Multiple-prism dispersion theory

The first description of multiple-prism arrays, and multiple-prism dispersion, was given by Isaac Newton in his book Opticks, also introducing prisms as beam expanders. Prism pair expanders were introduced by David Brewster in 1813. A modern mathematical description of the single-prism dispersion was given by Max Born and Emil Wolf in 1959. The generalized multiple-prism dispersion theory was introduced by F. J. Duarte and Piper in 1982.

Generalized multiple-prism dispersion equations 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

∂ ϕ 2 , m ∂ λ = H 2 , m ∂ n m ∂ λ + ( k 1 , m k 2 , m ) − 1 ( H 1 , m ∂ n m ∂ λ ± ∂ ϕ 2 , ( m − 1 ) ∂ λ ) {\displaystyle {\frac {\partial \phi _{2,m}}{\partial \lambda }}=H_{2,m}{\frac {\partial n_{m}}{\partial \lambda }}+(k_{1,m}k_{2,m})^{-1}{\bigg (}H_{1,m}{\frac {\partial n_{m}}{\partial \lambda }}\pm \ {\frac {\partial \phi _{2,(m-1)}}{\partial \lambda }}{\bigg )}}

which can also be written as

∇ λ ϕ 2 , m = H 2 , m ∇ λ n m + ( k 1 , m k 2 , m ) − 1 ( H 1 , m ∇ λ n m ± ∇ λ ϕ 2 , ( m − 1 ) ) {\displaystyle \nabla _{\lambda }\phi _{2,m}=H_{2,m}\nabla _{\lambda }n_{m}+(k_{1,m}k_{2,m})^{-1}{\bigg (}H_{1,m}\nabla _{\lambda }n_{m}\pm \nabla _{\lambda }\phi _{2,(m-1)}{\bigg )}}

using

∇ λ = ∂ ∂ λ {\displaystyle \nabla _{\lambda }={\frac {\partial }{\partial \lambda }}}

Also,

… excerpt ends here. Continue reading the full article.