PE1-2.1 Optical Fiber Waveguides

What this is and why it exists

A fibre is a thread of glass that keeps light inside it over tens of kilometres. It does that with one mechanism, and almost everything in this topic follows from that mechanism.

Light meets the boundary between a denser core and a lighter cladding. Beyond a certain angle it does not cross the boundary at all. It is entirely reflected, and it stays in.

The vocabulary

• Core and cladding — the inner glass that carries the light, and the surrounding glass of slightly lower refractive index.
• Total internal reflection — the complete reflection that happens beyond a critical angle at a boundary.
• Acceptance angle — the widest angle at the fibre end from which light will be guided.
• Numerical aperture — one number describing the fibre's light-gathering ability.
• Meridional ray — a ray that passes through the fibre axis.
• Skew ray — a ray that spirals along without ever crossing the axis.
• Mode — one field pattern the fibre can support, from the wave description.
• V-number — the single parameter deciding how many modes a fibre supports.
• Step index — a fibre whose refractive index changes abruptly at the boundary.
• Graded index — a fibre whose index falls smoothly from the centre outwards.
• Cut-off wavelength — the wavelength above which a fibre carries only one mode.
• Mode field diameter — how wide the light actually is in a single mode fibre.

The mental model

Start with the ray picture, because the first two results come straight out of it.

Light entering the flat end of a fibre is refracted, then travels to the core-cladding boundary. If it strikes that boundary beyond the critical angle it is totally reflected and continues down the fibre. Work backwards through the entrance refraction and you get the widest external angle whose ray survives, which is the acceptance angle. Its sine is the numerical aperture, and it turns out to depend only on the two refractive indices. Derive it once and you never need to memorise it.

Skew rays are the part the simple picture misses. Not every ray passes through the axis. Some spiral along the fibre, reflecting at the boundary without ever crossing the centre. They obey the same total internal reflection rule and they carry a real share of the power.

The ray picture then runs out. A fibre core is only a few wavelengths across, so the light has to be treated as a wave. A wave in a bounded structure supports only certain patterns. Those are the modes, exactly as in the waveguide topic elsewhere in this area.

One parameter decides how many. The V-number combines the core radius, the wavelength and the numerical aperture. A large V means many modes. Below a value of about 2.405 only one mode survives. That number is the first zero of a Bessel function, appearing in a practical specification.

Mode coupling is what happens in a real cable. Bends, small variations and stresses move power between modes as the light travels. After some distance the distribution settles into a steady pattern. That is why dispersion measured over a short piece can differ from dispersion over a long one.

Two index profiles, for one reason. In step index fibre the index changes abruptly. A ray bouncing at a steep angle travels a longer path than one going straight down, so it arrives later. In graded index fibre the index falls smoothly from the centre outwards. A steeper ray spends more of its journey in the lower-index outer region, where light travels faster, so it partly catches up. The path difference is largely cancelled by a speed difference. That is the fix for the spreading problem in the next topic.

Single mode fibre solves the same problem completely by leaving only one path. Making the core small enough drops the V-number below the threshold at the operating wavelength. There is a cut-off wavelength for each fibre, and above it only one mode propagates. This is why long haul links work at all.

Two parameters describe how that single mode actually sits. The mode field diameter is the width of the light, and it is wider than the core. A real fraction of the power travels in the cladding. The ray picture does not suggest that at all. The effective refractive index is the index the mode behaves as if it sees, somewhere between the core and cladding values. Both govern how much loss a splice or a bend causes, which makes them practical numbers rather than theoretical ones.

What you should now be able to explain or do

Derive acceptance angle and numerical aperture from total internal reflection. Say what skew rays are and why they matter. Compute a V-number and use it to say how many modes a fibre carries. Explain how a graded index reduces the spread in arrival times. Define cut-off wavelength and mode field diameter, and say why light travels in the cladding.

Check yourself