Impact of the pitch angle on the spin Hall effect of light weak measurement

Xiangxing Bai; Xiangxing Bai; Xiangxing Bai; Xiangxing Bai; Yang Liu; Yang Liu; Yang Liu; Yang Liu; Linlong Tang; Linlong Tang; Qing Zang; Qing Zang; Qing Zang; Jialu Li; Jialu Li; Jialu Li; Wenqiang Lu; Haofei Shi; Xiudong Sun; Xiudong Sun; Xiudong Sun; Xiudong Sun; Yueguang Lu; Yueguang Lu; Yueguang Lu

doi:10.1364/OE.392402

1. Introduction

As light is reflected and refracted at a plane interface, the reflected and refracted fields introduce in-plane and out-of-plane shifts depending on the incident polarization. These shifts, which are called Goos-H$\ddot{\textrm a}$nchen shift [1] and Imbert-Fedorov(IF) [2] shift, respectively, have been studied in a variety of systems, such as metamaterials [3,4], plasmonics [5], vortex beams [6] and anisotropic media [7,8]. In particular, as a variation of IF shift, the spin Hall effect of light (SHEL) originates from spin-orbit interactions and describes a spin-dependent phenomenon [9]. Given the incidence of a linearly polarized light beam, the left and right circularly polarized components of the reflected and refracted field are shifted transversely perpendicular to the refractive index gradient.

With further research, the physics parameters related to the properties of the incident light and interface, such as thickness [10,11], magneto-optical coefficient [12–14], polarization state [15], edge detection [16], particle displacement [17], chiral molecules [18,19], phase [20], refractive index [21,22] and Fermi energy of graphene [23], could be measured precisely by the SHEL. The tiny variations of these parameters are defined by transverse shifts with different incident angles (i.e., azimuth angle), and an effective metrological method for precision physics parameter sensing is confirmed. Remarkably, both the azimuth angle and the pitch angle are changed when an interface is rotated. However, the impact of the interface pitch angle has not received enough attention in previous studies. As an important interface parameter, the pitch angle not only has a great effect on subsequent metrology experimental studies of SHEL but also offers great advances in angle sensing of detection [24], magnetosphere [25], communication [26] and so on.

In this work, the impact of the pitch angle on the SHEL is investigated theoretically and experimentally. First, an analytical expression is established to describe the influence of the pitch angle on the spin shift of the SHEL. Then, the pitch angle is coupled with a modified pre-selected angle to realize the overlap between the pre-selected state and the post-selected state and enables the amplification of the spin shift to identify the pitch angle. Furthermore, the measurement region could be modulated by modulating the post-selection. Finally, an application of accurate interface angle sensing is presented for the pitch angle.

2. Theoretical model

A schematic illustration of the SHEL reflection at an interface is presented in Fig. 1(a). When a linear polarized Gaussian beam is reflected at the interface, it would be divided into left and right circularly polarized spin components, the centroids of which would shift reversely away from the incident plane. As the interface is rotated $\alpha$ around the x-axis (i.e., the pitch angle is $\alpha$), the modified incident angle $\theta '$ and the polarization angle $\gamma$ are changed accordingly to the following geometrical relationship

(1)$${\cos \theta ' = \cos \theta \cos \alpha },$$

(2)$${\cos \gamma = \frac{{\sin \theta }}{{\sqrt {{{\sin }^2}\theta + {{\tan }^2}\alpha } }}},$$

where $\theta$ is the initial incident angle. As shown in the inset in Fig. 1(a), because of the interface rotation, the initial horizontal polarization Gaussian wave packet in the coordinates $\left ( {{x_i},{y_i},{z_i}} \right )$ should be written as

(3)$${{f_H}(\theta ',\gamma ) = \frac{{{w_0}}}{{\sqrt {2\pi } }}\exp ( - \frac{{R(k_{{x_i}}^2 + k_{{y_i}}^2)}}{{2{k_0}}})\left( {\cos \gamma \textbf{{h}} + \sin \gamma \textbf{{v}}} \right)},$$

where ${w_0}$ is the beam waist; $R = {k_0}w_0^2/2$ is the Rayleigh length of the beam; ${k_x}_{_i}$ and ${k_y}_{_i}$ represent the transverse wave vector of the beam; ${k_0}$ is the central wave vector in a vacuum; and $\textbf {{h}}$ and $\textbf {{v}}$ are unit vectors along the horizontal and vertical polarizations for the rotated interface, respectively. Under the reflection at the rotated interface, the SHEL process occurs with the reflected field of the following form in the coordinate frame $({x'_r},{y'_r},{z'_r})$:

(4)$${\left| {{R_H}(\theta ',\gamma )} \right\rangle = \left| {\textrm{{h}}({\textbf{{k}}_{\textbf{{r}}'}},\theta ')} \right\rangle + \left| {v({\textbf{{k}}_{\textbf{{r}}'}},\theta ')} \right\rangle }.$$

Here, ${\textbf {{k}}_{\textbf {{r}}'}}$ represents the wave-vector of the reflected light beam when the pitch angle is $\alpha$; $\left | \textrm {{h}} \right \rangle$ and $\left | \textrm {{v}} \right \rangle$ are the general expressions of the reflected angular spectra of the horizontal polarization and vertical polarization incidence, respectively, with the following forms [27]:

(5)$${\left| {h({\textbf{{k}}_{\textbf{{r}}'}},\theta ')} \right\rangle = [\frac{1}{{\sqrt 2 }}({r_p} + i\delta )\left| + \right\rangle + \frac{1}{{\sqrt 2 }}({r_p} - i\delta )\left| - \right\rangle ]\cos (\gamma )},$$

(6)$${\left| {v({\textbf{{k}}_{\textbf{{r}}'}},\theta ')} \right\rangle = [\frac{1}{{\sqrt 2 }}( - i{r_s} + \delta )\left| + \right\rangle + \frac{1}{{\sqrt 2 }}(i{r_s} + \delta )\left| - \right\rangle ]\sin (\gamma )}.$$

Where $\delta = {k_{r'y}}({r_p} + {r_s})\cot (\theta ')/{k_0}$ is the spin shift in frequency space for different spin states; ${k_0}$ is the central wave vector; ${k_{r'y}}$ represents the transverse wave vector of the beam; ${r_p}$ and ${r_s}$ are the Fresnel coefficients of parallel and perpendicular polarization incidence, respectively, and $\left | \textrm {{+}} \right \rangle$ and $\left | \textrm {{-}} \right \rangle$ represent the left and right circular polarization states, respectively. To evaluate the spin shift of the SHEL, the reflected wave function is transformed into the position space and the barycentre of the left and right circular polarization light field could be calculated. Considering the first-order approximation of the Fresnel coefficient ${r_p} = {r_p} + {k_{ix}}\chi /{k_0}$ with $\chi = \partial {r_p}/\partial \theta$[28], the transverse spin shift of the left and right circular polarization components can be expressed as

(7)$${\delta {y_\sigma } = \sigma \Delta {y_{line}} + {z'_r}\Delta {y_{angular}} }$$

with

(8)$${\Delta {y_{line}} ={-} \frac{{\cot \theta 'R(A + \cos 2\gamma C)}}{{{{\cos }^2}\gamma (B + {{\cot }^2}\theta 'A) + {{\sin }^2}\gamma (4{k^2}{R^2}{r_s}^2 + {{\cot }^2}\theta 'A)}}},$$

(9)$${\Delta {y_{angular}} ={-} \frac{{\cot \theta '\sin 2\gamma C}}{{{{\cos }^2}\gamma (B + {{\cot }^2}\theta 'A) + {{\sin }^2}\gamma (4{k^2}{R^2}{r_s}^2 + {{\cot }^2}\theta 'A)}}}.$$

Here, $A = 2kR{({r_p} + {r_s})^2} + {\chi ^2}$, $B = 2kR(2k{r_p}^2R + {\chi ^2})$ and $C = 2kR({r_p}^2 - {r_s}^2) + {\chi ^2}$.

Fig. 1. (a)Schematic illustration of the SHEL when a polarized Gaussian beam is reflected at an interface. The Z-axis of the laboratory Cartesian coordinate frame $\left ( {X,Y,Z} \right )$ is normal to the interface at $Z = 0$, and the $Z'$-axis of the laboratory Cartesian coordinate frame $(X',Y',Z')$ is normal to the interface at $Z' = 0$ after rotating interface $\alpha$ around the x-axis (i.e., the pitch angle is $\alpha$). $({x_i},{y_i},{z_i})$ and $({x_r},{y_r},{z_r})$ represent the incident and reflected beam coordinate frames when the pitch angle $\alpha = {0^ \circ }$, respectively. $({x'_r},{y'_r},{z'_r})$ represents the reflected coordinate frame when the pitch angle $\alpha$. Inset: The polarization states as the pitch angles are ${0^ \circ }$ $(H,V)$ and $\alpha$ $(h,v)$; (b)Theoretical calculations of the spin shift versus pitch angle for horizontal polarization incidence. The incident wavelength $\lambda = 632.8$ $nm$, the waist of the incident Gauss beam ${w_0} = 90$ $\lambda$, the incident angle $\theta = {50^ \circ }$ and the propagation distance of light $z = 250$ $mm$

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The barycentre transverse shift of the two spin components is described by Eq. (7). The first term $\sigma \Delta {y_{line}}$ is independent with z as the spatial shift and represents the spin-dependent splitting of the SHEL. $\sigma = \pm 1$ represent the left and right circular polarization components transversely shift towards the opposite direction, respectively. The second term, ${z'_r}\Delta {y_{angular}}$ denotes the angular shift, which increases with z and is independent of the light spin states. This results from the variation of polarization and describes the transverse shift of the total reflected field (i.e., the IF angular shift [29]). As an example, Fig. 1(b) shows the relationship between the spin shifts and the pitch angle calculated by Eq. (7) using the parameters detailed in the figure caption. The curve indicates that the magnitude of the angular shift (the green line) gradually increases as the pitch angle increases from ${0^ \circ }$, and the left and right spin components are located on the upper and lower sides of the angular shift, respectively, with an equal spatial shift of $\Delta {y_{line}}$.

3. Experimental observation

To define the relationship between the pitch angle and the spin shift, the weak measurement technique is introduced to amplify the spin shift of the SHEL [30]. The experimental setup is illustrated in Fig. 2(a). First, a 632.8 $nm$ polarized Gaussian beam is generated by a He-Ne laser, and the half-wave plate (HWP) is used to control the light intensity. Next, the Gaussian beam passes through a lens with a short focal length (L1) and a Glan polarizer (P1), by which a horizontal polarization beam is produced as the initial pre-selection of the experimental system. Then, spin-dependent splitting resulting from the spin-orbit interaction of light couples with the system. After that, the system is post-selected with vertical polarization by another Glan polarizer (P2). Finally, the light beam is collected by a lens with a long focal length (L2) and detected by a charge-coupled device(CCD).

Fig. 2. (a) Experimental setup: L1 and L2 are the lenses with effective focal lengths of 100 $mm$ and 250 $mm$, respectively; HWP is the half-wave plate; P1 and P2 are Glan laser polarizers; Laser is a He-Ne laser at 632.8 $nm$ (Thorlabs, HNL210L); and CCD is the charge coupled device (Coherent, LaserCam-HR). (b) the principle of transverse shift amplification.

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The weak measurement requires a small deviation angle between the pre-selection and the post-selection for the amplification of the spin shift, which is fulfilled by setting the two selection states to be nearly perpendicular, as in previous studies. In contrast, the initial pre-selection is exactly perpendicular to the post-selection in our experimental setup, and the reflected field after post-selection could be described as

(10)$${E_r = \cos (\gamma )\left| {v({{\textbf{{x}}'}_r},\theta ')} \right\rangle \textrm{{ - }}\sin (\gamma )\left| {h({{\textbf{{x}}'}_\textbf{{r}}},\theta ')} \right\rangle },$$

where ${\textbf {{x}}'_r}$ is the reflected light beam coordinate frame when the pitch angle is $\alpha$. Interestingly, the spin shift could also be amplified because our modified pre-selection would introduce a small deviation angle. Specifically, as illustrated in Fig. 2(b), for the initially horizontal polarization incidence (marked by the red line), when the pitch angle of the interface is changed, it would be decomposed into two polarization components with respect to the rotated interface (i.e., ${v_i}$ and ${h_i}$). Then, the reflected light would be rotated with a small modified angle $\Delta$ resulting from different reflection coefficients for the horizontal and vertical polarization components, as marked by the green lines in the figure. The modified angle $\Delta$ represents the overlap of the pre-selection and post-selection states, which could enable the amplification of the spin shift of the SHEL. In addition, $\Delta$ follows the relationship

(11)$${\tan (\Delta + \gamma ) = {v_{0r}}/{h_{0r}}},$$

where ${h_{0r}}$ and ${v_{0r}}$ are the reflected amplitudes of the centre wavevectors for horizontal and vertical polarization, respectively. The centroids of the light field distribution are calculated by Eq. (10), from which the amplified transverse shifts are extracted. The curves of the transverse shift versus the pitch angle at initial incident angles of ${40^ \circ }$, ${50^ \circ }$ and ${60^ \circ }$ are displayed in Fig. 3(a). In all three cases, the amplified transverse shift is zero when the pitch angle is ${0^ \circ }$. Such a result is expected because the pre-selection is perpendicular to the post-selection. As the pitch angle increases from ${0^ \circ }$, the transverse shifts quickly increase to an extreme value of approximately 440 $\mu m$ and then decrease, signifying that the transverse shift is especially sensitivity to the pitch angle around ${0^ \circ }$. The modified angle and the amplified factor ($Transverse$ $Shift/\Delta {y_{line}}$) versus the pitch angle are investigated when the incident angle is ${50^ \circ }$, as depicted in Fig. 3(b). As the pitch angle increases from ${0^ \circ }$, the magnitude of the modified angle gradually rises, indicating that the overlap of the pre-selection and post-selection states would be changed accordingly. As a result, the amplified factor of weak measurement would be changed with the pitch angle. We find that the amplified factor and the transverse shift have a similar tendency versus the pitch angle, suggesting that the pitch angle has a great effect on the amplified factor, which in turn results in the main variation of the transverse shift.

Fig. 3. (a) Calculated transverse shift of the barycentre versus the pitch angle; (b) modified angle and amplification factor versus the pitch angle for the case of ${50^ \circ }$.

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In the above discussions, we implicitly assumed that the CCD is moved with the pitch angle due to the shift of the reflected light beam. However, moving the CCD with the constantly changing pitch angle is difficult to achieve in practice. To solve such a problem, the reflected coordinate frames with different pitch angles $\alpha$ should be transformed to the reflected coordinate frame with a pitch angle $\alpha = {0^ \circ }$ as described in the Appendix. The transverse shift in different reflected coordinate frames is transformed to the transverse shift in the initial reflected coordinate frame $\left ( {{x_r},{y_r},{z_r}} \right )$ and measured by a fixed CCD sensor without concerning the light beam shift for rotated interface. Then, the modified theoretical prediction (solid lines) and experimental results (solid dots) are plotted in Fig. 4(a). The transverse shift is measured every $\textrm {{0}}\textrm {{.00}}{\textrm {{5}}^ \circ }$ from $\alpha = - {0.2^ \circ }$ to $\alpha = {0.2^ \circ }$, and the initial incident angles are chosen as ${40^ \circ }$, ${50^ \circ }$ and ${60^ \circ }$. The figure suggests that the experimental measurement fits well with the theoretical prediction. The transverse shift first steeply increases as the pitch angle $\alpha$ increases from ${0^ \circ }$ and then slows down. To further analyse this phenomenon, the intensity profiles of the reflected field are plotted at the CCD for pitch angles of $\alpha$ = ${0^ \circ }$, ${0.02^ \circ }$, ${0.05^ \circ }$, ${0.1^ \circ }$ and ${0.2^ \circ }$, as calculated by Eq. (10) in Fig. 4(b). It can be found that the experimental results agree well with the theoretical prediction. When the pitch angle $\alpha = {0^ \circ }$, the left circular polarized and right circular polarized light spots are symmetrically distributed in the CCD because of the SHEL. The variation of the light field can be divided into two parts as the pitch angle is changed. The first part derives from the light beam offset. The entire light field shifts along to $+y$ as $\alpha$ increases. The second part is caused by the weak measurement of the SHEL. As the pitch angle is changed (i.e., as the modified pre-selection angle is changed), the upper part gradually intensify, and the under part is gradually weakened, resulting in the barycentre shifting along the $+y$ direction. In this manner, the relationship between the pitch angle and the transverse shift is confirmed experimentally by the modified weak measurement technique. In particular, when the pitch angle is changed, the modified pre-selected angle is introduced and results in the overlap of the pre-selection and post-selection states. Therefore, the amplification of the spin shift has been realized.

Fig. 4. (a) The modified calculation and experimental result of the transverse shift when the initial incident angle is ${40^ \circ }$, ${50^ \circ }$ and ${60^ \circ }$. (b) the modified calculation and experimental result of the intensity profiles when the initial incident angle is ${50^ \circ }$.

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According to our previous discussion, when the post-selection is perpendicular to the pre-selection at first (i.e., the post-selection is perpendicular to the pre-selection as $\Delta = 0^\circ$), the most sensitive region is around the pitch angle $\alpha = {0^ \circ }$. Hence, a natural question arises: it is possible to adjust the most sensitive region to other angle ranges? Here, an effective solution is provided by introducing a modulated post-selected angle $\psi$ to modulate the most sensitive region. As illustrated in Fig. 5(a), if the post-selected angle is rotated with $\psi$ to obtain a modulated post-selection that is initially perpendicular to the modified pre-selection, the modified pre-selected angle should be increased from $\Delta$ to $\Delta +\psi$ (i.e., the pitch angle should be changed). For example, if the modulated post selected angle $\psi = + {1^ \circ }$, the pitch angle should be set as ${-0.167^ \circ }$ to adjust the modified pre-selection angle as ${+1^ \circ }$ at first, as shown by the red dot in Fig. 3(b), which indicates that the most sensitive region is around ${-0.167^ \circ }$ accordingly. Similarly, the most sensitive region is around ${+0.167^ \circ }$ when the modulated post selected angle $\psi = - {1^ \circ }$. The transverse shift versus pitch angle curve is plotted when the modulated post-selected angle $\psi = {0^ \circ }$, $\pm {1^ \circ }$ in Fig. 5(b). The experimental measurement agrees with the theoretical calculation. As a result, the most sensitive region could be modulated by the modulated post-selection angle.

Fig. 5. (a) The principle of adjusting the measurement region; (b) the calculated and experimental transverse shifts when the modulated post-selected angle $\psi$ = ${-1^ \circ }$, ${0^ \circ }$ and ${+1^ \circ }$.

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On the basis of the theoretical and experimental results, the transverse shift is very sensitive to the pitch angle, especially for a tiny angle. Moreover, the transverse spin-dependent shift is also sensitive to the incident angle around the Brewster angle, as analysed in previous research [28]. Therefore, the SHEL could be used in interface angle sensing about the azimuth angle and the pitch angle, as shown in Fig. 6(a). For example, if we want to accurately adjust the interface from an azimuth angle of ${0^ \circ }$ and a pitch angle of ${0^ \circ }$ to an azimuth angle of ${-1.023^ \circ }$ and a pitch angle of ${0.038^ \circ }$, the SHEL could help to monitor the angle variation in real time by relying on the same experimental setup shown in Fig. 2(a) with the horizontal polarization incidence at the Brewster angle ($\approx {56.572^\circ }$). First, the post-selection P2 is oblique to P1 with an angle of ${90^ \circ }$+${2^ \circ }$. Then, the azimuth angle of the interface is altered, as shown in Fig. 6(b): the transverse shift is adjusted from 0 to 432.3 $\mu m$ (red dot), which means that the incident angle changes from ${56.572^ \circ }$ to ${55.549^ \circ }$ (i.e., the azimuth angle is ${-1.023^ \circ }$). After that, we adjust P2 to be perpendicular to P1, and the light spot would become two symmetric parts again. Finally, the pitch angle is adjusted to make the transverse shift from 0 to 407.2 $\mu m$, as shown by the red dot in Fig. 6(c). Namely, the pitch angle is changed from ${0^ \circ }$ to ${0.038^ \circ }$. Therefore, the tiny variations of the azimuth angle and the pitch angle are clearly captured by the transverse shift of the SHEL, and accurate angle sensing has been realized. Based on these results, one can estimate the minimum observable angles of such an interface angle sensing method. When the precision of the barycentre measurement is 1 $\mu m$, the minimum observable azimuth angle is $1.4 \times {10^{ - 3}}^ \circ$, and the minimum observable pitch angle is $6.6 \times {10^{ - 5}}^ \circ$ in the experimental setup.

Fig. 6. (a) Schematic illustration of the interface angle sensing; the transverse shift variation when we change the (b) azimuth angle and the (c) pitch angle.

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4. Conclusion

In conclusion, we have revealed the impact of the pitch angle on the transverse shift of the SHEL. Changing the pitch angle introduces the SHEL in the IF angular shift with the asymmetric spin-dependent phenomenon. Based on the modified weak measurement, accurate pitch angle measurement is achieved by defining the transverse shift of the SHEL, and the sensitive region could be modulated by post-selection. Furthermore, by extending the results, interface sensing about the azimuth angle and the pitch angle is realized with minimum observable angles of $1.4 \times {10^{ - 3}}^ \circ$ and $6.6 \times {10^{ - 5}}^ \circ$, respectively. The results may provide a simple and precise method for accurately measuring and monitoring the pitch angle.

Appendix: Coordinate frame transformation

In this part, a detailed calculation is given to describe the coordinate frame transformation from different value $\alpha$ to the case of $\alpha = {0^ \circ }$.

First, we transform from the reflected coordinate frame $\left ( {{{x'}_r},{{y'}_r},{{z'}_r}} \right )$ when the pitch angle is $\alpha$ (marked red in Fig. 7(a)) around y-axis by $\theta '$ (the incident angle) to the reference coordinate frame $\left ( {X'',Y'',Z''} \right )$ (marked green in Fig. 7(a)):

(12)$${{\textrm{{m}}_{x_{r}^{\prime} y_{r}^{\prime} z_{r}^{\prime} } \to XYZ} = \left( {\begin{array}{ccc} {\cos (\pi - \theta ')} & 0 & {\sin (\pi - \theta ')}\\ 0 & 1 & 0\\ { - \sin (\pi - \theta ')} & 0 & {\cos (\pi - \theta ')} \end{array}} \right)}.$$

Fig. 7. (a)Schematic illustration of the coordinate frame $\left ( {{{x'}_r},{{y'}_r},{{z'}_r}} \right )$, $\left ( {X'',Y'',Z''} \right )$ and $\left ( X',Y',Z' \right )$ (pitch angle is $\alpha$). (b)Schematic illustration of the coordinate frame $\left ( {X,Y,Z} \right )$ and $\left ( {{x_r},{y_r},{z_r}} \right )$

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Secondly, we transform from the reference coordinate frame $\left ( {X'',Y'',Z''} \right )$ around z-axis by $\xi$ to the reference coordinate frame $\left ( {X',Y',Z'} \right )$ (marked blue in Fig. 7(a)):

(13)$${{\textrm{{m}}_{X^{\prime\prime}Y^{\prime\prime}Z^{\prime\prime} \to X'Y'Z'}} = \left( {\begin{array}{ccc} {\cos (\xi )} & { - \sin (\xi )} & 0\\ {\sin (\xi )} & {\cos (\xi )} & 0\\ 0 & 0 & 1 \end{array}} \right) },$$

where $\xi = \arccos (\frac {{\sin \theta }}{{\sqrt {{{\sin }^2}\theta + {{\sin }^2}\alpha {{\cos }^2}\theta } }})$.

Thirdly, we transform from the reference coordinate frame $\left ( {X',Y',Z'} \right )$ around x-axis by $\alpha$ to the reference coordinate frame $\left ( {X,Y,Z} \right )$ (marked black in Fig. 7(b)):

(14)$${{\textrm{{m}}_{X'Y'Z' \to XYZ}} = \left( {\begin{array}{ccc} 1 & 0 & 0\\ 0 & {\cos (\alpha )} & { - \sin (\alpha )}\\ 0 & {\sin (\alpha )} & {\cos (\alpha )} \end{array}} \right)}.$$

Finally, we transform from the reference coordinate frame $\left ( {X,Y,Z} \right )$ around y-axis by $\theta$ (the initial incident angle) to the reflected coordinate frame $\left ( {{x_r},{y_r},{z_r}} \right )$ where the pitch angle is zero (marked yellow in Fig. 7(b)).

(15)$${{\textrm{{m}}_{XYZ \to {x_r}{y_r}{z_r}}} = \left( {\begin{array}{ccc} {\cos [ - (\pi - \theta )]} & 0 & {\sin [ - (\pi - \theta )]}\\ 0 & 1 & 0\\ { - \sin [ - (\pi - \theta )]} & 0 & {\cos [ - (\pi - \theta )]} \end{array}} \right)}.$$

Therefore, for an arbitrary pitch angle $\alpha$, the relationship between $\left ( {{{x'}_r},{{y'}_r},{{z'}_r}} \right )$ and $\left ( {{x_r},{y_r},{z_r}} \right )$ could be written as

(16)$${{M_{{{x'}_r}}}_{{{y'}_r}{{z'}_r} \to {x_r}{y_r}{z_r}} = {m_{XYZ \to {x_r}{y_r}{z_r}}}{m_{X'Y'Z' \to XYZ}}{m_{X^{\prime\prime}Y^{\prime\prime}Z^{\prime\prime} \to X'Y'Z'}}{m_{{{x'}_r}}}_{{{y'}_r}{{z'}_r} \to X^{\prime\prime}Y^{\prime\prime}Z^{\prime\prime}}}.$$

Funding

National Natural Science Foundation of China (21973023, 91950117); Chongqing Research Program of Basic Research and Frontier Technology (cstc2017jcyjAX0038); West Light Foundation of the Chinese Academy of Sciences.

Disclosures

The authors declare no conflicts of interest.

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Impact of the pitch angle on the spin Hall effect of light weak measurement

Abstract

1. Introduction

2. Theoretical model

3. Experimental observation

4. Conclusion

Appendix: Coordinate frame transformation

Funding

Disclosures

References

Cited By

Figures (7)

Equations (16)

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