therefore this is a challenging broadband problem. On two quad-core AMD Opte- ron 2350 processors and a total of 32 GByte of RAM, the simulations for M = 20 discretized byλ0/10 and λ0/100 were solved in 1 minute 3 seconds and 42 min-
utes 51 seconds respectively.
As expected, figure 4.2(a) shows that the Luneburg lens is more accurately ap- proximated when the number of shells increases. Only for M = 16 we notice a very small increase of the relative error. Figure 4.2(b) displays the relative error between the numerical solution for the piecewise constant Luneburg lens as com- pared to its analytical counterpart. As we can see from figure 4.2(b), the relative error of the fine discretization is about a factor 100 better than the error of the rough discretization, which is in line with the expected convergence rates. For each discretization, the relative error remains more or less constant as a func- tion of the number of shells, because each shell is divided into segments with a length ofλ0/10 and λ0/100 respectively. The numerical results clearly show that
our technique is capable of correctly handling broadband problems and objects embedded inside other objects allowing the simulation of a wide range of applica- tions.
4.4
Conclusion
In this chapter electromagnetic scattering problems by a Luneburg lens were nu- merically solved by using boundary integral equations, discretized by means of the Method of Moments. The classical MLFMA was hybridized with the NPWM to allow the simulation of broadband problems. The numerical results for the Luneburg lens are in very good agreement with the analytical solutions. For an increasing number of shells in the discretized lens, the relative error between the analytical and numerical solution remains almost constant. This proves that our method is capable of handling such complex problems.
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5
Simulations of a Swiss Roll
Ensemble
Bart Michiels, Ignace Bogaert, Jan Fostier and Daniël De Zutter
Published in Microwave and Optical Technology Letters, vol. 53, no. 10, pp. 2268–2274, Oct. 2011.
Æ Æ Æ
This chapter investigates a magnetic metamaterial, built from so-called Swiss rolls, by means of full-wave simulations. After determining the resonance fre- quencies of a single Swiss roll, the macroscopic behavior of an ensemble of Swiss rolls is investigated. The macroscopic material parameters of this en- semble are determined by S-parameter retrieval, using a bianisotropic model, that does not assume reciprocity a priori. As a result, the macroscopic perme- ability, permittivity and magnetoelectric coupling coefficients are obtained as a function of frequency.
5.1
Introduction
Metamaterials have attracted considerable attention in recent years. In general, metamaterials are ensembles of microscopic (i.e. much smaller than the wave- length) structures that can be homogenized into a macroscopic medium with ef- fective material parameters. The microscopic structure can be designed to allow the construction of metamaterials with remarkable material parameters, e.g. chi- ral, negative permittivity, negative permeability and even negative refractive index
materials. However, the retrieval of meaningful material parameters from these microscopic metamaterial structures remains a challenging issue and a topic of high interest in the metamaterial research community[1–4].
In this chapter a metamaterial structure built from so-called Swiss rolls will be homogenized by means of full-wave simulations. Swiss rolls are rolled-up per- fectly electrically conducting (PEC) plates that, when arranged into a periodic lattice, form a two-dimensional (2D) magnetic metamaterial. This metamaterial was first proposed in[5–7] and continues to attract much interest [8–10]. An inci- dent transverse-electrically (TE) polarized plane wave induces a current along the surface of the Swiss roll and the magnetic field in the center exhibits resonant be- havior as a function of frequency, giving rise to a negative permeability in certain frequency ranges.
The full-wave method, used throughout this chapter to perform the simulations, is a Method of Moments (MoM) solver[11] using the Electric Field Integral Equa- tion (EFIE) accelerated with the Multilevel Fast Multipole Algorithm (MLFMA) [12]. Such solvers typically require much less unknowns and have a higher accu- racy compared to e.g. Finite Difference Time Domain (FDTD) and Finite Elements (FE) solvers, at the cost of being more mathematically involved. In order to solve the low-frequency breakdown of the MLFMA, the Normalized Plane Wave Method (NPWM)[13] is invoked. All these methods and algorithms are implemented in Nero2d, an open source, full-wave solver for 2D scattering problems[14]. A val- idation of the algorithms for complex structures is discussed in[15]. Performing simulations at frequencies close to the resonance frequencies of the Swiss rolls is a real challenge for MoM-MLFMA solvers, mainly due to the high condition num- ber of the MoM-matrix. However, the fact that the structures do not have to be physically built is a considerable advantage when compared to measurements. The material parameters are calculated using S-parameter retrieval, assuming that the Swiss roll metamaterial behaves as a bianisotropic material. This approach is similar to[16, 17], in which split-ring resonators are analyzed. However, in contrast with[16, 17], reciprocity is not assumed a priori by our model. Of course the reciprocity property will be checked in order to further validate the results of our model, as Swiss rolls are PEC objects and therefore reciproque. Other methods to retrieve the material parameters also exist, such as the field-averaging method [18]. A waveguide setup allows the material parameters to be retrieved while simulating only a single row of Swiss rolls stacked inside a waveguide. According to image theory, this is equivalent to a grid of Swiss rolls extending to infinity in the direction perpendicular to the waveguide, but it requires significantly less computing capacity.
The outline of the chapter is as follows. First, in section 5.2 a single Swiss roll is analyzed. Based on a frequency sweep, the resonance frequencies are determined, along with a sufficiently accurate discretization of the Swiss roll. This discretized Swiss roll is subsequently used to determine the homogenized material parame- ters. Second, in section 5.3, an 8× 8 and a 16 × 16 grid of Swiss rolls is simulated