Computer generated holograms: cylindrical, conical, and helical waves

In computer generation of holograms the wave propagation is numerically simulated. Usually this propagation is approximated by the discrete Fourier transform or by geometrical optics. Use of these methods restricts the kind of object which can be recorded synthetically. We demonstrate that the reconstruction of holographic optical elements and 3-D images can be composed from line segments by superimposing several analytic distributions in the hologram plane. To implement the algorithm, the wave equation is considered in cylindrical coordinates. A solution is given which describes propagating waves denoted cylindrical, conical, and helical waves. Since the complex amplitude is known as an analytic expression throughout space, calculating the transmission of holograms on curved surfaces is possible. In the reconstruction step the waves converge to line foci of arbitrary position and orientation even parallel to the optical axis. The length is controlled by the use of geometrical optics. The feasibility is demonstrated by an optical reconstruction of a 3-D object.