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

Gustafsson, Mats

Kristensson, Gerhard

Sohl, Christian

English

Lund University Publications

LUND UNIVERSITYPO Box 117221 00 Lund+46 46-222 00 00Physical limitations on broadband scatteringSohl, Christian; Gustafsson, Mats; Kristensson, Gerhard2007Link to publicationCitation for published version (APA):Sohl, C., Gustafsson, M., & Kristensson, G. (2007). Physical limitations on broadband scattering. 1-3. Paperpresented at International Symposium on Electromagnetic Theory, 2007, Ottawa, Canada.Total number of authors:3General rightsUnless other specific re-use rights are stated the following general rights apply:Copyright and moral rights for the publications made accessible in the public portal are retained by the authorsand/or other copyright owners and it is a condition of accessing publications that users recognise and abide by thelegal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private studyor research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portalRead more about Creative commons licenses: https://creativecommons.org/licenses/Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will removeaccess to the work immediately and investigate your claim.PHYSICAL LIMITATIONS ON BROADBAND SCATTERINGChristian Sohl, Mats Gustafsson, and Gerhard KristenssonDept. of Electroscience, Lund University, Box 118, S-211 00 Lund, Sweden{Christian.Sohl, Mats.Gustafsson, Gerhard.Kristensson}@es.lth.seAbstract: In this paper, physical limitations on broadband scattering are presented for heterogeneous an-isotropic scatterers of arbitrary shape. A measure of broadband scattering in terms of the extinction crosssection is derived based on the holomorphic properties of the forward scattering dyadic. An isoperimetricbound for isotropic material parameters is presented and ellipsoidal scatterers are discussed. Finally, thetheoretical results are illustrated numerically by a generic scatterer.INTRODUCTIONAs far as the author knows, a broadband measure for scattering of electromagnetic waves wasfirst introduced in Purcell[1] concerning absorption and emission of radiation by interstellar dust.Purcell derived the integrated extinction in the special case of homogeneous spheroidal scatterersvia the Kramers-Kronig relations. In this paper, Purcell’s results are generalized to a considerablelarger class of scatterers. The underlying mathematical description is strongly motivated by thestudy of causality and dispersion relations in the quantum mechanical scattering theory of particles.The scattering problem considered in this paper is Fourier-synthesized plane wave scattering bya bounded heterogeneous obstacle of arbitrary shape. The scatterer is modeled by the Maxwell’sequations with anisotropic and dispersive constitutive relations in terms of the electric and mag-netic susceptibility dyadics, χe and χm, respectively. The analysis includes the perfectly conduct-ing material model, as well as general dispersion with or without a conductivity term.THE INTEGRATED EXTINCTIONConsider a scatterer V subject to an incident plane-wave Ein(c0t − k̂ · x) impinging in the k̂-direction. The scattering properties of V are described by the far field amplitude F via the linearand causal time-translational invariant convolutionF (τ, x̂) =∫ τ−∞St(τ − τ ′, k̂, x̂) · Ein(τ ′) dτ ′.Introduce the forward scattering dyadic S as the Fourier transform of St evaluated in the directionx̂ = k̂. Further, let E0 denote the Fourier amplitude of the incident field and introduce theelectric and magnetic polarizations, p̂e = E0/|E0| and p̂m = k̂ × p̂e, respectively. The function(k) = p̂∗e · S(k, k̂) · p̂e/k2 is then holomorphic in the upper half-plane Im k > 0 for a large classof dyadics St. Cauchy’s integral theorem applied to  can be proved to yield(iε) =∫ π0(iε − εeiφ)2πdφ +∫ π0(iε + Reiφ)2πdφ +∫ε<|k|<R(k + iε)2πikdk, (1)where k in the last integral is real-valued. Relation (1) is implicitly subject to the limits ε → 0 andR → ∞.EMTS 2007 International URSI Commission B - Electromagnetic Theory Symposium • July 26-28, 2007 • Ottawa, ON, CanadaThe left hand side of (1) and the integrand in the first integral on the right hand side are given bythe electric and magnetic polarizability dyadics, γe and γm, respectively, viz.,(ε) =14π(p̂∗e · γe · p̂e + p̂∗m · γm · p̂m) + O(ε) as ε → 0. (2)The second integral on the right hand side of (1) approaches zero in the limit R → ∞ due tothe extinction paradox (k) = −A(k̂)/(2πik) + O(|k|−2) as |k| → ∞, where A is a real-valuedconstant. Further, it is assumed that  is sufficiently regular to extend the contour in the thirdintegral to the real-axis. Under this assumption, the real part of (1) yields(0) =12(0) +12π∫ ∞−∞Im (k)kdk =12(0) +18π2∫ ∞−∞σext(k)k2dk (3)where k is real-valued and the optical theorem σext(k) = 4πk Im (k) have been used. Recall thatσext is non-negative. Relation (2) inserted into (3) using λ = 2π/k finally yields∫ ∞0σext(λ) dλ = π2 (p̂∗e · γe · p̂e + p̂∗m · γm · p̂m) . (4)Note that the integrated extinction (4) is independent of any temporal dispersion, depending onlyon the long wavelength limit properties of V .AN ISOPERIMETRIC BOUND ON BROADBAND SCATTERINGVarious isoperimetric bounds on the integrated extinction is discussed in Sohl et al.[2]. An examplefor isotropic material parameters, χe = χeI and χm = χmI, is given byπ2∫Vχe(x)χe(x) + 1+χm(x)χm(x) + 1dVx ≤∫ ∞0σext(λ) dλ ≤ π2∫Vχe(x) + χm(x) dVx. (5)Since the right hand side of (5) is bounded from above by π2|V | supx∈V (χe(x)+χm(x)) it followsthat the integrated extinction for any heterogeneous scatterer is at most equal to the integrated ex-tinction for the corresponding homogeneous scatterer with maximal susceptibilities supx∈V χe(x)and supx∈V χm(x). This observation leads to the fascinating conclusion that there is no fundamen-tal difference from a broadband point of view in scattering by heterogeneous and homogeneousobstacles.HOMOGENEOUS ELLIPSOIDAL SCATTERERSClosed-form expressions of γe and γm exist for the homogeneous ellipsoidal scatterers, viz., γe =|V |χe · (I + L · χe)−1 and γm = |V |χm · (I + L · χm)−1, where L is the depolarizing dyadic.The eigenvalues of L are the depolarizing factors Lj . Closed-form expressions of Lj exist inthe special case of the prolate and oblate spheroids. For example, the integrated extinction forthe homogeneous sphere is equal to 3π2|V |∑i=e,m χi/(χi + 3) independent of the electric andmagnetic polarizations. In the long wavelength PEC limit (χe → ∞ and χm → −1) the integratedextinction approaches 3π2|V |/2 for the sphere.EMTS 2007 International URSI Commission B - Electromagnetic Theory Symposium • July 26-28, 2007 • Ottawa, ON, Canada10 20 30 40 50 60 7001234561 1.5 2 2.5 3 3.50123456f/GHzf/GHz¾ext=¼a2{¾ext=¼a2{Figure 1: The averaged extinction cross section σ̄ext in units of πa2 as function of the frequency for a Lorentzdispersive cylinder with volume-equivalent sphere of radius a = 1 cm. The right figure is a close-up of the2GHz peak and illustrates the physical limitations presented in Sohl et al.[2].The integrated extinction for the PEC needle of length 2a with semi-axis ratio ξ is∫ ∞0σext(λ) dλ =4π3a33f(θ)ln 2/ξ − 1 + O(ξ2) as ξ → 0, (6)where θ denotes the angle of incidence and f(θ) = sin2 θ for the TM-polarization and f(θ) = 0 forthe TE-polarization. Since the extinction cross section is non-negative, (6) implies that σext van-ishes almost everywhere except on a set of measure zero consisting of the denumerable resonancesfor which an integer multiple of λ/2 approximately coincides with the length 2a of the needle.A NUMERICAL EXAMPLE: THE LORENTZ DISPERSIVE CYLINDERThe extinction cross section averaged over k̂ and p̂e for a homogeneous isotropic Lorentz disper-sive cylinder is depicted in Figure 1. The ratio of the cylinder length  to its radius b is /b = 2.The cylinder is non-magnetic with electric susceptibility given by the Lorentz dispersion modelχe(ω) = ω2p/(ω20 − ω2 − iων). The Lorentz parameters ωp, ω0 and ν are chosen such that allthe three curves in the left figure have the same long wavelength susceptibilities χe(0) = 1 andhence the same averaged integrated extinctions, see Sohl et al.[2]. The first two curves correspondto Lorentz resonances ω0/2π at 2 GHz and 10 GHz, while the third curve is for a non-dispersiveelectric susceptibility. The boundary curve of the box in the right figure corresponds to an artificialscatterer with an averaged extinction cross section vanishing everywhere outside the box. The av-eraged integrated extinction for the boundary of the box and the three curves in the left hand sideof Figure 1 coincide.REFERENCES[1] E. M. Purcell. “On the absorption and emission of light by interstellar grains”. J. Astrophys.,vol. 158, pp. 433–440, 1969.[2] C. Sohl, M. Gustafsson, and G. Kristensson. “Physical limitations on broadband scattering byheterogeneous obstacles”. Submitted to Journal of Mathematical Physics.EMTS 2007 International URSI Commission B - Electromagnetic Theory Symposium • July 26-28, 2007 • Ottawa, ON, Canada

Physical limitations on broadband scattering

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