Quantum Optics for Engineers

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)

${\nabla }_{\text{λ}}{\text{ϕ}}_{2,m}=\pm {\mathscr{H}}_{2,m}{\nabla }_{\text{λ}}{n}_m\pm {\left(k_{1,m}{k}_{2,m}\right)}^{-1}\left({\mathscr{H}}_{1,m}{\nabla }_{\text{λ}}{n}_m(\pm ){\nabla }_{\text{λ}}{\text{ϕ}}_{2,(m-1)}\right)$

(D.1)

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

${\nabla }_{\text{λ}}{\text{ϕ}}_{2,m}={\mathscr{H}}_{2,m}{\nabla }_{\text{λ}}{n}_m+{\left(k_{1,m}{k}_{2,m}\right)}^{-1}\left({\mathscr{H}}_{1,m}{\nabla }_{\text{λ}}{n}_m\pm {\nabla }_{\text{λ}}{\text{ϕ}}_{2,(m-1)}\right)$

(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 ...