Local signal decomposition based methods for the calculation of three-dimensional scalar optical diffraction field due to a field given on a curved surface

Abstract

A three-dimensional scene or object can be optically replicated via the threedimensional imaging and display method holography. In computer-generated holography, the scalar diffraction field due to a field given on an object (curved surface) is calculated numerically. The source model approaches treat the building elements of the object (such as points or planar polygons) independently to simplify the calculation of diffraction field. However, as a tradeoff, the accuracies of fields calculated by such methods are degraded. On the other hand, field models provide exact field solutions but their computational complexities make their application impractical for meaningful sizes of surfaces. By using the practical setup of the integral imaging, we establish a space-frequency signal decomposition based relation between the ray optics (more specifically the light field representation) and the scalar wave optics. Then, by employing the uncertainty principle inherent to this space-frequency decomposition, we derive an upper bound for the joint spatial and angular (spectral) resolution of a physically realizable light field representation. We mainly propose two methods for the problem of three-dimensional diffraction field calculation from fields given on curved surfaces. In the first approach, we apply linear space-frequency signal decomposition methods to the two-dimensional field given on the curved surface and decompose it into a sum of local elementary functions. Then, we write the diffraction field as a sum of local beams each of which corresponds to such an elementary function on the curved surface. By this way, we increase the accuracy provided by the source models while keeping the computational complexity at comparable levels. In the second approach, we firstly decompose the three-dimensional field into a sum of local beams, and then, we construct a linear system of equations where we form the system matrix by calculating the field patterns that the three-dimensional beams produce on the curved surface. We find the coefficients of the beams by solving the linear system of equations and thus specify the three-dimensional field. Since we use local beams in threedimensional field decomposition, we end up with sparse system matrices. Hence, by taking advantage of this sparsity, we achieve considerable reduction in computational complexity and memory requirement compared to other field model approaches that use global signal decompositions. The local Gaussian beams used in both approaches actually correspond to physically realizable light rays. Indeed, the upper joint resolution bound that we derive is obtained by such Gaussian beams.