Objectives:

• To understand the refraction of light; refractive index;  $n = \frac{c}{v}$ ;  $n \sin \theta = constant$ at a boundary where  $\theta$  is the angle to the normal
• To identify techniques and procedures used to investigate refraction and total internal reflection of light using ray boxes, including transparent rectangular and semi-circular blocks
• To appreciate what the critical angle is and the formal relating to it;  $\sin C = \frac{1}{n}$ ; total internal reflection for light

When studying light, the refractive index is  $n$  of a material is used to describe the how light travels through that medium. It is defined as;

$n = \frac{c}{v}$

where
$n$  is the refractive index, it is a dimensionless number, this is shown below
$v$  is the speed of light in that medium, measured in  $ms^{-1}$
$c$  is the speed of light in a vacuum, measured in  $ms^{-1}$

There are no units for refractive index, this can be shown be substituting units in the equation above;

$n = \frac{c}{v}$

$n \rightarrow \frac{ms^{-1}}{ms^{-1}}$

The units cancel to leave no unit.

As mentioned in the page on refraction there are some laws by which light sticks to when undergoing refraction;

1. The incident ray, the refracted ray and the normal are all in the same plane.
2. The ratio of the sine of the angle of incidence to the sine of the angle of refraction is the same for all rays travelling between a given boundary – this is known as Snell’s law;  $\frac{\sin{i}}{\sin{r}} = constant$

Snell’s Law

If light is travelling through a boundary between two materials, you use the law of refraction to calculate unknown angles or refractive indices.

Take the following image showing refraction;

Snell’s full law of refraction for a boundary between two mediums is given by :

$n_{1} \sin{\theta_{1}} = n_{2} \sin{\theta_{2}}$

where
$n_{1}$  is the refractive index of the initial medium
$n_{2}$  is the refractive index of the final medium
$\theta_{1}$ is the incident angle
$\theta_{2}$ is the refracted angle

One thing to be careful when dealing with refraction is the refractive index. There are two different ‘types’, the absolute refractive index and the refractive index of a boundary

Absolute refractive index

The absolute refractive index of a material is a property of that material only.

Using  $n = \frac{c}{v}$ we can see that in a medium light travels at the speed  $c$  and as such  $v = c$  and so  $n = \frac{c}{c} = 1$ . This tells us that the refractive index of a vacuum is  $n_{vacuum} = 1$ . Similarly, the speed of light in air is only a tiny bit smaller that that in a vacuum. So the refractive index of air is very similar to the refractive index of a vacuum,  $n_{air} = 1.0003 \approx 1$

The refractive index of a boundary

The refractive index between two materials is the ratio of the speed of light in material 1 to the speed of light in material 2, it is occasionally written as  $_{1}n_{2}$ .

Snell’s law has been written in two formats so far;

$\frac{\sin{i}}{\sin{r}} = constant$     (1)

$n_{1} \sin{\theta_{1}} = n_{2} \sin{\theta_{2}}$     (2)

If we rearrange equation (2) to get  $\frac{\sin{i}}{\sin{r}}$  on one side;

$\frac{\sin{\theta_{1}}}{\sin{\theta_{2}}} = \frac{n_{2}}{n_{1}}$

This shows us that $\frac{n_{2}}{n_{1}}$  is equal to the  $constant$  in equation (1).

This constant is the refractive index of a boundary, we can therefore write;

$_{1}n_{2} = constant = \frac{n_{2}}{n_{1}}$

$_{1}n_{2} = \frac{n_{2}}{n_{1}}$

Sadly, confusion will arise because many books will write Snell’s law as  $\frac{\sin{\theta_{1}}}{\sin{\theta_{2}}} = n$  and will not be clear on whether the refractive index here is an absolute refractive index or the refractive index of a boundary – so be cautious.

For more information of the refractive index, how it can change with temperature and the cool mirage effects that be seen click here.

The Critical Angle

When a light ray, travelling from a more dense medium to a less dense medium, strikes the  boundary between the mediums, one of three things can occur;

1. The light can refract through the boundary and change direction – because it is going from a more dense to less dense medium it will refract away from the normal – as shown in the far left image.
2. The light reflects off the surface and can remain in the more dense medium – as shown in the far right image.
3. At a specific angle between the ray either reflecting or refracting, the light can change direction such that it ends up travelling along the boundary of the two mediums – as shown in the middle image.

Using a known critical angle value, depending on what the angle of incidence is will ultimate result in either reflection, refraction, or light travelling along the boundary of the two mediums;

• If  $i < C$  refraction occurs
• If  $i > C$ reflection occurs – in this instance all the light will reflect and so we say that total internal reflection occurs (because all the light is contained inside the medium).

The formula to determine the critical angle can be derived (click here to see a derivation) and is:

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

where
$C$ is the angle of incidence which cause light to ultimately travel along the boundary between the two mediums
$n$ is the refractive index of the more dense medium.

The critical angle is useful so that a user/ manufacturer  can ensure that light is always totally internally reflected.

Total Internal Reflection, TIR

Total internal reflection, abbreviated to TIR, is being used more and more in technology these days. Light, which can be used for communication, is shone down one end of an optical fibre lets say, because all the light is totallyinternally, reflected an observed can see all the light coming out of the other end. Computers can use this principle for communication too. If light is shone down one end of an optical fibre, it will be seen at the other end and so a computer may recognise this as a signal and say that it is ‘on’, it can then assign the number  $1$ to it. When the light turns off, it will see no signal and can then assign a $0$ to it. Using this, $1's$ and $0's$ can be populated by a computer and a code formatted creating images, sound or videos, this (in principle) is how our internet works.

Broadband can be setup using electrical wires (transferring electricity down them for the $1's$ and $0's$ , or fibre optics can be used. Nothing travels faster than the speed of light and see increasing the speed of a users internet connection can be done by changing to fibre optic broadband!