We investigated spatio-temporal dynamics of a homogeneously broadened single-mode when diffraction is taken into account. In order to do this, we combined the temporal approach based on non-linear dynamical systems theory with the spatial approach which mainly uses spatial Fourier spectra. The Maxwell-Bloch equations describing this laser have progressive plane wave stationary solutions. To numerically integrate the equations, we developed a first numerical method based on Picard iterates. For the requirements of the dynamical analysis (longtime series), a second method has been implemented as a chain of oscillators coupled by the diffraction term. Hereafter, the effects of the pump profile and the boundary conditions on the stability of the solutions are studied. For gaussian and super-gaussian profiles, the instability domain disappears.
Dynamical regimes of the laser are characterized using a topological analysis. We showed that the dynamics provided by the two integration techniques are topologically equivalent. The influence of the diffraction parameter on the dynamics is then analyzed, and an effect similar to a noise perturbation is observed when this parameter is increased.
Above the second threshold, we investigated the spatio-temporal dynamics of the laser in a "good cavity" configuration. A particular intermittency, structured around few unstable periodic orbits always visited in the same order and according to a given sequence of wave vectors, was observed. Evolution of this dynamics versus the pumping parameter and the detuning are fully described.
When the active medium is dense, dipole-dipole interactions between atoms become important and the Maxwell-Bloch equations are modified by the local field corrections. These interactions, quantified by constant b, modify the stability of the solutions. In the instability domains, we showed that the intermittency subsists for small values of b but cannot remain for higher values.
Finally, using a reductive perturbation method, we established evolution equations for the envelopes of the progressive wave solutions close to laser threshold. Non-linear transport equations were obtained, and the analysis revealed the existence of source points, transition points where progressive waves disappear, and periodic patterns. We showed that such structures also exist when local field correction is taken into account. The latter leads to an intensity-dependent modulation of the spatial period of the patterns.
This Ph’D thesis can be downloaded here