1. Introduction

The concept of weak measurement was originally proposed by Aharonov, Albert, and Vaidman [1]. It was pointed out that with appropriate pre- and post-selections in a quantum measurement, the eigenvalue of a quantum state can be amplified by the weak value, which is now known as the weak value amplification (WVA) technique. The intriguing property of weak measurement has attracted widespread discussion and sparked many novel thoughts in quantum physics [2–8]. The interpretation of the weak value still remains an open question [9–13]. In a practical sense, weak measurement has also shown great power in precision parameter estimations since a tiny signal can be greatly amplified to be detectable. For example, the position and momentum displacements [14–16], the frequency and time shifts [17–20], the phase difference [21,22], and even the angular rotations [23] have effectively been amplified by the weak value.

In 2008, Hosten and Kwiat found that apart from the weak value, a propagation factor could be arranged to realize an additional amplification in weak measurement [24], such that the spin Hall effect of light, which results in a spatial displacement on the scale of nanometers (far below the pixel resolution of a scientific camera), can be successfully observed. From then on, the propagation amplification (PA) has been jointly applied in many precision measurement scenarios. Reports have shown that an angular deflection of a mirror down to 400 frad [25], an optical rotation angle of 0.2 mdeg [26], and a phase difference on the order of $10^{-7}$ rad [27] can be measured under the WVA and the PA.

In this work, we take into account the orbital angular momentum (OAM) that could accommodate a new dimension for amplification. Several studies have preliminarily demonstrated the feasibility of applying OAM to achieve an amplification in weak measurements, which, however, neglected the interaction of OAM with other amplifications [28–31]. We use the vortex beam that carries OAM to measure the spin Hall effect of light which can lead to many practically meaningful applications [26,32,33]. The results provide a compelling evidence that the OAM amplification can work together with the WVA and the PA.

2. Theoretical analysis

The experimental setup is shown in Fig. 1. We use a He-Ne laser with the wavelength of $\lambda =633$ nm as the Gaussian light source. The first polarizer (P1) makes the beam in the horizontal polarization state, which matches the working axis of the following phase-only spatial light modulator (SLM). The half wave plate (HWP1) can be used to adjust the light intensity. We load a fork-shaped grating in the SLM, such that a vortex beam is transformed in the first diffraction order. A 4$f$ system, which is made up of two lenses (L1 and L2) with the focal lengths both being 100 mm and an iris diaphragm, is built to filter out the first diffraction order. The wave function of the generated vortex beam can be written as [34]

 

Fig. 1. Schematic diagram of the experimental setup. We use a He-Ne laser as the light source, a half wave plate (HWP1), three polarizers (P1, P2 and P3), a phase-only spatial light modulator (SLM), four lenses (L1, L2, L3 and L4), an Iris diaphragm (Iris), a prism, and a charge coupled device (CCD).

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The second polarizer (P2) is placed to pre-select the polarization state in

When the light beam is reflected by the prism, the component in the state $|{L}\rangle$ will be shifted along the transversal direction ($y$-axis) by a value of $g$, and the component in $|{R}\rangle$ will be shifted along the opposite direction by the same value. This effect is termed the spin Hall effect of light, which is an analogue of the Hall effect. In the Hall effect, if a perpendicular magnetic field is applied to a conductor, the positive and negative charges are accumulated in opposite surfaces. While in the spin Hall effect of light, the left-handed and right-handed circular polarization components are spatially separated due to the refractive index gradient on the surface of the prism. It can be derived that $g=-\frac {\cot \theta }{k}(1+\frac {r_p}{r_s})$, where $\theta$ is the incident angle, $k=2\pi /\lambda$ is the wave number, $r_p$ and $r_s$ are the Fresnel coefficients of the prism [35]. In our experiment, the incident angle is set to $\theta \approx 30^\circ$ and the prism is fabricated from N-BK7. So it follows that $r_s\approx -0.246$, $r_p\approx 0.164$ and thereby, $g\approx 58$ nm.

In the back focal plane of L4, we have the wave function that is obtained by the Fourier transform of the reflected light field on the prism, which indicates that the spatial shift will be transformed to an angular shift of the value $\gamma =kg/f_2$. The anisotropic angular shifts can be well represented by the unitary transformation

Under these circumstances, it follows the wave function in the back focal plane of L4 that

We utilize a charge coupled device (CCD) to detect the intensity distribution, which is given by

It can be found that the change of $\gamma$ is directly reflected in the change of $I(x,y)$ along the $y$ direction. Therefore, it is convenient to choose the centroid of light spot as the pointer variable, the explicit expression of which reads

3. Experimental results

In the experiment, we initially set $f_1=50$ mm and $f_2=100$ mm. By rotating P3 to make $\alpha$ ranging from $-4.8^\circ$ to $4.8^\circ$, we record each $I(x,y)$ and then calculate $\bar {y}$. As the OAM number is respectively taken $m=1$ and $m=3$, the results are depicted in Fig. 2. The curves and the dots are corresponding to the theoretical expectations and the experimental data, which exhibit a good agreement. It can be seen that the centroid shifts of $m=3$ are larger than that of $m=1$, implying an amplification brought by OAM. However, when $\alpha$ approaches zero, the centroid shifts of $m=1$ and $m=3$ are identical, implying that $\bar {y}$ is independent of $m$. This could be readily account for if we look at Eq. (9). Consider that $\alpha \rightarrow 0$, $\cot \alpha$ becomes extremely large such that $(|m|+1)\gamma ^2w^2\cot ^2\alpha (\frac {f_2}{f_1})^2\gg 2$. Equation. (9) is simplified to $\bar {y}\approx 2\alpha /\gamma$, which indicates that $\bar {y}$ is inversely proportional to $\gamma$ and there is no amplification of $\gamma$. Therefore, we should make $\alpha$ small and maintain the relation $(|m|+1)\gamma ^2w^2\cot ^2\alpha (\frac {f_2}{f_1})^2\ll 2$ in the meantime. It immediately follows that

Consequently, the spatial shift $g$, which is due to the spin Hall effect of light, is amplified by three factors, the OAM number $|m|$, the so-called propagation factor $kw^2f_2/f_1^2$, and the imaginary weak value $A_w=i\cot \alpha$. If the input light field $\psi _0(x,y)$ is just a Gaussian wave function ($m=0$), then Eq. (10) will degrade into the standard form that has been widely applied in weak measurements.

 

Fig. 2. The centroid shift $\bar {y}$ in a function of the post-selected angle $\alpha$. The red dots and blue dots are corresponding to experimental results when the OAM number $m=1$ and $m=3$, respectively. The red and blue fitting curves indicate the theoretical expectation. Three intensity distributions are inset.

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To intuitively demonstrate the amplification, we measure the centroid shifts with respect to the OAM number ranging from 0 to 5. Figure 3 shows the experimental results when the post-selected angle $\alpha$ is taken $3^\circ$ and $4^\circ$, respectively. For each OAM number, we use a CCD to record 10 images of the intensity distribution, with the time interval of 1 second. Then the averages and the standard deviations of $\bar {y}$, which correspond to the dots and the error bars, can be calculated. The fitting lines is plotted based on the theoretical expectation Eq. (10). It can be found that the experimental results of $\bar {y}$ are larger than the expectations when $m=0$ and $m=5$. Whereas the experimental results of $m=1, 2, 3,$ and 4 are all slightly smaller. We may infer that the discrepancies between experiment and theory mainly come from the modulation error of SLM. The gradient of line increases as the post-selected angle reduces. The ratio of the two gradients can be used to compare the amplifications when two different post-selected angles are chosen. We can obtain that the ratio of $\alpha =3^\circ$ and $\alpha =4^\circ$ reads 1.27, which is close to the theoretical value $\cot 3^\circ /\cot 4^\circ \approx 1.33$.

 

Fig. 3. The centroid shift $\bar {y}$ in a function of the OAM number $m$. The magenta and purple dots are corresponding to the experimental results when the post-selected angle $\alpha =0.0419$ rad and $\alpha =0.0559$ rad, respectively. The fitting lines indicate the theoretical expectation.

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We further investigate the effect of propagation factor on the centroid shift. The post-selected angle is fixed at $3^\circ$, and the focal length of L4 is manually made to be $f_2=100$ mm and $f_2=150$ mm, respectively. For each OAM number, we also record 10 images of the intensity distribution by setting the time interval to be 1 second. Then we plot the experimental results in Fig. 4 with dots. It can be obtained that the ratio of the two gradients corresponding to $f_2=100$ mm and $f_2=150$ mm is 1.52 in experiment, which agrees well with the theoretical expectation $f_2/f_1=1.50$. It is worthy of noting that similar to the results in Fig. 3, the experimentally measured values of $\bar {y}$ are all larger than that in the fitting line when $m=0$ and $m=5$, but become smaller when $m=1, 2, 3,$ and 4. This feature again confirms that the discrepancies primarily result from the modulation error of SLM.

 

Fig. 4. The centroid shift $\bar {y}$ in a function of the OAM number $m$. The green and blue dots are corresponding to the experimental results when the focal length of L4 $f_2=100$ mm and $f_2=150$ mm, respectively. The fitting lines indicate the theoretical expectation.

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It is known that the precision in measurement is ultimately limited by the shot noise of detector, which scales as $\sqrt {N_d}$ with $N_d$ denoting the photon number received by detector. In our scheme, it follows that $N_d=|\langle {f}|{i}\rangle |^2N$ as $N$ photons are initially incident. So the signal-to-noise ratio (SNR) is proportional to $(|m|+1)/\sqrt {|\langle {f}|{i}\rangle |^2N}$, which means that the OAM amplification can also increase the SNR.

4. Conclusion

In summary, we have shown that OAM can be utilized to achieve a super amplification aside from the WVA and the PA in weak measurement. We have proposed a scheme to measure the spatial displacement that is induced by the spin Hall effect of light. The experimental results showed that the centroid shift can be not only amplified by the weak value and propagation factor, but also proportional to the OAM number. OAM has the ability to offer another dimension for amplifying a weak signal, which could find important applications in a variety of precision measurement scenarios.