) \nonumber \]

Display mask geometry and diffraction pattern: \( A \equiv 5\)

\[ \begin{matrix} N = 100 & \Delta p = 12 & j = 0 .. N & k = 0 .. N & px_j = - \Delta p + \frac{2 \Delta p j}{N} & py_k = - \Delta p + \frac{2 \Delta p~ k}{N} \end{matrix} \nonumber \]

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

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