We apply a simple harmonic expansion method to the single-mode laser equations to analyze their dynamic properties. First, we extend the well-known small signal analysis to predict the transient pulsations of the relaxation oscillations. Such transients are characteristic of the laser signal relaxing towards its long-term solution, at any level of excitation, both beyond and below the instability threshold. Secondly, we extend the method to a strong harmonic expansion to analyze the properties of the long-term solutions. These properties are derived for typical examples, extending well beyond the boundary region of the instability domain, for which the laser field amplitude undergoes regular pulsations around zero-mean values.
A simple harmonic expansion method is applied to investigate some aspects of the dynamic properties of the integro-differential 'Maxwell-Bloch' equations that describe the self-pulsing regime of a single-mode inhomogeneously broadened laser. First, we show that the usual small perturbative methods only describe the transient relaxation oscillations, characteristic of the signal relaxing towards its permanent state, either stable or unstable. Second, we show that an adapted strong harmonic expansion analysis applied to the permanent pulsing solutions yields an accurate evaluation of the pulsing frequencies.
Abstract:We have applied harmonic expansion to derive an analytical solution for the Lorenz-Haken equations. This method is used to describe the regular and periodic self-pulsing regime of the single mode homogeneously broadened laser. These periodic solutions emerge when the ratio of the population decay rate ℘ is smaller than 0 11. We have also demonstrated the tendency of the Lorenz-Haken dissipative system to behave periodic for a characteristic pumping rate "2C P " [7], close to the second laser threshold "2C 2 "(threshold of instability). When the pumping parameter "2C " increases, the laser undergoes a period doubling sequence. This cascade of period doubling leads towards chaos. We study this type of solutions and indicate the zone of the control parameters for which the system undergoes irregular pulsing solutions. We had previously applied this analytical procedure to derive the amplitude of the first, third and fifth order harmonics for the laser-field expansion [7,17]. In this work, we extend this method in the aim of obtaining the higher harmonics. We show that this iterative method is indeed limited to the fifth order, and that above, the obtained analytical solution diverges from the numerical direct resolution of the equations.
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