Tunable in-plane photonic spin Hall effect for vortex structured light on a ReS<sub>2</sub>-graphene heterostructure

Yiyu Shi; Zhiwei cui; Junting He; Wenqian Gong; Xinxing Zhou

doi:10.3788/COL202523.100002

1. Introduction

The photonic spin Hall effect (PSHE) refers to the in-plane and transverse beam centroid shifts of the reflected or refracted left-handed circularly polarized (LCP) and right-handed circularly polarized (RCP) components compared to the incident linearly polarized spatial confined light in gradient-index media^[1]. Such a phenomenon originates from the spin-orbit interaction at the optical interface^[2]. In this process, the polarization vector of each angular spectrum component rotates and interferes in momentum space, then results in the redistribution of the light intensity, which manifests itself as the spin-dependent shift of the beam centroid in real space^[3–5]. Besides, many works point out that the spatial beam mode can significantly impact such beam shifts^[6–8]. In contrast to the mainly studied Gauss mode, the vortex structured light, such as the Laguerre–Gauss mode^[9–11] possessing the intrinsic orbital angular momentum, will experience the orbital Hall effect at the optical interface aside from the spin Hall effect (SHE)^[12–14]. It brings extra unilateral orbital-angular-momentum-dependent shifts and provides additional freedom to control the PSHE, which raises more and more research interest recently^[15–20]. When the shifts of LCP and RCP components coincide, the in-plane shift is equivalent to the Goos–Hänchen shift and has the advantage of direct observation compared with the transverse shift. It has potential applications in optical sensing and metrology^[21–24]. Most of these applications rely on the effective modulation of the in-plane shift. Therefore, exploring the active and dynamical method to manipulate the in-plane shift of PSHE for LGM is desirable.

There have been some works devoted to this topic. Li et. al studied the tunable in-plane shift of PSHE in layered dielectric structure^[25]. Lin et. al investigated the thickness-dependent in-plane shift of PSHE in an anisotropic medium^[26]. Subsequently, Qin et. al explored the adjustable in-plane shift of PSHE based on the surface plasmon resonance^[27]. The methods to control the in-plane PSHE in these studies concern varying the thickness of materials, which is complicated and hard to apply in practice. In recent years, the ${ReS}_{2}$ , as a representative two-dimensional (2D) layered transition metal dichalcogenide, has captivated the scientific community’s attention due to its excellent optoelectronic properties. Owing to the formation of the Re-Re chain, the ${ReS}_{2}$ has in-plane low symmetry, exhibiting polarization-dependent photo-response^[28], orientation-dependent polarized direct optical transition^[29], anisotropic transient absorption^[30], anisotropic conductivity^[31], crystalline-axis-dependent electronic change^[32], and so on. Such in-plane anisotropic optical property offers additional freedom to control the light field. Via the interlayer van der Waals coupling engineering, the heterostructures (HSs) composed of ${ReS}_{2}$ also promise exotic properties and promote the development of the optoelectronic devices. Lately, there have been some works devoted to manipulating the beam shifts via the HSs composed of ${ReS}_{2}$ . Li et al. and Yan et al. successively conducted investigations on adjustable Goos-Hänchen shifts based on the Au- ${ReS}_{2}$ -graphene HS^[33] and ${ReS}_{2}$ -black phosphorus HS^[34]. These works indicate that HSs composed of ${ReS}_{2}$ could be used to manipulate the in-plane shifts of PSHE for LGM by rotating the HS flexibly rather than changing the physical properties of the HS.

Here, we conduct a comprehensive investigation of the in-plane shifts of PSHE for LGM at the glass-air interface coated with ${ReS}_{2}$ -graphene HS. First, the in-plane spatial and angular shifts for arbitrary linearly polarized LGM are derived. Then, we numerically simulate the in-plane spatial and angular shifts for different incident angles, polarization angles, topological charges of incident LGM, and the rotation angles of the ${ReS}_{2}$ . It is found that varying the topological charge or rotation angle can amplify or suppress the in-plane PSHE for the case of different incident angles. It is worth noting that the anomalous enhancement in the in-plane PSHE far from the critical angle will occur with specific polarization angles. The mechanism behind this phenomenon originates from the near-zero value of the reflection coefficient for the circularly polarized components.

2. Results and Discussion

As demonstrated in Fig. 1, we establish a reflection model to describe the in-plane shifts of PSHE for LGM at the glass-air interface coated with ${ReS}_{2}$ -graphene HS. When a linearly polarized LGM reflects at this interface, the in-plane spatial $δ_{\pm}^{x}$ and angular $Δ_{\pm}^{x}$ shifts of left- and right-handed circularly polarized components will emerge. Here, the $z$ axis of the global coordinate system $(x, y, z)$ is normal to the glass-air interface. The local coordinates $(x_{i}, y_{i}, z_{i})$ and $(x_{r}, y_{r}, z_{r})$ are respectively used to describe the incident and reflected light. The angular spectrum of the incident LGM is ${\tilde{E}}_{i} (k_{i x}, k_{i y}) = \frac{w_{0}^{2}}{4 π} {\frac{w_{0} [- i k_{i x} + sign (l) k_{i y}]}{\sqrt{2}}}^{| l |} e^{- (k_{i x}^{2} + k_{i y}^{2}) w_{0}^{2} / 4},$ (1)where $w_{0}$ is the beam waist, $| l |$ is the absolute value of topological charge $l$ , $sign (\cdot)$ is the sign function, and $k_{i x}$ and $k_{i y}$ are the incident wave vector $k_{0}$ components along $x$ and $y$ axes, respectively. The polarization states of the beam can be expressed by the Jones vector ${(\cos β, \sin β)}^{T}$ , where $β$ equals 0° or 90°, denoting the $p$ - or $s$ -polarization. The reflected angular spectrum is calculated under the paraxial case as^[35] $[\begin{matrix} {\tilde{E}}_{r}^{p} \\ {\tilde{E}}_{r}^{s} \end{matrix}] = [\begin{matrix} r^{p} - \frac{k_{r x}}{k_{0}} \frac{\partial r^{p}}{\partial θ_{i}} & \frac{k_{r y}}{k_{0}} \cot θ_{i} (r^{p} + r^{s}) \\ - \frac{k_{r y}}{k_{0}} \cot θ_{i} (r^{p} + r^{s}) & r^{s} - \frac{k_{r x}}{k_{0}} \frac{\partial r^{s}}{\partial θ_{i}} \end{matrix}] [\begin{matrix} {\tilde{E}}_{i}^{p} \\ {\tilde{E}}_{i}^{s} \end{matrix}],$ (2)where $r^{s}$ and $r^{p}$ represent the overall Fresnel reflection coefficients for $s$ - and $p$ -polarization, respectively, and $θ_{i}$ is the incident angle. Using the relationships $k_{r x} = - k_{i x}$ , $k_{r y} = k_{i y}$ , and ${\tilde{E}}_{r σ} = ({\tilde{E}}_{r}^{p} + i σ {\tilde{E}}_{r}^{s}) / \sqrt{2}$ , we can get the reflected electric field $E_{r σ} (x_{r}, y_{r}, z_{r})$ of the LCP ( $σ = 1$ ) and RCP ( $σ = - 1$ ) as $E_{r σ} = \iint_{\infty} {\tilde{E}}_{r σ} e^{i [k_{r x} x_{r} + k_{r y} y_{r} - z_{r} (k_{r x}^{2} + k_{r y}^{2}) / (2 k_{0})]} d k_{r x} d k_{r y} = [\begin{matrix} a (\frac{- i | l | sign (l) x_{r} + | l | y_{r}}{r^{2}} + \frac{i k_{0} x_{r}}{z_{R} + i z_{r}}) \\ + b (\frac{- | l | x_{r} - i | l | sign (l) y_{r}}{r^{2}} + \frac{i k_{0} y_{r}}{z_{R} + i z_{r}}) + c \end{matrix}] Φ,$ (3)where $a = - k_{0}^{- 1} \partial r^{p} / \partial θ_{i} \cos β + i σ k_{0}^{- 1} \partial r^{s} / \partial θ_{i} \sin β,$ (4a) $b = i σ k_{0}^{- 1} (r^{p} + r^{s}) \cot θ_{i} e^{- i σ β}, c = r^{p} \cos β - i σ r^{s} \sin β,$ (4b)and $Φ = {[\frac{i sign (l) y_{r} - x_{r}}{w}]}^{| l |} {[\frac{\sqrt{2} (1 - i z_{r} / z_{R})}{\sqrt{1 + {(z_{r} / z_{R})}^{2}}}]}^{| l |} \frac{e^{- \frac{(x_{r}^{2} + y_{r}^{2}) / w_{0}^{2}}{1 + i z_{r} / z_{R}}}}{1 + i z_{r} / z_{R}},$ (5)with $w = w_{0} \sqrt{1 + {(z / z_{R})}^{2}}$ and $z_{R} = k_{0} w_{0}^{2} / 2$ . It can be seen that the reflected electric field and in-plane shift depend not only on the absolute value of $l$ but also on the sign of $l$ . The total in-plane beam shifts of PSHE can be derived as $δ_{σ}^{x} + Δ_{σ}^{x} = \frac{\iint_{\infty} x_{r} {| E_{r σ} |}^{2} d x_{r} d y_{r}}{\iint_{\infty} {| E_{r σ} |}^{2} d x_{r} d y_{r}} = \frac{A - | l | B}{C - | l | D + | l |^{2} E},$ (6)in which $A = w^{2} [2 ρ Re (c) + 2 τ Im (c)] S_{1} / 8,$ (7a) $B = [2 χ Re (c) - 2 sign (l) ξ Im (c)] S_{2},$ (7b) $C = S_{1} k_{0}^{2} w^{2} (| a |^{2} + | b |^{2}) / (z_{R}^{2} + z_{r}^{2}) / 8 - | c |^{2} S_{4},$ (7c) $D = 2 {[ρ - sign (l) κ] χ + [γ + sign (l) τ] ξ} S_{2},$ (7d) $E = 8 S_{3} (χ^{2} + ξ^{2}) / w^{2},$ (7e)with $ρ = k_{0} \frac{Re (a) z_{r} - Im (a) z_{R}}{z_{R}^{2} + z_{r}^{2}}, γ = k_{0} \frac{Re (b) z_{r} - Im (b) z_{R}}{z_{R}^{2} + z_{r}^{2}},$ (8a) $τ = k_{0} \frac{Re (a) z_{R} + Im (a) z_{r}}{z_{R}^{2} + z_{r}^{2}}, κ = k_{0} \frac{Re (b) z_{R} + Im (b) z_{r}}{z_{R}^{2} + z_{r}^{2}},$ (8b) $χ = sign (l) Im (a) - Re (b), ξ = sign (l) Im (b) + Re (a),$ (8c) $S_{1} = \sum_{s = 0}^{| l |} C_{| l |}^{s} H_{2 s + 2} (0) H_{2 | l | - 2 s} (0),$ (8d) $S_{2} = \sum_{s = 0}^{| l | - 1} C_{| l | - 1}^{s} H_{2 s + 2} (0) H_{2 | l | - 2 s - 2} (0),$ (8e) $S_{3} = \sum_{s = 0}^{| l | - 1} C_{| l | - 1}^{s} H_{2 s} (0) H_{2 | l | - 2 s - 2} (0),$ (8f) $S_{4} = \sum_{s = 0}^{| l |} C_{| l |}^{s} H_{2 s} (0) H_{2 | l | - 2 s} (0),$ (8g)with $C_{n}^{m}$ being the combinatorial number and $H_{n}^{m}$ being the Hermite polynomial. If $l = 0$ , Eq. (6) can degenerate to the form of the Gauss mode^[35].

Figure 1.(a) Schematic diagram of the in-plane spatial and angular shifts of PSHE for LGM. (b) Configuration of glass-air interface coated with ReS₂-graphene HS. The relationships between angular shifts and deflections are Δ_±^x = z_rϕ_±^x. Θ is the rotation angle between the incident light’s polarization plane and the Re-Re chain.

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In this work, the overall $r^{s}$ and $r^{p}$ in the multilayered structure can be written as^[36] $r^{p (s)} = \frac{r_{01}^{p (s)} + r_{12}^{p (s)} e^{2 i k_{1} d_{1}} + e^{2 i k_{2} d_{2}} [e^{2 i k_{1} d_{1}} + r_{01}^{p (s)} r_{12}^{p (s)}] r_{23}^{p (s)}}{1 + e^{2 i k_{2} d_{2}} r_{12}^{p (s)} r_{23}^{p (s)} + e^{2 i k_{1} d_{1}} r_{01}^{p (s)} [r_{12}^{p (s)} + e^{2 i k_{2} d_{2}} r_{23}^{p (s)}]},$ (9)where $d_{j}$ is the thickness of the medium with relative refractive index $n_{j}$ , $k_{j}$ denotes the wave numbers in different media, and $r_{i j}^{p (s)}$ represents the reflection coefficients between different media, with subscripts $i, j = 0, 1, 2, 3$ standing for glass, ${ReS}_{2}$ , graphene, and air, respectively. Among them, the complex relative refractive index $n_{1}$ of the ${ReS}_{2}$ is^[33] $n_{1} = 1 / \sqrt{\cos^{2} Θ / n_{a} + \sin^{2} Θ / n_{b}},$ (10)in which $Θ$ is the angle between the polarization plane of the incident light and the Re-Re chain, called the rotation angle. $n_{a} = 4.20 + 0.77 i$ and $n_{b} = 3.93 + 0.39 i$ are the complex refractive indices of the parallel and orthogonal Re-Re chain directions of ${ReS}_{2}$ at the incident wavelength $λ = 0.6328 μm$ . It can be seen that the anisotropy of ${ReS}_{2}$ can be expressed from both the real and imaginary parts of the complex refractive indices. The imaginary part is related to absorption. Hence, the absorption coefficient of ${ReS}_{2}$ is obviously polarization-dependent, exhibiting highest and lowest when the polarization direction of the incident light is parallel or orthogonal to the Re-Re chain, respectively.

In the following numerical simulations, if not otherwise stated, we assume that topological charge $l = 1$ , the incident wavelength in air $λ = 0.6328 μm$ , the beam waist $w_{0} = 27 μm$ , the relative refractive index of graphene $n_{2} = 3 + 1.15 i$ , the relative refractive index of glass $n_{0} = 2.115$ ^[37] (the matched critical angle $θ_{c} = 28.21 °$ ), the thickness of ${ReS}_{2}$ $d_{1} = 0.70 nm$ , the thickness of graphene $d_{2} = 0.34 nm$ , and the propagation distance $z_{r} = 250 mm$ for angular shift. We plot the $δ_{+}^{x} - δ_{-}^{x}$ and $Δ_{+}^{x} - Δ_{-}^{x}$ in the following figures to represent the degree of in-plane PSHE, which show the differences between the spatial shifts and angular shifts of LCP and RCP components, respectively.

According to previous work^[35], the in-plane spatial and angular shifts at the glass-air interface will be enhanced near the critical angle and the impacts of physical parameters on them are more obvious than those far from the critical angle. Thus, we examine the in-plane shifts near the critical angle $θ_{c} = 28.21 °$ first. Figure 2 depicts the differences between in-plane spatial shifts $δ_{\pm}^{x}$ and angular shifts $Δ_{\pm}^{x}$ of LCP and RCP components with different rotation angles $Θ$ , incident angles $θ_{i}$ , and polarization angles $β$ at the glass-air interface coated with ${ReS}_{2}$ -graphene HS or not, near the critical angle $θ_{c}$ . When $θ_{i} < θ_{c}$ , the introduction of ${ReS}_{2}$ -graphene HS dramatically enhances the $Δ_{+}^{x} - Δ_{-}^{x}$ , which is always zero at the glass-air interface^[35]. The reason is that the total Fresnel reflection coefficients are purely real in the glass-air model when $θ_{i} < θ_{c}$ , so the spin-dependent components in the analytical expression of angular shift are zero. However, the introduction of ${ReS}_{2}$ -graphene HS leads to the complex total Fresnel reflection coefficients, and these spin-dependent components are no longer zero any more. In contrast to that in the glass-air model, the magnitude of $δ_{+}^{x} - δ_{-}^{x}$ is weakened to some extent. It is worth noting that the sign of $δ_{+}^{x} - δ_{-}^{x}$ turns from negative to positive for all $β$ as the $Θ$ changes from 0° to 90°, accompanied by the magnitude of $δ_{+}^{x} - δ_{-}^{x}$ being suppressed and then amplified. For the case of $θ_{i} > θ_{c}$ , the $δ_{+}^{x} - δ_{-}^{x}$ is obviously enhanced by the introduction of ${ReS}_{2}$ -graphene HS but becomes robust against $Θ$ . Additionally, the $Δ_{+}^{x} - Δ_{-}^{x}$ can be tuned from a positive value to a negative value for all $β$ by increasing $Θ$ from 0° to 90°. These phenomena indicate that the in-plane PSHE can be amplified, suppressed, and reversed by changing the $Θ$ .

Figure 2.Differences between in-plane spatial shifts δ_±^x and angular shifts Δ_±^x of LCP and RCP components with different rotation angles Θ, incident angles θ_i, and polarization angles β at the glass-air interface coated with ReS₂-graphene HS or not, near the critical angle θ_c. (a), (b) δ₊^x−δ₋^x; (c), (d) Δ₊^x−Δ₋^x.

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To obtain a clear physical picture concerning the impact of the rotation angle on the in-plane PSHE of LGM, Fig. 3 is introduced to reveal the normalized field distribution $| E_{r σ} |^{2}$ of the reflected LCP and RCP components at the glass-air interface coated with ${ReS}_{2}$ -graphene HS for different rotation angles $Θ$ near the critical angle $θ_{c}$ . In addition, the propagation distance $z_{r} = 250 mm$ , incident angle $θ_{i} = 28.22 °$ , and polarization angle $β = 78 °$ are fixed. Under this condition, the angular shifts take dominance in beam displacements. For $Θ = 90 °$ , the reflected LCP and RCP components both have $x$ -direction displacements, and the $Δ_{+}^{x} - Δ_{-}^{x} = - 0.17 mm$ .

Figure 3.Normalized field distribution |E_rσ|² of the reflected LCP and RCP components at the glass-air interface coated with ${ReS}_{2}$ -graphene HS for different rotation angles Θ near the critical angle θ_c, with propagation distance z_r = 250 mm, incident angle θ_i = 28.22°, and polarization angle β = 78°. The dashed lines denote the x_r = 0 and y_r = 0, and the solid lines represent the y_r = y_c (y_c is the horizontal position of the reflected beam centroid). (a)–(c) LCP and (d)–(f) RCP components.

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At the same time, the transverse beam displacements are too tiny to observe directly. If $Θ$ is varied to 45°, the displacements of the reflected LCP and RCP components are increased and decreased, respectively, and the $Δ_{+}^{x} - Δ_{-}^{x}$ is amplified with the sign switched. When $Θ = 0 °$ , we can clearly see that the reflected LCP and RCP components separate into opposite directions, which means that the corresponding spin-dependent displacements are very large. In this case, the $Δ_{+}^{x} - Δ_{-}^{x}$ is enhanced to 0.75 mm. These phenomena are all consistent with those in Fig. 2(d).

Next, the impacts of topological charge $l$ on the differences between in-plane spatial $δ_{\pm}^{x}$ and angular $Δ_{\pm}^{x}$ shifts of LCP and RCP components with different incident angles $θ_{i}$ and polarization angles $β$ at the glass-air interface coated with ${ReS}_{2}$ -graphene HS near the critical angle $θ_{c}$ are depicted in Fig. 4. It can be seen that the $δ_{+}^{x} - δ_{-}^{x}$ will be enhanced by increasing the value of $| l |$ when $β$ is close to 90°. By reversing the sign of $l$ from positive to negative, the magnitude of $δ_{+}^{x} - δ_{-}^{x}$ will be amplified when $θ_{i} < θ_{c}$ but suppressed if $θ_{i} > θ_{c}$ . For the $Δ_{\pm}^{x}$ , with increasing the value of $| l |$ , the $Δ_{+}^{x} - Δ_{-}^{x}$ can be enhanced with arbitrary $β$ . Meanwhile, the $Δ_{+}^{x} - Δ_{-}^{x}$ is not sensitive to the sign switch of $l$ . These phenomena verify the validity of manipulating the in-plane PSHE for LGM via changing the topological charge.

Figure 4.Impacts of topological charge l on the differences between in-plane spatial shifts δ_±^x and angular shifts Δ_±^x of LCP and RCP components with different incident angles θ_i and polarization angles β at the glass-air interface coated with ReS₂-graphene HS for the case of rotation angle Θ = 0°, near the critical angle θ_c. (a), (b) δ₊^x−δ₋^x; (c), (d) Δ₊^x−Δ₋^x.

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Aside from the enhancements near the critical angle $θ_{c}$ , we find that there exist anomalous enhanced in-plane spatial $δ_{\pm}^{x}$ and angular $Δ_{\pm}^{x}$ shifts of PSHE for LGM far from the critical angle $θ_{c}$ at the glass-air interface coated with ${ReS}_{2}$ -graphene HS, as plotted in Fig. 5. When $Θ = 0 °$ , the $δ_{+}^{x} - δ_{-}^{x}$ will be sharply enhanced near the $θ_{i} \approx 58.24 ° ≫ θ_{c}$ with $β \approx 72.46 °$ . It is worth noting that if we vary the $β$ when $θ_{i}$ is fixed at 58.24°, the sign of $δ_{+}^{x} - δ_{-}^{x}$ will reverse near $β \approx 72.46 °$ , which also does not appear in the glass-air model. If $Θ$ turns from 0° to 45°, $δ_{+}^{x} - δ_{-}^{x}$ is amplified near $θ_{i} \approx 63.81 °$ and $β \approx 84.22 °$ , also accompanied by an abrupt sign reversal across $β$ . Then we proceed to increase $Θ$ to 90°, and the $θ_{i}$ and $β$ corresponding to the enhancement of $δ_{+}^{x} - δ_{-}^{x}$ are increased to 67.54° and 81.36°. Analogous to the $δ_{+}^{x} - δ_{-}^{x}$ , $Δ_{+}^{x} - Δ_{-}^{x}$ is also amplified near these special $θ_{i}$ and $β$ . Differently, the sign reversal of the enhanced $Δ_{\pm}^{x}$ emerges across the $θ_{i}$ rather than the $β$ , which is similar to the enhancement of the in-plane PSHE across the critical angle. These phenomena imply that the introduction of ${ReS}_{2}$ -graphene HS provides an additional incident angle to enhance the in-plane PSHE, and this angle can be tailored by altering the rotation angle.

Figure 5.Anomalous enhanced in-plane spatial shifts δ_±^x and angular shifts Δ_±^x of PSHE for LGM with different rotation angles Θ, incident angles θ_i, and polarization angles β at the glass-air interface coated with ReS₂-graphene HS, far from the critical angle θ_c. (a)–(c) δ₊^x−δ₋^x; (d)–(f) Δ₊^x−Δ₋^x.

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To explore the underlying mechanisms behind the anomalous enhanced in-plane spatial $δ_{\pm}^{x}$ and angular $Δ_{\pm}^{x}$ shifts presented in Fig. 5, we plot the magnitude and phase of terms $k_{0} a$ , $k_{0} b$ , and $c$ in Eq. (6) with different incident angles $θ_{i}$ and polarization angles $β$ at the glass-air interface coated with ${ReS}_{2}$ -graphene HS in Fig. 6. Among them, the term $c$ represents the reflection coefficient of the LCP or RCP components for the arbitrary linearly polarized incident LGM. For convenience, we only analyze the cases for LCP component $σ = 1$ and rotation angle $Θ = 0 °$ . From Figs. 6(a) and 6(b), we can see that the magnitude of term $c$ is sharply decreased, accompanied by an abrupt phase jump near $β \approx 72.46 °$ when $θ_{i} \approx 58.24 °$ . Meanwhile, the magnitudes and phases of the terms $k_{0} a$ and $k_{0} b$ are steady. In actuality, the term $| c |^{2}$ is dominant in the denominator of Eq. (6) and the $δ_{\pm}^{x} \propto ϕ (c) / | c |$ under these conditions. Therefore, it leads to the enhancement of the $δ_{\pm}^{x}$ and the sign reversal across $θ_{i} \approx 58.24 °$ in Fig. 5(a), which is analogous to the enhanced transverse shifts of PSHE near the Brewster angle.

Figure 6.(a), (c) Magnitude and (b), (d) phase of the terms k₀a, k₀b, and c in Eq. (6) with different incident angles θ_i and polarization angles β at the glass-air interface coated with ReS₂-graphene HS, for LCP component σ = 1 and rotation angle Θ = 0°.

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A similar phenomenon can be seen in Figs. 6(c) and 6(d); the amplification and sign switch of $Δ_{\pm}^{x}$ in Fig. 5(d) can be ascribed to the near-zero value of term $c$ and phase jump of term $c$ around $θ_{i} \approx 58.24 °$ if $β \approx 72.46 °$ .

3. Conclusion

In summary, we have conducted a comprehensive investigation on the in-plane PSHE of an arbitrary linearly polarized LGM at the glass-air interface coated with ${ReS}_{2}$ -graphene HS. The general expressions of the in-plane spatial and angular shifts of PSHE for the LGM were derived. It was shown that the variations of topological charge of the LGM and the rotation angle of the ${ReS}_{2}$ all have obvious influences on the in-plane shifts of PSHE near the critical angle, including enhancing or weakening the magnitude of shifts and reversing their signs. Besides, the anomalous enhancement in the in-plane PSHE far from the critical angle with specific polarization angles was found. It rises from the sharp decrease of the reflection coefficient for the circularly polarized components. These findings may be valuable for the adjustment of the in-plane shifts of PSHE and provide valid guidance for the development of the optical sensor based on vortex structured light.

Special Issue: SPECIAL ISSUE ON STRUCTURED LIGHT: FUNDAMENTALS AND APPLICATIONS

Received: Apr. 23, 2025

Accepted: May. 23, 2025

Published Online: Sep. 8, 2025

The Author Email: Zhiwei cui (zwcui@mail.xidian.edu.cn), Xinxing Zhou (xinxingzhou@hunnu.edu.cn)

DOI:10.3788/COL202523.100002

CSTR:32184.14.COL202523.100002