A theoretical study of electromagnetic guided eigenwaves in two- and three-layered plates with metallized surfaces is accomplished. The appropriate dispersion equations are explicitly analyzed on the basis of some deiscretization, first introduced in Mindlin's theory of Lamb acoustic waves. It is shown that the dispersion branches of independent eigenmode families cross each other in the nodes of some grid formed by two infinite series of bond lines. The latter represent the dispersion curves for homogeneous plates with permittivities varepsilon₁ or varepsilon₂ coinciding with those for the layers of the wavegiude. It is proved that, beyond nodes of the grid, the dispersion curves may not intersect bond lines, which thus provide definite "corridors" for these curves. The dispersion lines have a wavy ("zigzag") form in the grid zone and remain smooth beyond the grid. The crossing branches have coinciding cutoff frequencies. In the dimensionless coordinates "slowness (S) vs frequency (f)" the branches S_l(f) have two asymptotic levels: S=sqrt(varepsilon₁)sqrt and S=sqrt(varepsilon₂)sqrt. At the lower one, the spectrum forms a step-like terracing pattern with a progressive closing to the asymptote of a succession of dispersion curves, which change each other at this level with further going up to the next asymptote. An extension to anisotropic waveguides with layers made of uniaxial crystals is considered. PACS: 41.20.Jb, 42.25.Gy

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