Introduction

Surface plasmon polaritons (SPPs) are charge density waves propagating along the metal–dielectric interface caused by the coupling between photons and electrons on the metal surface [1,2,3].Their electromagnetic fields decay exponentially in the direction perpendicular to the metal surface. The result is that SPPs can overcome the diffraction limit and have the ability to modulate light in nanoscale [4, 5]. Therefore, many kinds of photonic devices have been investigated, such as filters [6, 7], absorbers [8], all-optical switches [9], and sensors [10,11,12,13,14,15]. Compared with traditional devices, plasmonic devices have a smaller size to be suitable for integration [4].

Among the plasmonic waveguides, metal–insulator–metal (MIM) waveguides have low transmission loss and are more advantageous of strongly confined the light [16,17,18]. The sensors based on these structure are basically composed of MIM waveguide coupled with resonators [18,19,20,21,22,23]. Sensitivity and figure of merit (FOM) are usually used to evaluate the quality of optical sensors. In order to obtain a higher sensitivity and FOM sensor, many photonic devices based on Fano resonance have been proposed [18,19,20,21,22,23]. Fano resonance is quite sensitive to the change in refractive index; the refractive index sensors can achieve a higher sensitivity [18]. At the same time, Fano resonance has a typically asymmetric and sharp spectral line shape, to achieve a narrower full width at half maximum (FWHM) [20,21,22,23,24,25,26]. However, Fano resonance is unstable and easily disturbed by environmental changes. Some works with simple and tunable structure to improve sensing performance have been proposed [27,28,29,30].

In this paper, the plasmonic refractive index sensors with a structure composed of MIM waveguide coupled with concentric ring and disk resonator (CRDR) are proposed. In the near-infrared region, a high sensitivity of 1039 nm/RIU with ultra-high FOM of 401 is realized. The influence of the structural parameters of the plasmonic coupling system on sensing properties has also been investigated.

Structural model and computational method

Figure 1 illustrates the schematic of the 2D metallic micro-cavity composed of the MIM waveguide and the CRDR. The radius of the cavity is expressed as R, the coupling distance between the MIM waveguide and the CRDR is g, and the gap width between the concentric ring and disk is G. The widths of the MIM waveguide and the ring cavity are fixed at w = 50 nm, to ensure that only the fundamental transverse magnetic mode is supported in the MIM waveguide [31].

Fig. 1
figure 1

2D schematic of the plasmonic sensor filled with air

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The white areas and gray areas represent dielectric and metal, respectively. The medium of the dielectric is assumed to be air, and the metal is set to be silver. The dispersion equation of the fundamental TM mode in the MIM structure can be described as follows [32, 33]:

$$\frac{{\varepsilon }_{i}p}{{\varepsilon }_{m}k}=\frac{1-{e}^{kw}}{1+{e}^{kw}}$$
$$k=\sqrt{{\beta }_{spp}^{2}-{\varepsilon }_{i}{k}_{0}^{2}}$$
$$p=\sqrt{{\beta }_{spp}^{2}-{\varepsilon }_{m}{k}_{0}^{2}}$$
(1)

where w is the width of waveguide, \({\beta }_{spp}\) is the propagation constant of SPPs, \({n}_{eff}={\beta }_{spp}/{k}_{0}\) means effective refractive index of SPPs, \({k}_{0}=2\pi /\lambda \) refers to wave number, \(\lambda \) is the wavelength of incident light, \({\varepsilon }_{i}\) and \({\varepsilon }_{m}\) are relative permittivity of dielectric and metal. The relative permittivity of silver can be characterized by the Drude model [34, 35]:

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

here \({\omega }_{P}\) is the bulk plasma frequency; the bulk plasma frequency ωp depends on the carrier (holes) density Nh as given in the equation \(\omega_{p} = \left( {N_{h} /\varepsilon_{0} m_{h} } \right)^{1/2e}\), where mh is the hole effective mass, approximated to be 0.8m0; m0 is the electron mass.\(\Upsilon \) is damping frequency of the electron oscillation, and \(\omega \) is the incident electromagnetic radiation angular frequency. The parameters of Eq. (2) are \({\omega }_{P}=1.37\times {10}^{16}\) Hz and \(\Upsilon=3.21\times {10}^{13}\) Hz.

For a single ring resonator, the resonating wavelength can be obtained theoretically by the equation [28]:

$$ \frac{{J_{n}^{\prime } \left( {kr_{a} } \right)}}{{J_{n}^{\prime } \left( {kr_{i} } \right)}} - \frac{{N_{n}^{\prime } \left( {kr_{a} } \right)}}{{N_{n}^{\prime } \left( {kr_{i} } \right)}} = 0 $$
(3)

where \(k=\omega {\left({\varepsilon }_{0}{\varepsilon }_{r}{\mu }_{0}\right)}^\frac{1}{2}\), \({\mu }_{0}\) refers to the permeability in the air, \({\varepsilon }_{r}={n}_{eff}^{2}/{\mu }_{0}\) is the frequency-dependent effective relative permittivity. \({J}_{n}\)  and \({J}_{n}^{^{\prime}}\) are the first kind Bessel function with the order n and its derivative to the argument (kr); \({N}_{n}\)  and \({N}_{n}^{^{\prime}}\)  are the second kind Bessel function with the order n and its derivative to the argument (kr), respectively.

The resonating wavelength of the disk-shaped nanocavity can be obtained theoretically by the equation [29]:

$$ k_{i} \frac{{H_{n}^{\left( 1 \right)} \left( {k_{m} r} \right)}}{{H_{n}^{\left( 1 \right)} \left( {k_{m} r} \right)}} - k_{m} \frac{{J_{n}^{^{\prime}} \left( {k_{i} r} \right)}}{{J_{n} \left( {k_{i} r} \right)}} = 0 $$
(4)

here \({k}_{i,m}=k{\left({\varepsilon }_{i,m}\right)}^\frac{1}{2}\) refers to the wave vectors in the dielectric disk or metal, \({\varepsilon }_{i}\) and \({\varepsilon }_{m}\) are the relative permittivity of the dielectric and metal, and \(k\) means the wave number. \(r\) stands for the radius of the cavity. \({J}_{n}\)  and \({J}_{n}^{^{\prime}}\) mean the first kind Bessel function with the order n and its derivative; \({H}_{n}^{(1)}\) and \({{H}_{n}^{\left(1\right)}}^{,}\) refer to the first kind Hankel function with the order n and its derivative, respectively.

In this study, commercial software (COMSOL Multiphysics) of the finite element method (FEM) is used to research the transmission spectra. The two-dimensional (2D) FEM with perfectly matched layer (PML) boundary condition is adopted to simulate this structure and investigate the properties of the refractive index sensing. The transmittance of power is defined to be \(T={({S}_{21})}^{2}\), where \({S}_{21}\) is the transmission coefficient from \({P}_{in}\) to \({P}_{out}\)[3].

Results and discussion

In this part, the transmission spectra of the MIM waveguide coupled with CRDR are simulated under varying parameters of the structure. Figure 2a shows the simulated transmission spectrum of this structure with n = 1, R = 350 nm, g = 10 nm and G = 30 nm. n is the refractive index of the dielectric. Mode 1 and mode 2 can be considered as supermodes in which the inner disk and outer ring can resonate simultaneously [27]. The FWHM of mode 1(\({\uplambda }_{1}\)=965 nm) is 17.86 nm, and that of mode 2(\({\uplambda }_{2}\)=1038 nm) is 2.59 nm; the Q factors are 54, 401, respectively. Figure 2b and c plots the steady-state magnetic field \({H}_{z}\) distributions of the plasmonic coupling system of mode 1 and mode 2. The ratio of energy confined in the inner disk and the outer ring in mode 1 is 2.78 \(\times {10}^{-3}\), and in mode 2 it is 52.1. Energy of CRDR is mainly confined in the inner disk in mode 2, in the out ring in mode 1. The ratio of energy confined in the inner disk and the outer ring is defined as: \(R=\int {H}_{in}^{2}\mathrm{d}{S}_{in}/\int {H}_{\mathrm{out}}^{2}\mathrm{d}{S}_{out}\), where Hin is the magnitude in the inner disk, Hout is the magnitude in the outer ring, Sin is the area in the inner disk, and Sout is the area in the outer ring.

Fig. 2
figure 2

a Transmission spectra of MIM waveguide coupled with CRDR with n = 1, R = 350 nm, g = 10 nm and G = 30 nm; steady-state magnetic field \({H}_{z}\) distributions of the MIM waveguide coupled with CRDR at wavelengths of (b) 964.88 nm; c 1038.49 nm

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The coupled mode theory [36] is used to explain the principle of high Q factor formation of mode 2 in this paper. Figure 3a shows the transmission spectra of the proposed plasmonic refractive index sensor with inner disk resonator, outer ring resonator or CRDR. Figure 3b plots the steady-state magnetic field \({H}_{z}\) distributions of the \({TM}_{6}\) mode of the outer ring resonator. Figure 3c shows the steady-state magnetic field \({H}_{z}\) distributions of the \({TM}_{2}\) mode of the inner disk resonator. The transmission spectrum of the CRDR is almost the superposition of the outer ring transmission spectra and inner disk transmission spectra. In order to meet the angular momentum matching condition, the strong coupling is more prone to take place between the same order modes between the inner disk and outer ring. Figure 4 presents the diagram of mode coupling between the \({TM}_{2}\) modes of the inner disk and outer ring of the CRDR. Mode 2 is corresponded to quasi-in-phase CRDR mode. Owing to the asymmetry between the inside and outside of the CRDR structure, the resonance frequency difference between different order modes of inner disk and outer ring becomes smaller as shown in our results. Thus, \({TM}_{6}\) mode of the outer ring resonator has an impact on the quasi-in-phase CRDR mode. The final simulation result shows that Quasi-in-phase mode 2 has a narrow resonance dip and exhibits a little red shift. This phenomenon can be explained by the fact that the quasi-in-phase mode is closer to the inner disk mode and the energy of electromagnetic wave mainly confines in the inner disk. Because of the outer ring, the energy in the inner disk is more difficult to loss and is beneficial for energy storage. Therefore, the resonance dip induced by the quasi-in-phase mode 2 is very narrow which leads to ultra-small FWHM and ultra-high Q factor.

Fig. 3
figure 3

a Transmission spectra of the proposed plasmonic refractive index sensor with inner disk resonator, outer ring resonator or CRDR. b Steady-state magnetic field \({H}_{z}\) distributions of \({TM}_{6}\) mode of the outer ring resonator at wavelengths of 948.19 nm. c Steady-state magnetic field \({H}_{z}\) distributions of \({TM}_{2}\) mode of the inner disk resonator at wavelengths of 1034.96 nm

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Fig. 4
figure 4

Schematic of the mode coupling between the \({TM}_{2}\) modes of the inner disk and outer ring of the CRDR

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To research the effect of the refractive index of dielectric on the structure, we use different media to fill the structure; the transmission spectra are presented in Fig. 5a. The n is increased from 1 to 1.08 by the step 0.02. A red shift the transmission spectrum can be observed with an increase in n. The increase in the resonance wavelength is in line with the theoretical calculation result of formula 4 [29]. In this study, we use S and FOM to measure the sensing performance. They are defined as follows [37, 38]:

Fig. 5
figure 5

a Transmission spectra for the MIM waveguide coupled with CRDR with changing n while R = 350 nm g = 10 nm and G = 30 nm; b fitting curve between refractive index and resonant wavelength

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$$S=\frac{\Delta \lambda }{\Delta n}$$
(5)
$$FOM=\frac{S}{FWHM}$$
(6)

As shown in Fig. 5b, the sensitivity can be calculated by linear fitting. For mode 1, S is 971 nm/RIU; FOM is 54. For mode 2, S is 1039 nm/RIU; FOM is 401. Mode 2 is further investigated then in view of its excellent sensing performance in the near-infrared region.

To investigate the influences of different cavities on the transmission spectra and sensor performance, R is varied from 330 to 370 nm at intervals of 10 nm with n = 1, g = 10 nm and G = 30 nm. Figure 6a plots the transmission spectra of the structure of R = 330, 340, 350, 360 and 370 nm. With the increasing radius of the cavity, the transmission spectrum exhibits a red shift as shown in Fig. 6a. The increase in the resonance wavelength of mode 1 and mode 2 also accords with the theoretical calculation results of formulas 3 and 4 [29, 32]. As the radius increases, the shape of the transmission spectrum remains nearly constant. Figure 6b shows S and FOM of the plasmonic sensor for different values of R. Obviously, S and FOM of the plasmonic sensor can be improved by increasing the size of CRDR. When R = 370 nm, the sensor gets the maximum S and FOM, S = 1107 nm/RIU, FOM = 434.

Fig. 6
figure 6

a Transmission spectra of the MIM waveguide coupled with CRDR with changing radius of the cavity; b sensitivity and figure of merit of the sensor structure for different values of R

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The coupling distance between the MIM waveguide and the CRDR is changed to study its influences on the transmission spectra and sensor performance. The coupling distance g is increased from 6 nm to 14 in steps of 2 nm while the other parameters are fixed at n = 1, R = 350 nm and G = 30 nm. The transmission spectra of different coupling distance are shown in Fig. 7a. The simulation results show that for mode 2, when g increases from 6 to 14 nm, the transmittance of the resonance dip is significantly increased. This outcome can be explained the increase in coupling distance allows less energy to enter the CRDR, resulting in a higher transmittance. At the same time, the transmission spectrum also has a slight blue shift, which is consistent with the results in reference [29]. Figure 7b shows S and FOM of the plasmonic sensor for different values of g. The sensitivity of the plasmonic sensor is almost unchanged as the g increases, but the value of FOM can be significantly increased because of the narrower resonance dip. Maximum sensitivity of 1040 can be reached when g = 6 nm/RIU, and the value of the maximum FOM is 433 when g = 14 nm.

Fig. 7
figure 7

a Transmission spectra of the MIM waveguide coupled with CRDR with changing coupling distance between the MIM waveguide and the CRDR; b sensitivity and figure of merit of the sensor structure for different values of g

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For the investigation of the effect of the different gap width between the concentric ring and disk on the transmission spectra and sensor performance, G is increased from 20 to 40 nm at 5-nm intervals, with n = 1, R = 350 nm and g = 10 nm. The transmission spectra of the structure with the different gap width of G are shown in Fig. 8a. With the increasing gap width, blue shifts in the transmission spectrum can be observed. The blue shift of the mode 2 resonance wavelength is caused by the decrease in the radius of the inner circular cavity. And for the mode 2, the transmittance of the dip is significantly increased as G increasing. This phenomenon can be explained that the increase in the gap width allows less energy to enter the inner circular cavity, which leads a higher transmittance. Figure 8b presents S and FOM of the plasmonic sensor with increasing G. Although the sensitivity shows a decrease as G increases, the FOM still improves because of the narrower resonance dip. There is a compromise between high sensitivity and high FOM. The maximum sensitivity of the structure when G = 20 nm is 1085 nm/RIU, and the maximum FOM is 471 when G = 40 nm. With the increasing gap width, blue shifts in the transmission spectrum can be observed. And for the mode 2, the transmittance of the dip is significantly increased as G increasing. This phenomenon can be explained that the increase in the gap width allows less energy to enter the inner circular cavity, which leads a higher transmittance. When G is nearly to 0, the mode 1 with energy concentrated in the outer ring disappears, and the transmission spectra become the transmission spectra of the disk resonator.

Fig. 8
figure 8

a Transmission spectra of the MIM waveguide coupled with CRDR with changing gap width between the concentric ring and disk; b sensitivity and figure of merit of the sensor structure for different values of G

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To discuss the application of the proposed plasmonic sensor in bio-sensing, another simulation is performed near the refractive index of 1.33. The n is increased from 1.30 to 1.38 with the step 0.02 while the other parameters are fixed at R = 350 nm g = 10 nm and G = 30 nm; the transmission spectra are presented in Fig. 9. It can be seen that the sensitivity and FOM of the plasmonic sensor near the refractive index of water can still reach 1035 nm/RIU and 234, showing excellent sensing performance. Table 1 presents the comparison of the S and the FOM in our work and in some other recent works; our work shows ultra-high FOM.

Fig. 9
figure 9

Transmission spectra for the MIM waveguide coupled with CRDR with changing n while R = 350 nm g = 10 nm and G = 30 nm

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Table 1 Comparison of the S and the FOM in our work and in some other recent works
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Conclusions

In this work, a plasmonic refractive index sensor based on MIM waveguides coupled with CRDR is proposed. Our proposed plasmonic structure presents ultra-high Q factor in the quasi-in-phase mode. With the increase in the radius of the cavity, both sensitivity and the value of FOM show an obvious increase. As the coupling distance increases, the sensitivity of the plasmonic sensor remains nearly constant, but the value of FOM can be significantly improved. With an increase in the gap width between the concentric ring and disk, the FOM of the plasmonic sensor shows an increase, but sensitivity decreases. High sensitivity with ultra-high FOM can be reached in the near-infrared region. The max refractive index sensitivity of the sensor is 1107 nm/RIU, and the FOM can reach 471. In addition, this plasmonic structure has great potential in bio-sensing and can be easily integrated with other photonic devices.