CANONICAL FORMULATION FOR THE GEOMETRICAL-OPTICS OF CONCAVE GRATINGS

Dragt presented a method for characterizing optical systems that uses the concept of Lie transformations [J. Opt. Soc. Am. 72, 372 (1982)]. This method is applied to the geometrical optics of concave gratings that have plane symmetry, and the Lie transformation is derived for holographic gratings and for ruled gratings. This transformation corresponds to the canonical transformation in analytical mechanics. Unlike the systems treated by Dragt, concave grating systems do not have axial symmetry, and consequently odd-order Lie transformations are required. This formulation enables us to derive analytical expressions for the image coordinates. Applications of this formulation to compound optical systems are given.