Appendix D: Multiple-Prism Dispersion Series

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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)

$\nabla_{λ} ϕ_{2, m} = \pm ℋ_{2, m} \nabla_{λ} n_{m} \pm {(k_{1, m} k_{2, m})}^{- 1} (ℋ_{1, m} \nabla_{λ} n_{m} (\pm) \nabla_{λ} ϕ_{2, (m - 1)})$ $\nabla_{λ} ϕ_{2, m} = \pm ℋ_{2, m} \nabla_{λ} n_{m} \pm {(k_{1, m} k_{2, m})}^{- 1} (ℋ_{1, m} \nabla_{λ} n_{m} (\pm) \nabla_{λ} ϕ_{2, (m - 1)})$

(D.1)

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

$\nabla_{λ} ϕ_{2, m} = ℋ_{2, m} \nabla_{λ} n_{m} + {(k_{1, m} k_{2, m})}^{- 1} (ℋ_{1, m} \nabla_{λ} n_{m} \pm \nabla_{λ} ϕ_{2, (m - 1)})$ $\nabla_{λ} ϕ_{2, m} = ℋ_{2, m} \nabla_{λ} n_{m} + {(k_{1, m} k_{2, m})}^{- 1} (ℋ_{1, m} \nabla_{λ} n_{m} \pm \nabla_{λ} ϕ_{2, (m - 1)})$

(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 engineering-type equations. For instance, for increasing values of m, for the very special case of r identical prisms deployed at the same angle of incidence (i.e., ϕ_1,1 = ϕ_1,2 = … = ϕ_1,m and ψ_1,1 = ψ_1,2 = … = ψ_1,m) and orthogonal beam exit (i.e., ϕ_2,1 = ϕ_2,2 = … = ϕ_2,m = 0 and ψ_2,1 = ψ_2,2 = … = ψ_2,m = 0), Equation D.2 reduces to a simple power series (Duarte and Piper, 1982; Duarte, 1990):

$\nabla_{λ} ϕ_{2, r} = tan ψ_{1, 1} \nabla n_{1} (1 \pm k_{1, 1}^{- 1} \pm k_{1, 1}^{- 2} \pm k_{1, 1}^{- 3} \pm \dots \pm k_{1, 1}^{- (r - 1)})$ $\nabla_{λ} ϕ_{2, r} = tan ψ_{1, 1} \nabla n_{1} (1 \pm k_{1, 1}^{- 1} \pm k_{1, 1}^{- 2} \pm k_{1, 1}^{- 3} \pm \dots \pm k_{1, 1}^{- (r - 1)})$

(D.3)

Also, as shown in Chapter 5, for orthogonal beam exit, Equation D.2 reduces to the explicit series:

$\nabla_{λ} ϕ_{2, r} = \sum_{m = 1}^{r} (\pm 1) ℋ_{1, m} {(\prod_{j = m}^{r} k_{1, j})}^{- 1} \nabla_{λ} n_{m}$ $\nabla_{λ} ϕ_{2, r} = \sum_{m = 1}^{r} (\pm 1) ℋ_{1, m} {(\prod_{j = m}^{r} k_{1, j})}^{- 1} \nabla_{λ} n_{m}$

(D.4)

which was disclosed in print, in its double-pass version by Duarte (1985). This simple explicit equation obviously can be expressed in its long-hand version (Duarte, 2012):

$\nabla_{λ} ϕ_{2, r} = \pm ℋ_{1, 1} {(k_{1, 1} k_{1, 2} \dots k_{1, r})}^{- 1} \nabla_{λ} n_{1} \pm ℋ_{1, 2} {(k_{1, 2} \dots k_{1, r})}^{- 1} \nabla_{λ} n_{2} \pm \dots \pm ℋ_{1, r} {(k_{1, r})}^{- 1} \nabla_{λ} n_{r}$ $\nabla_{λ} ϕ_{2, r} = \pm ℋ_{1, 1} {(k_{1, 1} k_{1, 2} \dots k_{1, r})}^{- 1} \nabla_{λ} n_{1} \pm ℋ_{1, 2} {(k_{1, 2} \dots k_{1, r})}^{- 1} \nabla_{λ} n_{2} \pm \dots \pm ℋ_{1, r} {(k_{1, r})}^{- 1} \nabla_{λ} n_{r}$

(D.5)

These examples are included here to illustrate that the generalized dispersion Equation D.1 leads directly to easy-to-use explicit results that might appear as “new” to some.

References

Duarte, F. J. (1985). Note on achromatic multiple-prism beam expanders. Opt. Commun. 53, 259–262.

Duarte, F. J. (1990). Narrow linewidth pulsed dye laser oscillators. In Dye Laser Principles (Duarte, F. J. and Hillman, L. W., eds.). Academic Press, New York, Chapter 4.

Duarte, F. J. (2009). Generalized multiple-prism dispersion theory for laser pulse compression: higher order phase derivatives. Appl. Phys. B 96, 809–814.

Duarte, F. J. (2012). Tunable organic dye lasers: Physics and technology of high-performance liquid and solid-state narrow-linewidth oscillators. Prog. Quant. Electron. 36, 29–50.

Duarte, F. J. and Piper, J. A. (1982). Dispersion theory of multiple-prism beam expander for pulsed dye lasers. Opt. Commun. 43, 303–307.

Duarte, F. J. and Piper, J. A. (1983). Generalized prism dispersion theory. Am. J. Phys. 51, 1132–1134.

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Table of Contents for Appendix D: Multiple-Prism Dispersion Series

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Table of Contents for
Appendix D: Multiple-Prism Dispersion Series