Analysis of optical waveguides with arbitrary index profile using an immersed interface method

A numerical technique is described that can efficiently compute solutions in interface problems. These are problems with data, such as the coefficients of differential equations, discontinuous or even singular across one or more interfaces. A prime example of these problems are optical waveguides and as such the scheme is applied to Maxwell's equations as they are formulated to describe light confinement in Bragg fibers. It is based on standard finite differences appropriately modified to take into account all possible discontinuities across the waveguide's interfaces due to the change of the refractive index. Second and fourth order schemes are described with additional adaptations to handle matrix eigenvalue problems, demanding geometries and defects

Perturbations of Dark Solitons

A method for approximating dark soliton solutions of the nonlinear Schrodinger equation under the influence of perturbations is presented. The problem is broken into an inner region, where core of the soliton resides, and an outer region, which evolves independently of the soliton. It is shown that a shelf develops around the soliton which propagates with speed determined by the background intensity. Integral relations obtained from the conservation laws of the nonlinear Schrodinger equation are used to approximate the shape of the shelf. The analysis is developed for both constant and slowly evolving backgrounds. A number of problems are investigated including linear and nonlinear damping type perturbations