Single-Slit Diffraction

A slit of width a, or a hair of the same width, and a wall L metres away. The page computes the light on the wall from the wave theory, with no far-field shortcut, and draws under it where ray optics would put the light: a band exactly as wide as the slit. Drag the slit width and watch the two disagree.

Worked example — Single-Slit Diffraction

Direction of the first dark point	0.532°

These are the values the calculator shows when the page opens. Change any input above to recompute them.

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Lesson

The theory — Single-Slit Diffraction

Single-slit diffraction: light through a slit of width a spreads, and in the far field its brightness at angle θ is I(0)(sin β/β)² with β = πa sin θ/λ. The first dark points are at sin θ = ±λ/a, so on a wall at distance L the central band between them is 2λL/a wide. Whether a slit is in the far field is decided by its Fresnel number N = a²/λL: well below 1, the pattern is this one; well above 1, the light lands roughly where ray optics puts it.

What each symbol means

a: the width of the slit, or the thickness of the hair
λ: the wavelength of the light
L: the distance from the slit to the wall
N: the Fresnel number a²/λL, which says which kind of pattern the wall shows

Assumes: A long, straight slit lit head-on by an even beam, so the pattern varies across the wall only; small angles. For the hair, a laser beam 1 mm across, taken as uniform across the hair itself, which is good to 0.5% at the edges of a 70 µm hair.

Narrow a fine slit and the light spreads wider 🖖

Press A 70 µm slit. Ray optics says a slit lets through a band of light as wide as itself, 0.070 mm. On the wall 2 m away the dark points either side of the centre are 37.14 mm apart, 530.6 times wider. Now press Half as wide: 74.29 mm. The band is 2λL/a, so halving a doubles it. The first dark point is where light from the top half of the slit and light from the bottom half arrive half a wavelength out of step, pair by pair: a sin θ = λ. A narrower slit needs a steeper angle to make that half wavelength. The light also dims, faster than it spreads: half as much gets through and it is spread twice as wide, so the centre falls from 0.377% of the beam to 0.094%, a quarter. This holds for a fine slit, and the page says how fine: while the Fresnel number a²/λL is below 2, a slit narrower than 1.61 mm for red light at 2 m, every narrower slit gives a wider band. At 70 µm the number is 0.0038. Press A 5 mm slit: its Fresnel number is 19.2308, the light lands where the rays say, rippled near the edges, and there narrowing the slit narrows the light, as ray optics expects. Between the two the band passes through a narrowest width. A round opening, like a telescope's, does the same in two dimensions, which is the next page in this chapter: the diffraction limit.

The side bands are not halfway between the dark points 🖖

Press A 70 µm slit and then Boost ×10, which multiplies the display so the faint side bands show. The dark points are evenly spaced, 18.57 mm apart here, at β = π, 2π, 3π where β = πax/λL. It is natural to put the bright side bands halfway between them. They are not: the first sits at 1.4303 spacings from the centre, not 1.5, the second at 2.4590 and the third at 3.4709, creeping towards halfway without arriving. The brightness is (sin β/β)². The sin² part does peak halfway, but 1/β² is still falling there, and it pulls each peak back towards the centre, to where tan β = β. The first side band is 4.719% as bright as the centre, about a twenty-first, the second 1.648% and the third 0.834%. Put the first one halfway and you get 1/(π²·1.5²) = 4.503%, the figure OpenStax's University Physics works out at β = 3π/2, where it says the maximum is expected; the real peak is higher and nearer the centre. All of the side bands together hold under a tenth of the light: 90.28% falls inside the central band. The table under the results lists them for the slit on the page.

A hair makes the same pattern as a slit 🖖

Press A 70 µm hair. Ray optics gives an opaque strip a shadow 0.070 mm wide and nothing more. Instead, either side of the laser's own bright spot, the dark points are 37.14 mm apart, exactly as for the 70 µm slit. A slit and a strip of the same width together block nothing, so the waves they let past add up to the bare beam. Away from the beam's own spot, about 2 mm across on the wall, the bare beam is dark, so the hair's wave and the slit's are equal and opposite and their brightness is the same. Babinet put it in 1837, explaining coronae, the coloured rings round the Sun and the Moon: the effect of a small opaque body placed near the line of sight, but off it, "sera exactement le même que celui d'une ouverture toute pareille", exactly the same as that of an opening of the same shape. That is how to measure a hair with a laser pointer. The page gives the gap for a given hair; the worked problem runs it backwards.

Problem solved in full

Measure a hair with a laser pointer 3 steps
You tape a hair across a red 650 nm laser pointer and shine it at a wall 2 m away. The dark points either side of the bright spot are 31.2 mm apart, measured with a ruler good to 1 mm. How thick is the hair, and how well do you know it? Press A 70 µm hair first: the page gives the gap for a 70 µm hair at the same distance.

Open this problem in the tool
1. The dark points of a hair are a slit's, 2λL/a apart, so the thickness is a = 2λL/gap:
  
  $a = \frac{2 λ L}{gap} = \frac{2 \times 650 \times 1 0 ^{- 9} m \times 2 m}{0 . 0312 m} = 83.33 μ m$
2. Or by proportion from the page's own figure, since the gap goes as 1/a:
  
  $a = 70 μ m \times \frac{37 . 14 mm}{31 . 2 mm} = 83.33 μ m$
3. A gap of 31.2 ± 1 mm brackets the thickness:
  
  $\frac{2 λ L}{32 . 2 mm} = 80.75 μ m, \frac{2 λ L}{30 . 2 mm} = 86.09 μ m$
Answer

About 83 µm, give or take 3 µm. A 1 mm error on a 31 mm gap is about 3%, and it carries straight into the thickness. Type 83.3 into the width with Hair selected and the page gives 31.21 mm. Moving the wall further away widens the gap and shrinks the error: at 4 m the same hair gives 62.4 mm.

$a = 83 \pm 3 μ m; L = 4 m : gap = \frac{2 \times 650 \times 1 0 ^{- 9} \times 4}{83 . 33 \times 1 0 ^{- 6}} = 62.4 mm$

Learning path

Wave optics

References (3)

Babinet's statement of the principle, in his memoir on coronae, the coloured rings round the Sun and the Moon: a small opaque body placed near the line from a luminous point to the eye, but off it, has an effect "exactement le même que celui d'une ouverture toute pareille", exactly that of an identical opening, so that it gives as much light as it seems to take away; with the explanation from interference and the colours of spider threads and thin metal wires among its applications: M. Babinet (as the Comptes rendus print him), Mémoires d'optique météorologique, II: Sur les couronnes. Comptes rendus hebdomadaires des séances de l'Académie des sciences 4, 638–647 (1837); the principle on pp. 643–644.
The single-slit intensity (sin β/β)² with β = πa sin θ/λ, the minima at a sin θ = mλ, and the textbook's estimate of the first side maximum at β = 3π/2, I₁ ≈ 0.045 I₀, which the page's second insight block measures against the true peak: S. J. Ling, J. Sanny and W. Moebs, University Physics Volume 3. OpenStax, 2016. Section 4.2, Intensity in Single-Slit Diffraction.
The Fresnel integrals C(z) = ∫₀ᶻ cos(½πt²) dt and S(z) = ∫₀ᶻ sin(½πt²) dt, and their limits of ½, in the convention the page's computation uses: NIST Digital Library of Mathematical Functions, §7.2(iii) Fresnel Integrals. National Institute of Standards and Technology, release of 2026-09-15.

By the 42Math team · Reviewed against published sources. Corrections welcome.