1. Introduction

The remarkable capability of surface plasmon polaritons (SPPs) for light manipulation on a sub-wavelength scale beyond the diffraction limit [1] has prompted extensive theoretical and experimental research on highly integrated photonic systems [2–10]. SPPs can be excited on the surface of metals [11–13] and various metal-like materials [14–16]. Among the various plasmonic structures, the metal-dielectric-metal (MDM) waveguide has been recognized as a good integrated photonic device owing to the advantages including the support for SPPs propagation at the metal-dielectric boundary, light manipulation at a sub-wavelength scale, etc [17]. Due to these potential features, simple structure, ease of construction, and industrial fabrication, the MDM waveguide structure has been extensively investigated for designing optical filters [12,18,19], switches [20–22], sensors [23–25], and slow-light devices [26,27].

Matsuzaki et al. originally proposed a stub(or stub pair)-coupled MDM waveguide for wavelength-selective filtering [2]. When a single stub(or stub pair) is extended to a periodic stub, due to the bandgap structure of the transmission spectrum after multiple filtering [12,28–32], the periodic-stub-coupled MDM waveguide structure supports broad stopband/passband filtering function and a sub-wavelength broadband slow-light guided mode [33]. When a single-set-periodic-stub-coupled MDM waveguide is expanded to a two-set-periodic-stub-coupled structure with two different stub depths, a narrow-passband filter function is achieved due to the overlap of the bandgap edges of the two periodic structures under appropriate parameters [18,34], i.e., the heterostructured waveguide can achieve single plasmon-induced transparency (PIT) effects. However, when heterostructures are used for achieving multiple PIT effects, the waveguide becomes more complicated. Moreover, when N single-mode stub resonators with graded depths are side-coupled to an MDM waveguide, N-1 PIT peaks will appear in the transmission spectrum due to the phase-coupled effect between two adjacent stubs with detuned resonant wavelengths, achieving a multichannel filtering function [7,35]. Besides, a bright-dark-dark waveguide system can achieve dual PIT spectral response due to the destructive interference between two adjacent resonator modes [20,36,37]. Nevertheless, among the above waveguide structures, it is necessary to couple N resonators to generate N-1 PIT peaks due to the phase coupling effect [7,35,38] or destructive interference [20,36,37] between the adjacent single-mode resonators, and three resonators result in dual PIT peaks. However, a simpler structure is preferred. To the best of our knowledge, there are few reports on the realization of dual PIT effects with two stub-coupled MDM waveguides.

In this work, based on the basic coupled mode theory (CMT) [26], we introduce the multimode-coupled mode theory (M-CMT) to understand the coupling effects and analyze the spectral responses in an MDM waveguide structure side-coupled with a single-mode stub and a multimode stub. Moreover, the structure can achieve double PIT peaks by suitably adjusting these structural parameters. Compared to the previous waveguide structures with the same functions [20,35,38], the simplicity of our proposed structure is a significant advantage, and it can achieve more PIT peaks compared to a similar structure [26]. The theoretical calculation results are consistent with the simulation results, and this study provides basic analysis ideas for single-mode and multimode resonator-coupled systems. Because of the simple structure and ease of fabrication, the proposed system has great potential for applications in multichannel filters, sensors, and switches.

2. Structural model and theoretical analysis

The proposed MDM waveguide system, which includes a single-mode stub and a multimode stub, is shown in Fig. 1. For simplicity, the left stub is marked as stub1, while the right stub is marked as stub2. The structural parameters include the width w of the main waveguide, width w₁ and depth d₁ of stub1, width w₂ and depth d₂ of stub2, and the coupling distance L₁₂ between stub1 and stub2. Besides, L is the distance between the light source and the monitor. In the simulation, we respectively select air and silver as the insulator and metal of the system. The dielectric constant of silver, which is regulated by the frequency of the light wave, is expressed by the Drude mode [39]: ε_m(ω) = ε_∞-ω_p/(ω² + iωγ_p), where ω, ω_p(1.37 × 10¹⁶ rad/s), ε_∞(3.7), and γ_p(4.08 × 10¹³ rad/s) are the angular frequency, plasma frequency, relative dielectric constant at infinite frequency, and damping rate of the incident wave, respectively.

 

Fig. 1 Two-dimensional schematic of the MDM waveguide side-coupled with a single-mode stub and a multimode stub.

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The spectral response features of the system are qualitatively analyzed by two-dimensional finite-difference time-domain(FDTD) simulation, where the boundary conditions are set as perfectly matched layers with a mesh grid (Δx = Δy = 2 nm and t = Δx/2c, c is the speed of light in vacuum). When a Gaussian beam is input to the left of the main waveguide, SPPs, which propagate along the surface, are generated at the dielectric-metal interface, coupling part of the energy into the stubs and exciting the resonant modes.

In the simulations, the parameters are set as follows: d₁ = 250 nm, d₂ = 355 nm, L = 680 nm, L₁₂ = 200 nm, w = w₁ = 60 nm, and w₂ = 80 nm. Figure 2(a) displays the transmission spectra of the system coupled with only one stub: stub1(or stub2). For stub1, a transmission dip is formed at wavelength of 527 nm, as indicated by the black curve. Conversely, for stub2, two transmission dips are formed at wavelengths of 430 nm and 654 nm, respectively, as indicated by the blue curve, and there is a relatively wide transmission window between the two dips. It can be seen that the incident wave excites a resonant mode in stub1 and stimulates two resonant modes in stub2. The resonator, where the incident wave excites only one resonant mode, is called single-mode resonator. The resonator, where the incident wave stimulates two or more resonant modes, is called multimode resonator. Hence, stub1 and stub2 correspond to single-mode and multimode resonators, respectively. Two typical PIT peaks can be observed in the transmission spectrum of the two stub-coupled system, as displayed in Fig. 2(b). To explore the formation mechanism of the dual PIT shapes, the magnetic field distributions of the single stub-coupled structure at the three resonant dips are exhibited in Figs. 2(c)–2(e), respectively. TM_mn indicates the resonant modes, where m and n are non-negative integers representing the number of standing-wave nodes along the x-direction and y-direction, respectively. The resonant mode of stub1 corresponds to TM₀₁ (λ = 527 nm), while those of stub2 correspond to TM₀₁ (λ = 654 nm) and TM₀₂ (λ = 430 nm), respectively. When the two stubs are coupled to the main waveguide simultaneously, direct and indirect couplings occur between the stub resonators [26]. Physically, the coupling effects occur between the resonant modes, rather than between the stubs, i.e., the resonant mode of the single-mode stub interacts with each resonant mode of the multimode stub, respectively. Furthermore, direct coupling occurs between any two resonant modes of the multimode-stub resonator [40]. Thus, the transmission window of stub2 is split into two transmission windows, exhibiting a special spectral response with double PIT peaks, as depicted in Fig. 2(b). And the two peaks are recorded as P₁₂ (λ = 467 nm) and P₁₁ (λ = 564 nm), respectively. So, based on the coupling of TM₀₁ and TM₀₂ of stub2, the interaction between the TM₀₁ modes of the two stubs produces P₁₁, and the coupling between TM₀₁ of stub1 and TM₀₂ of stub2 produces P₁₂. The above conclusion is further confirmed by the magnetic field distributions of the PIT peaks displayed in Figs. 2(f) and 2(g).

 

Fig. 2 (a) and (b) Transmission spectra of the MDM waveguide side-coupled with a single-mode stub (stub1, black curve), multimode stub (stub2, blue curve), and single-mode and multimode stubs (stub1 and stub2, red curve) with w = 60 nm, w₁ = 60 nm, w₂ = 80 nm, L₁₂ = 200 nm, d₁ = 250 nm, and d₂ = 380 nm. (c)–(e) Magnetic field distributions of the individual single-mode and multimode stub-coupled structure at the resonant dips. (f) and (g) Magnetic field distributions of the coupled structure at the two PIT peaks.

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Next, in order to explore the coupling effects in the single-mode and multimode resonator-coupled system, and study the spectral response behaviors of the system in detail, based on the direct coupling among the resonant modes of the multimode resonator [40,41], considering the direct and indirect couplings between the resonant modes of different stubs, we introduce an extended M-CMT. In Fig. 1, $S_{\pm ,in}^{(N)}(N=1,2)$ and $S_{\pm ,out}^{(N)}(N=1,2)$ represent the incoming and outgoing waves in the main waveguide, respectively. The subscript ± indicates the propagating directions of the waveguide modes, and a_n (n = 1, 2, 3) is the energy amplitude of the nth resonant mode. Thus, the extended M-CMT can be expressed as follows:

When a wave beam propagates between the two stubs, it causes a phase shift,$\phi =\mathrm{Re}({\beta }_{spp})L_{12}=\omega \mathrm{Re}(n_{eff})L_{12}/c$, where L₁₂, β_SPP, and n_eff are the coupling distance, propagation constant, and effective refractive index of the SPPs [42], respectively. Based on the principle of conservation of energy, the following relationships can be obtained:

Theoretically, combining the boundary condition of $S_{-in}^{(2)}=0$ with Eqs. (1)–(5), the amplitude transmittance of the proposed system can be deduced as follows:

3. Results and discussion

Figures 3(a) and 3(b) show the FDTD simulated and M-CMT calculated transmission spectra of the proposed system for different d₂ values, and the other parameter values are same as in Fig. 2. The PIT peaks are recorded as P₁₁ or P₁₂ near d₂ = 355 nm, and P₁₂ or P₁₃ near d₂ = 555 nm. The simulation results are consistent with the calculated ones, not only proving the correctness of the theoretical model but also providing a basic idea for further researches. Figures 3(c)–3(e) depict respectively the wavelengths, transmission, and Q-factors of the PIT peaks for different d₂ values. Figure 3 displays three interesting results: (i) When d₂ = 355 nm (or 555 nm), the transmission spectrum exhibits substantially symmetric double PIT peaks; (ii) As d₂ increases, both PIT peaks exhibit a significant red shift as shown in Figs. 3(a)–3(c); (iii) As d₂ increases, the transmission of the right PIT peak slowly increases and its bandwidth widens, becoming a wide transmission window with low Q-factors, while the left PIT peak changes in the opposite direction and eventually disappears near d₂ = 471 nm as shown in Figs. 3(d) and 3(e). The above phenomena can be ascribed to the red shift of the resonant modes with the gradual increase in d₂.

 

Fig. 3 (a) and (b) Transmission spectra of the numerical and theoretical calculations, respectively, for various d₂ values. (c) Wavelengths, (d) transmission, and (e) Q-factors of the PIT peaks, respectively.

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Figure 4(a) displays the simulated transmission spectra of the system coupled only with the multimode stub for different d₂ values, and the other parameters are same as in Fig. 2. As d₂ increases, the transmission spectra undergo a red-shift and more dips appear, i.e., more resonant modes are stimulated in the multimode stub. When d₂ increases to 475 nm, three resonant modes are excited in the wavelength range of 400−1100 nm. The transmission valleys are recorded as dip1, dip2, and dip3, respectively. Figures 4(c)–4(e) display the magnetic field distributions of dip1, dip2, and dip3, corresponding to TM₀₁, TM_02, and TM₀₃, respectively. Figure 4(b) depicts the linear relationship between the resonant wavelengths and d₂. When d₂ = 471 nm, the resonant wavelength of TM₀₂ is equal to that of TM₀₁ of stub1 with d₁ = 250 nm. When d₂ < 471 nm, the resonant wavelength of the stub1 is between those of TM₀₁ and TM₀₂ of stub2. Hence, based on the direct coupling between the modes of stub2, TM₀₁ of stub1 couples with TM₀₁ and TM₀₂ of stub2, respectively, producing two PIT peaks recorded as P₁₁ and P₁₂, respectively. As d₂ increases, the wavelength detuning between the TM₀₁ modes of stub1 and stub2 increases, resulting in an increase for the transmission and FWHM of P₁₁, as shown in Figs. 3(a), 3(b), and 3(d), thereby decreasing the Q-factor of P₁₁, as illustrated in Fig. 3(e). However, the wavelength detuning between TM₀₁ of stub1 and TM₀₂ of stub2 reduces continuously, resulting in a decrease in the transmission and FWHM for P₁₂, as illustrated in Figs. 3(a), 3(b), and 3(d), and continuously increasing in Q-factor, as illustrated in Fig. 3(e); then P₁₂ completely disappears at d₂ = 471 nm, as depicted in Fig. 3(d). When d₂ > 471 nm, TM₀₃ is excited in the short wavelength band, and the resonant wavelength of TM₀₁ of stub1 is between those of TM₀₂ and TM₀₃ of stub2, as shown in Fig. 4(b), resulting in a large wavelength detuning between the two TM₀₁ modes of stub1 and stub2. So, the coupling between them shows a wide transmission window, which is not considered in the work. Next, we only consider the coupling effects that TM₀₁ of stub1 interacts with TM₀₂ and TM₀₃ of stub2, respectively. Similarly, as d₂ increases, the detuning of the resonant wavelength between TM₀₁ and TM₀₂ increases, resulting in enlarging for the transmission and FWHM of P₁₂. Whereas the detuning of the resonant wavelength between TM₀₁ and TM₀₃ exhibits an opposite trend with the increase in d₂, reducing the transmission and FWHM of P_13.

 

Fig. 4 (a) Transmission spectra of the MDM waveguide side-coupled with the multimode stub for various d₂ values. (b) Resonant wavelengths of dip1, dip2, and dip3 for various d₂ values. (c)−(e) Magnetic field distributions of the MDM waveguide side-coupled with the multimode stub at these resonant dips.

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According to the above analysis, as d₂ increases, P₁₁ slowly evolves into a broad transmission window, while P₁₂ undergoes a special evolution process of gradual narrowing, disappearing, and gradual widening, before becoming a broad transmission window, as depicted in Figs. 3(a), 3(b), and 3(d). If d₂ increases continuously, P₁₃ will also undergo the same evolution process, and a new PIT peak will appear in the transmission spectra in the short-wavelength band.

In the above, the modulation effect of d₂ on the two PIT peaks has been analyzed. Besides, other structural parameters, such as the depth of the single-mode stub and coupling distance, can well regulate the transmission features of the system. Next, we briefly discuss the regulation characteristics of d₁ on PIT peaks. In the case of d₂ = 355 nm and d₁ = 250 nm, the resonant wavelength of stub1 is between the ones of stub2, and the transmission spectrum exhibits substantially symmetric dual PIT peaks, as shown in Fig. 2(b). As d₁ increases, the resonant wavelength of TM₀₁ of stub1 has a red-shift while the resonant wavelengths of TM₀₁ and TM₀₂ of stub2 remain constants. So, the wavelength detuning between the TM₀₁ modes of stub1 and stub2 decreases with an increase in d₁, resulting in a decrease for the transmission and FWHM of P₁₁. However, the wavelength detuning between TM₀₁ of stub1 and TM₀₂ of stub2 increases continuously, leading to an increase in the transmission and FWHM of P₁₂. Thus, the two PIT peaks exhibit opposite evolution trends with the change in d₁, where the tendencies are completely opposite to those with the increase in d₂. In addition, it should be noted that multiple modes will be excited when the depth of stub1 is too large, so the depth of stub1 cannot be too large.

Finally, we investigate the sensing features because of the sensitivity of the optical transmission characteristics to the environment, and the application potential in sensors. Figure 5(a) illustrates the simulated spectra of the proposed system for various refractive indexes n, and the other parameters are same as in Fig. 2. In Fig. 5(a), with the increase of n, the transmission spectra show significant red-shift, maintaining nearly the same spectral shape. Figure 5(b) exhibits the relationship between the wavelengths of the PIT peaks (or dips) and the refractive index. The sensitivity is defined as S = Δλ_m/Δn [43], where Δn and Δλ_m are the changes in the refractive index and the corresponding wavelength shift of the PIT peaks (or dips), respectively. Because Fig. 5(b) shows good linear relationships, the sensitivities are equal to the slopes of the lines. By linear fitting, we obtain the corresponding slopes, and the slopes (sensitivities) of dip1, P₁₁, dip2, P₁₂ and dip3 are 625.51 nm/RIU, 526.77 nm/RIU, 476.91 nm/RIU, 413.34 nm/RIU, and 359.94 nm/RIU, respectively. These good linear relationships may be attributable to the relationship between the n_eff and the refractive index. The n_eff of the SPPs in an MDM waveguide with width w is expressed as follows [44,45]:

 

Fig. 5 (a) Transmission spectra for various refractive indices n, when d₁ = 250 nm and d₂ = 355 nm. (b) Wavelengths of the dips and peaks with respect to refractive indices n. (c)Relationship between Re(n_eff) and the wavelength, for n = 1.00, 1.03, and 1.06, respectively. (d) Relationship between Re(n_eff) and n, for λ = 430 nm, 527 nm, and 654 nm, respectively.

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Where ε_d and ε_m are the relative dielectric constants of the dielectric and metal, respectively, and k₀ is the wave vector in vacuum. When n = 1, 1.03, 1.06, and w = 60 nm, Fig. 5(c) depicts the relationship between the real part of the effective index Re(n_eff) of the SPPs and the wavelength of TM mode. When λ = 430 nm, 527 nm, and 654 nm, Fig. 5(d) displays obvious linear relations between Re(n_eff) and refractive indexes. These linear relationships are also satisfied for other wavelengths. In addition, because the resonant wavelength is proportional to Re(n_eff) of the SPPs, the resonant wavelengths and the wavelengths of PIT peaks are necessarily proportional to the refractive index of the medium, as revealed in Fig. 5(b). Thus the proposed structure can be utilized as a sensor for monitoring the refractive index of the environment, the environmental temperature, and liquid concentration.

4. Conclusions

In this study, a simple MDM waveguide structure side-coupled with a single-mode stub and a multimode stub at different positions have been presented. This system can achieve dual PIT peaks and can be used as a dual-channel filter. Moreover, its spectral responses are numerically and theoretically investigated. In order to comprehend the coupling effects of the proposed system, we have originally introduced an extended M-CMT, which considers not only the coupling effect between the modes in the multimode resonator but also the coupling effects between the modes from different resonators. Besides, our proposed structure is simpler than the previous structures with the same function. Furthermore, we have studied the sensing features of the proposed system by analyzing the relationships between the wavelengths of the dips (or peaks) and the refractive index, established that a very good linear relationship exists between them. This excellent linear relationship can serve as the foundation for sensor design.