The refractive index of a transparent material is generally not exactly the same for every wavelength. This wavelength dependence is called optical dispersion.

Write the refractive index as

$$n=n(\lambda).$$

Because

$$n=\frac{c}{v},$$

different wavelengths can have different phase speeds inside the same material.

Frequency does not change at an interface

When light crosses from one stationary medium into another, its frequency remains fixed by the source. Since

$$v=f\lambda,$$

a change in propagation speed therefore changes the wavelength inside the material.

Dispersion means that this speed change itself depends on wavelength.

Wavelength-dependent refraction

Snell's law becomes

$$n_1(\lambda)\sin\theta_1=n_2(\lambda)\sin\theta_2.$$

Therefore two wavelengths entering the same interface at the same incident angle can leave at different refracted angles.

For many transparent materials in the visible range, shorter visible wavelengths have a slightly larger refractive index than longer wavelengths. This common regime is called normal dispersion. Blue light then bends somewhat more strongly than red light when entering such a material from air.

Why a prism separates colors

A prism has two nonparallel refracting surfaces. At the first surface, wavelengths refract by slightly different amounts. At the second surface, they refract again, and the angular separation accumulates.

White light, which contains a range of visible wavelengths, can therefore emerge spread into a spectrum rather than as one ray.

Worked example

Suppose light enters from air at

$$\theta_1=45.0^\circ$$

into a material whose refractive indices are approximately

$$n_{\rm red}=1.50,$$

$$n_{\rm blue}=1.52.$$

Taking $n_{\rm air}\approx1$, Snell's law gives for red light

$$\sin\theta_{\rm red} =\frac{\sin45^\circ}{1.50} \approx0.4714,$$

so

$$\theta_{\rm red}\approx28.1^\circ.$$

For blue light,

$$\sin\theta_{\rm blue} =\frac{\sin45^\circ}{1.52} \approx0.4652,$$

so

$$\theta_{\rm blue}\approx27.7^\circ.$$

The difference at one interface is small, but a prism can turn such wavelength-dependent refraction into a clearly separated spectrum.

Dispersion and pulses

A wave packet or optical pulse contains a range of frequencies. If those components propagate differently through a dispersive medium, the pulse can change shape or spread in time. Thus dispersion matters not only for colors and prisms but also for optical communication and ultrashort pulses.

The simple statement that a material has 'a refractive index' is therefore an approximation valid when the wavelength range is sufficiently narrow. More generally, refractive index is a frequency- or wavelength-dependent property of the material.