Metamaterials o� er great potential for the development of new technologies and to make existing devices smaller, faster and more e� cient. Metamaterials produce electromagnetic phenomena that are not seen in natural materials, and come in a wide variety of types. When developing a new metamaterial, simulation can be used both to analyze both the bulk property of the material and individual element. This article will show how CST STUDIO SUITE can be used to simulate and design metamaterials through the entire workfl ow, from the design of the resonator to the implementation in a full device.

Metamaterials are structures designed to have electromag-netic properties that do not exist in nature. The application of metamaterials to industrial device design has great potential in the fi elds of electronics and optics. Researchers have pro-posed a huge range of devices using metamaterials, which range from miniaturized, low-noise versions of existing com-ponents such as waveguides, fi lters and antennas to brand new devices that were previously impossible, such as cloaks and superlenses that can overcome the diff raction limit.

There are several types of material that are broadly described as metamaterials, including:

■ Double negative (DNG) materials (materials which behave as if they have negative permittivity and perme-ability, and therefore negative refractive index)

■ Electronic band gap (EBG) structures (materials with frequency bands that do not support EM wave propagation)

■ Artifi cial magnetic conductors (AMC) (conductors with very small tangential magnetic fi eld even with a large electric fi eld along the surface).

These are illustrated in Figure 1.

Figure 1: Applications of metamaterials: (left) A compact horn with a DNG focusing component, (right top) a gapped EBG power plane for PCB noise mitigation and

(right bottom) a low-profi le antenna using AMC technology.

WHITEPAPERMoDelinG AnD SiMulAtion of MetAMAteriAl-BAseD DeVices for InDustriAl APPlicAtions

CST AG WHITEPAPER MODELING AND SIMULATION OF METAMATERIAL-BASED DEVICES FOR INDUSTRIAL APPLICATIONS 2

OVerVieW

A typical metamaterial device consists of an array of reso-nators at the millimeter or microscopic scale, embedded in or around a macroscopic structure. The design of these res-onators and the interaction between them mean that, on a large enough scale, the metamaterial should act like a bulk material with the desired properties.

To develop a metamaterial, both the individual unit cell and the full-scale system need to be considered. While treating the metamaterial as a bulk material with negative refractive index is useful at early stages of the development, the fi nal design needs to take into account the properties of the meta-material at the scale of individual cells.

Electromagnetic simulation can be used to verify theories and develop metamaterial concepts before constructing any prototypes. However, there are several special considerations that the researcher should be aware of when modeling metamaterials. Metamaterials comprise sub-wavelength el-ements, but these elements are often arranged in electrically large arrays. This means that a variety of solver technologies are required to develop a metamaterial device effi ciently. CST  STUDIO  SUITE can be used to simulate metamaterials at every step of the design process. (Figure 2)

Figure 2: Workfl ow for modeling the behavior of a metamaterial system, from a single

unit cell to a whole device.

CST AG WHITEPAPER MODELING AND SIMULATION OF METAMATERIAL-BASED DEVICES FOR INDUSTRIAL APPLICATIONS 3

Unit Cell DesiGn

There are numerous metamaterial topologies in the literature. While they operate on diff erent principles, they all need to be carefully engineered so that they give the correct behavior at the frequencies of interest. Simulation of the unit cell allows physical values such as transmission, absorption and refl ec-tion coeffi cients to be calculated in a straightforward way.

For EBG applications, their behavior can be characterized by a single element of the periodic structure (Figure 4) by means of the dispersion diagram. The element can be modeled as a unit cell and simulated using the eigenmode solver. The eigenmode solver can handle the lossy metal properties of the resonators and the dispersive properties of the substrate, and can use periodic boundary conditions to model an entire array based on a single element. 1

The dispersion diagram is produced by performing a param-eter sweep over the phase relations in the transverse direc-tions, as an example between the x and y periodic bound-aries, and plotting the frequencies of each mode across the Brillouin zone (Figure 3). After calculating the modes across all phases, the band gaps are given by the absence of propa-gating modes in certain frequency bands. CST STUDIO SUITE includes a built-in template that sets up the parameters and can generate dispersion diagrams automatically for a meta-material model.

For problems which are driven by an excitation source, the dispersion diagram can be obtained from the scattering pa-rameters (S-parameters). This is most commonly done in transmission line based devices like leaky wave antennas. To generate these, either the transient solver or the frequency do-main solver can be used. Just as with a classical material, the S-parameters show how much power is refl ected, how much is transmitted, and how much is absorbed or re-radiated at diff erent frequencies.

Frequency domain simulation also allows the use of Floquet ports. These can be used with the unit cell structure to calcu-late the phase refl ection diagram for any angle of incidence. This phase relationship diagram off ers an alternative approach for calculating the band gap for an artifi cial magnetic conduc-tor (AMC). The band gap corresponds to those frequencies where the refl ection phase is between ± 90°. In this region, the magnitude of the surface impedance exceeds the free space impedance, which means that antenna elements can lie very close to the ground plane without being shorted out.

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Figure 4: A single gapped plane element representing the EBG

power plane in Figure 1.

Figure 3: Band gaps over the irreducible Brillouin zone (Γ-Χ-Μ-Γ) in the EBG structure

shown in Figure 4.

CST AG WHITEPAPER MODELING AND SIMULATION OF METAMATERIAL-BASED DEVICES FOR INDUSTRIAL APPLICATIONS 4

BulK ProPertY EXtrAction

Once the element has been designed, the next step of the metamaterial design workfl ow (Figure 2) is to extract its ef-fective electromagnetic properties. The bulk properties of a metamaterial can be calculated using the Drude model for permittivity and the Lorentz model for permeability:

This gives seven variables that need to be calculated: ε∞ (per-mittivity at infi nite frequency), μ∞ (permeability at infi nite frequency), ωp (plasma frequency), ω0 (resonance frequen-cy), νc (collision frequency), δ (damping factor) and μs (static permeability). To fi nd the values that best fi t the metama-terial, a new 3D model of the system is created: a similar unit cell, in which the detailed metamaterial structure has been replaced by a single bulk material. This bulk material is modeled as a Drude-Lorentz material, with the unknown variables replaced by parameters in CST STUDIO SUITE.

Figure 6: Substitution of metamaterial with Drude-Lorentz bulk material in a wedge horn.

Figure 5: (left) The S-parameters of the optimized material compared to the S-parameters of the element unit cell. (right) The bulk material model as a unit cell.

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S1,1 - bulk DNGS1,1 - unit cellS2,1 - bulk DNGS2,1 - unit cell

Drude-Lorentz material1

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CST AG WHITEPAPER MODELING AND SIMULATION OF METAMATERIAL-BASED DEVICES FOR INDUSTRIAL APPLICATIONS 5

To fi nd the value of these parameters, an automotive optimi-zation is then carried out in order to “fi t” the Drude-Lorentz parameters. The S-parameters calculated for the unit cell are used as the goal of the optimization. The optimizer then searches the parameter space for the combination of values in the Drude-Lorentz model that best replicates the behaviour of the unit cell, by iteratively simulating the material and tweaking the variables. (Figure 5)

The bulk material can then be substituted into the full-sized structure (Figure 6) to allow the behavior of the device to simulated effi ciently (Figure 7). Replacing the detailed model with the bulk material can reduce the number of mesh cells and increase the time step, speeding up the simulation, but may neglect the contribution of edge eff ects and inter-actions with the rest of the structure. For this reason, it is sometimes worthwhile carrying out a fi nal simulation with the full detailed model.

For a standard metamaterial-based device, individual ele-ments can be a fraction of a wavelength in size, while the device itself is many times larger. This means that a very fi ne but very large mesh is required to accurately model the system. The time domain solver with hexahedral meshing can deal with these large, complex structures in an effi cient way, especially when used together with high-performance computing techniques such as GPU computing or MPI cluster computing.2 The time solver calculates a wide range of fi eld results, including the near-fi eld behavior in both the time and frequency domains, the farfi eld and phase center of the device and its transmission and refl ection coeffi cients.

Figure 7: Electric fi eld values in the metamaterial wedge lens from Figure 3, (top) in the double-positive region and (bottom) in the double-negative region.

CST AG WHITEPAPER MODELING AND SIMULATION OF METAMATERIAL-BASED DEVICES FOR INDUSTRIAL APPLICATIONS 6

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About CST

Founded in 1992, CST offers the market’s widest range of 3D electromagnetic field simulation tools through a glob-al network of sales and support staff and representatives. CST develops CST STUDIO SUITE, a package of high-per-formance software for the simulation of electromagnet-ic fields in all frequency bands, and also sells and sup-ports complementary third-party products. Its success is based on a combination of leading edge technology, a user-friendly interface and knowledgeable support staff. CST’s customers are market leaders in industries as diverse as telecommunications, defense, automotive, electronics and healthcare. Today, the company enjoys a leading position in the high-frequency 3D EM simulation market and employs 250 sales, development, and sup-port personnel around the world.

CST  STUDIO SUITE is the culmination of many years of research and development into the most accurate and efficient computational solutions for electromagnetic designs. From static to optical, and from the nanoscale to the electrically large, CST STUDIO SUITE includes tools for the design, simulation and optimization of a wide range of devices. Analysis is not limited to pure EM, but can also include thermal and mechanical effects and circuit simu-lation. CST STUDIO SUITE can offer considerable product to market advantages such as shorter development cycles, virtual prototyping before physical trials, and optimiza-tion instead of experimentation.

Further information about CST is available on the web at https://www.cst.com

Author

Arnab Bhattacharya, Application Engineer, CST AG

Conclusion

Developing a metamaterial for a particular application requires careful choice of the type and dimensions of each element. With electromagnetic simulation using CST STUDIO SUITE, the behavior of a potential metamaterial can be investigated prior to the prototyping stage. This makes it possible to extract the properties of a given design and to optimize the element for better performance. Metamaterials can be simulated both in isolation and in situ, and depending on the project requirements, considered either as a bulk ma-terial or an ensemble of individual elements.

References

1 A. Ciccomancini Scogna, “Electromagnetic Bandgap Structures for PPW Noise mitigation in PWR/GND plane pairs”, DesignCon 2007.

2 F.Hirtenfelder, G.Lubkowski, “3D Field Simulations using FI Time Domain Technique of Wedge- and Parabolic-Shaped Left Handed Materials (LHM)”, International Workshop on Antenna Technology (IWAT) 2007.