Geometric theory of the ellipsoidal grating.

A third-order geometric aberration theory of the ellipsoidal grating has been developed by analytically following an exact ray-tracing formalism with the aid of power series expansions. The theory takes into account all the possible aberrations up to third order and provides analytic formulas for the spot diagram of a spectral image formed by a modified or a nonmodified ellipsoidal grating with any of the groove patterns producible by means of mechanical ruling or conventional holographic recording. The present analytic formulas and other analytic ray-deviation formulas used in designing grating instruments have been evaluated in comparison with exact ray tracing. The results show the validity of the present theory and the limitation of the ray-deviation formulas based on the light path function and wave-frontaberration theory.