Fourier optics

To explain the basic principle of Fourier optics, it might be useful to consider the analogue with acoustics: If an arbitrary sound is analyzed through a spectrum analyzer, you will get all the different frequencies and their individual amplitudes that, together, form the sound. You can imagine that in a female soprano voice there are much more high frequency components than in a male bass voice.

The Fourier spectrum of an object is its diffraction pattern after having passed through a lens. If the object is at infinity, the diffraction pattern is in the focal plane. Small details in the object (e.g. sharp edges) correspond to high "frequencies" in the diffraction pattern (which is mathematically the Fourier transform of the object-function).

These high spatial "frequencies" can be found at larger distances to the optical axis than low spatial "frequencies", corresponding to the larger structures in the object. Imagine a fence consisting of vertical bars of constant distance. Its diffraction pattern would ideally be two dots at equal distance horizontally on both sides of the optical axis.

When changing the distance of the bars, these dots would move towards or off the optical axis in horizontal direction, i.e. the closer the bars, the larger the distance of the spots (the higher the spatial frequency). Fourier optics influences the diffraction pattern by, e.g., weakening or cancelling certain spatial "frequencies". If you, e.g., want to enhance the edges in an image you have to attenuate the central regions in the diffraction pattern while passing the higher spatial frequencies (outer regions of the diffraction pattern) in their full intensity before imaging the diffraction pattern through a lens.

More reading on this subject: Hecht /Zajac OPTICS, Addison-Wesley Publishing Comp.