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Design and Characterization of Chiral Metamaterials using Numerical Simulations Tools

Ernesto Martґin, Juan Mun~oz, Aґ ngel J. Garcґia-Collado, Member, IEEE, Joseґ Margineda, Member, IEEE, Ismael Barba, Alvaro Goґmez, and Gregorio J. Molina-Cuberos

Abstract--A short review of the work developed by the authors concerning the numerical simulation of chiral structures using commercially available simulators during the last decade is presented. Several software packages, such as MEFiSTo, COMSOL Multiphysics, CST Studio Suite or EMPro, have been used to explore the interaction of metallic and dielectric structures with electromagnetic waves. These numerical packages are a very useful tool in the design stage to optimize the structure response, before building the real prototype. Therefore cost and effort can be reduced significantly. They also allow to contrast the experimental measurements obtained in the laboratory and to analyse the electromagnetic response at frequencies beyond of the experimental set-up. Index Terms--Metamaterials, chiral media, numerical simulations.

I. INTRODUCTION

C HIRAL metamaterials (CM) are artificial media capable to affect the polarization of a propagating wave by

rotating the polarization angle (known as electromagnetic

rotatory dispersion) and changing the polarization from linear

to elliptical (knows as circular dichroism). Macroscopically, a

CM can be described by including a new parameter, usually

called chirality , into the constitutive relations [?]:

D = E - j0µ0H,

(1)

B = µH + j0µ0E,

(2)

where and µ are the electric permittivity and magnetic permeability, respectively, and the chirality parameter produces a coupling between the electric and magnetic fields. The first technique to build CM at microwave frequencies was the random distribution of elements with chiral symmetry in a dielectric substrate. Later, new techniques based on ordered distribution of metallic elements were shown as very useful alternatives [?]. Now the new technique provided new parameters for material design which enhanced the gyrotropic

Ernesto Martґin (email: [email protected]), Juan Mun~oz (email: [email protected]), Joseґ Margineda (email: [email protected]), and Gregorio J. Molina-Cuberos (email: [email protected]) are with the Dpto. Electromagnetismo y Electroґnica, Universidad de Murcia, Campus Espinardo, 30100, Murcia, Spain. Angel J. Garcґia-Collado is with Dpto. de Ciencias Politeґcnicas, Universidad Catoґlica San Antonio, Guadalupe, 30107, Murcia, Spain, (email: [email protected]). Ismael Barba is with the Dpto. de Electricidad y Electroґnica, Universidad de Valladolid, 47002, Valladolid, Spain (e-mail: [email protected]). Aґ lvaro Goґmez is with the Dpto. de Ingenierґia de Comunicaciones, Universidad de Cantabria. Plaza de la Ciencia S/N, 39005, Santander, SPAIN. (e-mail: [email protected]). Manuscript submitted 20th March 2013; revised mes dia, an~o.

effects and were even able to produce negative refractive index. The negative refraction is not achieved by making use of two set of resonant structures for electric and magnetic response, as in traditional metamaterials. For CMs only one resonant structure with high electromagnetic coupling is required and negative refraction can be achieved for circularly polarized waves [?]. During the last decade, we have been using several commercially available simulators to explore the electromagnetic behaviour of metallic and dielectric structures with electromagnetic waves. The use of numerical simulation tools has allowed us to optimize the design of new materials with uncommon properties before constructing the real prototype. There are a high variety of numerical suites able to model the interaction of electromagnetic waves with metallic and dielectric structures. Here we present some of our experience using commercial simulation packages, such as MEFiSTo, EMPro, COMSOL Multiphysics or CST Studio Suite. II. DESIGN OF CHIRAL MEDIA Fig. 1 shows some of the structures designed to study the electromagnetic activity at microwave frequencies. They are built by patterning a 2D periodical distribution of threedimensional particles on printed circuit boards. All the cells possess C4 symmetry on the perpendicular axis and, consequently, present uniaxial chirality for a normal incident TEM wave. The unit cell is patterned on a FR-4 board with copper metallization, using both sides and connecting the segments through vias, and the dimensions of the structure were chosen to resonate into the X-Band to be contrasted with experimental results. The basic structure is in the form of a crank, with metallization in both sides of the FR-4 board. Cranks can be packed forming four pairs, the eight crank resonator [?][?] or packed in a compact distribution to maximize the number density of cranks, the four crank resonator [?]. Cranks can be modified by adding a segment in the two sides of the board, producing the five segments cranks [?]. All these three configurations clearly lack of specular symmetry, i. e. they have chiral symmetry, and are expected to produce electromagnetic activity. In order to check the importance of chirality, we have slightly changed the four crank resonator to break the chiral symmetry by extending the metallization of the segments to connect with nearest neighbour, forming a parallelepiped with the twelve metallized edges.

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a) Eight crank resonator

b) Four crank resonator

c) Five segments cranks d) Parallelepiped structure Fig. 1: Structures analysed in this paper. Three of them have chiral symmetry: the eight crank resonator a), the four crank resonator b), and the five segment crank resonator c). Only the parallelepiped d) has non-chiral symmetry. III. MODELING A. Numerical Simulations Numerical simulations were mainly developed using COMSOL Multiphysics and CST Microwave Studio. These simulation suites represent two different point of views for modelling physical Systems and are oriented to different kind of users. COMSOL Multiphysics is a broad scope numerical tool designed to solve almost any system of partial Differential Equations, while CST Microwave Studio is totally focussed to electromagnetic systems. Other numerical packages as MEFiSTo and EMPro have been also used, however their application to our study is more limited. MEFiSTo, from Faustus Scientific Corporation, is the first software we have used during our study. It is a time domain simulator, based on TLM method, which allows the graphic visualization of the electromagnetic field propagation and its interaction with materials and boundaries, during the simulation. Agilent EMPro is the software we have been testing during the last months. It is not too expensive, allowing the possibility of having licenses enough to be used for teaching purposes with courses of around 20 students. EMPro has different integration engines (FDTD, FEM and FEM eigenmode). We normally use FEM technique for waveguide analysis and FEM eigenmode for cavity resonators. Although

we have not extensively used EMPro for our metamaterial systems, we have checked that FDTD can be used with plane wave excitation, although the post processing analysis is not as simple and fruitful as with COMSOL or CST. CST Microwave Studio has six different solvers, but in this work we have mainly made use of the Transient Solver. This method solves the integral formulation of Maxwell Equations using the Finite Integration Technique (FIT) [?] to calculate the time evolution of the electromagnetic field. The method is complemented by the use of the Perfect Boundary Approximation to mesh the system with special care into the details of the structure. Only Hexahedral mesh is allowed by the Transient Solver. We normally use the expert system based automatic mesh generator which, at a first level, can be adequately controlled by just a few settings allowing enough accuracy. To study periodic 2D (x-y) systems, the geometry of a single cell has to be drawn accompanied by periodic boundary conditions in both X and Y directions. Excitation is done by plane waves (linear or circularly polarized) with absorbing boundary conditions in Z. In order to extract information about the reflection and transmission coefficients, point probes can be situated to know any field component at the desired locations. In addition, field monitors can be defined to get information in volume or plane regions with time animation capabilities. Three-dimensional frequency-domain harmonic propagation mode in RF module of COMSOL is used. To simulate periodic boundary conditions, we have applied Floquet-PBC, the incident wave is modelled by using a Port Boundary Condition, which will both launch the incident wave, as well as absorb the reflected wave. The computational domain is truncated by using Perfectly Matched Layer with Cartesian coordinates. Two Cut Point probes are placed far from the CM for sampling the x-and y-components of the fields at each frequency of the working range and further manipulation of their values in order to characterize the sample. Predefined adaptive mesh up to steeps of refinement (2Ч104 elements) is normally used. Frequency-domain STATIONARY-DIRECT-PARDISO parametric solver for 41 frequency values in the range 817 GHz is used for fields computation at each frequency, at a rate of about two minutes per frequency.

B. Wave Propagation in Chiral Media

We have modelled normal incidence of a linearly polarized

TEM plane wave travelling through free space upon one layer

of CM. By splitting the linear incident wave into the two

propagation modes in unbound chiral media, i.e. right (RCP)

and left (LCP) circularly polarized waves, the electromagnetic

coupling is overcome and each propagation mode verifies a

constitutive equation without the coupling parameter [?].

The modes spread through the medium as if it were an

isotropic one with wave number and refractive index that can

be expressed as:

n± = n ± = rµr (1 ± r) = n (1 ± r) , (3)

k±

=

c

r µr

(1

±

r )

=

k0n

(1

±

r )

,

(4)

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Fig. 2: A simulation of the travelling wave at the resonance frequency for a four crank resonator (COMSOL). The incident wave is linearly polarized (y-axis) and the transmitted one has changed the polarization to elliptical and the polarization direction (x-axis)

where tively,

n+=and -rµrreipsretsheenrtetfhraecRtivCePinadnedx

LCP and

wra=ves, /respreµcr-

the relative chirality. It is interesting to highlight that the dif-

ference between the two refractive indices (n+, n-) increases when the value of r increases; in fact if r is large enough, either n+ or n- becomes negative.

The CM can be characterized once the transmission and

reflection coefficients are known [?]. Fig. 2 shows the

travelling wave at the resonance frequency for a four crank

resonator obtained using COMSOL. We can observe that the

wave is rotated 90o after transmission, and the polarization of

the wave change from linear to elliptical.

IV. RESULTS AND DISCUSSION The numerical modelling makes possible the examination of the field and current distribution inside the node. For example, it is possible to observe strong enhancement of the field and the induction of an electric charge in the end of every metallic cranks. The magnetic field rotates around the cranks due to the induced currents. First, we have modelled, using MEFiSTo, the behaviour of our eight cranks resonator (Fig. 1a). Fig. 3 represents the electric field around the two cranks which form the upper-left corner of the resonator, they are excited with polarization along the x-axis and at the first resonance frequency [?]. The electric field means there is an induced electric charge distributed along the x-axis, positive at left and negative at right i.e., corresponding to the first resonance mode. The five segment crank (Fig. 1c) have been simulated by CST and measured in the laboratory. Fig. ?? shows the experiment and simulated results. Clearly, two resonances at 8.2 GHz and 15.3 GHz are found. The first one is also obtained in the experiment while the second one is out-of the range.

Fig. 3: Field distribution around a two cranks sub cell: Electric field on the lower xy plane (top); electric field on the upper xy plane (middle); electric field on the xz plane (y=0) (bottom).

1

0.8

Magnitude

0.6

0.4

|T | |T |

0.2

RL

[deg]

0

75

4

6

8

10

12

14

16

50

25

0

-25

-50

-75

4

4

6

8

10

12

14

16

2

Chirality ()

0

-2 -4 2

real() imag()

4

6

8

10

12

14

16

18

Frequency (GHz)

Fig. 4: Simulation (dashed lines) and experimental (dot marks) results of the transmission coefficients for RCP and LCP waves (top), rotation () and ellipticity () angles (middle) and chirality (bottom) for the five-segments cranks (Fig. 1c).

The numerical results agree quite well with the experimental ones, which is a guarantee of the simulation process. At both resonances the transmitted wave is perpendicular to the incident one and the linearly polarized incident wave is distorted after transmission and becomes elliptical. The chiral symmetry of the particles is fundamental to

JOURNAL OF LATEX CLASS FILES, VOL. 6, NO. 1, JANUARY 2007

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CST 1

COMSOL 1

0.8

0.8

Magnitude

0.6

0.6

0.4

0.2

|T | |T | |R| RL

0.4

0.2

|T | |T | |R| RL

Angle [deg]

0

0

8

10

12

14

16

8

10

12

14

16

15

15

10

10

5

5

0

0

-5

-5

-10

-10

-15

-15

8

10

12

14

16

Frequency [GHz]

8

10

12

14

16

Frequency [GHz]

Fig. 5: Transmission coefficients and rotation angle obtained by COMSOL and CST for the cubic structure.

produce electromagnetic activity. If the chiral symmetry is broken, for example by slightly extending the segment length of the crank to contact with the via of the nearest neighbour, the electromagnetic activity of the sample disappears. Fig. ?? shows the numerical results obtained for the parallelepiped structure (Fig. 1d) using CST (left) and COMSOL (right). We observe a slight difference in the transmission coefficients of the RCP and LCP waves obtained by CST and, therefore, the wave is rotated after transmission, which is not coherent with the symmetry of the parallelepiped. These errors probably arise from the numerical method used in the simulation and can be decreased by a finer mesh or by illuminating the sample with RCP and LCP waves (which requires a different retrieval algorithm). Maximum efficiency of COMSOL algorithm is reached in frequency domain by solving the electromagnetic problem frequency by frequency. As a consequence, this method is slower when compared with CST, the solution takes around two minutes per frequency (two minutes for the whole frequency range with CST Transient Solver). V. CONCLUSIONS CST Microwave Studio is very attractive because it can be applied to the study of many diverse systems and its use is very simple and intuitive. We would emphasize that CST allows a very fast global design (geometry plus physics) of the system under study. This, sometimes, has as drawback that the user pay low attention to details that could be important. In a certain sense we could say that it is more oriented to an engineering use of the tool. In addition, the use of the Transient Solver (time domain) allow a very fast execution obtaining results in the frequency domain in a way that is transparent to the user. Finally, but very important, CST is very a expensive software and has to be renewed every year; if not, you lose the license. By contrast, COMSOL, that also is very attractive because of its versatility, requires a more precise control of the details. As an example, for periodic boundary conditions, you have to pay special attention to fitting the mesh in the same way at the

two boundaries involved. The frequency solver is much slower than the transient solver (in general, the COMSOL frequency solver requires the same time for a single frequency than the one required by CST for the whole frequency range). And finally, again very important, COMSOL is less expensive and if you do not renew the license you can keep using the last version you have bought, forever. Working with COMSOL and getting the best results from it requires a high degree of involvement from the user side but the effort worth as the user eventually gets control of the process. This can be hard in the beginning and technical support, which is excellent, is usually necessary. By contrast, you can get familiar with CST much more easily. COMSOL package has an standard Transient Solver but maximum efficiency is achieved with the Frequency Domain solver. It is based in the Element Finite Method (FEM) and frequency by frequency operation. Thus tetrahedral mesh which fits properly to all kind of geometries is allowed. The disadvantage is execution lasting more but the user is allowed to control the mesh. COMSOL Multiphysics is also a real multiphysics environment able to be used in many different topics of research by coupling their equations. ACKNOWLEDGEMENT This work has been partially supported by the Spanish Ministry of Science and Innovation through the research projects TEC2010-21496-C03-01, TEC2010-21496-C03-02, CONSOLIDER CSD2008-00066 EMET and by the European Commission (ERDF). REFERENCES [1] Lindell, I.V., A.H. Sihvola, S.A. Tretyakov, and A.J. Vitanen, Electromagnetic Waves in Chiral Media, Boston, USA: Artech House, 1994. [2] Barba, I., A.C. Cabeceira, A.J. Garcґia-Collado, G.J. Molina-Cuberos, J. Margineda and J. Represa, "Quasiplanar Chiral materials for microwave frequencies," in Electromagnetic Waves / Book 2, A. Kishk Ed. InTech open access Publisher, 97-116, 2010. [3] Pendry, J.B., "A Chiral Route to Negative Refraction," Science, Vol. 306, 13531355, 2004. [4] Barba, I., A.C.L. Cabeceira, A. Goґmez and J. Represa, "Chiral Media Based on Printed Circuit Board Technology: A Numerical Time-Domain Approach," IEEE Transactions on Magnetics, vol. 45, No. 3, pp. 1170 1173, 2009. [5] Molina-Cuberos, G. J., A.J. Garcia-Collado, I. Barba, and J. Margineda, "Chiral Metamaterials With Negative Refractive Index Composed by an Eight-Cranks Molecule" IEEE Antennas and Wireless Propagation Lett., Vol. 10, 14881490, 2011. [6] Garcґia-Collado, A.J., G.J. Molina-Cuberos , M.J. Nuґn~ez , E. Martґin, and J. Margineda, "Negative Refraction of Chiral Metamaterial Based on Four Crank Resonators", Journal of Electromagnetic Waves and Applications, 26:7, 986-995, 2012 [7] Molina-Cuberos, G.J., I.J. Martґinez-Soler, A.J. Garcґia-Collado, M.J. Nuґn~ez, and J. Margineda "Chiral Media based on Five Segments Cranks", Metamaterials '2012: The Sixth International Congress on Advanced Electromagnetic Materials in Microwaves and Optics, San Petersburg, Russia, 17-22 September, 2012. [8] Clemens, M., T.Weiland, "Discrete Electromagnetism with the finite Integration Technique", Progress in Electromagnetics Research, PIER 32, 65-87, 2001. [9] Bohren, C.F., "Light scatering by an optically active sphere," Chemical Phys. Let., Vol. 29, 458462, 1974. [10] Margineda, J., G.J. Molina-Cuberos, M.J. Nuґn~ez, A.J. Garcґia-Collado and E. Martґin Electromagnetic Characterization of Chiral Media, in Solutions and Applications of Scattering, Propagation, Radiation and Emission of Electromagnetic Waves, edited by Ahmed Kishk, InTech, November 14, 2012.

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