Abstract

The Imbert-Fedorov (IF) shift is defined as the transverse shift of barycenter of the entire beam when a circular or elliptically polarized incident beam is reflected. In this work, we examine the IF shift of Gaussian beam at the Brewster angle. Interestingly, the spin Hall effect of light takes place in the IF shift at the same time. Furthermore, this interesting phenomenon is experimentally observed using weak measurements. These findings may have useful applications in spin optics.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

When a bounded beam is reflected or refracted at a plane dielectric interface, it does not perfectly follow the geometrical-optics evolution. There usually exist four basic deviations from the geometrical-optics picture, which are in-plane (parallel to the plane of incidence) and out-of-plane (perpendicular to the plane of incidence) spatial and angular shifts, respectively. The in-plane (out-of-plane) spatial and angular shifts are known as the spatial and angular Goos-Hänchen shifts [Imbert-Fedorov (IF) shifts] [1–3]. A variant of the IF shift is the so-called Spin Hall effect of light (SHEL) [4–6]. As a result of spin-orbit coupling corresponding to geometrical Berry phase, the SHEL manifests itself as the spin-dependent shifts upon reflection or refraction. [7,8].

Recently, the SHEL has been widely researched in different physical systems such as optical physics [9–12], plasmonics [13, 14], semiconductors physics [15], metamaterial [16, 17], and even topological insulators [18,19]. Moreover, the SHEL has also drawn significant attention due to its potential application in precision metrology. For example, the SHEL has been used to measure the thickness of nanometal film [20], graphene layers [21], magneto-optical constant of Fe films [22,23], optical rotation of chiral solution [24], and the refractive index of media [25,26].

In this work, we theoretically and experimentally reveal the SHEL in IF shift in the case of a circular or elliptically polarized Gaussian beam reflected from the air-prism interface at the Brewster angle. The internal physical mechanism of this phenomenon is the spin-orbit coupling in IF shift. Particularly, the IF shift changes with the incident polarization state, while the spin splitting of the SHEL in IF shift is unchanged.

2. Theoretical model

Figure 1 schematically illustrates the SHEL in IF shift when the beam is reflected at the air-prism interface. The z axis of the Cartesian coordinate system (x, y, z) is perpendicular to the reflection interface at z = 0. The coordinate systems (xi, yi, zi) and (xr, yr, zr) denote the incident and reflected beams, respectively.

 figure: Fig. 1

Fig. 1 Schematic of the SHEL in IF shift. A beam is reflected from the air-prism interface at the Brewster angle θB. δ+ and δ indicate the magnitude of the spin splitting of the SHEL in IF shift, respectively.

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Consider that the incident beam is an elliptical polarization state |ψi〉 = cos α|H〉 + i sin α|V〉, where α is the azimuth angle (the angle between the long axis of elliptical polarization beam and horizontal direction), |H〉 and |V〉 denote the horizontally and vertically polarized states, respectively. The total wave function is given by

|Ψi=dkxidkyiΦ(kxi,kyi)|kxi|kyi|ψi,
where kxi, kyi are the transverse wave vectors in the x and y directions, respectively. We assume that Φ(kxi, kyi) is the transverse spatial distribution of Gaussian beam. Under the reflection at the air-prism interface, the initial state of system evolves to
|ψr=M^r|ψi,
where
M^r=[rpkyr(rp+rs)cotθik0kyr(rp+rs)cotθik0rs]
is the Jones matrix relating the incident and reflected polarization states in the beam coordinate frames [27,28]. θi is the incident angle. rp and rs are the Fresnel reflection coefficients for parallel and perpendicular polarizations, respectively. And k0 = 2π/λ with λ being the wavelength of light in air. According to the transversality, we can obtain kyr = kyi. Note that the incident angle is fixed at the Brewster angle θB in our paper, so the in-plane spread of wave vectors should be taken into account [27]. The Fresnel reflection coefficient of parallel polarization can be expanded to the first-order approximation of Taylor series,
rp=rpθi+χkxik0,
where χ = ∂rp/∂θi is the first-order derivative of the Fresnel reflection coefficient of parallel polarization. The total wave function evolves to
|Ψr=dkxrdkyrΦ(kxr,kyr)|kxr|kyr|ψr.
The transverse shift of barycenter of left- and right-handed circular polarization relative to the origin of the reflection coordinate system, which can be regarded as the initial spin splitting, is obtained with
Δσ=Ψrσ|ikyr|ΨrσΨrσ|Ψrσδcotα+σδ,
and the transverse shift of total barycenter of reflected field (i.e., the IF shift) can also be obtained with
ΔIF=σ=±Ψrσ|ikyr|Ψrσσ=±Ψrσ|Ψrσδcotα.
Here,
|Ψrσ=dkxrdkyrΦ(kxr,kyr)|kxr|kyr|ψrσ,
|ψrσ=Hr+iσVr2|σ=σrssinα2exp[iσkyr]exp[ikyδcotα]|σ,
|Ψrσ is the total wave function in the spin basis. σ = ± denotes the left- and right-handed circular polarization, respectively. Hr and Vr denote the horizontally and vertically polarized components from Eq. (2), respectively. In Eqs. (6), (7), and (9), δ = cot θB/k0, and we provide that tan α ≫ (χkxr)/(rsk0).

In Eq. (9), the term exp[−iσkyrδ] stands for the spin-orbit coupling (δ is the coupling strength), while the term exp[ikyrδ cot α] stands for the IF shift (−δ cot α is the magnitude of the IF shift). Putting them together, Eq. (9) shows the spin-orbit coupling in IF shift. We call this phenomenon as the SHEL in IF shift. Interestingly, from Eq. (9), we also find that the intensity distributions of left- and right-handed circularly polarized components are the same. In such case the IF shift can be regarded as the average value of the initial spin splitting. The result of Eq. (7) also proves this point. After obtaining the initial spin splitting and IF shift, we definite the relative shift between the initial spin splitting and IF shift as the spin splitting of the SHEL in IF shift, which can be expressed as

δσ=ΔσΔIF=σδ.

From Eq. (10), we know that the spin splitting of the SHEL in IF shift represents a symmetric property, which is different from the asymmetric spin-dependent splitting [29]. Next, we plot Fig. 2 to clearly reveal the SHEL in IF shift. As shown in Fig. 2(a), the magnitude of IF shift decreases with the increase of azimuth angle, and equals to zero at α = 90°, as described by Eq. (7). Moreover, the initial spin splitting always changes with IF shift. However, as shown in Fig. 2(b), the spin splitting of the SHEL in IF shift is a fixed value, as described by Eq. (10). Note that the spin splitting of the SHEL in IF shift is different from the spin splitting of the usually SHEL where the barycenter of both incident and reflected fields are in the same position (i.e., the origin of the Cartesian coordinate system) [9]. However, in our scheme, as shown in Fig. 1, there exists the IF shift between the barycenter of incident and reflected fields.

 figure: Fig. 2

Fig. 2 (a) shows the initial spin splitting and IF shift changing with the azimuth angle. (b) shows that the spin splitting of the SHEL in IF shift is a fixed value.

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3. Experimental observation

Since the spin splitting of the SHEL in IF shift is an exceedingly small value (about 66.5nm), it is extremely challenging to accurately measure it. However, weak measurements [30, 31] based on the signal enhancement technique have proven to be successful to resolve this problem. In 2008, this technique was used to observe the SHEL for the first time by Hosten and Kwiat [9]. Afterwards, this technique was widely used to estimate a variety of small physical parameters such as beam deflections [32], phase shifts [33,34], frequency shifts [35], and even temperature shifts [36].

The experimental setup is shown in Fig. 3. A Gaussian beam at 632.8 nm generated by a He-Ne laser passes through a half-wave plate for adjusting the light intensity. Then, the beam is focused by a lens (L1) and preselected by a Glan polarizer (P1) and a quarter-wave plate (QWP). And then, the beam is reflected at the air-prism interface where the SHEL in IF shift takes place. The incident angle is chosen as Brewster angle. Finally, the reflected beam is postselected by a Glan polarizer (P2) and collimated by a lens (L3). In the measurement, the transverse shift of barycenter of beam is recorded by CCD.

 figure: Fig. 3

Fig. 3 Schematic drawing of the experimental setup. Light source : He-Ne laser at 632.8nm (Thorlabs HNL210L). HWP: half-wave plate. L1 and L2 are lenses with effective focal length 50mm and 250mm, respectively. P1 and QWP are the Glan polarizer and quarter-wave plate for preselection, respectively. P2 is the Glan polarizer for postselection. CCD is the charge-coupled device for recording the amplified shift (Thorlabs BC106N-VIS/M). Inset: the process of preselection and postselection.

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More concretely, the preselection of the system is achieved with an elliptical polarization state |ψi〉 by P1 and QWP. Then, upon reflection at the air-prism interface, the weak coupling between an observable (i.e., spin operator σ̂3 = |+〉〈+| − |−〉〈−|) of the system and a meter variable (i.e., transverse wave-vector component of the wave packet kyi) leads to a spin splitting of the SHEL in IF shift δ, which can be regarded as a consequence of the geometrical Berry phase exp[−ikyrδσ̂3], see Eq. (9). The evolution of the total wave function can also be expressed as

|Ψr=dkxrdkyrΦ(kxr,kyr)|kxr|kyrexp[ikyδσ^3]R^|ψi,
where = diag (rp, rs) describes the change of polarization state of incident beam on reflection.

Finally, the postselection of the system is achieved with a linear polarization state |ψf〉 = cos β|H〉 + sin β|V〉 by P2, where β is the postselection angle. After postselection, the small spin-dependent transverse shift is convert into a large transverse shift of the whole light field. The final wave function is given by

|Ψf=ψf|Ψr=dkxrdkyrΦ(kxr,kyr)|kxr|kyr×exp[i(kxr2+kyr2)z2k0]ψf|exp[ikyδσ^3]|ψi,
where exp[i(kxr2+kyr2)z2k0] denotes the free evolution of the system (z is the free propagation distance) and |ψ′i〉 = |ψi〉 is the modified preselection after reflection. Provided that σ̂32 = 1, the geometrical Berry phase can be expanded to all orders with exp [−ikyrδσ̂3] = cos (kyrδ)iσ̂3 sin (kyrδ). The equation (12) can be rewritten as
|Ψf=ψf|Ψr=dkxrdkyrΦ(kxr,kyr)|kxr|kyr×exp[i(kxr2+kyr2)z2k0]ψf|ψi[cos(kyrδ)iAwsin(kyrδ)],
where
Aw=ψf|σ3|ψiψf|ψi
denotes the weak value. The amplified shift can be obtained with
y=Ψf|ikyr|ΨfΨf|ΨfFAwFacδ,
where
F=zR0
and
AwFac=sin2β[(χcotα)22k0R0rs2](χcotα)2(1+cos2β)+2k0R0rs2(1cos2β)
are the propagation amplification and weak value amplification factor (WVAF), which are decided by the free evolution of the system and weak value, respectively. R0=k0ω02/2 is the Rayleigh distance (ω0 is the focused beam waist). Note that we consider that |β| ≫ |δ/ω0| (i.e., the condition of the linear approximation in weak measurements [30]) in Eq. (17). The whole amplification factor from weak measurements is defined as
Λ=yδ=FAwFac

Figure 4(b) shows the experimental results of crossed polarization intensity profiles at different azimuth angles. These images captured by a color CCD are in agreement with the numerical simulations by calculating the intensity profile after P2, see Fig. 4(a). We see that the intensity profile of bilateral symmetry is gradually transformed into central symmetry, and finally transformed into longitudinal symmetry with the increase of azimuth angle. This is because in-plane and out-plane wave vectors (i.e., kxr and kyr) play different roles at different azimuth angles. When azimuth angle is small, χ cot α in Eq. (17) is large. From Eq. (3), we know that in-plane wave vector is closely related with the first-order derivative χ. Therefore, a large χ cot α shows that in-plane wave vector plays a leading role in the intensity distribution, which gives arise to the bilateral symmetry. With the increase of azimuth angle, χ cot α becomes small. The effect of in-plane wave vector on the intensity distribution is reduced. When α = 65°, both in-plane and out-plane wave vectors have the same effect on the intensity distribution, which gives arise to the central symmetry. With the further increase of azimuth angle, out-plane wave vector plays a leading role in the intensity distribution, which gives arise to the longitudinal symmetry.

 figure: Fig. 4

Fig. 4 Intensity profiles of the reflected beam passing through a crossed polarizer as a function of the azimuth angle α. (a) and (b) denote the theoretical and experimental results, respectively.

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Figures 5(a)–5(e) show the amplified shifts of spin splitting of the SHEL in IF shift as a function of the postselection angle. Here, the azimuth angles are selected as α = 30°, 45°, 60°, 75° and 90°. The black solid lines and red solid circles represent the theoretical predictions and experimental results, respectively. Note that there is no any approximate condition in Eq. (15) when the preselection and postselection are nearly orthogonal [37]. We can see that the amplified shifts first quickly increase with the postselection angle. Then, after reaching the peak value, the amplified shifts decrease gradually. From Eq. (7), we know that the IF shift is only decided by the azimuth angle. For different azimuth angles corresponding to different IF shifts, both the peak value of amplified shift and its corresponding postselection angle are different. Figure 5(f) shows the amplified shift and its corresponding spin splitting of the SHEL in IF shift as a function of azimuth angle at postselection angle β = 1.8°. The red solid line and the green dash line represent the theoretical predictions derived from Eqs. (15) and (10), respectively. The red hollow circles correspond to the amplified shift derived from measurement. The green solid diamonds correspond to the spin splitting in IF shift derived from the amplified shift divided by the amplification factor in Eq. (18). These experimental results are good agreement with the theoretical calculation.

 figure: Fig. 5

Fig. 5 (a)–(e) shows the amplified shifts of spin splitting in IF shift change with postselection angle at azimuth angle α = 30°, 45°, 60°, 75° and 90°. (f) shows the amplified shift and its corresponding spin splitting of the SHEL in IF shift as a function of azimuth angle at postselection angle β = 1.8°.

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In addition, although we have measured the spin splitting of the SHEL in IF using weak measurements and confirmed that the amplified shift is influenced by the IF shift, the IF shift has still not been measured in our experiment. This is because weak measurements are only helpful to the IF shift in some specially incident polarization states (e.g., the incident polarization state is a 45° linear polarization state.) [38], they are helpless to the IF shift when the incident polarization state is an elliptical polarization state in our paper. However, it is very possible to solve this issue using the standard lock-in amplifier based on heterodyne detection [39,40].

4. Conclusion

In conclusion, we have revealed the SHEL in IF shift when a circular or elliptically polarized Gaussian beam is reflected from the air-prism interface at the Brewster angle. The IF shift changes with the incident polarization state, while the spin splitting of the SHEL in IF is a fixed value. We found that this phenomenon is attributed to the spin-orbit coupling in IF shift. Moreover, the weak measurement method is employed to measure the spin splitting of the SHEL in IF shift. These findings not only provide a pathway for modulating the SHEL but also are useful for deep understanding of beam shifts.

Funding

The National Key R&D Program of China (Grant No. 2017YFB0405704); The National Natural Science Foundation of China (Grant Nos. 11674234 and 11604095); The Hunan Provincial Natural Science Foundation of China (Grant No. 2017JJ3209); The Fundamental Research Funds for the Central Universities (Grant No. 2012017yjsy143).

References

1. A. W. Snyder and J. D. Love, “Goos–Hänchen shift,” Appl. Opt. 15, 236–238 (1976). [CrossRef]   [PubMed]  

2. C. Imbert, “Calculation and Experimental Proof of the Transverse Shift Induced by Total Internal Reflection of a Circularly Polarized Light Beam,” Phys. Rev. D 5, 787–796 (1972). [CrossRef]  

3. F. I. Fedorov, “To the theory of total reflection,” J. Opt. 15, 014002 (2013). [CrossRef]  

4. M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. 93, 083901 (2004). [CrossRef]   [PubMed]  

5. K. Y. Bliokh and Y. P. Bliokh, “Conservation of Angular Momentum, Transverse Shift, and Spin Hall Effect in Reflection and Refraction of an Electromagnetic Wave Packet,” Phys. Rev. Lett. 96, 073903 (2006). [CrossRef]   [PubMed]  

6. X. Ling, X. Zhou, K. Huang, Y. Liu, C.-W. Qiu, H. Luo, and S. Wen, “Recent advances in the spin Hall effect of light,” Rep. Prog. Phys. 80, 066401 (2017). [CrossRef]   [PubMed]  

7. V. S. Liberman and B. Y. Zel’dovich, “Spin-orbit interaction of a photon in an inhomogeneous medium,” Phys. Rev. A 46, 5199–5207 (1992). [CrossRef]   [PubMed]  

8. K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interaction of light,” Nat. Photo. 9, 796 (2015). [CrossRef]  

9. O. Hosten and P. Kwiat, “Observation of the Spin Hall Effect of Light via Weak Measurements,” Science 319, 787–790 (2008). [CrossRef]   [PubMed]  

10. Y. Qin, Y. Li, H. He, and Q. Gong, “Measurement of spin Hall effect of reflected light,” Opt. Lett. 34, 2551–2553 (2009). [CrossRef]   [PubMed]  

11. S. Goswami, M. Pal, A. Nandi, P. K. Panigrahi, and N. Ghosh, “Simultaneous weak value amplification of angular Goos–Hänchen and Imbert–Fedorov shifts in partial reflection,” Opt. Lett. 39, 6229–6232 (2014). [CrossRef]   [PubMed]  

12. S. Goswami, S. Dhara, M. Pal, A. Nandi, P. K. Panigrahi, and N. Ghosh, “Optimized weak measurements of Goos–Hänchen and Imbert–Fedorov shifts in partial reflection,” Opt. Express 24, 6041–6051 (2016). [CrossRef]   [PubMed]  

13. Y. Gorodetski, A. Niv, V. Kleiner, and E. Hasman, “Observation of the Spin-Based Plasmonic Effect in Nanoscale Structures,” Phys. Rev. Lett. 101, 043903 (2008). [CrossRef]   [PubMed]  

14. Y. Gorodetski, K. Y. Bliokh, B. Stein, C. Genet, N. Shitrit, V. Kleiner, E. Hasman, and T. W. Ebbesen, “Weak Measurements of Light Chirality with a Plasmonic Slit,” Phys. Rev. Lett. 109, 013901 (2012). [CrossRef]   [PubMed]  

15. J.-M. Ménard, A. E. Mattacchione, M. Betz, and H. M. van Driel, “Imaging the spin Hall effect of light inside semiconductors via absorption,” Opt. Lett. 34, 2312–2314 (2009). [CrossRef]   [PubMed]  

16. X. Yin, Z. Ye, J. Rho, Y. Wang, and X. Zhang, “Photonic Spin Hall Effect at Metasurfaces,” Science 339, 1405–1407 (2013). [CrossRef]   [PubMed]  

17. W. Zhu, M. Jiang, H. Guan, J. Yu, H. Lu, J. Zhang, and Z. Chen, “Tunable spin splitting of Laguerre–Gaussian beams in graphene metamaterials,” Photon. Res. 5, 684–688 (2017). [CrossRef]  

18. X. Zhou, J. Zhang, X. Ling, S. Chen, H. Luo, and S. Wen, “Photonic spin Hall effect in topological insulators,” Phys. Rev. A 88, 053840 (2013). [CrossRef]  

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33. X.-Y. Xu, Y. Kedem, K. Sun, L. Vaidman, C.-F. Li, and G.-C. Guo, “Phase Estimation with Weak Measurement Using a White Light Source,” Phys. Rev. Lett. 111, 033604 (2013). [CrossRef]   [PubMed]  

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References

  • View by:

  1. A. W. Snyder and J. D. Love, “Goos–Hänchen shift,” Appl. Opt. 15, 236–238 (1976).
    [Crossref] [PubMed]
  2. C. Imbert, “Calculation and Experimental Proof of the Transverse Shift Induced by Total Internal Reflection of a Circularly Polarized Light Beam,” Phys. Rev. D 5, 787–796 (1972).
    [Crossref]
  3. F. I. Fedorov, “To the theory of total reflection,” J. Opt. 15, 014002 (2013).
    [Crossref]
  4. M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. 93, 083901 (2004).
    [Crossref] [PubMed]
  5. K. Y. Bliokh and Y. P. Bliokh, “Conservation of Angular Momentum, Transverse Shift, and Spin Hall Effect in Reflection and Refraction of an Electromagnetic Wave Packet,” Phys. Rev. Lett. 96, 073903 (2006).
    [Crossref] [PubMed]
  6. X. Ling, X. Zhou, K. Huang, Y. Liu, C.-W. Qiu, H. Luo, and S. Wen, “Recent advances in the spin Hall effect of light,” Rep. Prog. Phys. 80, 066401 (2017).
    [Crossref] [PubMed]
  7. V. S. Liberman and B. Y. Zel’dovich, “Spin-orbit interaction of a photon in an inhomogeneous medium,” Phys. Rev. A 46, 5199–5207 (1992).
    [Crossref] [PubMed]
  8. K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interaction of light,” Nat. Photo. 9, 796 (2015).
    [Crossref]
  9. O. Hosten and P. Kwiat, “Observation of the Spin Hall Effect of Light via Weak Measurements,” Science 319, 787–790 (2008).
    [Crossref] [PubMed]
  10. Y. Qin, Y. Li, H. He, and Q. Gong, “Measurement of spin Hall effect of reflected light,” Opt. Lett. 34, 2551–2553 (2009).
    [Crossref] [PubMed]
  11. S. Goswami, M. Pal, A. Nandi, P. K. Panigrahi, and N. Ghosh, “Simultaneous weak value amplification of angular Goos–Hänchen and Imbert–Fedorov shifts in partial reflection,” Opt. Lett. 39, 6229–6232 (2014).
    [Crossref] [PubMed]
  12. S. Goswami, S. Dhara, M. Pal, A. Nandi, P. K. Panigrahi, and N. Ghosh, “Optimized weak measurements of Goos–Hänchen and Imbert–Fedorov shifts in partial reflection,” Opt. Express 24, 6041–6051 (2016).
    [Crossref] [PubMed]
  13. Y. Gorodetski, A. Niv, V. Kleiner, and E. Hasman, “Observation of the Spin-Based Plasmonic Effect in Nanoscale Structures,” Phys. Rev. Lett. 101, 043903 (2008).
    [Crossref] [PubMed]
  14. Y. Gorodetski, K. Y. Bliokh, B. Stein, C. Genet, N. Shitrit, V. Kleiner, E. Hasman, and T. W. Ebbesen, “Weak Measurements of Light Chirality with a Plasmonic Slit,” Phys. Rev. Lett. 109, 013901 (2012).
    [Crossref] [PubMed]
  15. J.-M. Ménard, A. E. Mattacchione, M. Betz, and H. M. van Driel, “Imaging the spin Hall effect of light inside semiconductors via absorption,” Opt. Lett. 34, 2312–2314 (2009).
    [Crossref] [PubMed]
  16. X. Yin, Z. Ye, J. Rho, Y. Wang, and X. Zhang, “Photonic Spin Hall Effect at Metasurfaces,” Science 339, 1405–1407 (2013).
    [Crossref] [PubMed]
  17. W. Zhu, M. Jiang, H. Guan, J. Yu, H. Lu, J. Zhang, and Z. Chen, “Tunable spin splitting of Laguerre–Gaussian beams in graphene metamaterials,” Photon. Res. 5, 684–688 (2017).
    [Crossref]
  18. X. Zhou, J. Zhang, X. Ling, S. Chen, H. Luo, and S. Wen, “Photonic spin Hall effect in topological insulators,” Phys. Rev. A 88, 053840 (2013).
    [Crossref]
  19. W. J. M. Kort-Kamp, “Topological Phase Transitions in the Photonic Spin Hall Effect,” Phys. Rev. Lett. 119, 147401 (2017).
    [Crossref] [PubMed]
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  21. X. Zhou, X. Ling, H. Luo, and S. Wen, “Identifying graphene layers via spin Hall effect of light,” Appl. Phys. Lett. 101, 251602 (2012).
    [Crossref]
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    [Crossref]
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  32. P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, “Ultrasensitive Beam Deflection Measurement via Interferometric Weak Value Amplification,” Phys. Rev. Lett. 102, 173601 (2009).
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  33. X.-Y. Xu, Y. Kedem, K. Sun, L. Vaidman, C.-F. Li, and G.-C. Guo, “Phase Estimation with Weak Measurement Using a White Light Source,” Phys. Rev. Lett. 111, 033604 (2013).
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  34. X. Qiu, L. Xie, X. Liu, L. Luo, Z. Li, Z. Zhang, and J. Du, “Precision phase estimation based on weak-value amplification,” Appl. Phys. Lett. 110, 071105 (2017).
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  38. G. Jayaswal, G. Mistura, and M. Merano, “Observation of the Imbert–Fedorov effect via weak value amplification,” Opt. Lett. 39, 2266–2269 (2014).
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2018 (2)

T. Tang, J. Li, L. Luo, P. Sun, and J. Yao, “Magneto-Optical Modulation of Photonic Spin Hall Effect of Graphene in Terahertz Region,” Adv. Opt. Mater. 6, 1701212 (2018).
[Crossref]

X. Zhou, L. Sheng, and X. Ling, “Photonic spin Hall effect enabled refractive index sensor using weak measurements,” Sci. Rep. 8, 1221 (2018).
[Crossref] [PubMed]

2017 (5)

L. Xie, X. Qiu, L. Luo, X. Liu, Z. Li, Z. Zhang, J. Du, and D. Wang, “Quantitative detection of the respective concentrations of chiral compounds with weak measurements,” Appl. Phys. Lett. 111, 191106 (2017).
[Crossref]

W. Zhu, M. Jiang, H. Guan, J. Yu, H. Lu, J. Zhang, and Z. Chen, “Tunable spin splitting of Laguerre–Gaussian beams in graphene metamaterials,” Photon. Res. 5, 684–688 (2017).
[Crossref]

W. J. M. Kort-Kamp, “Topological Phase Transitions in the Photonic Spin Hall Effect,” Phys. Rev. Lett. 119, 147401 (2017).
[Crossref] [PubMed]

X. Ling, X. Zhou, K. Huang, Y. Liu, C.-W. Qiu, H. Luo, and S. Wen, “Recent advances in the spin Hall effect of light,” Rep. Prog. Phys. 80, 066401 (2017).
[Crossref] [PubMed]

X. Qiu, L. Xie, X. Liu, L. Luo, Z. Li, Z. Zhang, and J. Du, “Precision phase estimation based on weak-value amplification,” Appl. Phys. Lett. 110, 071105 (2017).
[Crossref]

2016 (3)

2015 (2)

2014 (4)

G. Jayaswal, G. Mistura, and M. Merano, “Observation of the Imbert–Fedorov effect via weak value amplification,” Opt. Lett. 39, 2266–2269 (2014).
[Crossref] [PubMed]

S. Goswami, M. Pal, A. Nandi, P. K. Panigrahi, and N. Ghosh, “Simultaneous weak value amplification of angular Goos–Hänchen and Imbert–Fedorov shifts in partial reflection,” Opt. Lett. 39, 6229–6232 (2014).
[Crossref] [PubMed]

J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, “Colloquium: Understanding quantum weak values: Basics and applications,” Rev. Mod. Phys. 86, 307–316 (2014).
[Crossref]

X. Qiu, X. Zhou, D. Hu, J. Du, F. Gao, Z. Zhang, and H. Luo, “Determination of magneto-optical constant of Fe films with weak measurements,” Appl. Phys. Lett. 105, 131111 (2014).
[Crossref]

2013 (5)

X. Zhou, J. Zhang, X. Ling, S. Chen, H. Luo, and S. Wen, “Photonic spin Hall effect in topological insulators,” Phys. Rev. A 88, 053840 (2013).
[Crossref]

K. Y. Bliokh and A. Aiello, “Goos–Hänchen and Imbert–Fedorov beam shifts: an overview,” J. Opt. 15, 014001 (2013).
[Crossref]

X.-Y. Xu, Y. Kedem, K. Sun, L. Vaidman, C.-F. Li, and G.-C. Guo, “Phase Estimation with Weak Measurement Using a White Light Source,” Phys. Rev. Lett. 111, 033604 (2013).
[Crossref] [PubMed]

X. Yin, Z. Ye, J. Rho, Y. Wang, and X. Zhang, “Photonic Spin Hall Effect at Metasurfaces,” Science 339, 1405–1407 (2013).
[Crossref] [PubMed]

F. I. Fedorov, “To the theory of total reflection,” J. Opt. 15, 014002 (2013).
[Crossref]

2012 (3)

Y. Gorodetski, K. Y. Bliokh, B. Stein, C. Genet, N. Shitrit, V. Kleiner, E. Hasman, and T. W. Ebbesen, “Weak Measurements of Light Chirality with a Plasmonic Slit,” Phys. Rev. Lett. 109, 013901 (2012).
[Crossref] [PubMed]

X. Zhou, Z. Xiao, H. Luo, and S. Wen, “Experimental observation of the spin Hall effect of light on a nanometal film via weak measurements,” Phys. Rev. A 85, 043809 (2012).
[Crossref]

X. Zhou, X. Ling, H. Luo, and S. Wen, “Identifying graphene layers via spin Hall effect of light,” Appl. Phys. Lett. 101, 251602 (2012).
[Crossref]

2011 (2)

H. Luo, X. Zhou, W. Shu, S. Wen, and D. Fan, “Enhanced and switchable spin hall effect of light near the brewster angle on reflection,” Phys. Rev. A 84, 043806 (2011).
[Crossref]

X. Zhu, Y. Zhang, S. Pang, C. Qiao, Q. Liu, and S. Wu, “Quantum measurements with preselection and postselection,” Phys. Rev. A 84, 052111 (2011).
[Crossref]

2010 (1)

D. J. Starling, P. B. Dixon, A. N. Jordan, and J. C. Howell, “Precision frequency measurements with interferometric weak values,” Phys. Rev. A 82, 063822 (2010).
[Crossref]

2009 (3)

2008 (2)

Y. Gorodetski, A. Niv, V. Kleiner, and E. Hasman, “Observation of the Spin-Based Plasmonic Effect in Nanoscale Structures,” Phys. Rev. Lett. 101, 043903 (2008).
[Crossref] [PubMed]

O. Hosten and P. Kwiat, “Observation of the Spin Hall Effect of Light via Weak Measurements,” Science 319, 787–790 (2008).
[Crossref] [PubMed]

2006 (2)

K. Y. Bliokh and Y. P. Bliokh, “Conservation of Angular Momentum, Transverse Shift, and Spin Hall Effect in Reflection and Refraction of an Electromagnetic Wave Packet,” Phys. Rev. Lett. 96, 073903 (2006).
[Crossref] [PubMed]

X. Yin and L. Hesselink, “Goos–Hänchen shift surface plasmon resonance sensor,” Appl. Phys. Lett. 89, 261108 (2006).
[Crossref]

2004 (1)

M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. 93, 083901 (2004).
[Crossref] [PubMed]

2002 (1)

1992 (1)

V. S. Liberman and B. Y. Zel’dovich, “Spin-orbit interaction of a photon in an inhomogeneous medium,” Phys. Rev. A 46, 5199–5207 (1992).
[Crossref] [PubMed]

1988 (1)

Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin −1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351–1354 (1988).
[Crossref] [PubMed]

1976 (1)

1972 (1)

C. Imbert, “Calculation and Experimental Proof of the Transverse Shift Induced by Total Internal Reflection of a Circularly Polarized Light Beam,” Phys. Rev. D 5, 787–796 (1972).
[Crossref]

Aharonov, Y.

Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin −1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351–1354 (1988).
[Crossref] [PubMed]

Aiello, A.

K. Y. Bliokh and A. Aiello, “Goos–Hänchen and Imbert–Fedorov beam shifts: an overview,” J. Opt. 15, 014001 (2013).
[Crossref]

Albert, D. Z.

Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin −1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351–1354 (1988).
[Crossref] [PubMed]

Amaya, W.

Barrera, D.

Betz, M.

Bliokh, K. Y.

K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interaction of light,” Nat. Photo. 9, 796 (2015).
[Crossref]

K. Y. Bliokh and A. Aiello, “Goos–Hänchen and Imbert–Fedorov beam shifts: an overview,” J. Opt. 15, 014001 (2013).
[Crossref]

Y. Gorodetski, K. Y. Bliokh, B. Stein, C. Genet, N. Shitrit, V. Kleiner, E. Hasman, and T. W. Ebbesen, “Weak Measurements of Light Chirality with a Plasmonic Slit,” Phys. Rev. Lett. 109, 013901 (2012).
[Crossref] [PubMed]

K. Y. Bliokh and Y. P. Bliokh, “Conservation of Angular Momentum, Transverse Shift, and Spin Hall Effect in Reflection and Refraction of an Electromagnetic Wave Packet,” Phys. Rev. Lett. 96, 073903 (2006).
[Crossref] [PubMed]

Bliokh, Y. P.

K. Y. Bliokh and Y. P. Bliokh, “Conservation of Angular Momentum, Transverse Shift, and Spin Hall Effect in Reflection and Refraction of an Electromagnetic Wave Packet,” Phys. Rev. Lett. 96, 073903 (2006).
[Crossref] [PubMed]

Boyd, R. W.

J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, “Colloquium: Understanding quantum weak values: Basics and applications,” Rev. Mod. Phys. 86, 307–316 (2014).
[Crossref]

Capmany, J.

Chen, S.

X. Zhou, J. Zhang, X. Ling, S. Chen, H. Luo, and S. Wen, “Photonic spin Hall effect in topological insulators,” Phys. Rev. A 88, 053840 (2013).
[Crossref]

Chen, Z.

Dhara, S.

Dixon, P. B.

D. J. Starling, P. B. Dixon, A. N. Jordan, and J. C. Howell, “Precision frequency measurements with interferometric weak values,” Phys. Rev. A 82, 063822 (2010).
[Crossref]

P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, “Ultrasensitive Beam Deflection Measurement via Interferometric Weak Value Amplification,” Phys. Rev. Lett. 102, 173601 (2009).
[Crossref] [PubMed]

Dressel, J.

J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, “Colloquium: Understanding quantum weak values: Basics and applications,” Rev. Mod. Phys. 86, 307–316 (2014).
[Crossref]

Du, J.

L. Xie, X. Qiu, L. Luo, X. Liu, Z. Li, Z. Zhang, J. Du, and D. Wang, “Quantitative detection of the respective concentrations of chiral compounds with weak measurements,” Appl. Phys. Lett. 111, 191106 (2017).
[Crossref]

X. Qiu, L. Xie, X. Liu, L. Luo, Z. Li, Z. Zhang, and J. Du, “Precision phase estimation based on weak-value amplification,” Appl. Phys. Lett. 110, 071105 (2017).
[Crossref]

X. Qiu, L. Xie, X. Liu, L. Luo, Z. Zhang, and J. Du, “Estimation of optical rotation of chiral molecules with weak measurements,” Opt. Lett. 41, 4032–4035 (2016).
[Crossref] [PubMed]

X. Qiu, X. Zhou, D. Hu, J. Du, F. Gao, Z. Zhang, and H. Luo, “Determination of magneto-optical constant of Fe films with weak measurements,” Appl. Phys. Lett. 105, 131111 (2014).
[Crossref]

Ebbesen, T. W.

Y. Gorodetski, K. Y. Bliokh, B. Stein, C. Genet, N. Shitrit, V. Kleiner, E. Hasman, and T. W. Ebbesen, “Weak Measurements of Light Chirality with a Plasmonic Slit,” Phys. Rev. Lett. 109, 013901 (2012).
[Crossref] [PubMed]

Fan, D.

H. Luo, X. Zhou, W. Shu, S. Wen, and D. Fan, “Enhanced and switchable spin hall effect of light near the brewster angle on reflection,” Phys. Rev. A 84, 043806 (2011).
[Crossref]

Fedorov, F. I.

F. I. Fedorov, “To the theory of total reflection,” J. Opt. 15, 014002 (2013).
[Crossref]

Gao, F.

X. Qiu, X. Zhou, D. Hu, J. Du, F. Gao, Z. Zhang, and H. Luo, “Determination of magneto-optical constant of Fe films with weak measurements,” Appl. Phys. Lett. 105, 131111 (2014).
[Crossref]

Genet, C.

Y. Gorodetski, K. Y. Bliokh, B. Stein, C. Genet, N. Shitrit, V. Kleiner, E. Hasman, and T. W. Ebbesen, “Weak Measurements of Light Chirality with a Plasmonic Slit,” Phys. Rev. Lett. 109, 013901 (2012).
[Crossref] [PubMed]

Ghosh, N.

Gilles, H.

Girard, S.

Gong, Q.

Gorodetski, Y.

Y. Gorodetski, K. Y. Bliokh, B. Stein, C. Genet, N. Shitrit, V. Kleiner, E. Hasman, and T. W. Ebbesen, “Weak Measurements of Light Chirality with a Plasmonic Slit,” Phys. Rev. Lett. 109, 013901 (2012).
[Crossref] [PubMed]

Y. Gorodetski, A. Niv, V. Kleiner, and E. Hasman, “Observation of the Spin-Based Plasmonic Effect in Nanoscale Structures,” Phys. Rev. Lett. 101, 043903 (2008).
[Crossref] [PubMed]

Goswami, S.

Guan, H.

Guo, G.-C.

X.-Y. Xu, Y. Kedem, K. Sun, L. Vaidman, C.-F. Li, and G.-C. Guo, “Phase Estimation with Weak Measurement Using a White Light Source,” Phys. Rev. Lett. 111, 033604 (2013).
[Crossref] [PubMed]

Hamel, J.

Hasman, E.

Y. Gorodetski, K. Y. Bliokh, B. Stein, C. Genet, N. Shitrit, V. Kleiner, E. Hasman, and T. W. Ebbesen, “Weak Measurements of Light Chirality with a Plasmonic Slit,” Phys. Rev. Lett. 109, 013901 (2012).
[Crossref] [PubMed]

Y. Gorodetski, A. Niv, V. Kleiner, and E. Hasman, “Observation of the Spin-Based Plasmonic Effect in Nanoscale Structures,” Phys. Rev. Lett. 101, 043903 (2008).
[Crossref] [PubMed]

He, H.

Hesselink, L.

X. Yin and L. Hesselink, “Goos–Hänchen shift surface plasmon resonance sensor,” Appl. Phys. Lett. 89, 261108 (2006).
[Crossref]

Hosten, O.

O. Hosten and P. Kwiat, “Observation of the Spin Hall Effect of Light via Weak Measurements,” Science 319, 787–790 (2008).
[Crossref] [PubMed]

Howell, J. C.

D. J. Starling, P. B. Dixon, A. N. Jordan, and J. C. Howell, “Precision frequency measurements with interferometric weak values,” Phys. Rev. A 82, 063822 (2010).
[Crossref]

P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, “Ultrasensitive Beam Deflection Measurement via Interferometric Weak Value Amplification,” Phys. Rev. Lett. 102, 173601 (2009).
[Crossref] [PubMed]

Hu, D.

X. Qiu, X. Zhou, D. Hu, J. Du, F. Gao, Z. Zhang, and H. Luo, “Determination of magneto-optical constant of Fe films with weak measurements,” Appl. Phys. Lett. 105, 131111 (2014).
[Crossref]

Huang, K.

X. Ling, X. Zhou, K. Huang, Y. Liu, C.-W. Qiu, H. Luo, and S. Wen, “Recent advances in the spin Hall effect of light,” Rep. Prog. Phys. 80, 066401 (2017).
[Crossref] [PubMed]

Imbert, C.

C. Imbert, “Calculation and Experimental Proof of the Transverse Shift Induced by Total Internal Reflection of a Circularly Polarized Light Beam,” Phys. Rev. D 5, 787–796 (1972).
[Crossref]

Jayaswal, G.

Jiang, M.

Jordan, A. N.

J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, “Colloquium: Understanding quantum weak values: Basics and applications,” Rev. Mod. Phys. 86, 307–316 (2014).
[Crossref]

D. J. Starling, P. B. Dixon, A. N. Jordan, and J. C. Howell, “Precision frequency measurements with interferometric weak values,” Phys. Rev. A 82, 063822 (2010).
[Crossref]

P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, “Ultrasensitive Beam Deflection Measurement via Interferometric Weak Value Amplification,” Phys. Rev. Lett. 102, 173601 (2009).
[Crossref] [PubMed]

Kedem, Y.

X.-Y. Xu, Y. Kedem, K. Sun, L. Vaidman, C.-F. Li, and G.-C. Guo, “Phase Estimation with Weak Measurement Using a White Light Source,” Phys. Rev. Lett. 111, 033604 (2013).
[Crossref] [PubMed]

Kleiner, V.

Y. Gorodetski, K. Y. Bliokh, B. Stein, C. Genet, N. Shitrit, V. Kleiner, E. Hasman, and T. W. Ebbesen, “Weak Measurements of Light Chirality with a Plasmonic Slit,” Phys. Rev. Lett. 109, 013901 (2012).
[Crossref] [PubMed]

Y. Gorodetski, A. Niv, V. Kleiner, and E. Hasman, “Observation of the Spin-Based Plasmonic Effect in Nanoscale Structures,” Phys. Rev. Lett. 101, 043903 (2008).
[Crossref] [PubMed]

Kort-Kamp, W. J. M.

W. J. M. Kort-Kamp, “Topological Phase Transitions in the Photonic Spin Hall Effect,” Phys. Rev. Lett. 119, 147401 (2017).
[Crossref] [PubMed]

Kwiat, P.

O. Hosten and P. Kwiat, “Observation of the Spin Hall Effect of Light via Weak Measurements,” Science 319, 787–790 (2008).
[Crossref] [PubMed]

Li, C.-F.

X.-Y. Xu, Y. Kedem, K. Sun, L. Vaidman, C.-F. Li, and G.-C. Guo, “Phase Estimation with Weak Measurement Using a White Light Source,” Phys. Rev. Lett. 111, 033604 (2013).
[Crossref] [PubMed]

Li, J.

T. Tang, J. Li, L. Luo, P. Sun, and J. Yao, “Magneto-Optical Modulation of Photonic Spin Hall Effect of Graphene in Terahertz Region,” Adv. Opt. Mater. 6, 1701212 (2018).
[Crossref]

Li, Y.

Li, Z.

L. Xie, X. Qiu, L. Luo, X. Liu, Z. Li, Z. Zhang, J. Du, and D. Wang, “Quantitative detection of the respective concentrations of chiral compounds with weak measurements,” Appl. Phys. Lett. 111, 191106 (2017).
[Crossref]

X. Qiu, L. Xie, X. Liu, L. Luo, Z. Li, Z. Zhang, and J. Du, “Precision phase estimation based on weak-value amplification,” Appl. Phys. Lett. 110, 071105 (2017).
[Crossref]

Liberman, V. S.

V. S. Liberman and B. Y. Zel’dovich, “Spin-orbit interaction of a photon in an inhomogeneous medium,” Phys. Rev. A 46, 5199–5207 (1992).
[Crossref] [PubMed]

Ling, X.

X. Zhou, L. Sheng, and X. Ling, “Photonic spin Hall effect enabled refractive index sensor using weak measurements,” Sci. Rep. 8, 1221 (2018).
[Crossref] [PubMed]

X. Ling, X. Zhou, K. Huang, Y. Liu, C.-W. Qiu, H. Luo, and S. Wen, “Recent advances in the spin Hall effect of light,” Rep. Prog. Phys. 80, 066401 (2017).
[Crossref] [PubMed]

X. Zhou and X. Ling, “Unveiling the photonic spin Hall effect with asymmetric spin-dependent splitting,” Opt. Express 24, 3025 (2016).
[Crossref] [PubMed]

X. Zhou, J. Zhang, X. Ling, S. Chen, H. Luo, and S. Wen, “Photonic spin Hall effect in topological insulators,” Phys. Rev. A 88, 053840 (2013).
[Crossref]

X. Zhou, X. Ling, H. Luo, and S. Wen, “Identifying graphene layers via spin Hall effect of light,” Appl. Phys. Lett. 101, 251602 (2012).
[Crossref]

Liu, Q.

X. Zhu, Y. Zhang, S. Pang, C. Qiao, Q. Liu, and S. Wu, “Quantum measurements with preselection and postselection,” Phys. Rev. A 84, 052111 (2011).
[Crossref]

Liu, X.

L. Xie, X. Qiu, L. Luo, X. Liu, Z. Li, Z. Zhang, J. Du, and D. Wang, “Quantitative detection of the respective concentrations of chiral compounds with weak measurements,” Appl. Phys. Lett. 111, 191106 (2017).
[Crossref]

X. Qiu, L. Xie, X. Liu, L. Luo, Z. Li, Z. Zhang, and J. Du, “Precision phase estimation based on weak-value amplification,” Appl. Phys. Lett. 110, 071105 (2017).
[Crossref]

X. Qiu, L. Xie, X. Liu, L. Luo, Z. Zhang, and J. Du, “Estimation of optical rotation of chiral molecules with weak measurements,” Opt. Lett. 41, 4032–4035 (2016).
[Crossref] [PubMed]

Liu, Y.

X. Ling, X. Zhou, K. Huang, Y. Liu, C.-W. Qiu, H. Luo, and S. Wen, “Recent advances in the spin Hall effect of light,” Rep. Prog. Phys. 80, 066401 (2017).
[Crossref] [PubMed]

Love, J. D.

Lu, H.

Luo, H.

X. Ling, X. Zhou, K. Huang, Y. Liu, C.-W. Qiu, H. Luo, and S. Wen, “Recent advances in the spin Hall effect of light,” Rep. Prog. Phys. 80, 066401 (2017).
[Crossref] [PubMed]

X. Qiu, X. Zhou, D. Hu, J. Du, F. Gao, Z. Zhang, and H. Luo, “Determination of magneto-optical constant of Fe films with weak measurements,” Appl. Phys. Lett. 105, 131111 (2014).
[Crossref]

X. Zhou, J. Zhang, X. Ling, S. Chen, H. Luo, and S. Wen, “Photonic spin Hall effect in topological insulators,” Phys. Rev. A 88, 053840 (2013).
[Crossref]

X. Zhou, Z. Xiao, H. Luo, and S. Wen, “Experimental observation of the spin Hall effect of light on a nanometal film via weak measurements,” Phys. Rev. A 85, 043809 (2012).
[Crossref]

X. Zhou, X. Ling, H. Luo, and S. Wen, “Identifying graphene layers via spin Hall effect of light,” Appl. Phys. Lett. 101, 251602 (2012).
[Crossref]

H. Luo, X. Zhou, W. Shu, S. Wen, and D. Fan, “Enhanced and switchable spin hall effect of light near the brewster angle on reflection,” Phys. Rev. A 84, 043806 (2011).
[Crossref]

Luo, L.

T. Tang, J. Li, L. Luo, P. Sun, and J. Yao, “Magneto-Optical Modulation of Photonic Spin Hall Effect of Graphene in Terahertz Region,” Adv. Opt. Mater. 6, 1701212 (2018).
[Crossref]

L. Xie, X. Qiu, L. Luo, X. Liu, Z. Li, Z. Zhang, J. Du, and D. Wang, “Quantitative detection of the respective concentrations of chiral compounds with weak measurements,” Appl. Phys. Lett. 111, 191106 (2017).
[Crossref]

X. Qiu, L. Xie, X. Liu, L. Luo, Z. Li, Z. Zhang, and J. Du, “Precision phase estimation based on weak-value amplification,” Appl. Phys. Lett. 110, 071105 (2017).
[Crossref]

X. Qiu, L. Xie, X. Liu, L. Luo, Z. Zhang, and J. Du, “Estimation of optical rotation of chiral molecules with weak measurements,” Opt. Lett. 41, 4032–4035 (2016).
[Crossref] [PubMed]

Malik, M.

J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, “Colloquium: Understanding quantum weak values: Basics and applications,” Rev. Mod. Phys. 86, 307–316 (2014).
[Crossref]

Mattacchione, A. E.

Ménard, J.-M.

Merano, M.

Miatto, F. M.

J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, “Colloquium: Understanding quantum weak values: Basics and applications,” Rev. Mod. Phys. 86, 307–316 (2014).
[Crossref]

Mistura, G.

Murakami, S.

M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. 93, 083901 (2004).
[Crossref] [PubMed]

Nagaosa, N.

M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. 93, 083901 (2004).
[Crossref] [PubMed]

Nandi, A.

Niv, A.

Y. Gorodetski, A. Niv, V. Kleiner, and E. Hasman, “Observation of the Spin-Based Plasmonic Effect in Nanoscale Structures,” Phys. Rev. Lett. 101, 043903 (2008).
[Crossref] [PubMed]

Nori, F.

K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interaction of light,” Nat. Photo. 9, 796 (2015).
[Crossref]

Onoda, M.

M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. 93, 083901 (2004).
[Crossref] [PubMed]

Pal, M.

Pang, S.

X. Zhu, Y. Zhang, S. Pang, C. Qiao, Q. Liu, and S. Wu, “Quantum measurements with preselection and postselection,” Phys. Rev. A 84, 052111 (2011).
[Crossref]

Panigrahi, P. K.

Pruneri, V.

Qiao, C.

X. Zhu, Y. Zhang, S. Pang, C. Qiao, Q. Liu, and S. Wu, “Quantum measurements with preselection and postselection,” Phys. Rev. A 84, 052111 (2011).
[Crossref]

Qin, Y.

Qiu, C.-W.

X. Ling, X. Zhou, K. Huang, Y. Liu, C.-W. Qiu, H. Luo, and S. Wen, “Recent advances in the spin Hall effect of light,” Rep. Prog. Phys. 80, 066401 (2017).
[Crossref] [PubMed]

Qiu, X.

X. Qiu, L. Xie, X. Liu, L. Luo, Z. Li, Z. Zhang, and J. Du, “Precision phase estimation based on weak-value amplification,” Appl. Phys. Lett. 110, 071105 (2017).
[Crossref]

L. Xie, X. Qiu, L. Luo, X. Liu, Z. Li, Z. Zhang, J. Du, and D. Wang, “Quantitative detection of the respective concentrations of chiral compounds with weak measurements,” Appl. Phys. Lett. 111, 191106 (2017).
[Crossref]

X. Qiu, L. Xie, X. Liu, L. Luo, Z. Zhang, and J. Du, “Estimation of optical rotation of chiral molecules with weak measurements,” Opt. Lett. 41, 4032–4035 (2016).
[Crossref] [PubMed]

X. Qiu, X. Zhou, D. Hu, J. Du, F. Gao, Z. Zhang, and H. Luo, “Determination of magneto-optical constant of Fe films with weak measurements,” Appl. Phys. Lett. 105, 131111 (2014).
[Crossref]

Rho, J.

X. Yin, Z. Ye, J. Rho, Y. Wang, and X. Zhang, “Photonic Spin Hall Effect at Metasurfaces,” Science 339, 1405–1407 (2013).
[Crossref] [PubMed]

Rodríguez-Fortuño, F. J.

K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interaction of light,” Nat. Photo. 9, 796 (2015).
[Crossref]

Salazar-Serrano, L. J.

Sales, S.

Sheng, L.

X. Zhou, L. Sheng, and X. Ling, “Photonic spin Hall effect enabled refractive index sensor using weak measurements,” Sci. Rep. 8, 1221 (2018).
[Crossref] [PubMed]

Shitrit, N.

Y. Gorodetski, K. Y. Bliokh, B. Stein, C. Genet, N. Shitrit, V. Kleiner, E. Hasman, and T. W. Ebbesen, “Weak Measurements of Light Chirality with a Plasmonic Slit,” Phys. Rev. Lett. 109, 013901 (2012).
[Crossref] [PubMed]

Shu, W.

H. Luo, X. Zhou, W. Shu, S. Wen, and D. Fan, “Enhanced and switchable spin hall effect of light near the brewster angle on reflection,” Phys. Rev. A 84, 043806 (2011).
[Crossref]

Snyder, A. W.

Starling, D. J.

D. J. Starling, P. B. Dixon, A. N. Jordan, and J. C. Howell, “Precision frequency measurements with interferometric weak values,” Phys. Rev. A 82, 063822 (2010).
[Crossref]

P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, “Ultrasensitive Beam Deflection Measurement via Interferometric Weak Value Amplification,” Phys. Rev. Lett. 102, 173601 (2009).
[Crossref] [PubMed]

Stein, B.

Y. Gorodetski, K. Y. Bliokh, B. Stein, C. Genet, N. Shitrit, V. Kleiner, E. Hasman, and T. W. Ebbesen, “Weak Measurements of Light Chirality with a Plasmonic Slit,” Phys. Rev. Lett. 109, 013901 (2012).
[Crossref] [PubMed]

Sun, K.

X.-Y. Xu, Y. Kedem, K. Sun, L. Vaidman, C.-F. Li, and G.-C. Guo, “Phase Estimation with Weak Measurement Using a White Light Source,” Phys. Rev. Lett. 111, 033604 (2013).
[Crossref] [PubMed]

Sun, P.

T. Tang, J. Li, L. Luo, P. Sun, and J. Yao, “Magneto-Optical Modulation of Photonic Spin Hall Effect of Graphene in Terahertz Region,” Adv. Opt. Mater. 6, 1701212 (2018).
[Crossref]

Tang, T.

T. Tang, J. Li, L. Luo, P. Sun, and J. Yao, “Magneto-Optical Modulation of Photonic Spin Hall Effect of Graphene in Terahertz Region,” Adv. Opt. Mater. 6, 1701212 (2018).
[Crossref]

Torres, J. P.

Vaidman, L.

X.-Y. Xu, Y. Kedem, K. Sun, L. Vaidman, C.-F. Li, and G.-C. Guo, “Phase Estimation with Weak Measurement Using a White Light Source,” Phys. Rev. Lett. 111, 033604 (2013).
[Crossref] [PubMed]

Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin −1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351–1354 (1988).
[Crossref] [PubMed]

van Driel, H. M.

Wang, D.

L. Xie, X. Qiu, L. Luo, X. Liu, Z. Li, Z. Zhang, J. Du, and D. Wang, “Quantitative detection of the respective concentrations of chiral compounds with weak measurements,” Appl. Phys. Lett. 111, 191106 (2017).
[Crossref]

Wang, Y.

X. Yin, Z. Ye, J. Rho, Y. Wang, and X. Zhang, “Photonic Spin Hall Effect at Metasurfaces,” Science 339, 1405–1407 (2013).
[Crossref] [PubMed]

Wen, S.

X. Ling, X. Zhou, K. Huang, Y. Liu, C.-W. Qiu, H. Luo, and S. Wen, “Recent advances in the spin Hall effect of light,” Rep. Prog. Phys. 80, 066401 (2017).
[Crossref] [PubMed]

X. Zhou, J. Zhang, X. Ling, S. Chen, H. Luo, and S. Wen, “Photonic spin Hall effect in topological insulators,” Phys. Rev. A 88, 053840 (2013).
[Crossref]

X. Zhou, Z. Xiao, H. Luo, and S. Wen, “Experimental observation of the spin Hall effect of light on a nanometal film via weak measurements,” Phys. Rev. A 85, 043809 (2012).
[Crossref]

X. Zhou, X. Ling, H. Luo, and S. Wen, “Identifying graphene layers via spin Hall effect of light,” Appl. Phys. Lett. 101, 251602 (2012).
[Crossref]

H. Luo, X. Zhou, W. Shu, S. Wen, and D. Fan, “Enhanced and switchable spin hall effect of light near the brewster angle on reflection,” Phys. Rev. A 84, 043806 (2011).
[Crossref]

Wu, S.

X. Zhu, Y. Zhang, S. Pang, C. Qiao, Q. Liu, and S. Wu, “Quantum measurements with preselection and postselection,” Phys. Rev. A 84, 052111 (2011).
[Crossref]

Xiao, Z.

X. Zhou, Z. Xiao, H. Luo, and S. Wen, “Experimental observation of the spin Hall effect of light on a nanometal film via weak measurements,” Phys. Rev. A 85, 043809 (2012).
[Crossref]

Xie, L.

L. Xie, X. Qiu, L. Luo, X. Liu, Z. Li, Z. Zhang, J. Du, and D. Wang, “Quantitative detection of the respective concentrations of chiral compounds with weak measurements,” Appl. Phys. Lett. 111, 191106 (2017).
[Crossref]

X. Qiu, L. Xie, X. Liu, L. Luo, Z. Li, Z. Zhang, and J. Du, “Precision phase estimation based on weak-value amplification,” Appl. Phys. Lett. 110, 071105 (2017).
[Crossref]

X. Qiu, L. Xie, X. Liu, L. Luo, Z. Zhang, and J. Du, “Estimation of optical rotation of chiral molecules with weak measurements,” Opt. Lett. 41, 4032–4035 (2016).
[Crossref] [PubMed]

Xu, X.-Y.

X.-Y. Xu, Y. Kedem, K. Sun, L. Vaidman, C.-F. Li, and G.-C. Guo, “Phase Estimation with Weak Measurement Using a White Light Source,” Phys. Rev. Lett. 111, 033604 (2013).
[Crossref] [PubMed]

Yao, J.

T. Tang, J. Li, L. Luo, P. Sun, and J. Yao, “Magneto-Optical Modulation of Photonic Spin Hall Effect of Graphene in Terahertz Region,” Adv. Opt. Mater. 6, 1701212 (2018).
[Crossref]

Ye, Z.

X. Yin, Z. Ye, J. Rho, Y. Wang, and X. Zhang, “Photonic Spin Hall Effect at Metasurfaces,” Science 339, 1405–1407 (2013).
[Crossref] [PubMed]

Yin, X.

X. Yin, Z. Ye, J. Rho, Y. Wang, and X. Zhang, “Photonic Spin Hall Effect at Metasurfaces,” Science 339, 1405–1407 (2013).
[Crossref] [PubMed]

X. Yin and L. Hesselink, “Goos–Hänchen shift surface plasmon resonance sensor,” Appl. Phys. Lett. 89, 261108 (2006).
[Crossref]

Yu, J.

Zayats, A. V.

K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interaction of light,” Nat. Photo. 9, 796 (2015).
[Crossref]

Zel’dovich, B. Y.

V. S. Liberman and B. Y. Zel’dovich, “Spin-orbit interaction of a photon in an inhomogeneous medium,” Phys. Rev. A 46, 5199–5207 (1992).
[Crossref] [PubMed]

Zhang, J.

W. Zhu, M. Jiang, H. Guan, J. Yu, H. Lu, J. Zhang, and Z. Chen, “Tunable spin splitting of Laguerre–Gaussian beams in graphene metamaterials,” Photon. Res. 5, 684–688 (2017).
[Crossref]

X. Zhou, J. Zhang, X. Ling, S. Chen, H. Luo, and S. Wen, “Photonic spin Hall effect in topological insulators,” Phys. Rev. A 88, 053840 (2013).
[Crossref]

Zhang, X.

X. Yin, Z. Ye, J. Rho, Y. Wang, and X. Zhang, “Photonic Spin Hall Effect at Metasurfaces,” Science 339, 1405–1407 (2013).
[Crossref] [PubMed]

Zhang, Y.

X. Zhu, Y. Zhang, S. Pang, C. Qiao, Q. Liu, and S. Wu, “Quantum measurements with preselection and postselection,” Phys. Rev. A 84, 052111 (2011).
[Crossref]

Zhang, Z.

L. Xie, X. Qiu, L. Luo, X. Liu, Z. Li, Z. Zhang, J. Du, and D. Wang, “Quantitative detection of the respective concentrations of chiral compounds with weak measurements,” Appl. Phys. Lett. 111, 191106 (2017).
[Crossref]

X. Qiu, L. Xie, X. Liu, L. Luo, Z. Li, Z. Zhang, and J. Du, “Precision phase estimation based on weak-value amplification,” Appl. Phys. Lett. 110, 071105 (2017).
[Crossref]

X. Qiu, L. Xie, X. Liu, L. Luo, Z. Zhang, and J. Du, “Estimation of optical rotation of chiral molecules with weak measurements,” Opt. Lett. 41, 4032–4035 (2016).
[Crossref] [PubMed]

X. Qiu, X. Zhou, D. Hu, J. Du, F. Gao, Z. Zhang, and H. Luo, “Determination of magneto-optical constant of Fe films with weak measurements,” Appl. Phys. Lett. 105, 131111 (2014).
[Crossref]

Zhou, X.

X. Zhou, L. Sheng, and X. Ling, “Photonic spin Hall effect enabled refractive index sensor using weak measurements,” Sci. Rep. 8, 1221 (2018).
[Crossref] [PubMed]

X. Ling, X. Zhou, K. Huang, Y. Liu, C.-W. Qiu, H. Luo, and S. Wen, “Recent advances in the spin Hall effect of light,” Rep. Prog. Phys. 80, 066401 (2017).
[Crossref] [PubMed]

X. Zhou and X. Ling, “Unveiling the photonic spin Hall effect with asymmetric spin-dependent splitting,” Opt. Express 24, 3025 (2016).
[Crossref] [PubMed]

X. Qiu, X. Zhou, D. Hu, J. Du, F. Gao, Z. Zhang, and H. Luo, “Determination of magneto-optical constant of Fe films with weak measurements,” Appl. Phys. Lett. 105, 131111 (2014).
[Crossref]

X. Zhou, J. Zhang, X. Ling, S. Chen, H. Luo, and S. Wen, “Photonic spin Hall effect in topological insulators,” Phys. Rev. A 88, 053840 (2013).
[Crossref]

X. Zhou, X. Ling, H. Luo, and S. Wen, “Identifying graphene layers via spin Hall effect of light,” Appl. Phys. Lett. 101, 251602 (2012).
[Crossref]

X. Zhou, Z. Xiao, H. Luo, and S. Wen, “Experimental observation of the spin Hall effect of light on a nanometal film via weak measurements,” Phys. Rev. A 85, 043809 (2012).
[Crossref]

H. Luo, X. Zhou, W. Shu, S. Wen, and D. Fan, “Enhanced and switchable spin hall effect of light near the brewster angle on reflection,” Phys. Rev. A 84, 043806 (2011).
[Crossref]

Zhu, W.

Zhu, X.

X. Zhu, Y. Zhang, S. Pang, C. Qiao, Q. Liu, and S. Wu, “Quantum measurements with preselection and postselection,” Phys. Rev. A 84, 052111 (2011).
[Crossref]

Adv. Opt. Mater. (1)

T. Tang, J. Li, L. Luo, P. Sun, and J. Yao, “Magneto-Optical Modulation of Photonic Spin Hall Effect of Graphene in Terahertz Region,” Adv. Opt. Mater. 6, 1701212 (2018).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (5)

L. Xie, X. Qiu, L. Luo, X. Liu, Z. Li, Z. Zhang, J. Du, and D. Wang, “Quantitative detection of the respective concentrations of chiral compounds with weak measurements,” Appl. Phys. Lett. 111, 191106 (2017).
[Crossref]

X. Zhou, X. Ling, H. Luo, and S. Wen, “Identifying graphene layers via spin Hall effect of light,” Appl. Phys. Lett. 101, 251602 (2012).
[Crossref]

X. Qiu, X. Zhou, D. Hu, J. Du, F. Gao, Z. Zhang, and H. Luo, “Determination of magneto-optical constant of Fe films with weak measurements,” Appl. Phys. Lett. 105, 131111 (2014).
[Crossref]

X. Qiu, L. Xie, X. Liu, L. Luo, Z. Li, Z. Zhang, and J. Du, “Precision phase estimation based on weak-value amplification,” Appl. Phys. Lett. 110, 071105 (2017).
[Crossref]

X. Yin and L. Hesselink, “Goos–Hänchen shift surface plasmon resonance sensor,” Appl. Phys. Lett. 89, 261108 (2006).
[Crossref]

J. Opt. (2)

K. Y. Bliokh and A. Aiello, “Goos–Hänchen and Imbert–Fedorov beam shifts: an overview,” J. Opt. 15, 014001 (2013).
[Crossref]

F. I. Fedorov, “To the theory of total reflection,” J. Opt. 15, 014002 (2013).
[Crossref]

Nat. Photo. (1)

K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interaction of light,” Nat. Photo. 9, 796 (2015).
[Crossref]

Opt. Express (2)

Opt. Lett. (7)

Photon. Res. (1)

Phys. Rev. A (6)

X. Zhou, J. Zhang, X. Ling, S. Chen, H. Luo, and S. Wen, “Photonic spin Hall effect in topological insulators,” Phys. Rev. A 88, 053840 (2013).
[Crossref]

V. S. Liberman and B. Y. Zel’dovich, “Spin-orbit interaction of a photon in an inhomogeneous medium,” Phys. Rev. A 46, 5199–5207 (1992).
[Crossref] [PubMed]

X. Zhou, Z. Xiao, H. Luo, and S. Wen, “Experimental observation of the spin Hall effect of light on a nanometal film via weak measurements,” Phys. Rev. A 85, 043809 (2012).
[Crossref]

D. J. Starling, P. B. Dixon, A. N. Jordan, and J. C. Howell, “Precision frequency measurements with interferometric weak values,” Phys. Rev. A 82, 063822 (2010).
[Crossref]

X. Zhu, Y. Zhang, S. Pang, C. Qiao, Q. Liu, and S. Wu, “Quantum measurements with preselection and postselection,” Phys. Rev. A 84, 052111 (2011).
[Crossref]

H. Luo, X. Zhou, W. Shu, S. Wen, and D. Fan, “Enhanced and switchable spin hall effect of light near the brewster angle on reflection,” Phys. Rev. A 84, 043806 (2011).
[Crossref]

Phys. Rev. D (1)

C. Imbert, “Calculation and Experimental Proof of the Transverse Shift Induced by Total Internal Reflection of a Circularly Polarized Light Beam,” Phys. Rev. D 5, 787–796 (1972).
[Crossref]

Phys. Rev. Lett. (8)

M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. 93, 083901 (2004).
[Crossref] [PubMed]

K. Y. Bliokh and Y. P. Bliokh, “Conservation of Angular Momentum, Transverse Shift, and Spin Hall Effect in Reflection and Refraction of an Electromagnetic Wave Packet,” Phys. Rev. Lett. 96, 073903 (2006).
[Crossref] [PubMed]

W. J. M. Kort-Kamp, “Topological Phase Transitions in the Photonic Spin Hall Effect,” Phys. Rev. Lett. 119, 147401 (2017).
[Crossref] [PubMed]

Y. Gorodetski, A. Niv, V. Kleiner, and E. Hasman, “Observation of the Spin-Based Plasmonic Effect in Nanoscale Structures,” Phys. Rev. Lett. 101, 043903 (2008).
[Crossref] [PubMed]

Y. Gorodetski, K. Y. Bliokh, B. Stein, C. Genet, N. Shitrit, V. Kleiner, E. Hasman, and T. W. Ebbesen, “Weak Measurements of Light Chirality with a Plasmonic Slit,” Phys. Rev. Lett. 109, 013901 (2012).
[Crossref] [PubMed]

P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, “Ultrasensitive Beam Deflection Measurement via Interferometric Weak Value Amplification,” Phys. Rev. Lett. 102, 173601 (2009).
[Crossref] [PubMed]

X.-Y. Xu, Y. Kedem, K. Sun, L. Vaidman, C.-F. Li, and G.-C. Guo, “Phase Estimation with Weak Measurement Using a White Light Source,” Phys. Rev. Lett. 111, 033604 (2013).
[Crossref] [PubMed]

Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin −1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351–1354 (1988).
[Crossref] [PubMed]

Rep. Prog. Phys. (1)

X. Ling, X. Zhou, K. Huang, Y. Liu, C.-W. Qiu, H. Luo, and S. Wen, “Recent advances in the spin Hall effect of light,” Rep. Prog. Phys. 80, 066401 (2017).
[Crossref] [PubMed]

Rev. Mod. Phys. (1)

J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, “Colloquium: Understanding quantum weak values: Basics and applications,” Rev. Mod. Phys. 86, 307–316 (2014).
[Crossref]

Sci. Rep. (1)

X. Zhou, L. Sheng, and X. Ling, “Photonic spin Hall effect enabled refractive index sensor using weak measurements,” Sci. Rep. 8, 1221 (2018).
[Crossref] [PubMed]

Science (2)

O. Hosten and P. Kwiat, “Observation of the Spin Hall Effect of Light via Weak Measurements,” Science 319, 787–790 (2008).
[Crossref] [PubMed]

X. Yin, Z. Ye, J. Rho, Y. Wang, and X. Zhang, “Photonic Spin Hall Effect at Metasurfaces,” Science 339, 1405–1407 (2013).
[Crossref] [PubMed]

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Figures (5)

Fig. 1
Fig. 1 Schematic of the SHEL in IF shift. A beam is reflected from the air-prism interface at the Brewster angle θB. δ+ and δ indicate the magnitude of the spin splitting of the SHEL in IF shift, respectively.
Fig. 2
Fig. 2 (a) shows the initial spin splitting and IF shift changing with the azimuth angle. (b) shows that the spin splitting of the SHEL in IF shift is a fixed value.
Fig. 3
Fig. 3 Schematic drawing of the experimental setup. Light source : He-Ne laser at 632.8nm (Thorlabs HNL210L). HWP: half-wave plate. L1 and L2 are lenses with effective focal length 50mm and 250mm, respectively. P1 and QWP are the Glan polarizer and quarter-wave plate for preselection, respectively. P2 is the Glan polarizer for postselection. CCD is the charge-coupled device for recording the amplified shift (Thorlabs BC106N-VIS/M). Inset: the process of preselection and postselection.
Fig. 4
Fig. 4 Intensity profiles of the reflected beam passing through a crossed polarizer as a function of the azimuth angle α. (a) and (b) denote the theoretical and experimental results, respectively.
Fig. 5
Fig. 5 (a)–(e) shows the amplified shifts of spin splitting in IF shift change with postselection angle at azimuth angle α = 30°, 45°, 60°, 75° and 90°. (f) shows the amplified shift and its corresponding spin splitting of the SHEL in IF shift as a function of azimuth angle at postselection angle β = 1.8°.

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

| Ψ i = d k x i d k y i Φ ( k x i , k y i ) | k x i | k y i | ψ i ,
| ψ r = M ^ r | ψ i ,
M ^ r = [ r p k y r ( r p + r s ) cot θ i k 0 k y r ( r p + r s ) cot θ i k 0 r s ]
r p = r p θ i + χ k x i k 0 ,
| Ψ r = d k x r d k y r Φ ( k x r , k y r ) | k x r | k y r | ψ r .
Δ σ = Ψ r σ | i k y r | Ψ r σ Ψ r σ | Ψ r σ δ cot α + σ δ ,
Δ IF = σ = ± Ψ r σ | i k y r | Ψ r σ σ = ± Ψ r σ | Ψ r σ δ cot α .
| Ψ r σ = d k x r d k y r Φ ( k x r , k y r ) | k x r | k y r | ψ r σ ,
| ψ r σ = H r + i σ V r 2 | σ = σ r s sin α 2 exp [ i σ k y r ] exp [ i k y δ cot α ] | σ ,
δ σ = Δ σ Δ IF = σ δ .
| Ψ r = d k x r d k y r Φ ( k x r , k y r ) | k x r | k y r exp [ i k y δ σ ^ 3 ] R ^ | ψ i ,
| Ψ f = ψ f | Ψ r = d k x r d k y r Φ ( k x r , k y r ) | k x r | k y r × exp [ i ( k x r 2 + k y r 2 ) z 2 k 0 ] ψ f | exp [ i k y δ σ ^ 3 ] | ψ i ,
| Ψ f = ψ f | Ψ r = d k x r d k y r Φ ( k x r , k y r ) | k x r | k y r × exp [ i ( k x r 2 + k y r 2 ) z 2 k 0 ] ψ f | ψ i [ cos ( k y r δ ) i A w sin ( k y r δ ) ] ,
A w = ψ f | σ 3 | ψ i ψ f | ψ i
y = Ψ f | i k y r | Ψ f Ψ f | Ψ f F A w Fac δ ,
F = z R 0
A w Fac = sin 2 β [ ( χ cot α ) 2 2 k 0 R 0 r s 2 ] ( χ cot α ) 2 ( 1 + cos 2 β ) + 2 k 0 R 0 r s 2 ( 1 cos 2 β )
Λ = y δ = FA w Fac

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