Diffraction and the Diffraction Grating

Stand a little way back from a doorway and listen to people talking in the next room. You can hear them clearly even though the wall is solid and you cannot see them — the sound has bent round the edge of the doorway and spread out into the room you are in. Watch sea waves roll through the narrow mouth of a harbour and you will see the same thing: beyond the gap the straight rows of waves fan out in gentle arcs, reaching parts of the water the opening does not point at.

This spreading of a wave as it passes through a gap or round an obstacle is called diffraction. It happens to every wave — water, sound, light, radio — and it is one of the clearest signs that something really is a wave. This page builds from that single idea up to one of the most precise instruments in all of physics: the diffraction grating, which uses diffraction to measure the wavelength of light to a handful of nanometres.

Diffraction: waves spread through a gap

When a plane wave meets a barrier with a gap in it, the part of the wave that gets through does not simply carry on as a straight beam the width of the gap. Instead it spreads out into the region beyond, curving into the "shadow" on either side. How much it spreads depends on one comparison: the size of the gap w against the wavelength \lambda.

Gap much wider than the wavelength (w \gg \lambda): the wave passes almost straight through, with only a little fraying at the edges. A beam of light through an open window barely spreads at all, because the window is millions of wavelengths wide.
Gap about equal to the wavelength (w \approx \lambda): the spreading is greatest. The wave fans out through almost a half-circle, as if the gap were a fresh point source of waves.

That is why you can hear round a doorway but not see round it. Sound in a room has a wavelength of roughly a metre — about the width of the doorway — so it diffracts strongly and floods the next room. Light has a wavelength of well under a thousandth of a millimetre, so against the same doorway it is nowhere near the w \approx \lambda condition and travels on in straight rays, casting a sharp-edged shadow.

A single slit: a bright centre and dim fringes

Send monochromatic light (a single wavelength, such as a laser) through one narrow slit and catch it on a screen. You do not get a single blob of light the shape of the slit. You get a broad, bright band in the middle — the central maximum — flanked by a series of much dimmer bands, the side maxima, separated by dark gaps where the light cancels out completely.

The pattern arises because every point across the open slit acts as a little source, and the wavelets they send out interfere: in some directions they arrive in step and reinforce (bright), in others exactly out of step and cancel (dark). Two features are worth fixing in your mind, because the grating builds directly on them:

The central maximum is by far the brightest and is twice as wide as the side fringes — most of the energy goes straight ahead.
Making the slit narrower makes the whole pattern spread out wider (more diffraction), because you are pushing w closer to \lambda. A wider slit gives a tighter, brighter central band.

From one slit to thousands: the diffraction grating

A diffraction grating is a piece of glass or plastic ruled with a huge number of extremely fine, equally spaced parallel lines — typically a few hundred to a few thousand lines per millimetre. Each clear strip between the ruled lines is a slit, so light passes through many thousands of identical, equally spaced slits at once.

Shine monochromatic light straight at it and, instead of the broad fuzzy fringes of a single slit, you see a set of sharp, bright spots (maxima) separated by wide dark regions. In certain special directions the wavelets from every slit arrive exactly in step and add up to an intense beam; in almost every other direction the contributions from the thousands of slits cancel out. The more slits are contributing, the more perfectly the in-between directions cancel — which is why a grating's maxima are so much sharper and brighter than a two-slit pattern.

In the figure the light comes in from the left and strikes the grating. The undeviated beam carries straight on as the zero order (n = 0), and bright orders n = \pm 1, \pm 2, \dots emerge at angles either side of it. Drag the sliders: increase the wavelength, or pack in more lines per millimetre (a smaller slit spacing), and watch every order swing out to a larger angle.

The grating equation

Fix the geometry. The slits are a distance d apart — the slit spacing (or grating spacing). Consider the light leaving the grating at an angle \theta to the straight-through direction (the normal). Compared with its neighbour, the wave from each slit has to travel an extra path of length d\sin\theta to reach a distant screen in that direction.

For all those thousands of wavelets to arrive in step — and so build a bright maximum — that extra path must be a whole number of wavelengths. If it were anything else, the small phase differences would march steadily round and, added over thousands of slits, wash out to nothing. So the condition for a bright order is

d\sin\theta = n\lambda, \qquad n = 0, 1, 2, \dots

For light of wavelength \lambda striking a grating of slit spacing d along the normal, bright maxima occur at angles \theta given by

d\sin\theta = n\lambda.

d — the slit spacing in metres, equal to d = 1/N where N is the number of lines per metre.
\theta — the angle of the maximum from the straight-through (zero-order) direction.
n — the order of the maximum, a whole number. n = 0 is the undeviated central beam; n = 1 the first bright spot either side, and so on.

Because \sin\theta can never exceed 1, the equation puts a hard ceiling on how many orders you can ever see: n \le d/\lambda. There is a maximum order, and it is the largest whole number that keeps \sin\theta \le 1.

Worked examples

Example 1 — the spacing from lines per millimetre. A grating is ruled with 300 lines per millimetre. Find the slit spacing d.

There are 300 lines in every millimetre, so there are 300{,}000 = 3.00\times10^{5} lines per metre. The spacing is the reciprocal:

d = \frac{1}{N} = \frac{1}{3.00\times10^{5}} = 3.33\times10^{-6}\ \text{m} = 3330\ \text{nm}.

Example 2 — the angle of an order. Green laser light of wavelength \lambda = 500\ \text{nm} passes through that same grating (d = 3.33\times10^{-6}\ \text{m}). At what angle is the first-order (n = 1) maximum?

\sin\theta = \frac{n\lambda}{d} = \frac{1 \times 500\times10^{-9}}{3.33\times10^{-6}} = 0.150,

\theta = \sin^{-1}(0.150) = 8.6^\circ.

The second order sits at \sin\theta = 0.300, i.e. \theta = 17.5^\circ — always a larger angle for a higher order.

Example 3 — measuring a wavelength. This is what gratings are for. Light through a grating with d = 2.00\times10^{-6}\ \text{m} gives a first-order maximum at \theta = 18.0^\circ. What is the wavelength?

\lambda = \frac{d\sin\theta}{n} = \frac{2.00\times10^{-6}\times\sin 18.0^\circ}{1} = \frac{2.00\times10^{-6}\times 0.309}{1} = 618\ \text{nm}.

A single angle measurement, read off a rotating table, pins the wavelength to the nearest few nanometres — orange-red light.

Example 4 — how many orders? For that grating (d = 2.00\times10^{-6}\ \text{m}) with red light \lambda = 650\ \text{nm}, what is the highest order that can be seen?

Set \sin\theta = 1 (the extreme) and solve for n:

n \le \frac{d}{\lambda} = \frac{2.00\times10^{-6}}{650\times10^{-9}} = 3.08.

n must be a whole number, so the highest is n = 3. Counting the zero order and both sides, you see orders 0, \pm1, \pm2, \pm3 — seven bright spots in all. (An "n = 3.08" order would need \sin\theta > 1, which is impossible, so it simply does not appear.)

Three traps that quietly swallow marks in exam questions:

d is the spacing between neighbouring slits, not the number of lines. It is the reciprocal of the lines-per-metre: d = 1/N. Convert lines per millimetre to lines per metre first — a grating of 500 lines/mm has N = 5\times10^{5} lines/m, so d = 2\times10^{-6}\ \text{m}, not 1/500. Getting d a thousand times too big is the single most common slip on this topic.
\sin\theta cannot exceed 1. If your working gives \sin\theta = 1.4, that order does not exist — you have not made an arithmetic error, you have simply run past the maximum order. Always check n \le d/\lambda.
A grating gives far sharper, brighter maxima than a double slit. Two slits give broad, gently varying fringes; thousands of slits make every non-maximum direction cancel almost perfectly, so the bright spots become needle-sharp. That sharpness is exactly what makes a grating a precision measuring instrument — more slits means better-defined angles.

Tilt a CD or DVD under a lamp and its surface blazes with bands of colour. That is diffraction, not pigment. The disc stores data along a spiral of microscopic pits, and the turns of that spiral are spaced only about 1.6\ \mu\text{m} apart on a CD — a spacing comparable to the wavelength of light. The rows of pits act as a reflection grating: white light bouncing off obeys d\sin\theta = n\lambda just as a transmission grating does, and since \theta depends on \lambda, each colour comes off at its own angle. Blue is thrown one way, red another, and you see the spectrum fanned out. A DVD packs its tracks even closer (\approx 0.74\ \mu\text{m}), so it spreads the colours through a wider angle still. The same physics paints the shimmering colours on a beetle's shell, a butterfly's wing, and the sheen of a peacock feather — fine, regular structures acting as natural gratings.

Astronomers turn this into a superpower. Pass a star's light through a grating and it splits into a spectrum crossed by dark lines — each line the fingerprint of one chemical element absorbing its own wavelengths. By reading which wavelengths are missing, and using d\sin\theta = n\lambda to pin them down, we learn what a star is made of, how hot it is, and — from how far the lines are shifted — how fast it is rushing towards or away from us, all without ever leaving Earth.