Performance analysis of grazing incidence imaging systems

An exact expression relating the coordinates of a point on the incident ray, a point of reflection from an arbitrary surface, and a point on the reflected ray is derived. The exact relation is then specialized for the case of grazing incidence, and first order and third order systematic analyses are carried out for a single reflective surface and then for a combination of two surfaces. The third order treatment yields a complete set of primary aberrations for single element and two element systems. The importance of a judicious choice for a coordinate system in showing field curvature to clearly be the predominant aberration for a two element system is discussed. The validity of the theory is verified through comparisons with the exact ray trace results for the case of the telescope.