Neil V. Budko

Spectrum of the Volume Integral Operator of Electromagnetic Scattering

The spectrum of the volume integral operator of three-dimensional electromagnetic scattering is analyzed. The operator has both continuous essential spectrum, which dominates at lower frequencies, and discrete eigenvalues, which spread out at higher ones. The explicit expression of the operators symbol provides an exact outline of the essential spectrum for any inhomogeneous anisotropic scatterer with Holder continuous constitutive parameters. Geometrical bounds on the location of discrete eigenvalues are derived for various physical scenarios. Numerical experiments demonstrate good agreement between the predicted spectrum of the operator and the eigenvalues of its discretized version.

Characterization of a two-dimensional subsurface object with an effective scattering model

The problem of the location and characterization of a two-dimensional (2-D) subsurface object is formulated as the inverse problem for an effective scattering model. An arbitrary finite-sized buried object is described as a subsurface circular cylinder with the radius, permittivity, and position of its center to be determined. The inversion is performed in a nonlinearized way, minimizing the discrepancy between the actual scattered field and that of the effective scattering model. A simple and quick solution for a circular cylinder embedded in a lossy half space is introduced. As far as numerical efficiency is concerned, the obtained approximate algorithm is comparable to the free-space solution. The location algorithm has been tested with the two-dimensional models of plastic antipersonnel land mines.

Electromagnetic inversion using a reduced-order three-dimensional homogeneous model

Nonlinearized electromagnetic inversion of a three-dimensional homogeneous model of arbitrary but known support is considered. We propose a reduced-order representation of the cost functional based on the shift invariance property of the Arnoldi decomposition. Numerical experiments demonstrate acceleration up to a factor of 100 with respect to the usual repetitive solution of the forward scattering problem. We also discuss the applicability of a homogeneous model for estimating the effective constitutive parameter of an inhomogeneous target.

Observation of locally negative velocity of the electromagnetic field in free space.

Since the 1983 definition of the speed of light in vacuum as a fundamental constant with the exact value of 299792458 m/s the question remains as to what apart from the wavefront travels at that speed. It is commonly assumed that the entire wave-packet or an impulse of the electromagnetic radiation in free space does. Here it is shown, both theoretically and experimentally, that there exists a region close to the source, where, while the wave-front travels at the speed of light, the individual impulses comprising the body of the wave-packet appear to slow down and even go backwards in time. This three-dimensional near-field late-time effect may also explain some of the free-space superluminal measurements.

Electromagnetic radiation in a time-varying background medium

Analytical solutions are presented for the electromagnetic radiation by an arbitrary pulsed source into a homogeneous time-varying background medium. In the constant-impedance case an explicit radiation formula is obtained for the synchronous permittivity and permeability described by any positive function of time. As might be expected, such a medium introduces significant spectral shifts and spatiotemporal modulation, which are analyzed here for the linear and exponential time variations of the medium parameters. In the varying-impedance case the solution is obtained for the fourth-order polynomial time dependence of the permittivity. In addition to the spectral shifts and modulation this spatially homogeneous medium scatters the field introducing causal echoes at the receiver location.

Two-Dimensional Object Characterization With an Effective Model

The problem of a two-dimensional object location and characterization is considered and solved via the concept of an effective model. The nonlinearized 2D inverse problem is solved employing an effective model of a homogeneous circular cylinder. Promising numerical results are obtained for the spatial location, and determination of size and effective constitution of the objects of different topology using poor a priori information and limited measurement data.

Classification of electromagnetic resonances in finite inhomogeneous three-dimensional structures.

We present a simple and unified classification of macroscopic electromagnetic resonances in finite arbitrarily inhomogeneous isotropic dielectric 3D structures situated in free space. By observing the complex-plane dynamics of the spatial spectrum of the volume integral operator as a function of angular frequency and constitutive parameters, we identify and generalize all the usual resonances, including complex plasmons, real laser resonances in media with gain, and real quasistatic resonances in media with negative permittivity and gain.

Estimation of the average contrast of a buried object

The problem of estimation of the average contrast of a buried object is formulated as a nonlinear inverse scattering problem. An effective scattering model of reduced dimensionality is applied in order to make the inversion scheme numerically fast and well-posed. It is shown that a good estimate of the objects average constitution is obtained if L1-type metric is employed and the two-media background is replaced with the average homogeneous one. The effective scattering model is used in combination with Synthetic aperture radar (SAR) imaging, adding significant value to conventional SAR images.

Transverse Electric Scattering on Inhomogeneous Objects: Spectrum of Integral Operator and Preconditioning

The domain integral equation method with its FFT-based matrix-vector products is a viable alternative to local methods in free-space scattering problems. However, it often suffers from the extremely slow convergence of iterative methods, especially in the transverse electric (TE) case with large or negative permittivity. We identify very dense line segments in the spectrum as being partly responsible for this behavior and the main reason why a normally efficient deflating preconditioner does not work. We solve this problem by applying an explicit multiplicative regularizing operator, which on the operator level transforms the system to the form “identity plus compact.” On the matrix level this regularization reduces the length of the dense spectral segments roughly by a factor of four while preserving the ability to calculate the matrix-vector products using the FFT algorithm. Such a regularized system is then further preconditioned by deflating an apparently stable set of eigenvalues with largest magnitudes, which results in a robust acceleration of the restarted GMRES under constraint memory conditions.

Singular modes of the electromagnetic field

We show that the mode corresponding to a point of the essential spectrum of the electromagnetic scattering operator is a vector-valued distribution representing the square root of the three-dimensional Dirac’s delta function. An explicit expression for this singular mode in terms of the Weyl sequence is provided and analysed. The essential resonance occurs if the permittivity of an object gets close to zero, which is often the case in plasmas and negativepermittivity metamaterials. Such resonance would lead to a perfect localization (confinement) of the electromagnetic field. Simultaneously, however, a portion of electromagnetic energy is removed from the Hilbert space and therefore the whole process may be viewed as absorption. PACS numbers: 03.50.De, 41.20.−q, 12.10.−g