arXiv:1001.1212v1 [physics.optics] 8 Jan 2010 Electromagnetic modes in cavities made of negative-index metamaterials Jan Wiersig, Julia Unterhinninghofen Institut f¨ur Theoretische Physik, Universit¨at Magdeburg, Postfach 4120, D-39016 Magdeburg, Germany Henning Schomerus Department of Physics, Lancaster University, Lancaster LA1 4YB, United Kingdom Ulf Peschel Institute for Optics, Information and Photonics, University Erlangen-Nuremberg, D-91058, Erlangen, Germany Martina Hentschel Max-Planck-Institut f¨ur Physik komplexer Systeme, N¨othnitzer Str. 38, D-01187 Dresden, Germany (Dated: November 20, 2018) We discuss electromagnetic modes in cavities formed by metamaterials with negative refraction and demon- strate that the straightforward approach to substitute negative values of the electric permittivity and the magnetic permeability leads to quasi-bound states with a negative quality factor. To ensure positive quality factors and a consistent physical interpretation of the quasi-bound states it is essential to include the frequency dispersion of the permittivity and the permeability, as required by positive field energy and causality. The basic mode equa- tion and the boundary conditions including linear frequency dispersion are derived. As an example we consider a disk-like cavity with deformed cross sectional shape. The transition from the unphysical nondispersive case with negative quality factors to the dispersive case with positive quality factors is demonstrated numerically and in an analytical perturbative treatment. PACS numbers: 42.25.-p, 42.60.Da, 78.67.Pt Negative-index metamaterials (NIMs) are artificial com- posites characterized by simultaneously negative values of the electric permittivity ε and the magnetic permeability µ [1–3]. These materials were theoretically predicted already in 1968 by Veselago [4]. In present days there is a strong interest in such materials because of potential applications, such as elec- tromagnetic cloaking [5], subwavelength imaging [6] and fo- cussing of light [7]. The paper is organized as follows. Sections II and III pro- vide a brief review on Maxwell’s equations for monochro- matic fields and on electromagnetic cavities made of conven- tional materials. The boundary conditions for NIM cavites and the appearance of negative quality factors are discussed in Sec. IV. Section V deals with the basic properties of the frequency dispersion of ε and µ . A modified mode equation including linear dispersion and a numerical solution of this equation is presented in Sec. VI. A discussion of the effects of the dispersion is given in Sec. VII. Finally, Sec. VIII con- tains the conclusions. formed cross-sectional shape. Cavities which confine electromagnetic waves in all three spatial dimensions have also attracted considerable attention in the recent years, in particular in the optical regime [8]. The strong interest is partly due to the numerous future applica- tions, such as single-photon emitters [9] and ultralow thresh- old lasers [10–12], and partly due to the possibility to ad- dress fundamental questions of light-matter interaction [13] and ray-wave correspondence [14–16]. The source-free Maxwell’s equations in the frequency do- main are The fabrication of electromagnetic cavities made of NIMs is rather challenging with nowadays technology. The lit- erature in this field is therefore limited to a few theoreti- cal studies, e.g., on one-dimensional cavities made of dis- tributed Bragg reflectors [17] and two-dimensional supersc at- terers [18]. These works, however, consider scattering of plane waves and do not treat the electromagnetic modes as quasi-bound states with finite lifetime. In this paper we fill this gap and discover a subtle difficulty when defining quasi- bounded states in the canonical way. We show that for a con- sistent physical interpretation of modes in NIMs the frequency dispersion of ε and µ is crucial. Our consideration is gen- eral and applies to all sorts of NIM cavities. For illustration we present results for two-dimensional disk cavities with de- where c is the speed of light in vacuum. The constitutive re- lations for monochromatic fields with frequency ω are in the isotropic and linear regime with electric permittivity ε and magnetic permeability µ . The boundary conditions at an interface between a material 1 and II. MAXWELL’S EQUATIONS I. INTRODUCTION ∇ × ⃗ E = i ω c ⃗ B , ∇ × ⃗ H = − i ω c ⃗ D , (1) ∇ · ⃗ D = 0 , ∇ · ⃗ B = 0 , (2) ⃗ D ( ⃗ r, ω ) = ε ( ⃗ r, ω ) ⃗ E ( ⃗ r, ω ) , (3) ⃗ B ( ⃗ r, ω ) = µ ( ⃗ r, ω ) ⃗ H ( ⃗ r, ω ) (4)