Combined amplitude and phase filters for increased tolerance to spherical aberration

Analysis of the expression for Strehl ratio for a circularly symmetric pupil allows one to design complex filters that offer reduced sensitivity to spherical aberration. It is shown that filters that combine hyper-Gaussian amplitude transmittance with hyper-Gaussian phase modulation provide five-fold reduction in sensitivity to spherical aberration. Furthermore, this is achieved without the introduction of zeros into the modulation transfer function and deconvolution can restore the transfer function to that of a diffraction-limited imager. The performance of the derived combined amplitude and phase filter is illustrated through the variation of its axial intensity versus spherical aberration. This technique is applicable to imaging in the presence of significant amounts of spherical aberration as is encountered in, for example, microscopy.