Polarization helicity and the optical spin-orbit Hall effect

Tiegen Song; Huajie Hu; Hehe Li; Xinzhong Li

doi:10.1364/OE.509737

1. Introduction

Optical angular momentum includes the spin angular momentum (SAM) and the orbital angular momentum (OAM) [1,2], which are closely related to the circular polarization state and vortex phase of the light. Generally, the SAM and the OAM are independent and don't interact each other during the light propagation in the free space. Though, when the light propagates in some specific optical background, such as the inhomogeneous medium [3–6], the anisotropic medium [7,8], the optical tight focusing [9], the SAM and the OAM can interact each other, which is named the spin-orbit coupling and will lead to some interesting observable phenomena, such as the Berry geometrical phase [10], the spin Hall effect [11–17], orbital angular momentum Hall effect [18], the spin-orbital Hall effect [19,20] and the spin-to-orbital angular momentum conversion [21–25] etc.. Because these effects have widely applied in the field of precision metrology [26,27] and optical tweezers [28] etc., the study on the optical angular momentum interaction has attracted much interest in recent years.

The spin Hall effect of light manifests a spin-dependent split of light trajectory [11–17], and occurs during the propagation of the scalar light, such as the linearly polarized light, which can be considered as a linear superposition of the right-handed and left-handed circularly polarized light, its polarization state can be expressed as $|{{\psi_E}} \rangle = |R \rangle + |L \rangle $, $|R \rangle = ({\hat{x} + i\hat{y}} )/\sqrt 2 $ and $|L \rangle = ({\hat{x} - i\hat{y}} )/\sqrt 2 $ denote the right-hand and left-hand circular polarization states. In other words, because of the spin-orbit coupling, the different circularly polarized light is deflected in the opposite direction, the linear superposition state (the linear polarization state) degenerate to two observable and independent states (the right-hand and left-hand circular polarization states) in the spin Hall effect. It's a similar physical principle in the case of the orbital angular momentum Hall effect [18]. The vector beams possess the spatial inhomogeneous polarization state distribution, the broken symmetry of polarization state distribution also can lead to the spin-orbit coupling, for instance, the spin Hall effect can be obtained in the tight focusing of the fractional order radially polarized beam [29,30], the power-exponent azimuthal-variant vector beam [31], and the grafted polarization vector beam [32].

It knows that the cylindrical vector beam, such as the radially polarized beam, doesn't possess the SAM and the OAM. Moreover, the radial polarization state can't be simply considered as a linear superposition of the circular polarization state and the vortex state, it should be considered as the entangled state of the SAM and the OAM of light. Interestingly, Fu etc. show that, when a radially polarized light propagates in uniaxial crystal along the optical axis, the SAM and the OAM are separated and follow a paraboliclike evolution (firstly increase and then decrease with the change of the polarization state initial phase), which is called the optical spin-orbit Hall effect [19]. Unlike the spin Hall effect of light, the optical spin-orbit Hall effect can be obtained for the radially polarized light and well controlled by the initial phase of the polarization state [19,20]. The initial phase leads to the twist of the polarization state distribution, how to represent the effect of the initial phase in the optical spin-orbit Hall effect? And, whether the optical spin-orbit Hall effect can be obtained in the propagation of the cylindrical vector beams with the high-order polarization state? References [19,20] do not offer the analytical discussion on the evolution of the SAM and the OAM in the optical spin-orbit Hall effect. What is the real evolution principle of the SAM and the OAM in the optical spin-orbit Hall effect? These issues need to be clarified.

In this paper, we investigate the propagation of the radially polarized beam in uniaxial crystal along the optical axial, and present a theoretical analysis on the optical spin-orbit Hall effect. We firstly introduce a concept of the “polarization helicity” to characterize the effect of the initial phase of the polarization states in the optical spin-orbit Hall effect. Especially, the polarization helicity of the radial polarization state is changed from $- 1$ to $+ 1$ by modulating the initial phase of the polarization state, the polarization helicity is zero for the other polarization order. By means of the perturbation theory of the optical filed in uniaxial crystal, we derive the analytical expressions of the radially polarized beams in uniaxial crystal, and show the separation of the SAM and the OAM is controlled by the initial phase of the polarization state and reach the maximum value when the initial phase of the polarization state equal to $\pi /4$ (or $- \pi /4$). Moreover, the sign of the SAM and the OAM are determined by the polarization helicity and the anisotropy of uniaxial crystal. We find the separation of the SAM and the OAM follow a sinusoidal evolution, firstly increase and then decrease with the change of the polarization state initial phase from 0 to $\pi /2$ (or 0 to $- \pi /2$). We also know the polarization state during the light propagation can be modulated by changing the initial phase of the incident polarization state. Our studies further deepen the understanding of the spin-orbit coupling of the vector beam, provide a potential technique for modulating the polarization state of the radially polarized light in uniaxial crystal.

2. Polarization helicity

The traditional cylindrical vector beams [33], such as the radially polarized light, it can be considered as the superposition of two antipodal circular polarization vortex states, its polarization state can be expressed as $|{{\psi_m}} \rangle = |{{R_m}} \rangle + |{{L_m}} \rangle $, $|{{R_m}} \rangle = \exp ({ - im\varphi } )({\hat{x} + i\hat{y}} )/\sqrt 2 $ and $|{{L_m}} \rangle = \exp ({im\varphi } )({\hat{x} - i\hat{y}} )/\sqrt 2 $ denote two antipodal circular polarization vortex states, m is the polarization order. The radial polarization state just evolves along the equatorial line on the Poincaré sphere, as shown in Fig. 1 (a), it can be described by the following coefficient matrix [34],

(1)$${\mathbf u} = \left[ {\begin{array}{c} {\cos (m\phi + {\varphi_0})}\\ {\sin (m\phi + {\varphi_0})} \end{array}} \right],$$

where ${\varphi _0}$ denotes the initial phase of polarization state. Figure 1 shows that, with the change of the initial phase ${\varphi _0}$, there are the different evolution properties of the polarization state for the different polarization order: when the polarization order $m = 1$, the polarization state is changed from the radial polarization state to azimuthal polarization state with the change of the initial phase ${\varphi _0}$; yet when the polarization order is greater than 1 ($m = 2$), the polarization state distribution just rotates around the optical axis in whole and its property is unchanged. These evolution properties can be explained by rewriting the expression of the polarization state of the radially polarized light, just as the following form

(2)$${\mathbf u} = R({\varphi _0})\left[ {\begin{array}{c} {\cos (m\phi )}\\ {\sin (m\phi )} \end{array}} \right] = \left[ {\begin{array}{cc} {\cos {\varphi_0}}&{ - \sin {\varphi_0}}\\ {\sin {\varphi_0}}&{\cos {\varphi_0}} \end{array}} \right]\left[ {\begin{array}{c} {\cos (m\phi )}\\ {\sin (m\phi )} \end{array}} \right].$$

Fig. 1. The Poincaré sphere representation for the polarization state of the cylindrical vector beams, the polarization order (a) m = 1, (b) m = 2.

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Obviously, $R({\varphi _0})$ is the rotation matrix, and it indicates a rotation operation on the basis vector of polarization state. Because of the rotational symmetry of the radially polarized light, the change of the polarization state initial phase doesn't affect the whole property of the cylindrical vector beams with the high-order polarization state ($m \ge 2$).

It is well-known that, for the radially polarized beam ($m = 1$), the change of initial phase leads to the polarization state changing from the radial to the azimuthal. Alternatively, the polarization state distribution of the radially polarized light will generate a kind of “twist” which is well controlled by the initial phase, as shown in Fig. 2. To further characterize the effect of the initial phase of the polarization states, especially the twist of the polarization state distribution induced by its initial phase, we introduce a definition of the polarization helicity for the radial polarization state as the following form,

(3)$${h_p} = \int_V {(\nabla \times {\mathbf u})} \cdot {\mathbf k}dV,$$

where ${\mathbf k}$ is the wave vector, ${\mathbf u}$ is the basis vector of polarization state. According to Eq. (1), the curl of the polarization state $\nabla \times {\mathbf u} = m\sin [(m - 1)\phi + {\varphi _0}]{\widehat {\mathbf e}_z}$, then the polarization helicity of the radially polarized light is determined by the polarization order and the initial phase of polarization state. It indicates that, when the polarization order $m = 1$, the curl of the polarization state $\nabla \times {\mathbf u} = \sin {\varphi _0}{\widehat {\mathbf e}_z}$, the polarization helicity ${h_p} > 0$ ($0 < {\varphi _0} < \pi /2$) and ${h_p} < 0$ ($- \pi /2 < {\varphi _0} < 0$); Specifically, the polarization helicity of the radially polarized light (${\varphi _0} = 0$) is 0, and the polarization helicity of the azimuthally polarized light (${\varphi _0} = \pi /2$ or ${\varphi _0} ={-} \pi /2$) is same and equal to $- 1$ (or $+ 1$). Yet, for the cylindrical vector beams with the high-order polarization state ($m \ge 2$), the curl of the polarization state is closely related to the sinusoidal function, then its polarization helicity always is zero. The initial phase of the polarization state plays a key role in the polarization helicity, it will directly affect the evolution of the radially polarized light in the uniaxial crystal.

Fig. 2. The polarization helicity for the radially polarized light, (a) the polarization helicity ${h_p} > 0$, the initial phase of the polarization state $- \pi /2 < {\varphi _0} < 0$, (b) the polarization helicity ${h_p} = 0$, the initial phase of the polarization state ${\varphi _0} = 0$, (c) the polarization helicity ${h_p} < 0$, the initial phase of the polarization state $0 < {\varphi _0} < \pi /2$.

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Then, let us consider the propagation of the radially polarized paraxial light in uniaxial crystal along the optical axis. The propagation of paraxial light in uniaxial crystal along the optical axis can be described by the coupling wave equations [35],

(4)$$i\frac{{\partial {E_x}}}{{\partial z}} + \frac{1}{{2{k_0}{n_o}}}\left( {\frac{{n_o^2}}{{n_e^2}}\frac{{{\partial^2}}}{{\partial {x^2}}} + \frac{{{\partial^2}}}{{\partial {y^2}}}} \right){E_x} ={-} \frac{\Delta }{{2{k_0}{n_o}}}\frac{{{\partial ^2}{E_y}}}{{\partial x\partial y}},$$

(5)$$i\frac{{\partial {E_y}}}{{\partial z}} + \frac{1}{{2{k_0}{n_o}}}\left( {\frac{{{\partial^2}}}{{\partial {x^2}}} + \frac{{n_o^2}}{{n_e^2}}\frac{{{\partial^2}}}{{\partial {y^2}}}} \right){E_y} ={-} \frac{\Delta}{{2{k_0}{n_o}}}\frac{{{\partial ^2}{E_x}}}{{\partial x\partial y}},$$

where ${E_x}$ and ${E_y}$ denote the transverse field components of the paraxial light, ${k_0}$ is the wave number in vacuum, ${n_o}$ and ${n_e}$ are the ordinary and extraordinary refractive indices, $\Delta= n_o^2/n_e^2 - 1$ is a coefficient associated with the degree of anisotropy of the medium. The transverse field components ${E_x}$ and ${E_y}$ are coupled during its propagation, which originates from the anisotropy of medium. This coupled wave equations can be solved approximatively, and its analytical solution can be expressed in the power-series expression [35],

(6)$${{\mathbf E}_ \bot } = \left[ {1 + \frac{{iz\Delta}}{{2{k_0}{n_o}}}\mathrm{\hat{T}}\sum\limits_{n = 1}^\infty {\frac{1}{{n!}}} {{\left( {\frac{{iz\Delta}}{{2{k_0}{n_o}}}\nabla_ \bot^2} \right)}^{n - 1}}} \right]{\mathbf E}_ \bot ^{(0)}({{\mathbf r}_ \bot },\frac{\boldsymbol z}{{{n_o}}}),$$

where $\mathrm{\hat{T}}$ is the operator tensor,

(7)$$\mathrm{\hat{T}} = \left[ {\begin{array}{cc} {\partial_x^2}&{\partial_{xy}^2}\\ {\partial_{xy}^2}&{\partial_y^2} \end{array}} \right],$$

and ${{\mathbf E}_ \bot } = {E_x}{\widehat {\mathbf e}_x} + {E_y}{\widehat {\mathbf e}_y}$ is the transverse field, ${\mathbf E}_ \bot ^{(0)}$ is the slowly varying transverse field which satisfies the paraxial wave equation $2i{k_0}\partial {\mathbf E}_ \bot ^{(0)}/\partial z + \nabla _ \bot ^2{\mathbf E}_ \bot ^{(0)} = 0$, then $|{{{(\nabla_ \bot^2)}^{n + 1}}{\mathbf E}_ \bot^{(0)}} |< < |{{{(\nabla_ \bot^2)}^n}{\mathbf E}_ \bot^{(0)}} |$. Thus, it is valid in the first-order approximation in our calculation. For the radially polarized light, the transverse field is expressed as ${\mathbf E}_ \bot ^{(0)} = {E_0}[{\cos (m\phi + {\varphi_0}){{\widehat {\mathbf e}}_x} + \sin (m\phi + {\varphi_0}){{\widehat {\mathbf e}}_y}} ]$, ${E_0}$ is the amplitude satisfying the paraxial wave equation. As a result of cumbersome but direct calculation, the optical field in uniaxial crystal along the optical axis can be obtained as following,

(8)$${{\mathbf E}_ \bot } = ({{E_0} + iM} )\left[ {\begin{array}{c} {\cos (m\phi + {\varphi_0})}\\ {\sin (m\phi + {\varphi_0})} \end{array}} \right] + iN\left[ {\begin{array}{c} {\cos (2\phi - m\phi - {\varphi_0})}\\ {\sin (2\phi - m\phi - {\varphi_0})} \end{array}} \right],$$

where

(9)$$M = \frac{{z\Delta}}{{4{k_0}{n_o}}}\left( {\frac{{{\partial^2}}}{{\partial {r^2}}} + \frac{1}{r}\frac{\partial }{{\partial r}} - \frac{{{m^2}}}{{{r^2}}}} \right){E_0},$$

(10)$$N = \frac{{z\Delta}}{{4{k_0}{n_o}}}\left( {\frac{{{\partial^2}}}{{\partial {r^2}}} + \frac{{2m - 1}}{r}\frac{\partial }{{\partial r}} + \frac{{{m^2} - 2m}}{{{r^2}}}} \right){E_0}.$$

It shows that, when the radially polarized paraxial light propagates in uniaxial crystal along the optical axis, its evolution is closely related to the polarization order, the initial phase of polarization state, and the anisotropy of uniaxial crystal. And, the parameter functions M and N are equal when the polarization order takes one (m = 1), the polarization state can be directly modulated by changing the initial phase. Based on the analytical expression of the transverse field, the evolution property of the SAM and the OAM during the propagation of the radially polarized beam in uniaxial crystal will be investigated in next.

3. Analytical study of optical spin-orbit Hall effect

Though the radially polarized light can be considered as the superposition of two antipodal circular polarization vortex states, it doesn’t possess the SAM and OAM. For simplicity, we consider the amplitude ${E_0} \to {E_0}\exp (ikz)$ which satisfies the paraxial wave equation, both the parameter functions M and N are real function. According to the definition of the SAM and OAM density [36], ${J_S} \propto Im[{E_x^\ast {E_y}} ]$ and ${J_O} \propto Im[{E_x^\ast {\partial_\phi }{E_x}} ]$, we obtain

(11)$$\begin{array}{c} {J_S} \propto Im[({MN + i{E_0}N} )\cos (m\phi + {\varphi _0})\sin (2\phi - m\phi - {\varphi _0})\\ + ({MN - i{E_0}N} )\sin (m\phi + {\varphi _0})\cos (2\phi - m\phi - {\varphi _0})], \end{array}$$

(12)$$\begin{array}{c} {J_O} \propto Im[i(m - 2)N{E_0}\cos (m\phi + {\varphi _0})\sin (2\phi - m\phi - {\varphi _0})\\ + imN{E_0}\sin (m\phi + {\varphi _0})\cos (2\phi - m\phi - {\varphi _0})\\ + (m - 2)NM\cos (m\phi + {\varphi _0})\sin (2\phi - m\phi - {\varphi _0})\\ - mNM\sin (m\phi + {\varphi _0})\cos (2\phi - m\phi - {\varphi _0})]. \end{array}$$

For the cylindrical vector beams with the high-order polarization state ($m \ge 2$), the parameter function $M \ne N$, because both the SAM and the OAM density are closely related to the trigonometric function, the integral of SAM and OAM density are zero, it means there is no SAM and OAM during its propagation in uniaxial crystal along the optical axis. Furthermore, for the cylindrical vector beams with the high-order polarization state, its polarization helicity is zero, the initial phase just leads to the whole rotation of the polarization state distribution, and the polarization state is unchanged. Then, the initial phase doesn’t affect the field evolution during the propagation of the cylindrical vector beams with the high-order polarization state.

When the polarization order $m = 1$, the parameter functions M and N take the same value, Eq. (8) can be rewritten as the following form

(13)$${{\mathbf E}_ \bot } = {E_0}\left[ {\begin{array}{l} {\cos (\phi + {\varphi_0})}\\ {\sin (\phi + {\varphi_0})} \end{array}} \right] + 2iM\cos {\varphi _0}\left[ {\begin{array}{c} {\cos \phi }\\ {\sin \phi } \end{array}} \right],$$

and

(14)$$M = N = \frac{{z\Delta}}{{4{k_0}{n_o}}}\left( {\frac{{{\partial^2}}}{{\partial {r^2}}} + \frac{1}{r}\frac{\partial }{{\partial r}} - \frac{1}{{{r^2}}}} \right){E_0}.$$

We can see that, when the initial phase of the incident polarization state takes zero, ${\varphi _0} = 0$, the optical field ${{\mathbf E}_ \bot } = ({E_0} + 2iM){[{\cos \phi ,\sin \phi } ]^T}$, it means the polarization state of incident beam doesn’t change during the propagation in the uniaxial crystal, and it still maintain the radial polarization state; when the initial phase of the incident polarization state takes $\pi /2$, ${\varphi _0} = \pi /2$, the incident beam becomes the azimuthally polarized light, and its polarization state also doesn’t change during the propagation in the uniaxial crystal. Both the radially and azimuthally polarized light don’t possess the SAM and the OAM, then it means that there is no the optical angular momentum conversion in these two cases. If the initial phase of the incident polarization state doesn’t equal to 0 and $\pi /2$ (or $- \pi /2$), the polarization helicity of incident vector light doesn’t equal to 0 and $- 1$ (or 1), the polarization state evolution the uniaxial crystal will be influenced by the polarization state initial phase of incident beam.

According to Eqs. (11) and (12), for the first-order radially polarized beam ($m = 1$), the longitudinal SAM and OAM density can be expressed as follows,

(15)$${J_S} \propto{-} N{E_0}\sin (2{\varphi _0}),$$

(16)$${J_O} \propto N{E_0}\sin (2{\varphi _0}).$$

Obviously, when initial phase of the incident polarization state ${\varphi _0} = 0$ (or ${\varphi _0} = \pi /2$), the SAM and the OAM density are zero during the propagation of the radially (or azimuthally) polarized beam. Because the incident radially polarized beam doesn't possess the optical angular momentum, then the total optical angular momentum is always conserved during its propagation, ${J_S} + {J_O} = 0$. As the change of the polarization state initial phase, the evolution of the SAM and the OAM density follow the sinusoidal function but with opposite signs, as shown in Fig. 3. We find that, both the SAM and the OAM increase with opposite signs gradually, when the initial phase ${\varphi _0} = \pi /4$, the separated SAM and OAM achieve the maximum value. By modulating the polarization state's initial phase of the incident radially beam, the evolution SAM and the OAM are reversed when the polarization helicity is reversed, it means the polarization helicity will directly affect the spin-orbit coupling of the vector beam in the uniaxial crystal. Just like the spin Hall effect of light, the separation of the SAM and OAM is a manifestation of the spin-orbit Hall effect of vector light, and it is called the optical spin-orbital Hall effect.

Fig. 3. The separation evolution of the spin and orbital angular momentum with the change of initial phase ${\varphi _0}$, (a) ${n_o} > {n_e}$, (b) ${n_o} < {n_e}$.

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Fig. 4. The polarization evolution with the change of the initial phase of the polarization state, the propagation distance $z = 0.1m$, the first row $\Delta> 0$, ${n_o} = 2.2154$, ${n_e} = 1.9929$, the second row $\Delta< 0$, ${n_o} = 1.9929$, ${n_e} = 2.2154$; the initial phase of the polarization state ${\varphi _0} ={-} \pi /2$, $- \pi /4$, 0, $\pi /4$, $\pi /2$ (columns 1-5, respectively).

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The SAM of light is closely related to its circular polarization state. Figure 4 shows the polarization evolution with the change of the initial phase of the incident polarization state, the numerical calculation parameters are: ${E_0} = r/{w_0}\exp ( - {r^2}/w_0^2)$, the wave number ${k_0} = 2\pi /\lambda $, the wave length $\lambda = 632.8nm$. We see that, if the incident vector beam is the radially polarized light (${\varphi _0} = 0$), the polarization state is unchanged during the propagation; if the incident vector beam is the azimuthally polarized light (${\varphi _0} = \pi /2$), the polarization state maintains unchanged. In the case of the anisotropy parameter $\Delta> 0$ (${n_o} > {n_e}$): when the incident radially polarized light possesses the positive polarization helicity ($0 < {\varphi _0} < \pi /2$), it will evolve into the left-handed elliptically polarized light (the red ellipse in Fig. 4 (b1)), when the incident radially polarized light possesses the negative polarization helicity ($- \pi /2 < {\varphi _0} < 0$), it will evolve into the right-handed elliptically polarized light (the green ellipse in Fig. 4 (d1)). In the case of the anisotropy parameter $\Delta< 0$(${n_o} < {n_e}$), the polarization state evolution is reversed. The evolution property of the polarization state corresponds to the occurrence of the SAM in the optical spin-orbital Hall effect.

Figure 5 shows the polarization evolution with the change of the propagation distance. We see that, the different polarization helicity and the anisotropy of the uniaxial crystal determine the different polarization state evolution. When the incident beam has the same polarization helicity, take the first and the second rows in Fig. 5 as the example, the polarization helicity always is negative during the propagation (the twist of the polarization state distribution doesn’t change), yet for the case $\Delta> 0$, the right-hand elliptical polarization state will gradually appear, and for the case $\Delta< 0$, the left-hand elliptical polarization state will gradually appear. It means there will be the different SAM during the propagation in the different uniaxial crystal. Meanwhile, if the incident beams possess the different polarization helicity (as the first and the third rows in Fig. 5), the evolution of the polarization state is different. Moreover, Fig. 5 describes an evolution trend of the polarization state of the incident beam in uniaxial crystal. We know that, as the increase of the propagation distance, the elliptical polarization state firstly appears in the central region, then it spread from the central region to the outer region.

Fig. 5. The polarization evolution with the change of the propagation distance, the first and the second rows, the polarization helicity ${h_p} < 0$, the third and the fourth rows, the polarization helicity ${h_p} > 0$; the first and the third rows $\Delta> 0$, ${n_o} = 2.2154$, ${n_e} = 1.9929$, the second and fourth rows $\Delta< 0$, ${n_o} = 1.9929$, ${n_e} = 2.2154$; the propagation distance $z = 0m$, $z = 0.01m$, $z = 0.1m$, $z = 0.5m$ (columns 1-4, respectively).

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4. Conclusion

Generally, there are the spin Hall effect or the SAM-to-OAM conversion when the scalar light propagates in the uniaxial crystal [16,24,25]. In this paper, we focus on the propagation of the radially polarized beam in uniaxial crystal along the optical axial theoretically, and give an analytical interpretation on the optical spin-orbit Hall effect in the uniaxial crystal. In analogy with the definition of the helicity in the hydromechanics, we firstly introduce the polarization helicity to describe the difference of the radial and the high-order polarization state. We found that, only the polarization helicity of the radial polarization state isn’t zero, it can be negative or positive and be well modulated by the initial phase of its polarization state, the polarization helicity of the cylindrical vector beams with the high-order polarization state always is zero. We showed that, though the incident radially polarized light doesn't possess the optical angular momentum, the SAM and the OAM can be controlled by modulating the polarization helicity, its maximum values also can be obtained when the initial phase of the polarization state equal to $\pi /4$ (or $- \pi /4$). According to the results we obtained, We have known the separation of the SAM and the OAM follow a sinusoidal evolution, firstly increase and then decrease with the change of the polarization state initial phase from 0 to $\pi /2$ (or 0 to $- \pi /2$). Correspondingly, the polarization state of the incident radially polarized light can evolve into the left-hand (or right-hand) elliptical polarization state as the change of the polarization helicity of incident light. Our studies further deepen the understanding of the spin-orbit coupling of the vector beams, and provide a potential technique for modulating the polarization state of the light in uniaxial crystal.

Funding

National Natural Science Foundation of China (11974101, 12204155).

Disclosures

The authors declare no conflicts of interest.

Data availability

No data were generated or analyzed in the presented research.

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Polarization helicity and the optical spin-orbit Hall effect

Abstract

1. Introduction

2. Polarization helicity

3. Analytical study of optical spin-orbit Hall effect

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (5)

Equations (16)

Optics Express