## 3 Analytical Modeling
Hence, based on wavelength, \(\lambda\), \(\Gamma\) and \(E\), the phase, \(\varpi\), [31] is defined as  
Figure 5:4  
Here, \(n\) represents refractive index. In terms of linear and nonlinear phenomena, (13) is written as  
Figure 5:5  
The 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, \(\theta_{0}\), Ref. [32] is given as  
Figure 5:6  
The SPM is a type of RIrNLs which causes fluctuations in the transmitted signal. In addition, XPM is another major RIrNL which happens when multi optical signals propagate over fiber simultaneously. For this effective refractive index, \(n_{e}\) is expressed as  
Figure 5:7  
Due to XPM, propagation constant \(H\) becomes power dependent, and is defined as  
Figure 5:8  
where \(H_{0}\) is linear propagation constant and \(H_{r}\) is nonlinear propagation constant. Furthermore, the nonlinear phase shift, \(\varpi_{r}\), is calculated as  
Figure 5:9  
When multiple channels are transmitted simultaneously, the nonlinear phase shift in a signal is related with co-channels. Therefore,  
\[\Omega_{P}=\varepsilon_{0}\kappa^{{}^{\prime}}E+\varepsilon_{0}\kappa^{{}^{ \prime\prime}}E^{2}+\varepsilon_{0}\kappa^{{}^{\prime\prime\prime}}E^{3}. \tag{1}\]0  
where \(\rho_{v0}\) and \(\rho_{v1}\) are the powers of two co-channels. This equation demonstrates that XPM degrades the system reliability several folds as compared to SPM. From (1) and (2), it is described that induced polarization composes of both linear and RIrNLs 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. To calculate the effect of FWM, the induced polarization is written as  
\[\Omega_{P}=\varepsilon_{0}\kappa^{{}^{\prime}}E+\varepsilon_{0}\kappa^{{}^{ \prime\prime}}E^{2}+\varepsilon_{0}\kappa^{{}^{\prime\prime\prime}}E^{3}. \tag{1}\]1 \[\Omega_{P}=\varepsilon_{0}\kappa^{{}^{\prime}}E+\varepsilon_{0}\kappa^{{}^{ \prime\prime}}E^{2}+\varepsilon_{0}\kappa^{{}^{\prime\prime\prime}}E^{3}. \tag{1}\]2 \[\Omega_{P}=\varepsilon_{0}\kappa^{{}^{\prime}}E+\varepsilon_{0}\kappa^{{}^{ \prime\prime}}E^{2}+\varepsilon_{0}\kappa^{{}^{\prime\prime\prime}}E^{3}. \tag{1}\]3 \[\Omega_{P}=\varepsilon_{0}\kappa^{{}^{\prime}}E+\varepsilon_{0}\kappa^{{}^{ \prime\prime}}E^{2}+\varepsilon_{0}\kappa^{{}^{\prime\prime\prime}}E^{3}. \tag{1}\]4 \[\Omega_{P}=\varepsilon_{0}\kappa^{{}^{\prime}}E+\varepsilon_{0}\kappa^{{}^{ \prime\prime}}E^{2}+\varepsilon_{0}\kappa^{{}^{\prime\prime\prime}}E^{3}. \tag{1}\]5  
The primary portion in the above equation represents SPM and XPM, while the secondary part denotes FWM. It explains that FWM occurs due to the mixing of different frequencies. As RIrNLs are the main causes of power penalty and system performance deduction, the proposed system mitigates their effects by employing NRZ duo-binary modulation, large effective area and uneven channel spacing. Moreover, RIrNLs can be limited by controlling nonlinear dispersion through a dispersion compensation fiber (DCF). The BER is used for the analysis of RIrNLs and phase noise parameters; this means that if these issues exist, then we do not know how many bits can be transferred including low error size per bit.