Reflection and the critical angle

​​In a nutshell

Reflection occurs when a wave changes direction at a boundary and remains in the initial medium. Total internal reflection occurs when light hits a boundary with an optically less dense material and its' angle of incidence is greater than the critical angle. Total internal reflection is used in optical fibres which use cladding with a low refractive index. Signal degradation is caused by absorption and dispersion.

Equations

Variable definitions

Reflection

Reflection occurs when a wave changes direction at a boundary and remains in the initial medium.

The law of reflection states that the angle of incidence is always equal to the angle of reflection. This applies whenever a wave is reflected. 

The frequency and wavelength, and therefore wave speed, of a wave does not change when it is reflected.

Total internal reflection (TIR)

Total internal reflection occurs due to light changing speeds in different mediums. This happens if the change in speed is enough to bend the light ray at an angle more than $$90 \degree$$ from the normal. It is therefore reflected back into the initial medium.

​​Refraction	​​Tir

Total internal reflection requires two conditions:

1. The light goes from an optically more dense material to an optically less dense material.
   
2. The light has an angle of incidence, $$\theta_1$$, greater than the critical angle, $$C$$​.

​​Critical angle

The critical angle occurs when light is refracted along the boundary of the materials. This is because it is the point between the light being refracted into the second materials and the light being reflected back into the first material.

As the light travel along the boundary, the angle of refraction, $$\theta_2$$, is $$90\degree$$. Using Snell's law, a new equation for calculating the angle of incidence (the critical angle, $$C$$) can be derived.

$$n_1\sin{C} = n_2 \sin{90} = n_2 \times 1 \\ \ \newline \sin{C} = \frac{n_2}{n_1}$$

Note: This derivation uses $$\sin{90} = 1$$ to go from Snell's law to the critical angle equation.

When the second medium is air, for example a light ray going from water to air, the critical angle can instead be found using

$$\sin{C} = \frac{1}{n_1}$$

Note: This can be derived assuming that the refractive index of air, $$n_2$$, is equal to $$1$$.

Uses of total internal reflection

The most common use of TIR is in optical fibres. An optical fibre is a flexible, thin tube made from glass or plastic fibre. They are useful for carrying light signals over long distances and around corners.

1. Cladding: $$n_2$$​​ 2. Core: $$n_1$$​​

This optical fibre is surrounded by cladding. The cladding has a lower refractive index that allows TIR to take place. Scratches to the core would allow light to escape, so the cladding provides protection.

Dispersion and absorption

Signal degradation occurs when a signal is affected by absorption or dispersion. This could results in some of the signal's information being lost. This can be reduced by using an optical fibre repeater which boosts the signal.

Absorption causes a decrease in the amplitude of the signal. This is because some of the signal's energy is absorbed by the material of the fibre.

Dispersion causes pulse broadening. Pulse broadening widens the signal, which can overlap with other broadened pulses and cause information loss.

There are two types of dispersion: modal and material.

Modal dispersion is caused by rays entering the fibre at different angles and taking different paths through the fibre. The longer paths take a longer amount of time for the light ray to reach the end. This can be prevented by using a single-mode (narrow) fibre which only allows one path.

Material dispersion is caused by light travelling at different speeds through the fibre due to differing wavelengths. The wavelengths that travel faster will reach the end before the others. This can be prevented by using monochromatic (single-wavelength) light.

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