Total Internal Reflection: The Angle Where Light Stops Escaping

Snell's law and the angle where refraction runs out

When light crosses the boundary between two transparent media, Snell's law relates the angles on either side to the two refractive indices: n1·sin(θ1) = n2·sin(θ2). Going from a denser medium (higher n1, like glass or water) into a less dense one (lower n2, like air), the refracted ray bends away from the normal, so θ2 is always larger than θ1. As θ1 increases, θ2 grows even faster, and at some incidence angle θ2 reaches exactly 90 degrees — the refracted ray skims along the boundary itself. Push θ1 past that point and there is no longer any real solution for θ2 at all: no light can refract through, and all of it reflects back into the denser medium. That threshold is the critical angle.

sin(theta_c) = n2 / n1     // n1 > n2, light going from dense to less-dense medium
theta_c = glass-to-air (n=1.5)   ~= 41.8 deg
theta_c = water-to-air (n=1.33)  ~= 48.6 deg
theta_c = diamond-to-air (n=2.42) ~= 24.4 deg

Total, not partial — and that's the whole point

Below the critical angle, a boundary between two transparent media always does both: some light refracts through, some reflects, in proportions given by the Fresnel equations, which depend on polarisation and angle. What makes total internal reflection (TIR) special is that beyond the critical angle, reflection is not merely dominant — it is exactly 100%, with zero transmission, as long as the second medium is genuinely lossless and non-absorbing. That perfect efficiency, with no partial leakage to account for, is what makes TIR useful as an engineering tool rather than just an optical curiosity.

Fresnel reflectance: what happens before you reach the critical angle

Even at normal incidence, a glass-air interface reflects a small fraction of light — about 4% for ordinary glass, from the Fresnel formula R = ((n1−n2)/(n1+n2))² at zero degrees. As the incidence angle increases toward the critical angle, the reflectance for the two polarisation components diverges and then both curves rise steeply and merge at exactly 100% right at θ_c, where they stay for every angle beyond it. This gradual approach, followed by a sudden clamp to total reflection, is exactly what you can sweep through on the simulation on this page.

Fibre optics: TIR as a data pipe

An optical fibre is a thin glass or plastic core surrounded by cladding of slightly lower refractive index. Light injected at a shallow enough angle strikes the core–cladding boundary beyond the critical angle every time it hits, so it reflects perfectly and bounces down the length of the fibre for kilometres with essentially no loss from the walls — the residual attenuation in real fibre comes from absorption and scattering in the glass itself, not from imperfect reflection. This single trick, total internal reflection guiding light along a bent path instead of a laser needing a straight line of sight, underlies the entire internet backbone, endoscopes, and decorative fibre-optic lamps alike.

Why diamonds sparkle

Diamond has an unusually high refractive index (about 2.42), which by sin(θc) = n2/n1 gives a very small critical angle of roughly 24 degrees. Gem cutters exploit this deliberately: a brilliant-cut diamond's facets are angled so that light entering through the top strikes the back facets at well beyond that small critical angle, reflects internally multiple times, and exits back out the top rather than leaking out the bottom or sides — trapping and redirecting the light into the sparkle and fire the cut is known for. Cut a diamond's facets at the wrong angles and light escapes out the back, leaving it looking dull no matter how well-polished the stone is.

Frequently asked questions

Try it live

Everything above runs in your browser — open Total Internal Reflection and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.