## 3 Results and Discussion  
### Single-Crystal Quartz  
Five quartz samples cut normal to the crystal axis (Z-cut) were measured, yielding THz optical parameters for the ordinary ray (o-ray, traveling parallel to the crystal axis) transmission. Excellent reproducibility among tested samples was seen: variation in obtained optical parameters was found to be comparable with the measurement uncertainty. In addition, one Y-cut crystal was also measured (cut parallel to the crystal axis) in order to obtain the optical parameters for the extraordinary ray (e-ray, traveling normal to the crystal axis) transmission. The refractive indices for ordinary and extraordinary rays in quartz crystal are presented in Figure 2, and the absorption coefficients in Figure 3. Quartz is a trigonal uniaxial crystal, and therefore birefringent and dichroic. Figure 3: The ordinary (\(\alpha_{\mathrm{o}}\)) and extraordinary (\(\alpha_{\mathrm{e}}\)) absorption coefficients of quartz crystal. The uncertainties are shown by light blue shading. The optical phonon resonance in the ordinary spectrum is at 3.855 \(\pm\) 0.005 THz. Figure 2: The ordinary (\(n_{\mathrm{o}}\)) and extraordinary (\(n_{\mathrm{e}}\)) refractive indices of quartz crystal. The uncertainties are shown by light blue shading. Birefringence is defined as the difference between extraordinary and ordinary refractive indices, and it is evident in Figure 2 that it is positive and increasing with frequency. The values of birefringence at a few selected frequencies are given in Figure 4a, where it is seen to rise from 0.048 at 1 THz to 0.054 at 5 THz (the values at 3.5 THz and 4 THz are omitted due to anomalous dispersion in the o-ray data). In contrast, in the visible range, the birefringence of quartz is 0.009, i.e., a factor of five smaller than at THz frequencies. The well-known optical phonon resonance [3,8,11] in the ordinary ray spectrum is seen to be at 3.855 \(\pm\) 0.005 THz. The absorption maximum is 10.7 \(\pm\) 0.1 cm\({}^{-1}\), and the peak FWHM (full width at half-maximum) is 145 \(\pm\) 5 GHz. Figure 4b shows a Voigt fit to the absorption peak. The Voigt function accounts for the broadening that may arise due to the presence of crystal defects. It is seen that the peak is asymmetric: whereas on the low-frequency side the fit is good, on the high-frequency side there is an additional broad wing, caused by the rising edge of the strong optical phonon at 11.8 THz. An important motivation for this work was to establish accurate values of THz and MW optical parameters for quartz and silica, in the light of significant variation found in the published data. This variation is especially large in the case of absorption coefficients, as seen in the figures below. Figures 5 and 6 present the results of this work together with the data reported in the literature [3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. Different types of instrumentation were used for these measurements. Early workers employed Fourier Transform spectrometers (FTS) [3, 4, 5, 6, 11], and a grating spectrometer [7,8]; in recent years, THz TDS has dominated the field [7,10, 11, 12]. Figure 4: (**a**) Birefringence of quartz, defined as \(n_{\mathrm{e}}\)–\(n_{\mathrm{o}}\). (**b**) Voigt fit to the phonon resonance peak at 3.855 \(\pm\) 0.005 THz. Figure 5: _Cont._  
Figure 5: THz optical parameters for the ordinary ray transmission in quartz crystal. Data reported in the literature are shown together with the results of this work. (**a**) refractive index; (**b**) absorption coefficient. Figure 6: THz optical parameters for the extraordinary ray transmission in quartz crystal. Data reported in the literature are shown together with the results of this work. (**a**) refractive index; (**b**) absorption coefficient. Our refractive index data and the values of birefringence show excellent agreement with the majority of published results. In the case of ordinary absorption, our data are in good agreement with the literature at frequencies up to about 4 THz.