Specular Reflection as the Universal Formulation for N-dimensional Diffraction Gratings, N$=$1- 3

Textbooks in Optics introduce the subject by the familiar 1-d grating formula, a[$\alpha $ -- $\alpha_{\mathrm{i}}$\textbraceright $=$ n$_{\mathrm{x}}\lambda $ , here `a' is the grating constant and $\lambda $ is wavelength. Ever since the development of precession ruling engines by Rowland, 1- dimensional optical diffraction gratings have become ubiquitous, and workhorse in optical devices. Optical cross gratings (2-d) with lines ruled in both x {\&} y directions are treated \textit{mutatis mutandis} by a pair of 1-d grating formula. In 1912, Max von Laue, Nobel Physics for 1914, proposed his three fundamental equations for 3-d, x-ray grating as: a[$\alpha $ -- $\alpha _{\mathrm{i}}$\textbraceright $=$ n$_{\mathrm{x}}\lambda $; b[$\beta $ -- $\beta_{\mathrm{i}}$\textbraceright $=$ n$_{\mathrm{y}}\lambda $ and c[$\gamma $ -- $\gamma_{\mathrm{i}}$\textbraceright $=$ n$_{\mathrm{z}}\lambda$, here $\alpha $, $\beta $ {\&} $\gamma $ ($\alpha_{\mathrm{i}}$, $\beta_{\mathrm{i}}$ {\&} $\gamma _{\mathrm{i}}$ ) are the direction cosines of the outgoing (incoming) x-ray beam. Furthermore for simplicity an orthorhombic crystal structure with lattice constants a, b {\&} c, oriented along each Cartesian axis respectively, were assumed. However, Laue's grating theory was soon superseded by Lawrence Bragg's namesake formula 2dSin($\theta )=$ n$\lambda $. Peter Ewald's reciprocal lattice construction demonstrated that when certain conditions, 3-d diffraction process reduces to Bragg's reflection law. We show that reflection is a generic or universal treatment for one, two or three-dimensional gratings.