Theories to design a three-dimensional superresolution filter (TDSF) for confocal microscopy are proposed that can obtain a globally optimal solution through linear programming. The designed TDSF is proved to be a phase-only element introducing a phase delay of 0 or π. Five design examples of the TDSF are presented to demonstrate the validity of these theories. Regardless of transverse superresolution, a curve of S~(eu)((G_(a))~(±)) defined as the maximum value of Strehl ratio S under the axial resolving power of (G_(a))~(±) is calculated to set the fundamental limits of axial optical superresolution. Finally, what is to our knowledge a novel analytic expression of S~(eu)((G_(a))~(±)) is deduced.
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