An optic fibre is a thin strand of glass or plastic with two layers. The core in the middle has a high refractive index, n₁. The cladding around it has a lower one, n₂. Light that goes into the end of the core hits the wall between the two layers. If the angle of incidence there is bigger than the critical angle, all of the light is reflected (total internal reflection). It zigzags down the fibre and cannot escape, even round bends. This is how a fibre carries telephone, internet and TV signals, and light into the body for an endoscope.

sin θ_c = n₂ / n₁

Why the cladding is needed. Total internal reflection only happens when the light is going towards a material with a lower refractive index, so the cladding must have a lower index than the core. If it does not (try “Same index” and “Cladding higher” in the Advanced tab), the light is not trapped and goes straight out through the wall.

The acceptance angle. The laser must also go in at a small enough angle. Light entering the end face from a material of index n₀ (air is 1.00) bends towards the axis (Snell’s law): n₀ sin θ₀ = n₁ sin θ₁. Inside the core it hits the wall at 90° − θ₁ to the normal, which must be at least θ_c. Putting these together gives the biggest angle that is accepted:

NA = √(n₁² − n₂²), sin θ_max = NA / n₀

NA is the numerical aperture. A bigger difference between the core and the cladding gives a bigger NA, so the fibre accepts light from a wider cone. (In the Advanced tab, choose “A spread of light” to see the cone: beams inside the angle are trapped and beams outside it leak out at the first wall.)

Why a bend lets light out. Where the fibre bends, the light hits the outer wall at a smaller angle of incidence than in a straight piece. A beam that travels along the axis hits the outer wall of a bend of radius R at an angle i given by sin i = R/(R + a), where a is the radius of the core. If i is smaller than the critical angle, part of the beam escapes through the wall. So the smallest radius the bend can have is:

R_min = a n₂ / (n₁ − n₂)

Real fibres have n₁ and n₂ very close (for example 1.48 and 1.46), so this radius is about 70 times the radius of the core. Because a real core is only 5 to 50 micrometres wide, that is still a bend of a few millimetres to a few centimetres, and fibres are never kinked. Beams that are not along the axis usually need an even bigger radius, and how much bigger depends on where the beam is in its zigzag when it reaches the bend (a beam that happens to be near the outer wall can survive a tighter bend).

What the picture leaves out. The fibre is drawn far wider than it is (and the cladding is thin) so you can see the light. The program follows rays in two dimensions, with Snell’s law and the Fresnel equations at every wall. When the angle of incidence is a little below the critical angle, part of the light reflects and part leaks out, and the program follows both. It leaves out the small loss of light when it enters the end face, absorption in the glass, the fact that light is a wave (modes and dispersion), and rays that spiral round the fibre.

Things to try. (1) In the Simple tab, tighten the bend until the light first leaks out. Where does it leak out? (2) Aim the laser steeply (“Aimed too steeply”). Why does the light leak at the very first wall? (3) In the Advanced tab, lower n₂. What happens to the critical angle, to NA and to the acceptance angle? (4) Why can a fibre with a bigger difference between n₁ and n₂ be bent more tightly, but carries a signal less well in practice (the beams take very different times to arrive)? (5) Put the fibre in water. What happens to the acceptance angle?