Thought Toys · Waves & rhythm · Exhibit 70
A beam of light inside glass, aimed at the surface. Tilt it and the escaping ray bends further and further over, the reflection underneath growing brighter — until one exact angle. Past it, the escaping ray doesn't dim. It stops existing, and the surface becomes a perfect mirror. Every fiber-optic cable on Earth is built on the far side of that angle.
The surface, and how much light crosses it the beam overall s-polarised p-polarised
—
—
A side view of the surface between glass (below) and air (above), with a beam of light inside the glass aimed up at it. At every tilt the beam splits in two: a refracted ray that escapes into the air, and a reflected ray that bounces back down. The brightness of each ray on screen is exactly its share of the light — no dramatisation. Straight on, glass lets 96% through and reflects 4%, which is why a window at night is a faint mirror. The graph on the right tracks the reflected share across every tilt at once. (The two thin curves are the two polarisations — the two directions the light wave's vibration can point. The green one falls to zero around 34°: at that tilt, Brewster's angle, glass simply cannot reflect that component. Polarised sunglasses exploit exactly this to erase glare.)
Now drag the tilt up and watch two things at once. The escaping ray bends further and further away from the normal — always more tilted than the beam inside, because light entering thinner stuff swings outward. And the reflected share climbs: slowly at first, then steeply. At 41.8° the escaping ray lies flat along the surface itself. One half-degree more and there is no direction left for it to go — refraction would need the sine of an angle to exceed 1, and no angle has one. The light doesn't mostly reflect. It all reflects: exactly 100%, to the last photon. No silvering, no coating — a perfect mirror made of pure geometry.
That perfection is the trick behind fiber optics. A glass thread carries light for kilometers by letting it bounce off the walls thousands of times per meter — survivable only because each bounce past the critical angle loses nothing at all. An ordinary household mirror eating 5% per bounce would go black in centimeters. Diamond plays the same trick on your eye: press Diamond, 30° and see how low its wall sits — 24.4°. Almost any ray inside a cut diamond is past it, so light entering the stone rattles around the facets before finding a way out. The sparkle is trapped light, escaping on the jeweller's schedule.
Then press send the light in from outside. Same surface, same physics, opposite direction — and the wall is gone. Going into denser stuff, light bends toward the normal, so no entry tilt — not even 89.5°, skimming the surface — can be refused. The trap is strictly one-way: easy to enter, and past one angle, impossible to leave.
improve/verify/70-total-internal-reflection.js): energy is
conserved (reflected + transmitted = 1) to 10⁻¹² across 36,214 cases and all 50,760 slider-reachable states;
the critical angle is sharp (T > 0 just below, R = 1 exactly at and above); Brewster's angle falls out of
the same formulas unasked (Rp < 10⁻²⁰ at atan(n₂/n₁)); and going into the
denser medium never traps at any tilt. Three negative controls fail as they must: the classic
amplitude-vs-energy mistake (using t² as the transmitted share) breaks conservation by 379%; the wrong-formula
critical angle acos(n₂/n₁) claims 45° glass-to-air light escapes when Snell would need
sin θ₂ = 1.06; and a hardcoded 45° wall misclassifies both water (48.8°) and glass (41.8°).
Past the wall a faint evanescent skin of field does extend into the air a few hundred
nanometers deep — but it carries no energy away, and the total reflection stands.
Also in Waves & rhythm: Fourier epicycles →
← the cabinet · Thought Toys — a cabinet of explorable explanations. Exhibit 70.