Aberration theory and a design method of double-element optical systems

To establish the theoretical basis for providing a better design method of synchrotron radiation beamline optics, we have developed a third-order aberration theory of a double-grating system and derived analytic formulas for spot diagrams and aberration curves. We describe an analytic merit function and a new definition of the resolving power. The former closely represents the variance of a very large number of ray-traced spots in the image plane and takes into account the dimensions of a source and optical elements. The latter rakes into account the effects of an asymmetric line profile and a finite exit-slit width. The equations of aberration curves, merit function, and the definition of the resolving power are evaluated by taking three designs of a Monk-Gillieson varied-spacing plane grating monochromator as testing optics. The analytic merit function is used for these designs. The results show that the features of ray-traced spot diagrams can be analyzed by means of aberration curves and that realistic resolving power can be predicted by the new definition. The results also show the relation between the formation of coma-free images and the choice of design wavelengths. All these findings support the effectiveness of the analytic merit function in the design work.