Diffraction grating

A diffraction grating is a surface ruled or patterned with grooves at a constant spacing $d$, usually quoted as a groove density in lines per millimeter. Each groove scatters the incident light, and the scattered waves add in phase only in directions where the path difference between neighboring grooves is a whole number of wavelengths. For light incident at angle $\alpha$ and diffracted at $\beta$, both measured from the grating normal, that condition is the grating equation,

with $m$ the integer diffraction order. It is diffraction from a periodic structure, and the Bragg condition is its counterpart for a structure that is periodic through its thickness rather than across a surface. For a 600 line/mm grating at normal incidence and 632.8 nm, the first order leaves at 22.31° and the second at 49.41°, and there is no third, because $\sin\beta$ would exceed one.

Dispersion and resolution

Differentiating the grating equation at fixed incidence gives the angular dispersion, $d\beta/d\lambda = m/(d\cos\beta)$, which is what separates wavelengths in a spectrometer. The resolving power, the smallest resolvable wavelength difference as a fraction of the wavelength, is $\lambda/\delta\lambda = mN$ for $N$ illuminated grooves; it depends on the width of the illuminated grating, not on the groove density alone. A 600 line/mm grating illuminated over 50 mm has $N$ = 30 000, so in first order it can in principle resolve 0.052 nm at 1550 nm. Higher orders multiply both dispersion and resolution, but orders overlap: first-order light at $\lambda$ leaves at the same angle as second-order light at $\lambda/2$, so spectrometers that cover a broad range use order-sorting filters. Echelle gratings take this to its limit, working in orders of tens to hundreds with a cross-disperser to separate them.

Littrow configuration and blaze

When the diffracted order returns along the incident beam, $\alpha = \beta$, the grating equation becomes $m\lambda = 2d\sin\theta$. A 600 line/mm grating in first-order Littrow at 1550 nm sits at 27.71°, with an angular dispersion of 0.039°/nm. This retroreflection is the feedback element of a Littrow external-cavity laser: rotating the grating tunes the wavelength returned to the diode. The Littman-Metcalf arrangement adds a mirror so the output beam does not move as the laser tunes.

Which order receives the light is controlled by the groove profile. A blazed grating has sawtooth grooves whose facets act as small mirrors, putting most of the energy into one order near the blaze wavelength; ruled gratings are blazed by the ruling tool, holographic gratings by ion etching, and holographic gratings generally scatter less stray light. Efficiency also depends on polarization, which is the origin of much of the polarization dependence specified for a grating-based optical spectrum analyzer. The same equation governs waveguide gratings that couple light out of a chip, in the grating coupler and the emitters of an optical phased array, and the phased-array grating that an arrayed waveguide grating builds from waveguides of stepped length.

References: C. Palmer, Diffraction Grating Handbook, 8th ed. (MKS Instruments, 2020); E. Hecht, Optics, 5th ed. (Pearson, 2017).