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Abstract
Surface plasmon polaritons have been extensively studied owing to the promising characteristics of near fields. In this paper, the cascade coupling of graphene surface plasmon polaritons (GSPPs) originating from cascading excitation and multiple coupling within a composite graphene–dielectric stack is presented. GSPPs confined to graphene layers are distributed in the entire stack as waveguide modes. Owing to the near-field enhancement effect and large lifetime of the GSPPs, the terahertz wave–graphene interaction is significantly enhanced, which induces an ultra-extraordinary optical transmission (UEOT) together with the reported negative dynamic conductivity of graphene. Furthermore, owing to cascade coupling, the UEOT exhibits considerable transmission enhancement, up to three orders of magnitude, and frequency and angle selections. Based on the key characteristics of cascade coupling, the mode density and coupling intensity of GSPPs, the dependences of the number of graphene layers in the stack, the thickness of dielectric buffers, and the effective Fermi levels of the graphene on the UEOT are also analyzed. The proposed mechanism can pave the way for using layered plasmonic materials in electric devices, such as amplifiers, sensors, detectors, and modulators.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Surface plasmon polaritons (SPPs) are electromagnetic waves that are induced by the collective oscillations of free electrons in noble metals [1–4]. Due to the larger wave vectors compared with plane waves in vacuum, SPPs are confined to the metal surface, thereby inducing a significant field enhancement effect [1,5]. Hence, SPPs are promising media for light–matter interactions in the visible light or ultraviolet region, which is useful for a large number of applications in modern science and technology [6–9]. In the terahertz (THz) region, owing to the lack of natural plasmonic materials, artificial meta-materials are usually used as platforms for light-matter interaction, such as a meta-surface realized by the oscillation of an effective inductance-capacitance (LC) resonance and sub-wavelength array supporting spoof surface plasmon polaritons (SSPPs) [10–15]. Based on these structures, numerous applications have been realized and characteristic phenomena have been reported [16–19]. For example, extraordinary optical transmission (EOT) with a high transmission originating from SSPPs in subwavelength hole arrays has been extensively studied owing to its potential applications for sensors and monolithic color filters [14,20].
Graphene, a two-dimensional material with various interesting properties such as tunable carrier density, is a promising platform for tightly confined graphene surface plasmon polaritons (GSPPs) in THz spectral range, which provides the possibility for realizing ultra-extraordinary optical transmission (UEOT) [21–26]. In addition, many published works reported that unique linear dispersion of graphene can lead to negative photo-induced conductivity under external illumination, which is related to the amplification of THz waves [27–36]. These remarkable properties make graphene a perfect material for achieving UEOT even above 1, compared to the traditional EOT in the THz region. However, the THz-wave–graphene interaction is restricted by the ultra-thin nature of the monolayer graphene, which may be enhanced by stacking, which is related to the characteristic of effective anisotropy permittivity according to the effective medium theory. Capolino theoretically demonstrated that a hyperbolic meta-material can enhance the near-field interaction between light and graphene [37]. After that, Jia et.al experimentally demonstrated a graphene-based hyperbolic meta-material that achieved a broadband light absorption of 85% [38].
In this paper, an UEOT induced by the cascade coupling of GSPPs is expected by the rigorous electromagnetic theory of transfer matrix. GSPPs can be excited by incident THz waves and delivered from one graphene layer to the next as a cascade with multiple couplings between all the layers. Through this process, the incident THz waves penetrate the stack, which induces UEOT together with light pumping by the enhanced THz-wave–graphene interaction, originating from the field enhancement effect and large lifetime of GSPPs. Hence, the proposed mechanism may pave the way for layered plasmonic materials to develop novel electric devices.
2. GSPPs cascade coupling
The schematic of the composite graphene–dielectric stack is shown in Fig. 1, in which the graphene layers are separated by dielectric buffers. The stack is sandwiched between a prism and a substrate. For an individual graphene, GSPPs are confined to the layer and attenuated exponentially along the normal direction. However, in the composite stack, when the thickness of the dielectric buffer is comparable to or smaller than the penetration depth ${D_P}$ of the GSPPs, the excited GSPPs penetrate the buffer and generate another GSPPs on the next layer. This process continues as a cascade up to the last layer. The excitation of GSPPs on one layer is also influenced by multiple coupling between all adjacent graphene layers. Thus, GSPPs cascade coupling is induced, which can be described by a transfer matrix [39]:
where ${M_i}^b$ and ${M_i}^p$ are the boundary and propagating matrices for the ith graphene layer and dielectric buffer, respectively, and ${M_{top}}^p$ are the propagating matrices in the top buffer.
Fig. 1. Schematic of the GSPPs cascade coupling in the stack. The composite stack comprises graphene layers separated by dielectric buffers, and is sandwiched between a prism and a substrate with ${\varepsilon _{pri}}$ of 9.8. The thickness of the dielectric buffer is 2 µm, and ${\varepsilon _{buf}}$ = 2.4.
To show the key to cascade coupling, the confinement factor and penetration depth of GSPPs in the composite stack calculated by Eq. (3) are presented in Fig. 2. As the mode density of GSPPs increases with the number of layers, the GSPPs are shifted to the region with a smaller confinement factor N, leading to a larger penetration depth ${D_P}$, as shown in Fig. 2(a) and 2(b), respectively. A larger penetration depth indicates more fields in the dielectric buffer, and yields a stronger multiple coupling. Accordingly, when ${D_P}$ is comparable with or larger than the buffer thickness, the cascade excited GSPPs can coupled with those on the adjacent graphene layers, yielding a cascade coupling. This can be confirmed by the contour maps calculated from Eq. (4), as shown in Fig. 3.
Fig. 2. Dependences of GSPPs (a) confinement factor in the composite stack and (b) penetration depth on the number of graphene layers. The conductivity of the graphene is obtained by Eq. (6) with an effective Fermi energy of 5 meV and relaxation time of 10 ps.
Fig. 3. Contour maps of GSPP cascade coupling at 1 THz in the stack. Contour maps of (a, b) modes 1 and 2 for two graphene layers, (c, d) modes 1 and 2 for five graphene layers, and (e) mode 2 for 10 graphene layers.
Figure 3 demonstrates the formation of cascade coupling as the number of graphene layers increase. For the stack with two layers, the excited GSPPs are confined mainly to the graphene surfaces with the classical modes of surface waves, as shown in Fig. 3(a) and 3(b). This is contributed to the weak coupling due to the small penetration depth.
For the stack with five graphene layers, an intermediate status of cascade coupling is observed. First, for mode 1 shown in Fig. 3(c), the fields are distributed as a transverse magnetic (TM)1-like mode in a dielectric waveguide as a whole. Five maximum values on the surfaces of the graphene are observed, which correspond to the confined GSPPs. This indicates that the presented TM1-like mode originates from the GSPPs. For this mode, the penetration depth ${D_P}$ reaches 2 µm, comparable to the buffer thickness. The fields are then dragged into the buffer by strong multiple coupling, thereby inducing GSPP cascade coupling. Second, for mode 2 shown in Fig. 3(d), the fields are confined mainly to the graphene surface. However, a TM2-like mode is still formed in the stack as a whole, as the increased number of graphene layers also enables multiple coupling even when the penetration depth is only 1.3 μm. The excited GSPPs are then distributed as an intermediate pattern between the classical surface wave and waveguide mode.
Therefore, a remarkable cascade coupling can be induced by more graphene layers owing to the increased penetration depth and coupling candidates, corresponding to a larger coupling intensity and a higher GSPPs mode density, which are important for cascade coupling. As shown in Fig. 3(e), (a) TM2-like waveguide mode is formed in the composite graphene–dielectric stack with 10 graphene layers.
3. Ultra-extraordinary optical transmission
Owing to cascade coupling, incident THz waves can be transmitted through the stack in the form of GSPPs, which may induce EOT. Furthermore, the reported negative dynamic conductivity makes the pumped graphene layers gain media. Accordingly, UEOT with a transmission larger than 1 could be obtained by cascade coupling. To ensure the excitation of GSPPs, N should be reduced to below $\sqrt {{{{\varepsilon _{pri}}} / {{\varepsilon _{buf}}}}}$ for phase matching, which indicates more than 80 graphene layers in the stack for the given parameters as shown in Fig. 2(a).
Considering the cascade coupling, the transmission and reflection coefficients, ${T_{UEOT}}$ and ${R_{UEOT}}$ can be obtained:
To ensure the generation of GSPPs, a composite stack with 100 graphene layers is studied as an illustration. The dependence of ${\alpha _{EN}}$ on the incident angle and frequency, obtained by Eq. (7), is shown in Fig. 4(a). The amplified transmission is dramatically enhanced at three bands in the region with an incident angle larger than 30°. At point A, ${\alpha _{EN}}$ reaches 3104 at 0.967 THz and an incidence angle of 48.2°. The field intensity in Fig. 4(b) shows that the fields are concentrated on the graphene layers and are distributed as a TM1-like mode in the whole stack. These characteristics indicate that the incident waves excite GSPPs on the graphene layers, penetrate the stack in a regular pattern, and are transformed into output waves in the substrate by leaky wave modes. On the one hand, the incident waves are enhanced on the graphene layers by the field confinement effect of GSPPs. On the other hand, the lifetime of the excited the GSPPs leads to a large interaction time. Thus, the THz-wave–graphene interaction is significantly strengthened by the GSPPs cascade coupling, which lead to a UEOT with dramatic amplification enhancement ratio${\alpha _{EN}}$. Accordingly, the UEOT bands coincide with the dispersion lines of the GSPPs (green dotted lines in Fig. 4(a)) and presents the characteristics of frequency and incident angle selections by the phase matching of GSPPs excitation.
Fig. 4. UEOT of the stack consisting of pumped graphene with an effective Fermi energy of 5 meV, $T$ = 77 K, and $\tau$ = 10 ps. (a) Dependence of the transmission ratio ${\alpha _{EN}}$ on frequency. The inset shows the transmission of individual graphene (the transmission reaches only 1.0004). Contour maps at points (b) A and (c) B and corresponding enlarged views of the field intensities.
Notably, ${\alpha _{EN}}$ in the left region of the buffer light line is induced by the oscillation of the incident waves in the stack. The distribution agrees with the dispersion lines of dielectric waveguides. This can be clarified by Fig. 4(c), in which the fields are not concentrated on the graphene layers. Accordingly, the THz-wave–graphene interaction is considerably weaker than that in the UEOT and ${\alpha _{EN}}$ reaches only 40 at point B.
To clarify the relation between the GSPPs and the enhanced transmission, the spectra of ${\alpha _{EN}}$ at the TM1-like mode and the corresponding GSPPs cascading excitation probability, defined by the reciprocal of Eq. (3) [39], are presented in Fig. 5. The trend of ${\alpha _{EN}}$ agrees well with the excitation probability as more fields are concentrated on the graphene sheets by the stronger GSPPs, thereby inducing stronger THz-wave–graphene interaction, which leads to a futher increase in ${\alpha _{EN}}$. For GSPPs with a smaller group velocity that could be excited more efficiently, the excitation probability increases with frequency, which leads to an increase in ${\alpha _{EN}}$. However, higher frequency GSPPs have a larger confinement factor, indicating a smaller penetration depth. When the depth is considerably smaller than the thickness of the dielectric buffer, the multiple coupling is weakened. This decreases the excitation probability and breaks cascade coupling, thereby leading to a sharp decrease in ${\alpha _{EN}}$. As shown in Fig. 5, the maximum ${\alpha _{EN}}$ is obtained at 0.962 THz, at which maximum GSPPs cascading excitation probability is observed.
Fig. 5. (a, b) Transmission ratio ${\alpha _{EN}}$ and excitation probability for the TM1-like mode, respectively. The incident angle is adjusted at each point for GSPPs cascade coupling.
Based on the mechanism above, ${\alpha _{EN}}$ is strongly dependent on cascade coupling, which is determined by the GSPPs mode density and the multiple coupling intensity, corresponding to the number of graphene layers and magnitude relationship between the buffer thickness ${h_i}$ and the penetration depth ${D_P}$.
First, when N is smaller than $\sqrt {{{{\varepsilon _{pri}}} / {{\varepsilon _{buf}}}}}$, cascade coupling can be generated by the incident THz waves. Accordingly, Fig. 6(a) shows several thresholds of layer numbers for different UEOT modes, such as 41 layers for the TM1-like mode. The increasing layer numbers enable the cascade coupling to gradually reach the perfect excitation points and then deviate, which leads to the maximum ratios for different modes. As shown in Fig. 6(a), ${\alpha _{EN}}$ reaches 1855 and 10004 at TM2-like and TM3-like modes for 86 and 138 layers, respectively.
Fig. 6. Ratio as functions of the (a) number of graphene sheets, (b) buffer thickness, and (c) effective Fermi level. All incident angles are adjusted to obtain the GSPPs cascade coupling.
Second, when ${h_i}$ is too small, the GSPPs on the graphene layers mainly act as that on the individual graphene. Thus, a small ${\alpha _{EN}}$ is induced at a smaller ${h_i}$, as shown in Fig. 6(b). With the increase in ${h_i}$, the GSPPs in the stack are dominated by cascade coupling and perfect excitation points are gradually reached for TM2-like and TM1-like modes at ${h_i}$ of 2.39 and 3.07 µm, corresponding to maximum ${\alpha _{EN}}$ values of 2766 and 2374, respectively, as shown in Fig. 6(b). This is because the higher-order TM-like mode has a smaller penetration depth ${D_P}$, demanding a smaller buffer thickness ${h_i}$ for GSPPs cascade coupling. When the thickness is considerably larger than ${D_P}$, multiple coupling is weakened again owing to the exponential attenuation, thereby inducing a sharp decrease in ${\alpha _{EN}}$, as shown in Fig. 6(b).
Owing to a similar reason, the multiple coupling intensity is influenced by the effective Fermi level of the pumped graphene. A larger Fermi level leads to a shift in the GSPPs dispersion lines to a higher-frequency region, which yields a larger${D_P}$. A higher-order cascade coupling mode is observed at a larger effective Fermi level. As shown in Fig. 6(c), ${\alpha _{EN}}$ reaches 15480 at a TM2-like mode for an effective Fermi level of 3.27 meV.
4. Summary
In summary, an UEOT induced by GSPPs cascade coupling in a composite graphene–dielectric stack is presented here. GSPPs are excited by the incident waves on the first graphene layer, and penetrate the dielectric buffer to excite another GSPPs on the next layer up to the last layer, and couple with each other. Induced by this cascade coupling within the stack, the excited GSPPs are distributed in a pattern as a waveguide mode. In this process, the incident fields are confined to the graphene layer as GSPPs and penetrate the stack by the cascade coupling, which enhances the THz-wave–graphene interaction by both near-field enhancement effect and large lifetime of the GSPPs. Thus, the UEOT can be induced together with the reported negative dynamic conductivity of the pumped graphene layers. The transmission coefficient and frequency of UEOT are determined by the GSPPs excitation probability and dispersion relation, respectively. Therefore, the enhancement ratio exceeded three orders of magnitude by GSPPs cascade coupling. The UEOT exhibits obvious characteristics of frequency and incident angle selections. Hence, the proposed mechanism paves the way for enhancements in the light–matter interactions of layered plasmonic materials, which could be significant for the development of electric devices, such as amplifiers, sensors, detectors, and modulators.
Appendix: The discussion about the loss in the stack
The revealed mechanism shows that the UEOT originates from the enhanced graphene-terahertz interaction induced by the cascade coupling of GSPPs in the stack. Accordingly, when the stack is lossy, the absorption will also be enhanced. Take the unpumped graphene for example, the stack is lossy due to the positive conductivity of the graphene. And then an ultra-absorption due to the cascade coupling in the stack can be obtained by
where T and R are the transmission and refraction coefficients. And the graphene layers are separated by the dielectric buffer with permittivity 2.4, and thickness 4 um. And the chemical potential of the graphene is 50 meV. It is found that the absorption is reached to 1 in Fig. 7(a), which is much larger than that without saccade coupling.Funding
National Natural Science Foundation of China (61771327, 61901093, 61921002, 61931006); Sichuan Province Science and Technology Support Program (2020JDRC0028); Key Technologies Research and Development Program (2018YFB1801503).
Acknowledgments
This work is supported by Sichuan Science and Technology Program under Contract No. 2020JDRC0028; Key Technologies Research and Development Program under Contract No. 2018YFB1801503; National Natural Science Foundation of China under Contract Nos. 61931006, 61901093, 61921002 and 61771327.
Disclosures
The authors declare no conflicts of interest.
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