Critical angle

The critical angle θ_c is the angle of incidence, measured from the normal, above which total internal reflection takes place when a wave travels from a denser medium (n₁) into a less dense medium (n₂ < n₁). It is defined by Snell's law with the transmitted angle set to 90°: n₁ sin θ_c = n₂ · 1, giving θ_c = arcsin(n₂/n₁). At exactly θ_c the refracted ray grazes the interface; beyond θ_c no refracted ray exists.

Typical values illustrate the geometric consequences. Water to air: n₁ = 1.33, θ_c ≈ 48.6° — fish see the entire above-water world compressed into a cone of half-angle 48.6° directly overhead (Snell's window). Ordinary optical glass to air: n ≈ 1.5, θ_c ≈ 42° — the angle at which light inside a glass block starts to fail to escape. Diamond to air: n = 2.42, θ_c ≈ 24.4° — remarkably shallow. This is the geometrical reason diamonds "sparkle": once light enters, almost any internal angle at a face results in TIR, and the light bounces multiple times through the faceted interior before exiting through a face at near-normal incidence, producing high internal reflectivity and strong coloured dispersion along the way. Lapidaries cut brilliants with facet angles calculated precisely so that most entering rays undergo TIR twice before exiting from the crown.