Temperature-dependent index of refraction Sellmeier model for crystalline and polycrystalline materials

M. E. Thomas; M. E. Thomas

doi:10.1364/AO.511188

Applied Optics
Vol. 63,
Issue 10,
pp. 2477-2486
(2024)
•https://doi.org/10.1364/AO.511188

Temperature-dependent index of refraction Sellmeier model for crystalline and polycrystalline materials

M. E. Thomas

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M. E. Thomas, "Temperature-dependent index of refraction Sellmeier model for crystalline and polycrystalline materials," Appl. Opt. 63, 2477-2486 (2024)

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Abstract

The temperature dependence of optical window materials remains an important issue for a variety of applications from spacecraft, laser components, to high-speed aircraft. Concerning the refractive index in regions of transparency, current models are empirically based polynomial fits for the Sellmeier model strength and mode location parameters. These polynomial fit functions limit the ability to accurately extrapolate beyond the experimental range used to develop the fit functions. Thus, the development of a physics-based model as a function of temperature is an important goal for these critical materials. Such a model will allow extrapolation to higher and lower temperatures as long as the physical mechanisms do not change. For vibrational modes, a thermal average of the anharmonically shifted energy levels is investigated and compared to experimental data. The first anharmonic term can be estimated using the Morse potential based on a multiphonon absorption model. Experimentally, these modes redshift, and this is consistent with the developed temperature-dependent index of refraction Sellmeier model. This redshifting phenomena can also be applied to electronic transition shifts. In addition, the temperature-dependent oscillator number density can be obtained from known expansion coefficient models and experimental data. Other model parameters, in particular the electronic and vibrational mode polarizability, still need experimental grounding for a given material. The method is incorporated into a modified Sellmeier model format.

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Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Material	$θ_{D}$ [K]	$α_{e x} (T_{0})$ [1/K]	${C T E}_{1}$ [K]	$ν_{1}$ [ ${c m}^{- 1}$ ]	${C T E}_{2}$	$ν_{2}$ [ ${c m}^{- 1}$ ]
Sapphire c-axis	1030	5.06E-06	40.0	3130.0	300.0	4730.0
Sapphire a-axis	1030	5.50E-06	0.0	3130.0	100.0	4730.0
${B a F}_{2}$	373	1.79E-05	65.0	1599.0	60.0	2294.0
ZnS	540	6.80E-06	90.0	2363.0	0.0
LiF	735	3.44E-05	0.0		0.0
ZnSe	440	7.10E-06	20.0	2300.0	0.0

Material	$D_{0} / h c$ [ ${c m}^{- 1}$ ]	$n_{v i b} (T = 296 K)$ [ ${c m}^{- 1}$ ]	$A_{e l}$ [K]	$B_{e l}$ [K]	$ν_{e l} (T = 0 K)$ [ ${c m}^{- 1}$ ]	$a$ [1/(K cm)]	$b$ [K]
Sapphire c-axis	9800	498.2	180.0	40.0
Sapphire a-axis	14000	568.1	440.0	70.0	84320.0	8.60	900
GaAs $E_{0}$	9750	268.7	450.0	40.0	12980.0	8.60	572
GaAs $E_{1}$	9750	268.7	510.0	50.0	25240.0	8.60	572
${B a F}_{2}$	4300	188.0	210.0	20.0	91820.0	8.60	200
ZnS	12000	304.4	290.0	60.0	30660.0	8.60	380
LiF	6500	354.0	250.0	10.0	30770.0	8.60	382
ZnSe	8000	205.0	220.0	20.0	26750.0	8.60	330

Material	$N$	$Δ α_{e l 1} (T_{0})$ [ $F {c m}^{2}$ ]	$Δ α_{e l 2} (T_{0})$ [ $F {c m}^{2}$ ]	$α_{1 e l 2}$ [ $K^{- 1}$ ]	$α_{2 e l 2}$ [ $K^{- 2}$ ]	$α_{3 e l 2}$ [ $K^{- 3}$ ]	$α_{4 e l 2}$ [ $K^{- 4}$ ]	$Δ α_{v i b 3} (T_{0})$ [ $F {c m}^{2}$ ]	$α_{1 v i b 3}$ [ $K^{- 1}$ ]	$α_{2 v i b 3}$ [ $K^{- 2}$ ]	$α_{3 v i b 3}$ [ $K^{- 3}$ ]	$α_{4 v i b 3}$ [ $K^{- 4}$ ]
${B a F}_{2}$	$1.678 \times 10^{22}$	$1.016 \times 10^{- 23}$	$2.222 \times 10^{- 24}$	$- 4.66 \times 10^{- 5}$	$- 2.1 \times 10^{- 8}$	$8.8 \times 10^{- 11}$	$- 4.3 \times 10^{- 13}$	$9.381 \times 10^{- 29}$	$4.0 \times 10^{- 5}$	$1.5 \times 10^{- 8}$	$- 1.0 \times 10^{- 11}$	$2.0 \times 10^{- 13}$
LiF	$6.169 \times 10^{22}$	0	$2.5 \times 10^{- 24}$	$- 1.28 \times 10^{- 6}$	$3.25 \times 10^{- 8}$	$- 2.0 \times 10^{- 11}$	$6.0 \times 10^{- 13}$	$9.225 \times 10^{- 29}$	$2.0 \times 10^{- 5}$	$6.0 \times 10^{- 8}$	$5.1 \times 10^{- 12}$	$- 2.0 \times 10^{- 12}$
ZnSe	$2.196 \times 10^{22}$	$4.697 \times 10^{- 24}$	$1.764 \times 10^{- 25}$	$4.7 \times 10^{- 4}$	$5.0 \times 10^{- 8}$	$- 1.6 \times 10^{- 10}$	$1.95 \times 10^{- 12}$	$5.286 \times 10^{- 29}$	$- 3.0 \times 10^{- 5}$	$2.0 \times 10^{- 7}$	0	0

Material	$θ_{D}$ [K]	$α_{e x} (T_{0})$ [1/K]	${C T E}_{1}$ [K]	$ν_{1}$ [ ${c m}^{- 1}$ ]	${C T E}_{2}$	$ν_{2}$ [ ${c m}^{- 1}$ ]
Sapphire c-axis	1030	5.06E-06	40.0	3130.0	300.0	4730.0
Sapphire a-axis	1030	5.50E-06	0.0	3130.0	100.0	4730.0
${B a F}_{2}$	373	1.79E-05	65.0	1599.0	60.0	2294.0
ZnS	540	6.80E-06	90.0	2363.0	0.0
LiF	735	3.44E-05	0.0		0.0
ZnSe	440	7.10E-06	20.0	2300.0	0.0

Material	$D_{0} / h c$ [ ${c m}^{- 1}$ ]	$n_{v i b} (T = 296 K)$ [ ${c m}^{- 1}$ ]	$A_{e l}$ [K]	$B_{e l}$ [K]	$ν_{e l} (T = 0 K)$ [ ${c m}^{- 1}$ ]	$a$ [1/(K cm)]	$b$ [K]
Sapphire c-axis	9800	498.2	180.0	40.0
Sapphire a-axis	14000	568.1	440.0	70.0	84320.0	8.60	900
GaAs $E_{0}$	9750	268.7	450.0	40.0	12980.0	8.60	572
GaAs $E_{1}$	9750	268.7	510.0	50.0	25240.0	8.60	572
${B a F}_{2}$	4300	188.0	210.0	20.0	91820.0	8.60	200
ZnS	12000	304.4	290.0	60.0	30660.0	8.60	380
LiF	6500	354.0	250.0	10.0	30770.0	8.60	382
ZnSe	8000	205.0	220.0	20.0	26750.0	8.60	330

Abstract

Data availability

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Figures (12)

Tables (3)

Equations (22)

Applied Optics