Plasmon analysis and homogenization in plane layered photonic crystals and hyperbolic metamaterials
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Plasmon Analysis and Homogenization in Plane Layered Photonic Crystals and Hyperbolic Metamaterials M. V. Davidovich* Saratov State University, Astrakhanskaya ul. 83, Saratov, 410012 Russia *e-mail: [email protected] Received February 6, 2016
Abstract—Dispersion equations are obtained and analysis and homogenization are carried out in periodic and quasiperiodic plane layered structures consisting of alternating dielectric layers, metal and dielectric layers, as well as graphene sheets and dielectric (SiO2) layers. Situations are considered when these structures acquire the properties of hyperbolic metamaterials (HMMs), i.e., materials the real parts of whose effective permittivity tensor have opposite signs. It is shown that the application of solely dielectric layers is more promising in the context of reducing losses. DOI: 10.1134/S106377611611025X
1. BACKGROUND AND STATEMENT OF THE PROBLEM When considering hyperbolic metamaterials (HMMs), one usually associates their properties with the real tensor of effective permittivity and does not take into account dissipation. An ideal (nondissipative) HMM is a uniaxial photonic crystal and is characterized by the effective permittivity tensor
⎡ε ⊥ 0 0 ⎤ (1) εˆ = ⎢ 0 ε ⊥ 0 ⎥ , ⎢ ⎥ ⎣⎢ 0 0 ε ⎦⎥ where ε⊥ε|| < 0 [1, 2]. Here the axis of the photonic crystal coincides with the z axis. The electromagnetic properties of such a medium were first considered in [3]. The absence of dissipation leads to real values of the components of tensor (1). Then Fresnel’s equation [4] (2) ε ||−1k ⊥2 + ε −⊥1k||2 = k 02 demonstrates (if ε⊥ and ε|| are constant) a hyperbolic dispersion law of the extraordinary wave, because the components of the wavenumbers are also real and bounded. Formally, the conditions k ⊥2 ≫ k 02 and k2 ≫
k 02 are possible, and the unbounded values of wavenumbers specify a number of interesting properties of HMMs [1–3] such as the Purcell effect, the possibility of anisotropic reflection (including ultralow reflection), anisotropic heat radiation and heat transfer, as well as a number of other properties that allow the use of HMMs in nanolithography, near-field microscopy, subwavelength resolution image transmission, and the
design of lenses and masking coatings. It is assumed, in particular, that the transverse components of the wave vector may have unbounded values. However, the time (frequency) and spatial dispersion lead to the dependence of tensor (1) on the wavenumber k0 and the wave vector k: εˆ = εˆ (k0, k), as well as to complex values on the left-hand side of Eq. (2). Dissipation restricts the values of the wave vector components and is related to dispersion, including spatial dispersion; in this case, the isofrequency surface is no longer a hyperboloid of revolution and is closed. For example, taking account of dissipation may lead to almost completely absorbing HMMs. Note that dispersion and dissipation are phenomena related by the causality principle (the Kramers–Kronig relations) [4]. In this paper, we consider a simplest HMM
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