Foci of Axially Symmetrical Filters

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Abstract
The pupil functions of axially symmetrical filters with significant focusing properties are shown to be periodic functions of the square of the distance from the axis. The representation of the diffraction field derived by Arsenault and Boivin is adapted for the periodic pupil function. According to this representation the diffraction field of a filter is expressed by the superposition of the diffraction fields of the lenses, whose pupil functions form the Fourier expansion of the filter transmissivity. Near the foci of the lenses the diffraction fields superpose in such a way that the diffraction field of a lens becomes dominant near its focus. Thus, the foci of an axially symmetrical filter can be defined as the points in the vicinity of which its diffraction field behaves as the diffraction field of the lens near the focus. Finally, with the use of the scalar theory of the optical diffraction it is shown that the energy transmitted by the filter is entirely distributed among the foci of the filter.

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