Last updated: 2026-06-16

Diffraction gratings

A diffraction grating is an optical element containing a large number of equally spaced parallel slits or grooves, typically several hundred to several thousand per millimetre. When coherent light strikes the grating, each slit acts as an independent wave source. The waves from all slits travel outward simultaneously and interfere with one another. At most angles the crests and troughs cancel, but at specific angles determined by the grating equation the waves from every slit arrive exactly in phase and reinforce — producing bright diffraction maxima. Because these angles depend on wavelength, the grating separates polychromatic light into its component colours.

The grating equation

The condition for constructive interference between adjacent slits separated by distance $d$ is that the path difference equals a whole number of wavelengths:

where $\theta$ is the angle of the diffracted beam measured from the grating normal, $m$ is the integer diffraction order and $\lambda$ is the wavelength. The slit spacing relates to the ruling density $N$ (lines per millimetre) by

Solving for the diffraction angle gives $\theta = \arcsin\!\left(\dfrac{m\lambda}{d}\right)$.

Formula table

Worked example

Consider a diffraction grating ruled at 600 lines/mm illuminated by sodium yellow light at 550 nm.

The slit spacing is

For the first order (order $m\!=\!1$):

The first-order maximum appears about 19.3° from the straight-through direction.

Diffraction orders and the maximum order

The zeroth order ($m\!=\!0$) satisfies $\sin\theta = 0$, so it always appears straight ahead at $\theta = 0°$ regardless of wavelength. It carries no spectral information. The first order is most commonly used for spectroscopy because it is bright and well-separated from zero. Higher orders exist at larger angles but grow progressively dimmer.

An order is physically observable only when $\sin\theta \leq 1$, which requires $m \leq d/\lambda$. The highest visible order is therefore

For the 600 lines/mm grating at 550 nm, $m_{\max} = \lfloor 1667/550 \rfloor = 3$. Attempting to observe the fourth order yields $\sin\theta_4 = 4 \times 550/1667 \approx 1.32 > 1$, which has no solution — the calculator flags this order as unavailable.

Applications in spectroscopy

Diffraction gratings are the dispersing element of choice in most modern spectrometers, monochromators and spectrophotometers. Their angular dispersion $d\theta/d\lambda = m/(d\cos\theta)$ is large and nearly linear across the spectrum, which makes wavelength calibration straightforward. Echelle gratings use high diffraction orders (up to $m \approx 100$) to achieve very high spectral resolution in compact instruments, such as the cross-dispersed spectrographs used in astronomy to record entire optical spectra at once.

Limits of the model

The grating equation assumes perfectly coherent, monochromatic plane waves and ideal uniformly spaced slits with no manufacturing defects. Real gratings have a finite width that limits the spectral resolving power to $R = mN$, where $N$ is the total number of illuminated slits. The blaze angle, a deliberate tilt of the grating facets, concentrates intensity into a preferred order. The calculator does not account for grating efficiency, polarisation effects or the finite bandwidth of real light sources.