Harmonic modulation of radiation of an external-feedback semiconductor laser

The appearance of the harmonic modulation regime at the Hopf bifurcation point is described analytically for a delayed-feedback semiconductor laser. The second-order delay differential equation with complex coefficients is derived. The frequency of oscillations appearing at the Hopf bifurcation point is determined by the solution of two relatively simple transcendental equations, from which the bifurcation point itself is found. These equations contain dependences on all the control parameters of the problem. The exact upper and lower limits of the oscillation frequency are found. A comparison with numerical results shows that the modulation frequency is preserved almost constant in a broad range of feedback phases. A procedure is proposed for determining the parameters of the laser providing the presence of bifurcations with a passage to oscillations with the specified frequency. The results obtained in the paper are of interest for WDM communication systems.