## 4 Thickness Gauging  
In order to investigate the precision of the ECOPS TDS system in thickness measurements, we chose a piece of an un-doped wafer of high-resistivity silicon (CrysTec GmbH, Berlin, Germany) as our sample. The silicon wafer has negligible dispersion, negligible absorption [34] and two plane-parallel surfaces. Owing to the high refractive index of the material (\(n\sim\) 3.4), both the front and the back surface of the wafer strongly reflect the incident terahertz pulses. We extract the sample thickness via time-of-flight measurements, i.e., from the time difference between the zero-crossings of the pulse \"echoes\" from either side. Figure 7 presents the results of thickness measurements carried out with the ECOPS TDS system (at 1600 pulse traces per second), the TeraFlash pro (measurement speed 10 traces per second), and a manual micrometer gauge (Holex 421505, Hoffmann GmbH, Munich, Germany).  
In order to quantify the reproducibility of the thickness readings, we repeated the terahertz-based measurements 10,000 times both with ECOPS and the TeraFlash pro, and 50 times with the micrometer gauge. The error bars in Figure 7 denote the standard deviation of the results. With ECOPS, the TeraFlash pro, and the micrometer gauge, we obtain thickness values of 527 \(\pm\) 1.5 \(\upmu\)m, 527.94 \(\pm\) 0.04 \(\upmu\)m, and 527.5 \(\pm\) 2.5 \(\upmu\)m, respectively. We find that all three methods yield consistent results within a 1-sigma error margin. Note, however, that the scatter of the TeraFlash pro results is approx. 40 times smaller than that of the ECOPS measurement. We attribute this to the difference in measurement speed between the two systems: the ECOPS system is 160 times faster, which comes at a cost in measurement accuracy. On the other hand, a repeatability on the 1 \(\upmu\)m level is sufficient for the majority of industrial applications. Moreover, as shown in the previous section, averaging successive pulse traces offers a trade-off between measurement speed and accuracy. For instance, extracting thickness data from  
Figure 7: Thickness measurement results of a single point on the silicon wafer, obtained with ECOPS, the TeraFlash pro and the micrometer gauge. Measurements with ECOPS and the TeraFlash pro were repeated 10,000 times each, and the micrometer gauge measurement was repeated 50 times. The error bars represent the standard deviation of the results.  
10 averaged traces reduces the standard deviation to \(\sim\)0.5 \(\upmu\)m (corresponding to an increase in the TD-SNR by a factor of 3 \(\approx\) 10 dB, see Figure 4), yet the ECOPS system remains 16 times faster than the TeraFlash pro.  
In order to validate the suitability of the system for a fast \"real-world\" thickness measurement, we moved the silicon sample at speeds similar to that of actual production lines. Our mock-up conveyor belt consisted of a vinyl-record player, on which the wafer was placed. The rotation speed of 45 rpm translates into a lateral velocity of about 700 mm/sec. The ECOPS measurement speed enables us to evaluate the thickness of the wafer with a spatial resolution of about 0.4 mm (not considering the actual focus size). Figure 8 depicts the ECOPS-based thickness readings across the wafer (red dots). The ruler shown in the photograph illustrates the horizontal axis. For comparison, we also examined selected locations on the same sample with the manual micrometer gauge. We note that the thickness varies along the sample by as much as 10 \(\upmu\)m, with ECOPS and micrometer-gauge results being in good agreement.  
We note that besides measurement speed, another important performance parameter is the minimum layer thickness an instrument can resolve. Preliminary results (not shown here) indicate that the thickness of layers as thin as \(\sim\)30 \(\upmu\)m can still be accurately determined. We surmise that even thinner layers can be characterized using more sophisticated data post-processing algorithms.