28 research outputs found
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