Nonparaxial theory of curved holograms

Nonparaxial imaging by holograms on a spherical surface, with a limited circular cross section, is formulated mathematically. Starting from a point-shaped object, the image point is redefined such that the deviation in the positions of the actual image-forming wave front and a spherical wave front centered around this point contains no terms of the first order in the hologram coordinates. Furthermore, their second-order contribution, averaged over the hologram, vanishes. The remaining terms up to the fourth order describe the primary aberrations. Compared with earlier theories, this formulation yields a more accurate position for the holographic image. In addition, expressions are obtained for the primary aberrations, which not only consist of astigmatism, coma, and spherical aberration but also contain three additional contributions, which occurred as higher-order aberrations in previous literature. Experiments support the conclusions.