Structure of modes of a smoothly irregular integrated-optical four-layer three-dimensional waveguide

The asymptotic method and the method of coupled waves used to study an integrated-optical multilayer three-dimensional waveguide satisfying the conditions of a continuously variable effective refractive index are considered. Three-dimensional fields of smoothly deforming modes of a four-layer integrated-optical waveguide are described analytically. Explicit dependences of the contributions of the first order of smallness to the electric and magnetic field amplitudes of quasi-waveguide modes are obtained. The canonical type of quasi-wave equations describing the structure of quasi-TE and quasi-TM modes in a smoothly irregular four-layer integrated-optical three-dimensional waveguide is presented for the asymptotic method. By using the perturbation theory, shifts of complex propagation constants are obtained in an explicit form for these modes. The elaborated theory can be used to analyse structures from dielectric, magnetic and metamaterials in a rather broad wavelength range of electromagnetic waves.