## Abstract

Coherent absorption, as the time-reversed counterpart to laser, has been widely proposed recently to flexibly modulate light-matter interactions in two-dimensional materials. However, the multiband coherent perfect absorption (CPA) in atomically thin materials still has been elusive. We exploit the multiband CPA in vertically stacked metal/dielectric/graphene heterostructures via ultraconfined acoustic plasmons which can reduce the photon wavelength by a factor of about 70 and thus enable multiple-order resonances on a graphene ribbon of finite width. Under the illumination of two counter-propagating coherent beams, the two-stage coupling scheme is used for exciting multispectral acoustic plasmon resonances on the heterostructure simultaneously, thereby contributing to the ultimate multiband CPA in the mid-infrared region. The strong dependence of the nearly linear dispersion of acoustic plasmons on the chemical potential in graphene and the separation between the metal and the graphene allows the tunability in spectral positions of absorption peaks. Intriguingly, the absorption of each resonant peak is continuously tuned by varying the relative amplitude of two counter-propagating beams, and even their phase difference, respectively. The maximum modulation depth of 4.46*105 is observed. The scattering matrix is employed to demonstrate the principle of CPA and the finite-difference time-domain (FDTD) simulations are used for elucidating the flexible tunability. More importantly, the multiband coherent absorber is robust to the incident angle, and thus undoubtedly benefits extensive applications on optoelectronic and engineering technology areas for modulators and optical switches.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

As a pioneer in monolayer materials, graphene [1], a conjugated *sp*^{2} carbon sheet tightly packed into two-dimensional hexagonal lattices, has received a steadily increasing attention in recent years for its extraordinary transport properties. The peculiar electronic band structure [2] of the linear energy-momentum relation rather than quadratic allows graphene to be a springboard for a plethora of applications in novel optoelectronic devices with some successful experimental demonstrations [3]. However, the light-graphene interaction accessible to a system is mainly limited by the small value of the fine-structure constant [4], as well as by the small thickness of a single atomic layer relative to the operating wavelength. The absorption of light in graphene with the thickness of 0.34 nm thus is about 2.3% [5], which is defined by the fine-structure constant *α* = *e*^{2}/*ћc*. Here, -*e* is the electron charge, *ћ* is the Planck’s constant divided by 2*π*, and *c* is the velocity of light in vacuum. This drawback constrains some of potential graphene-based applications in modulators [6], field effect transistors [7], and photodetectors [8] over a very broad spectral range covering from visible to terahertz regions. Interestingly, in the infrared and terahertz range, the dynamical surface conductivity of graphene is dominated by the intraband transitions of electrons and thus is well described by the Drude model. As a consequence, the doped graphene exhibits a negative permittivity with ohmic losses, which enables the excitation of surface plasmon polaritons (SPPs) [9–12] that are localized surface electromagnetic waves originating from the coupling between light and plasma oscillations. Based on the excitation of graphene plasmons (GPs) that generate the extreme near-field confinement and enhancement, the absorption of light in graphene has been significantly improved, especially at the patterned graphene [13,14]. Only utilizing the plasmon effect without auxiliary structural design, however, the absorption in graphene monolayer can be improved to 20%∼30% [15,16] and the perfect absorption is inaccessible.

In order to obtain total absorption of light in graphene, in addition to the composite structure of graphene and one-dimensional photonic crystal [17,18], the combination of graphene plasmonic metamaterials and the critical coupling effect has been extensively investigated [19–22]. Generally, the graphene plasmonic metamaterial absorber is constructed to be a single-port system. The transmission channel is suppressed by a lossless metallic mirror or a multilayer dielectric Bragg reflector, separated from the graphene metamaterials by an optical spacer with the thickness satisfying the quarter-wavelength condition. Under the illumination of one beam, the perfect absorption of light occurs at the graphene metamaterials when the coupling rate of the system equals its dissipative rate [23–24]. In the single-port absorber, however, the tunability of absorption intensity in graphene is limited and passive, because the coupling and dissipative rates are associated to the collective response of system parameters [25], including the structural and material parameters, as well as the external field. Coherent absorption [26], a new method for flexibly controlling absorption through coherent illumination of two counter-propagating beams, has received burgeoning amount of interest since it is first experimentally realized in a silicon slab cavity by Wan’s group [27]. CPA relies on the coherent interference of light where two coherent optical beams shine on the graphene structure from opposite sides. The reflected light of one optical beam destructively interferes with the transmitted light of the other, and vice versa [28]. Once the ideal interference trap for two beams is achieved, the scattering is completely dissipated by the absorption in graphene structure. Hence, the change in the phase or amplitude of one optical beam will give rise to the disturbance of the interference pattern, which eventually varies the absorption of the other. In other words, the absorption of the system can be dynamically tuned by controlling the relative intensity and phase difference between two counter-propagating beams, respectively, without passively changing the structural parameters. Based on the unusual method, the single-band coherent perfect absorber has been realized on unpatterned graphene monolayer [29] and nanostructured graphene resonators, including the graphene with a periodical array of holes [30], a graphene split-ring resonator [31], and graphene ribbons [32]. To realize multiband or broadband graphene-based coherent perfect absorbers that play a significant role in wavelength-selective optoelectronic devices, the traditional method of frequency stacking by harnessing multiple nanostructured graphene resonators with gradually changed geometric sizes into a unit cell is usually utilized [33,34]. Such configuration, however, not only is quite technologically demanding but also suffers from the disadvantage of the mutual coupling between multiple resonators. The mutual coupling will induce troublesome and out-of-control mode splitting. Thus, the independent multiple resonant modes in a single graphene-based resonator will be the ideal candidate for multiband coherent absorbers [35,36].

Herein, we investigate the novel vertically stacked metal/dielectric/graphene heterostructure in symmetric environments. When a metal plate is placed in proximity to the graphene with a thin dielectric spacer, the plasmons within graphene are balanced by the out-of-phase oscillations from image charges inside the metal surface [37–41]. The hybridization of GPs with their mirror image yields acoustic graphene plasmons (AGPs) characterized by a nearly linear dispersion and an ultraconfined electric field in the dielectric spacer between the graphene and the metal plate. The tightly confined AGPs strongly reduce the photon wavelength by a factor of about 70, which enables the mutually independent multiple-order Fabry-Pérot (F-P) resonances to be simultaneously supported on a graphene ribbon of finite width. Under the illumination of two counter-propagating coherent beams, the multiband CPA, as a result, is realized in the mid-infrared region. The spectral positions of absorption peaks are dynamically tuned by changing the gate-dependent chemical potential of graphene, as well as the distance between the graphene and the metal. Intriguingly, the absorption intensity of each absorption peak is not only continuously tuned from 99.22% to the incoherent absorption limit of 50% via varying the relative amplitude of two counter-propagating beams, but also is tuned from 99.89% to nearly 0 by controlling the phase difference between beams. The maximum modulation depth of 4.46*10^{5} is observed. The scattering matrix demonstrates the principle of CPA and the FDTD numerical simulations elucidate the tunability of absorption spectra. More importantly, the coherent absorber exhibits an excellent absorption stability over a wide range of incidence angles around ±60° at least. Our finding undoubtedly stimulates potential applications on mid-infrared detectors, transducers, modulators, and optical switches.

## 2. Structure and principle

Figure 1(a) shows the vertically stacked metal/dielectric/graphene heterostructure embedded into vacuum. The graphene ribbon is separated from the silver strip by a dielectric spacer of zinc sulphide (ZnS). The strip configuration of infinite length along the *z* direction allows us to treat this structure as a two-dimensional framework, as shown in Fig. 1(b). The period along the *x* direction is *P*. The width of the upper silver strip is *W*. The ZnS and graphene strips have the same width of *L*. The *t* and *d* denote the thicknesses of silver strip and ZnS interlayer, respectively. In the investigated mid-infrared region ranging from 5.3 to 6.5 µm, the graphene conductivity (*σ _{g}*) is dominated by the intraband transition of electrons, because the chemical potential of the doped graphene is much larger than the half of the photon energy [42]. The surface conductivity determined by the Kubo formula [43] thus is approximately reduced to the Drude equation [44,45]

Here, -*e* is the electron charge, *ћ* is the Planck’s constant divided by 2*π*. The *µ _{c}* is the chemical potential of graphene and the

*ω*is the angular frequency. The momentum relaxation time

*τ*is govern by $\tau = \mu {\mu _c}/e{v_f},$ where

*µ*and

*v*mark the carrier mobility and the Fermi velocity in graphene, respectively. The Drude equation enables graphene to exhibit a negative permittivity with ohmic losses, thereby allowing the existence of GPs at wavelengths of interest. Under the illumination of the transverse-magnetic-polarized plane wave with its electric field parallel to the

_{f}*x*axis, the GPs will be excited on the metal-free region of the graphene ribbon by the edge scattering of the upper silver strip. At the metal-coupled region, the propagating GPs cannot penetrate the upper silver plate which behaves as the perfect electrical conductor at mid-infrared wavelengths. However, the presence of a mirror plane within tens of nanometers from the graphene enables the polariton mode within graphene to be mirrored, but with an inverted charge distribution at the metal surface [46,47], as indicated by + and – symbols in Fig. 1(b). The coupling between GPs and their mirror image at the metal surface with out-of-phase oscillations leads to the AGPs in the metal/dielectric/graphene configuration. Similar to the plasmons in two-dimensional electron gases [48] near conductive substrates, the AGPs own the nearly linear dispersion and extreme field confinement several times tighter than the GPs. Meanwhile, most of the electric field is squeezed into the spacer between the silver and the graphene.

To better understand the physical image of AGPs, we numerically simulate near-field distribution of the free-standing metal/dielectric/graphene heterostructure at the wavelength of *λ*_{0}* *= 6432 nm by utilizing the commercially available software of Lumerical FDTD Solutions. The *t* = 40 and *d* = 10 nm are selected. The graphene is modelled as an ultra-thin isotropic film with the thickness of Δ = 1 nm, and its equivalent permittivity is calculated by ${\varepsilon _g} = 1 + i{\sigma _g}{\eta _0}/({{k_0}\Delta } )$[49], where the vacuum impedance is *η*_{0} ≈ 377 Ω and ${k_0} = 2\pi /{\lambda _0}.$ The *µ *= 10000 cm^{2}/(V·s), *v _{f}* = 10

^{6}m/s and

*µ*= 0.8 eV, are chosen to calculate the graphene conductivity. Herein, the thickness of the monolayer graphene theoretically assumed to be 1 nm is reasonable, although its real thickness is 0.34 nm. Because the equivalent permittivity of graphene is thickness-dependent, the different values of the Δ will give rise to the corresponding equivalent permittivity and these extremely small values of the thickness can lead to similar simulation results [49]. The permittivity of silver is characterized by the Drude model [50]: ${\varepsilon _m}(\omega )= {\varepsilon _\infty } - \omega _p^2/({\omega ^2} + i\omega \gamma ),$ with

_{c }*ɛ*

_{∞}_{ }= 3.7,

*ω*= 9.1 eV, and

_{p }*γ*= 0.018 eV. The ZnS is assumed to be non-dispersive dielectric with the refractive index of 2.2 [51]. A dipole point source with the electric field perpendicular to the graphene plane to excite the propagating AGPs. The point source with the wavelength of

*λ*

_{0}= 6432 nm is placed 2 nm below the graphene monolayer. The perfectly matched layer absorption conditions are employed in the

*x*and

*y*directions. For comparison, removing the metal plate and the dielectric spacer, the spatial distribution of GPs on a free-standing graphene sheet is also presented. Simulated results are shown in the insets of Fig. 2(a). The comparison of contour plots in the upper inset suggests that the AGPs offer a smaller wavelength and tighter confinement than the conventional GPs. The reduced wavelengths of

*λ*

_{AGPs}= 90 nm and

*λ*

_{GPs}= 420 nm are extracted from the electric field image. The AGPs wavelength thus is reduced by a factor of about 70 compared with the free-space wavelength of

*λ*

_{0}

*= 6432 nm. The enlarged image of AGPs shown in the lower inset reveals that the opposing charge distributions between the original polaritons in graphene and the mirror polaritons on the silver surface contribute to electric fields tightly concentrated inside the 10-nm-thick ZnS interlayer between the graphene and the silver. The near-field profiles perpendicular to the graphene surface are depicted in the Fig. 2(b), extracted along the dashed white line in the upper inset of Fig. 2(a). For the free-standing graphene monolayer, the propagating GPs with antisymmetric near-field distributions are observed. In strong contrast, for the AGPs on the heterostructure, the asymmetric near-field distribution with most of electric fields trapped into the gap between the silver and the graphene is witnessed. Inside the ZnS gap, the electric field nearly stays constant across the layer because of the antisymmetric charge distribution. Meanwhile, the out-of-plane decay length of AGPs on the heterostructure is reduced to 14.32 nm obtained by*

*δ*

_{AGPs}=

*λ*

_{AGPs}/2π, compared with the 66.65 nm for the free-standing graphene. It implies the tighter field confinement is achieved for AGPs at the vertically stacked metal/dielectric/graphene heterostructure.

Certainly, the field confinement can also be described by the effective refractive index (*N _{eff}*) of plasmons. The wave vector of GPs is described by ${\beta _{GPs}} = {k_0}\sqrt {1 - {{({2/{\sigma_g}{\eta_0}} )}^2}}$[44,52]. The dispersion relation of AGPs is implicitly expressed as [53]

*ɛ*are the dielectric constants, corresponding to the silver (

_{i}*i*= 1), ZnS (

*i*= 2), graphene (

*i*= 3), and vacuum (

*i*= 4), respectively. The

*k*is the wave vector at each layer. The

_{i}*β*is the wave vector of AGPs. The

_{AGPs}*N*numerically calculated by the Mode Solver in the commercial software of Lumerical FDTD Solutions is depicted in the Fig. 2(a). Compared with the GPs in the free-standing graphene, the fantastic AGPs on the heterostructure own larger

_{eff}*N*due to the stronger field confinement. For instance, at the wavelength of

_{eff}*λ*

_{0}

*= 6432 nm, the*

*N*equals 69.53 for the AGPs, which in turn implies the incident wavelength is reduced to 92.5 nm and thus corroborates our simulated results shown in the inset of Fig. 2(a).

_{eff}Since the AGPs can tightly reduce the photon wavelength, the graphene ribbon of finite width will support more acoustic plasmon resonances, compared with the conventional GPs. In our work, the strip configuration is constructed and the width of the upper metal ribbon is slightly smaller than that of the bottom graphene ribbon, as sketched in the Fig. 1(b). On the one hand, this unique design is contributed to realize the two-stage coupling scheme. The free-space light is first coupled to GPs at the metal-free region by the edge scattering of the metal ribbon, and the GPs are then coupled to AGPs at the metal-coupled region by the mirror image. On the other hand, at the metal-free region, the reflection phase of propagating GPs varies by *Φ* ≈ -π [54] at an abrupt edge of the graphene ribbon, without changing their amplitude. Therefore, the propagating AGPs along the graphene ribbon will bounce back and forth between its two edges. The normal standing-wave resonance and thereby the F-P resonator will be obtained on the individual graphene ribbon. Importantly, driven by the ultraconfined AGPs, the single graphene ribbon of finite width will simultaneously support multiple-order F-P resonances. Correspondingly, the multiband non-interfering absorption enhancement in graphene occurs.

Based on the possibility of multiband absorption enhancement in graphene, we theoretically investigate the coherent control of absorption intensity. When only one beam *I*_{1} shines on the heterostructure along the *y*-axis direction, the incoherent absorption of the system at the condition that the total thickness of the heterostructure along the *y* direction is much smaller than the wavelength of the incident light can be derived as [55]

*R*and

*T*are the reflectivity and transmittance of the system, respectively;

*r*and

*t*are the corresponding reflection and transmission coefficients. In the symmetric environment (the same refractive indices for the upper and lower sides of the heterostructure), the combined reflection and transmission coefficients are expressed as

*r*=

*η*and

*t*= 1+

*η*, respectively [24], where the

*η*is the self-consistent amplitude, hinging on the material and structural parameters of the heterostructure, as well as on the wavelength and angle of the incident beam. The incoherent absorption thus is reduced as

*A*= −2

*η*

^{2}- 2

*η*. For simplicity, the vertically stacked heterostructure embedded into vacuum is considered. On the basis of Eq. (5), the maximum incoherent absorption of

*A*= 0.5 is obtained at

_{max}*η*= −0.5 (namely

*R*=

*T*= 0.25), termed the incoherent absorption limit. The absorption limit of 0.5 from one beam incident can be understood by a symmetric argument: a single beam incident from one side can be decomposed into a superposition between an even mode (where equal amplitude beams with the same phase illuminate the heterostructure from both sides) and an odd mode (where equal amplitude beams with the 180° out of phase shine on the heterostructure from both sides). Because the odd mode has vanishing fields at the heterostructure, only the even mode contributes to the absorption in fact. The beam incident from one side has equal power in the even and odd modes, and therefore, the maximum absorption of the heterostructure under illumination of only one beam cannot exceed 0.5 [25].

When two coherent optical beams (*I*_{1} and *I*_{2}) impinge on the heterostructure from opposite sides, defined as the coherent illumination, the relationship between the input (*I*_{1} and *I*_{2}) and output beams (*O*_{1} and *O*_{2}) is described by the scattering matrix [30]

*r*

_{11},

*r*

_{22}and

*t*

_{12},

*t*

_{21}are complex reflection and transmission coefficients at each port, respectively. Under the symmetric environment with the symmetric coherent beams, the scattering matrix has the same complex reflection and transmission coefficients respectively at two ports, namely,

*r*

_{11 }=

*r*

_{22}=

*r*´ and

*t*

_{12}=

*t*

_{21 }=

*t*´. The combination of Eq. (6) and

*I*

_{1}=

*αe*

^{i}^{Δφ}

*I*

_{2}allows the coherent absorption of the heterostructure to be expressed as

Here, the *α* is the relative intensity of two counter-propagating beams and Δ*φ* is their phase difference. Once the incoherent absorption limit, namely *r*´ = −0.5 and *t*´ = 0.5, is satisfied, the Eq. (7) can be rewritten as

The Eq. (8) shows that the CPA of *A _{co-max}* = 1 can be realized at

*α*= 1 and Δ

*φ*= 2

*N*π, where the

*N*is an integer. Meanwhile, the coherent absorption

*A*can be dynamically tuned by changing the

_{co}*α*or Δ

*φ*. Specifically, when the

*α*= 1 is chosen, the coherent absorption

*A*can be tuned from 0 to 1 by varying the Δ

_{co}*φ*. Conversely, when Δ

*φ*is fixed to be 2

*N*π, the coherent absorption

*A*can be modulated from 0.5 to 1 by varying the

_{co}*α*.

## 3. Simulations and discussion

Rooted in theoretical understandings, a series of FDTD simulations are performed. The optimized structural parameters of *P* = 1800, *W* = 1400, *L* = 1500, *t* = 40, and *d* = 10 nm are selected. The material parameters are unchanged. In the implementation, periodic boundary conditions are applied along the *x* directions and perfectly matched layer absorption conditions are employed in the *y* directions. To save the storage space and computing time, the non-uniform mesh is implemented. The minimum mesh size inside graphene equals 0.1 nm and gradually increases outside graphene ribbons. The simulated transmission and reflection spectra of the heterostructure under the illumination of only one beam are shown in Fig. 3(a). As expected, the multiple incoherent illumination limit of *T* = *R* = 0.25 is observed at the investigated mid-infrared region. The maximum incoherent absorption of *A _{max}* = 0.5 is obtained at the spectral positions with

*T*=

*R*= 0.25, as shown in Fig. 3(b). Under the illumination of two counter-propagating coherent beams with the same intensity and phase, the five-band CPA is obtained, as shown in Fig. 3(c). The simulated results match well with the predication from Eq. (8). To reveal the underlying physics associated with the novel multiband CPA, the electric field

*E*distributions at CPA wavelengths of 5439 and 6432 nm are presented in Figs. 3(d) and 3(e), respectively. Comparison of Figs. 3(d) and 3(e) suggests that the fantastic AGPs with most of electric fields tightly confined in the gap between the graphene and the metal indeed are excited. Resulting from the edge reflection, the normal standing-wave resonance yields at the graphene ribbon. The acoustic plasmonic standing-wave resonances is theoretically described by a picture of an effective F-P cavity satisfying with the resonant equation:$2{N_{eff}}L = m{\lambda _0}\textrm{ (}m\textrm{ = 1, 3, 5, 7, } \cdot{\cdot} \cdot )$[56,57]. Here, the order (

_{y}*m*) of the F-P resonance is defined by the number of the standing wave node. Considering the symmetry of the heterostructure along the

*x*axis and the free boundary conditions at both ends of each unit cell, the symmetrical centre of the heterostructure is always a node and both ends are anti-nodes at the electric field distribution. As a result, the order

*m*should only be an odd number. The 39th and 31th-order F-P resonances in fact occur at CPA wavelengths of

*λ*

_{0}= 5439 and 6432 nm, respectively. Due to the F-P resonance with Lorentzian line shape, the reflected light of one optical beam destructively interferes with the transmitted light of the other, and vice versa. The scattering is dissipated by the total absorption in the heterostructure. On the contrary, without the realization of the normal F-P resonance at non-CPA wavelengths such as

*λ*

_{0}= 5526 nm, the main energy escapes from the heterostructure and forms as standing wave with the incident light by the constructive interference, thereby giving rise to the weak absorption, as shown in Fig. 3(f).

The combination of the dispersion equation of AGPs and the F-P resonant equation indicates that the spectral positions of CPA directly depend on the heterostructure parameters, especially the thickness of the ZnS spacer and also the gate-tuning chemical potential of graphene. In other word, the CPA wavelength can be controlled by changing the thickness of the ZnS spacer and the chemical potential, respectively. The calculated *N _{eff}* with different thicknesses of the ZnS spacer has been shown in Fig. 4(a). As the thickness

*d*decreases, the

*N*of AGPs increases due to the further enhanced field confinement. Thus, the spectral positions of multiband CPA will tend to exhibit a red shift with the thickness

_{eff}*d*decreasing. Simulated absorption spectra verify the spectral redshift, as shown in Fig. 4(b). Similarly, the decrease in the chemical potential of graphene enabled by tuning the external gate voltage gives rise to the increase of

*N*of AGPs, as suggested in Fig. 4(c). The absorption peak with same order of the F-P cavity thus shifts to longer wavelengths with decreasing the chemical potential, as described in Fig. 4(d). Note that, with the chemical potential decreasing, the lower-order resonant peaks are visible and their absorption intensity decreases, as indicated by the grey region with

_{eff}*µ*= 0.7 eV. We attribute this behaviour to that the self-consistent amplitude

_{c}*η*changes and thereby the incoherent absorption limit of

*R*=

*T*= 0.25 is broken, as demonstrated in the bottom plot of Fig. 4(d). Certainly, the spectral tunability of CPA wavelengths can also be droved by changing the width of the graphene ribbon and the refractive index of the dielectric spacer between the graphene and the metal, respectively. The tunability of CPA wavelengths afforded by changing the refractive index of the dielectric spacer will allow such absorber to find utility in highly sensitive optical sensing, thanks to the strong field enhancement between the graphene and the metal.

Intriguingly, the Eq. (8) suggests that the absorption intensity of each resonant peak can be dynamically controlled via varying the relative intensity (*α*) and phase difference (Δ*φ*) between two counter-propagating coherent beams, respectively, without changing the heterostructure parameters. The theoretical predication is well captured by our FDTD simulations. The phase difference is fixed to be Δ*φ* = 2*N*π and the absorption of the heterostructure with the relative intensity varying from *α* = 0 to 1 is shown in Fig. 5(a). The absorption intensity of five resonant peaks is simultaneously modulated by changing the *α*. Concretely, when the Δ*φ* = 0 is fixed, the absorption intensity of the heterostructure at the wavelength of *λ*_{0} = 6432 nm is tuned from the incoherent absorption limit of A = 0.5 to the maximum of A = 0.9922 by varying the *α* from 0 to 1, as shown in Fig. 5(b). Similarly, when the *α* = 1 is fixed, the absorption intensity of five resonant peaks can also be simultaneously modulated by changing the phase difference Δ*φ*, as shown in Fig. 6(a). At the wavelength of *λ*_{0} = 6432 nm, the absorption intensity is tuned from A = 99.89% to nearly 0 by changing the phase difference from Δ*φ* = 0 to *π*, as depicted in Fig. 6(b). Meanwhile, the absorption tunability followed the cosine law is clearly observed, resulting from the scattering (*O*_{1} and *O*_{2}) related to the phase difference (Δ*φ*) by the cosine function, as shown in Fig. 6(c). The magnetic field *H _{z}* distributions of the system at Δ

*φ*= 0 and

*π*are shown in the Fig. 6(d). The comparison of colour plots suggests that the CPA is realized at Δ

*φ*= 0, whereas the coherent perfect transparency (CPT) occurs at Δ

*φ*=

*π*where the substantial reduction in the total absorption is due to the constructive interference of scattering fields escaping from the heterostructure. It means that we can continuously switch the heterostructure from the strongly absorption to highly transparency only by simply varying the phase difference between two coherent counter-propagating beams. Such unique property will allow the proposed structure to find utility in optical modulators and switcher. The modulation depth as a function of the wavelength of the incident light is shown Fig. 7(a). It is calculated by the ration of maximum to minimum absorption intensity obtained by varying the phase difference. The maximum modulation depth of 4.46*10

^{5}is observed. So far, based on the ultraconfined AGPs on the vertically stacked metal/dielectric/graphene heterostructure, not only do achieve the mutually non-interfering multiband perfect absorber, but we also can dynamically tune the spectral position and absorption intensity of each resonant peak, respectively.

Finally, the absorption property of the multiband coherent absorber under oblique incidence is investigated. The color-coded coherent absorption of the heterostructure as a function of incident angles is shown in Fig. 7(b). The spectral positions and absorption intensity of multiband CPA remaining unchanged as the incident angle increases are observed intuitively. The excellent absorption stability is realized with the incident angle up to 60^{°} at least. We explain such excellent absorption stability by the two-stage coupling scheme for exciting AGPs at metal-coupled regions and the extreme field confinement in the spacer between the metal and the graphene. The free-space light is first coupled to GPs at metal-free regions by the edge scattering of the upper metal strip and the GPs are then coupled to AGPs at metal-coupled regions by the mirror image. Even under the illumination of oblique incidence, the two-stage coupling scheme is unaffected. The wavelengths of absorption peaks still depend on the F-P resonance with all incident energy tightly confined in the spacer between the metal and the graphene. Thus, the angle-independent property is observed, suggesting the possibility of a practical application of such absorber. By the way, we notice a similar structure has been proposed for multiband absorption enhancement in graphene [58]. Although the multiband absorption is obtained by the composite structure of metal/dielectric/graphene heterostructure on a metal substrate, the simultaneous perfect absorption at each resonance is inaccessible for the single-port system based on the critical coupling effect. More importantly, the fundamental physics of original AGPs on the metal/dielectric/graphene heterostructure, resulting from the hybridization of plasmons in graphene with their mirror image on a neighboring metal surface, have not been considered and discussed there.

## 4. Conclusions

In summary, we have theoretically and numerically investigated the vertically stacked metal/dielectric/graphene heterostructure. All results suggest that the metal plate in the vicinity of graphene do strongly influence the Coulomb interactions and collective excitation. The hybridization of plasmons in graphene with their mirror image on a metal surface contributes to the ultraconfined AGPs with the photon wavelength reduced to *λ*_{0}/70 at least, which allows the graphene ribbon of finite width to simultaneously support multiple-order F-P resonances. When the heterostructure in the symmetric environment is illuminated by two counter-propagating coherent beams, the five-band CPA is observed in the mid-infrared region. The spectral positions of CPA wavelengths are dynamically tuned by changing the gate-dependent chemical potential of graphene, as well as by varying the thickness of dielectric interlayer. Intriguingly, the absorption intensity of each resonant peak is not only continuously tuned from 99.22% to the incoherent absorption limit of 50% through tuning the relative intensity of two counter-propagating beams, but also is tuned from 99.89% to nearly 0 via controlling their phase difference. The maximum modulation depth of 4.46*10^{5} is observed. All FDTD simulations are well captured by theoretical analyses. Our finding is contributed to describe the polariton property in other Van der Waals heterostructures. The tunable angle-independent multiband coherent absorber undoubtedly finds utility in mid-infrared detectors, transducers, modulators, and optical switches.

## Funding

Fundamental Research Funds for the Central Universities (JZ2019HGTB0091); National Natural Science Foundation of China (11904096, 61805064).

## Disclosures

The authors declare no conflicts of interest.

## References

**1. **K. S. Novoselov, A. K. Geim, S. V. Morozov, D. A. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science **306**(5696), 666–669 (2004). [CrossRef]

**2. **A. H. Castro Neto, F. Guinea, N. M. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. **81**(1), 109–162 (2009). [CrossRef]

**3. **A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. **6**(3), 183–191 (2007). [CrossRef]

**4. **R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. R. Peres, and A. K. Geim, “Fine structure constant defines visual transparency of graphene,” Science **320**(5881), 1308 (2008). [CrossRef]

**5. **K. F. Mak, M. Y. Sfeir, Y. Wu, C. H. Lui, J. A. Misewich, and T. F. Heinz, “Measurement of the optical conductivity of graphene,” Phys. Rev. Lett. **101**(19), 196405 (2008). [CrossRef]

**6. **M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature **474**(7349), 64–67 (2011). [CrossRef]

**7. **M. C. Lemme, T. J. Echtermeyer, M. Baus, and H. Kurz, “A graphene field-effect device,” IEEE Electron Device Lett. **28**(4), 282–284 (2007). [CrossRef]

**8. **C. Chakraborty, R. Beams, K. M. Goodfellow, G. W. Wicks, L. Novotny, and A. Nick Vamivakas, “Optical antenna enhanced graphene photodetector,” Appl. Phys. Lett. **105**(24), 241114 (2014). [CrossRef]

**9. **A. Bostwick, F. Speck, T. Seyller, K. Horn, M. Polini, R. Asgari, A. H. MacDonald, and E. Rotenberg, “Observation of plasmarons in quasi-freestanding doped graphene,” Science **328**(5981), 999–1002 (2010). [CrossRef]

**10. **M. Farhat, S. Guenneau, and H. Bağcı, “Exciting graphene surface plasmon polaritons through light and sound interplay,” Phys. Rev. Lett. **111**(23), 237404 (2013). [CrossRef]

**11. **K. J. Ooi, H. S. Chu, C. Y. Hsieh, D. T. Tan, and L. K. Ang, “Highly efficient midinfrared on-chip electrical generation of graphene plasmons by inelastic electron tunneling excitation,” Phys. Rev. Appl. **3**(5), 054001 (2015). [CrossRef]

**12. **H. J. Li, L. L. Wang, J. Q. Liu, Z. R. Huang, B. Sun, and X. Zhai, “Investigation of the graphene based planar plasmonic filters,” Appl. Phys. Lett. **103**(21), 211104 (2013). [CrossRef]

**13. **Z. Fang, S. Thongrattanasiri, A. Schlather, Z. Liu, L. Ma, Y. Wang, P. M. Ajayan, P. Nordlander, N. J. Halas, F. Javier, and G. de Abajo, “Gated tunability and hybridization of localized plasmons in nanostructured graphene,” ACS Nano **7**(3), 2388–2395 (2013). [CrossRef]

**14. **W. Gao, G. Shi, Z. Jin, J. Shu, Q. Zhang, R. Vajtai, P. M. Ajayan, J. Kono, and Q. Xu, “Excitation and active control of propagating surface plasmon polaritons in graphene,” Nano Lett. **13**(8), 3698–3702 (2013). [CrossRef]

**15. **Z. Fang, Y. Wang, A. E. Schlather, Z. Liu, P. M. Ajayan, F. J. de Abajo, P. Nordlander, X. Zhu, and N. J. Halas, “Active tunable absorption enhancement with graphene nanodisk arrays,” Nano Lett. **14**(1), 299–304 (2014). [CrossRef]

**16. **S. Ke, B. Wang, H. Huang, H. Long, K. Wang, and P. Lu, “Plasmonic absorption enhancement in periodic cross-shaped graphene arrays,” Opt. Express **23**(7), 8888–8900 (2015). [CrossRef]

**17. **X. Wang, X. Jiang, Q. You, J. Guo, X. Dai, and Y. Xiang, “Tunable and multichannel terahertz perfect absorber due to Tamm surface plasmons with graphene,” Photonics Res. **5**(6), 536–542 (2017). [CrossRef]

**18. **J. Wu, L. Jiang, J. Guo, X. Dai, Y. Xiang, and S. Wen, “Tunable perfect absorption at infrared frequencies by a graphene-hBN hyper crystal,” Opt. Express **24**(15), 17103–17114 (2016). [CrossRef]

**19. **P. C. Wu, N. Papasimakis, and D. P. Tsai, “Self-affine graphene metasurfaces for tunable broadband absorption,” Phys. Rev. Appl. **6**(4), 044019 (2016). [CrossRef]

**20. **M. S. Jang, V. W. Brar, M. C. Sherrott, J. J. Lopez, L. Kim, S. Kim, M. Choi, and H. A. Atwater, “Tunable large resonant absorption in a midinfrared graphene Salisbury screen,” Phys. Rev. B **90**(16), 165409 (2014). [CrossRef]

**21. **R. Alaee, M. Farhat, C. Rockstuhl, and F. Lederer, “A perfect absorber made of a graphene micro-ribbon metamaterial,” Opt. Express **20**(27), 28017–28024 (2012). [CrossRef]

**22. **X. He, “Tunable terahertz graphene metamaterials,” Carbon **82**, 229–237 (2015). [CrossRef]

**23. **H. Li, C. Ji, Y. Ren, J. Hu, M. Qin, and L. Wang, “Investigation of multiband plasmonic metamaterial perfect absorbers based on graphene ribbons by the phase-coupled method,” Carbon **141**, 481–487 (2019). [CrossRef]

**24. **S. Thongrattanasiri, F. H. Koppens, and F. J. G. De Abajo, “Complete optical absorption in periodically patterned graphene,” Phys. Rev. Lett. **108**(4), 047401 (2012). [CrossRef]

**25. **J. R. Piper and S. Fan, “Total absorption in a graphene monolayer in the optical regime by critical coupling with a photonic crystal guided resonance,” ACS Photonics **1**(4), 347–353 (2014). [CrossRef]

**26. **Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. **105**(5), 053901 (2010). [CrossRef]

**27. **W. Wan, Y. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-reversed lasing and interferometric control of absorption,” Science **331**(6019), 889–892 (2011). [CrossRef]

**28. **Y. Ye, D. Hay, and Z. Shi, “Coherent perfect absorption in chiral metamaterials,” Opt. Lett. **41**(14), 3359–3362 (2016). [CrossRef]

**29. **Y. Fan, F. Zhang, Q. Zhao, Z. Wei, and H. Li, “Tunable terahertz coherent perfect absorption in a monolayer graphene,” Opt. Lett. **39**(21), 6269–6272 (2014). [CrossRef]

**30. **J. Zhang, C. Guo, K. Liu, Z. Zhu, W. Ye, X. Yuan, and S. Qin, “Coherent perfect absorption and transparency in a nanostructured graphene film,” Opt. Express **22**(10), 12524–12532 (2014). [CrossRef]

**31. **X. Hu and J. Wang, “High-speed gate-tunable terahertz coherent perfect absorption using a split-ring graphene,” Opt. Lett. **40**(23), 5538–5541 (2015). [CrossRef]

**32. **X. Feng, J. Zou, W. Xu, Z. Zhu, X. Yuan, J. Zhang, and S. Qin, “Coherent perfect absorption and asymmetric interferometric light-light control in graphene with resonant dielectric nanostructures,” Opt. Express **26**(22), 29183–29191 (2018). [CrossRef]

**33. **J. Si, Z. Dong, X. Yu, and X. Deng, “Tunable polarization-independent dual-band coherent perfect absorber based on metal-graphene nanoring structure,” Opt. Express **26**(17), 21768–21777 (2018). [CrossRef]

**34. **G. Yao, F. Ling, J. Yue, C. Luo, J. Ji, and J. Yao, “Dual-band tunable perfect metamaterial absorber in the THz range,” Opt. Express **24**(2), 1518–1527 (2016). [CrossRef]

**35. **G. Nie, Q. Shi, Z. Zhu, and J. Shi, “Selective coherent perfect absorption in metamaterials,” Appl. Phys. Lett. **105**(20), 201909 (2014). [CrossRef]

**36. **W. Lv, J. Bing, Y. Deng, D. Duan, Z. Zhu, Y. Li, C. Guan, and J. Shi, “Polarization-controlled multifrequency coherent perfect absorption in stereometamaterials,” Opt. Express **26**(13), 17236–17244 (2018). [CrossRef]

**37. **P. Alonso-González, A. Y. Nikitin, Y. Gao, A. Woessner, M. B. Lundeberg, A. Principi, N. Forcellini, W. Yan, S. Vélez, A. J. Huber, K. Watanabe, T. Taniguchi, F. Casanova, L. E. Hueso, M. Polini, J. Hone, F. H. L. Koppens, and R. Hillenbrand, “Acoustic terahertz graphene plasmons revealed by photocurrent nanoscopy,” Nat. Nanotechnol. **12**(1), 31–35 (2017). [CrossRef]

**38. **I. H. Lee, D. Yoo, P. Avouris, T. Low, and S. H. Oh, “Graphene acoustic plasmon resonator for ultrasensitive infrared spectroscopy,” Nat. Nanotechnol. **14**(4), 313–319 (2019). [CrossRef]

**39. **V. Despoja, D. Novko, I. Lončarić, N. Golenić, L. Marušić, and V. M. Silkin, “Strong acoustic plasmons in chemically doped graphene induced by a nearby metal surface,” Phys. Rev. B **100**(19), 195401 (2019). [CrossRef]

**40. **A. Principi, R. Asgari, and M. Polini, “Acoustic plasmons and composite hole-acoustic plasmon satellite bands in graphene on a metal gate,” Solid State Commun. **151**(21), 1627–1630 (2011). [CrossRef]

**41. **Y. Francescato, V. Giannini, and S. A. Maier, “Strongly confined gap plasmon modes in graphene sandwiches and graphene-on-silicon,” New J. Phys. **15**(6), 063020 (2013). [CrossRef]

**42. **F. H. Koppens, D. E. Chang, F. Javier, and G. de Abajo, “Graphene plasmonics: a platform for strong light-matter interactions,” Nano Lett. **11**(8), 3370–3377 (2011). [CrossRef]

**43. **G. W. Hanson, “Dyadic Green’s functions and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. **103**(6), 064302 (2008). [CrossRef]

**44. **B. Wang, X. Zhang, X. Yuan, and J. Teng, “Optical coupling of surface plasmons between graphene sheets,” Appl. Phys. Lett. **100**(13), 131111 (2012). [CrossRef]

**45. **M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B **80**(24), 245435 (2009). [CrossRef]

**46. **D. A. Iranzo, S. Nanot, E. J. C. Dias, I. Epstein, C. Peng, D. K. Efetov, M. B. Lundeberg, R. Parret, J. Osmond, J.-Y. Hong, J. Kong, D. R. Englund, N. M. R. Peres, and F. H. L. Koppens, “Probing the ultimate plasmon confinement limits with a van der Waals heterostructure,” Science **360**(6386), 291–295 (2018). [CrossRef]

**47. **M. B. Lundeberg, Y. Gao, R. Asgari, C. Tan, B. Van Duppen, M. Autore, P. Alonso-González, A. Woessner, K. Watanabe, T. Taniguchi, R. Hillenbrand, J. Hone, M. Polini, and F. H. L. Koppens, “Tuning quantum nonlocal effects in graphene plasmonics,” Science **357**(6347), 187–191 (2017). [CrossRef]

**48. **E. H. Hwang and S. D. Sarma, “Plasmon modes of spatially separated double-layer graphene,” Phys. Rev. B **80**(20), 205405 (2009). [CrossRef]

**49. **A. Vakil and N. Engheta, “Transformation optics using graphene,” Science **332**(6035), 1291–1294 (2011). [CrossRef]

**50. **H. J. Li, L. L. Wang, and X. Zhai, “Plasmonically induced absorption and transparency based on MIM waveguides with concentric nanorings,” IEEE Photonics Technol. Lett. **28**(13), 1454–1457 (2016). [CrossRef]

**51. **D. P. Edward and I. Palik, * Handbook of optical constants of solids* (Elsevier, 1985).

**52. **C. H. Gan, H. S. Chu, and E. P. Li, “Synthesis of highly confined surface plasmon modes with doped graphene sheets in the midinfrared and terahertz frequencies,” Phys. Rev. B **85**(12), 125431 (2012). [CrossRef]

**53. **H. Xu, W. Lu, W. Zhu, Z. Dong, and T. Cui, “Efficient manipulation of surface plasmon polariton waves in graphene,” Appl. Phys. Lett. **100**(24), 243110 (2012). [CrossRef]

**54. **A. Y. Nikitin, T. Low, and L. Martín-Moreno, “Anomalous reflection phase of graphene plasmons and its influence on resonators,” Phys. Rev. B **90**(4), 041407 (2014). [CrossRef]

**55. **Y. Ning, Z. Dong, J. Si, and X. Deng, “Tunable polarization-independent coherent perfect absorber based on a metal-graphene nanostructure,” Opt. Express **25**(26), 32467–32474 (2017). [CrossRef]

**56. **M. S. Jang, S. Kim, V. W. Brar, S. G. Menabde, and H. A. Atwater, “Modulated resonant transmission of graphene plasmons across a λ/50 plasmonic waveguide gap,” Phys. Rev. Appl. **10**(5), 054053 (2018). [CrossRef]

**57. **H. Li, B. Chen, M. Qin, and L. Wang, “Strong plasmon-exciton coupling in MIM waveguide-resonator systems with WS_{2} monolayer,” Opt. Express **28**(1), 205–215 (2020). [CrossRef]

**58. **Z. Wang and Y. Hou, “Ultra-multiband absorption enhancement of graphene in a metal-dielectric-graphene sandwich structure covering terahertz to mid-infrared regime,” Opt. Express **25**(16), 19185–19194 (2017). [CrossRef]