Why does light spread out after squeezing through a narrow slit?

How to use this simulation

1. Start with the defaults: slit width a = 5.0 µm, wavelength λ = 600 nm, screen at D = 1.0 m. The left panel shows the wave arriving from the left as straight crest lines and spreading out after the slit. The strip on the right is the screen, and the blue curve beside it is the intensity at each height on that screen.
2. Drag the slit-width slider to the left. The light behind the slit fans out more and the central band on the screen gets wider. Drag it right and the beam stays narrow. The readout for the central maximum follows 2λD/a.
3. Drag the wavelength slider. The colour of the wave changes with it. Longer wavelengths (red) spread more than shorter ones (blue), so the central band is wider for red light.
4. Drag the screen-distance slider. The screen moves farther from the slit in the drawing, and the centimetre readouts grow in proportion to D, while the angles stay the same.
5. Keep "Mark the dark fringes" ticked. Red dotted lines run from each dark fringe on the screen to the curve, where the intensity touches zero. The labels m = 1, 2, 3 count them, and the dashed rays from the slit mark the first minima at the angle θ₁.
6. Untick "Show the wave" to see only the graph and screen. Press play and pause to freeze the crest lines and study the shape of the wave behind the slit.

The shadow that is not sharp

Stand outside a classroom with the door half open and you can hear the teacher, even though you cannot see her. Sound has bent around the corner. Light does the same thing, but much less, which is why a shadow looks sharp to the eye. Look closely enough, with the right slit, and the edge of a light patch is not sharp either.

Diffraction is the spreading of a wave when it passes through a gap or around an edge. It is not a special property of light. Water waves do it in a harbour mouth, sound does it at a doorway, and radio waves do it over a hill. The only question is how big the gap is compared with the wavelength.

This page makes that question visible. You control the gap, a, and the wavelength, λ, and the simulation shows what the wave does next.

How big must the slit be?

Diffraction is strong when the slit width is about the size of the wavelength, and weak when it is much larger. Sound has wavelengths of about a metre, so a doorway is the right size and it bends around the corner. Light has wavelengths of about half a micrometre, so an ordinary door is millions of times too wide and the shadow looks sharp.

The ratio a/λ tells you everything. In the simulation, the readings panel shows it. At the defaults, it is 8.3. Drag a to its smallest value and the ratio falls towards 1.4, and the wave behind the slit becomes nearly circular. Drag it to the largest and the ratio climbs towards 25, and the beam behind the slit hardly spreads.

Every point of the slit is a source

Christiaan Huygens had a neat idea: every point on a wavefront acts like a tiny source of circular wavelets, and the wave a moment later is the sum of all of them. For a plane wave in free space, the wavelets add up to another plane wave, and the wave goes straight on. When a barrier blocks most of the wavefront, only the wavelets from the slit survive, and nothing is left to cancel the sideways ones.

The simulation does exactly this. It places a row of point sources across the slit, adds up the wavelets they send to every point of the picture, and colours the result. Nothing else is programmed in, so the spreading you see is the sum itself.

Notice that the wave picture is drawn with the slit at its true size relative to the wavelength: a/λ = 8.3 at the defaults. So the angle at which the wave fans out in the picture is the real angle. Only the distance to the screen is shortened to fit on the page.

Why the dark fringes sit where they do

Take a direction θ away from straight ahead. Divide the slit into two halves, upper and lower. For every point in the upper half there is a partner in the lower half exactly a/2 below it. Light from the two partners travels paths that differ by (a/2) sin θ.

If that path difference is half a wavelength, λ/2, the two wavelets arrive in opposite phase and cancel. Every upper point has a lower partner doing the same, so the whole slit cancels. That gives the first dark fringe: (a/2) sin θ = λ/2, or a sin θ = λ.

Divide the slit into four quarters instead and the same argument gives a sin θ = 2λ. Six pieces give 3λ. The general rule is the condition for dark fringes. Notice that it looks like the double-slit bright-fringe rule but is the opposite: here the equation marks darkness.

The intensity formula

A full calculation adds up all the wavelets at every angle and gives a surprisingly compact result. The intensity at an angle θ is I = I₀ (sin β / β)², where I₀ is the intensity straight ahead and β = π a sin θ / λ.

The curve on the right of the stage plots this against the position on the screen. At the centre β = 0, and (sin β / β) tends to 1, so I = I₀. Whenever β is a multiple of π, sin β is zero, and the intensity is zero. That is the same set of minima: β = mπ is exactly a sin θ = mλ.

Between two minima there is a small hump, a side maximum. The first side maxima are near β = 1.43π and 2.46π, with intensities of 4.7 % and 1.6 % of the centre. They are about 21 times fainter than the central band, which is why the middle of the pattern seems to take almost all the light.

The width of the central maximum

The central band is the bright region between the two first minima. For a screen a distance D from the slit, the first minimum is at y₁ = D tan θ₁. For small angles tan θ ≈ sin θ = λ/a, so y₁ ≈ λD/a and the full width is twice that.

With the defaults, the width is 2 × 600 nm × 1.0 m ÷ 5.0 µm = 24.0 cm. The readings panel prints the same number. Notice that the width is directly proportional to λ and D and inversely proportional to a. It is the one place in this chapter where the formula is easy to reason about from the picture.

The first table changes only the slit width. Halve the slit and the band doubles. The last column is the exact width from D tan θ₁; the two agree for wide slits and drift apart as the slit gets narrow, because the small-angle shortcut stops working.

Colour, distance and the position of the minima

The wavelength enters the width in the same way as the distance. Red light, with a longer wavelength, makes a wider central band than blue. Shine white light through the slit and the minima fall at different places for each colour, so the edges of the pattern turn coloured and the centre stays white.

The screen distance multiplies the whole pattern. Move the screen twice as far away and every dimension doubles, but the angles stay the same, which is why the two tables below look like simple scalings of the defaults.

Single slit and double slit side by side

Students mix these two up because both give bright and dark bands. They are different effects with a different spacing, and in a real Young's experiment they appear together.

Diffraction by each slit decides how bright the light is at each angle. Interference between the two slits slices that pattern into closely spaced fringes. The single-slit curve is the envelope, and the double-slit fringes live inside it.

A concrete case: slits 0.04 mm wide and 0.20 mm apart, lit by 600 nm light on a screen 1 m away. The fringes inside the pattern are λD/d = 3.0 mm apart. The envelope's central band is 2λD/a = 3.0 cm wide, so about 10 bright fringes sit under it. Each fringe is a double-slit effect; the fading of their brightness towards the edge is single-slit diffraction.

Solved problems

Eight problems in the exam style, solved with the same formulas as the simulation.

Mistakes that cost marks

Check this list before an exam. Each one has taken marks from someone.

• Using a sin θ = mλ for bright fringes. In a single slit it gives the dark ones.
• Counting m = 0 as a dark fringe. It is the central maximum.
• Saying a narrower slit gives a narrower pattern. It is the opposite: narrower slit, wider pattern.
• Forgetting the factor of 2 in the central width. The band runs from the first minimum on one side to the first on the other.
• Mixing up the slit width a with the slit separation d, which belongs to the double slit.
• Using nanometres for λ and micrometres for a without converting to the same unit.
• Treating all side bands as equally bright. The side maxima drop quickly: 4.7 %, then 1.6 %, then 0.8 %.

Where you meet it in real life

Hold a strand of hair in front of a laser pointer on a wall and you get a row of dots with a bright middle. A hair is an obstacle rather than a slit, but the physics is the same, and measuring the dot spacing tells you its thickness to a fraction of a micrometre.

Diffraction sets a limit on every optical instrument. A camera lens, a telescope and your own pupil all work as round openings. The smaller the opening, the more the light spreads, and that blur limits how fine a detail can be. It is why bigger telescopes see sharper pictures, and why a tiny pinhole camera is never razor-sharp.

Radio waves are metres long, so they bend around hills and buildings where light cannot. FM radio and mobile signals reach valleys for the same reason. The same idea, with X-rays and crystals, is how the structure of DNA was found.

Quick revision

The whole chapter on one page.

• Diffraction is the spreading of a wave through a gap or around an edge; it is strongest when a is close to λ.
• Single-slit dark fringes: a sin θ = mλ, m = ±1, ±2, …
• Intensity: I = I₀ (sin β / β)², with β = π a sin θ / λ.
• Central maximum width: 2λD/a. Narrower slit, longer wavelength or farther screen means a wider band.
• Side maxima are far fainter than the centre: about 4.7 % and 1.6 %.
• In a double slit, the single-slit curve is the envelope of the fringes.

Frequently asked questions