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

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    [Crossref] [PubMed]
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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.

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]

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).
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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).
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Kwiat, P.

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

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]

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.

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).
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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).
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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, 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]

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).
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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).
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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).
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Murakami, S.

M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. 93, 083901 (2004).
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M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. 93, 083901 (2004).
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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).
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K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interaction of light,” Nat. Photo. 9, 796 (2015).
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Onoda, M.

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

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).
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Rho, J.

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

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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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.

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]

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, 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]

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).
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J. Opt. (2)

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

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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)

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| 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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