Goos-H&#x00E4;nchen shifts for Airy beams impinging on graphene-substrate surfaces

Weiming Zhen; Dongmei Deng

doi:10.1364/OE.400939

1. Introduction

The Airy wave packet, a solution to the the potential-free Schrödinger equation, was predicted for the first time theoretically by Balazs and Berry in 1979 [1], which carries infinite energy and thus is not realizable physically. The first finite energy Airy beams were reported theoretically and verified experimentally by Siviloglou and Chiristodoulides in 2007 [2,3]. Airy beams have been extensively applied in various areas, such as optical routing [4], optical cleaning of the micro particles [5], light bullet and soliton pairs generation [6,7], bio-optical tweezers [8], vacuum electron acceleration [9,10] and so on, owing to their novel and unique properties such as self-acceleration [2,3], self-healing [11], and nondiffraction [12].

On the other hand, graphene, a one-atom-thick layer of carbon atoms with a hexagonal honeycomb lattice [13,14], has attracted tremendous interest, due to its unique optical and electronic properties [15,16], and its broad applications spanning from Klein tunneling [17], tunable plasmonic devices [18], polarizers [19], and modulators [20]. Moreover, the Fermi energy of graphene can be adjusted by varying the gate voltage or the doping concentration [21].

In geometric optics, as we all know, reflection and refraction of light on an interface are strictly governed by the Fresnel equations and Snell’s law [22]. When the wave properties of the physical beam are taken into account, the nonsecular reflection phenomena will occur and be observed, the most important ones are the Goos-Hänchen (GH) shift [23] and Imbert-Fedorov (IF) shifts [24], which occur in the plane of incidence and the plane normal to the plane of incidence, respectively. In addition, there have been lots of investigations about the photonic spin Hall effect (SHE), which is also the IF shift connected with the law of conservation of angular momentum, in the past few years [25–28]. These beam shifts have been studied for different kinds of incident beams, including Gaussian beams [29–34], Hermite-Gaussian beams [35–37], Laguerre-Gaussian beams [38,39], nondiffracting Bessel beams [40], Airy beams [41–44], and vortex beams [45,46]. Meanwhile, there have been numerous intriguing materials as reflection surfaces to investigate beam shifts as well, such as graphene [32,37,47–61], weakly absorbing media [62–64], photonic crystals [65–67], ferromagnets [68], and epsilon near-zero materials [69–71]. Importantly, these beam shifts have many practical applications in physics. Using the quantum weak measurement techniques, the GH shift and the photonic SHE hold great potential for precision metrology, such as measuring precisely the optical conductivity of graphene [56] and identifying the layer of graphene [57,58] as well as topological phase transitions [59]. Moreover, unique applications of the GH shift in optical differential operation and image edge detection have been reported recently [60]. However, the GH shift value always has the same order of magnitude as the incident wavelength, which is too extremely small to be easily observed and measured in experiment. Thus, the weak measurements have been introduced, which can enhance the magnitude of shift effectively [57,58,72–74]. Furthermore, giant GH shifts have been produced in graphene theoretically and experimentally. Cheng et al. have predicted a huge GH shift with a graphene layer at the excited surface plasmon resonance [48]. Li et al. have presented experimentally a giant GH shift using a beam splitter scanning method [49]. In addition, Airy beams can also raise the magnitude of GH shift, which is clearly seen in the analytical solution of the shift derived by Gao et al. [44], compared to the case of Gaussian beams. Whether the GH shifts are greatly enhanced for Airy beams impinging on graphene coated surfaces? There is still no any investigation on it, where some intriguing phenomena may occur.

The purpose of this paper is to study theoretically the spatial GH shift (GHS) and the angular one (GHA) for Airy beams impinging on weakly absorbing media coated with the monolayer graphene. The dependence of the GH shift on the incident angle, the incident wavelength, the Fermi energy, and the decay factors of Airy beams is investigated. A huge GH shift which is up to three orders of wavelengths is obtained.

This paper is organized as follow. In Sec. 2, we present the analytical expressions of the GHS and the GHA for Airy beams and the Fresnel reflection coefficients in the graphene-substrate system. The results and discussions of the simulations are presented in Sec. 3. We conclude some useful results in Sec. 4.

2. Theoretical model

To start our analysis, we consider a monochromatic, paraxial electric field impinging on a weakly absorbing semi-infinite medium coated with the monolayer graphene, as illustrated in Fig. 1. $\varepsilon _{1}=1$ and $\varepsilon _{2}=\varepsilon _{2r}+\varepsilon _{2i}i$ represent the relative permittivity of air and the absorbing medium. The laboratory frame $\{\hat {\mathbf {x}},\hat {\mathbf {y}},\hat {\mathbf {z}}\}$ is established at the graphene-coated surface, where the $\hat {\mathbf {z}}$ axis is normal to the interface pointing to the absorbing medium. $\{\hat {\mathbf {x}}_i,\hat {\mathbf {y}}_i,\hat {\mathbf {z}}_i\}$ and $\{\hat {\mathbf {x}}_r,\hat {\mathbf {y}}_r,\hat {\mathbf {z}}_r\}$ denote the incident and the reflected field coordinates, respectively.

The electric field intensity of position $\mathbf {r}$ for the incident and the reflected beams can be expressed as [75]:

(1)$$\textbf{E}_\mu(\textbf{r}_\mu)=\frac{1}{2\pi}{\iint}\widetilde{\textbf E}_\mu(\textbf{k}_\mu)e^{i(\textbf{k}_\mu{\cdot}\textbf{r}_\mu)}dk_{x_{\mu}}dk_{y_{\mu}},$$

where the subscript $\mu =I,R$ denotes the incident or the reflected coordinate system, $\textbf {k}_I=k_{x_I} \hat {\mathbf {x}} +k_{y_I} \hat {\mathbf {y}} +k_{z_I} \hat {\mathbf {z}}$ represents the wave vector of the incident coordinate system. Meanwhile, $U=k_{x_I}/k_1$, $V=k_{y_I}/k_1$, and $W=k_{z_I}/k_1$, the dimensionless components of the $\mathbf {k}_I$ are introduced in the incident frame, where $k_1=|\textbf {k}_I|=2 \pi / \lambda$ with the wavelength $\lambda$ of incident beams, and let $\hat {\mathbf {X}}_I=k_1 \hat {\mathbf {x}}_I, \hat {\mathbf {Y}}_I=k_1 \hat {\mathbf {y}}_I, \hat {\mathbf {Z}}_I=k_1 \hat {\mathbf {z}}_I$. According to the law of specular reflection [76], the relationship between $\hat {\mathbf {k}}_I$ and $\hat {\mathbf {k}}_R$ is $\hat {\mathbf {k}}_R=\hat {\mathbf {k}}_I-2 \hat {\mathbf {z}} (\hat {\mathbf {z}} \cdot \hat {\mathbf {k}}_I)$= $k_1(-U\hat {\mathbf {x}}_R+V\hat {\mathbf {y}}_R+ W\hat {\mathbf {z}}_R)$. Thus, the electric fields of the incident and the reflected beams can be described as

(2)$$\textbf{E}_I(k_1\textbf{r}_I)=\frac{k_1^2}{2\pi}{\iint}\widetilde{\textbf E}_I(U,V;\theta)e^{i(U\hat{\textbf X}_I+V\hat{\textbf Y}_I+W\hat{\textbf Z}_I)}dUdV,$$

(3)$$\textbf{E}_R(k_1\textbf{r}_R)=\frac{k_1^2}{2\pi}{\iint}\widetilde{\textbf E}_R(U,V;\theta)e^{i(-U\hat{\textbf X}_R+V\hat{\textbf Y}_R+W\hat{\textbf Z}_R)}dUdV,$$

whose the angular spectrums can be written as

(4)$$\widetilde{\textbf E}_\mu(U,V;\theta)=\sum_{\lambda}^{p,s}\hat{\textbf e}_\lambda(\textbf{k}_\mu)\alpha_\lambda(U,V;\theta)\widetilde{\textbf A}_{\mu}(U,V;\theta),$$

in which $\lambda =p,s$ denote p or s polarization, and $\hat {\mathbf {e}}_\lambda (\mathbf {k}_\mu )$ represents the polarization unit basis vectors written as

(5)$$\hat{\mathbf{e}}_p(\mathbf{k}_\mu)=\frac{\hat{\mathbf{e}}_s(\mathbf{k}_\mu){\times}\mathbf{k}_\mu}{|\hat{\mathbf{e}}_s(\mathbf{k}_\mu){\times}\mathbf {k}_\mu|},$$

(6)$$\hat{\mathbf{e}}_s(\mathbf {k}_\mu)=\frac{\hat{\mathbf{z}}{\times}\mathbf{k}_\mu}{|\hat{\mathbf{z}}{\times}\mathbf{k}_\mu|}.$$

The polarized vector spectral amplitudes $\alpha _\lambda (U,V;\theta )=\hat {\mathbf {e}}_\lambda (\mathbf {k}_I) \cdot \hat {f}$, where $\hat {f}=f_p\hat {\mathbf {x}}_I+f_s\hat {\mathbf {y}}_I$, and $f_p=a_p$, $f_s=a_se^{i\eta }$ with the phase difference $\eta$ between p and s polarization. The initial incident and the reflected angular spectrums are $\widetilde {A}_I(U,V;\theta )=\widetilde {A}(U,V;0)$, and $\widetilde {A}_R(U,V;\theta )=r_\lambda (U,V;\theta )\widetilde {A}(U,V;Z_I^0)$, respectively, where $Z_I^0$ corresponds to the transmission distance before reflecting along $Z_I$, and $r_\lambda (U,V;\theta )=R_\lambda e^{i\phi _\lambda }$ is the Fresnel reflection coefficients with the modulus value $R_\lambda$ as well as the phase $\phi _\lambda$. Using Taylor expansion method at $U=0, V=0$, the Fresnel reflection coefficients can also be expressed as follow [22]:

(7)$$r_\lambda(U,V;\theta){\simeq}r_\lambda+Ur^{\prime}_\lambda+\frac{1}{2}U^2r^{\prime\prime}_\lambda+\frac{1}{2}V^2r^{\prime}_\lambda,$$

where $r_\lambda^{\prime}$ and $r_\lambda^{\prime\prime}$ represent the first and the second order differentials of ordinary reflection coefficients to the incident angle $\theta$ respectively. In this case, the angular spectrum of the finite energy Airy beams is expressed as

(8)$$\widetilde{A}(U,V)=\frac{w_0^2}{2\pi}\mathrm{exp}(\frac{\alpha^3+\beta^3}{3})\mathrm{exp}(-\frac{\alpha U^2+\beta V^2}{\vartheta^2}) \mathrm{exp}[i(\frac{U^3+V^3}{3\vartheta^3}-\frac{\alpha^2U+\beta^2V}{\vartheta})],$$

where $\alpha$ and $\beta$ are the decay factors, $\vartheta =1/(k_1w_0)$, and $w_0$ is the length parameter of Airy beams.

Fig. 1. Schematic plot of the GH shift at the graphene-coated surface between air ($z<0$) and a semi-infinite weakly absorbing medium ($z>0$). The single graphene layer (characterized by its optical conductivity $\sigma$) is located on the interface at $z = 0$. $\varepsilon _{1}=1$ and $\varepsilon _{2}=\varepsilon _{2r}+\varepsilon _{2i}i$ represent the relative permittivity of air and the absorbing medium, respectively.

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According to the definition of the GH shift in Refs. [44,77], the shift can be written as

(9)$$\bar{X}=\bar{X}_R-\bar{X}_I,$$

where $\bar {X}_R$ and $\bar {X}_I$ can be expressed in the Fourier form:

(10)$$\bar X_R=\frac{{\iint}\mathrm{Im}[\widetilde{\mathbf{E}}_R^*\frac{\partial}{\partial U}\widetilde{\mathbf{E}}_R]dUdV}{{\iint}|\widetilde{\mathbf{E}}_R|^2dUdV}-(Z_R+Z_I)\frac{{\iint}\frac{U}{W}{|\widetilde{\mathbf{E}}_R|^2dUdV}}{{\iint}|\widetilde{\mathbf{E}}_R|^2dUdV},$$

(11)$$\bar X_I=\frac{-{\iint}\mathrm{Im}[\widetilde{\mathbf{E}}_I^*\frac{\partial}{\partial U}\widetilde{\mathbf{E}}_I]dUdV}{{\iint}|\widetilde{\mathbf{E}}_I|^2dUdV}-Z_I\frac{{\iint}\frac{U}{W}{|\widetilde{\mathbf{E}}_I|^2dUdV}}{{\iint}|\widetilde{\mathbf{E}}_I|^2dUdV}.$$

The GHS and the GHA can be written separately as follows:

(12)$$\Delta_{\mathrm{GH}}=\frac{{\iint}\mathrm{Im}[\widetilde{\mathbf{E}}_R^*\frac{\partial}{\partial{U}}\widetilde{\mathbf{E}}_R]dUdV}{k_1{\iint}|\widetilde{\mathbf{E}}_R|^2dUdV} +\frac{{\iint}\mathrm{Im}[\widetilde{\mathbf{E}}_I^*\frac{\partial}{\partial V}\widetilde{\mathbf{E}}_I]dUdV}{k_1{\iint}|\widetilde{\mathbf{E}}_I|^2dUdV},$$

(13)$$\Theta_{\mathrm{GH}}=\partial\bar{X}_R/\partial{Z_R}=-\frac{{\iint}\frac{U}{W}{|\widetilde{\mathbf{E}}_R|^2dUdV}}{{\iint}|\widetilde{\mathbf{E}}_R|^2dUdV}.$$

After straightforward calculations, the GHS and the GHA can be modified as [44]

(14)$$\Delta_{\mathrm{GH}}^\mathrm{Airy}=\frac{1}{\Lambda}(\Delta_{\mathrm{GH}}^g+\frac{\vartheta}{8\alpha^2k_1}\sum_{\lambda}^{p,s}\omega_{\lambda_{1}}),$$

(15)$$\Theta_{\mathrm{GH}}^\mathrm{Airy}=\frac{1}{4\alpha\Lambda}\Theta_{\mathrm{GH}}^g,$$

where $\Delta _{\mathrm {GH}}^g=\omega _p\partial \phi _p/\partial \theta +\omega _s\partial \phi _s/\partial \theta$ is the GHS for Gaussian beams, with the fractional energy in each polarization $w_\lambda =a_\lambda ^2R_\lambda ^2/(a_p^2R_p^2+a_s^2R_s^2)$, $\Lambda =1+\frac {\vartheta ^2}{4\alpha }\sum _{\lambda }^{p,s}\omega _{\lambda _{1}}+\frac {\vartheta ^2}{4\beta }\sum _{\lambda }^{p,s}\omega _{\lambda _{2}}$ with $\omega _{\lambda _{1}}=a_\lambda ^2(R_\lambda ^2+\mathrm {Re}[r^{\prime\prime}_\lambda r^*_\lambda ])/(a_p^2R_p^2+a_s^2R_s^2)$ as well as $\omega _{\lambda _2}=a_\lambda ^2(\mathrm {cot}\theta \mathrm {Re}[r^{\prime}_\lambda r^*_\lambda ]+\mathrm {cot}^2\theta R^2_{\bar \lambda })/(a_p^2R_p^2+a_s^2R_s^2)$, $\bar \lambda =s,p$ when $\lambda =p,s$, and $\Theta _{\mathrm {GH}}^g=-2\vartheta ^2(\omega _p\frac {\partial }{\partial \theta }\mathrm {ln}R_p+\omega _s\frac {\partial }{\partial \theta }\mathrm {ln}R_s)$ is the GHA of Gaussian beams.

On the other hand, the transfer matrix method is employed to calculate the Fresnel reflection coefficient in this system with the monolayer graphene [78,79]. We consider the p polarized incident light propagating in the absorbing medium, whose magnetic field is polarized along $\hat {\mathbf {y}}$ direction, written as:

(16)$$H_{1y}=(a_1 e^{k_{1z}z}+b_1 e^{-k_{1z}z})e^{k_{1x}x} \quad (z<0),$$

(17)$$H_{2y}=(a_2 e^{k_{2z}z}+b_2 e^{-k_{2z}z})e^{k_{2x}x} \quad (z>0),$$

where $k_{jx} (k_{jz})$ $(j=1,2)$ is the $x (z)$ componet of the wave vector $\mathbf {k}_j$, with $|\mathbf {k}_j|= \sqrt {\varepsilon _{j}} \omega /c$ having the angular frequency $\omega$ of the incident beam and the speed $c$ of light in vacuum; $a_j$ and $b_j$ are the field coefficients. Meanwhile, it is obvious that from the well-known Snell’s law [22]:

(18)$$k_{1x}=k_{2x}.$$

The transmission matrix of the graphene surface can be derived with the help of the boundary conditions at $z=0$ and Ohm’s law, as follow [79]:

(19)$$D_{p(1\rightarrow2)}=\frac{1}{2} \left[ \begin{array}{cc} 1+\eta_p+\xi_p & 1-\eta_p-\xi_p \\ 1-\eta_p+\xi_p & 1+\eta_p-\xi_p \\ \end{array} \right],$$

for p polarization, where $\eta _p=(\varepsilon _{1}k_{2z})/(\varepsilon _{2}k_{1z})$ and $\xi _p= (\sigma \omega k_{2z})/(\varepsilon _{0}\varepsilon _{2} \omega )$, as well as

(20)$$D_{s(1\rightarrow2)}=\frac{1}{2} \left[ \begin{array}{cc} 1+\eta_s+\xi_s & 1-\eta_s+\xi_s \\ 1-\eta_s-\xi_s & 1+\eta_s-\xi_s \\ \end{array} \right],$$

for s polarization, where $\eta _s=k_{2z}/k_{1z}$ and $\xi _p=\sigma \mu _0\omega /k_{1z}$; $\sigma$ is the optical conductivity of graphene, $\varepsilon _{0}$ and $\mu _0$ are the permittivity and permeability in vacuum, respectively. The propagation matrix in this system is

(21)$$P(0)= \left[ \begin{array}{cc} 1 & 0 \\ 0 & 1 \\ \end{array} \right].$$

The transfer matrix of the graphene-coating system can be written as

(22)$$P(0)= M_{p,s}=D_{p,s(1\rightarrow 2)} P(0)=D_{p,s(1\rightarrow2)} .$$

Hence, the Fresnel reflection coefficients for p and s polarizations can be derived as follow [79]:

(23)$$r_p=\frac{M_{p,21}}{M_{p,11}}=\frac{\varepsilon_2/k_{2z}-\varepsilon_1/k_{1z}+\sigma/(\varepsilon_0 \omega)}{\varepsilon_2/k_{2z}+\varepsilon_1/k_{1z}+\sigma/(\varepsilon_0\omega)},$$

(24)$$r_s=\frac{M_{s,21}}{M_{s,11}}=\frac{k_{1z}-k_{2z}-k_1\sigma/(\varepsilon_0 c)}{k_{1z}+k_{2z}+k_1\sigma/(\varepsilon_0 c)}.$$

In addition, according to the semiconductor theory, the optical conductivity of graphene can be written as the form [80]:

(25)$$\sigma(\omega,E_f)=\frac{e^2 E_f}{\pi \hbar^2} \frac{i}{\omega+i \tau^{-1}}+ \frac{e^2}{4 \hbar^2} \Bigg [H(\hbar \omega-2 E_f)+\frac{i}{\pi} \ln \bigg|\frac{\hbar \omega-2 E_f}{\hbar \omega+2 E_f} \bigg| \Bigg],$$

where $H(x)$ denotes the step function, $e=1.60 \times 10^{-19}$ C and $\hbar =h/(2\pi )=1.05 \times 10^{-34} \, \mathrm {J \cdot s}$ represent the elementary charge and the reduce Plank’s constants, accordingly, $E_f$ is the Fermi energy, $\tau =(\mu E_f)/(e v_f^2)$ indicates the electron-phonon relaxation time with the mobility $\mu =10^4\, \mathrm {cm^2 V/s}$ and the Fermi velocity $v_f=10^6$ m/s. Moreover, $E_f= \hbar v_f \sqrt {\pi n_{2D}}$ can be controlled by changing the charge density $n_{2D}$, manipulated by the external gate voltage, which gives an easy tool to control the GH shift in experiment.

3. Results and discussions

3.1 Spatial GH shift

In the following simulation, the case of p polarization is only considered, due to the extremely small value of the GHS for s polarizaion. First, to illustrate the meaning of this study, we plot the comparison of the GHSs ($\Delta _{\mathrm {GH}}$) for four different cases: in Fig. 2(a) Gaussian beam without graphene (solid line), Gaussian beam with monolayer graphene (dashed line), Airy beam without graphene (dotted line); in Fig. 2(b) Airy beam with the monolayer graphene, at fixed parameters: $\varepsilon _2=2+0.02i$, $\lambda =850$ nm, $E_f=0.5$ eV, and $\alpha =\beta =0.1$. It is clear that using the monolayer graphene or the Airy beam can enhance the GHS, compared to the case of the Gaussian beam without graphene. Meanwhile, the effect of the former is a little larger than the latter. It is should be noted that the GHS without graphene is maximal at the Brewster angle $\theta _B=54.74^\circ$, while it increases to $\theta _B=55.43^\circ$ in the case with the monolayer graphene, owing to the fact that the position of $\theta _B$ is determined by the real part of the graphene conductivity [54], where Re$(\sigma )=0.6\times 10^{-4}$ S/m at $E_f=0.5$ eV, as shown below in Fig. 4(b). Interestingly, in Fig. 2(b), the GHS for the Airy beam with the monolayer graphene can be enhanced significantly, which far exceeds the above two cases, achieving about 390 times of the incident wavelength $\lambda$.

Next, in Eq. (25), the surface conductivity $\sigma$ of graphene depends strongly on the incident wavelength $\lambda$, which has also influence on the GHS value. In Fig. 3, we plot the dependence of GHS ($\Delta _{\mathrm {GH}}$) of the monolayer graphene on $\theta$ with different incident wavelengths: $\lambda =325$ nm (solid line), $\lambda =488$ nm (dashed line), $\lambda =633$ nm (dotted line), and $\lambda =850$ nm (dash-dotted line) for two cases of Fig. 3(a) Gaussian beams and Fig. 3(b) Airy beams. It is indicated that for a bigger value of the wavelength $\lambda$, a larger maximal value of the GH shift is obtained. Meanwhile, the difference of the maximal GHSs between Gaussian and Airy beams also increases with the increasing $\lambda$, by comparing Figs. 3(a) and 3(b). Therefore, enlarging $\lambda$ of the Airy beam impinging on the graphene-coating surface can produce a larger GH shift. In the next discussion, we only consider $\lambda =850$ nm in order to obtain a larger GHS.

Fig. 2. The GHS ($\Delta _{\mathrm {GH}}$) as a function of the incident angle $\theta$ for (a) Gaussian beam without graphene (solid line), Gaussian beam with the monolayer graphene (dashed line), Airy beam without graphene (dotted line), and (b) Airy beam with the monolayer graphene. Here, $\varepsilon _2=2+0.02i$, $\lambda =850$ nm, $E_f=0.5$ eV, and $\alpha =\beta =0.1$.

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Fig. 3. The GHS ($\Delta _{\mathrm {GH}}$) with the monolayer graphene in dependence on the incident angle $\theta$, with different incident wavelengths: $\lambda =325$ nm (solid line), $\lambda =488$ nm (dashed line), $\lambda =633$ nm (dotted line), and $\lambda =850$ nm (dash-dotted line) for two cases of (a) Gaussian beams and (b) Airy beams. Here, $\varepsilon _2=2+0.02i$, $E_f=0.5$ eV, and $\alpha =\beta =0.1$.

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Fig. 4. (a) GHS ($\Delta _{\mathrm {GH}}$) as a function of the incident angle $\theta$ and the Fermi energy $E_f$ for Airy beams with the monolayer graphene. (b) The real part (solid line) and imaginary part (dashed line) of the conductivity $\sigma$ of graphene as the function of $E_f$. Here, $\varepsilon _2=2+0.02i$, $\lambda =850$ nm, and $\alpha =\beta =0.1$.

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Moreover, the surface conductivity $\sigma$ of graphene has a strong dependence on the Fermi energy $E_f$ as well. Figure 4(a) depicts the GHS ($\Delta _{\mathrm {GH}}$) for Airy beams with the monolayer graphene as a function of the incident angle $\theta$ and the Fermi energy $E_f$. Far away from $\theta _B$, the GHS is always small but non-zero. Near $\theta _B$, the GHS rises sharply. It is found that with the increasing $E_f$, the variations of $\Delta _{\mathrm {GH}}$ are not monotonous both at $\theta _B=55.43^\circ$ and $\theta _B=54.74^\circ$. The maximal value of $\Delta _{\mathrm {GH}}$ at $\theta _B=55.43^\circ$ appears at $E_f=0.58$ eV while one at $\theta _B=54.74^\circ$ exists at $E_f=0.82$ eV. The former one is a little bigger than the latter one. Figure 4(b) shows the conductivity $\sigma$ of graphene in dependence on the Fermi energy $E_f$. It is clear that the real part of the graphene conductivity is Re$(\sigma )=0.6\times 10^{-4}$ S/m at $E_f<0.73$ eV, while it becomes 0 at $E_f>0.73$ eV, leading to the two different $\theta _B=55.43^\circ$ and $\theta _B=54.74^\circ$ at both sides of 0.73 eV. Therefore, we can control a considerable GHS from $\theta _B=55.43^\circ$ to $\theta _B=54.74^\circ$ easily by adjusting the Fermi energy from $E_f=0.58$ eV to $E_f=0.82$ eV.

The above discussions in Figs. 3 and 4 involve only the influence of the monolayer graphene on the GH shift. Now, we consider the effect of Airy beams on the GHS. The dependences of the GHSs on the incident angle $\theta$ and the decay factors $\alpha$ $(\beta =\alpha )$ of Airy beams without graphene and with the monolayer graphene are demonstrated in Figs. 5(a) and 5(b), respectively. In fact, as the decay factor $\alpha$ decreases, the value of 1/$\Lambda$ in Eq. (14) also decreases discussed in the recent paper [44]. However, the decrease of $\alpha$ results in an increase of $\Delta _{\mathrm {GH}}$ at $\theta _B$ both in cases of with and without graphene, so the maximal value of $\Delta _{\mathrm {GH}}$ at $\theta _B$ exists at $\alpha =0.02$. It is revealed that the second term in the brackets in Eq. (14) is dominant to $\Delta _{\mathrm {GH}}$ when $\alpha$ is extremely small. Obviously, we find from Figs. 5(a) and 5(b) that using graphene coating can promote the enhancement of $\Delta _{\mathrm {GH}}$ with the decreasing $\alpha$ of Airy beams. Further, by comping Figs. 3–5, using Airy beams to enhance $\Delta _{\mathrm {GH}}$ is more strongly than inserting the monolayer graphene, which can arrive at $10^3$ $\lambda$.

Fig. 5. The GHS ($\Delta _{\mathrm {GH}}$) as a function of the incident angle $\theta$ and the decay factor $\alpha$ $(\beta =\alpha )$ of Airy beams (a) without graphene and (b) with the monolayer graphene ($E_f=0.3$ eV). Here, $\varepsilon _2=2+0.02i$, and $\lambda =850$ nm.

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3.2 Angular GH shift

The GHA ($\Theta _{\mathrm {GH}}$) is depicted in Fig. 6, and the initial beam width for Gaussian and Airy beams is $w_0=1$ mm. Other parameters are the same as those in Fig. 2. There are a pair of positive and opposite GHA peaks for all cases. Similar to the case of the GHS, using the Airy beam with no coating and inserting graphene coating for the Gaussian beam can also promote the GHA, compared to the case for the Gaussian beam without graphene (shown in Fig. 6(a)). In addition, the enhancement of the GHA for the Airy beam with graphene is more significant than the above two cases, by comparing Figs. 6(a) and 6(b). However, it is still an extremely small value of the GHA ( $\simeq 10^{-5}$ rad) which can not be easily observed and measured in experiment. Hence, there is not any more discussion about the GHA in this case.

Fig. 6. The GHA ($\Theta _{\mathrm {GH}}$) as a function of the incident angle $\theta$ for (a) Gaussian beam without graphene (solid line), Gaussian beam with the monolayer graphene (dashed line), Airy beam without graphene (dotted line), and (b) Airy beam with the monolayer graphene. Here, $w_0=1$ mm, $\varepsilon _2=2+0.02i$, $\lambda =1550$ nm, $E_f=0.3$ eV, and $\alpha =\beta =0.1$.

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4. Conclusion

In conclusion, we have reported the GHS and the GHA for the p polarized Airy beams impinging onto weakly absorbing media coated with a single layer of graphene for the first time. The influence of the GHS on the incident angle, the incident wavelength, the Fermi energy, and the decay factors of Airy beams is studied and discussed. The simulation results show that the GHS can be significantly enlarged, compared to the case of only using Airy beams or the monolayer graphene. The maximal value of the GHS reaches about three orders of wavelengths, which can be experimentally observed and measured without difficulty. As we all know, Airy beams and graphene are hot research topics in the optical field, and thus employing them can become an easy way to generate a giant GHS in experiment.

Funding

National Natural Science Foundation of China (11374108, 11775083).

Disclosures

The authors declare no conflicts of interest.

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Goos-Hänchen shifts for Airy beams impinging on graphene-substrate surfaces

Abstract

1. Introduction

2. Theoretical model

3. Results and discussions

3.1 Spatial GH shift

3.2 Angular GH shift

4. Conclusion

Funding

Disclosures

References

Cited By

Figures (6)

Equations (25)

Optics Express