Equivalency of the Casimir and the Landau-Lifshitz approaches to continuous-media electrodynamics and optical activity on reflection

Bungay, A.R.; Svirko, Yu. P.; Zheludev, Nikolay I.

doi:10.1103/physrevb.47.11730

Cited by 38 publications

(38 citation statements)

References 9 publications

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“…Reciprocity relations are implicit in linear response theory, which were used by Halperin 5 to demonstrate that the Kerr effect requires time-reversal symmetry breaking beyond simple dissipation. However, controversy existed before the Halperin paper 6-12 and continued to linger after it was published [13][14][15][16][17] .A source of the confusion can be traced to a subtlety in the boundary conditions on the electromagnetic field at the boundary between two media with different gyrotropic constants. Specifically, at the interface between two isotropic media, the requisite boundary conditions are:…”

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confidence: 99%

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Erratum: Kerr effect as evidence of gyrotropic order in the cuprates [Phys. Rev. B87, 115116 (2013)]

Hosur¹,

Kapitulnik²,

Kivelson³

et al. 2015

Phys. Rev. B

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Recent analysis has confirmed earlier general arguments that the Kerr response vanishes in any time-reversal invariant system which satisfies the Onsager relations. Thus, the widely cited relation between natural optical activity (gyrotropy) and the Kerr response, employed in Hosur et al, Phys. Rev. B 87, 115116 (2013), is incorrect. However, there is increasingly clear experimental evidence that, as argued in our paper, the onset of an observable Kerr signal in the cuprates reflects pointgroup symmetry rather than time-reversal symmetry breaking.Measurements of the rotation of the polarization of normal-incidence light upon reflection (Kerr effect) provide an extremely useful probe of any order which breaks time-reversal symmetry and all mirror symmetries, e.g. ferromagnetic order with a component of the moment perpendicular to the surface. However, arguments have been put forward that a combination of gyrotropy (handedness) and dissipation can lead to a Kerr response, even in the absence of time-reversal symmetry breaking. In Hosur et al. 1 we adopted these arguments and on this basis interpreted the onset of the Kerr effect below a welldefined, doping-dependent onset temperature in cuprate superconductors in terms of spontaneous generation of handedness induced by charge ordering. As outlined below, we are now persuaded by a review of the literature and by our own explicit calculations, that Kerr rotation is forbidden for a system subject to the combined constraints of linear response, thermal equilibrium, and time-reversal symmetry. We therefore wish to retract our suggestion that chiral charge density wave order, by itself, provides a basis for understanding the Kerr rotation observed in the cuprates.Assuming linear response, the Kerr effect can be inferred from the dielectric tensor of the material. It was a key element of the design of the Sagnac Interferometer employed 2-4 in the experiments in question, that it should reject all "non-reciprocal" effects. Reciprocity relations are implicit in linear response theory, which were used by Halperin 5 to demonstrate that the Kerr effect requires time-reversal symmetry breaking beyond simple dissipation. However, controversy existed before the Halperin paper 6-12 and continued to linger after it was published [13][14][15][16][17] .A source of the confusion can be traced to a subtlety in the boundary conditions on the electromagnetic field at the boundary between two media with different gyrotropic constants. Specifically, at the interface between two isotropic media, the requisite boundary conditions are:with α = 1/2, where ∆E , ∆B , and ∆γ are, respectively, the discontinuity in the parallel components of the electric and magnetic fields, and of the gyrotropy. The term proportional to ∆γ can be viewed as the contribution of an induced surface current, which for α = 1/2, but only for this value cancels the contributions of induced bulk currents to the Kerr effect. We will sketch below an analysis in one simple limit in which the value of α = 1/2 can be derived...

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Erratum: Kerr effect as evidence of gyrotropic order in the cuprates [Phys. Rev. B87, 115116 (2013)]

Hosur¹,

Kapitulnik²,

Kivelson³

et al. 2015

Phys. Rev. B

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“…The general underlying physics of bi-anisotropy has been discussed in the literature 25,26 (alternatively, the behaviour can be expressed in terms of a nonlocal response 27 ). Here, we are not interested in the general tensor formulation 25 and can restrict ourselves to the 1D situation.…”

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Photonic metamaterials by direct laser writing and silver chemical vapour deposition

et al. 2008

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Metamaterials are artificial materials that-unlike natural substances-enable magnetism to be achieved at optical frequencies 1-3 . The vast majority of photonic metamaterials 4,5 has been fabricated by electron-beam lithography and evaporation of metal films, both of which are well-established twodimensional (2D) technologies. Although stacking of three 6 or four 7 functional layers made using these methods has been reported, a truly 3D fabrication approach would be preferable for 3D photonic metamaterials. Here, we report first steps in this direction by using a combination of direct laser writing 8,9 and silver chemical vapour deposition 10,11 -the 3D analogues of electron-beam lithography and evaporation, respectively. The optical characterization of a planar test structure composed of elongated split-ring resonators is in good agreement with theory. Retrieval of the effective optical parameters reveals the importance of bi-anisotropy. Once suitable theoretical blueprints are available, our fabrication approach will enable rapid prototyping of truly 3D photonic metamaterials.The emerging field of photonic metamaterials is driven by the fascinating visions of perfect lenses 12 , optical cloaking 13,14 or quantum levitation 15 . Realizing these visions clearly ultimately requires large-scale three-dimensional (3D) metamaterials operating at optical frequencies. One of the major challenges concerns the (realistic) design and the nanofabrication of such structures. The latter is related to the fabrication of 3D (dielectric) photonic crystals, for which extensive literature is available (see, for example, the references in a recent review 16 ). It has become clear that polymeric templates made using holography and/or direct laser writing (DLW) can subsequently be converted into, for example, silicon structures using chemical vapour deposition (CVD) and/or atomic-layer deposition (ALD). In the context of metamaterials, however, metals are crucial; silver leads to the lowest losses at optical frequencies 17 . Yet, it has been unclear whether structures made using DLW and silver CVD would ever have sufficient quality and whether achievable feature sizes would result in metamaterials operating at optical frequencies. Fortunately, we report here that the answer is positive in both regards. Combined with suitable theoretical blueprints, which still need to be developed, this result means that one of the major hurdles in the field of photonic metamaterials is starting to disappear.Lithography of 3D polymeric templates by DLW with lateral feature sizes in the 100 nm range has become routine 8,9,18 and is even available commercially 19 . Our vision is to take such a template and coat it with a silver film of a few tens of nanometres in thickness, for example leading to the 3D arrangement of magnetic split-ring resonators (SRR) shown in Fig. 1a. Several questions arise immediately. (1) Silver films made using these methods are obviously always inherently connected-in sharp contrast to those made by electron-beam lithography...

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“…They, as well as G ij , have the dimensions of a volume. This multipole expansion forms the basis of the so-called Casimir formulation of electrodynamics of continuous media (15,29,30) where optical activity (as shown by, e.g., an isotropic ensemble of chiral molecules) is associated with the tensors G ij and A ijk (28). Instead, Landau and Lifshitz describe optical activity in terms of a wavevector-dependent permittivity (17).…”

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Metamaterials and the Landau–Lifshitz permeability argument: Large permittivity begets high-frequency magnetism

Merlín

2009

Proc. Natl. Acad. Sci. U.S.A.

137

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Homogeneous composites, or metamaterials, made of dielectric or metallic particles are known to show magnetic properties 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 show that these arguments do not apply to composites made of substances with Im ͌ S Ͼ Ͼ /ഞ or Re ͌ S ϳ /ഞ ( S and ഞ are the complex permittivity and the characteristic length of the particles, and Ͼ Ͼ ഞ 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 S and the dynamic magnetic properties of metamaterials. For ͦ ͌ Sͦ Ͻ Ͻ /a (a is the ring cross-sectional radius), the composites exhibit very weak magnetic activity, consistent with the Landau-Lifshitz argument and similar to that of molecular crystals. In contrast, 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 show high permittivity at wavelengths in the range 0.3-3000 m. Calculations for a system of spherical inclusions made of these materials, using the magnetic counterpart to Lorentz-Lorenz formula, uncover large magnetic effects the strength of which diminishes with decreasing wavelength.effective medium theory ͉ electromagnetic scattering ͉ negative refraction ͉ split rings M etamaterials 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 properties 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 novel antennas, beam steerers, sensors, and cloaking devices (4, 5). The refractive index of a material is negative if both the effective-medium permittivity and permeability 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 are extremely rare in nature, it should not come as a surprise that, with the possible exception of La 2/3 Ca 1/3 MnO 3 (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 bee...

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Equivalency of the Casimir and the Landau-Lifshitz approaches to continuous-media electrodynamics and optical activity on reflection

Cited by 38 publications

References 9 publications

Erratum: Kerr effect as evidence of gyrotropic order in the cuprates [Phys. Rev. B87, 115116 (2013)]

Erratum: Kerr effect as evidence of gyrotropic order in the cuprates [Phys. Rev. B87, 115116 (2013)]

Photonic metamaterials by direct laser writing and silver chemical vapour deposition

Metamaterials and the Landau–Lifshitz permeability argument: Large permittivity begets high-frequency magnetism

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