## 1 Introduction  
Over the years, there has been continual exponential growth in the demand for optical fiber transmission for applications in the areas of inter-island hopping and coastal festoon systems; additionally, the use of single ultra-long span terrestrial links in the range of several hundred kilometers is rising. Over medium-range transmission distances, ultra-long span unrepeetered systems that use distributed Raman amplification represent an uncomplicated and cost-effective solution that does not require in-line active components. Improvements in fiber optic core design and material characteristics [1], including the availability of large effective core area fibers [2] and very-low loss fibers [3], have enhanced the transmission performance and extended the propagation distance of unrepeetered systems without additional in-line active devices. In order to restore signal power at the receiver end after propagating several hundred kilometers, unrepeetered ultra-long single-span systems require high Raman gains in both forward and backward directions to compensate for very large accumulated fiber attenuation [4; 5]. There are two major limiting factors in such systems: the first factor is nonlinear Kerr effect-induced nonlinearity, such as self-phase modulation in a single-channel systems, cross-phase modulation, and four-wave mixing for a multi-channel systems; the second factor is optical signal-to-noise ratio (OSNR) degradation due to spontaneous Raman noise accumulation caused by high Raman pump power from both the forward and backward directions [6; 7]. Figure 1 illustrates the typical profile of signal power transmission along an ultra-long span based on [8] with modifications and recalculations for various pump power ratios over a very-long span length (in this case 390 km); additional details will be provided in the following sections. Figure 1 also shows the upper and lower signal power boundaries as two limiting factors that need to be addressed to improve transmission quality and provide an achievable propagation distance. The change of pump power ratio, which is the ratio between the forward pump power over the total pump power, will only shift the signal power profile in the vertical-axis direction, which means a trade-off between the Kerr nonlinear limit and the Raman noise limit will still exist. Therefore, to improve transmission performance, a solution that can modify the gain profile of bidirectional distributed Raman amplification and simultaneously reduce the Kerr nonlinear limit and the Raman noise limit is required. In this article, a hybrid fiber effective core area consisting of a three-segment structure is proposed as an alternative to the conventional single fiber effective core area in a single-segment structure. The introduction of a three-segment hybrid fiber effective core area is expected to increase flexibility in the Raman gain profile structure, reduce the negative impact of the nonlinear Kerr effect due to high signal power, and improve the OSNR by means of the reduction of Raman optical pump-induced amplified spontaneous emissions. In order to directly analyze the influence of the two factors, we built an analytical model covering mathematical formulation, analysis of bidirectional Raman amplification, and data transmission simulation on various data rates. The current model using a single-channel signal amplified by single-wavelength Raman pumps, which can be expanded to a multi-channel and multi-wavelength Raman pumps system based on the same principle. The system design for the overall length is based on the previous reports on the ultra-long span systems. The span configuration without an in-line active device has been reported as over 304 [2], 333.6 [9], and 365 km [3] for various bit rates. The span configuration with in-line active support has been reported as over 402 km [10] using in-line remote optically pumped amplifier (ROPA), and as over 500 km [11] using parallel fibers for pump power delivery.