TOTAL INTERNAL REFLECTION

Hold a glass of water above your head and look up through the underside. The surface you are looking at is a piece of glass-on-air — and from below, beyond a certain tilt, it is a perfect mirror. No silver coating. No aluminum backing. Just water, and then air, and an angle steep enough that light refuses to cross the boundary. You are watching total internal reflection. Every photon that arrives at the surface bounces back. Losses are essentially zero. Corner-cube retroreflectors on the Moon still beam laser pulses back to Earth half a century after Apollo — reflecting not off metal, but off glass corners geometrically arranged so every ray hits each face above the critical angle. A mirror built from nothing except refractive index.

The rule is a two-line consequence of Snell's law. When a ray travels from a dense medium (index n₁) into a less-dense one (n₂ < n₁), Snell demands n₁ sin θ_i = n₂ sin θ_t. The refracted angle θ_t grows faster than the incidence angle θ_i, because we are dividing by the smaller index. At some specific incidence angle the right-hand side pins against sin θ_t = 1 — the refracted ray would graze the surface. Push θ_i one degree further and the equation demands sin θ_t > 1, which is geometrically impossible. There is no refracted ray to produce. 100% of the light reflects.

The incidence angle where refraction just barely disappears is the critical angle θ_c. Setting sin θ_t = 1 in Snell's law and solving:

Two numbers are worth memorising. Glass-to-air (crown glass, n₁ = 1.5; air, n₂ = 1.0) gives θ_c = arcsin(1/1.5) ≈ 41.81°. Every 45°-45°-90° glass prism is designed around this fact — the two slanted faces are 45° incidence surfaces, which is steeper than 41.81°, so light entering through the hypotenuse retroreflects twice off the back faces and emerges out the opposite side. Water-to-air gives θ_c = arcsin(1/1.33) ≈ 48.75° — wider, because water is less of a refractive trap than glass. A fish looking up sees the whole hemisphere above squeezed into a disc with half-angle 48.75°; outside that disc the fish only sees the bottom of the pond reflected off the underside of the water.

If n₁ ≤ n₂ there is no critical angle at all — the argument of arcsin would exceed unity. Light going air-to-glass refracts at every incidence. TIR is strictly a dense-to-sparse phenomenon.

Drag the slider. Below θ_c you see the amber refracted ray tilt further and further from the normal while a weak Fresnel reflection comes back into the glass. At θ_c itself the refracted ray lies flat along the surface with infinitesimal amplitude. One nudge past and it is gone. The reflection is now total — unity magnitude on the Fresnel coefficient, zero transmission, an angle that has become a mirror.

"Total" is mildly misleading. Past the critical angle, zero energy crosses the interface — but the electromagnetic field does not vanish on the air side. Solving Maxwell's equations at the boundary under TIR conditions produces a wave whose phase travels along the interface and whose amplitude decays exponentially into the forbidden medium:

This is the evanescent wave. The decay length d is roughly one wavelength — for 550 nm light hitting a glass-air interface well past the critical angle, d ≈ 0.1–0.5 µm. Poke a needle into the field within that distance and you will pull light out of it. Touch the TIR face of a prism with your finger and you will see the fingerprint — the ridges of your skin come within a wavelength of the glass, and the evanescent field couples out into your oily lipids. That is how fingerprint scanners work.

The evanescent wave is why TIR optics require surgical cleanliness. A speck of dust within d of a TIR surface looks, to the field, like a finger. Light leaks out. In laser cavities and optical fibre launches this leakage is called evanescent-field loss, and engineers go to absurd lengths (cleanroom assembly, helium atmospheres, optical contacting) to keep the forbidden zone genuinely empty.

Now bring a second slab of glass up to the TIR interface. As its surface slides into the evanescent field — which is to say, as the gap shrinks below a wavelength — something genuinely strange happens. Some of the evanescent field couples into the second slab and becomes a real, propagating ray on the far side. Light has tunnelled across a gap that Snell's law declared forbidden.

This is frustrated total internal reflection, and it is the closest optical analog of a genuinely quantum-mechanical process. The field in the gap is the same exponentially-decaying envelope; bringing the receiver close "shortens the barrier" and the transmission grows as

Toggle frustrated mode and sweep the gap. At gap ≈ 0 the second slab is optically touching the first and the ray walks through as if the interface weren't there. At gap ≈ d about 14% of the intensity makes it. At gap ≈ 3d transmission is essentially zero. Commercial beam-splitter cubes for laser interferometry and variable-ratio TIR attenuators exploit this arithmetic directly — the ratio between reflected and transmitted beams is tuned by regulating a sub-micron gap.

Optical fibres are the unambiguous winners of TIR as a design principle. A glass core (n ≈ 1.48) is clad in a slightly less-dense glass (n ≈ 1.46), giving θ_c ≈ 81° at the core-cladding wall. Any ray zig-zagging down the core with an internal angle shallower than 9° from the axis reflects off every wall collision and propagates forever — the industrial meaning of "no losses" being a few tenths of a decibel per kilometre. The whole internet runs on rays that aren't allowed to leave their glass.

Sweep the input fan angle. The rays inside the acceptance cone (internal angle ≤ 90° − θ_c) bounce their way across without leaking; rays outside it refract straight through the wall and are lost within one pass. The ratio √(n₁² − n₂²) is the fibre's numerical aperture — the sine of the widest outside-air angle the fibre can capture. This is §09.10's main character; here it is a preview.

Back in the pond, the fish's bright disc overhead — the Snell's window — is TIR seen from below. Beyond the disc's edge the water surface turns into a mirror reflecting the pebbles on the bottom. Glass prisms steer binoculars and rifle scopes by the same rule. Diamonds sparkle because their critical angle is only 24°, so facets cut well above that reflect almost every ray that enters, trapping light inside the gem until it leaves through a specific face aimed at your eye. An angle, written down by Snell in 1621, became in turn: an optical communication network, a fingerprint sensor, and the reason your ring glitters.