Variation of polarization distribution of reflected beam caused by spin separation

The variation of polarization distribution of reflected beam at specular interface and far field caused by spin separation has been studied. Due to the diffraction effect, we find a distinct difference of light polarization at the two regions. The variation of polarization distribution of reflected light provides a new method to measure the spin separation displacement caused by Spin Hall Effect of light.


Introduction
The Spin Hall Effect of light (SHEL) at the interface has attracted much attention for its potential application in quantum information and optical information processing [1][2][3][4][5]. SHEL was first proposed in 2004, and it describes the opposite transverse displacements of two spin components of photonics at the interface [1]. The SHEL effect can be seen as the consequence of Berry phase, which corresponds to the spin-orbit interaction. Weak measurement has been applied to measure the displacement [4,5]. Recently, the in-plane spin separation (IPSSL) has also been reported, which can be seen as essential composition of the theory of spin separation [6]. Polarization variation of light at the interface caused by SHEL has some crucial features and has been mentioned in Refs. [3,6]. Previous studies mainly focus on the polarization distribution at the interface, however the polarization property between the interface and far field are intrinsically different due to diffraction effect of two separated spin polarizations. The effect of beam propagation is implied in the formalism developed in [7], however this has not been made explicit.
In this paper, we have systematically studied the polarization property of the reflected beam and demonstrated the variation of polarization distribution from the interface to far field. The polarization distribution at interface is determined by spin separation (SHEL and IPSSL), while the polarization variation during the propagation is caused by diffraction effect. In our theory, upon the reflection at an air-glass interface, a linearly polarized incident light becomes a beam with elliptically polarized distribution. However, during the propagation after reflected, the polarization ellipticity angle gradually vanish and the major axis rotate. At far field, the reflected beam eventually evolves to linearly polarized distribution. This feature provides a feasible way to measure the displacement caused by SHEL. Our experimental results show a perfect agreement with the theoretical prediction.

General description of the reflected beam
To ensure the integrity of our theory, it is necessary to review the general description of spin separation at interface [4,6]. The general SHEL experimental setup and coordinate systems are shown in Fig.1. We consider the wave-packets containing distribution of wave vector ( ) Under the action of the interface, horizontally polarized incident beam H and vertically polarized incident beam V evolve as: Δ and V Δ account for in-plane spin separation. When a horizontally polarized light with a certain intensity distribution incident on the interface, from Eq.(1.a), the reflected beam then can be expressed as: are the wave functions of incident and reflected beam at interface in momentum space, respectively. The above equation can also be expressed in spin basis to get more physical meaning as described in Refs. [4,6]. Then Fourier transformation gives the spatial distribution of reflected beam at the interface as: where, ψ initial x y and ( , ) ψ final x y are the wave functions of incident and reflected beam at spatial space, respectively.
Following the same procedure, the description of reflected beam with vertical polarization can also be obtained, and the result is similar.

Variation of polarization distribution of the reflected beam
In this section, we only focus on the horizontally polarized incident beam and discuss its polarization properties. The result of vertically polarized is quite similar. Arbitrary polarization beam can be obtained by the linear combination of horizontal and vertical polarization. For realistic, the incident beam has a Gaussian shape as cot( ) arc χ ε = , where ε is the ellipticity defined as the major-to-minor-axis ratio. Therefore γ and χ can be calculated by the following equations: , tan (2 ) . After the reflect beam propagates for a long distance z R ( 2 0 / (2 ) 2 I R k w z π = ), by using Fraunhofer diffraction theory, we can describe its property as follows: F denotes Fourier transformation. Using Eqs. (2) and (5), the polarization properties of reflected beam at far field can be revealed and they are shown in Fig.2(b) and (d). Unlike the results at the interface, the polarization distribution of reflected beam becomes pure linear at a certain position of the beam cross section. The polarization at y=0 is still horizontally polarized, while the polarization at other position has rotated a small angle. And the directions of rotation are different between two sides of y =0. To show how the polarization of light evolves from interface to far field, we use Fresnel diffraction to describe the propagation of reflected beam. This gives: Then we accomplish an experiment to verify our theoretical prediction. The experimental setup is shown in Fig.1. The light with 632.8nm generated by He-Ne laser passes through HWP and P1, and becomes horizontally polarized. When the beam reflects from the interface, it suffers from polarization evolution as discussed above. We use a charge-coupled device (CCD) to record different intensity profiles when we rotate the P2. The results are shown in Fig.4(d, e, f). Clearly, the dark fringe moves as θ Δ changes. We then measure the shift of dark fringe along the y-axis, y Δ , by processing the pictures we get from CCD. The experimental data are showing in Fig.5. The solid lines are the theoretical calculation based on Eq.(8) by using our experimental parameters, where δ H at different incident angles are calculated from Eq.(1). The experimental data and theory agree perfectly.
It should be mentioned that previous measurement of SHEL displacement requires position sensitive detector (PSD) to measure the gravity center of the reflected light. In our experiment, we measure the displacement by just using CCD to record pictures and analyze shift of dark fringe along the y-axis.

Conclusion
The variation of polarization distribution of reflected beam at the interface and far field caused by spin separation has been studied. We find a distinct difference of light polarization between the two regions due to the diffraction effect. The polarization evolution of light also provides a new method to measure the spin separation displacement caused by Spin Hall Effect of light. Our experimental results exhibit good agreement with the theoretical prediction.