Refractive Index and Total Internal Reflection

Refractive index (n) — the ratio of the speed of light in a vacuum to its speed in the medium:

n = c / v

Since light travels fastest in a vacuum, n is always greater than or equal to 1. Air is about 1.0003, usually taken as 1; water is about 1.33; glass about 1.5.

Total internal reflection (TIR) — all the light is reflected back into the denser medium, with none refracted out. Two conditions must both hold:

the light is travelling from a medium of higher refractive index towards one of lower refractive index, and

the angle of incidence is greater than the critical angle.

Critical angle (C) — the angle of incidence at which the refracted ray travels exactly along the boundary, at 90° to the normal. Substituting θ₂ = 90° into Snell's law gives, for a boundary with air:

sin C = 1 / n

and in general sin C = n₂ / n₁ for light going from medium 1 into medium 2.

Example: for glass of refractive index 1.5, sin C = 1 ÷ 1.5 = 0.667, so C = 41.8°. A ray striking the inside of the glass surface at more than 41.8° to the normal cannot escape at all.

A higher refractive index means a smaller critical angle — so light is trapped more easily in a denser medium. Diamond has n ≈ 2.4 and a critical angle of about 24°, which is why light entering a cut diamond bounces around inside many times before emerging, producing its characteristic sparkle.

Optical fibres — a core of high refractive index surrounded by cladding of slightly lower refractive index. Light entering within a narrow cone strikes the core–cladding boundary above the critical angle and is totally internally reflected along the fibre, however much it bends.

Why cladding is used rather than air — it protects the core surface from scratches and dirt, which would otherwise let light escape, and it prevents light crossing into a touching fibre. It also raises the critical angle slightly, narrowing the range of angles accepted and so reducing multipath dispersion, where rays taking different paths arrive at different times and blur the signal.