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

2014, Bulletin of the American Physical Society

Optics introduce the subject by the familiar 1-d grating formula, a[α-α i }= n x λ , here 'a' is the grating constant and λ 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 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[α-α i }= n x λ; b[ββ i }= n y λ and c[γ-γ i }= n z λ, here α, β & γ (α i , β i & γ 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 However, Laue's grating theory was soon superseded by Lawrence Bragg's namesake formula 2dSin(θ) = nλ. 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.