1. Introduction

At macroscopic scales, geometric optics gives a complete description for common optical phenomena. However, in the subwavelength, it is necessary to consider the relation between light’s spatial and polarization degrees of freedom. Owing to the optical spin-orbit interaction, light beam will not perfectly follow the geometrical-optics picture. When a linearly polarized finite width beam impinges on an interface between two media with different refractive indices, the left- and right-handed circularly polarized components will split in the opposite direction perpendicular to the incident plane. This phenomenon is called the photonic spin Hall effect (SHE) [1–6], and has drawn extensive attention due to its potential application in precision metrology [7–14]. Moreover, the photonic SHE has also been investigated in different physical systems such as nano-metal film [15,16],plasmonics [17–19], metamaterial [20–23], image edge detection [24], and even graphene [25–30].

Generally, the spin splitting in photonic SHE shows symmetry properties only when the incident beam is linear polarization state and reflected or refracted from an air-prism interface [1–3]. In a more general case, when an arbitrary polarized light beam is reflected or refracted from an isotropic interface, the spin splitting often exhibits asymmetry properties [31–33]. However, weak measurements based on the signal enhancement technique are only helpful to symmetric spin splitting [34–36], which are invalid to asymmetric spin splitting. How to transform asymmetric spin splitting into symmetric spin splitting is an important issue in photonic SHE.

Here, we put forward a phase compensation scheme to achieve the transformation from asymmetric spin splitting to symmetric spin splitting in photonic SHE. First, we theoretically establish the general expressions of spin splitting without and with phase compensation when an arbitrary polarized light beam is reflected from an isotropic interface. The spin splitting represents an asymmetric distribution before phase compensation and is transformed into a symmetric distribution after phase compensation. However, whether or not phase compensation, it does not influence the transverse shift of total barycenter of reflected field [i.e., the Imbert-Fedorov (IF) shift][37,38]. Then, under the condition of total internal reflection, we experimentally obtain the spin splitting after phase compensation with weak measurements. Finally, a conclusion is given.

2. Theoretical model

To reveal the transformation from asymmetric spin splitting to symmetric spin splitting with phase compensation in photonic SHE, we first establish the general propagation model to describe the reflection of light beam at the medium-medium interface. As shown in Fig.1, we consider a paraxial Gaussian incident beam is an arbitrary polarization state expressed as follows

Here $\eta$ is an azimuth angle and $\phi _{o}$ is the phase difference between horizontal polarization state $|H\rangle$ and vertical polarization state $|V\rangle$. When $\eta =\pi /4$, $\phi _{o}=\pm \pi /2$ denotes the left- and right-handed circular polarization states $|\pm \rangle$, respectively. The total wave function is given by

Here, $m=1$ and $2$ correspond to the cases $1$ and $2$, respectively. The transverse shift of barycenter of left- and right- handed circular polarization (i.e., the initial spin splitting) is obtained with

Here,

According to the above theoretical analysis, we can obtain the expressions of $\delta _m^\sigma$ and $\delta _m^{IF}$:

Here, $M=2{R_p^2}\cos ^2\eta +2{R_s^2}\sin ^2\eta$, $N=-2{R_p}{R_S}\cos \phi _o\sin (\phi _p-\phi _s)+({R_p^2}+{R_s^2})\sin \phi _o$ and $\Delta =(\cot \theta _i/k_0)(\cos \phi _c+2{R_p}{R_s}\cos \phi _o/M)$. Note that the IF shift is the transverse shift of total barycenter of reflected field while the initial spin splitting is the barycenter shift of left- and right- handed circular polarization relative to the coordinate origin.

Now we have obtained the general expressions of photonic spin splitting without and with phase compensation when an arbitrary polarized light beam is reflected from an isotropic interface. The situation of refraction can also be derived by similar method. From Eq. (10), in the case of 1 (i.e., without phase compensation), the spin operator $\sigma$ exists in both numerator and denominator. It leads to that the left- and right-handed circular polarization components represent an asymmetric distribution, and their intensities are also completely different by calculation, see Fig. 1(c). From Eq. (11), in the case of 2 (i.e., with phase compensation), the spin operator $\sigma$ is only associated with $\Delta$. It means that the left- and right-handed circular polarization components represent a symmetric distribution with respect to the IF shift $\delta ^{IF}$, and their intensities are exactly the same by calculation, see Fig. 1(d). The underlying physics stems from the redistribution of light field after phase compensation. Note that the asymmetric spin splitting is transformed into symmetric spin splitting after phase compensation. This property is very vital because weak measurements only work for symmetric spin splitting. In other words, the immeasurable spin splitting can be measured with weak measurements by phase compensation. From Eq. (12), whether or not phase compensation, it does not influence the IF shift.

 

Fig. 1. (a) shows the schematic of asymmetric spin splitting in photonic SHE. (b) shows the asymmetric spin splitting is transformed into symmetric spin splitting after phase compensation. (c) and (d) show the intensity distributions along $y_r$ direction without and with phase compensation, respectively.

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Next, we will explore a special case that can be easily done experimentally, which is the spin splitting of 45 degrees linear polarization under the condition of total internal reflection. In this situation, $|R_p|=1$, $|R_s|=1$, $\phi _o=0$ and $\eta =\pi /4$. Equations (10)–(12) can be simplified as

Here, $\Delta =(\cot \theta _i/k_0)[1+\cos (\phi _p-\phi _s)]$. Figures 2(a) and 2(b) show the asymmetric and symmetric spin splitting of 45 degrees linear polarization under the condition of total internal reflection. It is easy to find the left- and right-handed circular polarization components are redistributed after phase compensation. However, from Fig. 2(b), we cannot observe the symmetric spin splitting clearly. In order to show the symmetry properties clearly, we definite the relative shift between the initial spin splitting and IF shift as the spin splitting in IF shift, which is $\Delta _m^\sigma =\delta _m^\sigma -\delta ^{IF}$. As shown in Fig. 2(c) and 2(d), the asymmetric spin splitting in IF shift becomes symmetrical after phase compensation.

 

Fig. 2. (a) and (b) : the initial spin splitting without and with phase compensation changes with incident angle; (c) and (d): the spin splitting in IF shift without and with phase compensation changes with incident angle. The wavelength of the incident light is 632.8 nm, and the refractive indexes of medium 1 and medium 2 are 1.515 and 1, respectively.

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3. Experimental observation

From Fig. 2(d), the spin splitting in IF shift is on the order of nanometers, thus it is extremely difficult to accurately measure it using conventional method. However, weak measurements based on the signal enhancement technique can resolve this question effectively [34–36]. Since the spin splitting in photonic SHE was measured with weak measurements by Hosten and Kwiat [3], the technique has been widely employed for precision measurement, such as beam deflections [40], phase shifts [41], and even magneto-optical Kerr signals [42].

The schematic illustration of the experimental setup is shown in Fig. 3. A Gaussian beam generated by a He-Ne laser passes through a half-wave plate for attenuating the light intensity. Then, the short-focus lens (L1) and the long-focus lens (L2) are used to focus and collimate the beam. A 45 degrees linear polarization beam $\left [|\psi _i\rangle =1/\sqrt {2}(|H\rangle +|V\rangle )\right ]$ is preselected by the first Glan polarizer (P1). And then, the total internal reflection takes place at the prism-air interface. Here, the weak coupling between an observable (i.e., the spin operator $\hat {\sigma }_3$ of the system) and a meter variable (i.e., the transverse wave-vector component) leads to the photonic SHE. Meanwhile, the asymmetric spin splitting [Fig. 1(a)] is also produced. Next, the beam enters a quarter-half-quarter wave-plate combination for compensating the phase change between horizontal polarization state and vertical polarization state. In this step, the asymmetric spin splitting is transformed into the symmetric spin splitting in IF shift [Fig. 1(b)]. Finally, the postselection of the system is achieved with a linear polarization state $|\psi _f\rangle =\cos (45^{\circ }-\gamma )|H\rangle -\sin (45^{\circ }-\gamma )|V\rangle$ by the second polarizer (P2), and both the intensity profile and barycenter position are recorded by CCD.

 

Fig. 3. Schematic diagram of experimental setup. Light source: He-Ne laser at 632.8 nm (Thorlabs HNL210); HWP: half-wave plate; QWP: quarter-wave plate; L1 and L2: lenses of focal length 50 mm and 300 mm, respectively; P1 and P2: Glan polarizers; CCD: charge-coupled device (Thorlabs BC106N-VIS/M). Inset: (a) shows the schematic of total internal reflection; (b) and (c) show the polarization direction of Glan polarizers for preselection and postselection, respectively.

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According to the above theoretical analysis, phase compensation is the core idea in this paper. Hence how to effectively compensate the phase change between horizontal polarization state and vertical polarization state becomes a considerable problem in this experiment. Specifically, the implementation strategy of phase compensation can be described by three steps. First, the optical axis of P1 is fixed to $45^{\circ }$ from the $x_i$ axis. Meanwhile, the optical axis of P2 is fixed to $-45^{\circ }$ from the $x_r$ axis. Afterwards, in the quarter-half-quarter wave-plate combination, the optical axis of the first QWP is fixed to $45^{\circ }$ from the $x_r$ axis, which is used to transform the elliptical polarization state into linear polarization state. The optical axis of the second QWP is also fixed to $45^{\circ }$ from the $x_r$ axis, which is used to attenuate the light intensity. Finally, the HWP is used to adjust the polarization direction of linear polarization state by rotating the optical axis. When the intensity profile of bilateral symmetry appears, the phase change has been fully compensated.

Figure 4(b) demonstrates the experimental results of intensity profiles at different postselection angles. These images captured by CCD are in good agreement with the numerical simulation, see Fig. 4(a). When postselection angle $\gamma =0$, the intensity profile presents a distribution of bilateral symmetry. This is because in-plane wave vector plays a leading role with a tiny postselection angle [32]. With the increase of postselection angle, the intensity profile gradually shows a Gaussian distribution. This is because postselection angle $|\gamma |\gg |\Delta /\omega _0|$ satisfies the condition of the linear approximation in weak measurements [35]. Here, $\omega _0$ is the focused beam waist. According to the principle of weak measurements [35], one can obtain the spin splitting in IF shift through experimental measurements:

 

Fig. 4. Intensity profiles of the reflected beam passing through P2 as a function of the postselection angle at incident angle $\theta _i=43.5^\circ$. (a) and (b) denote the numerical simulation and experimental result, respectively.

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Here, $\Delta _{amp}$ is the amplified displacement measured by the CCD. $|A_w|=|\langle \psi _f|\hat {\sigma }_3|\psi _i\rangle /\langle \psi _f|\psi _i\rangle =|\cot \gamma |$ denotes the weak value. $F=z/R_0$ denotes the propagation amplification factor, where $z$ is the effective focal length of L2 and $R_0=k_0\omega _0^2/2$ is the Rayleigh distance. In our experiment, $F\approx 82.93$, with the beam waist $\omega _0\approx 27\mu {m}$.

Figure 5 shows the amplified shift of spin splitting in IF shift as a function of the postselection angle. The black solid line and red solid box represent the theoretical prediction and experimental results, respectively. To estimate the spin splitting in IF shift, the postselection angle $\gamma$ satisfying the condition of the linear approximation in weak measurements is chosen as $2.048^\circ$. The corresponding weak value $|A_w|=26.66$. According to Eq. (16) and the experimental results of Fig. 5, we can obtain the spin splitting in IF shift $\Delta =205.8nm$ when the incident angle $\theta _i=43.5^\circ$, which is close to the theoretic prediction of Eq. (14) $(\Delta =195.8nm)$.

 

Fig. 5. The amplified shift as a function of postselection angle $\gamma$.

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In addition, we also note that the weak measurement of the spin splitting in IF shift is similar to that of GH shift [43]. However, the GH shift shows symmetry properties only when the incident polarization state is 45 degrees linear polarization. In a more general case, the GH shift also exhibits asymmetry properties. The transformation from asymmetric GH shift to symmetric GH shift with phase compensation is in progress.

4. Conclusion

We have investigated the transformation from asymmetric spin splitting to symmetric spin splitting with phase compensation in photonic SHE. A proof-in-principle experiment based on weak measurements has been performed to verify the validity of our theory. These findings not only build a bridge for more photonic SHE experiments, but also provide a more comprehensive understanding for beam shift.