## 4 Simulation Model of DQPSK Modulated Data Transmission  
In this study, computer simulation based on MATLAB programming was used to mathematically analyze and numerically simulate the data transmission in an unrepeated ultra-long span system. In a single mode optical fiber, the complex field envelope of the slow-varying pico-second range optical pulse is governed by the non-linear Schrodinger equation (NLSE), a nonlinear partial differential equation that provides an analytical solution only in specific cases. The numerical method most commonly used to investigate the dispersive nonlinear effect of optical wave propagation along a transmission fiber is the split-step Fourier method (SSFM) due to its simplicity for implementation and high computational efficiency [29]. The efficiency of the SSFM depends on both the time-domain or frequency-domain resolution and on the distribution of step sizes along the fiber. In simulations of optical-fiber transmission systems, the time and frequency resolutions are determined by the bandwidth of the signal and the number of bits that are to be propagated through the system respectively. In the SSFM model, the fiber is divided into many small sections that are short enough that the effect of Kerr nonlinearity and dispersion \begin{table}
\begin{tabular}{l c c} \hline \hline
**Structure** & **NLPS (rad)** & **OSNR (dB)** \\ \hline Conventional single fiber core & 3.44 & 27.06 \\ Three-segment hybrid fiber core & 2.15 & 27.37 \\ Improvement & 1.29 & 0.31 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Summary of performance improvement result. Figure 6: Optical signal-to-noise ratio (OSNR) of the three-segment hybrid fiber core structure compared to the conventional single fiber core structure. The hybrid fiber core structure has a higher delivered OSNR at the received point, which enhances the signal performance. The inset figure shows the profile along the entire propagation span. can be calculated separately. In the first step, the NLSE is used to solve the nonlinear Kerr effect term by ignoring the dispersion term and in the next step, the NLSE is used to solve the dispersion term by ignoring the nonlinear Kerr effect term. The interplay between nonlinearity and dispersion is thus indicated for each step after the completion of each computation pair. The introduction of symmetrized-SSF model can improve the computational efficiency requirement for the simulation and optimization of fiber-optic communication systems [30]. The optical pulse field envelope \(U(z,t)\) propagating at normalized distance \(z\) in single mode optical fiber is governed by the NLSE and can be expressed as [23]  
\[\frac{\partial U(z,t)}{\partial z}+i\frac{\beta_{2}}{2}\frac{\partial^{2}U(z,t )}{\partial t^{2}}-i\gamma(z)|U(z,t)|^{2}U(z,t)=\frac{g(z)}{2}U(z,t), \tag{13}\]  
where \(t\) represents the normalized time in the retarded frame co-moving with the group velocity of the envelope. The optical characteristics of the transmission fiber depend on the group velocity dispersion \(\beta_{2}\), the nonlinear coefficient \(\gamma(z)\) of the fiber due to the Kerr self-focusing effect, and the distributed gain \(g(z)\) obtained from the distributed Raman amplification where the passive loss of the propagation fiber is already included. Even though Equation (13) does not provide a complete physical description of a system, it is the basis for modeling optical-fiber communications systems. Because a bidirectional pumped distributed Raman amplifier is used, the optical pulse propagation along the transmission fiber can be modeled by integrating the calculation of the impact of dispersion, the effect of Kerr nonlinearity, the fiber loss, and the Raman gain profile. Each orthogonal branch of a DQPSK modulator encodes information on two sets of binary phase changes between two adjacent bits, and uses differential precoding as a simple alternative compared to more complex encoding and decoding scheme in the coherent system.