1. Introduction

In the past few years, there has been a growing interest in artificially structured materials, termed metamaterials, showing properties not found in nature. One of the most attractive features of these composites is the possibility of obtaining negative refraction over a certain frequency band. A possible way to achieve a negative refractive index is to have a negative real part of the permittivity (ε=ε’+ iε”) and of the permeability (µ=µ’+iµ”) at the same frequency. However, this is not a necessary condition. Strictly, the relation that must be satisfied is a more relaxed one [1]: ε”µ’+µ”ε’<0. It is generally desirable to have simultaneous negative real parts of ε and µ so that losses become small. Since a metamaterial with simultaneous ε’ and µ’ using split ring resonators (SRR) [2] and wires was demonstrated in the microwave regime [3], much attention has been put in extending negative index of refraction metamaterials to optical wavelengths [4]. The first idea was to scale down the artificial “atoms” to increase the resonance frequency at which the negative refractive index occurred. Unfortunately, there exist limits that prevent from increasing the frequency to an arbitrary high value as the metal stops to behave as an ideal one [5]. Another approach is to use metallo-dielectric stacks, a design with which negative refraction and superlensing have been numerically demonstrated in the visible [6]. Up to now, the best experimental designs are based on the so-called fishnet structure composed of two perforated metal layers separated by a thin dielectric, rising wavelengths as small as 772 nm [7,8]. Here we propose a metamaterial composed of a single metal layer, which exhibits negative index of refraction for both polarizations and with low losses even in the visible range of the spectrum.

2. Negative index metamaterial

Metamaterials presenting resonant behavior in response to an incident magnetic field are often used to achieve effective negative µ. This is the case of a single SRR, which can be seen as a capacitance in series with an inductance, displaying negative permeability (unless stated explicitly, we refer to the real parts when we talk about permittivity, permeability or index of refraction) at the resonance frequency. Many other designs, such as parallel metal plates or stripes, have the same underlying physical mechanism [9,10]. This is the main reason why a periodic lattice of closed rings does not exhibit negative permeability since there is no capacitance to give the resonant behavior. Nevertheless, we can get negative µ’ with closed square rings under certain conditions as in the structures depicted in Fig. 1. We start from a lattice of metal stripes (Fig. 1(a)) with thickness t and width w. The stripes are interrupted periodically by gaps of length s. The incidence is normal to the structure (see Fig. 1(e)) with the E field along the stripes and the H field perpendicular to them. As far as the electric response is concerned, the metamaterial acts as a dilute plasmonic medium, i. e., a metal with a lower plasma frequency than that in bulk. In addition, the cuts in the stripes give rise to a resonance in the permittivity. According to Faraday’s law, the incident magnetic field induces an emf, which accounts for a current flow (thanks to a large thickness t of the stripes) in an open loop normal to the H field with opposite directions at each side of the gap. Since the current is interrupted by the gaps, we can consider the structure as an equivalent circuit which consists of an inductance in series with two capacitors, resulting in a resonant permeability that becomes negative in a certain band. This anti-symmetric mode generates a magnetic field that opposes the incident one above the resonance frequency, where the current phase is reversed, as Fig. 2 shows. Moreover, the permittivity is still negative in the magnetic resonance region giving rise to a negative index of refraction. Obviously, the metamaterial is sensible to polarization, in fact, it is almost transparent if we swap E and H. We can make it polarization independent by adding cut stripes normal to the original ones as in Fig. 1(d). The result is a symmetric medium made up of crosses very close to each other. We can go a step further and replicate the stripes (Fig. 1(b)). By doing so, we reinforce the magnetic resonance and shift it slightly. Note that the gap between the upper and lower stripes is of the same length as the one between horizontal stripes. Finally, we add double stripes parallel to H and obtain square rings exhibiting negative refraction in both polarizations.

 

Fig. 1. (a) Basic metamaterial lattice. All other designs are derived from this one. (b) Double stripe metamaterial. (c) Square rings metamaterial (double stripes crossing in normal directions). (d) Cross-hair metamaterial (adding perpendicular stripes to structure in (a). (e) Unit cell of (c) and incident wave (same orientation for all structures).

Download Full Size | PDF

 

Fig. 2. Calculated currents and fields near the gap for design 1(a). Dimensions are the same as in Fig. 3. (a) Currents below magnetic resonance (450 THz). (b) Currents above magnetic resonance (510 THz) for the same phase of the incident wave. (c) Electric field (510 THz). (d) Magnetic field (510 THz). In all cases, the incident wave has the direction drawn in (c)

Download Full Size | PDF

3. Numerical analysis

Numerical calculations are performed using a commercial 3D electromagnetic solver (CST Microwave Studio). We simulate a unit cell with periodic conditions along the dimensions normal to propagation and obtain the transmission (S21) and reflection (S11) parameters. Due to its low losses in the optical region, silver is chosen for the simulations. In addition, measured silver epsilon values [11] agree well with the Drude model for this metal in the range of interest, so the latter can be employed in the calculations. The plasma frequency for silver is ω_p=1.37×10¹⁶ s^-1 and the collision frequency is chosen to match data from [11]. In order to check the validity of the Drude model we also use a best-fit first order Debbie approximation matching with the above-mentioned experimental values, obtaining very similar results. Current and field distributions are calculated below and above the magnetic resonance frequency and are shown in Fig. 2. In addition to the current loop generated by the magnetic field, there is another current directed along the stripes due to the electric field. As expected, the fields are concentrated in the gaps and we can see how the displacement current closes the loop.

To extract the effective n and z from the calculated S11 and S21, the traditional retrieval method [12,13] is used. Then, ε and µ are obtained as n=(εµ) ^1/2 and z=(µ/ε) ^1/2. As an example, we simulate all structures with t=150 nm, l=106 nm, w=54 nm and s=30 nm (in designs 1(a) and 1(b) the length of the stripes is 2l+2w and their periodicity in the direction parallel to H is chosen to be the same as in their symmetric counterparts). The results are depicted in Fig. 3. The use of an effective-medium model is justified since the structures dimension in the propagation direction is, depending on the design, from four to six times smaller than the wavelength. Moreover, the amplitude of high-order modes is negligible compared to the plane-wave one.

 

Fig. 3. S parameters and real parts of the permittivity and permeability of structures in Fig. 1.

Download Full Size | PDF

Both the S parameters and extracted ε and µ are very similar for all designs verifying that the stripes parallel to the electric field are the ones responsible for the negative index behavior. There is a dip in S21 around 0.95 µm (no transmission) due to the resonance in ε, which is a consequence of the stripes not being continuous. The real part of the permeability has a strong resonance around 0.64 µm in all structures. At that frequency, the permittivity shows a characteristic antiresonant behavior [14] and is negative, except for the design 1(d) (although n’ is negative). However, it is possible to adjust the geometrical parameters of this structure to make ε’ and µ’ negative in the same region as shown below. For the double stripe metamaterials (Figs. 1(b) and 1(c)), the permittivity becomes more negative, or equivalently, the effective plasma frequency gets higher. This can be ascribed to the higher metal filling factor in the direction normal to the electric field since the larger the filling factor is, the more the material resembles bulk metal. At this point, it is interesting to know how the negative index band shifts in frequency with dimensions scaling, being the most important variables the magnetic resonance frequency (f_res) and the effective plasma frequency (f_p). Since all structures have a very similar response, we will focus on the simplest one (structure in Fig. 1(a)). It is appropriate to note that fabrication of the designs presented above is not straightforward due to the high ratio between thickness t and width w, which is of the order of 3:1. To overcome this difficulty we can increase the width w of the stripes and make it comparable to the thickness. Thus, we will scale the structure 1(a) with t=w and depict the evolution of f_res and f_p with a geometrical scaling factor (S). As shown, f_res does not vary linearly with S as would occur with ideal metals or any metal at low frequencies. On the contrary, there exists saturation because the magnetic energy no longer dominates the kinetic one and both become comparable [5]. The effective plasma frequency varies almost in a linear fashion with S within this frequency range.

 

Fig. 4. (a) Dependence of fres and fp on scaling. The base structure (S=1) has dimensions t=w=220 nm, s=60 nm and a total stripe length equal to 440 nm. (b) Dependence of f_res and f_p on s. Dimensions are t=w=110 nm and 220 nm stripe length.

Download Full Size | PDF

There is a region (S ^-1<0.8) where f_res is larger than f_p and therefore the refractive index is not negative. In Fig. 4(b) we can see that f_res increases when the spacing s grows whilst f_p remains almost unchanged. Hence, we have a way to invert the previous situation and make f_res<f_p by decreasing s. The thickness t and width w also have influence in these two parameters and could be adjusted in order to tune the negative index band. As we scale down the structure s becomes too small, complicating its fabrication. To avoid this, we can make s larger and compensate the resonance shift by scaling up the metamaterial. For instance, if we take the configuration of Fig. 4(b) with s=30 nm, increase it to 50 nm and then apply S ≈ 1.1, f_res remains in the same location. Regarding losses, the factor of merit (FOM) defined as FOM=|n’/n”| is usually taken as a measure of how good the metamaterial behaves. We show in Fig. 5 the extracted n’, ε’, µ’ and FOM for structure 1(a) with t=w=110 nm, s=60 nm and a stripe length equal to 220 nm. In this case, the FOM is larger than 6 at the wavelength where n ’=-1 (464 nm). To our knowledge, this is the first metamaterial exhibiting negative refractive index at such high frequencies with only one metal layer.

 

Fig. 5. Retrieved parameters for structure 1(a) with t=w=110 nm, s=60 nm and length 220 nm. Left: Factor of merit and real part of the refractive index. Right: Real parts of permittivity and permeability.

Download Full Size | PDF

Finally, we suggest a 3D extension of the cross-shape metamaterial (see Fig. 6), which is expected to be isotropic. Its complexity is higher because of the fact that the magnetic field will also induce a current loop near the gap between the arms placed along the propagation direction, something that will influence the total magnetic response. In principle, the electric response will remain almost unchanged since the electric field is normal to the added arms. The involved fabrication difficulty is the major drawback of this structure.

 

Fig. 6. Proposed 3D metamaterial.

Download Full Size | PDF

4. Conclusion

In summary, we have presented a single-layer metamaterial based on cut stripes, which displays a negative index of refraction in a band that can be easily tuned over a wide range covering two regions of relevance, namely, the telecommunication (1550 nm) and visible regions of the spectrum, attaining frequencies as high as blue light or even greater. The metamaterial presents several advantages such as polarization independence and high factor of merit. Moreover, the simplicity of the cross-shape design with height equal or smaller than lateral dimensions makes it compatible with current techniques such as conventional lift-off process.