To establish a theoretical basis for providing a better design method of multielement optical systems

we have developed a third-order geometric theory of a plane-symmetric multielement optical system that consists of a planar light source

an arbitrary number of ellipsoidal gratings

and an image plane. Analytic formulas of spot diagrams are derived for the system by analytically following a ray-tracing formalism. With these formulas

coma

spherical aberration

and resultant aberration are discussed. To make the theory practical

we determine the aberration coefficients numerically

rather than analytically

with the aid of ray tracing that takes into account the angular distribution of rays originating from a given light source. A merit function is defined so as to represent closely the variance of the spots formed when an infinite number of rays are traced and to take into account the dimensions of the source and the last optical element. The theory is also applicable to mirror-grating or mirror systems.