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We theoretically investigate the broadband light absorption in the THz range by canceling the strong coupling in an array of graphene ribbons at subwavelength scale. A series of resonators with different absorption frequencies can achieve a broadband absorber, however, the suppression of absorption always accompanies since the mutual coupling between resonators cause the mode splitting. By adjusting the near- and far-field coupling between the plasmon resonances of the graphene ribbon array to the critical point, the absorption linewidth is broadened for almost one magnitude larger than that of individual graphene ribbon, to be ~1 THz. Our study provides not only insight understanding but also new approaches towards the broadband graphene absorber.
© 2016 Optical Society of America
1. Introduction
Recently, graphene has shown its promising potentials in photonics and optoelectronic applications because of their unique band structures of Dirac Fermions [1,2]. However, due to the relatively low carrier concentration, this atomically thin graphene is almost transparent to optical waves and the absorption is only about 2.3% [3], which is considerably low for optoelectronic devices such as photodetector and solar cells. Accordingly, the enhancement of the light-matter interaction in graphene is one of the important issues to be solved. By adopting a grating coupler, a photonic crystal, a metamaterial on graphene or by patterning graphene itself into nanodisks or nanoribbons [4–15], total absorption of graphene has been demonstrated both theoretically and experimentally [4,8].
Besides the absorption enhancement, the light-matter interaction in a wide band range is also important for the devices such as sensors and photovoltaics, however, which is still a challenges for graphene devices [16,17]. One intuitive solution is to integrate the nanostructured graphene with gradually changed geometric sizes. In such configuration, broadband can be achieved provided that the radiation rate of the sustained plasmon resonances can be modified. However, in most cases, this design suffers from the fact that these resonances interfere mutually, manifesting as the coherent phenomena such as plasmon induced transparency, coherent perfect absorption, and superradiance between these modes [18–21]. Such mutual coupling is disadvantageous to the broadband absorption because it induces mode splitting. As a result, how to suppress or even avoid the coupling is the key point for broadband absorption of graphene [16]. In this study, we theoretically study the effects of the coherent coupling on the plasmon resonance characteristics. We achieve the “zero coupling” by compensating the near field (“direct” path) and far field (“indirect” channel) coupling between the graphene ribbons. Under the condition of critical point, broadband absorption is achieved, which linewidth is at almost one order of magnitude larger than that of individual graphene ribbon.
2. Theory
The temporal coupled mode theory (CMT) provides a very useful theoretical framework to study the interaction of resonators with external waves. From the CMT, the dynamics for two optical resonators with each frequency at ωi (i = 1, 2) are described as follows [22,23]:
where ai (i = 1,2) is the normalized amplitudes of the resonators, s+ the amplitudes of the incident light, Γi the absorption rate, and γi the radiation rates. In the equations, κ’ and κ” are the real and imaginary part of the coupling coefficient κ which comprises the contributions from both the near-field coupling κn and the far-field interactions κf between the two resonators. As a result, κ’ = κn + Ιm (κf) = κn + (γ1γ2)1/2sinφ and κ” = Re (κf) = (γ1γ2)1/2cosφ, where φ is the phase of the propagating wave [23,24]. It is clearly seen that κ’ is dependent on both the near-field coupling and the far-field interaction. For simplicity, we first assume γ1 = γ2 = γ0 and Γ1 = Γ2 = 0, let s+ = 0 in Eqs. (1) and (2), we obtain the quasi-eigenfrequency:Neglecting the term (ω1-ω2)2 when the resonant frequencies of the two resonators are close to each other, Eq. (3) can be reduced to: ω ± ≈(ω1 + ω2 ± 2κ')/2 + i (γ0 ± κ”). Clearly, two hybridized modes are formed as a result of the interactions between the original resonators [25]. The mode with the broadened linewidth γ0 + κ” is “bright” while the mode has a small linewidth (γ0 - κ”) and thus is “dark” to the incident plane wave. The splitting width between the two modes is 2κ'. In order to achieve a mode with broad bandwidth, the two splitting mode requires to be degenerated by satisfying the condition of κ' = 0 (i.e., critical point), which can be achieved by canceling the near-field coupling (direct path) by far-field coupling (indirect path). We emphasize here that the two resonators cannot be understood as “decoupled” at the critical point. The finite value of κ” indicates the modification of the bright mode’s radiation rate. Next, we consider the two resonators with intrinsic loss. The total absorption of the coupled system is Α = 2Γ1|a1/s+|2 + 2Γ2|a2/s+|2, from which the total absorption comprises the contribution of the two individual resonators [23]. At the critical point, the ratio of the normalized energy stored in the two resonator is |a1|2/|a2|2 = [Γ02 + (ω-ω2)2]/ [Γ02 + (ω-ω1)2] in which Γ1 ≈Γ2 = Γ0. At the resonant frequency ω1 (ω2), one resonator’s absorption is maximum while that of the other one becomes minimum. Such results indicate that the total absorption primarily depends on the single resonator excited selectively at the critical point. The absorption of two coupled resonators with slightly different resonance frequencies can be regard as the sum of absorptions of the individual resonators with their own uncoupled resonance frequency.3. Results and discussions
Now, we adopt the above idea to a plasmonic system consisting of two graphene ribbons in order to obtain broadband absorption, as shown in Fig. 1(a). As an example, two graphene ribbons with the width w1 = 900 nm and w2 = 920 nm are placed on a dielectric substrate with a dielectric constant ε = 2.1 and a thickness t = 50 nm. The period is set to be 14 μm. The optical properties of graphene in our simulations are defined by the complex permittivity ε(ω) = 1 + ιηs/ε0ωΔ, where the thickness Δ = 1 nm for graphene and ηs is the DC surface conductivity under the random phase approximation with an intrinsic relaxation time τ = μΕF/eν2 F [26–28]. We choose the DC mobility μ = 10000 cm2/Vs, and the Fermi velocity νF ≈c/300. The doping level of the two ribbons is assumed to be 1.2 eV [9]. Numerical simulations (CST MICROWAVE STUDIO) show that the resonance frequencies of individual graphene ribbon are ω1 = 20.88 THz and ω2 = 20.68 THz under the illumination of the plane wave at normal incidence, respectively [Fig. 1(b)]. The value of Γi and γi is extracted from the simulations. The absorption of the coupled ribbons calculated from the CMT is plotted as a function of the frequency and the real part of coupling coefficient, as shown in Fig. 1(c). The broad absorption peak (bright mode) at lower frequency and narrow absorption peak (dark mode) at higher frequency get close to each other when tuning κ' from a negative value to the critical point. As κ' = 0, there is only one absorption peak with a broadened linewidth [29]. When the κ' increases from zero to a positive value, similar spectral splitting behavior for the absorption spectra is observed.
Fig. 1 (a) Scheme of the geometry under consideration. Two graphene ribbons lay on the substrate. A beam is incident at normal direction onto the ribbons, with polarization along the transverse direction of the ribbons. (b) Absorption spectra for two uncoupled ribbons with the same geometries as Fig. 1(a). Open circles are simulation and the solid lines are CMT. (c) Absorption as a function of frequency and real part of coupling coefficient
To numerically demonstrate the absorption with different κ’, the two coupled graphene ribbons with different spacing are simulated [Fig. 2(a)], which agrees well with the theoretical results. When the spacing increases from 2 μm to 3.95 μm, the splitting absorption peaks with broaden linewidth and narrow linewidth get closer to each other. When the distance is 3.95 μm, κ’ crosses the critical point, where the near field coupling is canceled by the indirect coupling. However, the radiation rate (linewidth) in the two-ribbon configuration is modified compared with a single ribbon resonator. As the separation increases beyond 3.95 μm (κ’ > 0), the effect of indirect coupling becomes dominating. The dashed red lines which shown in Fig. 2(a) indicates the quasi-eigenfrequencies of this system. By adding the optically opaque substrate underneath, e.g., gold layer, the absorption can be enhanced to be near 100% [30]. However, the coupling strength of the graphene ribbons in the sandwiched structure is changed, resulting in a modified κ’. The interspacing between the graphene ribbons which satisfies the critical condition is therefore changed, to be 3.6 μm, as shown in the inset depicted in Fig. 2(a). A closer inspection on the near field coupling and the far field coupling clearly show the variation of the κn, Im (κf), κ’ and κ” as a function of separation distances [Fig. 2(b)]. At a small separation distance, κ’ exhibits a negative value as a result of the dominating contribution by the near field coupling. However, the absolute value of the near field coupling decreases rapidly as the separation distance increases, while the indirect coupling increasing slowly, leading to κ’ = 0 at 3.95 μm and a positive value at lager distance. Figures 2(c) and 2(d) are the energy investigations at the critical point by the CMT and the numerical simulations, respectively. From Fig. 2(c), the ribbon with the width w1 reach its maximum energy at the frequency of ω1 = 20.88 THz, (green line), while the energy stored in the ribbon with the width w2 is suppressed to its minimum (black line). At the frequency of ω1 = 20.68 THz, the stored energy have opposite behaviors. The numerical results of the electric field on the graphene are shown in Fig. 2(d), which are consistent with our theoretical analysis from the CMT.
Fig. 2 (a) Absorption spectra for the two coupled graphene ribbons with the increasing of space between them. Open circles are simulation and the solid line is the CMT. The blue line represents the case of κ' = 0 and two dashed red line indicate the eigenfrequencies of the system. (b) The extracted imaginary part of far-field interactions (black square) and near-field coupling (red circle) of coupling coefficient between the two ribbons, and the real part (green triangle) and imaginary part (blue diamond) of coupling coefficient between the two ribbons as a function of their edge to edge separation s. (c) The stored energy in each ribbon as a function of frequency in coupled system when the space is 3.95 μm. (d) The spectral response of electric fields (probe is placed 1 nm above the center of each ribbons). The green line and black line represent the ribbons. Inset, absorption spectra for the sandwiched structure with different κ'. The thickness of the dielectric layer is assumed to be 1.4 μm in order to achieve total absorption.
Understanding the above physical picture, the absorption linewidth can be further broadened by aligning together more resonators fulfilling the critical condition. This result can be explained as follows. The coupled two resonators aforementioned can be considered as a “new resonator” with a broadened linewidth. The “new resonator” can interact with the neighboring resonator at the critical point condition, achieving a further boarded linewidth. Finally, the absorption linewidth reaches a significant large value. For example, we extend the two-ribbon system to four-ribbon configuration [the inset of Fig. 3(a)]. The ribbons have widths ranging from 900 nm (left side) to 960 nm (right side) with an increment of 20 nm. The spacing between the nearest neighboring ribbon is 3.95 μm and the period here is 28 μm. In this configuration, the critical spacing in the two-ribbon configuration is used by realizing that interaction, both spatially and spectrally, between the non-neighboring ribbons is weak. As expected [Fig. 3(a)], this configuration gives broader absorption spanning from 20.01 to 20.93 THz. In four-resonator case, the absorption bandwidth is considerably broadened compared to the two-resonator configuration. However, unlike the two-ribbon configuration, the imposed oscillations are observed on the broad peak, which can be ascribed to the fact that the coupling between the separated ribbons actually plays a secondary role in the coupling. Such conclusion can be obtained by investigating the near field for each resonator [Fig. 3(b)]. Unlike the stored energy for the two-ribbon configuration, the maximum value in one resonator does not correspond to the minimum value of the resonators away, which hints they are coupled with each other. As a result, when the number of the ribbon is large (e.g., n > 4), the critical point condition extended from two-ribbon is not valid any more, which manifests larger oscillations or even splitting imposed on the bright mode.
Fig. 3 (a) Absorption spectrum of graphene ribbons array for normal incidence and polarization along the transverse direction of the ribbons. The inset is the top view of schematic of graphene ribbons array consisting of four ribbons with gradually varying widths. (b) The spectral response of electric fields (probe is placed 1 nm above the center of each ribbons).
4. Conclusion
In summary, we have shown that the total absorption of the graphene ribbons is broadened remarkably in the THz regime by fulfilling the critical point condition. The near-field and far-field coupling compensate at the critical point, avoiding the mode splitting and manifesting the broadband absorption. Such method is appealing in the explorations of the novel optoelectronic devices where broadband is highly desired.
Funding
National Natural Science Foundation of China (NSFC) (11474220, 11304038, 11474221); Fundamental Research Funds for the Central Universities (CQDXWL-2014-Z005).
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