Merit function for the design of grating instruments

A merit function that is closely correlated with the rms spread of an infinite number of ray-traced spots by means of analytic spot-diagram formulas is critically evaluated in comparison with exact ray tracing. The analytic spot-diagram formulas are found to generate spot diagrams that are almost indistinguishable from ray-traced ones. The rms spread of ray-traced spots about the mean shows varying degrees of statistical dispersion depending on the number of rays traced, and it approaches the value given by the merit function as the number of rays is increased. All the results clearly show that the merit function behaves as defined and provides enough accuracy and a sufficiently short computing time for the design of highly sophisticated grating instruments.