Topic 15. Mirrors, part 2. The law of reflection

When a light wave is incident on the boundary between two media, the “angle of incidence” is the angle between the incident rays and the normal to the surface at that point. The reflected ray emerges at the same angle with respect to the normal (the perpendicular to the surface at that point). The incident ray, the reflected ray and the normal all lie in the same plane. This result can be derived from Huygens’ principle using a simple geometrical construction of the incident wavefront.

Since the wave is incident on the boundary between the two media at some angle other than 0 degrees or 90 degrees, one part of the wavefront reaches the boundary before the rest of the wavefront. The reflected wavelets from that part start radiating backwards while the remainder of the wavefront is still moving towards the boundary. The reflected wavefront can be completed when the last part of the incident wave reaches the boundary. A simple geometrical construction shows that applying Huygens’ principle to this case results in the angle of incidence being equal to the angle of reflection.

If the reflecting surface is plane and smooth then a parallel beam of light produces a reflection whose rays are also parallel. This is “specular” reflection. If the surface is curved or not smooth, then the direction of the reflected beams will vary from point to point, and a parallel beam of light will produce a more complex reflection. In the limit of a reflecting surface that is very rough and irregular, the reflected light goes in all directions with more or less the same intensity. This is called “diffuse” reflection, in which the initial directivity of a parallel beam is completely lost. Note that this diffusion is due to the fact that the angle of incidence varies rapidly from point to point (even though the incident beam is composed of parallel rays) because the surface roughness means that the normal to the surface varies from point to point.

The law of reflection relating the angle of incidence to the angle of reflection is true for all materials, but the amplitude of the light reflected in a particular direction depends on the angle of incidence, on the type of material (metal or dielectric) and on the polarization of the incident radiation. This variation turns out to be much more important for reflection from dielectrics than for reflection from metals.

For most dielectrics, the amplitude of the reflected light increases as the angle of incidence increases, and most dielectrics become almost 100% reflectors at large angles of incidence (approaching 90°). Furthermore, the amplitude of the reflected light depends on the polarization of the incident beam, and there are certain angles where the reflected amplitude goes to 0 for one polarization so that only other polarizations are reflected.

These principles are illustrated by reflection from a roadway. When the road is dry, its surface is very rough, and the reflection is diffuse. Incident light from any direction is reflected in all other directions with essentially equal amplitude, so that light reflected from the road reaches our eyes and we can see it at night. When the road is wet, the surface becomes much more regular, and the reflection tends to be specular rather than diffuse. Light from an overhead street light tends to be reflected back upward and not into our eyes (the angle of incidence is quite small and therefore the angle of reflection is small as well). Light from the headlights of the car tends to be reflected forward – both the angle of incidence and the angle of reflection are large in this case. Very little light is reflected back to our eyes both because reflection is specular. Since the reflection coefficient tends to approach 100% at large angles of incidence, oncoming traffic sees a large amount of light that is reflected from the roadway at large angles of incidence. Most of the light from the headlights is of almost no use in seeing the road and simply causes glare for the oncoming traffic.

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