  A new rigorous approach for precise and efficient calculation of light
propagation along non-uniform waveguides is presented. Resonant states of a
uniform waveguide, which satisfy outgoing-wave boundary conditions, form a
natural basis for expansion of the local electromagnetic field. Using such an
expansion at fixed frequency, we convert the wave equation for light
propagation in a non-uniform waveguide into an ordinary second-order matrix
differential equation for the expansion coefficients depending on the
coordinate along the waveguide. We illustrate the method on several examples of
non-uniform planar waveguides and evaluate its efficiency compared to the
aperiodic Fourier modal method and the finite element method, showing
improvements of one to four orders of magnitude. A similar improvement can be
expected also for applications in other fields of physics showing wave
phenomena, such as acoustics and quantum mechanics.
