Refraction and total internal reflection
Refraction at spherical surfaces and optical fibres.
Refraction
When light passes from one medium to another, it bends. Snell's law: n₁ sin i = n₂ sin r. The refractive index of medium 2 with respect to medium 1 is n₂₁ = n₂/n₁ = v₁/v₂, so light travels more slowly in a medium of larger refractive index; in glass of n = 1.5, v = (3 × 10⁸)/1.5 = 2 × 10⁸ m/s. Light bends towards the normal when entering a denser medium. A familiar effect is apparent depth: an object under water viewed from directly above appears raised; apparent depth = real depth / n. A tank 12 cm deep filled with water (n = 4/3) looks only 9 cm deep.
Total internal reflection
When light goes from a denser to a rarer medium, the refracted ray bends away from the normal. At the critical angle C, the refracted ray grazes the surface (r = 90°): sin C = 1/n (for a denser medium of index n in air). If the angle of incidence exceeds C, no light is refracted and all is reflected back: total internal reflection (TIR). Conditions: (1) light travels from denser to rarer medium; (2) angle of incidence exceeds the critical angle.
Examples: for n = √2, C = 45°. Diamond (n ≈ 2.42) has a small critical angle (about 24°), so light is trapped and reflected many times, making it sparkle. Mirages and totally reflecting prisms also use TIR.
Optical fibres
An optical fibre has a glass core surrounded by cladding of lower refractive index. Light entering at a suitable angle strikes the core–cladding boundary at more than the critical angle and is totally reflected again and again, travelling long distances with little loss. Fibres are used in communication and in endoscopes.
Refraction at a spherical surface
For a surface of radius R separating media n₁ (object side) and n₂: n₂/v – n₁/u = (n₂ – n₁)/R with the usual sign convention. This result leads to the lens maker's formula.
Key points
- n₁ sin i = n₂ sin r; v = c/n.
- sin C = 1/n; TIR needs denser → rarer and i > C.
- Optical fibres: core index higher than cladding.
- Apparent depth = real depth / n.
Exam tips
- In spherical-surface problems, take R positive when the centre of curvature is on the side of the outgoing light.