## Abstract

The photonic spin Hall effect (SHE) manifests itself as the spin-dependent splitting of light beam. Usually, it shows a symmetric spin-dependent splitting, i.e., the left- and right-handed circularly polarized components are equally separated in position and intensity for linear polarization incidence. In this paper, we theoretically propose an asymmetric spin-dependent splitting at an air-glass interface under the illumination of elliptical polarization beam and experimentally demonstrate it with the weak measurement method. The left- and right-handed circularly polarized components show expectedly unequal intensity distributions and unexpectedly different spin-dependent shifts. Remarkably, the asymmetric spin-dependent splitting can be modulated by adjusting the handedness of incident polarization. The inherent physics behind this interesting phenomenon is attributed to the additional spatial Imbert-Fedorov shift. These findings offer us potential methods for developing new spin-based nanophotonic applications.

© 2016 Optical Society of America

## 1. Introduction

Recently, similar to the spin Hall effect (SHE) in electronic system [1, 2], the photonic SHE manifesting itself as the spin-dependent splitting of left- and right-handed circularly polarized components of a spatially confined light beam reflecting or refracting at an optical interface has been theoretically and experimentally investigated in different physical systems such as optical physics [3–9], semiconductor physics [10], high-energy physics [11, 12], and plasmonics [13–15]. The photonic SHE is generally believed to be a result of an effective spin-orbit coupling which corresponds to geometric Berry phase. There are two types of geometric phases: the spin-redirection Berry phase that is associated with the variations of direction of light beam propagation and the Pancharatnam-Berry phase related to beam’s polarization state evolution [16,17]. The photonic SHE itself can be developed into a useful metrological tool for characterizing the structure parameter variations of different physical systems. For example, we can use the photonic SHE to probe spatial distributions of electron spin states [10], measure the thickness of nanometal film [18], identify graphene layers [19], detect the axion coupling in topological insulators [20], and determine the magneto-optical constant of Fe films [21].

Remarkably, the photonic SHE offers new opportunities for manipulating photons, which provides an additional spin degree of freedom for developing spin-controlled nanophotonic applications [14, 15, 22]. Generally, the spin-dependent splitting in photonic SHE exhibit symmetry properties where the left- and right-handed circularly polarized components are equally separated in position and intensity for linear polarization incidence. In this work, we establish a general propagation model to describe an asymmetric spin-dependent splitting in photonic SHE when an elliptical polarization beam impinges upon an air-glass interface. We find that the left- and right-handed circularly polarized components show distinct intensity distributions and unequal spin accumulations. Importantly, the asymmetric distributions of the spin-dependent splitting can be modulated by adjusting the handedness of incident polarization. Therefore, we can develop a tunable photonic SHE with any desired intensity distributions and spin-dependent splitting for generation and manipulation of spin-polarized photons. These findings offer us potential methods for developing new spin-controlled nanophotonic devices. The rest of the paper is organized as follows. First, we theoretically analyze the asymmetric spin-dependent splitting in photonic SHE under a slightly elliptical polarization beam illumination. Next, we focus our attention on the weak measurements experiment for observing this phenomenon. Under this condition, the weak value includes both of the real and imaginary parts. Finally, a conclusion is given.

## 2. Theoretical model

In this part, we first establish a general propagation model to describe the process of elliptical polarization beam reflected at an air-glass interface and theoretically investigate the corresponding asymmetric photonic SHE. Here, the electric field distributions and the spin-dependent shifts of the left- and right-handed circularly polarized components will be discussed. The asymmetric photonic SHE is schematically shown in Fig. 1. The first and second columns show the polarization and intensity distributions of incident light beam with linear and elliptical polarization, respectively. We can see that the electric field intensity distributions are symmetrical for linear polarization light [Figs. 1(b) and 1(e)]. However, for the elliptical polarization beam, the left- and right-handed circularly polarized components indicate asymmetric splitting [Figs. 1(h) and 1(k)]. The last column of Fig. 1 describes the intensity distributions and spin-dependent splitting of the left- (solid line) and right-handed (dashed line) circularly polarized components of reflected beam. When the incident light beam is linear polarization, the filed intensity (the height of Gaussian components) and spin-dependent shifts (*δ*_{±}) of these two spin components show equal values [the same heights and *δ*_{+} = *δ*_{−}, as shown in Figs. 1(c) and 1(f)].

As for the illumination of elliptical polarization beam, the asymmetric photonic SHE appears, resulting in the unequal separation in position and intensity of left- and right-handed circularly polarized components [with different heights and *δ*_{+} ≠ *δ*_{−}, as shown in Figs. 1(i) and 1(l)].

Next, we will quantitatively study the reflection process of elliptical polarization beam and the corresponding asymmetric photonic SHE. For simplicity, we choose the long and short axis of the elliptical polarization beam directed along the horizontal (H) and vertical (V) directions, respectively. The elliptical polarization beam can be decomposed into two orthogonal polarization states H and V. In the spin basis, the H and V polarization states can be expressed as
$|H\u3009=\left(|+\u3009+|-\u3009\right)/\sqrt{2}$ and
$|V\u3009=i\left(|-\u3009-|+\u3009\right)/\sqrt{2}$. We theoretically analyze the asymmetric photonic SHE by establishing the relationship between the incident and reflected fields. As for the incident light beam with elliptical polarization, the Jones vector can be written as (cos*β*, *e ^{iφ}* sin

*β*)

*. Here*

^{T}*β*represents the azimuth angle (the angle between the crystal axis of wave plate and horizontal axis) and

*φ*denotes the phase difference between the H and V polarization components. In the present study, we consider an elliptical polarization beam with its long and short axis along to the horizontal and vertical directions. Therefore, the Jones vector can be simplified to (cos

*β*, +

*i*sin

*β*)

*or (cos*

^{T}*β*, −

*i*sin

*β*)

*representing the left- or right-elliptical polarization in the case of*

^{T}*φ*= ±

*π*/2. It is noted that the azimuth angle

*β*(also the ellipticity) mentioned above is a tiny value allowing for a slightly elliptical polarization.

We consider the incident source is a monochromatic Gaussian beam whose spectrum is arbitrarily narrow and can be written as

*w*

_{0}is the beam waist. The polarization operator

*σ*= ±1 stand for left- and right-handed circularly polarized components, respectively. According to the transversality, we can obtain the reflected field of left- and right-handed circularly polarized components in the case of left-elliptical polarization incidence [7] (the case of right-elliptical polarization can be obtained in the similar way), with the long axis being along to the horizontal and vertical directions:

*r*and

_{p}*r*denote the Fresnel reflection coefficients for parallel and perpendicular polarizations, respectively. As shown in Eq. (2), we provide that

_{s}*k*≪ 1,

_{ry}δ^{H}*k*tan

_{ry}δ^{H}*β*≪ 1, and

*η*≪ 1. In Eq. (3), we provide that

*k*≪ 1,

_{ry}δ^{V}*k*cot

_{ry}δ^{V}*β*≪ 1, and

*ϑ*≪ 1. Here,

*δ*=(1 +

^{H}*r*) cot

_{s}/r_{p}*θ*

_{i}/k_{0},

*δ*=(1 +

^{V}*r*) cot

_{p}/r_{s}*θ*

_{i}/k_{0},

*η*=

*r*tan

_{s}*β/r*, and

_{p}*ϑ*=

*r*cot

_{p}*β/r*.

_{s}*k*

_{0}is the wave number in free space. The terms exp(±

*ik*) and exp(±

_{ry}δ^{H}*ik*) stand for the spin-orbit coupling and we find that the shifts induced by these terms are symmetrical, whereas the other electric field components exp(

_{ry}δ^{V}*ik*tan

_{ry}δ^{H}*β*) and exp(

*ik*cot

_{ry}δ^{V}*β*) are not the spin-orbit coupling terms. However, they can affect the spin-dependent spatial splitting in photonic SHE, which will be discussed in the follwing. From Eqs. (2) and (3), we can obtain the intensity distributions of left- and right-handed circularly polarized components as shown in Figs. 2(a)–2(d). It should be noted that the electric field components exp(±

*η*) and exp(±

*ϑ*) cause the asymmetric intensity distributions. For simplicity, we only consider the long axis of incident elliptical polarization beam along to the horizontal direction. Here, for example, the ellipticity are chosen as 0.2° and 5°. And the incident angles are selected as

*θ*= 30° and

_{i}*θ*= 45°, respectively. From the pictures, we find that the left- and right-handed circularly polarized components show unequal intensity distributions which can be modulated by changing the ellipticity and incident angles.

_{i}The photonic SHE is described for the left- and right-circularly polarized components undergoing spin-dependent shifts, so the reflected field centroid should be determined. In the present work, we only consider the spin-dependent splitting in the transverse direction. According to Eqs. (2) and (3), we can calculate the field centroid distribution. At any given plane *z* = *const.*, the transverse displacement of field centroid compared to the geometrical-optics prediction is given by

**k**=

_{ry}*k*

_{ry}**e**

*. The first and second terms in Eq. (4) stand for the spatial and angular shifts, which are independent of and dependent on*

_{ry}*z*, respectively. In fact, after calculating, we find that the photonic SHE shows pure spatial displacements which are independent of

*z*.

Evaluating Eq. (4), we can obtain the asymmetric spin-dependent shifts of elliptical polarization beam. Figures 2(e)–2(h) show the initial transverse shifts of left- and right-handed circularly polarized components changing with the ellipticity and handedness of incident polarization. Under this condition, we find that the left- and right-handed circularly polarized components show unequal spin-dependent shifts, which is different from the previous works [4,5,22]. Here, the ellipticity are also chosen as 0.2° and 5°. First, we consider the long axis of elliptical polarization beam along to the horizontal direction [shown in Figs. 2(e) and 2(f)]. In this case, we find that the left- and right-handed circularly polarized components represent unequal values of spin-dependent shifts which are extremely sensitive to the ellipticity variations. Remarkably, the asymmetric spin-dependent splitting can be modulated by adjusting the handedness of incident polarization. Figures 2(g) and 2(h) denote the condition of the long axis of elliptical polarization beam along to the vertical direction. Here, the transverse displacements of left- and right-handed circularly polarized components change slowly with the variations of ellipticity. However, this asymmetric spin-dependent splitting can also be adjusted when the handedness of incident polarization alters. So, we can develop a tunable photonic SHE with any desired electric field intensity and spin-dependent splitting.

We can attribute this asymmetric photonic SHE to the additional spatial Imbert-Fedorov shift [23, 24]. From the Eqs. (2) and (3), as for the left elliptically polarized light impinges at the media interface, we find that there exists a unified spatial displacement induced by the terms exp(*ik _{ry}δ^{H}* tan

*β*) and exp(

*ik*cot

_{ry}δ^{V}*β*) in addition to the normal symmetric spin-dependent shifts. After calculating Eq. (4), we can describe the final field centroid as follows:

## 3. Experimental observation and discussion

The asymmetric photonic SHE induced spin-dependent splitting are too small to be detected directly. However, the signal enhancement technique known as the weak measurements [25–30] can resolve this problem. The weak measurements based on preselection and postselection states has attracted a lot of attention and holds great promise for precision metrology such as beam deflection measurement [31], measuring small optical phase shift [32,33], direct detection of the quantum wavefunction [34], observing the average trajectories of single photons [35], and full characterization of polarization states of light [36]. It should be noted that the weak measurements method involved in the present work shows some adjustments based on the previous weak measurements, which will be discussed in the following.

There exists an amplified factor so-called weak value playing a great role in weak measurements. The weak value establishes the relationship between the observable and the shifts in measurement pointer’s mean position and mean momentum

|*i*〉 and |

*f*〉 stand for the preselection and postselection states. As shown in Fig. 3(a), the Bloch sphere can be introduced to describe the weak measurements process [37]. We consider an elliptical polarization beam incident and the long and short axis are along to the horizontal and vertical directions. Therefore the preselection state of the system can be written as:

*π*/2. Using Eq. (6) together with Eq. (7) and (8), we can calculate the real and imaginary parts of the weak value:

*β*and Δ. This is different from the previous work [5] in which the angle

*β*= 0 and the weak value is a pure imaginary number. In order to obtain the large output, we need to maximize the weak value including both of the real and imaginary parts. Figures 3(b) and 3(c) show the products of |〈

*f*|

*i*〉Re(

*A*)| and |〈

_{w}*f*|

*i*〉Im(

*A*)|. We can see that both of the ellipticity

_{w}*β*and the amplification angle Δ should be chosen as the tiny values for obtaining a large weak value. For simplicity, in the following, we replace the symbols Re(

*A*) and Im(

_{w}*A*) with

_{w}*A*and

_{Re}*A*.

_{Im}There also exists another amplification mechanism called propagation amplification [29] that produces the amplified factor *F* accompanying by the imaginary part of weak value, which leads to the possibility of even larger enhancements following the beam free evolution. Finally, we can obtain the amplified shifts [from Eq. (5)] by the weak value and propagation amplifications:

*σ̂*

_{3}which is corresponding to the spin projection along the central propagation direction. The weak value amplification process can convert the position displacements caused by the photonic SHE into a momentum shift, and it also converts the momentum shifts into a position shift. However, in our work, the contribution of the real part is much smaller than that in imaginary part due to two reasons: the tiny ellipticity and the beam free evolution. Therefore, the imaginary part plays a key role in the final amplification and we can rewrite the Eq. (11) as follows: This is the important result for the following experimental measurement. We can measure the amplified shifts by reading out the difference between two states |

*V*± Δ〉 or |

*H*± Δ〉 (the

*A*shows opposite signs).

_{Im}Our experimental setup shown in Fig. 4(a) is similar to that in [5, 8, 18]. A He-Ne laser is used to generate linear polarization Gauss beam which firstly impinges onto the HWP. This HWP can adjust the polarization of incident light beam, which is used to control the light intensity preventing the charge-coupled device (CCD) from saturation. Then, the light beam passes through a short focal length lens (L1) and is preselected as a slightly elliptical polarization state by P1 and QWP, which is slightly different from the previous work [5]. Here, the P1 is chosen as |*H*〉 or |*V*〉. When the beam reaches the sample (BK7 glass) interface, the asymmetric photonic SHE takes place allowing for the reflected beam separated into two unequal spin components. As the reflected beam splits by a fraction of the wavelength, the left- and right-handed circularly polarized components interfere destructively at the second polarizer (P2) [as shown in Fig. 4(b)].

From Eq. (12), we can measure the amplified shifts by discriminating the difference between the displacements for rotating the P2 as two states |*f*〉 = |*V* ± Δ〉 or |*H* ± Δ〉. Here, the Δ is a small angle that we called amplification angle. This process can be seen as the postselection. Using this method, we can deduce the initial symmetrical spin-dependent shifts
${\delta}_{\pm}^{H,V}$. By considering the additional spatial Imbert-Fedorov shift in Eq. (5), we can finally obtain the asymmetric spin-dependent displacements
${y}_{\pm}^{H,V}$. In our weak measurement experiment, we choose the amplification angle Δ = 0.8°. After passing through the polarizer P2, we use L2 to collimate the beam and make the beam shifts insensitive to the distance between L2 and the CCD, which will improve the measurement precision. Finally, a CCD is used to measure the amplified shift after L2.

In the case of elliptical polarization beam with its long axis along to the horizontal direction, we measure the amplified displacements of asymmetric photonic SHE on the BK7 glass every 0.5° from 52° to 60° [as shown in Fig. 5(a)]. After obtaining the amplified shifts, we can deduce the initial displacements [Fig. 5(c)]. In our experiment, we choose the ellipticity as *β* = 0.2° and the incident beam is fixed to left-elliptical polarization. As for the left-handed circularly polarized component of reflected light beam, the spin-dependent shift first increases with the incident angle. After reaching the peak value at the incident angle about *θ _{i}* = 56.3°, the shift decreases rapidly and then gets the negative maximum value. As for the right-handed circularly polarized component, the spin-dependent shift shows opposite trend. It first decreases with the incident angle and reaches the negative peak value. We should note that, under this condition, the spin-dependent shifts of left- and right-handed circularly polarized components in photonic SHE represent asymmetrical values which is different from the previous symmetrical photonic SHE.

We measure the shifts of the asymmetric photonic SHE every 5° from 30° to 85° under the condition of elliptical polarization beam with its long axis along to the vertical direction [Figs. 5(b) and 5(d)]. The ellipticity is also chosen as *β* = 0.2°. Limited by the large holders of experimental equipments, the spin-dependent shift at small incident angles can not be measured. Here, the ellipticity and the polarization of incident beam are chosen as the same as the above condition (long axis along to horizontal direction). The spin-dependent shifts of left- and right-handed circularly polarized components change slowly with the incident angles. Both of them exhibit a peak value but with opposite signs. We note that, in this condition, the degree of asymmetry is smaller than the above case. In fact, in the case of elliptical polarization beam with its long axis along to the horizontal direction, there exists a relatively large spatial Imbert-Fedorov shift [38].

## 4. Conclusions

In conclusion, we have examined the asymmetric photonic SHE when an elliptical polarization beam reflected at a glass interface. It was found that the left- and right-handed circularly polarized components in photonic SHE represent distinct intensity distributions and the corresponding spin-dependent shifts show unequal values. We can attribute this asymmetric photonic SHE to the additional spatial Imbert-Fedorov shift which is corresponding to the overall transverse displacement of light beam. We have also found that the asymmetric distributions of the spin-dependent splitting can be modulated by adjusting the handedness of incident polarization. The weak measurement technique was used to measure this asymmetric photonic SHE, and the experimental results are in good agreement with the theoretical calculations. These findings provide an additional degree of freedom for generation and manipulation of spin-polarized photons and thereby open the possibility of developing new spin-controlled nanophotonic devices.

## Acknowledgments

This research was partially supported by the National Natural Science Foundation of China (Grant No. 11447010) and the Natural Science Foundation of Hunan Province (Grant No. 2015JJ3026).

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