## 3 Analytical Modeling
For length, \(L\), the effective length, \(L_{e}\), of the fiber is given as  
\[\Omega_{P}=\varepsilon_{0}\kappa^{{}^{\prime}}E\text{cos}(\Phi t-zx)+ \varepsilon_{0}\kappa^{{}^{\prime\prime\prime}}E^{3}\text{cos}^{3}(\Phi t-zx). \tag{2}\]0  
From (9) and (10), \(L_{e}\) can be written as  
\[\Omega_{P}=\varepsilon_{0}\kappa^{{}^{\prime}}E\text{cos}(\Phi t-zx)+ \varepsilon_{0}\kappa^{{}^{\prime\prime\prime}}E^{3}\text{cos}^{3}(\Phi t-zx). \tag{2}\]1  
The effective length after placing optical amplifiers with \(L\) and amplifier spaced distance \(z\)[50], is defined as  
\[\Omega_{P}=\varepsilon_{0}\kappa^{{}^{\prime}}E\text{cos}(\Phi t-zx)+ \varepsilon_{0}\kappa^{{}^{\prime\prime\prime}}E^{3}\text{cos}^{3}(\Phi t-zx). \tag{2}\]2  
Cross sectional area, \(\xi_{eff}\), of fiber core has inverse relation to \(\iota\)[51] and can be estimated as  
\[\Omega_{P}=\varepsilon_{0}\kappa^{{}^{\prime}}E\text{cos}(\Phi t-zx)+ \varepsilon_{0}\kappa^{{}^{\prime\prime\prime}}E^{3}\text{cos}^{3}(\Phi t-zx). \tag{2}\]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. Broadening of spectrum is the initial impact of SPM, which further increases for high launch power. Hence, based on wavelength, \(\lambda\), \(L\), and \(E\), the phase, \(\varpi\), is defined as  
\[\Omega_{P}=\varepsilon_{0}\kappa^{{}^{\prime}}E\text{cos}(\Phi t-zx)+ \varepsilon_{0}\kappa^{{}^{\prime\prime\prime}}E^{3}\text{cos}^{3}(\Phi t-zx). \tag{2}\]4  
where \(n\) represents refractive index. In terms of linear and nonlinear phenomenon, (14) is written as  
\[\Omega_{P}=\varepsilon_{0}\kappa^{{}^{\prime}}E\text{cos}(\Phi t-zx)+ \varepsilon_{0}\kappa^{{}^{\prime\prime\prime}}E^{3}\text{cos}^{3}(\Phi t-zx). \tag{2}\]5The first portion of the above equation is called linear phase \(\varpi_{0}\), and the second portion is known as nonlinear phase shift \(\varpi_{r}\). For time dependent phase angle, the modulated optical carrier frequency, \(\theta_{c}\), based on frequency spectrum, \(\theta_{0}\), is given as  
\[\Omega_{P}=\varepsilon_{0}\kappa^{{}^{\prime}}E\text{cos}(\Phi t-zx)+ \varepsilon_{0}\kappa^{{}^{\prime\prime\prime}}E^{3}\text{cos}^{3}(\Phi t-zx). \tag{2}\]6  
Furthermore, the nonlinear phase shift, \(\varpi_{r}\), can be written as  
\[\Omega_{P}=\varepsilon_{0}\kappa^{{}^{\prime}}E\text{cos}(\Phi t-zx)+ \varepsilon_{0}\kappa^{{}^{\prime\prime\prime}}E^{3}\text{cos}^{3}(\Phi t-zx). \tag{2}\]7  
When multiple channels are transmitted, then the phase shift of the signal is generated due to co-channels. Therefore,  
\[\Omega_{P}=\varepsilon_{0}\kappa^{{}^{\prime}}E\text{cos}(\Phi t-zx)+ \varepsilon_{0}\kappa^{{}^{\prime\prime\prime}}E^{3}\text{cos}^{3}(\Phi t-zx). \tag{2}\]8  
where \(\rho_{v0}\) and \(\rho_{v1}\) are the powers of two co-channels. From (1) to (3), it is described that induced polarization composes of both linear and FWM terms. The FWM nonlinear impact is caused by the third order susceptibility. In this process, photons of transmitted signals annihilate and result in production of new photons at various frequencies. In contrast to SPM and XPM, FWM is independent of bit rate and dependent on channel spacing and fiber dispersion.