1. Introduction

It has been well-known that when totally internal reflection (TIR) occurs at an ordinary-dielectric interface, the secondary beams will shift a small distance from their individual geometric-optical position. The shift in the incident plane is called the Goos-Hänchen (GH) shift [1, 2] and that perpendicular to the incident plane is known as the Imbert-Fedorov (IF) shift [3]. The GH-shift can be explained by the stationary phase method, which is proportion to the derivation of the phase difference of the reflective or transmitted coefficient [4]. However, the IF-shift originating from the spin-orbit interaction is one of spin-Hall effects of light, governed by the conservation law of total angular momentum [5]. Both GH- and IF-shifts have been verified in experiments [1, 2] and have been discussed widely for the surface of various materials, such as metals [6], photonic crystals [7] and metamaterials [8]. In the general case, the optical loss from the optical absorption or scattering cannot be ignored in most materials, especially in metals or metamaterials including a metal constituent. The shifts are the same order as the wavelength of the incident light for ordinary dielectric interfaces. Therefore, in general they are not easy to be observed. Various fascinating structures were used to enhance the shifts in order to make a direct measurement, such as the left-handed metamaterials [9], metasurfaces [10], anisotropic metamaterials [11], and photonic crystals coated by the graphene [12]. These efforts lead to many interesting phenomena to be found and further broaden the window of application in nano-micro-optics and opto-electronics. Besides, adjustable GH- or IF-shift accompanied by high reflection is vital to the design of optical devices, and several workable control methods were put forward [13, 14]. For example, Wang et al. first present a proposal to manipulate the GH-shift of a light beam by a coherent control field and found that the lateral shifts can be easily controlled by adjusting the intensity and detuning of the control field [13]. Ziauddin et al. reported a coherent control of the GH shift in a fixed configuration or device via superluminal and subluminal wave propagation [14].

In recent years, the graphene was paid great attention owing to its unique electronic properties. More importantly, the optical response of graphene relies on the surface’s conductivity determined by the temperature, frequency and chemical potential [15]. In particular, the chemical potential can be tunable by changing the external gate voltage or the chemical doping without altering material-structure [16]. In virtue of the above merits, the graphene is a benefiting material for one to design tunable optical or electronic devices. Many works on various graphene-based nanostructures were made [17–22]. Zhou et al. realized the precise control of positive and negative GH-shifts in a graphene-substrate system [17]; Zheng et al. observed the enhanced and controllable GH-shifts from the graphene surface plasmon in a modified Otto configuration in the terahertz regime [18]; Li et al. measured the GH-shift on the surface of glass prism coating graphene layer by the beam splitter scanning method [19]. The IF-shift also can be enhanced if the orbital angular momentum (OAM) is considered in beam shifts [5]; Zhuo et al. investigated the IF-shifts and the OAM spectrum of reflected Laguerre-Gaussian (LG) beams by placing a monolayer graphene on an hBN and an optically thick gold as a mirror [20]. Xu et al. proposed a method for controlling the IF-shifts of a reflected beam at a graphene-coated chiral metamaterial (CMM) interface [21]. Luo et al investigated theoretically the tunable and enhanced GH shift for the TM-polarized reflected beam on the graphene-based hyperbolic metamaterials near the Brewster angle [22]. Hence, a controllable spatial shift can be achieved by tuning the interface properties, i.e., graphene-coated interface or graphene-based metamaterials. Recently, the graphene also is widely used in the storage energy device, such as the electrode of supercapacitor [23, 24], and some novel nanomaterials in THz are further developed [25–30].

The hexagonal boron nitride (hBN) is a kind of van der Waals crystal (vdW) and is a natural hyperbolic crystal with uniaxial anisotropy. The longitudinal and transverse components of the permittivity are opposite in sign. The two components have two separated reststrahlen bands (RB-I and RB-II), respectively. The total reflection at the hBN surface can be observed for an incident wave with any incident angle and any frequency in either RB. In addition, it has been proved that surface phonon polaritons (SPhPs) in the hBN exhibit high confinement and even lower loss compared to graphene surface plasmon polaritons (SPPs) [31]. Recently, the hyperbolic metamaterials (HMMs) were greatly noticed due to their hyperbolic dispersion which can support the propagation of polaritons with large wave vectors [32, 33]. Thus, many interesting properties were observed, for example the sub-wavelength imaging [34], negative refraction [35] et al, and lead to widely potential applications in nano- and micro-optics or technology fields. In a previous work, we studied the GH- and IF-shifts of reflective beam on the surface of hBN and found the large GH-shift at the critical or Brewster angles in or near the RBs [36]. In the following discussion, we will focus on the effect of graphene layers on the shifts by mainly tuning their chemical potential, volume fraction and also the surface orientation of the metastructure.

Based on the above research, we propose a metastructure composed of alternately hBN layers and graphene layers. The graphene is a wonderful electronic and optical material, especially its optical and electronic properties can be modulated with an external electric-field (chemical potential) in a large frequency range, from THz to infrared. The hBN is a naturally-hyperbolic material in the infrared range. They both are 2-dimensional materials. One has known that an electromagnetic metamaterial may exhibit unique optical properties that do not exist in any natural material. Unlike metal-based metamaterials, graphene-based hyperbolic metamaterials exhibit much lower dissipative loss. We will investigate the GH and IF-shifts of reflective beam at the surface of the graphene/hBN metamaterial in this paper to explore the influence of chemical potential, filling ratio, or orientation of surface (tilted angle) to the shifts. In addition, the plasma-phonon coupling in the metamaterial also can influences the shifts. This graphene/hBN structure can be produced by the chemical vapor deposition (CVD) technique [37] as well as molecular beam epitaxy (MBE) [38]. One main advantage of graphene-based metamaterials is the tunability of some optical properties by adjusting the gate voltage or via doping. We propose that the graphene/hBN metamaterial is put in a statical external electric-field vertical to graphene layers or the graphene is properly doped to achieve the voltage and finite chemical potential, respectively.

The present paper is organized as follows. In section 2, we first establish the theoretical model by isometrically inserting identical graphene sheets into an hBN crystal to form a periodical metamaterial and then discuss its effective dielectric properties. On this base, the analytic expressions for GH- and IF-shifts will be obtained, including those at the critical and Brewster angles. In sections 3 and 4, we present main numerical results for the GH- and IF-shifts. Finally, we summarize our main results in section 5.

2. Theoretical model

The model and coordinate system to be used are shown in Fig. 1, where the graphene sheets are periodically embedded in an hBN crystal. The surface of the graphene/hBN metamaterial is neither parallel nor perpendicular to constituent layers. The structure period-direction or the optical axis (OA) is in the x-z plane and is at an angle $\phi $ to the z-axis. We apply I and R to represent the central rays of incident and reflective beams, respectively. It can be seen that a spatial shift of the reflective beam occurs when the incident beam illuminates the surface of graphene/hBN metamaterials for incident angle $\theta$. ${\textrm{d}_\textrm{h}}$ and ${\textrm{d}_\textrm{g}}$ indicate the thicknesses of hBN and graphene layers, respectively. For the hBN layers, the lattice vibration in the layer plane possesses rotational symmetry and the transverse vibration along the c-axis. Therefore, the two dielectric constants in the layer plane both are ${\varepsilon _ \bot }$ and opposite to the transverse one ${\varepsilon _{/{/}}}$ in sign. The permittivity of hBN in the principal-axis frame is a diagonal matrix ${\varepsilon _\textrm{h}} = \textrm{diag}({\varepsilon _ \bot },{\varepsilon _ \bot },{\varepsilon _{/{/}}})$ and that of the graphene is a scalar denoted by ${\varepsilon _\textrm{g}}$. The effective medium method is conventional and well-accepted for solving the effective permittivity or permeability of an electromagnetic metamaterial, but a necessary condition must be satisfied, i.e., the structure period of the metamaterial is much smaller than operating wavelength. Therefore, this method is available for subwavelength metamaterials or nano-structured metamaterials in visible and infrared ranges. The effective-medium method is an approximate method. The transfer-matrix method must be used if this necessary condition is fake. Whereas the frequency lies in the THz region in this work, the effective medium method can be applied for the graphene/hBN metamaterials. The effective-permittivity components in this metamaterial are expressed as $\varepsilon _ \bot ^{e} = (1 - {{f}_{h}}){\varepsilon _{g}} + {{f}_{h}}{\varepsilon _ \bot }$ and $\varepsilon _{/{/}}^{e} = {\varepsilon _{g}}{\varepsilon _{/{/}}}/[{{f}_{h}}{\varepsilon _{g}} + (1 - {{f}_{h}}){\varepsilon _{/{/}}}]$, where ${{f}_{h}} = {{d}_{h}}/({{d}_{h}} + {{d}_{g}})$ is the filling ratio of hBN. When the OA artificially orients in the incident plane, as shown in Fig. 1, the effective permittivity is a non-diagonal matrix in the present coordinate system, namely,

 

Fig. 1. Configuration of incidence-reflection and sketch of shifts wherein the optical axis of crystal lies in the x-z plane (the incident plane) and is at an angle $\phi$ relative to the z-axis, and the incident angle is indicated with $\theta$. $\Delta \textrm{x}$ and $\Delta \textrm{y}$ are the GH- and IF-shifts, respectively.

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$K_{ez}^{}$ and $K_{oz}^{}$ are the z-components of the o-wave and e-wave vectors, respectively. From Eq.(2), we can get the solutions of $K_{ez}^{}$ and $K_{oz}^{}$ which are expressed as

Based on Eq. (4), we can get the following relation,

We can easily judge from the Eq. (7) that the reflective coefficient only depends on the permittivity ${\varepsilon _{{yy}}}$ regardless of the titled angle which results in that the medium behaves like isotropic, so the GH shift is small. Even at the critical angle, the GH-shift is decided only by ${\varepsilon _{{yy}}}$, shown in Eq. (12). However, the reflective coefficient of p-wave is related to not only $\varepsilon _ \bot ^e$ and $\varepsilon _{/{/}}^e$ but also the tilted angle $\phi$. Consequently, the GH-shift can be large and achieve the peak values at both critical and Brewster angles. Therefore, in the following discussions we will only focus on the p-wave for the GH-shift.

Contrary to the longitudinal GH-shift, the IF shift is a transverse shift perpendicular to the incident plane. As we known, the IF-shift is attributed to the spin-orbit interaction in the reflective beam and contributed by both the reflective s- and p-waves. According to the Refs. [5, 42], when the helicity and linear polarization degree of the incident central wave are $\sigma { = }1$ and $\rho { = }0$ the IF-shift of the left circularly-polarized (LCP) beam can be written as

3. Goos-Hänchen shift of reflective beam

For the hBN layers in the mentioned structure in Fig. 1, the uniform expression of the two permittivity components is applied, i.e., ${\varepsilon _{h}} = {\varepsilon _\infty }[1 + (\omega _{{LO}}^2 - \omega _{{TO}}^2)/(\omega _{{TO}}^2 - \omega _{}^2 - {i}\omega {\tau _{h}})]$, where ${\varepsilon _\infty } = 4.95$, ${\omega _{LO}} = 825\textrm{cm}^{ - 1}$ and ${\omega _{TO}} = 760\textrm{cm}^{ - 1}$ for the longitudinal component ${\varepsilon _{/{/}}}$ and ${\varepsilon _\infty } = 4.52$, ${\omega _{LO}} = 1610\textrm{cm}^{ - 1}$ and ${\omega _{TO}} = 1{360c}{{m}^{ - 1}}$ for the transverse component ${\varepsilon _ \bot }$[31]. The damping coefficient of hBN is ${\tau _h} = 5\textrm{cm}^{ - {1}}$. We take ${f_t} = {\omega _{TO}} = 760\textrm{cm}^{ - 1}$ as the reference frequency. The hBN can be divided into two types. The type-I hyperbolic material (HM) corresponds to ${\varepsilon _ \bot } > 0$ and ${\varepsilon _{/{/}}} < 0$ and lies in the RB-I. On the contrary, the type-II one is situated in the RB-II where ${\varepsilon _ \bot } < 0$ and ${\varepsilon _{/{/}}} > 0$. On the other hand, the effective permittivity of graphene is indicated by ${\varepsilon _{g}} = 1 + {i}\sigma {/}{\varepsilon _0}\omega {{d}_{g}}$[12]. $\sigma$ is the conductivity of graphene and expressed by $\sigma = i{e^2}{\mu _c}/\pi {\hbar ^2}(\omega + i/{\tau _g})$ in the case of ignoring the interband transition, where ${e},{\mu _c},\omega ,{\tau _g}$ and $\theta$ are electronic charge, chemical potential, frequency, scattering time and Planck constant, respectively. The thickness of graphene is fixed at ${d_g} = 2{nm}$ and the damping coefficient of graphene is ${\tau _g} = 4{ps}$. In the following simulation, the effect of ${\mu _{c}}$, ${{f}_{h}}$ and $\phi$ on the shifts is investigated when the incident angle is taken as the critical or Brewster angle.

We first investigate the distribution of the RBs of graphene/hBN metamaterials versus the frequency and the chemical potential or the filling ratio. Four RBs are presented in Fig. 2(a) and (b), where $\varepsilon _ \bot ^e > 0$ and $\varepsilon _{{/{/}}}^e < 0$ or $\varepsilon _ \bot ^e < 0$ and $\varepsilon _{{/{/}}}^e > 0$. The yellow regions represent the front three RBs where the metamaterial is the type-I HMM ($\varepsilon _ \bot ^e > 0$ and $\varepsilon _{{/{/}}}^e < 0$) and the cyan one indicates RB4 where the metamaterial is the type-II HMM ($\varepsilon _ \bot ^e < 0$ and $\varepsilon _{{/{/}}}^e > 0$). In the other frequency regions, it is elliptical. Figure 2(a) shows the effect of ${\mu _c}$ on the RBs at ${f_h} = 0.7$. It is clearly seen that the frequency windows of RB3 and RB4 become wide and move to the higher frequency regions with the increasing ${\mu _{c}}$. But the frequency windows of RB1 and RB2 almost keep unchanged. Figure 2(b) indicates that the frequency windows of RBs will shrink versus the increasing ${f_h}$ at ${\mu _c}{ = 0}{.9eV}$. In addition, the RB1 and RB4 disappear once ${f_h} = 1.0$ which proves that they should originate from the effect of graphene. An interesting phenomenon should be noticed that a type transition of HM is occurred at ${f_h} = 0.976$ in RB2. The permittivity of graphene (${\varepsilon _{g}}$) and the effective permittivity ($\varepsilon _ \bot ^e$, $\varepsilon _{/{/}}^e$) versus ${\mu _c}$ at ${f_h} = 0.7$ is presented in Fig. 2(c), (d) and (e). The imaginary part of ${\varepsilon _{g}}$ is very large in the low-frequency range and approaches zero rapidly with the increase of frequency. An epsilon-near-zero (${\varepsilon _{{NZ}}}$) of the real part pointed by the arrow moves to the higher frequency with the increasing ${\mu _c}$, as shown in Fig. 2(c). In Figs.2(d) and (e), the imaginary parts of the effective permittivity are very large at the boundaries of RBs. Especially, the right boundary of RB4 exactly accords with the transition of the ${\varepsilon _{{NZ}}}$.

 

Fig. 2. RBs distribution diagram of graphene/hBN metamaterials versus chemical potential and frequency at $f_h = 0.7$ in (a) and versus filling ratio and frequency at $\mu _c = 0.9{\rm eV}$ in (b). Permittivity of graphene versus frequency at $\mu _c = 0.36{\rm eV},{\rm }0.9{\rm eV}$ in (c). Principal values of effective permittivity versus frequency at (d) $\mu _c = 0.36{\rm eV}$ and (e) $\mu _c = 0.9{\rm eV}$.

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In the following paragraphs, the GH-shift and reflectivity of the p-wave in the four RBs versus the tilted angle $\phi$ and frequency are examined in Fig. 3 when ${f_h}{ = }0.7$, ${\mu _c}{ = 0}{.9eV}$ and $\theta = {60^0}$. It can be found in Fig. 3(a) that the total reflection occurs in or near the four RBs, especially, the two boundaries of the RB3 and RB4 intersect at a point where the permittivity just jumps from negative to positive (see Fig. 2(b)). And it is easy to confirm that the reflectivity is zero at this special point, which is shown in the inset. Figure 3(b) presents the two mini bands possessing large GH-shift intersect at this point. Fitting the two mini bands with Eqs. (14) and (16), we find that one of the mini bands with the high reflectivity corresponding to the critical angle, and the other one is relative to the Brewster angle. Interestingly, the GH-shift around it is giant. In addition, the reflectivity is very high for larger tilted angles in the RB3 and for smaller ones in the RB4. We will mainly discuss the GH-shift around the Brewster angle in the RB4 since the similar conclusion can be made in the RB3. In Fig. 3(a) the blue regions represent the large power consumption since the metamaterial is semi-infinite and the reflective ratio is low there. Figure 3(c) provides the changes of reflectivity versus the frequency at $\phi { = }{0^0}, {45^0}, {90^0}$. It is obvious that the resonance absorption is the maximum and happens at the two resonant frequencies of the hBN permittivity, where the reflective ratio is the minimum, apart from this, a large absorption is attributed to a high transmission.

 

Fig. 3. Dependence of (a) reflectivity and (b) GH-shift on tilted angle and frequency at ${f_h}{ = }0.7$, ${\mu _c} = 0.9{\textrm{eV}}$ and $\theta = {60^0}$. (c) Reflectivity versus frequency and titled angle at ${f_h}{ = }0.7$, ${\mu _c} = 0.9{\textrm{eV}}$ and $\theta = {60^0}$.

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Figure. 4 present the GH-shift, reflectivity at the incident angle equal to the critical angle, and the critical angle as functions of frequency for different tilted-angle $\phi$. The maximum of the GH-shift reaches about one thousand times the vacuum wavelength at $\phi = {29.2^0}$, where the critical angle just is the same as the Brewster angle, as shown in Fig. 4(a). The peak-value shift quickly reduces to less than ten times when $\phi$ is decreased to ${0^0}$. It means that the closer the OA is to the z-axis, the smaller the GH-shift is. The GH-shift keeps positive in the range of ${0^0} < \phi < {29.2^0}$. However, it becomes negative for $\phi > {29.2^0}$ and disappears for $\phi \ge {34.5^0}$. Thus, $\phi = {29.2^0}$ is a critical titled angle for the transition from the positive GH-shift to negative one. The negative GH-shift for $\phi = {33^0}$ is shown by the inset in Fig. 4(a). We find from Fig. 4(b) that the reflectivity around the Brewster angle obviously decreases when frequency approaches the frequency corresponding to the Brewster angle. The right branches of the reflectivity curves almost coincide for $\phi = {0^0}$ and ${10^0}$. The inset is put into Fig. 4 (b) to show the slight difference between them. It can be seen that the considerable GH-shift with high reflectivity exist still in the range away from the Brewster angle, where the shift is at the critical angle. For example, at $\theta = {\theta _c} = {48.2^0}$ and $\omega /{{f}_t} = 17.037$, $\Delta {{x}_{p}}/{\lambda _0} = 116$ and ${R} = 0.4$. On the other hand, after the shift curves pass through the Brewster angle, the shift will attenuate quickly and tend to be placid with the increase of frequency regardless of the titled angle, and moreover the value of the GH-shift can be kept about ten times the vacuum wavelength in a wider frequency window.

 

Fig. 4. (a) GH-shift, (b) reflectivity and (c) critical angle versus frequency and tilted angle at ${f_h}{ = }0.7$ and ${\mu _c} = 0.9{\textrm{eV}}$. The intersection points of the curves with the vertical dot line in (c) correspond to the Brewster angle (also see Figs. 5(c) and 6(c)).

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It is an interesting topic to examine the effect of the graphene on the GH-shifts. Figure 5 shows the GH-shift and reflectivity versus frequency and critical angle for the various ${f_h}$ when $\phi = {25^0}$ and ${\mu _c}{ = 0}{.9eV}$. With the decreasing of ${f_h}$ the GH-shift is obviously enhanced at both the Brewster angle and the critical angle. In addition, the frequency window of evident GH-shift is also greatly expanded. The influence of ${f_h}$ on the reflectivity in the region of ${\theta _{c}} < {\theta _{b}}$ can be slight since the reflectivity remains unchanged. It can be realized that the GH-shift is improved without sacrificing the reflectivity by reducing the filling ratio of hBN appropriately. In the right region (${\theta _{c}} > {\theta _{b}}$) for ${f_h} < 0.8$, the GH-shift is larger still and is accompanied by a higher reflectivity in a relative wider frequency window. Another way of tuning the GH-shift is to change the chemical potential ${\mu _c}$ of the graphene, which can be easily controlled by an applied gate-voltage in the experiment. Figure 6 presents the GH-shift, reflectivity and the critical angle for various ${\mu _c}$ at $\phi = {25^0}$ and ${f_h} = 0.7$. With the increase of ${\mu _c}$, the frequency window of evident GH-shift moves to a higher frequency range and slightly become wider, which are consistent with Fig. 2(a). Moreover, the peak of GH-shift is obviously enhanced but the change of reflectivity dip is not evident. Thus, the frequency window of GH-shift can be selected according to the requirement in experiment. Finally, the GH-shift and reflectivity versus incident angle are investigated for different frequencies, as demonstrated by Fig. 7. Two shift-peaks can be observed in Fig. 7(a). Combing Figs. 5 and 6 with Fig. 7, we find that the higher peak corresponds to the Brewster angle and the other lower peak locates at the critical angle. For example, the green curve shows that the lower peak-value is $\Delta {x_p}/{\lambda _0} =$52 and corresponds to $\theta = {\theta _c} = {49.35^0}$ and $R =$0.78, and the higher peak-value is $\Delta {x_p}/{\lambda _0} =$512 corresponding to $\theta = {\theta _b} = {48.1^0}$ and $R \approx$0. Figure 7(b) illustrates the reflectivity related to the GH-shift. We find that the shift peak at the critical angle corresponds to the rapid change of the reflectivity but the peak at the Brewster angle does not.

 

Fig. 5. (a) GH-shift, (b) reflectivity and (c) critical angle with frequency and different filling ratio at $\phi = {25^0}$ and ${\mu _c} = 0.9{\textrm{eV}}$. The solid points in (c) correspond to the Brewster angle.

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Fig. 6. (a) GH-shift, (b) reflectivity and (c) critical angle versus frequency and different chemical potential at $\phi = {25^0}$ and ${f_h}{ = }0.7$. The solid points in (c) correspond to the Brewster angle.

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Fig. 7. (a) GH-shift and (b) reflectivity versus incident angle and different frequency at ${f_h}{ = }0.7$, $\phi = {25^0}$ and ${\mu _c} = 0.9{\textrm{eV}}$.

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In general, the GH-shift at the Brewster angle is very large, but the reflectivity is very small. Fortunately, the weak measurement can be established to observe the shifts in experiment [43, 44]. For hBN, we can promote the quality of the crystal to minimize the optical loss and further increase reflectivity and the shift. The imaginary parts of the principal values of the effective permittivity represent the loss of materials which are proportional to the damping of the graphene and hBN. From Eqs.(6) and (16) the GH-shift at the Brewster angle also can be observed with the decreasing of damping.

4. Imbert-Fedorov shift of reflective beam

The geometry of the IF-shift is also depicted in Fig. 1. The left circularly-polarized Gaussian beam illuminates the surface. Firstly, we consider the influence of titled angle $\phi$ on the IF-shift, as shown in Fig. 8(a). The two bands with extremely large shift-value are observed in RB3 and RB4, which are surrounded by the white dot lines. With the increase of $\phi$, the position and width of the band almost keep unchanged in RB3, but some significant changes take place in RB4. In the large-angle range, or the OA axis tends to parallel to the x-axis, it is easy to obtain the maximum of IF-shift. In the following discussion, we select the tilted angle of $\phi = {60^0}$ and the incident angle of $\theta = {10^0}$. The IF-shift versus frequency and the chemical potential is exhibited at ${f_h}{ = }0.7$ in Fig. 8(b). As clearly seen, two bands with extremely large shift-values in RB3 and RB4 exhibit the blue shift with the increase of ${\mu _{c}}$. Away from the two frequency bands, the IF-shift rapidly decreases and even is lower than the vacuum wavelength. Finally, we consider adjusting the IF-shift by changing the filling ratio of the graphene or hBN shown in Fig. 8(c) for ${\mu _c} = 0.{9eV}$. The width of either band with extremely large shift increases with the increase of the graphene filling-ratio in RB3 and RB4. But a relative narrow band with obvious IF-shift appears around the RB1 and RB2 at the large filling ratio of hBN. In order to more intuitively explain the condition of large IF-shift, we illustrate the reflectivity and the phases of reflective coefficients with the IF-shift together, as shown in Fig. 9. Obviously, the two peaks of IF-shift are represented in Fig. 9(a). If we check the reflectivity and phase difference between reflective coefficients in Figs.9(b) and (c), it is found that the conditions, $|{{r_s}} |= |{{r_p}} |$ and ${\varphi _s} - {\varphi _p} = 2j\pi$, are exactly met at the maximum value of IF-shift. But when $|{{r_s}} |= |{{r_p}} |$ and ${\varphi _s} - {\varphi _p} = (2j - 1)\pi$, the IF-shift is zero and independent of the incident angle. In this case, the reflective light changes into the right polarized state from the left polarized state.

 

Fig. 8. IF-shift for $\theta = {10^0}$ versus frequency and tilted angle in (a) at ${f_h}{ = }0.7$ and ${\mu _c} = 0.9{\textrm{eV}}$, versus frequency and chemical potential in (b) at ${f_h}{ = }0.7$ and $\phi = {60^0}$, and frequency and filling ratio in (c) at ${\mu _c} = 0.9{\textrm{eV}}$ and $\phi = {60^0}$.

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Fig. 9. (a) IF-shift, (b) reflectivity and (c) phase of reflective coefficient for p- and s-waves versus frequency at ${f_h}{ = }0.7$, ${\mu _c} = 0.9{\textrm{eV}}$, $\phi = {60^0}$ and $\theta = {10^0}$.

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According to the expression of the reflective coefficients, under the conditions of the peak values $|{{r_s}} |= |{{r_p}} |$ and ${\varphi _s} - {\varphi _p} = 2j\pi$, the peak values also depend on the incident angle, i.e., the incident angle also changes the peak values of IF-shift. Figure 10(a) shows the positions of peak-values of IF-shift in the $\omega - \theta$ plane for various tilted angle. Figure 10(b) depicts the positions in the $\omega - \theta$ plane for various chemical potential.

 

Fig. 10. Positions of IF-shift peaks in the $\omega - \theta$ plane. (a) For different tilted angles at ${\mu _c} = 0.9eV$, ${f_h}{ = }0.7$ and (b) for various chemical potential at $\phi = {60^0}$, ${f_h}{ = }0.7$. The incident beam is left circularly-polarized.

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In above analysis, the incident plane wave is supposed to be the central ray of the incident beam. Based on this assumption, we first obtained the reflective coefficients and then used the given formulae to get the GH and IF shifts. For the previous GH and IF results, there is a condition that the waist of light beam is large enough, otherwise the results change with the waist. Subsequently, we study the effect of the waist on the spatial shifts by using the beam simulation and compare the simulated results with the above ones. We consider a Gaussian beam with the width ${{w}_0}$ and TM-polarization (in-plane polarization) as the incident beam. The electric field in the k-space is expressed by ${E_i} = {w_0}\exp ( - w_0^2{\mathbf k}_ \bot ^2/4)/\sqrt {2\pi }$ where ${k_ \bot }$ is the difference from the central wave vector. The GH- and IF-shifts can be defined as [5]

 

Fig. 11. (a) GH-shift and (b) IF-shift versus beam waist, obtained with the beam simulation method. The other parameters are the same as ones in Fig. 6 and Fig. 9, respectively.

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5. Conclusions

To conclude, we have analytically and numerically calculated the GH- and IF-shifts of reflective beam incident on the surface of a metamaterial composed of hBN and graphene layers, where the orientation of the surface, the chemical potential of graphene and the filling ratios can be modulated. We found that the GH-shift of the p-wave is very large at the critical angle near the Brewster angle, and the corresponding reflectivity is also high, even more than 0.8. The position and width of frequency windows for large GH- and IF-shifts can be effectively manipulated by adjusting the chemical potential, filling ratio of the graphene and also the orientation of the surface. In order to obtain the largest IF-shift, we investigated the relation between it and the chemical potential, filling ratio, the titled angle or the incident angle. In addition, the desired polarization with a higher IF-shift can be realized by tuning the polarized state of the incident light beam. It can be seen that the theoretical results obtained are reasonable when the waist is large enough, compared with the wavelength, otherwise they will smoothly change with the waist. These results provide a pathway for modulating the GH- and IF-shifts and thereby offer the possibility for developing new nano-optical devices.