insight - Computational Complexity - # Spectrum of the Maxwell Operator Pencil for a Planar Interface

Core Concepts

The paper characterizes and classifies the entire spectrum, including eigenvalues and essential spectrum, of a non-self-adjoint operator pencil generated by the time-harmonic Maxwell problem with a nonlinear dependence on the frequency for the case of two homogeneous dispersive materials joined at a planar interface.

Abstract

The paper analyzes the spectral properties of a time-harmonic Maxwell pencil, where the frequency ω is the spectral parameter. The problem considers a composite material with an interface dividing the space into two half-spaces with different dispersive material properties.

Key highlights:

- When one of the materials is dispersive (dielectric constant depends on ω), the spectral problem is nonlinear in ω.
- When one of the materials is non-conservative (complex-valued dielectric constant), the problem is non-self-adjoint.
- An interesting phenomenon is the existence of surface plasmons, which are states localized at the interface.
- The paper considers both a one-dimensional and a two-dimensional reduction of the problem.
- The main contribution is the characterization and classification of the entire spectrum, including eigenvalues and essential spectrum, in the dispersive case.
- The authors introduce an auxiliary spectral parameter λ to properly define isolated eigenvalues in the nonlinear setting.
- The spectrum is described in terms of explicit sets determined by the function values of the dielectric constant.
- In the one-dimensional case, the spectrum consists of eigenvalues, essential spectrum, and Weyl spectrum, with the eigenvalues corresponding to surface plasmons.
- In the two-dimensional case, the discrete spectrum is empty, but the essential and Weyl spectra are still characterized.
- The authors also analyze the exceptional set where the dielectric constant vanishes, which can lead to eigenvalues of infinite multiplicity.

To Another Language

from source content

arxiv.org

Stats

The following sentences contain key metrics or figures:
The dielectric function is given by ϵ(x1, ω) = ϵ0(1 + χ̂(x1, ω)), where χ̂ is the linear electric susceptibility.
Two classical examples of χ̂m are the Drude model χ̂m(ω) = -cD/(ω^2 + iγω) and the Lorentz model χ̂m(ω) = -cL/(ω^2 + iγω - ω_*^2).
The exceptional set is defined as Ω0 = {ω ∈ D(ϵ) : ω^2 ϵ+(ω) = 0 or ω^2 ϵ-(ω) = 0}.
The sets M(k)± and N(k) are defined in (3.4) and (3.5) and play a key role in characterizing the spectrum.

Quotes

"The main contribution of the paper is the characterization and classification of the spectrum, which includes the non-trivial task of the definition of isolated eigenvalues in the nonlinear setting."
"The whole spectrum consists of eigenvalues and the essential spectrum, but the various standard types of essential spectra do not coincide in all cases."

Deeper Inquiries

Allowing the material permeability μ to depend on the spatial variable x1 would significantly complicate the spectral analysis of the Maxwell equations. The primary reason for this complexity arises from the fact that the dependence of μ on x1 introduces additional spatial variability into the problem, which alters the functional analytic setting. Specifically, the operator pencil P(ω; λ) would need to account for the spatially varying permeability, leading to a more intricate formulation of the curl-curl operator A(ω) and the multiplication operator B(ω).
In the current analysis, the dielectric function ε(x1, ω) is piecewise constant across the interface, which simplifies the spectral problem. However, if μ(x1) varies, the continuity conditions at the interface would become more complex, potentially leading to non-trivial boundary conditions that must be satisfied by the electromagnetic fields. This could result in a richer spectrum, including the possibility of additional eigenvalues and a more complicated essential spectrum.
Moreover, the introduction of a spatially varying permeability could lead to new physical phenomena, such as anisotropic wave propagation and the emergence of localized modes that are sensitive to the specific form of μ(x1). The mathematical tools used to analyze the spectrum, such as Weyl sequences and the definitions of essential spectra, would also need to be adapted to accommodate the new complexities introduced by the spatial dependence of μ.

The spectral properties derived from the analysis of the Maxwell operator pencil have significant implications for the time-dependent Maxwell problem and the propagation of electromagnetic waves in dispersive media. The existence of eigenvalues in the spectrum indicates the presence of localized modes, such as surface plasmons, which are critical for applications in photonics and nanotechnology. These modes arise due to the interaction of electromagnetic waves with the interface between different materials, particularly when one of the materials is dispersive.
The essential spectrum, on the other hand, provides insight into the behavior of radiation modes, which are non-localized solutions that propagate away from the interface. The presence of these modes suggests that electromagnetic waves can travel through the dispersive medium without being confined to the interface, leading to phenomena such as waveguide modes and radiation losses.
Furthermore, the classification of the spectrum into discrete and essential parts informs us about the stability and robustness of the solutions to the time-dependent Maxwell equations. For instance, isolated eigenvalues indicate stable modes that can persist under perturbations, while the essential spectrum reflects the continuous nature of the radiation modes, which may be more susceptible to changes in the medium or boundary conditions.
Overall, the spectral properties elucidate the complex interplay between localization and propagation in dispersive media, providing a framework for understanding how electromagnetic waves behave in various material configurations.

Yes, the techniques developed in this work can be extended to study the spectrum of the Maxwell operator pencil for more complex interface geometries beyond the planar case. The foundational mathematical framework established in the analysis of the planar interface, including the definitions of operator pencils, Weyl sequences, and the classification of spectra, can be adapted to accommodate more intricate geometrical configurations.
For instance, in the case of curved interfaces or multilayered structures, one could employ similar spectral analysis techniques by reformulating the Maxwell equations in a suitable coordinate system that captures the geometry of the interface. The key would be to ensure that the boundary conditions at the interface are appropriately defined, taking into account the curvature and material properties of the different regions.
Moreover, the use of numerical methods, such as finite element analysis or boundary element methods, could complement the analytical techniques, allowing for the exploration of complex geometries that may not yield to straightforward analytical solutions. These numerical approaches can provide insights into the spectral properties of the Maxwell operator pencil in scenarios where analytical methods are challenging to apply.
In summary, while the extension to more complex geometries presents additional challenges, the core techniques and concepts from the current work provide a robust foundation for investigating the spectral properties of the Maxwell operator pencil in a variety of geometrical contexts. This adaptability is crucial for advancing our understanding of electromagnetic wave propagation in real-world applications, such as metamaterials and photonic crystals.

0