## 3 Calibration of the Optoelectronic FDS-System  
### Frequency Calibration  
Frequency is one of the highest-precision quantities measurable by human endeavour. Frequency standards are ultimately traceable to the International Atomic Time Standard based on atomic clocks, providing precision down to \(10^{-16}\)[27; 28]. Accurate frequency determination is obviously important for spectroscopic measurements. Due to the broadband frequency range of the optoelectronic FDS-System, a frequency calibration standard must have multiple narrow spectral features. A common approach is to employ gas absorption lines, whose frequencies are known with high precision [29]. Atmospheric water vapour provides a convenient set of absorption lines in the 0.5-4 THz range that is widely used for frequency calibration of THz spectrometers. Figure 3a depicts the transmission spectrum through the ambient atmosphere showing water vapour absorption lines, measured by the three devices. Figure 3b plots the deviations of the measured lines from the frequencies listed by HITRAN [29], with results obtained by TeraFlash and TeraScan for comparison. For all three instruments, the deviations are randomly distributed and lie within the uncertainty of their frequency resolution, confirming the accuracy of their frequency scales. It is observed that T-Sweeper demonstrates the smallest deviations from the HITRAN data, due to its high frequency resolution (1 GHz) and low noise. TeraFlash has the largest deviations due to its lowest frequency resolution (5 GHz). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **T-Sweeper (FDS)** & **TeraFlash pro (TDS)** & **TeraScan (FDS)** \\ \hline Measuring Time & & & \\ (min) & \(\approx\)4 & \(\approx\)2.5 & \(\approx\)80 \\ \hline Frequency Range & & & \\ (THz) & 0.1–3 & 0.1–5.6 & 0.1–2.18 \\ \hline Frequency Resolution & & & \\ (GHz) & & & \\ \hline DR\({}_{\text{max}}\) (dB) & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of the parameters for the three used devices with the respective settings. Figure 2: (a) Spectral power and dynamic range using 1 to 100,000 averages. (b) The dynamic range at the peak and at four selected frequencies as a function of number of averages, showing the 10 dB per decade slope (dashed lines) and saturation above 20,000 averages. Water vapour lines have the drawback of being unevenly distributed with varying line strengths, sparse below 1 THz, and with no suitable lines below 0.5 THz. An alternative approach to frequency calibration is to employ a Fabry-Perot etalon [30,31]. An etalon has the advantage of an evenly spaced comb of equal-amplitude peaks across the whole spectral range of etalon transparency window, allowing systematic errors or drifts in frequency to be easily detected. However, the frequency resolution of an etalon is limited by its finesse, and commonly available etalons are low finesse, providing peaks that are 10-20 GHz FWHM (full width at half-maximum). Moreover, an etalon must be calibrated with a frequency standard, for example by using gas absorption lines. A high-resistivity silicon wafer can serve as a low-finesse etalon [30]. High-resistivity Si has negligible THz absorption and a near-constant refractive index of 3.42 in the range of measurement, and is compact and convenient to use. The free spectral range (FSR) of an etalon is the frequency spacing between successive etalon maxima (or minima). FSR is given by \(\text{FSR}=c/nd\), where \(n\) is the refractive index of the etalon material and \(d\) is its thickness. A Si wafer can thus be chosen to provide the desired FSR. The transmission spectrum of a Si wafer has been measured and the maxima identified. These occur at frequencies of \(f_{p}=2p\cdot\text{FSR}\), where \(p\) is the peak order. The FSR of the etalon can therefore be determined by plotting \(f_{p}\) versus \(p\) and applying a linear fit, as shown in Figure 4a.