# Refractive Indices of Ge and Si at Temperatures between 4-296 K in the 4-8 THz Region
## 2 Materials and Methods  
Commercial Si and Ge samples were purchased with thicknesses appropriate for the technique: 1.060 mm for Si and 0.575 mm for Ge. The Si sample has been reported on previously using the same technique, measured at room temperature [14]. Transmission through the samples was measured using a Bruker IFS125 HR FTS at a nominal resolution of 0.3 GHz (0.01 cm\({}^{-1}\)) in the range of 2-10 THz (67-334 cm\({}^{-1}\)) with the samples mounted in an Oxford Instruments continuous flow helium cryostat. The FTS instrument uses an \(f/8\) focused beam with the sample at the focus, which is the image of the source aperture. As such, the wavefront at the sample plane is approximately planar. The effect of off-axis beams in the sample is small, as confirmed by the constant amplitude of the oscillation over many interference orders (see Figure 1a). The radiation source in the FTS is a SiC (silicon carbide) glow bar, and the detector is a He-cooled composite Si bolometer. The cryostat windows are polypropylene and the bolometer window is high-density polyethylene. The reference scan for each sample was taken with the sample removed from the cryostat mount, but with the cryostat present in the system. Transmission spectra, i.e., frequency-dependent ratios of sample and reference signals were calculated from interferograms by the OPUS software package provided by the manufacturer, using standard processing methods. Absorption bands in the window of the detector caused sections of the measurement band to be dominated by noise (notably, the regions 3.8-4.2 THz, 7.7-8.2 THz) and so in the analysis, only the region 4.2-7.7 THz (140-257 cm\({}^{-1}\)) was used. To recover the real part of the refractive index, we used the Fourier-transform based methods which we previously reported in detail [14], with the additional details published under a GNU Open License (Version 1, 2018, University of Surrey, freely available source code) [16]. These methods improve on the traditional calculations that use fringe separation in that they also implement a Fourier windowing analysis together with the equations for the transmittance of an absorbing Fabry-Perot cavity. This combination of techniques enables us to obtain both real and imaginary parts of the complex refractive index (under defined experimental conditions), without recourse to reflection measurements or Kramers-Kronig analysis. The software package for extracting \(n\)(\(f\)) is robust to noise, to errors arising from insufficient frequency or amplitude resolution, and to amplitude errors where the transmission exceeds unity. The main limitation of these methods is the requirement of an initial condition (as explained below). We took advantage of the different assumptions of two techniques in the package to find good estimates for initial conditions. The \"fringe\" method finds the locations of peaks (\"fringes\") in the data and applies an extrapolation technique to deduce the absolute refractive index. This approach assumes that the refractive index remains constant between peak maxima, and produces an exact value in the absence of dispersion, i.e., when d\(n\)/d\(f\) = 0; whereas in the presence of dispersion the result is only approximate [14]. The initial condition (assumption) is therefore negligible dispersion d\(n\)/d\(f\). The \"phase\" method takes advantage of phase information encoded in the data to find d\(n\)/d\(f\), and numerically integrates from a known \(n\) at any \(f\) within the measurement bandwidth as the initial condition. We used the fringe method to estimate the initial starting point of the phase method (i.e., to determine \(n\) at some \(f\)), using as an assumed \"flat\" region a few fringes near 5 THz. Figure 1a shows an example of transmission for Ge at 4 K and 296 K, where Fabry-Perotfringes are clearly visible. The period of the fringes is smaller at 296 K than at 4 K due to higher refractive index. The amplitude of the fringes is also reduced at room temperature due to increased absorption losses.