1 Introduction

As is well known, guiding of light within a structure with subwavelength size has recently aroused people’s extensive attention, because surface plasmon polaritons (SPPs) can overcome the diffraction limit in conventional optics and manipulate the light at nanoscale domain [1, 2]. The guided modes are referred as the propagating SPPs along the metal-dielectric interface, which have been considered as the most promising solution to reduce the transverse dimension of optical devices [3, 4]. Among several plasmonic guiding structures, the metal-insulator-metal (MIM) plasmonic waveguide has attracted people’s considerable interest in the realization of nanoscale [5,6,7,8], due to the remarkable properties of easy fabrication, more compact, convenient integration, excellent propagation pathway, acceptable propagation length, and subwavelength light confinement [9, 10]. Until now, a number of subwavelength optical devices based on MIM plasmonic waveguide have been proposed and investigated [11,12,13,14,15], which can be achieved by different resonators coupled to MIM plasmonic waveguide. Nowadays, various cavity resonators are studied and reported from theory and experiment, such as rectangle [16], square [17], circle [18], racetrack [19], rectangular-ring [20], square-ring [21], circular-ring [22], and split-ring [23]. However, the shapes of these cavity resonators are symmetrical. Consequently, we propose sectorial-ring cavity resonator (SRCR) for the first time, which is simple configuration, compact integration and asymmetrical shape, and obtains multiple cavity and controllable or tunable transmission spectrum.

In this paper, a SRCR is first reported, and a novel nanoscale plasmonic resonator system is investigated that is composed of a left- or right-handed SRCR coupled to a MIM plasmonic waveguide. The spectral characteristic of the proposed system is studied by using the finite-difference time-domain (FDTD) method. The simulation displays that a transmission spectrum with two resonance modes is achieved in the novel system, whose left-handed SRCR and right-handed SRCR are discussed at the same time. Furthermore, when the refractive index of the dielectric, the width and center arc length of the cavity, and the gap distance between cavity and waveguide are fixed, the SRCR system with multiple cavity realizes a controllable transmission spectrum; and when they are unfixed, the SRCR system with multiple cavity realizes a tunable transmission spectrum. Additionally, the resonance wavelengths, minimum transmittances and magnetic field distributions of the two resonance modes are analyzed in detail. Consequently, we obtain the spectral characteristic of the SRCR system with a highly controllable or tunable transmission spectrum. This work provides a guideline to effectively control or accurately tune transmission spectrum in plasmonic nanodevices and a promising application for nanoscale and integration.

2 Structure and simulation

The two-dimensional (2D) schematic of the proposed system is illustrated in Fig. 1, whose z-axis is infinite, and blue and white areas, respectively, are metal and dielectric (it usually uses dielectric instead of insulator). It is consisted of a resonator and a waveguide, whose resonator is a single right-handed sectorial-ring cavity that is side-coupled to waveguide, and waveguide is a MIM plasmonic waveguide with a straight slot. Compared to these resonators in references [16,17,18,19,20,21,22,23], this SRCR is simple configuration by building with a single cavity, compact integration by single-sided coupling to the waveguide, and asymmetrical shape in the x and y directions. The main structural parameters are the central angle (θ), outer radius (R), inner radius (r), width (ws) and center arc length (lC) of the cavity, the width (w) of the waveguide, and the gap distance (g) between cavity and waveguide. The dielectric constant of metal (εm) is approximately described by the Drude model, which is defined as \({\varepsilon _{\text{m}}}={\varepsilon _\infty } - {\omega _{\text{p}}}^{2}/({\omega ^2}+i\omega \gamma )\), where ε∞ is the dielectric constant of metal at the infinite angular frequency, ω is the angular frequency of incident light, ωp is the natural frequency of bulk plasma, and γ is the damping frequency of electron collision. The dielectric constant of dielectric (εd) is usually a constant, which is related to the refractive index (n) of dielectric that is defined as \({\varepsilon _{\text{d}}}={n^2}/{u_{\text{d}}}\), where ud is the magnetic conductivity of dielectric.

Fig. 1
figure 1

The 2D schematic of the right-handed SRCR system and the power monitors of the proposed system

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The two power monitors are, respectively, used at the left and right of the waveguide to detect the incident power (Pin) and the transmitted power (Pout), as shown in Fig. 1. The transmittance (T) of the proposed system is calculated by

$$T={{{P_{{\text{out}}}}} \mathord{\left/ {\vphantom {{{P_{{\text{out}}}}} {{P_{{\text{in}}}}}}} \right. \kern-0pt} {{P_{{\text{in}}}}}},$$
(1)

and the FDTD method with perfectly matched layer (PML) absorbing boundary conditions is used to study numerically and theoretically the spectral characteristic of the proposed system. We implement the FDTD calculation with dimension (2D), boundary conditions (PML), time step (dt = dx/2c, c is the velocity of light in vacuum), mesh steps (dx = dy = 5 nm), source shape (Gaussian) and source injection axis (x-axis). To predigest the simulation, the metal and dielectric in the proposed system are selected to be silver and air, respectively. The parameters for the silver and air can be set as ε∞ = 3.7, ωp = 1.38 × 1016 Hz, γ = 2.73 × 1013 Hz and εd = 1 (n = 1) that are obtained by fitting experimental results [24], and other parameters for the cavity and waveguide are set to be 0° < θ ≤ 180° and w = 100 nm.

When an incident light injects along the x-axis in the waveguide, SPPs with the transverse-magnetic mode can be excited on the metal-dielectric interface and confined in the waveguide and then they are divided into the two parts of reflected SPPs and transmitted SPPs [16,17,18,19,20,21,22,23]. Based on the equivalence principle of the coupled mode theory (CMT) [25], a single cavity is regarded as a Fabry-Perot resonator. We define Δφ as the phase difference between reflected SPPs and transmitted SPPs, which is described as [25, 26]:

$$\Delta \varphi {\text{=}}\frac{{4\pi Re({n_{{\text{eff}}}}){l_{{\text{eff}}}}}}{{{\lambda _m}}}+\varphi ,$$
(2)

where Re(neff) is the real part of the effective refractive index (neff) of light in the waveguide (or cavity), leff is the effective length of SPPs in the cavity, φ is the phase shift caused by the reflection of SPPs on the metal-dielectric interface, and λm is the resonance wavelength of the cavity resonator at the m order (m = 0, 1, 2…) resonance mode. When the resonance condition is satisfied with \(\Delta \varphi =(2m+1) \cdot \pi\), λm at the minimum transmittance (or trough) of transmission spectrum is expressed as:

$${\lambda _m}=\frac{{4Re({n_{{\text{eff}}}}){l_{{\text{eff}}}}}}{{(2m+1) - {\varphi \mathord{\left/ {\vphantom {\varphi \pi }} \right. \kern-0pt} \pi }}}.$$
(3)

According to the CMT [25], the transmittance of the SRCR system is derived as [27]:

$$T=\frac{{{{(\omega - {\omega _0})}^2}+{{\left( {\frac{{{\omega _0}}}{{2{Q_0}}}} \right)}^2}}}{{{{(\omega - {\omega _0})}^2}+{{\left( {\frac{{{\omega _0}}}{{2{Q_0}}}+\frac{{{\omega _0}}}{{2{Q_{\text{w}}}}}} \right)}^2}}},$$
(4)

where ω0 is the resonance angular frequency of the cavity, Q0 and Qw, respectively, are the quality factors related to the intrinsic loss in cavity and the coupling loss between cavity and waveguide, and \(\omega ={{2\pi c} \mathord{\left/ {\vphantom {{2\pi c} \lambda }} \right. \kern-0pt} \lambda }\), \({\omega _0}={{2\pi c} \mathord{\left/ {\vphantom {{2\pi c} {{\lambda _m}}}} \right. \kern-0pt} {{\lambda _m}}}\), \({Q_0}=Re({n_{{\text{eff}}}})/2\operatorname{Im} ({n_{{\text{eff}}}})\) and \({Q_{\text{w}}}={{{\lambda _m}} \mathord{\left/ {\vphantom {{{\lambda _m}} {\Delta {\lambda _{{\text{FWHM}}}}}}} \right. \kern-0pt} {\Delta {\lambda _{{\text{FWHM}}}}}}\) [Im(neff) is the imaginary part of neff, ΔλFWHM is the full width at half maximum of transmission spectrum (\(\Delta {\lambda _{{\text{FWHM}}}}=({\lambda _{\text{R}}} - {\lambda _{\text{L}}})\left| {_{{1/2({T_{\hbox{max} }} - {T_{\hbox{min} }})}}} \right.\), where λR and λL, respectively, are the right-side wavelength and the left-side wavelength of transmission spectrum at resonance mode when \(T=1/2({T_{\hbox{max} }} - {T_{\hbox{min} }})\))]. Thus, at the ω0, the transmission spectrum has a trough with a minimal value, namely minimum transmittance, which is expressed as [27, 28]:

$${T_{\hbox{min} }}={{{{\left( {\frac{{{\omega _0}}}{{2{Q_0}}}} \right)}^2}} \mathord{\left/ {\vphantom {{{{\left( {\frac{{{\omega _0}}}{{2{Q_0}}}} \right)}^2}} {{{\left( {\frac{{{\omega _0}}}{{2{Q_0}}}+\frac{{{\omega _0}}}{{2{Q_{\text{w}}}}}} \right)}^2}}}} \right. \kern-0pt} {{{\left( {\frac{{{\omega _0}}}{{2{Q_0}}}+\frac{{{\omega _0}}}{{2{Q_{\text{w}}}}}} \right)}^2}}}.$$
(5)

3 Results and discussion

In general, the rotation direction in the SRCR has left-handed (the inset of Fig. 2a) and right-handed (Fig. 1) directions. By comparing the spectral characteristic of the left-handed SRCR system and right-handed SRCR system with n = 1, θ = 90°, R = 350 nm, r = 250 nm and g = 20 nm, we elucidate why only choose the right-handed SRCR. The transmission spectra of the left-handed SRCR system and right-handed SRCR system are shown in Fig. 2a, which are represented by the red and blue curves, respectively. It is found that the transmission spectrum with two resonance modes is achieved in the novel system, and the transmission spectrum is uniform at the left-handed and right-handed SRCR systems. It indicates that the left-handed and right-handed SRCR systems whose transmission spectra are coincidences of 100%, which are due to the identical structural parameters of the two resonators [28, 29]. Moreover, the mode 1 and mode 2 of the SRCR system occur at λm = 1271 nm with Tmin = 0.17 and λm = 657 nm with Tmin = 0.04, respectively. To further know spectral characteristic, we also analyze the magnetic field distribution. Figure 2b–e shows the magnetic field distributions of the left-handed SRCR system at mode 1 and mode 2, and the right-handed SRCR system at mode 1 and mode 2, respectively. It can be seen that for the left-handed SRCR system, the magnetic field distributions at mode 1 and mode 2 are all identical to that of the right-handed SRCR system, due to the identical propagation paths of SPPs in the two cavities [28, 29]. At the mode 1 and mode 2, almost all of the power is confined in the cavity, and no power is transported out. In other words, the magnetic field distributions at mode 1 and mode 2 are consistent with the transmission spectrum. Consequently, the right-handed SRCR is as an example at the full text. Additionally, the transmission (T), reflection (R) and absorption (A) spectra of the SRCR system at mode 1 and mode 2 are shown in Fig. 2f, g, respectively. It is found that the absorptances at mode 1 and mode 2 are 0.44 and 0.34, respectively. It reveals that the coupling losses between cavity and waveguide at mode 1 and mode 2 are relatively low, and the coupling loss at mode 2 is smaller than the mode 1. In a word, the SRCR system is a novel nanoscale plasmonic resonator system with relatively low loss.

Fig. 2
figure 2

Spectral characteristic of the SRCR system with n = 1, θ = 90°, R = 350 nm, r = 250 nm and g = 20 nm. a Transmission spectra of the left-handed and right-handed SRCR systems, and inset: the 2D structure diagram of the left-handed SRCR. Magnetic field distributions of the left-handed SRCR system at b mode 1 and c mode 2, and the right-handed SRCR system at d mode 1 and e mode 2. Transmission, reflection and absorption spectra of the SRCR system at f mode 1 and g mode 2

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We investigate the spectral characteristic of the SRCR system with various θ (0° < θ ≤ 180°), R and r when n = 1, ws = 100 nm (\({w_{\text{s}}}=R - r\)), lC = 150π nm (\({l_{\text{C}}}={{\theta \pi \left( {R+r} \right)} \mathord{\left/ {\vphantom {{\theta \pi \left( {R+r} \right)} {{{360}^{\text{o}}}}}} \right. \kern-0pt} {{{360}^{\text{o}}}}}\)) and g = 20 nm. For the sake of comparison, Fig. 3a shows the 2D structure diagrams of the SRCR with various values of θ, R and r [(θ, R, r) = (30°, 950 nm, 850 nm); (60°, 500 nm, 400 nm); (90°, 350 nm, 250 nm); (120°, 275 nm, 175 nm); (150°, 230 nm, 130 nm); (180°, 200 nm, 100 nm)]. It is found that the controllable sectorial-ring cavities are intuitively shown, which is good for us to observe structural changes. It can be seen that the cross-sectional areas of these cavities are equal, which helps us to compare spectral characteristics. The transmission spectra of the SRCR system with various values of θ, R and r are shown in Fig. 3b. It is found that the two resonance modes show almost no shift in the transmission spectrum of the novel system. It indicates that when n, ws, lC and g are constants, the novel system with various θ, R and r achieves an unchanged transmission spectrum. And the transmission spectra of the SRCR system are almost 100% superposed, except that the structure size (180°, 200 nm, 100 nm). The λm and Tmin of the mode 1 and mode 2 in the SRCR system as functions of θ (15° ≤ θ ≤ 180°) for various R and r are shown in Fig. 3c. It is clear that the resonance wavelengths of the mode 1 and mode 2 are almost unchanged, while the minimum transmittances of the mode 1 and mode 2 are also almost unchanged, except that the structure size (180°, 200 nm, 100 nm). And with the structure size (180°, 200 nm, 100 nm), the resonance wavelengths of the mode 1 and mode 2 increase slightly, while the minimum transmittances of the mode 1 and mode 2, respectively, decrease slightly and increase significantly. Figure 3d, e shows the magnetic field distributions of the SRCR system with various values of θ, R and r at mode 1 and mode 2, respectively. It reveals that the magnetic field distributions of the SRCR system with various θ, R and r at mode 1 or mode 2 are identical, which are attributed to the unchanged resonance wavelengths. Meanwhile, it also reveals that the magnetic field distributions at mode 1 and mode 2 are in accordance with the transmission spectrum.

Fig. 3
figure 3

Spectral characteristic of the SRCR system with various θ, R and r when n = 1, ws = 100 nm, lC = 150π nm and g = 20 nm. a 2D structure diagrams. b Transmission spectra. c λm and Tmin of the mode 1 and mode 2 as functions of θ. Magnetic field distributions at d mode 1 and e mode 2

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To elucidate this phenomenon, we analyze the λm of the two resonance modes in the transmission spectrum. Next, we need to analyze Re(neff) and leff, because they influence the size of λm (namely the transmission of SPPs is related to neff and leff). According to the dispersion relation, neff is defined as [30, 31]:

$${n_{{\text{eff}}}}=\sqrt {{\varepsilon _{\text{d}}}\left( {1 - \lambda /w\pi \cdot \sqrt {{\varepsilon _{\text{d}}} - {\varepsilon _{\text{m}}}} /{\varepsilon _{\text{m}}}} \right)}$$
(6)

where λ is the wavelength of incident light and w is the width (w) of the waveguide (or the width (ws) of the cavity). According to the arc length formula [32], the center arc length of the cavity in the SRCR stands for leff of the SPPs in the cavity [14], and leff is described to be

$${l_{{\text{eff}}}}={l_{\text{C}}}=\frac{{\theta \pi (R+r)}}{{{{360}^{\text{o}}}}}.$$
(7)

To understand how the refractive index (n) of the dielectric or the width (w) of the waveguide (or cavity) influences the Re(neff) and how the central angle (θ), outer radius (R) and inner radius (r) of the cavity influences the leff, the dependence of Re(neff) on the wavelength (λ) of incident light for various n or w and the dependence of leff on the θ, R and r are investigated. The single waveguide is shown in the inset of Fig. 4a, and it needs to be indicated that the region of the cavity with a width (ws) also fits the dependence of Re(neff) on n or w in the waveguide. The Re(neff) of light in the waveguide as a function of λ for various n of the dielectric (n = 1.0, 1.1, 1.2, 1.3, 1.4 when w = 100 nm) or w of the waveguide (w = 50, 75, 100, 125, 150 nm when n = 1) is drawed in Fig. 4a. It is found that the Re(neff) at various n or w has a identical variation tendency as λ increases, which first decreases nonlinearly and rapidly as λ increases from 400 nm to 900 nm and then becomes nearly saturated as λ increases from 900 to 2400 nm. And the Re(neff) takes on a trend of increasing linearly as n increases and decreases nonlinearly with a decrement as w increases. The leff of SPPs in the cavity as a function of θ (when R = 350 nm and r = 250 nm), R (when θ = 90° and r = 250 nm), or r (when θ = 90° and R = 350 nm) is drawed in Fig. 4b. It is clear that the relationship between leff and θ, R, or r is a linear relation. It also reveals that when R and r are various values, ws can be a constant; when θ, R and r are various values, lC can be a constant.

Fig. 4
figure 4

a Re(neff) as a function of λ for various n of the dielectric (when w = 100 nm) or w of the waveguide (when n = 1), and inset: the 2D structure diagram of the waveguide. b leff as a function of θ (when R = 350 nm and r = 250 nm), R (when θ = 90° and r = 250 nm), or r (when θ = 90° and R = 350 nm) of the cavity

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From the analysis above, there are two main ways to alter the size of λm [29]. The one way is to change the real part of the effective refractive index (Re(neff)) and the other way is to change the effective length (leff). To analyze how the property of the dielectric and the structure size of the cavity influence the transmission spectrum, the spectral characteristic of the SRCR system with various refractive indexs (n) (or dielectric constants (εd)) of the dielectric, or various geometric parameters (θ, R and r) of the cavity is investigated. Figures 5 and 6 investigate the influence of Re(neff) on the transmission spectrum by adjusting n or ws when lC and g are constants, and the influence of leff on the transmission spectrum by adjusting θ or (R + r) when Re(neff) and g are constants, respectively.

Fig. 5
figure 5

Transmission spectra of the SRCR system with various a n when ws = 100 nm, lC = 150π nm (θ = 90°, R = 350 nm and r = 250 nm) and g = 20 nm, or b ws when n = 1, lC = 150π nm (θ = 90° and R + r = 600 nm) and g = 20 nm

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Fig. 6
figure 6

Transmission spectra of the SRCR system with various a θ when n = 1, ws = 100 nm (R = 350 nm and r = 250 nm) and g = 20 nm, or b (R + r) when n = 1, ws = 100 nm (R − r = 100 nm and θ = 90°) and g = 20 nm

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Figure 5a, b shows the transmission spectra of the SRCR system with various values of n when ws = 100 nm, lC = 150π nm (θ = 90°, R = 350 nm and r = 250 nm) and g = 20 nm, and ws when n = 1, lC = 150π nm (θ = 90° and R + r = 600 nm) and g = 20 nm, respectively. It is obvious that when lC and g are constants, a changed transmission spectrum is achieved in the novel system, which is sensitive to n and ws. It is found from Fig. 5a that the mode 1 and mode 2 of the transmission spectrum undergo redshift as n increases from 1.0 to 1.4, which are attributed to λm increases by the increased Re(neff) and the unchanged leff (leff = lC = 150π nm), and it is in accordance with Eqs. (3), (6) and (7). It is found from Fig. 5b that the mode 1 and mode 2 of the transmission spectrum undergo blueshift as ws increases from 50 to 150 nm, which are attributed to λm decreases by the decreased Re(neff) and the unchanged leff (leff = lC = 150π nm), and it is in accordance with Eqs. (3), (6) and (7). Consequently, when the SRCR is adjusted with various n or ws, a highly tunable transmission spectrum of the SRCR system is obtained by changing the Re(neff) of the cavity and waveguide.

Figure 6a, b shows the transmission spectra of the SRCR system with various values of θ when n = 1, ws = 100 nm (R = 350 nm and r = 250 nm) and g = 20 nm, and (R + r) when n = 1, ws = 100 nm (R − r = 100 nm and θ = 90°) and g = 20 nm, respectively. Likewise, when Re(neff) and g are constants, a changed transmission spectrum is achieved in the novel system, which is sensitive to θ and (R + r). It can be seen from Fig. 6a that the mode 1 and mode 2 of the transmission spectrum undergo redshift as θ increases from 30° to 150°, which are due to λm increases by the unchanged Re(neff) and the increased leff (leff = lC = 50π, 100π, 150π, 200π and 250π nm, respectively), and it is in line with Eqs. (3), (6) and (7). It can be seen from Fig. 6b that the mode 1 and mode 2 of the transmission spectrum undergo redshift as (R + r) increases from 480 nm to 720 nm, which are due to λm increases by the unchanged Re(neff) and the increased leff (leff = lC = 120π, 135π, 150π, 165π and 180π nm, respectively), and it is in line with Eqs. (3), (6) and (7). Consequently, when the SRCR is adjusted with various θ or (R + r), a highly tunable transmission spectrum of the SRCR system is obtained by changing the leff of the cavity.

To elucidate this phenomenon, we also analyze the Tmin of the two resonance modes in the transmission spectrum. From reference [18], it can be used to alter the size of Tmin by changing the coupling strength between cavity and waveguide. To analyze how the gap distance between cavity and waveguide influence the transmission spectrum, the spectral characteristic of the SRCR system with various gap distances (g) between cavity and waveguide is investigated. Figure 7 investigates the influence of coupling strength on the transmission spectrum by adjusting g when Re(neff) and leff are constants. The transmission spectra of the SRCR system with various values of g when n = 1, ws = 100 nm and lC = 150π nm (θ = 90°, R = 350 nm and r = 250 nm) are shown in Fig. 7a. Similarly, a changed transmission spectrum is also achieved in the novel system when Re(neff) and leff are constants, which is sensitive to g. As g increases from 10 to 50 nm, the mode 1 and mode 2 of the transmission spectrum undergo slight blueshift, which are because the coupling strength between cavity and waveguide is slightly reduced by increasing the gap distance. The magnetic field distributions of the SRCR system with various values of g when n = 1, ws = 100 nm and lC = 150π nm (θ = 90°, R = 350 nm and r = 250 nm) at mode 1 and mode 2 are shown in Fig. 7b, c, respectively. Obviously, less SPPs energy can be stored in the cavity and more SPPs energy can be transported out. In the meanwhile, the ΔλFWHM of transmission spectrum decreases as g increases and the Tmin of transmission spectrum increases as g increases, which can be explained by the CMT. When only g is adjusted, \({\omega _0}/2{Q_{\text{0}}}\) can be regarded as almost unchanged, Δλ decreases as the weakened coupling between cavity and waveguide due to the increase in g, and \({\omega _0}/2{Q_{\text{w}}}\) decreases as Δλ decreases [18]. Thus, at the ω0, the Tmin increases as \({\omega _0}/2{Q_{\text{w}}}\) decreases, and it is consistent with Eq. (5). Consequently, when the SRCR is adjusted with various g, a highly tunable transmission spectrum of the SRCR system is also obtained by changing the coupling strength between cavity and waveguide.

Fig. 7
figure 7

Spectral characteristic of the SRCR system with various g when n = 1, ws = 100 nm and lC = 150π nm (θ = 90°, R = 350 nm and r = 250 nm). a Transmission spectra. Magnetic field distributions at b mode 1 and c mode 2

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Based on the result above, we can elucidate this phenomenon of unchanged transmission spectrum. On the basis of Figs. 5, 6 and 7, Re(neff) of the cavity and waveguide is unchanged due to the unchanged n (or εd) and ws, leff in the cavity is unchanged due to the unchanged lC, and the coupling strength between cavity and waveguide is unchanged due to the unchanged g. Thus, the transmission spectrum does not undergo a shift due to the unchanged λm and Tmin as θ, R and r change, and it is consistent with Eqs. (3), (5–7). However, with the structure size (180°, 200 nm, 100 nm), the transmission spectrum undergoes a slight redshift as θ increases to 180°, which is attributed to λm increases by the enhancing of the coupling strength between cavity and waveguide when the rotating side of the cavity is close to the waveguide. Consequently, when the SRCR is controlled with fixed n, ws, lC and g, a highly controllable transmission spectrum in the SRCR system is obtained with various θ (0° < θ ≤ 180°), R and r, due to the unchanging characteristcis of the Re(neff) of the cavity and waveguide, the leff in the cavity, and the coupling strength between cavity and waveguide.

A multiple cavity is easy conformation and convenient adjustment, which is formed by adjusting the n of the dielectric, the (R − r), θ or (R + r) of the cavity, or the g between cavity and waveguide, and plays an important role in the transmission spectrum. This modulation method is different from the one which is by changing the geometric parameters of the whole cavity or the partial cavity [16,17,18,19,20,21,22,23]. More importantly, it can prove a regulation that when the SRCR satisfies four conditions, the transmission spectrum of the SRCR system is not shifting by adjusting the central angle, outer radius and inner radius of the cavity. The four conditions are the fixed refractive index (n) of the dielectric, the fixed width (ws) of the cavity, the fixed center arc length (lC) of the cavity and the fixed gap distance (g) between cavity and waveguide, and n, ws, lC and g as follows:

$$n=\sqrt {{\varepsilon _{\text{d}}}{u_{\text{d}}}} {\text{ = constant,}}$$
(8)
$${w_{\text{s}}}=R - r{\text{ = constant,}}$$
(9)
$${l_{\text{C}}}=\frac{{\theta \pi (R+r)}}{{{{360}^{\text{o}}}}}{\text{ = constant,}}$$
(10)
$$g{\text{ = constant}}{\text{.}}$$
(11)

Such configuration may provide an alternate approach for designing optical devices in nanophotonic applications and offer a great flexibility for the design of controllable and tunable nanodevices in ultra-compact optical components and integrated optical circuits. Our proposed structure can be used as a novel plasmonic nanofilter or nanosensor, which is very different from the plasmonic nanofilters or nanosensors in previous researches, and it will have wide application in plasmonic nanodevices. Most of all, the unique feature of the SRCR system not only provides a promising way for realization of controllable or tunable transmission spectrum and for optimization of prospective structure size, but also has potential application in nanoscale optical devices and integrated optics devices. In particular, the structural model of the SRCR demonstrates a practical approach to design optical devices, which will satisfy different fabricating demands in future. Additionally, the optical technique with controllable or tunable transmission spectrum has also meaningful application in designing accurately the transmission spectrum in various dielectric/graphene metamaterials and metasurfaces [33,34,35,36].

4 Conclusion

In summary, we first propose a left- or right-handed SRCR coupled to a MIM plasmonic waveguide, and investigate the spectral characteristic of the novel system. It is found that the transmission spectrum is uniform at the left-handed and right-handed SRCR systems, and two resonance modes appear in the transmission spectrum of the SRCR system. And when the n, ws, lC and g are unfixed, the transmission spectrum can be highly tuned by adjusting the n, ws, θ, (R + r) or g. More importantly, when the n, ws, lC and g are fixed, it also can be highly controlled with various θ, R and r. The performances of the transmission spectrum are well elucidated by the resonance condition of the CMT, and the theoretical analysis agrees well with the FDTD numerical simulation. Consequently, the spectral characteristic of the SRCR system is obtained with a highly controllable or tunable transmission spectrum. The proposed structure is simple configuration, compact integration and asymmetrical shape, and realizes multiple and controllable or tunable cavity, which can become a novel plasmonic nanofilter or nanosensor. Our result indicates that the proposed structure has potential application in nanoscale and integration, and it may be useful for optical communication, optical information processing and other related areas. Our work demonstrates the unique feature of plasmonic nanodevices with effectively control or accurately tune transmission spectrum.