## 2 Propagation in Dielectric Media
 \tag{8}\]  
Using the last Expression (8), the frequency dependent incremental part of the time delay (3) can be derived analytically:  
\[\Delta\tau_{d}(f)=\frac{d}{2\pi}\bigg{(}\frac{2\pi\,f}{c}\bigg{)}^{2}\frac{1}{ \alpha^{2}+\beta^{2}}\bigg{[}\alpha\bigg{(}\frac{\epsilon^{\prime\prime}}{f}+ \frac{1}{2}\frac{d\epsilon^{\prime\prime}}{df}\bigg{)}+\beta\bigg{(}\frac{ \epsilon^{\prime}}{f}+\frac{1}{2}\frac{d\epsilon^{\prime}}{df}\bigg{)}\bigg{]} -\frac{d}{c} \tag{9}\]  
The above analytical Expressions (8) and (9) describe the three major physical phenomena of the electromagnetic field propagation in dielectric media: attenuation, phase dispersion, and the resulted group delay. Propagation of light as well as millimeter and sub-millimeter wave infra-red radiation in the atmosphere are usually studied using the complex refractive index \(n(f)\), which is written in terms of the refractivity, \(N(f)\) (given in ppm):  
\[n(f)=1+N(f)\times 10^{-6} \tag{10}\]  
The frequency dependent refractivity is complex and can be presented as:  
\[N(f)=N_{0}+N^{\prime}(f)-jN^{\prime\prime}(f) \tag{11}\]  
where the nondispersive part \(N_{0}\) is real and positive and the other two terms, the real \(N^{\prime}(f)\) and the imaginary \(N^{\prime\prime}(f)\) parts, are frequency dependent. The relations between the complex refractivity and complex permittivity can be obtained by using Equations (4), (5), (10) and (11):  
\[n(f)=\sqrt{\epsilon^{\prime}(f)-je^{\prime\prime}(f)}=1+[N_{0}+N^{\prime}(f)- jN^{\prime\prime}(f)]\times 10^{-6} \tag{12}\]  
resulting in an expression for the real part:  
\[\epsilon^{\prime}(f)=1+2[N_{0}+N^{\prime}(f)]\times 10^{-6} \tag{13}\]  
and for the imaginary part:  
\[\epsilon^{\prime\prime}(f)=2N^{\prime\prime}(f)\times 10^{-6} \tag{14}\]Since the dispersion models often use combinations of these quantities, permittivity, and refractivity, the above relations enable one to fuse between the models and present a generalized one. The media properties can be also presented in terms of the refractivity:  
\[ \begin{split}\alpha(f)=-\text{Im}\{k(f)\}=\tfrac{2\pi f}{c}N^{ \prime\prime}(f)\times 10^{-6}\\ \beta(f)=\text{Re}\{k(f)\}=\tfrac{2\pi}{c}\big{[}\big{(}1+N_{0} \times 10^{-6}\big{)}+N^{\prime}(f)\times 10^{-6}\big{]}\\ \tau_{d}(f)=\tfrac{d}{2\pi}\tfrac{d\beta}{df}=\tfrac{d}{c}\Big{\{} \big{(}1+N_{0}\times 10^{-6}\big{)}+\Big{[}N^{\prime}(f)+f\tfrac{dN^{\prime}}{df} \Big{]}\times 10^{-6}\Big{\}}\end{split}  \tag{15}\]  
For the study of millimeter waves propagation in the atmosphere, the MPM is employed [8; 9]. In these models, quantitative values of the dispersive complex refractivity (11) are given via the permittivity of the gases and water droplets composing the atmosphere. The refractivity is represented as a summation of five terms:  
\[N(f)=(N_{L}+N_{d}+N_{c})+N_{W}+N_{R} \tag{16}\]  
Here \(N_{L}(f)\) is moist air resonance contributions, \(N_{d}(f)\) is dry air non-resonant spectra, \(N_{c}(f)\) is water vapor continuum spectrum, \(N_{W}(f)\) is suspended water-droplet refractivity and \(N_{R}(f)\) is rain approximation. It is important to note that the different refractivity terms above are mutually independent and according to Equation (16) they are accumulated additively, contributing to the overall refractivity. A comparative study between the comprehensive MPM and the International Telecommunication Union (ITU) recommendation reveals that although the absorption peaks of water vapor and oxygen are not taken into account in the ITU model, there is good fit between the models [18]. From evaluation of the refractivity terms (16), it was found that the suspended water droplets had a major effect on the attenuation while the significant effect on the group delay was due to air humidity \(N_{L}(f)\)[19]. The dielectric permittivity for different values of relative humidity \(RH\) and water droplet concentration \(W_{0}\) is presented in Figure 1, for real \(\varepsilon^{\prime}(f)\) and imaginary \(\varepsilon^{\prime\prime}(f)\) parts.