Research work · 1988

Direction cosine space diagram

Douglas S. Goodman, IBM T. J. Watson

Annual Meeting Optical Society of America

Abstract

The direction cosine space representation of angles relates more directly to fundamental physics than the more common θ and ϕ specifications. Consequently, geometrical optics, radiometry, diffraction, and interference can be better unified. Directions and ranges of directions can be displayed with a simple diagram. If (α, β, γ) are the direction cosines, points in an (α, β) coordinate system represent directions and areas show ranges of direction. Many basic results can be obtained trivially with this diagram. For example, it is easily seen that brightness is conserved by refraction at plane surfaces and single-order diffraction by gratings. In addition, many quantities are actually in direction cosine space without this being realized. For example, the pupil function is properly represented in this space, as are computations involving it, for example, the evaluation of the OTF. The fundamental radiometric quantity ∫dΩ cosθ is an area in the cosine space diagram, as is π sin2θ, a radiometric quantity associated with imaging systems. The direction-cosine space diagram also facilitates understanding of multidimensional phase space, which can be represented by a diagram of diagrams involving both position and direction.

DOI: 10.1364/oam.1988.wo7