1. Introduction

The shift between the actual reflected light beam and the theoretical reflection of the light beam in geometric optics will be produced under the total reflection condition. In 1947, F. Goos and H. Hanchen observed this phenomenon for the first time through experiments and named Goos-Hanchen (GH) shift [1]. The GH shift was explained theoretically by Artmann in 1948 [2]. In recent years, it has been widely explored in optical difference operation and image edge detection [3]. However, in general structure, the GH shift is on the same order of the magnitude as the wavelength of the incident beam, which has created a great resistance and obstacle to observe and measure experimentally. In addition, more and more researchers tried to enhance and regulate effectively the GH shift in different structures with the development of novel materials and structures [4–8], such as photonic crystals [9,10], metamaterials [11–13], ENZ materials [14,15] and PT-symmetric structures [16–19]. To realize the enhancement of the GH shifts, the study of the GH shift in the structure of graphene has become a favored topic for researchers with the discovery of graphene as a zero-thickness interface [20–24].

In recent decades, due to the characteristic of the GH shift in the PT-symmetry structure, which has been paid much attention by researchers [16,17]. In 1998, Bender et al. proposed firstly that the PT-symmetry exhibited real eigenvalues [25]. In optics, the PT-symmetry needs to satisfy the refractive index distribution of $n(\vec {r})=n^{*}(-\vec {r})$, which can be used to design the optical PT-symmetry structure with balanced gain and loss [26]. The gain can be obtained by nonlinear two-wave mixing or doped Ge/Cr fiber [27], and the loss can be achieved by acoustic modulator [28]. In detail, the PT-symmetry can be achieved by time-alternating gain and loss of the optical amplifier and the amplitude modulator in the two loops, or by adding Si and germanium (Ge)/ Chromium (Cr) double-layer combinatorial mode effective indices consistent with real and imaginary modulation, respectively. The latter has been verified by the FDTD simulation [27,28]. The PT-symmetry has many related optical applications in the fields of coherent perfect laser absorbers, unidirectional invisibility, optical switches and asymmetric transmission, which provides more ideas for the research of optical devices [29–31]. Moreover, the PT-symmetric system has the exceptional points (EPs) when the system is located at the critical point of the phase transition between the symmetric phase and the broken phase, both the real and the imaginary parts of multiple eigenvalues degenerate [32]. It has been reported that the GH shifts became very large at the EPs in PT-symmetric photonic crystals [33]. The Bragg resonance usually occurs in the PT-symmetric periodic structures. The Bragg resonance refers to the phenomenon of total reflection caused by the interaction between the wave and the periodic structure in the propagation direction of a specific frequency [34,35]. Simultaneously, the large GH shifts may exist in PT-symmetric periodic dielectric multilayers [9] and non-Hermitian system constructed by incorporating graphene in dielectric multilayer [18]. However, there has been little research on using the EPs to control the GH shift in the PT-symmetric multilayer coating graphene structures.

In this work, we construct an asymmetric system coated with graphene on PT-symmetric multi-period structures. And we also investigate the control of the GH shift by the EPs and the GH shift of beams reflected from multi-period structures. Here, we show the values of the GH shifts are converted with the change of the EPs in the parameter space composed of the dielectric constant and the incident angle. Also, the EPs and the GH shifts are direction-dependent when the beam comes in from opposite sides. We further explore the control of the GH shifts by changing the Fermi energy of graphene under the condition of the dielectric constant fixed. In addition, the GH shifts enlarge with the number of periods in the multi-period case. In particular, the GH shift is independent of the incident direction at the high reflectivity due to the Bragg resonance.

2. Model and theory

We consider a Gaussian beam is incident from the air to the periodic PT-symmetric multilayer-structure coated with a graphene layer as shown in Fig. 1. The PT-symmetric structure consists of a gain medium layer and a loss medium layer without magnetism with the same thickness ${d}$. The dielectric constants of the gain and the loss media are $\epsilon _u=\epsilon _r-i\epsilon _i$ and $\epsilon _d=\epsilon _r+i\epsilon _i$, respectively. There are three Cartesian reference frames: the laboratory frame $\left \{x,y,z\right \}$ attached to the graphene-coated surface, and $\left \{x_i,y_i,z_i\right \}$ are the incident and $\left \{x_r,y_r,z_r\right \}$ the reflected field coordinates, respectively.

 

Fig. 1. Schematic of a periodic PT-symmetric system coated with a single graphene (a) when the light incident from the graphene side and (b) when the light incident from the side which is without graphene. (c) The graphene conductivity vs the Fermi energy. $\epsilon _u$ and $\epsilon _d$ represent the dielectric constant of the gain layer and the loss layer of the PT-symmetric structure, respectively. The thickness of each layer ${d}$ is 0.2 $\mu m$, and the wavelength $\lambda$ is 1550 $nm$.

Download Full Size | PDF

The reflection coefficients and transmission coefficients of the system can be calculated by the transfer matrix method (TMM) [17], which can be represented as

Here, $\sigma$ is the conductivity of graphene, $\varepsilon _{0}$ represents the dielectric constant in vacuum, and $\omega$ denotes the angular frequency of the incident light. $\eta _{0}=k_{0 z} / n_{0}^{2}$, where $k_{0 z}=k_{0} \cos \theta$ indicates the wave vector of the initial wave along the z-axis, $k_0=2 \pi / \lambda$ represents the wave vector in vacuum, and $n_{0}$ represents the refractive index in vacuum. Furthermore, according to semiconductor theory, the optical conductivity of graphene can be obtained as [36]

The total transfer matrix of the whole system when the beam passes through N periods is $M=\left (m_{u} m_{d}\right )^{N}$ when the beam passes through N periods, and the transmission matrix of the jth layer of the system can be expressed as follow

The reflection coefficient and the transmission coefficient of the system, in the case of only incident from the upper side, can be expressed as $r_{U}=H_{b}^{-} / H_{f}^{-}$, $t_{U}=H_{f}^{+} / H_{f}^{-}$. Conversely, in case of incident from the down side only, these coefficients are $r_{D}=H_{f}^{+} / H_{b}^{+}$, $t_{D}=H_{b}^{-} / H_{b}^{+}$. The subscript ${U(D)}$ denotes the beam incident from the upper(down) side. Consequently, the reflection coefficient and the transmission coefficient from the upper and down sides of the system can be derived as follow

Meanwhile, the reflectivity and the transmittance of the system are $R_{U, D}=\left |r_{U, D}\right |^{2}$, $T_{U, D}=\left |t_{U, D}\right |^{2}$, respectively. According to the characteristics of the PT structure, the dispersion matrix of the system can be written as

The eigenvalues of the S-matrix are $\beta _{\pm }=t \pm \left (r_{U} r_{D}\right )^{\frac {1}{2}}$, hence the exceptional points (EPs) of the eigenvalues occur at $\left (r_{U} r_{D}\right )^{\frac {1}{2}}=0$ which means $R_{U}=0$ or $R_{D}=0$. At the same time, as the reflectivity is zero, the reflected phase $\phi _{U, D}$ changes directly from -$\pi$ to $\pi$. These characteristics can be used to find the EPs.

Next, we can get the GH shift of the incident beam on both sides through the stationary phase theory, which can be expressed as [37]

The detailed derivation of Eq. (8) is shown in the appendix.

3. Results and discussions

We initial study the EPs and the GH shifts of Gaussian beams with the PT-symmetric structure (N = 1). The dielectric constants of the gain layer and the loss layer of the PT-symmetric structure are $\epsilon _u=\epsilon _r-0.03i$, $\epsilon _d=\epsilon _r+0.03i$, respectively. The thickness of each layer ${d}$ is 0.2 $\mu m$, and the wavelength $\lambda$ is 1550 $nm$. In the follow discussions, the ${p}$ polarization is only considered since the little effect of the system for the ${s}$ polarization.

Firstly, we investigate the influence of the reflectivity ($R_U$) and the phase of the reflection ($\phi _U$) on the GH shifts ($\Delta _U$) at $E_f$ = 0.3 eV, under the condition that the beam is incident from the upper side (shown in Fig. 1(a)) of the system. Figure 2(a) describes the reflectivity as a function of the incident angle and the real part of the dielectric constant. We know that the EPs occur when the system’s reflectivity approaches zero (see dashed lines). At the same time, the parts surrounded by the EPs are divided into two parts (I, II) with the dashed lines as the dividing line. With different incident angles and real parts of the dielectric constant, the phases of the reflection are shown in Fig. 2(b). Both in Part I and Part II, the GH shifts are meaningless when the direct jump from dark blue to yellow in the diagram, which corresponds to the saltation directly from -$\pi$ to $\pi$ in the phase of the reflection [18]. And the regions where the phase dislocates are all within the range surrounded by the EPs (see dashed lines). According to the Eq. (4), the GH shifts are affected by the reflection coefficient phase, and then Fig. 2(c) describes the GH shifts in the case of the incident angle $\theta$ and the real part of the dielectric constant $\epsilon _r$ in the Part I. By comparing Figs. 2(a) and 2(c), we can get the larger GH shifts near the EPs (see dashed lines), which gradually decrease to zero farther away from the EPs. Figure 2(d) illustrates the GH shifts in parametric space of the Part II. In this part, the GH shifts have large positive and negative values, and there are positive and negative inversion of the GH shifts at $\theta$ = 50.58$^{\circ }$, $\epsilon _r$ = 1.37. Based on the above discussion, the position of the EPs of this system can be changed by adjusting the real part of the dielectric constant, and the GH shifts occur only around the EPs (see dashed lines). Through observation, we can obtain slight positive and negative GH shifts in the Part I of EPs, and large shifts in the Part II.

 

Fig. 2. The beam is incident from the upper side (shown in Fig. 1(a)) of the system. The dashed line in the figure indicates the position of EPs. (a) The reflectivity $R_U$ in the parametric space of the incident angle $\theta$ and the real part of the dielectric constant $\epsilon _r$. In order to show the GH shift more completely, the reflectivity is divided into two parts according to the range surrounded by the EPs, where the EPs are represented by dashed lines in the figure. (b) Phase of reflection coefficient $\phi _U$ as a function of the incident angle $\theta$ and the real part of the dielectric constant $\epsilon _r$. (c) Dependence of the GH shifts on the incident angle $\theta$ and the real part of the dielectric constant $\epsilon _r$ in the Part I. (d) The GH shifts in the Part II. Here, N=1, $\epsilon _u=\epsilon _r-0.03i$, $\epsilon _d=\epsilon _r+0.03i$, and $E_f$ = 0.3 eV.

Download Full Size | PDF

With the asymmetric system we considered, next, we study the EPs and the GH shifts as a function of $\theta$ and $\epsilon _r$ for the beam incident from the down side (shown in Fig.1 (b)) of the structure. Figure 3(a) shows the reflectivity in the parameter space. By comparison of the Fig. 2, it can be found that the regions surrounded by the EPs in Fig. 3(a) are opposite to those in Fig. 2(a). For the phase, the phase mutation occurs when the yellow part jumps directly to the dark blue part in Fig. 3(b), which is outside the range surrounded by the EPs. Figures 3(c) and 3(d) represent the GH shifts in the Part I and II respectively. As can be seen from Fig. 3(c), the GH shifts only have positive values near the EPs (see dashed lines) and gradually decrease along the EPs with the increase of the incident angle and the real part of the dielectric constant. In addition, the GH shifts are zero in other regions. In Fig. 3(d), as the incident angle $\theta$ or the real part of the dielectric constant $\epsilon _r$ gradually increases, the GH shifts near the EPs (see dashed lines) rise from small values to larger ones. When $\theta$ = 52$^{\circ }$, $\epsilon _r$ = 1.73, significantly, the GH shifts change from negative values to positive ones, and then decrease gradually with the increasing of $\theta$ or $\epsilon _r$. Therefore, compared the incident beam from the down side of the system with the upper side (Figs. 2 and 3), the regions surrounded by the EPs are opposite, and the values of the GH shifts can be obtained are different accordingly.

 

Fig. 3. The beam is incident from the down side (shown in Fig. 1(b)) of the system. The dashed line in the figure indicates the position of EPs. (a) The reflectivity $R_D$ in parametric space of the incident angle $\theta$ and the real part of the dielectric constant $\epsilon _r$. In order to show the GH shifts more completely, the reflectivity is divided into two parts according to the range surrounded by the EPs, where the EPs are represented by dashed lines in the figure. (b) Phase of reflection coefficient $\phi _D$ as a function of the incident angle $\theta$ and the real part of the dielectric constant $\epsilon _r$. (c) Dependence of the GH shifts on the incident angle $\theta$ and the real part of the dielectric constant $\epsilon _r$ in the Part I. (d) The GH shifts in the Part II. Other parameters are the same as those in Fig. 2.

Download Full Size | PDF

Then, we discuss the influence of graphene on the GH shifts of the system which take the incident beam from the upper side (shown in Fig. 1(a)) as an example. We study the GH shifts as functions of the incident angle $\theta$ at different Fermi energy with the real part of the dielectric constant $\epsilon _r$ = 0.5, as shown in Figs. 4(a), (b) and (c). The GH shifts are always positive and rise with the increase of Fermi energy at $E_f$ < 0.4 eV. On the contrary, at 0.4 eV< $E_f$ < 0.48 eV , the GH shifts become negative and also increase with Fermi energy. When $E_f$ > 0.48 eV, the GH shifts, which are always positive, decrease with the rise of $E_f$. Meanwhile, the incident angles at which the maximum shifts occur are also changing. Figure 4(d) depicts the dependence of the GH shifts on the incident angle $\theta$ under three different conditions: at $E_f$ = 0.45 eV with graphene, at $E_f$ = 0.52 eV with graphene, and without graphene. Obviously, we find that in the presence of graphene, the incident angles of the maximum shifts are different at $E_f$ = 0.45 eV and $E_f$ = 0.52 eV, which are $\theta$ = 35.03$^{\circ }$ and $\theta$ = 35.14$^{\circ }$, respectively, and the sign of the GH shifts is opposite. In contrast, the GH shifts are almost zero without graphene. And the real and imaginary parts of the graphene dielectric constant vary with the Fermi energy as we can see from Fig. 4(e). This result shows that the graphene coating can greatly enhance the magnitude of the GH shifts compared with non-graphene. In addition, we can adjust the incident angle at which the GH shifts occur, as well as the sign and values of the GH shifts by changing the Fermi energy of graphene.

 

Fig. 4. The beam is incident from the upper side (shown in Fig. 1(a)). The GH shifts as a function of the incident angle $\theta$ (a) when $E_f<0.4 eV$, (b) when $0.4 eV<E_f<0.48 eV$, (c) when $E_f>0.48 eV$. (d) The GH shifts as a function of the incident angle $\theta$ for the system with graphene at $E_f$ = 0.45 eV (dashed blue line) and $E_f$ = 0.52 eV (solid red line) and without grapheme (dotted black line). (e) The real (dashed red line) and imaginary (solid blue line) parts of the graphene dielectric constant as a function of the Fermi energy. Here, N = 1, $\epsilon _u = 0.5 - 0.03i$, $\epsilon _d = 0.5 + 0.03i$.

Download Full Size | PDF

The above discussions are based on the PT-symmetric structure with a single period (N = 1). Finally, we consider the EPs and the GH shifts of the system influenced by the periodicity of the PT-symmetric structure. Figure 5 shows the reflectivity and the GH shifts when the beam is incident from the upper (solid line) and down (dashed line) sides, respectively, in the PT-symmetric structure with N = 10 and N = 15 at $E_f$ = 0.3 eV and $E_f$ = 0.39 eV. With the increase of the period N, the numbers and values of the GH shifts are increasing gradually, and the incident angles generating the maximal GH shifts are also shifting along with it. Moreover, in addition to the large GH shifts at the EPs, the GH shifts are also generated at the maximum reflectivity due to the Bragg resonance [17] (shown in the dashed lines in Figs. 5(a) and 5(b)), but the peak values at this time are smaller than that at the EPs. It is worth noting that the angles producing the Bragg resonance are the same regardless of the beam incident from the upper or the down side. Furthermore, as we can see from the insets of Figs. 5(c) and 5(d), the GH shifts generated by the beam on both sides completely overlap at the Bragg resonance. In contrast, at the EPs the GH shifts are opposite, indicating that the symmetry of the system has no effect on the GH shift at the Bragg resonance. On the other hand, as is shown in Figs. 5(e) and (f), the GH shifts from both sides are negative and some values increase obviously at $E_f$ = 0.39 eV. Therefore, the influence of graphene on the GH shifts of the system is still significant in the case of multiple periods. To sum up, we can adjust the position of the EPs of the system through the period number of the PT-symmetric structure and the direction of the incident at the dielectric constant remaining unchanged. Meanwhile we can also control the values and the sign of the GH shifts, and obtain the large GH shifts which can be positive or negative.

 

Fig. 5. Reflectivity and the GH shifts of the beam at (a), (c), (e) N = 10 and (b), (d), (f) N = 15 from the upper (shown in Fig. 1(a)) side (solid line) and the down (shown in Fig. 1(b)) side (dotted line) of the system, respectively. In addition, (c) and (d) show the GH shifts at $E_f$= 0.3 eV. (e) and (f) show the GH shifts at $E_f$= 0.39 eV. Here, $\epsilon _u = 0.5 - 0.03i$, $\epsilon _d = 0.5 + 0.03i$.

Download Full Size | PDF

4. Conclusion

In summary, we investigate the EPs and the GH shifts in the PT-symmetric multilayer structures with the graphene coating. We find that the position of the EPs can be adjusted by the real part of the dielectric constant and the incident angle, and the EPs can decide the value of the GH shifts. Moreover, we can obtain a large range of the value of the GH shifts at different EPs. For the PT-symmetric structure with single period, large positive and negative GH shifts occur only near the EPs, while the GH shifts in other regions are very small or even close to zero. Therefore, the large GH shifts only produce around the EPs, and the GH shifts can be controlled by the position of the EPs in the PT-symmetric systems. Under the dielectric constant fixed, the GH shifts can be adjusted by the Fermi energy of graphene. The GH shifts increase with the Fermi energy. The shifts are always positive in the case of $E_f$ < 0.4 eV and $E_f$ > 0.48 eV, while become negative in the case of 0.4 eV< $E_f$ < 0.48 eV. Meanwhile, the number of EPs and the value of the GH shifts also increase with the number of the periods of the PT-symmetric structure, and the maximum GH shifts can even reach $10^3$ wavelength. Also, the Fermi energy still has important influence on the GH shifts in the case of multiple periods. According to the Bragg resonance of the lattice, there are GH shifts at the higher reflectivity. The range of EPs and the value of the GH shift are different, when the beam is incident from the upper side of the structure and from the down side of the structure, while the GH shifts at the Bragg resonance are entirely coincident. Therefore, in our structure, the GH shifts can be controlled by adjusting different structural parameters. Our results have potential applications in the design and production of high sensitivity optical devices.

5. Appendix

The theoretical equation of the stationary phase method is:

(9)$$\psi(y)=\int_{-\infty}^{+\infty} A\left(k_{y}\right) \exp \left[i \phi\left(k_{y}, y\right)\right] d k_{y},$$

where $A\left (k_{y}\right )$ is unimodal positive definite function of $k_{y0}$, and $\phi \left (k_{y}\right )$ represents the total phase shift. The first order Taylor expansion of $\phi \left (\mathrm {k}_{y}, y\right )$ at $k=k_0$ can be expressed:

(10)$$\phi\left(k_{y}, y\right) \approx \phi\left(k_{y 0}, y\right)+\frac{\partial \phi\left(k_{y}, y\right)}{\partial k_{y 0}}\left(k_{y}-k_{y 0}\right),$$

then substitute Eq. (10) into Eq. (9) to get:

(11)$$\psi(\mathrm{y}) \approx \exp \left[i \phi\left(k_{y 0}, y\right)\right] \int_{-\infty}^{+\infty} A\left(k_{y}\right) \exp \left[i \frac{\partial \phi}{\partial k_{y 0}}\left(k_{y}-k_{y 0}\right)\right] d k_{y},$$

the condition for $\psi (\mathrm {y})$ to be at its maximum is

(12)$$\frac{\partial \phi\left(\mathrm{k}_{y,} y\right)}{\partial k_{y 0}}=0.$$

As shown in Fig. 6, we consider a beam in the air incident into the interface of the two media. The beam is uniformly distributed in the $z$ direction with an angular frequency of $\omega$, and $\theta_0$ is the incident angle corresponding to the central wave vector of the beam. The refractive index of the medium is $n_1$ and $n_2$, respectively.

 

Fig. 6. The GH shifts diagram of light beam with total reflection on the two dielectric interfaces with refractive index $n_1$ and $n_2$ respectively.

Download Full Size | PDF

Next, vector decomposition of the incident beam can be obtained:

(13)$$\psi_{\textrm{in }}(x, y)=\int_{-\infty}^{+\infty} A\left(k_{y}\right) \exp \left[i\left(k_{1 x} x+k_{y} y\right)\right] d k_{y},$$

when the beam is on the plane, which is $x=0$, we can get:

(14)$$\psi_{\textrm{in }}(0, y)=\int_{-\infty}^{+\infty} A\left(k_{y}\right) \exp \left(i k_{y} y\right) d k_{y},$$

it can be obtained by comparing Eq. (9):

(15)$$\phi\left(\mathrm{k}_{y}, y\right)=k_{y}y,$$

substituting Eq. (15) into Eq. (12), we can get that the peak value of incident light is located at $y_{in}=0$.

The incident beam is composed of a series of plane wave components, some of which incident at a critical angle greater than or less than total reflection, so the amplitude reflection coefficient of each plane wave component in the incident beam is:

(16)$$R\left(\mathrm{k}_{\mathrm{y}}\right)=\exp \left[i \varphi\left(\mathrm{k}_{\mathrm{y}}\right)\right].$$

In the actual reflection process, the plane wave component of the reflected beam is reflected with $R\left (\mathrm {k}_{\mathrm {y}}\right )$ as the reflection coefficient, so the actual reflected beam can be expressed as:

(17)$$\begin{aligned} \psi_{\mathrm{re}}&=\int A\left(\mathrm{k}_{\mathrm{y}}\right) R\left(\mathrm{k}_{\mathrm{y}}\right) \exp \left[i\left({-}k_{1 x} \mathrm{x}+k_{y} y\right)\right] d \mathrm{k}_{\mathrm{y}},\\ &=\int A\left(\mathrm{k}_{\mathrm{y}}\right) \exp \left[i\left( \varphi\left(\mathrm{k}_{\mathrm{y}}\right)-k_{1 x} x+k_{y} y\right)\right] d k_{y}, \end{aligned}$$

when the exit plane is at $x=0$, we can obtain $\phi \left (\mathrm {k}_{\mathrm {y}}, y\right )=\varphi \left (\mathrm {k}_{\mathrm {y}}\right )+\mathrm {k}_{\mathrm {y}} y$, and then by substituting it into Eq. (12) the peak value of the actual reflected beam can be obtained :

(18)$$\begin{aligned} &\frac{\partial\varphi\left(\mathrm{k}_{\mathrm{y}}\right)}{\partial\mathrm{k}_{\mathrm{y}}}+\mathrm{y}_{\mathrm{re}}=0,\\ &\mathrm{y}_{\mathrm{re}}={-}\frac{\partial\varphi\left(\mathrm{k}_{\mathrm{y}}\right)}{\partial\mathrm{k}_{\mathrm{y}}}, \end{aligned}$$

therefore the geometric displacement of the actual reflected beam is

(19)$$\Delta \mathrm{y}=\mathrm{y}_{\mathrm{re}}-\mathrm{y}_{\mathrm{in}}={-}\frac{\partial \varphi}{\partial k_{0}}.$$

The projection of the displacement on the direction perpendicular to the wave vector of the reflected beam is

(20)$$\begin{aligned} &\mathrm{~d}_{G H}=\Delta y \cos \theta={-}\frac{\partial \varphi}{\partial k_{y 0}} \cos \theta,\\ &k_{y 0}=k_{1} \sin \theta_{0},\\ &\mathrm{~d}_{G H}={-}\frac{\partial \varphi}{\partial k_{1} \sin \theta_{0}} \cos \theta_{0},\\ &\mathrm{~d}_{G H}={-}k_{1} \frac{\partial \varphi}{\partial \theta}. \end{aligned}$$

Thus, Eq. (8) is proven.

Funding

Science and Technology Program of Guangzhou City (2019050001); National Natural Science Foundation of China (11775083, 12174122).

Disclosures

The authors declare no conflicts of interest.

Data availability

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

References

1. F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 436(7-8), 333–346 (1947). [CrossRef]  

2. K. Artmann, “Berechnung der Seitenversetzung des totalreflektierten Strahles,” Ann. Phys. 437(1-2), 87–102 (1948). [CrossRef]  

3. D. Xu, S. He, J. Zhou, S. Chen, S. Wen, and H. Luo, “Goos–Hänchen effect enabled optical differential operation and image edge detection,” Appl. Phys. Lett. 116(21), 211103 (2020). [CrossRef]  

4. M. Gao and D. Deng, “Spatial Goos-Hänchen and Imbert-Fedorov shifts of rotational 2-D finite energy Airy beams,” Opt. Express 28(7), 10531–10541 (2020). [CrossRef]  

5. W. Zhen and D. Deng, “Giant Goos–Hänchen shift of a reflected spin wave from the ultrathin interface separating two antiferromagnetically coupled ferromagnets,” Opt. Commun. 474, 126067 (2020). [CrossRef]  

6. W. Zhen and D. Deng, “Goos–Hänchen shift for elegant Hermite–Gauss light beams impinging on dielectric surfaces coated with a monolayer of graphene,” Appl. Phys. B 126(3), 35 (2020). [CrossRef]  

7. W. Zhen and D. Deng, “Goos–Hänchen and Imbert–Fedorov shifts in temporally dispersive attenuative materials,” J. Phys. D: Appl. Phys. 53(25), 255104 (2020). [CrossRef]  

8. Q. Liu, W. Zhen, M. Gao, and D. Deng, “Goos-Hänchen and Imbert-Fedorov shifts for the rotating elliptical Gaussian beams,” Results Phys. 18, 103297 (2020). [CrossRef]  

9. L.-G. Wang and S.-Y. Zhu, “Giant lateral shift of a light beam at the defect mode in one-dimensional photonic crystals,” Opt. Lett. 31(1), 101–103 (2006). [CrossRef]  

10. S. Longhi, G. Della Valle, and K. Staliunas, “Goos-Hänchen shift in complex crystals,” Phys. Rev. A 84(4), 042119 (2011). [CrossRef]  

11. B. Zhao and L. Gao, “Temperature-dependent Goos-Hänchen shift on the interface of metal/dielectric composites,” Opt. Express 17(24), 21433–21441 (2009). [CrossRef]  

12. W. Zhen, D. Deng, and J. Guo, “Goos-Hänchen shifts of Gaussian beams reflected from surfaces coated with cross-anisotropic metasurfaces,” Opt. Laser Technol. 135, 106679 (2021). [CrossRef]  

13. R. Liu, G. Wang, D. Deng, and T. Zhang, “Spin Hall effect of Laguerre-Gaussian beams in PT symmetric metamaterials,” Opt. Express 29(14), 22192–22201 (2021). [CrossRef]  

14. Y. Xu, C. T. Chan, and H. Chen, “Goos-Hänchen effect in epsilon-near-zero metamaterials,” Sci. Rep. 5(1), 8681 (2015). [CrossRef]  

15. J. Wen, J. Zhang, L.-G. Wang, and S.-Y. Zhu, “Goos–Hänchen shifts in an epsilon-near-zero slab,” J. Opt. Soc. Am. B 34(11), 2310–2316 (2017). [CrossRef]  

16. Y.-L. Chuang and R.-K. Lee, “Giant Goos-Hänchen shift using PT symmetry,” Phys. Rev. A 92(1), 013815 (2015). [CrossRef]  

17. P. Ma and L. Gao, “Large and tunable lateral shifts in one-dimensional PT-symmetric layered structures,” Opt. Express 25(9), 9676–9688 (2017). [CrossRef]  

18. D. Zhao, S. Ke, Q. Liu, B. Wang, and P. Lu, “Giant Goos-Hänchen shifts in non-Hermitian dielectric multilayers incorporated with graphene,” Opt. Express 26(3), 2817–2828 (2018). [CrossRef]  

19. Y. Cao, Y. Fu, Q. Zhou, Y. Xu, L. Gao, and H. Chen, “Giant Goos-Hänchen shift induced by bounded states in optical PT-symmetric bilayer structures,” Opt. Express 27(6), 7857–7867 (2019). [CrossRef]  

20. S. Chen, C. Mi, L. Cai, M. Liu, H. Luo, and S. Wen, “Observation of the Goos-Hänchen shift in graphene via weak measurements,” Appl. Phys. Lett. 110(3), 031105 (2017). [CrossRef]  

21. X. Zhou, W. Cheng, S. Liu, J. Zhang, C. Yang, and Z. Luo, “Tunable and high-sensitivity temperature-sensing method based on weak-value amplification of Goos–Hänchen shifts in a graphene-coated system,” Opt. Commun. 483, 126655 (2021). [CrossRef]  

22. W. Zhen and D. Deng, “Goos-Hänchen shifts for Airy beams impinging on graphene-substrate surfaces,” Opt. Express 28(16), 24104–24114 (2020). [CrossRef]  

23. A. Pitilakis, D. Chatzidimitriou, T. V. Yioultsis, and E. E. Kriezis, “Asymmetric Si-slot coupler with nonreciprocal response based on graphene saturable absorption,” IEEE J. Quantum Electron. 57(3), 1–10 (2021). [CrossRef]  

24. D. Chatzidimitriou and E. E. Kriezis, “Optical switching through graphene-induced exceptional points,” J. Opt. Soc. Am. B 35(7), 1525–1535 (2018). [CrossRef]  

25. C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998). [CrossRef]  

26. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010). [CrossRef]  

27. L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2013). [CrossRef]  

28. A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity–time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012). [CrossRef]  

29. Y. Chong, L. Ge, and A. D. Stone, “PT-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106(9), 093902 (2011). [CrossRef]  

30. Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011). [CrossRef]  

31. M. Sakhdari, M. Hajizadegan, Q. Zhong, D. N. Christodoulides, R. El-Ganainy, and P.-Y. Chen, “Experimental observation of PT symmetry breaking near divergent exceptional points,” Phys. Rev. Lett. 123(19), 193901 (2019). [CrossRef]  

32. X. Zhou, X. Lin, Z. Xiao, T. Low, A. Alù, B. Zhang, and H. Sun, “Controlling photonic spin Hall effect via exceptional points,” Phys. Rev. B 100(11), 115429 (2019). [CrossRef]  

33. Ş. K. Özdemir, S. Rotter, F. Nori, and L. Yang, “Parity–time symmetry and exceptional points in photonics,” Nat. Mater. 18(8), 783–798 (2019). [CrossRef]  

34. P. St. J. Russell, “Bragg resonance of light in optical superlattices,” Phys. Rev. Lett. 56(6), 596–599 (1986). [CrossRef]  

35. Z. Tao, W. He, Y. Xiao, and X. Wang, “Wide forbidden band induced by the interference of different transverse acoustic standing-wave modes,” Appl. Phys. Lett. 92(12), 121920 (2008). [CrossRef]  

36. X. Zhou, S. Liu, Y. Ding, L. Min, and Z. Luo, “Precise control of positive and negative Goos-Hänchen shifts in graphene,” Carbon 149, 604–608 (2019). [CrossRef]  

37. L.-G. Wang, H. Chen, and S.-Y. Zhu, “Large negative Goos–Hänchen shift from a weakly absorbing dielectric slab,” Opt. Lett. 30(21), 2936–2938 (2005). [CrossRef]