Bandwidth is an important parameter in many metamaterial applications. It has been shown that Herglotz functions and sum rules offer a powerful methodology to analyze the trade-off between bandwidth and design parameters. Here, this approach is described for the temporal dispersion of constitutive relations and high-impedance surfaces

Gustafsson, Mats

Sjöberg, Daniel

English

Lund University Publications

LUND UNIVERSITYPO Box 117221 00 Lund+46 46-222 00 00Sum rules and physical limitations for passive metamaterialsGustafsson, Mats; Sjöberg, Daniel2011Link to publicationCitation for published version (APA):Gustafsson, M., & Sjöberg, D. (2011). Sum rules and physical limitations for passive metamaterials. Paperpresented at Metamaterials 2011: The Fifth International Congress on Advanced Electromagnetic Materials inMicrowaves and Optics, Barcelona, Spain.Total number of authors:2General 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.Sum rules and physical limitations for passive metamaterialsM. Gustafsson and D. Sj̈obergDept. Electrical and Information Technology Lund UniversityBox 118, SE-221 00 Lund, Swedenemail: mats.gustafsson@eit.lth.se and daniel.sjoberg@eit.lth.seAbstractBandwidth is an important parameter in many metamaterial app ications. It has been shown thatHerglotz functions and sum rules offer a powerful methodology to analyze the trade-off betweenbandwidth and design parameters. Here, this approach is describ d for the temporal dispersion ofconstitutive relations and high-impedance surfaces.1. IntroductionBandwidth is an important figure of merit in metamaterial applications such as high-impedance surfaces,cloaking, lenses, and antennas. In this paper, Herglotz functions and sum rules are used to derive con-straints on passive metamaterials [1, 2, 3]. The results are exemplified for the temporal dispersion of theconstitutive relations and the admittance of high-impedance surfaces.Metamaterials are temporally dispersive,e.g., the permittivity and permeability depend on frequency.The sum rule [1] relate weighted integrals of the constitutive parameter overall spectrum with the in-stantaneous and static response of the material model. Various sum rules are presented in [1] that con-strain the dispersion of the constitutive relations. The bounds constrain thetemporal dispersion over afrequency interval of the material parameter,e.g., how closeǫ(ω) can be to a constantǫm.In [2], a sum rule for high-impedance surfaces was introduced. It rela es frequency intervals havingimpedance above an arbitrary threshold with the static properties of the structure. The sum rule is validfor periodic structures composed by arbitrary dielectric and magnetic materials above a perfect conductor.The sum rule is used to derive physical bounds that show how the bandwidth depends on thickness, angleof incidence, polarization, and material properties.2. Temporal dispersion for constitutive relationsThe linear, causal, time translational invariant, continuous, isotropic, passive and non-magnetic constitu-tive relations areD(t) = ǫ0ǫ∞E(t) + ǫ0∫ t−∞χee(t− t′)E(t′) dt′ such that0 ≤∫ T−∞E(t) ·∂D(t)∂tdt (1)for all timesT and smooth fieldsE tending to zero ast → −∞. The Fourier transform (time dependencee−iωt) of (1) gives the frequency domain modelD(ω) = ǫ0ǫ(ω)E(ω) where the symbolsD andE arereused to denote the electromagnetic fields as functions of the angular frequencyω. Passivity restrictsthe permittivity ǫ such thathǫ = ωǫ(ω) is a Herglotz function [4, 5],i.e., hǫ(ω) is holomorphic andImhǫ(ω) ≥ 0 in the upper half planeImω > 0, see [3, 4].In [1], it is shown that a permittivity (1) satisfies the sum rule∫∞0Im{h1(ω)} dω =∫∞01πarg(ω(ǫ(ω)− ǫm)− ω0∆ω(ǫ(ω)− ǫm)+ ω0∆)dω =ω0∆ǫ∞ − ǫmfor ǫm < ǫ∞ (2)from which the following bound on the temporal dispersion aroundǫ(ω) ≈ ǫm is derivedmaxω∈[ω1,ω2]|ǫ(ω)− ǫm| ≥B1 +B/2(ǫ∞ − ǫm){1/2 lossy case1 lossless case.(3)whereB = (ω2 − ω1)/ω0 andω0 = (ω1 + ω2)/2. The bound (3) shows that the deviation|ǫ(ω) − ǫm|is proportional to the fractional bandwidthB and the differenceǫ∞ − ǫm.0.5 1 1.5 2-2-10120.5 1 1.5 200.20.40.60.810.5 1 1.5 2-2-10120.5 1 1.5 200.20.40.60.81Im ImImImRe Re²(!)²(!)!!!!a) b)d)c)¢ ¢-¢ -¢h (!)1 h (!)10.71Fig. 1: Illustrations of the Drude model withν = {0.1, 0.001} and sum rule (2) forǫ ≈ 0. a) thepermittivity ǫ(ω) for ν = 0.1. c) integrandIm{h1(ω)} in the sum rule (2) forν = 0.1. bd) correspondingcases forν = 0.001.As an example consider the Drude modelǫ(ω) = 1 + 2/(−iω(ν − iω)) whereω is a dimensionlessfrequency variable andν = {0.1, 0.001} see Fig. 1ab. It is close to zero forω ≈ ω0 ≈ 1.4. The sumrule (2) is evaluated forǫm = 0 and∆ = 1/2. The integrand in (2) is depicted in Fig. 1cd, where it isobserved that it has most of its area in the region aroundω0 ≈ 1.4, i.e., in the region where|ǫ(ω)| ≤ ∆.Moreover it is seen that the integral approaches a box as the losses decr ase. The area of this box isω0∆/(ǫ∞ − ǫm) ≈ 0.71 that gives the bandwidth (width of the box) as the height is 1 in the losslesscase. The additional factor of 2 for the lossy case in the bound (3) comesfrom potentially reducing theheight of the box to1/2, see [1].3. High-impedance surfacesHigh-impedance surfaces are artificial surfaces synthesized from periodic structures above a groundplane [6, 7]. The properties of the high-impedance surface depend onfrequency, polarization, and angleof incidence, and they have high impedance only over finite frequency bands [6, 7].In [2], a sum rule that relates wavelength intervals having admittance below an arbitrary threshold withthe static properties of the structure was introduced,i. .,∫∞0ReP∆(Y (λ)) dλ =∫∞01πarg(jY (λ)−∆jY (λ) +∆)dλ =(d+γ2A)2π∆, (4)whereλ denotes the wavelength,Y the admittance,d the height of the structure,γ the polarizability, andA the unit cell area. The sum rule is used to derive the bound [2]λ2 − λ1d≤ 4π∆{1 lossy case1/2 lossless case.where∆ = maxλ∈[λ1,λ2]|Y (λ)| (5)The sum rule (4) and bound (5) are illustrated in 2 for a mushroom structure [7]. The mushroom struc-ture modeled as PEC. This implies that the admittance is lossless below the first grating lobe and thepolarizability can be negative [2].2 4 6 8 10 12-2-10122 4 6 8 10 12-2-10121.8¸/d¢-¢Im Y Re P (Y)¢d=0.8``w=0.9`-¢¢0.9¸/da) b)Im Y Re P (Y)¢Fig. 2: Normalized admittanceY and compositionP∆(Y ) with ∆ = {1/π, 1/(2π)} for a mushroomstructure withℓ = 10mm, w = 9mm, andd = 8mm. a)∆ = 1/π with the lossless boundλ2−λ1 ≤ 2.b)∆ = 1/(2π) with the lossless boundλ2 − λ1 ≤ 1.4. ConclusionsIt is demonstrated that Herglotz functions and their associated integral identities (sum rules) provide apowerful methodology to analyze passive metamaterials. The sum rules areu ed to construct physicalbounds that evaluate the maximal bandwidth of metamaterials.References[1] M. Gustafsson and D. Sjöberg. Sum rules and physical bounds on passive metamaterials. New Journal ofPhysics, 12:043046, 2010.[2] M. Gustafsson and D. Sjöberg. Physical bounds and sum rules for high-impedance surfaces. Antennas andPropagation, IEEE Transactions on, 2011. in press.[3] Anders Bernland, Annemarie Luger, and Mats Gustafsson.Sum rules and constraints on passive systems.J.Phys. A: Math. Theor., 44(14):145205, 2011.[4] H. M. Nussenzveig.Causality and dispersion relations. Academic Press, London, 1972.[5] F. W. King. Hilbert Transforms, Volume 2. Cambridge University Press, 2009.[6] D. Sievenpiper, Lijun Zhang, R. F. J. Broas, and N. G. Alexopolous. High impedance electromagnetic surfaceswith a forbidden frequency band.IEEE Trans. Microwave Theory Tech., 47(11):2059–2074, 1999.[7] D. F. Sievenpiper. Artificial impedance surfaces for antennas. In C. A. Balanis, editor,Modern AntennaHandbook, pages 737–778. Wiley-Interscience, 2008.

Sum rules and physical limitations for passive metamaterials

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Sum rules and physical limitations for passive metamaterials

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