Rolf Schuhmann

Ab initio numerical simulation of left-handed metamaterials: Comparison of calculations and experiments

Using numerical simulation techniques, the transmission and reflection coefficients, or S parameters, for left-handed metamaterials are calculated. Metamaterials consist of a lattice of conducting, nonmagnetic elements that can be described by an effective magnetic permeability μeff and an effective electrical permittivity eeff, both of which can exhibit values not found in naturally occurring materials. Because the electromagnetic fields in conducting metamaterials can be localized to regions much smaller than the incident wavelength, it can be difficult to perform accurate numerical simulations. The metamaterials simulated here, for example, are based on arrays of split ring resonators (SRRs), which produce enhanced and highly localized electric fields within the gaps of the elements in response to applied time dependent fields. To obtain greater numerical accuracy we utilize the newly developed commercially available code MICROWAVE STUDIO, which is based on the finite integration technique with the per...

Stability of the FDTD algorithm on nonorthogonal grids related to the spatial interpolation scheme

In this paper we present a reformulation of the FDTD algorithm on nonorthogonal grids, which was originally proposed by Holland in 1983. Based on the matrix-vector notation of the finite integration technique (FIT), the new formulation allows to study a special type of instability, which is due to the spatial discretization and independent of the choice of the timestep. It is shown, that this type of instability can be avoided by a symmetric evaluation of the metric coefficients of the nonorthogonal grid. Two numerical examples demonstrate the stability properties and the high accuracy of the new method.

Conservation of Discrete Energy and Related Laws in the Finite Integration Technique - Abstract

We report some properties of the Finite Integration Technique (FIT), which are related to the definition of a discrete energy quantity. Starting with the well-known identities for the operator matrices of the FIT, not only the conservation of discrete energy in time and frequency domain simulations is derived, but also some important orthogonality properties for eigenmodes in cavities and waveguides. Algebraic proofs are presented, which follow the vector-analytical proofs of the related theorems of the classical (continuous) theory. Thus, the discretization approach of the FIT can be considered as the framework for a consistent discrete electromagnetic field theory.

Two-step Lanczos algorithm for model order reduction

The Pade-Via-Lanczos (PVL) algorithm proved to be a reliable technique for obtaining reduced-order models of electromagnetic devices. Its computational complexity is, however, quite large, since it involves inversion or factorization of a matrix which can be, for complex devices, on the order of hundreds of thousands. The present paper proposes a two-step approach based entirely on the Lanczos algorithm, meant to drastically reduce the computational complexity. In the first step, a Lanczos-based projection technique is used to reduce the un-inverted matrix to a manageable size, which can be dealt with by the PVL method in the second step. The computing time was thus reduced by a factor of ten, as compared to the classical PVL.

Dispersion and asymmetry effects of ADI-FDTD

In this paper, a generalized derivation of the alternating direction implicit finite-difference time-domain algorithm based on operator splitting is proposed. The formulation follows the notation of the finite integration technique. A straightforward proof of stability is given and the numerical dispersion formula is presented and verified by numerical experiments. As an additional parasitic effect, the asymmetric behavior of the algorithm even for exactly symmetric setups is revealed. Both the dispersion error and the asymmetry error are discussed in terms of the applicability of ADI for low-frequency problems.

A stable interpolation technique for FDTD on non-orthogonal grids

The application of the FDTD algorithm on generalized non-orthogonal meshes, following the basic ideas of Holland (1983), has been investigated by many authors for several years now, and detailed dispersion analysis as well as convergence studies have been published. Already in 1992 also a general stability criterion was given for the time integration using the standard leap-frog scheme (Lee et al.). Many authors, however, still propose some damped time stepping algorithms to work around unexpected instabilities in the discretization method. In this paper the origin of this type of instability is revealed, and a technique to obtain a stable discretization of Maxwells equations on non-orthogonal grids is proposed. To obtain more insight into the stability properties of the method, it is reformulated according to the matrix–vector notation of the Finite Integration Technique.

Effective Modeling of Double Negative Metamaterial Macrostructures

This paper presents a numerical procedure applied to the modeling of double negative metamaterial (MTM) structures. At the unit-cell level, the properties of the MTM are described by the extracted isotropic constitutive parameters, dispersion diagrams, and higher order modes analysis. The information gathered at the unit-cell level is used to characterize and homogenize the MTM macrostructure. By comparison of the simulation results for the effective and detailed macrostructures models, the applicability of this approach is confirmed and the significant savings of numerical costs are pointed out.

Long-time numerical computation of electromagnetic fields in the vicinity of a relativistic source

We propose an implicit scheme for the calculation of electromagnetic fields in the vicinity of short electron bunches moving in a transversally bounded domain, as it has place in particle accelerators. The scheme is able to accurately model curved boundaries and does not suffer from dispersion in direction of motion. It is based on splitting the discrete curl operator in its transversal and longitudinal parts. Unlike previous conformal schemes the new method has a second order convergence without the need to reduce the maximal stable time step of the conventional staircase approach. This feature allows the usage of a moving mesh easily. Several numerical examples are presented and the algorithm is compared to other approaches.

FDTD on nonorthogonal grids with triangular fillings

In this paper we present the combination of the FDTD method on nonorthogonal grids with the concept of triangular fillings. The resulting algorithm allows a more general mesh generation, where degenerated cells can usually be easily avoided, without additional computational cost or loss of modeling accuracy.

Subspace projection extrapolation scheme for transient field simulations

A subspace projection extrapolation (SPE) scheme is introduced for the start vector generation for the iterative solution of the algebraic systems of equations resulting from implicit time integration schemes for electromagnetic discrete field formulations. The SPE scheme yields optimal linear combinations from multiple available start vectors. Spectral components of the exact solution contained therein are optimally resolved which causes subsequently started conjugate gradient (CG) methods to converge according to a reduced effective condition number similar to deflated CG methods. Numerical tests show a significant improvement in numerical efficiency with the SPE scheme in comparison to established extrapolation methods.