Introduction

Recently, Fano resonance has attracted considerable attention in the field of plasmonics due to its unique characteristics. Unlike the conventional symmetric resonant Lorentzian line profile, the spectrum based on Fano resonance possesses an asymmetric one, which can provide high figure of merit (FOM) and sensitivity. Such a resonance phenomena is firstly discovered in the auto-ionizing states of atoms [1], and then is also found in the classical optical systems.

By taking advantage of surface plasmon polaritons (SPPs) which can overcome the classical optical diffraction limit, on-chip Fano resonance can be achieved in various plasmonic structures, showing great potential in the sensors, switching, nonlinear, and slow-light areas [2–6]. Specifically, the asymmetric spectral response can be obtained through the interference between a localized dark mode (corresponding to the discrete excited state) and a radiative bright mode (corresponding to the continuum state). For example, Fano resonance was observed in a dolmen-type slab structure consisting of a radiative element and a sub-radiant element [7]. Besides, plasmon rulers [8], core-shell structures [9], plasmonic nanoparticle clusters [10], composite cut-wire structures [11], planar plasmonic structure [12], and other structures [13–15] have been investigated theoretically or experimentally to obtain Fano-like resonances. It is worth to point out that plasmonic metal-dielectric-metal (MDM) waveguide structures have also received a lot of interest in studying Fano resonance due to the advantages of high integration, deep-subwavelength confinement of light, and easily manufactured [16–20]. Sharp and asymmetric Fano-line shapes were found in an MDM waveguide with multiple output ports, which were isolated by the baffles [21]. More and more MDM structures were developed to realize the Fano resonances, such as wide-gap and narrow-gap hybrid MDM structure [22], one-terminal-closed T-shaped MDM structure [23], and so on. In addition, a MDM bus waveguide with dual side-coupled slot cavities or dual parallel grooves were proposed to obtain the asymmetric spectrum with high sensitivity and FOM [24, 25]. According to demands of nanosensors, the FOM was further improved by employing an MDM T-shaped resonator [26].

In this paper, single or dual Fano resonance peaks are achieved by using a compact MIM waveguide structure, which is a composite of slot cavities and grooves. Since the slot cavities and the grooves can act as the Fabry-Perot (FP) resonators and the reflectors, thus bright modes with narrow bands and dark modes with broad bands are excited, respectively. Due to the interactions of the bright and dark modes, Fano resonances will then be generated. The spectral line profile and the peak wavelength can be manipulated by changing the lengths of the grooves and the slot cavities, respectively. The structure presents a high refractive index sensitivity and a FOM higher than the previous reported results.

Theory and Analysis

Figure 1 shows the scheme diagram of the plasmonic MDM waveguide structure. A groove is placed on one side of the MDM waveguide, while a side-coupled slot cavity is laid on the opposite side with a coupling distance d. When SPPs are launched into the MDM waveguide, they will be captured into the slot cavities and then oscillated around the groove, which is analogous to the classical Fano resonances of mechanical spring harmonic oscillator system [27]. Besides, the Fano-like resonance phenomenon can also be explained by the coherent coupling and interference between the bright mode and the dark mode. Since the slot cavity can act as a FP resonator, a narrow bright mode can be achieved at the wavelength, expressed as

Fig. 1
figure 1

Scheme diagram of the plasmonic MDM waveguide structure

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$$ \begin{array}{cc}\hfill \lambda =\frac{2{n}_{\mathrm{s}}L}{m-\phi /2\pi },\hfill & \hfill m=1,\kern0.5em 2,\kern0.5em 3,\kern0.5em \cdots \hfill \end{array} $$
(1)

where L is the length of the slot cavity, ϕ is the phase change due to the reflection at the FP facet, and m is an integer. n s is the real part of the effective index, which can be obtained from the dispersion equation [28]. Moreover, a broad dark mode will be exited by the groove due to the equivalent Bragg reflection. Due to the interaction of two modes, Fano resonance is generated to obtain a distinctly asymmetrical spectral profile, which is completely different from the Lorentzian resonance. The line shape of the transmission spectrum will be affected by the groove, while the wavelength of the Fano peak will be determined by the length of the slot cavity, as shown in Eq. (1). The sensitivity S and FOM, which are the two significantly factors to evaluate the performances, can be expressed as [29, 30]

$$ S=\frac{d\lambda }{dn\left(\lambda \right)} $$
(2)
$$ {\mathrm{FOM}}^{*}=\left|\frac{dT\left(\lambda \right)/dn\left(\lambda \right)}{T\left(\lambda \right)}\right| $$
(3)
$$ \mathrm{F}\mathrm{O}\mathrm{M}=\mathrm{Max}\left({\mathrm{FOM}}^{*}\right) $$
(4)

where T(λ) is the transmittance at the specific wavelength, and dT(λ)/dn(λ) is the transmittance change at fixed wavelength induced by a refractive index change. Based on Eqs. (3) and (4), it can be predicted that an ultra-low transmittance and a sharp increase of the transmittance induced by the index changes are preferred for obtaining a high FOM.

In the following, 2D FDTD simulation with perfect matched layer is used to investigate the performances, and the grid sizes are set to be 5 nm × 5 nm. PML layers with reflection of 0.0001 are set to surround the whole structure, and the fundamental TM mode of the plasmonic waveguide is excited by a plane wave incident from the input waveguide. The dielectric and metal are assumed to be air and silver, whose optical constant is obtained from the experiment [28]. The width and length of the slot cavity are defined as W 3 = 50 nm and L = 360 nm, respectively. The coupling distances from the slot cavity to the MDM bus waveguide is d = 25 nm, the length of the groove is H = 200 nm, and the widths of the MDM waveguide and the groove are W 1 = 50 nm and W 2 = 130 nm, respectively. To better understand the modes excited by the groove and the slot cavity, we firstly consider the MDM structure only with the groove or the slot cavity, respectively. When the MDM waveguide is only with the groove, the transmission spectrum is shown in Fig. 2 with a blue solid line. A broadband transmission dip with a negligible transmittance (low than −50 dB) arises at the wavelength of ∼970 nm. In contrast, the structure only with the slot cavity presents a much narrower transmission at the wavelength of ∼1179 nm with a slight higher transmittance (∼0.04), as shown in Fig. 2 with a red dot. When the MDM structure is with both the groove and the slot cavity, obviously, asymmetric Fano resonance peak is achieved at ∼1187 nm with a transmittance of ∼0.5 according to the modes interactions, as shown in Fig. 2 with a black dotted line. Besides, the quality factor (QF) of the Fano resonance is ∼18.

Fig. 2
figure 2

Transmission spectra with groove only (blue solid line), or with cavity only (red dotted), or with groove and slot cavity (black dotted line)

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Secondly, the length of the groove is changed as H = 290 nm to find more characteristics. The transmission spectrum is shown in Fig. 3, where asymmetrical spectral profiles are also achieved due to the Fano resonance effect. Comparing to the results in Fig. 2, the spectral profile presents an obvious transformation by changing H. The reversal mechanism of the asymmetric spectral profile can be attributed to the shift of the broad dark mode, which is determined by length H of the groove. For example, the right rising edge of the broad dark mode will interact with the narrow bright mode firstly. When the length of the groove is increased, the dark mode has a red shift and then its left rising edge will affect the resonance. In Fig. 3, a sharp transmission peak with a transmittance of ∼0.5 and a QF of ∼22 arises at 1170 nm, and the transmittance will decrease rapidly at the right side of the peak, resulting in a transmission dip with an ultra-low transmittance. Similar results are also obtained in Fig. 2 except for the dip, which emerges at the left side of the peak. Therefore, the line shape can be manipulated by changing the length of the groove.

Fig. 3
figure 3

Transmission spectra with H = 290 nm

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Figure 4 provides the magnetic field distributions of SPPs at the Fano peak wavelengths and the dip wavelengths that correspond to the results in Figs. 2 and 3. Specifically, SPPs at the peak wavelengths of 1170 and 1187 nm in Fig. 4a, c, respectively, can pass through the MDM waveguide. Strong field distributions are observed in the slot cavity that acts as a FP resonator, leading to a narrow transmission peak. Conversely, SPPs at the dip wavelengths (i.e., 1252 nm in Fig. 4b and 1106 nm in Fig. 4d) show a weak distribution in the slot cavity but a strong one in the groove, and thus, most of the SPPs are blocked by the groove. Due to the interaction of the narrow discrete bright mode (strong trapped resonance in the FP resonator) and the broad dark mode (strong reflection in the groove), sharp and asymmetric spectra, which are regarded as Fano resonances, can be obtained. These straightforward results in Fig. 4 provide the SPP power flows in our proposed structure in detail.

Fig. 4
figure 4

Magnetic field distributions of SPPs at the peak wavelengths and the dip wavelengths that correspond to the results in Figs. 2 and 3: a λ = 1170 nm, b λ = 1252 nm, c λ = 1187 nm, and d λ = 1106 nm

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According to Eq. (1), the peak wavelength should have a linear relationship with the length of the slot cavity. The analysis results are further confirmed by FDTD simulations, as shown in Fig. 5. By fixing the length of the groove as 200 or 290 nm and the dielectric as air, we firstly increase the length of the slot cavity from 330 to 380 nm with a step of 10 nm. Obviously, an average slope of the increasing peak wavelength is obtained as 2.89 and 2.91 by fixing H = 200 and 290 nm, respectively, as shown in the left side of Fig. 5 with blue lines. Then, one can easily manipulate the wavelength by adjusting the length of the slot cavity. Besides, a linear relationship between the wavelength and the refractive index of the dielectric is also investigated by fixing the lengths of the groove and the slot cavity. The details are presented by using the red line in the right side of Fig. 5, where the refractive index of the dielectric is increased from 1 to 1.07 with a step of 0.01. As a result, the sensitivity of the structure can be calculated as 1106 and 1131 nm/RIU with H = 200 and 290 nm, respectively. This is also the desired character for the sensors, which require a high sensitivity.

Fig. 5
figure 5

The variations of peak wavelengths with respect to the length of the slot cavity and the refractive index of the dielectric

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To better evaluate the performance of the proposed structure, the FOM is also studied in detail. Based on Eqs. (3) and (4), FOM* corresponding to the case in Figs. 2 and 3 (i.e., H = 200, 290 nm) is shown in Fig. 6 by changing the refractive index of the dielectric. The values of FOM are as high as 5.0 × 104 and 1.6 × 107 at 1251 and 1164 nm in Fig. 6a, b, respectively. These FOM values are significantly greater than that in the previous reports [22–26]. Due to this outstanding property, we consider that the proposed structure can work as a sensor well.

Fig. 6
figure 6

The calculated FOM* corresponding to Figs. 2 and 3

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In the next step, the slot cavity is proposed to be laid above the groove, as shown in Fig. 7. In this case, SPPs will propagate into groove and then be captured into the slot cavity, which is different from that in Fig. 1. However, the oscillation around the groove for SPPs will still occur. This phenomenon is also analogous to mechanical spring, while the slot cavity performs as the fixed end of the spring. Since the slot cavity and the groove will still provide a narrow dark mode and a broad bright mode, respectively, Fano resonance should be generated via the modes interaction. By setting the length of the groove as L = 560 nm, the coupling distance d = 30 nm, and other parameters to be unchanged, FDTD simulations are used to investigate the performances. The transmission spectra and the FOM* are shown in Fig. 8 by defining H = 200 and 150 nm.

Fig. 7
figure 7

The MDM waveguide structure with different slot cavity arrangement

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Fig. 8
figure 8

The transmission spectra and the FOM* with a H = 200 nm and b H = 150 nm

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Obviously, Fano resonances are observed in Fig. 8a, b, which show the sharp and asymmetrical peaks at 888 and 1187 nm with the QFs of ∼68 and ∼72, respectively. Comparing to the results in Fig. 2, the transmission peaks possess smaller full width at half-maximums for the reason that the second-order resonance mode of the slot cavity is used. In Fig. 8a with H = 200 nm, a steep slope emerges at the left side of the Fano peak, resulting in a dip with ultra-low transmittance. By changing the parameter H to be 150 nm, the transmittance decreases rapidly from the peak wavelength to the long wavelength, and then a transmission dip arises at the right side of peak. According to the FOM* plotted with a red dotted line, a large FOM of 5.0 × 104 or 2.3 × 104 is achieved at the dip with the wavelength of 871 or 900 nm in Fig. 8a, b, respectively.

More details about the magnetic field distributions are provided in Fig. 9. Specifically, Fig. 9a, c are the SPP propagation details at the Fano peaks with the wavelengths of 887 and 883 nm, while Fig. 9b, d are field distributions at the Fano dips with the wavelengths of 871 and 883 nm, respectively. Obviously, most of the SPPs at the peak wavelengths can pass through the MDM waveguide, but those at the dip wavelengths are almost completely stopped by the groove. The results agree well with those in Fig. 8.

Fig. 9
figure 9

Magnetic field distributions of SPPs at the peak wavelengths and the dip wavelengths that correspond to the results in Fig. 8: a λ = 887 nm, b λ = 871 nm, c λ = 883 nm, and d λ = 900 nm

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The variations of the peak wavelengths with respect to the length L of the slot cavity or the refractive index n of the dielectric are shown in Fig. 10. By increasing L from 520 to 600 nm with a step of 10 nm, we can obtain a uniform increase step of the wavelength as ∼13.4 nm with H = 150 and 200 nm. Likewise, a linear relationship between the wavelengths and n is also confirmed, and then a sensitivity of 840 and 847.5 nm/RIU is investigated with H = 150 and 200 nm, respectively.

Fig. 10
figure 10

The variations of peak wavelengths with respect to the length of the slot cavity and the refractive index of the dielectric

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According to the spectral line profiles in Figs. 2, 3, and 7, it is investigated that a Fano peak at the expected wavelength can be achieved by laying the slot cavity below or above the groove. It is interesting to explore more performances by adding two slot cavities in the structure, as shown in Fig. 11. According to the resonance conditions, two slot cavities with different lengths should support two specific bright modes. These two modes will be interacted with the dark mode from the groove, respectively, and actually, the two interaction processes are immune to each other because of the different wavelengths of the bight modes. As a result, dual Fano peak should be available in the new proposed structure. To perform the FDTD simulation, all the parameters are defined as L 1 = 560 nm, d 1 = 30 nm, L 2 = 360 nm, d 2 = 25 nm, H = 200 nm, W 1 = W 3 = 50 nm, and W 2 = 130 nm.

Fig. 11
figure 11

The MDM waveguide structure with two slot cavities

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Figure 12 shows the transmission spectrum, where two Fano peaks with the QFs of ∼61 and ∼35 arise at the wavelengths of 888 and 1187 nm, respectively. The steep slopes appear at the left sides of the peaks, resulting in the dips with ultra-low transmittances. These results are completely identical to the results in Figs. 2 and 8a, and the spectral profile seems to be their perfect synthesis because of the same parameters. This phenomenon is attributed to the independent interactions between the broad mode from the groove and the two narrow modes from two slot cavities, respectively. The FOM* are also provided in Fig. 12, where a large FOM of 2.2 × 104 and 2.9 × 104 is achieved at 873 and 1138 nm, respectively. Consequently, one can obtain dual Fano resonance peaks with comparable performances by using this composite structure. The magnetic field distributions of SPPs are also provided in Fig. 13, where sub-graphs (a) and (c) correspond to the peak wavelengths of 888 and 1187 nm, and sub-graphs (b) and (d) are the dip wavelengths of 873 and 1138 nm, respectively. The simulations are similar to that in Figs. 4 and 9, where most of the SPPs at the peak wavelengths can pass through the MDM waveguide while those at the dip wavelengths will be stopped.

Fig. 12
figure 12

The transmission spectra and the FOM* of the composite structure

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Fig. 13
figure 13

Magnetic field distributions of SPPs at the peak wavelengths and the dip wavelengths of a λ = 888 nm, b λ = 873 nm, c λ = 1187 nm, and d λ = 1138 nm

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Furthermore, the length of one slot cavity is changed while another one is fixed to find out more characters of the structure. Firstly, L 2 = 360 nm and L 1 = 560, 580, and 600 nm are employed in Fig. 14a–c. Obviously, the wavelength of the left Fano peak has a linear red shift along with the linear increase of L 1, but the right Fano peak always stays at the identical wavelength due to the same value of L 2. Likewise, the right Fano Peak increase linear proportionally with L 2 in Fig. 14c–e, where L 2 is defined from 360 to 400 nm with a step of 20 nm. Besides, the wavelength of left Fano peak is unchanged due to the same L 1 of 600 nm. Based on the simulations, the variations of the peak wavelengths with respect to the lengths of the cavities are further investigated. Therefore, one can easily design the specific Fano peak at the expected wavelength by changing the corresponding slot cavity but without any influence on the other peak. This character may offer great flexibility to design the device.

Fig. 14
figure 14

The transmission spectra with a L 1 = 560 nm, L 2 = 360 nm, b L 1 = 580 nm, L 2 = 360 nm, c L 1 = 600 nm, L 2 = 360 nm, d L 1 = 600 nm, L 2 = 380 nm, and e L 1 = 600 nm, L 2 = 400 nm

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Conclusion

In summary, Fano resonances have been investigated in our proposed MDM structure. According to the interactions between the narrow bright modes provided by the slot cavities and the broad dark mode from the groove, asymmetric line shapes were achieved. Single or dual Fano peaks, whose wavelengths were determined by the lengths of the slot cavities, could be obtained by designing the arrangements of the slot cavities. Besides, the line shape could be transformed by changing the length of the groove. The device could serve as an on-chip nanosensor due to the highest sensitivity and FOM as 1131 nm/RIU and 1.6 × 107, respectively.