1. Introduction

The physical mechanism of in-plane spin splitting (IPSS) originates from the photonic spin Hall effect (PSHE), which arises from the coupling of spin angular momentum and orbital angular momentum [1–3]. Similar to the Imbert-Fedorov (IF) shift along y-axis [4–6], IPSS induces the in-plane shift along x-axis, which fully considers the transverse wave vector component, has both spatial and angular characteristics in the geometrical space [7], enabling a distinctive method of optical spin control. The in-plane shift is equivalent to Goos–Hänchen (GH) shift when the shift of left- and right-circularly polarized components coincide, which has the advantage of direct observation compared with the transverse shift. It has been widely used in optical switches [8,9], temperature sensors [10,11], polarization beam splitters [12], and wavelength division multiplexer [13]. Naturally, a larger shift value is essential for improving the sensitivity of practical applications. Besides the conventional method of the Brewster effect [14–16], researchers have explored various optical medium for enhancing the in-plane shift, such as metal surface plasmon resonators [17,18], metamaterials [19–21], and two-dimensional materials [22–24].

More recently, the anisotropic medium has shown significant potential in exploring and aggravating the physical mechanism of spin-orbit interaction due to the extra manipulating degree of freedom in the refractive index provided by the strong anisotropy [25–27]. For example, tilted anisotropy plate [28–30], hyperbolic metamaterials [31–33], epsilon-near-zero materials [34] and two-dimensional anisotropic monolayer graphene surfaces [35] have been proposed to enable the realization of high-efficiency spin devices, optical switchers and optical sensors. Otherwise, as the study of single anisotropic interface is extended to the multilayer structure, such as layered nanostructure [36–39], photonic crystal [40,41] and bilayer graphene [42], research has still adapted the fixed anisotropic medium without considering the role parameter of its thickness. In the previous work of anisotropic multilayer structure, the thickness affects the spin shift of PSHE [43,44], especially we found that the interference effect caused by the multiple scattering could strongly enhance the transverse shift of PSHE [45].Therefore, for enhancing and manipulating IPSS, the influence of thickness on the IPSS in multilayer structure needs to be developed in-depth, which is beneficial to the comprehensive understanding of IPSS and broadens the precision measurement method of thickness.

In this work, we focus on the thickness-dependent in-plane shift in a three layered anisotropic structure near the Brewster angle and critical angle. Firstly, it is proved that when considering an isotropic medium as the second layer, the thickness can periodically modulate both in-plane spatial and angular shifts near the Brewster angle, while the enhancement near the critical angle is almost thickness-independent. Then, when the anisotropic medium is substituted for the isotropic medium, the thickness-dependent in-plane shift near the Brewster angle can be realized with a wider incident angle range due to the strong anisotropy. By adjusting the dielectric tensor of the anisotropic medium, the enhanced in-plane shift near the critical angle could be modulated periodically or linearly by thickness. Through discussing and analyzing this thickness-dependent characteristic, we infer that (ζ-ϕ_p) and |∂r_p/∂θ_i| of the Fresnel reflection coefficient r_p are the direct factors leading to the evolution process of in-plane shift near the above two incident angles respectively. Finally, as the incident polarization is extended to arbitrary linear polarization, we investigate the asymmetric in-plane shift, and find that the anisotropic medium can effectively improve the performance of the asymmetric degree, which further verify the advantages of the anisotropic medium in IPSS.

2. Theoretical model and numerical method

Firstly, we establish a general three layered [ɛ₁, ɛ₂, ɛ₃] structure model to reveal the PSHE, as illustrated in Fig. 1(a). The first layer is made of glass with a permittivity of ɛ₁= 1.515², while the third layer is air with a permittivity of ɛ₃= 1. Importantly, the glass is set as a hemisphere to ensure that the light beam can directly incident on the interface between the first and second layers at any incident angle [46]. Here, ɛ₁ > ɛ₂, the partial reflection occurs for the critical angle. Only a part of incident light beam reflects from the interface while the remainder continues to transmit. When a linearly polarized light beam reflects at the interface, the gravity center of left- and right-handed circularly polarized light beam will not only show the in-plane spatial shift $\mathrm{\delta }_\mathrm{\ \pm }^{{x}}$, but also the in-plane angular shift $\mathrm{\Delta }_\mathrm{\ \pm }^{{x}}$. To explore the light propagation clearly, the three layered structure in Fig. 1 (a) can be simplified and equivalent to the schematic in Fig. 1(b). When the light incidents into medium 2 from the glass layer, the in-plane spatial and angular shifts will occur at the interface between the glass and medium 2. Here d denotes the thickness of the medium 2.

 

Fig. 1. (a)Schematic diagram of the PSHE of Gaussian light beam at the three layered structure. (b) Schematics of light transmission and reflection of the three layered structure. Here, “E” and “H” represent the electric field and magnetic field, which respectively parallel and perpendicular to incident plane.

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Then, the propagation process of the light beam in the three layered structure must be deduced before calculating the in-plane shift. The incident light is a general Gaussian light beam with a wavelength of 632.8 nm. Based on the angular spectrum theory, it can be written as

At any given plane z = const, the in-plane shift of the field centroid is given as [47]:

Here z = 250 mm is the transmission distance.

The spatial and angular shift with H polarized incidence (transverse-magnetic, or p-polarized) can be derived based on the plane wave angular spectrum theory combining with enforcing boundary condition [48]:

Here ${{k}_{1}}{ = }\sqrt {{\mathrm{\varepsilon }_{1}}} \; {{k}_{0}}$ and ${{k}_{0}}\mathrm{\ =\ 2\pi /\lambda}$ denotes the wave vector. Only the shift of the right-handed circularly polarized component is considered while that of left-handed is the same with opposite sign. When the transmission distance is fixed, the in-plane angular shift here can be converted into distance, which can be normalized with the in-plane spatial shift about the wavelength. Then, the Fresnel reflection coefficient should be solved by first-order Taylor series expansion to distinguish the difference of plane wave components in different wave packets ${{r}_{{p, s }}}{ = }{{r}_{{p,s}}}{(}{{\theta }_{i}}{) + }\frac{{\partial {{r}_{{p, s}}}}}{{\partial {{\theta }_{i}}}}\frac{{{{k}_{{ix}}}}}{{{{k}_{0}}}}$ [49,50] with complex function ${{r}_{p}}{(}{{\theta }_{i}}{) = p \exp(i\; }{\phi _{p}}{)}$, ${{r}_{s}}{(}{{\theta }_{i}}{) = s\; \exp(i\; }{\phi _{s}}{)}$, $\partial {{r}_{p}}{/}\partial {{\theta }_{i}}{\ =\ \chi \;\exp(i\,\zeta ),}$ $\partial {{r}_{s}}{/}\partial {{\theta }_{i}}{\ =\ \eta \;\exp(i\,}{\epsilon }{)}$. In addition, when the polarization state of incident light beam is arbitrary, it should be represented by Jones matrix ${\mathrm{[cos\beta ,\,sin\beta ]}^\textrm{T}}$ (β here is the polarization angle). Combine with above, the in-plane spatial and angular shifts can be derived as:

Here, focusing on the PSHE in multilayer structure, the interference effect of p- and s- polarized waves should be considered to correctly calculate the radiative characteristic. Medium 3 (air) is regarded as a semi-infinite substrate. The overall Fresnel reflection coefficient of p- and s-polarized waves r_p,s based on the three layered structure can be rewritten as [40]:

Further, when the second layer medium is anisotropy, ɛ₂ should be written as a dielectric tensor: [ɛ_O, ɛ_O, ɛ_E], which represent the uniaxial relative permittivity components for ordinary and extraordinary waves respectively. In our work, the optical axis is set along z axis, then the Fresnel reflection coefficient should be re-deduced as [38]:

3. Results and discussion

For comparison, we first demonstrate the thickness-dependent in-plane shift in an isotropic medium, where the ɛ₂ is set as 1.432² (GaF₂). Here the Brewster angle and critical angle are approximately 32.76° and 41.304°, respectively. We plot the thickness-dependent in-plane shift according to Eqs. (3) and (4), as shown in Figs. 2(a) and (b). With the change of incident angle, both the in-plane spatial and angular shifts are obviously enhanced near the Brewster angle and critical angle. For the enhancement near the Brewster angle, as thickness increases, both in-plane spatial and angular shifts have a periodical intensity modulation with a period of approximately 0.4λ, which is the result of interference effect and satisfy the Fabry-Pérot resonance in the structure (also occurring when medium 2 is anisotropic) [52]. In Fig. 2(a), as the thickness increases within one period, the peak value of the in-plane spatial shift near the Brewster angle decreases from the maximum (approximately 20λ) firstly and then increases again to the maximum. Subsequently, after experiencing a zero gap, the peak value repeats the process with opposite direction. The incident angle with maximum shift also moves slightly and periodically near the Brewster angle with the change of thickness. In Fig. 2(b), as thickness increases, the periodical evolution process of the in-plane angular shift near Brewster angle is similar to that of in-plane spatial shift, however, the direction of in-plane angular shift is only altered with the motion of incident angle near Brewster angle, which is independent of the thickness. It should be noted that the in-plane spatial and angular shifts described in Eqs. (3) and (4) are directly related to the sin(ζ-ϕ_p) and cos(ζ-ϕ_p) (ζ=arg(∂r_p/∂θ_i), ϕ_p represents the phase of r_p), respectively. Therefore, we show the corresponding sin(ζ-ϕ_p) and cos(ζ-ϕ_p) in Figs. 2(c) and (d), which are consistent with the evolution of the in-plane spatial and angular shifts. Therefore, the (ζ-ϕ_p) is the direct factor leading to the evolution process of in-plane shift near the Brewster angle, and we can choose the corresponding thickness of the second layer medium to manipulate the shift value and direction of both the in-plane spatial and angular shifts.

 

Fig. 2. (a) In-plane spatial and (b) angular shift with different thickness and incident angles under [ɛ₁, ɛ₂, ɛ₃] = [1.515², 1.432², 1]. (c) and (d) shows the phase of Fresnel reflection coefficient of r_p.

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While for the enhancement near the critical angle, both the in-plane spatial and angular shifts show evident thickness-independent enhancement (fixed at about 20λ and 700λ respectively as the thickness increases). The sin(ζ-ϕ_p) and cos(ζ-ϕ_p) here keep constant, showing the thickness-independence. Even though the ideal in-plane spatial and angular shifts can always be obtained, the thickness insensitivity of the in-plane shift here still hinder its tunability.

Then, in order to obtain a more flexible thickness-dependent modulation and further explain the enhancement mechanism of PSHE near the two incident angles, the anisotropic medium is introduced into the three layered structure. Here, to exclude the influence of other factors and compare with the isotropic medium, we choose two conditions of [ɛ_O, ɛ_E] = [1, 1.432²] and [1.432², 1]. Figure 3 shows the three-dimensional relationship of the in-plane shift with incident angle and thickness, in which the density similar to Fig. 2 is plotted at the bottom. In Figs. 3(a) and (c), the in-plane spatial shift under [ɛ_O, ɛ_E] = [1, 1.432²] and [1.432², 1] both show the thickness-dependent enhancement near the Brewster angle. The similar behavior can also be found in Figs. 3(b) and (d) for the in-plane angular shift. However, unlike the nearly sine-shaped in Figs. 2(a) and (b), the enhancement of in-plane shift near the Brewster angle as thickness increases is fan-shaped. The enhancement range is no longer limited around the Brewster angle. It can even exceed the critical angle under [ɛ_O, ɛ_E] = [1, 1.432²] (Fig. 3(a)), while that in Fig. 3(c) is limited less than the critical angle, which is beneficial for the wide-angle enhanced PSHE. Besides, at some fixed thickness, as the incident angle increases, both in-plane spatial and angular shifts in Figs. 3(c) and (d) near the Brewster angle are enhanced more than once (twice for the spatial shift, even quartic for the angular shift), along with the change of direction.

 

Fig. 3. The relationship of in-plane shift with different thickness and incident angles. (a), (b) and (c), (d) respectively describe the in-plane shift under [ɛ_O, ɛ_E] = [1,1.432²] and [1.432²,1].

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While for the enhancement near the critical angle, both in-plane spatial and angular shifts exhibit obvious modulation as thickness increases, no longer keep almost constant as shown in Fig. 2. When [ɛ_O, ɛ_E] = [1, 1.432²] shown in Figs. 3(a) and (b), the in-plane spatial and angular shifts also unfold periodical intensity modulation as thickness increases, which is similar to the in-plane shift near the Brewster angle. Then under [ɛ_O, ɛ_E] = [1.432², 1] in Figs. 3(c) and (d), the in-plane spatial and angular shifts both show a nearly linear variation trend versus thickness. As the thickness increases gradually to 2λ, the in-plane spatial and angular shifts decrease slowly from 17λ and 700λ to approximately 12λ and 600λ, respectively.

In short, these observations illustrate that the introduction of strong anisotropy broadens the incident angle range for the thickness-dependently periodical enhancement of in-plane shift near the Brewster angle, besides makes the enhanced in-plane shift near the critical angle become thickness-dependently periodical to ameliorate the thickness insensitivity in the isotropic medium.

Then, we focus on the enhancement mechanism in the anisotropic medium near the Brewster angle and critical angle. Due to the property of (ζ-ϕ_p) and the fact that the in-plane angular shift only includes an additional multiplication term related to the transmission distance z, the mechanism of in-plane spatial and angular shifts is similar. Therefore, we focus our analysis solely on the in-plane spatial shift in order to thoroughly investigate the enhancement mechanism. Figures 4(a) and (c) display the sin(ζ-ϕ_p) with different thickness and incident angles under [ɛ_O, ɛ_E] = [1, 1.432²] and [1.432², 1], respectively. When near the Brewster angle, the sin(ζ- ϕ_p) directly corresponds to the in-plane spatial shift in Figs. 3(a) and (c). As the sin(ζ-ϕ_p) reaches the peak value of 1, the corresponding in-plane spatial shift in Figs. 3(a) and (c) also approaches the peak value. Simultaneously, the direction of sin(ζ-ϕ_p) also manipulates the direction of in- plane spatial shift. Therefore, these findings can further verify that (ζ-ϕ_p) is the main determinant of the in-plane shift near the Brewster angle. Certainly, compared with the evolution of the isotropic medium in Fig. 2, the influence of strong anisotropy on interference effect here changes the phase difference between these two reflected lights from the upper and lower interfaces, which ultimately leads to the wider incident range of thickness-dependent enhancement of the in-plane shift near the Brewster angle.

 

Fig. 4. [(a), (c)] sin(ζ-ϕ_p) and [(b), (d)] ϕ_p with different thickness and incident angles under [ɛ_O, ɛ_E] = [1, 1.432²] and [1.432², 1].

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However, it should be noted that near the critical angle shown in Figs. 4(a) and (c), sin(ζ-ϕ_p)=-1, which is irrelevant to the in-plane shift obviously. Then, we plot the corresponding ϕ_p in Figs. 4(b) and (d), which can further demonstrate this fan-shaped enhancement effect in the anisotropic medium. Under [1, 1.432²] in Fig. 4(b), the instantly jumps of ϕ_p could still exists when the incident angle exceeds the critical angle, that is to say the fan-shaped enhancement still occurs. Attention, the ϕ_p here still cannot directly explain the thickness-dependent modulation near the critical angle under different dielectric tensors of the anisotropic medium.

Therefore, for explaining the origin of the thickness-dependent enhancement of the in-plane shift near the critical angle, we then pay attention to on the χ=|∂r_p/∂θ_i| (Taylor’s first partial derivative of the Fresnel reflection coefficient r_p). When the incident angle exceeds the critical angle, here p = 1 (absolute value of r_p) and the χ can no longer be ignored. As a result, the in-plane spatial shift in Eq. (3) can be simplified as $\mathrm{\delta }_{x}^{H}{ = (k}{{w}^{2}}\mathrm{\chi )/\{\ }{{k}^{2}}{{w}^{2}}{ + }{\mathrm{\chi }^{2}}{ + 2[1 + \cos(}{\phi _{s}}{ - }{\phi _{p}}{)]}{{\cot}^{2}}{\mathrm{\theta }_{i}}{\} }{.}$ When ${[1 + \cos(}{\phi _{s}}{ - }{\phi _{p}}{)]}{\textrm{cot}^{2}}{\mathrm{\theta }_{i}}$ is small enough to be neglected, χ is the sole variable parameter to affect the in-plane spatial shift near the critical angle. Therefore, in the inset of Figs. 5(a) and (c), we show the relationship of χ versus thickness and incident angle, in which χ increases sharply near the critical angle. We then choose four different curves shown in Fig. 5 to describe the relationship between the in-plane spatial shift and χ near the critical angle. The black curve shows the conditions when the incident angle is slightly less than the critical angle (θ_i= 41.30°), the blue, green and red curves when exceeds the critical angle (θ_i= 41.31°, 41.32° and 41.33°, respectively).

 

Fig. 5. (a) and (c) show the χ=|∂r_p/∂θ_i| with different thickness under [ɛ_O, ɛ_E] = [1, 1.432²] and [1.432², 1]; inset show the three-dimensional image of χ with respect to thickness and incident angle. (b) and (d) illustrate the corresponding in-plane spatial shift with the change of thickness.

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In Fig. 5(a), it can be seen that under [ɛ_O, ɛ_E] = [1, 1.432²], due to the interference effect, χ exhibits a periodical evolution with the increase of thickness, while the peak value moves up. Obviously, the curve evolution of the in-plane spatial shift in Fig. 5(b) coincides with the corresponding χ in Fig. 5(a), which can confirm that χ is the direct result of thickness-dependently periodical modulation near the critical angle. When the incident angle exceeds the critical angle, the curve of χ gradually decreases with the increase of the incident angle, resulting in the decrease of in-plane spatial shift in Fig. 5(b). It should be noted that in Fig. 5(b), although the thickness-dependently periodical modulation always exits near the critical angle, the more obvious and distinguishable modulation only occurs when the incident angle exceeds the critical angle (the blue, green and red curves). The maximum in-plane spatial shift can approach the upper limit: half of beam waist (the blue curve of θ_i= 41.31°).

Then, under [ɛ_O, ɛ_E] = [1.432², 1], as thickness increases, the evolution of in-plane spatial shift and χ both keep flat near the critical angle, which absent the similar periodical effects in Figs. 5(a) and (b). As ɛ_E=ɛ₃= 1, the Fresnel reflection coefficient ${r}_{p}^{{23}}$ in Eq. (8) is simplified to a pure real number. Here ${r}_{p}^{{23}}$ cannot introduce additional phase when calculating the overall Fresnel reflection coefficient r_p in the three layered structure, eventually suppressing the interference effect near the critical angle. In addition, the in-plane spatial shift and χ show a good consistency when the incident angle exceeds the critical angle. They all gradually decrease with the increase of thickness, besides, their maximums gradually decrease with the increase of incident angle. The incident angle should be kept close to the critical angle (the blue curve of θ_i= 41.31° in Figs. 5(b) and (d)) to obtain the optimal in-plane spatial shift. Therefore, the above discoveries verify that χ=abs(∂r_p/∂θ_i) is the fundamental reason for the enhancement of in-plane shift near the critical angle, and it can be controlled by adjusting the dielectric tensor of the anisotropic medium to realize the periodical or linear thickness-dependent modulation near the critical angle. It can be concluded that the anisotropic medium has more valid and unique modulation effect than the isotropic medium, which provides new insights into the in-plane shift in multilayer structure.

Additionally, when the incident light has an arbitrary linear polarization state, the in-plane shifts of left- and right-circularly polarized components are not the same any more, which is the asymmetric shift. The asymmetric degree is described by the difference shift values, denoted by “$\mathrm{|\delta }_{ + }^{x}\mathrm{|-\ |\delta }_{ - }^{x}{|}$” and “$\mathrm{|\Delta }_{ + }^{x}\mathrm{|-\ |\Delta }_{ - }^{x}{|}$” for the in-plane spatial and angular shift respectively [53]. It is worth noting that at a single glass-air interface, the asymmetric degree disappears near the Brewster angle, where the in-plane spatial shifts of two polarization components possess the same magnitude but opposite direction, while for the in-plane angular shifts, they are always the same. Even near the critical angle, the asymmetric degree of both in-plane spatial and angular shifts is too weak to distinguish. In short, we attempt to improve the performance of these asymmetric degrees in the three layered structure.

Firstly, we demonstrate the asymmetric degree of in-plane shifts in an isotropic medium as stating before (ɛ₂= 1.432²) with different polarization angles, shown in Figs. 6(a) and (b). The left and right sides of each figure show the results near the Brewster angle (θ_i= 33.42°) and critical angle (θ_i= 41.32°). As can be seen in Figs. 6(a) and (b), near the Brewster angle (left side), both the value and direction of asymmetry degree exhibit the thickness-dependent characteristic with a period of approximately 0.4λ, but the large asymmetric degrees of both in-plane spatial and angular shifts only occur near β=0 (H polarization). While near the critical angle (right side), as thickness increases, although the asymmetry degrees can almost occur in the polarization angle range of 0-90° besides with thickness-dependently periodical modulation, the overall performance of these asymmetry degrees are still relatively weak (the peak values in right side of Figs. 6(a) and (b) are 1λ and 50λ respectively), which hinder the modulation effect of the thickness.

 

Fig. 6. asymmetric degree of [(a), (c), (e)] in-plane spatial and [(b), (d), (f)] angular shift with different polarization angles and thickness in the three layered structure, respectively.

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To address these, we replace the second layer medium with the anisotropic medium under [ɛ_O, ɛ_E] = [1, 1.432²] and [1.432², 1], as shown in Figs. 6(c)-(f). With the similar-evolution of thickness-dependently periodical modulation, near the Brewster angle (left side of Figs. 6(c)-(e)), the strong anisotropy can effectively expand the range of large asymmetric degree of in-plane shift, comparing with Figs. 6(a) and (b). It is worth noting that the range under [1.432², 1] (Figs. 6(e) and (f)) is much more than that under [1, 1.432²] (Figs. 6(c) and (d)). When set 2λ as the minimum discernible shift value, the widest range under [1.432², 1] could reach 70°, while 30° under [1, 1.432²]. The range in the isotropic medium (Figs. 6(a) and (b)) is only 20°. It can be concluded that this thickness-dependent asymmetric degree near the Brewster angle can be effectively controlled and expanded by further adjusting the dielectric tensor of the anisotropic medium.

Near the critical angle (right side of Figs. 6(c)-(f)), it is intuitive that the asymmetric degree of in-plane spatial and angular shifts is enhanced significantly (the peak values of asymmetric degree can reach 20λ and 600λ respectively), much more than that in the isotropic medium. Besides, in the right side of Figs. 6(e) and (f), the peak value of asymmetric degree under [1.432², 1] gradually decrease with the increase of thickness, while the period number is larger than that under [1, 1.432²] in Figs. 6(c) and (d). Therefore, the anisotropic medium reveals an efficient and flexible method to control the asymmetric degree of in-plane shift, and the thickness in multilayer structure should be fully considered to obtain an ideal asymmetric degree.

4. Conclusions

In conclusion, we have investigated the thickness-dependent in-plane spatial and angular shift in a multilayer structure near the Brewster angle and critical angle. On the basis of expression of in-plane shift with H polarized incidence, the phase and Taylor’s first partial derivative of the Fresnel reflection coefficient r_p can directly affect IPSS. For this thickness-dependent characteristic due to the interference effect, the enhancement of in-plane shift near the Brewster angle can be expanded with a wider range when introducing the anisotropic medium instead of the isotropic medium. By virtue of this strong anisotropy, the in-plane shift emerges a periodical or linear thickness-modulation near the critical angle, which ameliorates the almost thickness-independence in the isotropic medium. In addition, when considering the arbitrary linear polarization incidence, compared with the isotropic medium, the anisotropic medium not only expands the polarization angle range of large thickness-dependent asymmetric degree near the Brewster angle from 20° to 70°, but also enhances the asymmetric degree near the critical angle from 1λ and 50λ to 20λ and 600λ, respectively. These findings further reveal that the anisotropic medium-based platform can provide an efficient and flexible method for the thickness-dependent modulation, and may provide a perspective in probing the spin-orbit interaction.