There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

by J. Biggs and C. Tang, Maidenhead, England,
Open University Press, 2007, 360 pp., £29.99, ISBN-13: 978-0-335-22126-4

Teaching for quality learning at university

This book presents the separation-of-variables and T-matrix methods of calculating the scattering of electromagnetic waves by particles. Analytical details and computer programs are provided for determining the scattering and absorption characteristics of the finite-thickness slab, infinite circular cylinder (normal incidence), general axisymmetric particle, and sphere.The computer programs are designed to generate data that is easy to graph and visualize, and test cases in the book illustrate the capabilities of the programs. The connection between the theory and the computer programs is reinforced by references in the computer programs to equations in the text. This cross-referencing will help the reader understand the computer programs, and, if necessary, modify them for other purposes.

Light scattering by particles: Computational methods

In recent years we have learned to fabricate structures smaller than electromagnetic wavelengths, and to assemble them into metamaterials with exotic optical properties for previously unimaginable applications. One such property is perfect absorption of incident light, with no reflection or transmission, across many wavelengths. The authors review the physics, design principles, and classification of thin perfect absorbers, and outline avenues for progress.

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Thin perfect absorbers for electromagnetic waves: Theory, design, and realizations

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.

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Physical limitations on antennas of arbitrary shape

The extinction cross section, which quantifies the scattering and absorption of electromagnetic waves, is a function of the frequency of the external excitation. Due to causality of the scattered field in the forward direction and the optical theorem, the extinction cross section satisfies a sum rule. The result states that the weighted integral of the extinction cross section over all frequencies is related to the static polarizability dyadics of the object, irrespectively whether the permittivity or the permeability attain negative real parts at higher frequencies or not. From this sum rule, physical bounds on scattering are derived for general material models. The theory is illustrated with closed form expressions of the polarizability dyadics of both a temporally dispersive sphere and a chiral sphere, and compared with the outcome of some explicit numerical examples. 1 Theory — a scattering identity for general material models Recently, several intriguing identities concerning scattering of electromagnetic waves by finite objects have been developed [4–6]. The identities are examples of sum rules, which are well established tools in e.g., quantum scattering [3], but seem to be less known and exploited in electrodynamics and acoustics. A bewildering consequence of these sum rules is that the static properties of an scatterer (its polarizability dyadics) determine the weighted integral of the extinction cross section over all frequencies. The assumptions behind the sum rules are linearity, causality, energy conservation, and invariance under time translations. Due to these very general assumptions, the results are shown useful for a large class of scatterers — not just simple canonical geometries. In addition to being instrumental in understanding the broadband scattering properties of an object, these sum rules also give several new interesting results in antenna applications [1]. At a single frequency, when causality has no meaning, the material modeling of the scatterer is less critical. However, dealing with the broadband properties of an scatterer, it becomes important to use physically suitable dispersion models. The isotropic material models are well known and used for more than a century. In the 90’s, modeling of optical activity led to the introduction of chiral models (bi-isotropic materials), and recently much effort has focused on understanding the effects of negative real parts of the material parameters (metamaterials). In this paper we exploit a specific sum rule, which restricts the broadband scattering properties of an object. The sum rule depends only on the well defined static properties of the object, which are conveniently tractable by numerical computations. As a consequence, it does not matter whether the object shows metamaterial properties at higher frequencies or not, its weighted integral of the extinction cross section stays the same. Consider the direct scattering problem of a plane electromagnetic wave eeikk·x (time dependence e−iωt) of unit amplitude impinging in the k-direction on a target embedded in free space. The target can be a single scatterer (homogeneous or not) or it may consist of several parts. The material of the scatterer is modeled by a set of linear and passive constitutive relations which are assumed to be invariant under time translations (i.e., stationary constitutive relations). The scattering dyadic (amplitude) S is defined in terms of the scattered electric fieldEs and the polarization of the incident field e as [3] S(k; k x) · e = lim x→∞xe −ikxEs(k;x) where x = |x| denotes the magnitude of the position vector, and x = x/x. A target’s overall scattering properties are commonly quantified by the scattering cross section σs, defined as the total power scattered in all directions divided by the incident power flux. The extinction cross section σext = σs + σa is defined as the sum of the scattering and absorption cross sections, where the latter is a measure of the absorbed power in the target. The extinction cross section is also determined from the scattering dyadic in the forward direction, x = k, viz., σext(k; k, e) = 4π k Im { e∗ · S(k; k k) · e } (1) where an asterisk denotes the complex conjugate. Relation (1) is known as the optical theorem, and it is applicable to many different wave phenomena such as acoustic waves, electromagnetic waves, and elementary particles [3]. A sum rule for the combined effect of scattering and absorption of electromagnetic waves is derived in Ref. 4 from the holomorphic properties of the forward scattering dyadic. The result is valid for any scatterers satisfying the general assumptions made above. The sum rule, known as the integrated extinction, reads [4] ∫ ∞ 0 σext(k) k2 dk = π 2 ( e∗ · γe · e+ (k × e∗) · γm · (k × e) ) (2) Here, γe and γm denote the electric and magnetic polarizability dyadics, respectively [4]. The effects of (2) for metamaterials are exploited in this paper, and e.g., we immediately observe that all materials with the same static properties have identical integrated extinction. The integrand on the left-hand side of (2) is non-negative. Therefore, for any finite frequency interval K = k0[1 − B/2, 1 +B/2] with center frequency k0 and relative bandwidth B, the identity implies Bσext(κ) k0(1−B2/4) = ∫ K σext(k) k2 dk ≤ π 2 ( e∗ · γe · e+ (k × e∗) · γm · (k × e) ) (3) for some κ ∈ K. For all scatterers with the same static polarizability dyadics, this inequality shows that large scattering in a frequency interval is traded for smaller bandwidth, since the left-hand side of the inequality is bounded from above by the right-hand side. Corresponding inequalities can be obtained for the scattering and absorption cross sections, σs and σa, respectively. 2 Theory — material modeling with applications to chiral and dispersive spheres In this section, we illustrate the sum rule, (2), with two examples. The material of the scatterer is modeled by the following reciprocal, bi-isotropic constitutive relations [2] { D(ω, r) = 0 { (ω)E(ω, r) + iη0χ(ω) ·H(ω, r)} c0B(ω, r) = −iχ(ω)E(ω, r) + η0μ(ω)H(ω, r) where η0 = √ μ0/ 0 denotes the wave impedance of vacuum, and 0, μ0, and c0 are the permittivity, permeability and speed of light of vacuum, respectively. These constitutive relations contain three different material parameters, viz., the permittivity (ω), the permeability μ(ω), and the chirality χ(ω). Modeling realistic materials, these parameters have to satisfy additional requirements. In particular, if the object is passive, the following conditions have to be fulfilled: Im (ω) ≥ 0, Imμ(ω) ≥ 0, Im (ω) Imμ(ω) ≥ (Imχ(ω)) , for all ω > 0 (4) The first two inequalities are identical to the requirements of a passive isotropic material, and the third one is added for a bi-isotropic material. We observe that the imaginary part of the chirality cannot be chosen arbitrary, viz., a large imaginary part of the chirality in general creates an active material. In order to use the identity in (2), we have to know the polarizability dyadics γe and γm. For a chiral sphere of radius a embedded in free space these are known [2], viz., ⎪⎪⎪⎨ ⎪⎪⎪⎩ γe = 4πa ( ( (0)− 1)(μ(0) + 2)− χ2(0)) ( (0) + 2)(μ(0) + 2)− χ2(0) I− 12πχ(0)a ( (0) + 2)(μ(0) + 2)− χ2(0) k × I γm = 4πa ( ( (0) + 2)(μ(0)− 1)− χ2(0)) ( (0) + 2)(μ(0) + 2)− χ2(0) I+ 12πχ(0)a ( (0) + 2)(μ(0) + 2)− χ2(0) k × I where I denotes the unit dyadic in three dimensions. The right hand side of (2) then becomes e∗ · γe · e+ (k × e∗) · γm · (k × e) = 4πa ( (0)− 1)(μ(0) + 2) + ( (0) + 2)(μ(0)− 1)− 2χ(0) ( (0) + 2)(μ(0) + 2)− χ2(0) This is the general expression containing the polarizability dyadics for a chiral sphere of radius a with static values of the permittivity and the permeability, (0) and μ(0), and the chirality χ(0), respectively. Under normal conditions, the chiral effects disappear in the static limit, leading to χ(0) = 0. For this non-chiral static limit, the right hand side is the sum of the ordinary electric and magnetic polarizability contributions. For the dispersive material model, we adopt a Lorentz model, viz.,

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Physical bounds on scattering by metamaterials

This paper generalizes a result in the classical textbook by Stratton regarding the change in electrostatic potential energy due to the introduction of a conductor S 0 into a fixed system of n conductors of arbitrary shape. The change in electrostatic potential energy is rewritten as a surface integral over S 0 which connects the unperturbed problem (the electrostatic setting before S 0 is introduced) and the perturbed problem (the electrostatic setting after S 0 is introduced). The surface integral is verified by means of variable separation of Laplace's equation in bi-spherical coordinates.

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On the change in electrostatic potential energy due to the introduction of an additional conductor

Optimal limitations for scattering of spherical modes is considered where the geometry of the object is known but the temporal dispersion is unknown. Using Cauchy integral relations similar to Fano’s broadband matching theory, the optimal scattering limitations are detemined by the static response as well as the high frequency asymptotics of the reflection coefficient. It is conjectured that the problem of determining the scattering limitations of a dielectric sphere in the high-contrast limit is identical to the closely related, and yet different problem of finding the broadband tuning limits of the spherical modes. As in the Fano theory, the Cauchy integral relations yield a non-convex global optimization problem which in general is quite difficult to handle. Here, a relaxation of the Fano equations is used which is easily solved and which is especially useful in the regime of Rayleigh scattering.

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On the optimal limitations for scattering of spherical modes

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Brister och förslag på förbättringar i den nuvarande examinationsformen i elektromagnetisk fältteori

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Christian Sohl

Papers

Physical bounds on scattering by metamaterials

On the change in electrostatic potential energy due to the introduction of an additional conductor

On the optimal limitations for scattering of spherical modes

Brister och förslag på förbättringar i den nuvarande examinationsformen i elektromagnetisk fältteori

Är studenters syn på plagiering och otillåtet samarbete kulturellt betingad