Photon-like solutions of Maxwell's equations in dispersive media

2009 ◽  
Author(s):  
John E. Carroll ◽  
Joseph Beals IV
Keyword(s):  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Dispersive Media

3-D simulation of GPR surveys over pipes in dispersive soils

Geophysics ◽  
2000 ◽  
Vol 65 (5) ◽  
pp. 1560-1568 ◽  
Author(s):  
Tsili Wang ◽  
Michael L. Oristaglio
Keyword(s):  
Electrical Properties ◽  
Time Domain ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Displacement Vector ◽  
Electric Displacement ◽  
Field Vector ◽  
Dispersive Media ◽  
Dispersive Soils ◽  
Dispersive Soil

The finite‐difference time‐domain method is adapted to simulate radar surveys of objects buried in dispersive soils whose complex permittivity depends on frequency. The method treats dispersion through the constitutive relation between the electric field vector and the electric displacement vector, which is a convolution in the time domain. This convolution is updated recursively, along with Maxwell’s equations, after approximating the dispersion with a Debye (exponential) relaxation model. A novel feature of our work is the inclusion of dispersion in the perfectly‐matched layer formulation of Maxwell’s equations, which gives an absorbing boundary condition for dispersive media. We simulate 200-MHz ground‐penetrating radar surveys over metallic and plastic pipes buried at a depth of 2 m in soils whose electrical properties model are those of clay loams of different moisture contents. Radar reflections modeled for pipes in dispersive soil differ from those for pipes in soils whose electrical properties are constant (at the values of dispersive soil at the central frequency of the radar pulse). Because the permittivity decreases at higher frequencies in the soils modeled, energy in the reflections shifts toward the front of the waveform, and the amplitudes of trailing lobes in the waveform are suppressed. The effects are subtle, but become more pronounced in models of soils with 10% moisture content by weight.

A simplified Lax‐Wendroff correction for staggered‐grid FDTD modeling of electromagnetic wave propagation in frequency‐dependent media

Geophysics ◽  
1999 ◽  
Vol 64 (5) ◽  
pp. 1369-1377 ◽  
Author(s):  
Tim Bergmann ◽  
Joakim O. Blanch ◽  
Johan O. A. Robertsson ◽  
Klaus Holliger
Keyword(s):  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Numerical Dispersion ◽  
Staggered Grid ◽  
Dispersive Media ◽  
Frequency Dependent ◽  
Elegant Method ◽  
Constitutive Parameters

The Lax‐Wendroff correction is an elegant method for increasing the accuracy and computational efficiency of finite‐difference time‐domain (FDTD) solutions of hyperbolic problems. However, the conventional approach leads to implicit solutions for staggered‐grid FDTD approximations of Maxwell’s equations with frequency‐dependent constitutive parameters. To overcome this problem, we propose an approximation that only retains the purely acoustic, i.e., lossless, terms of the Lax‐Wendroff correction. This modified Lax‐Wendroff correction is applied to an O(2, 4) accurate staggered‐grid FDTD approximation of Maxwell’s equations in the radar frequency range (≈10 MHz–10 GHz). The resulting pseudo-O(4, 4) scheme is explicit and computationally efficient and exhibits all the major numerical characteristics of an O(4, 4) accurate FDTD scheme, even for strongly attenuating and dispersive media. The numerical properties of our approach are constrained by classical numerical dispersion and von Neumann‐Routh stability analyses, verified by comparisons with pertinent 1-D analytical solutions and illustrated through 2-D simulations in a variety of surficial materials. Compared to the O(2, 4) scheme, the pseudo-O(4, 4) scheme requires 64% fewer grid points in two dimensions and 78% in three dimensions to achieve the same level of numerical accuracy, which results in large savings in core memory.

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