1,011 research outputs found
A High-Order Numerical Method for the Nonlinear Helmholtz Equation in Multidimensional Layered Media
We present a novel computational methodology for solving the scalar nonlinear
Helmholtz equation (NLH) that governs the propagation of laser light in Kerr
dielectrics. The methodology addresses two well-known challenges in nonlinear
optics: Singular behavior of solutions when the scattering in the medium is
assumed predominantly forward (paraxial regime), and the presence of
discontinuities in the % linear and nonlinear optical properties of the medium.
Specifically, we consider a slab of nonlinear material which may be grated in
the direction of propagation and which is immersed in a linear medium as a
whole. The key components of the methodology are a semi-compact high-order
finite-difference scheme that maintains accuracy across the discontinuities and
enables sub-wavelength resolution on large domains at a tolerable cost, a
nonlocal two-way artificial boundary condition (ABC) that simultaneously
facilitates the reflectionless propagation of the outgoing waves and forward
propagation of the given incoming waves, and a nonlinear solver based on
Newton's method.
The proposed methodology combines and substantially extends the capabilities
of our previous techniques built for 1Dand for multi-D. It facilitates a direct
numerical study of nonparaxial propagation and goes well beyond the approaches
in the literature based on the "augmented" paraxial models. In particular, it
provides the first ever evidence that the singularity of the solution indeed
disappears in the scalar NLH model that includes the nonparaxial effects. It
also enables simulation of the wavelength-width spatial solitons, as well as of
the counter-propagating solitons.Comment: 40 pages, 10 figure
High-order numerical method for the nonlinear Helmholtz equation with material discontinuities in one space dimension
The nonlinear Helmholtz equation (NLH) models the propagation of
electromagnetic waves in Kerr media, and describes a range of important
phenomena in nonlinear optics and in other areas. In our previous work, we
developed a fourth order method for its numerical solution that involved an
iterative solver based on freezing the nonlinearity. The method enabled a
direct simulation of nonlinear self-focusing in the nonparaxial regime, and a
quantitative prediction of backscattering. However, our simulations showed that
there is a threshold value for the magnitude of the nonlinearity, above which
the iterations diverge. In this study, we numerically solve the one-dimensional
NLH using a Newton-type nonlinear solver. Because the Kerr nonlinearity
contains absolute values of the field, the NLH has to be recast as a system of
two real equations in order to apply Newton's method. Our numerical simulations
show that Newton's method converges rapidly and, in contradistinction with the
iterations based on freezing the nonlinearity, enables computations for very
high levels of nonlinearity. In addition, we introduce a novel compact
finite-volume fourth order discretization for the NLH with material
discontinuities.The one-dimensional results of the current paper create a
foundation for the analysis of multi-dimensional problems in the future.Comment: 47 pages, 8 figure
Determining Geological Properties by a Hybrid Seismic-Magnetotelluric Approach
This paper concerns the controlled source audio magnetotelluric technique (CSAMT) for imaging subsurface structure. Given the short time available, we limited our investigation to a simple 1D earth model where regional seismic and well logs suggest discrete layers, each with constant seismic velocity and constant electrical conductivity. In addition, the well logs provide rough estimates of velocity and conductivity for use as a starting point in the seismic and MT inversions
Invisibility and Inverse Problems
This survey of recent developments in cloaking and transformation optics is
an expanded version of the lecture by Gunther Uhlmann at the 2008 Annual
Meeting of the American Mathematical Society.Comment: 68 pages, 12 figures. To appear in the Bulletin of the AM
Higher-order finite element methods for the nonlinear Helmholtz equation
In this work, we analyze the finite element method with arbitrary but fixed polynomial degree for the nonlinear Helmholtz equation with impedance boundary conditions. We show well-posedness and (pre-asymptotic) error estimates of the finite element solution under a resolution condition between the wave number , the mesh size and the polynomial degree p of the form “ sufficiently small” and a so-called smallness of the data assumption. For the latter, we prove that the logarithmic dependence in from the case in [H. Wu, J. Zou, SIAM J. Numer. Anal. 56(3): 1338-1359, 2018] can be removed for . We show convergence of two different fixed-point iteration schemes. Numerical experiments illustrate our theoretical results and compare the robustness of the iteration schemes with respect to the size of the nonlinearity and the right-hand side data
Multiscale scattering in nonlinear Kerr-type media
We propose a multiscale approach for a nonlinear Helmholtz problem with possible oscillations in the Kerr coefficient, the refractive index, and the diffusion coefficient. The method does not rely on structural assumptions on the coefficients and combines the multiscale technique known as Localized Orthogonal Decomposition with an adaptive iterative approximation of the nonlinearity. We rigorously analyze the method in terms of well-posedness and convergence properties based on suitable assumptions on the initial data and the discretization parameters. Numerical examples illustrate the theoretical error estimates and underline the practicability of the approach
A survey on inverse problems for applied sciences
The aim of this paper is to introduce inversion-based engineering applications and to investigate some of the important ones from mathematical point of view. To do this we employ acoustic, electromagnetic, and elastic waves for presenting different types of inverse problems. More specifically, we first study location, shape, and boundary parameter reconstruction algorithms for the inaccessible targets in acoustics. The inverse problems for the time-dependent differential equations of isotropic and anisotropic elasticity are reviewed in the following section of the paper. These problems were the objects of the study by many authors in the last several decades. The physical interpretations for almost all of these problems are given, and the geophysical applications for some of them are described. In our last section, an introduction with many links into the literature is given for modern algorithms which combine techniques from classical inverse problems with stochastic tools into ensemble methods both for data assimilation as well as for forecasting
A complex ray-tracing tool for high-frequency mean-field flow interaction effects in jets
This paper presents a complex ray-tracing tool for the calculation of high-frequency Green’s functions in 3D mean field jet flows. For a generic problem, the ray solution suffers from three main deficiencies: multiplicity of solutions, singularities at caustics, and the determining of complex solutions. The purpose of this paper is to generalize, combine and apply existing stationary media methods to moving media scenarios. Multiplicities are dealt with using an equivalent two-point boundary-value problem, whilst non-uniformities at caustics are corrected using diffraction catastrophes. Complex rays are found using a combination of imaginary perturbations, an assumption of caustic stability, and analytic continuation of the receiver curve. To demonstrate this method, the ray tool is compared against a high-frequency modal solution of Lilley’s equation for an off-axis point source. This solution is representative of high-frequency source positions in real jets and is rich in caustic structures. A full utilization of the ray tool is shown to provide excellent results<br/
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