Diffraction Gratings

A diffraction grating — a slide with a very large number of equally spaced slits, typically hundreds per millimetre. Light passing through is diffracted at each slit, and the many beams interfere.

The grating equation —

d sinθ = nλ

where d is the slit spacing, θ the angle of the maximum from the straight-through direction, and n the order.

Finding d — gratings are specified by the number of lines per millimetre, so d is the reciprocal of the number of lines per metre.

Example: a grating with 500 lines per mm has 5.00 × 10⁵ lines per metre, so d = 1 ÷ (5.00 × 10⁵) = 2.00 × 10⁻⁶ m. For 600 nm light in the first order, sinθ = (600 × 10⁻⁹) ÷ (2.00 × 10⁻⁶) = 0.300, giving θ = 17.5°.

Orders — n = 0 is the straight-through maximum; n = 1 the first order either side, and so on. Since sinθ cannot exceed 1, the highest possible order is the largest integer n for which nλ ≤ d, which is why a question may ask how many orders are visible.

Why a grating beats two slits — with two slits, the intensity varies smoothly between maxima and minima. With thousands of slits, the many beams cancel almost everywhere except at the precise angles satisfying the grating equation, so the maxima become very sharp and bright with darkness between. Sharper maxima mean angles can be measured far more precisely, giving a much more accurate wavelength.

Spectra — white light gives a white zero-order maximum, since all wavelengths satisfy the equation at θ = 0, and a spectrum for each higher order. Within each spectrum red is deviated most, because it has the longest wavelength and a larger λ needs a larger θ.

Example: this is the opposite of a prism, where red is deviated least. Confusing the two is a common error — the mechanisms are entirely different, one being interference and the other dispersion by refraction.

Uses — measuring wavelengths accurately, and identifying elements from their line spectra in analytical chemistry and astronomy.