How can a clear piece of glass pull sunlight apart into seven colours?

How to use this simulation

1. Start in the Light in a prism scene. A white ray hits the left face at i = 50° of a prism with A = 60°. Six coloured rays leave the right face. The screen on the right shows where red and violet land.
2. Tick Exaggerate the colour spread at the bottom of the controls to see the fan clearly. The real spread is small, only 1.67° here, so the picture multiplies it. The readings on the right are always the true values.
3. Drag the angle of incidence. The whole spectrum swings up and down, and at one angle it stops and turns round. That turning point is the minimum deviation.
4. Drag the prism angle A. A bigger prism bends light more and spreads the colours further. Make A large and i small and one colour disappears: it has been totally reflected inside the glass.
5. Switch to Deviation against incidence. Every colour makes a U-shaped curve. The green dots mark the minimum, with its value and the angle of incidence at which it occurs.
6. Open Rainbow. Slide the angle on the drop and watch where red and violet leave. The graph shows why the rays pile up near 42°.

Newton’s trick with a piece of glass

Before 1666 most people believed a prism added colour to light. Newton made a small hole in his window shutter, let a thin beam of sunlight fall on a prism, and caught a band of colours on the wall. Then he did the clever part: he let one colour through a second prism and found that it did not change any further.

His conclusion was that sunlight is already a mixture of colours, and the prism only separates them. The separation is called dispersion, and the band of colours is the spectrum, from red at one end to violet at the other.

Dispersion is behind rainbows, the sparkle of a diamond and the coloured fringe around a bright object in a cheap lens. In exams it comes with a handful of formulas and one famous graph, all of which you can reproduce with the simulation.

Why the colours separate

When light enters glass it slows down, and that slowing is why its path bends. The amount of bending is set by the refractive index n = c/v. The key fact is that the index of glass is not the same for all colours. Violet is slowed a little more than red, so violet bends more.

For typical crown glass the index for red is 1.513 and for violet 1.532, a difference of only 0.019. That tiny gap, repeated at both faces of the prism, is enough to fan the colours out. Yellow, in the middle, is about 1.519.

These are typical values. Real glass types differ, and the numbers also depend a little on the exact wavelength you pick for red and violet. What matters is the order: n(red) < n(orange) < n(yellow) < n(green) < n(blue) < n(violet).

Following one ray through a prism

A prism has two faces that meet at the apex angle A. A ray hits the first face at angle of incidence i, bends toward the normal and travels inside at angle r₁. It then meets the second face at angle r₂ and leaves at angle e, bending away from the normal.

Geometry of the two normals gives r₁ + r₂ = A. Adding the bend at each face, the total turning of the ray, the angle of deviation, is δ = i + e − A. Snell’s law at each face connects the angles: sin i = n sin r₁ and sin e = n sin r₂.

Solved problem 1: trace red and violet

Crown glass has n = 1.513 for red and 1.532 for violet. White light hits a 60° prism at 50°. Find the deviation of each colour and the angle between them.

The deviation graph and the minimum

Keep the prism fixed and change only the angle of incidence. The deviation does not change steadily. It falls to a smallest value, the minimum deviation δₘ, and then rises again. The graph is a shallow U, and in the Deviation against incidence scene you can see one for every colour.

At the very bottom of the U the ray passes through the prism symmetrically: i = e and r₁ = r₂ = A/2. Nothing else gives that, and it is why a spectrometer is always set at minimum deviation: the beam is sharpest and any small error in i makes almost no change in δ.

The table fixes A = 60° and lists red and violet at six angles. Notice that the two columns reach their minimum at slightly different angles, and that violet is always deviated more. Between 28.9° and 30.3° of incidence red still leaves, but violet is already trapped inside, which is why the violet cell is empty in the first row.

Minimum deviation and the prism angle

Setting r₁ = A/2 and i = (A + δₘ)/2 in Snell’s law gives the formula at the bottom of the box above. It is the standard way to measure the refractive index of a glass: build a prism, find the minimum on a spectrometer, put A and δₘ in.

A bigger apex angle gives a bigger deviation, and also a bigger spread between colours. The table uses yellow light, n = 1.519, and shows both the minimum and the angle of incidence at which it occurs.

Solved problem 2: find the refractive index

A prism of angle 60° shows a minimum deviation of 39° for yellow light. Find the refractive index of the glass.

Thin prisms, dispersive power and the mean deviation

When the prism angle is small and the ray is nearly normal, every angle is small enough that sin x ≈ x. The deviation becomes δ = (n − 1) A. That is the thin-prism formula, used for spectacle prisms and in the optics of lenses.

Take A = 5°. Red deviates by 2.565°, violet by 2.660°, and the angular dispersion is only 0.095°. The mean deviation is 2.595°.

The dispersive power ω is the ratio of the angular dispersion to the mean deviation: ω = (n_v − n_r)/(n_y − 1). It describes the glass alone, whatever the prism angle, and for our crown glass it is 0.037. A glass with a bigger ω, such as flint, spreads colours more for the same bending.

When a colour disappears: total internal reflection

At the second face the ray goes from glass to air, so it can only leave if the angle r₂ is less than the critical angle, sin c = 1/n. For red in our crown glass c = 41.4°, and for violet c = 40.7°.

Violet has the smaller critical angle, so as you lower the angle of incidence violet gets trapped first. In the simulation set A = 60° and lower i below 30.3°: the violet ray stops at the right face and reflects back inside, while red is still on its way out until 28.9°.

The graph shades this zone red. In exam terms: a ray fails to emerge if the prism angle is too large, or the incidence too small, for the second face. For a given glass there is a largest prism angle for which a ray can ever pass through.

The rainbow is a prism with a drop of water

Sunlight enters a spherical raindrop and refracts, with the violet bending more than red, just as in the prism. It then reflects off the back of the drop and leaves through the front, refracting a second time. The path is refraction, one internal reflection, refraction.

The deviation of the ray is D = 180° + 2i − 4r, where r is the angle of refraction inside the drop. A ray that comes in near the edge (large i) and one near the centre (small i) are deflected by very different amounts. But there is an incidence where D reaches its minimum, and near it many different rays leave the drop in almost the same direction. They pile up there, and that is the bright bow.

For water, n = 1.331 for red and 1.343 for violet. The pile-up happens at i = 59.5° for red and 58.8° for violet, and the angle between the returning ray and the line from the Sun through your eye, 4r − 2i, reaches 42.4° for red and 40.6° for violet. Those are the roughly 42° and 40° quoted in textbooks.

This explains the order of colours. Red is at 42.4°, so it sits at the outside of the bow and violet, at 40.6°, on the inside. The bow is 1.7° wide. In the Rainbow scene, drag the slider until the dots sit at the top of each curve and note that moving i a little changes the angle very little.

Solved problem 3: the rainbow angle

A ray enters a raindrop at i = 59°. Using n = 1.331 for red and 1.343 for violet, find the viewing angle of each colour, and compare it with the maximum.

Mistakes that lose marks

Look for these before you finish an answer.

• Saying that the prism creates the colours. It only separates what was already in white light.
• Getting the order wrong. Violet bends most and is deviated most, so it comes out nearest the base.
• Using the thin-prism formula δ = (n − 1) A for a large prism.
• Forgetting that the minimum-deviation formula needs the half-angles (A + δₘ)/2 and A/2.
• Using glass indices for the raindrop. A raindrop is water, with n near 1.33.
• Placing the Sun in front of you when drawing the rainbow. It is always behind the observer.

Where dispersion matters

In a lens, dispersion causes chromatic aberration: red and blue focus at different points, so a bright object gets a coloured fringe. A camera lens fixes this by gluing together glasses with different dispersive powers.

Spectrometers use a prism or a grating to split light from a star or a flame and read its elements from the lines. The same idea is used to study the atmospheres of distant planets.

To see the underlying refraction without the colours, try the refraction of light animation first. For what lenses do with the rays, open the lens ray diagram animation next.

Frequently asked questions