1. Introduction

The reflection and transmission of a plane wave at a planar interface is presented by Snell’s Law and the Fresnel formulas. A bounded beam, a wave packet containing a set of plane wave components with different wave-vectors, however, may undergo the longitudinal Goos-Hanchen shift or/and the transverse Imbert-Fedorov (IF) shift. For the IF effect, the center of the reflected beam shows a small spatial shift perpendicular to the plane of incidence, relative to the position predicted by geometrical optics [1–4]. During total-reflection, a linearly polarized incident beam splits into two elliptically polarized rays that displace symmetrically [5]. In a uniaxial anisotropic media, the propagation of a beam has been investigated by taking into account of both the ordinary and the extraordinary rays [6–9]. Perez [10] analyzed the first-order nonspecular transverse effects of three dimensional Gaussian beams. Fadeyeva et al. [11] have found the asymmetric splitting of a high-order circularly polarized vortex-beam in a uniaxial crystal.

Recently, much attention has been paid to the spin Hall effect of light (SHEL) [12]. SHEL is a spin dependent transverse shift of the wave packet perpendicular to the gradient of the refractive index, which has been extensively studied theoretically [13–19] and experimentally [16,20–22]. This effect originates from the geometric phase gradient in the spin states, which arises from rotation of the coordinate frames attached to partial plane waves with different planes of incidence [15,21]. In fact, the conservation of the total angular momentum of light plays a key role in the phenomenon [14,23], and SHEL can be described as the result of the spin-orbit interaction [12,21]. Up to now, these researches have mostly focused on the homogenous isotropic planar interface. Here we study, theoretically and experimentally, the spin-dependent nanometer-sized displacements of the reflected light induced by SHEL from an air-uniaxial crystal interface that is extensively used in optics.

We demonstrate SHEL by considering the reflection from a planar interface of the uniaxial crystal sample (LiNbO₃). When reflected at the interface, a linearly polarized incident beam splits into two spin components that acquire opposite lateral displacements perpendicular to the incident plane: the component parallel ( = + 1, right-circularly polarized, denoted by) and antiparallel ( = −1, left-circularly polarized, denoted by) to the central wave vector, as shown in Fig. 1 . We use a laboratory Cartesian frame (x, y, z) attached to the interface, and employ the coordinate systems of individual beams, where (x_I, y, z_I) attached to the incident beam and (x_R, y, z_R) attached to the reflected beam. The z_I, z_R axes attach to the directions of the central wave vector of the incident and reflected beam as determined by the Snell’s law, respectively.

 

Fig. 1 Schematic of SHEL at an air–uniaxial crystal interface. ${\theta }_I$, incident angle; ${\delta }_{y|+>}$, the displacement of $|+>$ spin component. The subscripts I and R correspond to incident and reflected light, respectively.

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2. Theoretical analysis

The eigenstates of reflection are the linear p- and s-polarization states. For the air-uniaxial crystal interface, their Fresnel reflection coefficients vary with the ordinary and the extraordinary components, which is more intricate than that in the air-glass interface. Therefore, the displacements induced by SHEL depend not only on the incident polarization of the beam, but also on the orientation of the crystal optic axis.

An incident Gaussian beam can be considered as a wave-packet containing a distribution of wave-vectors ${\overrightarrow{k}}^{\left(I\right)}=k_I\left({\widehat{z}}_I+{\overrightarrow{\kappa }}^{\left(I\right)}\right)$ centered around$k_I{\widehat{z}}_I$, with ${\overrightarrow{\kappa }}^{\left(I\right)}=\left(k_{x_I}{\widehat{x}}_I+k_y\widehat{y}\right)/k_I$ $={\kappa }_{x_I}{\widehat{x}}_I+{\kappa }_y\widehat{y}$, $\left|{\overrightarrow{\kappa }}^{\left(I\right)}\right|<<1$, and ${\widehat{z}}_I$ is a unit vector along z_I axis and $k_I=2\pi /\lambda$ with λ being the wavelength of the light in the incident medium. For the reflected beam: ${\overrightarrow{k}}^{\left(R\right)}=k_R\left({\widehat{z}}_R+{\overrightarrow{\kappa }}^{\left(R\right)}\right)$ with ${\overrightarrow{\kappa }}^{\left(R\right)}$ $=\left(k_{x_R}{\widehat{x}}_R+k_y\widehat{y}\right)/k_R={\kappa }_{x_R}{\widehat{x}}_R+{\kappa }_y\widehat{y}$, $\left|{\overrightarrow{\kappa }}^{(R)}\right|<<1$, and $k_R$ = $k_I$ for Snell’s Law. For an arbitrary wave-vector, ${\widehat{k}}^{(I)}-\widehat{z}\left(\widehat{z}\cdot {\widehat{k}}^{(I)}\right)={\widehat{k}}^{(R)}-\widehat{z}\left(\widehat{z}\cdot {\widehat{k}}^{(R)}\right)$, connecting ${\overrightarrow{\kappa }}^{\left(I\right)}$ and ${\overrightarrow{\kappa }}^{\left(R\right)}$:$k_{x_I}=-k_{x_R}$ and $k_{y_I}=k_{y_R}=k_y$.

For the horizontally and the vertically polarized incident wave-packets denoted by $|\text{H}\left({\overrightarrow{k}}^{(I)}\right)>$ and $|V\left({\overrightarrow{k}}^{(I)}\right)>$, we can expand everything up to the first order in κ [21,22]:

2.1 Crystal optic axis is along the x-axis

When the crystal optic axis is along the x-axis, the linear p- and s-polarization states of the incident light beam can be written as:

Using Eqs. (1) and (2), the horizontally and vertically polarized beams are expressed by

Equation (5) shows that two spin components of the reflected light shift oppositely in y axis, with a displacement of ${\delta }_{y\alpha }^H$ or ${\delta }_{y\alpha }^V$ for the horizontal or the vertical incident polarization when the crystal optic axis is along the α-axis (α = x, y, z, and α = x in Eqs. (5) and (6)).

2.2 Crystal optic axis is along the y-axis

In this situation, $\frac{1}{\sin \xi \left({\overrightarrow{k}}^{(I)}\right)}\approx \frac{1}{\sin {\xi }_I}+\frac{\cot {\xi }_I}{\sin {\xi }_I}{\kappa }_{x_I}^{\left(I\right)}=1$ with ${\xi }_I=\frac{\pi }{2}$. Using the expressions: $\widehat{o}\left({\overrightarrow{k}}^{\left(I\right)}\right)=\widehat{y}\times {\widehat{k}}^{\left(I\right)}$ and $\widehat{e}\left({\overrightarrow{k}}^{\left(I\right)}\right)={\widehat{k}}^{\left(I\right)}\times \widehat{o}\left({\overrightarrow{k}}^{\left(I\right)}\right)$, we have

2.3 Crystal optic axis is along the z-axis

In this case, the linear s- and p-polarization states can be simply expressed as follows:

Equations (5), (8) and (11) present the analytical expressions for the evolution of the horizontally or the vertically polarized Gaussian beam after reflection from an air-uniaxial crystal interface. The reflected light beam experiences spin-dependent displacements along the y-axis. The amount of the splitting between the two spin components are given by Eqs. (6), (9) and (12). It shows that the displacement induced by SHEL depends not only on the incident polarization and the incident angle, but also on the orientation of the crystal optic axis. In fact, the p- or s-polarization state in Eqs. (2), (7) and (10) includes e- or o-component with a small amount of o- or e-component. If replacing the air-uniaxial crystal interface with an air-glass one, Eqs. (6), (9) and (12) reduce to the Eqs. (1) and (2) in the Ref [22]. since the difference between the ordinary and extraordinary rays disappears.

3. Experimental results and discussion

The spin-dependent shifts are measured by the experimental setup that is similar to that in Refs [21,22]. As shown in Fig. 2 , a Gaussian beam at 632.8 nm is generated by a He–Ne laser, passing through the first lens L1 with focal length of 25 mm and a Glan polarizer P1 to produce an initially linearly polarized focused beam. The beam splits into its two spin components when reflected at the interface of LiNbO₃ with the size of 10 × 5 × 4 mm, whose crystal optic axis is along the side of 4 mm. The refractive index for the ordinary and extraordinary rays are n_o = 2.232, n_e = 2.156 at 632.8 nm, respectively. Because the displacement is at the nanometer scale, the intensity distribution on the cross section of the reflected beam along the propagation direction is nearly unchanged. The second polarizer P2 is nearly orthogonal to the central wave vector of the reflected light. After P2, the two components of the splitting beam interfere. As a result, the light intensity redistributes and the position of the light spot moves [17]. This movement of the spot corresponds to an amplification of the relative displacement between the two splitting spin components. The second lens L2 with focal length of 125 mm is used to collimate the beam. Then, we use a position-sensitive detector (PSD) to measure the amplified displacement after L2 for calculating the original displacement induced by SHEL. As a matter of fact, this experimental setup, which is only sensitive to spatial displacements of the wave packets [22], converts the position displacements caused by SHEL into a momentum shift.

 

Fig. 2 Experimental setup. The He–Ne laser generates a Gaussian beam at 632.8 nm; HWP, half-wave plate for attenuating the intensity after P1 to prevent the position-sensitive detector (PSD) from saturating; L1 and L2, lenses with 25 and 125 mm focal lengths, respectively; P1 and P2, Glan polarizers.

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Figure 3 describes the displacement ${\delta }_{|+>}^V$ of the $|+>$ spin component induced by SHEL as a function of the incident angle, in the case of vertical polarization (the central electric-field-vector E//y). The blue, red, green curves indicate the theoretical prediction when the crystal optic axis is along x, y, z axis, respectively, and the dots, circles, triangles are the experimental results. The theory is nicely confirmed by the experimental data. Limited by the large holders of the optical elements and the small area of the interface, displacements at small and large incident angles were not measured. The negative displacement indicates that the $|+>$ spin component shifts along the −y direction at the interface. The displacement increases and reaches the maximum, then subsequently decreases to zero as the incident angle increases to 90 deg. The red and green curves almost overlap due to the similarity of the ratios $r_{p,ey}/r_{s,ey}$ and $r_{p,ez}/r_{s,o}$.

 

Fig. 3 Displacements of the $|+>$ spin component, ${\delta }_{|+>}^V$, as a function of incidence angle, ${\theta }_I$, in the case of vertical polarization. The dots, circles, triangles correspond to the cases that the crystal optic axis is along x, y, z axis respectively. The solid curves represent the theoretical predictions.

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We also measured the displacements for the horizontal polarization incidence. In this case, the displacement ${\delta }_{|+>}^H$ increases with the incident angle ${\theta }_I$, and rises rapidly when approaching Brewster angle ${\theta }_B$ (~65, 66 and 66 degree for the three different direction of crystal optic axis in our experiment), as illustrated in Fig. 4(a) . The inset shows the results for 0<θ_I <90°. When ${\theta }_I$>${\theta }_B$, the | + > spin component shifts oppositely and the displacement decreases with${\theta }_I$ as shown in Fig. 4(b). The details in the dashed frame are shown in the inset. There are obviously deviations from the theory in the proximity of ${\theta }_B$. The deviations may be caused by two reasons [22]: the Eqs. (6), (9) and (12) need to be modified near Brewster angle ($r_p\to 0$) and the intensity is too weak to accurately determine the position for PSD. When the incident angle is far from the Brewster angle, the experimental data are in agreement with the theory. For the horizontal incidence, the blue curve is overlapped with the red one in the most range because $r_{s,ex}$or $r_{s,o}$ are almost identical to $r_{p,ex}$ or $r_{p,o}$.

 

Fig. 4 Displacements ${\delta }_{|+>}^H$ as a function of incidence angle${\theta }_I$, in the case of horizontal polarization. The blue, red, green curves indicate the theoretical prediction when the crystal optic axis is along x, y, z axis, respectively, and the dots, circles, triangles are the corresponding experimental data. (a) ${\theta }_I$< Brewster angle${\theta }_B$. The inset shows the results for 0<${\theta }_I$<90°; (b) ${\theta }_I$> θ_B. The inset shows the enlarged image for 71° <${\theta }_I$<75°.

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4. Conclusions

In conclusion, we have calculated and measured the displacements induced by the spin Hall effect of the reflected light for an air-uniaxial crystal interface. Experimental results are in good agreement with the theory taking into account of the anisotropy of the crystal. Because the displacements depend not only on the incident polarization and the incident angle of the light beam, but also on the orientation of the crystal optic axis, it offers a flexible way to control the amount of the displacement.