We designed and analysed a new structure for the realization of left-handed metamaterials in the GHz region. The material is composed of pairs of metallic crosses and reveals improved polarization behavior. Left-handed properties can be observed as long as the electrical field vector is located in the plane of the crosses. Negative refraction as indication for simultaneous negative effective ε and μ is numerically verified by direct comparison of the wavefronts inside and outside the metamaterial at nonzero angles of incidence.
©2006 Optical Society of America
Veselago reported in his work from 1968 on the properties of media with negative parameters . He showed that in a material with simultaneous negative permittivity and permeability the field vectors and the wave vector form a left-handed system. Thus, the name of left-handed material was introduced to these media. Veselago also mentioned that such media can be described by a negative index of refraction, leading to deflection of incident light to the “wrong” side with respect to the normal of incidence. It took about thirty years to prove Veselago’s predictions. During the 90s Pendry et al. showed theoretically that materials built up by periodic metallic structures much smaller than the wavelength can act as composites with effective electromagnetic parameters depending only on the size and shape of the metallic metastructures. A lattice built up by thin wires behaves like a composite with low-frequency plasmons [2, 3] leading to negative permittivity and the use of so called split-ring-resonators (SRR) leads to a negative effective permeability . Only a short time after these publications the first left-handed material was fabricated  by combining these two structures. In the following years a lot of research was carried out on how to design and analyse these metamaterials [6–10]. For optical frequencies a structure consisting of pairs of metallic wires was recently introduced . However, the electromagnetic properties of this structure, also known as double wire structure, were analysed before in  and . During the last year other structures aiming at a negative index of refraction in the optical frequency regime were introduced [14, 15].
Many interesting applications arise concerning artificial materials with controllable parameters ε and μ as well as negative refraction. It is reported that a slab of left-handed material can act as a plane parallel lens . It is even predicted that lenses built from left-handed media could act as perfect lenses overcoming the diffraction limit due to the amplification of evanescent waves in materials with a negative index of refraction . This amplification was directly measured within a metallic waveguide driven beyond its cut-off frequency . Another remarkable result is the possibility to design artificial materials which can be “invisible” for electromagnetic radiation at certain frequencies, as shown experimentally in Ref. . The control of material parameters plays a key role in a possible realization of recently proposed cloaking devices which are able to render objects invisible [20, 21].
Nearly all of the aforementioned left-handed metamaterials, including the double wire structure, can only operate properly at a certain polarization of the electromagnetic wave. In order to resolve this problem we designed a metamaterial built up of pairs of metallic crosses, which are similar to the rosette design for artificial gyrotropic devices and left-handed materials . So, our structure can be looked upon as an extension of the double wire medium. By adding a second orthogonal wire pair, thus forming crosses, the polarization sensitive behavior is no longer present with the consequence, that the cross structure exhibits a left-handed transmission band as long as the electrical field vector is located in the plane of the metallic crosses.
2. Structure design
To design a left-handed metamaterial we need both a resonance due to coupling of the electric field as well as a resonance due to magnetic coupling. In our proposed structure in Fig. 1, the bars of the cross form an array of small electrical dipoles, thus leading to a negative effective permittivity as a result of stimulated oscillations of the conducting electrons. Together with the bar of the second cross within the elementary cell some kind of LC-circuit is formed where the inductance is built by the two opposing crossbars. Thus a magnetic field component perpendicular to the area between the crosses induces a current flowing in the metallic structure. The current loop is closed by the dielectric displacement current at the end sides of the crossbars [11, 21] and due to the bars of our structure acting as electrical capacitors the flowing current has a resonant character. This interaction between the structure and the magnetic field leads to a negative effective permeability. Our structure thus satisfies both conditions necessary to reach a negative index of refraction and these conditions are satisfied in whatever direction the electrical field vector points, as long as this vector is located in the x-y plane. The dimensions and an extract of three elementary cells of our structure are depicted in Fig. 1. Due to the geometric properties of our structure with an effective length of the dipoles of 5.5 mm we expect the resonance frequency to be at about 27 GHz. Another important fact is the very simple geometry of the crosses which should allow construction of the metamaterial using standard multilayer circuit-board lithography techniques as reported in Ref. . In our structure which is effectively only two dimensional no interconnections between the different layers are needed for the fabrication.
3. Simulation results
For our numerical simulations we use the commercial solver for Maxwell’s equations microwave studio from CST. The metallic crosses are modelled as perfect electrical conductors and the background material has a permittivity of one. The properties of the background material do not affect the basic functionality of the metastructure. The only effects of a background with constant complex e are a shift of the resonance frequency to smaller values  and increased losses . We apply periodic boundary conditions in the directions perpendicular to the direction of propagation thus ensuring a correct periodic continuation of the structure. In our example we simulate a slab of seven elementary cells in z-direction.
First we simulate the S-parameters of a structure with only one cross per elementary cell. Doing so, we eliminate the magnetic resonance and expect a rather broad stop band at a frequency of about 27 GHz (see Fig. 2) due to the negative effective permittivity of the material. Adding the second cross to allow also a magnetic coupling which leads to negative permeability, we expect at proper design the formation of a transmission band within the former stop band. Within this band ε and μ take on negative values and thus transmission is no longer forbidden as can be seen from the simulated transmittance curves also given in Fig. 2 for incidence in the z-direction. The left-handed transmission band with a bandwidth of about 1.2 GHz is surrounded by two stop bands with negative permittivity only. This behavior is due to the fact that the magnetic resonance is very sharp thus leading to a negative permeability only in a small frequency band. These S-parameter calculations have been performed for different orientations of the incident electrical field (in the x-y plane). As all results were identical, the basic properties of our structure turn out to be independent from the orientation of the electrical field vector within the x-y plane.
So far, wave propagation was only considered for “normal incidence” with respect to the x-y plane. For most of the potential applications of metamaterials, like plane parallel lenses, waves propagate at nonzero angles of incidence with respect to the normal of the boundary of the structure. So we recalculated the transmission data of our structure for different angles of incidence (plane of incidence: x-z plane) of the radiation to explore the functionality under the restriction that only a part of the magnetic field vector is oriented perpendicularly to the area between the crosses. In Fig. 3 the results of the calculations show that the spectral position and shape of the observed transmission band vary only slightly with the angle of incidence although the magnetic coupling decreases with larger angles of incidence.
The observation of this transmission band alone is only an indicator for a possible left-handed behavior of the structure but cannot be seen as a proof. We looked for a direct and straightforward method to show that the transmission indicated by the calculations above is indeed left-handed. So we calculated the field distribution for plane wave excitation under the angles of 15 and 30 degrees at a frequency of 27.11 GHz which is approximately in the center of the transmission band. From the inclination of the wave fronts inside the artificial structure one can directly see that the direction of refraction of the incident wave at the boundary to the metamaterial indicates an effective negative index of refraction. To further consolidate our results, we calculated the effective index of refraction using the method presented in  which yields an index of refraction of -1.31. For a wave incident under an angle of 15° this leads to an associated refraction angle of -11.2° and for one incident under 30° to an angle of -22.4°. In Figs. 4 and 5 the field distributions are depicted at a plane perpendicular to the y-axis at a position of y = 0.48mm. For reference the inclination of the phase fronts occurring in a homogenous medium with the calculated refractive index of -1.31 are included in Figs. 4 and 5 as dashed lines. The field inside the metallic structure matches these inclinations very well and there is no doubt that the transmission in the observed band is left-handed. However, it must be mentioned that the wave fronts for larger angles of incidence are not as homogenous as for small values. This may be a consequence of the decreased magnetic coupling for higher angles of incidence. However, the phenomenon of negative refraction can clearly be observed in both cases.
Up to now the inevitable losses in fabricated structures have not been considered. To show, that the observed properties of our metamaterial still hold for real metals, we recalculated the transmission coefficient, now using gold as material for the crosses. The simulated transmission curve is given in Fig. 6, where the curve for a perfect conductor is again included for reference. It can clearly be seen that, besides the damping of about 2 dB within the designated left-handed transmission band, no significant change is observed in comparison to the lossless case. This is a clear sign that the losses in our structure do not affect the left-handed transmission properties of our metamaterial.
In conclusion we presented a new structure to build up a metamaterial with a negative index of refraction in the microwave region. This new structure operates under improved polarization conditions and should easily be fabricated due to the simple form which is effectively only two dimensional. Negative refraction was numerically verified by calculation of the wavefronts both outside and inside the metamaterial. Even though tested in the microwave region we suppose that the structure should also be applicable in the terahertz region or even up to infrared frequencies because the basic principle of operation is similar to those of the double wire structure which was the first structure exhibiting a negative index of refraction at an optical wavelength. Of course at such high frequencies losses and the dispersive behavior of the used metals must be further considered for an optimized metamaterial design.
References and links
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