Abstract
In this paper, a method to measure the tiny spin splitting of the spin Hall effect of light (SHEL) using the almost-balanced weak measurement (ABWM) is presented. The ABWM technique uses two orthogonal post-selected states to record all of the information, which is a precise measurement method being different from the standard weak measurement (SWM). The theory model to describe the SHEL measurement based on ABWM is established. As results, the ABWM scheme has a larger amplification factor, reaching ∼105, which is nearly one order of magnitude higher than that of the SWM. When the post-selected angle is less than a certain value, the sensitivity and amplification factor of the ABWM scheme are higher than those of the SWM scheme, while the measurement precision and SNR of the ABWM technique are comparable to those of the SWM scheme. This research may have great potential for the precision metrology or sensing field based on the SHEL measurement.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The spin Hall effect of light (SHEL) describes a transverse spin-dependent splitting when a light beam reflected or refracted at an interface between two media with different refractive indices [1,2]. The SHEL is closely related to the physical nature of the media, such as, refractive index and thickness of the layer medium. Therefore, the SHEL has been widely applied in the field of precision measurement, such as, optical physics [3–5], semiconductor [6,7], plasmonics [8,9], nano-metal film [10], and magnetic film [11,12]. However, the SHEL is so weak that it is difficult to observe with conventional method.
Recently, the weak measurement which can realize signal enhancement has stepped into a public spotlight [13]. In 2008, Hosten and Kwiat used the weak measurement technique to observe the SHEL, and achieved an incredible resolution of 0.1 nm [3]. In addition, since the weak measurement can effectively improve the measurement precision limited by technical noises [14–17], it is has been widely used for precision measurement of small parameters [18–29]. However, the near orthogonality of the pre- and post-select states results in a large amount of statistical data being discarded, so standard weak measurement (SWM) is considered a ’harmful’ measurement [30–32].
In view of these problems, Strübi and Bruder reported a new method, the so-called almost-balanced weak measurement (ABWM), which collects all of the information by using two orthogonal post-selected states [33]. In this scheme, the intensities of two outputs are almost balanced and the differencing of them gives a SWM-like amplification in the meter. It has been shown that the ABWM scheme is robust against the alignment error and readout noise [33,34]. The ABWM scheme has also been experimentally validated in time delay estimation and provides a higher signal-to noise ratio (SNR) than SWM [35–37].
In this paper, we propose to measure the spin splitting of the SHEL based on the ABWM technique. The theory model of the SHEL measurement based on ABWM is established. In addition, the experiment based on the SWM technique is also presented for comparison. Our results show that the amplification factor and sensitivity to the original shift of the ABWM scheme are higher than that of the SWM technique.
2. Theory
First, we give general theories of parameter estimation based on SWM and ABWM. Consider a typical weak measurement procedure involving a two-level system $|\psi _{i}\rangle =\left (|0\rangle +|1\rangle \right )/\sqrt {2}$ and a meter state $|\phi _{m}\rangle =\int dF\varphi (F)|F\rangle$. Here, $|0\rangle$ and $|1\rangle$ are the eigenvectors of observable of the system $\hat {A}$, $\hat {A}=|0\rangle \langle 0|-|1\rangle \langle 1|$. $\varphi (F)$ represents the wave function of the continuous meter variable $F$. The interaction between the system and the meter at the time $t_{0}$ is described by the Hamiltonian $H=\gamma \delta (t-t_{0})\hat {A}\otimes \hat {F}$. Here, $\gamma$ stands for the coupling strength of the interaction, and $\hat {F}$ denotes the operator acting on the input meter variable. The system and meter are interrelated through interaction and evolve into an entangled state:
For the ABWM scheme, the system is post-selected by two almost balanced complementary states $|\psi _{f1,2}\rangle =\left (e^{i\epsilon }|0\rangle \pm i e^{-i\epsilon }|1\rangle \right )/\sqrt {2}$. Consequently, the final states evolve into
3. Experiment and results
We take the measurement of the spin splitting induced by SHEL as an example of our theory. Here, the weak coupling $\gamma$ is replaced by spin splitting. The transverse were vector $k_{y}$ and polarization degree of freedom are adopted as the meter variable and system, respectively. The experiment setup of the ABWM technique is shown in Fig. 1. A Gaussian beam with a wavelength of 632.8nm is emitted from the He-Ne laser. Then, the light beam passes through a half-wave plate (HWP) for adjusting the intensity of the incident light. After being slightly focused by the lens (L), the light beam transmits through a polarizer (P) which is used to control the incident polarization state of the light $|\psi _{i}\rangle$. Next, the light is reflected at the air-prism interface, and the SHEL takes place as a result of the spin-orbit interaction. The Wollaston prism (WP) is used to prepare two orthogonal post-selections $|\psi _{f1,2}^{AB}\rangle$, so the photons are spilt into two different post-selected states. Finally, the distributions of two almost balanced outputs are detected simultaneously by a charge coupled-device (CCD). The difference signal and the corresponding beam displacement are calculated by the computer. For the SWM technique, WP is replaced by the second polarizer whose polarization axis is nearly orthogonal to the optical axis of the first polarizer.
The theoretical models to describe the spin splitting estimation based on SWM and ABWM are established. First, the incident polarization of the light is pre-selected in
In the SWM scheme, the light beam is post-selected in $|\psi _{f}^{S}\rangle =|V\rangle =i(|+\rangle -|-\rangle )/\sqrt {2}$. Therefore, a large weak value is obtained with
For the ABWM procedure, the post-selected states are given by
The corresponding weak values are obtained asAccording to the experiment results, we calculated the signal-noise ratios (SNR) of the ABWM and SWM schemes, see Fig. 4. It can be seen that when the post-selected angle is close to 0, SNR of the ABWM technique is comparable to that of the SWM scheme. The ABWM scheme shows a relatively consistent SNR for all data points, see red dots in Fig. 4. For the SWM scheme, the SNR decreases with the post-selected angle reducing. For the ABWM scheme, since the light is divided into two beams after being post-selected by two different states, the technical noises of two measured signals are different. The ABWM technique is not robust to the non-common noises of two post-selelcted signal. Therefore, SNR of the ABWM scheme obtained in the experiment is lower than that of the SWM technique. However, if the experimental environment can be further optimized, ABWM can achieve higher SNR, which will find applications in precision measurement.
The measurement precision of the original shift is calculated as $\sigma _{ori}=\sigma ^{AB,S}/(|\partial \Delta y^{AB,S}_{0}/\partial \delta |)$ here, $\sigma ^{AB,S}$ is the standard deviation of the amplified shift calculated from the statistical value of repeated measurements, and $\partial \Delta y^{AB,S}_{0}/\partial \delta$ represents the sensitivity of the amplified displacement $\Delta y^{AB,S}_{0}$ to the original shift $\delta$. The sensitivities of two techniques changing with the post-selected angle are shown in Fig. 5. With the decrease of post-selected angle $\alpha$, $|\partial \Delta y^{AB}_{0}/\partial \delta |$ increases rapidly, see Fig. 5(a). It is interesting to note that $|\partial \Delta y^{S}_{0}/\partial \delta |$ reduces to zero, while the amplified displacement reaches the maximum value, see Fig. 5(b). it is notable that the sensitivity of the ABWM scheme is much higher than that of the SWM scheme. The experimental standard deviations of the amplified displacement were obtained from the statistics of 20 measurements. When choose a small post-selected angle ($\alpha <2\times 10^{-3} rad$), the measurement precision of the ABWM is about $3\times 10^{-3} um$, and that of SWM is $10^{-3} um$. As results, when the post-selected angle is small, the ABWM scheme can obtain higher amplification factor and sensitivity than the SWM strategy, while the SNR and measurement precision are comparable to the SWM scheme. Therefore, when it is necessary to detect more weak signals, the ABWM technique has more application prospects than the SWM scheme.
4. Conclusions
In conclusion, we present general theories for the parameter estimation based on ABWM and SWM schemes. Compared with the SWM technique, the ABWM scheme can obtain a larger meter shift when the two signals of different post-selections are almost equal. In addition, we study the SHEL measurement in both theory and experiment, with the ABWM and SWM schemes. As results, the ABWM technique can be widely used in sensing field, because it has a larger amplification factor and higher sensitivity than the SWM scheme. When a small post-elected angle is chose, the measurement precision and SNR of the ABWM technique are comparable to those of the SWM scheme. Therefore, ABWM could be used to measure more tiny shift of the light beam. We believe that the ABWM scheme has a promising application prospect in the precision measurement.
Funding
National Natural Science Foundation of China (11674234); Innovation project of Sichuan University (2018SCUH0021).
Disclosures
The authors declare no conflicts of interest.
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