1. Introduction

Amazing shifts of a physical light beam at a planar dielectric interface have been investigated for a long time. Among them, the in-plane Goos-Hänchen (GH) shift in the plane of incidence and the transverse Imbert-Fedorov (IF) shift perpendicular to the plane of incidence are well-known [1–4]. While the (linear) GH shift or the (linear) IF shift exhibits a linear positional shift of the beam center, the angular GH shift or the angular IF shift manifests the angular deviation of the beam axis, which can be considered as the shift in the wave vector space. Recently, the spin Hall effect of light (SHEL) has attracted much attention because it offer a promising tool in quantum and classical optical information processing applications [5–11]. As a result of an effective spin-orbit interaction, the reflected or transmitted beam slightly splits into its two spin components that acquire opposite transverse displacements perpendicular to the plane of incidence. This effect is also referred to as the optical Magnus effect of spinning particles [12,13]. SHEL has been studied at non-planar interfaces [14–18] and applied to probe the spatial distributions of electron spin states in semiconductors at the nanometer scale [19,20].

By analogy with the in-plane GH shifts and the transverse IF shifts, does exist the in-plane spin separation of light (IPSSL) different from the transverse spin separation induced by the SHEL when a linearly polarized beam is reflected or refracted at the planar dielectric interface? Here we report on the observation of the IPSSL. By considering the tiny in-plane spread of wave-vectors, we theoretically explain the experimental results, which exhibit excellent agreement. This observation thus not only consummates the completeness of beam shifts but also offers a way to control the direction and magnitude of the total spin separation.

2. Experimental setup

The experimental setup shown in Fig. 1 is similar to that in Refs.8-10. Along with the standard coordinate system (x, y, z) attached to the interface (z=0) and the incidence plane (y=0), we employ the beam coordinate systems (x_a, y, z_a), where a=I, R, T denotes incident, reflected, transmitted beams, respectively. The z_a axis attaches to the direction of the central wave vector.

 

Fig. 1 Experimental setup for observing the in-plane spin separation of light. 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; by replacing the PSD with a CCD, the intensity profiles of the beam after P2 can be captured. The insect shows the results when the incident angle θ_I= 30° and the polarization angle γ_I=30°.

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A He–Ne laser generates a Gaussian beam at 632.8 nm, passing through a lens L1 and a Glan polarizer P1 to produce an initially linearly polarized focused beam with a polarization angle γ_I (the angle between x _I and the central electric-field-vector). The reflected or refracted beam splits into its two spin components: the component parallel (σ=+1, right-circularly polarized, denoted by $|+>$) and antiparallel (σ=−1, left-circularly polarized, denoted by$|->$) to the central wave vector. We first focus on the reflected beam. The polarizer P2 is rotated by an angle ${\phi }_{p_2}$ from y in order to be crossed with the polarization direction of the central wave vector of the reflected light. Here, $\tan {\phi }_{p2}=\tan ({\gamma }_I)r_{s{\theta }_I}/r_{p{\theta }_I}$, where $r_{s{\theta }_I}$, $r_{p{\theta }_I}$ are the Fresnel reflection coefficients for s and p plane waves at the incident angle of ${\theta }_I$. A position-sensitive detector (PSD) is used to measure the amplified displacement after the collimation lens L2, and this allows for calculating the original separation induced by SHEL [7–10] and IPSSL. When the intensity profiles are captured, a color charge-coupled device (CCD) is used instead of the PSD.

3. Results

The experimental data are presented in Fig. 2 . In addition to the y-displacement of the $|+>$ spin component induced by the SHEL, ${\delta }_{y|+>}^{{\gamma }_{\text{I}}}$, as shown in Fig. 2(a), we observe the remarkable IPSSL along the x_R-direction, ${\delta }_{\text{x}|+>}^{{\gamma }_{\text{I}}}$, as shown in Fig. 2(b). For ${\theta }_I$ =30°(or 45°), the y-displacement decreases from 88.1 nm (or 224 nm) to 0 when ${\gamma }_I$ increases from 0 ($|H>$) to 39° (or 29°); then, the $|+>$ spin component shifts along the −y direction and increases to 58.5 nm (or 69.5 nm) at 90° ($|V>$). The zero values occur at ${\gamma }_I$ = ${\tan }^{-1}\sqrt{-r_{p{\theta }_I}/r_{s{\theta }_I}}$. For θ_I = 70°, however, the y-displacement monotonously decreases from 136 nm to 50.2 nm toward –y direction. The difference results from the sign of ${\beta }_{{\theta }_I}$ = $r_{s{\theta }_I}/r_{p{\theta }_I}$: when ${\theta }_I<{\theta }_B$, ${\beta }_{{\theta }_I}<0$; and if ${\theta }_I>{\theta }_B$, ${\beta }_{{\theta }_I}>0$, where ${\theta }_B$ is the Brewster angle (~57° in the experiment). The x_R-displacement is always in +x_R direction and reaches maximum at ${\gamma }_I^{}={\tan }^{-1}|r_{p{\theta }_I}/r_{s{\theta }_I}|$ that is comparable with the maximal y-displacement.

 

Fig. 2 The dependence of ${\delta }_{y|+>}$ (a) and ${\delta }_{x|+>}$ (b) of the $|+>$ spin component induced by SHEL and IPSSL on the polarization angle${\gamma }_I$. The dots, circles and triangles are experimental data at three typical incident angles: 30°, 45° and 70°. The curves are the theoretical results. The insets show the theoretical prediction for a period from 0° to 180°.

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In the following, we theoretically demonstrate that the IPSSL originates from the tiny in-plane spread in x _I direction of the wave-vector.

4. Theoretical analysis

4.1 Calculation of the transverse and in-plane spin separations at an interface

Here we derive the displacements induced by SHEL and IPSSL of the reflected beam at an interface, which are expressed in the spin basis [8,9]. Using the coordinate systems in Fig. 1, the incident 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 ${\overrightarrow{\kappa }}^{\left(R\right)}$:$k_{x_I}=e_R{k}_{x_R}$ and $k_{y_I}=k_{y_R}=k_y$ where $e_R=\cos {\theta }_R/\cos {\theta }_I=-1$ with ${\theta }_R=\pi -{\theta }_I$. In refraction, $e_{\text{T}}=\cos {\theta }_T/\cos {\theta }_I$ changes with the incident angle θ_I.

Because the eigenstates of the reflection and refraction are the linear p- and s-polarization states, $|p\left({\overrightarrow{k}}^{(I,R)}\right)>$ and $|s\left({\overrightarrow{k}}^{(I,R)}\right)>$, we start with a horizontally and vertically polarized incident wave-packet in the s-p basis and expend everything up to first order in κ:

These are the states that a generic polarizer would produce, and they are accomplished with the aid of the relations: $\widehat{s}\left({\overrightarrow{k}}^{(I,R)}\right)=\widehat{z}\times {\widehat{k}}^{\left(I,R\right)}/\sin ({\theta }_{I,R}({\widehat{k}}^{\left(I,R\right)}))$ and $\widehat{p}\left({\overrightarrow{k}}^{(I,R)}\right)=\widehat{s}\left({\overrightarrow{k}}^{(I,R)}\right)\times {\widehat{k}}^{\left(I,R\right)}$, in which $\frac{1}{\sin ({\theta }_{I,R}({\widehat{k}}^{\left(I,R\right)}))}\approx \frac{1}{\sin {\theta }_{I,R}}-\frac{\cot {\theta }_{I,R}}{\sin {\theta }_{I,R}}{\kappa }_{x_I,x_R}$.

We first calculate the spin separations of the reflected beam. After reflection, s- and p-polarizations evolve as: $|p\left({\overrightarrow{k}}^{(I)}\right)>\to r_p|p\left({\overrightarrow{k}}^{(R)}\right)>$ and $|s\left({\overrightarrow{k}}^{(I)}\right)>\to r_s|s\left({\overrightarrow{k}}^{(R)}\right)>$. We expand the Fresnel reflection coefficients $r_p$,$r_s$ for p and s plane waves about the central wave vector (corresponding to $r_{p{\theta }_I}$,$r_{s{\theta }_I}$at the incident angle ${\theta }_I$) [3, 7]:

The first derivative in Eq. (2) is not considered in Refs. 7-10, though it is responsible for the angular GH effect [1,3,4,21]. The correction proportional to $k_{x_I}^{}$ induces the IPSSL. Putting everything together, we can obtain:

In the spin basis set $|+>=\frac{1}{\sqrt{2}}\left(|H>+i|V>\right)$ and $|->=\frac{1}{\sqrt{2}}\left(|H>-i|V>\right)$, Eq. (3) are expressed by:

An arbitrary input polarization can be defined as follows [7–9]:

Here,

Equation (7) presents the analytical expression in the most general case of an arbitrarily polarized Gaussian beam. The reflected light beam experiences spin-dependent displacements not only along the y-axis (SHEL, ${\delta }_{y|\sigma >}^m$) but along the $x_R$-axis (IPSSL, ${\delta }_{x|\sigma >}^m$). In addition, there exist the angular shifts (momentum changes, related to the terms involving ${\mathrm{\Delta }}_{x|\sigma >}^m$, ${\mathrm{\Delta }}_{y|\sigma >}^m$) that are not measured in our experiment. As a matter of fact, the experimental setup is only sensitive to linear displacements of the wave packets [8]. This setup converts the position displacements caused by the SHEL and IPSSL into a momentum shift, and it also converts the momentum shifts into a position displacement which is small compared to the measured displacements in the experiment.

For a linearly incident polarization, which can be described by the polarization angle ${\gamma }_I$ from horizontal, Eq. (8) reduce to:

Equation (9.1) confirms the in-plane separation ${\delta }_{x|\sigma >}^{{\gamma }_I}{}_{}$in x_R direction in addition to the previously reported transverse separation ${\delta }_{y|\sigma >}^{{\gamma }_{\text{I}}}$ in y direction. In Refs. 8-10, the first derivatives were neglected, $r_p=r_{p{\theta }_I}$,$r_s=r_{s{\theta }_I}$, Δ^H,V = 0, as a result, only the SHEL was revealed. At the same time, there exist angular shift (momentum change) related terms in x_R and y directions that are not explicitly dependent on the spin σ, as described by Eqs. (9.2).

Equations (9) clearly demonstrate the correlation and difference between spin separations and angular shifts for a linearly polarized incidence. The transverse spin separation in SHEL is determined by ${\delta }^H$ and ${\delta }^V$or $r_{s{\theta }_I}/r_{p{\theta }_I}$while the angular IF shift is related to δ ^H-δ ^V. On the contrary, IPSSL is determined by Δ^H-Δ^V or the angle-of-incidence gradient of ln(r _s/r _p), while the angular GH shift is related to the Δ^H and Δ^V or the angle-of-incidence gradient of ln(r _s) and ln(r _p).

4.2 Calculation of the angular and linear GH/IF shifts

In the momentum-space representation, the reflected beam can be given by: $|{\psi }_R>=\mathrm{\Phi }(k_{x_R},k_y)|k_{x_R},k_y>|\varphi >$,with $\mathrm{\Phi }\left(k_{x_R},k_y\right)\propto \exp \left[-\frac{\mathrm{\Lambda }}{2k_I^2}\left(e_R^2{k}_{x_R}^2+k_y^2\right)-i\frac{k_{x_R}^2+k_y^2}{2k_I}{z}_R\right]$. From Eq. (7), we can also obtain the angular shifts, $<k_{x_R}^m>$ and $<k_{y_R}^m>$, and z_R-dependent parts of the linear shifts, $<x_R^m>$ and $<y_R^m>$, i.e., the expectation values of the momentum and position of the photons:

Equations (10) and (11) are in accordance with those in Refs. 2, 4, and 7.

4.3 The intensity profile of the reflected beam after P2

The intensity profile of the reflected beam after P2 for a linearly polarized incidence can be obtained from the Eqs. (7):

 

Fig. 3 Cross sections and polarization distributions of a reflected Gaussian beam. (a), (b), and (c) are schematics representing a linearly polarized beam, the reflected beam for |H> incidence and the reflected beam for an arbitrary linearly polarized incident beam, respectively. The yellow arrows and ellipses indicate the polarization distributions. (d), (e) and (f) are the corresponding intensity profiles after the beams going through a crossed polarizer. The spin separation, the ellipticity and the rotation angle of the elliptical polarizations in (b) and (c) are exaggerated for a better view.

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When both IPSSL and SHEL take place, the cross sections and polarization distributions of a reflected Gaussian beam change greatly as schematically demonstrated in Fig. 3. Figure 3(a) presents the cross section of a linearly polarized Gaussian beam with a uniform polarization distribution, in which the $|+>$ and $|->$ components fully overlap. For the |H> (or |V>) incidence as shown in Fig. 3(b), only SHEL can be observed, i.e., the reflected beam splits only along y-axis. The polarization varies mainly in y-axis with a minor change in x_R-axis due to the deflection in x_R direction that is related to the Eq. (4.3). For an arbitrary linearly polarized incident beam, the polarization distribution of the reflected beam is shown in Fig. 3(c) and Fig. 1. Figures 3(d), 3(e) and 3(f) are the corresponding intensity profiles after the beams going through a crossed polarizer. For |H> incidence, the profile shows a double-peak structure with a dark fringe at the center. As shown in Fig. 3(f), the double-peak profile rotates an angle of φ_df=${\tan }^{-1}(-{\delta }_{x|+>}^{}/{\delta }_{y|+>}^{})$ from x_R-axis when IPSSL occurs.

In the $k_{x_R}$-$k_y$ momentum space, the polarizations are linear with an angle of ${\tan }^{-1}[(k_y{\delta }^H+{\beta }_{{\theta }_I}(1+k_{x_R}{\mathrm{\Delta }}^V)\tan {\gamma }_I)/(1+k_{x_R}{\mathrm{\Delta }}^H-{\beta }_{{\theta }_I}{k}_y{\delta }^V\tan {\gamma }_I)]$from $k_{x_R}$-axis. For the |H> (or |V>) incidence, the angle is ${\tan }^{-1}[k_y{\delta }^H/(1+k_{x_R}{\mathrm{\Delta }}^H)]$ (or ${\tan }^{-1}[-(1+k_{x_R}{\mathrm{\Delta }}^V)/{\delta }^V{k}_y]$), so the dark fringe would move along y-axis by slightly rotating the polarizer.

Figure 4 demonstrates the experimental results of the rotated double-peak intensity profiles at the incident angle of 30° and 70°, respectively. These images are captured by a color CCD and in excellent agreement with the numerical simulations by calculating the intensity profile of the reflected beam after P2.

 

Fig. 4 Intensity profiles of the reflected beam passing through a crossed polarizer as a function of the polarization angle ${\gamma }_I$. The incident angle θ _I is 30° (a) and 70° (b), respectively. The experimental images of the intensity profiles are in excellent agreement with the numerical simulations in the upper row. The arrows indicate the total displacement of the $|+>$ spin component.

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IPSSL does not explicitly occur for |H> or |V> input polarization, but it covertly affects spin separations. At first glance, the two cases seem the same, but they actually differ from each other. From Figs. 2 and 4, it can be clearly seen that the dark fringe rotates 180° when ${\gamma }_I$ changes from |H> to |V> for ${\theta }_I$<${\theta }_B$. IPSSL plays a role in these two cases, resulting in the opposite direction of the spin displacement. When ${\theta }_I>{\theta }_B$, the total spin separation rotates counterclockwise and returns clockwise when ${\gamma }_I$ changes from |H> to |V>.

For the refracted beam, $k_{x_I}=e_T{k}_{x_T}$ and $k_{y_I}=k_{y_T}=k_y$. The corresponding expressions for the spin separations can be obtained by replacing $r_{s,p}$, $r_{s{\theta }_I,p{\theta }_I}$, $x_R$, $z_R$, $k_{x_R}$, ${\overrightarrow{k}}^{(R)}$, ${\psi }_R$, $w_{z_R}$, ${\gamma }_R$ and $e_R$with $t_{s,p}$, $t_{s{\theta }_I,p{\theta }_I}$, $x_T$, $z_T$, $k_{x_T}$, ${\overrightarrow{k}}^{(T)}$, ${\psi }_T$, $w_{z_T}$, ${\gamma }_T$ and $e_T$.

At the incident angle of 49.3°, ${\delta }_{y|+>}^{{\gamma }_{\text{I}}}$ monotonously decreases from 33.3 nm to 23.3 nm and ${\delta }_{\text{x}|+>}^{{\gamma }_{\text{I}}}$ grows from 0 to −8.9 nm and then returns to 0 when ${\gamma }_I$ changes from |H> to |V>. The experimental data are in excellent agreement with theoretical predictions as shown in Fig. 5 . The separations are smaller than those of reflected beam because of the smaller changes of the Fresnel transmission coefficients and their first-order derivatives.

 

Fig. 5 Displacements of the $|+>$ spin component of the refracted beam as the function of the polarization angle ${\gamma }_I$ at the incident angle of 49.3°, induced by SHEL (${\delta }_{y|+>}$, triangles) and IPSSL (${\delta }_{x|+>}$, dots), respectively. The error ranges are less than 2 nm. The curves indicate the theoretical results. The inset shows the theoretical prediction for a period from 0° to 180°.

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5. Conclusion

In conclusion, we have observed a new type of spin separation i.e. the in-plane spin separation, different from the transverse spin separation induced by the well-known SHEL, when a linearly polarized Gaussian beam is reflected or refracted at a planar dielectric interface. The measurements of displacements and the intensity profiles exhibit excellent agreement with the theoretical predictions by taking into account of the tiny in-plane spread of wave-vectors. Our study broadens the comprehensive understanding of beam shifts. As a result of the simultaneous SHEL and the new IPSSL, both the magnitude and direction of the total splitting of the two spin components can be controlled by the incident angle or the polarization angle of the incident linearly polarized light beam.