Laplace-Fourier analysis and instabilities of a gainy slab

The idealization of monochromatic plane waves leads to considerable simplifications in the analysis of electromagnetic systems. However, for active systems this idealization may be dangerous due to the presence of growing waves. Here we consider a gainy slab, and use a realistic incident beam, which is both causal and has finite width. This clarifies some apparent paradoxes arising from earlier analyses of this setup. In general it turns out to be necessary to involve complex frequencies $\omega$ and/or complex transversal wavenumbers $k_x$. Simultaneously real $\omega$ and $k_x$ cannot describe amplified waves in a slab which is infinite in the transversal direction. We also show that the only possibility to have an absolute instability for a finite width beam, is if a normally incident plane wave would experience an instability.