Introduction

Surface plasmon polaritons (SPPs), which originate from the interaction of incident photons and free electrons on the metal surface, propagate along with the metal–dielectric interface and have the potential to overcome the light diffraction limit, localization of light in subwavelength, and high level of integration capability [1,2,3]. SPPs have applications in optical devices such as switches [4, 5], sensors [6,7,8,9], integrated photonic devices [10,11,12], demultiplexers [13], and filters [14, 15]. One of the most promising waveguide structures is the metal–insulator-metal (MIM) waveguide, which has advantages such as low bending loss, simple structure, long propagation distance, deep sub-wavelength confinements, and easy integration [16,17,18,19,20].

Fano resonance is a fundamental resonance phenomenon caused by the interference between a continuum state and a discrete state and has typically sharp resonance peaks and asymmetrical line shapes [21, 22]. Fano resonance can be applied in lasers, slow light devices, and switches [23, 24]. Recently, Fano resonance based on the MIM waveguide structures has been widely studied and applied to sensors. For instance, Wang et al. investigated Fano resonances in a MIM waveguide with a baffle and a circular split-ring resonator cavity and realized the refractive index nanosensor with the maximum sensitivity of 1114.3 nm/RIU [20]. Chen et al. designed a MIM waveguide composed of a circular split-ring resonance cavity and a double symmetric rectangular stub waveguide to generate Fano resonance, and the sensitivity was up to 1180 nm/RIU with the figure of merit (FOM) of 5585.3 [25]. Liu et al. proposed a MIM waveguide structure consisting of a side-coupled rectangular cavity, a rightward opening semi-ring cavity, and a bus waveguide to realize independently tunable Fano resonances, and the sensitivity reached 1550.38 nm/RIU [26].

In this paper, we design a plasmonic structure of MIM waveguide consisting of a baffle waveguide and an r-shaped resonator and discuss its transmission characteristics and magnetic field distributions using the finite element method (FEM). Two Fano resonances are achieved by the interaction between a continuum state in the baffle waveguide and a discrete state in the r-shaped resonator. Then the influences of the geometrical parameters of the structure on the Fano resonances are investigated. Large sensitivity and FOM can be obtained by optimizing the geometrical parameters. Finally, the glucose concentration sensing characteristics of the structure are discussed.

Structural Model

The plasmonic structure of MIM waveguide is shown in Fig. 1, which consists of a single baffle waveguide and an r-shaped resonator. The r-shaped resonator is composed of a rectangular resonator and a quarter-ring resonator. The width W of the bus waveguide and r-shaped resonator is fixed to 50 nm to ensure that only the fundamental transverse magnetic mode exists in the MIM waveguide [27]. The width of the baffle is S, the coupling distance between the bus waveguide and the r-shaped resonator is g, the height of the rectangular resonator is H, and the effective radius of the quarter-ring resonator is defined as R' = (R + R0)/2, where R and R0 are the outer and inner radii of the quarter-ring resonator.

Fig. 1
figure 1

The schematic and geometrical parameters of the designed plasmonic structure

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In Fig. 1, the blue and white areas represent silver and air, respectively. Usually, the relative dielectric constant of air is \({\varepsilon }_{in}=1\), and the relative dielectric constant of silver is characterized by the Debye-Drude model [28] :

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

where \({\varepsilon }_{\infty }\) = 3.7 is the dielectric constant at infinite frequency, ω is the angular frequency of incident light in vacuum, \({\omega }_{p}\) = 1.38 × 1016 Hz is the bulk plasma frequency of free conduction electrons, and γ = 2.73 × 1013 Hz is the electron collision frequency. The fabrication of the plasmonic MIM waveguide structure begins with the preparation of an adequately thick Ag layer on a silicon substrate by using chemical vapor deposition (CVD) [29, 30]. Then, the r-shaped resonator and the bus waveguide can be etched on the Ag layer by electron beam etching.

According to the standing wave theory, constructive interference should occur when the resonance condition is satisfied: \(4\pi Re\left({n}_{eff}\right){l}_{eff}/\lambda +\phi =2m\pi , m=1, 2, 3, . . .\), where \({l}_{eff}\) is the effective length of the resonator [31,32,33]. Thus, the resonance wavelength can be derived as:

$$\uplambda =\frac{2Re\left({n}_{eff}\right){l}_{eff}}{m-\phi /2\pi }, m=1, 2, 3, . . .$$
(2)

where \(\phi\) is the phase shift introduced by the reflection in the resonance resonator, m is the resonant mode order, and \(Re\left({n}_{eff}\right)\) is the real part of the effective index \({n}_{eff}\).

The two-dimensional FEM is used to analyze the transmission characteristics of the plasmonic structure. We set the perfect matching layers at the top and bottom of the structure to absorb the escaping waves. Light is inputted from Port 1 and then outputted from Port 2. Therefore, the transmittance is expressed as \(\mathrm{T}={\left|{\mathrm{S}}_{21}\right|}^{2}\), where \({\mathrm{S}}_{21}\) is the transmission coefficient from Port 1 to Port 2 [34].

Results and Discussion

In order to investigate the optical properties of the plasmonic structure, the transmission spectra of the single baffle waveguide, the single r-shaped resonator, and the entire structure are numerically calculated and shown in Fig. 2. The geometrical parameters are set as R = 200 nm, R0 = 150 nm, H = 300 nm, S = 10 nm, and g = 10 nm. In Fig. 2, the baffle waveguide produces a continuum state, as indicated by the black line. Meanwhile, the r-shaped resonator coupled with the bus waveguide without the baffle produces a discrete state with two approximately symmetric Lorentzian-like troughs at 725 nm and 1275 nm, described by the red line. As a result, two Fano resonances are obtained by the interference between the continuum state and the discrete state, which are shown by the blue line in Fig. 2. To simplify the discussion, we name the left Fano resonance as FR1 and the right Fano resonance as FR2. In order to further investigate the two Fano resonances, the magnetic field intensity distributions of the plasmonic structure at λ = 720 nm (FR1 peak), 735 nm (FR1 dip), 1265 nm (FR2 peak), and 1305 nm (FR2 dip) are shown in Fig. 3. In Fig. 3a and c, the magnetic field intensity distributions at Fano resonance peaks are caused by the constructive interference between the continuum state and the discrete state, and the transmission of the structure is enhanced. However, at Fano resonance dips caused by the destructive interference [35], the energy is concentrated in the r-shaped resonator and cannot be transmitted through the structure, as shown in Fig. 3b and d.

Fig. 2
figure 2

Transmission spectra of the single baffle waveguide (black line), the single r-shaped resonator (red line), and the entire structure (blue line)

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Fig. 3
figure 3

Magnetic field intensity distributions of the plasmonic structure at a λ = 720 nm (FR1 peak), b λ = 735 nm (FR1 dip), c λ = 1265 nm (FR2 peak), and d λ = 1305 nm (FR2 dip)

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The transmission characteristics of the plasmonic structure can be tuned by varying the geometrical parameters. First, we investigate the influences of the coupling distance g and the baffle width S on the Fano resonances. Figure 4a shows the transmission spectra of the structure with different coupling distances. Here, the coupling distance is increased from 5 to 20 nm with an interval of 5 nm, and other parameters are set as R = 200 nm, R0 = 150 nm, H = 300 nm, and S = 10 nm. The increase of the coupling distance produces considerable blueshifts of the two Fano resonance wavelengths. The transmittance of the Fano resonances gradually decreases as the coupling distance increases, because the coupling strength between the r-shaped resonator and the baffle waveguide becomes weak. The relationship between the Fano resonance wavelength and the coupling distance is shown in Fig. 4b. It can be seen that FR1 and FR2 make blueshifts from 710 to 730 nm and from 1320 to 1235 nm, respectively. Figure 5a shows the transmission spectra of the plasmonic structure with different baffle widths. The baffle width is increased from 5 to 20 nm with an interval of 5 nm, and other parameters are set as R = 200 nm, R0 = 150 nm, H = 300 nm, and g = 10 nm. With the increase of the baffle width, the line shapes of the Fano resonances become sharper and more symmetric close to Lorentz resonance. As shown in Fig. 5b, the Fano resonance wavelengths hardly change with the baffle width. The resonance wavelengths of FR1 and FR2 change slightly from 715 to 720 nm and from 1260 to 1265 nm, respectively.

Fig. 4
figure 4

a Transmission spectra of the plasmonic structure with different coupling distances. b Relationship between the Fano resonance wavelength and the coupling distance

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Fig. 5
figure 5

a Transmission spectra of the plasmonic structure with different baffle widths. b Relationship between the Fano resonance wavelength and the baffle width

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Furthermore, the influences of the rectangular resonator height H and the outer radius of the quarter-ring resonator R on transmission spectra are also investigated. In Fig. 6a, the rectangular resonator height is increased from 290 to 320 nm with an interval of 10 nm, and other parameters are set as R = 200 nm, R0 = 150 nm, S = 10 nm, and g = 10 nm. The increase of the rectangular resonator height produces significant redshifts of the Fano resonance wavelengths, and the shift of FR2 is greater than that of FR1. Figure 6b shows the relationship between the Fano resonance wavelength and the rectangular resonator height. With the increase of the rectangular resonator height, FR1 and FR2 produce redshifts from 710 to 735 nm and from 1235 to 1325 nm, respectively. Transmission spectra of the plasmonic structure with different outer radii of the quarter-ring resonator are shown in Fig. 7a. With the increase of the outer radius, the transmittance of the Fano resonances gradually decreases. Meanwhile, FR1 and FR2 produce obvious redshifts from 700 to 755 nm and from 1225 to 1340 nm respectively, and the shift of FR2 is also greater than that of FR1, as shown in Fig. 7b.

Fig. 6
figure 6

a Transmission spectra of the plasmonic structure with different rectangular resonator height. b Relationship between the Fano resonance wavelength and the rectangular resonator height

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Fig. 7
figure 7

a Transmission spectra of the plasmonic structure with different outer radii. b Relationship between the Fano resonance wavelength and the outer radius

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Next, we investigate the refractive index sensing based on the effects of the refractive index of the r-shaped resonator on the resonance wavelengths. Generally, sensitivity (S) and FOM are crucial parameters for assessing the performances of sensors, which are defined as:

$$S=\frac{\Delta\uplambda }{\Delta n},$$
(3)
$$FOM=\frac{\Delta \mathrm{T}}{T\Delta n},$$
(4)

where Δλ is the resonance wavelength shift, Δn is the change of the refractive index, T is the transmittance, and ΔT is the transmittance change induced by Δn at a fixed wavelength [36].

The transmission spectra of the plasmonic structure with different refractive indices are shown in Fig. 8a, where the refractive index varies from 1.00 to 1.12 with an interval of 0.03 and other parameters are set as R = 200 nm, R0 = 150 nm, H = 300 nm, S = 10 nm, and g = 10 nm. It can be seen that the Fano resonance wavelengths exhibit significant redshifts with the increase of the refractive index, and the shift of FR1 is smaller than that of FR2. Figure 8b shows the approximately linear relationships between the Fano resonance wavelengths and the refractive index. The good linearity of FR1 and FR2 is of great importance for a high-performance sensor. Here, the slope of the straight line represents the sensitivity. Thus, the sensitivities of the refractive index sensing are about 667 nm/RIU for FR1 and 1250 nm/RIU for FR2, respectively.

Fig. 8
figure 8

a Transmission spectra of the plasmonic structure with different refractive indices. b Relationship between the Fano resonance wavelength and the refractive index

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Moreover, we investigate the effects of the coupling distance, the baffle width, the rectangular resonator height, and the outer radius of the quarter-ring resonator on sensitivity. The surrounding refractive index is increased from 1.00 to 1.12 with an interval of 0.03. Sensitivities of the plasmonic structure with different geometrical parameters are shown in Fig. 9. It can be seen that the geometrical parameters have different effects on sensitivity as described in Fig. 9a–d. The changes of geometrical parameters can affect the transmission characteristics of the structure and then adjust the sensing performances. By optimizing the geometrical parameters, we can achieve the maximum sensitivity of 1333 nm/RIU. Figure 10 shows the FOM of the plasmonic structure, and the surrounding refractive index is increased from 1.00 to 1.12 with an interval of 0.03. The maximum FOM is about 5876. The comparison of the sensitivity and FOM of this paper with other published papers is presented in Table 1.

Fig. 9
figure 9

Sensitivities of the plasmonic structure with different geometrical parameters. a The coupling distance, b the baffle width, c the rectangular resonator height, and d the outer radius of the quarter-ring resonator

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Fig. 10
figure 10

FOM for different refractive indices

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Table 1 Comparison of the sensitivity and FOM with other published papers
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Based on the refractive index sensing, the plasmonic structure can also be used to measure the biological parameters that can affect the refractive index of the waveguide, such as glucose concentration, glycerin concentration, and plasma concentration, in a much more exact and efficient manner due to the ability to detect small changes in refractive index [40]. The refractive index has unique characteristics for each material, and here we discuss the sensing of glucose solution concentration. The refractive index of glucose solution is described as [41] :

$$n=0.00011889\mathrm{C}+1.33230545$$
(5)

where C is the concentration of glucose solution. It can be seen that there is a linear relationship between the refractive index and the concentration of glucose.

The transmission spectra of the structure with different glucose solution concentrations are shown in Fig. 11a, where the concentration of glucose solution in the r-shaped resonator is increased from 0 to 150 g/L with an interval of 50 g/L. The glucose solution concentration can affect the Fano resonance wavelengths by changing the refractive index, and the resonance wavelengths of the two Fano resonances exhibit redshifts. Figure 11b shows the relationships between the resonance wavelengths and the concentration of glucose solution. The sensitivity of glucose solution concentration sensing is Sglucose = ∆λ/∆C. Thus, the sensitivities of FR1 and FR2 are about 0.1 nm·L/g and 0.16 nm·L/g, respectively.

Fig. 11
figure 11

a Transmission spectra of the structure with different glucose solution concentrations. b Relationship between the Fano resonance wavelength and the glucose solution concentration

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Conclusion

In this paper, we propose a plasmonic structure of MIM waveguide consisting of a baffle waveguide and an r-shaped resonator to produce Fano resonance and investigate its Fano transmission characteristics by using the FEM. Two Fano resonances can be produced and tuned by changing the geometrical parameters of the structure. The geometrical parameters can affect the sensitivity of refractive index sensing, and the maximum sensitivity is 1333 nm/RIU with the FOM of 5876. The application of the structure in glucose solution concentration sensing is also investigated, and the maximum sensitivity is about 0.16 nm·L/g. Based on the results, the structure may have potential applications in refractive index nano-sensing, bio-sensing, and nano-photonic devices.