Estimation of Kerr angle based on weak measurement with two pointers

Lan Luo; Lan Luo; Tong Li; Yinghang Jiang; Liang Fang; Liang Fang; Liang Fang; Bo Liu; Bo Liu; Bo Liu; Bo Liu; Zhiyou Zhang

doi:10.1364/OE.487363

1. Introduction

The magnetic properties and special effects of magnetic materials make them widely used in daily life, science, and technology [1,2]. The characterization and measurement of magnetic material parameters is an indispensable technique in the field of magnetism [3]. The commonly used measurement methods include vibrating sample magnetometer (VSM) [4–6], alternating gradient magnetometer (AGM) [7], magneto-optical Kerr magnetomete [8,9], etc. These methods are realized by measuring the magnetic moment and magnetic flux produced by the sample. Low dimensional magnetic materials with nanometer scale exhibit some unique effects, which opens up a new situation for many fields such as spintronic devices and magnetooptical switches, and information storage [10–13]. However, these traditional measurement methods make it difficult to measure two-dimensional magnetic samples because the magnetic moment is very small. The magneto-optical Kerr effect (MOKE) can be used to measure the magnetic properties of thin films with atomic layer thickness due to its advantages of high sensitivity, non-contact, and imageability. However, the traditional extinction and photometric magneto-optical Kerr magnetometers have low measurement precision, large system error, and slow measurement speeds. How to further improve the measurement precision of magneto-optical Kerr rotation angle is of great significance to expand the application of thin film magnetic materials.

As a high-precision measurement method in theory and experiment, weak measurement provides the possibility to solve such problems. Since the weak measurement was first proposed by Aharonov, Albert, and Vaidman in 1988 [14], it has been possible to improve the measurement precision limited by the technical noise by using the pre- and post-selections. Therefore, the weak measurement is widely used in the measurement of various small parameters, such as beam shifts [15–20], optical phases, [21–23] frequency shifts, refractive index variations [24], and Kerr non-linearity [25–28]. In recent years, weak measurement has also been widely used in the magneto-optical parameter measurement of thin film materials [29–32]. The weak measurement scheme based on spin Hall effect of light (SHEL) in Ref. [29–31] ignores the existence of ellipticity. In addition, the measurement dynamic range of these schemes is very small, and it is difficult to determine the magnetization saturation point.

In this paper, a weak measurement method for the Kerr rotation estimation using the double pointers (pointer shift and intensity of the post-selected light) is investigated. We establish a general weak measurement for the estimation of phase variation, and demonstrate that the product of the double pointers (the mean value shift and intensity of the post-selected light) is only related to the phase, and is robust to the amplitude error. Therefore, it is of great benefit to use the product of two pointers as the output when the amplitude noise is nonzero. This method is applied to estimate the magneto-optical Kerr angle of NiFe film. The product of amplified displacement shift and intensity of the post-selected light is sensitive to the Kerr angle (which is the phase variation between left- and right- polarized light) and independent of the ellipticity. The proposed scheme has a simple device and high precision and provides a new method for the measurement of magnetic parameters of two-dimensional magnetic materials.

2. Theoretical analysis

A general theoretical model of phase measurement based on weak measurement is established. Firstly, the whole state of the instrument and system can be expressed as

(1)$$\begin{aligned} |\left.\mathrm{\psi}_0\right\rangle\otimes|\left.\phi\right\rangle, \end{aligned}$$

where $\left |\psi _0\right \rangle =\left (e^{-i\alpha }\left |0\right \rangle +e^{i\alpha }\left |1\right \rangle \right )/\sqrt 2$ is the initial state of the light. $\left |0\right \rangle$ and $\left |1\right \rangle$ are the eigenstates of the observable operator ${\hat {A}}=\left |0\right \rangle \left \langle 0\right |-\left |1\right \rangle \left \langle 1\right |$. $\alpha <<1$ is the phase delay between $\left |0\right \rangle$ and $\left |1\right \rangle$. $|\phi \rangle =\int {dF|F\rangle \phi (F)}$, where $\phi \left (F\right )=\left (\pi \Delta F\right )^{-1/4}\exp {\left [-\left (F-F_0\right )^2/2\Delta F^2\right ]}$ is the distribution of the instrument. $F$ is input variables for the instrument. $F_0$ and $\Delta F$ are expected values and uncertainties, respectively.

The variable $F$ then is coupled feebly to the system observable operator ${\hat {A}}$ in the polarization degree of freedom via $U=\exp (-i\gamma \hat {A}\hat {F})$. $\gamma$ represents the coupling strength. There are some systematic errors in the measurement process. Here, we mainly analyze the influence of amplitude error which is expressed in the observable system: $U_{\epsilon }=\exp (-i\epsilon \hat {A}\hat {F})$, where $\epsilon$ is the amplitude error. Therefore, the state of the light beam evolves into

(2)$$\begin{aligned} |\Psi\rangle= U U_{\epsilon}|\left.\mathrm{\psi}_0\right\rangle\otimes|\phi\rangle=\frac{1}{\sqrt2}\left(e^{{-}i\gamma\hat{F}}e^{{-}i\alpha-\epsilon}|0\rangle+e^{i\gamma\hat{F}}e^{i\alpha+\epsilon}|1\rangle\right)|\phi\rangle. \end{aligned}$$

After post-selected in $|\psi _f\rangle =(|0\rangle -|1\rangle )/\sqrt 2$, the whole state of the light beam is given as

(3)$$\begin{aligned} |\Psi_f\rangle=\langle\mathrm{\psi}_f|U U_{\epsilon}|\psi_0\rangle\otimes|\phi\rangle =\langle\mathrm{\psi}_f|e^{{-}i\gamma\hat{A}\hat{F}} e^{{-}i\epsilon\hat{A}\hat{F}}|\psi_i\rangle\otimes|\phi\rangle. \end{aligned}$$

Since the average value $F_{0}$ of the initial distribution of the instrument is not necessarily zero (such as the frequency pointer). In order to make the calculation simpler, the modulation weak measurement is introduced [33]. Therefore, the instrument input variable and the pre-selected angle are transformed as follows:

(4)$$\begin{aligned} & F^\prime=F-F_0 \\ & \left|\psi_i\right\rangle\rightarrow \exp{\left({-}i\gamma\hat{A}F_0\right)} \exp\left({-}i\epsilon\hat{A}\hat{F}\right)\left|\psi_0\right\rangle. \end{aligned}$$

Therefore, Eq. (3) is rewritten as

(5)$$\begin{aligned} |\Psi_f\rangle=\langle\mathrm{\psi}_f|e^{{-}i\gamma\hat{A}\hat{F}^{\prime}}|\psi_i\rangle\otimes|\phi\rangle, \end{aligned}$$

where evolution operator is replaced by $U^{\prime }=\exp {\left (-i\gamma \hat {A}{\hat {F}}^\prime \right )}$,

Two pointers, the mean value shift and intensity of the post-selected light, as two kinds of conventional information carried by the post-selected light beam, which are respectively expressed as

(6)$$\begin{aligned} \delta F & =\frac{\int_{-\infty}^{\infty}\left\langle\Psi_f|F^\prime|\Psi_f\right\rangle}{\int_{-\infty}^{\infty}\left\langle\Psi_f|\Psi_f\right\rangle} \\ & ={-}\frac{\Delta F^2\gamma \sin(2\alpha+2\gamma F_0)}{\cos(2\alpha+2\gamma F_0)-e^{\gamma^2\Delta F^2}\cosh2\epsilon} \end{aligned}$$

and

(7)$$\begin{aligned} I_t=\int_{-\infty}^{\infty}\left\langle\Psi_f|\Psi_f\right\rangle=\frac{I_0}{2}\left[{-}e^{-\gamma^2\mathrm{\Delta}F^2}\cos(2\alpha+2\gamma F_0)+\cosh2\epsilon\right]. \end{aligned}$$

Here, $I_0$ is the initial intensity of the light beam without post-selection. Note that the mean value shift and post-selected light intensity will be affected by amplitude error. The mean value shift and intensity can be directly output by a detector (such as a CCD). After normalization by $I_0$, the product of post-selected light intensity and pointer shift is

(8)$$\begin{aligned} P=\frac{1}{I_0}I_t\delta F=\gamma \Delta F^2 e^{-\gamma^2\mathrm{\Delta}F^2} \sin(2\alpha+2\gamma F_0)/2. \end{aligned}$$

It is worth noting that the product of two pointers is only related to the phase variation. In addition, the normalized difference intensity of the left- and right- of the light distribution is obtained as

(9)$$\begin{aligned} \frac{1}{I_0}\left(\int_{0}^{\infty}\left\langle\Psi_{f}|\Psi_{f}\right\rangle-\int_{-\infty}^{0}\left\langle\Psi_{f}|\Psi_{f}\right\rangle\right)=\frac{\gamma\Delta F}{\sqrt\pi}\sin(2\alpha+2\gamma F_0) \propto P. \end{aligned}$$

The difference intensity of the left- and right- of the light distributions is proportional to the product of double pointers. Therefore, in phase measurement, Eqs. (8) and (9) apply to detectors with different outputs, i.e., Eq. (8) is applicable to CCD, which output the intensity and mean value, and Eq. (9) is applicable to single point detectors, which output the intensity.

The theoretical results of two pointers without any approximation are shown in Fig. 1. The coupling strength and the uncertainty are respectively set as $F_{0}=0$, $\gamma =1\times 10^{-5}$ and $\Delta F=1\times 10^{2}$. Here, we mainly analyze amplitude error independent of phase $\epsilon =0.001rad$, and amplitude error related to phase $\epsilon =\alpha$, $\epsilon =\alpha -0.001rad$. As shown in Figs. 1(a) and 1(b), the pointer of the mean value is very sensitive to the phase variation $\alpha$. The shaded area of Fig. 1(a) is the linear dynamic range of the phase estimation. When the amplitude error $\epsilon$ is very small, the mean value shift is independent of the amplitude error in the inversion region, see the red dotted line in Fig. 1(a). In the case of $\epsilon =\alpha -0.001$, the existence of amplitude error makes the curve of mean value shift unsymmetrical, see the green dotted line in Fig. 1(b). Figures 1(c) and 1(d) show the pointer of the intensity of the post-selected light changing with the phase variation under different amplitude errors. As shown in Fig. 1(d), when the amplitude error $\epsilon =\alpha -0.001$, the light intensity curve will shift and the lowest point of light intensity is no longer at $\alpha =0$. Therefore, in a certain dynamic range, the light intensity monotonically changes with phase variation $\alpha$, as seen in the shaded area of Fig. 1(d). The above analysis shows that the existence of amplitude error has a great impact on phase measurement.

Fig. 1. Theoretical results of two pointers under different amplitude errors. (a) and (b) show the mean value shift changing with the phase variation. (c) and (d) show the intensities of the post-selected light as functions of the phase variations.

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The existence of amplitude error will make the relationship between double pointers (mean value shift and intensity) and phase variation more complex. In order to reveal the underlying mechanism, we have analyzed the sensitivity of double pointers to the phase variation.

For the case of fixed amplitude error $\epsilon =0$ and $\epsilon =0.001$, the sensitivity of the post-selected light intensity to the phase decreases to 0 with the phase variation decreasing, see black and blue dotted lines in Fig. 2(a). For amplitude error related to phase ($\epsilon =\alpha$ and $\epsilon =\alpha -0.001$), the sensitivity $\partial I_t /\partial \alpha$ has changed. In the case of $\epsilon =\alpha -0.001$, the lowest sensitivity is no longer at the position of $\alpha =0$. As shown in Fig. 2(b), the sensitivity of the mean value shift to the phase variation is very complex. In the case of $\alpha =0$, with the increase of phase, the sensitivity $\partial \delta F/\partial \alpha$ decreases from the highest value to 0, then gradually increases, and then decreases. Several different amplitude noises cause very obvious changes in the sensitivity curve $\partial \delta F /\partial \alpha$, see Fig. 2(b).

Fig. 2. Theoretical results of the sensitivity of two pointers under different amplitude errors. (a) shows the sensitivity of the intensity of post-selected light to the phase variation as a function of the phase variation. (b) shows the sensitivity of the mean value shift to the phase variation as a function of the phase variation.

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The theoretical analysis of the product of two pointers (the intensity and the mean value) and its sensitivity are depicted in Fig. 3. The most interesting thing is that the product of two pointers is proportional to the phase delay $\alpha$, and completely independent of the amplitude error $\epsilon$. Comparing the results from Fig. 1, it is interesting to note that the dynamic range of the product of two pointers in the phase measurement is no longer limited. In addition, the sensitivity $\partial P/\partial \alpha$ is basically a constant, see Fig. 3(b). This is also the main advantage of this work. However, the measurement precision is affected by standard deviations of amplified displacement and light intensity. Therefore, when the phase to be measured is large, the measurement precision will also significantly decrease.

Fig. 3. Theoretical results of the product of double pointers under different amplitude errors. (a) shows the product of the double pointers changing with the phase variation. (b) shows the corresponding sensitivity of the product of double pointers to the phase variation as a function of the phase variation.

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3. Magneto Kerr angle measurement

The experiment setup for the Kerr rotation angle measurement is shown in Fig. 4. A monochromatic Gaussian light beam with the wavelength of $632.8 nm$ is emitted by the He-Ne laser. The half wave plate (HWP) is used to control the intensity of the incident light. Then, the light beam passes through the first polarizer (P1), whose optical axis is in the horizontal direction, to prepare the incident polarization. Next, the light is focused by the first lens (L1) and reflected on the surface of the magnetic sample (NiFe film) to excite SHEL. In our experiment, the magnetic NiFe film is ($d=30nm$) deposited on the surface of Si were prepared by using the vacuum thermal evaporation technique. After being collimated by the second lens (L2), the light beam is post-selected by the second polarizer (P2). Finally, the post-selected light beam is received by the CCD to output the amplified displacement and intensity.

Fig. 4. Experimental setup for magneto-optic Kerr angle measurement. Light source, He-Ne laser; HWP, half-wave plate; P1 and P2, Glan laser polarizers; L1 and L2, lenses with focal length 50 mm and 250 mm, respectively; CCD, Charge-coupled device.

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Firstly, the whole initial state is given by $|\left.\mathrm {\psi }_i\right \rangle \otimes |\left.\phi \right \rangle$, where $|\left.\mathrm {\psi }_i\right \rangle =(|+\rangle +|-\rangle )/\sqrt 2$ is the incident polarization state of the light. $|+\rangle =(|H\rangle +i |V\rangle )/\sqrt {2}$ and $|-\rangle =(|H\rangle -i |V\rangle )/\sqrt {2}$ are the left- and right-circularly polarized states. $|H\rangle$ and $|V\rangle$ are the horizontal and vertical polarization. $\left.|\phi \right \rangle =\int {dk_y\phi (k_y)\left.|k_y\right \rangle }$, and $\phi \left (k_y\right )=\left (w/\sqrt {2\pi }\right )\exp {\left (-w^2k_y^2/4\right )}$ is the transverse distribution of the pointer wave function. $k_y$ represents the wave vector in the direction $y$, and $w$ is the beam waist. After reflection on the surface of the magnetic sample, the state of the light beam is described as

(10)$$\left[\begin{matrix}E_r^H\\E_r^V\\\end{matrix}\right]={\hat{R}}_1\left[\begin{matrix}E_i^H\\E_i^V\\\end{matrix}\right],$$

where

(11)$$\hat{R}_{1}= \left[ \begin{array}{cc} r_{pp}-\dfrac{k_{ry}(r_{ps}-r_{sp})\cot\theta_{i}}{k_{0}} & r_{ps}+\dfrac{k_{ry}(r_{pp}+r_{ss})\cot\theta_{i}}{k_{0}}\\ r_{sp}-\dfrac{k_{ry}(r_{pp}+r_{ss})\cot\theta_{i}}{k_{0}} & r_{ss}-\dfrac{k_{ry}(r_{ps}-r_{sp})\cot\theta_{i}}{k_{0}}\\ \end{array} \right] $$

are the Jones matrix to describe the reflection on the surface of a magnetic materials. $r_{pp}$, $r_{ps}$, $r_{sp}$ and $r_{ss}$ are the complex Fresnel coefficients of the magnetic medium. Therefore, the state of the reflected light can be expressed as

(12)$$\begin{aligned} |\mathrm{\Psi}_i\rangle & =\frac{r_{pp}}{\sqrt2}\left[\left(1+ik_y\delta_H-i\frac{r_{sp}}{r_{pp}}\right)|+\rangle+\left(1-ik_y\delta_H+i\frac{r_{sp}}{r_{pp}}\right)|-\rangle\right]\left.|\phi\right\rangle \\ & \approx r_{pp} e^{ik_y\hat{\sigma_{3}}\delta_H}|\left.\psi_{pre}\right\rangle\left.|\phi\right\rangle \end{aligned}$$

where

(13)$$\begin{aligned} |\left.\psi_{pre}\right\rangle & =\frac{1}{\sqrt2}\left[\exp\left({-}i\frac{r_{sp}}{r_{pp}}\right)|+\rangle+\exp\left(i\frac{r_{sp}}{r_{pp}}\right)|-\rangle\right] \\ & =\frac{1}{\sqrt2}\left[\exp({-}i\theta_k-\varepsilon)|+\rangle+\exp(i\theta_k+\varepsilon)|-\rangle\right] \end{aligned}$$

is the pre-selected state of the weak measurement. $r_{sp}/r_{pp}=\theta _k-i\varepsilon$ , where $\theta _k<<1$ and $\varepsilon <<1$ are the magneto optic Kerr angle and ellipticity, respectively. The Kerr angle $\theta _k$ represents the phase variation between the left- and right-circularly polarized states, and the ellipticity $\varepsilon$ corresponds to the amplitude error discussed in the previous section. The exponential term $\exp (-ik_y\delta _H\hat {\sigma _{3}})$ is the coupling of the spin–orbit, which corresponds to the weak coupling in the weak measurement model. $\hat {\sigma _{3}}=\left |+\right \rangle \left \langle +\right |-\left |-\right \rangle \left \langle -\right |$ denote the spin operator of the photon. $\delta _H=\cot \theta _i(1+r_{ss}/r_{pp})/k_{0}$ is the spin splitting of the horizontal polarization. $\theta _i=70^{\circ }$ is the incident angle.

When adjusting the experimental working point, the magnetic field is zero, and the initial Kerr angle $\theta _{k0}$ and ellipticity $\varepsilon _0$ will also be introduced when the light is reflected on the surface of the magnetic medium. As the magnetic field changes, the pre-selected state is given by

(14)$$\begin{aligned} |\left.\psi_{pre}^{\prime}\right\rangle=\exp({-}i{\theta^\prime}_{k}-\varepsilon^\prime)|+\rangle+\exp(i{\theta^\prime}_k+\varepsilon^\prime)|-\rangle, \end{aligned}$$

where ${\theta ^\prime }_{k}=\theta _{k0}+\Delta \theta _k$ and $\varepsilon ^\prime =\varepsilon _0+\Delta \varepsilon$. $\Delta \theta _k$ and $\Delta \varepsilon$ are the Kerr angle and ellipticity varying with magnetic field. When the magnetic field is zero, by adjusting the post-selected state to make the output light field behave as a symmetric split-Gaussian. So the post-selected state is given by

(15)$$\begin{aligned} |\left.\psi_{post}\right\rangle=\frac{1}{\sqrt2}\left(e^{{-}i\theta_{k0}}|+\rangle-e^{i{\theta}_{k0}}|-\rangle\right). \end{aligned}$$

Here, this post-selected state can compensate for the initial Kerr angle $\theta _{k0}$. Therefore the whole state of the light is written as

(16)$$\begin{aligned} \left.|\mathrm{\Psi}_{f}\right\rangle & =r_{pp}\left\langle\psi_{post}|\right.e^{ik_y\delta_H\hat{\sigma_{3}}}|\left.\psi_{pre}^\prime\right\rangle\phi(p) \\ & =r_{pp}\left\langle\mathrm{\psi}_{post}|\left.{\mathrm{\psi}_{pre} ^\prime}\right\rangle\right.(\cos k_y\delta_H+i A_w \sin k_y\delta_H)\phi(p), \end{aligned}$$

where

(17)$$\begin{aligned} A_w=\frac{\langle\psi_{post}|\hat{\sigma_{3}}|\psi_{pre}\rangle}{\langle\psi_{post}|\psi_{pre}\rangle}=\frac{i\sin2\Delta\theta_k-\sinh2\varepsilon^{\prime}}{-\cos2\Delta\theta_k+\cosh2\varepsilon^{\prime}} \end{aligned}$$

is the weak value of the observable $\hat {\sigma _{3}}$. The amplified displacement shift and intensity of the post-selected light beam are as follows

(18)$$\begin{aligned} \delta y=\frac{\left\langle\Psi_f\right|i\partial k_y\left|\Psi_f\right\rangle}{\left\langle\Psi_f\middle|\Psi_f\right\rangle} =\frac{z}{r}\frac{\delta_H\sin2\Delta\theta_k}{(\cos2\Delta\theta_k-e^{2\delta^2/w^2}\cosh2\varepsilon^{\prime})}, \end{aligned}$$

(19)$$\begin{aligned} I_t=\left\langle\Psi_f\middle|\Psi_f\right\rangle =\frac{I_0}{2}r_{pp}^2({-}e^{{-}2\delta^2/ w^2}\cos2\Delta\theta_k+\cosh2\varepsilon^{\prime}). \end{aligned}$$

Here, $z=250mm$ is the free propagation distance and $r=kw^2/2$ represents the Rayleigh distance. $I_{0}=15mw$ is the initial intensity without the post-selection. The product of light intensity and amplified displacement is

(20)$$\begin{aligned} P=\dfrac{I_t\delta y}{I_{0}}={-}\frac{e^{{-}2\delta^2/ w^2} r_{pp}^2 z\delta_H\sin2\Delta\theta_k}{k w^2}\propto \Delta\theta_k. \end{aligned}$$

It worth to note that the product of two pointers $P$ is sensitive to the variation of the Kerr rotation $\Delta \theta _k$.

The amplified displacement and intensity of the post-selected light beam changing with the magnetic intensity variation are shown in Fig. 5. The magnetic field strength is from -1kGs to 1kGs, and then from 1kGs to -1kGs. The amplified displacement and light intensity are very sensitive to magnetic intensity. The curve of intensity (or amplified displacement) versus magnetic intensity is similar to a complete hysteresis loop. In Fig. 5(a), the light intensity decreases as the magnetic field increases until it stabilizes, indicating that the Kerr angle in this experiment is within the dynamic range of Fig. 1(d). It worth to note that, the hysteresis loop in Fig. 5(b) is significantly narrower than in Fig. 5(a) because the amplified displacement has reached its maximum value before the Kerr angle is saturated. This indicates that the measurement of Kerr angle exceeds its dynamic measurement range when using the amplified displacement as a pointer. According to the theoretical results of Eqs. (18) and (19), it is impossible to obtain an accurate Kerr angle cannot be obtained from the amplified displacement and intensity in the presence of ellipticity.

Fig. 5. (a) shows the intensity post-selected light versus magnetic intensity, (b) shows the relationship between amplified displacement and magnetic intensity. a and b stand for the positive saturation point and negative saturation point, respectively.

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The product of two pointers (amplified displacement and intensity of the post-selected light beam) is calculated by using the experimental results in Fig. 5. The curve of the product of two pointers changing with the magnetic field also shows a complete hysteresis loop, see Fig. 6. According to the theoretical analysis in Eq. (20), without any approximation, the relationship between the product of two pointers and Kerr angle is linear.

Fig. 6. The product of double pointers changing with magnetic intensity. a and b stand for the positive saturation point and negative saturation point, respectively.

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4. Conclusion

In conclusion, a weak measurement method using double pointers (the amplified displacement shift and intensity of the post-selected light beam) to estimate the magneto-optic Kerr angle is proposed. The product of the two pointers is proportional to the Kerr angle and independent of the ellipticity. It is demonstrated that the product of two pointers has a good linear relationship with the Kerr angle and a larger dynamic measurement range. The double pointers proposed in this paper enrich the research content and application field of quantum weak measurement. This technique has important research significance for characterizing the magnetic parameters of magnetic thin films and expanding applications.

Funding

National Natural Science Foundation of China (11674234).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Estimation of Kerr angle based on weak measurement with two pointers

Abstract

1. Introduction

2. Theoretical analysis

3. Magneto Kerr angle measurement

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Equations (20)

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