1. Introduction

Resonant optical responses of metamaterials have been intensively studied for applications ranging from optical switching to biosensing [1, 2]. Most of metamaterials comprise arrays of sub-wavelength metallic resonators. They suffer from high inherent resistive losses in the metals and the quality factor of resonances is limited to the order of ten. Thus lots of efforts have been made for the search for non-metallic materials for plasmonic and metamaterial applications [3]. High-refractive-index dielectrics offer an alternative to metals [4]. Dielectric particles display Mie resonances with very low absorption losses when incident photon energy is below the bandgap [5–7]. Moreover, the radiative loss can be suppressed by exploring the coupling effects [8–11]. Fano resonances have been demonstrated in individual semiconductor nanostructures [12], dielectric oligomers [13–15] and all-dielectric metamaterials comprising periodical arrays of dielectric elements [16]. Compared to their metallic counterparts, dielectric metamaterials can support resonances with unprecedented quality factors up to several hundreds or even higher [17]. As a result, there have been increasing interests in studying the optical properties and potential applications of all-dielectric metamaterials [18, 19] and exciting progresses have recently been made in the experimental realization of high quality factor dielectric photonic metamaterials [20–22].

While most of the electromagnetic fields are generally concentrated in the surrounding medium in metallic metamaterials, the fields in dielectric metamaterials are normally confined within the resonators due to the Mie resonance properties of dielectric particles. This limits the use of dielectric metamaterials for applications in sensing, nonlinear optics and luminescence enhancement of quantum dots. Recently, Yang et al has demonstrated that field enhancement and light-matter interactions can be significantly improved in resonant dielectric metamaterials designed with feed-gaps in the unit cells [17]. In this paper, we proposes a generalized and effective method to design nano gaps in dielectric resonant nanostructures for enhancement of ligh-matter interactions and systematically study a family of dielectric metamaterials using split resonators as building blocks. Silicon is chosen as the building material due to its relatively high refractive index and negligible absorption losses in the studied spectral range. The introduction of a nano sized feed-gap in the resonators leads to more than two order enhancement of local electric fields in the gap. Moreover, a large portion of the electromagnetic energy is confined within the gap, which can significantly enhance the interaction between light and the surrounding medium.

2. Dipole-resonant metamaterials

Figure 1(a) is the schematic of a dielectric metamaterial made of periodical arrays of bar resonators. Dielectric particles with different geometries have previously been employed as building blocks of dielectric metamaterials, such as nanospheres and cubes [6, 23]. Here dielectric rectangular cuboids are utilized, which can serve as dipole antennas and are easy to fabricate using top-down methods. Figures 1(b) and 1(c) are two dielectric metamaterials composed of split bar resonators.

 

Fig. 1 (a) Schematic of a dielectric metamaterial comprising periodical array of continuous bar resonators. (b) A dielectric metamaterial composed of split bar resonantors with a gap at the middle. (c) A dielectric metamaterial comprising pairs of parallel, geometrically asymmetric split bar resonators.

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We first consider a type of dielectric metamaterials made of uniform dipole antennas as shown in Figs. 1(a) and 1(b). Figures 2(a) and 2(b) are the top view of unit cells of the proposed metamaterials. One is made of continuous silicon bars and the other of split silicon bars. Figure 2(c) shows the simulated transmission spectra under the illumination of a normally incident plane wave with its electric field polarized along the bars. Numerical simulations are conducted using a fully three-dimensional finite element technique (in Comsol Multi-Physics). For simplification, silicon is assumed to be lossless with a refractive index of n=3.5. For continuous silicon bars, the metamaterial displays a resonance at around 1183 nm. The metamaterial with split silicon bars shows a very similar spectrum only with a slightly shorter resonance wavelength of about 1078 nm. The blue shift of the resonance wavelength is due to the reduction of the effective length of the silicon bars.

 

Fig. 2 Dipole resonances in dielectric metamaterials. (a) Top view and geometric parameters of a unit cell of the dielectric metamaterial consisted of continuous silicon bars; The silicon bars are 750 nm long and 200 nm wide. (b) A unit cell of the dielectric metamaterial consisted of split silicon bars with a 50 nm wide groove at the centre. (c) Numerically simulated transmission spectra of the two metamaterials. The metamaterials are illuminated with a x-polarized plane wave at normal incidence. The thickness of the silicon bars are both 150 nm.

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Figure 3(a) shows the z-component of electric field (Ez) distributions at the resonances, which indicate the distributions of polarization charges. Both the field distributions display typical dipole mode characteristics with opposite polarization charges accumulated at the two ends of the resonator. Moreover, the magnitudes of fields are nearly the same. This means that the metamaterial maintains its main resonant optical properties after the introduction of the feed-gap. However, there is a significant difference for the field distributions at the centre for the antennas with and without the gap. Opposite polarization charges are confined at the two sides of the gap. Similar to the gap effects in plasmonic structures, large enhancement of local field is realized at the gap [24]. This enhancement is due to the continuity of the x-component of electric displacement (Dx) across the interface between air and silicon, in a similar way as the strong field enhancement in slot waveguides [25, 26]. As shown in Fig. 3(b), the maximum field enhancement reaches about 11 for the metamaterial with split antennas which is about twice for the metamaterial with continuous antennas. At the same time, a large portion of electromagnetic energy density is now localized at the gap (see Fig. 3(c)). At the dipole resonance mode, the electric field and time averaged electromagnetic energy density (defined as $W=\frac{1}{4}\epsilon {\epsilon }_0{E}^2+\frac{1}{4}\mu {\mu }_0{H}^2$) reach their maximum at the centre of the continuous bar (see the top of Figs. 3(b) and 3(c)). So here it is important that the gap is located at the centre of the bar to realize maximum field enhancement.

 

Fig. 3 Local field enhancement and light localization in dielectric metamaterials at dipole resonances. (a) Electric field in z-direction. The field is normalized to the field amplitude (E₀) of the incident light and plotted at the x–y plane that is 5 nm above the top surface of the dielectric bars. (b) Normalized electric field and (c) Normalized time averaged energy density at the x–y plane bisecting the nanobars. The metamaterial is illuminated with a x-polarized plane wave at normal incidence. Top: metamaterial with continuous silicon bars. Bottom: metamaterial with split silicon bars.

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3. Fano-resonant metamaterials

The electric dipole resonance in dielectric metamaterials has relatively high radiation losses and the quality factor of the resonance is not very high. In plasmonic nanostructures, more than two order enhancement of field magnitude can be realized due to the strong polarizability even though the quality factors of plasmonic resonances are limited to the order of ten. The field enhancement in the gap of dielectric metamaterials can be increased by increasing the refractive index contrast between the dielectric and air. But the availability of low loss media with high refractive indices are limited (generally semiconductors such as silicon, germanium, and tellurium are among the best candidates). In order to realize stronger enhancement of electromagnetic field in dielectric metamaterials, one can exploit the high quality factor Fano-resonances.

Figure 4(a) is a type of all dielectric metamaterials that has been demonstrated to support a high quality factor Fano-resonance through a coupling between closely spaced, dissimilar dielectric nano-bars within the metamaterial unit cell [10, 16]. The 900 nm square unit cell comprises two silicon rods with same 200nm × 150nm cross-section in the y–z plane and different lengths (750 and 700 nm) in the x direction (Substrates are excluded from the numerical analyses presented here). Figure 4(b) is a similar structure with a gap (with a width of g) made in the centre of each silicon bar. Fig. 4(c) shows the transmission spectra of three metamaterials with gaps of g=100 nm, 50 nm and 0 (no gap), respectively. They show similar sharp Fano-resonances at around 1196 nm (Q ≈ 800), 1272 nm (Q ≈ 880) and 1451 nm (Q ≈ 970), respectively. Again, the blue shift of the resonance for the metamaterials comprising split antennas is because of the decrease of effective length and the slight reduction of quality factors is due to the increase of asymmetry between the short bar and the long bar [8,16]. In theory, the quality factor of the Fano-resonances in the proposed dielectric metamaterials can be further increased by reducing the asymmetry through structural design. However, practical values will be limited by material and manufacturing imperfections [16, 17].

 

Fig. 4 Fano-resonances in dielectric metamaterials. (a) Top view and geometric parameters of a dielectric metamaterial comprising pairs of parallel, geometrically dissimilar silicon nano-bars. (b) A similar dielectric metamaterial consisted of split silicon bars. (c) Numerically simulated transmission spectra of three metamaterials with different gaps. The width of the gaps are g=0 (no gap), 50 and 100 nm, respectively. The metamaterials are illuminated with a x-polarized plane wave at normal incidence. The thickness of the silicon bars are 150 nm.

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Figure 5(a) describes the z-component of electric fields at the resonances. The Fano-resonance arises from anti-phased oscillation of displacement currents in the two arms of the unit cell in dielectric structures. Figure 5(b) shows the normalized electric field. The maximum enhancement of the field amplitude is about 60 for the metamaterial comprising continuous silicon bars while the enhancement reaches more than 120 and 84 for the metamaterial made of the split antennas with a gap of g=50 and 100 nm, respectively. Such strong field enhancement is comparable to plasmonic nanostructures. Other resonant dielectric nanostructures such as micro- or nano- cavities can also realize local field enhancement [27–29], but they lack the spatial homogeneity in the metamaterials presented here. At the same time, the averaged electromagnetic energy density is enhanced by more than 7000 times inside the gap with a gap of g=50 nm (see Fig. 5(c)). So the introduction of a gap at the middle of the silicon bars not only leads to the localization of electromagnetic energy in the surrounding medium (i.e. inside the gap) but also increases the enhancement of local field. Figure 6 shows the dependence of the field enhancement on the width of the gap. As we mentioned above, the field enhancement in the gap can be explained microscopically by the opposite (polarization) charges at the two sides of the gap. So stronger field enhancement can be realized with a shorter gap (e.g. the maximum field enhancement reaches about 175 for a 25 nm gap). For practical applications, the influence of substrate should also be considered. For the metamaterial (g=50 nm) on a semi-infinite silica substrate (n=1.5), the resonance wavelength red-shifts to 1415 nm and our simulations show that the maximum enhancement of field amplitude at the resonance is reduced to about 65. The presence of a substrate will not only affect the field distributions but also increase the effective width of the gap. So a substrate with a low refractive index is preferred for strong field enhancement.

 

Fig. 5 Local field enhancement and light localization in dielectric metamaterials at Fano-resonances. (a) Electric field in z-direction normalized to the field amplitude (E₀) of the incident light. The field distribution is plotted at the x–y plane that is 5 nm above the top surface of the dielectric bars. (b) Normalized electric field and (c) Normalized time averaged energy density at the x–y plane bisecting the nanobars. The metamaterial is illuminated with a x-polarized plane wave at normal incidence. Top: metamaterial with continuous silicon bars. Middle: metamaterial with split silicon bars (g=50 nm). Bottom: metamaterial with split silicon bars (g=100 nm).

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Fig. 6 Dependence of the maximum field enhancement factor on gap size. The red curve is a fitted line.

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4. Optical refractive index sensor based on a metamaterial

The giant field enhancement and light localization in the gaps of the resonant elements of the metamaterial make it possible to realize strong interactions between light and surrounding medium. For some applications, such as nonlinear optics and enhanced Raman scattering, it is desirable to get strong field enhancement and one may use bar resonators with small gaps. For other potential applications of dielectric metamaterials, such as sensing and enhancing the luminescence of quantum dots, it requires more electromagnetic fields to interact with the surrounding media. So one needs a large proportion of electromagnetic field to be confined in the surrounding medium though field enhancement may be still important. At this situation, a relatively bigger gap is required as a larger proportion of electromagnetic energy can be located in the gap even though the maximum field reduces. We can optimize the size of gap according to specific applications.

As an example of applications, we propose an optical refractive index sensor based on the proposed metamaterial. The inset in Fig. 7(a) shows the schematic and geometric parameters of the metamaterial design. The two asymmetric bars are 700 and 750 nm long, respectively. Both of them are 300 nm wide and 200 nm thick. There is a 100 nm wide gap in the middle of each bar. The metamaterial is assumed to be on a semi-infinite silica substrate. The refractive index of substrate is n_s = 1.5. Here the width and thickness of the silicon bars are different from those in Fig. 2 and Fig. 4 in order to get resonances at longer wavelength and avoid diffraction. Figure 7(a) shows the simulated transmission spectra of the metamaterial when it is covered by liquids with different refractive indices. The liquid is assumed to fill the gaps and cover the top of the silicon bars. The resonance red shifts about 52.5 nm with the minimum transmission wavelength changing from 1600 nm to 1652.5 nm as the refractive index of the liquid increases from 1.3 to 1.4, corresponding to a sensitivity of S = 525 nm/RIU (RIU means refractive index unit).

 

Fig. 7 An optical sensor based on the proposed dielectric metamaterial. (a) Transmission spectra for a metamaterial on a silica substrate and covered by liquids with different refractive indices. The inset is the top view of a metamaterial unit cell. (b) Dependence of the minimum transmission wavelength on the refractive index of surrounding medium for metamaterials comprising dielectric bars with different gaps. The width of the gaps are g=0, 50 and 100 nm, respectively. The slope of the curves shows the sensitivity.

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The sensitivities of three dielectric metamaterial based sensors are shown in Fig. 7(b). All of them exhibit nearly linear sensitivities for the refractive index between 1.3 and 1.7. For the metamaterial made of continuous silicon bars (g=0), the averaged sensitivity is about 338 nm/RIU. For the metamaterial comprising split bars, the averaged sensitivity increases to 481.5 nm/RIU for g=50 nm and 522.5 nm/RIU for g=100 nm, respectively. With a larger gap, the sensitivity is higher as a bigger proportion of electromagnetic field can interaction with the surrounding media. The introduction of a 100 nm wide gap at the centre of the silicon bar resonators leads to an increase of more than 50% in sensitivity. However, it doesn’t keep increasing if the gap size is more than 100 nm. The sensitivity remains at about 530 nm/RIU when the gap size increases to 150 nm and 200 nm. This is because coupling between the two parts of the split bar and the field in the middle of the gap become weak for these large gaps. As a result, the proportion of electromagnetic field in the gap doesn’t continue to increase. At the same time, the quality factor of resonance and maximum field enhancement in the gap reduces as the gap size increases. So a gap of about 100 nm will be best here. The sensitivity of our proposed sensor is about 10% higher than the recently reported sensitivity of a Fano-resonant dielectric metamaterial [17]. In [17], crystalline silicon with a refractive index of n = 3.7 is employed and the substrate underneath the gap is over etched, which will be helpful for realizing high sensitivities as the sensing volume of the surrounding medium is increased. If similar strategies are applied to our design, the sensitivity of the metamaterial sensor can be further increased. Besides the sensitivity, linewidth or Q-factor of the resonance is also important for optical sensors. The overall performance of optically resonant sensors are characterized by the figure of merit (FOM). FOM is defined as FOM = S/Δλ, where S is the sensitivity and Δλ is the linewidth of the resonance. The resonance linewidth of our proposed dielectric metamaterial sensor is about 2 nm and the FOM is about 260. In practice, the linewidth may be broadened due to manufacturing imperfections and a linewidth of 5.2 nm will give a FOM comparable to that of the device reported in [17].

5. Conclusions

In conclusion, we proposed a type of all-dielectric metamaterials using split bar resonators as building blocks. Large field enhancement and light localization can be realized in the gaps. Fano-resonant dielectric metamaterials comprising pairs of asymmetric split bars are designed. The enhancement of electric field magnitude exceeds 120 while the electromagnetic energy density is enhanced by more than 7000 times at the resonance. Based on the proposed dielectric metamaterial, an optical refractive index sensor is designed with an sensitivity of 525 nm/RIU, which is comparable to and only slightly lower than the best sensors using localized surface plasmon resonances (LSPR) [30, 31]. Furthermore, the resonance quality factor of dielectric metamaterials can reach several hundreds or even higher while that of plasmonic nanostructures is generally limited to the order of ten. So the figure of merit of the proposed sensor can far exceed LSPR sensors. The proposed metamaterials can be fabricated by e-beam lithography. Currently, the quality factors of resonant dielectric metamaterials are mainly limited by sample imperfections induced by fabrication errors. Fano-resonant dielectric metamaterials with quality factors up to several hundreds have recently been experimentally demonstrated. With the progress of fabrication technology and with improved designs, the performance of the dielectric metamaterials can still be improved. The proposed metamaterial design provides an effective method to enhance light-matter interactions and will stimulate the research of dielectric metamaterials/metasurfaces for applications ranging from nonlinear optics and sensing to the realization of new types of active lasing [32].