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. To establish a theoretical basis for providing a better design method of multielement optical systems, we have developed 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 analyzing a ray-tracing formalism. With these formulas, coma, spherical aberration, and resultant aberrations are discussed. To make the theory practical, we determine the aberration coefficients numerically, rather than analyzed, 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 to closely observe 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 mirro r-grating or mirror systems.