Imaginary branches of SAW slowness curves

The imaginary branches of surface acoustic waves (SAW) slowness curves need to be known in many modal or spectral methods that have been developed to account for waveguides or diffraction based on the angular spectrum of waves approach. In the case of true, i.e., lossless, SAW, it is shown that the imaginary branches can be obtained by a search in the complex transverse slowness plane as a function of the propagation slowness. As a side result, the parabolic approximation is compared with the exact solution and it turns out that its quality depends dramatically on the particular material cut considered. When trying to extend the method to pseudo or leaky SAW, we face difficulties in the process of identifying a solution. These difficulties are two-fold, with possible problems in the partial waves selection or the appearance of amplification rather than attenuation of SAW in the effective permittivity computation.