# 19.2: Diffraction Patterns

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Laser diffraction experiments can be conducted using an optical bench, as shown below. Light from the laser (of wavelength *λ*) is diffracted by a mask (usually a small aperture or grating) and projected onto the screen, located at a large distance away, such that Fraunhofer geometry applies. The light on the screen is known as the diffraction pattern.

The form of the diffraction pattern from a single slit mask, of width w, involves the mathematical “sinc function”, where

\[\operatorname{sinc}(z)=\frac{\sin (z)}{z}\]

The observable pattern projected onto the screen (a distance *L* away) has an intensity pattern as follows, where *x* is the distance from the straight-through position:

\[I(x)=I_{o} \operatorname{sinc}^{2}\left(\frac{\pi x w}{\lambda L}\right)\]

Note that sinc(0) = 1.

Diffraction patterns can be calculated mathematically. The operation that directly predicts the amplitude of the diffraction pattern from the mask is known as a Fourier Transform (provided the conditions for Fraunhofer Diffraction are satisfied). The derivation of some simple patterns can be found here.

A diffraction grating is effectively a multitude of equally-spaced slits. The diffraction pattern from a complex mask such as a grating can be constructed from simplier patterns via the convolution theorem. The observed diffraction pattern is composed of repeated "sinc-squared" functions. Their positions from the central spot are determined by *s* (the spacing between slits) and their relative intensity is dependent on *w* (the width of individual slits).

More complicated masks, for example a periodic row of apertures, will show more intricate diffraction patterns, but still follow the same basic inverse relationship.