A numerical study of electromagnetic waves in periodic waveguides

Here are considered time‐harmonic electromagnetic waves in a quadratic waveguide consisting of a periodic dielectric core enclosed by conducting walls. The permittivity function may be smooth or have jumps. The electromagnetic field is given by a magnetic vector potential in Lorenz gauge, and defined on a Floquet cell. The Helmholtz operator is approximated by a Chebyshev collocation, Fourier–Galerkin method. Laurent's rule and the inverse rule are employed for the representation of Fourier coefficients of products of functions. The computations yield, for known wavenumbers, values of the first few eigenfrequencies of the field. In general, the dispersion curves exhibit band gaps. Field patterns are identified as transverse electric, TE, transverse magnetic, TM, or hybrid modes. Maxwell's equations are fulfilled. A few trivial solutions appear when the permittivity varies in the guiding direction and across it. The results of the present method are consistent with exact results and with those obtained by a low‐order finite element software. The present method is more efficient than the low‐order finite element method. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 490–513, 2014