Total Internal Reflection

Dive to the bottom of a swimming pool, roll onto your back and look up. Directly overhead you can see the sky, the diving board, the people at the edge — a bright circular window onto the world above. But tilt your gaze towards the shallow end and that window abruptly ends: past a certain angle the surface stops being a window and becomes a perfect silver mirror, showing you the pool floor and your own legs reflected back down. No paint, no silvering — just water and air. The light trying to escape upward is being thrown completely back into the water.

That flip from window to mirror is total internal reflection (TIR): the effect that pipes the internet across ocean floors, lets a surgeon see inside your body, and makes a diamond blaze. This page is about exactly when a boundary stops letting light through and starts trapping it entirely. It builds on reflection and refraction — so keep the normal and the rule "bends away from the normal when it speeds up" close to hand.

Refractive index: how much a material slows light

Light rips through empty space at c = 3.0\times10^{8}\ \text{m/s}, the cosmic speed limit. Push it into glass or water and it slows down, because the material keeps absorbing and re-emitting it along the way. We measure that slowing with a single number, the refractive index n:

n = \frac{c}{v},

where v is the speed of light in that material. Since nothing travels faster than c, we always have v \le c, so n \ge 1 for every material (and n = 1 exactly for a vacuum, very nearly so for air). A bigger n means light crawls more slowly — the material is optically denser. Some standard values:

vacuum / air — n \approx 1.00
water — n \approx 1.33
ordinary glass — n \approx 1.50
diamond — n \approx 2.42 (one of the highest of any clear material — remember this; it is the secret of the sparkle).

Worked example — find n from a speed. Light travels through a block of glass at v = 2.0\times10^{8}\ \text{m/s}. Its refractive index is

n = \frac{c}{v} = \frac{3.0\times10^{8}}{2.0\times10^{8}} = 1.5.

Turn it round and you can also recover the speed inside a material from its index: v = c/n. Diamond's n = 2.42 means light limps through it at only c/2.42 \approx 1.24\times10^{8}\ \text{m/s}, barely a third of its free-space speed.

Snell's law, and bending away from the normal

When a ray crosses a boundary between two materials its direction is fixed by Snell's law, which ties the two angles (both measured from the normal) to the two indices:

n_1 \sin\theta_1 = n_2 \sin\theta_2.

Read it as a see-saw: the material with the larger index takes the smaller angle. So when light goes from a denser material into a less dense one — glass into air, water into air — it moves into the region where it can go faster, and the ray bends away from the normal. The refracted angle \theta_2 is bigger than the angle of incidence \theta_1. Hold that picture: it is the whole engine of what comes next.

Now do a thought experiment. Sit inside the glass and slowly tilt the incident ray — raise \theta_1 from small to large. Every step, the escaping ray bends a little further from the normal, hugging closer and closer to the boundary surface. There is clearly a limit: the refracted ray cannot bend past 90^\circ, at which point it would skim along the surface. What happens when we reach it?

The critical angle

The angle of incidence at which the refracted ray grazes exactly along the boundary — that is, \theta_2 = 90^\circ — is called the critical angle C. Put \theta_1 = C and \theta_2 = 90^\circ into Snell's law (with the light going from the denser side n_1 to the less dense side n_2):

n_1 \sin C = n_2 \sin 90^\circ = n_2 \quad\Longrightarrow\quad \sin C = \frac{n_2}{n_1}.

For the very common case of a material of index n meeting air (n_2 = 1), this collapses to the formula worth memorising:

\sin C = \frac{1}{n}.

Push the angle of incidence past C and Snell's law would demand \sin\theta_2 = n_1\sin\theta_1 / n_2 > 1 — which is impossible; no real angle has a sine above 1. There is simply no refracted ray to be had. With nowhere to go, all of the light's energy is thrown back into the denser material, obeying the ordinary law of reflection (i = r). That is total internal reflection: not "mostly" reflected as at any everyday surface, but essentially 100% — which is why it makes such a flawless mirror.

Refractive index: n = \dfrac{c}{v} \ge 1, where v is the speed of light in the material.
Snell's law: n_1\sin\theta_1 = n_2\sin\theta_2, angles measured from the normal.
Critical angle: \sin C = \dfrac{n_2}{n_1}, which for a material against air/vacuum becomes \sin C = \dfrac{1}{n}.
Total internal reflection occurs only when both hold: the light is passing into a less dense medium (n_1 > n_2), and the angle of incidence exceeds the critical angle (\theta_1 > C).

Worked examples

1 — the critical angle of glass. Ordinary glass has n = 1.50 against air, so

\sin C = \frac{1}{1.50} = 0.667 \quad\Longrightarrow\quad C = \sin^{-1}(0.667) = 41.8^\circ.

Any ray inside the glass striking the surface at more than 41.8^\circ to the normal is trapped.

2 — water versus diamond. Repeat the sum for water and for diamond:

\text{water: } \sin C = \frac{1}{1.33} = 0.752,\ C = 48.8^\circ; \qquad \text{diamond: } \sin C = \frac{1}{2.42} = 0.413,\ C = 24.4^\circ.

Notice the pattern: the higher the refractive index, the smaller the critical angle. Diamond's tiny 24.4^\circ means light finds it very hard to escape — almost any ray inside is above the critical angle and bounces around by TIR. That is the physics of "fire" in a well-cut gem.

3 — does TIR happen? A ray inside water (n = 1.33) meets the water–air surface at 50^\circ to the normal. Water's critical angle is 48.8^\circ. The light is heading into a less dense medium (water \to air) and 50^\circ > 48.8^\circ, so both conditions are met: the ray is totally internally reflected. Drop the angle to 45^\circ — now 45^\circ < 48.8^\circ, so most of the light refracts out into the air and only a little reflects back. This is exactly the edge of the bright "window" you saw looking up from underwater.

See it flip

Below, a ray travels up through glass (the denser lower half) and strikes the boundary with air. Drag the angle of incidence. While it is below the critical angle (C \approx 42^\circ, marked by the faint dashed reference ray) a refracted ray escapes into the air, bending further and further from the normal as you push the angle up. The moment you cross C, the refracted ray vanishes and every bit of the light turns into the reflected ray, bouncing back down into the glass — total internal reflection. Watch the state label switch.

Where TIR earns its keep

Optical fibres. A hair-thin thread of very pure glass carries a light or laser signal down its core. Every time the light drifts towards the wall it hits at an angle well above the critical angle and is totally internally reflected, so it zig-zags kilometre after kilometre losing almost no energy. This is how broadband and the transoceanic internet travel — as pulses of light bouncing along glass, far faster and clearer than electricity in copper.
Endoscopes. Doctors thread a bundle of these fibres into the body; one set of fibres pipes light in and another carries the image out, all by TIR, so a surgeon can see inside a joint or a gut without major surgery.
Prisms in binoculars and periscopes. A right-angled glass prism reflects light through 90^\circ or 180^\circ by TIR off its internal faces. Because TIR reflects essentially 100% of the light, these prisms give a brighter, crisper image than a silvered mirror, which always absorbs a little.
The sparkle of a diamond. With n = 2.42 and a critical angle of only 24.4^\circ, a well-cut diamond traps light and ricochets it around inside by repeated TIR before letting it burst out of the top face — concentrating and splitting it into flashes of colour.

The three slips examiners bank on:

TIR needs BOTH conditions, not one. The light must be going into a less dense medium (denser \to less dense) and hit at an angle of incidence greater than the critical angle. Going the other way (air into glass) there is no critical angle at all — the ray always refracts in. And below C, even at a glass–air surface, the light mostly escapes.
A higher refractive index gives a SMALLER critical angle. Because \sin C = 1/n, as n goes up \sin C goes down, so C shrinks. That is why diamond (C = 24^\circ) sparkles far more than glass (C = 42^\circ): a smaller critical angle means more rays are trapped and reflected. Don't let the words "higher index → bigger anything" trick you.
\sin C = 1/n is only for a boundary with air (or vacuum). At a boundary between two dense materials you must use the full \sin C = n_2/n_1. For a glass-to-water boundary, for instance, \sin C = 1.33/1.50, giving a much larger critical angle than glass-to-air.

Under the Atlantic and the Pacific run cables no thicker than a garden hose, and inside each are strands of glass finer than a human hair. A message to a server on the other side of the world is first turned into a storm of light pulses — a laser flicking on and off billions of times a second — and fired into one of those glass strands. The core of the fibre is made of slightly denser glass than the cladding around it, so any pulse straying towards the edge meets the boundary above the critical angle and is totally internally reflected straight back into the core.

With no refracted ray leaking out at each bounce, the light loses almost nothing over enormous distances (a faint booster amplifier every 50–100 km tops it up). So your video call is not really electricity racing through wires — it is a pulse of light, ricocheting down a thread of glass by millions of total internal reflections, crossing an entire ocean in a few hundredths of a second. The same trick, in a bundle a few millimetres across, lets a doctor's endoscope light up and photograph the inside of a living knee.