Symmetric spin splitting of elliptically polarized vortex beams reflected at air-gold interface via pseudo-Brewster angle

Ze Chen; Hu Zhang; Xiaoguang Zhang; Hui Li; Jingxuan Yang; Wenbo Zhang; Lixia Xi; Xianfeng Tang

doi:10.1364/OE.403916

1. Introduction

Reflected light beams (or transmitted light beams) split into two spin components in the direction perpendicular to the incident plane (i.e., transverse direction) when a spatially bounded structured light is obliquely incident on an interface between two different media. This phenomenon is referred to as the spin Hall effect (SHE) of light [1,2]. The SHE has attracted significant attention from researchers [3,4]. It is known that the direction of spatially bounded structured light propagation can be affected by spatial variations in refractive indexes, meaning plane wave components with slightly different wave vectors will exhibit different geometric phases (Berry’s phase), leading to the spin-dependent redistribution of light intensity [3,5,6]. Similar to the SHE in electronic systems, a photon with a spin of −1 plays the role of an electron with a spin of −1 / 2 and refractive index gradients correspond to electric potential gradients [2,3].

The spin splitting of light occurs when a polarized beam is reflected or transmitted by various surfaces, such as glass [3,7–12], graphene [13,14], chiral metamaterials [15], epsilon-near-zero metamaterials [16–18], black phosphorus [19], and metacrystal surfaces [20]. Remarkably, the giant spin splitting of light beam can be observed by circularly polarized light beam reflected at metasurfaces [21]. Besides, the novel spin splitting of light beam can also be realized by other methods such as magnetic dipole [22] and helical traveling-wave nanoantennas [23].

The spin splitting consists of two components: spin-independent splitting and symmetric spin splitting (SSS) [3,6,12,13,17,20–23]. Here, the SSS, which means that the spin-1 splitting and spin- –1 splitting are equal in magnitude but in opposite directions. Generally speaking, the spin-independent splitting is on the same order of magnitude as the SSS. Besides, there is another type of spin splitting called asymmetric spin splitting (ASS) [10,11,14,16,18,19]. Therefore, the spin splitting of light can be divided into two types based on symmetry. From the perspective of light-matter interactions, the spin splitting of light originates from the conversion from intrinsic spin angular momentum (SAM) to extrinsic orbital angular momentum (EOAM) [24]. For an incident Laguerre Gaussian (LG) beam carrying intrinsic orbital angular momentum (IOAM, i.e., vortex beam) [25], the generation mechanism of SSS is more flexible based on the coupling interactions between three types of optical angular momentum (i.e., EOAM, IOAM, and SAM) [24,26]. Additionally, IOAM provides an additional spatial dimension for manipulating structured light [27,28]. Therefore, vortex beams are advantageous for manipulating structured light using IOAM compared to ordinary Gaussian beams (i.e., non-vortex beams). For the above reasons, SSS can be enhanced by IOAM, when incidence elliptically polarized vortex beam impinges on strong absorbing media surface. More recently, IAOM induced beam shifts in reflection can also be measured by the optimized weak measurement technique [29].

In general, the SSS of light occurs when linearly polarized light impinges on weak absorbing isotropic media interfaces. Recently, the tiny transverse spatial SSS can be detected using weak measurements method when elliptically polarized light is reflected on air-glass interface at the Brewster angle incidence [30]. A large angular SSS in the direction parallel to the plane of incidence (i.e., longitudinal direction) for reaching 1500 $\mu$m at the propagation distance of 300 mm can also be realized [31] when the elliptically polarized incident light beam with $\Delta \phi$=90° and $\alpha$=0.5°(it can also be viewed as a quasi-horizontal polarized light beam) is reflected on the glass surface via Brewster angle. Compared to the dielectric media with pure real refractive index, the strongly absorbing media have complex refractive index, and the relationship between real parts and imaginary parts of complex refractive index satisfy: $n_2^I \gg n_2^R$. The strongly absorbing media with large imaginary part of complex refractive index leads to interesting optical phenomena of which the most surprising is that of transverse spatial SSS for elliptically polarized light beam reflected. Besides, the strongly absorbing medium such as conventional metallic is strongly reflected and attenuated at optical or infrared frequencies [32], thereby intensities of the two spin components of the reflected light beam for the elliptically polarized light beam near pseudo-Brewster angle incidence are larger than that of the weakly absorbing media or dielectric media. Interestingly, real part of the Fresnel reflection coefficients along parallel to the incidence plane and imaginary part of the Fresnel reflection coefficients along perpendicular to the incidence plane almost disappear at near the pseudo-Brewster angle incidence. It is well known that the complex refractive index of the strong absorbing media plays a crucial role in producing the longitudinal angular SSS. Thereby, the IOAM induced the transverse spatial SSS associated with longitudinal angular SSS.

In this study, we analyzed the generation mechanism of the transverse spatial spin splitting of the arbitrary polarized LG beam reflected by strong absorbing media surfaces. First, closed expressions for total transverse spatial spin-dependent shifts were derived. Next, an ASS factor g is introduced to identify that whether the transverse spatial spin splitting is symmetric or not, thereby a quantitative description method for optical spin splitting under universal conditions was developed. Additionally, it is found that the transverse spatial SSS will occur in an elliptically polarized vortex beam with a phase difference of 90° impinging on an air-gold interface near the pseudo-Brewster angle. In this study, incident light beam with arbitrary polarized angles were generated by a rotated ideal polarizer and fixed quarter-wave plate [33]. Finally, we concluded that the transverse spatial SSS of an elliptically polarized vortex beam with a phase difference of 90° can be dynamically manipulated by modulating the magnitude of the polarized angle. These results could potentially be applied to precision measurements [34] and edge-enhanced imaging [35–37].

2. Theory and model

We consider that the transverse spatial spin splitting of an arbitrary polarized vortex beam occurs during reflection near the pseudo-Brewster angle on strong absorbing media surfaces. A geometrical schematic of beam reflection is presented in Fig. 1, where ${\theta ^i}$ is the incident angle. The unit vectors ${\hat{{\textbf {x}}}^a}$, ${\hat{{\textbf {y}}}^a}$, and ${\hat{{\textbf {z}}}^a}$ represent the bases of the central Cartesian coordinate frame of the a-th beam, where the superscripts a = i and r denote incident and reflected light beams, respectively.

Fig. 1. Schematic illustration of the spin splitting of an arbitrary polarized vortex beam reflected near the pseudo-Brewster angle on a strong absorbing media surface. $\sigma ={+} 1$ and $\sigma ={-} 1$ represent the right-hand and left-hand circularly polarized light field components of the reflected vortex beam, respectively.$\Delta _{\sigma ={\pm} 1,l}^{{\textbf {IF}},{y^r}}$ represents the spatial spin-independent shifts along the y^r axis and $\Delta _{\sigma ={\pm} 1,l}^{{\textbf {s}},{y^r}}$ represents the spatial spin-dependent shifts along the y^r axis.

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Here, a polarized incident LG beam with its waist located at the interface and a radial mode index equal to zero (i.e., p = 0) is considered. Therefore, the angular spectrum of the electric field for the LG beam can be written as follows:

(1)$$|{{{\tilde{{\textbf {E}}}}^i}(k_x^i,k_y^i)} \rangle = ({e_p}|H \rangle + {e_s}|V \rangle )\exp [ - \frac{{{{(k_x^i)}^2} + {{(k_y^i)}^2}}}{\textrm{4}}w_0^2]\tilde{u}_L^i,$$

where $\tilde{u}_L^i = {[(k_x^i + i{\mathop{\rm sgn}} (l) \cdot k_y^i)]^{|l |}}$, ${e_p}\textrm{ = }\cos \alpha$, ${e_s}\textrm{ = }\sin \alpha \exp (i\Delta \phi )$, and $\Delta \phi$ is the phase difference between the p and s components of the incident beam. $\alpha$ is the polarized angle of the incident beam.${w_\textrm{0}}$ and l denote the beam waist size and azimuthal mode index of the incident LG beam, respectively. The following sequential processes are considered: transformation from the incident central coordinate frame to the incident local coordinate frame, Fresnel formulas [38], ${r_{p,s}} = {r_{p,s}}({\theta ^i}) + ({\partial _{{\theta ^i}}}{r_{p,s}})k_x^i/k$, and transformation from the reflection local coordinate frame to the reflection central coordinate frame. Therefore, the angular spectrum representation of the transverse electric field for the reflected LG beam can be obtained:

(2)$${\tilde{{\textbf {E}}}^r} = \left( {\begin{array}{cc} {{r_p} - \kappa_x^r{\partial_{{\theta^i}}}{r_p}}&{({r_p} + {r_s})\kappa_y^r\cot {\theta^i}}\\ { - ({r_p} + {r_s})\kappa_y^r\cot {\theta^i}}&{{r_s} - \kappa_x^r{\partial_{{\theta^i}}}{r_s}} \end{array}} \right)\left( {\begin{array}{c} {{e_p}}\\ {{e_s}} \end{array}} \right)\exp [ - \frac{{{{(k_x^r)}^2} + {{(k_y^r)}^2}}}{\textrm{4}}w_0^2]\tilde{u}_L^r,$$

where $\kappa _x^r\textrm{ = }k_x^r/k$, $\kappa _y^r\textrm{ = }k_y^r/k$($k_x^r\textrm{ = } - k_x^i$,$k_y^r\textrm{ = }k_y^i$), and $\tilde{u}_L^r = {[( - k_x^r + i{\mathop{\rm sgn}} (l) \cdot k_y^r)]^{|l |}}$. By applying an inverse Fourier transform to Eq. (2), the relationships $|H \rangle = (|\textrm{ + } \rangle \textrm{ + }|- \rangle )/\sqrt 2$ and $|V \rangle ={-} i(|\textrm{ + } \rangle - |- \rangle )/\sqrt 2$ are obtained and the electric field for the reflected LG beam in the spin basis vector space can be written as follows:

(3)$$|{{{\textbf {E}}^r}} \rangle = u_L^r({x^r} - \tilde{x}_{\sigma ={+} 1}^r,{y^r} - \tilde{y}_{\sigma ={+} 1}^r,{z^r})|\textrm{ + } \rangle \textrm{ + }u_L^r({x^r} - \tilde{x}_{\sigma ={-} 1}^r,{y^r} - \tilde{y}_{\sigma ={-} 1}^r,{z^r})|- \rangle ,$$

where $\tilde{x}_\sigma ^r\textrm{ = }i({x_p} - i\sigma m{x_s})/(1 - i\sigma m)$, $\tilde{y}_\sigma ^r\textrm{ = }\tilde{x}_\sigma ^r|{_{{x_{p,s}} \to {y_{p,s}}}} $,$m = {r_s}{e_s}\textrm{/}{r_p}{e_p}$, ${x_{p,s}} ={-} ({\partial _{{\theta ^i}}}{r_{p,s}})/{r_{p,s}}/k$, ${y_p} = (\textrm{1} + {r_s}/{r_p})\cot {\theta ^i}({e_s}/{e_p})/k$, ${y_s} ={-} (\textrm{1} + {r_p}/{r_s})\cot {\theta ^i}({e_p}/{e_s})/k$, and $u_{L}^{r}\left(x^{r}, y^{r}, z^{r}\right)=\left[-x^{r}+i \operatorname{sgn}(l) \cdot y^{r}\right]^{|l |} \exp \left[-\frac{k}{2} \frac{\left(x^{r}\right)^{2}+\left(y^{r}\right)^{2}}{z_{R}+i z^{r}}\right]$. The spin-dependent shifts are essentially derived from the intensity centers of two spin components of the reflected light, namely the $\sigma ={+} 1$ component (right-hand circularly polarized, RCP) and $\sigma ={-} 1$ component (left-hand circularly polarized, LCP). Therefore, the expressions of the total transverse spatial spin-dependent shifts of the reflected LG beam can be obtained using the defined formula for ${\left\langle {{y^r}} \right\rangle _\sigma }({z^r} = 0) = \left\langle {{\textbf {E}}_\sigma^r} \right|{y^r}|{{\textbf {E}}_\sigma^r} \rangle \textrm{/}\left\langle {{{\textbf {E}}_\sigma^r}} \mathrel{|{\vphantom {{{\textbf {E}}_\sigma^r} {{\textbf {E}}_\sigma^r}}}} {{{\textbf {E}}_\sigma^r}} \right\rangle$ as follows:

(4)$$\Delta _{\sigma ,l}^{{y^r}} = \Delta _{\sigma ,l}^{{\textbf {IF}},{y^r}} + \Delta _{\sigma ,l}^{{\textbf {s}},{y^r}},$$

with $\Delta _{\sigma ,l}^{{\textbf {IF}},{y^r}} = \Delta _\sigma ^{{\textbf {IF}},{y^r}} + l\Theta _\sigma ^{{\textbf {GH}},{x^r}}{z_R}$,$\Delta _{\sigma ,l}^{{\textbf {s}},{y^r}} = \Delta _\sigma ^{{\textbf {s}},{y^r}} + l\Theta _\sigma ^{{\textbf {s}},{x^r}}{z_R}$.

Where ${z_R}$ is the Rayleigh length, $\Delta _\sigma ^{{\textbf {IF}},{y^r}} ={-} (y_p^I + {|m |^2}y_s^I)/{N_\sigma }$, $\Theta _\sigma ^{{\textbf {GH}},{x^r}} = (x_p^R + {|m |^2}x_s^R)/{N_\sigma }/{z_R}$, $\Delta _\sigma ^{{\textbf {s}},{y^r}} = \sigma [({m^R}( - y_p^R + y_s^R) - {m^I}(y_p^I + y_s^I)]/{N_\sigma }$, $\Theta _\sigma ^{{\textbf {s}},{x^r}} = \sigma [{m^R}( - x_p^I + x_s^I) + {m^I}(x_p^R + x_s^R)]/{N_\sigma }/{z_R}$, ${N_\sigma } = (1 + {|m |^2})(1 + \sigma g)$, $g = 2{m^I}/(1 + {|m |^2})$. Here, the case of $w_0^2 \gg {|{\tilde{x}_\sigma^r} |^2} + {|{\tilde{y}_\sigma^r} |^2}$ is considered in this paper. The superscripts R and I denote the real and imaginary parts of the complex numbers, respectively. Equation (4) represents the main result of our study. The second and forth terms in Eq. (4) represent IOAM-induced transverse spatial spin-dependent shifts. This shifts can essentially be considered as coupling interactions among three types of optical angular momentum (i.e., EOAM, IOAM, and SAM). To reveal the intrinsic physics of the spin splitting, a dimensionless physical quantity $g$ is introduced. When $|g |$ is closed to zero, the total transverse spatial spin splitting of reflected light can be considered as a superposition of the transverse spatial spin-independent splitting (i.e., $\Delta _{\sigma ={+} 1,l}^{{\textbf {IF}},{y^r}} = \Delta _{\sigma ={-} 1,l}^{{\textbf {IF}},{y^r}}$) and transverse spatial SSS (i.e., $\Delta _{\sigma ={+} 1,l}^{{\textbf {s}},{y^r}} ={-} \Delta _{\sigma ={-} 1,l}^{{\textbf {s}},{y^r}}$). This splitting can also be considered as SSS with respect to the point ($\Delta _{\sigma ,l}^{{x^r}}$,$\Delta _{\sigma ,l}^{{\textbf {IF}},{y^r}}$). Here, the longitudinal spatial SSS of the light beams can be ignored. In other words, the longitudinal spatial shifts of the two spin components of the reflected light satisfy the relationship $\Delta _{\sigma ={+} 1,l}^{{x^r}} = \Delta _{\sigma ={-} 1,l}^{{x^r}}$. Additionally, $\Delta _\sigma ^{{\textbf {IF}},{y^r}}$ is essentially equivalent to the so-called Imbert-Fedorov shifts of reflected beams (composed of $\sigma ={+} 1$ and $\sigma ={-} 1$ components). $\Theta _\sigma ^{{\textbf {GH}},{x^r}}$ and $\Theta _\sigma ^{{\textbf {s}},{x^r}}$ represent longitudinal angular spin-independent splitting(i.e., angular Goos-Hänchen shifts) and longitudinal angular SSS of the light beams, respectively. In particular, the perfect transverse spatial SSS of the light beams occurs when the magnitude of the transverse spatial SSS ($\Delta _{\sigma ={+} 1,l}^{{\textbf {s}},{y^r}} ={-} \Delta _{\sigma ={-} 1,l}^{{\textbf {s}},{y^r}}$) of light beams is much larger than that of the transverse spatial spin-independent splitting ($\Delta _{\sigma ={+} 1,l}^{{\textbf {IF}},{y^r}} = \Delta _{\sigma ={-} 1,l}^{{\textbf {IF}},{y^r}}$), i.e., $\Delta _{\sigma ={+} 1}^{{y^r}} \approx \Delta _{\sigma ={+} 1,l}^{{\textbf {s}},{y^r}} = - \Delta _{\sigma ={-} 1,l}^{{\textbf {s}},{y^r}} \approx{-} \Delta _{\sigma ={-} 1}^{{y^r}}$. Whereas, transverse spatial SSS of the light beams completely vanishes when the magnitude of transverse spatial spin-independent splitting is much larger than that of the transverse spatial SSS, i.e., $\Delta _{\sigma ={+} 1}^{{y^r}} \approx \Delta _{\sigma ={+} 1,l}^{{\textbf {IF}},{y^r}} = \Delta _{\sigma ={-} 1,l}^{{\textbf {IF}},{y^r}} \approx \Delta _{\sigma ={-} 1}^{{y^r}}$. However, when $|g |$ cannot be ignored compared to one, then the total transverse spatial spin splitting of a reflected beams can be considered as a superposition of the transverse spatial ASS along one direction and along opposite directions ($\Delta _{\sigma ={+} 1,l}^{{\textbf {IF}},{y^r}} \cdot \Delta _{\sigma ={-} 1,l}^{{\textbf {IF}},{y^r}} \ge 0$ and $\Delta _{\sigma ={+} 1,l}^{{\textbf {s}},{y^r}} \cdot \Delta _{\sigma ={-} 1,l}^{{\textbf {s}},{y^r}} \le 0$). Therefore, g can be considered as an asymmetric spin splitting factor. For a strong absorbing media interface (${n_2}^R \ll {n_2}^I$), the total transverse spatial spin-dependent shifts of polarized light beams with $\Delta \phi$ = 90° can be represented by simplifying Eq. (4) as follows:

(5)$$\begin{array}{l} \Delta _{\sigma ,l}^{{y^r}} ={-} \frac{{(\tan \alpha + \sigma )\cot {\theta ^i}\textrm{[1} + (1 + \sigma \tan \alpha ){{({r_s}/{r_p})}^R} + \sigma \tan \alpha {{|{{r_s}\textrm{/}{r_p}} |}^2}]}}{{k\textrm{(1} + {{|{{r_s}\textrm{/}{r_p}} |}^2}{{\tan }^2}\alpha )(\textrm{1} + \sigma g)}} - ,\\ l\frac{{[({\partial _{{\theta ^i}}}{{|{{r_p}} |}^2} + {{\tan }^2}\alpha {\partial _{{\theta ^i}}}{{|{{r_s}} |}^2})/2] + \sigma \tan \alpha {{[{\partial _{{\theta ^i}}}({r_p}^ \ast {r_s})]}^R}}}{{k{{|{{r_p}} |}^2}\textrm{(1} + {{|{{r_s}\textrm{/}{r_p}} |}^2}{{\tan }^2}\alpha )(\textrm{1} + \sigma g)}} \end{array}$$

The IOAM-dependent term in Eq. (5) is a simplified description of $(\Theta _\sigma ^{{\textbf {GH}},{x^r}} + \Theta _\sigma ^{{\textbf {s}},{x^r}}){z_R}$ for polarized light with $\Delta \phi$ = 90°. For horizontally polarized light, Eq. (5) can be further simplified as follows:

(6)$$\Delta _{\sigma ,l}^{{y^r},H} ={-} \frac{{\sigma \cot {\theta ^i}}}{k}[1 + {(\frac{{{r_s}}}{{{r_p}}})^R}] - \frac{l}{k}\frac{{{\partial _{{\theta ^i}}}{{|{{r_p}} |}^2}}}{{2{{|{{r_p}} |}^2}}},$$

Similarly, when vertically polarized light is reflected by a strong absorbing media surface, then $\Delta _{\sigma ,l}^{{y^r},V}$ can also be obtained by swapping the subscripts p and s (i.e., $p \to s,s \to p$) in Eq. (6).

(7)$$\Delta _{\sigma ,l}^{{y^r},V} ={-} \frac{{\sigma \cot {\theta ^i}}}{k}[1 + {(\frac{{{r_p}}}{{{r_s}}})^R}] - \frac{l}{k}\frac{{{\partial _{{\theta ^i}}}{{|{{r_s}} |}^2}}}{{2{{|{{r_s}} |}^2}}},$$

3. Numerical results and analysis

An air-gold interface was considered in this study as a typical example. Gold with a refractive index of ${n_2}\textrm{ = }$ 0.188 + 5.39i [39] at $\lambda \textrm{ = }$826.6 nm was selected. Figure 2 presents the changes in ${r_p}^R$, ${r_p}^I$, ${r_s}^R$, and ${r_s}^I$ with the incident light angle. The pseudo-Brewster angle of 79.16° (${r_p}^R \approx 0$) is clearly indicated in the inset of the Fig. 2 [see the red solid line]. One can see that $|{{r_s}^R} |$ is much greater than $|{{r_s}^I} |$ near the pseudo- Brewster angle (${\theta ^i}$=79.80°).

Fig. 2. Dependencies of the real and imaginary parts of Fresnel reflection coefficients on the incident light angle in the case of an air-gold interface.

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For polarized light ($\alpha$=45 °) with $\Delta \phi$=0°, 5°, 45 °,85° and 90° reflected at air-gold interface, a very small $|g|$ can occur at different incident angles, as shown in Fig. 3(a). Therefore, the transverse spatial SSS can be achieved by controlling phase difference of the incident polarized light. i.e., different incident polarization states correspond to different incident angles for g→0. In this work, for the case of an incident angle near the pseudo-Brewster angle is considered only. Interestingly, a very small |g| factor perfectly appears when incident polarized light ($\alpha$=45 °) with $\Delta \phi$ = 90° impinges on air-gold interface near the pseudo-Brewster angle. To further investigate the transverse spatial spin splitting behavior of light beams with different incident polarization state, changes of the |g| factor corresponding to polarized light for $\alpha$=1°, 5°, 45 °, 85° and 90° with a varying phase difference $\Delta \phi$ are shown in Fig. 3(b). It can be found that the values of |g| factor corresponding to polarized light with $\Delta \phi$=${\pm}$90° are closed to zero. Clearly, the above conclusions about |g| factor can be furtherly exhibited in Fig. 3(c). Therefore, the SSS of reflected light can be realized when incident polarized light with $\Delta \phi$ = 90° impinges on air-gold interface near the pseudo-Brewster angle. In particular, the values of $|g|$ for the incident polarized light with $\alpha =$1° or 89° (i.e., quasi-horizontal polarized light or quasi-vertical polarized light) are relatively small [Fig. 3(b)], the transverse spatial spin splitting of the reflected light can also be approximated as an SSS.

Fig. 3. Changes in the |g| factor for an air-gold interface. (a) Changes in the |g| factors corresponding to polarized light for different $\Delta \phi$($\alpha$ = 45°) with a varying incident angle. (b) Changes in the |g| factor corresponding to polarized light for different $\alpha$(${\theta ^i}$ = 79.8°), with a varying phase difference $\Delta \phi$. (c) Changes in the |g| factor corresponding to elliptically polarized light for $\Delta \phi$=90° (${\theta ^i}$ = 79.8°) with a varying the polarized angle.

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In the following discussion, we will analyze the transverse spatial SSS for an elliptically polarized beam with $\Delta \phi$=90° and ${w_0}\textrm{ = 20}\lambda \textrm{/}\pi$ based on Eq. (5). The g factor corresponding to an incident angle of ${\theta ^i}$=79.8° is almost zero [the blue solid line in Fig. 3]. Figure 4 presents the dependencies of spin-dependent shifts and spin-independent shifts on the polarized angle $\alpha$ at ${\theta ^i}$ = 79.8° for a Gaussian beam (l = 0) with $\Delta \phi$=90°. The transverse spatial spin-independent shift is of the same order of magnitude as the transverse spatial spin-dependent shift over a wide range of polarized angles, as shown in Fig. 4(a). The transverse spatial spin-independent shifts (i.e., spatial IF shift) for circularly polarized light with l = 0 can reach the values of 25 nm as shown in Fig. 4(a). Interestingly, the result predicted by us can be better verified indirectly base on the results reported by the previous experiments [40]. However, the longitudinal angular spin-independent shifts (i.e., angular GH shift) for circularly polarized light with l = 0 vanish, as shown in Fig. 4(b). On other hand, because both ${\partial _{{\theta ^i}}}{|{{r_p}} |^2}$ and ${\partial _{{\theta ^i}}}{|{{r_s}} |^2}$ almost disappear, as shown in the inset in Fig. 4(b) (purple and green solid lines), the longitudinal angular spin-dependent shift is far greater than the longitudinal angular spin-independent shift ($\Theta _\sigma ^{{\textrm{GH}},{x^r}} \approx 0$), as shown in Fig. 4(b). Therefore, the total longitudinal angular spin splitting exhibits SSS over the entire range of polarized angles. In particular, the longitudinal angular SSS for $\alpha$ = 0° is close to zero because $- \Theta _\sigma ^{{\textrm{GH}},{x^r}}{z_R}$ can be reduced to $- {\partial _{{\theta ^i}}}{|{{r_p}} |^2}/(2{|{{r_p}} |^2})/k$ [i.e., the last term in Eq. (6) for l = 1] near pseudo-Brewster angle incidence, which is very small. In contrast, ${|{{r_s}\textrm{/}{r_p}} |^2}$ is close to one near pseudo-Brewster angle incidence, meaning extremum of $\Theta _\sigma ^{s,{x^r}}$ is mainly determined by $- \sigma [\tan \alpha /\textrm{(1} + {\tan ^2}\alpha )]{[{\partial _{{\theta ^i}}}({r_p}^ \ast {r_s})]^R}/{|{{r_p}} |^2}/k$, which leads to peak values for longitudinal angular SSS near the polarized angles of ${\pm}$45° [blue solid and dotted lines in Fig. 4(b)]. And the longitudinal angular SSS achieved by the circularly polarized light beam reflection on the air-gold interface can reach 2000 μm at the propagation distance of 300 mm.

Fig. 4. Dependencies of spin-dependent shifts and spin-independent shifts on the polarized angle at ${\theta ^i}$ = 79.8° for a Gaussian beam (l = 0) with $\Delta \phi$ = 90°. (a) Spatial spin-dependent shifts perpendicular to the plane of incidence. (b) Angular spin-dependent shifts parallel to the plane of incidence.

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Based on Eq. (5), the dependencies of the total transverse spatial spin-dependent shifts and spin-independent shifts with $\Delta \phi$ = 90° on the polarized angle $\alpha$ at ${\theta ^i}$ = 79.8° for l = 0 and l = 1 are presented in Fig. 5. Firstly, the transverse spatial SSS for l = 0 ($\sigma$=+1 component for RCP light and $\sigma$=−1 component for LCP light) with values close to 25 nm are clearly shown in Fig. 5(a). Secondly, the transverse spatial SSSs appear perfectly when the polarized incident light with a phase difference of 90° for l = 1, l = 2 and l = 3 is reflected at air-gold interface near pseudo-Brewster angle, as shown in Figs. 5(b)–5(d). This result shows that the transverse spatial SSS can be enhanced by increasing l. Remarkably, the large transverse spatial SSS can reach about 400 nm for l=1, 800 nm for l=2 and 1200 nm for l=3. The occurrence of the large transverse spatial SSS can be explained by the following two aspects. One is that the occurrence of a large value for ${[{\partial _{{\theta ^i}}}({r_p}^ \ast {r_s})]^R}/{|{{r_p}} |^2}$ leads to the appearance of a large longitudinal angular SSS when incident angle is manipulated to be close to the pseudo-Brewster angle. On the basis of satisfying the above incident angle, the IOAM of the vortex beam is enhanced, which means that strong spin-orbit coupling is achieved. Furthermore, the achievement of strong spin-orbit coupling is manifested by the appearance of a large transverse spatial SSS as shown in Eq. (5). On the other hand, the large transverse spatial SSS can also be explained by the fact that the large longitudinal angular SSS appears at the pseudo-Brewster angle, which leads to a large IOAM-dependent transverse spatial SSS associated with longitudinal angular SSS [41]. Remarkably, this vortex-induced transverse spatial SSS can be regarded as a coupling interaction among three types of angular momentum (i.e., EOAM, IOAM, and SAM) of light. Specifically, the rotational symmetry of the bulk medium about the z-axis can be satisfied, which means that the z-component of the total angular momentum must be conserved for the incident, reflected and transmitted light beams. Thus the following relation can be obtained:

(8)$$L_I^i + L_S^i = ({Q^r}L_{I,\sigma ={\pm} 1}^r + {Q^r}L_{S,\sigma ={\pm} 1}^r + {Q^t}L_{I,\sigma ={\pm} 1}^t + {Q^t}L_{S,\sigma ={\pm} 1}^t) + {Q^r}L_{E,\sigma ={\pm} 1}^r + {Q^t}L_{E,\sigma ={\pm} 1}^t$$

where, $L_I^i$ and $L_S^i$ represent the mean IOAM and SAM of photons for the incident beam, $L_{I,\sigma ={\pm} 1}^{r,t}$, $L_{S,\sigma ={\pm} 1}^{r,t}$ and $L_{E,\sigma ={\pm} 1}^{r,t}$ represent mean IOAM, SAM and EOAM of photons for the reflection/transmission beam on the basis of circular polarizations, respectively. ${Q^{r,t}}$ is the energy reflection/transmission coefficient. More detailed discussions of the Eq. (8) are given by many researchers [1,24,26,41–44]. Obviously, it is impossible to determine two unknown values of SSS and ASS for reflected and transmitted light beams through one conservation law (8). However, conservation law (8) still helps to understand spin splitting of light beam from another physical point of view. Specifically, for the case of LG beam, changes in the SAM and IOAM of the two spin components of the reflected light beam which occur upon Snell–Fresnel reflection and transmission must be compensated by the nonzero EOAM of the two spin components for the reflected and transmitted beams. The nonzero EOAM of the two spin components must be manifested through transverse spatial SSS and IF shifts. Finally, to provide a clear physical representation, the intensity distributions of two opposite spin components of the reflected LG beam are presented in Fig. 6.

Fig. 5. Dependencies of transverse spatial spin-dependent shifts and spin-independent shifts on the polarized angle when an elliptically polarized beam (l = 0, l = 1, l=2, l=3) with $\Delta \phi$ = 90° is reflected at ${\theta ^i}$ = 79.8° by an air-gold interface.

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Fig. 6. Reflected intensity distributions of two spin components when an circularly polarized LG beam (l = 1) with $\Delta \phi$ = 90° strikes an air-gold interface with ${\theta ^i}$ = 79.8°: (a) $\alpha$= −45°, $\sigma$= +1, (b) $\alpha$ = 0°, $\sigma$=+1, (c) $\alpha$ = 45°, $\sigma$= +1, (d) $\alpha$ = -45°, $\sigma$= −1, (e) $\alpha$ = 0°, $\sigma$= −1, (f) $\alpha$ =+45°, $\sigma$= −1.

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4. Conclusions

In summary, a complete theoretical description of the transverse spatial spin splitting of an arbitrary polarized vortex beam reflected by a strong absorbing media surface was developed based on the angular spectrum method. An ASS factor g is introduced to identify that whether the transverse spatial spin splitting is symmetric or not, thereby the symmetry of the transverse spatial spin splitting of light beams can be characterized. Furthermore, vortex-induced transverse spatial SSS was demonstrated perfectly by coupling interactions between three types of optical angular momentum when an elliptically polarized beam with a phase difference of 90° was reflected by air-gold interface at incident angles near the pseudo-Brewster angle (${\theta ^i}$=79.80°). We expected that our results will help to improve the understanding of the mechanism of generation of spin-orbit interactions and orbit (intrinsic)-orbit (extrinsic) interactions in structured light-carrying IOAM.

Funding

Fund of State Key Laboratory of IPOC (BUPT) (IPOC2019ZZ02, IPOC2020ZT07); National Natural Science Foundation of China (61527820, 61571057, 61575082, 61875247).

Disclosures

The authors declare no conflicts of interest.

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Symmetric spin splitting of elliptically polarized vortex beams reflected at air-gold interface via pseudo-Brewster angle

Abstract

1. Introduction

2. Theory and model

3. Numerical results and analysis

4. Conclusions

Funding

Disclosures

References

Cited By

Figures (6)

Equations (8)

Optics Express