Mode dynamics in nonuniform waveguide arrays: a graph Laplacian approach

Abstract

Light propagation in nonlinear waveguide arrays displays rich phenomena with multiple applications. Technological advances present opportunities to study distinct dynamical features for a wide range of one-dimensional and two-dimensional arrays. The traditional research on arrays considers the individual elements to be identical. Here we study coupled arrays of non-identical waveguides in geometries that account for nonuniform coupling strengths. The model is now a generalized nonlinear Schrödinger equation where the usual Laplacian is replaced by the graph Laplacian. We show that since this matrix is symmetric, the dynamics is naturally described using its eigenvectors as a basis. When the eigenvalues are all distinct, we show that there is no resonant transfer of energy between the different eigenmodes. We illustrate this on a simple graph.