Refractive index and Descartes’ law

Reminder: refraction

When a ray of light passes from one transparent medium to another (for example, from air to glass), it changes direction: this is known as refraction. It is described by Descartes’ law (or Snell–Descartes’ law):

n1 * sin(i1) = n2 * sin(i2)

where n1 and n2 are the indices of refraction of the two media, and i1 and i2 are the angles measured relative to the normal to the surface.

The index of refraction depends on colour

For air, n is approximately 1. For glass, n is close to 1.5, but not exactly the same for all colours: this is the key to dispersion. In general, the index is higher for shorter wavelengths: n(violet) > n(red).

Consequences in the prism

Since n(violet) > n(red), Descartes’ law dictates that the violet ray must be deflected further than the red ray as it passes through the prism. This is why, at the exit, violet is always at the bottom (or at the top, depending on the orientation) and red on the other side: the order of the colours in the spectrum is not random; it directly reflects the order of the refractive indices.

Simplified numerical example

A piece of glass has a refractive index of n = 1.50 for red and n = 1.53 for violet. For the same angle of incidence, the violet ray will be deflected by a smaller angle of refraction (it moves closer to the normal), which, combined with the prism’s geometry, increases its total deflection at the exit.

Common pitfall

A medium such as a vacuum (or, approximately, air) disperses light to a negligible extent because its refractive index varies (almost) not at all with wavelength: without a change in n, there is no dispersion, even if the light is refracted.