Enhanced Fano resonance for high-sensitivity sensing based on bound states in the continuum

Guang Feng; Zhihui Chen; Yang Wang; Xin Liu; Yinshan Liu; Xiao Liu; Fei Sun; Yibiao Yang; Shuqi Chen

doi:10.3788/COL202321.031202

Abstract

Although previously reported terahertz absorbers can achieve high-sensitivity refractive index sensing, the resonant peak is too broad, which leads to a low figure of merit (FOM). Transmissive sensors based on bound states in the continuum (BIC) can achieve high FOM, but they have some limitations in high sensitivity. Herein, we propose a periodic triple parallel metal bars structure to obtain high quality, a strong field, and multiple hot spots by the Friedrich–Wintgen BIC. Numerical results show the sensitivity and FOM can reach 1877 GHz/RIU and 665, respectively. Compared to the previously reported transmissive sensors based on BIC, the sensitivity has been greatly improved.

Keywords

bound states in the continuum Fano resonance high-sensitivity sensing terahertz

1. Introduction

Terahertz (THz) sensing has become a promising technique because of its low energy and fingerprint for molecules^[1–3]. However, the sensitivity of THz sensing is limited due to the mismatch between the relatively long wavelengths of THz waves and the analyte sizes. Recently, metasurfaces have been used to overcome this limitation.

A number of THz absorbers and metamaterial-based refractive-index sensing have been reported, such as metal-dielectric-metal configuration^[4], planar structures based on breaking symmetry^[5], and graphene-based structures^[6-9]. Metamaterial THz absorbers are very sensitive to the changes of the refractive index of the surrounding medium. The THz absorber can achieve high sensitivity of 3.5 THz/RIU due to its cavity coupling to realize large-area field enhancement^[10]. This strong field enhancement between light with a microstructure^[11–13] allows for enhanced detection accuracy. Therefore, strong field enhancements are essential for high-sensitivity refractive-index label-free sensors^[14–16]. Moreover, THz absorbers of high-order resonant frequency can also achieve high sensitivity of 4.72 THz/RIU^[17]. This is because the frequency shift of the high order is larger than that of the lower order. For refractive index sensors, sensitivity and figure of merit (FOM) are two important indicators. To evaluate the actual performance of refractive index sensors, the FOM defined by the ratio of sensitivity to full width at half-maximum (FWHM) is used, which means the high-performance sensor requires not only considerable sensitivity but also a high quality (Q) factor^[18]. Although previously reported, many THz absorbers can achieve high-sensitivity refractive index sensing, the resonant peak is too broad, which leads to a low Q factor and FOM. Therefore, strong sensitivity does not ensure peak distinguishability if the resonant peak is too broad.

For transmissive sensors in the THz band, it is difficult to achieve high-sensitivity refractive index sensing due to small-area and low-field enhancement. Recently, owing to Fano resonance having a high Q and strong field enhancement, the refractive index sensor based on Fano resonance has been studied^[19]. Moreover, the Fano resonance of metasurfaces has also been linked to the bound states in the continuum (BIC)^[20–24] to obtain high Q. High Q is often accompanied by a strong field enhancement. Therefore, there has been much research on Fano resonance based on BIC to improve the sensitivity of refractive indices^[25–28]. However, the field enhancement is not proportional to the Q factor due to material dissipation^[29]. Most of the research based on BIC is to achieve a refractive index sensing by obtaining a high Q factor^[25,26,28]. Therefore, although the structure with the high Q factor can obtain high FOM, it cannot obtain the strongest local field enhancement^[20], which limits the high sensitivity. In addition, the interaction between the field and the analyte is reduced due to the small-area local-field enhancement. Therefore, the reported transmissive sensors based on BIC is difficult to achieve more than 1 THz/RIU in previous work. To enhance the interaction between light field and analyte, strong local-field enhancement and large-area local enhancement play an important role in improving refractive index sensitivity.

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Here, the Fano resonance of quasi-BIC is realized by tuning the length of triple copper rods. The Fano resonance with high Q and strongest near-field enhancement can be simultaneously obtained when the length of triple copper rods is 55 µm, and the multiple resonant structures can achieve multiple hot spots. The strong field and multiple hot spots are important in highly sensitive refractive index label-free sensors. Therefore, the theoretical sensitivity and FOM are, respectively, calculated to 1877 GHz/RIU and 665.

2. Structure and Methods

Figure 1(a) shows the designed triple metal rods structure, and Fig. 1(b) is a plan view of the periodic structure unit. Triple parallel metal bars have a length of L, the width of the metal strips is $W = 10 µm$ , the spacing between the metal strips is $S = 3 µm$ , and the period of the structure unit is $P = 70 µm$ . The metal material is copper, the electrical conductivity of the copper is $5.91 \times 10^{7} S / m$ , and the thickness is 0.2 µm. The material of the substrate is PDMS, the fixed refractive index is 1.4, and the thickness is 20 µm. The proposed structure is numerically calculated by using the finite-difference time-domain (FDTD) method. In the simulation, we set periodic boundary conditions in the $x$ and $y$ directions and perfectly matched layer (PML) boundary conditions in the $z$ direction. The plane wave is incident along the $z$ direction.

Figure 1.(a) Three-dimensional structure diagram; (b) plan view of structure unit.

Although we only study in simulation, the potential feasible preparation method is shown in Fig. 2. The structure can be fabricated by a standard optical lithography process. The detailed procedure of fabrication is illustrated in Fig. 2. The metal structure is fabricated on a PDMS substrate with a 20 µm thick layer and a refractive index of 1.4. First, radio-frequency magnetron sputtering is used to deposit a 200 nm thick copper film on the substrate. Subsequently, the 1 µm thick layer of positive photoresist is coated on copper film. Before lithography, we dry the photoresist to prevent the photoresist layer from deforming in the lithography process. Then, samples are prepared by photolithography and reactive ion etching. Finally, we can obtain the structure by using lift-off technology.

Figure 2.Fabrication for the proposed structure.

3. Results and Discussion

Figure 3(a) shows the transmission spectrum obtained by tuning the length L of the triple metal bars. The length L is from 50 to 70 µm. When the length L is 60 µm, the Fano resonance with a frequency of around 3.9 THz will disappear in Fig. 3(b). To prove that the resonance obtained is Fano resonance, the Fano linear formula is given^[26]: $T_{Fano} = {| a_{1} + j a_{2} + \frac{b}{ω - ω_{0} + j γ} |}^{2},$ where $a_{1}$ , $a_{2}$ , and $b$ are real constant factors, and $ω_{0}$ and $γ$ are the resonance frequency and the damping rate, respectively. By fitting the result of FDTD when the length is 56 µm, the $a_{1}$ , $a_{2}$ , $b$ , and $γ$ can be extracted; they are $- 0.0006931$ , 0.9747, 0.002271, and 0.0015, respectively. The results are consistent, as shown in Fig. 3(c).

Figure 3.(a) Color plot of the transmission spectra with different lengths of metal bars; (b) transmission when the lengths of the metal bars are from 50 to 66 µm; (c) fitting curve of Fano resonance when the length of L is 56 µm; (d) transmission of MICMT and simulation when the lengths of the metal bars are 62, 60, and 58 µm, respectively. By curve fitting, the decay time of the internal losses, respectively, are about τ₁₀ = 4000 fs, τ₂₀ = 8000 fs, and τ₃₀ = 500 fs. The decay time of coupling and the total coupling phases, respectively, are about τ₁ = 400 fs, τ₂ = 8000 fs, τ₃ = 250 s, φ₁ = 0.2π, φ₂ = −0.3π, φ₃ = −0.1π; τ₁ = 400 fs, τ₃ = 250 s, φ₁ = 0.2π, φ₃ = −0.1π; τ₁ = 400 fs, τ₂ = 8000 fs, τ₃ = 250 s, φ₁ = 0.2π, φ₂ = −0.3π, φ₃ = −0.1π. (e) Quadrupole modes and (f) dipole modes when the lengths of the metal bars are 55 µm.

Since the BIC is sensitive to geometric sizes, it can be determined that this is accidental BIC. The formation of BIC in Fig. 3(b) is the consequence of destructive interference between two radiating channels with different optical behaviors: the quadrupole and dipole modes in Figs. 3(e) and 3(f). In the vicinity of the BIC point, the quadrupole mode exhibits sharp linewidths; in contrast, the dipole mode shows a wide resonance spectrum, which leads to strong radiative losses^[30]. It is well known that for a system with two different resonance phenomena continuously changing with the parametric variation, at one particular set of parameters, there may occur an avoided crossing through which both the modes can interact and interfere destructively, resulting in the vanishing of one of the modes^[31]. The avoided crossing of two different resonances is the feature of forming Friedrich–Wintgen BIC (FW-BIC).

Figure 3(d) is the transmission spectrum when the lengths of the three metal bars are 62, 60, and 58 µm, respectively; the Q factor is 1186, $\infty$ , and 1503, respectively. Meanwhile, with the increase of lengths of the three metal bars, the peak of the Fano resonance is redshifted. The transmission peaks^[32] can be evaluated by $λ_{peak} = \frac{P}{\sqrt{i^{2} + j^{2}}} n_{sp}$ . Here, $λ_{peak}$ is the central wavelength to which the transmission peak corresponds, and the resonance frequency can be obtained by dividing the speed of light by the central wavelength, where $P$ is the physical periodicity, $n_{sp}$ is the refractive index of the surface propagating wave, and $i$ and $j$ are indices corresponding to the resonance order. As the lengths of the three metal bars incrases, $n_{sp}$ will increase, which will lead to the redshift of the resonance peak^[22]. Moreover, for multimode coupling, the phase and modulus of different resonant modes and resonators are related to each other. These coupling phases and modulus will affect the transmittance of the resonator system. To study the transmission characteristics of the triple metal rods resonator system, the multimode interference coupled mode theory (MICMT) equation is given. The MICMT is as follows^[33,34]: $T = {| t |}^{2} = {| \sum_{n = 1}^{3} \frac{2 γ_{n 1} e^{{j φ}_{n}}}{- j (ω - ω_{n}) τ_{n} + 2 + \frac{τ_{n}}{τ_{n 0}}} |}^{2},$ where $ω_{n}$ is the resonant frequency of the $n$ th resonant mode. The three resonant modes are at 2.82, 3.88, and 4.23 THz when the length is 58 µm, respectively. $τ_{n}$ is the decay time of the coupling between the resonator. $τ_{n 0}$ is the decay time of internal loss of the $n$ th resonant mode in the resonator, and $φ_{n}$ is the total coupling phase difference of the $n$ th resonant mode. $γ_{n 1}$ is the normalized coefficients ( $γ_{n 1} = 1$ in this paper). The MICMT result is given in Fig. 3(d); the MICMT result agrees well with the simulation result.

In Fig. 4, we investigated the effect of single, double, and triple metal bars on the FWHM, Fano intensity, Q, and $Q \times I$ (the amplitude difference between the resonant peak and the resonant valley is defined as I^[35]) of the Fano resonance. With the increase in the number of metal bars, the FWHM and Fano intensity have been significantly weakened in Figs. 4(b) and 4(d). The narrower FWHM can obtain a high Q factor, but the low Fano intensity can lead to a low $Q \times I$ . Therefore, by comparing the ratio of the Fano intensity to the FWHM in Fig. 4(c), the triple parallel metal bars based on multimode coupling can produce a high ratio, which results in the high Q factor and high $Q \times I$ in Fig. 4(d). Compared with single metal bars and double metal bars, the Q factor and $Q \times I$ of the triple parallel metal bars have been greatly improved. BIC can be obtained when the length of the metal bars is 60 µm, which theoretically will have an infinite Q, but it does not exist. It can achieve the highest Q factor when the length is 59 µm: the Q factor can be up to 1510. Because high Q is often accompanied by low resonance intensity, the highest Q factor will not get the highest $Q \times I$ . Therefore, when the length is 55 µm, the high Q of 1075 and the highest $Q \times I$ of 820 can be obtained simultaneously.

Figure 4.(a) Single, double, and triple metal bars structure unit; (b) FWHM under different lengths; (c) ratio of Fano intensity to FWHM under different lengths; (d) Fano intensity, Q factor, and Q × I under different lengths.

To obtain the maximum field enhancement, we investigate the electric and magnetic fields at different lengths in Fig. 5. As the simulations in Figs. 5(a) and 5(b) show, the resonances of the electric field and magnetic field enhancement disappear when the length is 60 µm. With the length L decreasing, the electric field enhancement increases. For the highest Q factor when the length is 59 µm, the electric field and magnetic field enhancements are not obvious. The maximum electric field enhancement of 71 and magnetic field enhancement of 0.12 are achieved when the length is 55 µm in Figs. 5(c) and 5(d). Meanwhile, the highest $Q \times I$ can also be obtained when the length is 55 µm in Fig. 4(d). The green dotted lines in Figs. 5(a) and 5(b) are where the electric and magnetic fields are enhanced, which are shown in Figs. 5(e) and 5(f), respectively. It shows the electric field enhancement in Figs. 5(e) and 5(f) has the same variation trend as the $Q \times I$ in Fig. 4(d).

Figure 5.Simulated field enhancement of the triple metal bars. (a) and (b) show color plots of the field enhancement for different lengths in the triple metal bars, respectively; (c) electric field enhancement and (d) magnetic field enhancement for the lengths of 53, 55, 59, and 60 µm in the triple metal bars, respectively; (e) electric field enhancement and (f) magnetic field enhancement corresponding to the green dotted lines in (a) and (b), respectively.

It is known that field enhancement is known to be proportional to $\sqrt{Q}$ for lossless material^[36]. However, the field enhancement is not proportional to the Q factor due to material dissipation^[29]. Meanwhile, the field enhancement is proportional to the square root of the intensity of the resonance. Owing to the presence of material dissipation, the intensity of the resonance decreases as it approaches BIC, which will lead to a decrease in field enhancement. Therefore, the field enhancement can be proportional to the product of the Q factor and the intensity of the resonance, which considers the trade-off between the Q factor and the resonance intensity.

In addition, the electric field, magnetic field, and surface current distribution of the structure with lengths of 59, 55, and 50 µm is investigated in Fig. 6. The three dimensions, respectively, represent the maximum Q, $Q \times I$ , and Fano intensity. It can be found that when the structure is at the maximum $Q \times I$ , the maximum electric and magnetic field enhancement can be obtained. The electromagnetic field enhancement is mainly due to the local surface plasmon resonance. The electric field is confined in the edge regions of the metal strip, leading to strong electric field enhancement. For the magnetic field, in addition to the edge position of the upper and lower metal strips, the enhanced position also includes the middle position of the middle metal strip. Therefore, multiple metal bar structures can provide multiple hot spots. The enhancement of electromagnetic fields and multiple hot spots can have important applications in biological detection. Moreover, the distribution of surface current is shown in Figs. 6(c), 6(f), and 6(i). The surface currents radiate energy into free space, but the energy generated by the opposite current is mutually repellent. In Figs. 6(c) and 6(i), the repellence of opposing currents becomes less, resulting in more energy radiating into free space. Therefore, a higher Fano intensity can be achieved. These results agree with the results of Fig. 4(d).

Figure 6.(a), (d), (g) Electric, (b), (e), (h) magnetic fields, and (c), (f), (i) surface current at the Fano resonance. (a), (b), (c) The length is 59 µm; (d), (e), (f) the length is 55 µm; (g), (h), (i) the length is 50 µm.

The strong field and multiple hot spots are all very important for highly sensitive refractive index label-free sensors. The enhancement of the electromagnetic field and multiple hot spots can lead to enhanced interaction between the electromagnetic field and the specific analyte, which can achieve high refractive index sensitivity for application in the biological and biochemical fields. The refractive index sensor is calculated in Fig. 7. An analyte layer is selected to be added to the structure. By varying the refractive index from 1.3 to 1.6 at a thickness of 70 µm, the Fano resonance peak is redshifted from 3.41 to 2.85 THz in Fig. 7(a). The linear relationship of Δf−n can be observed in Fig. 7(c), and the sensitivity S can be obtained by calculating the corresponding derivative of the Δf−n curve. Theoretical refractive index sensitivity can be defined as the ratio of the resonance peak frequency shift to the refractive index change. The refractive index sensitivity is 1877 GHz/RIU. This result is higher than other reported work in Table 1. In addition, the thickness of the analyte layer also affects the frequency shift. It can be seen from Fig. 7(b) that at a fixed refractive index of 1.6, the frequency shift $Δ f$ increases from 3.63 to 2.85 THz when the thickness increases from 10 to 70 µm. The effect of analyte layer thickness on $Δ f$ also shows a nonlinear trend in Fig. 7(d). Once it exceeds this value, the $Δ f$ would not increase significantly anymore, no matter how large the thickness is. Moreover, the FOM calculated as $FOM = S / FWHM$ is used to study the sensing performance^[18]. With the refractive index increasing from 1.3 to 1.6, the FOM increases from 665 to 998, much higher than BIC-based metasurface sensors in previous work. Although the FOM in Ref. [28] is very high, the structure is based on lossless silicon and high-order resonant frequency.

Structure	Index Sensitivity	FOM	Year
Two silicon cuboids	465.74 GHz/RIU	$\sim 10, 000$	2022^[28]
Double metal split ring resonators	455.7 GHz/RIU	$< 50$	2019^[37]
Three gold blocks	165 GHz/RIU	$< 50$	2021^[26]
Double metal split ring resonators	775.7 GHz/RIU	284	2022^[38]
Triple metal rods	1877 GHz/RIU	665	This work

Table 1. Properties of BIC-Based Metasurface Sensors

View all Tables

$(a) Dependence of transmission on the refractive index increased from 1.3 to 1.6 under the thickness of 70 µm; (b) dependence of transmission on the analyte thickness increased from 10 to 70 µm under the refractive index of 1.6; (c) frequency shift under different refractive indices of analyte extracted from (a); (d) frequency shift under different analyte thicknesses extracted from (b).$

Figure 7.(a) Dependence of transmission on the refractive index increased from 1.3 to 1.6 under the thickness of 70 µm; (b) dependence of transmission on the analyte thickness increased from 10 to 70 µm under the refractive index of 1.6; (c) frequency shift under different refractive indices of analyte extracted from (a); (d) frequency shift under different analyte thicknesses extracted from (b).

Meanwhile, the resonance characteristics have been investigated with the oblique incidence of TM. The simulated transmission of the triple metal bar of 55 µm length with oblique incidence (0° to 25°) of TM is displayed in Fig. 8(a). There is an obvious symmetry-protected BIC (SP-BIC) and FW-BIC that can be produced when the TM angle is 0° and 16.5°, respectively. The SP-BIC is formed due to the oblique incidence of the TM, resulting in symmetry breaking^[39]. From Figs. 8(b) and 8(c), it can be found that the resonance disappears due to the mode crossing of Mode 1 and Mode 2. The crossing and avoided crossing of resonance modes both are typical features of FW-BIC. Therefore, BIC can also be realized here by adjusting the incidence angle, which provides a new manipulation method for the application.

Figure 8.Angular response for the triple metal bar length of 55 µm: (a) simulated transmission contour plot under TM-polarized incidence; (b) simulated transmission curves with incidence angles from 10° to 24°; (c) field distribution of Mode 1 and Mode 2 at 10° and 24°, respectively.

4. Conclusions

Due to the mutual coupling between the resonators, the triple parallel metal bars structure can realize the FW-BIC. By tuning the structural parameters of triple rods, the high Q, strong field, and multiple hot spots can be obtained when the length of metal bars is 55 µm. Moreover, the sensitivity and FOM are calculated to 1877 GHz/RIU and 665, respectively. Compared to the previously reported transmissive sensors based on BIC, the sensitivity has been greatly improved. An angular scan can achieve the SP-BIC and FW-BIC, which provides a new manipulation method. Therefore, the Fano resonance structure with strong field and multiple hot spots achieves high sensitivity refractive index sensing for application in the biological and biochemical fields.

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