We study the propagation of electromagnetic waves through a transition layer between positive-index and negative-index media theoretically. We consider three models where both the dielectric permittivity epsilon and the magnetic permeability micro change linearly from positive values to negative values. At the positions where epsilon or micro vanishes, the mode conversion of the incident electromagnetic waves into longitudinal plasma oscillations can occur. Using the invariant imbedding theory of mode conversion in inhomogeneous media, we calculate the mode conversion coefficient and the electromagnetic field profile in a numerically exact manner. We find that strong mode conversion can occur for both s- and p-polarized incident waves. The dependence of mode conversion on polarization is influenced very sensitively by the spatial profiles of epsilon and micro. We also discuss the interaction effects between mode conversion phenomena occurring at two resonance points located nearby.