Asymptotic radiation field of asymmetric planar dielectric waveguide

In the spectral domain analysis of practical open integrated waveguiding structures, multiple non-removable branch points occur in the axial Fourier-transform plane. When the inversion integral for the distant radiation field is approximated asymptotically by deforming the real-axis integration contour (C) into a steepest-descent contour (SDC) all of the associated branch cuts are crossed twice. Passage from the top Riemann sheet to lower sheets and back to the top sheet occurs so both ends of the contour lie on the top sheet and C can be directly connected into SDC. If the observation aspect angle is sufficiently large, however, all but one of the cuts are crossed a third time and the two ends of the SDC lie on different sheets. Deformation of C into SDC then requires a fourth crossing of those cuts and integration around them to remain on the top sheet. A continuous spectrum contribution consequently augments the saddle-point term contributed by the SDC approximation. These wave phenomena are studied in detail for the asymmetric planar dielectric waveguide, which is the simplest canonical structure for which multiple non-removable branch points occur