Focal lines of arbitrary position and orientation even parallel to the optical axis may be used in a variety of situations, such as display of 3-D images, contouring of inclined surfaces and holographic optical elements. A synthetic generation using a computer is promising because of the flexibility. Then, the wave propagation has to be simulated. We present a calculation by use of the stationary phase method and show the analogy to the solution of the wave equation. A cube composed of line segments is recorded in a hologram and reconstructed optically. For each line segment, a second line foci occurs in the Fourier plane of the hologram which causes irritation when the image is visually observed. Superposition of a slight random phase variation removes the ambiguity and allows secure accomodation. Intensity variations along the line segment can be introduced by superposition of several distributions which reconstruct the same line segment. A further generalization may be achieved to reconstruct nonstraight focal lines in 3-D.
D . Leseberg,
"Free Positioned And Oriented Focal Lines From Computer-Generated Holograms", Proc. SPIE 0812, Progress in Holography, (6 October 1987); doi: 10.1117/12.941624; https://doi.org/10.1117/12.941624