Single-slit diffraction

When light passes through a slit whose width $a$ is a few to a few hundred wavelengths, it does not form a sharp image of the slit on a distant screen. It spreads into a bright central band, twice as wide as each of the side fringes, flanked by a series of much fainter bands separated by dark minima. For a 632.8 nm helium-neon laser, a 0.1 mm slit and a screen 1 m away, the central band is 12.7 mm wide and the first side fringes carry 4.7 % of the central peak intensity. Narrower slits give wider patterns: the angular scale is $\lambda/a$. It is the simplest case of diffraction and a standard undergraduate experiment.

Where the minima fall

By the Huygens-Fresnel principle, every point across the slit radiates a secondary wavelet. At an angle $\theta$ the wavelets from the two edges differ in path by $a\sin\theta$. When that difference is one wavelength, the slit can be divided into two halves whose wavelets pair off with a half-wave difference and cancel. The dark fringes therefore lie at

$$a \sin\theta_m = m\lambda, \qquad m = \pm1, \pm2, \dots$$

The central maximum spans $-\lambda/a < \sin\theta < \lambda/a$, so on a screen at distance $L$ its full width in the small-angle approximation is

$$w = \frac{2\lambda L}{a}.$$

Intensity pattern

In the far field the amplitude is the Fourier transform of the uniform slit, a sinc function, and the intensity is

$$I(\theta) = I_0 \left(\frac{\sin\beta}{\beta}\right)^2, \qquad \beta = \frac{\pi a \sin\theta}{\lambda}.$$

The secondary maxima sit where $\tan\beta = \beta$, at $\beta$ = 1.430π, 2.459π, 3.471π, slightly closer to the centre than the midpoints between minima. Their relative intensities are 4.72 %, 1.65 % and 0.83 %. The central lobe contains 90.3 % of the transmitted power.

Worked example

Take $\lambda$ = 632.8 nm, $a$ = 0.100 mm and $L$ = 1.00 m.

• First minimum: $\sin\theta_1 = \lambda/a$ = 6.328 × 10⁻³, so $\theta_1$ = 6.33 mrad and the minimum lies 6.33 mm from the centre.
• Central maximum width: $2\lambda L/a$ = 12.66 mm.
• First and second side maxima: 9.05 mm and 15.56 mm from the centre, at 4.7 % and 1.6 % of the peak.

The same slit at 1550 nm gives a central band 31.0 mm wide. Measuring the minimum spacing with a ruler and inverting $a = \lambda L/y_1$ gives the slit width to a few percent, which is the usual laboratory exercise.

Far-field condition

The sinc² pattern is the Fraunhofer, or far-field, result. It applies when the Fresnel number $F = (a/2)^2/(\lambda L)$ is much less than 1. In the worked example the half-width is 0.05 mm and $F$ = 0.0040 at 1 m, well into the far field; a distance of 1 mm would give $F$ = 4.0, and the pattern there is a Fresnel pattern with fringes inside the geometric image of the slit. A lens placed after the slit produces the far-field pattern in its focal plane at any slit width, with $L$ replaced by the focal length.

In practice

In grating spectrometers the entrance slit width controls the trade between throughput and spectral resolution, and the diffraction limit of the grating's own width sets the best attainable resolution. Slit diffraction also governs the divergence of light leaving narrow apertures such as the output facet of an edge-emitting laser diode, whose thin active layer makes the beam spread fastest in the direction normal to the junction. By Babinet's principle, a thin wire of diameter $a$ produces the same far-field pattern as a slit of width $a$ apart from the undiffracted beam, and laser diffraction gauges use this to measure wire and fiber diameters without contact.

The single-slit pattern is also the envelope of the double-slit experiment: two slits of width $a$ produce cos² fringes whose brightness is modulated by the sinc² of each slit, and a diffraction grating extends this to many slits. A circular aperture gives the two-dimensional analogue, the Airy disk, with its first minimum at 1.22 $\lambda/D$ in place of $\lambda/a$.

Pitfalls

The small-angle formula $y = m\lambda L/a$ underestimates positions by about $(m\lambda/a)^2/2$, 0.5% at $m\lambda/a$ = 0.1 and 2% at 0.2; for slits only a few wavelengths wide, the scalar theory itself fails and polarization matters. The illumination must be spatially coherent across the slit; an extended lamp source smears the minima unless it is first passed through a pinhole or narrow slit. Slit edges that are not parallel fill in the minima.

Common questions

What is the formula for single-slit diffraction minima?

$a\sin\theta = m\lambda$ with $m = \pm1, \pm2, \dots$; there is no minimum at $m = 0$, which is the central maximum.

Why is the central maximum twice as wide as the others?

It runs from the $m = -1$ minimum to the $m = +1$ minimum, a span of $2\lambda/a$ in $\sin\theta$, while each side fringe runs between consecutive minima separated by $\lambda/a$.

What happens when the slit is made narrower?

The pattern widens in inverse proportion: halving the slit width doubles the angle of the first minimum, and the total transmitted power falls by half.

The slit diffraction calculator draws the single-slit pattern for a chosen slit width, wavelength and screen distance, and adds more slits to show the change to interference fringes.

References: E. Hecht, Optics, 5th ed. (Pearson, 2017), Ch. 10; M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999), Ch. 8; J. W. Goodman, Introduction to Fourier Optics, 4th ed. (W. H. Freeman, 2017).