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.

Finding an unknown wavelength

The grating is most often used in reverse: measure the angle of a known order and solve for the wavelength. Rearranging the grating equation gives $\lambda=\dfrac{d\sin\theta}{m}$, so a measured first-order angle and a known slit spacing give the wavelength directly. Using a higher order improves precision, because the same wavelength is spread to a larger angle where an angular measurement error matters less. This is exactly how a spectrometer determines the wavelengths of the bright lines emitted by an excited gas, linking this topic to atomic spectra: the grating disperses the light, the angles are measured, and each line's wavelength (and hence the energy-level transition that produced it) is calculated.

Grating versus prism

Both a grating and a prism disperse white light into a spectrum, but they do so by different physics and in opposite orders. A prism relies on refraction, where the refractive index varies with wavelength, so violet (short wavelength) is bent most and red least. A grating relies on interference through $d\sin\theta=m\lambda$, so the long-wavelength red light is diffracted to the largest angle and violet to the smallest, the reverse of a prism. The grating also produces several orders and gives a wavelength that can be calculated exactly from the geometry, whereas a prism's dispersion must be calibrated. These differences are a frequent comparison question.

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.