Description of diffraction grating behavior in direction cosine space

It is well known that the angular separation of non-paraxial diffracted orders from a linear grating varies drastically with incident angle. Furthermore, for oblique incident angles (conical diffraction), it is rather cumbersome both analytically and graphically to describe the number and angular position of the various propagating orders. One can readily demonstrate that wide-angle diffraction phenomena (including conical diffraction from gratings) are shift-invariant with respect to incident angle in direction cosine space. Only when the grating equation is expressed in terms of the direction cosines of the propagation vectors of the incident beam and the diffracted orders can we apply the Fourier techniques resulting from linear systems theory. This formulation has proven extremely useful fur small-angle diffraction phenomena and in modern, image formation theory. New insight and an intuitive understanding of diffraction grating behavior results from a simple direction cosine diagram