Three-dimensional transfer functions in 4Pi confocal microscopes

We investigate three-dimensional transfer functions in order to understand image formation in the recently proposed 4Pi confocal microscope [ J. Opt. Soc. Am. A9, 2159 ( 1992);Opt. Commun.93, 277 ( 1992)]. Both nonfluorescence and fluorescence imaging methods are considered, and the effects of apodization functions are taken into account. It is shown that for a 4Pi confocal system of finite-sized aperture the cutoff spatial frequency in the axial direction is larger than that in the transverse direction; this behavior is opposite that of conventional confocal systems, explaining the observation of high axial resolution in 4Pi confocal systems. In addition, the three-dimensional optical transfer function for a conventional confocal fluorescence microscope of high aperture is presented.