Critical Angle for Total Internal Reflection

Take Snell's law, $n_1 \sin\theta_1 = n_2 \sin\theta_2$, with the light starting in the denser medium, and increase the angle of incidence. The refracted ray bends further and further from the normal until it is skimming along the surface at $\theta_2 = 90^\circ$, where $\sin\theta_2 = 1$ and the relation reduces to $\sin\theta_c = n_2/n_1$. Push past that and there is simply no solution — the sine of the refracted angle would have to exceed 1 — so no light refracts at all and every bit of it reflects back inside. The word "total" is precise here in a way it never is for a mirror: a silvered surface loses a few per cent on every bounce, and total internal reflection loses nothing.

Glass to air, with $n_1 = 1.5$ and $n_2 = 1.0$, gives $\theta_c = \arcsin(0.667) = 41.8^\circ$. That it falls below 45° is the reason a plain 45–45–90 glass prism reflects perfectly with no coating, and why binocular prisms, SLR pentaprisms and corner-cube retroreflectors are made the way they are. Water to air gives $\arcsin(1/1.333) = 48.6^\circ$. Diamond gives $\arcsin(1/2.417) = 24.4^\circ$, a remarkably small angle, so light entering a brilliant cut bounces repeatedly among the facets before it can escape — which, together with diamond's strong dispersion, is what a gemmologist means by fire.

Optical fibre is this equation as an industry. A step-index single-mode fibre has a core at $n = 1.4682$ and a cladding at $n = 1.4629$, a difference of about a third of one per cent, giving $\theta_c = \arcsin(1.4629/1.4682) = 85.1^\circ$. Measured from the normal, that means any ray travelling within 4.9° of the fibre axis is trapped, and it stays trapped for tens of kilometres between amplifiers. Note that the reflection happens at the core–cladding boundary, not at the glass–air surface: the cladding is what lets the fibre keep working when it is bundled, buried, bent and handled.

Three errors, in descending order of how often I see them. Order matters: $n_1$ must be the denser medium, the one the light is already in. If $n_2 > n_1$ there is no critical angle at all — light going from air into glass always refracts and never totally reflects — and asking for one is asking for the arcsine of a number greater than 1. The page refuses, and it is right to. Second, all angles here are measured from the normal, not from the surface, and a ray described as being "at 5° to the fibre axis" is at 85° to the normal. A good half of all critical-angle mistakes are this one substitution. Third, "no light escapes" is true of the propagating wave but not of the field: an evanescent wave extends about a wavelength beyond the surface, decaying exponentially, and it carries no energy away — unless you bring a second piece of glass within that distance, in which case light tunnels across the gap. Frustrated total internal reflection is a real effect with real products behind it, including optical fingerprint scanners and some beam splitters.