Directional Excitation of Acoustic Graphene Plasmons Using Oblique Incidences

In this work, the directional excitation of acoustic graphene plasmons (AGPs) are numerically studied using finite element methods. In our proposed hybrid graphene-metal structure under oblique incidences, not only are AGPs excited efficiently, but also they are unidirectional propagating along a graphene monolayer. Although the symmetry AGPs dispersion relations are broken by oblique incidence, both left- and right-moved AGPs are excited simultaneously at a resonant wavelength due to almost equaled wavenumbers of directional propagated AGPs. Based on the fact that great AGPs excitation efficiency can’t guarantee high EM energy propagating in one direction, we will focus on how the directional propagating net energy are affected by geometrical parameters. Due to the tunable graphene conductivity, AGPs propagation with great unidirectional net energy can be dynamically controlled by a relatively low externally applied bias voltage (electrostatic gating). The prototype structure may find applications in ultra-confined plasmon launchers and switchers in integrated optics.

Graphene, a revolutionary two-dimensional material with only one atom thickness, has been explored by tremendous researchers due to its many outstanding properties after firstly fabricated through mechanical exfoliation [1]. Since its imaginary part of complex conductivity is positive as it is doped by electrostatic or chemical doping techniques [2][3][4][5], tunable graphene surface plasmons (GSPs) are supported on graphene structures in the mid-infrared/THz frequency range [6]. Due to its strong field confinement and unique tunability, GSPs have attracted tremendous investigation and found various novel optical applications, including efficient mid-infrared/THz sources [7,8], modulators [9,10], broadband polarizer [11], plasmonic components [12,13], and optoelectronic components, such as photodetectors [14].
GSPs are bounded waves which strongly confined down to the atomic layer and thus can be useful for enhancing IR molecular signals, particularly for probing biomolecules [15].
However, the confinement of plasmons in the mid-IR frequency range still challenges the sensing of thin molecular layers or angstrom-thick material. Recently, a stronger confined plasmonic mode-acoustic graphene plasmons (AGPs) has been theoretically [16,17] and experimentally [18][19][20] demonstrated. In a graphene sheet placed shortly above a metallic substrate, AGPs are formed by the hybridization of GSPs and their mirror image, resulting in the electromagnetic field squeezed between graphene and the metallic substrate. Because of their ultraconfined optical field down to atomic level, AGPs have shown many superior properties compared to GSPs in graphene without metallic film, such as larger near field enhancement, less damped, lower controlled voltage, and better stability. Due to these outstanding properties, AGPs can further improve the performance in sensing applications, especially for probing small quantities of molecules and quantum effect exploration.
So far, the investigation of AGPs is limited to the realization of strong confinement, great coupling efficiencies and their fundament applications. However, the exploration of controlling the propagation direction of AGPs is still missing, which is useful for applications like ultra-confined plasmons launchers and switching devices for integrated optics [7]. Recently, several methods for directional excitation of conventional plasmonic waves (SPPs) have been explored [20,21]. Among 1 3 these proposed schemes, it is essential to break the symmetry in the structure or the incident light to excite the propagation plasmons along a desired direction. Besides, the grating is the main component for coupling free photons to SPPs. In the case of normal incidence, there are plasmonic resonances (standing wave modes) in each unit cell of a grating in which the counterpropagating SPPs interfere constructively. In other words, the unit cell of a grating operates as optical resonator, which largely confines the THz light into the nano-scale space. However, the method of using normally incident light has faced challenges in controlling the directionality of the SPPs.
Generally, the excitation of propagating SPPs in grating structures is dependent on the incident angle; this is because that the SPPs coupling condition is k spps = k 0 sinθ ± G m , where k 0 is a wave vector of incident light, θ is an incident angle, and G m = m 2 p is a reciprocal lattice vector with the integer m representing the order number. For a certain incident angle, there are two solutions of k spps that satisfy the resonance condition, thus corresponding to two unidirectional propagating SPPs. In other words, spectra splitting phenomenon would occur in the transmission or absorption spectra when oblique incidence is applied. In principle, it is not clear whether the hypothesis that AGPs behave like SPPs is valid, due to a negligibly small transverse wavenumbers at oblique incidence ( k 0 sinθ ≪ k AGPs ).
Inspired by these works, we designed and demonstrated AGPs launchers that overcome these limits using periodic nanoslits in a metal film under oblique incidence. In order to ensure graphene plasmons to be excited by external illumination, p-polarized plane waves are diffracted by the periodic metallic slit array, leading to phase-matching between the free-space photons and AGPs. The grating is appropriately set to compensate a parallel component wave-vector as Due to the oblique angle θ breaks the left-right symmetry, the difference of absolute of wave-vectors between the left and right propagating AGPs can be generated but far smaller than k AGPs , which still realize unidirectional excitation of AGPs. Although the coupling conditions for unidirectional AGPs excitation are satisfied, both left-and right-moved AGPs are effectively excited simultaneously due to almost equaled wavenumbers of directional propagated AGPs, which indicate that high excitation efficiency is not equal to the great unidrectional net propagating energy. In this work, we will consider on how the directional propagating net energy is affected by geometrical parameters. With the tunability of the graphene conductivity, the directionality of AGPs propagation can also be dynamically controlled by changing the doping level of the graphene. At last, based on nano-sized space between graphene and metal gate electrodes, low voltages are required to (1) AGPs = k 0 sinθ + m 2 p dynamically control directional propagating AGPs. The proposed scheme may find applications in ultra-confined plasmon launchers and switchers in integrated optics.

Model and Methods
The system we investigate is schematically depicted in Fig. 1. The proposed nanostructure consists of a graphene monolayer coated on a pattered metal thin film separated by a dielectric film with a narrow gap of g. The nanoslit array with a periodicity of p is milled in a metal film with thickness of t m . The nanoslit width denoted as w s is set to nanometer size to be avoid of breaking the forming condition of AGPs. Salisbury screen effects also are adopted to enhance the coupling efficiency of AGPs. So we use an electric conductor film as a back reflector at a distance of d away from the pattered metal film, which should specifically choose to form an oscillation antinode of a standing wave at the location of the graphene sheet. A coherent monochromatic transverse magnetic (p-polarized) plane waves with the incident angle of θ respective to -y axis are incident on the graphene-metal structure from the upside. The period of slit array, nanoslit width, metal film thickness, and gap g are set as p = 600 nm, w s = 30nm ,t m = 100nm, and g = 50nm , respectively, which are fixed unless otherwise specified. These values are well within reach of current nano-fabrication methods [22]. For simplicity, the space over the graphene layer is assumed to be air ( 1 = 1 ). For practical application, the insulating dielectric below the graphene layer is chose to be silica ( d = 2.25 ) due to the silica layer is easily manufactured using current techniques. We choose gold (Au) as the metallic material because gold has relatively greater stability compared with other metal such as Ag, Al, and Fe. The electric permittivity of Au is approximated as Drude model represented by expression , with the bulk plasma frequency P = 1.37 × 10 16 rad∕s, and damping rate m = 4.08 × 10 13 rad s , respectively, which behaves as perfect electric conductor at terahertz or infrared frequencies.
The optical feature of the graphene monolayer can be characterized by a complex valued surface conductivity of g . With the local random phase approximation, the conductivity of graphene sheet can be attributed to intraband and interband electronic transition [23,24]: σ g = + , which can be derived for where = 2 f is the angular frequency, E F is the Fermi level (or the chemical potential) representing the doping level of the graphene layer, k B is the Boltzmann's constant, T = 300 K is the room temperature,ℏ = h∕2 is the reduced Planck's constant, e is the elementary charge, is the electron-phonon scattering rate with the graphene carries mobilityu = 1m 2 ∕Vs , and the Fermi-velocity v F = 1 × 10 6 m∕s . The first term inter describes the intraband electron-photon scattering and the last term inter corresponds to interband transitions of charge carriers in graphene.
To save important computational resources and time, the graphene sheet is modeled as a surface conductive boundary [25], which generates a surface current intensity of J = σ g E with a response to in-plane electric field E. In the proposed hybrid structure, voltages can be directly add to both the graphene sheet and the metal film to serve as a gate electrode, resulting in the graphene conductivity can be dynamically controlled by gate voltage. It should be mentioned that since the mid-infrared TM light cannot "see" the effect of the nano-sized metallic structure, the nano-slit perforated metal film proposed here can be treated as a transparent conducting electrode for mid-infrared p-polarized signal transmission. Periodic boundary conditions are set in the x direction, and perfectly matched layers (PML) are adopted in the y direction for the truncation of open boundaries. In the simulation, reflection (R) and absorption (A = 1 − R) of such a hybrid structure are calculated due the transmission channel are blocked by the metallic back reflector. We address that since the intrinsic losses of the dielectric and the metal are not included or negligible small in our model, we denote the absorption A simply as the excitation efficiency of plasmonic modes which mainly dissipate on graphene as Joule heat.

Results and Analysis
We start this work with the dispersion curve of AGPs in graphene/dielectric/metal structure, as shown in Fig. 2a. Considering a graphene sheet lying over a metal film, we calculated the refractive index of plasmonic mode propagating on a graphene monolayer supported by a metal film separated by nano-sized dielectric layer with thickness of g = 50 nm, where graphene are doped at the different Fermi energy of 0.3,0.5.0.7, and 0.9 eV. One note that the real part of the mode refractive constants for all doped the graphene almost linearly reduces in long-wavelength regime, which confirms that AGPs are indeed propagate in our proposed system. However, the refractive index exponentially increases as the Fermi level of graphene changes from 0.3 to 0.9 eV at the fixed operated wavelength, indicating higher doping graphene supports AGPs with larger wavenumber (or stronger optical field confinement). Besides, the imaginary part of the effective mode index for AGPs is related to the attenuation of propagating waves. One can obtain the propagation length through the formula of L AGPs = 1 2Im(k AGPs ) = k 0 2Im(n AGPs ) . As shown in Fig. 2a, Im(n AGPs ) marked with dotted lines could find their minimum value at specific wavelength, which indicate the longest propagating-length mode in the plasmonic modes. Another interesting feature is that the propagation loss of AGPs get smaller with increasing Fermi energy which is contrary with that of general GSPs. Im(n eff ) are also calculated to identify the optimal guided modes in the corresponding structure. A larger FOM means a shorter mode wavelength at the cost of a lower propagation loss. In Fig. 2b, the function FOM of the operating wavelength is shown for the Fermi energy E F changing from 0.3 to 0.9 eV. Obviously, FOM is 100 for 6-20 μm range, larger than GSPs (20) [26], indicating that AGPs show longer propagation distance compared to that with the same degrees of confinement. Besides, as the Fermi level increases, the FOM increases accordingly, and the larger FOM tend to appear at shorter wavelengths, e.g., the FOM experiences its maximum as high as 190 at the operating wavelength of 6 μm.
The absorption spectra of the hybrid structure are calculated for different incident angle as shown in Fig. 3a. In the case of normal incidence ( θ = 0 • ), there are three absorption peaks near 6.51, 8.39, and 15.22 μm . When an oblique incident wave with the angle θ = 45 • is applied, each absorption peaks splits into two narrow peaks, one below the original wavelength and one above. For a comparison, oblique incidence of −45 • is also used to excite the plasmonic mode of the hybrid structure; we find out that the absorption   Fig. 3a) of |m| = 1 , with red arrows indicating the local Poynting vector. Here, the graphene is doped at E F = 0.7 eV spectrum overlap with the one for 45 • incidence due to the symmetry of the considered structure.
Utilizing the mode analysis solver (by COMSOL Multiphysics software), AGPs are numerically obtained in a monolayer graphene-insulator-metal film of the same geometrical parameters as in Fig. 2. Substituting Re( AGPs ) into Eq. (1), the resonant peaks corresponding to different diffraction order number m are listed in Table 1, in which m is rounded due to its value and is very close to the corresponding integer. Under normal incidence, each peak in the absorption spectrum corresponds to two diffraction order numbers of ±|m|(m = 1, 2, 3, respectively) due to the structural symmetry, indicating the free photons can coupled to AGPs in both directions. While for oblique incidence waves with angle of 45 • , each peak represents the AGPs excited by only one diffraction order with m= −1or1, −2or2, −3or3, respectively, in which the positive (or negative) m represents the forward (or back) direction of the phase compensation for the excitation of unidirectional moved AGPs.
As shown in Fig. 3b, we also calculate the power flux along x axis (defined as ∫ y2 y1 S x dy , here S x is the x component of Poynting vector). There are indeed no net power flux propagating at both direction at normal incidence; while in the case of oblique incident light, one can easily find that the splitting peaks originated from the main peak propagate in opposite directions, as indicated by circlemarked red curve. Besides, the direction of excitation can be reversed by the oblique incidence wave of −45 • as indicated by the triangle-marked blue curve. As a result, one can easily realize directional excitation of AGPs at specific wavelength with oblique incidences. It should be mentioned that the net power propagating in one direction is substantially large. This feature is very different from the conventional metal SPPs which are scatted strongly by the edge of the grating unit cell, leading to most of EM energy oscillating in one unit cell of grating. However, the AGPs are hardly scattered by the edge of metallic slits due to their great optical field confinement. As a result, the way of using the oblique incident wave is very suitable for directional excitation of AGPs.
At normal incidence, a tightly-confined AGPs are formed in graphene-metal film structure with the field distribution of E y for the m = 1 mode plotted in Fig. 3c. One could easily see that the counter-propagating AGPs resonantly interfere with each other in one unit cell to form a standing wave mode. The Poynting vectors, indicated by red arrows, reveal that the mode is symmetric, i.e., there is none net EM energy flow propagating along the graphene monolayer in the x direction. In the case of θ = 45 • , the field distributions for the m = ±1 modes are plotted in Fig. 3d and e, with Poynting vectors indicating that generated AGPs modes are directional, with a right-moving (in the + x direction) mode at the shorter wavelength, and a left-moving (in the -x direction) one at the longer wavelength.
Generally speaking, oblique incident waves induce the asymmetry in phase matching condition, which result in directionally excite AGPs at different resonant wavelengths. To illustrate the relation between splitting process and the incident angle, we calculate the absorption and the power flux along x for different incident angles, as shown in Fig. 5. From the absorption spectra in Fig. 4a in which we choose three representative angle of θ = 0, 45°, and 80°, the location of absorption peaks indicating effectively formed AGPs change slightly with the incident angle, which can be attribute to the fact that the incident wave-vector is far smaller than that of AGPs. But the amplitude of absorption peaks change a lot due to the change of the optical path length for Fabry-Perot (FP) resonance, named as "Salisbury screen" effect discussed next. More importantly, as shown in Fig. 4b, the power flux along x is greatly dependent on the incident angle, e. g., for the |m|= 1 mode, the maximum power flux along x is achieved at θ = 40°, while at θ = 20° for |m|= 2 and |m|= 3 mode. Absorption (shown in Fig. 4c) and the power flux along x (shown in Fig. 4d) versus incident angles are also plotted when the operating wavelengths are fixed at absorption peaks for incident angle 45°. The unidirectional AGP excitation efficiency of different orders increases with increasing incident angle firstly and then approaches to their maximum and reduces at last. The power flux along x of each mode possesses an optimal incident angle for transmitting EM energy in one direction. As a result, one note that the net power flux along x of AGPs behave in a different way with their excitation efficiency.
To clearly understand the directional excitation behavior of such hybrid structure, the dependence of AGPs on the structural parameters, such as the sit periodicity p and the screen distance d are also studied in detail, as shown in Fig. 5. We firstly investigate the influences of the slit periodicity on the behaviors of AGPs. At 45° incidence, Fig. 5a-b presents the absorption and the power flux along x, respectively, with slit array periodicity p increases from 400 to 800 nm. When p is increased, the resonant wavelength of unidirectional propagating AGPs has an apparent redshift; this is because the metallic slit grating is used to compensate for mismatched wavenumber by multiples of 2 p . As mentioned before, to further enhance the absorption, Salisbury screens play an important role in such hybrid structure, which have been adopted by several other reports [27,28]. We deposited the hybrid structure over a metallic back reflector, which creates Salisbury screen effect to greatly enhance the absorption in the graphene/metal structure. Based on the FP model theory, the maxima absorption can be achieved under the condition of an odd multiples of quarter wavelength fulfilled for optical distance, e.g., the minimum distance of 2.5 μm for the free space light with operating wavelength of 10 μm . In particular, a great phase delay would occur when incident waves transmit through the graphene-covered metallic slit arrays. Besides, the distance d would substantially decrease for oblique incidence in comparison with that under normal incidence, since the normal component of the wave-vector for oblique light is smaller than the normal vector for normal incidence. Based on two factors mentioned above, the distance d for our proposed structure would be as small as several hundreds of nanometers (e.g. 300 nm). As illustrated in Fig. 5c, the resonant absorption peaks of the hybrid nanostructure hold the location of resonant wavelength as the distance d increase from 300 to 700 nm. Obviously, the intensity of absorption peaks induced by the directional AGPs is sensitive to the distance of d, indicating the proposed structure can be designed as a ultra-compact system for the absorption of TM mid-infrared light based on unidirectional AGPs.
To illustrate the tunability of the directionality of AGPs, we focused on the mode of m=± 1. As shown in Fig. 6a, the left-and right-moving AGPs are easy affected by the doping level of the graphene represented by the Fermi level E F . For instance, not only the location of the two directional propagating mode shift to the shorter wavelength, but also the intensity of the power flux get larger as the Fermi level increase from 0.5 to 0.9 eV. This feature indicates the directionality of AGPs mode can be controlled dynamically by the Fermi level which can be realized by a gate voltage. From Fig. 6b, one clearly note that the right-moving AGPs can be reversed as Fermi levels change from 0.7 to 0.8 eV.
The Fermi level of graphene E F can be changed by static electric field applying a biased gate electrode. The Fermi level of E F = ℏv F ( n g ) 1∕2 can be estimated as function of the total carrier density n g = 0 d V∕eg , with d and V are the relative electric constant of the dielectric material filled in the gap space and the bias voltage between the two electrodes, respectively. Besides, g is the thickness of the insulating dielectric film. As a result, the relationship between the Fermi level and the gate voltage is derived as where C = 0 d ∕g is the formed electrostatic capacitance per unit area. From Eq. (4), it is obvious that when the electrostatic capacitance per unit area C increases (e.g., by reducing the distance between the two electrodes g or using dielectric with higher refractive index), a lower voltage can be used to achieve the same doping level of graphene. One could use a gate dielectric material for graphene to protect the high mobility of graphene, such as hexagonal boron nitride (h-BN) film [29][30][31] which has a large band gap of around 6 eV and performs well as a perfect substrate material to maintain higher carrier mobility of graphene.
So, the applied voltage is another import factor for design an opto-electronic components. Figure 7a presents the absorption of the hybrid structure at fixed excited wavelength as a function of the biased gate voltage with different gap g. It is evidently observed that the absorption line oscillate more frequently as the gap get narrower. This phenomenon can be attributed to two factors, one is that the effective refractive index of AGPs gets larger; the other is that the Fermi level of graphene changes a lot with the gap size. However, the net power flux along x do not obey the same trend as the absorption curve; however, the direction of AGPs excitation can be reversed, while the gate voltage increase from 15 to 40 V. This is because that AGPs are more confined by the metal film and the mode are more sensitive to the edge of the nanoslit, which would reflect AGPs to form FP resonant mode in one unit cell, resulting (4) E F = ℏv F √ CV∕e Fig. 6 a Power flux along x of the first-order (m = 1) AGPs vary with incident wavelength for the doping level from 0.5 to 0.9 eV. b Power flux along x of the AGPs at the wavelength 14 μm versus E F with the incident angle is fixed to 45° no net energy propagate in one direction. More importantly, unidirectional propagating AGPs are still realized as the gap size shrinks to several nanometers. Besides, the voltage for AGPs propagation reversal is as low as 20 V at g = 20 nm. This feature is suited for deigning opto-electronic plasmonic switch with low power consumption.

Conclusions
In conclusion, we have proposed a kind of hybrid nanostructure, consisting of graphene monolayer over metallic slit array with narrow separation, for directional excitation of AGPs with oblique incidence. Firstly, unidirectional AGPs can be excited using metallic grating with different phase compensation for left-and right-moved waves, and then the excitation efficiency can be further enhanced with Salisbury effects. With the tunability of the graphene conductivity, the directionality of AGPs propagation is also dynamically controlled by changing the doping level of the graphene. The dependence of directionality on the structural parameters has also been demonstrated in detail. At last, based on nanosized space between graphene and metal gate electrodes, low voltages are required to dynamically control directional propagating AGPs. The proposed scheme may offer great opportunities in low-voltage controlled mid-infrared plasmonic devices, including ultra-confined plasmonic launcher, switchers, and nano-sensors. Fig. 7 a Absorption spectra versus gate bias voltage for hybrid structures with different gaps g = 10, 20, 50 nm. b Power flux along x in the hybrid structure versus the gate voltage. Here the operating wavelength is fixed to be 10 μm