Dispersion

Level 1 - Physics topic page in Light Propagation.

Principle

Dispersion occurs when refractive index depends on wavelength. Different colors then travel at different speeds and refract by different amounts.

Notation

\(n(\lambda)\)

wavelength-dependent refractive index

\(\lambda\)

wavelength

\(\mathrm{m}\)

\(v\)

wave speed in the material

\(\mathrm{m\,s^{-1}}\)

\(\theta\)

ray angle measured from the normal

\(\mathrm{rad}\)

\(\Delta n\)

index difference between colors

Method

Derivation 1: Link color to wavelength

Visible colors correspond to different wavelengths. Violet light has shorter wavelength than red light.

Material speed

\[v(\lambda)=\frac{c}{n(\lambda)}\]

Derivation 2: Use Snell's law for each wavelength

Each wavelength obeys Snell's law with its own refractive index.

Color-dependent refraction

\[n_1\sin\theta_1=n_2(\lambda)\sin\theta_2(\lambda)\]

Derivation 3: Interpret ordinary dispersion

In ordinary transparent materials, shorter wavelengths usually have larger refractive index. Violet light bends more strongly than red light in a prism.

Rules

Dispersive index

\[n=n(\lambda)\]

Speed in material

\[v(\lambda)=\frac{c}{n(\lambda)}\]

Snell by wavelength

\[n_1\sin\theta_1=n_2(\lambda)\sin\theta_2(\lambda)\]

Examples

Question

In ordinary glass, red light has

\[n=1.51\]

and violet light has

\[n=1.53\]

Which travels slower?

Answer

Violet light travels slower because

\[v=c/n\]

and violet has the larger refractive index.

Checks

Dispersion needs refractive index to depend on wavelength.
Frequency remains fixed when light crosses a boundary.
Wavelength changes because the speed changes.
A prism separates colors because different wavelengths refract by different amounts at its surfaces.

Internal Reflection

Polarization