Preventing Overfitting in Infrared Ellipsometry using Temperature Dependence: Fused Silica as a Case Study

Materials with vibrational resonances such as fused silica, quartz, and silicon carbide can be used for mid-infrared absorbers and thermal emitters [1], [2]. Here, we investigated fused-silica glass; while fused silica is perhaps the most-studied optical material in the visible and near-infrared, its mid-infrared properties are less known, with almost no data available above room temperature [3]. Apart from the use of this data in high-temperature applications of fused silica, temperature-dependent ellipsometry can provide a new source of information with which to evaluate oscillator models, in addition to the conventionally used goodness of fit.<br/><br/>We conducted temperature-dependent spectroscopic ellipsometry measurements on three different grades of silica from Corning: HPFS 7980 standard grade (-OH content 800–1000 ppm, metallic impurities &lt;1000 ppb), 7979 IR grade (-OH content &lt;1 ppm, metallic impurities &lt;100 ppb), and 8655 ArF grade (-OH content &lt;1 ppm, metallic impurities &lt;10 ppb). For each sample, we performed temperature-dependent variable-angle ellipsometry measurements at wavelengths of 5–25 µm.<br/><br/>Subsequently, we performed fits on the experimental ellipsometric data, ψ and Δ. We observed 3 main features centered around the wavelengths of 9 mm, 12.5 mm, and 22 mm, known to be due to vibrational resonances [3]. A previous work that looked at reflectance data (rather than ellipsometry) at room temperature used 8 distinct Gaussian oscillators to fit the experimental data [3]. Similarly, we were able to achieve excellent fits at each temperature (7 temperatures total for each sample) using 7 Gaussian oscillators, observing that Gaussian oscillators fitted the data better than Lorentz oscillators, likely due to inhomogeneous broadening in the amorphous material. However, when we plotted the fitting parameters as a function of temperature, we observed nonphysical nonmonotonic temperature dependence of the oscillator positions, widths, and amplitudes, implying that our model could be improved.<br/><br/>After simplifying the model to 6 oscillators, we again fitted the experimental data independently at each temperature for each sample, observing a slight (but not very significant) reduction in the goodness of fit (the normalized mean-squared error (MSE) for Ψ and Δ as defined in ref. [4] was 1.3 with 7 oscillators and 1.6 with 6 oscillators, both indicating excellent fits). In our new fit with fewer oscillators, the oscillator parameters that contributed most to the optical properties changed monotonically with temperature, confirming the validity of our fits. By comparing the temperature-dependence trends among the oscillator parameters, we can effectively mitigate overfitting of the experimental ellipsometric data.<br/>In addition to our demonstration of the use of temperature-dependent measurements to prevent overfitting of materials parameters, we generated highly precise and accurate datasets for the temperature-dependent mid-infrared optical properties of various grades of fused silica, which can be used for modeling infrared absorbers, thermal emitters, and other photonic structures. These datasets will be made publicly available.<br/><br/>[1] J. D. Caldwell <i>et al.</i>, <i>Nanophotonics</i>, vol. 4, no. 1, pp. 44–68, 2015.<br/>[2] J. long Kou et al., ACS<i> Photonics</i>, vol. 4, no. 3, pp. 626–630, 2017.<br/>[3] R. Kitamura et al., <i>Applied Optics</i>, Vol. 46, Issue 33, pp. 8118-8133, 2007.<br/>[4] Hiroyuki. Fujiwara, <i>Spectroscopic ellipsometry: principles and applications</i>. John Wiley & Sons, 2007.