Fourier optics
The Fourier model for image formation provides a simple and elegant explanation of the limits of resolution of an optical system and of the nature of the image itself. The underlying principle is the same as that used for the analysis of sounds or electronic signals. The whole of sound reproduction theory is based on the principle that any sound, no matter how complex, is made up of sinusoidal (wave-form) components of various amplitudes, frequencies, and phases. The converse, that any sound pattern can be built up from an appropriate spectrum of sinusoids, is equally true. This is called respectively Fourier analysis and Fourier synthesis, after Joseph Fourier (1768-1830), who first worked out the mathematical principles.
The precise optical analogy is that the light beam emitted (reflected or transmitted) by any object, no matter how complex, is identical with the pattern of diffraction that would be transmitted by a spectrum of sinusoidal diffraction gratings of various transmittances, spatial frequencies, and spatial phases, and, for a two-dimensional object, orientations. Similarly, any desired light beam can be constructed by superimposing appropriate sinusoidal gratings on a plane wavefront. One small extension is necessary. Because (unlike a sonic or electrical signal) a transmittance cannot be negative, there is always a ‘d.c.’ component present in an optical grating, equal to half the amplitude of the grating.
The Fourier approach treats imaging as a diffraction phenomenon, and a lens as a device for performing an optical Fourier analysis (in two dimensions) of the sinusoidal functions that together create the shape of the wavefront from the object. The principle is approached by considering a simple sinusoidal grating (Fig. 1). This produces a single pair of diffracted plane wavefronts. If the grating is situated at the front focal plane of the lens, these wavefronts will converge towards the rear focal plane, forming two spots; this can be readily demonstrated using laser light. The electrical field in this plane is the Fourier transform (i.e. the spatial frequency spectrum) of the object grating, and represents—as it must—a single spatial frequency.
When the object is more complicated, the diffraction pattern at the rear focal plane is also more complicated. But it still consists of pairs of spots. Each pair indicates a specific amplitude, spatial frequency, and orientation of the originating sinusoidal grating by (respectively) their intensity, separation, and angle. Indeed, simple objects can be identified from their diffraction patterns. The letter V of (Fig. 2) shows the characteristic pattern of two rectangles at an angle to one another. One thing the visible pattern does not show is the spatial phases of the gratings; but they are nevertheless present in the electric field.
A fundamental property of a Fourier transform is that it obeys the commutative law. This implies that if a Fourier transform is carried out on a Fourier transform, the result is just the original function. So a second lens positioned with its front focal plane in the rear focal plane of the first will produce a replica of the original object at its own rear focal plane (Fig. 3). This, of course will not be news to a photographer: it is part of traditional geometrical optics. What was not generally appreciated, and not actually seen before lasers appeared, was the presence of the diffraction pattern that proved the existence of the optical Fourier transform in the principal focal plane of the lens. (It is, of course, present even with white light, but is smeared out by dispersion and lack of coherence.)
One important outcome of the Fourier principle is that it indicates the absolute limit of resolution of a lens. Because the highest spatial frequencies in the object wavefront are diffracted to the largest angle, they may miss the lens altogether (Fig. 4). Closing the lens aperture down progressively cuts off the higher spatial frequencies, just as applying a high-frequency blocking filter removes the highest sound frequencies from a hi-fi amplifier. The effect was first noticed in the early 1950s, when it was seen that the image of a Sayce grating (a rectangular intensity-profile grating, Fig. 5a) became markedly sinusoidal in profile at the fine-detail end as the aperture of the lens under test was closed down, indicating that the higher spatial frequencies present in the bars were not being recorded (Fig. 5b).

Fig.-1

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Fig.-3

Fig.-4

Fig.-5a

Fig.-5b
— Graham Saxby
Bibliography
- Steward, E. G., Fourier Optics: An Introduction (1989).
- Saxby, G., The Science of Imaging: An Introduction (2002)


































