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. . .
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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. . .
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1 Nature versus engineering
(Picture by N. Zheludev)
Metamaterials, dispersion, SRR. . .
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1 Metamaterial concept
Metamaterials, dispersion, SRR. . .
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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. . .
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1 First ”original metamaterial designs”
J. Brown, 1953; W. Rotman, 1961; J. Pendry, 1996.
Metamaterials, dispersion, SRR. . .
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1 First DNG/Veselago material
R.A. Shelby, et al., Science, vol. 292, pp. 77-79, 2001.
Metamaterials, dispersion, SRR. . .
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1 (Geometrical) classification
Nanostructures Optically dense Optically sparse Optically dense in (q·a<1) (q·a >1) one direction, while either optically sparse or with extended inclusions in other direction(s) 3D, bulk Bulk MTM of small Photonic crystals Wire media, inclusions and quasi- multilayer optical crystals, fishnet structures, Bulk nanostructured Optically sparse alternating solid materials without random plasmonic and useful and unusual composites dielectric electromagnetic nanolayers properties 2D, surface Metasurfaces / Plasmonic Artificial metafilms diffraction grids, impedance surfaces optical band -gap with long inclusions nanostructured surfaces and or slots optically dense optical frequency surfaces without selective surfaces useful and unusual electromagnetic properties 1D, linear Metawaveguides Not yet investigated, but possible
Metamaterials, dispersion, SRR. . .
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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. . .
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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 Z∞ = − (τ)E(t − τ) dτ = (τ)E(t − τ) dτ
∞ 0
Metamaterials, dispersion, SRR. . .
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1 Fourier and inverse Fourier transforms
Z∞ f (ω) = f (t) exp(−jωt) dt −∞ Z∞ 1 f (t) = f (ω) exp(jωt) dω 2π −∞
Metamaterials, dispersion, SRR. . .
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1 Time- and frequency-domain relations
Z∞ D(t) = (τ)E(t − τ) dτ
0
Fourier transform of a convolution integral:
D(ω) = (ω)E(ω)
Z∞ (ω) = (τ) exp(−jωτ) dτ
0
Frequency dispersion. . .
Metamaterials, dispersion, SRR. . .
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1 Spatial dispersion
Let us define the induction vectors as
Jind D = E + P = E + , H = µ−1B 0 0 jω 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. . .
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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 | 0 → 0. i V ij j ij |r−r |→∞
I Ω > λ/2 – strong spatial dispersion (SD) I Ω λ – weak SD I Ω is negligibly small – no SD
Metamaterials, dispersion, SRR. . .
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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 = jω aijEj + aijk(∇kEj) + aijkl(∇l∇kEj)
Metamaterials, dispersion, SRR. . .
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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 = jω aijEj + aijk(∇kEj) + aijkl(∇l∇kEj)
Metamaterials, dispersion, SRR. . .
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1 Second-order spatial dispersion
Constitutive relations:
Di = (0δij + aij)Ej + aijk(∇kEj) + aijkl(∇l∇kEj)
−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. . .
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1 Isotropic materials Field transformation
D = E + α∇ × E + β∇∇ · E + γ∇ × ∇ × E −1 H = µ0 B The Maxwell equations are invariant with respect to transformation
D0 = D + ∇ × Q, H0 = H + jωQ Indeed, ∇ × H = jωD ⇒ ∇ × H0 − jω∇ × Q = jωD0 − jω∇ × Q ⇒ ∇ × H0 = jωD0
Metamaterials, dispersion, SRR. . .
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1 First-order spatial dispersion
−1 D = E + α∇ × E, H = µ0 B α Transformation with Q = − 2 E. The constitutive relations transform as α α D0 = E + ∇ × E = E − jω B 2 2 α H0 = H = µ−1B − jω 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. . .
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1 Isotropic media Second-order spatial dispersion
D = E − jξB + β∇∇ · E + γ∇ × (∇ × E) −1 H = µ0 B − jξE Because ∇ × E = −jωB, we can try to transform the relations in such a way that the curl would disappear.
Transforming again, with Q = −γ∇ × E = jωγB, we get
D = E − jξB + β∇∇ · E, H = µ−1B − jξE
µ0 µ = 2 1 − ω µ0γ Artificial “magnetism”. . .
Metamaterials, dispersion, SRR. . .
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1 Natural versus artificial magnetism
(Picture by A. Shipulin)
Metamaterials, dispersion, SRR. . .
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1 Reciprocity
∇ × E1 = −jωµ · H1, ∇ × H1 = jω · E1 + J1ext
∇ × E2 = −jωµ · H2, ∇ × H2 = jω · E2 + J2ext
Multiplying and subtracting. . .
E2 ·∇×H1 −E1 ·∇×H2 = jω(E2 ··E1 −E1 ··E2)+E2 ·J1ext −E1 ·J2ext
Conclusion: In reciprocal materials permittivity is a symmetric dyadic T =
Metamaterials, dispersion, SRR. . .
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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. . .
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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. . .
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1 Classification of bianisotropic materials
Table by A. Sihvola
Metamaterials, dispersion, SRR. . .
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