Total Internal Reflection and Fibre Optics

When light cannot escape

Most of the time, when light meets the boundary between two transparent materials, some passes through and some reflects back. But under the right conditions something dramatic happens: all of the light reflects, and none escapes at all. This is total internal reflection (TIR), and it is one of the most useful effects in physics. It is the reason a diamond sparkles, the reason binoculars can be compact and bright, and — most importantly — the reason the entire internet can travel through hair-thin threads of glass as flashes of light. To understand it, we start with refraction.

This lesson builds directly on the ideas in reflection and refraction of light, so a quick review of Snell's law and the refractive index will make everything here click into place.

A quick recap: refraction and Snell's law

Light slows down when it enters a denser, more optically "thick" material. The refractive index n of a material measures how much it slows light:

n = (speed of light in a vacuum) ÷ (speed of light in the material)

Air is about 1.00, water about 1.33, ordinary glass about 1.50, and diamond a hefty 2.42. When light crosses a boundary, the amount it bends is governed by Snell's law:

n₁ sin θ₁ = n₂ sin θ₂

where the angles θ are measured from the normal (the line perpendicular to the surface). Crucially, when light passes from a denser medium into a less dense one (glass into air, say), it bends away from the normal — and this is the key to everything that follows.

Building up to the critical angle

Picture a ray of light travelling inside glass (n₁ = 1.5), heading toward the boundary with air (n₂ = 1.0). Start with the ray nearly straight on, then gradually increase the angle of incidence and watch the refracted ray that escapes into the air:

• At a small angle of incidence, the ray refracts out into the air, bent away from the normal.
• As you increase the angle of incidence, the refracted ray bends further and further, hugging closer to the surface.
• At one special angle — the critical angle, θc — the refracted ray bends all the way to 90° from the normal, skimming exactly along the boundary.
• Push the angle of incidence even larger than θc, and there is no longer any angle for the refracted ray to take. Refraction becomes impossible, so 100% of the light reflects back into the glass. That is total internal reflection.

The two conditions for TIR

Total internal reflection only happens when both of these are true:

1. Light is travelling from a denser medium (higher n) toward a less dense one (lower n).
2. The angle of incidence is greater than the critical angle.

If light is going the other way — from less dense to denser — it can never undergo TIR, because there is always a valid refracted ray. Both conditions matter.

Calculating the critical angle

We can find the critical angle straight from Snell's law. At the critical angle, the angle of refraction is exactly 90°, so sin θ₂ = sin 90° = 1:

n₁ sin θc = n₂ sin 90° n₁ sin θc = n₂ × 1 sin θc = n₂ / n₁

where n₁ is the denser medium the light starts in.

Worked example 1 — glass to air. Find the critical angle for glass (n₁ = 1.50) against air (n₂ = 1.00).

sin θc = n₂ / n₁ = 1.00 / 1.50 = 0.667 θc = sin⁻¹(0.667) = 41.8° ≈ 42°

So any ray inside this glass that hits the surface at more than about 42° is totally internally reflected.

Worked example 2 — water to air. For water (n₁ = 1.33):

sin θc = 1.00 / 1.33 = 0.752 θc = sin⁻¹(0.752) = 48.8° ≈ 49°

Notice the pattern: a higher refractive index gives a smaller critical angle, making TIR easier. Diamond's index of 2.42 gives a critical angle of just 24°, so light entering a cut diamond bounces around inside many times before escaping — which is why diamonds sparkle so intensely.

How an optical fibre traps light

An optical fibre is a thin strand of very pure glass, often thinner than a human hair. It is built in two layers:

• a central core of glass with a high refractive index, and
• an outer cladding of glass with a slightly lower refractive index.

Because the core is the denser medium and the cladding is less dense, condition 1 for TIR is automatically satisfied at the core-cladding boundary. The fibre is designed so that light fired into the core strikes that boundary at an angle greater than the critical angle every time. The result: the light bounces off the boundary by total internal reflection, again and again, zig-zagging its way along the length of the fibre without ever leaking out. Even when the fibre bends, the light follows the curve, because each bounce still exceeds the critical angle.

Because TIR reflects essentially 100% of the light (unlike a metal mirror, which absorbs a little each time), a signal can travel for tens of kilometres before fading enough to need a boost.

Fibre optics in the real world

The internet and communications. Your messages, videos, and calls travel as ultra-fast pulses of laser light down fibre-optic cables. A flash of light might mean a 1 and darkness a 0, encoding digital data. Fibres can carry vastly more information than copper wires, with far less signal loss, which is why fibre forms the backbone of the internet — including the cables laid across entire ocean floors connecting continents.

Medicine: the endoscope. Doctors use bundles of optical fibres in an endoscope to see inside the body without major surgery. One set of fibres carries light in to illuminate, say, the stomach, and another carries the image back out to a camera — all guided by total internal reflection.

Prisms in optics. Good binoculars, periscopes, and SLR cameras use glass prisms that rely on TIR instead of mirrors to redirect light. Because TIR loses almost no light, the image stays bright and crisp.

Try it yourself! 🧪

Experiment 1 — Bend light down a stream of water. This classic demonstration shows light trapped by TIR, just as in a fibre. Take a clear plastic bottle and poke a small hole near the bottom of one side. In a dark room, cover the hole with tape and fill the bottle with water. Shine a bright torch or, better, a laser pointer through the opposite side of the bottle so the beam lines up with the hole. Now uncover the hole over a sink. As water arcs out in a smooth stream, the light becomes trapped inside the stream of water, bouncing along it by total internal reflection and lighting up where the water lands — the light follows the curving water instead of going straight. You have built a working "liquid optical fibre". (Never aim a laser at anyone's eyes; do this over a sink with the beam pointing safely downward.)

Experiment 2 — Find the critical angle of water. In a darkened room, add a few drops of milk to a clear tank of water so a beam will show up. Shine a laser pointer from inside the water (enter through a flat side of the tank near the bottom) up toward the underside of the water surface. Slowly increase the angle of the beam toward the surface. At shallow angles the beam refracts out into the air above; as you steepen it past about 49° from the normal, the beam suddenly stops escaping and reflects entirely back down into the water — you are watching the critical angle of water in action, the very effect that runs the internet beneath the sea.

From the sparkle of a diamond to a message crossing an ocean in milliseconds, total internal reflection is light that simply cannot get out — and we have learned to put that trapped light to extraordinary use.