, and we have where denotes the convolution operation, then we also have we can write the aperture function as The amplitude is then given by the Fourier transform of this expression as: Intensity The intensity is given by: The diagram shows the diffraction pattern for a grating with 20 slits, where the width of the slits is 1/5th of the slit separation. The size of the main diffracted peaks is modulated with the diffraction pattern of the individual slits. Other gratings The Fourier transform method above can be used to find the form of the diffraction for any periodic structure where the Fourier transform of the structure is known. Goodman uses this method to derive expressions for the diffraction pattern obtained with sinusoidal amplitude and phase modulation gratings. These are of particular interest in holography. Extensions Non-normal illumination If the aperture is illuminated by a monochromatic plane wave incident in a direction , the first version of the Fraunhofer equation above becomes: The equations used to model each of the systems above are altered only by changes in the constants multiplying and , so the diffracted light patterns will have the form, except that they will now be centred around the direction of the incident plane wave. The grating equation becomes Non-monochromatic illumination In all of the above examples of Fraunhofer diffraction, the effect of increasing the wavelength of the illuminating light is to reduce the size of the diffraction structure, and conversely, when the wavelength is reduced, the size of the pattern increases. If the light is not mono-chromatic, i.e. it consists of a range of different wavelengths, each wavelength is diffracted into a pattern of a slightly different size to its neighbours. If the spread of wavelengths is significantly smaller than the mean wavelength, the individual patterns will vary very little in size, and so the basic diffraction will still appear with slightly reduced contrast. As the spread of wavelengths is increased, the number of "fringes" which can be observed is reduced. See also Kirchhoff's diffraction formula Fresnel diffraction Huygens principle Airy disc Fourier optics References Diffraction Fourier analysis