University of Groningen New methods for the numerical solution of Maxwell's equations Kole, Joost Sebastiaan IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check the document version below. Document Version Publisher's PDF, also known as Version of record Publication date: 2003 Link to publication in University of Groningen/UMCG research database Citation for published version (APA): Kole, J. S. (2003). New methods for the numerical solution of Maxwell's equations s.n. Copyright Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons). Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons the number of authors shown on this cover page is limited to 10 maximum. Download date: 07-12-2018
Wovon man nicht sprechen kann, darüber muß man schweigen. L. Wittgenstein, Tractatus Logico-Philosophicus So I hope you can accept Nature as She is absurd. R. Feynmann, QED, The strange theory of light and matter
Chapter 1 Introduction Electromagnetic phenomena play a very prominent role in the modern age.the number of electric machines one uses on a daily basis without thinking about it is large, and only becomes imminent during an unfortunate power failure.physicists have succeeded quite well in formulating the laws to which these phenomena must adhere.in the late 1860s, J.C. Maxwell constructed the mathematical framework combining the phenomenological findings of his predecessors concerning electromagnetism.this can be viewed as the birth of mathematical physics.maxwell s achievement has stimulated many other people since, to construct similar basic sets of equations to describe other fields in physics, but it turned that not every field of physics could be as nicely and elegantly described as electromagnetism. The set of basic equations Maxwell constructed became known as the Maxwell Equations, and are given in their differential formulation, using MKS (meter,kilo,second) units, by 1 [1,2] t H(t) = 1 µ E(t), t E(t) = 1 ( H(t) J(t)), ε (1.1b) div εe(t) = ρ(t), div µh(t) = 0. (1.1a) (1.1c) (1.1d) Here E is the electric field (in volts/meter). H is the magnetic field (in amperes/meter) J is the total electric current density (in amperes/meter 2 ), and equates J = J source + σe, wherej source satisfies the continuity equation J source = ( / t)ρ. µ is the magnetic permeability (in henrys/meter).in this thesis, we assume that the magnetic permeability does not depend on time (or consequently frequency). ε is the electric permittivity (in farads/meter), in vacuum we have c = (µ 0 ε 0 ) 1/2.The electric permittivity is also assumed not to depend on time. σ is the electric conductivity (in siemens/meter), its value is non-negative for dissipative structures. ρ is the charge density (in coulomb/meter 3 ). 1 here we omit the spatial dependency r of all the variables.
4 Introduction Figure 1-1: Unit cell of the Yee grid. The variety of applications based on electromagnetism is enormous.unfortunately, in the case where a solution of the Maxwell Equations is required, it is usually not possible to solve them analytically.however, if the problem itself is well formulated, for example when the material parameters are all known, one can turn to numerical solutions.in case of the Maxwell Equations, two different approaches exist to solve the equations numerically. Firstly, we have the time-independent approach, where the time-dependence is assumed to be harmonic (i.e. e iωt ); this is a very reasonable assumption if for example all other parameters are time independent or vary harmonically with time.if the calculation of a frequency spectrum 2 is required, these methods are particularly useful.however, known disadvantages of these plane-wave expansion methods [3] are convergence problems [4] and the almost compulsory use of periodic boundary conditions. The second approach does not assume harmonically varying fields and solves the timedependent Maxwell Equations (TDME) directly.a well-known and widely used algorithm for this purpose is the Yee algorithm [5].A wide variety of finite-difference time-domain methods (FDTD) has descended from it [6,7].It gained much popularity due to its flexibility, robustness and speed.since its introduction, the original algorithm has been refined and optimized for specific problems and purposes to a high degree, but the core has essentially stayed the same.the main features of the original algorithm are the specific grid setup (the Yee space lattice), the leapfrog timestepping (meaning alternatingly updating the electric and magnetic field components) and the conditional stability. Yee chose the grid setup such that each E and H component is placed in the center of four circulating H resp. E components, as indicated in figure 1-1.The coordinates are labeled according to (i, j, k) = (i x, j y, k z), where x, y and z are the mesh sizes for each dimension in this uniform, cubic lattice.by choosing this specific arrangement, the three dimensional space lattice is effectively an interlinked array of Faraday s Law and 2 we will use the term spectrum to indicate the density of states (DOS) or eigenmode distribution. The procedure to calculate the DOS is explained in detail in appendix B.
Ampere s Law contours, since each field component is surrounded by exactly those other (four) field components that affect its time evolution, if a central-difference method is used for the spatial derivative.some additional advantages should be noted here.firstly, this alignment of the grid, combined with the central-difference method for the spatial derivative, ensures that both the differential form and integral form of Maxwell s Equations are discretized in space in a highly efficient manner [6].Secondly, in the absence of charges, thedivergenceofboththee and H fields implicitly vanishes for the Yee algorithm (see Ref.[6]), as required by Maxwell s Equations.In the original Yee algorithm one employs, just like the spatial derivative, a central-difference method to discretize the time derivative. The implementation of the Yee algorithm is straightforward, since it exactly prescribes how each component should be updated at each instant of time. So, why is there any need for new algorithms? A notable disadvantage of the FDTD algorithms is the conditional stability.for the basic Yee algorithm it can be proven that the Courant factor, defined as S = c t/ ( x) 2 + ( y) 2 + ( z) 2, must be smaller than one to guarantee stability of the algorithm.we will return to this point in more detail in Chapter 3.The Courant limit implies that for a given mesh size, the maximum timestep is fixed. For some physically relevant applications, like bioelectromagnetics and VLSI design [6,8,9], this fact has some undesirable consequences, namely that the finest mesh size necessary to resolve the spatial structure is much smaller than the shortest wavelength of a significant spectral component occurring in the source, and secondly that the total simulation time is related to this smallest wavelength.these two facts combined with the Courant limit on the maximum timestep require that a very large number of timesteps is necessary to arrive at the total simulation time.in such situations, one needs algorithms that are unconditionally stable.the largest timestep is then only limited by the desired accuracy of the solution. An advancement of the FDTD technique has been developed for this purpose.instead of the Yee leapfrog timestepping one uses alternating-direction-implicit (ADI) timestepping [10], and recently this method has revived due to an implementation in three dimensions [11 13].In the ADI timestepping approach, all field components are defined on identical time instances instead of staggered-in-time, and are updated by making use of the values of the other fields defined at two other time instances, rather than one as with the leapfrog timestepping.due to the implicit formulation of the update equations, the implementation requires to solve tridiagonal matrices in a direction alternate from the direction of the field component which is updated.the ADI algorithm can be proved to be unconditionally stable [14, 15], but is not very accurate for some applications [13].This fact emphasizes that there is still need for other unconditionally stable algorithms. Another point, not mentioned so far, is that there is no algorithm available yet that can perform a time-dependent simulation and compute a spectrum.the FDTD algorithms are not very suitable for the calculation of spectra as they do not conserve the electromagnetic energy density.on the other hand, the time independent methods to solve the Maxwell Equations that are mostly used to calculate photonic band structures, are subject to the restriction of periodic boundary conditions. To overcome the problems stated above, we introduce in this thesis a new class of unconditionally stable algorithms that do not have these restrictions.additionally, we derive an algorithm that is not based on timestepping to perform the time integration, but approximates the solution for a specific time instance.the resulting one-step algorithm is very accurate and more efficient than any other algorithm if high accuracy is required. 5
6 Introduction 1.1 Overview Theoutlineofthisthesisisasfollows: Chapter 2.In this chapter, we will construct new algorithms for the numerical solution of the TDME.The main motivation is that formally, the solution of the TDME can be written as the matrix exponential of a skew-symmetric matrix, which is an orthogonal transformation.in order to preserve the orthogonality, the skew-symmetric properties of the TDME must be conserved during spatial discretization and time integration. The time integration itself can be performed by timestepping or a direct approximation of the matrix exponential for a specific time instance.as a spin-off, we show how the conventional Yee algorithm can be improved with minor effort.furthermore, higher order (time stepping) algorithms in space and time will be developed and we will incorporate a grid with a variable mesh, absorbing boundary conditions and a current source. Chapter 3.Here we present a numerical analysis of the newly constructed algorithms, with emphasis on stability and accuracy.the new algorithms are compared with each other and with the conventional Yee algorithm. Chapter 4.The main application of our algorithms is in the analysis of the properties of photonic crystals.we measure the density of states of various existing crystals, but also present some new ones.furthermore, the ability to measure the (local) density of states of finite-sized system makes it possible to measure the width and depth of photonic bandgaps for a particular system as a function of cluster size.in addition, we analyze the effect of imperfections in the structure on these quantities.finally, the results of a simulation of the scattering of a wavepacket on a grating are compared with the known grating formula. Chapter 5.Here we summarize the basic properties of the algorithms that solve the TDME, review the main results and discuss some remaining open issues. Chapter 6.The method to construct new algorithms to solve the TDME is extended to other wavefield phenomena.we construct algorithms to solve the elastodynamic equations, and show some results for typical seismic events.