Diffraction Grating Orders & Sub-Wavelength Limit Calculator

When a Grating Diffracts, and When It Is Just a Film

A surface-relief grating of pitch Λ either splits light into discrete orders or it does not, and the boundary is sharp. Above the cutoff wavelength only the zeroth order survives and the grating acts as a birefringent film; below it, energy leaves along directions no transfer-matrix method can produce.

The Grating Equation Sets the Boundary

Periodicity conserves the transverse wavevector only to within a multiple of \(2\pi/\Lambda\), so a grating couples the incident wave into a comb of orders. Order \(m\) propagates in a medium of index \(n\) only while its transverse component still fits inside that medium's light cone; outside it the order is evanescent and carries no flux away. As \(\Lambda\) shrinks the orders cut off one by one, and past the last cutoff the structure is optically a homogeneous but anisotropic film, with one index for the field along the lines and a lower one across them. That form birefringence is what a wire-grid polarizer and a sub-wavelength anti-reflection grating both exploit.

\(\lambda_c\) is the longest wavelength at which any order beyond the zeroth still propagates — note that oblique incidence pushes it up, so a grating that is sub-wavelength at normal incidence need not stay that way. The reflectance curves below are a transfer-matrix solve of the Rytov effective-medium film, \(\varepsilon_{\mathrm{TE}} = f\varepsilon_1 + (1-f)\varepsilon_2\) and \(\varepsilon_{\mathrm{TM}}^{-1} = f\varepsilon_1^{-1} + (1-f)\varepsilon_2^{-1}\), and they are drawn only where that stand-in is legitimate. Shorter than \(\lambda_c\) the answer is rigorous coupled-wave analysis, which is what the calculator runs behind a grating finish.

Interactive design tool

This page includes a live transfer-matrix calculator for this design. Adjust the parameters to recompute spectra, layer stack, and performance figures in the browser, then open the design in the full calculator.