Theory of concave gratings based on a recursive definition of facet positions

A general theory for concave gratings is presented that is based on a recursion formula for the facet positions and that differs from previous theories that are based on a power-series expansion of the light path function. In the recursion formula approach the facet positions are determined from a numerical solution for the roots of two constraint functions. Facet positions are determined in sequence, starting from the grating pole. One constraint function may be chosen to give a stigmatic point. A variety of grating designs are discussed, including a design that cannot be generated with the power-series approach.