Shout from one room and a friend in the next room hears you plainly, even though a solid wall stands between you and the only opening is a doorway off to one side. The sound does not travel in a tidy straight beam through the gap and stop dead at the edges of it — it bends round the edges and floods the whole next room. Now take a laser pointer, a beautifully collimated pencil of light, and send it through a slit scratched a fraction of a millimetre wide. On the far wall you do not get a thin bright line the shape of the slit. You get a broad smear of light, brightest in the middle and fading through a series of dark and bright bands out to the sides. The narrower you make the slit, the wider the smear spreads.

Both of these are the same phenomenon: diffraction — the bending and spreading of a wave as it passes an aperture or an edge. It is not a minor optical blemish. Diffraction is the hard limit on the sharpness of every telescope, microscope and camera ever built; it is the working principle of the spectrometers that tell us what stars are made of; and it is one of the cleanest pieces of evidence that light is a wave at all. This page teaches diffraction as a first-year physicist meets it: the qualitative picture, then the quantitative single slit, then the diffraction grating — and the sign trap that snares almost everyone who studies both at once.

What diffraction is, and when it matters

Treat every point on a wavefront as a little source of its own outgoing wavelets — that is Huygens' principle. When an unbroken plane wave rolls forward, those countless wavelets add up along the front and cancel everywhere else, so the wave marches on in a straight line. But put a barrier with a gap in it, and you slice out only a short strip of wavefront. The wavelets from that strip have no neighbours out to the sides to cancel against, so the wave curls into the geometric shadow and spreads beyond the gap.

How much it spreads is governed by a single comparison — the aperture width a against the wavelength \lambda:

• Aperture much wider than the wavelength (a \gg \lambda): the beam passes almost straight through with only a little fraying at the rims. A window is millions of wavelengths of light wide, so daylight through it casts a crisp-edged patch — barely any diffraction.
• Aperture comparable to the wavelength (a \approx \lambda): the spreading is dramatic, fanning out through a wide arc as though the gap were a single fresh point source.

This one comparison explains the opening puzzle. Everyday sound has a wavelength of order a metre — about the width of a doorway — so it diffracts strongly and fills the next room. Visible light has a wavelength under a thousandth of a millimetre, so against a doorway a/\lambda is enormous, diffraction is negligible, and light travels on as sharp rays. To make light spread the way sound does, you must shrink the aperture down toward the wavelength itself — which is exactly what a diffraction slit does.

The single slit: where the dark fringes come from

Send monochromatic plane waves (one wavelength \lambda) straight at a single slit of width a and collect the light far away, at an angle \theta from the straight-through direction. By Huygens, imagine the open slit divided into a great many tiny strips, each radiating a wavelet. The pattern on the screen is the sum — the interference — of all of them, and whether a given direction is bright or dark depends only on the path differences between those strips.

Here is the elegant argument for the dark fringes. The wavelet from the very top of the slit and the wavelet from the very bottom differ in path length by a\sin\theta. Now split the slit into two equal halves and pair up each strip in the top half with the strip a distance a/2 below it in the bottom half. Every such pair differs in path by (a/2)\sin\theta. If that half-slit path difference is exactly half a wavelength, every pair cancels — the whole slit goes dark. That gives the first minimum:

Splitting into four strips, six strips, and so on repeats the trick and yields a dark fringe whenever

Notice the shape of the result: the first minimum sits at \sin\theta = \lambda/a, so a narrower slit (smaller a) throws the fringes out to larger angles — the pattern spreads. That is diffraction's signature written into the algebra: squeeze the aperture toward \lambda and the light fans wide.

The full intensity pattern

The minima locate the dark bands, but adding up all the wavelets (a neat exercise with phasors) gives the complete brightness curve. Writing the half-phase across the slit as

the intensity at angle \theta is

Drag the slider below to change the slit width, measured in wavelengths (a/\lambda), and watch the pattern breathe. A wider slit pulls the central peak into a tight, tall spike with many faint side lobes crowded near the middle; a narrower slit flings the whole pattern wide open. The side lobes are strikingly faint — the first is only about 4.5\% of the central peak — which is why the middle of a single-slit pattern so dominates the eye.

Worked examples — the single slit

Example 1 — angle of the first minimum. Red laser light, \lambda = 633\ \text{nm}, passes through a slit of width a = 0.0200\ \text{mm} = 2.00\times10^{-5}\ \text{m}. Where is the first dark fringe? Use a\sin\theta = m\lambda with m = 1:

The central bright band therefore stretches from -1.81^\circ to +1.81^\circ — an angular width of 3.6^\circ, twice the half-angle, just as promised.

Example 2 — width on a screen. That pattern falls on a wall L = 3.0\ \text{m} away. For small angles \sin\theta \approx \tan\theta = y/L, so the half-width of the central maximum is

The full central band is about 19\ \text{cm} wide — a broad glow, from a slit narrower than a hair.

Example 3 — squeeze the slit. Halve the slit to a = 1.00\times10^{-5}\ \text{m} with the same light. The first minimum moves to \sin\theta = \lambda/a = 0.0633, \theta = 3.63^\circ — double the angle. Narrowing the aperture widens the spread, exactly the inverse relationship \sin\theta \propto 1/a built into a\sin\theta = m\lambda.

From one slit to thousands: the diffraction grating

A diffraction grating replaces the single slit with a great many identical, equally spaced slits — a plate ruled with hundreds or thousands of fine parallel lines per millimetre, each clear strip a slit. Call the centre-to-centre spacing between neighbouring slits d, the grating spacing (with N lines per metre, d = 1/N).

Look at light leaving the grating at angle \theta. Each slit's wavelet travels an extra d\sin\theta compared with its neighbour. For all those thousands of wavelets to arrive in step and build an intense beam, that extra path must be a whole number of wavelengths. Anything else and the marching phase differences, summed over thousands of slits, cancel to nothing. Hence the principal maxima:

The many slits do two things a single slit cannot. First, they make each maximum needle-sharp: with N slits contributing, the bright orders narrow roughly as 1/N, so a grating of ten thousand slits gives pinpoint spots against wide dark gaps. Second, because the angle of each order depends on \lambda, the grating splits white light into a spectrum, throwing each colour to its own angle — the foundation of spectroscopy.

The sharper the maxima, the finer the wavelengths a grating can tell apart — its resolving power. Qualitatively, two nearby wavelengths are resolvable when their peaks are narrower than their separation; since peak width falls as 1/N, more illuminated slits means finer resolution. In symbols the resolving power is \lambda/\Delta\lambda = mN — it improves with both the order m and the number of slits N the beam covers. This is precisely why a grating, not a single slit, is the instrument of choice for measuring the wavelengths in starlight.

Worked examples — the grating

Example 1 — spacing from lines per millimetre. A grating has 500 lines per millimetre. That is N = 5.00\times10^{5} lines per metre, so

Example 2 — angles of the orders. Green light, \lambda = 500\ \text{nm}, hits that grating. First order (m = 1):

Second order sits at \sin\theta = 0.500, \theta = 30.0^\circ; third order at \sin\theta = 0.750, \theta = 48.6^\circ — higher orders always at wider angles.

Example 3 — how many orders? Same grating, red light \lambda = 650\ \text{nm}. Set \sin\theta = 1:

Orders 0, \pm 1, \pm 2, \pm 3 appear — seven bright spots. The would-be "m = 3.08" is not an arithmetic slip; it just means the fourth order has run off past \sin\theta = 1 and vanished.

Example 4 — measuring a wavelength. This is what gratings are for. Light through a grating with d = 1.67\times10^{-6}\ \text{m} (a 600 lines/mm grating) gives a first-order maximum at \theta = 21.1^\circ:

One angle read off a rotating table pins the wavelength to a few nanometres — orange light.

Single slit versus grating — read the equation carefully

The two headline equations look almost identical, and that is the whole danger. They describe opposite things:

• Single slit: a\sin\theta = m\lambda locates the dark fringes (minima), with m = \pm 1, \pm 2, \dots and m = 0 pointedly excluded — a is the width of the one slit.
• Grating: d\sin\theta = m\lambda locates the bright maxima, with m = 0, \pm 1, \pm 2, \dots and m = 0 firmly included — d is the separation between neighbouring slits.

Same shape, opposite meaning. The single-slit condition kills the light; the grating condition lights it up. And the two effects coexist: in a real grating each slit has a finite width a, so the single-slit envelope (a) modulates the brightness of the grating's sharp orders (d) — occasionally a grating order lands right on a single-slit minimum and is snuffed out, a "missing order".