Appendix D: Multiple-Prism Dispersion Series

D.1 Multiple-Prism Dispersion Series

In Chapter 5, the generalized multiple-prism dispersion equation, applicable to multiple-prism arrays of any geometry, configuration, or materials, is given as (Duarte, 2009)

${\nabla}_{\text{\lambda}}{\text{\varphi}}_{2,m}=\pm {\mathscr{H}}_{2,m}{\nabla}_{\text{\lambda}}{n}_{m}\pm \text{\hspace{0.17em}}{\left({k}_{1,m}{k}_{2,m}\right)}^{-1}\text{\hspace{0.17em}}\left({\mathscr{H}}_{1,m}{\nabla}_{\text{\lambda}}{n}_{m}(\pm ){\nabla}_{\text{\lambda}}{\text{\varphi}}_{2,(m-1)}\right)$ |
(D.1) |

For positive refraction only, this equation becomes (Duarte and Piper, 1982, 1983)

${\nabla}_{\text{\lambda}}{\text{\varphi}}_{2,m}={\mathscr{H}}_{2,m}{\nabla}_{\text{\lambda}}{n}_{m}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{\left({k}_{1,m}{k}_{2,m}\right)}^{-1}\text{\hspace{0.17em}}\left({\mathscr{H}}_{1,m}{\nabla}_{\text{\lambda}}{n}_{m}\pm {\nabla}_{\text{\lambda}}{\text{\varphi}}_{2,(m-1)}\right)$ |
(D.2) |

where the ± sign refers to either a positive (+) or compensating configuration (−).

In Chapter 5, Equation D.2 is expressed in a series format directly applicable to the geometry at hand. Duarte and Piper (1982) also provide further examples of simple special cases leading to explicit ...

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