Classification of Electromagnetic Resonances in Finite Inhomogeneous Three-Dimensional Structures

Abstract

We present a simple and unified classification of macroscopic electromagnetic resonances in finite arbitrarily inhomogeneous isotropic dielectric 3D structures situated in free space. By observing the complex-plane dynamics of the spatial spectrum of the volume integral operator as a function of angular frequency and constitutive parameters, we identify and generalize all the usual resonances, including complex plasmons, real laser resonances in media with gain, and real quasistatic resonances in media with negative permittivity and gain. DOI: 10.1103/PhysRevLett.96.023904 PACS numbers: 41.20.ÿq, 42.25.ÿp It is hard to overestimate the role played by macroscopic electromagnetic resonances in physics. Phenomena and technologies such as lasers, photonic band-gap materials, plasma waves and instabilities, microwave devices, and a great deal of electronics are all related or even entirely based on some kind of electromagnetic resonance. The usual way of analysis consists of deriving the so-called dispersion equation, which relates the wave vector k or the propagation constant jkj of a plane electromagnetic wave to the angular frequency !. The solutions of this equation may be real or complex. In the first case we talk about a real resonance, i.e., such that can be attained for some real angular frequency and therefore, in principle, results in unbounded fields. In reality, however, amplification of the fields is bounded by other physical mechanisms, e.g., nonlinear saturation. If the solution is complex, then we have a complex resonance and, depending on the sign of the imaginary part, the associated fields are either decaying or growing with time. This common approach is rather limited and does not include all pertaining phenomena. Indeed, more or less explicit dispersion equations can be obtained only for infinite (unbounded) homogeneous and periodic media, as often done in plasma and photonic studies. Other approaches impose explicit boundary conditions and can handle resonators and waveguides with perfectly conducting walls, and idealistic piecewise homogeneous objects (e.g., plane layered medium, circular cylinders, a sphere). On the other hand, very little can be said in the general case of a finite inhomogeneous dielectric object situated in free space. Because of the absence of an explicit dispersion equation and explicit boundary conditions, even the existence and classification of resonances in such objects is still an open problem.We describe here an alternative, mathematically rigorous approach to electromagnetic resonances, based on the volume integral formulation of the electromagnetic scattering, also known as the Green's function method and the domain integral equation method. This formulation is equivalent to the Maxwell's equations and is perfectly suited for bounded inhomogeneous objects in free space. Despite its generality, nowadays the volume integral equation is mostly used as a numerical tool, for instance, in near-field optics and geophysics. The main limitation seems to be the implicit ...

Singular modes of the electromagnetic field

“…In particular, it is not clear how exactly the energy is transferred (if it is transferred) between the normal modes of the point spectrum and the essential modes. Elsewhere, we show that it is impossible to associate any causal evolution with this process, and there may be a jump-like chaotic transformation [20].…”

“…In both plasmonic and metamaterial cases temporal dispersion leads to values of the dielectric permittivity close to zero. In [6] we show that in the latter case an essential resonance may occur. We also speculated that the modes associated with the essential spectrum of the scattering operator may be highly localized in space.…”

“…The concept of electromagnetic resonances and eigenmodes is a natural description of microwave resonators and waveguides [1], microstrip lines [2], and other simple, often, infinite, homogeneous, or periodic structures [3][4][5]. In a recent paper [6] we have generalized this idea for arbitrary dielectric objects of finite extent. The strongest point of our generalization was the incorporation of the full information about the spatial spectrum of the electromagnetic scattering operator [7], which has both discrete eigenvalues and an essential (continuous) part.…”

“…Our previous work on the essential spectrum [6,7] was based on Mikhlin's theory of singular integral operators [13] which does not concern the shape of the modes. In fact, there are very few examples of the analysis of the modes corresponding to the essential spectrum of operators.…”

Effective permittivity of finite inhomogeneous objects

“…45 If either an eigenvalue or the point of essential spectrum gets close to the zero of the complex plane, then a resonance is observed and most of the electromagnetic energy will be accumulated in the corresponding eigenfunction or a pseudomode, which there-fore will determine the spatial distribution of the total field on D. Since all eigenfunctions and pseudomodes are rapidly decaying, if continued outside D, the resonances will generally lead to an increase in the field strength inside the scatterer and a decreased field outside D-something one expects and observes. The difference between the eigenvaluebased and the essential-spectrum-based resonances is in their physical origins.…”