## 5 Visibility Measurement and Estimation of Water Droplet Concentration  
The visibility measurements were discussed elsewhere [17]. However, we will concentrate more on the experimental technique here. Two laser diode-based systems (ThorLabs CPS635) were simultaneously operated in different positions across the tunnel to measure the optical transmittance and test the homogeneity of the cloud during the entire course of measurements. The laser source at 635 nm was placed on one side of the tunnel and a Si Avalanche Photodetector (ThorLabs APD36) was placed 2.19 m away on its other side, as shown in Figure 10.  
Figure 10: The laser beam passing through the fog tunnel. Figure 9: Experiment system: (**a**) the Network Analyzer connected to two horn antennas outside the chamber; (**b**) the horn antennas. The transmittance over this distance can be measured using the Beer-Lambert Equation as follows:  
\[\frac{I_{t}}{I_{0}}=e^{-\alpha_{ext}W_{liq}d} \tag{10}\]  
where \(\alpha_{ext}\) is the averaged mass extinction coefficient (in \(m^{2}/g\)), \(W_{liq}\) is the concentration of the liquid water phase (in \(g/m^{3}\)), and \(I_{0}\) & \(I_{t}\) are the intensity signals measured at time zero when the tunnel is clean and at a time 't', respectively. To normalize for the laser intensity fluctuations, the p-polarized light was split near the source using a Brewster window to minimize reflections. Thus, about 5% of the output intensity was measured simultaneously by a reference APD detector located next to it. All optics were purged with fresh air to avoid condensation on the surfaces. The light intensity was modulated by varying the current periodically at a frequency of 129 Hz and the measured signal via the Ref/Signal APDs was plugged to a National Instruments A/D card (PCI-4472) equipped with a LabView lock-in amplifier software kit. According to Ref. [17], the instantaneous visibility at a 1 Hz is calculated via the following expression:  
\[Vis(t)=\frac{ln50\cdot d}{\ln\left(\frac{I_{0}}{I_{t}}\right)} \tag{11}\]  
Note that \(d=2.19\)\(m\) is the actual optical pass length through the cloud. Since reference APD is applied here, the explicit expression for the power ratio used in Equation (11) is as follows:  
\[\frac{I_{0}}{I_{t}}=\frac{I_{0}^{Sig}\cdot I_{t}^{Ref}}{I_{t}^{Sig}\cdot I_{0} ^{Ref}} \tag{12}\]  
where Sig and Ref indexes are the abbreviations for the APD types. The mass extinction coefficient \(\alpha_{ext}\) is calculated by weighting the relative contribution of the extinction efficiency \(Q_{ext}(r)\) from Mie theory taking into account the water refractive index \(n(635 nm)=1.331+i1.5\cdot 10^{-8}\) and the actual size distribution \(n(r)\) inside the tunnel (see), as follows:  
\[\alpha_{ext}=\frac{\int\frac{r^{2}Q_{ext}(r)n(r)dr}{4\beta\int\ r^{3}n(r)dr}} \approx 0.246\ m^{2}/g \tag{13}\]  
where \(\rho\) is the water density. When substituting the value of \(\alpha_{ext}\) from Equation (13) into Equation (10), the cloud concentration in the liquid phase \(W_{liq}\) can be derived vs. time. Figure 7a shows an example for the cloud concentration calculated during a full course of 12 propagation measurements. Panel (b) shows the associated visibility that was derived using Equation (11). We note, that while mmW propagation is affected by the total water concentration absorption in all phases, the propagation of Vis-IR radiation through foggy atmosphere is predominantly influenced by a Mie scattering, as described above. The point relevant to our estimation method for the cloud water concentration is that the extinction of the 635 nm light is mainly affected by the aerosol phase. Thus, the signal measured under Equation [10] is oblivious to the existence of a water concentration due to a vapor phase (\(W_{\text{gas}}\)).