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Metric-Signature Topological Transitions in Dispersive Metamaterials

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Metric-Signature Topological Transitions in Dispersive Metamaterials View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Biblioteca Digital del Sistema de Bibliotecas de la Universidad de Antioquia PHYSICAL REVIEW E 89, 033202 (2014) Metric-signature topological transitions in dispersive metamaterials E. Reyes-Gomez,´ 1 S. B. Cavalcanti,2 L. E. Oliveira,3 and C. A. A. de Carvalho4 1Instituto de F´ısica, Facultad de Ciencias Exactas y Naturales, Universidad de Antioquia UdeA, Calle 70 No. 52-21, Medell´ın, Colombia 2Instituto de F´ısica, Universidade Federal de Alagoas, Maceio,´ Alagoas 57072-970, Brazil 3Instituto de F´ısica, Universidade Estadual de Campinas - Unicamp, Campinas, Sao˜ Paulo 13083-859, Brazil 4CNPEM, Caixa Postal 6192, Campinas, Sao˜ Paulo 13083-970, Brazil (Received 1 October 2013; revised manuscript received 28 January 2014; published 14 March 2014) The metric signature topological transitions associated with the propagation of electromagnetic waves in a dispersive metamaterial with frequency-dependent and anisotropic dielectric and magnetic responses are examined in the present work. The components of the reciprocal-space metric tensor depend upon both the electric permittivity and magnetic permeability of the metamaterial, which are taken as Drude-like dispersive models. A thorough study of the frequency dependence of the metric tensor is presented which leads to the possibility of topological transitions of the isofrequency surface determining the wave dynamics inside the medium, to a diverging photonic density of states at some range of frequencies, and to the existence of large wave vectors’ modes propagating through the metamaterial. DOI: 10.1103/PhysRevE.89.033202 PACS number(s): 41.20.−q, 78.20.Ci, 42.50.Ct, 78.67.Pt I. INTRODUCTION a contravariant tensor Gαβ . Thus, the metric determines the Condensed matter systems may provide laboratory re- electromagnetic responses in the flat space reinterpretation. alizations of cosmological scenarios thanks to analogous This correspondence has inspired the design of a number mathematical descriptions that ultimately allow us to relate of experimental settings to test the transitions associated physical properties in those distinct settings [1]. Analogies in to metric signature changes in momentum space [5] and the mathematical descriptions of different physical systems topological transitions [6], the relationship between statistical have often been used as a tool to further our understanding. and cosmological time flow via the optical analog of a A celebrated example is that of photonic crystals [2,3], where big-bang-like event [7], Hawking radiation [8], cosmological much has been learned about photon dynamics by drawing on inflation [9], and the behavior of the vacuum in a strong the similarities with electron dynamics in crystalline solids. magnetic field [10]. In the case of photonic crystals, Maxwell’s equations in In the present work, we concentrate on the study of periodically arranged media of different electric permittivities the transitions caused by signature changes in metrics in () and magnetic permeabilities (μ)aresimilarinform reciprocal space. We perform a systematic analysis of the to Schrodinger’s¨ equation for electrons propagating in the cases that arise as we use various materials to induce such periodic potential of a crystalline solid. changes. We begin with anisotropic media that experience A more recent example [4] draws upon the analogous form different signs for their electric permittivities, and go beyond of Maxwell’s equations in curved space-time and in a related by including dispersive metamaterials in our analysis. It is material medium. The electric and magnetic responses of the important to point out that, in practice, an isotropic left-handed medium are obtained from the gravitational metric of the material (LHM) is still a great challenge for researchers, as the curved space-time, thus opening the possibility of testing a available LHM structures are intrinsically anisotropic. A study host of phenomena described by general relativity in laboratory of a semiconductor metamaterial with anisotropic dielectric experiments. All that is required is a reinterpretation of the response may be found in the work by Hoffman et al. [11], equation for electromagnetism in curved space whereas for general anisotropic metamaterials, we refer to the studies by Krishnamoorthy et al. [6], Soukoulis et al. [12], 1 √ 4π Wilson [13], and Zheludev [14]. √ ∂ ( −gFαβ ) = j α, (1) −g β c Artificial metamaterials [14–16] are also known as LHMs, due to their peculiar feature that light phase velocity travels as in such materials in the opposite direction of the group velocity. Metamaterial structures are designed to achieve un- αβ 4π α usual electromagnetic properties, such as negative refraction, ∂β G = J , (2) c superlenses, optical magnetism, and cloaking, among others. √ √ Periodic, quasiperiodic, fractal, and disordered arrangements αβ = − αβ α = − α upon identifying G gF and J gj . Equa- of layered one-dimensional positive-index material or meta- tion (2) can then be viewed as the equation for electromag- material stacks have been extensively studied, and also exhib- = 0i netism in a medium in flat space, if we define Di G ited unusual features, such as non-Bragg gaps, corresponding = 1 jk and Hi 2 ij k G . We can then relate these quantities to to zero average refractive index, plasmon-polariton gaps, and = = 1 Ei Fi0 and Bi 2 ij k Fjk. The constitutive√ relations are Anderson delocalization properties [17–24]. αβ αμ βν easily derived from the relation G = −gg g Fμν .Note The use of artificially nanostructured metamaterials to that the metric intervenes as (E,B) are defined in terms of a mimic gravitational environments will certainly add new covariant tensor Fαβ , whereas (D,H) are defined in terms of experimental situations to the ones already proposed in the 1539-3755/2014/89(3)/033202(8) 033202-1 ©2014 American Physical Society E. REYES-GOMEZ´ et al. PHYSICAL REVIEW E 89, 033202 (2014) literature, which have not included those materials where which determines the dispersion ω = ω(k). One may write both the electric and magnetic responses may be negative, Eq. (8)as thus leading to negative refraction indices. However, we ω2 ω4 ω2 k2 + k2 k2 + k2 k2 + k2 have restricted our discussion to studying the behavior of − 1 2 + 1 3 + 2 3 1 2 3 2 4 2 electromagnetic fields that experience changes in the signature c c c 3 2 1 of the metric in reciprocal space associated to their dispersion k2 k2 k2 relations, and the unusual phenomena that this brings about, + 1 + 2 + 3 k2 = 0, (9) such as the copious production of particles, dubbed “big 23 13 12 flashes,” that could also occur in Bose-Einstein condensates where k2 = k2 + k2 + k2. Here we focus our attention in or in gravitational systems. 1 2 3 uniaxial dielectric materials. In this case, 1 = 2 = ⊥ and As real metamaterials are intrinsically dispersive materi- 3 = , and Eq. (9) transforms into als, the present study analyzes the influence of dispersive 2 2 2 2 2 2 2 electric and magnetic responses on the metric signature 2 ω k ω k1 k2 k3 ω ⊥ − + + − = 0, (10) change phenomenon. The metric changes correspond to c2 ⊥ c2 ⊥ c2 looking at different frequency ranges for light propagating in the metamaterials. The work is organized as follows. which leads to the dispersion Section II presents the theoretical framework and numerical k2 ω2 results concerning nonmagnetic materials with dielectric = (11) ⊥ c2 anisotropy, whereas Sec. III is devoted to the study of metric signature topological transitions in dielectric and magnetic corresponding to the ordinary ray, and to the dispersion relation anisotropic materials. Our conclusions are in Sec. IV. k2 k2 k2 ω2 1 + 2 + 3 = (12) ⊥ c2 II. NONMAGNETIC MATERIALS WITH DIELECTRIC ANISOTROPY corresponding to the extraordinary ray. For isotropic materials one has ⊥ = , and Eq. (12) reduces to Eq. (11). In the absence of free charges and electric currents, the First we analyze the case of the extraordinary ray and Maxwell equations describing the behavior of the electromag- suppose that both ⊥ and are functions of the frequency netic field in a continuous medium may be written as ω. By introducing the temporal component k0 of the four- ω ↔ momentum kμ as k0 = ω/c,Eq.(12) may be rewritten as k × k × E(k,ω) = k × [μ ·H (k,ω)] (3) c q2 q2 q2 1 + 2 + 3 = 1, (13) and ⊥ ω ↔ k × k × H (k,ω) =− k × [ ·E(k,ω)], (4) where q = k /k (i = 1, 2, 3) are the three dimensionless c i i 0 spatial components of the four-momentum kμ. The above ↔ ↔ ↔ ↔ where μ = μ (k,ω) and = (k,ω) are the magnetic perme- equation allows to obtain the isofrequency surface in the k ability and electric permittivity tensors, respectively, associ- space at a given frequency ω. ated with the material medium. Let us now turn to Fig. 1, where we have depicted the Equations (3) and (4) may be used to study the metric isofrequency surfaces for various metrics, assuming a Drude- signature transition in a wide variety of optical materials. like electric response, i.e, Here, we first consider the case of nonmagnetic materials with 2 ↔ 0 ωeα μ = α(ω) = 1 − , (14) dielectric anisotropy. In this case I and Eqs. (3) and (4) α ω2 lead to with α =⊥, . Figures 1(a), 1(b), and 1(d) where obtained ω2 ↔ 0 = 0 = = = [k · E(k,ω)]k − k2E(k,ω) =− ·E(k,ω). (5) for ⊥ 1.21, 1, ωe⊥ 2π GHz, and ωe 6π GHz, at c2 the frequencies ν = ω/(2π) = 3.5 GHz, ν = 2 GHz, and ν = ↔ 0.5 GHz, respectively. Notice that < 0( > 0) if ν<ν Without loss of generality, one may assume that the tensor α α eα (ν>ν ), with ν = ω /(2π) and α =⊥, .Hereν ⊥ = 1 is diagonal.
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