## 2 Propagation in Dielectric Media  
An electromagnetic wave propagating in a dielectric medium such as the atmosphere is being affected by losses and time delay. These effects are described by the propagation expression for the electric field in the frequency domain. Propagating a distance \(d\) in a homogeneous, linear medium, the resulted field is:  
\[\widetilde{E}_{out}(f)=\widetilde{E}_{in}(f)\mathrm{e}^{-jk(f)\cdot d} \tag{1}\]  
where \(\widetilde{E}_{in}(f)\) and \(\widetilde{E}_{out}(f)\) are the transmitted and received fields respectively, presented as phasors in the frequency domain and:  
\[k(f)=-ja(f)+\beta(f) \tag{2}\]  
is a frequency-dependent propagation factor, composed of the attenuation per unit length \(\alpha(f)=-\mathrm{Im}\{k(f)\}\) and the wavenumber \(\beta(f)=\mathrm{Re}\{k(f)\}\). The group delay at a distance \(d\) can be found from the derivative of the wavenumber:  
\[\tau_{d}(f)=\frac{d}{2\pi}\frac{d\beta}{df}=\frac{d}{c}+\Delta\tau_{d}(f) \tag{3}\]  
where \(c\approx 2.997\times 10^{8}\) m/s is the speed of light in a vacuum. The group delay can be described as a summation of the constant delay \(\tau_{0}=d/c\) in vacuum and a frequency-dependent incremental part \(\Delta\tau_{d}(f)\). In the first stage of the model description, the properties of the atmosphere as a dielectric medium are presented in terms of its permittivity. In the second stage, the permittivity of a suspended droplets atmospheric medium as in foggy or cloudy conditions is discussed. Finally, we derive the parameters of millimeter wave propagation in such conditions. The propagation factor is commonly expressed via the dielectric medium properties, either relative permittivity \(\varepsilon_{r}(f)\) or index of refraction \(n(f)\):  
\[k(f)=\frac{2\pi f}{c}\underbrace{\sqrt{\varepsilon_{r}(f)}}_{n(f)} \tag{4}\]  
where \(c\approx 2.997\times 10^{8}\) m/s is the speed of light in a vacuum. The relative permittivity of the medium is a complex, frequency dependent quantity:  
\[\varepsilon_{r}(f)=\varepsilon^{\prime}(f)-je^{\prime\prime}(f) \tag{5}\]  
with real \(\varepsilon^{\prime}(f)\) and imaginary \(e^{\prime\prime}(f)\) parts. The attenuation \(\alpha(f)\) and the wavenumber \(\beta(f)\) are related to the permittivity via two coupled equations:\[ \begin{split}\alpha^{2}(f)-\beta^{2}(f)=-\Big{(}\frac{2\pi\,f}{c} \Big{)}^{2}\epsilon^{\prime}(f)\\ 2\alpha(f)\ \beta(f)=\Big{(}\frac{2\pi\,f}{c}\Big{)}^{2}\epsilon^{ \prime\prime}(f)\end{split}  \tag{6}\]  
The last equations have two pairs of solutions:  
\[ \begin{split}\alpha_{1,2,3,4}&=\pm\frac{1}{\sqrt{2}} \Big{(}\frac{2\pi\,f}{c}\Big{)}\sqrt{-\epsilon^{\prime}\pm\sqrt{\left( \epsilon^{\prime}\right)^{2}+\left(\epsilon^{\prime\prime}\right)^{2}}}\\ \beta_{1,2,3,4}&=\pm\frac{1}{\sqrt{2}}\Big{(}\frac {2\pi\,f}{c}\Big{)}\epsilon^{\prime\prime}\frac{1}{\sqrt{-\epsilon^{\prime} \pm\sqrt{\left(\epsilon^{\prime}\right)^{2}+\left(\epsilon^{\prime\prime} \right)^{2}}}}\end{split}  \tag{7}\]  
Since the attenuation \(\alpha(f)\) and the wavenumber \(\beta(f)\) are real and positive numbers, the physical solution should be [17]:  
\[ \begin{split}\alpha(f)=-\text{Im}\{k(f)\}=\frac{2\pi\,f}{c} \sqrt{\frac{\epsilon^{\prime}(f)}{2}\Bigg{[}\sqrt{1+\Big{(}\frac{\epsilon^{ \prime\prime}(f)}{\epsilon^{\prime}(f)}\Big{)}^{2}}-1\Bigg{]}}\\ \beta(f)=\text{Re}\{k(f)\}=\frac{2\pi\,f}{c}\sqrt{\frac{ \epsilon^{\prime}(f)}{2}\Bigg{[}\sqrt{1+\Big{(}\frac{\epsilon^{\prime\prime} (f)}{\epsilon^{\prime}(f)}\Big{)}^{2}}+1\Bigg{]}}\end{split}