Developing a Time-Domain Finite Element Method for the Lorentz Metamaterial Model and Applications

In this paper, we consider the time-dependent Maxwell's equations when Cole–Cole dispersive medium is involved. The Cole–Cole model contains a fractional time derivative term, which couples with the standard Maxwell's equations in free space and creates some challenges in developing and analyzing time-domain finite element methods for solving this model as mentioned in our earlier work [J. Li, J. Sci. Comput., 47 (2001), pp. 1–26]. By adopting some techniques developed for the fractional diffusion equations [V.J. Ervin, N. Heuer, and J.P. Roop, SIAM J. Numer. Anal., 45 (2007), pp. 572–591], [Y. Lin and C. Xu, J. Comput. Phys., 225 (2007), pp. 1533–1552], [F. Liu, P. Zhuang, V. Anh, I. Turner, and K. Burrage, Appl. Math. Comput., 191 (2007), pp. 12–20], we propose two fully discrete mixed finite element schemes for the Cole–Cole model. Numerical stability and optimal error estimates are proved for both schemes. The proposed algorithms are implemented and detailed numerical results are provided to justify our theoretical analysis.

Hybrid finite element (FE) solutions of transient electromagnetic field are proposed. Time dependent Maxwell's curl equations are solved separately using edge elements for the electric field approximation and nodal-based elements for the magnetic field approximation. Two different formulations are presented: the first adopts a unique mesh, the second two complementary FE meshes. The solution of the resulting equations systems is performed by explicit methods.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Using the Hankel transform, we obtain, in their most general form, the spinor solutions of Maxwell's equations in free space for a light beam propagating along 0z with cylindrical symmetry around 0z. Then we compute a paraxial approximation of these solutions and we discuss the case of Gaussian beams. A comparison of the spinor solutions with the scalar ones supplied by the parabolic equation is also given.

Biorthogonal interpolating functions have been applied to electromagnetic field modeling through the wavelet-collocation technique in the time domain, yielding a versatile first-principle algorithm for the solution of time dependent Maxwell's equations with inhomogeneous media. The resulting technique is rigorous and maintains high accuracy, while, by virtue of its sub-gridding capability, significant reduction of the computational requirement has been obtained particularly for the modeling of electrically large dielectric waveguides. The proposed technique has been applied to a two dimensional dielectric waveguide beam splitter, with dimensions typical of optical waveguide systems, and improvement in the computational efficiency has been demonstrated.

Dispersion analysis of finite difference and discontinuous Galerkin schemes for Maxwell's equations in linear Lorentz media

In the first part of this work, we introduce a monochromatic solution to the scalar wave equation in free space, defined by a superposition of monochromatic nondiffracting half Bessel-lattice optical fields, which is determined by two scalar functions; one is defined on frequency space, and the other is a complete integral to the eikonal equation in free space. We obtain expressions for the geometrical wavefronts, the caustic region, and the Poynting vector. We highlight that this solution is stable under small perturbations because it is characterized by a caustic of the hyperbolic umbilical type. In the second part, we introduce the corresponding solution to the Maxwell equations in free space.

Finite-element method for time-dependent Maxwell's equations based on an explicit-magnetic-field scheme

It is known that there is a close analogy, exhibited by the use of five-dimensional tensor notation, between the electromagnetic equations in free space and the first- and second-order equations of the quantum theory. In the present paper an analogy is traced between two tensors, one of which is related to the magnetic and electric moments of a doublet while the other represents the electric and magnetic field-strengths.

Parallel block preconditioners for virtual element discretizations of the time-dependent Maxwell equations

The aim of the present work is to introduce two monochromatic solutions to the scalar wave equation in free space, characterized by a caustic with a singularity of the hyperbolic umbilical type. The first solution, is a superposition of half-Mathieu beams, and the second one, is a superposition of parabolic beams. Since these solutions are determined by two particular complete integrals of the eikonal equation in free space, we compute their geometrical wavefronts, the caustic regions, and the corresponding Poynting vectors. Finally, we remark that, under certain conditions, these solutions describe three-dimensional accelerating beams in free space, propagating along semielliptical and parabolic paths, respectively.

A Fourier penalty method for solving the time-dependent Maxwell's equations in domains with curved boundaries

We present an efficient method for computing numerically the solution to the time- dependent Maxwell equations in an axisymmetric domain, with arbitrary (not necessarily ax- isymmetric) data. The method is an extension of those introduced in (20) for Poisson's equation, and in (4) for Maxwell's equations in the fully axisymmetric setting (i.e., when the data is also axisymmetric). It is based on a Fourier expansion in the azimuthal direction, and on an improved variant of the Singular Complement Method in the meridian section. When solving Maxwell's equations, this method relies on continuous approximations of the fields, and it is both H(curl)- and H(div)-conforming. Also, it can take into account the lack of regularity of the solution when the domain features non-convex edges or vertices. Moreover, it can handle noisy or ap- proximate data which fail to satisfy the continuity equation, by using either an elliptic correction method or a mixed formulation. We give complete convergence analyses for both mixed and non-mixed formulations. Neither refinements near the reentrant edges or vertices of the domain, nor cutoff functions are required to achieve the desired convergence order in terms of the mesh size, the time step and the number of Fourier modes used.

A high‐order, leapfrog integrator is presented and analyzed for Maxwell's time domain equations. The integrator is shown to be quite robust in terms of its efficiency and memory demands. Moreover, the integrator retains the original simplicity of the traditional second‐order, leapfrog integrator. Rigorous Fourier analyses are presented to quantify the dispersion, dissipation, and stability properties of the integrator. To limit the scope of the presentation, the 2×2, 2×4, 4×2, and 4×4 schemes are examined and compared. Here the first digit denotes temporal accuracy; the second digit denotes spatial accuracy. Numerical demonstrations, using the three‐dimensional rectangular waveguide as the object under test, are provided to further substantiate the theoretical analysis.

We construct exact solutions to the paraxial wave equation in free space characterized by stable caustics. First, we show that any solution of the paraxial wave equation can be written as the superposition of plane waves determined by both the Hamilton-Jacobi and Laplace equations in free space. Then using the five elementary stable catastrophes, we construct solutions of the Hamilton-Jacobi and Laplace equations, and the corresponding exact solutions of the paraxial wave equation. Therefore, the evolution of the intensity patterns is governed by the paraxial wave equation and that of the corresponding caustic by the Hamilton-Jacobi equation.

Continuous Galerkin methods for solving the time-dependent Maxwell equations in 3D geometries