Abstract
It has been known for some time that there are systems in hydrodynamics where perturbations are able to extract very efficiently energy from a background state even when such background state is asymptotically stable. This results in strong transient amplification of the perturbations that are not (and cannot be) predicted by standard linear stability analysis. Transient noise amplification and other discrepancies between linearization theories and experiments are explained by the fact that the eigenvectors of the linear stability operator are not orthogonal to one another [1]. Similar phenomena are known in optics as transient gain and are usually associated to laser systems with cavity geometries such that the cavity modes are not orthogonal. In this work we show that there can be transient growth of perturbations of a lasing state even for laser systems with orthogonal cavity modes. We show that whenever the background state is a lasing state, the linear stability operator has, in general, non–orthogonal modes. This means that the short term dynamics and the spectrum of the perturbations are always different from those predicted by a standard stability analysis. It does not, however, guarantee the presence of transient gain in the region where the background state is stable. We show the universality of this phenomenon by proving that the linear stability operator of the complex Swift-Hohenberg equations (CSH) for semiconductor lasers [2] always presents non–orthogonal eigenvectors when the background state is a lasing state.
© 2007 IEEE
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