## 2 Modeling of Generalized Mathematical Formulas for Multi-Segment Transmission Link  
In order to provide proper gain to overcome fiber attenuation and improve the noise performance of the optical transmission system, distributed Raman amplification has many advantages compared to conventional lump amplification, such as with erbium-doped fiber amplifiers or semiconductor optical amplifiers [1]. Distributed Raman amplifications can be placed on the transmitter side to work as forward pumped configurations and on the receiver side to work as backward pumped configurations. A bidirectional pumped configuration can be constructed by combining both forward and backward pumped configurations. The Raman gain depends primarily on the material and the optical characteristics of the transmission fibers, such as the Raman gain coefficient, the size of the effective core area, and the attenuation at the signal and pump wavelengths [4; 5]. Figure 2 is a schematic diagram of unrepeatered ultra-long single span transmission using a novel hybrid fiber effective core area structure that combines three segments of optical fiber consisting of different fiber effective core area and configured as a bidirectional pumped Raman amplified transmission span. In this model, the transmission span is divided into three segments and consists of three types of fiber cores. The choice of fiber for the individual fiber segment is based on the following design objective: the first segment (core #1) uses a large-sized core to reduce the nonlinear Kerr effect; the second segment (core #2) uses a medium-sized core to function as the major transmission medium and has the longest distance compared to the other segments; and the third segment (core #3) uses a small-sized core to optimize the Raman gain when the signal power is already weak after long-distance propagation. A special splicing technique is available to minimize signal distortion at the joining point between two different fiber core sizes. All fibers are selected from actual fibers available on the market, in this case, Vascade(r) EX3000, Vascade(r) EX1000 and SMF-28 ULL fibers (Corning Inc., Corning, NY, USA). The fibers follow step-index single-mode fiber (SMF) characteristics at the transmission signal wavelength where optical fiber normalized frequency (V parameter) has to value less than 2.405 and defined as [23]  
Figure 2: Schematic diagram of a bidirectional pumped Raman amplified span using three segments of optical fiber consisting of different fiber effective core area. Core #1 contributes to nonlinear Kerr effect reduction, core #2 works as a major transmission medium, and core #3 contributes to Raman gain optimization. \[\mathrm{V}=\frac{2\pi a}{\lambda}\left(n_{1}^{2}-n_{2}^{2}\right)^{1/2}<2.405, \tag{1}\]  
where \(n_{1}\) and \(n_{2}\) are the refractive index of core and cladding, \(a\) is the fiber core radius, and \(\lambda\) is the signal wavelength. To maintain the characteristics of a single mode, the larger the size of the core, the smaller the refractive index difference between the core and cladding so that it requires more precise thickness control during fiber manufacturing. The proposed three-segment hybrid fiber core structure is expected to increase flexibility in the Raman gain profile, reduce the negative impact of the nonlinear Kerr effect, and improve the OSNR that could be translated to the improvement on the link capacity of conventional single mode fiber. To overcome the issue of capacity limits in the optical fiber infrastructure, multicore fiber (MCF) is one of the most promising technologies for increasing the spatial efficiency within the available fiber cross-section and reducing the footprint of network equipment [24]. In general, distributed Raman amplification systems generate three types of optical noise: amplified spontaneous emission (ASE) noise, double Rayleigh backscattering (DRS) noise, and relative intensity noise (RIN). These three types of noise have distinct characteristics and different impacts on the system performance.