Dispersive prism

A ray trace through a prism with apex angle α. Regions 0, 1, and 2 have indices of refraction ${\displaystyle n_{0}}$, ${\displaystyle n_{1}}$, and ${\displaystyle n_{2}}$, and primed angles ${\displaystyle \theta '}$ indicate the ray's angle after refraction.

Ray angle deviation and dispersion through a prism can be determined by tracing a sample ray through the element and using Snell's law at each interface. For the prism shown at right, the indicated angles are given by

${\displaystyle {\begin{aligned}\theta '_{0}&=\,{\text{arcsin}}{\Big (}{\frac {n_{0}}{n_{1}}}\,\sin \theta _{0}{\Big )}\\\theta _{1}&=\alpha -\theta '_{0}\\\theta '_{1}&=\,{\text{arcsin}}{\Big (}{\frac {n_{1}}{n_{2}}}\,\sin \theta _{1}{\Big )}\\\theta _{2}&=\theta '_{1}-\alpha \end{aligned}}}$.

All angles are positive in the direction shown in the image. For a prism in air ${\displaystyle n_{0}=n_{2}\simeq 1}$. Defining ${\displaystyle n=n_{1}}$, the deviation angle ${\displaystyle \delta }$ is given by

${\displaystyle \delta =\theta _{0}+\theta _{2}=\theta _{0}+{\text{arcsin}}{\Big (}n\,\sin {\Big [}\alpha -{\text{arcsin}}{\Big (}{\frac {1}{n}}\,\sin \theta _{0}{\Big )}{\Big ]}{\Big )}-\alpha }$

If the angle of incidence ${\displaystyle \theta _{0}}$ and prism apex angle ${\displaystyle \alpha }$ are both small, ${\displaystyle \sin \theta \approx \theta }$ and ${\displaystyle {\text{arcsin}}x\approx x}$ if the angles are expressed in radians. This allows the nonlinear equation in the deviation angle ${\displaystyle \delta }$ to be approximated by

${\displaystyle \delta \approx \theta _{0}-\alpha +{\Big (}n\,{\Big [}{\Big (}\alpha -{\frac {1}{n}}\,\theta _{0}{\Big )}{\Big ]}{\Big )}=\theta _{0}-\alpha +n\alpha -\theta _{0}=(n-1)\alpha \ .}$

The deviation angle depends on wavelength through n, so for a thin prism the deviation angle varies with wavelength according to

${\displaystyle \delta (\lambda )\approx [n(\lambda )-1]\alpha }$.

Aligning multiple prisms in series can enhance the dispersion greatly, or vice versa, allow beam manipulation with suppressed dispersion.

As shown above, the dispersive behaviour of each prism depends strongly on the angle of incidence, which is determined by the presence of surrounding prisms. Therefore, the resulting dispersion is not a simple sum of individual contributions (unless all prisms can be approximated as thin ones).