Skip to main content
Chemistry LibreTexts

Grating Equation

  • Page ID
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    If we say: all angles are measured counter-clockwise, starting at the grating normal, then in the above drawings, α is positive and β is negative. For constructive interference to occur, we know that the phase shift in scattering from adjacent echellettes must be an integer number of wavelengths (so the crests of the scattered waves from the two facets are in-phase). This leads to The Grating Equation:

    \[nλ = d(\sin α + \sin β)\]

    with the sign convention as noted. n is the diffraction order and says by how many wavelengths the wavefront shifts between incidence and scattering. If the grating acts as a mirror, n = 0. Dispersion, different wavelengths giving constructive interference at different angles, requires that n be a non-zero integer. Since the sine of an angle is always of magnitude less than 1, the largest value of λ for which there is a solution for the grating equation is for n = 1, sin α = 1, sin β = 1, so λ = 2d.

    This page titled Grating Equation is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Contributor.

    • Was this article helpful?