Tunable in-plane and transverse spin angular shifts in layered dielectric structure

Lijuan Sheng; Lijuan Sheng; Linguo Xie; Linguo Xie; Jingjing Sun; Sixian Li; Yaodong Wu; Yu Chen; Xinxing Zhou; Xinxing Zhou; Zhiyou Zhang; Zhiyou Zhang

doi:10.1364/OE.27.032722

1. Introduction

The photonic spin Hall effect (SHE), an analogical phenomenon to electronic SHE [1], appears as an in-plane and transverse spin-dependent shifts when a light beam is reflected or refracted at the medium interface [2–9]. The physical mechanism of photonic SHE is due to spin-orbit coupling which describes the interaction between the photonic spin and beam trajectory [10,11]. In recent years, the photonic SHE has caused great concern. It has not only been widely studied in various research realms such as optical physics [12–16], plasmonics [17–19], semiconductor physics [20], anisotropic materials [21,22], and metamaterials [23–27], but also been demonstrated its value in practical application such as precision measurement [28,29] and spin-based nanophotonic devices [10]. At the same time, it has been found that the spin displacements generated by the photonic SHE consist of two parts, i.e., spatial shift and angular shift [15,30]. Among them, the angular shift is expressed that the displacement of the center of the beam varies with the propagation distance. Due to the special feature of spin angular shifts, many researchers focus on this topic. For example, in the complex medium, Aiello et al. have revealed the duality between the angular and spatial shifts [30]. By employing an array of subwavelength metallic structure, Kang et al. presented a polarization-dependent angular shifts [31]. When the light beam is reflected near the critical angle, the spin angular shift can be greatly enhanced [32]. Generally, the spin angular shifts are relatively larger than the spatial shifts. Thus, it holds great potential application in spin-optics.

However, there still lacks an effective way for manipulating the spin angular shift, especially for enhancing or suppressing its magnitude or switching its sign at will, limiting its practical application. In this paper, we propose an effective and simple method for modulating the spin angular shifts by introducing the layered structure. The general expressions of the in-plane and transverse spin angular shifts have been derived in this model. We find that the in-plane and transverse spin angular shifts can be effectively regulated by adjusting the structure parameters of layered model. In addition, the angular shifts can represent spin-dependent or spin-independent properties due to the different incident polarization states. Interestingly, as for the vertical polarization beam, the spin angular shifts can also be amplified. The rest of the paper is structured as follows. Firstly, we theoretically study the in-plane and transverse photonic spin angle splitting of the reflected beam in layered structure. Here, the general and simplified angular shift expressions are derived and analyzed in detail. Then, we explore some interesting phenomena of spin angular shift and reveal the corresponding physical mechanisms. Finally, a conclusion is given.

2. Theoretical analysis

We propose a five-layer structure for controlling the spin angular shifts as shown in Fig. 1. For this model, we consider the cases of both symmetry and asymmetry. The wavelength in this work is chosen as 632.8 nm. In the case of symmetry, it consists of two layers of BK7 plate glass (${n_2} = {n_4} = 1.515$ at 632.8 nm), two media of air (${n_1} = {n_5} = 1$ at 632.8 nm) and one layer of MgF₂ (${n_3} = 1.377$ at 632.8 nm) [33]. The upper layer and the lower layer are both air layers, the middle layer is MgF₂, and the BK7 glass plate is located between the air layer and MgF₂, as shown in Fig. 1(a). The non-symmetric case is formed by using an S-LAH79 slab (${n_4} = 1.996$ at 632.8 nm) instead of the second BK7 glass plate described above. The photonic SHE occurs when a bundle of arbitrary linearly polarized Gaussian beam is incident at the layered structure. Therefore, it produces in-plane and transverse angular shifts as shown in Figs. 1(b) and 1(c), respectively.

Fig. 1. (a) The photonic SHE of arbitrary linear polarization beam (with polarization angle $\beta$) in the reflection of a layered structure model; (b) The in-plane spin angular shifts; (c) The transverse spin angular shifts. Among them, [${n_1}$, ${n_2}$, ${n_3}$, ${n_4}$, ${n_5}$] and [${d_2}$, ${d_3}$, ${d_4}$] are the refractive indices and thicknesses of the respective media, respectively. ${\theta _i}$ is the incident angle. The subscripts + and - indicate the left- and right-handed circularly polarized beams, respectively. In addition, $\Delta $ and $\delta$ stand for angular and spatial shifts, respectively.

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We will conduct a theoretical analysis of the in-plane and transverse angle splitting of the above-mentioned layered structure. Firstly, we consider an incident beam with an arbitrary linearly polarized Gaussian distribution. Its angular spectrum can be written as [7]

(1)$${\tilde{E}_i}({k_{ix}},{k_{iy}}) = \frac{{{w_0}}}{{\sqrt {2\pi } }}\exp [ - \frac{{w_0^2({k_{ix}}^2 + {k_{iy}}^2)}}{4}], $$

where ${w_0}$ is the waist. ${k_{ix}}$ and ${k_{iy}}$ denote incident wave-vectors along the x direction and the y direction, respectively. After the coordinate transformation, the relationship between the established reflection field and the incident field is as follows

(2)$$\left( {\begin{array}{c} {\tilde{E}_r^H}\\ {\tilde{E}_r^V} \end{array}} \right) = \left( {\begin{array}{cc} {r_p} & {\frac{{{k_{ry}}({r_p} + {r_s})\cot {\theta_i}}}{{{k_0}}}}\\ { - \frac{{{k_{ry}}({r_p} + {r_s})\cot {\theta_i}}}{{{k_0}}}} & {r_s} \end{array}}\right)\left( {\begin{array}{c} {\tilde{E}_i^H}\\ {\tilde{E}_i^V} \end{array}} \right).$$

Here, ${r_p}$ and ${r_s}$ represent the Fresnel reflection coefficients under the layered structure, respectively. H and V stand for the horizontal and vertical polarizations. It should be noted that the above equation also needs to be multiplied by ${\left( {\begin{array}{ll} {\cos \beta }&{\sin \beta } \end{array}} \right)^T}$ to indicate arbitrary linearly polarized beam. Among them, $\beta$ is the polarization angle. The linearly polarized beam is decomposed into left- and right-handed circularly polarized components, namely $\tilde{E}_i^H = {{({{\tilde{E}}_{i + }} + {{\tilde{E}}_{i - }})} \mathord{\left/ {\vphantom {{({{\tilde{E}}_{i + }} + {{\tilde{E}}_{i - }})} {\sqrt 2 }}} \right.} {\sqrt 2 }}$ and $\tilde{E}_i^V = {{i({{\tilde{E}}_{i - }} - {{\tilde{E}}_{i + }})} \mathord{\left/ {\vphantom {{i({{\tilde{E}}_{i - }} - {{\tilde{E}}_{i + }})} {\sqrt 2 }}} \right.} {\sqrt 2 }}$. Combining Eqs. (1) and (2) gives a reflection angle spectrum. Then, the electric field expressions of the left- and right-handed circularly polarized components are obtained by Fourier transform. In the given z-plane, the in-plane and transverse angular displacements are obtained as follows [11]

(3)$$\Delta {x_\sigma } = \frac{z}{k}\frac{{\int_{ - \infty }^{ + \infty } {\int_{ - \infty }^{ + \infty } {{k_{rx}}{{\tilde{E}}_\sigma }\tilde{E}_\sigma ^ \ast d{k_{rx}}d{k_{ry}}} } }}{{\int_{ - \infty }^{ + \infty } {\int_{ - \infty }^{ + \infty } {{{\tilde{E}}_\sigma }\tilde{E}_\sigma ^ \ast d{k_{rx}}d{k_{ry}}} } }}, $$

(4)$$\Delta {y_\sigma } = \frac{z}{k}\frac{{\int_{ - \infty }^{ + \infty } {\int_{ - \infty }^{ + \infty } {{k_{ry}}{{\tilde{E}}_\sigma }\tilde{E}_\sigma ^ \ast d{k_{rx}}d{k_{ry}}} } }}{{\int_{ - \infty }^{ + \infty } {\int_{ - \infty }^{ + \infty } {{{\tilde{E}}_\sigma }\tilde{E}_\sigma ^ \ast d{k_{rx}}d{k_{ry}}} } }}. $$

Here $\sigma ={\pm} 1$ corresponds to the left- and right-handed circularly polarized components, respectively. $\Delta {x_\sigma }$ and $\Delta {y_\sigma }$ express the displacements that vary with the propagation distance in parallel and perpendicular to the plane of incidence, respectively. z is the propagation distance. We know that the spatial and angular shifts can be seen as spin-dependent displacements that are independent of z (propagation distance) and related to z, respectively. In fact, in this work, we define the angular deflection $\phi {x_ \pm }$ or $\phi {y_ \pm }$ into a spatial displacement $\Delta {x_ \pm }$ or $\Delta {y_ \pm }$ for the convenience of calculation and practical measurement, as shown in Fig. 1. The relationship between these two definitions are $\Delta {x_ \pm } = z\phi {x_ \pm }$ or $\Delta {y_ \pm } = z\phi {y_ \pm }$. The physical mechanism is the same.

Considering that we propose a five-layer structure model, the Fresnel coefficient is no longer simple, that is, there are reflection and refraction coefficients in each layer of medium, and it is not a simple superposition. First, we use the boundary conditions to find the general formula for the reflection and transmission coefficients of each layer of medium. Then, the reflection coefficient of the arbitrary wave-vector component is determined on the model using the transformation and transmission matrix [33]. Expanding its Taylor series to the first-order term leads to

(5)$${r_p} = {r_p}({{\theta_i}} )+ \frac{{\partial {r_p}}}{{\partial {\theta _i}}}\frac{{{k_{ix}}}}{k}, $$

(6)$${r_s} = {r_s}({{\theta_i}} )+ \frac{{\partial {r_s}}}{{\partial {\theta _i}}}\frac{{{k_{ix}}}}{k}. $$

Among them, the subscripts p, s denote the H and V directions, respectively. It should be emphasized that the Fresnel coefficient is a complex number in our model. Hence, the above factors need to be further expressed, that is, ${r_p}({\theta _i}) = p{e^{i\rho }}$, ${{\partial {r_p}} \mathord{\left/ {\vphantom {{\partial {r_p}} {\partial {\theta_i}}}} \right.} {\partial {\theta _i}}} = \chi {e^{i\xi }}$, ${r_s}({\theta _i}) = s{e^{i\gamma }}$, and ${{\partial {r_s}} \mathord{\left/ {\vphantom {{\partial {r_s}} {\partial {\theta_i}}}} \right.} {\partial {\theta _i}}} = \eta {e^{i\varepsilon }}$. Finally, for an arbitrary linearly polarized incident beam, the in-plane and transverse angular shifts of the left- and right-handed circularly polarized beams are obtained as follow:

(7)$$\Delta {x_\sigma } ={-} \frac{{z[{A{{\cos }^2}\beta + B{{\sin }^2}\beta + \sigma C\sin ({2\beta } )} ]}}{{D{{\cos }^2}\beta + E{{\sin }^2}\beta + \sigma F\sin ({2\beta } )}}, $$

(8)$$\Delta {y_\sigma } ={-} \frac{{z\cot {\theta _i}[{G\sin ({2\beta } )- \sigma H\cos ({2\beta } )} ]}}{{D{{\cos }^2}\beta + E{{\sin }^2}\beta + \sigma F\sin ({2\beta } )}}, $$

where $A = 2p\chi \cos ({\xi - \rho } )$, $B = 2s\eta \cos ({\gamma - \varepsilon } )$, $C = s\chi \sin ({\gamma - \xi } )+ p\eta \sin ({\varepsilon - \rho})$, $D = {k^2}{p^2}{w^2} + {\chi ^2}$, $E = {k^2}{s^2}{w^2} + {\eta ^2}$, $F = \eta \chi \sin ({\varepsilon - \xi } )+ {k^2}ps{w^2}\sin ({\gamma - \rho } )$, $G = {p^2} - {s^2}$, and $H = 2ps\sin ({\gamma - \rho } )$.

It can be seen from the above equations that the angular shifts include the spin-dependent and spin-independent quantities. Simultaneously, the angular shift is associated with the amplitude and phase of the Fresnel coefficient, the propagation distance, the incident angle, the polarization angle, the thickness, and the like. We will study the effects of changing these parameters on angular shift. In addition, when the polarization angle is 0 or 90 degree (H or V states), the angular shifts can be simplified as

(9)$$\Delta x_\sigma ^H ={-} \frac{{2zp\chi \cos ({\xi - \rho } )}}{{{k^2}{p^2}{w^2} + {\chi ^2}}}, $$

(10)$$\Delta y_\sigma ^H ={-} \sigma \frac{{2zps\cot {\theta _i}\sin ({\gamma - \rho } )}}{{{k^2}{p^2}{w^2} + {\chi ^2}}}, $$

(11)$$\Delta x_\sigma ^V ={-} \frac{{2zs\eta \cos ({\gamma - \varepsilon } )}}{{{k^2}{s^2}{w^2} + {\eta ^2}}}, $$

(12)$$\Delta y_\sigma ^V = \sigma \frac{{2zps\cot {\theta _i}\sin ({\gamma - \rho } )}}{{{k^2}{s^2}{w^2} + {\eta ^2}}}. $$

3. Results and discussion

Select the propagation distance $z = 250mm$, the beam waist ${w_0} = 27\mu m$ and the thickness of the BK7 glass plate or S-LAH79 prism layer ${d_2} = {d_4} = 0.2\mu m$. To begin with, we investigate the variation of the Fresnel coefficient with the incident angle and thickness by considering the symmetry and asymmetry of the layered structure. From this we get the Brewster angle and the Brewster-like angle, and through the angular shift expression, we have some conjectures about the angular shift. Then, the above guess is researched by taking the angular shift as a function of the angle of incidence and the angle of polarization. Furthermore, we find that the incident angle and the polarization angle are very strict for achieving the control of angular shifts. Finally, in order to achieve effective manipulation of the angular shifts, we discuss the angular shifts changing with structural parameters (thickness or refractive index) in detail.

To investigate the angular shift characteristics of a layered structure, we draw Fig. 2 to reveal the role of the incident angles and thickness under the layered structure. Note that the thickness of MgF₂ selected here is ${d_3} = 0.2\mu m$. Figures [2(a), 2(b)] and [2(d), 2(e)] depict the amplitude of Fresnel coefficients as functions of incident angle and thickness for symmetrical and asymmetrical conditions, respectively. In the case of symmetry, it can be seen from the Fig. 2(a) that there are two points with a reflection coefficient of zero, namely about 54.2° (red solid line) and 81.3° (blue dashed line). It should be borne in mind that the red and blue lines represent the Fresnel coefficients of horizontal and vertical polarization components, respectively. That is to say, in the case of H and V polarizations, there is a condition where the Fresnel coefficient is zero. Usually, this phenomenon occurs at the Brewster angle in the case of H polarization. Interestingly, under this structure, this phenomenon also occurs when the V polarized beam is incident near an angle of 81.3°. We call this angle of incidence Brewster-like angle. However, in the case of asymmetry, neither the H nor the V polarization has a situation in which the Fresnel coefficient is zero.

Fig. 2. The Fresnel coefficient changing with the angle of incidence and the thickness of MgF₂. (a) and (d) show the variation of the amplitude of Fresnel coefficient with the incident angle. (b) and (e) describe the amplitude of Fresnel coefficient changing with the thickness of MgF₂ at 81.3°. (c) and (f) represent the phase of Fresnel coefficient changing with the incident angle. [(a)-(c)] and [(d)-(f)] indicate symmetrical and asymmetrical conditions, respectively.

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In order to reveal the reason that the Fresnel coefficient of V polarization is 0, we fixed the incident angle to 81.3°, and further study the variation of Fresnel coefficient with MgF₂ thickness (d₃) as shown in Figs. 2(b) and 2(e). It can be seen that due to the resonance of Febry-Perot (F-P), it is sensitive to d₃ in both symmetrical and asymmetrical structures, and there is a sharp oscillation phenomenon. This also explains why there is a Brewster-like angle phenomenon for V polarization. At the same time, it can be observed that there is $s = 0$ at ${d_3} = 0.2\mu m$, which is consistent with Fig. 2(a). It is also found that under the asymmetric structure, the Fresnel coefficient can equal to zero by changing ${d_3}$. We know that the Fresnel coefficient here is not a pure real number, and it not only has an amplitude but also a phase. In the study of the phase, it is found that there are obvious mutations at 54.2° and 81.3° under the symmetrical structure [Fig. 2(c)], and there is no mutation in the case of non-symmetry [Fig. 2(f)]. From this we can speculate that the angular shift has a maximum value near these two incident angles (54.2° and 81.3°) in the symmetric case.

Obviously, from the above analysis we can conclude that the angular shift is closely related to the thickness, angle of incidence, and polarization angle. For a fixed thickness of the MgF₂ of 0.2µm, the angular shifts changing with the incident angle and the polarization angle are shown in Fig. 3. The left [Figs. 3(a)–3(d)] and the right images [Figs. 3(e)–3(h)] represent symmetrical and asymmetrical cases, respectively. As can be seen from the color bar in the figure, the in-plane angular displacement is relatively greater than transverse angular displacement. Moreover, it can be seen that the in-plane angular displacement has a significant increase in the proximity of the Brewter angle and the Brewster-like angle, while the transverse angular shift changes only rapidly near the Brewster angle. In fact, it can also be increased around the Brewster-like angle, but its change is not obvious. Equation (8) including a proportionality factor $\cot {\theta _i}$ shows that the magnification of the transverse angular displacement is not significant at a large angle of incidence (81.3°). In other words, the transverse angular shifts can also be amplified near the Brewster-like angle when the polarization state of the incident beam is V polarization. In the case of asymmetry, the right-handed angular shift has an amplification phenomenon around the Brewster angle while the left-handed angular shift has little change. This asymmetric spin splitting also exists in the case of anisotropic [34,35]. It is noteworthy that both in-plane and transverse angular shifts have positive, zero, and negative values. That is to say, the angular displacements can be adjusted by selecting different incident angles and polarization angles. However, the regulation of the angular shift is very demanding for the adjustment of the incident angle and the polarization angle.

Fig. 3. [(a),(e)] and [(c),(g)] are the relationship between the left-handed in-plane and transverse angular shifts with the incident angle and the polarization angle, respectively; [(b),(f)] and [(d),(h)] are the relationship between the right-handed in-plane and transverse angular shifts with the incident angle and the polarization angle, respectively. Among them, [(a)-(d)] and [(e)-(h)] are symmetrical and asymmetrical cases, respectively.

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To achieve a more simple and effective angular shift control, Figs. 4 and 5 are introduced to reveal the role of thickness in in-plane and transverse angular shift. Here, we fix the incident angle at 81.3 degree and select different polarization angles ($\beta = {0^{\circ}},{45^{\circ}},{90^{\circ}}$). As shown in Figs. 4 and 5, all of the angular shifts can show positive, zero, and negative values. Moreover, for arbitrary polarization, the angular shift varies drastically with the thickness. Therefore, adjusting the structural parameters can effectively achieve dynamic adjustment of angular shift. When the polarization angle is 0° or 90°, the red and blue lines are completely coincident in symmetrical and asymmetrical systems [see Figs. 4(a), 4(c), 4(d), and 4(f)], but they do not completely coincide in the Figs. 4(b) and 4(e). This means that changing the polarization angle allows for spin-independent to spin-dependent regulation. In addition, from Figs. 4(a) and 4(c), the in-plane angular shift is particularly large, up to about 900µm. From Fig. 5, when the polarization angle is 0° or 90°, the red and blue lines are symmetrical about the x-axis. These phenomena can also be confirmed from the Eqs. (9)–(12). From Eqs. (9) and (10), the changes of the left- and right-handed in-plane angular shifts of the H and V polarizations are exactly the same, i.e., they are independent of spin at this time. From Eqs. (11) and (12), the changes of the left- and right-handed transverse angular shifts of the H and V polarizations are opposite, i.e., they are dependent of spin in this situation.

Fig. 4. [(a),(d)], [(b),(e)], and [(c),(f)] show changes in the in-plane angular shift with MgF₂ thickness when the polarization angles are 0°, 45°, and 90°, respectively. [(a)-(c)] and [(d)-(f)] represent symmetrical and asymmetrical conditions, respectively. Among them, the signs + and - respectively indicate left- and right-handed circular polarization.

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Fig. 5. [(a),(d)], [(b),(e)], and [(c),(f)] show the changes in the transverse angular shift with MgF₂ thickness when the polarization angles are 0°, 45°, and 90°, respectively. [(a)-(c)] and [(d)-(f)] represent symmetrical and asymmetrical conditions, respectively. Among them, the signs + and - indicate left- and right-handed circular polarization, respectively.

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From Fig. 3, we can find that there is a significant increasement in the in-plane angular shift near the Brewster angle, however, the transverse angular shift is relatively small. Therefore, we focus on the amplification mechanism of the in-plane angular shift [33,36]. From Eqs. (9) and (11), it is obvious that the magnitude of the in-plane angular shift is closely related to the magnitude and phase of the Fresnel coefficients. When the incident light beam is horizontal polarization, the magnitude of reflected coefficient p is close to zero near the Brewster angle. Thus, the in-plane angular shift shows a very large value and its sign reversal is due to the phase jump of $\rho$. When the incident light beam is vertical polarization, the amplified and reversed angular shift near the Brewster-like angle is close to the decrease of the magnitude of reflected coefficient s and the phase jump of $\gamma$.

In fact, we have theoretically verified the feasibility of the layered structure to manipulate the spin angular shift in the photonic SHE. It is not only simple to be processed, but also highly adjustable. More importantly, the transition from spin-dependent to spin-independent can also be implemented by adjusting the polarization angle. Through Eqs. (7) and (8), we know that when $C\sin ({2\beta } )= F\sin ({2\beta } )= 0$ or $H\cos ({2\beta } )= F\sin ({2\beta } )= 0$, the in-plane angular shift or the transverse angular shift achieves this transition; and it can be seen that the A and D terms are only related to the H polarization, and the B and E terms are only related to the V polarization, while the $\sigma C$, $\sigma F$, G, and $\sigma H$ terms are related to both H and V polarizations. When the incident beam is H polarization, the in-plane and the transverse angular shifts are dominated by ${A \mathord{\left/ {\vphantom {A D}} \right.} D}$ and ${{\sigma H} \mathord{\left/ {\vphantom {{\sigma H} D}} \right.} D}$, respectively. When the incident beam is V polarization, the in-plane angular shift and the transverse angular shift are dominated by ${B \mathord{\left/ {\vphantom {B E}} \right.} E}$ and ${{\sigma H} \mathord{\left/ {\vphantom {{\sigma H} E}} \right.} E}$, respectively. No matter which one or more of the main functions, we can see that there are trigonometric functions in the simplified sub-forms that determine the angular shift, which also explains why the angular shift oscillates. When $\cos ({\xi - \rho } )> 0$ or $\cos ({\gamma - \varepsilon } )> 0$, the in-plane angular shift in the case of H polarization or V polarization will be a negative value. When $\sigma \sin ({\gamma - \rho } )> 0$ or $\sigma \sin ({\gamma - \rho } )< 0$, the transverse angular shift in the case of H or V polarization can be a negative value. Interestingly, anomalies of angular shifts have also been revealed even in the case of incident light with vertical polarization states due to the zero-reflection angle. Near this point, a large angular shift is detected. Therefore, the effective regulation of the angular shift can be achieved by changing the structural parameters, which may provide a possible way to apply angular shift to practical application.

4. Conclusion

In conclusion, we have theoretically investigated the controllability of the in-plane and transverse spin angular shifts in the layered structure. It was found that the spin angular shift can be effectively adjusted by changing the thickness parameter of the layered structure, especially, the in-plane angular shift can be modulated from the spin-independent to the spin-dependent mutual transition by changing the polarization angle. When the polarization angle is 0° or 90°, the value of the left- and right-handed transverse angular shift are the same in magnitude and opposite in direction. Furthermore, we have found that there is a near-zero reflection angle similar to Brewster's angle at vertical polarization. Under this condition, the spin angular shifts can be significantly amplified. These findings not only provide us with a smart way to steer the angular shift but also open up the possibility for the practical application of the photonic SHE such as spin-based nanophotonic devices.

Funding

National Natural Science Foundation of China (11604095); Science Foundation of Civil Aviation Flight University of China (JG2019-19); Hunan Provincial Innovation Foundation for Postgraduate (CX2018B291); School Project of Hunan Normal University (2017089).

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Tunable in-plane and transverse spin angular shifts in layered dielectric structure

Abstract

1. Introduction

2. Theoretical analysis

3. Results and discussion

4. Conclusion

Funding

References

Cited By

Figures (5)

Equations (12)

Optics Express