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Abstract
Achieving an ultra-narrow bandwidth analogue of electromagnetically induced transparency (EIT) in bright–bright mode coupling metasurface requires a large contrast of the Q factor and small wavelength detuning between the two coupled modes. Here, by coupling a toroidal dipole (TD) high-Q Fano resonance and a low-Q magnetic dipole (MD) mode, we numerically demonstrated a high Q factor analogue of EIT on an all-silicon metasurface in the terahertz regime. The Q factor of Fano resonance and consequent EIT can be easily adjusted by the spacing between the air holes. By adjusting the radii of the air holes, the thickness of the silicon wafer, or the lattice constant of the metasurface, EIT-like response exhibiting a very high group refractive index and a large group delay was achieved. The proposed EIT metasurface is easy to fabricate and has potential applications in the fields of narrowband filtering and slow-light based devices.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Electromagnetically induced transparency (EIT) is a phenomenon in which the quantum interference effect occurs in the atomic three energy level system, which produces a sharp transparent window in a wide absorption spectrum [1]. Later, a phenomenon similar to the EIT was found in metamaterial [2–5] as well. The EIT-like response in metamaterial is generally realized by the coherent coupling between a bright mode and a dark mode [6–12] or between two bright modes [13–20]. The strong dispersion characteristic of the EIT can be used in slow-light equipment [21–23]. The main index of the slow-light effect is the group refractive index or group velocity, which is closely related to the bandwidth or Q factor of the EIT transparency window; the higher the Q value of the EIT, the larger is the group refractive index that can be obtained.
To achieve high-Q EIT response in metasurfaces via bright-bright mode coupling, two resonances participating in coherent coupling are usually required to have large Q factor contrast (a high- and a low-Q resonance) and small wavelength detuning [24]. In recent years, high-Q resonances have been achieved by breaking the structural symmetry of all-dielectric metasurfaces [25–29], and high-Q Fano resonance related to BIC (Bound States in the Continuum) [30–38] gives more options. For example, periodically arranged dimer structure metasurfaces have been proposed, such as high refractive index nanodisks, perforated silicons, not only high-Q Fano resonances are demonstrated in the optical band, but also the Q value of the resonance is easily adjusted in a wide range by changing the distance or size of the dimers [26,27,35]. And Liu et al. demonstrated experimentally an ultra-high Q Fano resonance in a BIC metasurface [39].
On the hand, progress has been made in realizing high-Q EIT by coupling of a high- and a low-Q resonance in bilayer metasurfaces [17,40]. By use of the advantages of designing the two resonances in individual layer, the optical coupling of the two resonances is easily manipulated by changing the distance between the two layers, but the group refractive index in this case is not high due to the increased thickness of the layer, and the fabrication of the bilayer structure is complicated as well. However, achieving high-Q EIT in single layer metasurfaces remains challenging. Recently, D. R. Abujetas et al. proposed and demonstrated a high-Q EIT by coupling of a quasi-BIC and a low-Q magnetic dipole resonance from a disk array metasurface in microwave regime [41], nevertheless, it is difficult to be extended to terahertz or optical band because of the very high refractive index material used.
In this research, based on bright-bright mode coupling, we realized and numerically investigated an analog of EIT with a high Q factor in an all-silicon perforated dimer structure metasurface in the terahertz regime. The results show that under normal incidence, there are three excited resonance modes in the frequency range of 0.20–0.55 THz, including two high-Q toroidale dipole (TD) Fano resonances and one low-Q magnetic dipole (MD) resonance. By changing the distance of air holes, it is easy to adjust the Q value of the two TD resonances. In particular, by adjusting the size of the air holes, the thickness of the silicon wafer, or the lattice constant of the metasurface, both the detuning and coupling of the high-Q TD resonance and low-Q MD resonance can be easily manipulated, and ultra-narrow EIT with high group refractive index can be achieved. Such kind of EIT metasurface is of simple structure and ease fabrication, and has potential applications in slow-light devices and other photonic devices.
2. Design and resonances of the metasurface
The metasurface we investigated is composed of a periodic array of air holes in a silicon wafer [26,27], as shown in Fig. 1(a). A unit cell is displayed in Fig. 1(b), which shows two air holes with radius R and spacing D. The thickness H of the used low-loss high-resistance silicon wafer is 100 µm, and the lattice constants of the unit cell in the x and y directions are Λx = Λy = Λ = 380 µm. The fact that dielectric constant contrast between the scattering particles and the environment increases without a quartz substrate is beneficial to Mie’s resonance design. COMSOL Multiphysics software was used for the simulations; the refractive index of the silicon wafer was n = 3.45 and n = 1.0 for the air environment. Periodic boundary conditions are applied in both the x and y directions, and perfectly matched layers (PMLs) are used in the wave propagating direction z. A plane terahertz wave polarized along the y-axis is irradiated on the metasurface from normal incidence on the z-axis.
Fig. 1. (a) Schematic diagram of silicon metasurface consisting of air hole array supporting Fano resonances. (b) Unit cell of the metasurface, where R is radius of the air holes, D is distance between the two air holes; the lattice constants of the metasurface are Λx = Λy = 380 µm, and the thickness of silicon wafer H = 100 µm.
When D = 30 µm and R = 70 µm, we calculated the transmission spectrum of the metasurface in the range 0.20–0.55 THz as shown in Fig. 2(a). Three very strong resonances can be clearly seen in the transmission spectrum: two sharp Fano resonances near 0.33 THz and 0.44 THz, named Mode 1 and Mode 2, respectively, each can be regarded as the interference between a high-Q resonance and a much broader Fabry–Pérot (FP) resonance of the perforated silicon metasurface; and a broad resonance at 0.49 THz called Mode 3. The Q values of these three resonances are 802, 452, and 14 calculated by fitting the Fano resonance formula as follows [24]:
Fig. 2. (a): Calculated transmission spectrum of the metasurface when R = 70 µm and D = 30 µm (b–d): Normalized scattering power of different multipole moments at Modes 1, 2, and 3. (e, f): Distributions of electric near-field enhancements in the x–y plane at 0.33 THz and 0.44 THz, respectively, black arrows represent displacement currents. (g): Distributions of magnetic near-field enhancement in the x–y plane at 0.49 THz, black arrows represent magnetic field.
To better understand the microscopic properties of the three resonances, we use the multipole decomposition calculation in Cartesian coordinates [42,43] to obtain the contribution of the multipole scattering power. The normalized scattering power of the five leading multipoles at three resonances is shown in Figs. 2(b)–2(d), where Py, Mx, Ty, Qe, and Qm are the electric dipole ED along the y direction, magnetic dipole MD in the x direction, toroidal dipole TD along the y direction, electric quadrupole, and magnetic quadrupole, respectively. From Figs. 2(b)–2(d), Modes 1, 2, and 3 can be analyzed quantitatively. Ty contributing to Mode 1 accounts for the highest proportion; followed by Qm, which is only one-fifth of Ty. The other multipole components are much smaller; hence, Mode 1 belongs to TD resonance. Mode 2 is also a TD resonance dominated by Ty; but Qm is slightly higher, reaching one-third of Ty. As for Mode 3, we can observe that MD along the x direction is dominant; followed by Qe and Py, which are three-fifth and one-fourth of the MD, respectively; however, TD has little effect on Mode 3.
To perform a deeper analysis of the two different TD resonances, we evaluate the electric near-field distribution at the two resonances in the x–y plane, as shown in Figs. 2(e) and 2(f), respectively. From Fig. 2(e), it can be observed that the electric near-field at resonance of Mode 1 is mainly concentrated in the two holes and between the holes along the y direction. Moreover, clockwise and anticlockwise circular displacement currents are formed between the air holes of the adjacent unit cell in the y direction. Such a typical pair of displacement current loops often leads to the excitation of head-to-tail magnetic moment, i.e., toroidal dipole along the y axis Ty. However, unlike Mode 1, the electric near-field of Mode 2 is concentrated between the two air holes, and a pair of clockwise and counterclockwise displacement current loops are mainly formed inside the two air holes, as shown in Fig. 2(f), which leads to the excitation of head-to-tail magnetic moment, i.e., toroidal dipole along the y axis Ty. Different excitations of TD resonances of Modes 1 and 2 result in different resonance frequencies. For the MD resonance Mode 3, the magnetic near-field distribution in the x–y plane shown in Fig. 2(g) can determine that the MD is along the x direction.
3. High-Q Fano resonances
The spacing D between the air holes has a large influence on Modes 1 and 2. We calculated the transmission spectra of the metasurface at different values of D (30–50 µm) under a fixed air hole radius R = 70 µm (not shown in the figure.), from which the resonance frequency and Q value of the two TD resonances are obtained as shown in Figs. 3(a) and 3(b). It should be pointed out here that the metasurface with spacing D is the same as the metasurface with spacing 100−D; hence, in Figs. 3(a) and 3(b), when D changes from 30 µm to 50 µm or from 70 µm to 50 µm, the results are identical. It can be seen from the figure that when D increases from 30 µm to 50 µm, the resonant frequencies of Modes 1 and 2 (red lines) exhibit very small red shifts of 3 GHz and 4 GHz, respectively, but the Q factor (blue lines) rises quickly. In particular, both Q values of Modes 1 and 2 become infinite when D = 50 µm, i.e., the two resonances disappeared. At this time, the air holes are uniformly distributed along the x axis by period of Λx/2, which is called symmetry metasurface. Thus, slightly breaking the symmetry of the structure by a deviation of value of D from 50 µm or by slightly different hole radius will cause an extremely sharp decrease in the resonant Q factor of Modes 1 and 2 [26,27,35].
Fig. 3. Q value and resonance frequency with respect to spacing D for (a) Mode 1 and (b) Mode 2. Both lossless Si and lossy Si with loss tangent of 0.0001 are considered in calculation, the radius of the air holes is fixed at R = 70 µm.
It should be mentioned here that in the entire range of D, 30–50 µm, the Q value of Mode 3 varies very little, and its resonance frequency is slightly blue-shifted by 1 GHz.
Moreover, if a practical lossy HRFZ-Si is used as silicon wafer and its typical loss tangent of 0.0001 [44] is taken into account, we recalculated the Q factor with respect to D for Mode 1 and 2 and show also in Fig. 3. High Q value over 2 × 104 can still be reached for Mode 1, and around 1.0×104 for Mode 2. In addition, the refractive index of typical HRFZ-Si is 3.416, a little bit smaller than that we used (3.45), this will result in very small resonance shift (around 3–5 GHz) for the three modes.
4. EIT-like response
Similar to the spacing D, the radius of the air holes R also has a large influence on the resonances of the metasurface. We calculated the transmissions of the metasurface for four different radii, R = 60 µm, 70 µm, 80 µm, and 88 µm, as shown in Fig. 4(a). The corresponding spacing are D = 60 µm, 40 µm, 20 µm, and 4 µm. Contrary to the spacing D, the radius R has a large influence on the resonance frequency of the three modes. It can be seen from the figure that as R increases, the resonances of Modes 1–3, all have a relatively large blue shift. It is interesting to note that the resonance shift of Mode 2 is significantly larger than that of Mode 3 or Mode 1, this can be simply explained by the excitation of the Mode 2 in Fig. 2 which originated from the two holes, its resonance frequency is very sensitive to the diameter of the air holes. Thus, as R increases, the high-Q resonance of Mode 2 gradually approaches the low-Q resonance of Mode 3; the different optical coupling between them can be occurred and analyzed as follows. When R = 60 µm or 70 µm, the two resonances of Modes 2 and 3 are relatively far away, and the coupling between them is negligible. As R increases to 80 µm, although the detuning of the two resonances is still relatively large, strong coupling between them exists, resulting in obvious Fano resonance. When R is further increased to 88 µm, the detuning of the two resonances was very small, and the destructive interference of the two resonances occurred (bright–bright mode coupling), leading to a high-Q EIT-like resonance at ∼ 0.5 THz [17].
Fig. 4. (a) Transmission spectra of metasurface with different R. (b) Transmission transparency windows for different D when R = 88 µm. In all cases, H = 100 µm.
The Q value of this type of EIT resonance mainly depends on the high-Q bright mode of the two coupling modes, i.e., Mode 2. It is easy to adjust the Q factor of Mode 2 by changing D, thereby changing the Q factor of the EIT response. Figure 4(b) shows the EIT transmission spectra obtained for three different D = 4 µm, 8 µm, and 12 µm when R = 88 µm, here, D = 14 µm corresponds to symmetry metasurface. The corresponding Q values of the EIT resonances are 128, 386, and 3150, respectively. Furthermore, because the change in D will affect the resonance frequency of Mode 2 to a certain extent, the detuning of the two coupling modes is also observed to be altered. It can be seen from the figure that when D = 12 µm, the detuning of Modes 2 and 3 is the smallest, leading to the best symmetry of the EIT transparent window. And the detuning of the two modes can be further reduced by slightly adjusting R when D = 4 µm and 8 µm.
Another method to realize EIT resonance is by setting an appropriate thickness for the silicon wafer. We calculated the transmission spectra for the metasurface with different silicon thicknesses H = 100 µm, 110 µm, 120 µm, and 132 µm, when R = 70 µm and D = 30 µm. As shown in Fig. 5(a), as the thickness of the silicon wafer increases, the resonances of Modes 2 and 3 exhibit a relatively large red shift due to the increase in the effective refractive index [45]. Furthermore, the MD resonance is more sensitive to the thickness of the silicon wafer, i.e., the resonance shift of Mode 3 is much larger than that of Mode 2. Similar to the realization of the EIT response by changing R, different coupling between a high-Q Mode 2 and a low-Q Mode 3 can be observed when H increases from 100 µm to 128 µm. A high Q EIT resonance at 0.42 THz is achieved when the detuning of the two coupling modes is very small at H = 128 µm. Further, we slightly adjusted H to equal 132 µm and calculated the EIT transmissions for the metasurface with different values of D = 30 µm, 35 µm, and 45 µm. As shown in Fig. 5(b), the corresponding Q values of the EIT resonances are 515, 1009, and 9827, respectively. Recall that D = 50 µm corresponds to symmetry structure when R = 70 µm, this verifies again that a smaller asymmetry of the structure results in a higher Q of the TD resonance and thereafter higher Q of the EIT resonance.
Fig. 5. (a) Transmission spectra for metasurface with different silicon thicknesses H = 100 µm, 110 µm, 120 µm, and 132 µm. (b) EIT transparency windows for different D when R = 70 µm and H = 132 µm.
Interestingly, EIT-like response can also be realized by adjusting lattice constant Λ. Figure 6(a) displays the transmission spectra for the metasurface with different Λ = 380 µm, 350 µm, 320 µm, and 300 µm, when R = 60 µm and D = 20 µm. A high-Q EIT resonance at 0.54 THz is achieved when the detuning of the two coupling modes (Mode 2 and Mode 3) is very small at Λ = 300 µm. Figure 6(b) shows EIT transparency windows for different values of D = 20 µm, 24 µm, and 28 µm when Λ = 300 µm, the corresponding Q values of the EIT resonances are 590, 2538, and 16414, respectively.
Fig. 6. (a) Transmission spectra for metasurface with different lattice constants Λ = 380 µm, 350 µm, 320 µm, and 300 µm. (b) EIT transparency windows for different D when R = 60 µm and Λ = 300 µm. In all cases, H = 100 µm.
When EIT phenomenon occurs, the phase near the transparent window changes sharply. Accompanied by the severe dispersion phenomenon, the refractive index changes significantly with the change in frequency, resulting in a significant reduction in the propagation speed of electromagnetic waves in this medium, which gives rise to the slow-light effect. Group delay τg or group refractive index ng is an important indicator of the EIT-like response; it can be calculated according to the following formula [46]:
where φ represents the transmission phase, ω is the angular frequency, c is the speed of light in vacuum, ${v_g}$ is the group velocity of electromagnetic waves in the metasurface, and H is the thickness of the silicon wafer.The group refractive indices and group delays corresponding to the EIT window shown in Fig. 4(b) were calculated and are shown in Fig. 7(a). When D increases from 4 µm to 10 µm, the Q factor of the EIT resonance increases slowly; hence, the group refractive index and group delay also increase slowly. When D further increases, the Q factor of EIT increases sharply; the group refractive index and group delay increase accordingly and reach up to 5948 and 1982 ps, respectively, when D = 12 µm. The corresponding waveform of the group delay is shown in Fig. 7(b). Similarly, the group refractive indices and group delays corresponding to the EIT window shown in Fig. 5(b) were also calculated and are shown in Fig. 7(c). The group refractive index reaches up to 17220, and the group delay is over 7500 ps (waveform shown in Fig. 7(d)) when D = 45 µm. Such a large group delay is much longer than the maximum group delay (46 ps) obtained by plasmonic metasurfaces [47–51].
Fig. 7. (a), (c): Group refractive indices and group delays calculated from the EIT resonances shown in Fig. 4(b) and Fig. 5(b), respectively. (b), (d): group delay waveforms when D = 12 µm in (a) and D = 45 µm in (c).
5. Conclusion
Based on bright–bright mode coupling, we numerically demonstrated the analogue of EIT with a high Q factor in an all silicon metasurface in the terahertz regime. Two high-Q TD resonances located at 0.33 THz and 0.44 THz were excited by breaking the symmetry of the structure, whose Q values can be easily adjusted by the spacing between the air holes. By adjusting the radius of the air holes or the thickness of the silicon wafer, we achieved an EIT response exhibiting a very high group refractive index and large group delay by coupling a high-Q TD resonance and a low-Q MD resonance. Such kind of all-silicon metasurface can be fabricated by photolithography, deep reactive-ion etching, and thinning process on a commercially available $\ge $ 100µm thickness HRFZ-Si wafer [33,52,53]. The complexity of fabrication is greatly reduced without a bonding process, in addition, the influence of material loss is reduced without quartz or other substrate, which is beneficial for achieving high-Q resonance of the metasurface. The proposed EIT metasurface is easy to design and fabricate, and can be used in narrowband filtering and slow-light based devices.
Funding
National Natural Science Foundation of China (61875179, 61875251); Science Research Foundation of Zhejiang Province (LGG19F050004); Primary Research and Development Plan of Zhejiang Province (2019C03114).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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