Hopf bifurcation at the phase-locking point of an externally driven, homogeneously broadened laser

A model describing the dynamics of a white- and colored-noise injected-signal, single-mode, homogeneously broadened laser in the semiclassical limit is presented. The system has a Hopf bifurcation above the threshold where phase locking occurs. This bifurcation point is characterized by a simple relation between gain and loss parameters, detuning, and external-driving-field intensity. The stochastic normal form of the system near the bifurcation point is found and the contribution of the harmonics of the bifurcation frequency in the spectrum is thus analytically determined. As a result of gain saturation and phase-excursion phenomena the resonances become apparent before reaching the bifurcation point for low driving-force values. Above this bifurcation point, the spectrum is that of a frequency-locked laser in the presence of noise. These results are consistent with numerical experiments carried out with both white and colored noise. In the case of colored noise, resonances are wider than in the white-noise case, depending on the correlation time of the multiplicative and the additive terms into which noise has been decomposed.