Fourier Transform and Diffraction

Cuthbert Nyack

One area of application of the Fourier transform is diffraction, radiation patterns and image formation.
Here we look at diffraction from a single and double slits. The layout for the single slit is shown by the image below.
The Diffraction pattern is the Fourier Transform of f(x), the transmission function.
f(x) = 1.0 for -a/2 £ x £ +a/2 and zero elsewhere.

The coordinates x and k form a Fourier pair and they are related as shown below.

The intensity is F(k)² and is plotted below. The horizontal axis is given in terms of q and ranges from -p/2 and +p/2.

For a double slit with the centers of the slits separated by b + a, the pattern includes interference and diffraction effects. The expression is given below and its square is plotted in the applet below.

When "white" light rather than monochromatic is used, the pattern can be approximated by the applet below.

When N slits are used, the fringes become narrower with minor peaks between the major peaks. The pattern is shown below

and its square is shown in the applet below.

An approximation showing the Airy pattern for for diffraction from a circular aperture is shown below. This applies to image formation, radiation from a circular source(speaker) etc.
In the applets above the lower intensity fringes have been enhanced for visibility. However the intensity plots in white give the correct relation.
In the applet below the fringes are very faint. Setting Vis = 1.0 gives the correct intensity plot and fringes. However the fringes only become visible for Vis ~ 1.5. The effect of Vis on the intensity plot is shown.

For a Rectangular Aperture, the diffraction pattern is a 2D Fourier Transform obtained by multiplying the pattern of a horizontal slit with that of a vertical slit. The result is illustrated in the applet below. In this applet both vertical and horizontal axes have the same range of q.

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