## Abstract

Based on the theory of surface plasmon polaritons (SPPs), a metal–insulator–metal (MIM) waveguide structure consisting of a streamlined resonant cavity and a baffle is proposed. When the incident light waves enter into the waveguide structure, through the interaction between the three discrete states and the continuous state by the streamlined cavity and the baffle, three sharp asymmetric Fano resonance spectra can be formed. The transmission spectrum characteristics of the structure are simulated and analyzed by the finite element method (FEM), and the effects of structure parameters and refractive index on the transmission spectra characteristics are studied. By optimizing the structure parameters, it is found that the performance of the system can be adjusted and optimized flexibly by changing the structure parameters, the figure of merit (FOM) can reach 1.99 × 10^{6}, and the sensitivity is 2960 nm/RIU. This flexible Fano resonant-structure has several applications in micro-nano biosensor, nonlinear optics, and slow light devices.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## Corrections

15 April 2020: A typographical correction was made to the author listing.

## 1. Introduction

Surface plasmon polaritons (SPPs) have the ability to overcome the limit of optical diffraction and is one of the most promising choices for realizing of high-density integrated optical circuits [1]. In recent years, more and more attention has been paid to MIM structure waveguide based on SPPs. MIM structure has very important application value in high density integrated photon circuit [2–5] because of its high constraint, low loss, long propagation length and easy manufacture. Researchers have proposed and studied a large number of devices based on MIM waveguides to achieve various functions, such as filters [6–11], splitters [12,13], sensors [14–20] etc.

In metallic micro-nano structures, Fano resonance as a fundamental resonance effect is formed by the interference of continuous energy bands (bright modes of broad radiation) and discrete energy levels (dark modes of narrow non-radiation) [21,22]. Unlike Lorentzian resonance, Fano resonance has a typical sharp asymmetric transmission spectrum and is extremely sensitive to structure parameters and surrounding environment. With these characteristics, high sensitivity and figure of merit (FOM) can be obtained, and because Fano resonance has the advantages of enhancing biochemical sensing spectrum, it is combined with the advantages of MIM SPPs waveguide structures. Thus, it shows a good application prospect in biochemistry sensors.

In recent years, there have been many studies on the design of Fano resonance sensing structures using MIM surface plasmon polaritons waveguides. Yang et al [23] have proposed a side-coupled square cavity structure, which produces multistage Fano resonance with adjustable mode, and the sensitivity and FOM are up to 1120 nm/RIU and 1.7×10^{5} respectively. The M-type resonant cavity structure proposed by Qiao et al [24] have the highest FOM of 1.56 × 10^{5}, and its sensitivity is not high enough, which is 780 nm/RIU. The MIM waveguide and rectangular side-coupled cavity system proposed by Chen et al [25] has a sensor sensitivity of 1280 nm/RIU, and multiple Fano resonance peaks can be obtained by increasing the number of rectangular cavity to achieve multiple Fano resonances control. Studies have shown that when multiple Fano resonances are used simultaneously in sensing, the transmission spectrum can be modulated simultaneously in multiple bands to reduce radiation broadening and effectively remove some external environmental factors. Therefore, the study of multi-Fano resonance also opens up a new way for the design of multi-band sensors. All these structures excite dark states by changing the propagation direction of the incident light, resulting in partial reflection, and then couple with bright states to generate Fano resonance. The dark state can also be excited by breaking the symmetry of the structure, and the Fano resonance can be generated by coupling with the bright state, such as the symmetrical bilateral groove coupled cavity structure [26], the asymmetrical bilateral groove structure [13]. Fano resonance is also realized by superimposing resonators, such as end-coupled cascaded-ring [27] and coupled cascade ring structure [7].

These existing research results may be helpful to further study the principle and application of Fano resonance, and can also guide people to design some integrated surface plasmon polaritons optical waveguide devices based on Fano resonance, such as biosensor [28], all-optical switch [29], slow-light device [30], etc. However, in terms of the structure designed in the current literatures, both the sensitivity and the FOM of the transmission characteristic index have not reached a relatively high value at the same time. Therefore, based on the previous studies, which originated from the popular streamline style of scientific research and industrial production, a streamlined resonator structure based on baffle MIM waveguide coupling is proposed in this paper. When the light propagates in a waveguide with a metal baffle, it forms a broad continuous state energy band, which is reflected into a streamlined cavity and forms three narrower discrete state resonance troughs. Under the action of the near field, the continuous state and the discrete state are coupled to each other to form three modes of Fano resonance. The effects of structural parameters and refractive index of environmental media on transmission characteristics are studied by numerical simulation based on finite element method. Finally, by optimizing the parameters of the structure, high sensitivity and FOM are obtained. This structure has great potential applications in optical integrated circuits, optoelectronic devices, especially micro-nano biosensors.

## 2. Structural design and theoretical analysis

#### 2.1 Establishment of structural model

The coupled streamlined resonant cavity waveguide structure is composed of a MIM waveguide, a streamlined resonant cavity and a metal baffle. The structure schematic diagram is shown in Fig. 1, in which the blue and gray are metal Ag and dielectric layer air, respectively. The relative dielectric constant of air is ${\varepsilon _\textrm{d}} = 1$, and the relative dielectric constant of Ag is calculated by standard Drude model, which can be expressed as [31,32] :

_{2}, etc. Secondly, the coupled streamlined resonant cavity waveguide structure is etched on the metal layer by electron beam exposure and stripping or reactive ion etching, the cavity is filled with air.

In order to study the transmission characteristics of the structure, the COMSOL Multiphysics 5.2 software based on the finite element method (FEM) is used to simulate the transmission process of light in the structure. Firstly, the geometric model is established, and then the relevant calculation parameters are set up. In the simulation process, the upper and lower boundary of the waveguide structure adopts the perfectly matched layer (PML) absorption boundary condition, without incident field, and the scattering wave is the first-order plane wave. The light enters from the incident port, transmits through the waveguide structure, and finally outputs at the output port. The corresponding transmissivity and magnetic field distribution of the light in the waveguide structure are simulated and analyzed. In the simulation process, the mesh is divided into smaller triangular meshes, the largest element is 74 nm, the smallest element is 0.25 nm, the maximum element growth rate is 1.25, the curvature factor is 0.25, and the resolution of narrow area is 1, which ensures the accuracy of the calculation results.

The streamlined resonant cavity is composed of a semi-ellipse and a rectangle, where *a* and *b* are the lengths of the short axis and the long axis of the semi-ellipse respectively. The transmission spectra corresponding to different ratios of *a* and *b* are obtained by simulation, as shown in Fig. 2. As can be seen from the figure, when *a *< 0.5*b*, two obvious resonance peaks appear in this wavelength range, the sharp asymmetric Fano resonance on the left and the symmetric Lorentzian resonance on the right, and the resonance peaks shift blue with the increase of the ratio coefficient of *a* and *b*. When *a *> 0.5*b*, three obvious resonance peaks appeared. Only when *a *= 0.5*b*, the structural system can form two perfect Fano resonance spectrum. The length of the long axis of the half-ellipse is *b *= 1100 nm, the height of the rectangle is *l *= 535nm, the coupling distance between the streamlined cavity and the waveguide is *g *= 16 nm, and the width of the metal baffle is *t *= 25 nm. Since the wavelength is much larger than the width of the waveguide to ensure that only the transverse magnetic (TM) mode can propagate, the width of the waveguide is set to *w *= 50 nm. This relatively large structural volume also leads to higher FOM, which has good sensing performance [33–35]. The metal baffle in the waveguide structure causes the electromagnetic wave in the waveguide to be reflected partly, which disturbs the amplitude and phase of the wave propagating without the baffle, and thus, there is a complex interference phenomenon, resulting in the acutely asymmetric Fano resonance spectrum. Define the transmittance as [36]:

#### 2.2 Analysis of Fano resonance principle

When the incident light wave propagates in TM mode, SPPs will be generated on the metal surface of the sub-wavelength waveguide structure, which can break through the diffraction limit and propagate in the waveguide. The dispersion equation of TM mode in MIM waveguide is as follows [37]:

In the streamlined resonator cavity, the phase change of a period of wave propagation for one cycle is expressed as $\Delta \phi \textrm{ = }{{4\pi {n_{\textrm{eff}}}{L_{\textrm{eff}}}} \mathord{\left/ {\vphantom {{4\pi {n_{\textrm{eff}}}{L_{\textrm{eff}}}} \lambda }} \right.} \lambda } + 2\phi $, where ${L_{\textrm{eff}}}$ represents the effective resonance length of the cavity and $\phi $ is the phase change caused by the reflection of SPPs at the metal- insulator interface. When $\varDelta \phi \textrm{ = }2j\pi $ $(j = 0,1,2,3 \cdots )$, stable standing waves can be formed in the cavity, and $j = {2^{(m + n)}}$ represents the number of antinode of standing wave SPPs, where *m* and *n* represent the number of wave nodes, respectively. The mode (*m*, *n*) can be used to represent the different resonant modes of the resonator. Thus the resonance wavelength is defined as [38,39]:

In order to further understand the formation mechanism of Fano resonance, the steady-state magnetic field of the structural system is simulated and analyzed, as shown in Fig. 4(a)-(d). The three resonance modes of streamlined cavity structure are mode (1, 1), mode (1, 0) and mode (0, 1). Figure 4(a) shows the steady-state magnetic field distribution at FR1 resonance valley wavelength *λ*=1460 nm, in this case, there is less magnetic field distribution in the resonance cavity and a large number of magnetic field are concentrated at the incident end of the waveguide, this is because SPPs waves are coupled into the streamlined cavity and reflected, and when they are transmitted to the waveguide, they are opposite to the propagation phase of the waveguide, causing interference cancellation and zero transmission. Then we analyze FR1 and FR3, i.e. mode (1, 1) and mode (0, 1), and the steady-state magnetic field distribution at formant wavelengths *λ*=1608 nm and *λ*=3118 nm. As shown in Fig. 4(b) and 4(d), in mode (1, 1) (FR1), the antinode of the magnetic field in the cavity appears at the four corners of the cavity. The magnetic field of mode (0, 1) (FR3) appears at the top and bottom of the cavity in the antinode of the resonant cavity, so the bottom magnetic fields of these two modes are strong and easily leaked into the MIM waveguide. Moreover, SPPs are coupled into the streamlined cavity, reflected and transmitted to the waveguide with the same phase as the waveguide, causing interference grows, and causing a large number of SPPs wave pass through, thus a sharp asymmetric Fano resonance spectrum is formed. Figure 2(c) shows the steady-state magnetic field distribution at the resonance peak wavelength *λ*=2216 nm of FR2 and the approximate mode (1, 0). It can be seen from the figure that the magnetic field is mainly concentrated at the input port of the waveguide, and there is almost no magnetic field at the output port, so the transmittance of FR2 is relatively low and is close to zero transmission.

## 3. Analysis of structural transmission characteristics

The sensitivity S and the figure of merit FOM are used to evaluate the sensing performance of the system, which is expressed as [40]:

*n*changes one unit when the wavelength is fixed. For Fano resonance, the transmission spectrum is sharp asymmetric, and there is a sharp change from the resonance peak to the resonance valley, so a higher FOM can be obtained.

#### 3.1 Influence of parameters on transmission characteristics

Next, the effect of geometric parameters of the structure on the transmission spectrum is studied. Firstly, the influence of semi-elliptical long axis *b* is studied, *b* changes from 1060nm to 1140 nm, and other parameters remain unchanged (*l *= 535 nm, *t *= 25 nm, *g *= 16 nm). When *b* is 1060 nm, 1080 nm, 1100 nm, 1120 nm and 1140 nm, the transmission spectra are shown in Fig. 5(a). It can be seen from the figure that with the increase of the *b*, the transmission spectrum has a red shift. Figure 5(b) shows the relationship between the wavelength at the peak and valley of FR1 and FR3 and *b*. It can be seen from the figure that there is a good linear relationship between resonance wavelength and *b* (FR1:${{\varDelta \lambda } \mathord{\left/ {\vphantom {{\varDelta \lambda } {\varDelta b}}} \right.} {\varDelta b}} \approx 1.12$; FR3:${{\varDelta \lambda } \mathord{\left/ {\vphantom {{\varDelta \lambda } {\varDelta b}}} \right.} {\varDelta b}} \approx 1.8$). From formula (4), it can be seen that the effective propagation length of the cavity increases with the increase of *b*, so the resonant wavelength increases and the red shift occurs, which is in line with the change trend of the simulation results.

Then, we fix the length of the long axis of the semi-ellipse *b *= 1100 nm, change the height of the rectangle *l* from 495 nm to 575 nm, and keep other parameters unchanged. When *l* is 495 nm, 515 nm, 535 nm, 555 nm and 575 nm, the transmission spectrum is shown in Fig. 6(a). It can be seen from the figure that with the increase of *l*, the transmission spectrum also shows red shift, and the red shift of FR3 is more obvious. The linear relationship is shown in Fig. 6(b) (FR1:${{\varDelta \lambda } \mathord{\left/ {\vphantom {{\varDelta \lambda } {\varDelta l}}} \right.} {\varDelta l}} \approx 0.5$; FR3:${{\varDelta \lambda } \mathord{\left/ {\vphantom {{\varDelta \lambda } {\varDelta l}}} \right.} {\varDelta l}} \approx 1.85$). This is due to the effective length of the cavity will increase with the increase of *l*, and the resonance wavelength will increase, resulting in a red shift of the transmission spectrum. This also accords with the trend of simulation results. From the above comparison, it can be seen that the resonance wavelength is very sensitive to both the length of the long axis of the semi-ellipse *b* and the height of the rectangle *l*. Therefore, the structure system can slightly adjust its shape and the position of the wave peak by changing the length of the long axis of the semi-ellipse *b* or the height of the rectangle *l*. Figure 7(a) gives the FOM corresponding to different structure parameters *l*. The figure of merit of FR1 and FR3 are defined as FOM1 and FOM3, respectively. It can be seen from the figure that with the increase of *l*, FOM3 generally presented a change trend of increasing first and then decreasing, while FOM1 did not change obviously. When *l *= 355 nm, FOM1 reached the maximum value, whose value is FOM1 = 1150. The maximum value of FOM3 is 1.99×10^{6} when *l *= 435 nm. When the structure parameters *b *= 1100 nm, *l* = 435 nm, *g* = 16 nm, *t* = 25 nm, FOM3 is optimized to the maximum, as shown in Fig. 7(b) is the transmission and FOM values corresponding to different wavelengths of FR3 when the parameters are optimized. Here, FOM3 is 1.99×10^{6} and FOM1 is 1520.

#### 3.2 Influence of refractive index of medium on transmission characteristics

The sensing characteristics of the optimized structure are analyzed, the other parameters remain unchanged (*b *= 1100 nm, *l* = 535 nm, *g* = 16 nm, *t* = 25 nm), and the refractive index of the medium *n* is changed. When the value of *n* is 1.00, 1.01, 1.02, 1.03, 1.04 and 1.05, the transmission spectrum is shown in Fig. 8(a). As can be seen from the figure, with the increase of refractive index *n*, the Fano resonance peaks all red shift, and the peak of FR1 gradually decreases, but the peak value of FR3 has hardly changed. Formula (4) shows that the effective refractive index of the cavity increases with the increase of the refractive index of the medium, so the resonance wavelength increases, thus the red shift occurs. Moreover, there is a good linear relationship between the wave lengths of the peaks and dips and the refractive index of the medium, as shown in Fig. 8(b). The sensitivity S corresponding to the optimal FOM is 2960 nm/RIU. This performance index is higher than that of many waveguide structures. It shows that the change of refractive index of medium has a highly sensitive response, that is to say the structure system is very sensitive to environmental change and has a good sensing performance. The sensing performance compared to other structures are shown in Table 1.

## 4. Conclusion

In summary, based on the theory of surface plasmon polaritons (SPPs) and MIM waveguide, the Fano resonance of MIM waveguide structure composed of baffle and streamline cavity is studied. The simulation results show that the asymmetric sharp lines and resonance wavelength can be adjusted easily by changing the structure parameters and refractive index of the medium when light enters the waveguide structure. After parameter optimization, the sensitivity of the structure is 2960nm/RIU and the FOM is up to 1.99×10^{6}. Due to the high sensitivity, high figure of merit and controllable resonance mode of the waveguide structure, it will have a good application prospect in optical integrated circuits, especially in the design direction of biosensors.

## Funding

National Natural Science Foundation of China (No. 61631007); Fundamental Research Funds for the Central Universities (No.lzujbky-2018-K11).

## Acknowledgments

System numerical simulation was provided by COMSOL, Inc.

## Disclosures

The authors declare no conflicts of interest.

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