Refraction through a prism: deviation and minimum deviation

How a prism deviates light

A prism is a transparent block with two plane refracting faces that meet at an angle A, the angle of the prism. A ray entering the first face bends towards the normal. At the second face it bends away from the normal. Both bends turn the ray towards the base. The angle between the incident ray and the emergent ray is the angle of deviation, δ. As the angle of incidence i increases, δ first decreases, reaches a minimum δm, and then increases. Any other value of δ occurs for two values of i, which swap roles as i and e (the reversibility of light). At minimum deviation there is only one such value, so i = e and the ray passes symmetrically through the prism.

Derivation: the prism formula

A ray PQ falls on face AB at angle i. It refracts at angle r₁, meets face AC at angle r₂ from inside, and leaves along RS at angle e. The normals at Q and R meet at O.

1. 1
   In the quadrilateral AQOR, the angles at Q and R are 90°. So ∠A + ∠QOR = 180°.
2. 2
   In triangle QOR: r₁ + r₂ + ∠QOR = 180°.
3. 3
   Comparing the two: r₁ + r₂ = A.
4. 4
   The deviation at the first face is i − r₁. The deviation at the second face is e − r₂.
5. 5
   Total deviation δ = (i − r₁) + (e − r₂) = i + e − A. So i + e = A + δ.
6. 6
   At minimum deviation, δ = δm, i = e and r₁ = r₂ = r. So 2r = A, giving r = A/2.
7. 7
   Also 2i = A + δm, giving i = (A + δm)/2.
8. 8★
   Snell's law at the first face: μ = sin i / sin r, so μ = sin[(A + δm)/2] / sin(A/2).
9. 9
   So, measuring A and δm gives the refractive index of the prism material.

★ key step

How to draw the prism ray diagram and the i-δ graph

• Draw a triangle with the apex A at the top and the base at the bottom. Mark the angle A at the apex.
• Draw the incident ray PQ striking the left face at Q. Draw the normal at Q as a dotted line and mark i.
• Draw the refracted ray QR inside the prism, bent towards the normal and towards the base. Mark r₁.
• Draw the normal at R on the right face. Mark r₂ inside, then draw the emergent ray RS bent away from the normal. Mark e.
• Extend PQ forwards and SR backwards as dotted lines until they meet. Mark the angle between them as δ.
• Mark O where the two normals meet inside the prism.
• Graph: put i on the x-axis and δ on the y-axis. Draw a U-shaped curve. Mark its lowest point as δm, with the matching angle of incidence below it.

Worked example: the prism formula

Activity: prism, angle of incidence against deviation graph and refractive index

Find δm from an i-δ graph and use it to find the refractive index of the glass. You need a drawing board, white paper, a prism, pins and a protractor.

1. 1
   Fix white paper on the drawing board. Draw a long line AB for one face of the prism, and a normal at a point N on it.
2. 2
   Draw an incident line at i = 30° to the normal. Do not use i below 30°, as the ray may be totally reflected inside the prism.
3. 3
   Place the prism with one face on AB. Fix pins P₁ and P₂ on the incident line. Looking through the other face, fix pins P₃ and P₄ in line with the images of P₁ and P₂.
4. 4
   Mark the other face along a ruler and remove the prism. Join P₃ and P₄ for the emergent ray. Extend both rays until they meet, and measure δ and the angle A.
5. 5
   Repeat for i = 35°, 40°, 45°, 50°, 55° and 60°.
6. 6
   Observation: plot δ against i. The curve is U-shaped, and its lowest point gives δm.
7. 7
   Result: μ = sin[(A + δm)/2] / sin(A/2). For a 60° glass prism, δm is about 37° to 39°, so μ is about 1.5.