Diffraction by a Single Slit
Squeeze light through a slit and it spreads: the narrower the slit, the wider the spread.
Lesson at a glanceShow the key points
- Diffraction: wavelets from the unobstructed part of a wavefront superpose, so light bends round edges and forms fringes.
- Fraunhofer diffraction: source and screen effectively at infinity (lenses), plane waves. Fresnel: source or screen at a finite distance.
- Single slit of width b: pair each point with one b/2 below it; they cancel when b sin θ = λ. Minima at b sin θ = nλ (17.13).
- Intensity I = I₀ (sin β/β)², β = πb sin θ/λ (17.14–17.16). The first side maximum is only about 1/22 of the central one.
- Most light lies within sin θ = ±λ/b; a narrower slit spreads the light more; b ≫ λ gives almost no diffraction.
Squint at a street light through your eyelashes and it sprouts streaks and fringes. Light squeezed through narrow gaps doesn’t keep going straight: it spreads.
Diffraction
When part of a wavefront is blocked by an obstacle, only the uncovered part sends out Huygens wavelets. Their superposition gives a new wavefront with a different shape, so the light bends round edges into regions that would otherwise be in shadow. This is diffraction. With light it produces patterns of bright and dark fringes.
A diffraction set-up has three parts: a narrow source, the diffracting element (an obstacle or an opening), and a screen. Wavelets from the uncovered part of the wavefront interfere on the screen.
Source and screen effectively at infinity: plane waves arrive, and the waves meeting at each point on the screen left the aperture parallel. In the lab: lenses before and after the aperture, source and screen at their focal planes. Simpler to analyse.
Source or screen at a finite distance from the obstacle; the wavefronts are curved.
A single slit
A parallel beam falls normally on a long slit of width b. Every point of the slit sends out wavelets. A lens just after the slit collects the waves leaving at angle θ and brings them together at one point P on a screen at its focal plane.
The centre is bright
At θ = 0 every wavelet travels the same optical path to P₀, so all arrive in phase: the brightest point.
Where the first dark fringe is
At angle θ, the wave from the top edge and the wave from the middle of the slit differ in path by . When that equals λ/2 they cancel. And then every point in the top half cancels with the point b/2 below it in the bottom half. The whole slit cancels in pairs: darkness, at .
Dividing the slit into 4, 6, … strips in the same way gives the other dark fringes:
The bright fringes lie roughly midway between them. A full calculation gives the intensity:
Dashed: the same curve ×10, to show that the side bands are faint (the first is about 1/22 of the centre).
A wide central band
Most of the light falls between the first dark fringes, sin θ = ±λ/b. The first side band is only about 1/22 as bright as the centre. The spread is set by λ/b: a narrower slit spreads the light more widely; a slit much wider than λ lets the light through almost undiffracted. Once again, diffraction shows up clearly only when the opening is comparable to the wavelength.
A parallel beam of 500 nm light passes through a slit 0.25 mm wide. Find the angular spread containing most of the light.
- 1Diffraction: wavelets from the uncovered part of a wavefront superpose, bending light round edges.
- 2Fraunhofer: source and screen effectively at infinity; Fresnel: at finite distances.
- 3Single slit: bright centre; minima at b sin θ = nλ; I = I₀(sin β/β)², β = πb sin θ/λ.
- 4Most light lies within sin θ = ±λ/b; narrower slits spread light more.
Check yourself
Keep exploring
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