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Solving the holography equation has long been a numerical task. Whileeffective, the numeric approach has its own set of limitations. Relying solelyon numerical approaches often obscures the intricate interplay and influence ofthe individual terms within the equation. This not only hampers a deeperunderstanding of the underlying physics but also makes it challenging topredict or control specific outcomes. In this study, we address thesechallenges by leveraging our recently published updated Fraunhofer diffractionexpression. This approach allows us to derive an analytic solution forcomplex-valued phase disks in on-axis holography. This solution facilitates thedirect computation of each term's influence within the holographic equation,paving the way for a more profound comprehension and application of theholographic process. When compared to experimental results and the numericFresnel diffraction solution, our analytic approach shows impressive accuracy,considering the inherent approximations. Notably, it remains precise forFresnel numbers that extend well beyond the traditionally accepted boundariesof the Fraunhofer regime.

Gabor-Type Holography Solved Analytically for Complex-valued Phase Disks