Numerical calculation of the linewidth of Raman fiber lasers due to spontaneous Raman scattering

We calculate the spectral width of the Stokes output of Raman fiber lasers determined by spontaneous Raman scattering. The conventional model with zero bandwidths of the pump and Stokes lines is extended so that it describes a frequency-resolved Stokes band driven by the pump wave through both stimulated and spontaneous Raman scattering. To solve the boundary value problem for the Stokes spectra, a semi-analytic method is developed which reduces the spectrally resolved problem to an equivalent conventional model based on effective mirror reflectivities. The Stokes spectra are given in closed form in terms of the mirror reflection spectra. Spontaneous Raman scattering perceptibly broadens the RFL output spectra for pump powers around the lasing threshold.