Maxwell Operator Research Articles

An asymptotic formula for perturbations in an electromagnetic eigenvalue problem due to the presence of small deformable dielectric inclusion

AbstractWe consider a homogeneous electromagnetic material enclosing a bounded ‐domain with a small dielectric inclusion. We derive a rigorous asymptotic expansion of the eigenvalues of the Maxwell operator as the diameter of the inclusion goes to zero.

Maxwell operator in a cylinder

The Maxwell operator in a 3D cylinder is considered. The coefficients are assumed to be scalar functions depending on the longitudinal variable only. Such an operator is represented as a sum of a countable set of matrix differential operators of first order acting in L 2 ( R ) . Based on this representation we give a detailed description of the structure of the spectrum of the Maxwell operator in two particular cases: (1) in the case of coefficients stabilizing at infinity; and (2) in the case of periodic coefficients.

Analysis of Parametric Instability of Waves in Magnetic Nanogratings from the Bifurcation Points of the Nonlinear Maxwell Operator

Analysis of Parametric Instability of Waves in Magnetic Nanogratings from the Bifurcation Points of the Nonlinear Maxwell Operator

Numerical Technique for Determining the Bifurcation Points of the Nonlinear Maxwell Operator (Maxwell’s Equations Together with the Landau–Lifshitz Equation)

Numerical Technique for Determining the Bifurcation Points of the Nonlinear Maxwell Operator (Maxwell’s Equations Together with the Landau–Lifshitz Equation)

Spectral properties of the inhomogeneous Drude-Lorentz model with dissipation

We establish spectral enclosures and spectral approximation results for the inhomogeneous lossy Drude-Lorentz system with purely imaginary poles, in a possibly unbounded Lipschitz domain of R3. Under the assumption that the coefficients θe, θm of the material are asymptotically constant at infinity, we prove that spectral pollution due to domain truncation can lie only in the essential numerical range of a curlcurl0−f(ω) pencil.As an application, we consider a conducting metamaterial at the interface with the vacuum; we prove that the complex eigenvalues with non-trivial real part lie outside the set of spectral pollution. We believe this is the first result of enclosure of spectral pollution for the Drude-Lorentz model without assumptions of compactness on the resolvent of the underlying Maxwell operator.

Fast multi-source nanophotonic simulations using augmented partial factorization

Numerical solutions of Maxwell’s equations are indispensable for nanophotonics and electromagnetics but are constrained when it comes to large systems, especially multi-channel ones such as disordered media, aperiodic metasurfaces and densely packed photonic circuits where the many inputs require many large-scale simulations. Conventionally, before extracting the quantities of interest, Maxwell’s equations are first solved on every element of a discretization basis set that contains much more information than is typically needed. Furthermore, such simulations are often performed one input at a time, which can be slow and repetitive. Here we propose to bypass the full-basis solutions and directly compute the quantities of interest while also eliminating the repetition over inputs. We do so by augmenting the Maxwell operator with all the input source profiles and all the output projection profiles, followed by a single partial factorization that yields the entire generalized scattering matrix via the Schur complement, with no approximation beyond discretization. This method applies to any linear partial differential equation. Benchmarks show that this approach is 1,000–30,000,000 times faster than existing methods for two-dimensional systems with about 10,000,000 variables. As examples, we demonstrate simulations of entangled photon backscattering from disorder and high-numerical-aperture metalenses that are thousands of wavelengths wide.

A compactness result for the div-curl system with inhomogeneous mixed boundary conditions for bounded Lipschitz domains and some applications

For a bounded Lipschitz domain with Lipschitz interface we show the following compactness theorem: Any mathsf {L}_{}^{2}-bounded sequence of vector fields with mathsf {L}_{}^{2}-bounded rotations and mathsf {L}_{}^{2}-bounded divergences as well as mathsf {L}_{}^{2}-bounded tangential traces on one part of the boundary and mathsf {L}_{}^{2}-bounded normal traces on the other part of the boundary, contains a strongly mathsf {L}_{}^{2}-convergent subsequence. This generalises recent results for homogeneous mixed boundary conditions in Bauer et al. (SIAM J Math Anal 48(4):2912-2943, 2016) Bauer et al. (in: Maxwell’s Equations: Analysis and Numerics (Radon Series on Computational and Applied Mathematics 24), De Gruyter, pp. 77-104, 2019). As applications we present a related Friedrichs/Poincaré type estimate, a div-curl lemma, and show that the Maxwell operator with mixed tangential and impedance boundary conditions (Robin type boundary conditions) has compact resolvents.

The absence of eigenvalues for certain operators with partially periodic coefficients

The absence of eigenvalues is proved for the Schrödinger operator−Δ+V(x,y)-\Delta + V(x,y)in the Euclidean space whose potential is periodic in some variables and decays in the remaining variables faster than power11. A similar result for the Maxwell operator is established.

M-dissipative boundary conditions and boundary tuples for Maxwell operators

For Maxwell operators ▪ in Lipschitz domains, we describe all m-dissipative boundary conditions and apply this result to generalized impedance and Leontovich boundary conditions including the cases of singular, degenerate, and randomized impedance coefficients. To this end we construct Riesz bases in the trace spaces associated with the curl-operator and introduce a modified version of boundary triple adapted to the specifics of Maxwell equations, namely, to the mixed-order duality of the related trace spaces. This provides a translation of the problem to operator-theoretic settings of abstract Maxwell operators. In particular, we show that Calkin reduction operators are naturally connected with Leontovich boundary conditions and provide an abstract version of impedance boundary condition applicable to other types of wave equations. Taking Friedrichs and Krein-von Neumann extensions of related boundary operators, it is possible to associate m-dissipative Maxwell operators to arbitrary non-negative measurable impedance coefficients.

A limiting absorption principle for Helmholtz systems and time-harmonic isotropic Maxwell's equations

In this work we investigate the Lp−Lq-mapping properties of the resolvent associated with the time-harmonic isotropic Maxwell operator. As spectral parameters close to the spectrum are also covered by our analysis, we obtain an Lp−Lq-type Limiting Absorption Principle for this operator. Our analysis relies on new results for Helmholtz systems with zero order non-Hermitian perturbations. Moreover, we provide an improved version of the Limiting Absorption Principle for Hermitian (self-adjoint) Helmholtz systems.

Spectral theory of first-order systems: From crystals to Dirac operators

Let [Formula: see text] be a constant coefficient first-order partial differential system, where the matrices [Formula: see text] are Hermitian. It is assumed that the homogeneous part is strongly propagative. In the non-homogeneous case it is assumed that the operator is isotropic. The spectral theory of such systems and their potential perturbations is expounded, and a Limiting Absorption Principle is obtained up to thresholds. Special attention is given to a detailed study of the Dirac and Maxwell operators. The estimates of the spectral derivative near the thresholds are based on detailed trace estimates on the slowness surfaces. Two applications of these estimates are presented: Global spacetime estimates of the associated evolution unitary groups, that are also commonly viewed as decay estimates. In particular, the Dirac and Maxwell systems are explicitly treated. The finiteness of the eigenvalues (in the spectral gap) of the perturbed Dirac operator is studied, under suitable decay assumptions on the potential perturbation.

Maxwell operator in a cylinder with coefficients that do not depend on the cross-sectional variables

Maxwell operator in a cylinder with coefficients that do not depend on the cross-sectional variables

Stability in the higher derivative Abelian gauge field theories

We present an exact derivation of conserved tensors associated to the higher-order symmetries in the higher derivative Abelian gauge field theories. In our model, the wave operator of the derived theory is a n-th order polynomial expressed in terms of the usual Maxwell operator. Relying on this formalism and utilizing the extension of Noether's theorem, we acquire a series of conserved second-rank tensors which includes the standard canonical energy-momentum tensors. Moreover, with the aid of auxiliary fields, we succeed in obtaining the relations between the root decomposition of characteristic polynomial of the wave operator and the conserved energy-momentum tensors in the context of another equivalent lower-order representation. Under the certain conditions, although the canonical energy of the higher derivative dynamics is unbounded from below, the 00-component of the linear combination of these conserved quantities is bounded. By this reason, the original derived theory is considered stable. Finally, as an instructive example, we elaborate the third-order derived system and analyze the stabilities in different cases of root decomposition of the characteristic polynomial extensively.

Matrix‐free preconditioning for high‐orderH(curl) discretizations

AbstractThe greater arithmetic intensity of high‐order finite element discretizations makes them attractive for implementation on next‐generation hardware, but assembly of high‐order finite element operators as matrices is prohibitively expensive. As a result, the development of general algebraic solvers for such operators has been an open research challenge. Fast matrix‐free application of high‐order operators has received significant attention in the literature in the context of Poisson‐type problems, but preconditioners and solvers for inverting more general operators are not very well‐developed. In this paper, we consider the problem of preconditioning a definite Maxwell operator at high polynomial order without assembling a matrix. We show that given efficient preconditioners for high‐orderH1finite element problems on the same mesh, efficientH(curl)preconditioners can be constructed in an auxiliary space framework. We demonstrate the resulting preconditioners in a practical setting with tensor‐product basis functions on an unstructured mesh of quadrilaterals. Our approach uses a sparsifiedH1solver constructed on a low‐order mesh of the nodal points of the underlying high‐order space, and we show that the resultingH(curl)preconditioner is effective at very high polynomial orders for two‐dimensional model problems with complicated geometry, varying piecewise constant coefficients, and curved elements. The resulting preconditioner scales with nearly optimalO(pd + 1)floating point operation count and optimalO(pd)memory transfer requirements, outperforming existing Maxwell preconditioners in the high‐order regime.

Green's formulas and Poisson's equation for bosonic Laplacians

A bosonic Laplacian is a conformally invariant second‐order differential operator acting on smooth functions defined on domains in Euclidean space and taking values in higher‐order irreducible representations of the special orthogonal group. In this paper, we firstly introduce the motivation for study of the generalized Maxwell operators and bosonic Laplacians (also known as the higher spin Laplace operators). Then, with the help of connections between Rarita–Schwinger type operators and bosonic Laplacians, we solve Poisson's equation for bosonic Laplacians. A representation formula for bounded solutions to Poisson's equation in Euclidean space is also provided. In the end, we provide Green's formulas for bosonic Laplacians in scalar‐valued and Clifford‐valued cases, respectively. These formulas reveal that bosonic Laplacians are self‐adjoint with respect to a given L2 inner product on certain compact supported function spaces.

Nonuniqueness of the quasinormal mode expansion of electromagnetic Lorentz dispersive materials.

Any optical structure possesses resonance modes, and its response to an excitation can be decomposed onto the quasinormal and numerical modes of a discretized Maxwell operator. In this paper, we consider a dielectric permittivity that is an N-pole Lorentz function of the frequency. Even for discretized operators, the literature proposes different formulas for the coefficients of the quasinormal-mode expansion, and this comes as a surprise. We propose a general formalism, based on auxiliary fields, which explains why and evidences that there is, in fact, an infinity of mathematically sound possible expansion coefficients. The nonuniqueness is due to a choice of the linearization of Maxwell's equations with respect to frequency and of the choice of the form of the source term. Numerical results validate the different formulas and compare their accuracy.

On Maxwell’s Equations on Globally Hyperbolic Spacetimes with Timelike Boundary

We study Maxwell's equation as a theory for smooth $k$-forms on globally hyperbolic spacetimes with timelike boundary as defined by Ak\'e, Flores and Sanchez. In particular we start by investigating on these backgrounds the D'Alembert - de Rham wave operator $\Box_k$ and we highlight the boundary conditions which yield a Green's formula for $\Box_k$. Subsequently, we characterize the space of solutions of the associated initial and boundary value problem under the assumption that advanced and retarded Green operators do exist. This hypothesis is proven to be verified by a large class of boundary conditions using the method of boundary triples and under the additional assumption that the underlying spacetime is ultrastatic. Subsequently we focus on the Maxwell operator. First we construct the boundary conditions which entail a Green's formula for such operator and then we highlight two distinguished cases, dubbed $\delta\mathrm{d}$-tangential and $\delta\mathrm{d}$-normal boundary conditions. Associated to these we introduce two different notions of gauge equivalence and we prove that in both cases, every equivalence class admits a representative abiding to the Lorenz gauge. We use this property and the analysis of the operator $\Box_k$ to construct and to classify the space of gauge equivalence classes of solutions of the Maxwell's equations with the prescribed boundary conditions. As a last step and in the spirit of future applications in the framework of algebraic quantum field theory, we construct the associated unital $*$-algebras of observables proving in particular that, as in the case of the Maxwell operator on globally hyperbolic spacetimes with empty boundary, they possess a non-trivial center.

Construction of Special Solutions for the Maxwell Equations

In this paper, we consider homogeneous polynomial solutions for the classical Maxwell equations in a time-space domain with time variable $$t<t_0$$, where $$t_0$$ is an arbitrary constant. The technique is motivated by the study of the so-called generalized Maxwell operators, which are constructed as conformally invariant differential operators in the framework of Clifford analysis. We firstly show that solutions of the classical Maxwell equations are equivalent to null solutions of the generalized Maxwell operator, then a direct decomposition and a partially orthogonal polynomial basis for the space of homogeneous polynomial null solutions of the generalized Maxwell operator are provided. At the end, a basis of homogeneous polynomial solutions for the source-free Maxwell equations on a bounded time-space domain is given.

Linear and nonlinear electromagnetic waves in modulated honeycomb media

AbstractWave dynamics in topological materials has been widely studied recently. A striking feature is the existence of robust and chiral wave propagations that have potential applications in many fields. A common way to realize such wave patterns is to utilize Dirac points, which carry topological indices and is supported by the symmetries of the media. In this work, we investigate these phenomena in photonic media. Starting with Maxwell's equations with a honeycomb material weight as well as the nonlinear Kerr effect, we first prove the existence of Dirac points in the dispersion surfaces of transverse electric and magnetic Maxwell operators under very general assumptions of the material weight. Our assumptions on the material weight are almost the minimal requirements to ensure the existence of Dirac points in a general hexagonal photonic crystal. We then derive the associated wave packet dynamics in the scenario where the honeycomb structure is weakly modulated. It turns out the reduced envelope equation is generally a two‐dimensional nonlinear Dirac equation with a spatially varying mass. By studying the reduced envelope equation with a domain‐wall‐like mass term, we realize the subtle wave motions, which are chiral and immune to local defects. The underlying mechanism is the existence of topologically protected linear line modes, also referred to as edge states. However, we show that these robust linear modes do not survive with nonlinearity. We demonstrate the existence of nonlinear line modes, which can propagate in the nonlinear media based on high‐accuracy numerical computations. Moreover, we also report a new type of nonlinear modes, which are localized in both directions.

Maxwell's equations with arbitrary self-action nonlinearity in a waveguiding theory: Guided modes and asymptotic of eigenvalues

The paper treats a nonlinear eigenvalue problem that describes propagation of a transverse-magnetic wave in a plane dielectric waveguide having perfectly conducted walls at both sides. The dielectric's permittivity is characterised by an arbitrary monotonically increasing self-action nonlinearity. The full set of guided modes is described by eigenvalues of the corresponding (nonlinear) Maxwell operator with appropriate boundary conditions. We give a comprehensive analysis of this problem and develop an original approach to study its solvability and properties of solutions. Several results about existence of the eigenvalues are proved, their distribution and asymptotic are found; zeros of the eigenfunctions and their location are also determined; criterion of periodicity for the eigenfunctions is found, comparison theorem is derived, etc. It is shown that unbounded nonlinearities lead to appearance of nonlinearised solutions. This results in the existence of a novel guided regime that cannot be described within the framework of perturbation of linear guided wave regimes.