## Abstract

In this paper, a metal–insulator–metal (MIM) waveguide structure consisting of a side-coupled rectangular cavity (SCRC), a rightward opening semi-ring cavity (ROSRC), and a bus waveguide is reported. The finite element method is used to analyze the transmission characteristics and magnetic-field distributions of the structure in detail. The structure can support triple Fano resonances, and the Fano resonances can be tuned independently by altering the geometric parameters of the structure. Moreover, the structure can be applied in refractive index sensing and biosensing. The maximum sensitivity of refractive index sensing is up to 1550.38 nm/RIU, and there is a good linear relationship between resonance wavelength and refractive index. The MIM waveguide structure has potential applications in optical on-chip nano-sensing.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Fano resonance, a pervasive resonance phenomenon, was proposed initially by Ugo Fano in 1961, which occurs when a localized discrete state is coupled to a broad continuum state [1–7]. Unlike Lorentzian resonance, Fano resonance has a typically sharp and asymmetric line-shape which provides good performances in sensitivity to the surrounding environment [8,9]. Surface plasmon polaritons (SPPs), a form of electromagnetic wave that propagates along with the metal-dielectric interface, are excited by the coupling of external incident photons and free electrons on the surface of metal [10–12]. SPPs have the ability to overcome the conventional limit of diffraction [13,14]. Therefore, the combination of Fano resonance and SPPs would become a potential candidate to achieve ultra-compact functional optical components for highly integrated optics. Recently, many MIM waveguide structures with SPPs have been proposed as sub-wavelength devices such as sensors [15–24], filters [25], all-optical switches [26,27], slow-light devices [28,29], and perfect absorbers [30–32].

Over the past decade, the independent tunability of Fano resonance has become a topic of significant interest to researchers for its ability to greatly provide flexibility to plasmonic devices [33,34]. By way of example, Qi et al. reported an asymmetric plasmonic MIM structure composed of a bus waveguide and a rectangular cavity, which can support double Fano resonances. The double Fano resonances can be well tuned by changing the parameters of the rectangular cavity [12]. Wang et al. realized multiple Fano resonances in a plasmonic MIM waveguide with a circular split-ring cavity. The positions and line-shapes of the Fano resonances can be easily tuned by adjusting the direction angle of the metal-strip core in the circular cavity [35]. In addition, Sun et al. designed a plasmonic structure with a stub and a side-coupled split-ring resonator, and the Fano resonance wavelength and line-shape can be tuned by adjusting the opening direction of the split ring [36]. However, it is difficult to realize independent tunability for multiple Fano resonances resulting from the collective behavior of the total waveguide structure. The research on the independent tunability of Fano resonances should be given more attention since it may open up a pathway in photonics and provide prospects of integrated devices that manipulate and transmit light.

In this paper, a MIM waveguide structure is proposed, which is composed of a side-coupled rectangular cavity (SCRC), a rightward opening semi-ring cavity (ROSRC), and a bus waveguide with a silver-air-silver barrier (SASB). Then triple Fano resonances are obtained and investigated in the structure. Based on the inﬂuences of the geometric parameters of the structure on Fano resonances, the independent tunability of the triple Fano resonances can be achieved. Finally, refractive index sensing and biosensing characteristics of the structure are investigated.

## 2. Structure and theory

Figure 1 shows the proposed MIM waveguide structure, and the gray and white parts are defined as silver and air. The relative dielectric constant of air is *ε*_{i}* *= 1.0, and the frequency-dependent complex relative permittivity of silver is described by the Drude model [37–39]:

*ε*

_{∞}=3.7 is the dielectric constant at an infinite frequency,

*ω*

_{p}=9.1 eV is the plasma frequency of silver,

*ω*is the angular frequency of the incident wave, and

*γ*= 0.018 eV is the frequency of the electron collision. The MIM waveguide structure can be fabricated easily. The sufficiently thick Ag layer can be prepared on silicon substrate by chemical vapor deposition (CVD) [35]. After that, the SCRC, the ROSRC, and the bus waveguide with the SASB can be etched on the Ag layer by electron beam etching.

*L* is the length of the SCRC, and *G*_{1} is the distance between the SCRC and the bus waveguide. The width *w* of all waveguides is 50 nm so that only the fundamental transverse magnetic mode (TM_{0}) can exist and propagate in this structure. The bus waveguide filled with air is cut off by the SASB at the center, and *t* is the thickness of the barrier. In addition, *G*_{2} is the distance between the ROSRC and the bus waveguide, and *R*_{1} and *R*_{2} are the inner and outer radii of the ROSRC so the effective radius of the ROSRC can be defined as *R*=(*R*_{1}+*R*_{2})/2. In addition, the geometric centers of the SCRC, the ROSRC, and the bus waveguide with the SASB are on the same line.

Based on the standing wave theory, the resonance occurs when the following resonance condition is met [19,40]:

*λ*is the resonance wavelength,

*L*

_{eff}is the effective length of the resonator,

*ϕ*is the phase shift induced by the reflection, Re(

*n*

_{eff}) is the real part of the effective refractive index, and

*m*is the order of resonance mode.

## 3. Results and discussion

The finite element method (FEM) is used to analyze the propagation features of the SPPs. At the top and bottom of the structure, we set the perfect matching layers (PMLs) to absorb the escaping waves. The fine triangular meshes are selected to ensure the perfect segmentation of the simulation area, and the maximum triangular mesh size is 10 nm. In order to reduce the need of running memory and the runtime of simulation, two-dimensional waveguide structure is chosen and it is assumed that the size of the z direction is infinite.

In order to investigate the optical properties of the structure, the transmission spectra of the structure without the SASB, the bus waveguide with the SASB, and the whole structure are shown in Fig. 2. The geometric parameters of the structure are set as *L*=270 nm, *G*_{1}=*G*_{2}=10 nm, *t*=60 nm, and *R*=165 nm, and the thicknesses of the double silver barriers in the SASB are 5 nm. It is known that Fano resonance can be realized by the interference of the narrow discrete state and the broad continuum state. In Fig. 2, as shown in the red line, there are three transmission dips at 795 nm, 984 nm and 1554 nm in the transmission spectrum of the structure without the SASB, which could be treated as narrow discrete states. Meanwhile, the bus waveguide with the SASB can produce a broad continuum state indicated by the blue line. As a result of the interference of the discrete states and the continuum state, triple sharp and asymmetrical Fano resonance peaks (marked as FR1, FR2, and FR3) appear at the wavelengths of 786nm, 933nm, and 1542nm, shown in the black line. To further understand the mechanism of Fano resonances of the structure, the magnetic-field patterns of FR1, FR2 and FR3 are displayed in Figs. 3(a), 3(b) and 3(c), and the height expressions of the magnetic-field patterns are given in Figs. 3(d), 3(e) and 3(f). It can be seen that almost all the energy is confined in the ROSRC at 786 nm and 1542 nm, indicating that both FR1 and FR3 may be more sensitive to the ROSRC. The magnetic-field patterns show that the effective length of the resonator can be approximated as *L*_{eff}=(2×π×*R*)/2=π*R* for FR1 and FR3, and the orders of resonance mode of FR1 and FR3 are 2 and 1 respectively. Oppositely, there is almost no energy distribution in the ROSRC at 933 nm, and most of the energy of FR2 is distributed in the SCRC, meaning that FR2 may be more sensitive to the SCRC. In this case, the effective length of the resonator can be approximated as *L*_{eff}=*L* for FR2, and the order of resonance mode of FR2 is 1.

Next, we investigate the influences of the ROSRC and SCRC sizes on Fano resonances of the structure. In Fig. 4(a), the effective radius of the ROSRC increases from 160 nm to 175 nm with a step of 5 nm, and the width of the ROSRC is 50 nm. The approximately linear relationships between the effective radius of the ROSRC and the resonance wavelengths of FR1, FR2 and FR3 are shown in Fig. 4(b). In this case, the effective length of the resonator for FR1 and FR3 is changed, but is fixed for FR2. Therefore, with the increase of the effective radius, FR1 and FR3 make redshifts from 764 nm to 828 nm and from 1498 nm to 1630 nm respectively, but FR2 keeps fixed. Figure 5(a) shows the transmission spectra when the length of the SCRC increases from 260 nm to 290 nm with a step of 10 nm, and Fig. 5(b) shows the approximately linear relationships between the length of the SCRC and the resonance wavelengths. At this point, the effective length of the resonator for FR2 is changed, but is fixed for FR1 and FR3. As the length of the SCRC increases, FR2 makes a redshift from 906 nm to 990 nm. However, the resonance wavelengths of FR1 and FR3 are not affected. The results here demonstrate that FR1 and FR3 can be simultaneously tuned by altering the effective radius of the ROSRC, and FR2 can be tuned by changing the length of the SCRC, meaning that the triple Fano resonances can be tuned independently. Therefore, we can obtain different Fano resonances which may be applied to various optical sensing by changing the geometric parameters. However, this must be completed in the process of simulation, because it is impractical to change the geometric parameters once the structure is fabricated.

Based on the triple Fano resonances, we first discuss the refractive index sensing characteristics of the structure, and here the ROSRC and the SCRC have the same refractive index. The refractive index is increased from 1.25 to 1.35 with a step of 0.02, and the range of refractive index here is meaningful for liquid and biological sensing. The refractive indies of other parts are fixed. The corresponding transmission spectra are shown in Fig. 6(a). It can be seen that the resonance wavelengths of the triple Fano resonances all appear considerable redshifts. Figure 6(b) shows the approximately linear relationships between the refractive index of the ROSRC and SCRC and the resonance wavelengths. It can be seen that FR1, FR2, and FR3 all have good performances in linearity. The linear correlation coefﬁcients of FR1 and FR2 are 0.99969 and 0.99995, while the linear correlation coefﬁcient of FR3 is up to 0.99997. The sensitivity (*S*) of refractive index sensing is usually defined as the shift of the resonance wavelength (Δ*λ*) per unit variation of the refractive index (Δ*n*), i.e. *S* =Δ*λ*/Δ*n* [41–43]. Thus, the sensitivities are about 759.21 nm/RIU for FR1, 928.77 nm/RIU for FR2, and 1550.38 nm/RIU for FR3, respectively. It should be noted that compared with FR1 and FR2, FR3 has the largest sensitivity. From Eq. (2), it can be seen that the sensitivity is proportional to *L*_{eff}/*m*. FR1 and FR3 have the same effective length of the resonator, but FR1 has the larger order of resonance mode. Therefore, the sensitivity of FR1 is smaller than that of FR3. Meanwhile, FR2 and FR3 has the same order of resonance mode, but FR2 has the smaller effective length of the resonator, meaning that the sensitivity of FR2 is also smaller than that of FR3. The comparison of the sensitivity with other published papers is shown in Table 1 [11,13,14,37,38,41,44–47].

Furthermore, the figure of merit (*FOM*) is another key factor to describe the sensing performance, which is defined as *FOM* =Δ*T*/(*T*·Δ*n*), where *T* is the transmission of the structure and Δ*T* is the relative transmission variation (induced by the refractive index change Δ*n*) at a fixed wavelength [20,48]. The maximum *FOM* of the structure is as high as 7358 at 996 nm due to the sharp and asymmetric Fano line-shape and ultralow transmission.

The MIM waveguide structure can also be used for sensing of biological parameters such as glucose concentration and ethanol temperature, which can determine the refractive index. When the structure is invoked as glucose concentration or ethanol temperature sensing, the air in the ROSRC and SCRC could be replaced by glucose solution or ethanol. To simulate the practical sensing condition of biological parameters, the refractive index of glucose solution is described as [37,49]:

where*C*is the concentration of glucose solution. Equation (3) specifies the linear relationship between the concentration of glucose and the refractive index. In Fig. 7(a), the concentration of glucose solution in the ROSRC and SCRC increases from 0 g/L to 200 g/L with a step of 50 g/L. In this case, the refractive index of the ROSRC and SCRC is increased from 1.33231 to 1.35608 with a step of 0.00594. It can be seen that the resonance wavelengths of the triple Fano resonances exhibit redshifts. Figure 7(b) shows the approximately linear relationships between the concentration of glucose solution and the resonance wavelengths, and the linear correlation coefficients of FR1, FR2 and FR3 are 0.99964, 0.99838, and 0.99994, respectively. The sensitivity of glucose solution sensing is

*S*

_{glouse}=Δ

*λ*/Δ

*C*. Therefore, the sensitivities are 0.08nm·L/g for FR1, 0.1nm·L/g for FR2, and 0.186nm·L/g for FR3.

On the other hand, the relationship between the refractive index of ethanol and temperature is as follows [50,51],

where*T*

_{0}is the room temperature (20 °C) and

*T*is the ethanol temperature. To investigate the sensing properties of the structure, the temperature increases from 10°C to 50°C with a step of 10°C. In this case, the refractive index of the ROSRC and SCRC is decreased from 1.3992 to 1.2456 with a step of 0.0384. It can be seen from Fig. 8(a) that FR1, FR2, and FR3 all have evident blueshifts with the increase of temperature. Figure 8(b) shows the approximately linear relationships between the temperature of ethanol and the resonance wavelengths. The linear correlation coefﬁcients of FR1 and FR2 are 0.99965 and 0.99986, while the linear correlation coefﬁcient of FR3 is up to 0.99997. The sensitivity of ethanol temperature sensing is

*S*

_{ethanol}=Δ

*λ*/Δ

*T*. Therefore, the sensitivities are 2.96 nm/°C for FR1, 3.51 nm/°C for FR2, and 5.89 nm/°C for FR3. The results above show that the structure may have a number of applications in the field of biosensing.

In practice, the fabrication accuracy is one of the most important factors to obtain good sensing performances. As discussed above, the sensing performances of the structure depend heavily on the geometric parameters. Low fabrication accuracy can cause the geometric parameters to deviate from the predetermined values, and thus reduce the sensing performances such as sensitivity and linearity. Furthermore, effectively filling the structure with measured media such as glucose solution and ethanol is another difficulty for sensing. The uniformity of the filling can affect the distribution of refractive index in the structure. Therefore, the non-uniformity filling of measured media can produce large sensing errors, which should be avoided.

## 4. Conclusions

In this work, a MIM waveguide structure which is composed of a SCRC, a ROSRC and a bus waveguide with a SASB is proposed to achieve triple Fano resonances. The Fano resonances can be well tuned independently by altering the effective radius of the ROSRC and the length of the SCRC. The maximum sensitivity of refractive index sensing is 1550.38 nm/RIU, with the maximum *FOM* of 7358. Meanwhile, the linear correlation coefficient between the resonance wavelength and the refractive index is up to 0.99997. Based on the refractive index sensing, the structure can also be applied in sensing of some biological parameters such as glucose concentration and ethanol temperature. The maximum sensitivities of glucose solution concentration sensing and ethanol temperature sensing are 0.186nm·L/g and 5.89 nm/°C, respectively. These results may provide helpful references in designing on-chip plasmonic nano-sensing.

## Funding

Fundamental Research Funds for the Central Universities (2572019BC04); Natural Science Foundation of Heilongjiang Province (LH2019F041).

## 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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