## 2 Proposed Model Layout
#### Refractive Index Related Non-Linearities  
The induced non-linearities in optical systems are composed of two types; the first type relates to nonlinearities due to scattering phenomena [19], and the second type of nonlinearities are associated with refractive index of the optical fiber. The second type of nonlinearities are known as RIrNLs. FWM, XPM and SPM nonlinear factors are the main subparts of RIrNLS. A brief explanation of each nonlinear parameter is given below.  
Four wave mixing (FWM) is an intermodulation incident that occurs inside an optical fiber [20], whereby interaction among the channel frequencies generate new unwanted frequencies, as shown in Figure 3. It is particularly important for WDM-based OTN, where multiple wavelengths travel through the same fiber. From Figure 3, it is clear that \(\vartheta_{/00}\), \(\vartheta_{/01}\), and \(\vartheta_{/02}\) are induced unwanted pulses with original \(\vartheta_{/ji}\), \(\vartheta_{/jj}\) and \(\vartheta_{/kk}\) imparted optical signals, respectively. FWM nonlinearity is generated due to the third order susceptibility [21], when high power is launched to ensure the propagation of high capacity optical signals over long distances. The distortion caused by FWM is worse than the other two subtypes of RIrNLs.  
Another phenomena which is important for WDM-based systems is the cross phase modulation (XPM) which happens due to variation in refractive index of the optical fiber  
Figure 3: Induction of unwanted frequencies due to four wave mixing (FWM) effect.  
Figure 2: Procedure of frequency domain multiple in multiple out (FD-MIMO) equalizer.  
along the transmission length. Phase shift in the optical pulse is induced due to alternating refractive index of the fiber as the signal travels through it. This phenomenon leads to a change in optical signal frequency spectrum as shown in Figure 4. This effect is directly proportional to the number of channels passing inside the SMF [22]. Figure 4 depicts that the output signal is disturbed due to XPM factor. As the number of channels in a WDM framework changes dynamically, an adaptive equalization is required to cancel the XPM influence on optical signals.  
An associated kind of RItNLs is the self phase modulation (SPM), caused due to the self modulation of an optical signal in a SMF. It leads to spectral broadening [23] and phase delay [24], as shown in Figure 5. Though it is not directly caused by the WDM transmission, the effect of SPM increases with the launched power levels and the optical pulse spreads and tends to overlap the neighboring optical channels. Consequently, a particular signal of interest is not distinguished by the receiver for WDM channels. Moreover, the penalty in power is also caused on account of SPM.  
XPM, SPM and FWM are collectively addressed as RItNs, because these are generated due to the variation in the refractive index. As these effects have strong contribution in degradation of a WDM system, the proposed model aims to minimize them through mathematical modeling and verification through simulations. Features, such as duo-binary modulation, channel spacing, nonlinear effective area and refractive index, and nonlinear dispersion are discussed in this paper to investigate RItNLs. This section demonstrates the framework of the proposed work, while the next section concentrates on the analytical model and the proposed solution.