Dispersion Relations in Scattering and Antenna Problems

This dissertation deals with physical bounds on scattering and absorption of acoustic and electromagnetic waves. A general dispersion relation or sum rule for the extinction cross section of such waves is derived from the holomorphic properties of the scattering amplitude in the forward direction. The derivation is based on the forward scattering theorem via certain Herglotz functions and their asymptotic expansions in the low-frequency and high-frequency regimes. The result states that, for a given interacting target, there is only a limited amount of scattering and absorption available in the entire frequency range. The forward dispersion relation is shown to be valuable for a broad range of frequency domain problems involving acoustic and electromagnetic interaction with matter on a macroscopic scale. In the modeling of a metamaterial, i.e., an engineered composite material that gains its properties by its structure rather than its composition, it is demonstrated that for a narrow frequency band, such a material may possess extraordinary characteristics, but that tradeoffs are necessary to increase its usefulness over a larger bandwidth. The dispersion relation for electromagnetic waves is also applied to a large class of causal and reciprocal antennas to establish a priori estimates on the input impedance, partial realized gain, and bandwidth of electrically small and wideband antennas. The results are compared to the classical antenna bounds based on eigenfunction expansions, and it is demonstrated that the estimates presented in this dissertation offer sharper inequalities, and, more importantly, a new understanding of antenna dynamics in terms of low-frequency considerations. The dissertation consists of 11 scientific papers of which several have been published in peer-reviewed international journals. Both experimental results and numerical illustrations are included. The General Introduction addresses closely related subjects in theoretical physics and classical dispersion theory, e.g., the origin of the Kramers-Kronig relations, the mathematical foundations of Herglotz functions, the extinction paradox for scattering of waves and particles, and non-forward dispersion relations with application to the prediction of bistatic radar cross sections

Physical limitations on G and B for antennas

In this paper, physical limitations on the partial gain G and relative bandwidth B are derived for antennas of arbitrary shape based on holomorphic properties of the forward scattering dyadic. The limitation on the performance of G and B is shown to be bounded from above by the long wavelength response of the antenna in terms of the electric and magnetic polarizability dyadics. The special case of ellipsoidal geometries are discussed

Physical limitations on D/Q for antennas

In this paper, physical limitations on the directivity D and Q-factor Q are derived for antennas of arbitrary shape. The quotient D/Q is shown to be bounded from above by the antenna volume and certain shape coefficients in terms of the eigenvalues of the high-contrast polarizability dyadic. The theory is exemplified by numerical results for the half-wave antenna with astonishing agreement

Physical limitations on broadband scattering

In this paper, physical limitations on broadband scattering are presented for heterogeneous anisotropic scatterers of arbitrary shape. A measure of broadband scattering in terms of the extinction cross section is derived based on the holomorphic properties of the forward scattering dyadic. An isoperimetric bound for isotropic material parameters is presented and ellipsoidal scatterers are discussed. Finally, the theoretical results are illustrated numerically by a generic scatterer

Physical limitations on broadband scattering by heterogeneous obstacles

In this paper, new physical limitations on the extinction cross section and broadband scattering are investigated. A measure of broadband scattering in terms of the integrated extinction is derived for a large class of scatterers based on the holomorphic properties of the forward scattering dyadic. Closed-form expressions of the integrated extinction are given for the homogeneous ellipsoids, and theoretical bounds are discussed for arbitrary heterogeneous scatterers. Finally, the theoretical results are illustrated by numerical computations for a series of generic scatterer

Physical limitations on antennas of arbitrary shape

In this paper, physical limitations on bandwidth, realized gain, Q-factor and directivity are derived for antennas of arbitrary shape. The product of bandwidth and realizable gain is shown to be bounded from above by the eigenvalues of the long-wavelength, high-contrast polarizability dyadics. These dyadics are proportional to the antenna volume and are easily determined for an arbitrary geometry. Ellipsoidal antenna volumes are analysed in detail, and numerical results for some generic geometries are presented. The theory is verified against the classical Chu limitations for spherical geometries and shown to yield sharper bounds for the ratio of the directivity and the Q-factor for non-spherical geometries

New physical limitations in scattering and antenna problems

The extinction cross section integrated over all frequencies is shown to be related to the electric and magnetic polarizability dyadics by exploiting the analytic properties of the forward scattering dyadic. This identity can be used threefold: 1) in scattering theory to bound the total scattering properties of an arbitrary scatterer, 2) in antenna theory to derive new physical limitations on antennas, and 3) in material modeling. The theory is illustrated by numerical simulations with excellent agreement