## 3 Analytical Modeling  
The proposed system aims to maintain system fidelity for long distance transmission, system volume and data rates. This section presents a detailed analytical model for  
Figure 4: Cross phase modulation (XPM) effects inside nonlinear optical medium. Figure 5: Pulse broadening due to self phase modulation (SPM) non-linearity. analyzing RIrNLs. The parameters which affect the amount of RIrNLs are cross-section area of SMF, amount of light intensity propagating through the fiber, number of channels in the WDM system, and data rate. These factors generate an-harmonic motions of photons inside the SMF [25], where the behavior of induced polarization \(\Omega_{P}\) no more remains linear, which is given as  
\[\Omega_{P}=\varepsilon_{0}\kappa^{{}^{\prime}}E+\varepsilon_{0}\kappa^{{}^{ \prime\prime}}E^{2}+\varepsilon_{0}\kappa^{{}^{\prime\prime\prime}}E^{3}. \tag{1}\]  
Here, \(\varepsilon_{0}\) describes free space permittivity, \(E\) means electrical field intensity, and \(\kappa^{{}^{\prime}}\), \(\kappa^{{}^{\prime\prime}}\) and \(\kappa^{{}^{\prime\prime\prime}}\) determine first, second and third order susceptibility, respectively. The effects of RIrNLs for optical medium are originated from the electric diplole polarization induction, which is mainly due to the \(\kappa^{{}^{\prime}}\), while the \(\kappa^{{}^{\prime\prime}}\) does not contribute for silica based propagation mediums [26] and \(\kappa^{{}^{\prime\prime\prime}}\) produces very low order nonlinear effects. Thus, by neglecting \(\kappa^{{}^{\prime\prime}}\), the induced polarization from (1) becomes  
\[\Omega_{P}=\varepsilon_{0}\kappa^{{}^{\prime}}E\text{cos}(\Phi t-zx)+ \varepsilon_{0}\kappa^{{}^{\prime\prime\prime}}E^{3}\text{cos}^{3}(\Phi t-zx). \tag{2}\]  
In (2), both linear and nonlinear parameters are included. For the phase wave explored in (2), the intensity \(\iota\) of pulse [27] is defined as  
\[\iota=\frac{1}{2}c\varepsilon_{0}n_{0}E_{0}^{2}. \tag{3}\]  
Here, \(c\) denotes the light velocity, and linear refractive index is represented by \(n_{0}\) at low electrical fields [28]. Thus, (2) simplifies to  
\[\Omega_{P}=\varepsilon_{0}\kappa^{{}^{\prime}}E\text{cos}(\Phi t-zx)+\frac{3}{ 2}\frac{\kappa^{{}^{\prime\prime\prime}}}{c\varepsilon_{0}n_{0}}\iota E\text{ cos}(\Phi t-zx), \tag{4}\]  
The effective susceptibility, \(\kappa_{e}\), of the optical medium is calculated as  
\[\kappa_{e}=\kappa^{{}^{\prime}}+\frac{3}{2}\frac{\kappa^{{}^{\prime\prime \prime}}}{c\varepsilon_{0}n_{0}}\iota, \tag{5}\]  
and the effective refractive index, \(n_{e}\), is measured as  
\[n_{e}=n_{0}+n_{r}\iota. \tag{6}\]  
In (6), \(n_{r}\) represents nonlinear refractive index, which is given as  
\[n_{r}=\frac{3}{4}\frac{\kappa^{{}^{\prime\prime\prime}}}{c\varepsilon_{0}n_{0 }^{2}}. \tag{7}\]  
In addition, RIrNLs enlarge with the increase in length of fiber. As length of fiber gets larger, the interaction of a signal with the fiber and other traveling signals increases, which results in enhancing the effect of RIrNLs [28]. This optical power, in terms of length of fiber, \(\Gamma\), and attenuation, \(\alpha\), is written as  
\[\rho_{r}=\rho_{i}\exp^{-\alpha\Gamma}, \tag{8}\]  
where \(\rho_{i}\) and \(\rho_{r}\) determine input and output powers. For length, \(\Gamma\), the effective length, \(\Gamma_{e}\), of the fiber is given as  
Figure 5:0  
From (8) and (9), \(\Gamma_{e}\) is written as  
Figure 5:1The effective length after installing optical amplifiers with \(\Gamma\) and amplifier spaced distance \(z\)[29], is defined as  
Figure 5:2  
Cross sectional area, \(\xi_{eff}\), of fiber core has inverse relation to \(t\)[30] and can be estimated as  
Figure 5:3  
where \(m\) and \(b\) are the polar coordinates. Due to the SPM, different parts of the optical signal undergo different phase shifts, which results in frequency chirping incident. Broadening of spectrum is the initial impact of SPM, which gets larger for high launch power.