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Directional control of propagating graphene plasmons by strain engineering

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Abstract

Control of propagating surface plasmon on a scale beyond the diffraction limit is important for the development of integrated nanophotonic circuits and optical information technology. In this paper, a strain-based modulation mechanism for directional control of propagating graphene plasmons was proposed. We demonstrated numerically that the GPs can be directionally controlled by the implementation of strain on graphene. The topologies of GPs excited by a z-polarized optical emitter in unstrained and strained graphene were illustrated both in real space and momentum space. When imposing strain engineering to graphene in different directions with a different modulus, multi-dimensional control of GPs in any direction can be realized. The simulated propagation length ratio η of the GPs can reach 3.5 when the strain with a modulus of 0.20 is applied along or perpendicular to the zigzag direction of graphene. Besides, the effect of PDMS on GPs was investigated finally for the experiments to be carried out and we show that the PDMS does not affect the generation of directional GPs under strain engineering. Our proposed directional control of GPs not only has the advantages of wide operating wavelength but does not require additional coupling mechanisms, which is beneficial to the design of integrated photonic devices.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Information transmission and processing in nanoscale is an important subject in scientific research. As an excellent supporter for photon manipulation and integration in nanoscale, surface plasmons (SPs) have made outstanding achievements in nano-photonics circuits [13]. And the key to prompting its application is how to modulate the propagation direction of SPs more effectively. The directional control of SPs in noble metal has been studied extensively. By breaking the symmetry of the structure, the unidirectional excitation and modulation of SPs can be realized by using a decorated nanoslit [48] or asymmetric double slits [9]. Besides, by using a circularly or elliptically polarized beam [10,11], two interference beams with a particular phase difference [12], or oblique illumination [13,14], the directional SPs can also be excited and controlled by breaking the symmetry of the incident light beams. What’s more, it has been reported that photo-sensitive, temperature-sensitive, magneto-optical or nonlinear substrate can be developed as an external factor for the directional control SPs [1520]. However, the Ohmic heat effect in metal results in a large loss of plasmons, which makes the design of the nano-photonic devices based on the metal SPs encounter a bottleneck [21,22]. The bottleneck has been broken by the successful preparation of graphene [23]. It has shown that doped graphene can also support the SPs, and the SPs in graphene are mainly in the mid-infrared to terahertz range [24], which broadens the operating wavelength of SPs-based photonic devices. Compared with the traditional metal SPs, graphene plasmons(GPs) have the advantages of high active controllability, extreme localization of electromagnetic field, longer plasmon life, etc. Therefore, the directional control of GPs has received extensive attention.

Similar to SPs in metal, the primary physical mechanism for directional control of propagating GPs is symmetry breaking. But the configurations in metal are not viable owing to the one atomic layer of graphene. The modulation of propagating GPs usually requires a design of composite structures or an additional coupling method. For graphene composite structures, the directional propagating GPs can be excited and controlled by linearly polarized beam [25], circularly or elliptically polarized beam [26], or the interference of two linearly polarized light beams [27] that are incident on the metal-graphene composited structure. What’s more, graphene-based meta-surface has also been designed to control the GPs by tuning the chemical potential of graphene [2830]. Besides, GPs can also be controlled by changing the properties of the surrounding medium. According to transformation optics, Ashkan et al. [31]designed different microstructures on graphene substrates to modulate the propagation path of GPs through voltage tuning. Soljacic et al. [32]designed graphene-boron nitride heterostructure to realize full-angle negative refraction by tuning the carrier mobility of graphene or changing the thickness of boron nitride, thus realizing active and coordinated modulation of GPs. Shimura etal [33]also achieved unidirectional launching of propagating GPs only by designing asymmetric nano-ridges within the graphene itself. For additional coupling methods, the GPs can be successfully modulated by integrating graphene and waveguide [34] or Bragg reflector [35]together. Besides, magneto-optic [36] or nonlinear coupling [37] has also been found to be able to control the propagating direction of GPs. However, for composite structures, once the structure is given, the propagation direction of GPs is also determined, which makes it difficult to achieve the multi-dimensional or dynamic control of GPs. While for the method based on additional coupling, the complex coupling means lower efficiency in general. Moreover, these researches are mainly limited to light, electricity or magnetism modulation methods. Due to the good flexibility and other excellent mechanical properties of graphene, strain has become an effective way to tune graphene, the advantage of using strain to graphene is that it offers high repeatability [3840]. In addition, the optical properties of graphene show anisotropy under strain engineering [41], which enables graphene to achieve more novel plasmon-based devices, such as the design of new flexible optoelectronic elements. However, in the mid-infrared to terahertz band, the modulation of GPs by strain engineering is rarely involved. Therefore, the in-depth study of a new modulation mechanism-strain engineering will provide a new platform for the development of GPs based devices.

In this paper, a strain-based modulation mechanism for directional control of propagating GPs was proposed. Considering graphene supported on a flexible substrate, we demonstrate that the propagating GPs in a particular direction can be excited by an emitter when strain engineering was imposed on the configuration. Without destroying the graphene lattice structure, we illustrate that multi-dimensional control of GPs can be realized by applying strain with different modulus in different directions of graphene. Furthermore, when the strain with a modulus of 0.20 is applied along or perpendicular to the zigzag direction of graphene, the direction ratio $\eta$ of the GPs propagation length can reach 3.5. Moreover, our results show that effective modulation can be achieved at any wavelength from mid-infrared to terahertz in support of GPs, and the modulation frequency can reach 300MHz, that is the proposed mechanism has the advantages of wide working wavelength and high modulation frequency. Besides, To provide more detailed guidance for experimental implementation, PDMS was taken as a flexible substrate of graphene, then how the PDMS affect the propagation properties of GPs were investigated extensively, and the results indicate that the generation of directional GPs was not influenced by the substrate. More importantly, our proposed configuration is simple in structure and does not require additional coupling mechanisms, which is beneficial to the design of integrated photonic devices. Our research results can provide detailed guidance for experimental implementation as well as theoretical support in micro-nano scale information transmission, biomedicine, wearable electronics, robot "skin" and other aspects, and further promote and broaden the practical application of the GPs based micro-nano photonics devices.

2. In-plane optical properties of graphene under strain engineering and details of FDTD simulation

Figure 1 displays our studied system and the axis orientation. A single-layer graphene was placed on a flexible substrate and the $x$-axis was chosen along the zigzag direction of the graphene. When the system is uniformly stretched along a prescribed direction in the range of elastic deformation, the strain tensor reads [42]

$$\overline{\kappa}=\kappa\begin{pmatrix} \cos^{2}\theta-\rho \sin^{2}\theta & (1+\rho)\cos\theta\sin\theta \\ (1+\rho)\cos\theta \sin\theta & \sin^{2}\theta-\rho\cos^{2}\theta \end{pmatrix}$$
where $\kappa$ is the strain modulus, without damaging the lattice structure of graphene, the maximum $\kappa$ is taken as 0.20 [38]. $\theta$ represents the angle between the stretched direction T and the $Ox$ axis (zigzag direction) and $\rho$=0.165 is the known experimental Poisson’s ratio of graphene [43]. Under strain engineering, the conductivity tensor of graphene modeled as [44]
$$\widetilde{\sigma}(\omega)=\sigma(\omega)(\overline{I}-2\gamma\overline{\kappa}+\gamma{\text{Tr}(\overline{\kappa})})$$
where $\gamma \simeq 2.37$ denotes the Grüneisen parameter [45]. $\sigma (\omega )$ is the isotropic optical conductivity of graphene calculated according to the well-known Kubo formula [46]
$$\begin{aligned} \sigma(\omega) = &\frac{2e^2T}{\pi\hbar}\frac{i}{\omega+i\tau^{{-}1}}\log[2\cosh(\frac{E_F}{2K_BT})]\\ &+\frac{e^2}{4\hbar}\Bigg[H(\omega/2)+\frac{4i\omega}{\pi}\int_{0}^{\infty}d\varepsilon\frac{H(\varepsilon)-H(\omega/2)}{\omega^2-4\varepsilon^2}\Bigg], \end{aligned}$$
where
$$H(\varepsilon)=\frac{\sinh(\hbar\varepsilon/k_BT)}{\cosh(E_F/k_BT)+\cosh(\hbar\varepsilon/k_BT)}.$$

Here $\tau$ is the electron-phonon relaxation time estimated from $\tau =\mu E_F/ev_F^2$, $v_F\approx c/300$ is the Fermi velocity, $E_F$ is the Fermi level and $\mu$ is the DC mobility of graphene. In this study, the Fermi level is set as 0.4 eV, the DC mobility is 1000 cm$^2$/Vs and $T$ is 300 K.

 figure: Fig. 1.

Fig. 1. Schematic diagram of strain acting on graphene. The graphene is uniformly stretched along a prescribed direction, where the angle between the tensile direction and the zigzag direction of the honeycomb lattice of graphene is $\theta$, here the zigzag direction was chosen along the $x$ axis in the Cartesian system. GPs were excited in the strained graphene for the case of tension T perpendicular to the zigzag direction ($x$-axis) by a $z$-oriented emitter located at $z_{0}$ = 10 nm above the surface.

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To begin with, the in-plane conductivity $\sigma _{xx}$ and $\sigma _{yy}$ of graphene with or without strain engineering were calculated to investigate the effect of strain on graphene. Figure 2(a) shows the real and imaginary part of graphene conductivity with a strain modulus of $\kappa$=0.20 for the case of the tension T perpendicular to the zigzag direction (i.e. $\theta =90^{\circ }$). At the same time, the conductivity for the unstrained graphene was calculated for comparison, as the black line described in Fig. 2(a). One can find that the conductivity decreases along the stretch direction, and it increases in the vertical direction compared to the conductivity of unstrained graphene. Since the properties of GPs depend mainly on the imaginary part of the in-plane conductivity, then how the imaginary part of the conductivity changes with angle $\theta$ was depicted in Fig. 2(c), without loss of generality, the strain modulus was taken as $\kappa$ = 0.20 at a wavelength of 11.6 $\mu$m. The imaginary part of $\sigma _{xx}$ (red dashed line) and $\sigma _{yy}$ (blue dashed line) were compared with the case of $\kappa$ = 0 (black line). One can see that the in-plane conductivity is anisotropic under strain engineering, both the Im($\sigma _{xx}$) and Im($\sigma _{yy}$) of graphene periodically alters with the change of angle $\theta$. In addition, the Im($\sigma _{xx}$) and Im($\sigma _{yy}$) change in the opposite direction with the same amplitude, and the greatest increase or decrease in conductivity can be seen when the graphene was stretched along or perpendicular to the zigzag direction. By taking advantage of the anisotropy conductivity produced by strain engineering, anisotropy propagating GPs can be excited and controlled.

 figure: Fig. 2.

Fig. 2. The anisotropic optical properity of graphene under strain engineering. The real and imaginary part of in-plane graphene conductivity (a) and permittivity (b) with a strain modulus of $\kappa$=0.20 for the case of tension T perpendicular to the zigzag direction (i.e. $\theta =90^{\circ }$). While the black line represents the conductivity $\sigma _{iso}$ and permittivity $\epsilon _{iso}$ without strain engineering. The change of imaginary part of conductivity (c) and the real part of permittivity (d) as a function of angle $\theta$ with a strain modulus of $\kappa$ = 0.20 at $\lambda _0$ = 11.6 $\mu$m.

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In this paper, the finite difference time domain (FDTD) method is used to simulate the properties of GPs, where the permittivity of graphene needs to be concerned. The permittivity of monolayer graphene can be expressed as $\epsilon _{g}(\omega )=1+i\widetilde {\sigma }(\omega )/(\omega \epsilon _0d)$, where d is the thickness of single-layer graphene sheet. Therefore, similar to Figs. 2(a) and 2(c), the in-plane permittivity of graphene with or without strain engineering was calculated in Figs. 2(b) and 2(d). By comparison, we find that the change of the dielectric constant is opposite to that of conductivity. The numerical calculations of GPs were performed by using the anisotropic permittivity from Figs. 2(b) and 2(d), the dimensions of the simulation region along $x$, $y$ and $z$ axis were set to 20 $\mu$m, 20 $\mu$m and 1 $\mu$m respectively. To avoid scattering caused by the boundary of the graphene sheet, the sheet extends out of the simulation region in the $x$-$y$ plane to mimic infinite graphene, and the thickness of graphene was taken as $d$ = 1 nm as a reasonable value. The point electric dipole was placed 10 nm above the center of the graphene layer, and the Perfect Matching Layers were chosen for all the sides of the simulation region. What’s more, the dimensions of the mesh covering the graphene layer and their near vicinity were set to 5 nm, 5 nm and 1 nm along the $x$, $y$ and $z$-axis respectively.

3. GPs on unstrained and strained graphene

Next, we turn to the excitation and modulation of propagating GPs supported by the unstrained and strained graphene. By using the FDTD method discussed above, the topologies of GPs excited by a $z$-polarized optical emitter in unstrained and strained graphene at a wavelength of 11.6 $\mu$m were presented in Fig. 3. Firstly, the electric field distribution $E_z$ and intensity $|E|$ on the unstrained ($\kappa$=0) graphene was shown in Figs. 3(a) and 3(c), we can see that the propagating GPs spread outward isotropically in $x$-$y$ plane with the same property. Then, considering the graphene was uniformly stretched along the $y$-axis (i.e.$\theta =90^{\circ }$), the electric field $E_z$ and $|E|$ on the strained graphene with a modulus $\kappa$=0.20 were simulated. As shown in Figs. 3(b) and 3(d), the GPs propagate anisotropically and most energy propagates along the $x$-axis on the surface. Applying tension to graphene can change the topology of propagating GPs from an isotropic circle to an anisotropic ellipse. The propagation distance of GPs will be significantly prolonged in the direction perpendicular to the tension. What’s more, It is obvious that by applying different strain modulus to graphene in different directions, the GPs propagating in the strained graphene can be controlled in multiple dimensions.

 figure: Fig. 3.

Fig. 3. The $E_z$ field and electric intensity $|E|$ of propagating GPs excited by a $z$-oriented optical emitter located at a distance $z_{0}$=10 nm above the unstrained (a, c) and strained (b, d) graphene for a wavelength of 11.6 $\mu$m at $E_{F}$ = 0.4 eV. Here the graphene was stretched along $Oy$ axis (i.e.$\theta = 90^{\circ }$) with a strain modulus of $\kappa$ = 0.20.

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After that, to further understand the propagation behavior of GPs on the same graphene with or without strain engineering, it is necessary to calculate the dispersion relation of GPs analytically. Without considering the dissipation of energy in graphene, the dispersion relation of the supported GPs can be expressed as [47]

$$\mu_0(k_{x}^{2}\text{Im}(\sigma_{xx})+k_{y}^{2}\text{Im}(\sigma_{yy}))^{2}(k_{x}^{2}+k_{y}^{2}-k_{0}^{2})-4\epsilon_0k_{0}^{2}(k_{x}^{2}+k_{y}^{2})^{2}=0$$

As a function of the conductivity, it can be concluded that different propagating GPs topologies can be generated with different $\sigma _{xx}$ and $\sigma _{yy}$ of graphene, that is, different $k$ vectors of GPs can be obtained. For unstrained graphene, taking $\text {Im}(\sigma _{xx})=\text {Im}(\sigma _{yy})=2.85\times 10^{-4}S$ at $\lambda _0 = 11.6 \mu$m, the calculated $k$-surface of GPs is plotted as the insert red curve shown in Fig. 4(a), where $k_{x} =k_{y}=18.7 k_{0}$, $\lambda _{x} = \lambda _{y}$ = 620 nm, the wavelength of GPs is compressed by 18.7 times, and there is no preferred direction for GPs. When the graphene is streched along the $y$-axis with $\kappa$ = 0.20, the isotropic conductivity of graphene changed to $\text {Im}(\sigma _{xx})=4.43\times 10^{-4}S$ and $\text {Im}(\sigma _{yy})=1.27\times 10^{-4}S$. The corresponding $k$-surface of GPs for the strained graphene turns into an ellipse, as the insert red curve shown in Fig. 4(b). Here the calculated wavevector $k$ and wavelength of GPs towards $x$-axis changed to $k_{x} = 11.8 k_{0}$, $\lambda _{x}$ = 983nm, while $k_{y}= 41.6 k_{0}$, $\lambda _{y}$ = 279 nm. Besides, performing Fourier transform on the electric field distribution Ez simulated in Fig. 3 through $\tilde {E}(k_x,k_y)=\int \int E(x,y)e^{-i(k_xx+k_yy)}dxdy$, the dispersion of GPs can be obtained, it can be found that the contour map coincides very well with the analytic results.

 figure: Fig. 4.

Fig. 4. The corresponding $k$ surface contours of isotropic (a) and anisotropic (b) GPs, which are achieved by the Fourier transform of $Ez$ and the red line insert are the theoretical results calculated by the dispersion relation.

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In the following, the normalized momenta and propagation length of the GPs modes in unstrained and strained graphene were calculated in Fig. 5 for comparison. The electric field of the electromagnetic eigenmode in graphene has the form $\textbf {E}(x,y)\text {exp}(i\beta k_0z)$, where $\textbf {E}(x,y)$ denotes the electric field distribution of GPs, $k_0 = \omega /c$ is the free-space wave vector. $\beta$ is the modal wave vector of GPs, here the real part of $\beta$ refer to the normalized momentum of the GPs mode, which can be written as $k/k_0=\text {Re}(\beta )$, while the imaginary part of $\beta$ provides the propagation length $L$ through $L/\lambda _0=1/[2\pi \cdot \text {Im}(\beta )]$, $L/\lambda _0$ is the called normalized propagation length of plasmons. Then eigensolutions of $\textbf {E}(x,y)$ and $\beta$ have been performed by using the MODE Solutions developed by Lumerical. Take the graphene was stretched along the $y$-axis (i.e. $\theta =90^{\circ }$) by a strain modulus of 0.20 as an example, the normalized momentum $k/k_0$ and normalized propagation length $L/\lambda _0$ in $x$ and $y$-axis as a function of wavelengths were obtained by solving the eigenmode. As shown in Figs. 5(a) and 5(b), the red and blue curves represent for GPs propagating along $x$ and $y$ direction respectively, while the black line represents the unstrained graphene. It can be seen that the normalized wave vector of GPs increases and the normalized propagation length decreases along the direction of tension, while in the direction perpendicularly direction, the normalized wave vector decreases, and the normalized propagation length increases. In addition, the change is consistent in trend for different wavelengths of the emitter. Specifically, the directional control of the propagating GPs in our proposed mechanism is mainly due to the anisotropy in-plane conductivity generated under strain engineering. Compared Fig. 5 with Fig. 2, we can intuitively see that an increase in conductivity leads to a decrease in the normalized wave vector and an increase in the propagation distance of GPs. For example, when the graphene sheet was stretched along the $y$-axis ( i.e. $\theta =90^{\circ }$ ), maximized Im$\sigma _{xx}$ and minimized Im$\sigma _{yy}$ obtained, resulting in the biggest increase of Lx and decrease of Ly. On the other hand, the zigzag direction of graphene can also be determined by studying the propagation property of GPs by stretching graphene in different directions.

 figure: Fig. 5.

Fig. 5. The plasmon modes in unstained and stained graphene. (a) The normalized momenta and (b) the normalized propagation distances of GPs at a different wavelength, the blue and red lines are the cases of the graphene stretched along $y$-axis (i.e. $\theta =90^{\circ }$) by a strain modulus of 0.20, while the black line represents for the unstrained graphene sheet. (c) The normalized momenta and (d) propagation distances of the GPs propagating along $x$-axis as a function of the angle $\theta$ for three different strain modulus at $\lambda _0$ = 11.6 $\mu$m.

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Furthermore, by using nonretarded approximation and electrostatic limit, one can also obtain the analytical dispersion relation of GPs by the formula [24]

$$\beta\simeq{i}\frac{(\varepsilon_1+\varepsilon_2)\omega}{4\pi\sigma}$$
where the open circle lines in Figs. 5(a) and 5(b) represent the results of theoretical calculation, which are in good agreement with the eigensolution results. Next, how the propagating GPs tuned by different stretching directions $\theta$ or various strain modulus $\kappa$ were theoretically calculated in Figs. 5(c) and 5(d). Take the GPs propagating along the $x$-axis as an example, Fig. 5(c) describes the normalized momenta as a function of different angle $\theta$ at a wavelength of 11.6 $\mu$m, the blue and red line represent the graphene stretched along $y$-axis by a strain modulus of 0.10 and 0.20 respectively and $\kappa =0$ for the unstrained case. Besides, the normalized propagation distance as a function of angle $\theta$ at a wavelength of 11.6 $\mu$m was present in Fig. 5(d). One can find that the normalized momenta and propagation distance of GPs change periodically with the angle $\theta$. Therefore, Fig. 5 render, with the same excitation wavelength of emitter near the graphene, the momenta and propagation distance of GPs can be changed by different modulus or stretching directions, that is to say, the properties of propagated GPs can be effectively tuned by strain in multiple dimensions.

To better describe the modulation of the directional GPs by strain stretching, the stretching direction $\theta$ dependent directional ratio $\eta$ of GPs in strained graphene was simulated in 2d polar coordinates, as shown in Fig. 6, the directional ratio was defined as $\eta =L_x/L_y$. Firstly, suppose a strain with a modulus of $\kappa$ = 0.20 was imposed to graphene, the ratio $\eta$ of GPs propagation distance in strained graphene as a function of $\theta$ was illustrated in Fig. 6(a), the red line and open circle represent the directional ratio $\eta$ at two different incident wavelengths of $\lambda$ = 11.6 $\mu$m and 4.6 $\mu$m respectively, while the black line represents the ratio $\eta$ of graphene in unstrained graphene. Two perfectly coincident curves can be observed when the graphene is stretched under the same condition. In other words, the propagating GPs modulated by strain engineering are independent of the excitation wavelengths of an emitter, namely the GPs can be controlled in a wide spectrum. Besides, maximized $\eta$ can obtain when the graphene was stretching along the $y$-axis. What’s more, according to the relation between $\sigma _{xx}$ and $\sigma _{yy}$ illustrated in Fig. 2(c), we can deduce that when the graphene was stretching along the zigzag direction (i.e. $\theta =0^{\circ }$), the propagation distance will increase in the $y$-axis and decreases in the $x$-axis with the same amplitude. Here the ratio $\eta$ should be defined as $\eta =L_x/L_y$, the max ratio $\eta$ will obtain at $\theta =0^{\circ }$, and the maximum value is still as high as 3.5. Then, chose an emitter with a wavelength of 11.6 $\mu$m and impose different modulus of strain on graphene, the propagation distance ratio $\eta$ of GPS as a function of $\theta$ for three different $\kappa$ was studied in Fig. 6(b). It can be found that the propagation ratio $\eta$ of GPs changes under different strain directions and modulus. Therefore, the propagation direction and propagation ratio of GPs can be effectively controlled by the direction and modulus of strain imposed on, so as to achieve multi-dimensional control of GPs.

 figure: Fig. 6.

Fig. 6. The directional ratio $\eta$ of the GPs as a function of the strain directions $\theta$ with different excitation wavelengths (a) and various strain modulus (b).

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It should be noted that all the above calculations are based on self-standing graphene, and it is well known that the GPs depend a lot on the dielectric environment due to their evanescent property. Take PDMS as a flexible substrate of graphene, where the optical properties of PDMS were taken from a Drude-Lorentz oscillator model with 15 oscillators, which is given by

$$\epsilon(\omega)=\epsilon_{\infty}+\sum_{k=1}^{N}[\frac{s_k}{1-(\frac{\omega}{\omega_k})^2-i\Gamma_k(\frac{\omega}{\omega_k})}]$$
where $\epsilon _{\infty }$=2.276, the value of $\omega _k$, $s_k$ and $\Gamma _k$ was taken from Ref. [48]. The PDMS-dependent momentum, propagation distance and direction ratio were shown in Fig. 7. The normalized momenta and propagation distance of GPs with and without PDMS as a substrate were compared at different wavelengths in Figs. 7(a) and 7(b), the blue and red lines are the cases of the graphene stretched along $y$-axis (i.e. $\theta =90^{\circ }$) with a strain modulus of 0.20, while the black line represents for the unstrained graphene sheet. One can see that the effect of the PDMS on GPs propagating at any direction in graphene sheet is synchronous. However, at an incident wavelength from 9 to 15 $\mu$m, the PDMS will greatly reduce the propagation distance of GPs due to the strong absorption of PDMS in this range [49]. Benefits from the effective modulation can be achieved at any wavelength from mid-infrared to terahertz in support of GPs, we can choose a wavelength with low PDMS loss. For example, take the incident wavelength of 4.6 $\mu$m and the strain modulus of $\kappa$=0.20, the directional ratio $\eta$ of the GPs as a function of $\theta$ with and without PDMS were compared in Fig. 7(c). We find that the lines coincide perfectly with whether graphene was placed on PDMS or not, which means that the PDMS does not affect the generation of directional GPs by strain engineering.

 figure: Fig. 7.

Fig. 7. The PDMS-dependent GPs under strain engineering. (a) the normalized momenta and (b) the normalized propagation distances of GPs with (dashed lines) and without(solid line) PDMS as a substrate compared at different wavelengths, the blue and red lines are the cases of the graphene stretched along $y$-axis (i.e. $\theta =90^{\circ }$) by a strain modulus of 0.20, while the black line represents for the unstrained graphene sheet. (c) The directional ratio $\eta$ of GPs as a function of $\theta$ with and without PDMS. (d) The PDMS dependent directional ratio $\eta$ of the GPs with various strain modulus.

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Last but not least, the modulation rate of strain is also estimated, which defined as $d\kappa$ / dt = $v_{g} / L_{0}$. In the formula, $\kappa$ = $(L-L_{0}) / L_{0}$ represents the deformation ratio between the elongation $L-L_{0}$ and the original length $L_{0}$ of graphene, $v_{g}$ has the form of $v_{g}$=$\sqrt {\frac {dY}{3(1-2\rho )s}}$ in graphene, where $d$ = 0.35 nm, Y = 0.95 TPa , $\rho$ = 0.165 are the thickness, Young’s modulus and Poisson’s ratio of graphene, s =0.76 mg/$\mathrm {m^{2}}$ denotes the surface density of graphene sheet [50], so the estimated value of $v_{g}$ is approximately 15 km/s. Therefore, several hundred megahertz modulation rate can be obtained by controlling the GPs through strain engineering.(e.g. $L_{0}$ = 50 $\mathrm {\mu }$m, $d\kappa$ / dt = 300 MHz)

4. Conclusion

In conclusion, by taking advantage of the unique mechanical properties of graphene, we demonstrate numerically that the GPs can be directionally controlled by imposing strain in graphene. Different topologies of GPs excited by a $z$-polarized optical emitter were observed in unstrained and strained graphene due to the anisotropy conductivity generated in graphene under strain engineering. How the directivity of the GPs changes with different excitation wavelengths, strain modulus or strain directions were calculated theoretically to further reveal the parameter dependence of the proposed strain-based GPs modulation mechanism. Our simulation results indicated that not only effective modulation can be achieved at a wide spectrum in support of GPs, but multi-dimensional control of GPs in any direction can be realized. Moreover, when the strain with a modulus of 0.20 was implemented along or perpendicular to the zigzag direction of graphene, a maximized propagation ratio $\eta$ as high as 3.5 was obtained. Considering the specific experiments to be carried out, the PDMS-dependent momentum, propagation distance and direction ratio were investigated extensively, the results prove that the PDMS has no influence on the generation of directional GPs under strain engineering. Therefore, our theoretical results provide detailed guidance for experimental implementation and a new path for the application of integrated photonic devices.

Funding

Research Project of Tianjin Municipal Education Commission (2020KJ088); The Science and Technology Development Fund of Tianjin Education Commission for Higher Education (2020ZD03); Tianjin Technical Expert Project under grant (20YDTPJC01770); Science and Technology Planning Project of Jin Nan District Tianjin under grant (20190111).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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13. H. Kim and B. Lee, “Unidirectional surface plasmon polariton excitation on single slit with oblique backside illumination,” Plasmonics 4(2), 153–159 (2009). [CrossRef]  

14. B. Wang, L. Aigouy, E. Bourhis, J. Gierak, J. P. Hugonin, and P. Lalanne, “Efficient generation of surface plasmon by single-nanoslit illumination under highly oblique incidence,” Appl. Phys. Lett. 94(1), 011114 (2009). [CrossRef]  

15. A. V. Krasavin, T. P. Vo, W. Dickson, P. M. Bolger, and A. V. Zayats, “All-plasmonic modulation via stimulated emission of copropagating surface plasmon polaritons on a substrate with gain,” Nano Lett. 11(6), 2231–2235 (2011). [CrossRef]  

16. J. Chen, Z. Li, S. Yue, and Q. Gong, “Highly efficient all-optical control of surface-plasmon-polariton generation based on a compact asymmetric single slit,” Nano Lett. 11(7), 2933–2937 (2011). [CrossRef]  

17. T. Jie, Y. R. Liu, L. J. Zhang, X. C. Fu, X. M. Xue, G. Qian, Z. Ning, and Z. Tong, “Flexible thermo-optic variable attenuator based on long-range surface plasmon-polariton waveguides,” Micromachines 9(8), 369 (2018). [CrossRef]  

18. Q. Li, C. Li, H. Xu, Z. Liu, and H. Wei, “Photothermal modulation of propagating surface plasmons on silver nanowires,” ACS Photonics 6(8), 2133–2140 (2019). [CrossRef]  

19. P. J. Song, I. S. Jin, H. K. Song, R. C. Song, R. S. Bong, and H. Joachim, “Ultracompact high-contrast magneto-optical disk resonator side-coupled to a plasmonic waveguide and switchable by an external magnetic field,” Phys. Rev. B 98(4), 041406 (2018). [CrossRef]  

20. R. Wang, L. Xu, J. Wang, L. Sun, Y. Jiao, Y. Meng, S. Chen, C. Chang, and C. Fan, “Electric fano resonance-based terahertz metasensors,” Nanoscale 13(44), 18467–18472 (2021). [CrossRef]  

21. J. B. Herzog, M. W. Knight, and D. Natelson, “Thermoplasmonics: quantifying plasmonic heating in single nanowires,” Nano Lett. 14(2), 499–503 (2014). [CrossRef]  

22. Z. J. Coppens, L. Wei, D. G. Walker, and J. G. Valentine, “Probing and controlling photothermal heat generation in plasmonic nanostructures,” Nano Lett. 13(3), 1023–1028 (2013). [CrossRef]  

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

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

25. Z. Ma, C. Wei, W. Lei, Y. Xiang, M. Ren, X. Zhang, and J. Xu, “Unidirectional excitation of graphene plasmons in au-graphene composite structures by a linearly polarized light beam,” Opt. Express 25(5), 4680 (2017). [CrossRef]  

26. B. Zhu, G. Ren, Y. Gao, B. Wu, C. Wan, and S. Jian, “Graphene circular polarization analyzer based on unidirectional excitation of plasmons,” Opt. Express 23(25), 32420 (2015). [CrossRef]  

27. L. Wang, W. Cai, X. Zhang, J. Xu, and Y. Luo, “Directional generation of graphene plasmons by near field interference,” Opt. Express 24(17), 19776 (2016). [CrossRef]  

28. H. Zhu, M. Deng, S. Chen, and L. Chen, “Graphene-based meta-coupler for direction-controllable emission of surface plasmons,” Opt. Lett. 44(13), 3382–3385 (2019). [CrossRef]  

29. H. Zhu, S. Chen, J. Wen, J. Wang, and L. Chen, “Graphene-based metasurfaces for switching polarization states of anomalous reflection and focusing,” Opt. Lett. 44(23), 5764–5767 (2019). [CrossRef]  

30. Z. Ju, M. Deng, J. Wang, and L. Chen, “Reconfigurable multifrequency and wide-angle directional beaming of light from a subwavelength metal slit with graphene metasurfaces,” Opt. Lett. 45(10), 2882–2885 (2020). [CrossRef]  

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

32. L. Xiao, Y. Yi, N. H. Rivera, J. J. Lopez, and M. Soljacic, “All-angle negative refraction of highly squeezed plasmon and phonon polaritons in graphene-boron nitride heterostructures,” Proc. Natl. Acad. Sci. U. S. A. 114(26), 6717–6721 (2017). [CrossRef]  

33. S. Vantasin, Y. Tanaka, and T. Shimura, “Launching and control of graphene plasmons by nanoridge structures,” ACS Photonics 5(3), 1050–1057 (2018). [CrossRef]  

34. 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]  

35. M. Wang, K. Wang, L. Wang, J. Li, and J. Liu, “Graphene-based terahertz tunable plasmonic directional coupler,” Appl. Phys. Lett. 105(8), 081903 (2014). [CrossRef]  

36. F. Liu, C. Qian, and Y. D. Chong, “Directional excitation of graphene surface plasmons,” Opt. Express 23(3), 2383 (2015). [CrossRef]  

37. T. J. Constant, S. M. Hornett, D. E. Chang, and E. Hendry, “All-optical generation of surface plasmons in graphene,” Nat. Phys. 12(2), 124–127 (2016). [CrossRef]  

38. C. Lee, X. Wei, J. W. Kysar, and J. Hone, “Measurement of the elastic properties and intrinsic strength of monolayer graphene,” Science 321(5887), 385–388 (2008). [CrossRef]  

39. M. Huang, H. Yan, C. Chen, D. Song, and H. Tony, “Phonon softening and crystallographic orientation of strained graphene studied by raman spectroscopy,” Proc. Natl. Acad. Sci. U. S. A. 106(18), 7304–7308 (2009). [CrossRef]  

40. K. Cao, S. Feng, Y. Han, L. Gao, and Y. Lu, “Elastic straining of free-standing monolayer graphene,” Nat. Commun. 11(1), 284 (2020). [CrossRef]  

41. F. M. D. Pellegrino, G. G. N. Angilella, and R. Pucci, “Strain effect on the optical conductivity of graphene,” Phys. Rev. B 81(3), 035411 (2010). [CrossRef]  

42. V. M. Pereira, A. H. Castro Neto, and N. M. R. Peres, “Tight-binding approach to uniaxial strain in graphene,” Phys. Rev. B 80(4), 045401 (2009). [CrossRef]  

43. O. L. Blakslee, D. G. Proctor, E. J. Seldin, G. B. Spence, and T. Weng, “Elastic constants of compression–annealed pyrolytic graphite,” J. Appl. Phys. 41(8), 3373–3382 (1970). [CrossRef]  

44. M. Oliva-Leyva and G. N. Gerardo, “Anisotropic ac conductivity of strained graphene,” J. Phys.: Condens. Matter 26(12), 125302 (2014). [CrossRef]  

45. M. Olivaleyva and G. G. Naumis, “Understanding electron behavior in strained graphene as a reciprocal space distortion,” Phys. Rev. B 88(8), 085430 (2013). [CrossRef]  

46. B. Wunsch, T. Stauber, F. Sols, and F. Guinea, “Dynamical polarization of graphene at finite doping,” New J. Phys. 8(12), 318 (2006). [CrossRef]  

47. J. S. Gomez-Diaz, M. Tymchenko, and A. Alu, “Hyperbolic plasmons and topological transitions over uniaxial metasurfaces,” Phys. Rev. Lett. 114(23), 233901 (2015). [CrossRef]  

48. A. Srinivasan, B. Czapla, J. Mayo, and A. Narayanaswamy, “Infrared dielectric function of polydimethylsiloxane and selective emission behavior,” Appl. Phys. Lett. 109(6), 061905 (2016). [CrossRef]  

49. Z. Ma, Z. Chen, J. Xu, W. Li, and L. Wang, “Investigation of a new graphene strain sensor based on surface plasmon resonance,” Sci. Rep. 10(1), 16870 (2020). [CrossRef]  

50. J. Jiang, J. Wang, and B. Li, “Young’s modulus of graphene: a molecular dynamics study,” Phys. Rev. B 80(11), 113405 (2009). [CrossRef]  

References

  • View by:

  1. M. Brongersma, R. Zia, J. Schuler, and A. Chandran, “Plasmonics–the new wave of chipscale technologies!?” in Integrated Photonics Research and Applications/Nanophotonics Technical Digest (CD) (Optical Society of America, 2006), paper NFA1.
  2. R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip-scale technology,” Mater. Today 9(7-8), 20–27 (2006).
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  6. J. Chen, Z. Li, and Q. Gong, “Efficient unidirectional generation of surface plasmon polaritons with asymmetric single-nanoslit,” Appl. Phys. Lett. 97(4), 041113 (2010).
    [Crossref]
  7. X. Huang and M. L. Brongersma, “Compact aperiodic metallic groove arrays for unidirectional launching of surface plasmons,” Nano Lett. 13(11), 5420–5424 (2013).
    [Crossref]
  8. W. Yao, S. Liu, H. Liao, Z. Li, C. Sun, J. Chen, and Q. Gong, “Efficient directional excitation of surface plasmons by a single-element nanoantenna,” Nano Lett. 15(5), 3115–3121 (2015).
    [Crossref]
  9. T. Liu, Y. Shen, W. Shin, Q. Zhu, S. Fan, and C. Jin, “Dislocated double-layer metal gratings: An efficient unidirectional coupler,” Nano Lett. 14(7), 3848–3854 (2014).
    [Crossref]
  10. S. Y. Lee, I. M. Lee, J. Park, S. Oh, W. Lee, K. Y. Kim, and B. Lee, “Role of magnetic induction currents in nanoslit excitation of surface plasmon polaritons,” Phys. Rev. Lett. 108(21), 213907 (2012).
    [Crossref]
  11. F. J. Rodríguez-Fortuño, G. Marino, P. Ginzburg, D. O’Connor, A. Martínez, G. A. Wurtz, and A. V. Zayats, “Near-field interference for the unidirectional excitation of electromagnetic guided modes,” Science 340(6130), 328–330 (2013).
    [Crossref]
  12. S. Y. Lee, W. Lee, Y. Lee, J. Y. Won, J. Kim, I. M. Lee, and B. Lee, “Phase–controlled directional switching of surface plasmon polaritons via beam interference,” Laser Photonics Rev. 7(2), 273–279 (2013).
    [Crossref]
  13. H. Kim and B. Lee, “Unidirectional surface plasmon polariton excitation on single slit with oblique backside illumination,” Plasmonics 4(2), 153–159 (2009).
    [Crossref]
  14. B. Wang, L. Aigouy, E. Bourhis, J. Gierak, J. P. Hugonin, and P. Lalanne, “Efficient generation of surface plasmon by single-nanoslit illumination under highly oblique incidence,” Appl. Phys. Lett. 94(1), 011114 (2009).
    [Crossref]
  15. A. V. Krasavin, T. P. Vo, W. Dickson, P. M. Bolger, and A. V. Zayats, “All-plasmonic modulation via stimulated emission of copropagating surface plasmon polaritons on a substrate with gain,” Nano Lett. 11(6), 2231–2235 (2011).
    [Crossref]
  16. J. Chen, Z. Li, S. Yue, and Q. Gong, “Highly efficient all-optical control of surface-plasmon-polariton generation based on a compact asymmetric single slit,” Nano Lett. 11(7), 2933–2937 (2011).
    [Crossref]
  17. T. Jie, Y. R. Liu, L. J. Zhang, X. C. Fu, X. M. Xue, G. Qian, Z. Ning, and Z. Tong, “Flexible thermo-optic variable attenuator based on long-range surface plasmon-polariton waveguides,” Micromachines 9(8), 369 (2018).
    [Crossref]
  18. Q. Li, C. Li, H. Xu, Z. Liu, and H. Wei, “Photothermal modulation of propagating surface plasmons on silver nanowires,” ACS Photonics 6(8), 2133–2140 (2019).
    [Crossref]
  19. P. J. Song, I. S. Jin, H. K. Song, R. C. Song, R. S. Bong, and H. Joachim, “Ultracompact high-contrast magneto-optical disk resonator side-coupled to a plasmonic waveguide and switchable by an external magnetic field,” Phys. Rev. B 98(4), 041406 (2018).
    [Crossref]
  20. R. Wang, L. Xu, J. Wang, L. Sun, Y. Jiao, Y. Meng, S. Chen, C. Chang, and C. Fan, “Electric fano resonance-based terahertz metasensors,” Nanoscale 13(44), 18467–18472 (2021).
    [Crossref]
  21. J. B. Herzog, M. W. Knight, and D. Natelson, “Thermoplasmonics: quantifying plasmonic heating in single nanowires,” Nano Lett. 14(2), 499–503 (2014).
    [Crossref]
  22. Z. J. Coppens, L. Wei, D. G. Walker, and J. G. Valentine, “Probing and controlling photothermal heat generation in plasmonic nanostructures,” Nano Lett. 13(3), 1023–1028 (2013).
    [Crossref]
  23. A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6(3), 183–191 (2007).
    [Crossref]
  24. F. H. Koppens, D. E. Chang, and F. Javier García de Abajo, “Graphene plasmonics: a platform for strong light-matter interactions,” Nano Lett. 11(8), 3370–3377 (2011).
    [Crossref]
  25. Z. Ma, C. Wei, W. Lei, Y. Xiang, M. Ren, X. Zhang, and J. Xu, “Unidirectional excitation of graphene plasmons in au-graphene composite structures by a linearly polarized light beam,” Opt. Express 25(5), 4680 (2017).
    [Crossref]
  26. B. Zhu, G. Ren, Y. Gao, B. Wu, C. Wan, and S. Jian, “Graphene circular polarization analyzer based on unidirectional excitation of plasmons,” Opt. Express 23(25), 32420 (2015).
    [Crossref]
  27. L. Wang, W. Cai, X. Zhang, J. Xu, and Y. Luo, “Directional generation of graphene plasmons by near field interference,” Opt. Express 24(17), 19776 (2016).
    [Crossref]
  28. H. Zhu, M. Deng, S. Chen, and L. Chen, “Graphene-based meta-coupler for direction-controllable emission of surface plasmons,” Opt. Lett. 44(13), 3382–3385 (2019).
    [Crossref]
  29. H. Zhu, S. Chen, J. Wen, J. Wang, and L. Chen, “Graphene-based metasurfaces for switching polarization states of anomalous reflection and focusing,” Opt. Lett. 44(23), 5764–5767 (2019).
    [Crossref]
  30. Z. Ju, M. Deng, J. Wang, and L. Chen, “Reconfigurable multifrequency and wide-angle directional beaming of light from a subwavelength metal slit with graphene metasurfaces,” Opt. Lett. 45(10), 2882–2885 (2020).
    [Crossref]
  31. A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011).
    [Crossref]
  32. L. Xiao, Y. Yi, N. H. Rivera, J. J. Lopez, and M. Soljacic, “All-angle negative refraction of highly squeezed plasmon and phonon polaritons in graphene-boron nitride heterostructures,” Proc. Natl. Acad. Sci. U. S. A. 114(26), 6717–6721 (2017).
    [Crossref]
  33. S. Vantasin, Y. Tanaka, and T. Shimura, “Launching and control of graphene plasmons by nanoridge structures,” ACS Photonics 5(3), 1050–1057 (2018).
    [Crossref]
  34. 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]
  35. M. Wang, K. Wang, L. Wang, J. Li, and J. Liu, “Graphene-based terahertz tunable plasmonic directional coupler,” Appl. Phys. Lett. 105(8), 081903 (2014).
    [Crossref]
  36. F. Liu, C. Qian, and Y. D. Chong, “Directional excitation of graphene surface plasmons,” Opt. Express 23(3), 2383 (2015).
    [Crossref]
  37. T. J. Constant, S. M. Hornett, D. E. Chang, and E. Hendry, “All-optical generation of surface plasmons in graphene,” Nat. Phys. 12(2), 124–127 (2016).
    [Crossref]
  38. C. Lee, X. Wei, J. W. Kysar, and J. Hone, “Measurement of the elastic properties and intrinsic strength of monolayer graphene,” Science 321(5887), 385–388 (2008).
    [Crossref]
  39. M. Huang, H. Yan, C. Chen, D. Song, and H. Tony, “Phonon softening and crystallographic orientation of strained graphene studied by raman spectroscopy,” Proc. Natl. Acad. Sci. U. S. A. 106(18), 7304–7308 (2009).
    [Crossref]
  40. K. Cao, S. Feng, Y. Han, L. Gao, and Y. Lu, “Elastic straining of free-standing monolayer graphene,” Nat. Commun. 11(1), 284 (2020).
    [Crossref]
  41. F. M. D. Pellegrino, G. G. N. Angilella, and R. Pucci, “Strain effect on the optical conductivity of graphene,” Phys. Rev. B 81(3), 035411 (2010).
    [Crossref]
  42. V. M. Pereira, A. H. Castro Neto, and N. M. R. Peres, “Tight-binding approach to uniaxial strain in graphene,” Phys. Rev. B 80(4), 045401 (2009).
    [Crossref]
  43. O. L. Blakslee, D. G. Proctor, E. J. Seldin, G. B. Spence, and T. Weng, “Elastic constants of compression–annealed pyrolytic graphite,” J. Appl. Phys. 41(8), 3373–3382 (1970).
    [Crossref]
  44. M. Oliva-Leyva and G. N. Gerardo, “Anisotropic ac conductivity of strained graphene,” J. Phys.: Condens. Matter 26(12), 125302 (2014).
    [Crossref]
  45. M. Olivaleyva and G. G. Naumis, “Understanding electron behavior in strained graphene as a reciprocal space distortion,” Phys. Rev. B 88(8), 085430 (2013).
    [Crossref]
  46. B. Wunsch, T. Stauber, F. Sols, and F. Guinea, “Dynamical polarization of graphene at finite doping,” New J. Phys. 8(12), 318 (2006).
    [Crossref]
  47. J. S. Gomez-Diaz, M. Tymchenko, and A. Alu, “Hyperbolic plasmons and topological transitions over uniaxial metasurfaces,” Phys. Rev. Lett. 114(23), 233901 (2015).
    [Crossref]
  48. A. Srinivasan, B. Czapla, J. Mayo, and A. Narayanaswamy, “Infrared dielectric function of polydimethylsiloxane and selective emission behavior,” Appl. Phys. Lett. 109(6), 061905 (2016).
    [Crossref]
  49. Z. Ma, Z. Chen, J. Xu, W. Li, and L. Wang, “Investigation of a new graphene strain sensor based on surface plasmon resonance,” Sci. Rep. 10(1), 16870 (2020).
    [Crossref]
  50. J. Jiang, J. Wang, and B. Li, “Young’s modulus of graphene: a molecular dynamics study,” Phys. Rev. B 80(11), 113405 (2009).
    [Crossref]

2021 (1)

R. Wang, L. Xu, J. Wang, L. Sun, Y. Jiao, Y. Meng, S. Chen, C. Chang, and C. Fan, “Electric fano resonance-based terahertz metasensors,” Nanoscale 13(44), 18467–18472 (2021).
[Crossref]

2020 (3)

Z. Ju, M. Deng, J. Wang, and L. Chen, “Reconfigurable multifrequency and wide-angle directional beaming of light from a subwavelength metal slit with graphene metasurfaces,” Opt. Lett. 45(10), 2882–2885 (2020).
[Crossref]

K. Cao, S. Feng, Y. Han, L. Gao, and Y. Lu, “Elastic straining of free-standing monolayer graphene,” Nat. Commun. 11(1), 284 (2020).
[Crossref]

Z. Ma, Z. Chen, J. Xu, W. Li, and L. Wang, “Investigation of a new graphene strain sensor based on surface plasmon resonance,” Sci. Rep. 10(1), 16870 (2020).
[Crossref]

2019 (3)

2018 (3)

P. J. Song, I. S. Jin, H. K. Song, R. C. Song, R. S. Bong, and H. Joachim, “Ultracompact high-contrast magneto-optical disk resonator side-coupled to a plasmonic waveguide and switchable by an external magnetic field,” Phys. Rev. B 98(4), 041406 (2018).
[Crossref]

T. Jie, Y. R. Liu, L. J. Zhang, X. C. Fu, X. M. Xue, G. Qian, Z. Ning, and Z. Tong, “Flexible thermo-optic variable attenuator based on long-range surface plasmon-polariton waveguides,” Micromachines 9(8), 369 (2018).
[Crossref]

S. Vantasin, Y. Tanaka, and T. Shimura, “Launching and control of graphene plasmons by nanoridge structures,” ACS Photonics 5(3), 1050–1057 (2018).
[Crossref]

2017 (2)

L. Xiao, Y. Yi, N. H. Rivera, J. J. Lopez, and M. Soljacic, “All-angle negative refraction of highly squeezed plasmon and phonon polaritons in graphene-boron nitride heterostructures,” Proc. Natl. Acad. Sci. U. S. A. 114(26), 6717–6721 (2017).
[Crossref]

Z. Ma, C. Wei, W. Lei, Y. Xiang, M. Ren, X. Zhang, and J. Xu, “Unidirectional excitation of graphene plasmons in au-graphene composite structures by a linearly polarized light beam,” Opt. Express 25(5), 4680 (2017).
[Crossref]

2016 (3)

L. Wang, W. Cai, X. Zhang, J. Xu, and Y. Luo, “Directional generation of graphene plasmons by near field interference,” Opt. Express 24(17), 19776 (2016).
[Crossref]

A. Srinivasan, B. Czapla, J. Mayo, and A. Narayanaswamy, “Infrared dielectric function of polydimethylsiloxane and selective emission behavior,” Appl. Phys. Lett. 109(6), 061905 (2016).
[Crossref]

T. J. Constant, S. M. Hornett, D. E. Chang, and E. Hendry, “All-optical generation of surface plasmons in graphene,” Nat. Phys. 12(2), 124–127 (2016).
[Crossref]

2015 (4)

F. Liu, C. Qian, and Y. D. Chong, “Directional excitation of graphene surface plasmons,” Opt. Express 23(3), 2383 (2015).
[Crossref]

J. S. Gomez-Diaz, M. Tymchenko, and A. Alu, “Hyperbolic plasmons and topological transitions over uniaxial metasurfaces,” Phys. Rev. Lett. 114(23), 233901 (2015).
[Crossref]

B. Zhu, G. Ren, Y. Gao, B. Wu, C. Wan, and S. Jian, “Graphene circular polarization analyzer based on unidirectional excitation of plasmons,” Opt. Express 23(25), 32420 (2015).
[Crossref]

W. Yao, S. Liu, H. Liao, Z. Li, C. Sun, J. Chen, and Q. Gong, “Efficient directional excitation of surface plasmons by a single-element nanoantenna,” Nano Lett. 15(5), 3115–3121 (2015).
[Crossref]

2014 (4)

T. Liu, Y. Shen, W. Shin, Q. Zhu, S. Fan, and C. Jin, “Dislocated double-layer metal gratings: An efficient unidirectional coupler,” Nano Lett. 14(7), 3848–3854 (2014).
[Crossref]

J. B. Herzog, M. W. Knight, and D. Natelson, “Thermoplasmonics: quantifying plasmonic heating in single nanowires,” Nano Lett. 14(2), 499–503 (2014).
[Crossref]

M. Wang, K. Wang, L. Wang, J. Li, and J. Liu, “Graphene-based terahertz tunable plasmonic directional coupler,” Appl. Phys. Lett. 105(8), 081903 (2014).
[Crossref]

M. Oliva-Leyva and G. N. Gerardo, “Anisotropic ac conductivity of strained graphene,” J. Phys.: Condens. Matter 26(12), 125302 (2014).
[Crossref]

2013 (5)

M. Olivaleyva and G. G. Naumis, “Understanding electron behavior in strained graphene as a reciprocal space distortion,” Phys. Rev. B 88(8), 085430 (2013).
[Crossref]

Z. J. Coppens, L. Wei, D. G. Walker, and J. G. Valentine, “Probing and controlling photothermal heat generation in plasmonic nanostructures,” Nano Lett. 13(3), 1023–1028 (2013).
[Crossref]

X. Huang and M. L. Brongersma, “Compact aperiodic metallic groove arrays for unidirectional launching of surface plasmons,” Nano Lett. 13(11), 5420–5424 (2013).
[Crossref]

F. J. Rodríguez-Fortuño, G. Marino, P. Ginzburg, D. O’Connor, A. Martínez, G. A. Wurtz, and A. V. Zayats, “Near-field interference for the unidirectional excitation of electromagnetic guided modes,” Science 340(6130), 328–330 (2013).
[Crossref]

S. Y. Lee, W. Lee, Y. Lee, J. Y. Won, J. Kim, I. M. Lee, and B. Lee, “Phase–controlled directional switching of surface plasmon polaritons via beam interference,” Laser Photonics Rev. 7(2), 273–279 (2013).
[Crossref]

2012 (1)

S. Y. Lee, I. M. Lee, J. Park, S. Oh, W. Lee, K. Y. Kim, and B. Lee, “Role of magnetic induction currents in nanoslit excitation of surface plasmon polaritons,” Phys. Rev. Lett. 108(21), 213907 (2012).
[Crossref]

2011 (5)

A. V. Krasavin, T. P. Vo, W. Dickson, P. M. Bolger, and A. V. Zayats, “All-plasmonic modulation via stimulated emission of copropagating surface plasmon polaritons on a substrate with gain,” Nano Lett. 11(6), 2231–2235 (2011).
[Crossref]

J. Chen, Z. Li, S. Yue, and Q. Gong, “Highly efficient all-optical control of surface-plasmon-polariton generation based on a compact asymmetric single slit,” Nano Lett. 11(7), 2933–2937 (2011).
[Crossref]

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

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]

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

2010 (2)

J. Chen, Z. Li, and Q. Gong, “Efficient unidirectional generation of surface plasmon polaritons with asymmetric single-nanoslit,” Appl. Phys. Lett. 97(4), 041113 (2010).
[Crossref]

F. M. D. Pellegrino, G. G. N. Angilella, and R. Pucci, “Strain effect on the optical conductivity of graphene,” Phys. Rev. B 81(3), 035411 (2010).
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2009 (6)

V. M. Pereira, A. H. Castro Neto, and N. M. R. Peres, “Tight-binding approach to uniaxial strain in graphene,” Phys. Rev. B 80(4), 045401 (2009).
[Crossref]

M. Huang, H. Yan, C. Chen, D. Song, and H. Tony, “Phonon softening and crystallographic orientation of strained graphene studied by raman spectroscopy,” Proc. Natl. Acad. Sci. U. S. A. 106(18), 7304–7308 (2009).
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J. Jiang, J. Wang, and B. Li, “Young’s modulus of graphene: a molecular dynamics study,” Phys. Rev. B 80(11), 113405 (2009).
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I. P. Radko, S. I. Bozhevolnyi, G. Brucoli, L. Martín-Moreno, and A. Boltasseva, “Efficient unidirectional ridge excitation of surface plasmons,” Opt. Express 17(9), 7228–7232 (2009).
[Crossref]

H. Kim and B. Lee, “Unidirectional surface plasmon polariton excitation on single slit with oblique backside illumination,” Plasmonics 4(2), 153–159 (2009).
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B. Wang, L. Aigouy, E. Bourhis, J. Gierak, J. P. Hugonin, and P. Lalanne, “Efficient generation of surface plasmon by single-nanoslit illumination under highly oblique incidence,” Appl. Phys. Lett. 94(1), 011114 (2009).
[Crossref]

2008 (1)

C. Lee, X. Wei, J. W. Kysar, and J. Hone, “Measurement of the elastic properties and intrinsic strength of monolayer graphene,” Science 321(5887), 385–388 (2008).
[Crossref]

2007 (2)

F. López-Tejeira, S. G. Rodrigo, L. Martín-Moreno, F. García-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, S. I. Bozhevolnyi, and M. González, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3(5), 324–328 (2007).
[Crossref]

A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6(3), 183–191 (2007).
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2006 (2)

R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip-scale technology,” Mater. Today 9(7-8), 20–27 (2006).
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B. Wunsch, T. Stauber, F. Sols, and F. Guinea, “Dynamical polarization of graphene at finite doping,” New J. Phys. 8(12), 318 (2006).
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1970 (1)

O. L. Blakslee, D. G. Proctor, E. J. Seldin, G. B. Spence, and T. Weng, “Elastic constants of compression–annealed pyrolytic graphite,” J. Appl. Phys. 41(8), 3373–3382 (1970).
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B. Wang, L. Aigouy, E. Bourhis, J. Gierak, J. P. Hugonin, and P. Lalanne, “Efficient generation of surface plasmon by single-nanoslit illumination under highly oblique incidence,” Appl. Phys. Lett. 94(1), 011114 (2009).
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Alu, A.

J. S. Gomez-Diaz, M. Tymchenko, and A. Alu, “Hyperbolic plasmons and topological transitions over uniaxial metasurfaces,” Phys. Rev. Lett. 114(23), 233901 (2015).
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Angilella, G. G. N.

F. M. D. Pellegrino, G. G. N. Angilella, and R. Pucci, “Strain effect on the optical conductivity of graphene,” Phys. Rev. B 81(3), 035411 (2010).
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Blakslee, O. L.

O. L. Blakslee, D. G. Proctor, E. J. Seldin, G. B. Spence, and T. Weng, “Elastic constants of compression–annealed pyrolytic graphite,” J. Appl. Phys. 41(8), 3373–3382 (1970).
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A. V. Krasavin, T. P. Vo, W. Dickson, P. M. Bolger, and A. V. Zayats, “All-plasmonic modulation via stimulated emission of copropagating surface plasmon polaritons on a substrate with gain,” Nano Lett. 11(6), 2231–2235 (2011).
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Boltasseva, A.

Bong, R. S.

P. J. Song, I. S. Jin, H. K. Song, R. C. Song, R. S. Bong, and H. Joachim, “Ultracompact high-contrast magneto-optical disk resonator side-coupled to a plasmonic waveguide and switchable by an external magnetic field,” Phys. Rev. B 98(4), 041406 (2018).
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Bourhis, E.

B. Wang, L. Aigouy, E. Bourhis, J. Gierak, J. P. Hugonin, and P. Lalanne, “Efficient generation of surface plasmon by single-nanoslit illumination under highly oblique incidence,” Appl. Phys. Lett. 94(1), 011114 (2009).
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Bozhevolnyi, S. I.

I. P. Radko, S. I. Bozhevolnyi, G. Brucoli, L. Martín-Moreno, and A. Boltasseva, “Efficient unidirectional ridge excitation of surface plasmons,” Opt. Express 17(9), 7228–7232 (2009).
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F. López-Tejeira, S. G. Rodrigo, L. Martín-Moreno, F. García-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, S. I. Bozhevolnyi, and M. González, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3(5), 324–328 (2007).
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M. Brongersma, R. Zia, J. Schuler, and A. Chandran, “Plasmonics–the new wave of chipscale technologies!?” in Integrated Photonics Research and Applications/Nanophotonics Technical Digest (CD) (Optical Society of America, 2006), paper NFA1.

Brongersma, M. L.

X. Huang and M. L. Brongersma, “Compact aperiodic metallic groove arrays for unidirectional launching of surface plasmons,” Nano Lett. 13(11), 5420–5424 (2013).
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R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip-scale technology,” Mater. Today 9(7-8), 20–27 (2006).
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Brucoli, G.

Cai, W.

Cao, K.

K. Cao, S. Feng, Y. Han, L. Gao, and Y. Lu, “Elastic straining of free-standing monolayer graphene,” Nat. Commun. 11(1), 284 (2020).
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Castro Neto, A. H.

V. M. Pereira, A. H. Castro Neto, and N. M. R. Peres, “Tight-binding approach to uniaxial strain in graphene,” Phys. Rev. B 80(4), 045401 (2009).
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Chandran, A.

R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip-scale technology,” Mater. Today 9(7-8), 20–27 (2006).
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M. Brongersma, R. Zia, J. Schuler, and A. Chandran, “Plasmonics–the new wave of chipscale technologies!?” in Integrated Photonics Research and Applications/Nanophotonics Technical Digest (CD) (Optical Society of America, 2006), paper NFA1.

Chang, C.

R. Wang, L. Xu, J. Wang, L. Sun, Y. Jiao, Y. Meng, S. Chen, C. Chang, and C. Fan, “Electric fano resonance-based terahertz metasensors,” Nanoscale 13(44), 18467–18472 (2021).
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Chang, D. E.

T. J. Constant, S. M. Hornett, D. E. Chang, and E. Hendry, “All-optical generation of surface plasmons in graphene,” Nat. Phys. 12(2), 124–127 (2016).
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F. H. Koppens, D. E. Chang, and F. Javier García de Abajo, “Graphene plasmonics: a platform for strong light-matter interactions,” Nano Lett. 11(8), 3370–3377 (2011).
[Crossref]

Chen, C.

M. Huang, H. Yan, C. Chen, D. Song, and H. Tony, “Phonon softening and crystallographic orientation of strained graphene studied by raman spectroscopy,” Proc. Natl. Acad. Sci. U. S. A. 106(18), 7304–7308 (2009).
[Crossref]

Chen, J.

W. Yao, S. Liu, H. Liao, Z. Li, C. Sun, J. Chen, and Q. Gong, “Efficient directional excitation of surface plasmons by a single-element nanoantenna,” Nano Lett. 15(5), 3115–3121 (2015).
[Crossref]

J. Chen, Z. Li, S. Yue, and Q. Gong, “Highly efficient all-optical control of surface-plasmon-polariton generation based on a compact asymmetric single slit,” Nano Lett. 11(7), 2933–2937 (2011).
[Crossref]

J. Chen, Z. Li, and Q. Gong, “Efficient unidirectional generation of surface plasmon polaritons with asymmetric single-nanoslit,” Appl. Phys. Lett. 97(4), 041113 (2010).
[Crossref]

Chen, L.

Chen, S.

Chen, Z.

Z. Ma, Z. Chen, J. Xu, W. Li, and L. Wang, “Investigation of a new graphene strain sensor based on surface plasmon resonance,” Sci. Rep. 10(1), 16870 (2020).
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Chong, Y. D.

Constant, T. J.

T. J. Constant, S. M. Hornett, D. E. Chang, and E. Hendry, “All-optical generation of surface plasmons in graphene,” Nat. Phys. 12(2), 124–127 (2016).
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Coppens, Z. J.

Z. J. Coppens, L. Wei, D. G. Walker, and J. G. Valentine, “Probing and controlling photothermal heat generation in plasmonic nanostructures,” Nano Lett. 13(3), 1023–1028 (2013).
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Czapla, B.

A. Srinivasan, B. Czapla, J. Mayo, and A. Narayanaswamy, “Infrared dielectric function of polydimethylsiloxane and selective emission behavior,” Appl. Phys. Lett. 109(6), 061905 (2016).
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Deng, M.

Devaux, E.

F. López-Tejeira, S. G. Rodrigo, L. Martín-Moreno, F. García-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, S. I. Bozhevolnyi, and M. González, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3(5), 324–328 (2007).
[Crossref]

Dickson, W.

A. V. Krasavin, T. P. Vo, W. Dickson, P. M. Bolger, and A. V. Zayats, “All-plasmonic modulation via stimulated emission of copropagating surface plasmon polaritons on a substrate with gain,” Nano Lett. 11(6), 2231–2235 (2011).
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Ebbesen, T. W.

F. López-Tejeira, S. G. Rodrigo, L. Martín-Moreno, F. García-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, S. I. Bozhevolnyi, and M. González, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3(5), 324–328 (2007).
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A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011).
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Fan, C.

R. Wang, L. Xu, J. Wang, L. Sun, Y. Jiao, Y. Meng, S. Chen, C. Chang, and C. Fan, “Electric fano resonance-based terahertz metasensors,” Nanoscale 13(44), 18467–18472 (2021).
[Crossref]

Fan, S.

T. Liu, Y. Shen, W. Shin, Q. Zhu, S. Fan, and C. Jin, “Dislocated double-layer metal gratings: An efficient unidirectional coupler,” Nano Lett. 14(7), 3848–3854 (2014).
[Crossref]

Feng, S.

K. Cao, S. Feng, Y. Han, L. Gao, and Y. Lu, “Elastic straining of free-standing monolayer graphene,” Nat. Commun. 11(1), 284 (2020).
[Crossref]

Fu, X. C.

T. Jie, Y. R. Liu, L. J. Zhang, X. C. Fu, X. M. Xue, G. Qian, Z. Ning, and Z. Tong, “Flexible thermo-optic variable attenuator based on long-range surface plasmon-polariton waveguides,” Micromachines 9(8), 369 (2018).
[Crossref]

Gao, L.

K. Cao, S. Feng, Y. Han, L. Gao, and Y. Lu, “Elastic straining of free-standing monolayer graphene,” Nat. Commun. 11(1), 284 (2020).
[Crossref]

Gao, Y.

García-Vidal, F.

F. López-Tejeira, S. G. Rodrigo, L. Martín-Moreno, F. García-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, S. I. Bozhevolnyi, and M. González, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3(5), 324–328 (2007).
[Crossref]

Geim, A. K.

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

Geng, B.

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).
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Gerardo, G. N.

M. Oliva-Leyva and G. N. Gerardo, “Anisotropic ac conductivity of strained graphene,” J. Phys.: Condens. Matter 26(12), 125302 (2014).
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Gierak, J.

B. Wang, L. Aigouy, E. Bourhis, J. Gierak, J. P. Hugonin, and P. Lalanne, “Efficient generation of surface plasmon by single-nanoslit illumination under highly oblique incidence,” Appl. Phys. Lett. 94(1), 011114 (2009).
[Crossref]

Ginzburg, P.

F. J. Rodríguez-Fortuño, G. Marino, P. Ginzburg, D. O’Connor, A. Martínez, G. A. Wurtz, and A. V. Zayats, “Near-field interference for the unidirectional excitation of electromagnetic guided modes,” Science 340(6130), 328–330 (2013).
[Crossref]

Gomez-Diaz, J. S.

J. S. Gomez-Diaz, M. Tymchenko, and A. Alu, “Hyperbolic plasmons and topological transitions over uniaxial metasurfaces,” Phys. Rev. Lett. 114(23), 233901 (2015).
[Crossref]

Gong, Q.

W. Yao, S. Liu, H. Liao, Z. Li, C. Sun, J. Chen, and Q. Gong, “Efficient directional excitation of surface plasmons by a single-element nanoantenna,” Nano Lett. 15(5), 3115–3121 (2015).
[Crossref]

J. Chen, Z. Li, S. Yue, and Q. Gong, “Highly efficient all-optical control of surface-plasmon-polariton generation based on a compact asymmetric single slit,” Nano Lett. 11(7), 2933–2937 (2011).
[Crossref]

J. Chen, Z. Li, and Q. Gong, “Efficient unidirectional generation of surface plasmon polaritons with asymmetric single-nanoslit,” Appl. Phys. Lett. 97(4), 041113 (2010).
[Crossref]

González, M.

F. López-Tejeira, S. G. Rodrigo, L. Martín-Moreno, F. García-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, S. I. Bozhevolnyi, and M. González, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3(5), 324–328 (2007).
[Crossref]

Guinea, F.

B. Wunsch, T. Stauber, F. Sols, and F. Guinea, “Dynamical polarization of graphene at finite doping,” New J. Phys. 8(12), 318 (2006).
[Crossref]

Han, Y.

K. Cao, S. Feng, Y. Han, L. Gao, and Y. Lu, “Elastic straining of free-standing monolayer graphene,” Nat. Commun. 11(1), 284 (2020).
[Crossref]

Hendry, E.

T. J. Constant, S. M. Hornett, D. E. Chang, and E. Hendry, “All-optical generation of surface plasmons in graphene,” Nat. Phys. 12(2), 124–127 (2016).
[Crossref]

Herzog, J. B.

J. B. Herzog, M. W. Knight, and D. Natelson, “Thermoplasmonics: quantifying plasmonic heating in single nanowires,” Nano Lett. 14(2), 499–503 (2014).
[Crossref]

Hone, J.

C. Lee, X. Wei, J. W. Kysar, and J. Hone, “Measurement of the elastic properties and intrinsic strength of monolayer graphene,” Science 321(5887), 385–388 (2008).
[Crossref]

Hornett, S. M.

T. J. Constant, S. M. Hornett, D. E. Chang, and E. Hendry, “All-optical generation of surface plasmons in graphene,” Nat. Phys. 12(2), 124–127 (2016).
[Crossref]

Huang, M.

M. Huang, H. Yan, C. Chen, D. Song, and H. Tony, “Phonon softening and crystallographic orientation of strained graphene studied by raman spectroscopy,” Proc. Natl. Acad. Sci. U. S. A. 106(18), 7304–7308 (2009).
[Crossref]

Huang, X.

X. Huang and M. L. Brongersma, “Compact aperiodic metallic groove arrays for unidirectional launching of surface plasmons,” Nano Lett. 13(11), 5420–5424 (2013).
[Crossref]

Hugonin, J. P.

B. Wang, L. Aigouy, E. Bourhis, J. Gierak, J. P. Hugonin, and P. Lalanne, “Efficient generation of surface plasmon by single-nanoslit illumination under highly oblique incidence,” Appl. Phys. Lett. 94(1), 011114 (2009).
[Crossref]

Javier García de Abajo, F.

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

Jian, S.

Jiang, J.

J. Jiang, J. Wang, and B. Li, “Young’s modulus of graphene: a molecular dynamics study,” Phys. Rev. B 80(11), 113405 (2009).
[Crossref]

Jiao, Y.

R. Wang, L. Xu, J. Wang, L. Sun, Y. Jiao, Y. Meng, S. Chen, C. Chang, and C. Fan, “Electric fano resonance-based terahertz metasensors,” Nanoscale 13(44), 18467–18472 (2021).
[Crossref]

Jie, T.

T. Jie, Y. R. Liu, L. J. Zhang, X. C. Fu, X. M. Xue, G. Qian, Z. Ning, and Z. Tong, “Flexible thermo-optic variable attenuator based on long-range surface plasmon-polariton waveguides,” Micromachines 9(8), 369 (2018).
[Crossref]

Jin, C.

T. Liu, Y. Shen, W. Shin, Q. Zhu, S. Fan, and C. Jin, “Dislocated double-layer metal gratings: An efficient unidirectional coupler,” Nano Lett. 14(7), 3848–3854 (2014).
[Crossref]

Jin, I. S.

P. J. Song, I. S. Jin, H. K. Song, R. C. Song, R. S. Bong, and H. Joachim, “Ultracompact high-contrast magneto-optical disk resonator side-coupled to a plasmonic waveguide and switchable by an external magnetic field,” Phys. Rev. B 98(4), 041406 (2018).
[Crossref]

Joachim, H.

P. J. Song, I. S. Jin, H. K. Song, R. C. Song, R. S. Bong, and H. Joachim, “Ultracompact high-contrast magneto-optical disk resonator side-coupled to a plasmonic waveguide and switchable by an external magnetic field,” Phys. Rev. B 98(4), 041406 (2018).
[Crossref]

Ju, L.

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]

Ju, Z.

Kim, H.

H. Kim and B. Lee, “Unidirectional surface plasmon polariton excitation on single slit with oblique backside illumination,” Plasmonics 4(2), 153–159 (2009).
[Crossref]

Kim, J.

S. Y. Lee, W. Lee, Y. Lee, J. Y. Won, J. Kim, I. M. Lee, and B. Lee, “Phase–controlled directional switching of surface plasmon polaritons via beam interference,” Laser Photonics Rev. 7(2), 273–279 (2013).
[Crossref]

Kim, K. Y.

S. Y. Lee, I. M. Lee, J. Park, S. Oh, W. Lee, K. Y. Kim, and B. Lee, “Role of magnetic induction currents in nanoslit excitation of surface plasmon polaritons,” Phys. Rev. Lett. 108(21), 213907 (2012).
[Crossref]

Knight, M. W.

J. B. Herzog, M. W. Knight, and D. Natelson, “Thermoplasmonics: quantifying plasmonic heating in single nanowires,” Nano Lett. 14(2), 499–503 (2014).
[Crossref]

Koppens, F. H.

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

Krasavin, A. V.

A. V. Krasavin, T. P. Vo, W. Dickson, P. M. Bolger, and A. V. Zayats, “All-plasmonic modulation via stimulated emission of copropagating surface plasmon polaritons on a substrate with gain,” Nano Lett. 11(6), 2231–2235 (2011).
[Crossref]

Krenn, J. R.

F. López-Tejeira, S. G. Rodrigo, L. Martín-Moreno, F. García-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, S. I. Bozhevolnyi, and M. González, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3(5), 324–328 (2007).
[Crossref]

Kysar, J. W.

C. Lee, X. Wei, J. W. Kysar, and J. Hone, “Measurement of the elastic properties and intrinsic strength of monolayer graphene,” Science 321(5887), 385–388 (2008).
[Crossref]

Lalanne, P.

B. Wang, L. Aigouy, E. Bourhis, J. Gierak, J. P. Hugonin, and P. Lalanne, “Efficient generation of surface plasmon by single-nanoslit illumination under highly oblique incidence,” Appl. Phys. Lett. 94(1), 011114 (2009).
[Crossref]

Lee, B.

S. Y. Lee, W. Lee, Y. Lee, J. Y. Won, J. Kim, I. M. Lee, and B. Lee, “Phase–controlled directional switching of surface plasmon polaritons via beam interference,” Laser Photonics Rev. 7(2), 273–279 (2013).
[Crossref]

S. Y. Lee, I. M. Lee, J. Park, S. Oh, W. Lee, K. Y. Kim, and B. Lee, “Role of magnetic induction currents in nanoslit excitation of surface plasmon polaritons,” Phys. Rev. Lett. 108(21), 213907 (2012).
[Crossref]

H. Kim and B. Lee, “Unidirectional surface plasmon polariton excitation on single slit with oblique backside illumination,” Plasmonics 4(2), 153–159 (2009).
[Crossref]

Lee, C.

C. Lee, X. Wei, J. W. Kysar, and J. Hone, “Measurement of the elastic properties and intrinsic strength of monolayer graphene,” Science 321(5887), 385–388 (2008).
[Crossref]

Lee, I. M.

S. Y. Lee, W. Lee, Y. Lee, J. Y. Won, J. Kim, I. M. Lee, and B. Lee, “Phase–controlled directional switching of surface plasmon polaritons via beam interference,” Laser Photonics Rev. 7(2), 273–279 (2013).
[Crossref]

S. Y. Lee, I. M. Lee, J. Park, S. Oh, W. Lee, K. Y. Kim, and B. Lee, “Role of magnetic induction currents in nanoslit excitation of surface plasmon polaritons,” Phys. Rev. Lett. 108(21), 213907 (2012).
[Crossref]

Lee, S. Y.

S. Y. Lee, W. Lee, Y. Lee, J. Y. Won, J. Kim, I. M. Lee, and B. Lee, “Phase–controlled directional switching of surface plasmon polaritons via beam interference,” Laser Photonics Rev. 7(2), 273–279 (2013).
[Crossref]

S. Y. Lee, I. M. Lee, J. Park, S. Oh, W. Lee, K. Y. Kim, and B. Lee, “Role of magnetic induction currents in nanoslit excitation of surface plasmon polaritons,” Phys. Rev. Lett. 108(21), 213907 (2012).
[Crossref]

Lee, W.

S. Y. Lee, W. Lee, Y. Lee, J. Y. Won, J. Kim, I. M. Lee, and B. Lee, “Phase–controlled directional switching of surface plasmon polaritons via beam interference,” Laser Photonics Rev. 7(2), 273–279 (2013).
[Crossref]

S. Y. Lee, I. M. Lee, J. Park, S. Oh, W. Lee, K. Y. Kim, and B. Lee, “Role of magnetic induction currents in nanoslit excitation of surface plasmon polaritons,” Phys. Rev. Lett. 108(21), 213907 (2012).
[Crossref]

Lee, Y.

S. Y. Lee, W. Lee, Y. Lee, J. Y. Won, J. Kim, I. M. Lee, and B. Lee, “Phase–controlled directional switching of surface plasmon polaritons via beam interference,” Laser Photonics Rev. 7(2), 273–279 (2013).
[Crossref]

Lei, W.

Li, B.

J. Jiang, J. Wang, and B. Li, “Young’s modulus of graphene: a molecular dynamics study,” Phys. Rev. B 80(11), 113405 (2009).
[Crossref]

Li, C.

Q. Li, C. Li, H. Xu, Z. Liu, and H. Wei, “Photothermal modulation of propagating surface plasmons on silver nanowires,” ACS Photonics 6(8), 2133–2140 (2019).
[Crossref]

Li, J.

M. Wang, K. Wang, L. Wang, J. Li, and J. Liu, “Graphene-based terahertz tunable plasmonic directional coupler,” Appl. Phys. Lett. 105(8), 081903 (2014).
[Crossref]

Li, Q.

Q. Li, C. Li, H. Xu, Z. Liu, and H. Wei, “Photothermal modulation of propagating surface plasmons on silver nanowires,” ACS Photonics 6(8), 2133–2140 (2019).
[Crossref]

Li, W.

Z. Ma, Z. Chen, J. Xu, W. Li, and L. Wang, “Investigation of a new graphene strain sensor based on surface plasmon resonance,” Sci. Rep. 10(1), 16870 (2020).
[Crossref]

Li, Z.

W. Yao, S. Liu, H. Liao, Z. Li, C. Sun, J. Chen, and Q. Gong, “Efficient directional excitation of surface plasmons by a single-element nanoantenna,” Nano Lett. 15(5), 3115–3121 (2015).
[Crossref]

J. Chen, Z. Li, S. Yue, and Q. Gong, “Highly efficient all-optical control of surface-plasmon-polariton generation based on a compact asymmetric single slit,” Nano Lett. 11(7), 2933–2937 (2011).
[Crossref]

J. Chen, Z. Li, and Q. Gong, “Efficient unidirectional generation of surface plasmon polaritons with asymmetric single-nanoslit,” Appl. Phys. Lett. 97(4), 041113 (2010).
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M. Brongersma, R. Zia, J. Schuler, and A. Chandran, “Plasmonics–the new wave of chipscale technologies!?” in Integrated Photonics Research and Applications/Nanophotonics Technical Digest (CD) (Optical Society of America, 2006), paper NFA1.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (7)

Fig. 1.
Fig. 1. Schematic diagram of strain acting on graphene. The graphene is uniformly stretched along a prescribed direction, where the angle between the tensile direction and the zigzag direction of the honeycomb lattice of graphene is $\theta$, here the zigzag direction was chosen along the $x$ axis in the Cartesian system. GPs were excited in the strained graphene for the case of tension T perpendicular to the zigzag direction ($x$-axis) by a $z$-oriented emitter located at $z_{0}$ = 10 nm above the surface.
Fig. 2.
Fig. 2. The anisotropic optical properity of graphene under strain engineering. The real and imaginary part of in-plane graphene conductivity (a) and permittivity (b) with a strain modulus of $\kappa$=0.20 for the case of tension T perpendicular to the zigzag direction (i.e. $\theta =90^{\circ }$). While the black line represents the conductivity $\sigma _{iso}$ and permittivity $\epsilon _{iso}$ without strain engineering. The change of imaginary part of conductivity (c) and the real part of permittivity (d) as a function of angle $\theta$ with a strain modulus of $\kappa$ = 0.20 at $\lambda _0$ = 11.6 $\mu$m.
Fig. 3.
Fig. 3. The $E_z$ field and electric intensity $|E|$ of propagating GPs excited by a $z$-oriented optical emitter located at a distance $z_{0}$=10 nm above the unstrained (a, c) and strained (b, d) graphene for a wavelength of 11.6 $\mu$m at $E_{F}$ = 0.4 eV. Here the graphene was stretched along $Oy$ axis (i.e.$\theta = 90^{\circ }$) with a strain modulus of $\kappa$ = 0.20.
Fig. 4.
Fig. 4. The corresponding $k$ surface contours of isotropic (a) and anisotropic (b) GPs, which are achieved by the Fourier transform of $Ez$ and the red line insert are the theoretical results calculated by the dispersion relation.
Fig. 5.
Fig. 5. The plasmon modes in unstained and stained graphene. (a) The normalized momenta and (b) the normalized propagation distances of GPs at a different wavelength, the blue and red lines are the cases of the graphene stretched along $y$-axis (i.e. $\theta =90^{\circ }$) by a strain modulus of 0.20, while the black line represents for the unstrained graphene sheet. (c) The normalized momenta and (d) propagation distances of the GPs propagating along $x$-axis as a function of the angle $\theta$ for three different strain modulus at $\lambda _0$ = 11.6 $\mu$m.
Fig. 6.
Fig. 6. The directional ratio $\eta$ of the GPs as a function of the strain directions $\theta$ with different excitation wavelengths (a) and various strain modulus (b).
Fig. 7.
Fig. 7. The PDMS-dependent GPs under strain engineering. (a) the normalized momenta and (b) the normalized propagation distances of GPs with (dashed lines) and without(solid line) PDMS as a substrate compared at different wavelengths, the blue and red lines are the cases of the graphene stretched along $y$-axis (i.e. $\theta =90^{\circ }$) by a strain modulus of 0.20, while the black line represents for the unstrained graphene sheet. (c) The directional ratio $\eta$ of GPs as a function of $\theta$ with and without PDMS. (d) The PDMS dependent directional ratio $\eta$ of the GPs with various strain modulus.

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

κ ¯ = κ ( cos 2 θ ρ sin 2 θ ( 1 + ρ ) cos θ sin θ ( 1 + ρ ) cos θ sin θ sin 2 θ ρ cos 2 θ )
σ ~ ( ω ) = σ ( ω ) ( I ¯ 2 γ κ ¯ + γ Tr ( κ ¯ ) )
σ ( ω ) = 2 e 2 T π i ω + i τ 1 log [ 2 cosh ( E F 2 K B T ) ] + e 2 4 [ H ( ω / 2 ) + 4 i ω π 0 d ε H ( ε ) H ( ω / 2 ) ω 2 4 ε 2 ] ,
H ( ε ) = sinh ( ε / k B T ) cosh ( E F / k B T ) + cosh ( ε / k B T ) .
μ 0 ( k x 2 Im ( σ x x ) + k y 2 Im ( σ y y ) ) 2 ( k x 2 + k y 2 k 0 2 ) 4 ϵ 0 k 0 2 ( k x 2 + k y 2 ) 2 = 0
β i ( ε 1 + ε 2 ) ω 4 π σ
ϵ ( ω ) = ϵ + k = 1 N [ s k 1 ( ω ω k ) 2 i Γ k ( ω ω k ) ]

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