Normal-form theory for a laser model with periodic signal and noise injection

A model for a laser with modulated signal and noise in the semiclassical approach allows the characterization of a Hopf bifurcation point and correspondingly of its spectrum. By means of the normal-form theory we have been able to give explicit normal-form equations that describe the system’s dynamic in the vicinity of the bifurcation point. At first order we determine the system’s behavior for the case when the signal modulation is coincident with the Hopf frequency and when it does not. In both cases, a characterization of the unfolding parameter is possible. We also consider an injected laser the Hopf frequency of which is an integer multiple of order n of the frequency modulated signal. In this case a perturbative theory of order n is necessary. We present also an example of noise for which this system does not present cooperative amplification. Numerical analysis is consistent with this prediction.