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

An asymptotic method is used for researching an integrated optical multilayer waveguide satisfying the condition of the smooth modification of the shape of the studied three-dimensional structure. Three-dimensional fields of smoothly deforming modes of the integrated optical waveguide are described analytically. An evident dependence of the contributions that the first order of smallness makes to the amplitudes of the electrical and magnetic fields of the quasi-waveguide modes is obtained. The canonical type of equations describing the propagation of quasi-TE and quasi-TM modes in the smoothly irregular part of a four-layer integrated optical waveguide is represented for an asymptotic method. With the help of the method of coupled waves and the perturbation theory method, the shifts of complex propagation constants for quasi-TE and quasi-TM modes are obtained in an explicit form. The elaborated theory is applicable for an analysis of similar structures of dielectric materials, magnetic materials, and metamaterials over a wide enough range of electromagnetic wavelengths.