Apply the diffraction grating equation to analyse the dispersion of light into spectra

What this dot point is asking

WACE wants you to use the grating equation, understand why many slits sharpen the maxima, and explain how a grating produces a spectrum. The grating is the precision instrument of wave optics.

The grating equation

For a grating with slit spacing $d$ (the distance between adjacent slits), bright maxima occur where

Here $m$ is the order of the maximum. The slit spacing is usually given as lines per millimetre $N$, so $d=1/N$ in millimetres, which you convert to metres. The zeroth order ($m=0$) is straight through; higher orders appear at larger angles.

Why many slits give sharp lines

With only two slits the bright fringes are broad and gradually fade. With thousands of slits, light from all of them must be in phase to give a maximum, which happens only at very precise angles. A tiny deviation from the exact angle causes waves from different parts of the grating to cancel, so the maxima become extremely narrow and bright. This sharpness is what makes gratings able to resolve closely spaced wavelengths.

Dispersing light into a spectrum

Because the diffraction angle depends on wavelength, each colour in white light emerges at a slightly different angle in each order. The result is a spectrum spread out by wavelength, with violet (short wavelength) deviated least and red (long wavelength) most. Unlike a prism, a grating spreads the colours in a predictable, calculable way, which is why gratings are used in spectrometers to measure the wavelengths emitted by light sources.

Limits on the orders

Since $\sin\theta$ cannot exceed $1$, only a finite number of orders exist for a given grating and wavelength. The highest visible order is the largest integer $m$ for which $m\lambda/d\le 1$. Long wavelengths and finely ruled gratings (small $d$) produce fewer orders.

Working with line densities

Convert lines per millimetre to a spacing $d$ in metres before substituting, and remember $m$ must be a whole number. If a calculation gives $\sin\theta>1$ for some order, that order does not exist for that wavelength.