Scattering under the Rayleigh–Debye–Gans condition for transparent glass ceramics

Max Whiteman; Stuart D. Jackson

doi:10.1364/AO.493018

Applied Optics
Vol. 62,
Issue 16,
pp. 4221-4227
(2023)
•https://doi.org/10.1364/AO.493018

Scattering under the Rayleigh–Debye–Gans condition for transparent glass ceramics

Max Whiteman and Stuart D. Jackson

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Max Whiteman and Stuart D. Jackson, "Scattering under the Rayleigh–Debye–Gans condition for transparent glass ceramics," Appl. Opt. 62, 4221-4227 (2023)

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Abstract

We reviewed the various theoretical relations that quantify light scattering under the Rayleigh–Debye–Gans (RDG) approximation and applied these relations to calculate scattering within transparent glass ceramics (TGCs) composed of large nanocrystals within a glass matrix. For a more realistic picture of scattering, we included material dispersion of the crystals and glasses across the transparency range of these materials by way of the Sellmeier equation. We first selected a number of crystal–glass sets that are near-index-matched in the visible and near-IR to fulfill one of the RDG criterion. We found that the various forms of scattering under the RDG approximation differ significantly across the visible and near-IR. We also found that the inclusion of material dispersion significantly changes the trends in the calculated scattering cross section across the studied wavelength range. Overall, we found that calculation of the scattering cross section is highly dependent on the chosen theoretical relation and that the inclusion of material dispersion is vital to better understand scattering loss in this new class of optical materials.

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Source	Scattering Cross Section, $C_{s c a}, u n i t s : m^{2}$	$x$ Range
Rayleigh [13]	$\frac{2 π^{5}}{3} \frac{d^{6}}{λ^{4}} \| \frac{m^{2} - 1}{m^{2} + 2} \|^{2}$	$≪ 0.2$
Apetz and Van Bruggen [8]	$\frac{8 π^{3} r^{4}}{λ_{m}^{2}} (\frac{Δ n}{n_{m e d i u m}})^{2}$ , $Δ n = n_{m e d i u m} - n_{s c a t t e r}$	0.5–2
RDG with limits [14]	Small, $\frac{32}{27} (m - 1)^{2} x^{4} G$ , Large, $2 (m - 1)^{2} x^{2} G$	0.5–1
Garcia-Lopez, et al. [10]	$\frac{9 V^{2}}{16 π^{2} λ^{4}} \| \frac{m^{2} - 1}{m^{2} + 2} \|^{2} \int_{0}^{π} P (θ) (1 + \cos^{2} θ) \sin θ d θ$	0.5–2
Full RDG [7]	$G (m - 1)^{2} (\frac{5}{2} + 2 x^{2} - \frac{\sin 4 x}{4 x} - \frac{7}{16 x^{2}} (1 - \cos 4 x)$	0.5–2
	$+ (\frac{1}{2 x^{2}} - 2) (γ + \log 4 x - C i (4 x)))$
Van de Hulst [7]	$G (2 - \frac{4}{ρ} \sin ρ + \frac{4}{ρ^{2}} (1 - \cos ρ))$	$> 0.5$
	$ρ = 2 x (m - 1)$

Material	Sellmeier Equation
Aluminosilicate [6]	$n^{2} = 1.414 + \frac{0.9504 λ^{2}}{λ^{2} - 0.0132} + \frac{0.9044 λ^{2}}{λ^{2} - 100}$
$B a F_{2}$ [6]	$n^{2} - 1 = \frac{0.6443 λ^{2}}{λ^{2} - 0.0033} + \frac{0.5068 λ^{2}}{λ^{2} - 0.012} + \frac{3.8261 λ^{2}}{λ^{2} - 2151.698}$
ZBLAN [6]	$n^{2} - 1 = \frac{1.2251 λ^{2}}{λ^{2} - {0.0897}^{2}} + \frac{1.529 λ^{2}}{λ^{2} - {21.3825}^{2}}$
GTZN [17]	$n^{2} = 1.679 + \frac{1.292}{λ^{2} - 0.0305} + \frac{0.5}{λ^{2} - 149.5} + \frac{1}{λ^{2} - 1}$
Sapphire [18]	$n^{2} - 1 = \frac{1.431 λ^{2}}{λ^{2} - 0.0053} + \frac{0.6505 λ^{2}}{λ^{2} - 0.0053} + \frac{0.6505 λ^{2}}{λ^{2} - 0.0142} + \frac{5.3414 λ^{2}}{λ^{2} - 325.02}$
$30 % {C a F}_{2} : 70 % P G$ [19]	$n^{2} = 1.4236 + \frac{0.0064}{λ^{2}} - \frac{0.001}{λ^{4}}$
$E u : C a F_{2}$ [19]	$n^{2} = 1.4231 + \frac{0.0113}{λ^{2}} - \frac{0.0024}{λ^{4}}$
LaSF 40 [20]	$n^{2} - 1 = \frac{1.9855 λ^{2}}{λ^{2} - 0.011} + \frac{0.2741 λ^{2}}{λ^{2} - 0.0475} + \frac{1.289 λ^{2}}{λ^{2} - 96.91}$
YAG [21]	$n^{2} - 1 = \frac{2.282 λ^{2}}{λ^{2} - 0.0119} + \frac{3.276 λ^{2}}{λ^{2} - 282.734}$

Source	Scattering Cross Section, $C_{s c a}, u n i t s : m^{2}$	$x$ Range
Rayleigh [13]	$\frac{2 π^{5}}{3} \frac{d^{6}}{λ^{4}} \| \frac{m^{2} - 1}{m^{2} + 2} \|^{2}$	$≪ 0.2$
Apetz and Van Bruggen [8]	$\frac{8 π^{3} r^{4}}{λ_{m}^{2}} (\frac{Δ n}{n_{m e d i u m}})^{2}$ , $Δ n = n_{m e d i u m} - n_{s c a t t e r}$	0.5–2
RDG with limits [14]	Small, $\frac{32}{27} (m - 1)^{2} x^{4} G$ , Large, $2 (m - 1)^{2} x^{2} G$	0.5–1
Garcia-Lopez, et al. [10]	$\frac{9 V^{2}}{16 π^{2} λ^{4}} \| \frac{m^{2} - 1}{m^{2} + 2} \|^{2} \int_{0}^{π} P (θ) (1 + \cos^{2} θ) \sin θ d θ$	0.5–2
Full RDG [7]	$G (m - 1)^{2} (\frac{5}{2} + 2 x^{2} - \frac{\sin 4 x}{4 x} - \frac{7}{16 x^{2}} (1 - \cos 4 x)$	0.5–2
	$+ (\frac{1}{2 x^{2}} - 2) (γ + \log 4 x - C i (4 x)))$
Van de Hulst [7]	$G (2 - \frac{4}{ρ} \sin ρ + \frac{4}{ρ^{2}} (1 - \cos ρ))$	$> 0.5$
	$ρ = 2 x (m - 1)$

Material	Sellmeier Equation
Aluminosilicate [6]	$n^{2} = 1.414 + \frac{0.9504 λ^{2}}{λ^{2} - 0.0132} + \frac{0.9044 λ^{2}}{λ^{2} - 100}$
$B a F_{2}$ [6]	$n^{2} - 1 = \frac{0.6443 λ^{2}}{λ^{2} - 0.0033} + \frac{0.5068 λ^{2}}{λ^{2} - 0.012} + \frac{3.8261 λ^{2}}{λ^{2} - 2151.698}$
ZBLAN [6]	$n^{2} - 1 = \frac{1.2251 λ^{2}}{λ^{2} - {0.0897}^{2}} + \frac{1.529 λ^{2}}{λ^{2} - {21.3825}^{2}}$
GTZN [17]	$n^{2} = 1.679 + \frac{1.292}{λ^{2} - 0.0305} + \frac{0.5}{λ^{2} - 149.5} + \frac{1}{λ^{2} - 1}$
Sapphire [18]	$n^{2} - 1 = \frac{1.431 λ^{2}}{λ^{2} - 0.0053} + \frac{0.6505 λ^{2}}{λ^{2} - 0.0053} + \frac{0.6505 λ^{2}}{λ^{2} - 0.0142} + \frac{5.3414 λ^{2}}{λ^{2} - 325.02}$
$30 % {C a F}_{2} : 70 % P G$ [19]	$n^{2} = 1.4236 + \frac{0.0064}{λ^{2}} - \frac{0.001}{λ^{4}}$
$E u : C a F_{2}$ [19]	$n^{2} = 1.4231 + \frac{0.0113}{λ^{2}} - \frac{0.0024}{λ^{4}}$
LaSF 40 [20]	$n^{2} - 1 = \frac{1.9855 λ^{2}}{λ^{2} - 0.011} + \frac{0.2741 λ^{2}}{λ^{2} - 0.0475} + \frac{1.289 λ^{2}}{λ^{2} - 96.91}$
YAG [21]	$n^{2} - 1 = \frac{2.282 λ^{2}}{λ^{2} - 0.0119} + \frac{3.276 λ^{2}}{λ^{2} - 282.734}$

Abstract

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Equations (7)

Applied Optics