Analytical transfer characteristics of axially symmetric 3D optical imaging systems

This paper presents a unified analytical framework to derive the three-dimensional transfer characteristics of axially symmetric optical imaging systems operating in the far field, with emphasis on coherence scanning interferometry, confocal microscopy, and focus variation techniques. While many 3D optical instruments are treated as linear systems, in practice inconsistencies remain in how their transfer functions are derived and interpreted, particularly across forward- and back-scatter geometries. Addressing this, we develop closed-form expressions for the 3D transfer function and point spread function for generic back- and forward-scatter systems under commonly applied apodization conditions (e.g., uniform, root-cosine, and cosine). These derivations clarify the spatial frequency support and resolution trade-offs intrinsic to each geometry and validate the characteristic "bowtie" and "umbrella" structures observed experimentally in the spatial frequency domain. Our analytical results not only resolve the ambiguities of previous numerical models but provide a means to validate and understand the applicability of approximate 3D and 2D numerical models. The formalism is robust, generalizable, and appropriate to the modeling and correction of real instruments in 3D surface metrology and optical tomography.