Refraction through a prism

A ray through a prism

Narrated video. Transcript:

Here is a triangular glass prism, seen from the end. The angle between its two refracting faces is the angle of the prism, A.

A ray PQ strikes the first face at Q with an angle of incidence i. It bends towards the normal: the angle of refraction is r₁.

Inside the glass the ray travels to the second face, meeting it at R with an angle of incidence r₂. It leaves bending away from the normal, at an angle of emergence e.

Draw the two normals. They meet at N. In the quadrilateral formed by the apex, Q, N and R, the angles at Q and R are right angles, so the angle A plus the angle QNR is 180°.

In the triangle QNR, r₁ plus r₂ plus the angle QNR is also 180°. Comparing the two, r₁ + r₂ = A.

Now extend the incident ray forwards and the emergent ray backwards. The angle between them is the total deviation δ: how much the prism has turned the light.

At Q the ray turns by i − r₁; at R it turns by e − r₂. Adding, δ = (i − r₁) + (e − r₂) = i + e − A.

Notice that the ray always bends towards the base of the prism. Both refractions turn it the same way.

So for a given prism, δ depends on the angle of incidence i. Let us measure how.

Minimum deviation

Narrated video. Transcript:

Turn the prism slowly so that i grows, and plot the deviation δ against i. The graph falls, reaches a lowest point, and then rises again.

Every value of δ, except the lowest, occurs at two angles of incidence. They are i and e swapped: the path of light is reversible.

At the lowest point only one i is possible, so here i = e. This is the angle of minimum deviation, D. Then r₁ = r₂, and since they add up to A, each equals A/2.

Inside the prism the ray now runs parallel to the base, symmetric through the prism.

At minimum deviation, D = 2i − A, so i = (A + D)/2. Snell’s law at the first face gives n₂₁ = sin i over sin r, which is sin((A + D)/2) over sin(A/2).

This is how refractive index is measured in the lab. Example: a prism of angle 60° gives a minimum deviation of 30°. Then n = sin 45° over sin 30° = 0.707 ÷ 0.5 = 1.414, which is √2.

The other way round: glass of n = 1.5 in a 60° prism. sin((60° + D)/2) = 1.5 × sin 30° = 0.75. So (60° + D)/2 = 48.6°, and D is about 37.2°.

For a thin prism, with A only a few degrees, the sines equal the angles. Then D = (n₂₁ − 1)A. A 5° prism of glass with n = 1.5 deviates light by just 2.5°.

Notice that A and n are all we need. A thin prism turns every ray near normal incidence by almost the same small angle, (n₂₁ − 1)A.

Try it: the δ–i graph

Interactive simulation. Things to try:

• Drag the amber dot, or tap the graph. Watch δ fall to a minimum and rise again.
• Press “Go to minimum deviation”. Check that r₁ = r₂ = A/2 and i = e, and the ray inside runs parallel to the base.
• The last readout uses D to compute sin((A + D)/2)/sin(A/2). It always returns the n you set.
• Make i small. At some point the ray is totally reflected at the second face and no light emerges.

Measure n

A prism of angle 60° gives an angle of minimum deviation of 30°. What is the refractive index of its glass?

1. 1.414
2. 1.5
3. 1.155
4. 2.0

A thin prism

A thin prism of angle 5° is made of glass with n = 1.6. By how much does it deviate a ray of light?

1. 3°
2. 8°
3. 5°
4. 0.6°

Summary

• Angles inside: r₁ + r₂ = A
• Deviation: δ = i + e − A
• Minimum deviation: n₂₁ = sin((A + D)/2) / sin(A/2)
• Thin prism: D = (n₂₁ − 1)A

• A prism bends light towards its base, twice.
• δ against i falls to a minimum D and rises again; each other δ occurs for two values of i (i and e swapped).
• At minimum deviation: i = e, r₁ = r₂ = A/2, and the ray inside is parallel to the base.
• Measuring A and D gives the refractive index of the prism.