Critical-point behaviors of gratings in Littrow mountings

By using the recent theory of elemental scattering matrix, I study the critical-point behaviors of conically mounted and crossed gratings in Littrow mountings. As functions of the projected incident wave vector in the grating plane, the (m, n)th-order reflected principal diffraction efficiencies (the maximum and minimum diffraction efficiencies of a diffraction order for all possible incident polarizations) of a crossed grating have C₂ symmetry with respect to the (m, n)th-order Littrow point, and for a conically mounted grating, the symmetry type is increased to C₂. In a Littrow mounting, each one of the two principal diffraction efficiencies, independent of the other, can only be at one of the three possible states: a local maximum, a local minimum, or a saddle point of an even (most likely the second) degree. If the grating is mirror-symmetric with respect to a plane parallel to the grating's mean plane, the results hold for a transmitted diffraction order as well.