Asymptotics of the far field generated by a modulated point source in a planarly layered electromagnetic waveguide

In the present work, we analyze the electromagnetic field generated by a modulated point source in a planarly layered waveguide, in the far field region. On the basis of the two-dimensional stationary phase method, we obtain expressions for the asymptotics of the field at large distance from the source and a large value of the time. The analysis relies on the eigenfunctions and eigenvalues of an auxiliary one-dimensional spectral problem, which is intimately linked to the Helmholtz equation for inhomogeneous media. In addition, from the spectral parameter power series method [Math. Meth. Appl. Sci. 2010; 33(4): 459-468], we obtain an explicit representation for the dispersion relation of the waveguide, which leads us to the allowed propagation constants and the group velocities for the guided modes. Several examples show the spectral parameter power series approach of the present analysis.