Refraction

How the model works

Light slows down in a material by a factor equal to its refractive index n = c/v. At a boundary the ray bends according to Snell's law:

n₁ sin θ₁ = n₂ sin θ₂

Angles are measured from the normal, the line perpendicular to the surface. Going into a denser medium (n₂ > n₁), the ray bends towards the normal. Going into a less dense one, it bends away.

When n₁ > n₂ there is a critical angle θc = arcsin(n₂/n₁). Beyond it, no refracted ray exists and all the light reflects. This is total internal reflection, which keeps light inside optical fibres.

The fraction reflected comes from the Fresnel equations, averaged over both polarisations (unpolarised light). Ray brightness in the picture follows these fractions.

Ideal flat boundary, single wavelength (no dispersion), no absorption. Indices are typical values for visible light.

Guided experiments

1. Air into water. 45° from air into water. Prediction: sin θ₂ = sin 45°/1.33, so θ₂ ≈ 32.1°, bent towards the normal. Only about 3% reflects. Air → water at 45°
2. Critical angle in glass. From glass (1.5) into air, raise the angle past 41.8°. Prediction: at 41.8° the refracted ray skims along the surface, and beyond it the reflection is total. Glass → air at 45°
3. Diamond sparkle. Diamond (2.42) into air. Prediction: the critical angle is only about 24.4°, so light entering a cut diamond is trapped and bounces around. That is the sparkle. Diamond → air

Challenge

An optical fibre has a core of n = 1.48 and cladding of n = 1.46. What is the critical angle at the core–cladding boundary? Why are the two indices chosen to be so close?

References

• OpenStax, University Physics Volume 3, §1.3 “Refraction” and §1.4 “Total Internal Reflection”.
• Hecht, Optics, ch. 4 (Fresnel equations).