Abstract

Geometric transformations have been an area of continued research for the past fifteen years. A wide variety of applications, including scale rotation invariant pattern recognition and diffraction plane sampling, have been proposed. Recent advances in binary optics allow the construction of geometric transform elements which demonstrate a high diffraction efficiency and a large space-bandwidth product. Binary optics refers to computer-generated holograms made by multiple mask lithography. For a single mask (as is conventionally used in computer-generated holography) two possible phase levels are available. This leads to a maximum diffraction efficiency of 40%. Using N masks, 2^N−1 phase values are possible with a diffraction efficiency given by [sin(π/N)/ (π/n)]². (A binary optic constructed using three masks has a diffraction efficiency approaching 95%) To demonstrate the performance of a binary optic transformer, an element is constructed which generates a logarithmic polar coordinate transformation: each point (x, y) on the input plane is mapped to a point [ln(x² + y²)^1/2, tan⁻¹(y/x)] on the output plane.

© 1989 Optical Society of America