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Abstract
We investigate edge-coupling of twisted split-ring resonator (SRR) pairs in the terahertz (THz) frequency range. Using a simple coupled-resonator model we show that such a system exhibits resonance splitting and cross-polarization conversion. Numerical simulations and experimental measurements agree well with theoretical calculations, verifying the resonance splitting as a function of the coupling strength given by the SRR separation. We further show that a metal ground plane can be integrated to significantly enhance the resonance coupling, which enables the effective control of resonance splitting and the efficiency and bandwidth of the cross-polarization conversion. Our findings improve the fundamental understanding of metamaterials with a view of accomplishing metamaterial functionalities with enhanced performance, which is of great interest in realizing THz functional devices required in a variety of applications.
© 2017 Optical Society of America
1. Introduction
Control and manipulation of light is essentially altering its amplitude, phase and polarization states using materials with desirable properties. Metamaterials composed of periodic subwavelength metal/dielectric resonators (i.e. meta-atoms) can extend their electromagnetic properties beyond naturally occurring materials to accomplish exotic optical phenomena such as negative refraction [1–3]. While bulk metamaterials suffer from high losses and great fabrication challenges particularly in the optical frequency regime, their two-dimensional equivalents – metasurfaces – have attracted increasing attention because of the ease in fabrication using conventional planar methods and, at the same time, the capability of realizing novel functionalities [4–8]. Although the performance in single-layer plasmonic metasurfaces may be limited due to the low-efficiency in converting the incident waves to reflection/transmission with desirable polarization state [4,9–11], few-layer metasurfaces can address such an issue by taking advantage of coupling among individual metasurface layers, resulting in not only much higher efficiencies but also desirable functionalities that are beyond what individual metasurface layers can afford, such as perfect absorption [12, 13], antireflection [14, 15], and broadband polarization conversion [16–19]. On the other hand, when meta-atoms form meta-molecules, the properties may change dramatically, providing tremendous opportunities to explore new functionalities and applications of metasurfaces.
Among many subwavelength resonators that have been developed as basic building blocks of metamaterials, split-ring resonators (SRRs) [20] have played a special role serving as an important model structure, possessing simultaneously magnetic and electrical resonances as well as magnetoelectric coupling [21]. Earlier works in magnetoelectric coupling include the investigation of edge- and broadside-coupled SRRs exhibiting bi(iso/aniso)tropic behavior of metamaterials, and also resonator size-reduction to enable better effective media approximation [21, 22]. Coupled resonators have been widely used during the past few years to study plasmonic induced transparency (PIT) [23], which mimics the quantum phenomenon of electromagnetically induced transparency (EIT) [24]. Such a classical analog of EIT system typically consists of both bright and dark metallic resonators, where the incident electromagnetic waves efficiently excite the bright resonator with a broad resonant response, which is then resonantly coupled through near-field interactions to the dark resonator that cannot be directly excited and is only very weakly coupled to free space waves. The overall spectral response reveals a narrow transparent window within a broad opaque background, leading to an extremely strong dispersion and large group refractive index suitable for slow light applications [25]. On the other hand, the apparent PIT can be also observed in coupled resonators where both resonators radiate strongly to free space, i.e., there is no dark resonator. For instance, Zhang et al. investigated such a behavior using edge-coupled resonators [26–28], although the assignment of dark eigenmode is somehow questionable as the SRR radiates strongly to free space once excited and it was not directly excited due to the SRR’s ‘wrong’ orientation. Averitt et al. have investigated broadside-coupled SRRs that exhibit similar behavior [29, 30], which is essentially introduced by resonance splitting depending on the coupling strength between the two resonators. In the optical regime the coupling effects have been extensively investigated by Liu and Giessen with a variety of different SRR configurations [31, 32].
Here we investigate the edge-coupling between the two resonators within a twisted SRR pair in the terahertz (THz) frequency range. We observe the resonance splitting as a function of the coupling strength, accompanied by polarization conversion to the orthogonal direction. The experimental results agree with numerical simulations, and are consistent with results obtained from a coupled-resonator model where the excited dipoles are calculated both in co- and cross-polarization directions. We further show that a metal ground plane can be integrated to either significantly enhance the sharpness of the resonance splitting or increase the polarization conversion efficiency and bandwidth, depending on the coupling strength of the SRR pairs and their separation to the ground plane. Our findings in coupled metamaterial resonators may find interesting applications by providing suitable structures for optical polarization manipulation and switching, as well as enhancement of optical sensing and nonlinearity.
2. Resonance coupling and theoretical model
The schematic of the twisted SRR pairs is shown in Fig. 1. Similar structures have been adopted, for instance in [26, 31], to study coupling between the so-called bright and dark eigenmodes or magnetoinductive and electroinductive coupling in plasmonic metamolecules. The twisted SRR pair consists of two identical planar SRRs rotated by 90° with respect to each other. When the incident electric field is linearly polarized along x direction, the fundamental resonance mode of only the left-SRR is excited, where the circulating current, i1, creates a magnetic dipole m1, and the charges accumulated at the capacitive gap form an electric dipole p1 in x direction. The fundamental resonance mode of the right-SRR, on the other hand, cannot be directly excited due to its ‘wrong’ orientation. In such a configuration, the directly excited electric dipole is localized at the split gap of the left-SRR, which is relatively far away from the right-SRR, and the electric field between the two SRRs is negligible as we will show later, resulting in minimal electrical coupling to the right-SRR. This is in remarkable contrast to the case where two SRRs are arranged face-to-face resulting in strong capacitive coupling [33]. However, the SRR pair represents two inductively coupled electrical loops, where the excited magnetic field from m1 extends much beyond the left-SRR, allowing sufficient near-field coupling of the excitation to the right-SRR. The coupling excites the fundamental resonance mode in the right-SRR represented by the circulating current i2, magnetic dipole m2, and electric dipole p2 in y direction. Under normal incidence, the excited magnetic dipoles m1 and m2 do not radiate in the normal directions, and their in-plane radiation cancels out due to the subwavelength periodic arrangement. On the other hand, the excited electric dipoles radiate strongly to the normal directions, with p1 polarizing in x direction and p2 polarizing in y direction, causing co-polarized and cross-polarized reflection/transmission, respectively.
Fig. 1 Schematic of gold twisted resonator pairs on silicon substrate with geometrical parameters specified: Ax = 72 μm and Ay = 36 μm the periods in x and y directions, respectively, a = 24 μm, w = 4 μm, g = 2 μm, and s the separation between the two SRRs (s is defined as 0 in the structure within the left panel). The incident electric field E is linearly polarized in x direction.
The above qualitative analysis allows us to consider the subwavelength edge-coupled SRR pair as two lumped RLC resonators possessing mutual inductance L12. Based on the Kirchhoff’s voltage law and using an analysis similar to [30], the excited currents can be solved by using the following coupled equations:
where the subscripts ‘1’ and ‘2’ indicate left-SRR and right-SRR, respectively, Ein is the incident electric field, g is the gap size, and C1 = C2 = C, R1 = R2 = R, and L1 = L2 = L are the capacitance, resistance, inductance of the SRRs, with equal values due to their identical geometry dimensions. Solving the equations we obtain the excited circulating currents: where i is the imaginary unit, and ω is the angular frequency of the incident electromagnetic waves. The induced electric dipoles are proportional to the charge accumulation at the split gaps, i.e., p1,2 ∝ ∫ i1,2dt = i1,2/(−iω), and the corresponding electric fields are proportional to the second order derivative of electric dipole moments, i.e., Ex,y ∝ d2p1,2/dt2 = −ω2p1,2, which gives: where the tilde indicates complex values, distinguishing them from the amplitudes Ex and Ey.The inductance of the SRR is determined by its geometric dimensions, which can be estimated [34] as L = 50 pH. From the resonance frequency of ~0.725 THz for individual SRRs, the SRR gap capacitance is then estimated as C = 1 fF. The resistance can be estimated from the quality factor (Q ~ 6.5) of the resonance, which gives R = 34 Ω. The mutual inductance depends on the coupling strength and decreases when the separation between the two SRRs increases. It can be estimated in principle but here we select a few reasonable values varying from L12 = 12 pH to 6 pH. Taking these values of the lumped circuit elements for the SRRs we calculate the radiated field amplitude of the excited dipoles as shown in Fig. 2. The first feature observed from the calculations is the strong cross-polarization coupling. The radiation amplitude Ey from the electric dipole p2 at the right-SRR, which is indirectly excited through the magnetic near-field coupling, is comparable to the radiation amplitude Ex from the electric dipole p1 at the left-SRR directly excited by the incident electric field. Decreasing the mutual inductance narrows the amplitude spectra and slowly reduces the cross-polarized radiation Ey, while the co-polarized radiation Ex slightly increases.
Fig. 2 Amplitude spectra Ex radiated from the electric dipole p1 (red curves) directly excited by the incident field, and Ey from the electric dipole p2 (cyan curves) indirectly excited through magnetic coupling, as functions of the separation between the two SRRs represented by different values of mutual inductance L12: (a) 12 pH, (b) 10 pH, (c) 8 pH, and (d) 6 pH.
The second important feature is the splitting of the resonance spectra. When the coupling is relatively strong, both Ex and Ey spectra exhibit two resonance peaks. As it will be shown later in numerical simulations, at the lower resonance frequency, calculations show that Ex and Ey are almost out of phase, suggesting an asymmetric mode with opposite circulating current directions and antiparallel magnetic dipoles. At the higher resonance frequency, they are almost in phase with the same circulating current directions and parallel magnetic dipoles. With mutual inductance L12 = 12 pH, the two peaks of Ex are located at and (Δν(x) = 0.207 THz) as shown in Fig. 2(a); the splitting is reduced when the coupling strength decreases, with the two peaks moving to and (Δν(x) = 0.097 THz) for L12 = 6 pH shown in Fig. 2(d). The Ey spectra also exhibit resonance splitting, however, much narrower than that of Ex. The two peaks are located at and (Δν(y) = 0.147 THz) for L12 = 12 pH shown in Fig. 2(a) (here we just take the peak frequencies rather than through fitting), but become indistinguishable with reduced L21 = 6 pH as shown in Fig. 2(d). The obtained resonance splitting reminisces the energy level splitting and band structure formation when atoms form molecules and crystals, and such a resonance splitting and mode hybridization has also been extensively studies in plasmon response in complex nanostructures and metamaterials [32, 35]. The different resonance splitting between Ex and Ey might be counter-intuitive at first. However, the underlying physics can be qualitatively interpreted by the fact that the radiation of the cross-polarization Ey goes through a two-step process – the incident field first resonantly excites the left-SRR, which is then resonantly coupled to the right-SRR, thus narrowing the spectral response.
3. Numerical simulations and experimental results
Full-wave numerical simulations have been performed using the software package CST Microwave Studio. The twisted SRR pairs are on top of an intrinsic silicon substrate with a permittivity of εSi = 11.7 and negligible loss [36], and made of 200-nm-thick gold with Drude conductivity [37]. In all simulations, plane wave excitation is used with normal incidence, containing modes linearly polarized in either x or y direction. Periodic boundary conditions are used to simulate an infinite two-dimensional periodic array, with unit cell and structural dimensions specified in Fig. 1. The spacing parameter s is subject to variation and set to be 0, 2, 6, and 12 μm. The obtained S21 parameters are normalized to a bare substrate surface and plotted in Fig. 3. These curves represent transmissions, txx and tyx, that contain both x- and y-polarization components, respectively, excited by the incident field linearly polarized in x direction.
Fig. 3 Simulated transmission spectra polarized in x (red curves) and y (cyan curves) polarizations, for various values of SRR separation s: (a) 0 μm, (b) 2 μm, (c) 6 μm, and (d) 12 μm.
In these simulated results, we can clearly see the strong cross-polarized component tyx that decreases only slowly with increasing s. The amplitude spectra near the resonance frequencies suggest similar magnitude of the excited p1 and p2 and thereby a high coupling efficiency. There is a large resonance splitting when the two SRRs are in contact with each other or with small separation, and the resonance splitting is reduced quickly with increasing s. These results are in excellent agreement with the theoretical prediction presented in Fig. 2. The low-frequency cross-polarized peak is blue-shifted as compared the corresponding co-polarization dip, which is also consistent with the theoretical calculations. However, the high-frequency cross-polarization peak has a similar frequency as that of the co-polarization dip, which is contradictory to the theoretical prediction in Fig. 2. This discrepancy may come from the neglect of conducting coupling when the two SRRs are in contact with each other, or capacitive coupling originating from the small gap between the two SRRs [31, 32].
It is instructive to understand the resonance coupling by visualizing the excited current flow and electric field distribution. The excited current flow at the low resonance frequency is displayed in Fig. 4(a) and at the high resonance frequency in Fig. 4(b), where the separation of SRRs is s = 2 μm. For the low frequency resonance, the transient currents in the left- and right-SRRs flow in counterclockwise and clockwise directions, respectively, corresponding to the asymmetric resonance mode. For the high frequency resonance, the currents have the same circulating direction, corresponding to the symmetric resonance mode. These observations are consistent with the theoretical results discussed in the previous section. The distributions of electric field are given in Figs. 4(c) and 4(d) for the low and high resonance frequencies, respectively. It is obvious that in both cases the electric fields are highly localized in the gap areas while the field intensity is quite small between the two SRRs, which well justifies the neglect of electrical coupling between the two SRRs.
Fig. 4 (a,b) Surface currents and (c,d) electric field distributions at low (a,c) and high (b,d) resonance frequencies, with the separation of the twisted SRRs s = 2 μm.
Periodic arrays of such SRR pairs are fabricated on double-side-polished intrinsic silicon substrates with resistivity ρ > 10 000 Ω cm, using standard photolithographic methods, e-beam metal deposition of 5 nm/200 nm Ti/Au films, and a lift-off process. Example scanning electron microscopy (SEM) images are shown in Fig. 5(a) for the fabricated metasurface structures, along with their enlarged views illustrating the unit cell details. The SRR pair arrays occupy an area of 10 × 10 mm2, much larger than the focal diameter of the incident THz beam (~1 mm2). The metasurface samples are characterized under normal incidence using a standard terahertz time-domain spectroscopy (THz-TDS) setup incorporating four wire grid polarizers (WGPs) [17], schematically shown in Fig. 5(b). WGP1 is placed before the sample to ensure x -polarized incident THz waves. WGP2 is either parallel or perpendicular to WGP1, which allows either co- or cross-polarized transmission to be selected in the measurements. WGP3 is oriented at 45° so that equal fraction of the THz field is projected, and finally WGP4 is parallel to WGP1 and brings the THz polarization back to x direction, allowing for the highest THz detection efficiency using a 1-mm-thick <110> ZnTe crystal. All the THz transmission spectra are normalized to a co-polarized transmission spectrum through a blank silicon substrate.
Fig. 5 (a) SEM micrographs of two fabricated example metasurface structures with s = 0 (left) and s = 2 μm (right), as well as their enlarged view specified with measured gap size and separation. Scale bar: 100 μm. (b) Schematic of the experimental measurements using four wire grid polarizers (WGPs) with their directions specified. The incident THz is linearly polarized in x direction and the output THz field is also measured in x direction.
The measured transmission amplitude spectra are shown in Fig. 6. When the two SRRs are in contact with each other, the co-polarized transmission reveals a large resonance splitting with and (Δν(x) = 0.190 THz). The resonance frequencies slightly deviate from those obtained in simulations, which could be attributed to fabrication tolerance as well as the properties of materials being slightly different from the values used in simulations. As shown in the SEM images in Fig. 5(a), the fabricated samples have gap sizes varying from g = 1.5 μm to 1.8 μm in stead of the designed 2 μm and the line width varying from w = 4.2 μm to 4.5 μm as compared to the designed value of 4 μm, in addition to significant rounding at structure corners. The resonance splitting decreases rapidly with increasing SRR separation s, reduced to Δν(x) = 0.081 THz for s = 2 μm and Δν(x) = 0.051 THz for s = 6 μm. For s = 12 μm the resonance splitting becomes indistinguishable, only with a small bump in the transmission dip indicating the resonance splitting. The cross-polarized transmission also reveals resonance splitting, though slightly less pronounced as compared to numerical simulations. This less pronounced resonance splitting could be in part attributed to the non-ideal performance of the WGPs, resulting in leakage of the undesirable polarization component picked up by the THz detector. The splitting also decreases rapidly with increasing s and diminishes at s = 6 μm and 12 μm. The low-frequency cross-polarization peak has a blue-shift as compared to the corresponding co-polarized transmission dip. The maximal amplitude of the cross-polarization transmission varies from ~40% for s = 0 μm to ~25% for s = 12 μm, only slightly smaller than those obtained in numerical simulations. The experimental results reproduce all the important features observed in full-wave numerical simulations and predicted in theoretical calculations.
Fig. 6 Measured transmission spectra polarized in x (red curves) and y (cyan curves) polarizations, for various values of SRR separation s: (a) 0 μm, (b) 2 μm, (c) 6 μm, and (d) 12 μm.
4. Enhancing the resonance coupling and polarization conversion
The linear polarization conversion from the twisted SRR pairs shares the similarity with anisotropic resonators such as cut wires and split ring resonators with incident electric field linearly polarized at 45° with respect to their principle axes. In such a single-layer metasurfaces, the conversion efficiency is fundamentally limited by the facts that there are co- and cross-polarization components in both reflection and transmission directions. In the past few years it has been shown that few-layer metasurfaces can dramatically enhance the performance of metasurface functionalities [6, 14, 17–19]. A simple yet powerful approach is adding a metal ground plane separated from the resonators by a thin dielectric spacer. It can be intuitively explained using an over-simplified physics picture that the ground plane reflects the transmitted waves back to the metasurface, thereby allowing for additional interactions. A more precise model is that the metasurface and the ground plane form a Fabry-Pérot-like cavity [13], where tailoring the metasurface anisotropy, dispersion and spacer thickness enables a variety of functionalities including metamaterial perfect absorbers [12, 38], broadband highly efficient linear polarization converters [17], and designer phase for coding and digital metamaterials [39, 40].
This strategy is exploited here in order to enhance the resonance coupling and polarization conversion in the twisted SRR pairs. We perform full-wave numerical simulations using the structure schematically shown in the inset to Fig. 7(b). For simplicity and the purpose of only validating the feasibility, we assume lossless dielectric spacer with a dielectric constant of ε= 2.92 and twisted SRR pairs with the same dimensions as shown in Fig. 1. The SRR separation s and spacer thickness ts are variables to realize different resonant behaviors of the metasurface structures. When ts = 45 μm, the reflection amplitude spectra are shown in Figs. 7(a) and 7(b) for s = 0 and 6 μm, respectively. As compared to simple SRR pairs with results shown in Figs. 3(a) and 3(c), dramatically enhanced resonance splitting, represented by the sharp dips (for co-polarization) and peaks (for cross-polarization), is observed with much narrower line widths. It is also accompanied by the increased linear polarization conversion efficiency. Tuning the spacer thickness to ts = 30 μm leads to significant change of the reflection amplitude spectra, shown by the broadened resonance peaks and dips with much overlap in Fig. 7(c) for s = 0 μm and even disappeared resonance splitting in Fig. 7(d) for s = 6 μm, while the linear polarization conversion efficiency remains to be high. Note that the blue-shift of the resonance frequencies is because now the SRR pairs are on the spacer with a reduced dielectric constant as compared to that of silicon. These results clearly show that adding the ground plane, together with tuning the spacer thickness as well as the SRR separation, provides degrees of freedom to control the resonance coupling in twisted SRR pairs represented by the amount of resonance splitting and sharpness of spectral response. It also offers a mechanism to realize highly efficient narrowband, dual-band, or broadband linear polarization conversion.
Fig. 7 Enhanced resonance coupling and linear polarization conversion when integrating a metal ground plane. When the spacer thickness ts = 45 μm the reflection amplitude spectra are shown in (a) for s = 0 and (b) for s = 6 μm. When ts = 30 μm they are shown in (c) for s = 0 and (d) for s = 6 μm. Red curves: co-polarization; cyan curves: cross-polarization. Inset to (b): schematic of the unit cell structure.
5. Conclusion
To summarize, we have investigated the magnetic interactions of edge-coupled twisted SRR pairs as a function of their coupling strength represented by different values of the SRR separation. Using a coupled-resonator model by considering the SRRs as lumped RLC resonant circuits, we calculated the induced currents within the SRR loops, electric dipoles at the split gaps, and the corresponding radiation spectra for both co- and cross-polarizations. The theoretical calculations revealed that the strong resonance coupling results in comparable magnitude of electric dipoles, although one of the SRR cannot be excited directly but only through magnetic coupling. The calculations also showed significant resonance splitting which strongly depends on the coupling strength. Full-wave numerical simulations of ideal structures and experimental measurements based on the fabricated metasurfaces validated the theoretical predictions with good agreement, and the observed discrepancies were also discussed. We found that the resonance coupling can be dramatically enhanced through adding a metal ground plane separated from the SRR pairs by a thin dielectric spacer. This resulted in either sharpened or flattened resonance splitting depending on the metasurface geometric parameters, which can be exploited to tailor the bandwidth the linear polarization conversion with much improved efficiencies. Our findings advance the understanding of resonance coupling within complex metamolecules, which can lead to novel metamaterial/metasurface structures for advanced functionalities and applications.
Funding
National Natural Science Foundation of China (NSFC) (Grant No. 11574219); Los Alamos National Laboratory LDRD Program.
Acknowledgments
This work was performed, in part, at the Center for Integrated Nanotechnologies, a U.S. Department of Energy, Office of Basic Energy Sciences Nanoscale Science Research Center operated jointly by Los Alamos and Sandia National Laboratories. Los Alamos National Laboratory, an affirmative action/equal opportunity employer, is operated by Los Alamos National Security, LLC, for the National Nuclear Security Administration of the U.S. Department of Energy under Contract No. DE-AC52-06NA25396.
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