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Cross-coupling effect induced beam shifts for polarized vortex beam at two-dimensional anisotropic monolayer graphene surface

Ze Chen, Hu Zhang, Xiaoguang Zhang, Hui Li, Wenbo Zhang, and Lixia Xi

Open Access Open Access

Abstract

We investigated beam shifts for an arbitrarily polarized vortex beam reflected and transmitted at two-dimensional (2D) anisotropic monolayer graphene surface. And generalized expressions are theoretically derived for calculating beam shifts of vortex beam. Then, we presented the beam shifts associated with the self-isotropic (SI) effect, self-anisotropic (SA) effect and cross-coupling (XC) effect originated from self-isotropic interaction, self-anisotropic interaction and cross-coupling interaction between isotropic and anisotropic of two-dimensional media, respectively. More importantly, novel optical phenomena resulting from the XC effect are flexibly shown by manipulation OAM. We believe that our results can be extensively extended to 2D anisotropic Dirac semimetals and Weyl semimetals, and expect the results to be significant and contribute to the understanding of the spin and orbit Hall effect of the light.

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

1. Introduction

The reflection and transmission of a real light beam with a finite width (or with a structured wave-front) at a planar interface differ from that of plane waves, which obey the well-known Snell’s law and the Fresnel equations [1]. A structured wave-front beam can be considered as a set of plane waves with different wave vectors each obeying Snell’s law and the Fresnel equations. The reflected or transmitted wave can be considered as a superposition of all the reflected or transmitted plane waves that exhibits a different behavior from that of individual plane waves. These differences in behavior include the small linear shifts parallel and orthogonal to the incident plane relative to the position predicted by geometrical optics. The parallel shift is referred to as the linear Goos−Hänchen (GH) shift (longitudinal shift) [2,3] and the orthogonal shift is referred to as the linear Imbert−Fedorov (IF) shift (transverse shift) [4–6], which is closely related to the spin angular momentum or state of polarization (SOP) of the incident beam and hence, it is also called the spin Hall effect of light [7–9]. Several studies have focused on the angular GH and IF shifts, also regarded as shifts in momentum space (k-space), detected in the case of external reflection [10,11]. The magnitude of these shifts is determined by the SOP of the incident beam and the Fresnel coefficient corresponding to the central angle of incidence. Accurate analyses of the GH and IF shifts for Gaussian incident beam were carried out in Refs. [8,12,13] in isotropic media, and the results of these analyses were demonstrated experimentally in [9,14]. The discussion of these shifts recently become a hot topic because of their wide potential applications in nano-optics [15,16], plasmonics [17,18], and nonlinear optics [19,20].

Several reports have been published on the GH and IF shifts of Gaussian beam upon reflection and transmission on various complex media interfaces, such as air–glass [21–23], air–metal [14,24], air–uniaxial crystal [25], air–chiral metamaterials [26–28], air–graphene [29–33], and air–Weyl semimetal [34]. Interestingly, photonic spin Hall effect can also be understood as the beam shifts associated with the right-handed and left-handed circularly polarized (RCP and LCP) components, and can potentially be applied in precision metrology [9,35–37]. More recently, photonic spin Hall effect has also been investigated on the surface of 2D atomic crystals [38]. Moreover, it has been found that the shifts of a vortex beam with a large topological charge l, which can be arbitrarily large in theory [39], can be much larger than those of other types of light beam. Therefore, in addition to the SOP of the light and the Fresnel coefficients, the topological charge of the vortex beam also plays an extremely important role in manipulating the beam shifts. Theoretical and experimental studies on the GH and IF shifts have been extended to vortex beam carrying an intrinsic orbital angular momentum (OAM) [39–45]. On the other hand, beam shifts (including GH and IF shifts) induced by self-isotropic (SI) effect, self-anisotropic (SA) effect and cross-coupling (XC) effect between isotropic and anisotropic of media will appear, when the arbitrarily polarized vortex beam is reflected and transmitted at the two-dimensional (2D) anisotropic media. Remarkably, there is an important difference between beam shifts on the anisotropic media and isotropic media, which is caused by XC effect. However, beam shifts induced by XC effect on vortex beam reflection and transmission at the 2D anisotropic media surface are affected by SOP and topological charge, to our knowledge, which has not been reported previously.

Graphene, as a single layer of carbon atoms arranged in a hexagonal lattice, which has attracted much attention due to its extraordinary electronic and photonic properties. Based on the semi-classical approach, the anisotropic conductivity of graphene has been theoretically studied when subjected to a perpendicular static magnetic field [46]. Moreover, the 2D anisotropic graphene shows that the feasibility of simple and fast ambipolar doping together with a large Faraday rotation in a broad frequency range, which is a unique combination, but is still not presented in other known materials [47,48]. Because of the above reasons, it can be potentially applied to design fast tunable ultrathin magneto-optical devices based on the magneto-optical modulation of beam shifts. In this paper, we investigated beam shifts induced by SI, SA and XC effects for an arbitrarily polarized vortex beam reflected and transmitted at 2D anisotropic monolayer graphene surface based on the angular spectrum method. Firstly, expressions of the linear and angular GH shifts and IF shifts of reflection and transmission vortex beam are derived. Secondly, the novel spin-dependent GH and spin-independent IF shifts induced by XC effect are proposed. Finally, beam shifts with varying incident angle, magnetic field, SOP and topological charge l are simulated, when a polarized LG beam with narrow spectral width impinges from air to 2D anisotropic monolayer graphene surface. Simultaneously, the beam shifts caused by XC effect are also discussed in detail. The results can be potentially applied to optical switches [49,50], optical sensors [51], and in precision metrology [9,35–37].

2. Theory and model of vortex beam shifts

We firstly establish a theoretical model of the GH and IF shifts of the OAM light beam that is reflected and transmitted at 2D anisotropic graphene interface when subject to magnetic field B along z axis. Semi-classical conductivity tensors of the 2D anisotropic graphene are represented by [46–48]:

$${\sigma _{xx}} = {\sigma _{yy}} = \frac{{{e^2}|{{\mu_c}} |}}{{\pi {\hbar ^2}}}\frac{{i(\omega + i/\tau )}}{{[{{(\omega + i/\tau )}^2} - \omega _c^2]}},\;{\sigma _{xy}} ={-} {\sigma _{yx}} = \frac{{{e^2}|{{\mu_c}} |}}{{\pi {\hbar ^2}}}\frac{{{\omega _c}}}{{[{{(\omega + i/\tau )}^2} - \omega _c^2]}},$$
where ${\mu _c}$ is a chemical potential, $\tau$ is the relaxation scattering time, ${\omega _c}$ is the cyclotron frequency, and ${\omega _c} = eBv_F^2/{\mu _c}$, ${v_F} \approx {10^6}$m/s is the Fermi velocity, e is the elementary charge. Expressions of the semi-classical conductivity tensors are only valid for frequencies $\hbar \omega\;<\;2|{{\mu_c}} |$.

For convenience, we considered a paraxial Laguerre Gaussian (LG) beam with arbitrary SOP impinging on planar anisotropic graphene sheet placed along oxy plane at an angle ${\theta ^i}\textrm{ = }\arccos (\hat{{\mathbf k}}_c^i \cdot \hat{{\mathbf z}})$, as shown in Fig. 1. The planar anisotropic graphene sheet is sandwiched between two dielectric media with different refractive indices, ${n_1}$ and ${n_\textrm{2}}$. It is convenient to consider the coordinate system (o, xyz) attached to the interface with the z axis normal to the planar interface (z = 0). Furthermore, we introduce the central beam coordinate systems (oa, xayaza), the superscript a = i, r, t denotes the incident, reflected, and transmitted beams, respectively. In all coordinate systems, the direction of the ya axis coincides with that of the y axis, and they have a common origin of coordinates zero, oa=o. ${\mathbf k}_c^i = k{\kern 1pt} {\hat{{\mathbf z}}^i}$ and ${{\mathbf k}^i} = k^{\prime}{\kern 1pt} {\kern 1pt} {\hat{{\mathbf z}}^i}{{\kern 1pt} ^\prime }$ denote the central and non-central (local) wave vectors of the incident beam, respectively, with $k = k^{\prime}$. An arbitrarily wave vector (include central and non-central vectors) of the incident beam corresponding to incident angle can be defined as: $\theta _{{{\mathbf k}^i}}^i\textrm{ = }\arccos ({\hat{{\mathbf k}}^i} \cdot \hat{{\mathbf z}})$.

 figure: Fig. 1.

Fig. 1. (a) Geometrical relationships of the beams reflection and transmission at 2D anisotropic monolayer graphene subject to magnetic field along z axis; Projections the GH shift, $\Delta _{GH}^{a = r,t}\cos {\theta ^{a = r,t}}$, and the IF shift $\Delta _{IF}^{a = r,t}$, are shown by green and blue arrows, respectively. (b) Schematic illustration of the considered surface. The optical properties of 2D anisotropic monolayer graphene (purple) is characterized by its surface conductivity ${\sigma _{ij}}$ (ij = xx, xy, yx, yy). ${\mathbf k}_c^i$, ${\mathbf k}_c^r$ and ${\mathbf k}_c^t$ denote the central wave vectors of the incident, reflected and transmitted beams, respectively.

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We assume that the incident beam is a polarized LG beam with the waist located at the interface, and without radial index, i.e., p = 0, so that the electric field of the vortex beam has the form:

$${{\mathbf E}^i}({x^i},{y^i},{z^i}\textrm{ = 0}){\kern 1pt} {\kern 1pt} \textrm{ = }{\kern 1pt} {\kern 1pt} \textrm{(}{e_p}{\hat{{\mathbf x}}^i} + {e_s}{\hat{{\mathbf y}}^i})u_L^i({x^i},{y^i},{z^i}),$$
where $u_L^i({x^i},{y^i},{z^i} = 0)\textrm{ = }{C_{LG}}\exp {\{ } - k[{({x^i})^2} + {({y^i})^2}]/2{D^i}\} {({x^i} + i{y^i})^{|l |}}$, ${C_{LG}}$ is the normalized constant of the LG beam, ${D^i}\textrm{ = }kw_0^2\textrm{/2}$ is the Rayleigh length, and ${w_\textrm{0}}$ is the waist spot size of the incident beam. Through a Fourier transform, the angular spectrum (momentum or k-space) representation of the transverse electric field for the incident beam can be written as:
$${\tilde{{\mathbf E}}^i}(k_x^i,k_y^i) = \left( {\begin{array}{c} {{e_p}}\\ {{e_s}} \end{array}} \right)\tilde{u}_L^i(k_x^i,k_y^i),$$
where $k_x^i = {{\mathbf k}^i} \cdot {\hat{{\mathbf x}}^i}$, and $k_y^i = {{\mathbf k}^i} \cdot {\hat{{\mathbf y}}^i}$. ${({e_p},{e_s})^T}$ is the normalized Jones vector of the incident beam polarization in the ${\mathbf (}{\hat{{\mathbf x}}^i}$, ${\hat{{\mathbf y}}^i})$ basis, ${|{{e_p}} |^2} + {|{{e_s}} |^2} = 1$, and the scalar amplitude $\tilde{u}_L^i$ describes the k-space distribution of the incident LG beam. From the transformation between the central and non-central wave vector (local) coordinates, as well as the Fresnel equations, we obtain the representation of the transverse beam field reflected and transmitted in the k-space:
$${\tilde{{\mathbf E}}^a}(k_x^a,k_y^a){\kern 1pt} {\kern 1pt} \textrm{ = }{\hat{{\mathbf M}}^a}\hat{{\mathbf F}}{\hat{{\mathbf M}}^i}{\tilde{{\mathbf E}}^i}(k_x^i,k_y^i), $$
with

${\hat{{\mathbf M}}^a}\textrm{ = }\left( {\begin{array}{cc} 1&{ - k_y^a\cot {\theta^a}/{k^a}}\\ {k_y^a\cot {\theta^a}/{k^a}}&1 \end{array}} \right)$, $\hat{{\mathbf F}}\textrm{ = }\left( {\begin{array}{cc} {\tilde{f}_{pp}^a}&{\tilde{f}_{ps}^a}\\ {\tilde{f}_{sp}^a}&{\tilde{f}_{ss}^a} \end{array}} \right)$, ${\hat{{\mathbf M}}^i}\textrm{ = }\left( {\begin{array}{cc} 1&{k_y^i\cot {\theta^i}/k}\\ { - k_y^i\cot {\theta^i}/k}&1 \end{array}} \right),$ where a = r, t denote the reflected and transmitted beams, respectively. $\hat{{\mathbf F}}$ represents the Fresnel reflection and transmission matrix, where $\tilde{f}_{pp}^a$ and $\tilde{f}_{ss}^a$ are the direct-reflection and direct-transmission coefficients, respectively, $\tilde{f}_{ps}^a$ and $\tilde{f}_{sp}^a$ are the cross-reflection and cross-transmission coefficients, respectively.

We consider the Taylor expansion of the Fresnel coefficients up to first order in $k_x^i/k$,

$$\tilde{f}_{ij}^a\textrm{ = }f_{ij}^a + ({\partial _{{\theta ^i}}}f_{ij}^a)\frac{{k_x^i}}{k}, $$
where, ${\partial _{{\theta ^i}}} = \partial /\partial {\theta ^i}$, $f_{ij}^a$ is replaced by ${r_{ij}}$ and ${t_{ij}}$ for the reflection and transmission, respectively. ${r_{ij}}$ and ${t_{ij}}$ ($j{\kern 1pt} {\kern 1pt} \textrm{ = }pp,ps,sp,ss$) can be given by [52,53] :
$$\begin{aligned} {r_{pp}}{\kern 1pt} {\kern 1pt} \textrm{ = }{\kern 1pt} {\kern 1pt} [ & ({n_2}\cos {\theta ^i} - {n_1}\cos {\theta ^t} + {Z_0}{\sigma _{xx}}\cos {\theta ^i}\cos {\theta ^t})({n_1}\cos {\theta ^i} + {n_2}\cos {\theta ^t} + {Z_0}{\sigma _{yy}})\\ & - Z_0^2{\sigma _{xy}}{\sigma _{yx}}\cos {\theta ^i}\cos {\theta ^t}]/\Delta, \end{aligned}$$
$${r_{ps}}{\kern 1pt} {\kern 1pt} \textrm{ = }{\kern 1pt} {\kern 1pt} {r_{sp}}{\kern 1pt} {\kern 1pt} \textrm{ = }{\kern 1pt} {\kern 1pt} \textrm{2}{n_1}{Z_0}{\sigma _{xy}}\cos {\theta ^i}\cos {\theta ^t}\textrm{/}\Delta , $$
$$\begin{aligned} {r_{ss}}{\kern 1pt} {\kern 1pt} \textrm{ = }{\kern 1pt} {\kern 1pt} [ & ({n_2}\cos {\theta ^i} + {n_1}\cos {\theta ^t} + {Z_0}{\sigma _{xx}}\cos {\theta ^i}\cos {\theta ^t})({n_1}\cos {\theta ^i} - {n_2}\cos {\theta ^t} - {Z_0}{\sigma _{yy}})\\ & + Z_0^2{\sigma _{xy}}{\sigma _{yx}}\cos {\theta ^i}\cos {\theta ^t}]/\Delta, \end{aligned} $$
$${t_{pp}}{\kern 1pt} {\kern 1pt} \textrm{ = }{\kern 1pt} {\kern 1pt} 2{n_1}\cos {\theta ^i}({n_2}\cos {\theta ^i} + {n_1}\cos {\theta ^t} + {Z_0}{\sigma _{yy}})/\Delta , $$
$${t_{ps}}{\kern 1pt} {\kern 1pt} \textrm{ = }{\kern 1pt} {\kern 1pt} - {r_{ps}}\cos {\theta ^i}/\cos {\theta ^t}, $$
$${t_{sp}}{\kern 1pt} {\kern 1pt} \textrm{ = }{r_{ps}}{\kern 1pt} {\kern 1pt} \textrm{ = }{r_{sp}}, $$
$${t_{ss}}{\kern 1pt} {\kern 1pt} \textrm{ = }2{n_1}\cos {\theta ^i}({n_2}\cos {\theta ^i} + {n_1}\cos {\theta ^t} + {Z_0}{\sigma _{xx}}\cos {\theta ^i}\cos {\theta ^t})/\Delta , $$
where $\Delta \textrm{ = [(}{n_2}\cos {\theta ^i} + {n_1}\cos {\theta ^t} + {Z_0}{\sigma _{xx}}\cos {\theta ^i}\cos {\theta ^t})({n_1}\cos {\theta ^i} + {n_2}\cos {\theta ^t} + {Z_0}{\sigma _{yy}}) - Z_0^2{\sigma _{xy}}{\sigma _{yx}}$ $\cos {\theta ^i}\cos {\theta ^t}]$, ${Z_0} = 377\Omega $ is the impedance of vacuum. Using the relationships: $k_x^a{\kern 1pt} {\kern 1pt} \textrm{ = }{\gamma ^a}k_x^i$, $k_y^a{\kern 1pt} {\kern 1pt} \textrm{ = }k_y^i$, [54] and ${\gamma ^a}{\kern 1pt} {\kern 1pt} \textrm{ = }\cos {\theta ^i}\textrm{/}\cos {\theta ^a}$, from Eqs. (4) and (5) we obtain:
$${\tilde{{\mathbf E}}^a}(k_x^a,k_y^a) = \hat{{\mathbf M}}{\tilde{{\mathbf E}}^i}, $$
with
$$\hat{{\mathbf M}}\textrm{ = }\left( {\begin{array}{@{}cc@{}} {f_{pp}^a\textrm{ + [}{\gamma^{{a^{ - 1}}}}{\partial_{{\theta^i}}}f_{pp}^a\frac{{k_x^a}}{k} - (f_{ps}^a + f_{sp}^a{\gamma^{{a^{ - 1}}}})\frac{{k_y^a}}{k}]\cot {\theta^i}}&{f_{ps}^a\textrm{ + [}{\gamma^{{a^{ - 1}}}}{\partial_{{\theta^i}}}f_{ps}^a\frac{{k_x^a}}{k} + (f_{pp}^a - {\gamma^{{a^{ - 1}}}}f_{ss}^a)\frac{{k_y^a}}{k}]\cot {\theta^i}}\\ {f_{sp}^a\textrm{ + [}{\gamma^{{a^{ - 1}}}}{\partial_{{\theta^i}}}f_{sp}^a\frac{{k_x^a}}{k} - ( - {\gamma^{{a^{ - 1}}}}f_{pp}^a + f_{ss}^a)\frac{{k_y^a}}{k}]\cot {\theta^i}}&{f_{ss}^a\textrm{ + [}{\gamma^{{a^{ - 1}}}}{\partial_{{\theta^i}}}f_{ss}^a\frac{{k_x^a}}{k} + ({\gamma^{{a^{ - 1}}}}f_{ps}^a + f_{sp}^a)\frac{{k_y^a}}{k}]\cot {\theta^i}} \end{array}} \right).$$
Through an inverse Fourier transform of Eq. (6), the electric field for the reflected and transmitted beams in R-space can be written as
$${{\mathbf E}^a}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} ^a}) = \int\limits_{ - \infty }^{ + \infty } {\int\limits_{ - \infty }^{ + \infty } {{{\tilde{{\mathbf E}}}^a}} } (k_x^a,k_y^a)\exp (i{\varphi ^a})dk_x^adk_y^a, $$
where ${\varphi ^a}\textrm{ = }(k_x^a{x^a} + k_y^a{y^a})\textrm{ + }{k^a}[\textrm{1} - \frac{{{{(k_x^a)}^2} + {{(k_y^a)}^2}}}{{\textrm{2}{{({k^a})}^\textrm{2}}}}]{z^a}$. After simple calculations, Eq. (7) can be expressed as follows:
$$\begin{array}{l} {{\mathbf E}^a}({x^a},{y^a},{z^a})\textrm{ = \{ }[(f_{pp}^a + {\gamma ^{{a^{ - 1}}}}{\partial _{{\theta ^i}}}f_{pp}^a\frac{{ - i}}{k}{\partial _{{x^a}}}){e_p}\textrm{ + }(f_{ps}^a + {\gamma ^{{a^{ - 1}}}}{\partial _{{\theta ^i}}}f_{ps}^a\frac{{ - i}}{k}{\partial _{{x^a}}}){e_s}]\\ + [(f_{pp}^a{e_s} - f_{ps}^a{e_p}) - {\gamma ^{{a^{ - 1}}}}(f_{sp}^a{e_p} + f_{ss}^a{e_s}]\frac{{ - i\cot \theta }}{k}{\partial _{{y^a}}}\} u_L^a{{\hat{{\mathbf x}}}^a}\\ + {\{ }[(f_{sp}^a + {\gamma ^{{a^{ - 1}}}}{\partial _{{\theta ^i}}}f_{sp}^a\frac{{ - i}}{k}{\partial _{{x^a}}}){e_p}\textrm{ + }(f_{ss}^a + {\gamma ^{{a^{ - 1}}}}{\partial _{{\theta ^i}}}f_{ss}^a\frac{{ - i}}{k}{\partial _{{x^a}}}){e_s}]\\ + [(f_{sp}^a{e_s} - f_{ss}^a{e_p}) + {\gamma ^{{a^{ - 1}}}}(f_{pp}^a{e_p} + f_{ps}^a{e_s})]\frac{{ - i\cot \theta }}{k}{\partial _{{y^a}}}\} u_L^a{{\hat{{\mathbf y}}}^a}, \end{array}$$
with $u_L^a\textrm{ = }{C_{LG}}\exp [ - \frac{{{k^a}{{({x^a})}^2}}}{{2(D_x^a + i{z^a})}}]\exp [ - \frac{{{k^a}{{({y^a})}^2}}}{{2(D_y^a + i{z^a})}}]{[{\gamma ^a}{x^a} + i{\mathop{\rm sgn}} (l){y^a}]^{|l |}}$, where $D_x^a\textrm{ = }{\gamma ^{{a^{ - 2}}}}{D^a}$, $D_y^a\textrm{ = }{D^a}$, ${D^a}\textrm{ = }{k^a}w_0^2\textrm{/2}$ [23]. To derive universal expressions for the four basic beam shifts, we introduce complex shifts [41,42] and consider the case of an anisotropic medium. For convenience, we introduce the following symbols:
$$\tilde{x}_{ij}^a{\kern 1pt} {\kern 1pt} \textrm{ = }{\kern 1pt} {\kern 1pt} i{\gamma ^{{a^{ - 1}}}}\frac{\textrm{1}}{k}{\partial _{{\theta ^i}}}\ln f_{ij}^a,\textrm{for}\;ij\textrm{ = }pp,ps,sp,ss.$$
$$\tilde{y}_{ij = pp \leftrightarrow ss}^a{\kern 1pt} {\kern 1pt} \textrm{ = } \pm i\frac{{{e_{s \leftrightarrow p}}}}{{{e_{p \leftrightarrow s}}}}(\textrm{1} - {\gamma ^{{a^{ - 1}}}}\frac{{f_{ij = ss \leftrightarrow pp}^a}}{{f_{ij = pp \leftrightarrow ss}^a}})\cot {\theta ^i}/k,\textrm{for}\;ij\textrm{ = }pp( + ),ss( - ),$$
$$\tilde{y}_{ij\textrm{ = }ps \leftrightarrow sp}^a{\kern 1pt} {\kern 1pt} \textrm{ = } \mp i\frac{{{e_{p \leftrightarrow s}}}}{{{e_{s \leftrightarrow p}}}}(\textrm{1} + {\gamma ^{{a^{ - 1}}}}\frac{{f_{ij = sp \leftrightarrow ps}^a}}{{f_{ij = ps \leftrightarrow sp}^a}})\cot {\theta ^i}/k,\textrm{for}\;ij\textrm{ = }ps( - ),sp( + ).$$
Then, the transverse reflected and transmitted beams with complex shifts can also be expressed as
$${{\mathbf E}^a}({x^a},{y^a},{z^a}){\kern 1pt} {\kern 1pt} \textrm{ = }{\kern 1pt} {\kern 1pt} {{\mathbf G}^a}({x^a} - \tilde{x}_{ij}^a,{y^a} - \tilde{y}_{ij}^a,{z^a}) = \sum\limits_{j = p}^s {\sum\limits_{i = p}^s {\int\limits_{\textrm{ - }\infty }^{\textrm{ + }\infty } {\int\limits_{\textrm{ - }\infty }^{\textrm{ + }\infty } {f_{ij}^a{e_j}\tilde{u}_L^i\exp (i{\varphi ^a}^\prime )dk_x^adk_y^a{\mathbf e}_i^a} } } } ,$$
where, ${{\mathbf G}^a}({x^a} - \tilde{x}_{ij}^a,{y^a} - \tilde{y}_{ij}^a,{z^a}) = \sum\limits_{j = p}^s {\sum\limits_{i = p}^s {f_{ij}^a{e_j}u_L^a({x^a} - \tilde{x}_{ij}^a,{y^a} - \tilde{y}_{ij}^a,{z^a}){\mathbf e}_i^a} } ,\;{\mathbf e}_p^a = {\hat{{\mathbf x}}^a}$, ${\mathbf e}_s^a = {\hat{{\mathbf y}}^a}$, ${\varphi ^a}^\prime \textrm{ = }k_x^a({x^a} - \tilde{x}_{ij}^a) + k_y^a({y^a} - \tilde{y}_{ij}^a)\textrm{ + }{k^a}[\textrm{1} - \frac{{{{(k_x^a)}^2} + {{(k_y^a)}^2}}}{{\textrm{2}{{({k^a})}^\textrm{2}}}}]{z^a}$. To calculate the GH and IF shifts, we perform a first-order Taylor expansion of ${{\mathbf E}^a}$ about $x_{ij}^a{\kern 1pt} {\kern 1pt} \textrm{ = }y_{ij}^a{\kern 1pt} {\kern 1pt} \textrm{ = }0$:
$$\begin{array}{l} {|{{{\mathbf E}^a}({x^a},{y^a},{z^a})} |^\textrm{2}}\textrm{ = }{|{{{\mathbf G}^a}({x^a},{y^a},{z^a})} |^\textrm{2}}\\ - 2[\sum\limits_{i = p}^s {\sum\limits_{j = p}^s {{{|{f_{ij}^a{e_j}} |}^2}{\mathop{\rm Re}\nolimits} \tilde{x}_{ij}^a} } {\mathop{\rm Re}\nolimits} (u_L^{{a^ \ast }}{\partial _{{x^a}}}u_L^a) - \sum\limits_{i = p}^s {\sum\limits_{j = p}^s {{{|{f_{ij}^a{e_j}} |}^2}{\mathop{\rm Im}\nolimits} \tilde{x}_{ij}^a} } {\mathop{\rm Im}\nolimits} (u_L^{{a^ \ast }}{\partial _{{x^a}}}u_L^a)]\\ - 2[\sum\limits_{i = p}^s {\sum\limits_{j = p}^s {{{|{f_{ij}^a{e_j}} |}^2}{\mathop{\rm Re}\nolimits} \tilde{y}_{ij}^a} } {\mathop{\rm Re}\nolimits} (u_L^{{a^ \ast }}{\partial _{{y^a}}}u_L^a) - \sum\limits_{i = p}^s {\sum\limits_{j = p}^s {{{|{f_{ij}^a{e_j}} |}^2}{\mathop{\rm Im}\nolimits} \tilde{y}_{ij}^a} } {\mathop{\rm Im}\nolimits} (u_L^{{a^ \ast }}{\partial _{{y^a}}}u_L^a)]\\ - 2[\Delta x{\mathop{\rm Re}\nolimits} (u_L^{{a^ \ast }}{\partial _{{x^a}}}u_L^a) - \Delta x^{\prime}{\mathop{\rm Im}\nolimits} (u_L^{{a^ \ast }}{\partial _{{x^a}}}u_L^a)] - 2[\Delta y{\mathop{\rm Re}\nolimits} (u_L^{{a^ \ast }}{\partial _{{y^a}}}u_L^a) - \Delta y^{\prime}{\mathop{\rm Im}\nolimits} (u_L^{{a^ \ast }}{\partial _{{y^a}}}u_L^a)], \end{array}$$
with
$$\begin{array}{l} \Delta x{\kern 1pt} {\kern 1pt} \textrm{ = [}{\mathop{\rm Re}\nolimits} (f_{pp}^a{e_p}f_{ps}^{{a^ \ast }}e_s^ \ast ){\mathop{\rm Re}\nolimits} (\tilde{x}_{pp}^a + \tilde{x}_{ps}^a) + {\mathop{\rm Re}\nolimits} (f_{sp}^a{e_p}f_{ss}^{{a^ \ast }}e_s^ \ast ){\mathop{\rm Re}\nolimits} (\tilde{x}_{sp}^a + \tilde{x}_{ss}^a)]\\ - [{\mathop{\rm Im}\nolimits} (f_{pp}^a{e_p}f_{ps}^{{a^ \ast }}e_s^ \ast ){\mathop{\rm Im}\nolimits} (\tilde{x}_{pp}^a - \tilde{x}_{ps}^a) + {\mathop{\rm Im}\nolimits} (f_{sp}^a{e_p}f_{ss}^{{a^ \ast }}e_s^ \ast ){\mathop{\rm Im}\nolimits} (\tilde{x}_{sp}^a - \tilde{x}_{ss}^a)]\\ \Delta x^{\prime}{\kern 1pt} {\kern 1pt} \textrm{ = [}{\mathop{\rm Re}\nolimits} (f_{pp}^a{e_p}f_{ps}^{{a^ \ast }}e_s^ \ast ){\mathop{\rm Im}\nolimits} (\tilde{x}_{pp}^a + \tilde{x}_{ps}^a) + {\mathop{\rm Re}\nolimits} (f_{sp}^a{e_p}f_{ss}^{{a^ \ast }}e_s^ \ast ){\mathop{\rm Im}\nolimits} (\tilde{x}_{sp}^a + \tilde{x}_{ss}^a)]\\ + [{\mathop{\rm Im}\nolimits} (f_{pp}^a{e_p}f_{ps}^{{a^ \ast }}e_s^ \ast ){\mathop{\rm Re}\nolimits} (\tilde{x}_{pp}^a - \tilde{x}_{ps}^a) + {\mathop{\rm Im}\nolimits} (f_{sp}^a{e_p}f_{ss}^{{a^ \ast }}e_s^ \ast ){\mathop{\rm Re}\nolimits} (\tilde{x}_{sp}^a - \tilde{x}_{ss}^a)]\\ \Delta y{\kern 1pt} {\kern 1pt} \textrm{ = [}{\mathop{\rm Re}\nolimits} (f_{pp}^a{e_p}f_{ps}^{{a^ \ast }}e_s^ \ast ){\mathop{\rm Re}\nolimits} (\tilde{y}_{pp}^a + \tilde{y}_{ps}^a) + {\mathop{\rm Re}\nolimits} (f_{sp}^a{e_p}f_{ss}^{{a^ \ast }}e_s^ \ast ){\mathop{\rm Re}\nolimits} (\tilde{y}_{sp}^a + \tilde{y}_{ss}^a)]\\ - [{\mathop{\rm Im}\nolimits} (f_{pp}^a{e_p}f_{ps}^{{a^ \ast }}e_s^ \ast ){\mathop{\rm Im}\nolimits} (\tilde{y}_{pp}^a - \tilde{y}_{ps}^a) + {\mathop{\rm Im}\nolimits} (f_{sp}^a{e_p}f_{ss}^{{a^ \ast }}e_s^ \ast ){\mathop{\rm Im}\nolimits} (\tilde{y}_{sp}^a - \tilde{y}_{ss}^a)]\\ \Delta y^{\prime}{\kern 1pt} {\kern 1pt} \textrm{ = [}{\mathop{\rm Re}\nolimits} (f_{pp}^a{e_p}f_{ps}^{{a^ \ast }}e_s^ \ast ){\mathop{\rm Im}\nolimits} (\tilde{y}_{pp}^a + \tilde{y}_{ps}^a) + {\mathop{\rm Re}\nolimits} (f_{sp}^a{e_p}f_{ss}^{{a^ \ast }}e_s^ \ast ){\mathop{\rm Im}\nolimits} (\tilde{y}_{sp}^a + \tilde{y}_{ss}^a)]\\ + [{\mathop{\rm Im}\nolimits} (f_{pp}^a{e_p}f_{ps}^{{a^ \ast }}e_s^ \ast ){\mathop{\rm Re}\nolimits} (\tilde{y}_{pp}^a - \tilde{y}_{ps}^a) + {\mathop{\rm Im}\nolimits} (f_{sp}^a{e_p}f_{ss}^{{a^ \ast }}e_s^ \ast ){\mathop{\rm Re}\nolimits} (\tilde{y}_{sp}^a - \tilde{y}_{ss}^a)] \end{array}$$
where ‘Re’, ‘Im’, and the superscript ‘$\ast $’ denote the real, the imaginary part, and the complex conjugate, respectively. The last two parts of Eq. (11) represent the light field originated from the cross-coupling interaction between isotropic and anisotropic of the 2D anisotropic graphene, which is the generation origin of the beam shifts caused by XC effect. The intensity centroid of the reflected and transmitted beams is conveniently expressed as $\left\langle {{x^a}} \right\rangle {\kern 1pt} {\kern 1pt} \textrm{ = }\left\langle {{{\mathbf E}^a}} \right|{x^a}\left| {{{\mathbf E}^a}} \right\rangle /\left\langle {{{{\mathbf E}^a}}} \mathrel{\left | {\vphantom {{{{\mathbf E}^a}} {{{\mathbf E}^a}}}} \right. } {{{{\mathbf E}^a}}} \right\rangle $, $\left\langle {{y^a}} \right\rangle {\kern 1pt} {\kern 1pt} \textrm{ = }\left\langle {{{\mathbf E}^a}} \right|{y^a}\left| {{{\mathbf E}^a}} \right\rangle /\left\langle {{{{\mathbf E}^a}}} \mathrel{\left | {\vphantom {{{{\mathbf E}^a}} {{{\mathbf E}^a}}}} \right. } {{{{\mathbf E}^a}}} \right\rangle $. We assume that ${\mathbf G}$ approaches zero at infinity and both integrals of ${x^a}{|{\mathbf G} |^2}$ and ${y^a}{|{\mathbf G} |^2}$ over the beam cross-section ${x^a}{y^a}$ are approximately equal to zero. Thus, the follow expresses can be obtained:
$$\left\langle {{x^a}} \right\rangle {\kern 1pt} {\kern 1pt} \textrm{ = }{\kern 1pt} {\kern 1pt} - \frac{{{\mathop{\rm Im}\nolimits} (\tilde{x}_{\textrm{SD}}^a\textrm{ + }\tilde{x}_{\textrm{SC}}^a\textrm{ + }\tilde{x}_{\textrm{XC}}^a)}}{{N_\textrm{S}^a(1 + N_{\textrm{X/S}}^a)}} + {\gamma ^{{a^{ - 1}}}}l\frac{{{\mathop{\rm Re}\nolimits} (\tilde{y}_{\textrm{SD}}^a\textrm{ + }\tilde{y}_{\textrm{SC}}^a\textrm{ + }\tilde{y}_{\textrm{XC}}^a)}}{{2N_\textrm{S}^a(1 + N_{\textrm{X/S}}^a)}} + (1 + |l |)\frac{\textrm{1}}{{D_x^a}}\frac{{{\mathop{\rm Re}\nolimits} (\tilde{x}_{\textrm{SD}}^a\textrm{ + }\tilde{x}_{\textrm{SC}}^a\textrm{ + }\tilde{x}_{\textrm{XC}}^a\textrm{)}}}{{N_\textrm{S}^a(1 + N_{\textrm{X/S}}^a)}}{z^a},$$
$$\left\langle {{y^a}} \right\rangle {\kern 1pt} {\kern 1pt} ={-} \frac{{{\mathop{\rm Im}\nolimits} (\tilde{y}_{\textrm{SD}}^a\textrm{ + }\tilde{y}_{\textrm{SC}}^a\textrm{ + }\tilde{y}_{\textrm{XC}}^a)}}{{2N_\textrm{S}^a(1 + N_{\textrm{X/S}}^a)}} - {\gamma ^a}l\frac{{{\mathop{\rm Re}\nolimits} (\tilde{x}_{\textrm{SD}}^a\textrm{ + }\tilde{x}_{\textrm{SC}}^a\textrm{ + }\tilde{x}_{\textrm{XC}}^a)}}{{N_\textrm{S}^a(1 + N_{\textrm{X/S}}^a)}} + (1 + |l |)\frac{\textrm{1}}{{D_y^a}}\frac{{{\mathop{\rm Re}\nolimits} (\tilde{y}_{\textrm{SD}}^a\textrm{ + }\tilde{y}_{\textrm{SC}}^a\textrm{ + }\tilde{y}_{\textrm{XC}}^a)}}{{2N_\textrm{S}^a(1 + N_{\textrm{X/S}}^a)}}{z^a}, $$
where, $\tilde{x}_{\textrm{SI}{\kern 1pt} }^a\textrm{ = }{\gamma ^{{a^{ - 1}}}}({|{f_{pp}^a{e_p}} |^2}{\partial _{{\theta ^i}}}\ln f_{pp}^a + {|{f_{ss}^a{e_s}} |^2}{\partial _{{\theta ^i}}}\ln f_{ss}^a)/k$, $\tilde{x}_{\textrm{SA}}^a\textrm{ = }{\gamma ^{{a^{ - 1}}}}({|{f_{ps}^a{e_s}} |^2}{\partial _{{\theta ^i}}}\ln f_{ps}^a + {|{f_{sp}^a{e_p}} |^2}\;{\partial _{{\theta ^i}}}$ $\ln f_{sp}^a)\textrm{/}k,\ \tilde{x}_{\textrm{XC}}^a\textrm{ = }{\gamma ^{{a^{ - 1}}}}\textrm{[}\bar{\rho }(f_{pp}^af_{ps}^{{a^ \ast }}{\partial _{{\theta ^i}}}\ln f_{pp}^a + f_{sp}^{{a^ \ast }}f_{ss}^a{\partial _{{\theta ^i}}}\ln f_{ss}^a + f_{pp}^{{a^ \ast }}f_{ps}^a{\partial _{{\theta ^i}}}\ln f_{ps}^a\textrm{ + }f_{sp}^af_{ss}^{{a^ \ast }}{\partial _{{\theta ^i}}}\ln f_{sp}^a)/2+ i\bar{\sigma }(f_{pp}^{{a^ \ast }}f_{ps}^a{\partial _{{\theta ^i}}}$ $\ln f_{ps}^a + f_{sp}^af_{ss}^{{a^ \ast }}{\partial _{{\theta ^i}}}\ln f_{sp}^a - f_{pp}^af_{ps}^{{a^ \ast }}{\partial _{{\theta ^i}}}\ln f_{pp}^a - f_{sp}^{{a^ \ast }}f_{ss}^a{\partial _{{\theta ^i}}}\ln f_{ss}^a)/2]\textrm{/}k$, $\tilde{y}_{\textrm{SI}}^a{\kern 1pt} {\kern 1pt} = {\kern 1pt} {\kern 1pt} \cot {\theta ^i}{\{ }\bar{\rho }[{|{f_{pp}^a} |^2}(\textrm{1} - {\gamma ^{{a^{ - 1}}}}f_{ss}^a\textrm{/}f_{pp}^a) - {|{f_{ss}^a} |^2}(\textrm{1} - {\gamma ^{{a^{ - 1}}}}f_{pp}^a\textrm{/}f_{ss}^a)]\textrm{ + }i\bar{\sigma }[{|{f_{pp}^a} |^2}(\textrm{1} - {\gamma ^{{a^{ - 1}}}}f_{ss}^a\textrm{/}f_{pp}^a) + {|{f_{ss}^a} |^2}(\textrm{1} - {\gamma ^{{a^{ - 1}}}}$ $f_{pp}^a\textrm{/}f_{ss}^a)]{\} /}k$, $\tilde{y}_{\textrm{SA}}^a\textrm{ = }\cot {\theta ^i}{\{ }\bar{\rho }[ - {|{f_{ps}^a} |^2}(\textrm{1} + {\gamma ^{{a^{ - 1}}}}f_{sp}^a\textrm{/}f_{ps}^a)\textrm{ + }{|{f_{sp}^a} |^2}(\textrm{1} + {\gamma ^{{a^{ - 1}}}}f_{ps}^a\textrm{/}f_{sp}^a)]\textrm{ + }i\bar{\sigma }[{|{f_{ps}^a} |^2}(\textrm{1} + {\gamma ^{{a^{ - 1}}}}\;f_{sp}^a\textrm{/}f_{ps}^a) + {|{f_{sp}^a} |^2}(\textrm{1} + {\gamma ^{{a^{ - 1}}}}f_{ps}^a\textrm{/}f_{sp}^a)]\} \textrm{/}k$, $\tilde{y}_{\textrm{XC}}^a\textrm{ = 2}\cot {\theta ^i}\{ {|{{e_s}} |^2}[f_{ps}^{{a^ \ast }}(f_{pp}^a - {\gamma ^{{a^{ - 1}}}}f_{ss}^a) + f_{ss}^{{a^ \ast }}({\gamma ^{{a^{ - 1}}}}f_{ps}^a + f_{sp}^a)]\; - {|{{e_p}} |^2}[f_{pp}^{{a^ \ast }}(f_{ps}^a + {\gamma ^{{a^{ - 1}}}}f_{sp}^a) + f_{sp}^{{a^ \ast }}( - {\gamma ^{{a^{ - 1}}}}f_{pp}^a + f_{ss}^a)]\} \textrm{/}k$, $\bar{\rho }\textrm{ = }2{\mathop{\rm Re}\nolimits} (e_p^ \ast {e_s})$, $\bar{\sigma }\textrm{ = }2{\mathop{\rm Im}\nolimits} (e_p^ \ast {e_s})$, $N_\textrm{S}^a\textrm{ = }{|{f_{pp}^a{e_p}} |^2} + {|{f_{ps}^a{e_s}} |^2}\textrm{ + }{|{f_{sp}^a{e_p}} |^2} + {|{f_{ss}^a{e_s}} |^2}$, $N_{\textrm{X/S}}^a\textrm{ = }[\bar{\rho }{\mathop{\rm Re}\nolimits} (f_{pp}^af_{ps}^{{a^ \ast }} + f_{sp}^af_{ss}^{{a^ \ast }}) + \bar{\sigma }{\mathop{\rm Im}\nolimits} (f_{pp}^af_{ps}^{{a^ \ast }} + f_{sp}^af_{ss}^{{a^ \ast }})]/[{|{f_{pp}^a{e_p}} |^2} + {|{f_{ps}^a{e_s}} |^2}\textrm{ + }{|{f_{sp}^a{e_p}} |^2} + {|{f_{ss}^a{e_s}} |^2}]$.

The linear and angular GH and IF shifts are defined as

$$\Delta _{GH}^a{\kern 1pt} {\kern 1pt} \textrm{ = }{\kern 1pt} {\kern 1pt} \left\langle {{x^a}} \right\rangle ({z^a}{\kern 1pt} {\kern 1pt} \textrm{ = }0),\Theta _{GH}^a{\kern 1pt} {\kern 1pt} \textrm{ = }{\kern 1pt} {\partial _{{z^a}}}{\kern 1pt} \left\langle {{x^a}} \right\rangle, $$
$$\Delta _{IF}^a{\kern 1pt} {\kern 1pt} \textrm{ = }{\kern 1pt} {\kern 1pt} \left\langle {{y^a}} \right\rangle ({z^a}{\kern 1pt} {\kern 1pt} \textrm{ = }0),\Theta _{IF}^a{\kern 1pt} {\kern 1pt} \textrm{ = }{\partial _{{z^a}}}{\kern 1pt} {\kern 1pt} \left\langle {{y^a}} \right\rangle. $$
Then, the four basic beam shifts of the reflected and transmitted beams are as follows:
$$\Delta _{GH}^a{\kern 1pt} {\kern 1pt} \textrm{ = }{\kern 1pt} {\kern 1pt} \Delta _{\textrm{SI},GH}^a + {\kern 1pt} \Delta _{\textrm{SA},GH}^a + {\kern 1pt} \Delta _{\textrm{XC},GH}^a,$$
$$\Theta _{GH}^a{\kern 1pt} {\kern 1pt} \textrm{ = }{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \Theta _{\textrm{SI},GH}^a + {\kern 1pt} \Theta _{\textrm{SA},GH}^a + {\kern 1pt} \Theta _{\textrm{XC},GH}^a,$$
$$\Delta _{IF}^a{\kern 1pt} {\kern 1pt} \textrm{ = }{\kern 1pt} \Delta _{\textrm{SI},IF}^a + {\kern 1pt} \Delta _{\textrm{SA},IF}^a + {\kern 1pt} \Delta _{\textrm{XC},IF}^a,$$
$$\Theta _{IF}^a{\kern 1pt} {\kern 1pt} \textrm{ = }{\kern 1pt} \Theta _{\textrm{SI},IF}^a + {\kern 1pt} \Theta _{\textrm{SA},IF}^a + {\kern 1pt} \Theta _{\textrm{XC},IF}^a,$$
with
$$\Delta _{\textrm{M},GH}^a{\kern 1pt} {\kern 1pt} \textrm{ = }{\kern 1pt} - \frac{1}{{N_\textrm{S}^a(1 + N_{\textrm{X/S}}^a)}}{\mathop{\rm Im}\nolimits} (\tilde{x}_\textrm{M}^a) + {\gamma ^{{a^{ - 1}}}}l\frac{{{\mathop{\rm Re}\nolimits} (\tilde{y}_\textrm{M}^a)}}{{2N_\textrm{S}^a(1 + N_{\textrm{X/S}}^a)}},$$
$$\Theta _{\textrm{M},GH}^a{\kern 1pt} {\kern 1pt} \textrm{ = }{\kern 1pt} {\kern 1pt} (1 + |l |)\frac{\textrm{1}}{{D_x^a}}\frac{1}{{N_\textrm{S}^a(1 + N_{\textrm{X/S}}^a)}}{\mathop{\rm Re}\nolimits} (\tilde{x}_\textrm{M}^a\textrm{)},$$
$$\Delta _{\textrm{M,}IF}^a{\kern 1pt} {\kern 1pt} \textrm{ = }{\kern 1pt} {\kern 1pt} - \frac{{{\mathop{\rm Im}\nolimits} (\tilde{y}_\textrm{M}^a)}}{{2N_\textrm{S}^a(1 + N_{\textrm{X/S}}^a)}} - {\gamma ^a}l\frac{1}{{N_\textrm{S}^a(1 + N_{\textrm{X/S}}^a)}}{\mathop{\rm Re}\nolimits} (\tilde{x}_\textrm{M}^a\textrm{)},$$
$$\Theta _{\textrm{M},IF}^a{\kern 1pt} {\kern 1pt} \textrm{ = }{\kern 1pt} {\kern 1pt} (1 + |l |)\frac{\textrm{1}}{{D_y^a}}\frac{{{\mathop{\rm Re}\nolimits} (\tilde{y}_\textrm{M}^a)}}{{2N_\textrm{S}^a(1 + N_{\textrm{X/S}}^a)}}.$$

Equations (15)−(18) are the main results of the derivation. The subscript M replaced by SI, SA and XC represents the beam shifts caused by self-isotropic interaction, self- anisotropic interaction and the cross coupling interaction between isotropic and anisotropic of 2D media, respectively. Equations (15)−(18) describe four generalized beam shifts upon the reflection and transmission of an arbitrarily polarized vortex beam at anisotropic media interface. To the best of our knowledge, it is the first time to achieve the generalized expressions of the four basic beam shifts at anisotropic media interface. $N_{\textrm{X/S}}^a$ describes the contribution from XC effect to the normalization factor. For a pair of polarized incident beam (i.e., ${e_p}\textrm{ = cos(} \pm \alpha \textrm{)}$, ${e_s}\textrm{ = }\sin ({\pm} \alpha )\exp (i\Delta \phi )$) for polarized angle of ${\pm} \alpha$, the SI, SA and XC-induced beam shifts as a function of polarized angle is odd symmetric (about the coordinate origin) or even symmetric(about the about y axis), when $N_{\textrm{X/S}}^a$ is much less than 1. As a physical parameter, $N_{\textrm{X/S}}^a$ can be used to characterize degree of degeneration of the symmetry (even symmetry or odd symmetry) for the curves of beam shifts mentioned above. More importantly, the values of $N_{\textrm{X/S}}^a$ corresponding to the RCP and LCP incident light ($\alpha ={\pm} 45$°, $\Delta \phi =$90°) show profound physical significance. It means the degree of degeneration of the spin-dependent or spin-independent beam shifts induced by SI, SA and XC effects. When the anisotropy of the 2D material disappears (i.e., $N_{\textrm{X/S}}^a$=0), the following relationships for the Gaussian beam (l = 0) can be satisfied:

$$\Delta _{GH}^{ {\pm} a}{\kern 1pt} {\kern 1pt} \textrm{ = }{\kern 1pt} - \frac{1}{{k({{|{f_{pp}^a} |}^2} + {{|{f_{ss}^a} |}^2})}}{\mathop{\rm Im}\nolimits} [{\gamma ^{{a^{ - 1}}}}({|{f_{pp}^a} |^2}{\partial _{{\theta ^i}}}\ln f_{pp}^a + {|{f_{ss}^a} |^2}{\partial _{{\theta ^i}}}\ln f_{ss}^a)],$$
$$\Delta _{IF}^{ {\pm} a}{\kern 1pt} {\kern 1pt} \textrm{ = }{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{ \pm \cot {\theta ^i}}}{{2k({{|{f_{pp}^a} |}^2} + {{|{f_{ss}^a} |}^2})}}{\mathop{\rm Re}\nolimits} [{|{f_{pp}^a} |^2}(\textrm{1} - {\gamma ^{{a^{ - 1}}}}\frac{{f_{ss}^a}}{{f_{pp}^a}}) + {|{f_{ss}^a} |^2}(\textrm{1} - {\gamma ^{{a^{ - 1}}}}\frac{{f_{pp}^a}}{{f_{ss}^a}})],$$
where, ${\pm}$ represent RCP and LCP incident light, respectively. Obviously, Eqs. (23) and (24) mean that spin-independent beam shift appear in the ${x^{a = r,t}}$ direction and spin-dependent beam shift appear in the ${y^{a = r,t}}$ direction, which is completely consistent with that of the previously reported [8,54]. The latter shift is also as spin Hall effect of light [7,8] . And our results for the arbitrarily polarized vortex beam also agree well with previously obtained results [40,54]. When the anisotropy of the 2D material appears, the expresses of the beam shifts for the horizontally polarized, vertically polarized, RCP and LCP Gaussian beam can be further simplified by the Eqs. (15)–(18). And expresses of these beam shifts for the reflected Gauss beam are completely consistent with that proposed by Ye et al. [34]. Intriguingly, $\Delta _{\textrm{SA,}IF}^r{\kern 1pt} {\kern 1pt} \textrm{ = }{\kern 1pt} {\kern 1pt} 0$ and $\Theta _{\textrm{SA,}IF}^r{\kern 1pt} {\kern 1pt} \textrm{ = }{\kern 1pt} {\kern 1pt} 0$ due to ${r_{ps}} = {r_{sp}}$.

When $N_{\textrm{X/S}}^a$ is far less than 1, GH and IF shifts for the a pair of RCP ($\bar{\sigma }\textrm{ ={+} 1}$) and LCP ($\bar{\sigma }\textrm{ = } - \textrm{1}$) incident Gaussian beam (l = 0) induced by SI, SA and XC effects satisfy the following relationships:

$$\Delta _{\textrm{SI/SA},GH}^{ + ,l = 0,a}{\kern 1pt} {\kern 1pt} \textrm{ = }{\kern 1pt} \Delta _{\textrm{SI/SA},GH}^{ - ,l = 0,a}{\kern 1pt} {\kern 1pt}, $$
$$\Theta _{\textrm{SI/SA},GH}^{ + ,l = 0,a}{\kern 1pt} {\kern 1pt} \textrm{ = }{\kern 1pt} \Theta _{\textrm{SI/SA},GH}^{ - ,l = 0,a}{\kern 1pt} {\kern 1pt}, $$
$$\Delta _{\textrm{SI/SA},IF}^{ + ,l = 0,a}{\kern 1pt} {\kern 1pt} \textrm{ = }{\kern 1pt} - \Delta _{\textrm{SI/SA},IF}^{ - ,l = 0,a}{\kern 1pt} {\kern 1pt}, $$
$$\Theta _{\textrm{SI/SA},IF}^{ + ,l = 0,a}{\kern 1pt} {\kern 1pt} \textrm{ = }{\kern 1pt} - \Theta _{\textrm{SI/SA},IF}^{ - ,l = 0,a}{\kern 1pt} {\kern 1pt}, $$
$$\Delta _{\textrm{XC},GH}^{ + ,l = 0,a}{\kern 1pt} {\kern 1pt} \textrm{ = }{\kern 1pt} - \Delta _{\textrm{XC},GH}^{ - ,l = 0,a}{\kern 1pt} {\kern 1pt}, $$
$$\Theta _{\textrm{XC},GH}^{ + ,l = 0,a}{\kern 1pt} {\kern 1pt} \textrm{ = }{\kern 1pt} - \Theta _{\textrm{XC},GH}^{ - ,l = 0,a}{\kern 1pt} {\kern 1pt}, $$
$$\Delta _{\textrm{XC},IF}^{ + ,l = 0,a}{\kern 1pt} {\kern 1pt} \textrm{ = }{\kern 1pt} \Delta _{\textrm{XC},IF}^{ - ,l = 0,a}{\kern 1pt} {\kern 1pt}, $$
$$\Theta _{\textrm{XC},IF}^{ + ,l = 0,a}{\kern 1pt} {\kern 1pt} \textrm{ = }{\kern 1pt} \Theta _{\textrm{XC},IF}^{ - ,l = 0,a}{\kern 1pt} {\kern 1pt}. $$
From the Eqs. (25)–(32), we can find that a pair of RCP and LCP light whose corresponding GH shifts induced by XC effect are equal in magnitude but opposite in direction (i.e., spin-dependent spatial and angular GH shifts), and corresponding IF shifts induced by XC effect are equal in magnitude and coincide in direction (spin-independent spatial and angular shifts). For the vortex beam, the spatial GH shift ($\Delta _{\textrm{XCE},GH}^{ {\pm} \bar{\sigma },l,a}{\kern 1pt} {\kern 1pt} \textrm{ = }{\gamma ^{{a^{ - 1}}}}l\Theta _{\textrm{XCE},IF}^{ {\pm} \bar{\sigma },l\textrm{ = 0},a}{\kern 1pt} {\kern 1pt} {D^a}$) associated with the spin-independent angular IF shift is induced by OAM. And there also exists a similar effect that the spatial IF shift ($\Delta _{\textrm{XCE},IF}^{ {\pm} \bar{\sigma },l,a}{\kern 1pt} {\kern 1pt} \textrm{ = } - {\gamma ^{{a^{ - 1}}}}l\Theta _{\textrm{XCE},GH}^{ {\pm} \bar{\sigma },l\textrm{ = 0},a}{\kern 1pt} {\kern 1pt} {D^a}$) associated with the spin-dependent angular GH shift is induced by OAM. Through the above derivation, it is not difficult to find that XC effect are a physical origin of the beam shifts difference for an arbitrarily polarized beam impinged at anisotropic media and the isotropic media interface. Interestingly, some novel optical phenomena resulting from the XC effect are flexibly shown by manipulation OAM, when the vortex beam is reflected and transmitted at the anisotropic media interface.

3. Numerical results and analysis

To provide a clearly physical picture, we perform numerical analysis of four basic beam shifts with varying SOP and topological charge l when a polarized LG beam with narrow spectral width impinges from air to 2D anisotropic monolayer graphene surface. Simultaneously, we also focus on the discussion that the beam shifts associated with SI, SA and XC effects are affected by incident angle, magnetic field and SOP.

Firstly, we discuss the dependences of $N_{\textrm{X/S}}^a$ for the RCP and LCP incident light on the incident angle and magnetic field. Figures 2(a) and 2(b) show the values of $N_{\textrm{X/S}}^{r,t}$ corresponding to the RCP and LCP incident light. The $N_{\textrm{X/S}}^r$ for B = 7T is closer to zero than the $N_{\textrm{X/S}}^t$ at grazing incidence as shown in Fig. 2(a). The main reason is that ${|{{r_{pp}}} |^2} + {|{{r_{ss}}} |^2}$ is much larger than ${|{{t_{pp}}} |^2} + {|{{t_{ss}}} |^2}$, as shown in Fig. 2(c). The $N_{\textrm{X/S}}^r$ and $N_{\textrm{X/S}}^t$ curves show a peak where B approaching to 2T, respectively, when ${\theta ^i}$ is equal to 80° as Fig. 2(b) exhibited. The appearances of the $N_{\textrm{X/S}}^r$ and $N_{\textrm{X/S}}^t$ peaks are closely related to the peaks of the $|{{r_{ps}}} |$ and $|{{t_{ps}}} |$ curves, respectively, as can clearly be seen in Fig. 2(d). Moreover, the off-diagonal conductivity of the 2D anisotropic monolayer graphene disappear at B = 0, thereby resulting in the vanishing cross-Fresnel reflection and transmission coefficients. Therefore, the values of $N_{\textrm{X/S}}^r$ and $N_{\textrm{X/S}}^t$ reach zero at B = 0. We can clearly see that the variation trends of the $N_{\textrm{X/S}}^r$ and $N_{\textrm{X/S}}^t$ are mainly determined by the $|{{r_{ps}}} |$ and $|{{t_{ps}}} |$, respectively, because of the changes of ${|{{r_{pp}}} |^2} + {|{{r_{ss}}} |^2}$ and ${|{{t_{pp}}} |^2} + {|{{t_{ss}}} |^2}$ quite slowly with increasing of the magnetic field for B > 7T.

 figure: Fig. 2.

Fig. 2. The values of $N_{\textrm{X/S}}^a$ corresponding to the RCP and LCP incident light, Quantities related to the Fresnel coefficients (QRFC). (a) The values of $N_{\textrm{X/S}}^a$ as a function of incident angle for a fixed magnetic field of the B = 7T, (b) The values of $N_{\textrm{X/S}}^a$ as a function of magnetic field for a fixed incident angle ${\theta ^i}$=80°, (c) QRFC as a function of incident angle for a fixed magnetic field of the B = 7T, and (d) QRFC as a function of magnetic field for a fixed incident angle ${\theta ^i}$=80° at an air-anisotropic monolayer graphene interface. In the calculations, the following parameters are used: ${n_\textrm{1}}\textrm{ = 1}$, ${n_\textrm{2}}\textrm{ = 3}\textrm{.415}$, $\mu$ = 1, $f$=1Thz, ${\mu _c} = 0.2$meV, ${v_F} = 0.95 \times {10^6}$m/s, $\tau = \textrm{0}\textrm{.1} \times {10^{\textrm{ - 12}}}$s.

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Next, we will consider the simplest case of beam shifts, i.e., the beam shifts of horizontally polarized incident light for l = 0. For the case of reflection, $\Delta _{\textrm{SA,}GH}^r{\kern 1pt} {\kern 1pt}$, $\Delta _{\textrm{SI,}GH}^r{\kern 1pt} {\kern 1pt}$ and $\Delta _{\textrm{XC,}GH}^r{\kern 1pt} {\kern 1pt}$ can be reduced to ${|{{r_{pp}}} |^2}[{\mathop{\rm Im}\nolimits} ({\partial _{{\theta ^i}}}\ln {r_{pp}})]/[k({|{{r_{pp}}} |^2}\textrm{ + }{|{{r_{sp}}} |^2})]$, ${|{{r_{sp}}} |^2}[{\mathop{\rm Im}\nolimits} ({\partial _{{\theta ^i}}}\ln {r_{sp}})]/[k\;({|{{r_{pp}}} |^2}\textrm{ + }{|{{r_{sp}}} |^2})]$ and $0$, respectively. Similarly, $\Delta _{\textrm{SI,}IF}^r{\kern 1pt} {\kern 1pt}$, $\Delta _{\textrm{SA,}IF}^r{\kern 1pt} {\kern 1pt}$ and $\Delta _{\textrm{XC,}IF}^r{\kern 1pt} {\kern 1pt}$ can be reduced to 0, 0 and ${\kern 1pt} - \textrm{2}\cot {\theta ^i}\{ {\mathop{\rm Im}\nolimits} {\kern 1pt} [r_{ps}^ \ast ({r_{pp}} + {r_{ss}})]\} /k({|{{r_{pp}}} |^2}\textrm{ + }{|{{r_{sp}}} |^2})$. Figure 3 shows how the SI, SA and XC-induced beam shifts of the horizontally polarized incident light vary with incident angle, when B = 7T. The spatial GH shift induced by SI effect and IF shift induced by XC effect show a peak near ${\theta _B} =$63.37° due to the presence of the minimum of ${r_{pp}}$ at the Brewster angle, as shown in Figs. 3(a) and 3(b). The beam shifts can be considered as a superposition of the beam shifts generated by the RCP and LCP components of reflected light with different weights, when horizontally polarized incident light is reflected at an air-anisotropic monolayer graphene interface. Therefore, the reflected light will undergo the large asymmetric spin splitting along the directions of ${x^r}$ and ${y^r}$-axes near the Brewster angle and became to the symmetric spin splitting far away from the Brewster angle. The similar behavior can also be found in monolayer black phosphorus [55]. Remarkably, the two new light fields are acquired by RCP and LCP components of horizontally polarized incident light reflected at an air-anisotropic monolayer graphene interface, which is no longer orthogonal in general. Thus, the beam shifts cannot generally be understood to be a superposition of the beam shifts generated by the RCP and LCP components of horizontally polarized incident light.

 figure: Fig. 3.

Fig. 3. Dependences of beam shifts for the horizontal polarized incident light on the incident angle reflected at an air-anisotropic monolayer graphene interface. (a) SI, SA and XC-induced spatial GH shifts; (b) SI, SA and XC-induced spatial IF shifts for l = 0. In the calculations, the following parameters are used: B = 7 T. Other parameters are the same as in Fig. 2.

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To predict the spin Hall effect similar to that presented in the isotropic media [8],we further analyze the beam shifts of RCP and LCP incident light for ${\theta ^i}$=80°,l = 0. For the case of reflection, $\Delta _{\textrm{SI,}GH}^r{\kern 1pt} {\kern 1pt}$, $\Delta _{\textrm{SA,}GH}^r{\kern 1pt} {\kern 1pt}$ and $\Delta _{\textrm{XC,}GH}^r{\kern 1pt} {\kern 1pt}$ can be reduced to $\{ {|{{r_{pp}}} |^2}[{\mathop{\rm Im}\nolimits} ({\partial _{{\theta ^i}}}\ln {r_{pp}})] + {|{{r_{ss}}} |^2}$ $[{\mathop{\rm Im}\nolimits} ({\partial _{{\theta ^i}}}\ln {r_{ss}})]\} /[2kN_\textrm{S}^r(1 + N_{\textrm{X/S}}^r)]$, ${|{{r_{ps}}} |^2}[{\mathop{\rm Im}\nolimits} ({\partial _{{\theta ^i}}}\ln {r_{ps}})]/[kN_\textrm{S}^r(1 + N_{\textrm{X/S}}^r)]$ and $\bar{\sigma }\{ {\mathop{\rm Im}\nolimits} [i{r_{ps}}\;(r_{pp}^ \ast \textrm{ + }r_{ss}^ \ast )\;{\partial _{{\theta ^i}}}\ln {r_{ps}}\; - ir_{ps}^ \ast ({r_{pp}}{\partial _{{\theta ^i}}}\ln {r_{pp}} + {r_{ss}}{\partial _{{\theta ^i}}}\ln {r_{ss}})]\} \textrm{/[2}kN_\textrm{S}^r(1 + N_{\textrm{X/S}}^r)]$, respectively. Similarly, $\Delta _{\textrm{SI,}IF}^r{\kern 1pt} {\kern 1pt}$, $\Delta _{\textrm{SA,}IF}^r{\kern 1pt} {\kern 1pt}$ and $\Delta _{\textrm{XC,}IF}^r{\kern 1pt} {\kern 1pt}$ can be reduced to $- \bar{\sigma }\textrm{[}{\mathop{\rm Im}\nolimits} ({\kern 1pt} i{\kern 1pt} \cot {\theta ^i}{|{{r_{pp}} + {r_{ss}}} |^2}\textrm{)]/[2}kN_\textrm{S}^r(1 + N_{\textrm{X/S}}^r)]$, $0$ and 0, respectively, where $N_\textrm{S}^r$ and $N_{\textrm{X/S}}^r$ can also be reduced to ${|{{r_{pp}}} |^2} + 2{|{{r_{ps}}} |^2} + {|{{r_{ss}}} |^2}$, $2\bar{\sigma }\;[{\mathop{\rm Im}\nolimits} ({r_{pp}}r_{ps}^ \ast{+} {r_{ps}}r_{ss}^ \ast )]/({|{{r_{pp}}} |^2} + 2{|{{r_{ps}}} |^2} + {|{{r_{ss}}} |^2})$, respectively. Next, we analyze the impact of the magnetic field on beam shifts. Figure 4 shows the characteristic behaviors of SI, SA and XC-induced beam shifts changing with respect to magnetic field for ${\theta ^i}$=80°. Figures 4(a) and 4(b) can clearly illustrate that the XC-induced spatial GH shift obviously larger than the SI and SA-induced spatial GH shifts in a wide-range of magnetic field, the SI-induced spatial IF shift appears and the SA and XC-induced spatial IF shifts vanish. Intriguingly, the XC-induced spatial GH shift is more sensitive to a change magnetic field than the SI-induced spatial IF shift. Therefore, variation of the magnetic field B would certainly impact the spin-dependent spatial GH shift induced by XC effect, which provides an extra degree of freedom for manipulation spin-dependent beam shifts. More importantly, when magnetic field is greater than 4T, the novel spin Hall effects along the directions of both ${x^r}$ and ${y^r}$-axes can be predicted due to the disappearances of the non-XC-induced spatial GH shifts and XC-induced spatial IF shift.

 figure: Fig. 4.

Fig. 4. Dependences of beam shifts for the RCP and LCP incident light reflected at an air-anisotropic monolayer graphene interface on the magnetic field. (a) SI, SA and XC-induced spatial GH shifts; (b) SI, SA and XC-induced spatial IF shifts for l = 0. In the calculations, the following parameters are used: ${\theta ^i} =$80°. Other parameters are the same as in Fig. 2.

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Now, we focus on the beam shifts of elliptically polarized incident light. The beam shifts of the reflected light as a function of the polarized angle (–90°<$\alpha$<90°) is discussed here, when a light beam with $\Delta \phi \textrm{ = }\pi \textrm{/2}$ at incident angle ${\theta ^i}$=80° impinges on the interface between air and anisotropic monolayer graphene. We firstly consider the case of gauss beam with l = 0. It is well known that theoretical curves of the GH and IF shifts as a function of incident polarized angle are always symmetric about y axis (even symmetric) and coordinate origin (odd symmetric), respectively. However, the simultaneous occurrence of the SI, SA and XC-induced beam shifts will break the symmetry of beam shift curve mentioned above. Obviously, $\tilde{x}_{\textrm{SI}}^r{\kern 1pt} {\kern 1pt}$, $\tilde{x}_{\textrm{SA}}^r$ and $\tilde{x}_{\textrm{XC}}^r$ can be reduced to $- ({\cos ^2}\alpha {|{{r_{pp}}} |^2}{\partial _{{\theta ^i}}}\ln {r_{pp}} + {\sin ^2}\alpha {|{{r_{ss}}} |^2}\;{\partial _{{\theta ^i}}}\ln {r_{ss}}){/}k$, $- {|{{r_{ps}}} |^2}{\partial _{{\theta ^i}}}\ln {r_{ps}}\textrm{/}k$, and $- i\sin (2\alpha )[{r_{ps}}(r_{pp}^ \ast \textrm{ + }r_{ss}^ \ast ){\partial _{{\theta ^i}}}\ln {r_{ps}} - r_{ps}^ \ast ({r_{pp}}{\partial _{{\theta ^i}}}\ln {r_{pp}} + {r_{ss}}{\partial _{{\theta ^i}}}\ln {r_{ss}})]{/}2\textrm{/}k$, respectively. Thus, the spatial GH shifts caused by SI effect is mainly determined by $- [{\cos ^2}\alpha {|{{r_{pp}}} |^2}{\mathop{\rm Im}\nolimits} ({\partial _{{\theta ^i}}}{\phi _{pp}}) + {\sin ^2}\alpha {|{{r_{ss}}} |^2}{\mathop{\rm Im}\nolimits} ({\partial _{{\theta ^i}}}{\phi _{ss}})]\textrm{/}k$, where ${\phi _{ij = pp,ss}} = \arg {r_{ij = pp,ss}}$. The spatial GH shifts caused by SA effect is close zero due small ${|{{r_{ps}}} |^2}$. And the follow relations are satisfied for ${\theta ^i} = 80$°, ${\partial _{{\theta ^i}}}|{{r_{ij}}} |\gg {\partial _{{\theta ^i}}}{\phi _{ij}}$[see Figs. 5(a) and 5(b)], ${\mathop{\rm Re}\nolimits} ({r_{ij = pp,ss}}) \gg \;{\mathop{\rm Re}\nolimits} ({r_{ij = ps,sp}})$, ${\mathop{\rm Re}\nolimits} ({r_{ij = pp,ss}}) \gg \;{\mathop{\rm Im}\nolimits} ({r_{ij = pp,ps,sp,ss}})$,[see Fig. 5(c)]. As a result, the spatial GH shift caused by XC effect is mainly determined by $- \sin (2\alpha )\{ {\mathop{\rm Re}\nolimits} (r_{pp}^ \ast \textrm{ + }r_{ss}^ \ast )({\partial _{{\theta ^i}}}|{{r_{ps}}} |) - {\mathop{\rm Re}\nolimits} (r_{ps}^ \ast )[({\partial _{{\theta ^i}}}|{{r_{pp}}} |) + {\partial _{{\theta ^i}}}|{{r_{ss}}} |]\} /2\textrm{/}k$. Furtherly, the spatial GH shifts caused by SI and XC effects appear but the spatial GH shifts caused by SA effect vanishes, as shown in Fig. 6(a). As a result, the total spatial GH shift curve exhibits an asymmetry, as shown in Fig. 6(b) (purple). Coincidentally, $N_{\textrm{X/S}}^r$ is closed to zero and non-XC-induced spatial GH shift for the polarized angles of ${\pm}$45° disappears, which leads to the occurrence of the spin-dependent beam shifts as shown in Fig. 6(a). Thus, a perfect the odd symmetric spatial GH shift is shown for the polarized angles of ${\pm}$45°($\Delta _{\textrm{SD/SC},GH}^{ + ,l = 0,r}{\kern 1pt} {\kern 1pt} \textrm{ = }{\kern 1pt} \Delta _{\textrm{SD/SC},GH}^{ - ,l = 0,r}{\kern 1pt} {\kern 1pt} \textrm{ = 0}$). This indicates that a novel spin-dependent beam shifts occurs in the ${x^r}$ direction. Comparing the angular GH shift with spatial GH shift, there is a little difference between the two scenarios, which is shown in Fig. 6(c). The XC-induced angular GH shift is obviously smaller than the case for the SI-induced angular GH shift in a wide-range of polarized angle. Thus, the total angular GH shift remains an even symmetric feature, which is depicted in Fig. 6(d) (purple solid line). Similarly, $\tilde{y}_{\textrm{SI}}^r{\kern 1pt} {\kern 1pt}$, $\tilde{y}_{\textrm{SA}}^r$ and $\tilde{y}_{\textrm{XC}}^r$ can be reduced to ${\kern 1pt} i{|{{r_{pp}} + {r_{ss}}} |^2}\;{\kern 1pt} \cot {\theta ^i}\sin (2\alpha )\textrm{/}k$, $0$ and $\textrm{2}r_{ps}^ \ast ({r_{pp}} + {r_{ss}})\;\cot {\theta ^i}\cos (2\alpha )\textrm{/}k$, respectively. Likewise, one can find from Fig. 6(f) (purple solid line) that the total spatial IF shift curve remains the odd symmetric feature because of the spatial IF shifts caused by SI arising and spatial IF shifts caused by SA and XC effects vanishing. However, it is worth emphasizing that an even symmetry is shown in Fig. 6(h) (see purple solid line). This feature is responsible for the absent of the angular IF shift induced by SI and SA effects. And then, we will discuss beam shifts effect in the case of the LG beam, which carries OAM. The OAM-dependent angular GH and IF shifts can be enhanced a factor of $\textrm{1 + }|l |$ as described by Eqs. (20) and (22). Thus, the angular GH and IF beam shifts are the same for light beam carrying topological charge + l and – l, which are indicated by overlapping curves in Figs. 6(d) and 6(h). On the other hand, from Eq. (19), we can find that there should exist the small OAM-dependent spatial GH shift, since the angular IF shift is small. This means that the spatial GH shifts show tiny differences in magnitude for the incident beam carrying different the value of l, as shown in Fig. 6(b). However, the OAM-dependent spatial IF shifts curves exhibit large changes since the larger angular GH shift (including larger SI and small SA and XC-induced angular GH shifts) exist. Indeed, the OAM-dependent beam shifts are quite small, which makes it very difficult to detect these beam shifts. Fortunately, very recently developed weak measurement method will be quite helpful to detect such tiny beam shift [56].

 figure: Fig. 5.

Fig. 5. The values of ${\partial _{{\theta ^i}}}|{{r_{ij}}} |$, ${\partial _{{\theta ^i}}}{\phi _{ij}}$ and ${r_{ij}}$ (including real and imaginary parts). (a) The values of ${\partial _{{\theta ^i}}}|{{r_{ij}}} |$. (b) The values of ${\partial _{{\theta ^i}}}{\phi _{ij}}$ and (c) ${r_{ij}}$ as function of magnetic field at fixed incident angle ${\theta ^i} =$80°. Other parameters are the same as in Fig. 2.

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 figure: Fig. 6.

Fig. 6. Dependences of beam shifts for the reflected beam on the incident polarization angle at an air-anisotropic monolayer graphene interface. (a) SI, SA and XC- induced spatial GH shifts, (c) SI, SA and XC- induced angular GH shifts, (e) SI, SA and XC- induced spatial IF shifts and (g) SI, SA and XC-induced angular IF shifts for l = 0. (b) total spatial GH shifts, (d) total angular GH shifts, (f) total spatial IF shifts and (h) total angular IF shifts for l = 0; l=${\pm} \textrm{1}$; l=${\pm} \textrm{2}$; l=${\pm} \textrm{3}$. In the calculations, the following parameters are used: B = 7 T and ${\theta ^i}$=80°. Other parameters are the same as in Fig. 2.

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In the following, the beam shifts of transmitted are also clearly described. The beam shifts of transmitted light as a function of the polarized angle (–90°<$\alpha$<90°) is shown in Fig. 7, when an incident beam with $\Delta \phi \textrm{ = }\pi \textrm{/2}$ at angle ${\theta _B}$ impinges on the interface between air and anisotropic monolayer graphene. The variation characteristic of beam shifts is similar to that of the reflected beam, which can also be obtained. Remarkably, there is a large difference for the GH and IF shifts curves between the reflected beam and transmitted beam, i.e., the beam shifts no longer maintain strictly even or odd symmetric distribution. The reason is that $N_{\textrm{X/S}}^t$ is obviously larger than $N_{\textrm{X/S}}^r$. And the $\tilde{y}_{\textrm{SA}}^t$ no longer disappears completely, due to ${\gamma ^{{t^{ - 2}}}} \ne 1$, which leads to appearance of the small $\Delta _{\textrm{SA,}IF}^t{\kern 1pt} {\kern 1pt}$. For the OAM-dependent beam shifts, both XC-induced spatial GH shift and SD-induced spatial IF shift can also be obviously enhanced by increasing the value of l.

 figure: Fig. 7.

Fig. 7. Dependences of beam shifts for the transmitted beam on the incident polarization angle at an air-anisotropic monolayer graphene interface. (a) SI, SA and XC-induced spatial GH shifts, (c) SI, SA and XC-induced angular GH shifts, (e) SI, SA and XC- induced spatial IF shifts and (g) SI, SA and XC-induced angular IF shifts for l = 0. (b) total spatial GH shifts, (d) total angular GH shifts, (f) total spatial IF shifts and (h) total angular IF shifts for l = 0; l=${\pm}$1; l=${\pm}$2; l=${\pm}$3. In the calculations, the following parameters are used: B = 7 T and ${\theta ^i}$=80°. Other parameters are the same as in Fig. 2.

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4. Conclusions

Closed form expressions of four basic beam shifts (linear GH shift, angular GH shift, linear IF shift, and angular IF shift) arising from the reflection and transmission of an arbitrarily polarized vortex beam at 2D anisotropic monolayer graphene surface were derived based on the angular spectrum method. And then, the novel spin-dependent GH shifts and spin-independent IF shifts induced by XC effect are proposed for the ordinary Gaussian beam. Remarkably, the XC-induced beam shifts can also be flexibly manipulated by changing OAM for the vortex beam. Furthermore, we numerically analyzed the beam shifts for the reflection and transmission of the polarized vortex beam upon 2D anisotropic monolayer graphene, and novel optical phenomena resulting from the XC effect are flexibly shown by manipulation OAM. We believe that our results can be extensively extended to 2D anisotropic Dirac semimetals and Weyl semimetals.

Funding

National Natural Science Foundation of China (61527820, 61571057, 61575082).

Disclosures

The authors declare no conflicts of interest.

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Figures (7)
Fig. 1.
Fig. 1. (a) Geometrical relationships of the beams reflection and transmission at 2D anisotropic monolayer graphene subject to magnetic field along z axis; Projections the GH shift, $\Delta _{GH}^{a = r,t}\cos {\theta ^{a = r,t}}$ , and the IF shift $\Delta _{IF}^{a = r,t}$ , are shown by green and blue arrows, respectively. (b) Schematic illustration of the considered surface. The optical properties of 2D anisotropic monolayer graphene (purple) is characterized by its surface conductivity ${\sigma _{ij}}$ (ij = xx, xy, yx, yy). ${\mathbf k}_c^i$ , ${\mathbf k}_c^r$ and ${\mathbf k}_c^t$ denote the central wave vectors of the incident, reflected and transmitted beams, respectively.

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Fig. 2.
Fig. 2. The values of $N_{\textrm{X/S}}^a$ corresponding to the RCP and LCP incident light, Quantities related to the Fresnel coefficients (QRFC). (a) The values of $N_{\textrm{X/S}}^a$ as a function of incident angle for a fixed magnetic field of the B = 7T, (b) The values of $N_{\textrm{X/S}}^a$ as a function of magnetic field for a fixed incident angle ${\theta ^i}$ =80°, (c) QRFC as a function of incident angle for a fixed magnetic field of the B = 7T, and (d) QRFC as a function of magnetic field for a fixed incident angle ${\theta ^i}$ =80° at an air-anisotropic monolayer graphene interface. In the calculations, the following parameters are used: ${n_\textrm{1}}\textrm{ = 1}$ , ${n_\textrm{2}}\textrm{ = 3}\textrm{.415}$ , $\mu$  = 1, $f$ =1Thz, ${\mu _c} = 0.2$ meV, ${v_F} = 0.95 \times {10^6}$ m/s, $\tau = \textrm{0}\textrm{.1} \times {10^{\textrm{ - 12}}}$ s.

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Fig. 3.
Fig. 3. Dependences of beam shifts for the horizontal polarized incident light on the incident angle reflected at an air-anisotropic monolayer graphene interface. (a) SI, SA and XC-induced spatial GH shifts; (b) SI, SA and XC-induced spatial IF shifts for l = 0. In the calculations, the following parameters are used: B = 7 T. Other parameters are the same as in Fig. 2.

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Fig. 4.
Fig. 4. Dependences of beam shifts for the RCP and LCP incident light reflected at an air-anisotropic monolayer graphene interface on the magnetic field. (a) SI, SA and XC-induced spatial GH shifts; (b) SI, SA and XC-induced spatial IF shifts for l = 0. In the calculations, the following parameters are used: ${\theta ^i} =$ 80°. Other parameters are the same as in Fig. 2.

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Fig. 5.
Fig. 5. The values of ${\partial _{{\theta ^i}}}|{{r_{ij}}} |$ , ${\partial _{{\theta ^i}}}{\phi _{ij}}$ and ${r_{ij}}$ (including real and imaginary parts). (a) The values of ${\partial _{{\theta ^i}}}|{{r_{ij}}} |$ . (b) The values of ${\partial _{{\theta ^i}}}{\phi _{ij}}$ and (c) ${r_{ij}}$ as function of magnetic field at fixed incident angle ${\theta ^i} =$ 80°. Other parameters are the same as in Fig. 2.

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Fig. 6.
Fig. 6. Dependences of beam shifts for the reflected beam on the incident polarization angle at an air-anisotropic monolayer graphene interface. (a) SI, SA and XC- induced spatial GH shifts, (c) SI, SA and XC- induced angular GH shifts, (e) SI, SA and XC- induced spatial IF shifts and (g) SI, SA and XC-induced angular IF shifts for l = 0. (b) total spatial GH shifts, (d) total angular GH shifts, (f) total spatial IF shifts and (h) total angular IF shifts for l = 0; l= ${\pm} \textrm{1}$ ; l= ${\pm} \textrm{2}$ ; l= ${\pm} \textrm{3}$ . In the calculations, the following parameters are used: B = 7 T and ${\theta ^i}$ =80°. Other parameters are the same as in Fig. 2.

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Fig. 7.
Fig. 7. Dependences of beam shifts for the transmitted beam on the incident polarization angle at an air-anisotropic monolayer graphene interface. (a) SI, SA and XC-induced spatial GH shifts, (c) SI, SA and XC-induced angular GH shifts, (e) SI, SA and XC- induced spatial IF shifts and (g) SI, SA and XC-induced angular IF shifts for l = 0. (b) total spatial GH shifts, (d) total angular GH shifts, (f) total spatial IF shifts and (h) total angular IF shifts for l = 0; l= ${\pm}$ 1; l= ${\pm}$ 2; l= ${\pm}$ 3. In the calculations, the following parameters are used: B = 7 T and ${\theta ^i}$ =80°. Other parameters are the same as in Fig. 2.

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Equations (44)

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σ x x = σ y y = e 2 | μ c | π ℏ 2 i ( ω + i / τ ) [ ( ω + i / τ ) 2 − ω c 2 ] , σ x y = − σ y x = e 2 | μ c | π ℏ 2 ω c [ ( ω + i / τ ) 2 − ω c 2 ] ,
E i ( x i , y i , z i  = 0 )  =  ( e p x ^ i + e s y ^ i ) u L i ( x i , y i , z i ) ,
E ~ i ( k x i , k y i ) = ( e p e s ) u ~ L i ( k x i , k y i ) ,
E ~ a ( k x a , k y a )  =  M ^ a F ^ M ^ i E ~ i ( k x i , k y i ) ,
f ~ i j a  =  f i j a + ( ∂ θ i f i j a ) k x i k ,
r p p  =  [ ( n 2 cos ⁡ θ i − n 1 cos ⁡ θ t + Z 0 σ x x cos ⁡ θ i cos ⁡ θ t ) ( n 1 cos ⁡ θ i + n 2 cos ⁡ θ t + Z 0 σ y y ) − Z 0 2 σ x y σ y x cos ⁡ θ i cos ⁡ θ t ] / Δ ,
r p s  =  r s p  =  2 n 1 Z 0 σ x y cos ⁡ θ i cos ⁡ θ t / Δ ,
r s s  =  [ ( n 2 cos ⁡ θ i + n 1 cos ⁡ θ t + Z 0 σ x x cos ⁡ θ i cos ⁡ θ t ) ( n 1 cos ⁡ θ i − n 2 cos ⁡ θ t − Z 0 σ y y ) + Z 0 2 σ x y σ y x cos ⁡ θ i cos ⁡ θ t ] / Δ ,
t p p  =  2 n 1 cos ⁡ θ i ( n 2 cos ⁡ θ i + n 1 cos ⁡ θ t + Z 0 σ y y ) / Δ ,
t p s  =  − r p s cos ⁡ θ i / cos ⁡ θ t ,
t s p  =  r p s  =  r s p ,
t s s  =  2 n 1 cos ⁡ θ i ( n 2 cos ⁡ θ i + n 1 cos ⁡ θ t + Z 0 σ x x cos ⁡ θ i cos ⁡ θ t ) / Δ ,
E ~ a ( k x a , k y a ) = M ^ E ~ i ,
M ^  =  ( f p p a  + [ γ a − 1 ∂ θ i f p p a k x a k − ( f p s a + f s p a γ a − 1 ) k y a k ] cot ⁡ θ i f p s a  + [ γ a − 1 ∂ θ i f p s a k x a k + ( f p p a − γ a − 1 f s s a ) k y a k ] cot ⁡ θ i f s p a  + [ γ a − 1 ∂ θ i f s p a k x a k − ( − γ a − 1 f p p a + f s s a ) k y a k ] cot ⁡ θ i f s s a  + [ γ a − 1 ∂ θ i f s s a k x a k + ( γ a − 1 f p s a + f s p a ) k y a k ] cot ⁡ θ i ) .
E a ( r ⇀ a ) = ∫ − ∞ + ∞ ∫ − ∞ + ∞ E ~ a ( k x a , k y a ) exp ⁡ ( i φ a ) d k x a d k y a ,
E a ( x a , y a , z a )  = {  [ ( f p p a + γ a − 1 ∂ θ i f p p a − i k ∂ x a ) e p  +  ( f p s a + γ a − 1 ∂ θ i f p s a − i k ∂ x a ) e s ] + [ ( f p p a e s − f p s a e p ) − γ a − 1 ( f s p a e p + f s s a e s ] − i cot ⁡ θ k ∂ y a } u L a x ^ a + { [ ( f s p a + γ a − 1 ∂ θ i f s p a − i k ∂ x a ) e p  +  ( f s s a + γ a − 1 ∂ θ i f s s a − i k ∂ x a ) e s ] + [ ( f s p a e s − f s s a e p ) + γ a − 1 ( f p p a e p + f p s a e s ) ] − i cot ⁡ θ k ∂ y a } u L a y ^ a ,
x ~ i j a  =  i γ a − 1 1 k ∂ θ i ln ⁡ f i j a , for i j  =  p p , p s , s p , s s .
y ~ i j = p p ↔ s s a  =  ± i e s ↔ p e p ↔ s ( 1 − γ a − 1 f i j = s s ↔ p p a f i j = p p ↔ s s a ) cot ⁡ θ i / k , for i j  =  p p ( + ) , s s ( − ) ,
y ~ i j  =  p s ↔ s p a  =  ∓ i e p ↔ s e s ↔ p ( 1 + γ a − 1 f i j = s p ↔ p s a f i j = p s ↔ s p a ) cot ⁡ θ i / k , for i j  =  p s ( − ) , s p ( + ) .
E a ( x a , y a , z a )  =  G a ( x a − x ~ i j a , y a − y ~ i j a , z a ) = ∑ j = p s ∑ i = p s ∫  -  ∞  +  ∞ ∫  -  ∞  +  ∞ f i j a e j u ~ L i exp ⁡ ( i φ a ′ ) d k x a d k y a e i a ,
| E a ( x a , y a , z a ) | 2  =  | G a ( x a , y a , z a ) | 2 − 2 [ ∑ i = p s ∑ j = p s | f i j a e j | 2 Re x ~ i j a Re ( u L a ∗ ∂ x a u L a ) − ∑ i = p s ∑ j = p s | f i j a e j | 2 Im x ~ i j a Im ( u L a ∗ ∂ x a u L a ) ] − 2 [ ∑ i = p s ∑ j = p s | f i j a e j | 2 Re y ~ i j a Re ( u L a ∗ ∂ y a u L a ) − ∑ i = p s ∑ j = p s | f i j a e j | 2 Im y ~ i j a Im ( u L a ∗ ∂ y a u L a ) ] − 2 [ Δ x Re ( u L a ∗ ∂ x a u L a ) − Δ x ′ Im ( u L a ∗ ∂ x a u L a ) ] − 2 [ Δ y Re ( u L a ∗ ∂ y a u L a ) − Δ y ′ Im ( u L a ∗ ∂ y a u L a ) ] ,
Δ x  = [ Re ( f p p a e p f p s a ∗ e s ∗ ) Re ( x ~ p p a + x ~ p s a ) + Re ( f s p a e p f s s a ∗ e s ∗ ) Re ( x ~ s p a + x ~ s s a ) ] − [ Im ( f p p a e p f p s a ∗ e s ∗ ) Im ( x ~ p p a − x ~ p s a ) + Im ( f s p a e p f s s a ∗ e s ∗ ) Im ( x ~ s p a − x ~ s s a ) ] Δ x ′  = [ Re ( f p p a e p f p s a ∗ e s ∗ ) Im ( x ~ p p a + x ~ p s a ) + Re ( f s p a e p f s s a ∗ e s ∗ ) Im ( x ~ s p a + x ~ s s a ) ] + [ Im ( f p p a e p f p s a ∗ e s ∗ ) Re ( x ~ p p a − x ~ p s a ) + Im ( f s p a e p f s s a ∗ e s ∗ ) Re ( x ~ s p a − x ~ s s a ) ] Δ y  = [ Re ( f p p a e p f p s a ∗ e s ∗ ) Re ( y ~ p p a + y ~ p s a ) + Re ( f s p a e p f s s a ∗ e s ∗ ) Re ( y ~ s p a + y ~ s s a ) ] − [ Im ( f p p a e p f p s a ∗ e s ∗ ) Im ( y ~ p p a − y ~ p s a ) + Im ( f s p a e p f s s a ∗ e s ∗ ) Im ( y ~ s p a − y ~ s s a ) ] Δ y ′  = [ Re ( f p p a e p f p s a ∗ e s ∗ ) Im ( y ~ p p a + y ~ p s a ) + Re ( f s p a e p f s s a ∗ e s ∗ ) Im ( y ~ s p a + y ~ s s a ) ] + [ Im ( f p p a e p f p s a ∗ e s ∗ ) Re ( y ~ p p a − y ~ p s a ) + Im ( f s p a e p f s s a ∗ e s ∗ ) Re ( y ~ s p a − y ~ s s a ) ]
⟨ x a ⟩  =  − Im ( x ~ SD a  +  x ~ SC a  +  x ~ XC a ) N S a ( 1 + N X/S a ) + γ a − 1 l Re ( y ~ SD a  +  y ~ SC a  +  y ~ XC a ) 2 N S a ( 1 + N X/S a ) + ( 1 + | l | ) 1 D x a Re ( x ~ SD a  +  x ~ SC a  +  x ~ XC a ) N S a ( 1 + N X/S a ) z a ,
⟨ y a ⟩ = − Im ( y ~ SD a  +  y ~ SC a  +  y ~ XC a ) 2 N S a ( 1 + N X/S a ) − γ a l Re ( x ~ SD a  +  x ~ SC a  +  x ~ XC a ) N S a ( 1 + N X/S a ) + ( 1 + | l | ) 1 D y a Re ( y ~ SD a  +  y ~ SC a  +  y ~ XC a ) 2 N S a ( 1 + N X/S a ) z a ,
Δ G H a  =  ⟨ x a ⟩ ( z a  =  0 ) , Θ G H a  =  ∂ z a ⟨ x a ⟩ ,
Δ I F a  =  ⟨ y a ⟩ ( z a  =  0 ) , Θ I F a  =  ∂ z a ⟨ y a ⟩ .
Δ G H a  =  Δ SI , G H a + Δ SA , G H a + Δ XC , G H a ,
Θ G H a  =  Θ SI , G H a + Θ SA , G H a + Θ XC , G H a ,
Δ I F a  =  Δ SI , I F a + Δ SA , I F a + Δ XC , I F a ,
Θ I F a  =  Θ SI , I F a + Θ SA , I F a + Θ XC , I F a ,
Δ M , G H a  =  − 1 N S a ( 1 + N X/S a ) Im ( x ~ M a ) + γ a − 1 l Re ( y ~ M a ) 2 N S a ( 1 + N X/S a ) ,
Θ M , G H a  =  ( 1 + | l | ) 1 D x a 1 N S a ( 1 + N X/S a ) Re ( x ~ M a ) ,
Δ M, I F a  =  − Im ( y ~ M a ) 2 N S a ( 1 + N X/S a ) − γ a l 1 N S a ( 1 + N X/S a ) Re ( x ~ M a ) ,
Θ M , I F a  =  ( 1 + | l | ) 1 D y a Re ( y ~ M a ) 2 N S a ( 1 + N X/S a ) .
Δ G H ± a  =  − 1 k ( | f p p a | 2 + | f s s a | 2 ) Im [ γ a − 1 ( | f p p a | 2 ∂ θ i ln ⁡ f p p a + | f s s a | 2 ∂ θ i ln ⁡ f s s a ) ] ,
Δ I F ± a  =  ± cot ⁡ θ i 2 k ( | f p p a | 2 + | f s s a | 2 ) Re [ | f p p a | 2 ( 1 − γ a − 1 f s s a f p p a ) + | f s s a | 2 ( 1 − γ a − 1 f p p a f s s a ) ] ,
Δ SI/SA , G H + , l = 0 , a  =  Δ SI/SA , G H − , l = 0 , a ,
Θ SI/SA , G H + , l = 0 , a  =  Θ SI/SA , G H − , l = 0 , a ,
Δ SI/SA , I F + , l = 0 , a  =  − Δ SI/SA , I F − , l = 0 , a ,
Θ SI/SA , I F + , l = 0 , a  =  − Θ SI/SA , I F − , l = 0 , a ,
Δ XC , G H + , l = 0 , a  =  − Δ XC , G H − , l = 0 , a ,
Θ XC , G H + , l = 0 , a  =  − Θ XC , G H − , l = 0 , a ,
Δ XC , I F + , l = 0 , a  =  Δ XC , I F − , l = 0 , a ,
Θ XC , I F + , l = 0 , a  =  Θ XC , I F − , l = 0 , a .
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