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We present symmetric and asymmetric couplings within a pair of split-ring resonators (SRRs). The former shows a single transmittance dip, following the equivalent circuit model; yet, the latter introduces an additional transmittance peak, stemming from an asymmetrically coupled resonance (ACR) between the subradiant and superradiant modes. The mechanism of such induced transparency is elucidated well by the suppression of induced currents within the SRR element with a lower quality factor. Finally, the excitation of ACR is further associated with remarkable confinement of electromagnetic field, providing a compelling sensing performance based on its excellent sensitivity and figure of merit.
©2009 Optical Society of America
1. Introduction
Metamaterials are a new class of artificial structures constructed by the sub-wavelength building elements, enabling unprecedented electromagnetic properties in nature, for example, artificial magnetism [1], artificial plasma [2,3], super lenses [4,5], magnetic surface plasmon [6,7] and invisibility of cloak [8]. Foremost among these properties to tailor new physics is the artificial magnetic response, even allowing the realization of negative refractive index media [9–12]. Among them, the split-ring resonators (SRRs) are the most frequently used elements for achieving effective negative permeability [12–14], excited from the time-varying magnetic field (H-field) vertically to the SRRs plane.
In addition to the excitation from the H-field [12–14], under the electric excitation (E-field) the SRR structures demonstrate multiple-mode plasmonic resonances, which are classified in two groups of eigenfrequencies for electric and magnetic resonances with respect to the polarization of the illuminating E-field [15–18]. Specifically, the pure electric responses are excited once the external E-field is polarized perpendicular to the gap side of SRRs and can be manifested as even order plasmonic modes. On the other hand, the odd order plasmonic modes all contribute to the magnetic responses that are excited by the polarization of E-field along the gap side of SRRs. In addition, the fundamental mode of SRRs can be also regarded as the collective responses from individual LC resonators, in which the gap and the metallic ring of SRR units represent equivalent capacitance (C) and inductance (L), respectively [14]. Nevertheless, once the spacing between neighboring metallic elements is not far enough, the interaction among them will unambiguously influence the resonant properties and the impact of coupling effect can no longer be neglected. In particular, by breaking the geometrical symmetry of the coupled planar metamaterials, the exceptional resonance with a very high quality factor can be realized [19]. Such a method provides the additional freedom to manipulate the electromagnetic properties beyond the original resonant responses of metamaterials. In fact, the concept of breaking symmetry has also been applied on other metallic nanostructures, such as nanowire pairs [20,21] and ring/disk cavity [22], resulting in Fano-type resonances caused by the interaction of narrow dark modes with broad bright modes [23].
As a consequence, in this study we introduce an asymmetrically coupled resonance (ACR), from two different SRR structures in a unit cell whose resonant frequencies are overlapped but their quality factors are manipulated at the respective small and great values. By comparing the asymmetrically coupled SRRs from the symmetrically coupled ones, it is evident to observe an additional transmittance peak with an enhanced quality factor rather than the original transmittance dip with a low quality factor. Such sharp induced transparency by the ACR effect originates from strong coupling between a narrow subradiant mode and a broad superradiant mode in SRR structures, and is significantly modulated by the spacing of two SRR constituents. Finally, based on various SRR structures including single SRRs, symmetric SRR pairs and asymmetric SRR pairs, the refractive-index sensitivity and corresponding sensing performance are further evaluated.
2. Design of SRR units
Referring to the model of standing-wave plasmonic resonances, the multi-mode resonant frequencies of an SRR element can be quantitatively evaluated by its total length [18]. Consequently, we readily design two kinds of geometry-different SRRs with partially overlapped resonant frequency regions as shown in the insets of Fig. 1 . Notice that the total length of the narrow SRR element (SRRn) is 1.88 μm, slightly shorter than the wide SRR one (SRRw; length= 1.92 μm), leading the SRRn to a higher resonant frequency. As expected, the respective resonant frequencies of SRRn and SRRw center at 40.9 THz and SRRw 39.3 THz in Fig. 1(a) and (b), simulated by a commercial electromagnetic solver (Microwave Studio, CST) in which the permittivity of gold is treated by the Drude dispersion model [24]. Such partially overlapped resonant frequency regions allow us to clarify the coupling effect from two distinct plasmonic resonances as discussed in next section.
Fig. 1 The simulated transmission spectrum of (a) SRRn and (b) SRRw. The insets show the illustration of a unit cell of SRRs with related structural parameters, W=0.1 μm, T=0.05 μm, L (total length)=1.88 μm in SRRn and 1.92 μm in SRRw. (c) The distribution of induced currents at resonant frequency of SRRn and SRRw, respectively.
The spectral positions of these two SRR elements are similar, but their quality factors (Q-factors) are much different– 13.6 for the SRRn and 3.3 for the SRRw. To interpret these distinct resonance characteristics, we plot the distribution of the induced currents in the SRR elements at resonance as presented in Fig. 1(c). Two SRR structures respond similarly to introduce circulating currents, in which the anti-parallel currents in both side arms cancel each other giving rise to nearly zero dipole moment. Therefore, the net dipole moments excited from the external E-field are mainly contributed from the bottom arm of SRRs, leading a longer bottom arm in the SRRw to a broader bandwidth (defined as FWHM) of resonance due to the fact that the radiation damping increases with the particle size [25,26]. In contrast, the SRRn with a shorter bottom arm contributes to a less radiation damping, resulting in a smaller value of FWHM, as demonstrated earlier in Fig. 1(a) and (b).
3. Coupled resonances in symmetric and asymmetric planar SRR pairs
First, we investigate the case of symmetric coupling in an SRR pair. As shown in Fig. 2(a) and (b) , the symmetric SRR pair is comprised of two identical SRR structures (the SRRn and SRRtw, respectively) opposite to each other with a fixed spacing (d) of 100 nm [27,28]. In addition, the simulated spectra for theses two symmetric SRR pairs are shown in the spectra of Fig. 2, demonstrating a slight blue shift with respect to the resonance frequency of the individual SRR structures. To reveal the origin of the symmetrically coupled resonance (SCR), we analyze the distribution of induced currents at resonance frequency. As shown in the insets of Fig. 2(a) and (b), the circulating currents are induced and oscillate in parallel direction within both SRR elements. Thus, viewing the symmetric SRR pair as an equivalent LC circuit, these two resonantly circuit loops construct an inductive coupling, leading to the slight blue shift of resonance frequencies [27,28]. In short, the SCR from the interaction in the symmetric SRR pairs demonstrates the similar resonant type with that in the single SRR constituents, but owning a slight blue-shifting resonant frequency.
Fig. 2 The illustration of symmetrically coupled SRR structures and its transmission spectrum for (a) SRRn and (b) SRRw (solid line). The transmission spectra of corresponding SRR units are also shown (dash line). In short, the resonances from symmetrically coupled SRR structures demonstrate the similar resonant type with that in the unit SRR structures, but owning a slight blue-shifting resonant frequency. The insets show the distribution of induced currents at their resonant frequencies, respectively.
Second, we investigate the electromagnetic response in the coupled SRR system with symmetry breaking. Two distinct SRR units (the SRRn and SRRw) face oppositely with a spacing of d =100 nm, presenting an asymmetric coupling system as displayed in Fig. 3(a) . Here the SRRn and SRRw units are designed to resonate at approximate frequencies, whereas their quality-factors (Q-factors) differ in an order of magnitude. Therefore, the excited resonance in the SRRn units with large Q-factor behaves a less damped resonant type to serve as a subradiant mode and on the other hand, the resonance in the SRRw units is comparably lossy due to the resulting small Q-factor to render as a superradiant mode. While these subradiant and superradiant modes coupling with each other, there emerges an extraordinary electromagnetic response in contrast to the previous symmetric cases– two splitting transmission dips and a sharp transmission peak as shown in Fig. 3(b) rather than a single transmission dip as shown in Fig. 2.
Fig. 3 (a) The illustration of asymmetrically coupled SRR structures and its transmission spectrum is shown in (b). It demonstrates two splitting transmission dips (denoted as ω1 and ω2) and a sharp transmission peak (denoted as ACR) in a transmission spectrum rather than a single transmission dip in the symmetrically coupled SRR structures. (c) The distribution of induced currents at resonance ω1 and ω2, respectively. (d) Retrieved effective permittivity (εeff) and permeability (μeff) of the asymmetrically coupled SRR structures.
In the asymmetric SRR pairs, two splitting transmission dips can be interpreted by the concept of plasmon hybridization model [29,30]. We calculate the distribution of induced currents at ω1 and ω2, as depicted in Fig. 3(c). The low-energy mode (ω1) possesses the opposite induced currents in two SRR constitutes, analog to the bonding mode in a hybridized molecular system. The high-energy mode (ω2), rather, exhibits the induced currents in the similar direction which can be directly associated with the antibonding mode in a hybridized molecular system. Besides, the effective permittivity (εeff) and permeability (μeff) of the coupled SRR structures are also retrieved in Fig. 3(d), following the method in the ref. 31. The retrieved real part of εeff at two hybridized modes (ω1 and ω2) demonstrate the nature of plasmonic resonances in a Lorentzian type, which is relatively dispersive since these two hybridized resonances are rather close in frequency regions.
In addition to the plasmon hybridization, a sharp transmission peak within two hybridized modes is activated through the coupling in the asymmetric SRR elements, referred to as asymmetrically coupled resonance (ACR). Different from the recent reported electromagnetic induced transparency (EIT) in metamaterials that is excited by the interaction between dark (non-radiative) and bright (radiative) modes [32–34], such an ACR response is introduced from the coupling between two radiative plasmonic modes instead, and its resonant strength depends on the spacing of two asymmetric SRR elements. As shown in Fig. 3(b), 4(a) and 4(b), as increasing the spacing from 100 to 400 and 700 nm, the ACR turns weaker due to the diminishing coupling between two radiative plasmonic modes so that the corresponding transmittance of ACR is suppressed to 84%, 78% and 35%, respectively. Moreover, the bandwidth of ACR, associated with the energy splitting of two hybridized modes (Δω=ω2-ω1), can be also determined by the spacing of two asymmetric SRR elements– the larger spacing, the narrower bandwidth (i.e., greater quality factor).
Fig. 4 The simulated transmission spectrum of asymmetrically coupled SRRs with the spacing of (a) 400 nm and (b) 700 nm. (c) The distribution of induced currents at resonance of coupled SRR pairs with the spacing of 100, 400, 700 nm and uncoupled SRRw, respectively. The net induced currents in the SRRw are suppressed due to the coupling from the SRRn and significantly dependent on the spacing (d) of SRRw and SRRn.
4. Induced transparency caused by the suppressed currents due to the destructive interference between two radiative modes
To interpret the induced transparency from ACR response, the distribution of induced currents in the asymmetrically integrated SRRs with four different spacings (d) is plotted as shown in Fig. 4(c). First, while the spacing (d) is 100 nm, the coupling effect from the SRRn modulates the resonant condition in the SRRw most and indeed, the net excitation of oscillating currents within the SRRw displays the weakest one among four cases due to the destructive interference between resonantly interacting eigenmodes, leading the suppressed net currents to greater transparency [32]. Be specific, the oscillating charges in the SRRw are excited through two pathways: one undergoes the direct resonance in the SRRw by the external E-field, giving rise to the induced currents in a counterclockwise direction, the other responds to the clockwise induced currents caused by the coupling in the SRRn, and both pathways introduce anti-parallel oscillating charges in the SRRw and in turn suppress the net excitation of induced currents. Next, as increasing the spacing (d) from 100 nm to 400 nm, 700 nm and infinite to diminish the coupling from the SRRn, it is clear to observe the stronger induced currents excited in the SRRw owing to weaker destructive interference between two radiative modes, leading the fading ACR response to less transparency. Notice that without the coupling effect from the neighboring SRRn (d=∞), the SRRw resonantly introduces currents in a counterclockwise direction solely excited by the external E-Field, exhibiting a transmission dip instead as shown in Fig. 1(b). In short, the induced transparency from ACR response is excited in case of strong asymmetric coupling between a narrow subradiant mode with a broad superradiant mode from the SRRn and SRRw of the SRR pairs, respectively.
Finally, we also calculate the frequency of ACR response versus the difference of resonant frequencies between the uncoupled SRRn (ωn) and SRRw (ωw). To unveil the role of the SRRn within the coupled system, the ωn is carefully controlled by varying the entire length of the SRR ring whereas the ωw remains fixed. As shown in Fig. 5 , the frequency of ACR response is specifically determined by ωn, indicating that the excitation of ACR is mediated by the resonance supported by the SRRn element. The linear dependence here particularly provides an applicable guideline to design the frequency of exciting ACR response. In short, this spacing-dependent induced transparency from two radiative modes reveals that the ACR response originates from the plasmonic couplings in the asymmetric SRR elements, and further suggests a way to realizing the high Q-factor resonance.
Fig. 5 The frequency of ACR versus the difference of resonant frequencies between the uncoupled SRRn (ωn) and SRRw (ωw) (i.e., ωn-ωw). It demonstrates an obvious linear relationship, providing an applicable guideline to design the frequency of ACR response.
5. A scalable refractive-index based sensor with the excellent sensitivity and figure of merit by ACR
Following the above discussions, the excitation of ACR stems from the interaction between subradiant and superradiant modes. Once these two eigenmodes are strongly coupled, there appears an intensive localization of electromagnetic energy inside the coupled nanostructures [22], which essentially behaves as a nano-scale cavity with an excellent quality factor and can be employed as a scalable refractive-index based sensor because its spectral position substantially depends on the dielectric condition of surrounding media. Here we evaluate the sensing performance of five distinct SRR structures by testing several dielectric layers (n=1-4) with the same thickness of 50 nm.
The result is shown in Fig. 6 . All structures manifest an explicit linear relationship between the resonance wavelengths and the refractive indices of the covering thin layers in a wide detection range of refractive index [18], which can be readily employed as excellent zero-order refractive-index based sensors. Foremost among those five distinct SRR nanostructures is the asymmetric SRR pairs, whose ACR response possesses the greatest sensitivity calculated in terms of Δλ/Δn, where Δλ and Δn represent the shift of the resonance wavelengths and the change of the refractive indices from covering thin layers, respectively. Due to the remarkable localization of electromagnetic energy, the asymmetric SRR pair with the high-Q factor reaches the sensitivity up to 1440 nm/RIU, approximately 150 % greater than the case of unit SRR structures, and over 200 % greater than the case of symmetric SRR pairs. More importantly, the frequency shift at ACR is about 30 times higher than the experimental results from conventional double SRR structures when considering the similar detection layers [35]. In addition, a practical concept of figure of merit (FOM) [36] is applied to further evaluate the sensing performance as below,
The corresponding FOM values from different SRR structures are presented in Table 1 to summarize that the asymmetric SRR structures own not only the highest sensitivity, but also the best FOM among various SRR-based designs owing to its advantageous ACR response.Fig. 6 The dependence of wavelength change at resonance to the refractive index of covering layer by examining different SRR structures. Notice that except the asymmetric SRR pairs, all other SRR structures resonantly reflect the incident light and possess a transmission dip.
Table 1. Evaluation of sensing performance by figure of merit (FOM). Clearly, the asymmetricSRR structures own the best FOM among various SRR-based designs.
6. Conclusion
In conclusion, we introduce the coupling of plasmonic resonances in the SRR pairs, especially in the asymmetric one that supports an extraordinary electromagnetic response referred to as asymmetrically coupled resonance (ACR). By artificially mimicking the subradiant and superradiant modes in a plasmonic manner, we observe that the ACR response is excited in case of strong coupling between a narrow subradiant mode with a broad superradiant mode, and this ACR can be modulated by varying the spacing of two SRR constituents. The underlying mechanism of such induced transparency is elucidated well by the suppression of induced currents within the SRRw element. Finally, the excitation of ACR is further associated with excellent sensitivity and narrow bandwidth, leading a remarkable optical sensing technique of freeing from label agents and optical couplers but possessing great values of FOM, to benefit practical applications of chemical and biological detection.
Acknowledgments
The authors gratefully acknowledge the financial support from National Science Council (NSC 95-2112-M-007 048 MY3) and the National Nano Device Laboratories (NDL-94S-C142) for this study.
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