In this paper, strong longitudinal coupling of the Tamm plasmon polaritons (TPPs) is investigated in a graphene/DBR/Ag slab hybrid system. It is found that TPPs can be excited at both the top graphene and the bottom silver slab interface, which can strongly interact with each other in this coupled structure. Numerical simulation results demonstrate that the vertical Tamm plasmon coupling can be either tuned by adjusting the geometric parameters or actively controlled by the Fermi energy in graphene sheet as well as the incident angle of light, allowing for strong light-matter interaction with a tunable dual-band perfect absorption. Moreover, the coupling strength of the hybrid modes exhibits a large tuning range, from a large Rabi splitting to an extremely narrow induced transparency in this coupled regime. Coupled mode theory has been employed to explain the strong coupling phenomenon. The controllable TPP coupling with an ultrahigh dual-band absorption capability offered by this simple layered structure opens new avenues for developing a broad range of graphene-based active optoelectronic and polaritonic devices.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Tamm plasmon polariton (TPP) is a unique electromagnetic surface mode existing at the interface between metal film and one-dimensional (1D) photonic crystal (PhC) [1–4]. As an analogue to the electron surface state at the edge of truncated periodic atomic potential predicted by Tamm , it exhibits many intriguing characteristics. Unlike surface plasmon polariton (SPP), with the dispersion within the light cone, TPP can be directly excited by both TM- and TE- polarized light from free space, without the need of prisms or diffraction gratings for the in-plane momentum compensation. So far, this easily-realizable resonant surface mode has been successfully used to enhance the light-matter interaction in applications such as solar cells , confined Tamm Plasmon Lasers , single photon emission , perfect light absorber [9,10], large Goos–Hänchen shift , exciton-polaritons coupling , nonlinear optical effects [13–15].
Coupling of TPP with other resonant modes can produce interesting hybridization of the resonance polaritons, which can be utilized to tune the resonance frequency and guide the design of the nanostructures [16,17]. It has recently attracted increasing interests. For instance, TPP has been exploited to couple with magnetic plasmons  or localized SPPs  for large field enhancement, with microcavity resonance for narrowband thermal emission , with optical defect states for induced transparency/reflection [21,22], with semiconductor excitons for development of polaritonic devices [12,23]. Nevertheless, the resonance frequencies of these coupled structures are solely determined by the geometric parameters and lack of active tunability. Moreover, much of the TPP light energy can be dissipated in the form of heat in the metal layer, which inevitably will reduce the photoelectric conversion efficiency, especially for the optoelectronic devices.
Graphene, as a pioneering 2D material , has shown plasmonic response in mid- and far-infrared region [25,26], featuring tremendously enhanced electric field and ultra-fast optical tunability. These extraordinary plasmonic properties have substantially injected new vitality to realize the enhanced light-matter interaction with tunability at the nanoscale . By replacing the metal film in conventional TPP structure with a monoatomic graphene sheet, a new type of TPP may be excited at the boundary between graphene and dielectric Bragg reflector (DBR) [10,28], which is referred to as graphene TPP (GTPP). Accordingly, the resonance properties of GTPP can be actively modulated by graphene Fermi energy with mitigated heat effects. In absent of mode coupling, the resonantly enhanced GTPP field will be only confined below the graphene layer. Now, suppose this GTPP structure is placed on a metal substrate, graphene Tamm plasmon may longitudinally penetrate the dielectric multilayer and reach the DBR/metal interface at the bottom. As such, the conventional metal TPP (MTPP) may be excited at surface of the metal layer, which may in turn strongly interact with GTPP in this hybrid regime. To our best knowledge, TPP coupling between graphene and metal has not been reported before. It can be expected that the GTPP-MTPP coupling may enable strong light energy exchange between graphene and metal interfaces in this system. The hybrid plasmonic fields may emerge in either of the top graphene and the bottom interfaces, which may be actively tuned for selectively enhancing light-matter interaction. Moreover, engineering of the mode coupling with an optimized geometry can also be exploited to achieve intense  and even broadband light absorption [30,31]. All these promising properties in this TPP hybrid structure are much appealing for a large variety of graphene optoelectronic devices.
In this work we investigate the strong coupling of the two TPPs in a graphene/DBR/silver slab coupled structure. It is found that the GTPP mode exhibits an asymmetric Fano resonance line shape, which can strongly couple with the conventional metal TPP at the DBR/silver interface. Moreover, our simulation results indicate the vertical Tamm plasmon coupling can be adjusted by changing the geometric parameters or actively controlled by tuning the Fermi energy in graphene as well as the incident angle, allowing for tunability of this strong light-matter interaction. Temporal coupled mode theory has been employed to theoretically analyze such strong coupling of the two TPP modes. Active control of the longitudinal TPP coupling in this simple layered structure will be very beneficial for the development of various graphene optoelectronic devices.
2. Theoretical model and principle
The proposed structure for coupling of TPP modes is schematically illustrated in Fig. 1(a). It consists of a graphene monolayer covering a DBR structure, which is placed on a thick silver substrate. DBR structure comprises N = 11 pairs of alternative poly 4-methyl-pentene (TPX)/silica layers with a silica spacer on the top. Here TPX and silica layer in each unit has thickness of da = 53.45 μm and db = 40.79 μm, which satisfy the Bragg condition at the central frequency around 0.968 THz. The spacer thickness is initially set as d0 = 31.3 μm. The refractive index of TPX and silica are chosen as na = 1.46 and nb = 1.9, respectively; the frequency-dependent permittivity of silver is calculated by the Drude model with the following parameters: high-frequency constant of = 3.4, plasma frequency of = 1.39 × 1016 rad/s and scattering rate of = 2.7 × 1013 rad/s [32,33]. Graphene is modeled by an effective surface conductivity sheet, whose permittivity can be expressed as the formula , where ω, and are the angular frequency, the permittivity of vacuum, and the surface conductivity of graphene. Over the far-infrared region of interest, the isotropic graphene conductivity is governed by the intra-band contribution in Kubo formula [35–37]
Figure 1(c) illustrates a F-P cavity resonance model, which has been proposed to understand the Tamm plasmon resonance . Therein, the travelling light could be resonantly trapped between the two imaginary cavity mirrors, corresponding to TPP modes. By using the transfer matrix method (TMM), its condition can be readily derived by solving the eigenmode equation for the field at metal/DBR interface as
Here, DBR is designed with an approximately complete photonic bandgap over the frequency range of [0.93, 1.01] THz (Fig. 1(d)), wherein TPP modes could be created. A coupled TPP resonator is formed by simply sandwiching the DBR between a graphene sheet and a thick silver substrate. Due to the metallic properties of graphene in this spectral region, GTPP can be excited at the graphene/DBR interface. Despite its longitudinal decay, the tail of this resonant surface mode may reach the DBR/silver interface below. Consequently, the silver TPP (STPP) mode can be excited near the boundary of silver substrate, which may in turn strongly interact with the GTPP, rendering the mode hybridization. The hybrid polariton dispersion can be described by using a coupled oscillator model 39]
Figure 2(a) shows the simulated absorption spectra for the graphene/DBR and the DBR/silver slab structures, where the DBR parameters are the same as those in Fig. 1(d). For the graphene/DBR structure, we see that a low-Q resonance (Q = 24) with A = 100% perfect absorption creates near 0.964 THz. The E field at the peak is mainly confined below the Graphene/DBR interface, corresponding to the GTPP mode. Meanwhile, absorption spectra for the DBR/silver structure exhibits an extremely narrow linewidth (Q = 1923), whose peak is around 0.961 THz. In this case, the E field is resonantly enhanced near the surface of silver substrate, being related to the STPP mode. If the graphene sheet and the silver slab share the same DBR, strong GTPP-STPP coupling is expected in the graphene/DBR/silver structure. Figure 2(b) indicates the absorption spectra of this coupled system, where a notable frequency splitting can be observed, corresponding to the strong longitudinal TPP coupling.
Intriguingly, we see the absorption spectra of GTPP mode exhibits an asymmetric Fano line shape with a broader line-width. The asymmetry of the absorption spectra can be attributed to the interference effect by the band edge mode of DBR. According to the Fano resonance model , the frequency-dependent absorption formula can be described asEq. (7), the theoretically fitted absorption spectra for GTPP and STPP modes are shown in Fig. 2(a), coinciding excellently with the simulations. The fitting parameters for GTPP mode are F = −3.7, f0 = 0.965 THz, γ = 0.108 THz and = 14.27; while those for STPP are chosen as F = 0, f0 = 0.961 THz, γ = 1.8 × 10−3 THz and = 2.5, corresponding to a Lorentzian resonance.
Strong coupling of two optical modes can be theoretically described by the temporal coupled mode theory (CMT) . For simplicity, here we just focus on the coupling of GTPP and STPP modes. The absorption spectra of the coupled mode have been fitted by CMT in Fig. 2(b), which matches well with the simulated results in the anticrossing spectral region. The discrepancy in lower frequency can be ascribed to the influence of the DBR band edge mode. Due to the fact that the resonance frequency of edge mode is far from those of TPP resonance, its impact on TPP coupling can be safely ignored. To further gain the physical insights on the strong TPP coupling, E and H field distributions at the absorption peaks and dip marked with I, III and II for this hybrid mode are provided in Fig. 3. We clearly see, the electromagnetic field distributions at peaks of I (Figs. 3(a) and 3(d)) and III (Figs. 3(c) and 3(f)) exhibit the distinct hybridization features, namely, the enhanced EM fields are confined at both graphene/DBR and silver/DBR interfaces, corresponding to the GTPP-TPP coupled mode. While, those at dip II are mainly localized nearly the surface of silver slab, displaying the typical characteristics of the STPP mode (Figs. 3(b) and 3(e)), which essentially acts as a dark mode to destructively interfere with the GTPP, forming a very deep transparent window.
3. Results and discussion
3.1. Dependence of TPP mode coupling on the spacer thickness
According to the phase condition of Tamm plasmon resonance of Eq. (3), TPP resonance frequency depends on the phase shift of the wave travelling across the virtual micro cavity under normal illumination. Figure 4(a) provides the simulated GTPP resonance frequency as a function of the spacer thickness d0 for Gaphene/DBR structure. As d0 increases, GTPP resonance frequency will red-shift to low frequency.
Meanwhile, for the Graphene/DBR/silver coupled structure, the mapping of absorption spectra as a function of d0 is shown in Fig. 4(b). In the simulation, the geometrical parameters except d0 are the same as those in Fig. 2(a). A notable anticrossing can be clearly observed in absorption spectra near the STPP resonance frequency of 0.961 THz, corresponding to the strong coupling of Tamm plasmon in the vertical direction. Essentially, at the zero detuning of the resonance frequency for these two modes, the light energy strongly exchanges between the top graphene and the bottom silver interfaces via the GTPP-STPP hybrid mode. Particularly, as d0 is decreased (< 30 μm), the left branch of hybrid mode bends towards the low frequency direction, exhibiting the STPP-like characteristics. While with increasing d0, the hybrid mode gradually changes its nature and the GTPP-like features appear. From the theoretical viewpoint, the eigen-frequency of the coupled polariton mode can be described as [42,43]Eq. (8), the analytical resonance frequencies for this hybrid mode are plotted with the white dashed lines in Fig. 4(b), which are in excellent agreement with the simulated results.
3.2. Tuning the Tamm plasmon coupling via graphene Fermi energy and incident angle
As well known, the graphene Fermi energy can be actively changed by chemical  or electrical  doping, enabling a dynamically tunable optical response in graphene sheet. Besides, resonance frequency of TPP mode can also be modulated by changing the incident angle [4,45]. Accordingly, active tuning of the Tamm plasmon coupling can be expected in this graphene coupling structure. Here we investigate the vertical Tamm plasmon coupling between GTPP and STPP modes by varying the Fermi energy in the graphene sheet and the incident angle of light, as shown in Figs. 5 and 6. The structural parameters of the model are kept the same as those used in Fig. 2(b) with TM-polarized illumination.
With the geometric parameters of DBR and silver slab being fixed, the resonance frequency of STPP mode remains at 0.961 THz. Specially, at μc = 0.7eV, the resonance frequencies of GTPP and STPP modes nearly coincide with each other , enabling a striking frequency splitting shown in Fig. 5(b). In this case, it can be observed that the strong TPP coupling gives rise to a dual-band 100% perfect absorption in this TPP coupled system. Moreover, as the graphene Fermi energy goes down from 0.7eV to 0.3 eV, the resonance peak of GTPP mode will gradually move away from that of STPP mode towards the lower frequency direction. Due to the increased detuning for resonance frequencies of these two coupled modes, the absorption spectra exhibit a typical Fano-like feature, which should be attributed to the interference of a discrete state (STPP) with a continuum background (GTPP). Surprisingly, we note that the resonant light absorption of this coupled system with different graphene Fermi energy can maintain at a high level (>83.5%). Furthermore, by fixing μc = 0.7eV, the influence of incident angle on the coupled TPP mode of this coupled system has been provided in Fig. 6. It is indicated that the resonance frequencies of the hybrid mode will blueshift to the higher frequency, as the incident angle θ increasing. From the TPP phase resonance condition of Eq. (3), it is easy to understand the resonance wavelength should decrease to maintain the phase shift of as a constant for TPP generation, where d is the effective length of virtual cavity, leading to a blueshift. It is found that dual-band 100% perfect absorption can retain over the angle range from 0° to 36°.
The results above clearly demonstrate that the resonance frequencies of these coupled mode in this regime can be flexibly modulated by graphene gating as well as incident angle. Moreover, the spectral line shape can actively be changed from the symmetric Rabi splitting to the asymmetric Fano resonance with a high light absorption level, holding the potential in many tunable optoelectronic devices.
3.3. Influence of the DBR pair number on coupling strength of Tamm plasmon
In this multilayer coupled structure, graphene Tamm plasmon is initially generated at the top graphene/DBR interface. As a resonant surface mode, it will naturally decay in the DBR multilayer. Nevertheless, the tails of this resonant surface mode may reach the surface of silver substrate to create the STPP mode. The vertical interaction of these Tamm plasmons eventually produces the GTPP-STPP hybrid modes. Considering the nature of the evanescent field coupling of these coupled modes, its coupling strength shall closely rely on the DBR thickness in this coupled system.
The influences of the DBR pair number N on the coupling strength of these two types of TPP modes have been numerically studied, which is shown in Fig. 7. Figure 7(a) displays the absorption curves for the structures with different N. Clearly, as N is reduced from 15 to 7, Rabi splitting energy increases markedly, as illustrated in Fig. 7(b). Specifically, for the case of small N = 7, the Rabi energy can reach 0.2 meV in the far-infrared spectral band in this coupled regime, which satisfies the condition of for strong coupling . Physically, for structure with smaller N, the resonantly enhanced field of the GTPP mode can pass through the DBR and reach the DBR/silver interface with a relatively high intensity. Consequently, the silver Tamm plasmon can be efficiently excited to intensively interact with the GTPP mode, leading to a large Rabi splitting. Due to the strong longitudinal coupling of Tamm plasmon, the light energy can intensively exchange between graphene and silver interfaces, which can significantly render an enhanced light-matter interaction in either of the top and bottom in this coupled regime.
Meanwhile, for the DBR structure with a larger N, especially for N >13, it can be seen that the absorption spectra exhibit a distinctive line shape of electromagnetic induced transparency (EIT) [17,47,48], where opens an extremely narrow transparent window in the spectral background of the GTPP resonance. Essentially, this EIT-like phenomenon can be understood by coherent coupling of a discrete state to a wide continuous state under a relatively weak coupling strength. Moreover, considering the sensitivity of surface modes to the environmental media [49,50], it is believed that the generated hybrid TPP surface modes could provide attractive opportunities in sensing applications.
In conclusion, we numerically and theoretically investigated the strong longitudinal coupling of two TPPs in a hybrid structure of a graphene monolayer covered dielectric Bragg mirror on a thick metal substrate. We found the Graphene TPP mode exhibits a typical Fano resonance line shape, which can strongly couple with the Tamm plasmon excited at the dielectric/metal interface. Our investigations further reveal that the vertical plasmon coupling can be either tuned by adjusting the geometric parameters or actively controlled by modulating the Fermi energy in graphene sheet as well as the incident angle, allowing for the strong light-matter interaction with a dual-band perfect light absorption. Both large Rabi splitting and EIT with a narrow transparent window can be achieved in the far-infrared band with this coupled regime. A temporal coupled mode theory has been employed to explain the mechanism of the coupling behavior for these two types of TPP modes. With a combination of a controllable polaritonic coupling and ultrahigh dual-band absorption capability in this simple geometry, one could envision the design of more efficient, tunable hybrid optoelectronic devices, such as photodetectors, sensors and nano light sources.
National Key Research and Development Program of China (2016YFC0102401); National Natural Science Foundation of China (NSFC) (U1532113, 61422506, 61805064, 61875052); Anhui Provincial Natural Science Foundation (1708085MA24).
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