Orthogonal solutions for asymmetric strongly coupled waveguide arrays: an elegant, analytical approach

Abstract

Silicon-on-insulator (SOI) waveguides are the focus of present-day photonics due to their smaller device footprint and compatibility with CMOS technology. Coupled waveguide arrays have been widely analyzed by the simple yet intuitive coupled-mode theory. However, for asymmetric strongly coupled waveguide arrays, the supermodes obtained by coupled-mode theory are not orthogonal solutions, thereby introducing error in the power distribution in the array. We present an elegant, completely generalized three-step analytical methodology for obtaining accurate orthogonal supermodes. Once the propagation characteristics of the individual waveguides are obtained, an orthogonal basis set, which is formed by a linear combination of the modal fields of the individual waveguides, is generated using the Gram–Schmidt orthogonalization procedure. The Ritz–Galerkin variational method, with the trial field as an expansion in terms of the newly formed orthogonal basis set, is then used to obtain the modes of the coupled waveguide array. The procedure is illustrated by use on strongly coupled longitudinally homogenous 2D asymmetric SOI waveguide arrays. The TE modal solutions obtained for the waveguide arrays are exactly orthogonal to one another. Since the new orthogonal basis set is essentially a linear combination of the modal solutions of the individual waveguides, the quantities involved in the analysis are analytically identical to those defined as coupling coefficients and overlap integrals in the conventional coupled-mode theory. The excitation and power distribution along the propagation distance obtained from the proposed method is simple and accurate. The theory presents itself as a strong alternative to numerically intensive and time-consuming techniques that are frequently employed for the analysis of such structures.