Integral form expansion of fiber Raman amplifier problem

We propose a novel fiber Raman amplifier (FRA) modeling method based on the expansion of the complicated Raman ordinary differential equation (ODE) into a closed integral/matrix form. By taking the effective length as the interim solution and updating its value along the iteration axis, Raman gain and pump/signal evolution can be calculated with orders of magnitude increase in convergence speed at the equivalent accuracy when compared to the previous approaches based on the direct numerical method for the ODE. Application of this formalism to the problem of (i) gain prediction with a given parameters, (ii) gain spectrum engineering for the search of optimum pump power set under the given constraints, (iii) gain clamping problem for the channel reconfiguration, and (iv) derivation of analytic formula for the faster FRA dynamic control have been addressed.