## 3 Analytical Modeling
To calculate the effect of FWM, the induced polarization is written as:  
\[\Omega_{P}=\varepsilon_{0}\kappa^{{}^{\prime}}E\text{cos}(\Phi t-zx)+ \varepsilon_{0}\kappa^{{}^{\prime\prime\prime}}E^{3}\text{cos}^{3}(\Phi t-zx). \tag{2}\]9 \[\Omega_{P}=\varepsilon_{0}\kappa^{{}^{\prime}}E\text{cos}(\Phi t-zx)+\frac{3} {4}\varepsilon_{0}\kappa^{{}^{\prime\prime\prime}}E^{3}\text{cos}^{3}(\Phi t- zx). \tag{3}\]0 \[\Omega_{P}=\varepsilon_{0}\kappa^{{}^{\prime}}E\text{cos}(\Phi t-zx)+\frac{3} {4}\varepsilon_{0}\kappa^{{}^{\prime\prime\prime}}E^{3}\text{cos}^{3}(\Phi t- zx). \tag{3}\]1 \[\Omega_{P}=\varepsilon_{0}\kappa^{{}^{\prime}}E\text{cos}(\Phi t-zx)+\frac{3} {4}\varepsilon_{0}\kappa^{{}^{\prime\prime\prime}}E^{3}\text{cos}^{3}(\Phi t- zx). \tag{3}\]2 \[\Omega_{P}=\varepsilon_{0}\kappa^{{}^{\prime}}E\text{cos}(\Phi t-zx)+\frac{3} {4}\varepsilon_{0}\kappa^{{}^{\prime\prime\prime}}E^{3}\text{cos}^{3}(\Phi t- zx). \tag{3}\]3  
The primary portion in the above equation represents SPM and XPM, while the secondary part denotes FWM. It explains that FWM occurs due to mixing of different frequencies. As FWM are the main causes of power penalty and system performance degradation, the proposed system mitigates their effects by employing NRZ, DB modulation, large effective area and uneven channel spacing limited by controlling nonlinear dispersion through a dispersion compensation fiber (DCF). FWM causes generation of new unwanted mixed frequencies, \(f_{new}\), in WDM optical transmission framework in the range of original frequencies as shown in \[\Omega_{P}=\varepsilon_{0}\kappa^{{}^{\prime}}E\text{cos}(\Phi t-zx)+\frac{3} {4}\varepsilon_{0}\kappa^{{}^{\prime\prime\prime}}E^{3}\text{cos}^{3}(\Phi t- zx). \tag{3}\]4 The unwanted signals also known as beat frequencies can be written as  
\[\Omega_{P}=\varepsilon_{0}\kappa^{{}^{\prime}}E\text{cos}(\Phi t-zx)+\frac{3} {4}\varepsilon_{0}\kappa^{{}^{\prime\prime\prime}}E^{3}\text{cos}^{3}(\Phi t- zx). \tag{3}\]5  
where \(f_{l}\), \(f_{m}\) and \(f_{n}\) are the frequencies of the input signals and \(f_{new}\) is the new generated frequency. (19) and (20) generate new idlers, which interface nearby signals and introduce new frequencies, which can be written as:  
\[\Omega_{P}=\varepsilon_{0}\kappa^{{}^{\prime}}E\text{cos}(\Phi t-zx)+\frac{3} {4}\varepsilon_{0}\kappa^{{}^{\prime\prime\prime}}E^{3}\text{cos}^{3}(\Phi t- zx). \tag{3}\]6  
\[\Omega_{P}=\varepsilon_{0}\kappa^{{}^{\prime}}E\text{cos}(\Phi t-zx)+\frac{3} {4}\varepsilon_{0}\kappa^{{}^{\prime\prime\prime}}E^{3}\text{cos}^{3}(\Phi t- zx). \tag{3}\]7 \[\Omega_{P}=\varepsilon_{0}\kappa^{{}^{\prime}}E\text{cos}(\Phi t-zx)+\frac{3} {4}\varepsilon_{0}\kappa^{{}^{\prime\prime\prime}}E^{3}\text{cos}^{3}(\Phi t- zx). \tag{3}\]8  
The power of the FWM effect for continuous spectrum in long range fiber, \(P_{\text{FWM}}\), is given as  
\[\Omega_{P}=\varepsilon_{0}\kappa^{{}^{\prime}}E\text{cos}(\Phi t-zx)+\frac{3} {4}\varepsilon_{0}\kappa^{{}^{\prime\prime\prime}}E^{3}\text{cos}^{3}(\Phi t- zx). \tag{3}\]9 \[t=\frac{1}{2}\varepsilon_{0}n_{0}E_{0}^{2}, \tag{4}\]0 \[t=\frac{1}{2}\varepsilon_{0}n_{0}E_{0}^{2}, \tag{4}\]1  
where \(w=\lambda^{2}B^{2}DL(1-CR)(N_{ch})^{2}/16c\), \(l=\lambda^{2}\psi_{B}^{2}DL(1-CR)(N_{ch})^{2}/2c\), \(N_{ch}\) is the number of channels used in WDM-based optical fiber system, \(\gamma\) represents nonlinear coefficient and is given as  
\[t=\frac{1}{2}\varepsilon_{0}n_{0}E_{0}^{2}, \tag{4}\]2  
where \(D\) is used for dispersion of signal, \(\alpha\) explores the attenuation of optical fiber, \(L_{e}\) is the effective fiber length, \(L\), is the span of the installed fiber, number of span is denoted by \(N_{s}\), continuous frequency range is described by \(CR\), \(c\) is speed of light, \(\lambda_{eff}\) means effective wavelength and \(\psi_{B}\) depicts the bandwidth.