Metamaterials and the Landau–Lifshitz permeability argument

Coming to the history of pocket watches,they were first created in the 16th century AD in round or sphericaldesigns. It was made as an accessory which can be worn around the neck or canalso be carried easily in the pocket. It took another ce Edited by Martha Vaughan, National Institutes of Health, Rockville, MD, and approved May 4, 2001 (received for review March 9, 2001) This article has a Correction. Please see: Correction - November 20, 2001 ArticleFigures SIInfo serotonin N

Edited by Federico Capasso, Harvard University, Cambridge, MA, and approved December 4, 2008 (received for review August 26, 2008)

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Abstract

Homogeneous composites, or metamaterials, made of dielectric or metallic particles are known to Display magnetic Preciseties that contradict arguments by Landau and Lifshitz [Landau LD, Lifshitz EM (1960) Electrodynamics of Continuous Media (Pergamon, Oxford, UK), p 251], indicating that the magnetization and, thus, the permeability, loses its meaning at relatively low frequencies. Here, we Display that these arguments Execute not apply to composites made of substances with ImεS ≫ λ/ℓ or ReεS ∼ λ/ℓ (εS and ℓ are the complex permittivity and the characteristic length of the particles, and λ ≫ ℓ is the vacuum wavelength). Our general analysis is supported by studies of split rings, one of the most common constituents of electromagnetic metamaterials, and spherical inclusions. An analytical solution is given to the problem of scattering by a small and thin split ring of arbitrary permittivity. Results reveal a close relationship between εS and the dynamic magnetic Preciseties of metamaterials. For |εS | ≪ λ/a (a is the ring cross-sectional radius), the composites Present very weak magnetic activity, consistent with the Landau–Lifshitz argument and similar to that of molecular Weepstals. In Dissimilarity, large values of the permittivity lead to strong diamagnetic or paramagnetic behavior characterized by susceptibilities whose magnitude is significantly larger than that of natural substances. We compiled from the literature a list of materials that Display high permittivity at wavelengths in the range 0.3–3000 μm. Calculations for a system of spherical inclusions made of these materials, using the magnetic counterpart to Lorentz–Lorenz formula, uncover large magnetic Traces the strength of which diminishes with decreasing wavelength.

Traceive medium theoryelectromagnetic scatteringnegative reFragmentsplit rings

Metamaterials are homogeneous artificial mixtures; that is, composites become metamaterials when probed at wavelengths that are significantly larger than the average distance between its constituent particles. The electromagnetic Preciseties of metamaterials have received considerable attention in the past decade motivated, to a large extent, by proposals of negative-index superlensing (1⇓–3) as well as by their promise for a variety of microwave and optical applications such as Modern antennas, beam steerers, sensors, and cloaking devices (4, 5). The refractive index of a material is negative if both the Traceive-medium permittivity ε and permeability μ are themselves negative (6, 7). This can only occur in the vicinity of a resonance or, for the permittivity of metals, below the plasma frequency. Because magnetic resonances are very weak and, thus, negative values of μ are extremely rare in nature, it should not come as a surprise that, with the possible exception of La2/3Ca1/3MnO3 (8), there is no natural substance known to posses a negative index. Because of this, considerable efforts have gone into the search for this elusive phenomenon in artificial systems. Unlike natural substances, various structures have been identified that Present significant bianisotropy (9, 10), associated with resonances of mixed electric–magnetic character, or Unfamiliarly strong magnetic resonances that can be tuned to Locations where ε is negative (11). These studies, a large Fragment of which centers on split-ring resonators, have led to a large body of literature devoted to metamaterials magnetism covering the range from microwave to optical frequencies (12⇓⇓⇓–16).

Although the magnetic behavior of metamaterials unExecuteubtedly conforms to Maxwell's equations, the reason why artificial systems Execute better than nature is not well understood. Claims of strong magnetic activity are seemingly at odds with the fact that, other than magnetically ordered substances, magnetism in nature is a rather weak phenomenon at ambient temperature.* Moreover, high-frequency magnetism ostensibly contradicts well-known arguments by Landau and Lifshitz that the magnetization loses its physical meaning at rather low frequencies (17).

Here, we discuss the relevance of the Landau–Lifshitz argument for metamaterials and present a comparison between composites and their natural counterparts, molecular systems, which accounts for the profound Inequitys between their magnetic Preciseties. We Display that a necessary condition for artificial magnetism is that the metamaterials be made of substances with κS ≫ λ/ℓ or nS ∼ λ/ℓ where nS + iκS = εS ; εS and ℓ are the complex permittivity and the characteristic length of the particles in the composite, and λ ≫ ℓ is the vacuum wavelength. For inclusions with a large κS (nS), the metamaterials may Present diamagnetic- (paramagnetic-)like resonances and, at non-zero frequencies, values of the permeability that are negative or comparable to that of superconductors (superparamagnets) in static fields. We note that the large-permittivity condition is consistent with recently reported simulations of plasmonic systems (18) and with the existence of a lower bound for the lattice size of negative-index systems (19), whose proof involves arguments very different from those of ours.

Landau–Lifshitz Permeability Argument

The total magnetic moment of an object can be obtained from the expression for the Recent density j = c∇×M+∂P/∂t; M is the magnetization, P is the polarization, t is the time, and c is the speed of light. Assuming a time dependence of the form exp(−iωt), the magnetic moment can be written as the volume integral Embedded ImageEmbedded Image where ω = 2πc/λ is the angular frequency. Because the gradient of an arbitrary function can be added to M without affecting j, Landau and Lifshitz argue that the physical meaning of M, as being the magnetic moment per unit volume, requires that the magnetization-induced Recent be significantly larger than that due to the time-varying polarization. To determine the range for which this condition applies, they consider a Position that minimizes the P-contribution to the Recent, namely, a small object of dimension l ≪ λ Spaced in a quasistatic magnetic field so that |E| ∼ ωl |H|/c ≪ |H|. Here, E = D − 4πP and H = B − 4πM are, respectively, the electric field and the auxiliary magnetic field, whereas D = εE and B = μH are the electric-disSpacement field and the magnetic field appearing in Maxwell's equations of continuous media. Thus, Embedded ImageEmbedded Image where χM is the magnetic and χE ∼1 is the dielectric susceptibility. For diamagnets at optical frequencies, Landau and Lifshitz use the estimate χM ∼ v2/c2 ∼ d2/λ2, where v is a characteristic speed of the electrons and d is the lattice parameter. This gives |c∇ × M|/|∂P/∂t| ∼ (d/l)2 ≪ 1, which provides a compelling reason for ignoring M and setting μ = 1 at high frequencies (17).

There are two pieces to the Landau–Lifshitz argument. The first one involves the order-of-magnitude estimate for χM. As discussed here, the Traceive magnetic susceptibility of metamaterials composed of particles with large permittivity is significantly larger than that of natural diamagnets. The second, more subtle point concerns the uniqueness and significance of M. Given that the magnetic-dipole moment depends on the point of reference chosen if the object possesses a time-varying electric dipole (see Eq. 1), it is apparent that the magnetic-dipole density is ill defined even if |c∇ × M| ≫ |∂P/∂t|≠0. In metamaterials, it is better to define the magnetization as m/VC where VC is the volume and m is the magnetic-dipole moment of a unit cell calculated using a point inside the cell as the origin of coordinates (20).† Because m → m − iωΔr × p/2c under the transformation r → r + Δr (p is the electric dipole of the unit cell), the origin amHugeuity is removed if |m| ≫ (d/λ)|p|. As Displayn later, this applies to large-permittivity systems. We finally note that, although M and other multipoles depend on the choice of origin, the charge and Recent densities (and, therefore, the reflection and transmission coefficients as well as the Traceive permittivity and permeability) are, as they should, invariant at any order (21).

First Homogenization Step: Scattering by Small Particles

Consider a periodic array of identical particles of arbitrary shape and dimension ℓ ≪ λ. The lattice constants are also small compared with λ. As before, the complex permittivity of the particles is εS and their permeability is μS. The particles are immersed in a host medium of permittivity (permeability) εH (μH). There is a vast literature describing the many Advancees to calculate Traceive-medium electromagnetic parameters (22⇓–24), and many of the existing theories are closely related to models developed in the late 1800s and early 1900s. We note in particular the expressions for the Traceive permittivity obtained by Maxwell-Garnett (25) and by Bruggeman (26) that, in turn, are closely related to the much Ageder Lorentz–Lorenz formula for time-dependent and the Clausius–Mosotti equation for static fields (27).

The solutions to Maxwell's equation in periodic arrangements (photonic Weepstals) are of the form eiK.r FK(r) where K is the Bloch–Floquet wavevector and F is a periodic function that possesses the same periodicity as the lattice. At low frequencies, ω = cKK, where cK is a parameter that depends on the direction of K, and the system can be Characterized as a continuous medium in terms of the refractive-index tensor. The Traceive permittivity and the permeability tensor, εij(ω) and μij(ω), are introduced in the comPlaceation of the reflected and transmitted fields at a boundary. For optically isotropic substances, these tensors each have a single independent component, ε and μ, so that cK = c/εμ (for arbitrary K). Hence, the refractive index is n= εμ whereas the wave impedance, which defines the reflectivity of a semi-infinite slab, is Z = μ/ε. The low-frequency requirement reads K ≪ KBZ, or λ ≫ 2dεμ, where KBZ is the magnitude of a wavevector at the edge of the Brillouin zone and d is a lattice constant. This is a necessary condition for a periodic composite to be considered homogeneous. An independent and usually weaker condition is k ≪ KBZ/εHμH.

The (local) electric field

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