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Abstract
An analog of electromagnetically induced transparency (EIT) in an asymmetric E-shaped all-dielectric metasurface was proposed and numerically demonstrated in the near infrared spectral region. The E-shaped metasurface supports a strong toroidal dipolar resonance with high quality (Q) factor, which is verified by decomposed scattered powers for multipole moments using a Cartesian coordinate system. A high transmission EIT-like optical response was achieved, and clearly interpreted by the destructive interference between the dark toroidal dipolar moment and bright magnetic dipolar mode through the asymmetric metasurface. The bandwidth of the transparency window can be easily designed by changing the asymmetric parameter of the structure. The proposed E-shaped all-dielectric metasurface gives a new way to realize toroidal dipolar response and has potential applications in bio-chemical sensing, narrowband filters, optical modulations, and slow light based devices.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Electromagnetically induced transparency (EIT) is a coherent process originally occurred in three energy atomic levels due to the quantum destructive interference between two different excitation pathways [1–4], the EIT phenomenon drastically alters light-matter interactions such that an optically opaque medium becomes transparent. Because of the highly dispersive nature and transparent features, the EIT phenomenon provided a potential solution to realize slow-light devices, optical-sensors and enhanced nonlinearity. In recent years, the EIT-like response in plasmonic metamaterials has attracted much attention [5–14], but the main limitations of the plasmonic structures to achieve high transmission, high Q factor, or large group index, especially in optical wavelengths [5,8,10] are that they unavoidably suffer from ohmic loss of the metal and radiative loss of surface modes of plasmonic metasurfaces. All-dielectric resonators, however, possess the advantages of low radiative losses due to large resonance mode volume and low absorption loss because of the eliminated use of metallic material [15–19], which implies that strong and high Q resonance is possible. Owing to effective light confinement, dielectrics with high permittivity such as germanium and silicon can generate magnetic dipoles, electric dipoles and higher order dipoles due to Mie resonances with low absorption loss [16,18]. Consequently, it becomes a more advisable choice to replace the plasmonic metasurface by all-dielectric one in some way to increase the Q factor of the EIT or Fano resonance [20–25]. Realization of the all-dielectric EIT-like response in metamaterials or metasurfaces is commonly achieved by bright-dark mode coupling [21,23], where magnetic or electric dipolar resonances are used as “bright” and “dark” mode. Toroidal dipolar resonance in metamaterials was firstly proposed and experimentally observed in 2010 [26,27]. Since then, much attention has been paid to explore toroidal resonant responses in both plasmonic [28–37] and dielectric metamaterials where dielectric disks, spheres, wires, or circularly arranged rods are usually used as metamolecules [38–43]. Since the toroidal and electric dipoles have identical scattering patterns in the far field, it offers a possibility to the cancellation between them, leading to the excitation of so called nonradiating anapole modes [35–37,40–43]. The excitation of anapole modes is significant for many applications such as high efficiency radiations [44,45] and high Q factor devices [35,36,41] due to the nonradiating nature and the strong energy confinement. Moreover, low-loss EIT-like responses have been achieved in plasmonic metasurfaces based on destructive coupling between toroidal resonances and electric or magnetic resonances [46,47], where the toroidal response serves as the dark mode. However, such toroidal dipole based EIT-like response has not been reported in all-dielectric metasurfaces.
In this work, different from the reported plasmonic toroidal metamaterials consisting of double E structured metamolecules [32,36], we proposed and numerically demonstrated the excitation of toroidal dipolar response with high Q factor in one E-shaped all dielectric metasurface. Furthermore, EIT-like response was achieved by destructive interference between this toroidal dipole moment and magnetic resonance through asymmetric E structure. By changing the asymmetrical parameter of the metasurface, the bandwidth of the transparency window can be easily designed.
2. Structure of all-dielectric metasurface
The schematic of the designed E-shaped all-dielectric metasurface is shown in Fig. 1(a), where an E-shaped silicon array (refractive index n = 3.7) with thickness h = 200 nm is deposited on a quartz substrate (n = 1.48). Figure 1(b) displays one unit cell of the metasurface which consist of three parallel and one vertical connected silicon rods, the geometrical parameters are as follows: l1 = 510 nm, l2 = 730 nm, w = 120 nm, Δg is defined as Δg = g2-g1 and Δg ≠ 0 means the E structure is asymmetric. The unit cells are periodically arranged along the x and y directions with the same lattice constants px = py = 900 nm. Numerical simulations of the metasurface are carried out by using the commercial CST software. In simulations, periodic boundary conditions are applied in both x and y directions, and perfectly matched layers (PMLs) are used in the wave propagating direction z, and the metasurface is illuminated by a normally incident plane wave polarized along the x or y axis. The transmission which we showed below is calculated by the co-polarized transmission Tii (i = x or y).
3. Toroidal resonance with high Q factor in symmetric metasurface
When the electric field of the incident wave is along the x-axis (the inset, Fig. 2(a)), the transmission spectrum of a symmetric E-shaped structure (Δg = 0, l2 = 750 nm) in the wavelength range of 1450 ~1620 nm is given in Fig. 2(a). It is interestingly found that a very sharp Fano–like resonant response is observed at wavelength of 1540 nm with a Q factor of ~6900. Here, Q value of the resonance is calculated by fitting the spectrum with a Fano formula [48,49],
where F is the Fano parameter, γ and ω0 represent the width and position of the resonance. Then the Q value is calculated by ω0 /γ. In this case, the fitting parameter are F = 1, ω0 = 2π × 194.78 THz, γ = 2π × 0.028 THz, respectively. The corresponding distributions of electric and magnetic field in the x-y plane and y-z plane at the wavelength of 1540 nm are illustrated in Fig. 2(b) and 2(c), respectively. We notice that there are opposite circular displacement currents around the upside and downside of the E structure simultaneously in Fig. 2(b), each of them could produce a magnetic field, the corresponding z-component magnetic field of upside is along -z axis, and + z axis for downside, these induced magnetic field patterns generate a circular magnetic moment (see Fig. 2(c)). This head-to-tail magnetic moment M formed in the E structure indicates the excitation of toroidal dipole moment T along the x-direction [26,27], which is more clearly seen in Fig. 2(d).
Fig. 2 (a) Transmission spectra of symmetric E-shaped metasurface (l2 = 750 nm) when electric field of the incidence is along the x-axis shown in the inset. (b),(c) Distributions of the electric and magnetic field in the x-y plane and y-z plane at λ = 1540 nm, respectively. (d) Schematic of T (toroidal dipole) excitation in the unit cell, M represents the head-to-tail magnetic moment.
To further access the role of the toroidal excitation in the observed response, decomposed scattered powers for multipole moments are calculated based on density of the induced current inside of the metamolecules by using a Cartesian coordinate system [38,50,51]. Figure 3 plots the five strongest scattering powers of multipoles around resonance wavelength of 1540 nm, which includes the electric dipole P, magnetic dipole M, toroidal dipole T, electric quadrupole Qe, and magnetic quadrupole Qm, higher order dipoles can be ignored due to the extremely weak influence on the scattered intensity. Above and below the resonant wavelength of ~1540 nm, the electric dipole dominates as expected, this means that the metasurface is electrically excited and hard to couple with a magnetic component of the incoming wave, leading to very weak magnetic dipole. However, in the vicinity of the resonant wavelength at 1540 nm, the strength of electric dipole decreases dramatically. At same time, the toroidal dipole increases remarkably and dominates other multipoles in the far-field scattering power, which confirms the resonant mode as toroidal dipole. Specifically, at the resonance, the toroidal dipole T is about 8 times stronger than the electric quadrupole Qe and magnetic quadrupole power Qm, around 4 orders more of magnitude stronger than the electric dipole P and ~730 times stronger than the magnetic dipole M. Note also that in spite of the remarkable increase of the electric and magnetic quadrupole Qe and Qm at the resonance, they almost overlap and cancel each other, which in combination with the dramatic decrease of P, explains the sharp Fano-like resonance in the transmission [25,30,52]. Since the power radiated by the toroidal dipole is much larger than the power of the electric dipole at 1540 nm, destructive interference between the electric dipole and toroidal dipole does not occur, this means that anapole mode is not excited in the metasurface, which is different from previous works in which extremely high Q resonance was obtained by the excitation of anapole modes [35,36,41].
Fig. 3 The full multipole decomposition of the first contributing five multipole moments of the E-shaped metasurface: electric dipole (P), magnetic dipole (M), toroidal dipole (T), electric quadrupole (Qe) and magnetic quadrupoles (Qm). The log scale in the y axis is chosen so as to reveal more clearly the contribution of the multipole terms as well.
In order to get a further insight into the resonance properties, we studied the resonance wavelength and Q factor of the toroidal dipole for various geometric parameters. The Q value and resonant wavelength as a function of the l2, w for E-shaped metasurface are shown in Fig. 4(a) and Fig. 4(b), respectively. In Fig. 4(a), it is revealed that the resonant wavelength of toroidal dipole exhibits a monotonic red-shift when the rod's length l2 increases from 680 nm to 780 nm. However, the corresponding Q factor has a peak value when l2 = 730 nm, reaching up more than ten thousands at resonant wavelength of about 1525 nm. Moreover, the results of Cartesian multipole expansion at different rod's length l2 indicate again that the toroidal dipole still dominants the contributions of the scattering power in the vicinity of the resonant wavelength. The wavelength and Q factor dependences of toroidal response on the parameter w shown in Fig. 4(b) are quite similar as that on l2. In addition, the wavelength of the toroidal resonance also shows red-shifts when the rod's length l1 or thickness h increases, but the Q value of the resonance will not change (not shown in the figure). We should indicate here that high Q toroidal dipolar response is sensitive to the loss of the metasurface. Although the material loss of the quartz and silicon in our considered wavelength are very small, fabrication processes will introduce some loss. If an equivalent loss tangent value of 0.001 is used for the silicon in simulation [53], the Q value of the toroidal dipolar response shown in Fig. 2(a) will decrease remarkably to 1200.
Fig. 4 Q factor and wavelength dependences of toroidal resonance on the rod’s length l2 (w = 120 nm) (a) and width w (l2 = 730 nm) (b).
4. Analog of EIT in asymmetric metasurface
When the incident wave is polarized along the y-axis instead of x-axis, the transmission spectrum of the symmetric E-shaped metasurface (Δg = 0, l2 = 730 nm) is calculated and shown in Fig. 5(a). It is clear to see a broad resonance dip at wavelength of 1520 nm with a low Q factor of ~15. Figure 5(b) and 5(c) show the distributions of electric and magnetic field at 1520 nm in the x-y plane and y-z plane, respectively. The displacement current forms loops in the x-y plane, while the magnetic field is polarized mainly along the z axis in the center of the structure, which corresponds to magnetic dipole resonance. At same time, circular magnetic field can be observed inside upper and lower rods of the E-shaped structure, showing electric response also contributes to this broad resonance. From the results of multipole expansion (Fig. 5(d)), the magnetic dipole moment dominants the contributions of the scattering power in the full simulated range. The electric dipole also contributes to the scattering power, but 5 times weaker than the magnetic dipole in the vicinity of the resonance. However, the toroidal dipole, in this case, is so weak that it could be ignored. It is worth to mention that the broad magnetic resonance in transmission will disappear in closed E-shaped (or 8-like) metasurface, thus the following EIT-like response cannot be realized.
Fig. 5 (a) Transmission spectra of symmetric E-shaped metasurface (Δg = 0, l2 = 730 nm) when the incident wave is polarized along the y-axis shown in the inset. (b), (c) Distributions of the electric and magnetic field in the x-y plane and y-z plane at λ = 1520 nm, respectively. (d) Multipole expansion results of the scattered power in the Cartesian coordinate system for broad magnetic resonance.
However, once an asymmetric E-shaped metasurface (ex. Δg = 20 nm) is used and illuminated by the same y-polarized incident wave, a fairly sharp EIT-like window with high transmission about 98% at 1525 nm is observed (shown in Fig. 6(a)). The full width at half maximum of the transparency window is 1.31 nm, and the corresponding Q value is 1160. We notice that the narrow transparency window is just located around the dip of the broad magnetic dipole resonance shown in Fig. 5(a), in addition, it is fully coincidence with the sharp toroidal resonance shown in Fig. 4(a) at l2 = 730 nm when the metasurface is illuminated by the x-polarized incident wave. Therefore, the EIT-like transmission can be clearly interpreted by destructive interference between bright and dark modes, in which the broad magnetic resonance serves as the bright mode, and the sharp toroidal resonance serves as the dark mode. And the asymmetric structure provides a pathway for exciting the dark toroidal mode [54,55]. The small wavelength detuning and large difference of Q factor between the bright mode and dark mode are necessary to achieve this sharp EIT-like response [49].
Fig. 6 (a) Transmission of asymmetric E-shaped metasurface (Δg = 20 nm) when the electric field of the normal incidence is along the y-axis shown in the inset. (b)~(d): Electric field distributions in the x-y plane (e)~(g): Magnetic field distributions in the x-z plane bisecting the metasurface. (b),(e): dip I (λ = 1514 nm); (c),(f): peak II (λ = 1525 nm); and (d),(g): dip III (λ = 1528 nm), respectively.
In order to further verify the excitation of the dark toroidal mode in the EIT-like effect, we analyze the electric field distributions in the x-y plane (Fig. 6(b)-6(d)) and the magnetic field distributions in the x-z plane (Fig. 6(e)-(g)) at transmission dip I, peak II and dip III (marked in Fig. 6(a)), respectively. As shown in Fig. 6(b), 6(d), a typical circular displacement current in the x-y plane is induced around the upside of the structure at dip I, but downside at dip III, whereas the corresponding magnetic field in the x-z plane (see Fig. 6(e), 6(g)) shows localization in the center of the upside and downside of the structure. These field patterns prove the excitation of the magnetic resonances [15], which are caused from the hybridization between the bright magnetic mode and dark toroidal mode, producing the bonding and anti-bonding modes corresponding to dip I and dip III, respectively. For the transmission peak II, however, the pattern of displacement currents (Fig. 6(c)) and magnetic field (Fig. 6(f)) shows a typical toroidal dipole response, which further proves the excitation of the dark toroidal mode.
Furthermore, the impact of asymmetric E structure on EIT-like property is analyzed. Figure 7(a) shows the calculated transmissions of the metasurface in the EIT window at different Δg. The EIT window becomes narrower with decreasing Δg due to the decrease of the coupling coefficient between the bright and dark modes [21]. The corresponding Q factor of the EIT-like resonance is also calculated and shown in Fig. 7(b). As Δg decreases from 100 nm to 10 nm, the Q value of the EIT resonance increases from 60 to 4480 (56 to 940 when loss tangent value of 0.001 is considered) and the transmission remains high at the same time. We should mention here that the high Q value of the EIT-like resonance is not only determined by the asymmetric parameter Δg of the metasurface, but also limited by the Q value of the dark toroidal resonance. In addition, transmissions of the metasurface are also calculated at different l1 and shown in Fig. 7(c). As the rod length l1 increases from 490 nm to 530 nm, the EIT-like responses can be observed at different wavelengths, this means that the resonant wavelength of the bright mode almost keeps pace with that of the dark mode as the length l1 varies.
Fig. 7 (a) Transmissions of asymmetric metasurface with various Δg. (b) Q value dependence of the EIT-like window on Δg. (c) Transmissions of asymmetric metasurfaces (Δg = 20 nm) at different parameter of l1.
The sharp profile of the EIT window is potential to achieve strong dispersion and highly-confined slow light device with large group refractive index. The slow-light metasurface can trap photons for a long time inside the structure, which is useful for enhancing light-matter interactions. The corresponding group index ng of the metasurface in the transparency window can be evaluated by the formula [12,56]:
where ne denotes the real part of the effective refractive index, and ω is the angle frequency of resonance. In order to calculate the group index, we firstly extracted the effective refractive index of the structure from the transmission and reflection data by using the standard S-parameter retrieval algorithm [57], and here, a loss tangent of 0.001 is also considered in calculation. As shown in Fig. 8(a), the real and imaginary part of the refractive index is retrieved for the proposed metasurface when Δg = 100 nm. In the transparent window, the imaginary part of the effective refractive index has a dip (almost zero), meaning that a low-loss metasurface EIT is obtained. Figure 8(b) gives out the calculated group index at transparency peak wavelength of 1525 nm with varying values of Δg. With the decrease of Δg, the group index significantly increases because the EIT window gets sharper.Fig. 8 (a) The retrieved effective refractive index for the proposed structure when Δg = 100 nm. (b) The extracted group index at transparency wavelength of 1525 nm with respect to Δg.
5. Conclusion
In conclusion, we have proposed and numerically demonstrated EIT-like response in an E-shaped all-dielectric metasurface in the near infrared spectral region. Strong and high Q toroidal dipolar response is obtained and verified by both Cartesian multipole expansion and near field electric/magnetic field distribution in the metasurface, and dependences of wavelength and Q value of the toroidal resonance on geometrical parameters of the E structure are given. Furthermore, a sharp EIT-like response with high transmission is achieved by optical coupling between the dark toroidal dipole and a bright magnetic resonance through asymmetric E structure. And its Q value can be designed by changing the asymmetric parameter of the E structure, but limited by the sharp toroidal dipolar response. The proposed E-shaped all-dielectric metasurface gives a new way for realizing EIT-like transparency window based on toroidal dipolar response.
Funding
National Natural Science Foundation of China (NSFC) (Grant Nos. 61377108 and 61405182).
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