Critical Angle, Total Internal Reflection and Optical Fibres

1A water surface that shines like silver

Fill a clear drinking glass with water and hold it a little above your eyes. Look up through the side of the glass at the underside of the water surface. From below and at a slant, the surface shines like a polished mirror, and if you dip a finger into the water you can see its reflection in it. No silver is there: the water surface is reflecting all of the light that reaches it from inside. This lesson finds out when that happens, and how the same effect carries telephone calls and internet data through glass threads thinner than a hair.

2Experiment: a ray inside a semicircular block

Apparatus: a semicircular glass or Perspex block, a ray box with a single slit, white paper, a protractor and a pencil. If a laser pointer is used, follow the laser safety rules from lesson 1.

1. Draw round the block and mark the centre O of its flat face. Draw the normal through O.
2. Aim the ray through the curved face at O. A ray aimed at the centre travels along a radius, so it meets the curved surface along the normal and does not bend there. All the bending happens at the flat face.
3. Start with a small angle of incidence at O and slowly increase it. Watch the ray that leaves the flat face and the weak ray reflected back into the glass.

For this block, $n=1.50n\; =\; 1.50$. At $i={30}^{\circ }i\; =\; 30^\backslash circ$ most of the light escapes into the air, bent away from the normal to 49°, and a weak ray is reflected. As $ii$ increases, the escaping ray swings closer to the surface and gets fainter, while the reflected ray gets brighter. At $i={41.8}^{\circ }i\; =\; 41.8^\backslash circ$ the escaping ray grazes along the surface: the angle of refraction is 90°. At any larger angle, such as 55°, no light escapes at all. The surface reflects all of it, and the reflected ray obeys the law of reflection.

3Calculating the critical angle

For light leaving a medium of refractive index $nn$ into air, lesson 1 gave $\sin {\theta }_a=n\sin {\theta }_m\backslash sin\; \backslash theta\_a\; =\; n\; \backslash sin\; \backslash theta\_m$. At the critical angle, ${\theta }_m=c\backslash theta\_m\; =\; c$ and ${\theta }_a={90}^{\circ }\backslash theta\_a\; =\; 90^\backslash circ$, with $\sin {90}^{\circ }=1\backslash sin\; 90^\backslash circ\; =\; 1$:

The larger the refractive index, the smaller the critical angle, and the easier it is to trap light inside the material.

4Uses of total internal reflection

Prisms that turn light

A right-angled glass prism with two 45° angles makes an excellent reflector. Light entering one face along the normal meets the long face at 45°, which is greater than glass's critical angle of 41.8°, so it is totally reflected.

Turning light through 90° twice is how a prism periscope works; turning it through 180° is used in binoculars, which are shorter than the telescope they replace because the light path folds back on itself. Prisms are better than plane mirrors here because total internal reflection reflects practically all the light, there is no silvering to tarnish or peel, and there are no faint extra images from the front surface of a glass mirror. Bicycle reflectors and road studs use the same idea with three faces at right angles, which send light back towards the headlamp it came from.

Optical fibres

An optical fibre is a thin strand of very pure glass. Its central core is surrounded by a cladding of glass with a slightly lower refractive index. Light entering one end meets the core and cladding boundary at more than the critical angle, so it is totally internally reflected again and again, and follows the fibre even round gentle bends.

For a core of $n=1.48n\; =\; 1.48$ and cladding of $n=1.46n\; =\; 1.46$, the critical angle at their boundary is ${\sin }^{-1}\left(\frac{1.46}{1.48}\right)={80.6}^{\circ }\backslash sin^\{-1\}\backslash left(\backslash frac\{1.46\}\{1.48\}\backslash right)\; =\; 80.6^\backslash circ$, so only rays travelling almost parallel to the fibre are trapped. The cladding also protects the core surface: a scratch or a speck of dust touching a bare core would let light leak out.

• Telecommunication: pulses of light in fibres carry telephone calls, television and internet data between towns and under the oceans. Unit 9 returns to this.
• Medicine: an endoscope is a flexible tube that a doctor passes into the body, for example into the stomach. One bundle of fibres carries light in, and a second bundle, with its fibres kept in the same order at both ends, carries the image back out, so the doctor can see inside without surgery.

Sparkling diamonds and mirages

Diamond's critical angle is only 24.4°, so light entering a well-cut diamond is reflected inside it several times before it escapes through the top. That makes a diamond sparkle far more than a glass copy of the same shape, whose critical angle is about 41°.

On a hot, sunny afternoon a tarmac road far ahead can seem to hold pools of water that vanish as you approach. The air just above the hot road is hotter and less dense than the air higher up, so it has a slightly smaller refractive index. Light from the sky heading down towards the road at a shallow angle passes into ever less dense air, bends gradually away from the normal, and is finally totally reflected upwards into your eyes. You see an image of the sky on the road, and the brain reads it as water. This is a mirage.