Quality factors of an aberrant Gaussian beam diffracted through a convex grating spectrometer

We analyze the evolution of a complex-source-point spherical wave through a concentric catoptric system that is made up of one convex grating and two concave mirrors. First we trace the paraxial paths of a chief ray propagating from a complex source point through the system to complex image points of different orders. Then we find an expression of the fourth order-corrected Gaussian beam which includes Seidel-type aberrations. Finally, we analyze the quality of an aberrant Gaussian beam diffracted through the system. The results presented here will be useful in dealing with the propagation of a focused Gaussian beam through a diffractive optical system and the beam quality degradation due to aberration of the system.