High-<i>Q</i> resonances in terahertz all-silicon metasurface with imperforated air-hole array

Qing Wang; Pengfei Wang; Jianjun Liu; Fangzhou Shu; Guiming Pan; Zhongwei Jin; Xufeng Jing; Zhi Hong

doi:doi:10.3788/COL202321.113601

Chinese Optics Letters, 2023, 21 (11): 113601, Published Online: Nov. 6, 2023

High-Q resonances in terahertz all-silicon metasurface with imperforated air-hole array

论文大纲

Qing Wang Pengfei Wang Jianjun Liu Fangzhou Shu ^*Guiming Pan Zhongwei Jin Xufeng Jing Zhi Hong ^**

Author Affiliations

Centre for THz Research, China Jiliang University, Hangzhou 310018, China

bound state in the continuum all-silicon metasurface high-quality factor terahertz

Abstract

We propose and experimentally demonstrate a high quality (Q)-factor all-silicon bound state in the continuum (BIC) metasurface with an imperforated air-hole array. The metasurface supports two polarization-insensitive BICs originated from guided mode resonances (GMRs) in the frequency range of 0.4 to 0.6 THz, and the measured Q-factors of the two GMRs are as high as 334 and 152, respectively. In addition, the influence of the thickness of the silicon substrate on the two resonances is analyzed in detail. The proposed all-silicon THz metasurface has great potential in the design and application of high-Q metasurfaces.

1. Introduction

High quality ( $Q$ )-factor resonance is pursued in photonics to increase light–matter interactions, which is important for fundamental and applied research. In recent years, bound state in the continuum (BIC) has attracted extensive attention due to the infinite $Q$ -factor in theory. It is a physical phenomenon in which the energy is perfectly bound to the unlimited size of the structure without any radiation^[1–4]. However, due to the finite size of the structure, the absorption of the material, and other external perturbations, BIC tends to collapse to a Fano resonance with a limited radiative $Q$ -factor. Such a resonance is known as quasi-BIC^[5,6]. A metasurface provides an excellent platform to explore the BICs. Thus far, the kinds of BIC have been proposed based on multiple mechanisms, including symmetry-protected BIC^[7–10], Friedrich–Wintgen BIC^[11–14], resonance-trapped BIC^[7,15], guided mode resonances (GMRs), and surface lattice resonance-related BIC in dimer or multimer configurations^[16–22]. Owing to the ultra-high $Q$ -factor, BIC metasurfaces have been applied in high-sensitivity sensing^[23,24], ultra-low threshold lasers^[7], and nonlinear harmonic generations^[8,25]. Moreover, the working wavelength of BIC metasurfaces can cover from the visible to the microwave region by adjusting the structural parameters.

The terahertz (THz) band is an important region in the electromagnetic spectrum and has widespread applications in communication, sensing, and imaging. Recently, BIC metasurfaces have also been studied in the THz band. By breaking the symmetry of the structure or employing the mode coupling, BICs are achieved in metallic metasurfaces^[14,26–30]. However, due to the Ohmic loss of the metal, the measured $Q$ -factors are limited. The highest $Q$ -factor in THz metallic BIC metasurfaces reported up to now is 227^[26]. In contrast, dielectric structures lack the Ohmic loss, and electromagnetic fields are well confined in the interior of dielectric structures^[31], which has great potential for achieving high- $Q$ BIC metasurfaces^[8,32–35]. However, the measured highest $Q$ -factor in THz all-dielectric metasurfaces before 2022 is only 250 and this was achieved by designing silicon cuboids on quartz substrates^[15]. In fact, a resonance with an arbitrary high- $Q$ value can be easily designed in BIC metasurfaces theoretically by breaking the symmetry of the structure or by selecting structural parameters that deviate from those for ideal BICs. It remains a challenge to achieve this experimentally, especially at the THz band, because of the material’s absorption loss, fabrication difficulty, and high spectral resolution measurements. In previous investigations of all-dielectric THz metasurfaces, most of them are realized by designing dielectric building blocks on the substrate, and interface bonding of the two bulk materials is required in device fabrication, which will not only increase the fabrication complexity but also bring about bonding defects. Thus, an all-dielectric THz metasurface made of a single material has great advantages. One direct method for generating a high- $Q$ metasurface made of a single material is to design a complementary metasurface consisting of an air-hole array without a substrate or with a substrate made of same material. However, high- $Q$ resonances were not observed experimentally, and the underlying physics was not interpreted^[36,37]. Recently, all-silicon BIC metasurfaces consisting of a perforated air-hole array without a substrate were reported^[38,39], and the quasi-BIC resonance originated from the GMR with a record high- $Q$ value of 1049 was experimentally achieved^[38]. However, the quality of the perforated surface is difficult to control in the deep etching process.

Here, we propose and experimentally investigate an all-silicon THz BIC metasurface consisting of an imperforated air-hole array with a Si substrate, which is easier to fabricate than that without the substrate. We demonstrate that there are two quasi-BIC resonances originating from the GMR in the frequency range of 0.4 to 0.6 THz. Furthermore, the influence of thickness of the silicon substrate on the two GMRs is analyzed and compared with the metasurface without the substrate in detail. Finally, all-silicon metasurfaces with different air-hole parameters were fabricated by combining photolithography and deep reactive-ion etching, and the measured results show that $Q$ -factors of the two GMRs are as high as 334 and 152, respectively.

2. Design and Simulation Results of All-Silicon Metasurface

2.1. Structure of an all-silicon metasurface

The schematic diagram of an all-silicon metasurface with an imperforated air-hole tetramer array is shown in Fig. 1(a). The unit cell of the metasurface contains four circular air-holes, as presented in Fig. 1(b). The structural parameters are as follows: periods of the unit cell in $x$ - and $y$ -directions are P_x = P_y = 300 µm, radii of the air-holes at the two diagonals are represented by $r_{1}$ and $r_{2}$ , respectively, and the distance between the centers of the two circles along the $x$ - and $y$ -directions $s$ is fixed as $s = P / 2 = 150 μm$ . The thickness of the metasurface is $t = t_{1} + t_{2} = 200 μm$ , where $t_{1}$ is the depth of the air-holes, and $t_{2}$ is the thickness of the silicon substrate. The all-silicon metasurface was fabricated by combining photolithography and deep reactive-ion etching, and a scanning electron microscope (SEM) image of one fabricated metasurface is shown in Fig. 1(c). Actually, when $t_{2} = 0$ , the designed metasurface becomes a metasurface with a perforated air-hole array as in Ref. [38].

(a) Schematic diagram of an all-silicon terahertz metasurface consisting of periodic imperforated air-hole tetramers when normally illuminated by a terahertz wave polarized along the x-direction. (b) Unit cell of the metasurface. The periods of the unit cell are Px = Py = P = 300 µm. The depth of air-holes is denoted as t1, and the thickness of the substrate is t2. (c) SEM picture of one fabricated metasurface when r1 = 25 µm, r2 = 55 µm, and t1 = 150 µm.

Fig. 1. (a) Schematic diagram of an all-silicon terahertz metasurface consisting of periodic imperforated air-hole tetramers when normally illuminated by a terahertz wave polarized along the x-direction. (b) Unit cell of the metasurface. The periods of the unit cell are P_x = P_y = P = 300 µm. The depth of air-holes is denoted as t₁, and the thickness of the substrate is t₂. (c) SEM picture of one fabricated metasurface when r₁ = 25 µm, r₂ = 55 µm, and t₁ = 150 µm.

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2.2. Simulation results of all-silicon metasurfaces

First, we studied the optical properties of the all-silicon metasurfaces with imperforated air-hole arrays using the finite element method (COMSOL Multiphysics software). In simulation, periodic boundary conditions are employed in the $x$ - and $y$ -directions, whereas a perfectly matched layer (PML) is used in the $z$ -direction. The metasurface is illuminated by the terahertz wave polarized along $x$ -axis at normal incidence, and the dielectric constant of the silicon is set to be 11.67. Figure 2(a) shows the calculated transmission spectra of the metasurface with different $r_{1}$ , where $r_{2}$ and $t_{1}$ are fixed at 55 µm and 150 µm, respectively. When $r_{1} = 25 μm$ , we can find that there are two strong Fano resonances located at 0.489 THz and 0.521 THz, respectively, which are labeled as $f_{1}$ and $f_{2}$ . Actually, the resonances are polarization-insensitive according to the geometric symmetry of the structure. The frequency and $Q$ -factor of the Fano resonance are obtained by fitting the transmittance curve $T$ with the following Fano formula^[40–42]: $T (ω) = T_{0} + A_{0} \frac{{[q + 2 (ω - ω_{0}) / γ]}^{2}}{1 + {[2 (ω - ω_{0}) / γ]}^{2}},$ (1)where $q$ is the Fano fitting parameter that determines the asymmetry of resonance curve; $ω_{0}$ and $γ$ represent resonance angular frequency and bandwidth, respectively; $T_{0}$ is transmittance baseline; and $A_{0}$ is the coupling coefficient, and therefore $Q = ω_{0} / γ$ .

(a) Simulated transmittances of the metasurface with different r1, when r2 = 55 µm. (b)–(e) Near-field electric and magnetic field distributions of resonances f1 and f2 at 0.489 THz and 0.521 THz, when r1 = 25 µm. (f) Resonance frequencies of f1 and f2 with respect to r1. (g) Q-factors of f1 and f2 versus r1.

Fig. 2. (a) Simulated transmittances of the metasurface with different r₁, when r₂ = 55 µm. (b)–(e) Near-field electric and magnetic field distributions of resonances f₁ and f₂ at 0.489 THz and 0.521 THz, when r₁ = 25 µm. (f) Resonance frequencies of f₁ and f₂ with respect to r₁. (g) Q-factors of f₁ and f₂ versus r₁.

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To understand the properties of the two resonances, we calculated the near-field electric field and magnetic field distributions ( $E_{z}$ and $H_{z}$ ) at frequencies of the resonances $f_{1}$ and $f_{2}$ . As shown in Figs. 2(b)–2(e), $H_{z} \approx 0$ for resonance $f_{1}$ , while $E_{z} \approx 0$ for $f_{2}$ . Therefore, resonance $f_{1}$ is a transverse magnetic (TM) mode, while $f_{2}$ is a transverse electric (TE) mode. In addition, the electromagnetic fields of $f_{1}$ are well confined within the air-hole layer, while more electromagnetic fields of $f_{2}$ are distributed within the substrate. It can also be seen from Fig. 2(a) that both the resonances $f_{1}$ and $f_{2}$ blue shift with the increase of $r_{1}$ . The variation of the resonance frequencies of $f_{1}$ and $f_{2}$ , as a function of $r_{1}$ , is illustrated in Fig. 2(f). When $r_{1}$ increases from 25 µm to 75 µm, the blue shift of the resonance $f_{1}$ is about 107 GHz, while the blue shift is only 36 GHz for the resonance $f_{2}$ . This can be explained as the electromagnetic field of resonance $f_{1}$ being mainly distributed inside the air-hole structure layer. Hence, it is very sensitive to $r_{1}$ , while the electromagnetic field of $f_{2}$ is distributed in both the structure layer and the substrate. Thus, it is less sensitive to $r_{1}$ . It can also be seen that the two curves cross, but the coupling of the two modes does not occur. We also retrieve the $Q$ -factor of resonances $f_{1}$ and $f_{2}$ with a different $r_{1}$ according to the calculated transmission spectra, as shown in Fig. 2(g). We can find that when $r_{1}$ increases from 25 µm to 55 µm, both $Q$ -factors of the resonances $f_{1}$ and $f_{2}$ increase quickly, whereas they decrease rapidly with the increase of $r_{1}$ when $r_{1}$ is larger than 55 µm. In particular, when $r_{1} = r_{2} = 55 μm$ , the two resonances $f_{1}$ and $f_{2}$ disappear in the transmission spectrum, and the $Q$ -factors of resonances $f_{1}$ and $f_{2}$ approach infinite, which is a typical characteristic of BICs.

In order to further understand this phenomenon, we calculated the dispersion curves of the metasurface when $r_{1} = r_{2} = 55 μm$ . As shown in Fig. 3(a), there are four eigenmodes related to the resonances $f_{1}$ and $f_{2}$ . Two TM modes near 0.53 THz are denoted as TM 1 and TM 2, while two TE modes near 0.55 THz are denoted as TE 1 and TE 2. $Q$ -factors of the four eigenmodes are all close to infinity at $Γ = 0$ and off $Γ$ as well. We also calculated the electromagnetic field distributions of the four modes at $Γ = 0$ , as shown in Fig. 3(b). TM 1 and TM 2 are two TM guide modes propagating along 45° and $- 45 °$ relative to the $x$ -axis, respectively, which are induced by (1, 1) Rayleigh diffraction of 2D metagrating with a period of $P$ ^[17,38,43]. Similar to the two TM modes, TE 1 and TE 2 are two TE guide modes induced by (1, 1) diffraction of 2D metagrating. Therefore, when $r_{1} = r_{2} = 55 μm$ , both resonances $f_{1}$ and $f_{2}$ are ideal BICs that originated from the GMR^[17,44]. When the lattice symmetry is broken ( $r_{1} \neq r_{2}$ ), the two BIC modes are transformed into two quasi-BIC modes with a finite $Q$ -factor, which can be observed experimentally. From above calculations, we can get that the two resonances $f_{1}$ and $f_{2}$ of the metasurface consisting of the air-hole array with or without the substrate are essentially the same, except for different frequencies and Q values.

(a) Dispersion curves of the four related TE and TM eigenmodes, as a function of k P/2π, when r1 = r2 = 55 µm, where k is a propagating constant. The inset shows the first Brillouin zone of the square lattice. (b) Electric (Ez) and magnetic (Hz) near-fields in the xy plane of the four modes at Γ = 0.

Fig. 3. (a) Dispersion curves of the four related TE and TM eigenmodes, as a function of k P/2π, when r₁ = r₂ = 55 µm, where k is a propagating constant. The inset shows the first Brillouin zone of the square lattice. (b) Electric (E_z) and magnetic (H_z) near-fields in the xy plane of the four modes at Γ = 0.

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2.3. Effect of substrate on resonance characteristics

In previous calculations, the depth of the air-holes $t_{1}$ is fixed at 150 µm and the thickness of the substrate $t_{2}$ is fixed at 50 µm. In the following, we discuss the effect of the substrate on the two GMRs. We calculated the transmission spectra of the metasurface, where the depth of the air-holes $t_{1}$ is changed from 130 µm to 200 µm, and $t$ is fixed at 200 µm. The variation of the resonance frequencies of $f_{1}$ and $f_{2}$ , as a function of $r_{1}$ , is illustrated in Figs. 4(a) and 4(c), respectively. As $t_{1}$ increases from 130 µm to 200 µm, i.e., the thickness of the substrate $t_{2}$ decreases from 70 µm to 0 µm, both resonances $f_{1}$ and $f_{2}$ have blue shifts, as is usually the case^[13]. Since $f_{2}$ is more sensitive to the thickness of the substrate than $f_{1}$ , the blue shift of $f_{2}$ is larger than that of $f_{1}$ . For example, when $r_{1} = 30 μm$ , the resonance $f_{1}$ has only a blue shift of about 17 GHz, while the blue shift for $f_{2}$ is 45 GHz. In addition, the frequency of the resonance $f_{2}$ gradually becomes sensitive to $r_{1}$ , as the thickness of the substrate decreases. Figures 4(b) and 4(d) show the influence of the depth of air-holes $t_{1}$ on the $Q$ -factors of the two resonances. The $Q$ -factors of resonances $f_{1}$ and $f_{2}$ increase quickly with the increase of $t_{1}$ , that is, the electromagnetic field confinements of the two resonances are enhanced, leading to a decrease in the radiation loss of the two modes. Further, when $t_{1}$ is less than 150 µm, the $Q$ -factor of $f_{2}$ is very low, but it rises rapidly when $t_{1}$ increases from 150 µm to 200 µm, because much more electromagnetic field distribution of $f_{2}$ switches from the substrate to the air-hole layer. The results show that the addition of the substrate greatly reduces the Q values of the two GMRs.

(a), (c) Influence of air-hole depth t1 on the frequency and (b), (d) Q-factor of the resonances f1 and f2, when thickness of metasurface t = 200 µm.

Fig. 4. (a), (c) Influence of air-hole depth t₁ on the frequency and (b), (d) Q-factor of the resonances f₁ and f₂, when thickness of metasurface t = 200 µm.

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3. Experimental Results

To demonstrate previous simulated results, we fabricated and measured the transmission spectra of the all-silicon metasurfaces with an imperforated air-hole array. In experiments, the designed metasurfaces were fabricated in a high-resistivity silicon wafer ( $> 5000 Ω \cdot cm$ ) with a thickness of 200 µm because of its low loss and mature fabrication technique. According to the structure shown in Fig. 1(a), four all-silicon metasurfaces with a size of $15 mm \times 15 mm$ were fabricated through photolithography and followed by deep reactive ion etching (Bosch process), where $t_{1} = 150 μm$ , $r_{2} = 55 μm$ , and $r_{1}$ was chosen as 25 µm, 30 µm, 35 µm, and 40 µm, respectively. An SEM picture of one of the fabricated samples ( $r_{1} = 25 μm$ ) is shown in Fig. 1(c). We used a high spectral resolution (140 MHz) terahertz frequency-domain spectroscopy system (TeraScan 1550 from Toptica) to measure the transmission spectra of the four samples. In measurements, the terahertz wave radiated from the photoconductive antenna by photomixing is collimated and focused by 2.54 cm off-axis parabolic lenses with 75 mm focal length and normally incident on the metasurfaces. The measurements were done at room temperature and in dry air conditions (humidity ≤1%) to eliminate the effects of water vapor absorption, and the integration constant was set to be 300 ms to enhance the signal-to-noise ratio (SNR) of the measurement.

Considering both the error of the air-hole radius in the fabrication and the loss caused by defects, we recalculated the transmission spectra of the metasurface by adding 2 µm to the designed air-hole radius and a loss tangent of 0.0003 to the equivalent material loss of the silicon, as shown in Fig. 5(a). The measured transmission spectra are also shown in Fig. 5(b). Figures 5(c) and 5(d) are enlarged views of the measured resonances $f_{2}$ and $f_{1}$ and their Fano fitting curves, respectively. It can be seen that a strong resonance $f_{2}$ is observed in all four samples, meanwhile resonance $f_{1}$ is clearly measured in three samples, but it is not observed in the sample $r_{1} = 40 μm$ because of its high- $Q$ factor (1370 calculated). It is well known that the higher the $Q$ -factor of the resonance, the more susceptible it is to the material’s absorption, fabrication defects, or device size. The measured frequencies of the two resonances are in good agreement with the recalculated results. In addition, the $Q$ -factors of the two resonances $f_{1}$ and $f_{2}$ increase with the increase of $r_{1}$ as expected. For resonance $f_{2}$ , the measured $Q$ -factors are very close to those recalculated with equivalent material loss. This is because the $Q$ -factor of $f_{2}$ is low ( $\leq 175$ calculated and $\leq 152$ measured), and it will not be largely influenced by the material loss and fabrication defects, while the $Q$ -factors of resonance $f_{1}$ are much larger than those of resonance $f_{2}$ . For example, for the three metasurfaces $r_{1} = 25 μm$ , 30 µm, and 35 µm, the calculated $Q$ -factors are 412, 616, and 955, respectively, but the measured values are only 115, 133, and 334, smaller than the calculated ones. Compared with the previous all-dielectric THz BIC metasurfaces, which design dielectric building blocks on the substrate, the measured $Q$ -factor is improved based on the all-silicon THz BIC metasurface containing an air-hole array as done in this work. However, it is smaller than that we reported in the corresponding all-silicon metasurface with a perforated air-hole array without using a substrate^[38].

(a) Calculated and (b) measured transmittances of four metasurfaces. (c) and (d) Enlarged views of measured resonance f2 for the sample r1 = 40 µm (blue) and f1 for the sample r1 = 35 µm (yellow) and their Fano fitted curves (pink).

Fig. 5. (a) Calculated and (b) measured transmittances of four metasurfaces. (c) and (d) Enlarged views of measured resonance f₂ for the sample r₁ = 40 µm (blue) and f₁ for the sample r₁ = 35 µm (yellow) and their Fano fitted curves (pink).

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4. Conclusion

In summary, we propose and experimentally demonstrate an all-silicon BIC metasurface consisting of an air-hole tetramer array with a Si substrate. The metasurface supports two polarization-insensitive BIC resonances originated from the GMR in the frequency range of 0.4–0.6 THz, and their $Q$ -factors can be adjusted flexibly by changing the air-hole radius. The characteristics of the two GMRs with different thickness of the substrate are numerically investigated. The results show that $Q$ -factors of the two resonances without the substrate are much larger than those with the substrate. In addition, they can be improved by increasing the depth of the air-holes. Finally, the designed all-silicon metasurfaces are fabricated by combining photolithography and deep reactive-ion etching, and the measured $Q$ -factors of the two GMRs are as high as 334 and 152, respectively. As high- $Q$ silicon devices are very sensitive to changes in refractive index or absorption of silicon materials, the proposed metasurface can be used as efficient THz switches or modulators by exploring its photon induced carrier absorption effect, and the optical pump power can be remarkably reduced.

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1. Introduction

2. Design and Simulation Results of All-Silicon Metasurface

2.1. Structure of an all-silicon metasurface

2.2. Simulation results of all-silicon metasurfaces

2.3. Effect of substrate on resonance characteristics

3. Experimental Results

4. Conclusion

Qing Wang, Pengfei Wang, Jianjun Liu, Fangzhou Shu, Guiming Pan, Zhongwei Jin, Xufeng Jing, Zhi Hong. High-Q resonances in terahertz all-silicon metasurface with imperforated air-hole array[J]. Chinese Optics Letters, 2023, 21(11): 113601.

High-Q resonances in terahertz all-silicon metasurface with imperforated air-hole array

1. Introduction

2. Design and Simulation Results of All-Silicon Metasurface

2.1. Structure of an all-silicon metasurface

2.2. Simulation results of all-silicon metasurfaces

2.3. Effect of substrate on resonance characteristics

Fig. 4. (a), (c) Influence of air-hole depth t₁ on the frequency and (b), (d) Q-factor of the resonances f₁ and f₂, when thickness of metasurface t = 200 µm.

3. Experimental Results

Fig. 5. (a) Calculated and (b) measured transmittances of four metasurfaces. (c) and (d) Enlarged views of measured resonance f₂ for the sample r₁ = 40 µm (blue) and f₁ for the sample r₁ = 35 µm (yellow) and their Fano fitted curves (pink).

4. Conclusion

Article Outline

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High-Q resonances in terahertz all-silicon metasurface with imperforated air-hole array

1. Introduction

2. Design and Simulation Results of All-Silicon Metasurface

2.1. Structure of an all-silicon metasurface

2.2. Simulation results of all-silicon metasurfaces

2.3. Effect of substrate on resonance characteristics

Fig. 4. (a), (c) Influence of air-hole depth t1 on the frequency and (b), (d) Q-factor of the resonances f1 and f2, when thickness of metasurface t = 200 µm.

3. Experimental Results

Fig. 5. (a) Calculated and (b) measured transmittances of four metasurfaces. (c) and (d) Enlarged views of measured resonance f2 for the sample r1 = 40 µm (blue) and f1 for the sample r1 = 35 µm (yellow) and their Fano fitted curves (pink).

4. Conclusion

Article Outline

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Fig. 4. (a), (c) Influence of air-hole depth t₁ on the frequency and (b), (d) Q-factor of the resonances f₁ and f₂, when thickness of metasurface t = 200 µm.

Fig. 5. (a) Calculated and (b) measured transmittances of four metasurfaces. (c) and (d) Enlarged views of measured resonance f₂ for the sample r₁ = 40 µm (blue) and f₁ for the sample r₁ = 35 µm (yellow) and their Fano fitted curves (pink).