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The lens as a Fourier Transform System
Figure 1 shows how a Fourier Transform and inverse transform can be achieved optically. A collimated beam is projected through a signal, f(x,y), contained on a transparency. The transform lens causes parallel bundles of rays to converge in the back focal plane of the lens. This back focal plane is known as the Fourier transform plane. In this plane the spatial image is transformed into a spatial frequency spectra. In affect the lens has carried out a two-dimensional Fourier transform at the speed of light. A far field diffraction pattern is observed by placing a screen in the transform plane. The intensity of the pattern is related to the square of the amplitude of the Fourier Transform of the input signal. By placing a stop at a particular frequency lobe in this plane a spatial frequency can be removed from the image. Typically all but the zero order diffraction is removed, thus removing the noise from the image. This cleaning of the beam is achieved by using a spatial filter. The diameter and quality of the lens limit the upper frequency bandwidth. The lower bandwidth is limited by the ability of the user to discriminate all but the zero order diffraction information. This analogue optical system is difficult to adapt into alternative kinds of filters and so is not very versatile. The aperture of the lens limits the resolution of the Fourier transform. The second lens forms the inverse transform and recovers the original signal.
If the input signal is a sinusoidal grating, figure 2, there will be two spots either side of the central DC component. The two spots correspond to the spatial frequency content of the input signal. The radial distance between these spots and the DC term represents the spatial frequency of the input signal. There will be a row of spots in the transform plane of the ‘square wave’ bar grating, indicating the presence of harmonics of the fundamental frequency.
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