Project Title: **Advanced Graphics for Computational
Electromagnetics on PCs**

Summary:

The graphic representation of electromagnetic fields in the three-dimensional space is a very demanding task because the field consists of six or more components, covers the entire space, and varies in time. Projections on one or several planes may give some insight but more sophisticated techniques are required for obtaining a deeper insight in electrodynamic phenomena. Modern personal computers are powerful enough for advanced animations, provided that the total number of pictures is sufficiently small because of limited storage. This condition can be met especially when the time dependence of the field is periodic. On one hand side, the periodicity allows to save memory, but on the other hand side, it requires more complex tools for playing such kind of movies. In the graphic platform of the 3D MMP code we have implemented a special meta language that simplifies the production of complicated movies consisting of several sequences that are repeated when the movie is played. In addition, we are developing a special movie player and archive tools for MMP movies. 3D MMP also provides features for simulating particle-field and particle-particle interactions. PCs are powerful enough for the direct visualization of the movement of particles. In addition to 'real' particles, we have implemented 'virtual' particles that allow to draw field lines and to study 'fictitious worlds' based on user-defined laws.

Contacts: Ch.Hafner, H.U. Gerber, IFH, ETZG95, ETH Zentrum, CH-8092 Zurich

Electronic Contacts: hafner@ifh.ee.ethz.ch

Index Terms: Graphics, Animation

Project Title: **MMP Computation of Periodic Structures**

Summary:

The 3D MMP code for computational electromagnetics is a powerful tool for very accurate and reliable solutions of scattering and antenna problems. It includes a large library of basis functions that are analytic solutions of the Maxwell equations. Gratings and bi-periodic structures are of considerable interest in optics, radio science, and nano science. Most of the available codes for periodic structures are very time-consuming, not very reliable, or adapted to special cases. Previous attempts of MMP computations based on the Floquet theory were also rather inefficient. We now have implemented periodic boundary conditions as a new type of boundary conditions and Rayleigh expansions as a new type of periodic basis functions. Together with appropriate transformations, these features allow to compute the electromagnetic field in one cell of the periodic structure in exactly the same way as in ordinary scattering problems. This technique allows us to compute arbitrary gratings and bi-periodic structures with 3D MMP. Comparisons with results obtained with the well-known modal method have shown, that 3D MMP is not only extremely efficient but also reliable and accurate. Since 3D MMP includes Surface Impedance Boundary Conditions (SIBC), one can also obtain simplified solutions when materials with a good conductivity are present. In order to illustrate the capability of 3D MMP, we have simulated an anisotropic chiral slab by a bi-periodic array of helical structures. Our results are in excellent agreement with the theory of chiral media.

Contacts: Ch.Hafner, IFH, ETZG95, ETH Zentrum, CH-8092 Zurich

Electronic Contacts: hafner@ifh.ee.ethz.ch

Index Terms: Computational Electromagnetics, Gratings, Chiral media

Project Title: **Genetic Algorithms for Computational
Electromagnetics**

Summary:

In computational electromagnetics, one is often confronted with very demanding non-linear optimization problems. The optimization of the shape of an insulator in high voltage technique and the solution of inverse scattering problems are typical examples. There are also similar 'internal' optimization problems within codes for computational electromagnetics, for example, the solution of non-linear eigenvalue problems, the pole-setting procedure in the 3D MMP code and similar optimizations of the expansion of the electromagnetic field. Finite Difference (FD) and other well-known algorithms can easily be generalized by introducing weights (MEI method). This and further generalizations usually lead to the problem of an optimal selection of the parameters introduced by the generalization. It seems that Genetic Algorithms (GA) provide a powerful instrument for such tasks. It is important to recognize that GAs require a lot of experience also some commercial GA codes are available. Moreover, usual GAs are not powerful enough for most of our applications. It seems that 1) a mixture of GAs with other strategies and 2) an optimization of the parameters of the Gas are required in most cases. The latter leads to the concept of self-optimizing GAs. Also we did not yet implement such a GA, our experience is very encouraging. We were able to drastically improve the performance of eigenvalue computations and of the computation of the frequency dependence of electromagnetic phenomena by an internal optimization of the MMP code.

Contacts: Ch.Hafner, J. Froehlich, IFH, ETZG95, ETH Zentrum, CH-8092 Zurich

Electronic Contacts: hafner@ifh.ee.ethz.ch

Index Terms: Computational Electromagnetics, Genetic Algorithms, Optimization

Project Title: **Parameter Estimation
Technique for Dense Matrix Methods**

Summary:

In computational electromagnetics one can observe a growing interest in Finite Difference Time Domain (FDTD) algorithms. FDTD is an implicit sparse matrix method and works with one iteration per time step which makes the method numerically efficient. The Method of Moments (MoM) and other dense matrix methods like 3D MMP require much more iterations when an iterative matrix solver, e.g. the method of Conjugate Gradients (CG) is applied. Since iterative matrix solvers are the key to efficient solutions of large problems, it is extremely important for dense matrix methods to find a technique that allows to drastically reduce the number of CG iterations. Dense matrix methods typically work in the frequency domain. Although the solution of a previous time step is a good estimate for the solution of the actual time step, this knowledge usually is wasted when dense matrix methods are applied. The Parameter Estimation Technique (PET) uses the information obtained in one or several previous time steps for obtaining an excellent initial guess for the parameters to be computed with CG algorithms. The rectangular MMP matrices often have a large condition number. This provides hard problems for CG. It has been found that higher order PET with power series extrapolation does not allow to reduce the number of CG iterations. Although a linear PET is helpful, one can generalize the PET by using arbitrary basis functions. When the PET basis is optimized with a genetic algorithm, one can obtain a drastic reduction of the CG iterations (factor>50).

Contacts: Ch.Hafner, IFH, ETZG95, ETH Zentrum, CH-8092 Zurich

Electronic Contacts: hafner@ifh.ee.ethz.ch

Index Terms: Computational Electromagnetics, Matrix Methods, Optimization

Supported by: SNF

Project Title: **Investigations on the Condition Number of
Matrix Methods**

Summary:

The condition number is a well-known measure of matrices. Large condition numbers cause inaccuracies in the matrix solvers. One of the most useful matrix solvers for high condition numbers is based on Givens plane rotations. Although this algorithm is included in LINPACK, it has been discarded in LAPACK because many scientists think that one should avoid techniques that lead to large condition numbers. A detailed study of the MMP code for computational electromagnetics has shown that it is more likely to obtain accurate results when the condition number is high - provided that the matrix solver does not cause a significant loss of the accuracy. Since this surprising result might be unique for MMP, generalized Fourier analysis and other techniques for the approximation of functions (based on non-orthogonal basis functions) have been studied. This study supports the surprising observation: The probability of obtaining accurate results is higher when the condition number of the corresponding matrix is higher. Therefore, one should not try to reduce the condition number when one is designing a numerical method and one should develop matrix solvers that allow to efficiently handle ill-conditioned matrices. It has also been observed that methods based on ill-conditioned matrices are powerful for the extrapolation of functions known in a finite interval and for the recognition of the type of a measured function.

Contacts: Ch.Hafner, IFH, ETZG95, ETH Zentrum, CH-8092 Zurich

Electronic Contacts: hafner@ifh.ee.ethz.ch

Index Terms: Condition Number, Matrix Methods, Accuracy

Project Title: **Transition from Maxwell to Transmission Line
Theory**

Summary:

The computation of the propagation of electromagnetic waves on transmission lines is based on electrostatic and magnetostatic computations and on additional formula describing dynamic effects (skin effect, proximity effect). Although this procedure is numerically powerful, it is theoretically not consistent for several reasons. For example, the static computations implicitly assume different propagation constants in different media. It seems to be reasonable to clarify this situation by a rigorous computation of transmission line modes based on an analytic solution of Maxwell's equations. We have studied transmission line modes by the MMP code for computational electromagnetics. This code is based on a method that is very close to the analytic solution, because it uses basis functions that are analytic solutions of the Maxwell equations. The code allows to compute the propagation constant of guided modes by solving a nonlinear eigenvalue problem and gives important information on the accuracy of the results. It has been found that one can accurately compute both the electromagnetic field and the propagation constant of any mode when the frequency is sufficiently high. At lower frequencies, i.e. in the domain of the transmission line theory, the computation of the propagation constant becomes completely inaccurate. Nonetheless one can accurately compute the field. Therefore, the initial (wrong) assumption of different propagation constants in different media does not cause inaccuracies in the field computations.

Contacts: Ch.Hafner, IFH, ETZG95, ETH Zentrum, CH-8092 Zurich

Electronic Contacts: hafner@ifh.ee.ethz.ch

Index Terms: Maxwell Theory, Transmission Line Theory

Project Title: **Numerical Solution of Complex Eigenvalue
Problems**

Summary:

In electrodynamics one describes guided waves and waves in resonators as solutions of nonlinear eigenvalue problems. As long as one has no losses, the eigenvalues are real valued. The introduction of losses is important from the technical point of view. Lossy waveguides are usually described by complex propagation constants and lossy resonators are described by complex resonance frequencies. Although this approach is usual, it causes theoretical problems because a lossy structure is no longer energetically closed. Nonetheless, one can easily design a numerical code for solving such complex eigenvalue problems. If the structure is geometrically simple, one can even find analytic solutions. Analytic solutions of complex eigenvalue problems usually lead to transcendent equations that have to be solved numerically. We have extensively studied lossy structures with rotational symmetry, both with a code based on the analytic solution and with the MMP code for computational electromagnetics. Since the results were often very surprising, the very good agreement of the two solutions (>6 digits) was important. We found different types of transitions of the corresponding modes of a loss-free structure and a transition from guided to evanescent modes. It is well known that evanescent modes (below the cutoff frequency of the corresponding guided mode) have an imaginary propagation constant and exist even in the limit frequency -> 0. However, the correspondent 'evanescent' modes on lossy structures show a cutoff frequency.

Contacts: Ch.Hafner, L. Novotny, IFH, ETZG95, ETH Zentrum, CH-8092 Zurich

Electronic Contacts: hafner@ifh.ee.ethz.ch

Index Terms: Numerical Methods, Eigenvalue Problems

Project Title: **Numerical Simulation in Nearfield Optics**

Summary:

Nearfield optics is an interesting branch of nano science. One of the promising instruments in nearfield optics is the Scanning Nearfield Optical Microscope (SNOM). The SNOM allows to optically observe objects that are considerably smaller than the wavelength. This is possible because the electromagnetic field can be concentrated in an area that is considerably smaller than the wavelength. This is well known in high voltage technology and other branches of electrical engineering but the design and fabrication of optical structures with an extremely localized field is very difficult. At the same time, the numerical simulation of such structures is very difficult because one has to model parts that are small compared with the wavelength as well as large parts. I.e., a code is required that can handle small and large parts at the same time. In a first step we considered 2D models of SNOM tips with objects at different positions near the tip. An attempt with a FDTD code did not lead to useful results within two months but the results with our MMP code were very encouraging. It seems that MMP is superior to FDTD and MoM codes for SNOM and other problems of nearfield optics. Therefore, we started considering 3D SNOM models with MMP. Although some results have been obtained, severe problems have been observed. These problems mainly are caused by infinite boundaries in the model of the substrate. In order to solve these problems, we are implementing new 'multidomain' basis functions in the code.

Contacts: Ch.Hafner, L. Novotny, IFH, ETZG95, ETH Zentrum, CH-8092 Zurich

Electronic Contacts: hafner@ifh.ee.ethz.ch

Index Terms: Numerical Simulation, Nearfield Optics, SNOM

In collaboration with: IBM Rueschlikon