The central-ray aberrations of the two-mirror plane-grating spectrograph are studied in three dimensions. Two mounts are discussed. Astigmatism is found to be reduced in both mounts by a reduction of the ruled length of the grating and/or by reduction of the mirror-tilt angles. When the rulings are orthogonal to the meridional plane, the coma-free condition from the two-dimensional theory is found not to eliminate horizontal coma completely; the residual coma is reduced by the same arrangements that will reduce astigmatism and by placing the mirrors close to each other. For the case in which the rulings are parallel with the meridional plane, an expression for complete elimination of horizontal coma in the central image is given. In both mounts, the conditions for elimination of coma should be satisfied by variation of the curvatures of the mirrors while the tilt angles should be kept as small as possible. Criteria for elimination of astigmatism and spherical aberration are also given.
PETER LINDBLOM, "Theory of the Two-Mirror Plane-Grating Spectrograph," J. Opt. Soc. Am. 62, 756-762 (1972)