On approximate analytical solutions of rate equations for studying transient spectra of injection lasers

We discuss the feasibility of constructing approximate, analytical solutions of the rate equations for an injection laser with several longitudinal modes. By adjusting a few empirical parameters it is possible to reproduce the principal features of the exact numerical solutions for the electron and photon densities and to provide explicit expressions for the frequency and time decay rates of the relaxation oscillations. The solutions show that the relative powers of the modes change continually during the transient period. Initially, many longitudinal modes begin to oscillate with nearly equal amplitudes which decay at different rates until they settle down to form a Lorentzian, steady-state distribution. In short lasers ( m) this behavior permits a single mode to establish itself rapidly, leading to single wavelength operation. The decay rates of the longitudinal laser modes are found to be inversely proportional to their corresponding steady-state mode amplitudes.