Three-dimensional Coherent Transfer Functions for Confocal Imaging with Unequal Annular Lenses

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Abstract
We report, in this paper, the three-dimensional (3D) coherent functions for a confocal scanning system consisting of two unequal annular lenses. Analytical expressions for the 3D coherent transfer functions are derived for both transmission and reflection cases. Various numerical plots of the 3D coherent transfer function are given for different confocal systems including dark field microscopes. Our method is then generalized to the fibre-optical confocal system in which optical fibres rather than pinholes are employed for illumination and collection. In this case, the 3D coherent transfer function in transmission can also be obtained analytically, whilst that in reflection cannot be expressed by analytical functions, but by a single integral. Finally, images of a grating of periodic square structures are simulated to reveal the significance of the derived 3D coherent transfer functions.