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

Gustafsson, Mats

Kristensson, Gerhard

Sohl, Christian

English

Lund University Publications

LUND UNIVERSITYPO Box 117221 00 Lund+46 46-222 00 00New physical limitations in scattering and antenna problemsKristensson, Gerhard; Sohl, Christian; Gustafsson, Mats2007Link to publicationCitation for published version (APA):Kristensson, G., Sohl, C., & Gustafsson, M. (2007). New physical limitations in scattering and antenna problems.1-3. Paper presented 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.NEW PHYSICAL LIMITATIONS IN SCATTERING AND ANTENNAPROBLEMSGerhard Kristensson, Christian Sohl, and Mats GustafssonDept. of Electroscience, Lund University, Box 118, SE-211 00 Lund, Sweden{Gerhard.Kristensson, Christian.Sohl, Mats.Gustafsson}@es.lth.seAbstract: The extinction cross section integrated over all frequencies is shown to be related to the electric and magneticpolarizability dyadics by exploiting the analytic properties of the forward scattering dyadic. This identity can be usedthreefold: 1) in scattering theory to bound the total scattering properties of an arbitrary scatterer, 2) in antenna theoryto derive new physical limitations on antennas, and 3) in material modeling. The theory is illustrated by numericalsimulations with excellent agreement.THE INTEGRATED EXTINCTIONLet an incident plane wave impinge on a bounded scatterer of arbitrary shape in the direction k̂. The materialproperties of the scatterer are in general both inhomogeneous and dispersive, and not necessarily isotropic.The dominant contribution of the scattered field at large distance, r, in the forward direction k̂ is expressedby the scattering dyadic, S(k, k̂), viaEs(k, rk̂) =eikrrS(k, k̂) · E0 as r → ∞where k is a complex variable in the upper half plane withRe k = ω/c0. Here,E0 is the (Fourier) amplitudeof the incident wave evaluated at the origin.The analytic properties of S are employed to determine an integral identity for the extinction cross section,σext, by the use of the Hilbert transform. For this purpose, introduce the analytic functionκ(k) = −p̂e · S(k, k̂) · p̂∗eik+A(k̂)2π, Im k ≥ 0where A(k̂) is real-valued, and p̂e = E0/|E0| is a complex-valued vector, independent of k, that representsthe electric polarization, and, moreover, satisfies p̂e · k̂ = 0. The function κ(k) is assumed to vanish in thelimit |k| → ∞ in the upper half plane. This assumption is correct for a large class of scatterers and supportedby the Physical Optics approximation for a perfectly conducting scatterer. Furthermore, the function κ isassumed to be square integrable in k on the real axis, and, due to causality of the scattering dyadic in theforward direction, its inverse Fourier transform vanishes on the negative real axis, i.e.,κt(τ) =12π∫Rκ(k)e−ikτ dk = 0 for τ < 0Under these assumptions, it follows by Titchmarsh’s theorem that κ satisfies the Hilbert transformκ(k) =1πiP∫Rκ(k′)k′ − kdk′ (1)where k is real-valued and P denotes Cauchy’s principle value. Based on the imaginary part of (1), it isstraightforward to derive the integrated extinction (λ = 2π/k)2π∫∞0σext(k)k2dk =∫∞0σext(λ) dλ = π2 (p̂∗e · γe · p̂e + p̂∗m · γm · p̂m) (2)EMTS 2007 - International URSI Commission B - Electromagnetic Theory Symposium • July 26-28, 2007 • Ottawa, ON, Canada5 10 15 2000.511.522.533.544.5ka¾ext=¼a2{Figure 1: The extinction cross section averagedover all incident directions for a prolate (dashed)and oblate (dotted) non-magnetic spheroid with per-mittivities ε = 2 and semi-axis ratios ξ = 1/2. Thesolid curve depicts the volume-equivalent sphere ofradius a.20 40 60 80 10002468101214¾ext=¼a2f/GHz{Figure 2: The extinction cross section averaged over allincident directions for a raindrop of volume-equivalentradius a = 2mm. The smooth curve shows a dis-persive material (Debye-model), while the oscillatorycurve presents the corresponding a non-dispersive case(ε = 80.2), with identical long wavelength permittivity.via the optical theorem σext(k) = −4π Re κ(k), and the low-frequency behavior of the forward scatteringdyadic, S(k, k̂), see Kleinman et al. [1], viz.,κ(0) = (p̂∗e · γe · p̂e + p̂∗m · γm · p̂m) /4πHere, p̂m = k̂ × p̂e denotes the magnetic polarization, and γe and γm are the electric and magnetic po-larizability dyadics, respectively. These dyadics are real-valued and symmetric, and they scale with thevolume of the scatterer. A full derivation of (2) is presented in Sohl et al. [2]. The sum of the scatteringand absorption cross sections is the extinction cross section, σext = σs + σa. The three cross sections areby definition real-valued and non-negative. A direct consequence of (2) is that any two scatterers with thesame polarizability dyadics share the same value of the integral.APPLICATIONSThree major applications of the result presented in the previous section are natural, and these topics arebriefly commented below. The full analyses are found in Sohl et al. [2] and in two forthcoming papers.Scattering. In this application we concentrate on the scattering cross section. The first, most obvious,application of (2) is to find physical bounds on the scattering properties of a scatterer. To this end, use theestimate ∫Λσs(λ) dλ ≤∫∞0σext(λ) dλ = π2 (p̂∗e · γe · p̂e + p̂∗m · γm · p̂m)and we see that the broadband scattering properties of a scatterer in any wavelength interval Λ ∈ [0,∞),are bounded by the static polarizability properties of the scatterer. We illustrate this identity in Figure 1 forthree scatterers with different polarizability dyadics. Using the results presented in Jones [3], it is seen thatamong all isotropic, homogeneous scatterers of equal volume and susceptibilities, the spherical scattererminimizes the averaged integrated extinction, and consequently, the solid curve has a lower integrated ex-tinction compared to the other two. Several more numerical examples that verify and illustrate these resultsare presented in Sohl et al. [2], as well as additional lower and upper bounds on the integrated extinction.EMTS 2007 - International URSI Commission B - Electromagnetic Theory Symposium • July 26-28, 2007 • Ottawa, ON, CanadaAntennas. In the antenna application the absorption problem is in focus. Using similar arguments as in thescattering case, but retaining the absorption cross section, which is related to the partial realized gain GΛ ofthe antenna, we obtainGΛB ≤4πλ30∫Λσa(λ) dλ ≤4πλ30∫∞0σext(λ) dλ =4π3λ30(p̂∗e · γe · p̂e + p̂∗m · γm · p̂m)where B denotes the relative bandwidth of the antenna. This inequality bounds the product of the partialrealized gain and the relative bandwidth, and it shows several similarities with the classic results by Chu [4],which gives bounds on the directivity and theQ-value of an antenna. However, the results by Chu hold onlyfor spherical inclusions. The antenna result presented here has no such limitations, but is instead valid for anantenna of arbitrary shape and electrical size. Additional advantages are that the present analysis includesa description of the material as well as the polarization properties of the antenna. Moreover, no resonancemodel is assumed, and no introduction of a quality factor Q is needed.Material modeling. The integrated extinction (2) relates the scattering properties at all frequencies to thestatic properties of the scatterer. As a consequence, all materials that have the same static properties, yieldthe same integrated extinction. These effects are clearly seen in Figure 2, where the areas under the twocurves are identical, despite the fact that the material properties of the two objects are different at all non-zero frequencies. As a result of (2), the larger the scattering, the smaller the bandwidth (higher Q-value).It is therefore not possible to obtain arbitrary high scattering at a certain frequency interval without payingthe price of more sharp resonances. As an additional consequence, an object that shows bi-isotropic effectshas an integrated extinction that is identical to the corresponding isotropic scatterer with the same staticproperties. Since all chiral effects vanish in the low-frequency limit, it is clear that the introduction ofchirality cannot enhance the integrated extinction. Similar arguments apply to materials that show negativepermittivity and/or permeability.CONCLUSIONSIt is shown in this paper that one of the most fundamental physical properties, viz., causality, implies that theextension cross section integrated over all wavelengths is identical to the sum of the electric and magneticpolarizability properties of the scatterer. This identity leads to a number of interesting bounds on scattering,antennas, and material modeling. It is believed that there are more physical quantities that apply to thistheory.REFERENCES.[1] R. E. Kleinman and T. B. A. Senior. “Rayleigh scattering”. In Low and high frequency asymptotics,V. V. Varadan and V. K. Varadan, eds., Amsterdam: Elsevier Science Publishers, vol. 2 of Acoustic,Electromagnetic and Elastic Wave Scattering, chap. 1, pp. 1–70. 1986.[2] C. Sohl, M. Gustafsson, and G. Kristensson. “Physical limitations on broadband scattering by hetero-geneous obstacles”. Tech. Rep. LUTEDX/(TEAT-7151)/1–25/(2006), Lund University, Department ofElectroscience, P.O. Box 118, S-211 00 Lund, Sweden, 2006. http://www.es.lth.se.[3] D. S. Jones. “Low frequency electromagnetic radiation”. J. Inst. Maths. Applics., vol. 23(4), pp. 421–447, 1979.[4] L. J. Chu. “Physical limitations of omni-directional antennas”. Appl. Phys., vol. 19, pp. 1163–1175,1948.EMTS 2007 - International URSI Commission B - Electromagnetic Theory Symposium • July 26-28, 2007 • Ottawa, ON, Canada

New physical limitations in scattering and antenna problems

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New physical limitations in scattering and antenna problems

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