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A method of analysis for electromagnetic wave phenomena in multilaiyered media consisting of bianisotropic slabs with periodically inhomogeneous material parameters is presented. The analysis relies on a transformation of Maxwell's equations into an equivalent eigenoperator equation for the transverse components of the field vectors, which is Fourier transformed into an algebraic eigenvalue equation in spectral domain. The fields in the inhalmogeneous layers are then expressed in terms of eigenmodes, where the expansion coefficients are determined by imposing internal and external boundary conditions. The proposed method lis verified by comparing numerical results for ordinary isotropic-media problems with corresponding results obtained previously by different methods. The applicability and convergence behavior of the approach is illustrated for typical sample applications involving scattering- and guided-wave problems.