Fano resonances based on multimode and degenerate mode interference in plasmonic resonator system

Shilei Li; Yilin Wang; Rongzhen Jiao; Lulu Wang; Gaoyan Duan; Li Yu

doi:10.1364/OE.25.003525

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]. And many plasmonic structures have been designed to realize the Fano resonances, such as the array structures ranging from particle lattices and oligomers to nano-wire lattices and split-ring type structures [2–8]. In our previous work [9, 10], the multiple Fano resonances with high figure of merit (FOM*) are also achieved in plasmonic resonator system. Different from the Lorentzian resonances emerging in universal resonators, the Fano resonances exhibit sharp and asymmetric spectral line-shapes together with strong field enhancements, which have potential applications in the areas of sensors, splitters, nonlinear and slow-light devices [11–17]. Compared with the array structures, the unit-cell plasmonic structures are much compact and easy to be integrated into chips [18]. Thus, the investigations of the response line-shapes and the physical mechanism in plasmonic resonator systems are of importance for designing complex functional plasmonic devices as well as for improving their performances.

In this paper, three Fano resonances based on three different physical mechanisms are theoretically predicted in a plasmonic resonator system, comprised of two circular cavities. According to MICMT, one of the Fano resonances originates from the interference between different resonant modes of one resonator, the other is induced by the interference between the resonant modes of different resonators, and the third kind of Fano resonance is attributed to the interference between degenerate modes. The degenerate interference coupled mode theory (DICMT) is proposed to explain the third kind of Fano resonance. Because the three Fano resonances originate from three different mechanisms, each Fano resonance can be tuned independently or semi-independently by changing the radii of the two circular cavities. The sensitivity and FOM* (figure of merit) of the three Fano resonances can be as high as 840 nm/RIU and 100, respectively. The devices based on our tunable Fano resonances can be used in highly integrated plasmonic devices with excellent performance, such as sensors, splitters, filters and optical switches.

2. Structures and MICMT analysis

The structure in Fig. 1(a) is composed of two MIM waveguides (S₁ and S₂) and a circular cavity (A). The change (ΔH) of the relative positions between the axis of the MIM waveguide and the circular cavity in y direction is used to describe the symmetry-breaking. Another small circular cavity B is added to the structure shown in Fig. 1(a), thereby forming a new plasmonic resonator system shown in Fig. 1(b). The gap between circular cavity A and B is g, the dielectric around the resonator is silver. In order to excite the SPPs, the input light is set to be transverse magnetic (TM) wave. And the transmittance of the system is numerically calculated using finite difference time domain (FDTD) method.

Fig. 1 Schematic diagram of the plasmonic resonator system composed of (a) two MIM waveguides and one circular cavity and (b) two MIM waveguides and two circular cavities. And the distribution of the normalized magnetic field (Hz) of the resonant modes (c) TM_0,4 mode at 652 nm, (d) TM_0,3 mode at 812 nm, (e) TM_1,0 mode at 854 nm, (f) TM_0,2 mode at 1094 nm in circular cavity A with R_A = 500 nm, respectively.

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When the influences of waveguides on the resonator are very slight, the total field can be expressed as the field superposition of the resonant modes in plasmonic resonator system. For single mode coupling [19], if the choice of the reference plane is appropriate, the coupling phase between the resonant mode and waveguide will be zero and has no effect on the transmittance of the plasmonic resonator system. But for multiple modes coupling, the phases and modulus of different resonant modes correlate with each other, and the coupling phases and modulus between waveguide and each resonant mode of the resonator are also different. The transmittance of the plasmonic resonator system can be influenced by these coupling phases and modulus, so that the relationship between these coupling phases and modulus of different resonant modes need to be considered. In order to investigate the properties of coherent transmission of the plasmonic resonator system, the MICMT equations including coupling phases and modulus are given as follows

\frac{d a_{n}}{d t} = (- j ω_{n} - \frac{1}{τ_{n 0}} - \frac{1}{τ_{n 1}} - \frac{1}{τ_{n 2}}) a_{n} + κ_{n 1} s_{n, 1 +} + κ_{n 2} s_{n, 2 +}

s_{1 -} = - s_{1 +} + \sum_{n} κ_{n 1}^{*} a_{n}, κ_{n 1} = \sqrt{\frac{2}{τ_{n 1}}} e^{j θ_{n 1}}

s_{2 -} = - s_{2 +} + \sum_{n} κ_{n 2}^{*} a_{n}, κ_{n 2} = \sqrt{\frac{2}{τ_{n 2}}} e^{j (θ_{n 2} - ϕ_{n})}

s_{n, 1 +} = γ_{n 1} e^{j φ_{n 1}} s_{1 +}, s_{n, 2 +} = γ_{n 2} e^{j φ_{n 2}} s_{2 +}

Where

a_{n}

and

ω_{n}

are the field amplitude and resonant frequency of the nth resonant mode, respectively.

τ_{n 0}

is the decay time of internal loss of the nth resonant mode in resonator.

τ_{n 1}

and

τ_{n 2}

are the decay time of the coupling between the resonator and waveguides (S1 and S2),

θ_{n 1}

and

θ_{n 2}

are the coupling phases of the nth resonant mode.

γ_{n 1} e^{j φ_{n 1}}

and

γ_{n 2} e^{j φ_{n 2}}

are the normalized coefficients (

γ_{n 1} = γ_{n 2} \approx 1

in this paper).

ϕ_{n}

is the phase difference between output port and input port of the nth resonant mode.

s_{i \pm}

are the field amplitudes in each waveguide (

i = 1, 2

, for outgoing (-) or incoming ( + ) from the resonator). According to MICMT equations, when

s_{2 +} = 0

, the complex amplitude transmission coefficient from waveguide S1 to waveguide S2 can be expressed as follows

t = \frac{s_{2 -}}{s_{1 +}} = \sum_{n} \frac{γ_{n 1} | κ_{n 1} | | κ_{n 2} | e^{j φ_{n}}}{- j (ω - ω_{n}) + \frac{1}{τ_{n 0}} + \frac{1}{τ_{n 1}} + \frac{1}{τ_{n 2}}}, φ_{n} = φ_{n 1} + ϕ_{n} + θ_{n 1} - θ_{n 2}

Then, the corresponding transmittance of the plasmonic system is

T = {| t |}^{2}

. And

φ_{n}

is the total coupling phase difference of the nth resonant mode.

In theory, there are a number of resonant modes in plamonic 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) wave, the resonant modes can be classified using TM_q,m, q, m are integers and indicate the radial and angular resonant orders, respectively. Within the spectral range of 750 nm ~1000 nm, the transmission response of the plasmonic resonator system show in Fig. 1(a) is mainly related to the four resonant modes TM_0,2, TM_1,0, TM_0,3, and TM_0,4 mode. The four resonant modes, of which the normalized magnetic field (Hz) distributions are given in Figs. 1(c)-1(f), are expressed as a₁, a₂, a₃ and a₄. If waveguides S1 and S2 with the widths are equal are symmetrical about circular cavity A, then $τ_{n 1} = τ_{n 2} = τ_{n}$ and $θ_{n 1} = θ_{n 2}$ , the transmittance formula of the plasmonic system is simplified as

T = {| t |}^{2} = {| \sum_{n = 1}^{4} \frac{2 γ_{n 1} e^{j φ_{n}}}{- j (ω - ω_{n}) τ_{n} + 2 + \frac{τ_{n}}{τ_{n 0}}} |}^{2}, φ_{n} = φ_{n 1} + ϕ_{n}

3. Simulation results and discussions

Before the plasmonic resonator system in Fig. 1(b) is investigated, the plasmonic resonator system in Fig. 1(a) needed to be investigated first. The parameters of the structure in Fig. 1(a) are R_A = 500 nm, D₁ = D₂ = 50 nm, and the total coupling phase difference $φ_{n} (ω)$ can be approximately considered as a constant. The simulation result and MICMT result of the transmittance of this system with ΔH = 0 are given in Fig. 2(a). Here, the right Fano resonance is called FR1, and the left Fano resonance is called FR2. For closely separated FR1 and FR2, it impossible to identify the location of exact trough of the Fano lineshape. So, we specify the position of the peak and trough of FR1 are at 849 nm and 869 nm, and the position of the peak and trough of FR2 are at 815 nm and 835 nm, respectively. In order to facilitate the calculation, the complex amplitude transmission coefficient of each resonant mode in the plasmonic resonator is set as $t_{n} = 2 e^{j φ_{n}} / [- j (ω - ω_{n}) τ_{n} + 2 + τ_{n} / τ_{n 0}]$ , then $t = \sum_{n = 1}^{4} t_{n}$ . The meaning of this formula is that the transmission response of the plasmonic resonator system can be understood as the ‘interference’ between the transmission coefficients $t_{n}$ of the four resonant modes [10].

Fig. 2 The simulation results (blue lines) and MICMT results (red lines) of spectral transmittance of the plasmonic resonator system with R_A = 500 nm and (a) ΔH = 0, (c) ΔH = 50 nm, (d) ΔH = 100 nm. The change of ΔH almost has no influence on the internal losses of the four resonant modes. By curve fitting, it can be obtained that the decay time of the internal losses respectively are about τ₁₀ = 68fs, τ₂₀ = 190fs, τ₃₀ = 71fs, τ₄₀ = 38fs. The decay time of coupling and the total coupling phases respectively are about: (a) τ₁ = 31fs, τ₂ = 150fs, τ₃ = 31fs, τ₄ = 36fs, φ₁ = 0, φ₂ = 0.3π, φ₃ = −0.7π, φ₄ = 0; (c) τ₁ = 31fs, τ₂ = 160fs, τ₃ = 71fs, τ₄ = 176fs, φ₁ = −0.45π, φ₂ = 0.85π, φ₃ = −0.46π, φ₄ = 0.8π; (d) τ₁ = 31fs, τ₂ = 170fs, τ₃ = 180fs, τ₄ = 36fs, φ₁ = 0, φ₂ = −0.2π, φ₃ = 0.24π, φ₄ = 0; (b) Argand diagram of the transmission coefficient t_n of the four resonant modes at peak (red lines) and trough (blue lines) of FR1 in the plasmonic system shown in Fig. 1(a) with R_A = 500 nm and ΔH = 0.

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Here, the phase of the transmission coefficient $t_{1}$ is taken as the reference zero point, and the Argand diagram of the transmission coefficients $t_{n}$ at the peak and trough of FR1 is given in Fig. 2(b). Figure 2(b) shows that FR1 is the result of the interaction between the transmission coefficients $t_{n}$ of the four resonant modes. The major contribution to FR1 are the change of $t_{2}$ , not only is the change of its modulus (about 0.28) obvious but also the change of its phase respectively is about 0.55π, and the changes of the modulus and phases of $t_{3}$ are much smaller than that of $t_{2}$ , while the modulus and phases of $t_{1}$ and $t_{4}$ almost remain unchanged. The reason is that the changes of the modulus and phases of $t_{2}$ are much more drastic than that of $t_{1}$ , $t_{3}$ and $t_{4}$ near the resonant wavelength of TM_1,0 mode. The simulation and MICMT results of the transmittance of this system with ΔH = 50 nm and ΔH = 100 nm are given in Figs. 2(c) and 2(d), respectively. The decay time and coupling phases of the coupling between the resonator and waveguides can be influenced by the change of ΔH. As shown in Figs. 2(c) and 2(d), there is some deviation between MICMT result and simulation result, the reason of which is mode degeneracy is removed when the symmetry of this system is broken in y direction, so as to induce the degenerate interference Fano resonance. The removal of mode degeneracy is not considered in Eqs. (5) and (6), thereby the deviation appears in MICMT results. The influence, which is induced by the interference between degenerate modes, on the transmission response of this system will be discussed below.

In circular resonator, each mode has two degenerate modes except the modes of TM_q,0. Based on MICMT, the DICMT equations including degenerate coupling phases are given as follows

\frac{d a_{n}^{l}}{d t} = (- j ω_{n} - \frac{1}{τ_{n 0}} - \frac{1}{τ_{n 1}} - \frac{1}{τ_{n 2}}) a_{n}^{l} + κ_{n 1}^{l} s_{n, 1 +} A_{n 1}^{l} + κ_{n 2}^{l} s_{n, 2 +} A_{n 2}^{l}, l = ↓, ↑

s_{1 -} = - s_{1 +} + \sum_{n} (κ_{n 1}^{↓ *} a_{n}^{↓} + κ_{n 1}^{↑ *} a_{n}^{↑})

s_{2 -} = - s_{2 +} + \sum_{n} (κ_{n 2}^{↓ *} a_{n}^{↓} + κ_{n 2}^{↑ *} a_{n}^{↑})

A_{n 1}^{↓} + A_{n 1}^{↑} = 1, A_{n 2}^{↓} + A_{n 2}^{↑} = 1

where

a_{n}^{↓}

(

a_{n}^{↑}

) is the field amplitude of the nth degenerate mode with orbital angular momentum along the negative (positive) direction of z-axis,

A_{n 1}^{l}

and

A_{n 2}^{l}

are degenerate normalized coefficients.

Because of waveguides S1 and S2 with the widths are equal are symmetrical about circular cavity A, so $τ_{n 1}^{↓} = τ_{n 1}^{↑} = τ_{n 2}^{↓} = τ_{n 2}^{↑} = τ_{n}$ . In this case, the transmittance of the plasmonic system can be expressed as

T = {| t |}^{2} = {| \sum_{n} \frac{2 γ_{n 1} e^{j φ_{n}} (e^{j ϕ_{n}} A_{n 1}^{↓} + e^{- j ϕ_{n}} A_{n 1}^{↑})}{- j (ω - ω_{n}) τ_{n} + 2 + \frac{τ_{n}}{τ_{n 0}}} |}^{2}, ϕ_{n} = 2 m \cdot arc \cos (\frac{Δ H}{R_{A}})

Here, m is the angular resonant order.

A_{n 1}^{↓}

is the function of angular frequency, the approximate fitting expression is

A_{n 1}^{↓} = 0.5 + e^{j φ_{n d}} / [- j (ω - ω_{n d}) τ_{n d} + 2 + δ_{n}]

,

φ_{n d}

is the coupling phase difference between degenerate modes of the nth mode,

ω_{n d}

is the angular frequency of central position of the removal of degeneracy,

τ_{n d}

and

δ_{n}

are related to the degree of the removal of degeneracy.

Figures 3(c)-3(g) show the simulation results of the transmittance of this system with ΔH = 50 nm ~300 nm. It can be seen from Figs. 3(c)-3(g) that the position of degenerate interference Fano resonance (DIFR) remains unchanged when ΔH changes from 50 nm to 300 nm, but the coupling coefficients between TM_0,2 mode and waveguides are different. When ΔH is near 188 nm, the coupling between TM_0,2 mode and waveguides almost drop to the minimum and most of the energy is reflected back to waveguide S1, therefore the transmittance of this system is almost zero near the resonant wavelength of TM_0,2 mode.

Fig. 3 The simulation results (blue lines) and DICMT results (red lines) of spectral transmittance of the plasmonic resonator system with R_A = 500 nm and (a) ΔH = 50 nm, (b) ΔH = 100 nm. The parameters of degenerate coupling respectively are about $φ_{1 d} = - 0.32 π$ , $ω_{1 d} = 1.69 \times 10^{15} r a d / s$ , $τ_{1 d} = 203 f s$ and $δ_{1} = 0.1$ . The coupling parameters of TM_0,1 mode respectively are about $τ_{60} = 69 f s$ and (a) $τ_{6} = 58 f s$ , $φ_{6} = 1.1 π$ ; (b) $τ_{6} = 48 f s$ , $φ_{6} = - 0.66 π$ . (c-g) The simulation results of spectral transmittance of the plasmonic resonator system with different ΔH.

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The transmission response of the plasmonic resonator system shown in Fig. 1(b) will be investigated next. Figure 4 shows the simulation and MICMT results of the transmittance and the distribution of the normalized magnetic field (Hz) of the plasmonic system in Fig. 1(b). As shown in Figs. 4(c) and 4(d), a new resonant mode appears in the range of 750 nm ~1000 nm after the circular cavity B is added to the plasmonic structure in Fig. 1(a), of which the resonant wavelength has significant relationship with circular cavity B. Here, the new added resonant mode is considered as the fifth resonant mode a₅ of the plasmonic system. According to MICMT, the number of the resonant modes increases to 5, that is $n = 1, 2, \dots, 5$ . The simulation and MICMT results of the transmittance are shown in Fig. 4(d), and the Fano resonance on the right of FR1 is called FR3.

Fig. 4 The distribution of the normalized magnetic field (Hz) of the resonant modes at (a) 812 nm, (b) 854 nm and (c) 1001 nm in the plasmonic resonator system with ΔH = 0, R_A = 500 nm, R_B = 250 nm, g = 10 nm, respectively. (d) The simulation result (blue line) and MICMT result (red line) of the transmission response of the plasmonic resonator system.

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The resonant wavelength and radius of the circular cavity satisfy the relationship of $λ_{q, m} = c_{q, m} R$ . Here, $λ_{q, m}$ is the resonant wavelength of TM_q,m mode, $c_{q, m}$ is a proportional constant. When ΔH = 0, g = 10 nm are fixed value, the influence of the structure parameters on the transmission response is studied in detail and the results are shown in Fig. 5. As shown in Figs. 5(a)-5(e), when R_A increases from 480 nm to 520 nm, the positions of FR1, FR2 and TM_0,2 mode increase linearly, but the position of FR3 slightly fluctuates near 996 nm. Figures 5(f)-5(j) show that, with R_B increases from 240 nm to 260 nm, the position of FR3 increases from 965 nm to 1023 nm, while the positions of FR1, FR2 and TM_0,2 mode have no change, so this kind of tuning is called ‘independent tuning’ [20].

Fig. 5 The dependence of transmission response on the structure parameters, when ΔH = 0, g = 10 nm. (a-e) Different R_A = 480 nm ~520 nm with fixed R_B = 250 nm; (f-j) Different R_B = 240 nm ~260 nm with fixed R_A = 500 nm.

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The figure of merit (FOM) is a key parameter for Fano resonance, which can be defined as $F O M = Δ T / (T Δ n)$ at a fixed wavelength and $F O M^{*} = \max | F O M |$ near a fixed wavelength [21], where $T = 0.5 \times [T (n = 1.00) + T (n = 1.02)]$ . Figure 6(a) shows the transmission spectra for different refractive indexes of the dielectric in the waveguides and cavities with the structure parameters of R_A = 500 nm, R_B = 250 nm, g = 10 nm, ΔH = 0. With the increasing of n from 1.00 to 1.02, the peaks of FR1, FR2 and FR3 are all red shift, the peaks of FR 1 red shifts from about 850 nm to 867 nm, the peaks of FR 2 red shifts from about 816 nm to 831 nm and the peaks of FR3 red shifts from about 995 nm to 1014 nm. The FOM curve of this plasmonic system is given in Fig. 6(b), and the FOM* of FR1 are remarkable and as high as 100. For the Fano resonances in the plasmonic nanocavity, the transmittance of the SPPs varies sharply from the valley to the peak with only a significantly decreased wavelength shift. This reveals that the wavelength shift required for a completely on/off variation is significantly reduced, implying a high sensitivity to the index variations of a nearby or surrounding medium [17, 22]. For example, when a detected sample (e.g., gas, liquid, or biomolecule) is filled in the plasmonic system, the FR1 would have a corresponding wavelength shift [23–25] with a sensitivity of about 840 nm/RIU (700 nm/RIU in [18]). That is because the strong field confinement and low leaky loss in our structure, making it more sensitive to the refractive index of the material.

Fig. 6 (a) The transmission spectra of the plasmonic resonator system with R_A = 500 nm, R_B = 250 nm, g = 10 nm, ΔH = 0, and different refractive indices of the dielectric in the waveguides and cavities, n = 1.00 (blue line), n = 1.02 (red line). (b) The FOM curve of this plasmonic resonator system.

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4. Conclusion

In summary, a plasmonic resonator system with small footprint (≈2μm²) has been numerically and theoretically demonstrated, in which the reasons for the formation of the three kinds of Fano resonances are explained by using the proposed MICMT and DICMT. The three kinds of Fano resonances can be tuned by changing the parameters of the two plasmonic resonators, so that the coupling coefficients and coupling phases between waveguides and different resonant modes can be changed, and then the transmission response of the plasmonic system can be controlled. This made it have more functional and complex applications. This also may pave new routes to realizing Fano resonances in on-chip plasmonic nanostructures. The sensitivity and FOM* of these Fano resonances can be as high as 840 nm/RIU and 100, respectively. This small footprint together with the tunable spectral response can actualize active devices for fundamental study and applications in sensors, splitters, lasing and slow-light devices.

Funding

Ministry of Science and Technology of China (2016YFA0301300); National Natural Science Foundation of China (NSFC) (11374041, 61571060, 11574035, 11404030).

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Fano resonances based on multimode and degenerate mode interference in plasmonic resonator system

Abstract

1. Introduction

2. Structures and MICMT analysis

3. Simulation results and discussions

4. Conclusion

Funding

References and links

Cited By

Figures (6)

Equations (11)

Optics Express