Dispersion (Optics): Definition, Formula & Examples

Dispersion is a phenomenon associated with the refraction of light. Though it can occur with any type of wave and with any wavelength of light, it is often discussed with respect to visible light. Dispersion, after all, is the reason for rainbows!

Dispersion, sometimes more specifically called ​chromatic dispersion​, occurs when the velocities of different components of a light wave depend on the wavelengths of those components. Specific types of chromatic dispersion are defined by what causes the dependence of velocity on wavelength.

For ​material dispersion​, this means the index of refraction in a material differs slightly depending on wavelength. (Recall that the index of refraction is n = c/v, where ​c​ is the speed of light in a vacuum and ​v​ is the speed of light in the given medium.)

How much a material actually disperses light is measured by a parameter called Abbe's number. To calculate Abbe's number, you need to measure several indices of refraction in the material that come from the characteristic light emissions of certain elements; these light emissions happen only at certain exact wavelengths, creating individual lines at each of those wavelengths in the spectrum, and the pattern of these lines is unique to each element.

The indices of refraction needed to calculated Abbe's number are: the index of the blue ​F​ line of hydrogen, the yellow ​D​ lines of sodium and the red C line of hydrogen. These are three different wavelengths of light that will all have different indices of refraction in the medium, and the Abbe's number for the medium is then calculated using the following equation:

\(v=\frac{n_D-1}{n_F-n_C}\)

If a material has a lower Abbe's number, it will have more dispersion over the visible spectrum.

​Waveguide dispersion​ is when the velocity of a light wave in a waveguide depends on its frequency because of the geometry of the waveguide's structure. In an optical fiber, both material and waveguide dispersion are usually present.

Material dispersion occurs due to different indices of refraction in a medium depending on wavelength; waveguide dispersion occurs due to the structure of the waveguide causing light of different wavelengths to travel at different speeds. Another type of dispersion is called ​polarization mode dispersion​, where a light wave's speed depends on its polarization in the medium.

A slightly more complex type of dispersion is ​group velocity dispersion​. Light waves can travel in "wave packets," also known as "signals" or "pulses," and the velocity of those pulses is called a group velocity. Within the pulses are wave components of different frequencies; group velocity dispersion occurs when these components start to separate due to having different velocities in the medium. The pulse then begins to "spread out," losing information. This degradation of signal is a big problem in optical communications systems that use optical fibers.

How much light bends when passing from one medium into another is determined by Snell's law. For an angle of incidence ​_θ_i​ and angle of refraction ​θ_r_​,

\(n_i\sin{\theta_i}=n_r\sin{\theta_r}\)

where ​_n_i​ is the refractive index of the incident medium and ​n_r_​ is the refractive index of the second medium.

If a light wave travels from a medium with a high refractive index to one with a much lower index, with a large-enough angle of incidence, Snell's law requires the sine of the angle of refraction to be greater than 1. This is technically impossible, and it means that the light wave won't reflect at all; in fact, it will entirely reflect off of the boundary between the two mediums. This is called total internal reflection.

Several scientists independently discovered this law over the course of history, including Rene Descartes.

The index of refraction in materials tends to be higher for shorter wavelength blue light and lower for longer wavelength red light. This means blue light will travel more slowly through a dispersive medium than red light will.

When white light is incident on a triangular prism, for example, the dispersing of the different wavelengths causes separation of light by different color, creating a rainbow.