1. Introduction

Most naturally existing transparent materials possess small positive refractive index, ranging approximately from 1 to 4. In the last decade, exploring materials with refractive index of high value or even negative value has attracted research interest and bonus lots of applications. The interest in exploiting artificial materials with unusual effective electromagnetic responses mainly focuses on designing subwavelength resonances in metallic structures to control refractive-index [1–4]. However, most of the previous researches on metamaterials in the terahertz region concentrated on realizing the negative refractive index materials [5]. On the contrary, the opposite side-high refractive index materials attracts far less attentions. As proven by Pendry et. al, improving the refractive index will lead to a higher resolution application [12–14]. Recently, high refractive index metamaterials were numerically and experimentally investigated [6–11] and demonstrated very promising applications. However, previous approaches were not successful in realizing high refractive index with polarization-independent and broadband properties. The previously proposed metamaterials exhibit polarization dependency because of the structural anisotropy of the unit cell.

According to the Maxwell's equations, the refractive index of materials is determined by effective permeability and effective permittivity (). So in order to achieve high effective refractive index, it is required to increase either the effective permittivity or the effective permeability, or both of them. To increase the effective permittivity using artificial atoms (or molecules), we need to increase the dipole moment of the artificial atom. I-shaped metallic patches proposed in previous work meet all the requirements and provide the high effective permittivity [9]. By periodically arranging I-shaped metallic patches with narrow gaps in-between, the capacitance of the constituting sub-wavelength scale capacitors (I-shaped metallic patches) can be increased. As the gap gets narrower, the capacitance diverges rapidly, leading to the accumulation of large numbers of charges at the end of the I-shaped metallic patches. This huge accumulation of charges, in turn, results in extreme polarization density, and therefore the huge effective permittivity. Furthermore, in order to minimize the diamagnetic effect that gives rise to the decrease in effective permeability, the thickness of the metallic structure needs to be reduced to decrease the metallic volume fraction. In this letter, following the same design idea and aiming at improving its symmetry, we demonstrate that a double-layer metamaterial consisting of I-shaped metallic patches to realize polarization-independent high refractive index. Because of the particularity of the structure, a dual-band refractive index behavior is also investigated. Finally, the high refractive index behavior is extended to extremely broadband by a four-layer metamaterial design.

2. Metamaterial design and analysis

The unit cell of the proposed high refractive index metamaterial is shown in Fig. 1(a), with the definition of the TE (the electrical fields pointing in the y direction). Two layers of thin “I” shaped metallic patches are embedded in a dielectric substrate and orthogonal to each other. The dielectric substrate is selected as polyimide $(n_{poly}=1.8+0.04)$, and the metal used to construct the metamaterial is gold. As a demonstration of the properties of this high refractive index metamaterial, the electromagnetic response of the structure is simulated using 3D finite element modeling COMSOL Multiphysics software. The thin I-shaped metallic patches are considered as transition boundary condition, and illuminated with a normal incident plane wave. The complex dielectric constant of gold for the studied frequency range is described using Drude model with a plasma frequency of ${\omega }_p=1.37\times {10}^{16}{s}^{-1}$ and collision frequency of $\text{γ}=4.07\times {10}^{13}{s}^{-1}$.

 

Fig. 1 Schematic view of unit cell structure of the (a) double-layer and (b) four-layer high-index metamaterial.

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The geometric parameters used in the simulation are set as L = 40 μm, a = 39 μm, w = 3 μm, d = 1μm. The thickness of both gold patches is 100 nm. The gap width between the metallic patches is defined by g = L-a. It’s worth noting that both “I” shaped metallic structures share the same geometry and parameters.

The simulated transmission/reflection spectra for TE polarization is plot in Fig. 2(a). Two resonances in reflection and corresponding minima in transmission in between 0.1 and 4 THz are observed in the spectra. The reflection peaks are at frequencies of 2.26 THz and 0.65 THz. In order to explain the appearance of these two resonances, an effective parameter retrieving method was used to extract relevant material parameters and plot in Fig. 2(b) [16]. There are two peaks in the real part of reflective index, i.e., n = 33.1 at 0.65 THz and n = 55.1 at 2.26 THz. The refractive index is much higher compared to previous reported results and proves that our structure can provide a dualband high refractive index. To study the physical origin of the high refractive index, the electric and magnetic field distributions around the metallic patch in the unit cell at 0.65 THz and 2.26 THz are plot in Fig. 2(c) and Fig. 2(d). We can find that, the electric field was strongly concentrated in the gap along the long side between each ‘I’-shaped metallic structure on the top layer at 0.65 THz. This strong electric field arises extreme polarization density to provide huge effective permittivity. At the same time, another strong electric field distribution along the short side is observed at the bottom layer at 2.26 THz. As a result, the double layer metamaterial structure exhibits high refractive index at these two different frequencies. Obviously, because of the negligible metallic volume fraction, the magnetic field at both frequencies penetrates deeply into the unit cell, as shown in Fig. 2(e) and Fig. 2(f). Therefore, it can be directly inferred that the top layer played a leading role at lower frequency of 0.61THz while the bottom layer dominated at higher frequency of 2.1THz. The transmission/refraction spectra and the retrieved refractive index (not shown in Fig. 2) for TM polarization almost overlap with those for TE polarization, proving that the proposed structure is highly isotropic.

 

Fig. 2 (a) Simulated transmission (T) and reflection (R) and absorption (A) spectra of the structure. (b) Numerically extracted values of complex refractive index (n) obtained from the S-parameter retrieval method. (c) Saturated electric of upper layer at 0.65THz. (d) electric field of under layer at 2.25THz. (e) magnetic field of upper layer at 0.65THz. (f) magnetic field of under layer at 2.25THz for a double-layer metamaterial, respectively.

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3. Metamaterial geometry optimization methodologies

Previous work has proved that a dual-band high refractive index can be realized via a double-layer metamaterial. However, both refractive index peaks are pretty narrow. In addition, the loss for the double-layer metamaterial is rather high with the largest FOM of less than 100. To solve these two problems, a three dimensional metamaterial composed of four layers “I” shaped metallic patches is proposed. The unit cell of the simulated structure is presented in Fig. 1(b). Figure 3(a) represents the simulated transmission/reflection spectra, while the retrieved complex value of refractive index is plotted in Fig. 3(b). The obtained refractive index differs greatly from the results of double-layers metamaterial, due to the coupling between each layer [17,18]. The refractive index does not change greatly at low frequency, while at high frequency the high refractive index exhibits a very broadband behave with full-width at half-maximum (FWHM) of more than 2.24 THz. Moreover, double refractive index peaks of 24.9 at 0.5 THz and 67.9 at 2.1 THz are realized. The numbers are much higher than the previous result. Also, for most of the frequencies, as showed in Fig. 3(c), the figure of merit (FOM = Re (n)/Im (n)) maintains a rather high value, with a peak value exceeding 800 at 1.46 THz. This indicates that this four-layer metamaterial has low loss.

 

Fig. 3 Simulated transmission (T) and reflection (R) spectra of the structure for (a) TE polarization and (d) TM polarization, respectively. Numerically extracted values of complex refractive index (n) obtained from the S-parameter retrieval method for (b) TE polarization and (e) TM polarization. Numerically obtained values of figure of merit (FOM) for (c) TE polarization and (f) TM polarization.

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As the proposed high-index metamaterial is a symmetrical structure, the refractive index of the metamaterial should be weakly polarization dependent. As expected, when the normal incident wave changes from TE polarization to TM polarization, the refractive index does not change a lot. Figure 3(d) and Fig. 3(f) represent the simulated transmission (T) /reflection (R) spectra and numerically extracted values of complex refractive index (n) of the structure for TM polarization. Similarly, there are two deeps in the transmission spectrum around 0.5 THz and 2 THz due to electrical resonances. Accordingly, two refractive index peaks appear at 0.48 THz with n = 39.6 and at 2.16 THz with n = 66.9, respectively. The refractive index still keeps the feature of broadband and low loss with the largest FOM value over 200. There are still some differences between refractive index at both polarizations, which is mainly due to different longitudinal locations of the metallic patches.

The value of the gap width (g) and the central beam width (w) are two key geometrical parameters for increasing the refractive index. In order to verify the dependence of refractive index on these two parameters and guide further experimental study, the effective refractive index is numerically retrieved for designs having different g and w values. The dependency of refractive index on w is plotted in Fig. 4(a). For this simulation, the parameters of the structure are set as L = 40 μm, d = 2 μm, a = 39 μm. We can find that the effective refractive index increases gradually as the central beam width decreases. This is because a larger area of current loop will lead to a greater diamagnetic response. It is also worth noting that, the central beam width control can also be used to tune the electric resonance frequency of the metamaterial, which is caused by the change in the inductance of the metallic patch. Figure 4(b) shows the frequency dependence of refractive index as a function of g. As the gap width decreases, there is a great increase of the capacitive due to coupling between unit cells. As a result, refractive index is drastically increased. So it can be easily predicted that we can get an even higher refractive index through reducing the width of the gap.

 

Fig. 4 Geometrical parameter-dependent effective refractive index. (a) Effective refractive index plotted as a function of central beam width w. (b) Effective refractive index plotted as a function of gap width g.

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4. Conclusion

In summary, we designed two polarization-independent metamaterials with extremely high refractive index in the terahertz region. The obtained dependence of the effective refractive index on the geometric parameters provides us with a general recipe for designing such metamaterials. Moreover, we can predict that via rational design, high-performance high-index metamaterials can also be realized in even higher frequency ranges, such as mid-infrared frequencies. Creating a high refractive index metamaterial with polarization-independence will benefit its applications in lithography and imaging, where the numerical apertures are proportional to the refractive index.