Introduction

Plasmonics at terahertz (THz) frequencies has sparked a wealth of research interest driven by its capabilities of enhancing near-field effects, improving imaging resolution and making nonlinear interactions effectives [1–3]. These features render it promising for developing the next-generation nanophotonics devices in THz bio-sensing, ultra-high resolution imaging, and ultrafast communications fields [4–6]. In such an explosion of interesting fields, classic metallic nanostructures [7–9] play a pivotal role in controlling and manipulating the terahertz electromagnetic wave at scales substantially smaller than its wavelength. However, at terahertz frequencies, plasmon on metal surface suffers great losses and limited tunability, severely hindering the development of THz plasmonic devices.

Graphene that emerged as a fascinating two-dimensional atomic film with extraordinary electrical and optical properties, opens up a brand-new opportunity to address these challenges [10, 11]. As a kind of revolutionary plasmonics material, graphene supports intrinsic transverse-magnetic (TM) plasmon modes with extremely tight confinement, relatively long lifetime, and very low losses, making it possible to manipulate the THz electromagnetic wave in the single atomic plane. In particular, its plasmon frequencies can be tuned electrically [12], which offers an innovative solution to break through the bottlenecks of the conventional terahertz devices. These unprecedented properties point to a promising future of graphene as an ideal platform for carbon-based optoelectronics across the THz spectrum region. Very recently, great progresses have been made on the investigation of the basic characteristics of graphene surface plasmonics (GSP) modes. For instance, the GSP eigenmode is predicted theoretically and observed experimentally [13–15]. The excitations of GSP wave are demonstrated in both patterned graphene (graphene nano-disks [16], micro-ring [17], micro-ribbon [18], and anti-dot arrays [19]) and continuous graphene film (based on prism configuration [20] and diffraction grating [21–23]). And the distinctive propagation characteristics of GSP wave have been implemented for designing various optical switch [18], interferometer [24], optical waveguide [25], directional coupler [26], and so forth.

However, previous researches mainly focus on the physical nature, excitation, and propagation of the GSP modes, further investigation about the effect of near-field perturbation on the coupling of GSP modes remains unrevealed, which is an inevitable problem when designing an active plasmonics device [27]. For a classic dielectric-graphene-dielectric plasmonics system [13], a small change in the geometric structure of the periodical nanostructure will yield a perturbation within the near-field region, which certainly affects the coupling between GSP modes and evanescent wave modes in the nanostructure, leading to a considerable variation of absorbance spectra and mode patterns of the system. Thus, it raises interesting and urgent questions: how the near-field perturbation influences the performance of the system and what is the underlying mechanism behind this effect? Understanding these working principles will play a fundamental role in building basic blocks for the development of the GSP-based THz devices and systems.

Herein, we explore the coupling effect of GSP modes induced by near-field perturbation in one-dimensional grating-spacer-graphene hybrid systems. First, the influence of the near-field perturbation on the coupling between GSP modes and evanescent wave modes in dielectric and metallic hybrid systems are investigated systematically. Then, the physical mechanism behind the coupling effect is revealed using effective medium theory and rigorous coupled wave theory. Finally, the penetration of GSP waves into the dielectric and metallic gratings is also discussed in detail. Our study may promote innovative methods for confining and controlling terahertz wave and provide fundamental insight for the designs of plasmonics devices at terahertz frequencies, such as ultrasensitive terahertz biosensor, ultrafast terahertz modulators, and super-resolution imaging.

Structure and Results

As schematically depicted in Fig. 1, the grating-spacer-graphene hybrid systems are proposed to investigate the coupling of GSP modes induced by near-field perturbation. In order to systematically investigate the coupling behaviors in the hybrid systems, the systems are categorized into the dielectric grating-spacer-graphene (DGSG) hybrid system and the metallic grating-spacer-graphene (MGSG) hybrid system considering the different permittivity characteristics of the dielectric grating (Re(ε) > 0) and metallic grating (Re(ε) < 0), as shown in Fig. 1a, c, respectively.

Fig. 1
figure 1

Schematic of the DGSG hybrid system (a) and the MGSG hybrid system (c). The gray and yellow regions are dielectric grating and metallic grating whose period, height, and groove width are Λ, h, and w, respectively. The violet region is a dielectric spacer with thickness of t, and under the spacer is a monolayer graphene. b The cross section of the hybrid system shows the physical process of the coupling between GSP modes and evanescent wave modes

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The grating contains the following geometric parameters: period Λ, height h, and occupation ratio η. A dielectric spacer with thickness of t is sandwiched between grating and graphene, which can be employed to eliminate the non-uniform doping and strains of the graphene induced by the grating. Without explicit statement, the permittivity of dielectric grating and spacer used for calculations are assumed to be ε r  = 11.8 and ε 2 = 2.25. The permittivity of metal (Au) is given by the Drude model [28]. Graphene is modeled as a monolayer with thickness of 0.34 nm. Simulations are carried out by finite element method employing Comsol mutiphysics. The mesh size inside graphene layer along the Z-axis is 0.03 nm, and the mesh size gradually increases outside the graphene.

As a transverse magnetic (TM) wave irradiates vertically from the top of the grating, it will be scattered into various diffraction orders which are evanescent waves. Once the in-plane wavevector of a GSP wave matches the wavevector of a diffraction order, these two kinds of evanescent waves couple together to excite a highly confined fundamental GSP mode in graphene as illustrated in Fig. 1b. The coupling strength is determined by the overlap area between a GSP mode and an evanescent wave mode [29]. In such process, the variation of distance t between grating and graphene and the change of the occupation ratio η of grating will yield near-field perturbation to the coupling of GSP modes. This leads to the variation of the overlap area, and consequently, the energy exchange between GSP modes and evanescent wave modes, which results in the changes of the essential features in GSP modes. These features are characterized by the evolution of the absorption curves in the spectral domain and the energy redistribution of GSP modes in the spatial domain. Therefore, to investigate systematically, the effect of near-field perturbation caused by the variation of t and η on GSP modes, the resonant frequency, absorbance, and spatial mode patterns are calculated and compared in the DGSG and MGSG hybrid systems, respectively.

Coupling of GSP Modes Induced by Perturbation in Dielectric Nanostructure

In this section, the coupling effect of the GSP modes induced by near-field perturbation as t decreases is investigated. Figure 2a illustrates the evolution of the absorption spectra of the DGSG hybrid system when the occupation ratio and the period of the grating are fixed as η = 50 % and Λ = 200 nm. Once the wavevector matching condition is satisfied, the normal-incidence free space light is coupled into GSP wave and highly confined in graphene. Then, a characteristic absorption peak can be observed in the absorption spectrum, and the incident light is converted into GSP wave with maximum excitation efficiency at the resonant frequency. As t decreases from 45 to 5 nm, the resonant frequency of GSP wave redshifts from 17.72 to 13.93 THz, along with the increase of absorbance from 7.25 to 41.97 % due to the enhanced coupling between GSP modes and evanescent wave modes in the grating. It means that the coupling strength can be modulated by the thickness of the spacer.

Fig. 2
figure 2

a The evolution of the absorption spectra with varying distance t between the graphene and the dielectric grating when η = 50 % and Λ = 200 nm. b The variation of the electrical field distribution of the fundamental GSP modes with a fixed frequency of 15 THz when t = 45, 35, 15, and 0 nm. c Normalized |Ex| intensity on the graphene layer with different t

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The near-field perturbation also impacts the spatial energy distribution of GSP modes in this process. To visually observe the energy redistribution, the GSP mode patterns are simulated with a fixed frequency of 15 THz when t equals 45, 35, 15, and 0 nm, respectively. As indicated in Fig. 2b, energy of GSP modes redistributes in the sub-wavelength region, exhibiting a concentration of GSP energy from the grating gap toward the grating ridge as the dielectric grating gets close to the graphene layer. To further illustrate this tendency, the near-field normalized |Ex| intensity along the graphene layer is extracted from Fig. 2b and depicted in Fig. 2c. Evidently, the maximum |Ex| intensity under the grating ridge becomes much stronger than that under the grating trench as t decreases.

The coupling induced by another kind of near-field perturbation due to the variation of occupation ratio η is also explored fixing t = 20 nm and Λ = 200 nm. It is observed from the absorption spectra in Fig. 3a that the resonant frequency blueshifts from 15.84 to 18.18 THz continuously as η decreases from 95 to 25 %. However, the absorbance no longer varies monotonously; instead, it initially exhibits a sharp increasing behavior from only 13.12 % (η = 95 %) to a maximum value of 41.97 % (η = 75 %) and then decreasing to merely 6.9 % (η = 25 %). This phenomenon indicates that the occupation ratio η of grating has a strong ability to modulate the absorbance of the hybrid system. Meanwhile, it is observed that the energy of GSP modes shows a concentration tendency toward the grating ridge as the occupation ratio η reduces (see Fig. 3b). This variation tendency can be further confirmed by Fig. 3c, which illustrates that the normalized |Ex| at the point X = 0 decreases from 0.98 to 0.21 as η decreases from 95 to 35 %.

Fig. 3
figure 3

a The evolution of the absorption spectra with different occupation ratio η in one dielectric grating period when t = 20 nm and Λ = 200 nm. b The variation of the electrical field distribution of the fundamental mode with a fixed frequency of 16.22 THz when η = 95, 50, 40, and 35 %. c Normalized |Ex| intensity on the graphene layer with different η

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Coupling of GSP Modes Induced by Perturbation in Metallic Nanostructure

Further investigation focuses on the effect of the near-field perturbation on the coupling between GSP modes and evanescent wave modes in the metallic grating. Simulations are performed using a gold grating with the same geometric parameters as the dielectric grating and the spacer with the same permittivity (ε 2 = 2.25). Fig. 4a presents the absorption spectra with varying distance t between the metallic grating and the graphene. It illustrates that the shift of the resonant frequency of the MGSG hybrid system is approximately twice larger than that of the DGSG hybrid system. Meanwhile, the corresponding absorption efficiency is higher than that in the DGSG hybrid system. These results indicate that the coupling effect between the GSP modes and evanescent wave modes in the MGSG hybrid system is stronger than that in the DGSG hybrid system.

Fig. 4
figure 4

a The evolution of the absorption spectra with varying distance t between the graphene and the metallic grating when η = 50 % and Λ = 200 nm. b The variation of the electrical field distribution of the fundamental mode with a fixed frequency of 30 THz when t = 50, 35, 15, and 0 nm. c Normalized |Ex| intensity on the graphene layer with different t

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In contrast to the DGSG hybrid system, a completely opposite feature is observed that the energy of the GSP mode under the ridge is gradually suppressed as the metallic grating approaches to the graphene, yielding a concentration of energy on the graphene layer under the grating gap instead of the ridge (displayed in Fig. 4b). Specially, when the grating contact with graphene directly (t = 0), the GSP wave under the ridge is suppressed thoroughly. In the situation, as illustrated in Fig. 4c, the energy is mainly confined in the grating gap and the magnitude of |Ex| at the metallic-graphene interface under the ridge is nearly zero.

Finally, we investigated the evolutions of the coupling behaviors when the occupation ratio η of metallic grating varies. It is found that the MGSG hybrid system still maintains a high absorbance even when its occupation ratio is large. As indicated in the absorption spectra in Fig. 5a, the absorbance of the MGSG hybrid system still remains at a high level (∼49.1 %) when the occupation ratio η is larger than 75 %, while that of the DGSG hybrid system drops by a large margin to only 13.12 % (Fig. 3a). To further understand this behavior, the GSP modes profiles with varying η at 16.22 THz are visualized in Fig. 5b. Obviously, a surface plasmon wave with high intensity is generated in the grating gap when η equals 95 %, leading to a considerable confinement of the incident wave. In addition, as shown in Fig. 5c, most portion of the two |Ex| normalized intensity distribution curves when η equals 50 and 95 % are well overlapped, which means that a metallic grating with large occupation ratio exhibits a stronger ability to modulate the incident light than a dielectric grating with the same geometry.

Fig. 5
figure 5

a The evolution of the absorption spectra with different occupation ratio η in one metallic grating period when t = 20 nm and Λ = 200 nm. b The variation of the electrical field distribution of the fundamental mode with a fixed frequency of 16.22 THz when η = 95, 50, 40, and 35 %. c Normalized |Ex| intensity on the graphene layer with different η

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Physical Model and Mechanism

From the results presented above, it is clear that the coupling induced by the near-field perturbation results in the shift of the resonant frequency and the variation of absorbance. We first build a grating-spacer-graphene physical model to quantitatively unveil the shift of the resonant frequency using effective medium theory. For simplicity, we approximate the grating as an effective medium with relative permittivity of ε 1 = [1 + ηε r + (1 − η)ε g]/2 [30], where ε r and ε g are the permittivity of the grating ridge and the grating trench, respectively. In our considering terahertz frequencies, the conductivity of graphene is dominated by the intraband transitions, which can be represented by a Drude model [18]. The dispersion relation of GSP for TM modes in a continuous monolayer graphene in the hybrid system is approximately given by the following

$$ q\left(\omega \right)=\frac{\pi {\hslash}^2{\varepsilon}_0\left({\varepsilon}_3+\overline{\varepsilon}\right)}{e^2{E}_f}\left(1+\frac{i}{\omega \tau}\right){\omega}^2 $$
(1)

with \( \overline{\varepsilon} \) defined as

$$ \overline{\varepsilon}\equiv {\varepsilon}_2\frac{\left({\varepsilon}_1+{\varepsilon}_2\right)\cdot \exp \left(2 qt\right)+\left({\varepsilon}_1-{\varepsilon}_2\right)}{\left({\varepsilon}_1+{\varepsilon}_2\right)\cdot \exp \left(2 qt\right)-\left({\varepsilon}_1-{\varepsilon}_2\right)}. $$
(2)

In Eq. 2, ω is the angular frequency of the incident light, e is the elementary charge, τ = 0.64 ps is the relaxation time, ħ is the reduced Planck’s constant, E f  = 0.1 eV is the Fermi energy of graphene, and ε 3  = 1 is the permittivity of the background (air). In Eq. 3, \( \overline{\varepsilon} \) is the permittivity of the spacer and grating when they are taken as an effective uniform medium, and ε2 is the permittivity of the dielectric spacer. A GSP wave can be resonantly excited once the reciprocal lattice vectors of the grating matches the wave vectors of the GSP mode

$$ Re\left[q\left(\omega \right)\right]=n\frac{2\pi }{\varLambda }, $$
(3)

where n is an integer and n ⋅ 2π/Λ is the reciprocal lattice vector of the grating. In this work, the plasmonic wave is excited by the first diffraction order, and hence n = 1 in Eq. 3. The resonant frequency ω GSP can be expressed as

$$ {\omega}_{\mathrm{GSP}}=\sqrt{\frac{2{e}^2\times {E}_f}{\hslash^2{\varepsilon}_0\left({\varepsilon}_3+\overline{\varepsilon}\right)\varLambda }}. $$
(4)

Based on the derivation above, the predicting resonant frequencies calculated using Eq. 4 with varying t and η are provided in Fig. 6a, b, respectively. Meanwhile, they are compared with the simulated results extracted from the absorption spectra in Fig. 2a, Fig. 3a, Fig. 4a, and Fig. 5a. One can clearly observe that the theoretical resonant frequencies show good agreement with the simulated ones. The shift of a resonant frequency is the result of the variation of the effective permittivity (i.e., Eq. 2) induced by the near-field perturbation. Specifically, the decrease of t (η) gives rise to the increase (decrease) of the effective permittivity around graphene, resulting in the redshift (blueshift) of resonant frequency. The effective medium physical model is further verified with varying grating period Λ and permittivity ε 2 of the spacer. The dependence of the absorption spectra of the DGSG hybrid system on Λ and ε 2 are calculated numerically and illustrated in Fig. 6c, d. The red light line represents the resonant frequency. One can visually see that the resonant frequency redshifts to the low frequency region as Λ and ε 2 increases due to the decrease of the effective permittivity around the graphene. Meanwhile, the analytical resonant frequencies are also calculated using Eq. 4 and shown as the white hollow sphere lines. Clearly, simulation results and theoretical ones are in well accordance. Therefore, our physical model is trustworthy in predicting the shift of the resonant frequency induced by the near-field perturbation in both two hybrid systems.

Fig. 6
figure 6

Comparison between theoretical resonant frequencies and simulation results with varying distance t (a) and occupation ratio η (b), respectively, when Λ = 200 nm. c Simulated absorbance mapping of the DGSG hybrid system with varying grating period Λ and incident frequency. d Simulated absorbance mapping of the DGSG hybrid system with varying permittivity \( \varepsilon \) 2 of spacer and incident frequency when t = 20 nm and η = 50 %. The white hollow sphere lines are theoretical resonant frequencies as a function Λ and \( \varepsilon \) 2

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Besides, we revealed that the variation of absorbance in the coupling process is attributed to the change of the intensity of the first diffraction order in the grating. For the case when the distance t decreases, the coupling between the GSP modes and evanescent wave modes becomes stronger due to the exponential decay of intensity of the first diffraction order along the direction of the grating height (z-axis in Fig. 1a, b). This results in the monotonous increase of the absorbance (Fig. 2a and Fig. 4a). While for the case when η decreases, we find that the absorbance of the system is also modulated by the first diffraction order. In Fig. 7a, b, the intensity of the first diffraction order in both dielectric and metallic gratings are calculated using rigorous coupled wave theory and compared with the absorbance of the system. It shows that when the occupation ratio of the gratings increases, the absorbance exhibit a similar variation tendency with the intensity of the evanescent wave in both DGSG and MGSG hybrid systems. This signifies that the absorbance of the hybrid systems is mainly determined by the intensity of the first order evanescent wave. We also note that the two curves in Fig. 7b exhibits slightly different tendency after the occupation ratio of metallic grating exceeds 60 %. This attributes to the fact that the absorbance is also dependent on the overlap area between the plasmonics mode and the evanescent wave mode [29], which varies slightly as the occupation ratio of metallic grating increases. The above quantitative revelation of the evolution of the spectra characteristics provides a guidance to design THz GSP devices with high-coupling efficiency.

Fig. 7
figure 7

Comparison between the numerical resonant absorbance and the first order evanescent wave intensities calculated by rigorous coupled wave theory in the hybrid systems with the occupation ratio η of the dielectric grating (a) and metallic grating (b) vary from 30 to 95 %. The red hollow star lines represent the variation of the absorbance of the hybrid system while the blue solid star lines represent the variation of the intensity of the first order evanescent wave

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Discussion

The above analysis has unveiled the evolution of the absorption spectra induced by the near-field perturbation. In this section, we reveal the evolution of the GSP modes in such coupling process. We found that the opposite distribution tendency of the GSP mode energy in the hybrid systems as the grating approaches the graphene (Fig. 2b and Fig. 4b) is attributed to the different penetration abilities of GSP waves into the dielectric and metallic nano-gratings. To intuitively observe the penetration of the GSP wave into the grating, the GSP mode profiles of the DGSG and MGSG hybrid systems when t = 10 nm, η = 50 %, and Λ = 200 nm are given in the insets in Fig. 8a, b, respectively, and the corresponding normalized electric field amplitudes |Ex| along the green lines (z-axis) are also extracted. For the DGSG hybrid system, |Ex| first drops to 42 % at the spacer/dielectric interface (at the position of Z = 10 nm), and then slowly falls to 35 % at the position of Z = 20 nm in the dielectric grating. Whereas, |Ex| rapidly reduces to only 1.5 % at the spacer/Au interface and becomes negligible at Z = 20 nm in the MGSG hybrid system. These indicate that the GSP wave possesses a much stronger ability to penetrate into the dielectric grating than into the metallic grating.

Fig. 8
figure 8

The distributions of electric field amplitudes along the green lines in the GSP modes for dielectric gratings (a) and Au gratings (b), respectively. c The dependence of the absorbance of the DGSG hybrid system and the MGSG hybrid system on grating heights h when it varies from 5 to 200 nm. Graphene is at the position of zero, dielectric and Au gratings are 10 nm away from graphene

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To quantitatively measure the penetration ability, the decay length [21] into the nano-grating of a GSP wave is employed, which is defined as where the field amplitude drops to 1/e of that at the spacer/grating interface. According to this definition, the decay length of GSP waves in dielectric grating is ∼60 nm as can be seen from Fig. 8a, which means the GSP wave can penetrate into the dielectric grating with a considerable length. Meanwhile, refractive index (n r = 3.4) of the grating ridge is larger than that of the grating gap (n g = 1) and the spacer (n spa = 1.5). Thus, the effective refractive index under the grating ridge increases as the grating approaches the graphene, resulting in the increase of the field intensity and a concentration tendency of GSP waves toward the grating ridge. On the other hand, the decay length of GSP waves in metallic grating is extremely small (only ∼5 nm, see Fig. 8b), greatly suppressing the penetration of the GSP wave into the metallic grating; and hence, the GSP wave energy should concentrate to the grating gap instead of the grating ridge when the spacer thickness t is reduced. Especially for the case when the grating touches the graphene, the continuous GSP mode is completely suppressed at the metal-graphene interface, resulting in a localized mode confined in the grating gap. It means that in order to avoid localized GSP modes, we need to employ a dielectric spacer in the MGSG hybrid system. Based on these discussions, we can conclude that the different decay lengths of the GSP wave in the dielectric and metallic grating contribute to the opposite spatial energy distribution of GSP modes as the grating approaches the graphene.

In addition, we found that the different decay lengths of the GSP wave also determine the dependence of the absorbance of the hybrid systems on the grating height. We calculated the absorption spectra in both DGSG and MGSG system at different grating heights and extracted the resonant absorbance from the spectra. The results are shown in Fig. 8c. When h is larger than the decay length of GSP waves in dielectric grating (i.e., 60 nm), the resonant absorbance is nearly unchanged in the DGSG system. Nevertheless, when h is smaller than 60 nm, the energy of GSP wave cannot be completely confined in the system, and a portion of GSP wave is converted into free space wave, resulting in a sharp decay of the resonant absorbance to merely ∼10 % (h = 10 nm). In contrast, for the MGSG hybrid system, its high absorbance barely changes even when h decreases to 5 nm, benefited from its extraordinary small decay length. This result indicates that the energy of GSP wave can be effectively trapped as long as the grating height is larger than the decay length, and the ultra-thin metallic grating exhibits stronger ability to confine and absorb the incident light than the ultra-thin dielectric grating.

The revealed penetration characteristics of GSP waves into dielectric and metallic grating promote innovative methods for controlling and confining terahertz wave. On one hand, the redistribution of GSP spatial energy in a sub-grating region offers a possible solution for ultra-high resolution imaging [31]; on the other hand, ultra-thin metal nanostructure with strong ability to confine the incident light allows for realizing the highly integration of three-dimensional stacked terahertz elements [32].

Conclusion

In conclusion, we systematically investigate the coupling effect between the GSP modes and evanescent wave modes induced by the near-field perturbation using finite element method. The underlying physical mechanism of the coupling effect is quantitatively revealed based on effective medium theory and the decisive factor that dominates the absorbance of the hybrid system is investigated using rigorous coupled wave theory, which exhibits good agreement with the simulated results. Furthermore, the redistribution and effective trapping of energy in the hybrid systems is found to be closely related to the penetration ability of the GSP wave into the nanostructure. These understandings facilitate further engineering of the next-generation GSP terahertz devices and system.