Diffraction and the single slit

Spreading at a gap

Pass a wave through a gap in a barrier and the wavefronts beyond it curve at the edges. The wave bends into the region that straight-line travel would leave empty. This spreading is diffraction, and it happens at every gap and every edge; the only question is how much.

The controlling comparison is gap size against wavelength. A gap much larger than λ$\lambda$λ lets the wave through nearly unchanged. As the gap narrows, the angular spreading grows, becoming conspicuous once gap and wavelength are comparable; by the time the gap is about a wavelength across, the wave is spreading into nearly every forward direction. You can hear a conversation through an open door without seeing the speakers for exactly that reason. Sound wavelengths run to about a metre, comparable to the doorway, while light wavelengths are millions of times smaller and the light simply passes straight through.

Huygens' construction, and why a gap spreads a wave

The paragraph above says what happens without saying why it happens. The mechanism has a name and a picture, and it is Edexcel that examines them, so treat this section as an examinable construction if Edexcel is your specification and as the explanation behind the last figure if it is not.

Christiaan Huygens' idea was to stop treating a wavefront as one object. Take any wavefront and imagine every point on it as a source of its own tiny circular wave, a secondary wavelet, spreading forwards at the wave speed. Let a short time pass, so that every wavelet has grown to the same radius, and then draw the surface that just touches all of them. That surface, the envelope of the wavelets, is where the wavefront has got to. Repeat and you have propagated the wave.

In open water the construction does nothing surprising, which is the point of trying it there first. The wavelet centres lie along a straight line and every wavelet has the same radius, so their common tangent is another straight line, parallel to the first and one radius further on. A plane wave stays plane, and it stays plane because each wavelet's sideways bulge is buried inside its neighbours' circles and never reaches the envelope.

Now put a barrier with a gap in the way. The barrier does one thing to the construction: it deletes every wavelet source it covers, leaving only those inside the gap. Across the middle of the gap nothing has changed, because those wavelets still have neighbours on both sides and their envelope is still flat. At each edge, though, the last surviving wavelet has no neighbour beyond it, and so nothing buries its outer flank. The envelope has no choice but to wrap round that flank, into the region straight-line travel would have left dark. That wrapping is diffraction, and it is happening at the edges, which is why an edge on its own diffracts a wave just as a gap does.

The size of the gap now decides everything, because it decides how many wavelets are left. A gap many wavelengths across leaves a long row of them, most with neighbours, so the new wavefront has a long flat middle and only a slight curl at each side: the wave carries on nearly straight. Narrow the gap towards λ$\lambda$λ and hardly more than one wavelet survives, with nothing either side of it to trim anything, and a single wavelet is simply a circle. The wave leaves such a gap spreading into every forward direction, which is exactly the picture the figure above draws and the reason the spreading keeps growing as the gap narrows, reaching its fullest once the gap is down at about a wavelength.

An obstacle is the same argument with the deletion in the middle instead of at the sides. The obstacle removes the wavelets across its own width, so the wavefront either side of it advances as usual, but the wavelets at the two exposed edges have nothing further in to bury their inner flanks and each curls into the shadow. If the obstacle is small compared with λ$\lambda$λ the two curls meet behind it and the shadow closes over completely, which is why a mooring post leaves no strip of still water downstream of it and why long-wave radio reaches around a hill.

The single-slit pattern

Shine monochromatic light, light of a single wavelength, through a slit not much wider than the wavelength, and the screen beyond does something richer than a simple smear. It shows a bright central maximum, then darkness, then much fainter fringes alternating with dark minima on each side.

Two features identify the pattern in an exam. The central maximum is twice the width of every other maximum, and it is far brighter than any of them. No equation is required here; what you need is the shape and how it responds when something changes.

Make the slit narrower and the pattern spreads wider, though dimmer, because less light gets through. Use a longer wavelength and the pattern also spreads wider, since the gap is now closer to λ$\lambda$λ. Red light therefore makes a wider pattern than blue through the same slit.

White light

White light is every visible wavelength at once, and each wavelength builds its own pattern with its own width. All of them share the middle of the screen, so the centre stays white. Away from the centre the patterns separate: each side fringe becomes a little spectrum, with violet on its inner edge, closest to the centre, and red on its outer edge, because the longer wavelengths spread more. Further out still, the overlapping spectra wash out.