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Metamaterials Frequency Dispersion and Spatial Dispersion Split Rings and Related Structures U Metamaterials Frequency dispersion and spatial dispersion Split rings and related structures U. Jena, June 2012 Sergei Tretyakov sergei.tretyakov@aalto.fi 1 June 15, 2012 Lecture plan I Introduction: Metamaterials I Frequency dispersion and spatial dispersion I Bi-anisotropy (chirality) and artificial magnetism I Split rings, dual bars, etc. I History and basic properties I Circuit model I Lorentz dispersion Metamaterials, dispersion, SRR. page 2 1 Definition: Metamaterial Meta- denotes position behind, after, or beyond, and also something of a higher or second-order kind. Metamaterial is an arrangement of artificial structural elements, designed to achieve advantageous and unusual electromagnetic properties. More precisely, properties that cannot be achieved at the atomic or molecular level are achieved through the electromagnetic properties of “particles” formed at levels much higher than the atomic level but whose dimensions are small compared to the wavelength of operation. Metamaterials, dispersion, SRR. page 3 1 Nature versus engineering (Picture by N. Zheludev) Metamaterials, dispersion, SRR. page 4 1 Metamaterial concept Metamaterials, dispersion, SRR. page 5 1 Imitating nature. Artificial dielectrics Artificial chiral materials The right figure from J.C. Bose, On the rotation of plane of polarization of electric waves by twisted structure, Proc. Royal Soc., vol. 63, pp. 146-152, 1898. Metamaterials, dispersion, SRR. page 6 1 First ”original metamaterial designs” J. Brown, 1953; W. Rotman, 1961; J. Pendry, 1996. Metamaterials, dispersion, SRR. page 7 1 First DNG/Veselago material R.A. Shelby, et al., Science, vol. 292, pp. 77-79, 2001. Metamaterials, dispersion, SRR. page 8 1 (Geometrical) classification Nanostructures Optically dense Opticallysparse Opticallydensein (q·a<1) (q·a >1) onedirection,while either optically sparseor with extendedinclusions inother direction(s) 3D,bulk BulkMTMofsmall Photoniccrystals Wiremedia, inclusions andquasi- multilayer optical crystals, fishnetstructures, Bulk nanostructured Opticallysparse alternatingsolid materialswithout random plasmonic and usefulandunusual composites dielectric electromagnetic nanolayers properties 2D,surface Metasurfaces / Plasmonic Artificial metafilms diffractiongrids, impedancesurfaces opticalband -gap withlonginclusions nanostructured surfaces and or slots opticallydense opticalfrequency surfaceswithout selectivesurfaces usefulandunusual electromagnetic properties 1D,linear Metawaveguides Notyet investigated,but possible Metamaterials, dispersion, SRR. page 9 1 Metamaterial “road map” N. Zheludev, The Road Ahead for Metamaterials, Science, 328, 583, 2010; see also http://www.metamorphose-vi.org/ Metamaterials, dispersion, SRR. page 10 1 Frequency dispersion: Delayed response of materials to fields In vacuum: D(t) = 0E(t) In some electrically polarizable medium: D(t) = 0E(t) + P(t) Stationary system: Zt Zt 0 0 0 0 0 0 P(t) = 0 χ(t − t )E(t ) dt ; D(t) = (t − t )E(t ) dt −∞ −∞ Z0 Z1 = − (τ)E(t − τ) dτ = (τ)E(t − τ) dτ 1 0 Metamaterials, dispersion, SRR. page 11 1 Fourier and inverse Fourier transforms Z1 f (!) = f (t) exp(−j!t) dt −∞ Z1 1 f (t) = f (!) exp(j!t) d! 2π −∞ Metamaterials, dispersion, SRR. page 12 1 Time- and frequency-domain relations Z1 D(t) = (τ)E(t − τ) dτ 0 Fourier transform of a convolution integral: D(!) = (!)E(!) Z1 (!) = (τ) exp(−jωτ) dτ 0 Frequency dispersion. Metamaterials, dispersion, SRR. page 13 1 Spatial dispersion Let us define the induction vectors as Jind D = E + P = E + ; H = µ−1B 0 0 j! 0 Polarization is induced by electric field, thus Z Jind(r) = K(r; r0) · E(r0) dV0 V No natural magnetic fraction: no need to include response directly on B. Spatial dispersion — non-local response of the medium Metamaterials, dispersion, SRR. page 14 1 Strong and weak spatial dispersion y x r ’ W r r ’’ |Kij (r - r ’’)|<<| Kij (r - r ’)| V R 0 0 0 Figure: J = K (r − r )E (r ) dV . Generally K j 0 ! 0. i V ij j ij jr−r j!1 I Ω > λ/2 – strong spatial dispersion (SD) I Ω λ – weak SD I Ω is negligibly small – no SD Metamaterials, dispersion, SRR. page 15 1 Weak spatial dispersion Z Z Jind(r) = K(r; r0) · E(r0) dV0 ≈ K(r; r0) · E(r0) dV0; kΩ < 1 V Ω 0 0 1 0 0 E(r ) = E(r) + (@αE) (rα − rα) + (@β@αE) (rα − rα)(r − rβ) + ··· r 2 r β Taking into account spatial derivatives up to the second order: h i ind Ji = j! aijEj + aijk(rkEj) + aijkl(rlrkEj) Metamaterials, dispersion, SRR. page 16 1 Weak spatial dispersion Z Z Jind(r) = K(r; r0) · E(r0) dV0 ≈ K(r; r0) · E(r0) dV0; kΩ < 1 V Ω 0 0 1 0 0 E(r ) = E(r) + (@αE) (rα − rα) + (@β@αE) (rα − rα)(r − rβ) + ··· r 2 r β Taking into account spatial derivatives up to the second order: h i ind Ji = j! aijEj + aijk(rkEj) + aijkl(rlrkEj) Metamaterials, dispersion, SRR. page 17 1 Second-order spatial dispersion Constitutive relations: Di = (0δij + aij)Ej + aijk(rkEj) + aijkl(rlrkEj) −1 H = µ0 B (J.W. Gibbs, 1882) These definitions imply unusual boundary conditions. We want to transform the fields to let them satisfy the usual ones. Metamaterials, dispersion, SRR. page 18 1 Isotropic materials Field transformation D = E + αr × E + βrr · E + γr × r × E −1 H = µ0 B The Maxwell equations are invariant with respect to transformation D0 = D + r × Q; H0 = H + j!Q Indeed, r × H = j!D ) r × H0 − j!r × Q = j!D0 − j!r × Q ) r × H0 = j!D0 Metamaterials, dispersion, SRR. page 19 1 First-order spatial dispersion −1 D = E + αr × E; H = µ0 B α Transformation with Q = − 2 E. The constitutive relations transform as α α D0 = E + r × E = E − j! B 2 2 α H0 = H = µ−1B − j! E 0 2 Material relations for isotropic chiral media: 0 0 −1 D = E − jξB; H = µ0 B − jξE (the chirality parameter ξ = ωα/2) Metamaterials, dispersion, SRR. page 20 1 Isotropic media Second-order spatial dispersion D = E − jξB + βrr · E + γr × (r × E) −1 H = µ0 B − jξE Because r × E = −j!B, we can try to transform the relations in such a way that the curl would disappear. Transforming again, with Q = −γr × E = jωγB, we get D = E − jξB + βrr · E; H = µ−1B − jξE µ0 µ = 2 1 − ! µ0γ Artificial “magnetism”. Metamaterials, dispersion, SRR. page 21 1 Natural versus artificial magnetism (Picture by A. Shipulin) Metamaterials, dispersion, SRR. page 22 1 Reciprocity r × E1 = −j!µ · H1; r × H1 = j! · E1 + J1ext r × E2 = −j!µ · H2; r × H2 = j! · E2 + J2ext Multiplying and subtracting. E2 ·r×H1 −E1 ·r×H2 = j!(E2 ··E1 −E1 ··E2)+E2 ·J1ext −E1 ·J2ext Conclusion: In reciprocal materials permittivity is a symmetric dyadic T = Metamaterials, dispersion, SRR. page 23 1 Reciprocal and non-reciprocal anisotropic media T T + − = + = + 2 2 symmetric antisymmetric Reciprocal: = symmetric = uuu + vvv + www Non-reciprocal: = symmetric + antisymmetric = uuu + vvv + www + g × I Metamaterials, dispersion, SRR. page 24 1 Bianisotropic materials D = · E + a · H B = µ · H + b · E Reciprocity T T T = ; µ = µ ; a = −b Alternative set of parameters: T T a = χ − jκ ; b = χ + jκ Reciprocity again: T a = −b ) χ − jκ = −χ − jκ Conclusion: In reciprocal media χ = 0. χ is the measure of non-reciprocity: the non-reciprocity parameter. Metamaterials, dispersion, SRR. page 25 1 Classification of bianisotropic materials Table by A. Sihvola Metamaterials, dispersion, SRR. page 26 1.
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