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Fig.1 shows a beam of light incident on an interface, which is a boundary between two different optical materials in which the speed of light is different. The boundaries between air and glass or air and water form common examples. The rules governing reflection are very simple, which is that the reflected light leaves at the same angle to the normal as the incident light. (The normal is a line at right angles to the interface). That makes sense as the velocity and wavelength do not change on reflection.

Reflection is also independent of wavelength, so there is no difficulty where colored images are involved.

The transmitted case is rather different. Assuming that the light encounters a denser medium, the frequency of the light doesn’t change, but the velocity does, and the wavelength must also change. This means that the incident wave front will only be coherent at a new angle consistent with the shorter wavelength. The beam of light will be bent towards the normal as it enters the denser medium.

The bending is called refraction and it can be predicted from the relative speed of light in the two materials. The refractive index specifies the speed of light in a material compared to the speed in vacuo. In most cases, the speed of light is not constant but varies with frequency. This means that the angle of refraction will be a function of wavelength. This is how the traditional prism demonstration can split visible white light up into its constituent colors. For applications such as chandeliers, glass which is highly dispersive will be used, whereas lenses require just the opposite characteristic.

It is perhaps interesting to consider how the refracted light knows where to go. Why does it prefer that crooked path? Fig.2 shows a variety of paths through an interface between two points, along with the time taken for the light to make the journey from A to B. There is a low point in the curve of propagation time where the slope of the curve is horizontal, meaning that over a small range of paths, the time taken is practically constant and coherent addition of wavelets is possible. For other paths, the time would vary too much for wavelets to be coherent.

This concept is known as Fermat’s principle of least time. Richard Feynman came up with a very good analogy, which is that at point A is someone on a beach and that at point B is someone in the water who needs help. A physicist at point A would know he could run a lot faster than he could swim, so would take the path of least time. To bring the analogy up to date, a politician at point A would conclude that the situation could not be monetized and would simply state that the beach was perfectly safe.

Consideration of Figures 1 and 2 suggests that as light is always bent towards the normal when entering a denser medium, there must be some angles in that medium at which light could never enter. Fig.3 shows the situation in which light arriving over a subtended angle of 180 degrees is refracted into a cone. Outside that cone the only light that can arrive must have come from inside the medium and suffered total internal reflection.

In other words, beyond a certain angle from the normal, light simply cannot leave a medium and the interface acts like a perfect mirror reflecting the light back in. The reflecting surface does not need to be silvered. Most binoculars obtain their characteristics through the use of internal reflection in glass prisms. Telescopes are physically long and some architectures invert the image. Using internally reflecting prisms, the light path of the telescope can be folded to make a more compact device and the image can be erected if necessary.

In a single lens reflex camera, the viewfinder looks through the taking lens via a mirror that swings up when the picture is taken. The mirror corrects the inversion of the lens, but leaves the image laterally inverted. The inversion is commonly cancelled by the use of internal reflection in a roof pentaprism. Unlike binoculars, the angles of a pentaprism do not cause total internal reflection, so the reflecting surfaces may need to be coated.

A common example of internal reflection is the rainbow. Raindrops are near-spherical and sunlight entering a raindrop is refracted. As water is dispersive, the angle of refraction is a function of wavelength. The refracted light is then internally reflected and is refracted once more as it leaves the raindrop. Although this process takes place over a wide range of wavelengths, we can only see the effect in the visible spectrum which is that white light is split into the colors that the HVS can see.

To see a rainbow, an observer must look away from the sun and it must be raining down sun from the observer. It need not be raining on the observer. Light leaves the raindrops at about 42 degrees to the angle of incidence so the observer sees a cone of reflected light at that angle from the shadow of his head on the ground. There are an infinite number of rainbows and no two observers can see the same one. The rainbow is an optical effect and no corresponding object exists, nor is it possible to reach the end of a rainbow as any movement by the observer will result in the rainbow moving with him.

Another thing that Fig.1 does not show is the relative amounts of incident light that are reflected and refracted. This is quite a complex subject because it involves polarization. Light is an electrical oscillation and a magnetic oscillation mutually at right angles. The direction of the electric oscillation defines the polarization. At the macroscopic scale, most light sources consist of billions of discrete quantum events that are essentially random, so the light coming out is polarized uniformly at every possible angle.

It does not clarify things when such light is described as un-polarized, when it should be described as randomly polarized. An interface between two media will treat the electric and magnetic components of the light differently, so there will always be an effect on the polarization, both in reflection and refraction.

This was first explained by Fresnel who produced the graphs shown in Fig.4. On the left at zero degrees, light is reflected in the state of polarization in which it arrived. As the angle from the normal increases, the responses to Tp, the polarization with the electric field in the plane of incidence and Ts, the polarization at right angles to that, begin to diverge.

At the Brewster angle, the incident light with Tp polarization is 100% transmitted, meaning the reflected light Rp is zero. At the Brewster angle, the reflected and transmitted rays are orthogonal so the polarized light cannot radiate in that direction and all of it must be transmitted. The only reflected light is Rs, meaning that in real life, when sunlight is reflected from horizontal surfaces, it tends to have the electric field horizontal. Such light causes glare that can be reduced by looking through a vertically polarized filter. That is the principle of Polaroid sunglasses.

Light from the sun reaching the Earth is randomly polarized, but the after passing through the atmosphere sunlight will be polarized in a plane at right angles to the arriving sunlight. Photographers use polarizing filters to increase contrast and saturation in outdoor shots. This only works fully when the camera is aimed transversely to the sun and the angle of the filter is adjusted to match the angle of elevation of the sun. Polarizing filters for cameras are always supplied in rotating mounts so that adjustment is possible.

Sky polarization is also useful for other purposes. For example, even after sunset the sky is still illuminated by the sun and using polarization it is possible to work out the angle of the sunlight and thus tell the time. Navigation by the sun is frustrated by overcast conditions, but using polarizing filters it is possible to locate the sun even when it cannot be seen.

Some naturally occurring crystals such as Iceland spar act as polarizing filters and it has been speculated that the Vikings used these so-called sunstones for navigational purposes on their long voyages.

Returning to Fig.4 it will be seen that as light becomes more oblique, reflection becomes stronger and transmission falls. For opaque substances there is no transmission but the increase in reflection will be noted. Strong reflection at grazing angles will be seen over water, glass, polished floors and paint, glossy magazines and so on. The glass display cabinet is an oxymoron as the contents will often be concealed by reflections in the glass.