Last updated: 2026-06-16

Single-slit diffraction

Single-slit diffraction is the spreading and interference of light waves after passing through an aperture of finite width. When coherent light falls on a narrow slit of width $a$, each point across the slit acts as a secondary wavelet source. These wavelets travel outward at all angles and interfere with one another, producing a characteristic pattern on a distant screen: a bright central maximum flanked by alternating dark fringes and weaker bright fringes.

How the minima form

The dark fringes arise because wavelet contributions from opposite halves of the slit cancel in pairs. For the first minimum, a wavelet from the top edge of the slit travels exactly half a wavelength farther than one from the centre; together they cancel. The same pairing argument extends to every pair of wavelets separated by $\tfrac{a}{2}$, so the entire slit contributes zero net amplitude at that angle. For the second minimum the slit is divided into four zones, and so on for higher orders.

Formula

The condition for the $m$-th dark fringe is

where $a$ is the slit width, $\lambda$ is the wavelength, $m$ is the integer fringe order (counting from the centre outward on either side) and $\theta_m$ is the angle from the straight-through direction to that minimum. Solving for the angle gives

The formula is only valid when $m\lambda/a \leq 1$; if the ratio exceeds 1 the minimum does not physically exist.

Worked example

A laser with wavelength 550 nm illuminates a slit 20 µm wide. The first-order minimum (order $m\!=\!1$) occurs at

The second minimum (order $m\!=\!2$) is at $\sin\theta_2 = 0.055$, giving $\theta_2 \approx 3.153°$.

Central maximum width

The central bright fringe spans from $-\theta_1$ to $+\theta_1$, so its full angular width is $2\theta_1$ — twice the width of any side lobe. For a screen at distance $L$ the physical half-width of the central maximum is approximately $\lambda L / a$ in the small-angle limit ($\sin\theta \approx \tan\theta \approx \theta$). The full width is therefore $2\lambda L/a$.

Because the central maximum occupies twice the angular space it contains a disproportionate share of the diffracted intensity, making it far brighter than the side lobes.

Applications

Single-slit diffraction sets a fundamental limit on optical resolution. The Rayleigh criterion for a circular aperture of diameter $D$ gives a minimum resolvable angle of $1.22\lambda/D$. This governs the resolution of telescopes, microscopes, and camera lenses. Diffraction also appears in X-ray crystallography, where atomic spacings act as the "slit" and the pattern reveals crystal structure, and in spectroscopy, where diffraction gratings separate wavelengths by the same underlying physics.

Limits of the model

The formula assumes Fraunhofer (far-field) diffraction, where both the source and the observation screen are effectively at infinity — in practice, at a distance much greater than $a^2/\lambda$. At shorter distances Fresnel diffraction applies and the pattern becomes more complex. The derivation also assumes a perfectly monochromatic plane wave and an infinitely long slit; real slits are finite in height, producing a two-dimensional pattern. Polarisation effects are neglected.