Electromagnetic signals in a waveguide filled with an inhomogeneous time-variant medium

The problem of arbitrary signal propagation in inhomogeneously filled waveguides (with factorized permittivity and permeability) is solved by the partial separation of variables in the time domain (the Evolutionary Waveguide Equations Approach). We introduce two self-adjoined operators and then the electromagnetic field is expanded into a series of eigenmodes of these operators with dependence on z and t modal coefficients sought for. Such an approach allows one to consider time-variant and nonlinear media as well. Assuming a time-harmonic signal, one can use this technique to find the dispersion characteristic of the waveguide without any need to solve the boundary problem for every frequency (lossy media can be treated in this way without extra difficulties). After some generalization this approach can be applied to fiber optics (dielectric waveguides).