Multiple-prism dispersion theory

The first description of multiple-prism arrays, and multiple-prism dispersion, was given by Newton in his book Opticks. Prism pair expanders were introduced by Brewster in 1813. A modern mathematical description of the single-prism dispersion was given by Born and Wolf in 1959. The generalized multiple-prism dispersion theory was introduced by Duarte and Piper in 1982.

The generalized mathematical description of multiple-prism dispersion, as a function of the angle of incidence, prism geometry, prism refractive index, and number of prisms, was introduced as a design tool for multiple-prism grating laser oscillators by Duarte and Piper, and is given by

which can also be written as

using

Also,

Here, ${\displaystyle \phi _{1,m}}$ is the angle of incidence, at the mth prism, and ${\displaystyle \psi _{1,m}}$ its corresponding angle of refraction. Similarly, ${\displaystyle \phi _{2,m}}$ is the exit angle and ${\displaystyle \psi _{2,m}}$ its corresponding angle of refraction. The two main equations give the first order dispersion for an array of m prisms at the exit surface of the mth prism. The plus sign in the second term in parentheses refers to a positive dispersive configuration while the minus sign refers to a compensating configuration. The k factors are the corresponding beam expansions, and the H factors are additional geometrical quantities. It can also be seen that the dispersion of the mth prism depends on the dispersion of the previous prism (m - 1).

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