1. Introduction

When a polarized light beam with bounded width is reflected or refracted from a planar interface, its left and right spin components will split in the direction perpendicular to the refractive index gradient. This interesting physical phenomenon is called the Photonic spin Hall effect (PSHE) [1–4]. There are four kinds of spin splitting in PSHE, which are in-plane (parallel to the plane of incidence) spatial and angular spin splitting [5–8], and out-of-plane (perpendicular to the plane of incidence) spatial and angular spin splitting [4,5,8,9]. Because the state of spin splitting is susceptible to the characteristic parameters of the incident beam and reflection/refraction interface, that has excellent application prospects in precision measurement [10–13], highly sensitive detection and sensing [14–24]. Such as identifying graphene layer numbers [10], precision measurement of the optical conductivity [11], magnetic properties of thin films [12], highly sensitive real-time detection of chemical reaction rate [15] and phase change process of phase change materials [16], highly sensitive refractive index sensing [23,24] and so on. More importantly, as a counterpart of the electronic spin Hall effect, the PSHE is promising to develop a new area of research—spinoptics [25,26]. For the above reasons, the PSHE has attracted more and more interest [27–36].

The PSHE is generally believed to be a result of an effective spin-orbital interaction, which describes the mutual influence of the spin (polarization) and the trajectory of the light beam [25,26,37]. Many articles have pointed out or confirmed that PSHE is affected by incident polarization [3–7,13,31,38,39]. However, the research on the effect of incident polarization on PSHE is not in-depth and systematic, and many issues are not very clear. Such as, does the incident polarization have the same or similar effects on in-plane spin splitting (IPSS) and out-of-plane spin splitting (OPSS)? What is the difference between angular and spatial shifts of IPSS and OPSS affected by incident polarization? When the reflection interface changes from a non-absorbing medium interface to an absorbing medium interface, what happens to the impact of incident polarization on the four kinds of spin splitting? Does the incident polarization have the same effect on the left and the right spin component under the same incident conditions? And so on.

In this article, we will deeply and systematically study the effect of incident polarization on PSHE, and the above questions will be clearly answered. We first established the relationship model between incident polarization and the four kinds of spin splitting of PSHE, then systematically studied the effect of incident polarization on those spin splitting, and summarized the law and characteristics of the effect of incident polarization on PSHE.

2. Establishment of the theoretical model

To better analyze the effect of incident polarization on PSHE, we first model the theoretical relationship between the two. Assume that a monochromatic Gaussian beam with an arbitrarily linear polarization state is reflected from a planar interface, as shown in Fig. 1. The incident coordinate frame K_i= (x_i,y_i,z_i) and the reflected coordinate frame K_r= (x_r,y_r,z_r) are attached to the incident and reflected beams, respectively. The laboratory coordinate frame K = (x,y,z) is fixed on the reflection interface, and the xz-plane represents the plane of incidence,

 

Fig. 1. shows the schematic diagram of PSHE when a beam is reflected from an interface. (a) is an overall schematic diagram of PSHE. (b) is a front view, and (c) is a top view. The red beam represents the incident beam, and the green and blue beams represent the reflected beam's left-handed and right-handed components. $\delta _x^ \pm$ and $\Theta _x^ \pm$ denote the in-plane spatial and angular spin-splitting shifts, respectively. $\delta _y^ \pm$ and $\Theta _y^ \pm$ denote the out-of-plane spatial and angular spin-splitting shifts, respectively. θ_i and θ_r represent the angle of incidence and reflection, respectively. ε₁ and ε₂ represent the dielectric constants of medium 1 and 2, respectively.

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Considering the incident Gaussian beam with an arbitrary linearly polarized state, and its angular spectrum expression ${\tilde{{\mathbf {\mathbf E}}}_i}$ can be written as [10]:

The expression of the angular spectrum of the reflected beam can be obtained by using the boundary conditions of the electric field and the coordinate transformation [40]:

From the Snell’s law, one can obtain ${k_{xi}} ={-} {k_{xr}}$, ${k_{yi}} = {k_{yr}}$. In the spin basis set, $|\sigma \rangle \textrm{ = }{{({|H \rangle + i\sigma |V \rangle } )} / {\sqrt 2 }}$, $\sigma \textrm{ = } \pm 1$ denote the left and right spin components, respectively. Combined with Eqs. (1)-(3), the angular spectrum expression of left and right spin components of the reflected electric field can be obtained:

Then, at arbitrary linearly incident polarization, the four kinds of spin splitting shifts of left and right spin components of the reflected beam can be calculated by the following equation [41]:

After tedious calculations, the relationship model expressions between the incident polarization and PSHE can be obtained, as shown below:

Generally, when the light beam reflects from the interface of non-absorbing medium (no total reflection occurs), its reflection efficient is a real number, that is, the reflection phase ${\phi _{p,s}}$ is zero or π. When the light beam reflects from the interface of absorbing medium, its reflection coefficient is complex number, that is, the reflection phase ${\phi _{p,s}}$ is not zero or π. Because the reflection phase parameter is included in the model, it indicates that the reflection coefficient in the model can be either a real number or a complex number, and it also indicates that the model is applicable to the reflection of the beam on the interface of non-absorbing medium as well as on the interface of absorbing medium.

3. Results and discussion

To intuitively show and conveniently observe the influence characteristics and laws of incident polarization on PSHE, we next draw the graphs between the four kinds of splitting shift ($\delta _x^ \pm$, $\Theta _x^ \pm$, $\delta _y^ \pm$ and $\Theta _\textrm{y}^ \pm$) and incident polarization when a beam with an arbitrary linearly polarized state is reflected from the absorbing medium interface and the non-absorbing medium interface. If the conventional drawing method is followed, namely, to draw $\delta _x^ +$, $\delta _x^ -$, $\Theta _x^ +$, $\Theta _x^ -$, $\delta _y^ +$, $\delta _y^ -$, $\Theta _y^ +$ and $\Theta _y^ -$ respectively when the reflective interfaces are an absorbing medium interface and a non-absorbing medium interface, 16 graphs need to be drawn. This will not only make the page lengthy and complicated, but also not conducive to the observation and comparative analysis of the phenomenon. If the spin splitting shift under the absorbing medium interface and the non-absorbing medium interface are displayed on one graph at the same time, the number of graphs to be drawn can be reduced by half, and it is also conducive to comparing and observing the characteristics of PSHE. According to Eqs. (13) to (16), the parameter that reflects whether the reflecting interface is an absorbing medium or a non-absorbing medium is the reflection coefficient. If the reflection coefficient is a real number, it means a non-absorbing medium interface, and if it is a complex number, it means an absorbing medium interface. It is well known that when a beam is reflected at the wave-density/wave-sparse medium interface, if the incident angle is less than the critical angle, the reflection coefficient is a real number, and if the critical angle is large, the reflection coefficient is a complex number. Therefore, we can take advantage of this feature to display spin splitting shifts under the interface of both absorbing and non-absorbing media on a single graph. Figure 2 shows the spatial and angular IPSS shifts, and Fig. 3 shows the angular and spatial OPSS shifts.

 

Fig. 2. shows the spatial and angular IPSS shifts versus the incident polarization around the critical angle. (a) and (b) describe the spatial shift of the reflected beam's left and right spin components, respectively. (c) and (d) illustrate their angular shifts, respectively.

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Fig. 3. shows the spatial and angular OPSS shifts versus the incident polarization around the critical angle. (a) and (b) describe the spatial shift of the reflected beam's left and right spin components, respectively. (c) and (d) illustrate their angular shifts, respectively.

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Because this work is mainly based on the wave dense/wave sparse medium interface to study the effect or influence characteristics of incident polarization on PSHE, we only require the reflecting interface to be the wave dense/wave sparse medium interface that can generate total reflection. Therefore, the parameters in our paper are not specific. In our research process, the parameters we selected are commonly used in the current study of PSHE. For example, the beam wavelength is 632.8 nm, the wave dense/wave sparse medium interface is glass/air interface (the refractive index of glass is 1.515 at the wavelength of 632.8 nm). Then, it is easy to calculate that the critical angle is 41.305°. In our previous study, we found that the size the of beam waist had no influence on the behavior characteristics of PSHE [42], so there is no specific requirement for this parameter. Therefore, the size the of beam waist we used this time is 27µm that we used in the previous study on PSHE.

It can be clearly seen from Figs. 2 and 3 that the impact of incident polarization on spatial and angular IPSS and OPSS shift has both similarities and differences, respectively. Whether total reflection occurs or not, incident polarization affects left and right spin components differently. It mainly has the following four characteristics.

(1) When polarization angle γ_i = 0° or ±90°, then $\delta _x^ +{=} \delta _x^{{\kern 1pt} - }$, $\Theta _x^ +{=} \Theta _x^{{\kern 1pt} - }$, $\delta _y^ +{=} - \delta _y^{{\kern 1pt} - }$, $\Theta _y^ +{=} - \Theta _y^{{\kern 1pt} - }$. Furthermore, when θ_i, θ_T, $\delta _x^ +{=} \delta _x^ -{\equiv} 0$, $\Theta _y^ +{=} \Theta _y^ -{\equiv} 0$, when θ_i<θ_T, $\Theta _x^ +{=} \Theta _x^ -{\equiv} 0$, as shown in Fig. 4. Characteristic (1) means that when γ_i = 0° or ±90°, no matter whether the reflection interface is an absorbing medium interface or a non-absorbing medium interface, the left and right spin components of the reflected beam will not split in the in-plane, only split in the out-of-plane, and the split is symmetrical. Although no IPSS occurs when the beam is reflected at the interface, this does not mean that the left and right spin components of the reflected beam are not shifted. The IPSS didn't happen because the left and right spin components moved in exactly the same size and direction. However, it is worth mentioning that when the reflection interface is a non-absorbing medium interface, the left and right spin components do not move in the in-plane position space and the out-of-plane angular space. When the reflection interface is an absorbing medium, they do not move in the in-plane angular space.

 

Fig. 4. illustrates the relationship curves between incident angle and spin splitting shift at different polarization angles. (a) and (b) are about the spatial and angular IPSS shifts, respectively. (c) and (d) are about the spatial and angular OPSS shifts, respectively. The solid and dotted lines correspond to the left and right spin components, respectively. The black dashed lines indicate that the incident angle is the critical angle.

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We can understand why this phenomenon occurs from Eqs. (9) to (16). By substituting γ_i = 0° into Eqs. (13) to (16), we can obtain that $\chi _{Re }^ + \textrm{ = }\chi _{Re }^ - \textrm{ = }{{({\varphi _p}\sin {\phi _p} + {\rho _p}\cos {\phi _p})} / {{R_p}}}$, $\chi _{{\mathop{\rm Im}\nolimits} }^ + \textrm{ = }\chi _{{\mathop{\rm Im}\nolimits} }^ - \textrm{ = }{{\textrm{(}{\varphi _p}\cos {\phi _p} - {\rho _p}\sin {\phi _p})} / {{R_p}}}$, $\xi _{Re }^ +{=} - \xi _{Re }^ -{=} {{\cot \theta [{R_p} + {R_s}\cos ({\phi _p} - {\phi _s})]} / {{R_p}}}$, and $\xi _{{\mathop{\rm Im}\nolimits} }^ +{=} - \xi _{{\mathop{\rm Im}\nolimits} }^ -{=} - {{\cot \theta {R_s}\sin ({\phi _p} - {\phi _s})} / {{R_p}}}$. Then, we can quickly obtain that $\delta _x^ +{=} \delta _x^{{\kern 1pt} - }$, $\Theta _x^ +{=} \Theta _x^{{\kern 1pt} - }$, $\delta _y^ +{=} - \delta _y^{{\kern 1pt} - }$ and $\Theta _y^ +{=} - \Theta _y^{{\kern 1pt} - }$ according to Eqs. (9) to (12). In addition, when θ_i, θ_T, ϕ_p,s and φ_p,s are both zero, as shown in Fig. 5, the numerator of Eq. (9) and (12) can be reduced to $\chi _{{\mathop{\rm Im}\nolimits} }^ +{=} \chi _{\textrm{Im}}^ - \textrm{ = }{{\textrm{(}{\varphi _p}\cos {\phi _p} - {\rho _p}\sin {\phi _p})} / {{R_p}}}\textrm{ = }0$ and $\xi _{{\mathop{\rm Im}\nolimits} }^ +{=} - \xi _{{\mathop{\rm Im}\nolimits} }^ -{=} - {{\cot \theta {R_s}\sin ({\phi _p} - {\phi _s})} / {{R_p}}}\textrm{ = }0$, respectively. Then one can easily deduce that $\delta _x^ +{=} \delta _x^ -{=} 0$ $\Theta _y^ +{=} \Theta _y^ -{\equiv} 0$. When θ_i>θ_T, the numerator of Eq. (10) can be reduced to $\chi _{Re }^ + \textrm{ = }\chi _{Re }^ - \textrm{ = }{{({\varphi _p}\sin {\phi _p} + {\rho _p}\cos {\phi _p})} / {{R_p}}}\textrm{ = }0$, and then we can easily deduce $\Theta _x^ +{=} \Theta _x^ -{\equiv} 0$. The analysis process is similar when γ_i =±90°, so we will not repeat it here.

 

Fig. 5. illustrate the value of R_p,s (a), ϕ_p,s (b), ρ_p,s (c) and ψ_p,s (d) around the critical angle. The black dashed lines indicate that the incident angle is the critical angle.

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In fact, the above performances of PSHE under different incident polarization and different reflection interfaces are due to the changes of the Rytov-Vladimirskii-Berry (RVB) and Pancharatnam-Berry (PB) phases of the beam during its propagation. The RVB phase is associated with the propagation direction of the wave vector, and the PB phase is related to the polarization evolution of light [30,43]. When the propagation trajectory of the paraxial beam changes, the direction of each angular spectrum component rotates in momentum space to acquire different RVB phases. Then, a geometric phase gradient is obtained, which manifests as the spatial spin splitting shift. Nevertheless, when the polarization state of the incident light is changed, a spatially varying PB phase can be obtained in real space and lead to an angular spin splitting shift.

(2) If polarization angle γ_i ≠ 0° and ±90°, when θ_i, θ_T, $\delta _x^ + ({{\gamma_i}} )={-} \delta _x^{{\kern 1pt} - }({{\gamma_i}} )$, $\Theta _x^ + ({{\gamma_i}} )= \Theta _x^{{\kern 1pt} - }({{\gamma_i}} )$, $\delta _y^ + ({{\gamma_i}} )={-} \delta _y^{{\kern 1pt} - }({{\gamma_i}} )$ and $\Theta _y^ + ({{\gamma_i}} )= \Theta _y^{{\kern 1pt} - }({{\gamma_i}} )$. This is because when θ_i, θ_T, ϕ_p,s and φ_p,s are both zero, then we can obtain $\chi _{{\mathop{\rm Im}\nolimits} }^ + ({\gamma _i}) ={-} \chi _{\textrm{Im}}^ - ({\gamma _i})$, $\chi _{Re }^ + ({\gamma _i}) = \chi _{Re }^ - ({\gamma _i})$, $\xi _{Re }^ + ({\gamma _i}) ={-} \xi _{Re }^ - ({\gamma _i})$ and $\xi _{{\mathop{\rm Im}\nolimits} }^ + ({\gamma _i}) = \xi _{{\mathop{\rm Im}\nolimits} }^ - ({\gamma _i})$ according to Eqs. (13) to (16). However, when θ_i>θ_T, these equations above no longer hold. Furthermore, $|{\delta_x^ + ({{\gamma_i}} )} |\ne |{\delta_x^{{\kern 1pt} - }({{\gamma_i}} )} |$, $|{\Theta _x^ + ({{\gamma_i}} )} |\ne |{\Theta _x^{{\kern 1pt} - }({{\gamma_i}} )} |$, $|{\delta_y^ + ({{\gamma_i}} )} |\ne |{\delta_y^{{\kern 1pt} - }({{\gamma_i}} )} |$ and $|{\Theta _y^ + ({{\gamma_i}} )} |\ne |{\Theta _y^{{\kern 1pt} - }({{\gamma_i}} )} |$. These indicate that when the reflection interface is a non-absorbing medium, the reflected beam will only undergo symmetric spatial IPSS and OPSS, and will not undergo angular spin splitting, but the left and right components will move the same distance in the same direction in the angular space. When the reflection interface is an absorbing medium interface, the reflected beam will undergo asymmetric spin splitting in both in-plane and out-of-plane position and angular space, as shown in Fig. 4.

(3) $\delta _x^ + ({{\gamma_i}} )= \delta _x^{{\kern 1pt} - }({ - {\gamma_i}} )$ and $\Theta _x^ + ({{\gamma_i}} )= \Theta _x^{{\kern 1pt} - }({ - {\gamma_i}} )$, but $\delta _y^ + ({{\gamma_i}} )={-} \delta _y^{{\kern 1pt} - }({ - {\gamma_i}} )$ and $\Theta _y^ + ({{\gamma_i}} )={-} \Theta _y^{{\kern 1pt} - }({ - {\gamma_i}} )$. This is because $\chi _{Re }^ + ({{\gamma_i}} )= \chi _{Re }^ - ({ - {\gamma_i}} )$, $\chi _{{\mathop{\rm Im}\nolimits} }^ + ({{\gamma_i}} )= \chi _{{\mathop{\rm Im}\nolimits} }^ - ({ - {\gamma_i}} )$, $\xi _{Re }^ + ({{\gamma_i}} )={-} \xi _{Re }^ - ({ - {\gamma_i}} )$ and $\xi _{{\mathop{\rm Im}\nolimits} }^ + ({{\gamma_i}} )={-} \xi _{{\mathop{\rm Im}\nolimits} }^ - ({ - {\gamma_i}} )$ can be known from Eqs. (13) to (16). These equations show that for in-plane shifts, the shift of the left spin component is symmetric with the shift of the right spin component about γ_i = 0°. But for out-of-plane shifts, it is symmetric with the inverse shift of the right spin component about γ_i = 0°. It does not matter whether the reflection interface is an absorbing medium interface or a non-absorbing medium interface. To better observe this phenomenon, we plotted the relationship curves between incident polarization and spin shift at different incident angles, as shown in Fig. 6.

 

Fig. 6. illustrates the relationship curves between incident polarization and spin splitting shift at different incident angles. (a) and (b) are about the spatial and angular IPSS shifts, respectively. (c) and (d) are about the spatial and angular OPSS shifts, respectively. The solid and dotted lines correspond to the left and right spin components, respectively.

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Furthermore, when θ_i, θ_T, we can easily obtain that $\delta _x^\sigma ({{\gamma_i}} )={-} \delta _x^\sigma ({ - {\gamma_i}} )$, $\Theta _x^\sigma ({{\gamma_i}} )= \Theta _x^\sigma ({ - {\gamma_i}} )$, $\delta _y^\sigma ({{\gamma_i}} )= \delta _y^\sigma ({ - {\gamma_i}} )$ and $\Theta _y^\sigma ({{\gamma_i}} )={-} \Theta _y^\sigma ({ - {\gamma_i}} )$ according to Eqs. (9) to (16). We can also see these relations intuitively from Figs. 2, 3 and 6. Combined with the above equations, we can further conclude that $\delta _x^\sigma ({{\gamma_i}} )={-} \delta _x^\sigma ({ - {\gamma_i}} )= \delta _x^{{\kern 1pt} - \sigma }({ - {\gamma_i}} )$, $\Theta _x^\sigma ({{\gamma_i}} )= \Theta _x^\sigma ({ - {\gamma_i}} )= \Theta _x^{{\kern 1pt} - \sigma }({ - {\gamma_i}} )$, $\delta _y^\sigma ({{\gamma_i}} )= \delta _y^\sigma ({ - {\gamma_i}} )={-} \delta _y^{{\kern 1pt} - \sigma }({ - {\gamma_i}} )$ and $\Theta _y^\sigma ({{\gamma_i}} )={-} \Theta _y^\sigma ({ - {\gamma_i}} )={-} \Theta _y^{{\kern 1pt} - \sigma }({ - {\gamma_i}} )$ when θ_i, θ_T. From these equations above, one can clearly and systematically understand the shifting relationship of the incident polarization and spin components. It can also provide theoretical guidance for photon manipulation and precise measurement.

In addition, it can be clearly seen from Fig. 4 that both the spatial and angular in-plane spin dependent shifts will change sharply near the critical angle. Both can be effectively suppressed and enhanced by slightly adjusting the incident angle. The maximum in-plane spatial and angular spin dependent shift can both reach its upper limit, which are one-half of the beam waist [44] and one-half of the beam divergence angle [45]. Therefore, PSHE can be enhanced and suppressed based on this. This modulation method has more advantages than the method based on surface plasmon resonance (SPR) and Brewster angle. Because the latter two require the incident beam with a specific polarization state (horizontal polarization, i.e., γ_i =±90°). In addition, the incident angle when PSHE is enhanced is near the SPR angle and Brewster angle, and the reflectivity of the horizontally polarized beam is almost zero at this time, hence, only a small part of the incident power will be reflected. The modulation method mentioned in this article is near the critical angle, which is basically independent of the polarization state, and the reflectivity near the critical angle is close to or equal to 100% as shown in (a) and (b) of Fig. 5, so there is almost no loss of light intensity in the process of reflection.

It can be clearly seen from the above introduction and Fig. 4 that the IPSS and OPSS on both sides of the critical angle have completely different manifestations. For example, for spatial IPSS, the spin splitting on one side of the critical angle is symmetrical, and the other side is asymmetric. For angular IPSS, spin splitting occurs on one side of the critical angle, but not on the other side. Then, the switching between symmetric splitting and asymmetric splitting, splitting and non-splitting of the beam can be easily realized by slightly adjusting the incident angle at the critical angle. These exciting findings can provide new ideas and means for photonic manipulation and device fabrication.

4. Conclusions

We derived and established a theoretical relationship model between incident polarization and the spatial and angular IPSS and OPSS shifts of PSHE, which is applicable to both non-absorptive and absorptive reflective interfaces. Then, the effect of the polarization state of the incident beam on the four kinds of spin splitting shifts is systematically studied, and several characteristics of the relationship between the incident polarization and the four kinds of spin splitting shifts are found. The relationship between them is revealed. In addition, it is also found that the in-plane spin dependent shift will be significantly enhanced and suppressed near the critical angle under arbitrary incident polarization. The splitting state of the reflected beam, such as symmetric or asymmetric splitting, splitting or non-splitting, can be easily switched by slightly adjusting the incident angle at the critical angle. These findings will help to further deepen the understanding of PSHE, and can also provide new ideas and methods for precision metrology, photonic manipulation, and photonic device fabrication.