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Refraction Calculations and Total Internal Reflection
So we know that waves slow down when they enter optically denser materials, and bend towards the normal line.
But can we predict how far waves will change direction?
If we label the angle of incidence as i and the angle of refraction as r, then it can be shown that when travelling from a vacuum into a material, the ratio remains constant for all values of i and r. This is Snell's Law.
We call the constant from Snell's law the refractive index, n.
All transparent materials have a refractive index. It shows how well the material refracts light. The index is always given for the case of light crossing from a vacuum into the material.
You can show that the refractive index can be given by:
Where c is the speed of light in the vacuum or material, and is the wave length in the vacuum or material.
The refractive index always has a value greater than 1.
Note: That when the light leaves a medium and crosses back to a vacuum, the light refracts the other way - away from the normal.
So, to apply the equation here you need to invert it (turn it upside down)!
So, what happens if you are travelling from a material that isn't a vacuum into another material?
Well it's simple:
If the wave is travelling from material 1 into material 2, the ratio of the sine of the angles is still constant, but now we use the relative refractive index, 1n2.
When light leaves a material (let's say, glass) and speeds up, it refracts away from the normal. If the angle of incidence is too great, the refracted ray will (according to the equation) try to refract through an angle greater than 90°. That's not possible, because if it remains in the glass, it won't change speed and it's the speed change that causes it to refract in the first place!
The effect we see here is called Total Internal Reflection. The inside surface of the glass behaves like a mirror. The Law of Reflection applies, so the angle of incidence equals the angle of reflection inside of the material.
The critical angle is the angle of incidence at which the angle of refraction is 90°.
Any angle of incidence greater than the critical angle will cause total internal reflection.
Remembering that this is for light leaving a slower material it can be shown that...
This is a useful way of measuring the refractive index for a material.
Uses of TIR:
There are many uses for TIR starting from the simple replacement of mirrors with prisms in periscopes to the complicated world of fibre optics.
Optical fibres use TIR to send light pulses down glass fibres.
Layer 1 is a protective layer, usually made of some form of plastic to protect the inner glass layers.
Layers 2 and 3 are glass with different values of refractive index. Depending on the glass used in each layer, the relative refractive index can be altered to change the critical angle for the boundary. This alters the amount of waves that are internally reflected along the cable.
Optical fibres are used in endoscopes to look inside of the human body and in communication networks. It is far more efficient to send light pulses down glass fibres than to send electrical signals along copper cables.