# Fourier Transform, Diffraction and Interference.

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.

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

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.
The case of a single slit is shown by Fn = 1 in the applet below. sW sets the slit width and a very simplified indication of the colors can be seen by varying l.
Fn = 1 to 4 show special cases of Fn = 1.
Fn = 5 is the case of a single slit with the source having 2 wavelengths.
Fn = 6 to 7 show special cases of Fn = 5.
Fn = 8 is the case of a single slit with the source having 3 wavelengths.
Fn = 9 is the case of a double slit.
Fn = 10 to 12 show special cases of Fn = 9.
Fn = 13 is the case of a double slit with the source having 3 wavelengths.
Fn = 14 to 16 show special cases of Fn = 13.
Fn = 17 is the case of a N slits.
Fn = 18 to 21 show special cases of Fn = 17.
Fn = 22 is the case of a Rectangular Aperture.
Fn = 23 to 26 show special cases of Fn = 22.
Fn = 27 is the case of a Circular Aperture.
Fn = 28 to 31 show special cases of Fn = 27.
Fn = 32 is the case of a Circular Aperture with the source having 3 wavelengths.
Fn = 33 to 36 show special cases of Fn = 32.
Fn = 37 is the case of a single slit with the source having 3 wavelengths.
Fn = 38 show a special case of Fn = 37.
Fn = 39 is the case of a double slit with the source having 3 wavelengths.
Fn = 40 to 41 show special cases of Fn = 39.