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5.21: Multiple Slit Diffraction and the Fourier Transform

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    150546

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    The American Journal of Physics published a translation of Claus Jonssonʹs paper ʺElectron Diffraction at Multiple Slitsʺ in American Journal of Physics 42, 4‐11 (1974). The following calculation is in agreement with the diffraction pattern reported by Jonsson.

    A four slit geometry is created. This represents the coordinate space wave function. It is Fourier transformed into the momentum representation to generate its diffraction pattern.

    \[ \begin{matrix} \text{Number of slits:} & n = 4 & \text{Slit positions:} & j = 1 .. n & x_j = j & \text{Slit width:} & \delta = .2 \end{matrix} \nonumber \]

    \[ \Phi (p) = \frac{ \sum_{j=1}^n \int_{x_j - \frac{ \delta}{2}}^{ x_j + \frac{ \delta}{2}} \frac{1}{ \sqrt{2 \pi}} exp (-i px) \frac{1}{ \sqrt{ \delta}} dx}{ \sqrt{n}} \nonumber \]

    Screen Shot 2019-05-09 at 8.25.53 PM.png

    The momentum wave function is Fourier transformed back to coordinate space to generate the spatial wave function or slit geometry.

    \[ \begin{matrix} x = 0, .01 .. 5 & \Psi (x) = \frac{1}{ \sqrt{2 \pi}} \int_{-30}^{30} exp (ipx) \Phi (p) dp \end{matrix} \nonumber \]

    Screen Shot 2019-05-09 at 8.27.18 PM.png

    This page titled 5.21: Multiple Slit Diffraction and the Fourier Transform 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.