1. Introduction

Fano resonance has emerged as an important area in the field of plasmonics over the recent past, because surface plasmon polaritons (SPPs) can overcome the diffraction limit and confine light in deep sub-wavelength dimensions [1]. Many plasmonic structures have been designed to realize the Fano resonance [2–5]. J. Wang et al. [6] reported a planar plasmonic structure to obtain double Fano resonances originated from interplay of electric and magnetic plasmon modes. D. Wang [7] realized tuning multiple Fano and plasmon resonances in plasmonic-photonic nanostructures. In our previous work [8], the multiple Fano resonances with high figure of merit(FOM) is also achieved in coupled plasmonic resonator system. Due to the advantage for enhanced bio-chemical sensing, spectroscopy, and multicolor nonlinear processes, the multiple Fano resonances become more important and have gained much attention [9–13]. Among all the nanostructures, the MIM waveguide structures have attracted many researchers attention because these structures exhibiting more suitable for the highly integrated optical circuits due to their deep-sub-wavelength confinement of light [14, 15]. And based on the MIM waveguides, a large number of devices, such as splitters [12, 16], sensors [17, 19], optical switches [20], nonlinear [21] and slow-light devices [22–24]. Thus, investigations of the response line-shapes and the physical mechanism in the coupled-resonator systems are of importance for designing complex functional plasmonic devices as well as for improving their performances [25]. For many applications, the tunability of the Fano resonances is very important. However, because the multiple Fano resonances are caused by the collective behavior of the total plasmonic systems, it is difficult to get an independent and precise tailoring of the multiple Fano resonances.

In this paper, independently or semi-independently tunable triple Fano resonances are realized in asymmetric coupled plasmonic resonator system, and the multimode interference coupled mode theory (MICMT) including coupling phases is proposed based on single mode coupled mode theory (CMT). MICMT is used for describing and explaining the multiple Fano resonances phenomenon in coupled plasmonic resonator systems. Firstly, an asymmetric plasmonic structure composed of a rectangular cavity and two MIM (metal-insulator-metal) waveguides is investigated. The double Fano resonances produced by this structure originate from the interference between different resonant modes according to MICMT, one is induced by the interference between the symmetric modes, and another originates from the interference between the symmetric and anti-symmetric modes. Another rectangular cavity is added to this structure, so as to form a new coupled plasmonic resonator system including new resonant modes, which can support triple Fano resonances originating from three different mechanisms. The third Fano resonance originates from the interference between the resonant modes of the added and original rectangular cavities. Because the triple Fano resonances originate from three different mechanisms, each Fano resonance can be tuned independently or semi-independently by changing the height and length of the two rectangular cavities. And during the tuning process, the FOMs (figure of merit) of the three Fano resonances remain keep large values, all larger than 725. Based on the large FOM and independent tunability of the triple Fano resonances, our tunable triple Fano resonances plasmonic structure can be used in highly integrated plasmonic devices with excellent performance, such as splitters, filters, bio-chemical sensors and optical switches.

2. Structures descriptions

The coupled plasmonic resonator system in Fig. 1(b) is the aim of our research, before which is investigated the coupled plasmonic resonator system in Fig. 1(a) needed to be investigated first. The structure in Fig. 1(a) is composed of two MIM waveguides (S₁ and S₂) and a rectangular cavity (A). The height and length of rectangular cavity A are H and LA respectively, and the waveguides S₁ and S₂ with the widths are both D₁ are symmetric about rectangular cavity A. The change of the relative positions between the axis of the MIM waveguide and the rectangular cavity in y direction $\Delta H=H_2-H_1$, as shown in Fig. 1(a), is used to describe the symmetry-breaking. Another small rectangular cavity B is added to the structure shown in Fig. 1(a), thereby forming a new coupled plasmonic resonator system shown in Fig. 1(b). The width and length of rectangular cavity B are D₂ and LB, the gap between rectangular cavity A and rectangular cavity B is g. In consideration of the application in biotechnology, the dielectric in the waveguides and cavities are chosen to be water (n = 1.33). The metal is sliver, of which the permittivity is characterized by the Drude model [26]: ${\epsilon }_m={\epsilon }_{\infty }-{\omega }_p^2/\left({\omega }^2+j\gamma \omega \right)$ with ${\epsilon }_{\infty }=3.7,{\omega }_p=9.1eV,\gamma =0.018eV$.

 

Fig. 1 Schematic diagram of the asymmetric plasmonic structure composed of (a) two MIM waveguides and one rectangular cavity and (b) two MIM waveguides and two rectangular cavities.

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In order to investigate the optical responses of the proposed structure, the transmittance are numerically calculated using commercial software (COMSOL Multiphysics) of the finite-element analysis method (FEM) [4]. The transmittance of the SPPs in plasmonic structure is defined as the quotient between the SPP power flows (obtained by integrating the Poynting vector over the cross section of the MIM waveguide) of the observing port with cavities and without cavities, that is $T=P/P_0$ [18, 19]. In order to excite the SPPs, the input light is set to be transverse magnetic (TM) plane wave obtained from boundary mode analysis.

3. MICMT analysis

In coupled plasmonic resonator system, the total field can be expressed as the field superposition of the resonant modes. For single mode coupling, if the choice of the reference plane is appropriate, the coupling phase between waveguide and resonant cavity will be zero and has no effect on the transmittance of the coupled plasmonic resonator system. For multiple modes coupling, the coupling phases of different modes in waveguide and resonant cavity are different at the same reference plane. Because of the interference between different modes, the transmittance of the coupled plasmonic resonator system will be influenced by these coupling phases, so that the relationship between these coupling phases need be considered. Based on single mode coupled mode theory(CMT), the MICMT equations including coupling phases are given as follows

In theory, there are a number of resonant modes in plasmonic resonator. Here, we only consider the interference between the modes of which the resonant wavelengths are in and adjacent to the spectral range that we are interested in, and ignoring the influences of other modes. Because of the input light is set to be transverse magnetic (TM) plane wave, the resonant modes can be classified using TM_m,n, m, n are integers and indicate the x- and y-directional resonant orders, respectively. The resonant wavelengths of two modes TM_0,3 and TM_1,0 are in the spectral range of 600 nm ~900 nm that we are interested in, as shown in Figs. 2(a) and 2(b). Two modes of which the resonant wavelengths are adjacent to the spectral range of 600 nm ~900 nm are TM_0,2 and TM_0,4, as shown in Figs. 2(c) and 2(d). So within the spectral range of 600 nm ~900 nm, the transmission response of the plasmonic structure shown in Fig. 1 (a) is mainly related to the four resonant modes. The four resonant modes TM_0,2, TM_0,3, TM_1,0 and TM_0,4, of which the magnetic field (Hz) distributions are given in Fig. 2, are expressed as a₁, a₂, a₃ and a₄. The resonant wavelengths of the four resonant modes are about ${\lambda }_1=1105nm$, ${\lambda }_2=788nm$, ${\lambda }_3=734nm$ and ${\lambda }_4=593nm$, respectively. Waveguides S₁ and S₂ with the widths both are D₁ are symmetrical about rectangular cavity A, so ${\tau }_{n1}={\tau }_{n2}={\tau }_n$ and ${\theta }_{n1}={\theta }_{n2}$, the transmittance formula of the plasmonic system is simplified as

 

Fig. 2 The distribution of the normalized magnetic field (Hz) of the resonant modes (a) TM_1,0 mode at 734 nm, (b) TM_0,3 mode at 788 nm, (c) TM_0,2 mode at 1105 nm and (d) TM_0,4 mode at 593 nm respectively in the rectangular cavity A with LA = 220 nm, H = 760 nm in Fig. 1(a).

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4. Simulation results and discussions

Before the coupled plasmonic resonator system in Fig. 1(b) is investigated, the coupled plasmonic resonator system in Fig. 1(a) needed to be investigated first. When the parameters of the structure are LA = 220 nm, H = 760 nm, ΔH = 0 and D₁ = 50 nm, the system is symmetric in y-direction. In this case, the y-directional asymmetric mode(TM_0,3) cannot be excited, only the y-directional symmetric modes exist in this system. In the four modes that we are concerned about, there are three y-directional symmetric modes TM_0,2, TM_1,0 and TM_0,4 of which the resonant wavelengths are 1105 nm, 734 nm and 593 nm, respectively. Thus, the subscript m,n in the transmittance formula Eq. (4) and Eq. (5) just take 1,3,4. The decay time of the coupling and internal loss can be obtained by curve fitting, which are ${\tau }_1=10.3fs$, ${\tau }_3=179fs$, ${\tau }_4=21fs$ and ${\tau }_{10}=911fs$, ${\tau }_{30}=496fs$, ${\tau }_{40}=53.8fs$, respectively. In order to facilitate the calculation, the coupling phase difference ${\phi }_n\left(\omega \right)$ can be approximately considered as a constant in very narrow wavelength range. Then, the coupling phase differences are about ${\phi }_1=-0.3\pi$, ${\phi }_3=0.58\pi$ and ${\phi }_4=0.89\pi$ near the resonant wavelength of 734 nm. The simulation curve and theoretical curve of the transmittance of this plasmonic system are given in Fig. 3(a).

 

Fig. 3 The FEM simulation results (blue lines) and MICMT results (red lines) of the transmission spectrum of the asymmetric plasmonic system with H = 760 nm, LA = 220 nm and (a) ΔH = 0, (b) ΔH = 15 nm, (c) ΔH = 30 nm. The phase curves of the complex amplitude transmission coefficients of the four modes TM_0,2, TM_0,3, TM_1,0 and TM_0,4 in the asymmetric plasmonic system with H = 760 nm, LA = 220 nm and (d) ΔH = 0, (e) ΔH = 15 nm, (f) ΔH = 30 nm.

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When the symmetry-breaking parameter of the structure respectively is $\Delta H=15nm$ and $\Delta H=30nm$, the simulation curve and theoretical curve of the transmittance are given in Figs. 3(b) and 3(c). Here, the left Fano resonance is called FR1, and the right Fano resonance is called FR2 [5]. When the symmetry-breaking of the plasmonic structure was induced, the y-directional asymmetric mode TM0,3 at about 788 nm in the rectangular cavity A is excited, which is apparent in symmetrical plasmonic structure. The FR 2 originates from the interference between the asymmetric mode TM_0,3 and the symmetric modes TM_0,2, TM_1,0, TM_0,4. By curve fitting, it can be obtained that ${{\tau }_2|}_{\Delta H=15nm}=1565fs$, ${{\tau }_2|}_{\Delta H=30nm}=1565fs$ and${\tau }_{20}=478fs$. The coupling phase differences are about ${{\phi }_2|}_{\Delta H=15nm}=-0.11\pi$, ${{\phi }_2|}_{\Delta H=30nm}=-0.06\pi$. Through comparing Figs. 3(b) and 3(c), it can be seen that the transmittance difference between the peak and dip of FR2 when $\Delta H=15nm$ is smaller than that when $\Delta H=30nm$, the reason will be illustrated in the analysis of Argand diagram latter. Moreover, the reason of the deviation between the simulation and theoretical curves is that the coupling phase difference ${\phi }_n\left(\omega \right)$ is approximately considered as a constant, actually it is the function of angular frequency(ω).

In order to facilitate the theoretical analysis, the complex amplitude transmission coefficient of each mode is set as $t_n=2e^{j{\phi }_n}/\left[j\left(\omega -{\omega }_n\right){\tau }_n+2+{\tau }_n/{\tau }_{n0}\right]$. From Eq. (5), it is known that the transmission response of the plasmonic system can be expressed as the ‘interference’ between the transmission coefficients $t_n$ of the four resonant modes. The phase curve ${\varphi }_n$ of the transmission coefficient $t_n$ is given in Figs. 3(d)-3(f). As shown in Figs. 3(e) and 3(f), the phase mutation occurred in $t_3$ is about π near the resonant wavelength of 734 nm, while the phases of $t_1$, $t_2$ and $t_4$ almost remain unchanged; the phase mutation occurred in $t_2$ is also about π near the resonant wavelength of 788 nm, while the phases of $t_1$, $t_3$ and $t_4$ almost remain unchanged. In this case, the coherence properties (destructive or constructive) of transmission coefficients are reverse near the opposite sides of the two resonant wavelengths, thereby forming the asymmetric double Fano resonances line-shape.

The Argand diagrams of the transmission coefficients $t_n$ at the peak and dip of FR1 and FR2 are given in Fig. 4. Here, the phase ${\varphi }_1$ of the transmission coefficient $t_1$ is taken as the reference zero point. As shown in Figs. 4(a)-4(d), FR1 is mainly the result of the interference between the transmission coefficients $t_1$, $t_3$ and $t_4$, the influence of the transmission coefficient $t_2$ on FR1 is so little that it can be neglected. Figures 4(d) and 4(e) show that the transmission coefficients of the four modes all have effect on FR2. And we have already mentioned that the transmittance difference between the peak and dip of FR2 when $\Delta H=15nm$ is smaller than that when $\Delta H=30nm$, the reason is that the amplitudes and phases of the transmission coefficients $t_1$, $t_3$ and $t_4$ when $\Delta H=15nm$ are basically identical with those when $\Delta H=30nm$ and only the phase change of the transmission coefficient $t_2$ is obvious. The amplitudes of the transmission coefficients ${t_2|}_{\Delta H=15nm}$ are 0.22 and 0.27 at the peak and dip of FR2 respectively, and the amplitudes of the transmission coefficients ${t_2|}_{\Delta H=30nm}$ are 0.25 and 0.27, they are basically identical. While the phase ${\varphi }_2$ of ${t_2|}_{\Delta H=15nm}$ are 0.34π and 0.78π at the peak and dip of FR2 respectively, and that of ${t_2|}_{\Delta H=30nm}$ are 0.31π and 0.90π respectively. This is the reason why the difference of the transmittance between the peak and dip of FR2 induced by the change of ΔH is different.

 

Fig. 4 Argand diagrams of the transmission coefficients t₁, t₂, t₃ and t₄ of the four resonant modes TM_0,2, TM_0,3, TM_1,0 and TM_0,4 at the peak (red lines) and dip (blue lines) of FR1 and FR2 in the plasmonic system shown in Fig. 1(a) with H = 760 nm, LA = 220 nm and (a, e) ΔH = 15nm, (c, f) ΔH = 30 nm. (b) and (d): The detailed transmission coefficients t₁, t₂ and t₃ in the areas selected by rectangular boxes in (a) and (c), respectively.

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The transmission responses of the asymmetric plasmonic system shown in Fig. 1(b) will be investigated next. The parameters of the plasmonic structure are ΔH = 30 nm, g = 10 nm, H = 760 nm, LA = 220 nm, LB = 500 nm and D₁ = D₂ = 50 nm, respectively. Figure (5) shows the simulation and theoretical curves of the transmittance and the distribution of the normalized magnetic field (Hz) in the plasmonic system shown in Fig. 1(b). As shown in Figs. 5(c) and 5(d), a new resonant mode appears in the range of 600 nm ~900 nm after the rectangular cavity B is added to the plasmonic structure in Fig. 1(a), of which the resonant wavelength has significant relationship with rectangular cavity B and the upper part of rectangular cavity A. Here, the new added resonant mode is considered as the fifth resonant mode a₅ of the plasmonic system, and its resonant wavelength is about 672 nm. According to MICMT, the numbers of the resonant modes increase to 5, that is $n=1,2,\cdots ,5$. The simulation and theoretical curves of the transmittance is shown in Fig. 5(d), and the Fano resonance on the left of FR1 is called FR3. It can be known from Fig. 5(c) that the resonant wavelength of the fifth resonant mode a₅ has significant relationship with LA and LB.

 

Fig. 5 The distribution of the normalized magnetic field (Hz) of the resonant modes at (a) 791 nm, (b) 788 nm and (c) 1105 nm in the asymmetric plasmonic system respectively, when ΔH = 30 nm, H = 760 nm, LA = 220 nm, LB = 500 nm, g = 10 nm and D₁ = D₂ = 50 nm. (d) The FEM simulation results (blue lines) and MICMT results (red lines) of the transmission spectrum of the asymmetric plasmonic system.

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The figure of merit (FOM) is a key parameter for Fano resonance, which can be defined as $FO{M}^{*}=\Delta T/T\Delta n$ at a fixed wavelength and $FOM=\max \left|FO{M}^{*}\right|$ near a fixed wavelength [27], where T denotes the transmittance in the proposed structure. Figure 3(a) shows the transmission spectra for different refractive indexes of the dielectric in the waveguides and cavities with the structure parameters of ΔH = 30 nm, g = 10 nm, H = 760 nm, LA = 220 nm, LB = 500 nm and D₁ = D₂ = 50 nm. With the increasing of n from 1.33 to 1.34, the peaks of FR1, FR2 and FR3 are all red shift, the peaks of FR 1 red shifts from about 733 nm to 739 nm, the peaks of FR 2 red shifts from about 791 nm to 796 nm and the peaks of FR3 red shifts from about 672 nm to 677 nm. The FOM*curve of this plasmoic system is given in Fig. 6(b), and the maximum FOMs of FR1, FR2 and FR3 are about 3803, 816 and 2947 respectively.

 

Fig. 6 (a) The transmission spectra of the plasmonic structure with H = 760 nm, LA = 220 nm, LB = 500 nm, D₁ = D₂ = 50 nm, g = 10 nm, ΔH = 30 nm, and different refractive indexes of the dielectric in the waveguides and cavities, n = 1.33 (blue line), n = 1.34 (red line). (b) The FOM* curve of this plasmonic structure.

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When ΔH = 30 nm, g = 10 nm and D₁ = D₂ = 50 nm are fixed value, the influence of the structure parameters on the transmission spectrum profile is studied in detail and the results are shown in Fig. 7. Figure 7(a) depicts that the position of FR2 can be tuned by the height (H) of the rectangular cavity A. As shown in Fig. 7(a), with H increasing from 730 nm to 770 nm, the position of FR2 changes from 763 nm to 800 nm, while the positions of FR1 and FR3 almost have no change(< 1 nm), this kind of tuning is called ‘independent tuning’. Figure 7(b) shows that the position of FR3 can be tuned by the length (LB) of the rectangular cavity B. As shown in Fig. 7(b), with LB increasing from 490 nm to 530 nm, the position of FR3 changes from 667 nm to 687 nm, while the positions of FR1 and FR2 also almost have no change(< 1 nm), so the tuning of FR2 is also ‘independent tuning’. Figure 7(c) depicts that the position of FR1 can be tuned by the length (LA) of the rectangular cavity A. As shown in Fig. 7(c), with LA increasing from 215 nm to 227 nm, the position of FR1 changes from 722 nm to about 750 nm, and the position of FR3 changes from 665 nm to about 682 nm, while the positions of FR2 has a little change about 3 nm. So, the tuning of FR1 has little effect on FR2 but evident on FR3, this kind of tuning is called ‘semi-independent tuning’. And as shown in Fig. 7(d), with the change of the height (H) of rectangular cavity A, the FOMs of FR1, FR2 and FR3 change slightly, no more than 25%. In Fig. 7(e), with the change of the length LB, the FOMs of FR1 and FR2 almost have no change, no more than 9%, but the FOM of FR3 increase by about 55%. In Fig. 7(f), with the change of the length LA, the FOMs of FR2 and FR3 change a little, no more than 27%, but the FOM of FR1 increase by about 1.2 times. This means that the FOMs of the triple Fano resonances keep high value in the tuning process.

 

Fig. 7 The dependence of transmittance on the structure parameters, when ΔH = 30 nm, g = 10 nm, n = 1.33 and D₁ = D₂ = 50 nm. (a) Different H with fixed LA = 220 nm and LB = 500 nm; (b) different LB with fixed H = 760 nm, LA = 220 nm; (c) different LA with fixed H = 760 nm and LB = 500 nm. (d) ~(f) The change curves of FOM of FR 1 (blue line), FR 2 (green line) and FR 3 (red line) with the change of the structure parameters H, LB and LA, respectively.

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As shown in Fig. 8, attribute to the tunability of FR 1 and FR 2 and the reversed ‘direction’ of Fano asymmetry, the transmission window can be constructed with the position and the full width at half maximum (FWHM) can be adjusted freely. And because FR3 is independently tunable, after the position and FWHM of the first Lorentzian-like line-shape transmission window is determined, then the tuning for FR3 is proceeded, so that the ‘M’ type double Lorentzian-like line-shape transmission windows of which the peak interval and FWHMs are respectively less than 20nm and 8nm can be constructed. And the Lorentzian-like line-shape on the right is called LL1, the Lorentzian-like line-shape on the left is called LL2. As shown in Fig. 8(a), the position of the ‘M’ type double Lorentzian-like line-shape transmission window can be adjusted freely, and Fig. 8(b) shows that the FWHM of each line-shape can also be adjusted freely. This kind of transmission window, whose position and FWHM can be adjusted freely, can be used in filters, enhanced bio-chemical sensors, nonlinear and slow-light devices.

 

Fig. 8 The dependence of the ‘M’ type of double Lorentzian-like line-shape transmittance on the structure parameters, when ΔH = 30 nm, g = 10 nm, n = 1.33 and D₁ = D₂ = 50 nm are fixed values. H = 709 nm, LA = 225 nm, LB = 600 nm (blue line); H = 698 nm, LA = 221 nm, LB = 590 nm (green line); H = 708 nm, LA = 225 nm, LB = 610 nm (red line).

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5. Conclusion

In summary, we report an asymmetric plasmonic structure composed of two MIM waveguides and two rectangular cavities. And the multimode interference coupled mode theory including coupling phases is proposed based on single mode coupled mode theory. The MICMT is used for describing and explaining the origin of the multiple Fano resonances in coupled plasmonic resonator systems. Our proposed structure can support triple Fano resonances originating from three different mechanisms. One mechanism originates from the interference between the symmetric modes, and another is induced by the interference between the symmetric and anti-symmetric modes. The third one originates from the coupling between the resonant modes of the two cavities. The triple Fano resonances can be well tuned independently or semi- independently by changing the parameters of the two rectangular cavities, and the FOMs of the triple Fano resonances remain keep large values, all larger than 725. By tuning the positions of the three Fano resonances, the transmission spectra can be well tailored and the narrow ‘M’ type of double Lorentzian-like line-shape transmission windows can be constructed. Benefit from the different ‘direction’ of Fano asymmetry, the large FOM, independent tunability and the compactness, the triple Fano resonances can find widely applications in switches, bio-chemical sensors, slow-light devices and modulator of high density nano-photonic integrated circuits devices with excellent performance.