Refractive index

Light travels at different speeds in different substances. The refractive index of a substance compares the speed of light in a vacuum with its speed in that substance:

Here, $$n$$ is the refractive index, $$c$$ is the speed of light in a vacuum, and $$c_s$$ is the speed of light in the substance. The two speeds must be in the same units, so refractive index has no unit.

A larger refractive index means that light travels more slowly in the substance: as $$c_s$$ decreases, the ratio $$c/c_s$$ increases. Air has a refractive index of approximately $$1$$, so light travels through air at approximately the same speed as it does in a vacuum.

Snell's law

When light crosses a flat boundary between two media, its speed may change. If the light enters at an angle to the normal, this change in speed also changes its direction. The change in direction is called refraction.

The normal is an imaginary line perpendicular to the boundary at the point where the ray meets it. Both the angle of incidence and the angle of refraction are measured between the ray and the normal, not between the ray and the boundary.

The relationship between the angles

Each medium has a refractive index, written as $$n$$. A larger refractive index corresponds to a lower speed of light in that medium. Snell's law relates the refractive indices to the angles on either side of the boundary:

Here, $$n_1$$ and $$\theta_1$$ are the refractive index and angle of incidence in the first medium; $$n_2$$ and $$\theta_2$$ are the refractive index and angle of refraction in the second medium.

Direction of bending

• If the second medium has a larger refractive index, light slows down and bends towards the normal. The angle of refraction is smaller than the angle of incidence.
• If the second medium has a smaller refractive index, light speeds up and the refracted ray bends away from the normal. The angle of refraction is larger than the angle of incidence.

A ray entering along the normal has an angle of incidence of zero. It does not change direction at the boundary, even if its speed changes.

Total internal reflection

When light travels from a medium with a higher refractive index into one with a lower refractive index, the refracted ray bends away from the normal, the line perpendicular to the boundary. Angles of incidence and refraction are measured from this line.

As the angle of incidence increases, the angle of refraction also increases. At one particular angle of incidence, the refracted ray is at $$90^\circ$$ to the normal and travels along the boundary. This angle of incidence is the critical angle, $$\theta_c$$.

For a first medium with refractive index $$n_1$$ and a second medium with refractive index $$n_2$$, where $$n_1 > n_2$$, the critical angle is given by:

If the angle of incidence is greater than the critical angle, no light is refracted into the second medium. The light is reflected back into the first medium. This is total internal reflection.

Both conditions are necessary: the light must travel towards a medium of lower refractive index, and its angle of incidence must exceed the critical angle.

Step-index optical fibres

A step-index optical fibre guides light through a central core surrounded by cladding. The core has a higher refractive index than the cladding. The term step-index means that the refractive index changes abruptly at the core–cladding boundary, rather than gradually.

Light travelling in the core can meet this boundary and be reflected back into the core. If its angle of incidence, measured from the normal to the boundary, is greater than the critical angle, the light undergoes total internal reflection. Repeated reflections guide the light along the fibre.

The lower refractive index of the cladding makes total internal reflection possible at the boundary. It also helps keep light confined to its own core when fibres are close together, reducing the transfer of light between them.

Material dispersion

An optical pulse usually contains light over a range of wavelengths rather than at one exact wavelength. In an optical fibre, the refractive index of the material depends on wavelength, so light at different wavelengths travels through the material at different speeds.

The wavelength components of a pulse therefore reach the end of the fibre at slightly different times. The pulse becomes spread out in time as it travels. This spreading is called material dispersion; it increases with the distance travelled through the fibre.

If neighbouring pulses spread enough to overlap, a receiver may struggle to distinguish them. Pulses must then be sent farther apart in time, limiting the rate at which information can be transmitted over a given distance.

Modal dispersion

A short pulse of light entering an optical fibre can travel along different modes: permitted paths through the fibre. If these paths take different times to reach the far end, the pulse becomes spread out in time. This spreading is called modal dispersion.

In a step-index fibre, light travelling at a greater angle to the fibre’s axis follows a longer path than light travelling closer to the axis. The different path lengths mean that parts of the same input pulse arrive at different times. As a result, the output pulse is broader than the input pulse.

Modal dispersion and pulse overlap

The longer the fibre, the greater the difference between the arrival times of light travelling along the different paths. If successive pulses spread enough to overlap, a receiver may no longer be able to distinguish them reliably. Modal dispersion therefore limits how quickly separate pulses can be sent over a given distance.

Absorption in optical fibres

As light travels along an optical fibre, the fibre material absorbs some of its energy. Less energy remains in the light, so the optical signal becomes less intense with distance. This reduction in signal strength is called attenuation.

Over a long distance, absorption can make the signal too weak for a receiver to detect reliably. This limits the distance over which it can be transmitted unless the signal is restored along the way.

Absorption and dispersion are different effects. Absorption reduces the strength of the signal, whereas dispersion spreads light pulses out in time. A weaker pulse and a broader pulse are not the same thing.