## 2 Atmospheric mmW Propagation in the Presence of Water Suspension
The propagation factors, namely the attenuation \(\alpha\)(f) and the wavenumber \(\beta\)(f), can be expressed in terms of the complex refractive index as follows:  
\[\alpha(f)=\frac{2\pi f}{c}N^{\prime}(f)\cdot 10^{-6} \tag{2}\] \[\beta(f)=\frac{2\pi f}{c}\big{\{}1+[N_{0}+N^{\prime}(f)]\cdot 10^{-6} \big{\}}\]  
The total attenuation along a propagation distance d is calculated via \(\alpha\)(f):  
\[L(f)=e^{2\alpha d}=\exp\bigg{[}\frac{4\pi f}{c}d\cdot N^{\prime\prime}(f)\cdot 1 0^{-6}\bigg{]} \tag{3}\]  
and the corresponsive phase shift of the field is determined by the wave number \(\beta\)(f):  
\[\phi(f)=-\beta(f)\cdot d=\underbrace{\frac{2\pi f}{c}d}_{\phi_{0}(f)}- \underbrace{\frac{2\pi f}{c}d\cdot[N_{0}+N^{\prime}(f)]\cdot 10^{-6}}_{\Delta \phi(f)} \tag{4}\]  
Defining the phase shift in vacuum as:  
\[\phi_{0}(f)=-2\pi f\cdot\frac{d}{c} \tag{5}\]  
the incremental phase shift caused by the atmosphere due to its incremental refractivity is then:  
\[\Delta\phi(f)=\phi(f)-\phi_{0}(f)=\phi_{0}(f)\cdot[N_{0}+N^{\prime}(f)]\cdot 1 0^{-6} \tag{6}\]  
In the current work, emphasis is given to the study of foggy atmospheric conditions, where droplets of water are suspended in the air. The corresponding refractivity in the mmW's is derived from the total moisture concentration in the cloud, which independently analyzed by atmospheric transmission measurements in the visible light (635 nm), taking into account the Mie scattering theory. The typical droplet diameter in natural fog is about Rp\(\sim\)10 \(\upmu\)m.  
The attenuation L calculated in [dB/km] as a function of frequency for the millimeter and sub-millimeter regime is shown in Figure 1a. Fog visibility is defined as the distance for which visible light is attenuated to 2% of its maximum intensity in clear sky. According to the International Commission of Illumination, this definition presents the visual range where the contrast ratio for a black target (of a \"reasonable\" size) against the horizon falls to 2% of the range obtained in the clear sky [26]. Graphs are drawn for various visibility conditions. When visibility is relatively high \(\geq\)10 m, pure rotational transitions of water vapor and molecular oxygen can be clearly seen in the region of 22 GHz and 60 GHz, respectively. The additional phase shift \(\Delta\phi(f)/\phi_{0}\) as a function of frequency is shown in Figure 1b. As expected, the change in attenuation due to oxygen absorption at 60 Hz is also accompanied by a change in the calculated phase shift; this in accordance with the Kramer-Kronig relations between the imaginary and the real part of the refractive index [20]. Figure 1: Transmission of millimeter waves in various fog visibilities in: (**a**) attenuation \(L(f)\) in [dB]; (**b**) incremental phase shift \(\Delta\phi(f)/\phi_{0}\); (**c**) incremental group delay \(\Delta t_{d}(f)\). A comparative study between the results obtained from the comprehensive MPM [15] with those calculated using the International Telecommunication Union (ITU) [27] reveals that, although the absorption peaks of water vapor and molecular oxygen are not taken into account in the ITU model, there is a good fit between the two models [28]. The corresponding graphs for the incremental group delay \(\Delta\tau_{d}(f)\) (in [ps/km]) are shown in Figure 1c. The ITU model does not refer to the time delay.