Abstract

The fast Fourier transform, although efficient, for calculating computer-generated holograms (CGHs) of objects in the form of 2-D arrays, is less useful with other types of object. In the work reported here, the interest was in developing an efficient algorithm for calculating Fourier transform CGHs of objects composed of a number of line segments. First, the object is restructured by approximating each line-segment angle ϕ by the closest angle ϕ′ in a set of standard angles whose arctangents are ratios of integers. (This significantly reduces the number of trigonometric functions to be performed.) Next, the line segments are grouped by ϕ′ and by center to exploit mathematical symmetries inherent in the sinc function (the Fourier transform of a single line segment). The Fourier transform is then performed on each segment, and the final hologram, represented by amplitude and phase matrices, is a summation of the individual line-segment transforms. The CGHs are encoded using the binary detour phase technique, printed with a laser printer, and then photoreduced into a 35-mm slide film. The reconstructed images of 2-D line-segment objects show multiple orders due to slight oversampling of the Fourier transform. The reconstructions of 3-D ot,ects, composed of 2-D line-segment objects spaced along the optical axis, show real images at different locations near the focal point of the reconstruction beam. Alternatively, virtual images may be seen by addition of a viewing lens placed after the reconstruction lens. The application of this algorithm to real-time metrology of optical surfaces and to 3-D holographic displays is currently being investigated.

© 1988 Optical Society of America