## 3 Analytical Modeling  
The proposed system aims to maintain system fidelity for long distance transmission at high data rates. This section presents a detailed analytical model for analyzing FWM. The parameters that affect the severity of FWM are the SMF cross sectional area, magnitude of laser light, multi channel WDM system with high data rate speed. Inside SMF the inharmonic movements of photons are induced by reason of these mentioned factors [42], and gives nonlinear induced polarization \(\Omega_{P}\)[43], which is given as:  
\[\Omega_{P}=\varepsilon_{0}\kappa^{{}^{\prime}}E+\varepsilon_{0}\kappa^{{}^{ \prime\prime}}E^{2}+\varepsilon_{0}\kappa^{{}^{\prime\prime\prime}}E^{3}, \tag{1}\]  
where \(\varepsilon_{0}\) is free space permittivity, \(E\) is electrical field intensity and \(\kappa^{{}^{\prime}}\), \(\kappa^{{}^{\prime\prime}}\) and \(\kappa^{{}^{\prime\prime\prime}}\) are first, second and third order susceptibility, respectively. The effects of FWM for optical medium are originated from the electric dipole polarization induction, which is mainly due to the \(\kappa^{{}^{\prime}}\) while the \(\kappa^{{}^{\prime\prime}}\) does not contribute for silica based propagation mediums [44] and \(\kappa^{{}^{\prime\prime\prime}}\) produces very low order nonlinear effects. Thus, by neglecting \(\kappa^{{}^{\prime\prime}}\)[45] the induced polarization from (1) becomes:  
\[\Omega_{P}=\varepsilon_{0}\kappa^{{}^{\prime}}E\text{cos}(\Phi t-zx)+ \varepsilon_{0}\kappa^{{}^{\prime\prime\prime}}E^{3}\text{cos}^{3}(\Phi t-zx). \tag{2}\]  
By applying the trigonometric property of expansion on (1) and neglecting constant terms and \(3\Phi\) terms, we get:  
\[\Omega_{P}=\varepsilon_{0}\kappa^{{}^{\prime}}E\text{cos}(\Phi t-zx)+\frac{3} {4}\varepsilon_{0}\kappa^{{}^{\prime\prime\prime}}E^{3}\text{cos}^{3}(\Phi t- zx). \tag{3}\]  
In (3), both linear and nonlinear parameters are included. For the phase wave explored in (2), the intensity \(t\) of pulse [46] is defined as  
\[t=\frac{1}{2}\varepsilon_{0}n_{0}E_{0}^{2}, \tag{4}\]  
where \(c\) denotes the light velocity, and linear refractive index is represented by \(n_{0}\) at low electrical fields [47]. Thus, (3) simplifies to  
\[\Omega_{P}=\varepsilon_{0}\kappa^{{}^{\prime}}E\text{cos}(\Phi t-zx)+\frac{3} {2}\frac{\kappa^{{}^{\prime\prime\prime}}}{c\varepsilon_{0}n_{0}}tE\text{cos} (\Phi t-zx). \tag{5}\]  
Figure 4: Distortion of propagated optical signal due to phase mode dispersion. The effective susceptibility [48], \(\kappa_{e}\), of the optical medium is calculated as  
\[\kappa_{e}=\kappa^{{}^{\prime}}+\frac{3}{2}\frac{\kappa^{\prime\prime\prime}}{c \varepsilon_{0}n_{0}}, \tag{6}\]  
and the effective refractive index, \(n_{e}\), is measured as  
\[n_{e}=n_{0}+n_{r}t, \tag{7}\]  
where \(n_{r}\) represents nonlinear refractive index, which is given as  
\[n_{r}=\frac{3}{4}\frac{\kappa^{\prime\prime\prime}}{c\varepsilon_{0}n_{0}^{2}}. \tag{8}\]  
In addition, FWM increases with the increase in length of fiber. As length of fiber increases, the interaction of a signal with the fiber and other traveling signals increases, which results in enhancing the effect of FWM [49]. This optical power, in terms of length of fiber, \(L\), and attenuation, \(\alpha\), is written as  
\[\rho_{r}=\rho_{i}\exp-\alpha L, \tag{9}\]  
where \(\rho_{i}\) and \(\rho_{r}\) are input and output powers, respectively.