Calculating diffraction patterns

Abstract

Following Marcella’s approach to the double-slit experiment (Marcella T V 2002 Eur. J. Phys. 23 615–21), diffraction patterns for two-dimensional masks are calculated by Fourier transform of the Mask geometry into momentum space. Iw ish to describe a simple extension of Marcella’s [1] recent analysis of the double-slit experiment to two dimensions. The essential point Marcella makes in his unique treatment of this well-known experiment is that the diffraction pattern at the detection screen is actually a measurement of the momentum distribution of the diffracted particles. Therefore the calculate dd iffraction pattern is simply obtained from the Fourier transform of the coordinate space wavefunction (the doubleslit geometry) into momentum space. Marcella considered two spatial models: (model 1) infinitesimally thin slits represented by Dirac delta functions; and (model 2) slits of finite width. About 60 years ago Sir Lawrence Bragg [2] proposed the optical transform as an aid in the interpretation of the x-ray diffraction patterns of crystals. This required the fabrication of two-dimensional masks of various crystal or molecular geometries and the generation of the diffraction pattern using visible electromagnetic radiation. Present-day laser technology has made the generation of suc hd iffraction patterns routine, even in the classroom. In addition, Marcella’s computational approach makes calculating the diffraction patterns conceptually and mathematically straightforward. If one considers the mask as consisting of point scatterers (model 1), the coordinate space wavefunction is a linear superposition of the scattering positions: |� �= 1 √ N