1 Introduction

Because of the potential of overcoming the diffraction limit of light and confining it in sub wavelength scales, surface plasmon polaritons (SPPs) have concerned much interest for creating photonic on chip circuits (Barnes et al. 2003; Anker et al. 2010). SPPs are the waves that confine at the boundary of two materials with the different sign of dielectric constants such as metal and dielectric (Hutter and Fendler 2004; Maier 2007; Maier and Atwater 2005). Over recent years, SPP-based Fano resonances (FRs) have been extensively investigated since the development in the nanostructure fabrication allows studying these resonances beyond the classical limit (Lukyanchuk et al. 2010).

FR is a fundamental resonant phenomenon that shows a typically sharp resonance peak and asymmetric line-shape (Miroshnichenko et al. 2010), which is a result of coupling and quantum interference of a discrete state with a continuum of propagation modes (Lukyanchuk et al. 2010). FR features promise applications in sensors (Tavousi et al. 2018), logic gates (Moradi et al. 2019), and so on. Because of the distinctive characteristics of FR-based sensors, they provide great features in sensitivity and figure of merit (FOM) (Wen et al. 2015; Lee et al. 2018; Mesch et al. 2018). In subwavelength structures, the FRs can be found through the interaction of the narrow and broad modes. In addition, Fano-like resonances in some configurations such as nanoring arrays (Li et al. 2018), plasmonic arrays of particles (Bakhti et al. 2016), nano loops (Vahedian and Naser-Moghadasi 2018), split rings (Zarrabi et al. 2018), and so on (Wang et al. 2013; Zhang et al. 2011) have been studied. In this regard, metal–insulator–metal (MIM) waveguide based FRs is broadly investigated, as well (Chen et al. 2013a, b; Qi et al. 2014). MIM waveguides based on SPPs have considerable usages due to the sub-wavelength field confinements and low bending losses (Rakhshani and Mansouri-Birjandi 2016a, b, 2017a; 2018). Beside MIM waveguides, FRs have been broadly investigated in various configurations such as quantum dot systems (Schuray et al. 2017), photonic crystals (Wang et al. 2019), and hybrid oligomers (Lepeshov et al. 2017).

Localization of an electromagnetic field can be achieved by optical resonators (Rakhshani and Mansouri-Birjandi 2012, 2013a; Mansouri-Birjandi and Rakhshani 2013; Khani et al. 2018). Optical resonators can develop interactions between light and matter and have features such as high quality factor and deep light confinement. These features make various applications for optical resonators such as sensing, switching, and multiplexing (Kwon 2013; Rakhshani and Mansouri-Birjandi 2013b; 2014 Naser-Moghadasi et al. 2016).

In order to develop applications in biomedical (Vafapour 2019; Keshavarz and Vafapour 2018, 2019a; Wu and Soleimani 2019; Vafapour and Ghahraloud 2018), chemical (Yan et al. 2019) and detecting fields (Demirkap et al. 2019), sensor structures are being widely studied. Recently, various plasmonic metamaterials and metasurface structures have been proposed for sensing and detecting applications (Amin et al. 2013; Christofi et al. 2018; Zhu et al. 2018; Vafapour 2018a, b, c; Bagci and Akaoglu 2018, 2019; Keshavarz and Vafapour 2019b; Alipour et al. 2018; Xia et al. 2018; Arju et al. 2015).

In this paper, a structure with square resonator and two MIM waveguides is designed to achieve the FRs. A square resonator is proposed near the ends of MIM waveguides that are symmetrically placed at both sides. The FR wavelength and its line shape rely on the resonator internal and external lengths. The proposed plasmonic sensor structure can provide good sensitivity and FOM of about 6400 nm/RIU and 1 × 104, respectively.

2 Structures and basics

A schematic illustration of our proposed sensor and its related parameters is depicted in Fig. 1a. The square resonator is symmetrically situated near to the ends of input and output waveguides. The silicon is chosen as a substrate, because of its relative low loss (Khurgin 2015). In MIM waveguides, the refractive index affects the guided modes that obtained the following relation (Dionne et al. 2006):

$$\varepsilon_{i} k_{z2} + \varepsilon_{m} k_{z1} \coth \left( { - \frac{{ik_{z1} W}}{2}} \right) = 0$$
(1)

where k 2z1,2  = εi,mk 20  − (k0neff)2 is the propagation constant in a transverse direction, k0 = 2π/λ, εm and εi are permittivity’s of the metal and insulator, respectively, and W is a waveguide thickness. Here, air and silver are supposed for dielectric and metal, respectively.

Fig. 1
figure 1

a Representation of the plasmonic sensor based on the square resonator and MIM waveguides, b Transmission spectrum with silver, gold, and aluminum as metal layer

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In plasmonic MIM configuration, compared to the Bragg reflection, broad bright modes are obtained (Lin and Huang 2008). Also, using the waveguide-resonator structure, narrow dark modes can be achieved (Ma and Lee 2013; Wen et al. 2013). By merging the characteristics of these ideas, the FR will be produced in the proposed structure. Unlike Lorentzian resonance, the FR shows a specifically asymmetrical and sharp spectrum. Resonator length specifies the FR wavelength, extracted by the following equation:

$$\lambda = \frac{{2n_{s} L}}{m - \phi /2\pi }, \quad m = 1,2,3, \ldots$$
(2)

where ns, m, L, and ϕ are the effective refractive index, resonance order, the resonator effective length, and phase shift, respectively.

Furthermore, sensitivity and FOM are important features extensively used to study the sensor performance. The sensitivity to the refractive index is expressed as S = dλ/dn, where dn is a variation of refractive index and dλ is the change of wavelength due to a dn (Becker et al. 2010; Lu et al. 2012). Also, at fixed wavelength λ0, a transmittance change dT(λ)/dn(λ) is achieved and then FOM* and FOM are expressed as (Becker et al. 2010; Lu et al. 2012):

$$FOM^{*} = \left| {\frac{dT(\lambda )/dn(\lambda )}{T(\lambda )}} \right|$$
(3)
$$FOM = { \hbox{max} }(FOM^{*} )$$
(4)

where T and dT/dn are the transmission and change of transmission owning to dn at a specific wavelength, respectively, and λ0 is the corresponding wavelength in FOM calculation. It can be expected from Eq. (3), for obtaining a large value of FOM, a sharp Fano line shape and very low transmittance are preferred.

3 Simulations and discussions

The transmission spectrum is numerically produced by Finite-Difference Time-Domain (FDTD) method with mesh dimensions of 0.5 nm × 0.5 nm. The perfectly matched layers (PML) as the absorbing boundary condition for all the boundaries are used. Input and output waveguide widths (W) are set to be 50 nm, the internal and external lengths of the resonator are s = 220 nm and h = 380 nm, respectively, and d is set to be 15 nm. The background material is silver and with a permittivity determined as a function of wavelength by solving Eq. (1). The permittivity characteristics and detail values for silver are given in Ref. (Rakhshani and Mansouri-Birjandi 2016a). Two metals, silver and gold, are commonly used in plasmonic applications. Silver has a smaller absorption loss and is preferred for sensing applications. Also, among all the metallic elements, silver (Ag) has the smallest damping constant and is the best performing choice at optical frequencies.

The effect of the bulk metal materials on the transmission is studied. For this purpose, silver, gold, and aluminum are used, as illustrated in Fig. 1b. As can be seen in Fig. 1b, strong Fano resonance exists for silver. Hence, silver is a good candidate for the proposed structure.

First, the inner length of the resonator is increased with a step of 4 nm in range of s = 216–228 nm to study its effect on the output spectrum. The simulated output spectrum (Fig. 2a) demonstrates that all curves have asymmetrical spectral shapes. Also, the transmittance is about 0.6 and at left side drop off quickly with very low transmitted power. The full widths at half-maximums (FWHMs) are about 11, 12, 13, and 14 nm. These are also the preferred features for the sensors (Wen et al. 2015). Besides, the peak and dip wavelengths are 680.05 and 667.2 nm, or 689.5 and 674.9 nm, or 699.08 and 682.8 nm, or 708.6 and 690.6 nm when the inner lengths of the resonator s are 216, 220, 224, or 228 nm, respectively. Figure 2b shows a linear relationship between the s and λresonance. Here, the peak and dip wavelength growth are about 9.5 and 7.8 nm, respectively. In this case, by tuning the resonator inner length, one can easily control the wavelength.

Fig. 2
figure 2

a Transmission spectra with different inner lengths of resonator and b the peak-wavelength and dip-wavelength variations with the inner lengths of resonator

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Figure 3 demonstrates the SPPs field distributions at the wavelength of the dip and peak for s = 216 nm (corresponding to Fig. 2a). It is observed that the SPPs can transmit through the resonator only at the peak wavelengths while they are entirely stopped in the input waveguide at the dip wavelengths. These behaviors give the SPP power flow details in the designed sensor.

Fig. 3
figure 3

Magnetic field intensity distributions of SPPs at the a peak and b dip wavelengths that correspond to the results in Fig. 2a for s = 216 nm

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In addition, Fig. 2a results signify that the shifts of wavelength from the peaks “on” to the valleys “off” are about 12.9, 14.6, 16.3, and 18 nm, respectively. Such a slight variation in wavelength due to the index changes can offer a high sensitivity value. Hence, the resonator with a different n is utilized to study the output spectrum. Herein, the resonator inner length is fixed to 220 nm, and other parameters are fixed, as illustrated in Fig. 2a.

The transmission spectrum in Fig. 4 shows that the peaks transmittances are about 0.57, 0.59, 0.61, 0.63, 0.65, and 0.67 at 680.05, 688.4, 696.7, 705.05, 713.4, and 721.7 nm when n = 1, 1.01, 1.02, 1.03, 1.04, and 1.05, respectively. The λresonance increases linearly by changing the refractive index, and we can see an obvious red-shift. Therefore, the sensitivity value is obtained to be 833 nm/RIU.

Fig. 4
figure 4

Transmission spectra based on different refractive indices of resonator

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Next, the resonator outer length h is investigated to study the output characteristic while keeping other parameters the same as above. By defining h and s as (375 nm, 215 nm), (380 nm, 220 nm), (385 nm, 225 nm), and (390 nm, 230 nm), we attain the output spectrums in Fig. 5a–d, respectively. The line-shape of the output characteristic shows an asymmetrical and sharp transformation by varying h and s parameters. As can be seen, by increasing the s and h, the resonance peaks have a linear red-shift. The dips arise at the left side of the peaks and then rise quickly. To demonstrate the sensing performance of the designed configuration, the FOM value is calculated. By solving Eqs. (3) and (4), by varying the refractive index of the square resonator, FOM* value is shown in Fig. 6. The FOM value is about 1 × 104, which is considerably greater than that of previously reported papers (Wen et al. 2015; Wang et al. 2013; Qi et al. 2014; Rakhshani and Mansouri-Birjandi 2016b, 2017a; Kwon 2013; Lu et al. 2012).

Fig. 5
figure 5

Transmission spectra with different inner and outer lengths of resonator

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Fig. 6
figure 6

The calculated FOM* corresponding to Fig. 4

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By cascading several square resonators, the multi-resonances outputs can be recognized. For cascade two and three square resonators as depicted in Figs. 7a and 8a, there will be double and triple resonance as depicted in Figs. 7b and 8b, respectively. The related H-field distributions are illustrated in the inset of Figs. 7b and 8b. This result demonstrates that cascade M square resonators can produce M resonance modes.

Fig. 7
figure 7

a Proposed configuration with cascade two resonators. b The output characteristics and corresponding H-field distributions

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Fig. 8
figure 8

a Proposed configuration with cascade three resonators. b The output characteristics and corresponding H-field distributions

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4 Application as glucose sensor

SPP sensors can be used as diverse biomedical analytes sensing like DNA (deoxyribonucleic acid) and biomolecular interactions (Rakhshani and Mansouri-Birjandi 2017b, 2018; Belushkin et al. 2018; Sharma et al. 2009; Chen et al. 2016). SPPs are a significant candidate for finding the glucose value much more efficiently and more precisely (Zhang et al. 2018) because of its ability to detect the little variations of the refractive index. Water is the most important fluid of the human organs and is the vital element of blood plasma, etc. Understanding the optical properties of water is significant for the solution of problems in medical optics.

The refractive index is a specific characteristic of each material and bio-medicine deals with the unique biomaterials refractive index (Li et al. 2000). Thus, here, by utilizing the mathematical model of water refractive index, we aim to develop a sensor structure to detect glucose concentrations. Different samples, with varying glucose concentrations, are filled into the square resonator at 20 °C (room temperature). Each sample, have specific concentrations, thus the SP should resonant at varied wavelengths owning to its refractive index.

Daimon and Masumura (2007) experimentally found the refractive index of water based on glucose concentration value in a 300–1000 nm spectral region. They used the following mathematical model to calculate water refractive index (nw) in different wavelengths:

$$n_{w} = 1 + \frac{{B_{1} \lambda^{2} }}{{\lambda^{2} - D_{1} }} + \frac{{B_{2} \lambda^{2} }}{{\lambda^{2} - D_{2} }} + \frac{{B_{3} \lambda^{2} }}{{\lambda^{2} - D_{3} }} + \frac{{B_{4} \lambda^{2} }}{{\lambda^{2} - D_{4} }}$$
(5)

where λ represent the wavelength, and B1,2,3,4 and D1,2,3,4 are experimentally determined coefficients listed in Table 1 (Daimon and Masumura 2007). Figure 9 depicts the nw for T = 20 °C.

Table 1 Values and units of model parameters at T = 20 °C
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Fig. 9
figure 9

The nw for T = 20 °C

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The changing of the refractive index with glucose concentration is expressed as follow:

$$\Delta n(\lambda ) = \frac{dn}{dc}(\lambda )C$$
(6)

where C is the glucose concentration. The dn/dC was determined to be a constant for the studied C value.

To better confirm the feasibility of this theoretical model, different glucose concentrations of the tested water were considered. Figure 10 depicts the nw for C = 2, 4, 6, 8, and 10 g at 20 °C. As shown, all curves have linear relations with the C, and increasing the C increases the nw.

Fig. 10
figure 10

The nw for different glucose concentration (g/100 ml)

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For glucose sensing, water samples are positioned in the square resonator. Figure 11 illustrates the output characteristics of the proposed sensor with varied C (0–6 g/100 ml) in the resonator. The λresonance for different C are 912 nm, 917.5 nm, 923 nm, and 928 nm, respectively, which so clearly are apart from each other. As depicted in Fig. 11, the resonance wavelengths are red-shifted with an increscent of C. In this case, the sensitivity has a high value of about 6400 nm/RIU. This value of sensitivity is considerably greater than that in reported in previous works. The comparisons among some published sensitivity values for diverse MIM-based sensing devices are listed in Table 2. The high quantity of sensitivity attained in this device provides a path toward optical on-chip sensors.

Fig. 11
figure 11

a Transmission and b resonance wavelengths of sensor in a case that different C (= 0–6 g/100 ml) are subjected in resonator

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Table 2 Comparisons of reported sensitivity values
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5 Conclusions

The FR based plasmonic sensor has shown the capability in determining the water glucose concentration in the MIM square resonator configuration. Simulation results of the proposed plasmonic sensor demonstrate that FR can be controlled by varying the internal and external lengths of the square resonator and different resonance modes can be achieved. This proposed sensor has the most important characteristics, such as low sample consumption, high speed, small size, and high sensitivity for monitoring the different analytes. By investigating the transmission spectrums by the Finite Difference Time Domain technique, our SPP based sensor produces the sensitivity of 6400 nm/RIU and figure of merit of about 1 × 104. Therefore, it can find numerous applications, mainly in nano-medicine.