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Total Internal Reflection

Drag the yellow points to bend the pipe, and the torch to shine the light in at a slant.

GlassAir

Trapped! Total internal reflection: every time the light meets the side, it is more slanted than the critical angle, so all of it bounces back in, round every bend, and comes out at the far end.

Closest to head-on
57°
Critical angle
42°
Bounces
4
Light reaching the end
all of it
Pipe

What’s happening

Shine a torch into one end of a bent glass rod and the other end lights up, though light travels in straight lines. When light inside glass or water reaches the surface, it bends as it goes out into the air, more the more slanted it is, until at the critical angle it would only skim along the surface. Any more slanted and it can’t get out at all: all of it bounces back in, better than any mirror. This is total internal reflection. Light that keeps meeting the side at a slant bounces from side to side all the way along the pipe, even round a bend. Bend it too sharply, though, and the light meets the side more head-on, and some of it escapes.

Try this

  • How sharply can you bend the glass rod before light starts to leak out?
  • Where does the light escape: on the inside of a bend, or the outside?
  • Switch to the stream of water. Is it better or worse at keeping the light in? Why?
  • Shine the light straight in. Does it still bounce?
  • At home, over a sink in a dark room: make a hole low in the side of a plastic bottle full of water, and shine a torch through the bottle at the hole. Does the stream glow?
Learn more: the critical angle, leaky bends, and optical fibres

The critical angle. Light going from a material with refractive index n1n_1 into one with a smaller index n2n_2 bends away from the normal (the line standing straight out of the surface). At the critical angle θc\theta_c, measured from the normal, it would leave exactly along the surface:

sin⁡θc=n2n1\sin\theta_c = \frac{n_2}{n_1}

Into air, where the index is 1, that is just

sin⁡θc=1n\sin\theta_c = \frac{1}{n}

so 42° for glass (index 1.5) and 49° for water (1.33). Meeting the surface more slanted than that (a bigger angle from the normal), none of the light can get out. It only works going from slower stuff into faster stuff, never the other way round (see Refraction).

Leaks. Below the critical angle, light isn’t all lost at once: some of it still bounces back in, more the closer it is to the critical angle. In the drawing, the light gets fainter at each leak, and the readout shows how much reaches the end.

Bends. Each bounce only depends on the angle where the light meets the side right there, not on the shape of the whole pipe. A gentle bend hardly changes that angle; a sharp one makes the light hit the outside of the bend more head-on, and it leaks. Real optical fibres have a smallest bend they are allowed to make for just this reason (it’s called “bend loss”).

Real optical fibres are glass threads about as thick as a hair. The glass in the middle, the core, is wrapped in another layer of glass, the cladding, with a slightly smaller index (something like 1.48 inside 1.46). Then fingerprints, dust and scratches can’t touch the surface where the light bounces and let it out. With the two so alike, the critical angle is over 80°, so the light has to run almost along the fibre to stay in. How wide a cone of light a fibre can take in is its numerical aperture:

NA=n12−n22\text{NA} = \sqrt{n_1^2 - n_2^2}

A fibre with a very thin core (about 9 thousandths of a millimetre) lets the light travel only one way, straight along it (single-mode), which is what long internet cables use. A thicker core lets it zigzag many ways at once (multi-mode): easier to shine light into, but the zigzags take different times, so a quick flash spreads out on a long trip.

  • The internet: flashes of light through glass fibres carry most of the world’s internet and phone calls, under the oceans too.
  • Doctors’ endoscopes: a bundle of fibres goes round the body’s bends to carry light in and a picture back out.
  • Fibre lamps and light-up fountains: the light follows the fibres, or the stream of water, and shines out at the tips.
  • Diamonds: with a critical angle of only 24°, light bounces around inside a cut diamond before it gets out of the top, which is why it sparkles.

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