Introduction

Surface plasmon polaritons (SPPs) are the electromagnetic waves, which can be generated by the interaction between incident light and metal structure [1, 2]. The amplitude of SPP optical field decays exponentially in the direction perpendicular to the metal-insulator interface [3, 4]; therefore, SPPs only exist on the interface between metal and dielectric. Because the SPPs has the excellent characteristics of breaking the diffraction limit [5, 6], it has very important applications in nanolevel optical manipulation, transmission, processing, and control in high-density photonic integrated circuits [7,8,9,10]. In recent years, the various devices based on SPPs, such as the metal-insulator-metal (MIM) waveguide structures, have received more attention and have been extensive researched because of their strong local area capability and acceptable propagation length [11,12,13].

At present, people have conducted extensive research on Fano resonance [14, 15] and electromagnetic induced transparency (EIT) [16, 17] effect based on the interaction between incident light and metal nanostructure. Fano resonance initially originated from the interference between continuous state and discrete state in the ionization process of condensed matter physics, and its resonance line shape has an obvious asymmetry type [13]. Fano resonance shows good sensitivity to the changes of structural parameter and environmental refractive index, and has important applications in biosensing, lasers, and optical modulators [18, 19]. Several sensors based on MIM waveguide structure have been studied and reported now, which indicate that Fano resonance–based refractive index sensors can be implemented in MIM waveguide structure [20,21,22].

In this paper, a MIM waveguide structure is designed, which is coupled with two stub resonators and a ring resonator. By the numerical simulation based on the finite element method, the transmission spectrum and magnetic field distribution are analyzed. Changing the structure parameters of the MIM waveguide and the refractive index of the filling medium of the ring resonator cavity, the simulation results show that the functions of refractive index sensor and band-stop filter based on multi-Fano resonance can be realized simultaneously.

Model and Calculation Method

Figure 1 shows a two-dimensional (2D) schematic diagram of a coupled system consisting of a MIM waveguide with two stub resonators and a ring resonator. When the depth of the z-direction is deep enough, the simulation results of the three-dimensional model are the same as that of the two-dimensional model and the calculation complexity is greatly reduced; therefore, the two-dimensional model is used in the whole calculation [23].

Fig. 1
figure 1

Two-dimensional schematic diagram of the MIM waveguide system

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In Fig. 1, the blue part represents silver and the white part represents air (n = 1.0). The width of the bus waveguide is W. As can be seen from Fig. 1 of reference [24], the SPPs can be excited by the input light, which is the transverse magnetic (TM) field mode obtained from boundary mode analysis. The coupling distance between the bus waveguide and the ring cavity is g. The lengths of the two stubs are expressed as H1 and H2, and their widths are W1 and W2, respectively. The distance between the left stub and the ring resonator is L1, the distance between the right stub and the ring resonator is L2, and L1 is not equal to L2. The inner radius and outer radius of the ring resonator are denoted by r and R, respectively. Initially, the filled medium is air and its refractive index is n = 1.0.

In the calculation process, the relative dielectric constant of the silver material we chose can be defined by the Debye-Ruder dispersion model [25, 26]:

$$\varepsilon \left(\omega \right)={\varepsilon }_{\infty }-\frac{{\omega }_{\mathrm{p}}^{2}}{\omega (\omega +i\gamma )}$$
(1)

In Eq. 1, the plasma frequency of silver is ωp = 1.37 × 1016 s−1, the damping constant is γ = 2.73 × 1013 s−1, and the dielectric constant of the infinite frequency is ε∞ = 3.7.

For the proposed waveguide coupled system, the sensing performance is studied by analyzing the resonance wavelength shifts. The resonant wavelength and the real part of the MIM waveguide structure’s effective refractive index based on the standing wave theory [27] can be represented as Eqs. (2) and (3), respectively.

$$\lambda_m=\frac{2\mathrm{Re}(n_{\mathrm{eff}})L}{\mathrm m-\psi/\pi}m=1,2,3\cdots$$
(2)
$$\mathrm{Re}\left({n}_{\mathrm{eff}}\right)={\left[{\varepsilon }_{m}+{(k/{k}_{0})}^{2}\right]}^{1/2}$$
(3)

where L represents the effective length of the cavity, ψ represents the phase shift caused by the reflection of SPPs at the interface of the metal and insulator, and the positive integer m represents the number of antinodes of the standing wave SPPs.

To illustrate the Fano resonance phenomenon by the temporal coupled mode theory [28] in detail, the amplitudes of the SPP wave of the structure are denoted by Si ± (i = 1, 2). The subscripts denote the input and output electromagnetic wave of the cavity (as shown in Fig. 1), respectively. The coupling coefficients between the input bus waveguide and the first stub, between the bus waveguide and the ring cavity, and between the output bus waveguide and the second stub are κ1, κ2, and κ3, respectively. When an optical wave with a certain frequency is launched only from the input port of the MIM waveguide (S2+  = 0), the time evolution amplitude of the stubs AS and the time evolution amplitude of the ring cavity AR can be derived as

$$\frac{\partial A_{\mathrm s}}{\partial t}=\left(j\omega_s-\kappa_1-\kappa_2-\kappa_3\right)A_{\mathrm s}+\kappa_1S_1+\kappa_2S_2+\kappa_3A_R$$
(4)
$$\frac{\partial A_{\mathrm R}}{\partial t}=\left(j\omega_{\mathrm R}-\kappa_3\right)A_{\mathrm R}+\kappa_3{\mathrm A}_S$$
(5)

where j is the imaginary unit (j2 = −1) and ωs and ωR are the resonance frequencies of the stubs and the ring cavities, respectively. Eqs. 6 and 5

In accordance with the energy conservation law, the amplitude of the input and output optical waves in the coupled waveguides are derived by Eq. 6

$${S}_{2-}={S}_{1+}+\sqrt[2]{{K}_{2}}{A}_{\mathrm{s}}$$
(6)

The transmittance T can be obtained as follows: Eq. 7

$$T=\left|\frac{s_{2-}}{s_{1+}}\right|^2=\left|\frac{\kappa_1\kappa_2\left[j\left(\omega-\omega_R\right)+\kappa_3^2\right]}{\left[j(\omega-\omega_{\mathrm S})+\kappa_1^2+\kappa_2^2{+\kappa}_3^2\right]\left[j(\omega-\omega_{\mathrm R})+\kappa_3^2\right]{-\kappa}_3^2}\right|^2$$
(7)

In the following section, the propagation characteristics of the coupling structure are simulated by COMSOL Multiphysics software. During the simulation, the absorbing boundary of the bottom and top of the coupled structure is set as the same PML condition.

Results and Discussion

Transmission Properties of the MIM Waveguide System

In order to investigate the transmission spectrum properties of this MIM waveguide system, it must simulate the MIM waveguide structure under varying parameters. The initial values of the MIM waveguide structural parameters are W = 50 nm, W1 = W2 = 100 nm, L1 = 560 nm, L2 = 500 nm, R = 300 nm, r = 160 nm, H1 = H2 = 720 nm, and g = 20 nm.

Firstly, the transmission properties of the MIM structure are examined, and the results are shown in Fig. 2. It is clear that four Fano resonances with asymmetric lines can be obtained in the transmission spectrum of the MIM structure. These four Fano resonances are labeled as F1, F2, F3, and F4, and their resonance peak positions are λ1 = 829 nm, λ2 = 1039 nm, λ3 = 1312 nm, and λ4 = 1625 nm, respectively. The reflection losses of the four resonance positions are −27.1 dB, −21.5 dB, −20.2 dB, and −28.3 dB, respectively, and they are all less than −20 dB, which shows that our simulation process is appropriate. In addition, it can be found that a U shape appears in the transmission spectrum from 1039 to 1312 nm, so the designed structure achieves band-stop filtering function. Interestingly, this U-shaped transmission spectrum is composed of two opposite Fano lines, which are F2 and F3. The band-stop width is defined as the range of wavelengths with transmittance less than 1%. Therefore, the stop band formed by the U-shaped transmission spectrum begins at λ5 = 1092 nm and ends at λ6 = 1241 nm, respectively, so the stop band width is Δ = 149 nm.

Fig. 2
figure 2

Transmission characteristics of the MIM waveguide structure

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In order to better understand the specific resonance conditions of different Fano resonance, Fig. 3a–d show the magnetic field distributions of F1, F2, F3, and F4, respectively. Figure 3e, f are the magnetic field distributions at λ5 = 1092 nm and λ6 = 1241 nm. Firstly, we can easily find that both stubs in the MIM waveguide structure participate in the four Fano resonances. However, on the one hand, it can be found that the ring resonator participates in the resonance F1 and F4 as shown in Fig. 3a, d. On the other hand, the ring resonator does not participate in the resonance F2 and F3 as shown in Fig. 3b, c, those two resonance results are only caused by the two stubs. The ring resonator can be used to regulate the two resonances, F1 and F4. Secondly, as shown in Fig. 3e, f, the SPPs are blocked by the second stub, and normal transmission cannot be continued at λ5 = 1092 nm, while the SPPs can transmit normally at λ6 = 1241 nm. The two Fano resonances F2 and F3 on the U-shaped line both result from the coupling of the strong trapped resonance in the FP resonator and weak resonance in the tooth cavity. Therefore, the positions of the peaks can be controlled by the space between the teeth-shaped cavities, which means that the band-stop width of the U-shaped transmission spectrum can be regulated finely.

Fig. 3
figure 3

Magnetic field distributions of the MIM waveguide at (a) λ1 = 829 nm, (b) λ2 = 1039 nm, (c) λ3 = 1312 nm, (d) λ4 = 1625 nm, (e) λ5 = 1092 nm, and (f) λ6 = 1241 nm, respectively

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It can be found that the MIM waveguide structure proposed in this paper, which provides a way to realize multiple Fano resonances and a large band device with a certain band-stop filtering function. In order to understand the mechanism of Fano resonances and optimize the performance the band-stop filter, the transmission spectra of the MIM waveguide structure under different conditions are simulated.

Influence of Parameters W 1 and W 2 on the Transmission Spectrum of the MIM Waveguide

As shown in Fig. 4, the effect of the width of two stubs on the transmittance of the MIM waveguide was studied. As shown in Fig. 4a, for F1 and F4 resonance, it can be seen that the resonance positions and resonance intensities have not changed when W1 and W2 increase at the same time. This behavior is also similar to independent increase of W1 or W2, as Fig. 4b, c show. The resonance positions and resonance intensities of F1 and F4 are not adjusted by the stubs because they are determined by the ring resonator. When W1 and W2 increase at the same time or increase alone, F2 resonance blue shifts significantly, and its intensity decreases. However, the decrease of F2 resonance intensity when W1 and W2 increase together is significantly less than the decrease when W1 or W2 increased alone. For F3 resonance, when W1 and W2 increase at the same time, it will undergo a significant blue shift. However, when W1 or W2 is increased alone, the position of F3 resonance will have a very weak blue shift, which can be regarded as unchanged basically. For the width of the stop band, when W1 and W2 increase at the same time, the blue shift degrees of F2 and F3 are similar, so the width of the stop band also blue shifts significantly while the stop width of the band remain unchanged. But when W1 or W2 increases alone, especially when W1 or W2 is greater than 200 nm, although the resonance valley of F3 remains unchanged, the F2 resonance mode basically disappears, which directly leads to the width of the stop band increases rapidly. It can be seen that increasing the widths of the two stubs W1 and W2 at the same time can change the band range of the band-stop filter, increasing the width of the two stubs W1 and W2 alone can expand the stopband bandwidth of the band-stop filter.

Fig. 4
figure 4

Influence of parameters W1 and W2 on the transmission spectrum of the MIM waveguide, when (a) W1 and W2 are changed, (b) only W1 is changed, and (c) only W2 is changed

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Influence of Parameters L 1 and L 2 on the Transmission Spectrum of the MIM Waveguide Structure

As shown in Fig. 5a, b, the influence of changing the horizontal distance L1 and L2 between the two stubs and the center of the ring on the transmittance of the MIM waveguide is studied. When L1 and L2 increase, respectively, the resonance positions and resonance intensities of F1 and F4 remain unchanged, because F1 and F4 are determined by the ring resonator and are not adjusted by the stubs. Both F2 and F3 resonances have a significant red shift with the increase of L1 or L2. However, when L1 is greater than 760 nm or L2 is greater than 700 nm, F2 resonance disappears basically. So, changing the value of L1 or L2 is another effective method to expand the stopband bandwidth of the band-stop filter.

Fig. 5
figure 5

Influence of parameters: (a) L1 and (b) L2 on the transmission characteristics of the MIM waveguide

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Influence of Parameters r and g on the Transmission Spectrum of the MIM Waveguide Structure

As shown in Fig. 6a, the change in the radius of the ring resonator r also has a significant effect on the transmittance of the MIM waveguide. As the inner radius r of the ring resonator increases from 140 to 180 nm, the resonance intensity of F1 and F4 increases rapidly and a significant redshift occurs, and the redshift degree of F4 is significantly greater than that of F1, indicating that F4 is more sensitive to the change of the inner ring. At the same time, it can also be found that the resonance position and resonance intensity of F2 and F3 remain unchanged, which also verifies the conclusion that changing the parameters of the ring will not affect F2 and F3 resonance. Therefore, the U-shaped transmission spectrum determined by F2 and F3 and the stopband filter with a stopband width of Δ = 149 nm have strong stability characteristics.

Fig. 6
figure 6

Influence of parameters (a) r and (b) g on the transmission characteristics of the MIM waveguide structure

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When the distance between the ring resonator and the waveguide g is changed, the transmittance of the MIM waveguide is also affected, as shown in Fig. 6b. At this situation, it can be found that when the value of g increases from 10 to 50 nm, the resonance positions of F1 and F4 are almost unchanged, and the modulation depth of F1 and F4 decreases, so the resonance intensity decreases. From Fig. 6b, it can be seen that as g further increases to 60 nm, the F1 and F4 resonances will disappear, and the ring resonator will no longer participate in the coupling. In this case, the transmission characteristic of the waveguide structure is the same as the transmission characteristic without the ring resonator, as shown by the dotted curve in Fig. 6b.

Influence of Refractive Index Environments on the Transmission of the MIM Waveguide Structure

In recent years, the MIM structure have been intensively studied as refractive index–sensitive sensors in the biological and chemical fields, such as the solution concentration and pH value, which can be measured via refractive index changes. In this part, the transmission spectra of the proposed MIM waveguide structure under different refractive index conditions are studied.

Figure 7a shows the sensitivity of the MIM waveguide structure as the variable refractive index media only fills the ring resonator. At this case, the results show that when the refractive index of the medium increases from 1 to 1.08, both F1 and F4 modes have a significant red shift. F2 and F3 modes did not change with the refractive index of the medium in the ring resonator. Figure 7b shows the sensitivity of the entire MIM waveguide structure when the variable refractive index medium fills the entire MIM waveguide. The results show that when the refractive index of the medium increases from 1 to 1.08, F1, F2, F3, and F4 modes all have obvious red shifts. Comparing Fig. 7a, b, it can be proved once again that the ring resonator can control the F1 and F4 modes.

Fig. 7
figure 7

Influence of environment refractive index on the transmission characteristics of the MIM waveguide structure. (a) The variable refractive index medium only fills the ring resonator. (b) The variable refractive index medium fills the entire MIM waveguide

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In sensing application region, the figure of merit (FOM) is an important factor to be considered to measure the sensing properties of the devices. The FOM is defined by the following equation [29] Eq. 8:

$$\mathrm{FOM}=\frac{S}{\mathrm{FWHM}}=\frac{\delta \lambda /\delta \mathrm{n}}{\mathrm{FWHM}}$$
(8)

where \({{S{ = }\delta \lambda } \mathord{\left/ {\vphantom {{S{ = }\delta \lambda } {\delta n}}} \right. \kern-\nulldelimiterspace} {\delta n}}\) is the refractive index sensitivity which is calculated by resonance peak shift per index refractive of external medium, \(\delta \lambda\) is the shift quantity of the resonance peak, \(\delta n\) is the change quantity of medium refractive, and FWHM is the full width at half maximum of the resonance line.

As shown in Table 1 and 2, the refractive index sensitivity and FOMs of the proposed MIM waveguide structure under different refractive index conditions are studied. In the first situation, when the media with different refractive indexes are only added to the ring resonator of the MIM waveguide structure, the refractive index sensitivity of F1 and F4 modes reached 800 nm/RIU and 1650 nm/RIU, respectively, and the FOM reached 88.9 and 117.8 in magnitude, respectively. In the second situation, when the same medium with different refractive index is added to the entire MIM waveguide structure, all four Fano resonances exhibit good refractive index sensitivity. Among F1, F2, F3, and F4, the refractive index sensitivity of F1 and F4 modes reached 800 nm/RIU and 1600 nm/RIU, which are basically similar to the first case. However, the FOM values are respective reached 100 and 123.2 in magnitude, which was slightly higher than the first situation that mainly due to the strengthening of the resonance of F1 and F4 at this time.

Table 1 Refractive index sensitivity and FOM of the MIM waveguide when the variable refractive index media only fills the ring resonator
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Table 2 Refractive index sensitivity and FOM of the MIM waveguide when the variable refractive index media fills the whole MIM waveguide
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In addition, when the entire MIM waveguide structure is immersed in a different refractive index environment, F2 and F3 modes will also produce good refractive index sensitivity to environmental change, reaching 1000 nm/RIU and 1300 nm/RIU, respectively. However, due to the fact that the FWHM of F2 and F3 mode is relatively large, which directly causes their FOM values to be very small, which can only reach 23.8 and 18.1 in magnitude, respectively.

Eventually, under two different conditions, the refractive index sensitivity of the four Fano resonances in the MIM waveguide is much greater than that of sensors based on plasmonic nanostructures [30, 31]. In the meantime, for the superior sensitivity and FOM, the MIM waveguide structure is very sensitive to refractive index changes in the close vicinity of the waveguide structure surface, which fully demonstrate that the MIM waveguide structure with a smaller volume has potential applications in the near-infrared refractive index sensor and biosensors.

Conclusion

In this paper, through the finite element method, the transmission spectrum characteristics of the MIM waveguide structure composed of two stubs resonator and a ring resonator are calculated. The results show that four Fano resonances can be realized in this MIM waveguide structure; furthermore, it can be confirmed that the Fano resonances F1 and F4 are adjusted by the ring resonator, and the Fano resonances F2 and F3 are controlled by the two stub resonators. Meanwhile, the two Fano resonances, F2 and F3, can form a stopband filter with a 149-nm stopband width. In addition, the MIM waveguide device has ideal sensitivity to changes in the refractive index of the surrounding environment, and can be used to identify materials with different refractive index characteristics. Therefore, the MIM waveguide structure we proposed can not only effectively excite and flexibly adjust multi-Fano resonances but also can be applied to optical devices that rely on Fano resonance, such as the stop-band filters and refractive index sensors devices.