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Abstract
CaF2, BaF2, and MgF2 are low-index, infrared-transparent materials that are extensively used in optical systems. Despite their technological importance, a systematic investigation into the temperature dependence of their optical properties is lacking. In this study, spectroscopic ellipsometry was used to obtain the refractive index of monocrystalline CaF2, BaF2, and MgF2 for wavelengths between 220 nm and 1700 nm, and for temperatures between 21 °C and 368 °C. The raw ellipsometric data was fit to a Sellmeier model with temperature-dependent oscillator terms to extract the real part of the refractive index of each material. The refractive index of CaF2 and BaF2 was observed to decrease linearly with increasing temperature, which can be largely attributed to a reduction in the mass density due to thermal expansion. In contrast, the refractive index of MgF2 was found to vary nonlinearly with temperature, which suggests competing effects from the material’s electronic polarizability. The temperature-dependent refractive index data reported here provide a finely-resolved mapping of the thermo-optic coefficient for these three materials, which could inform the development of optical devices operating at elevated or unsteady temperatures.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Due to their characteristically low refractive index and wide optical transparency spanning the ultraviolet to mid-infrared, the alkaline earth metal difluorides are critical optical components for applications like photolithography, thermal imaging and infrared spectroscopy [1,2]. Although the optical properties of these compounds are well known at room temperature (i.e., ∼25 °C) [3–5], the variation in these properties at elevated temperatures is poorly characterized. This temperature-dependence is central to the performance of optical devices that operate in unsteady or extreme thermal environments. Absorptive heating in windows for high-power laser radiation represents one widely appreciated issue, while an emerging application includes non-contact temperature sensors leveraging the thermo-optic effect [6,7]. In addition, there is growing interest in these compounds for developing selective thermal emitters based on nanophotonic structures, with potential applications including passive radiative cooling [8]. Recent works [9,10] have also identified these compounds as optimal materials for enhancing near-field radiative heat transfer using surface phonon polaritons, which has ramification for technologies like heat-assisted magnetic recording and thermophotovoltaic energy conversion for waste heat recovery.
In this report we characterize the temperature-dependent optical properties of monocrystalline CaF2, BaF2, and MgF2 for wavelengths between 220 nm and 1700 nm, and temperatures between 21 °C and 368 °C. These compounds were selected for this study because they are representative alkaline earth metal difluorides that are widely available and have broad technological relevance. CaF2 and BaF2 have a cubic crystal structure and are optically isotropic, while MgF2 has a tetragonal structure and is birefringent. How the crystal structure impacts the degree to which the optical properties vary with temperature is a major question that is addressed in this work. The measurements themselves were conducted by combining a heating stage with a variable angle spectroscopic ellipsometer (VASE). A similar approach has been employed to characterize the temperature-dependent optical properties of high index semiconductors and metals [11,12], but hardly for the fluoride compounds studied here.
Available datasets of the refractive index (n), thermo-optic coefficient (dn/dT), and wavelength dispersion (dn/dλ) of the fluoride compounds are sparse. Early reports [13,14] leveraged minimum-deviation refractometry to characterize dn/dT of CaF2, BaF2, and MgF2 in a wide temperature range between −180 °C and 200 °C. Yet this data is only available for a few wavelengths in the visible and infrared. Further, the accuracy of this data is questionable as there are major inconsistencies with data reported in subsequent studies, which include cryogenic measurements of CaF2 [15,16] and room-temperature measurements of MgF2 [17]. The few ellipsometric studies of these compounds at elevated temperatures are similarly limited. Firoz et al. [18] measured the refractive index of single crystal CaF2 up to temperatures of 1050 K, but only for a single wavelength (632.8 nm). Li et al. [19] employed a combined transmission-ellipsometry technique to characterize the refractive index and absorption index of polycrystalline CaF2, BaF2, and MgF2 for a few temperatures between 20 °C and 350 °C. The spectral range for these measurements was largely in the mid-infrared, and there was significant uncertainty in the measured index data.
The major utility of this study is that it maps the optical properties for each of these materials over a wide range of temperatures and wavelengths, and with high resolution (i.e., δT ≈ 20 °C, δλ = 5 nm). The overall result is a more detailed understanding of the thermo-optic effect than those based on sparse datasets.
2. Sample preparation
Single crystal substrates of <100> CaF2, < 100> BaF2 and <100> MgF2 were purchased from MTI Corporation (www.mtixtl.com). Both CaF2 and MgF2 substrates had a size of 10 mm × 10 mm × 0.5 mm. The BaF2 substrate was also 10 mm × 10 mm but had a thickness of 1 mm. The surface finish of each substrate came highly polished with an average roughness less than 25 Å. The backside of each substrate was then sandblasted with SiC particles (50 µm-size) in order to minimize measurement artifacts due to backside reflections. After sandblasting, the substrates were ultrasonically cleaned in acetone for 10 minutes, rinsed with deionized water, and dried using a nitrogen gun.
3. Experimental setup and modeling approach
To obtain the sample’s optical properties at elevated temperatures, the sample was mounted to a commercial heating stage (Linkham Scientific, Model HFSEL600) that was coated with a graphene carbon layer (SEMicro) to further reduce backside reflections. The Linkham stage, which has a temperature stability of 0.1 °C and a range up to 600 °C, was then mounted to the V-VASE ellipsometer (J. A. Woollam). The ellipsometric parameters (Ψ, Δ) of the sample were measured for wavelengths between 220 nm and 1700 nm, with a wavelength resolution of 5 nm. These measurements were repeated for set-point temperatures between 25 °C and 400 °C, in steps of 20 °C. The actual sample temperature was found to differ slightly from the stage’s set-point temperature, and this difference was quantified in a separate calibration experiment that is described in the Supplementary Information (Supplement 1, Section S1). For measurements at set-point temperatures above 25 °C, a cover with fused silica optical windows was connected to the heating stage so that the sample could be purged with nitrogen gas to prevent oxidation. With the cover in place, the angle-of-incidence of the VASE beam was fixed at 70°. At elevated temperatures, an iris was employed to reduce the size of the VASE beam impinging on the detector to avoid measurement artifacts due to thermal emission from the sample and stage (SI, Section S2), which previous ellipsometric studies have shown can be a significant source of error for temperatures above 500 K [11,12].
At each temperature, the raw data was processed using the commercial software WVASE32. Since the samples exhibit negligible absorption in the measured spectral region, the measured Δ values were very close to zero and were ignored. The Ψ data were fit to a Sellmeier model to extract the real part of the refractive index. For CaF2 and BaF2, we employed the three-oscillator Sellmeier formula (2 oscillators in the UV and 1 oscillator in the IR) given by Eq. (1):
Fig. 1. Illustration of the sample orientations for the measurement of MgF2. Optical axis is (a) perpendicular to, and (b) parallel to the plane of incidence.
4. Results and discussion
4.1 Calcium fluoride data
Key results for the CaF2 sample are presented in Fig. 2. For ease of viewing, the wavelength-dependence of Ψ and n are displayed at three selected temperatures (21, 186, and 368 °C) in Fig. 2(a) and 2(b), respectively. A step of ∼0.04° is observed in the raw Ψ data near a wavelength of 1000 nm for all Ψ data sets, which could potentially be attributed to a small misalignment between the collimator and detector within the ellipsometer. Yet errors in the refractive index due to this artifact are within the estimated measurement uncertainty (see Section 4.4 and SI, Section S3). The measured refractive index at 21 °C exhibits normal dispersion such that n drops with increasing wavelength, which is a general feature of transparent materials. Further, the n values are consistent with room-temperature data reported in the literature (see dashed line in Fig. 2(b)). At elevated temperatures, the refractive index at all wavelengths systematically drops since thermal expansion of the material leads to a reduced density. Figure 2(c) displays the variation of refractive index with temperature for three selected wavelengths (260, 530, and 1600 nm), which indicates that the n-T relationship is approximately linear over the entire temperature range. The thermo-optic coefficient (dn/dT) was obtained from the slope of the best-fit line to this data, and the results are plotted in Fig. 2(d) for each wavelength. The magnitude |dn/dT| increases by ∼45% from the ultraviolet to near infrared (dn/dT becomes more negative), and ultimately plateaus at a value of ∼1.57 × 10−5 K−1 at λ = 1700 nm. Literature values of dn/dT are comparable to our measured values for wavelengths near 632 nm [18].
Fig. 2. Measured data for CaF2. Wavelength-dependence of (a) the raw Ψ data, and (b) the refractive index n at three selected temperatures. The dashed curve in (b) represents literature values of n from Ref. [3]. (c) Variation of n with temperature for three selected wavelengths. (d) Variation of dn/dT with wavelength. Vertical error bars in (c) and (d) capture the combined uncertainty of the measurement and fitting approach (see Section 4.4). (e) Variation of the Sellmeier fitting parameters (S2 and λ2) with temperature.
Due to strong correlations between the four fitting parameters (Si and λi) for the 1st and 2nd Sellmeier oscillators, fitting the two parameters (S2 and λ2) for the 2nd Sellmeier oscillator was adequate to obtain the temperature-dependence of refractive index over the measured wavelength and temperature range. These fitting parameters are plotted as a function of temperature in Fig. 2(e). The value of S2 decreases with temperature but the value of λ2 exhibits a more complicated temperature dependence. Table S1(see SI, Section S4) provides a summary of all Sellmeier parameters used to model the refractive index of CaF2 in this temperature range. Finally, it should be noted that systematic temperature sweeps were conducted for CaF2 and no hysteresis in the measured data was observed.
4.2 Barium fluoride data
The key results for the BaF2 are presented in Fig. 3. The dispersion of Ψ and n are displayed at three selected temperatures in Fig. 3(a) and 3(b), respectively. Similar to the CaF2 sample, the refractive index of BaF2 exhibits normal dispersion. Further, the refractive index at all wavelengths is reduced at elevated temperatures. Figure 3(c) also reveals that the refractive index at each wavelength varies approximately linearly with temperature. The n-T curve for λ = 260 nm appears somewhat nonlinear compared to the curves at λ = 530 and 1600 nm, which can be attributed to larger measurement uncertainties in the ultraviolet. Details of this uncertainty analysis are provided in Section 4.4. The wavelength dependence of dn/dT is plotted in Fig. 3(d), and has a similar shape to the curve in Fig. 2(d) though the magnitude |dn/dT| is a few percent larger for BaF2. The magnitude |dn/dT| plateaus to a value of ∼1.66 × 10−5 K−1 at λ = 1700 nm, which is consistent with literature data [22]. The Sellmeier fitting parameters S2 and λ2 are displayed as a function of temperature in Fig. 3(e). While the oscillator strength S2 varies monotonically with temperature, the oscillator location λ2 appears to vary with temperature non-monotonically. Here, the values of S1 and λ1 are fixed. Table S2 (see SI, Section S4) summarizes the Sellmeier parameters for BaF2. Finally, differences in the refractive index between heating and cooling sweeps fall within the measurement uncertainty. As a result, it was concluded that hysteresis is not significant for BaF2 in the measured temperature range.
Fig. 3. Measured data for BaF2. Wavelength-dependence of (a) the raw Ψ data, and (b) the refractive index n at three selected temperatures. The dashed curve in (b) represents literature values of n from Ref. [5]. (c) Variation of n with temperature for three selected wavelengths. (d) Variation of dn/dT with wavelength. Vertical error bars in (c) and (d) capture the combined uncertainty of the measurement and fitting approach (see Section 4.4). (e) Variation of the Sellmeier fitting parameters (S2 and λ2) with temperature.
4.3 Magnesium fluoride data
Key results for the MgF2 are presented in Fig. 4 and Fig. 5. The dispersion of Ψ for two sample orientations are displayed at three selected temperatures (21, 186, and 368 °C) in Fig. 4(a) and 4(b). Specifically, the c-axis is parallel to (Fig. 4(a)) and perpendicular to (Fig. 4(b)) the plane of incidence of the VASE beam. Fitting these two data sets simultaneously, the dispersion of both ne and no was obtained. As shown in Fig. 4(c), no exhibits normal dispersion over the measured wavelength range. The temperature-dependence of no at elevated temperatures is shown in Fig. 4(d) to be nonlinear for the three selected wavelengths. As a result, the thermo-optic coefficients (dno/dT) were obtained from these temperature sweeps using a 3rd-order polynomial fit (see dashed curves in Fig. 4(d)). A 2D color map of dno/dT as a function of wavelength and temperature is provided in Fig. 4(e). The thermo-optic coefficient exhibits little wavelength-dependence, and has a room-temperature value of ∼8 × 10−6 K−1 that is consistent with literature data [17]. This plot also reveals that the thermo-optic coefficient has a more complicated temperature-dependence than that of CaF2 and BaF2. Specifically, the value of dno/dT changes sign at two temperatures: 50 °C and 315 °C. Between these temperatures, dno/dT becomes negative with a local minimum near 180 °C. It should be noted that this trend is qualitatively different from older data sets that are plagued by significant measurement uncertainties, and that only provide measurements up to 200 °C. Finally, the temperature-dependence of the Sellmeier fitting parameters (So1 andSo2) for no are plotted in Fig. 4(f) and tabulated in Table S3 (see SI, Section S4).
Fig. 4. Measured data and fitting results for MgF2. Wavelength-dependence of the raw Ψ data at three selected temperatures for two sample orientations: c-axis (a) parallel to, and (b) perpendicular to the plane of incidence of the VASE beam. (c) Variation of no with wavelength for three selected temperatures. The dashed line corresponds to literature values from Ref. [4]. (d) Variation of no with temperature for three selected wavelengths. Vertical error bars capture the combined uncertainty of the measurement and fitting approach (see Section 4.4). (e) Colormap of dno/dT plotted over all wavelengths and temperatures in the measured range. Gray dashed lines mark where dno/dT = 0. (f) Variation of the Sellmeier fitting parameters (So1 and So2) with temperature.
Fig. 5. Fitting results for MgF2. (a) Wavelength-dependence of the refractive index ne at three selected temperatures. The dashed line corresponds to literature values from Ref. [4]. (b) Variation of ne with temperature for three selected wavelengths. Vertical error bars capture the combined uncertainty of the measurement and fitting approach (see Section 4.4). (c) Colormap of dne/dT plotted over all wavelengths and temperatures in the measured range. Gray dashed lines mark where dne/dT = 0. (d) Variation of the Sellmeier fitting parameters (Se1 and Se2) with temperature. (e) Wavelength-dependence of the birefringence (Δn = ne – no) at three selected temperatures. (f) Temperature-dependence of Δn for three selected wavelengths.
It is also observed that ne exhibits normal dispersion (Fig. 5(a)), and varies nonlinearly with temperature (Fig. 5(b)) over the measured range. The 2D color map in Fig. 5(c) captures the variation of dne/dT with wavelength and temperature. dne/dT exhibits a qualitatively similar temperature-dependence as dno/dT. The measured dne/dT values are negative between 35 °C and 330 °C with a local minimum near 180 °C. Figure 5(d) plots the temperature-dependence of the Sellmeier fitting parameters (Se1 and Se2) for ne. The oscillator strength Se2 appears to vary with temperature based on a third-order polynomial fit in the measurement range, while Se1 remains nearly unchanged. Table S4 (see SI, Section S4) summarizes the Sellmeier parameters for ne. Finally, Fig. 5(e) and 5(f) plot the birefringence of MgF2, defined as Δn = ne - no, as a function of wavelength and temperature, respectively. Since the magnitude of dno /dT is overall larger than that of dne /dT, the temperature dependence of Δn is primarily determined by the temperature dependence of no at each wavelength. These plots illustrate that Δn of MgF2 is maximized at a temperature near 278 °C for each wavelength. It should also be noted that no hysteresis was observed in the systematic temperature sweeps for MgF2.
4.4 Uncertainty analysis
Systematic errors and random errors were considered to estimate the uncertainty in the measured refractive index of each sample. The uncertainty associated with the angle of incidence and wavelength shift of the VASE beam are the main sources of systematic errors. Random errors include fluctuations of the detector signal and beam intensity, as well as mechanical drift of the optical elements.
For the V-VASE ellipsometer (J. A. Woollam), the accuracy of the angle of incidence and wavelength resolution of the VASE system are reported as 0.01° and 2 nm, respectively. Therefore, systematic errors were estimated by introducing a global offset of 0.01° for the angle of incidence and introducing a wavelength shift of 2 nm into the model. The combined uncertainty can be expressed according to Eq. (2):
where σn,sys represents the total systematic error, σn,angle represents the uncertainty due to angle of incidence errors, and σn,wvl represents the uncertainty due to bandwidth errors of the spectrometer.Random errors were estimated using the Monte Carlo Fit-Result Analysis feature in the WVASE32 software. The raw Ψ data was added with random measurement noise and then fit to the Sellmeier equation. This procedure was repeated 100 times to generate a statistical population of refractive index data. The random errors were defined from a 90% confidence interval such that σn,rand = 1.65 × stdev, where stdev represents the standard deviation of refractive index statistics.
Finally, the total uncertainty of refractive index at each wavelength and temperature, σn,tot, was calculated by combining systematic errors and random errors according to Eq. (3):
The uncertainty of the thermo-optic coefficient (dn/dT) was also obtained using a Monte-Carlo method. Specifically, at each wavelength, random noise based on the total uncertainty σn,tot, was added to the refractive index at each temperature, and the corresponding dn/dT was obtained from the slope of the best-fit line to the updated data. The process was repeated 100 times to obtain a statistical population of dn/dT data, and the uncertainty of dn/dT was defined from a 90% confidence interval such that σthermo-optic = 1.65 × stdev, where stdev represents standard deviation of the dn/dT statistics.
4.5 Discussion
Based on the Clausius-Mossotti equation, the thermo-optic coefficient of cubic materials like CaF2 and BaF2 can be described in terms of the volumetric thermal expansion coefficient β and polarization coefficient φ according to Eq. (4) [18]:
The polarization coefficient, which arises due to anharmonic effects, is defined by φ = (dα/dT)(1/α), where α is the electronic polarizability. Since dn/dT for CaF2 and BaF2 is always negative, it is clear from Eq. (4) that the thermal expansion coefficient, which is generally a positive quantity, has a much higher value than the polarization coefficient in the measurement range.
The expression from Eq. (4) can also be generalized to anisotropic crystals. Specifically, the thermo-optic coefficient for tetragonal crystals like MgF2 can be described in terms of the pure temperature effect (i.e., the φ-term) and thermal expansion effects (i.e., the β-term, and the γ-term which vanishes for isotropic crystals) according to Eq. (5) [23,24]:
5. Conclusions
Key differences in the temperature-dependent optical properties were observed for the three single-crystal, transparent materials studied here. The refractive indices of CaF2 and BaF2 decrease linearly with increasing temperature from 21 °C to 368 °C, with thermo-optic coefficients that increase in magnitude at higher wavelengths. Because of this linear temperature-dependence and negligible hysteresis, CaF2 and BaF2 are potential candidates for thermoreflectance-based temperature sensing applications. For MgF2, both ordinary and extraordinary refractive indices exhibit a complex relationship with temperature, which can be modeled using a 3rd-order polynomial in the measurement range. The thermo-optic coefficients of MgF2 were found to be significantly smaller than that of CaF2 and BaF2, especially for the extraordinary axis. In fact, dne /dT is close to zero at temperatures near 35 °C and 330 °C. Similarly, dno/dT is close to zero at temperatures near 50 °C and 315 °C. As a result, MgF2 is a candidate material for optical devices if stability of the refractive index is desired.
Funding
National Science Foundation (2044788); Office of Naval Research (4720008832).
Acknowledgements
The authors gratefully acknowledge Mikhail Kats at UW-Madison and Tom Tiwald at J.A. Woollam Company for helpful discussions about fitting the ellipsometric data. The authors gratefully acknowledge use of facilities and instrumentation supported by NSF through the University of Wisconsin Materials Research Science and Engineering Center (DMR-1720415).
Disclosures
The authors declare no conflicts of interest.
Data availability
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.
Supplemental document
See Supplement 1 for supporting content.
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