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Linear and nonlinear waves in surface and wedge index potentials
Get PDFWe study optical beams that are supported at the surface of a medium with a linear index potential and by a piecewise linear wedge type potential. In the linear limit the modes are described by Airy functions. In the nonlinear regime we find families of solutions that bifurcate from the linear modes and study their stability for both self-focusing and self-defocusing Kerr nonlinearity. The total power of such nonlinear waves is finite without the need for apodization
Observation and Optical Tailoring of Photonic Lattice Filaments
No full textWe demonstrate, for the first time, that photonic lattices support a new type of laser filaments, called lattice filaments (LF). The LF attributes (length, width, and intensity) can be tailored by both varying the photonic lattice properties and also dynamically through the interaction between filaments. This opens the way for extensive all-optical control of the nonlinear propagation of intense ultrafast wave packets. Our approach is generic and applicable to all transparent media, with potential strong impact on various photonic applications
Sparse implicitization by interpolation: Characterizing non-exactness and an application to computing discriminants
Get PDFWe revisit implicitization by interpolation in order to examine its properties in the context of sparse elimination theory. Based on the computation of a superset of the implicit support, implicitization is reduced to computing the nullspace of a numeric matrix. The approach is applicable to polynomial and rational parameterizations of curves and (hyper)surfaces of any dimension, including the case of parameterizations with base points.
Our support prediction is based on sparse (or toric) resultant theory, in order to exploit the sparsity of the input and the output. Our method may yield a multiple of the implicit equation: we characterize and quantify this situation by relating the nullspace dimension to the predicted support and its geometry. In this case, we obtain more than one multiples of the implicit equation; the latter can be obtained via multivariate polynomial gcd (or factoring).
All of the above techniques extend to the case of approximate computation, thus yielding a method of sparse approximate implicitization, which is important in tackling larger problems. We discuss our publicly available Maple implementation through several examples, including the benchmark of bicubic surface.
For a novel application, we focus on computing the discriminant of a multivariate polynomial, which characterizes the existence of multiple roots and generalizes the resultant of a polynomial system.
This yields an efficient, output-sensitive algorithm for
computing the discriminant polynomial
Caustic design in periodic lattices
Get PDFWe study curved trajectory dynamics and design in discrete array settings. We find that beams with power law phases produce curved caustics associated with the fold and cusp type catastrophes. A parabolic phase produces a focus that suffers from spherical aberrations. More importantly, we find that by designing the initial phase or wavefront of the beam we can construct trajectories with pure power law caustics as well as aberration-free focusing of discrete waves
Energy consistent DG methods for the Navier Stokes Korteweg system
Get PDFWe design consistent discontinuous Galerkin finite element schemes for the approximation of the Euler Korteweg and the Navier Stokes Korteweg systems. We show that the scheme for the Euler Korteweg system is energy and mass conservative and that the scheme for the Navier Stokes Korteweg system is mass conservative and monotonically energy dissipative. In this case the dissipation is isolated to viscous efects, that is, there is no numerical dissipation. In this sense the methods is consistent with the energy dissipation of the continuous PDE systems
Error control for time-splitting spectral approximations of the semiclassical Schrödinger equation
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