Stability analysis of Raman propagation equations

Raman Fiber Amplifiers (RFA) are assuredly one of the most seriously renewed research subjects in the field of optical fiber communication systems. The amplification is based on stimulated Raman scattering (SRS), which occurs when there is enough pumping power within the fiber transferred to the signals. In the simplest situation, only the interaction between the pumping lasers and the probe signals needs to be accounted being the system power evolution along the distance given by a set of ordinary nonlinear ordinary differential equations (ODE). Although some consistent work has been done to provide numerical solutions for both forward and backward pumping configurations, the nonlinearity of the equations disable the computation of analytical solutions. However, efforts that lead to qualitative understanding of the solution rather than detailed quantitative information are quite valuable. This approach is geometrical and deals with the topic of stability. In this paper, we present a qualitative study of Raman equations valid for forward and backward pumping. The origin was proved to be an asymptotically stable node. The determination of the second equilibrium point is not straightforward for a generic number of equations, but some simplifications can be done.