We present metamaterial-induced transparency (MIT) phenomena with enhanced magnetic fields in hybrid dielectric metamaterials. Using two hybrid structures of identical-dielectric-constant resonators (IDRs) and distinct-dielectric-constant resonators (DDRs), we demonstrate a larger group index (ng~354), better bandwidth-delay product (BDP~0.9) than metallic-type metamaterials. The keys to enable these properties are to excite either the trapped mode or the suppressed mode resonances, which can be managed by controlling the contrast of dielectric constants between the dielectric resonators in the hybrid metamaterials.
©2012 Optical Society of America
Electromagnetically induced transparency (EIT), a coherent optical nonlinearity caused by quantum interference  in a multi-level atomic system , gives rise to a variety of fascinating phenomena, such as lasing without inversion , freezing light , and dynamic storage of light in a solid state system . So far, the EIT phenomenon has been manipulated successfully by ultracold atomic gases , doped solid-state materials  and “quantum dot”-based systems , but all of them require severe conditions such as low temperatures, specific materials or pumping lasers. To mimic EIT-like sharp spectra but to free from those burdens aforementioned, few researchers have been developing metamaterial-induced transparency (MIT), a classical analog of induced transparency by metamaterials accompanied with sharp dispersion, dramatic enhancement in group indices and nonlinear interactions [9–12]. The MIT effect can be interpreted by either “trapped mode resonance”  or “bright-dark (or call superadiant-subradiant)” eigenmodes coupling resonance , and thus several metallic metamaterials have been employed to realize the MIT effect, such as coupled fish-scale structures , coupled triple U-shape resonators , coupled cut wires , coupled asymmetric arcs  and coupled asymmetric split-ring resonators (SRRs) .
Nevertheless, the present metallic resonators produce an instance of significant conduction loss and strong anisotropic properties that tarnish the coupling performance. In contrast to metallic structures, dielectric resonators (DRs) possess the advantages of low absorption loss, high temperature stability and symmetric design, which guarantee a more promising approach to demonstrate MIT phenomenon in solid-state systems. As a consequence, in this study we introduce two new types of hybrid dielectric metamaterials to realize low-loss and highly symmetric MIT planar structure. One type of the hybrid dielectric metamaterials was composed of a pair of identical-dielectric-constant resonators (IDR) to assemble IDR-type MIT medium, and the other was made of a pair of distinct-dielectric-constant resonators (DDR) to assemble DDR-type MIT media, respectively. With these two types of hybrid dielectric metamaterials, we successfully demonstrate the MIT phenomenon that is associated with the responses of included sharp transparency peaks, large group indices, and strong magnetic response behaviors. Both the measured transmittance and reflectance spectra agree well with simulated results. Furthermore, the two types of metamaterials differed from each other regarding their respective magnetic response behavior within the frequency domain of MIT resonance. Therefore, in the end, by controlling the contrast of dielectric constants between the DRs in the hybrid metamaterials, eventually we discuss the group indices dependence between IDR-type and DDR-type dielectric metamaterials with resonant modes.
2. Design of the dielectric resonators and measurement methods
In this study, we produced three different DRs from commercial low-loss ceramics, including DR-A (the Al2O3 cube with the larger dimensions), DR-B (the Al2O3 cube with the smaller dimensions), and DR-C (the ZrO2 cube with the dimensions), as shown in Fig. 1(a) . Besides, the dielectric constant εr and the loss tangent for the Al2O3 cube correspond to 9.9 and 0.001, and are 33 and 0.004 for the ZrO2 cube, respectively. Then, we fabricated two types of MIT media by hybridizing two unique DRs: one type, known as IDR-type, comprised DR-A and DR-B, as shown in Fig. 1(b); and the other pair, known as the DDR-type, comprised DR-B and DR-C, as shown in Fig. 1(c). To measure the structures’ responses of transmission coefficient (S21), refection coefficient (S11) and group delay duration (τg), we employed a measurement setup containing an Agilent E8364A network analyzer connected with a WR-137 rectangular waveguide (cross section: ), which provided electromagnetic waves within a range extending from 5.85 to 8.20 GHz. Here, the dielectric structures were located in the center of the waveguide tube, with their edges parallel to the E (y-axis) and H (x-axis) fields. Such an experimental setup reflects the scattering results of the one-layer array consisting of an infinite number of dielectric resonators on the xy plane at the excitation of the TE10 mode in accordance to the mirror theory . Furthermore, we also numerically verified the measurements by using a commercial finite integral time domain (CST Microwave Studio).
3. Measured results and simulated analysis
3.1 Resonances in the uniform dielectric resonators
Referring to model of Mie theory and dielectric resonator theory , the resonant frequencies of a dielectric resonant element can be quantitatively evaluated by its geometric parameters and permittivity . Therefore, we carefully designed and fabricated three kinds of dielectric resonators with the desired dimension and permittivity, to work at the specific frequencies. The measured transmittance spectra () of three kinds of dielectric resonators, DR-A, DR-B and DR-C, are presented in Fig. 2(a) as black, red and blue solid lines, respectively. There exist three transmittance dips locating at 6.95, 7.21, and 7.44 GHz (marked as ωI, ωII and ωIII, respectively), agreeing well with the simulated results [see dash lines in Fig. 2(a)]. Although, they have a small deviation (about 0.1 GHz) compared to simulated results. We suggest that such a small deviation is possibly caused by the dispersive dielectric constant and the geometric inaccuracies of the DRs.
These three transmittance dips result from the magnetic dipolar resonances , which is excited by displacement current () . Therefore, the magnetic strength between the Al2O3 cubes and the ZrO2 cubes at resonances can be understood by their permittivities that cause obvious different magnetic responses [16,20]. For the Al2O3 and ZrO2 cubes, the quality factors of the transmittance dips (Qd) are respectively estimated to be 9 and 37, which could be determined from full width at half maximum (FWHM). Notice that between these three transmittance spectra, in Fig. 2(a), there are two frequency intervals respectively contributed from (1) the resonances frequencies of DR-A (ωI) and DR-B (ωII), and (2) the resonances of DR-B (ωII) and DR-C (ωIII). Two bands of the resonance frequencies defined as, and, are much smaller than their corresponding resonant frequency, i.e.,. These narrow bands are necessary to enable the sharp Fano resonance in accordance with the coupling theory [21,22], for instance, in regard to, the coupled DRs will exhibit MIT mode resonance. Thus, the MIT effects from two distinct coupling models will be discussed in next section.
3.2 Metamaterial induced transparency phenomenon by the hybrid dielectric resonators
First, we hybridize the DR-A and DR-B () together to assemble an IDR-type metamaterial [see Fig. 1(b)]. In this case, surprisingly, the original broad transmittance dip turns into a sharp transmittance peak at 6.9 GHz (marked as f2) as shown in Fig. 2(b). The transmittance peak performs an enhanced Q-factor (Qp = 71), yielding high transmittance () and low reflectance () to demonstrate MIT phenomenon. This enhanced transmittance peak is activated through the harmonic coupling resonance inside the elements of IDR-type metamaterial at the frequency (f2). Notably, based on the coupling theory , such an MIT phenomenon cannot be realized by uniform DRs (i.e., the same resonant frequency, ), which is also evidenced in Fig. 2(a).
In addition to the transmittance peak, there are two transmittance dips caused from the hybridized modes where the transmittance () declined to a value less than −18 dB [marked as f1 and f3 in Fig. 2(b)], which stemmed from the magnetic resonances [24,25]. Such magnetic resonances are different from the recent reported magnetic plasmon hybridization modes  in metallic metamaterials, which are excited by the interaction between dark (non-radiative) and bright (radiative) modes. The hybridized dielectric metamaterial are excited by individual DR, i.e., one of DRs resonates at lower energy mode, and another resonates at high energy mode. Therefore, the hybridized modes in the dielectric MIT metamaterial are regarded as a superposition from the individual DR’s magnetic resonant nature.
Next, we hybrid the DR-B and DR-C ( GHz) to fabricate a DDR-type metamaterial [see Fig. 1(c)], whose corresponding measured and simulated transmittance spectra are presented in Fig. 2(c). There are two transmittance dips () at f’1 (lower energy mode) and f’3 (higher energy mode) which corresponds to the nature of magnetic resonances in DR-B and DR-C, respectively. In addition, a sharp transmittance peak within two hybridized modes is activated through the coupling in the distinct dielectric resonators. The transmittance peak appeared between two hybridized modes exceeds −5 dB at resonance frequency (), meanwhile, the Qp of the transmission peak of the DDR-type metamaterial is estimated to be 59, which was lower than that of IDR-type metamaterial (Qp,IDR~71). Besides, both types of hybrid dielectric metamaterials exhibit larger Qp (71 and 59) beyond the highest record achieved by the asymmetric metallic metamaterials (max Qp~20) . Therefore, hybrid dielectric metamaterials provide sharper spectra than those by metallic metamaterials since less intrinsic loss, and can be readily employed as excellent zeroth-order refractive index sensors .
3.3 Strong magnetic field at the coherent coupling frequency
Recent reports about MIT were only explained its mechanism by coupled electric-dipoles response in the metallic metamaterials. Rather here we further demonstrated pronounced coupled magnetic-dipole modes in the hybrid dielectric metamateials. To clarify different resonant modes in the paired uniform DRs, IDR-type and DDR-type metamaterials, we calculate the x component of magnetic field (Hx) and the circulating displacement currents (jd’s) at their resonant frequencies accordingly. First, we examine the magnetic response in the paired uniform DRs [here we present the paired DR-B’s as a representative as shown in Fig. 3 (a) ]. At the frequency of the transmittance dip region (shown as in Fig. 2 (a)), the paired DRs co-oscillate in an out-of-phase fashion with respect to the excitation field, leading to negative permeability and thus forbidding the propagation of electromagnetic waves .
For the IDR-type metamaterial, in contrast, the coupling effect between both DRs causes enhanced anti-phase dipolar responses [24,25] by two intense displacement currents along opposite directions, as shown in Fig. 3(b). As the MIT phenomenon occurred, the magnitude of Hx in the IDR-type metamaterial is more than one order greater than that in the paired uniform DRs. Notably, such enhanced anti-phase magnetic dipoles (or jd’s) in both DRs present similar strength, which lead the magnetic fields to interfere destructively between DRs. Also, the collective electromagnetic response of anti-phase magnetic dipoles is weakly coupled to free space, leading to a higher Qp and lower radiation loss, yielding to a magnetic trapped mode resonance . Next, for the DDR-type metamaterial, as shown in Fig. 3(c), there appear similar anti-phase magnetic dipoles, but they own very different magnetic fields strength at the coherent coupling state (f’2). In this case, the anti-phase magnetic dipoles excited by two unequal jd’s are partially cancelled out, leading to a residue magnetic dipole.
The resonance mode of the DDR-type metamaterial is functioned as a suppressed mode resonance , possessing higher loss, lower transmittance and lower Qp compared to those in the IDR-type metamaterial (i.e., trapped mode resonance). Besides, at the MPR frequencies (f1, f3, and f’1, f’3), both IDR-type and DDR-type systems demonstrate magnetic dipolar mode in one element of paired DRs, presented by magnetic field distribution as shown in Fig. 3(d) and 3(e). For instance, Fig. 3(d) plots that the low-energy mode (f1) is treated the magnetic dipolar mode in the DR-A only, and the high-energy mode (f3), rather, demonstrates the similar resonance mode in the DR-B only. The DR is excited as the individual uncoupled DR, and owing weaker magnetic response than that in the coherent frequency (f2, or f’2).
The magnetic field enhancement at the coupling resonance of hybrid dielectric metamaterial is about one order greater than conventional uncoupled dielectric metamaterials. Such magnetic field enhancement currently appear an attractive subject , due to its potential applications in magnetic nonlinearity, magnetic probes  and others . Especially at optical frequencies, the magnetic response is typically so weak that this method of enhancing the magnetic field becomes quite valuable.
4. Large group index caused by the strong magnetic response due to the coherent resonance between two dielectric resonators
So far we manifested that the resonance is a key to excite the MIT phenomenon and strong magnetic response for the hybrid dielectric metamaterials and thus, we anticipate a dramatic dispersion and a corresponding large group index as resonance occurred, according to the relationship. In this part, we measure the group delay, and then calculate the group index of dispersion. As shown in Figs. 4(a) and 4(b), the maximum group delays approximately located at the coherent resonances (f2 and f’2) that correspond to 8 ns and 13 ns for the IDR-mode and DDR-mode metamaterials, respectively, which are more than one order of magnitude greater than those of the single Al2O3 or ZrO2 cubes. Based on the measured group delays, we further calculate the corresponding group indices (ng) by the definition,, where represents to the duration of the group delay and leff represents to the thickness of the metamaterial along the direction of the wave propagation. Accordingly, the calculated results show the group indices of the hybrid dielectric metamaterials are 218 for the IDR-mode and 354 for the DDR-mode resonance, respectively. Again, both values of group index are one order larger than the single Al2O3 and ZrO2 cubes. Such large group indices can benefit the efficiency of nonlinear process since the nonlinear phase sensitivity enhancement is proportional to group index square .
Since enhanced group indices were demonstrated by hybrid dielectric metamaterials, next, we therefore seek to engineer the group indices by various coupling models. The various hybrid dielectric metamaterials are functioned by adjusting different dielectric cubes (defined as DR-X) hybridized to DR-B. First, we define a parameter γ as, the contrast of dielectric constants between the constituent DR-X and DR-B in the hybrid metamaterials. As shown in Fig. 5 , the group indices are growing with the increasing γ, and finally achieves to a great group index of 532. Such simulated results agree that the larger degree of suppressed mode resonance inherent in the DDR-type metamaterial owns larger group indices than the IDR-metamaterial’s trapped mode resonance. We suggest that the large group indices (ng) are caused by the strong magnetic response, rested on the group indices-energy density relation : , in which the group indices (ng) relay on the magnetic energy density (σ) and the intensity of incident magnetic field (). Since the incident fields are constant in this study, the energy inherent in the amplified magnetic response system is quite intense at the coherent resonance states, as shown in Fig. 3(b) and 3(c), leading great group indices. Finally, we adopt bandwidth-delay product (BDP), an index of data storage ability to identify the slow-light effect as MIT occurs . Here, we compare the BDP values of our hybrid dielectric metamaterials with the coupled metallic metamaterials that were recently presented to allow MIT. As shown in Table 1 , it is clearly that our hybrid dielectric metamaterials should perform better slow-light effect than the reported coupled metallic metamaterials, promising practical applications of optical buffers, controllable delay lines and others .
In summary, by fabricating two types of hybrid dielectric metamaterials, identical-dielectric-constant resonators (IDR) and distinct-dielectric-constant resonators (DDR), we successfully demonstrate metamaterial-induced transparency phenomena associated with enhanced magnetic response. We observed that both types of dielectric metamaterials possess additional transmittance peaks with great Q-factor (Qp~71), large group indices (ng~354) and BDP parameter. By further comparing the group indices in these two hybrid systems, we conclude that the suppressed-mode resonance in the DDR-type metamaterial displays stronger magnetic responses, higher group indices than the trapped-mode resonance in the IDR-type metamaterial. Owing to their low-loss nature, dielectric metamaterials present better performance beyond conventional plasmonic metamaterials. Finally, the demonstrated performance in this work can be further enhanced by employing other dielectrics such as titanium dioxide, silicon carbide  and barium strontium titanate , and can be readily applied in various optical applications .
The authors would like to gratefully acknowledge the financial support from the National Science Council (NSC98-2112-M-007-002-MY3, NSC100-2120-M-010-001, and NSC100-2120-M-002-008), and from the Ministry of Education (“Aim for the Top University Plan” for National Tsing Hua University).
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