Abstract

We examine the spin-orbit interaction of light and photonic spin Hall effect on the surface of anisotropic two-dimensional atomic crystals. As an example, the photonic spin Hall effect on the surface of black phosphorus is investigated. The photonic spin Hall effect manifests itself as the spin-dependent beam shifts in both transverse and in-plane directions. We demonstrate that the spin-dependent shifts are sensitive to the orientation of the optical axis, doping concentration, and interband transitions. These results can be extensively extended to other anisotropic two-dimensional atomic crystals. By incorporating the quantum weak measurement techniques, the photonic spin Hall effect holds great promise for detecting the parameters of anisotropic two-dimensional atomic crystals.

© 2018 Chinese Laser Press

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References

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    [Crossref]
  5. K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9, 796–808 (2015).
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  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]
  7. 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]
  8. 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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  9. M. Merano, “Fresnel coefficients of a two-dimensional atomic crystal,” Phys. Rev. A 93, 013832 (2016).
    [Crossref]
  10. M. Merano, “Transverse electric surface mode in atomically thin boron-nitride,” Opt. Lett. 41, 2668–2671 (2016).
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    [Crossref]
  12. L. Cai, M. Liu, S. Chen, Y. Liu, W. Shu, H. Luo, and S. Wen, “Quantized photonic spin Hall effect in graphene,” Phys. Rev. A 95, 013809 (2017).
    [Crossref]
  13. W. J. M. Kort-Kamp, N. A. Sinitsyn, and D. A. R. Dalvit, “Quantized beam shifts in graphene,” Phys. Rev. B 93, 081410 (2016).
    [Crossref]
  14. W. Wu, S. Chen, C. Mi, W. Zhang, H. Luo, and S. Wen, “Giant quantized Goos–Hänchen effect on the surface of graphene in the quantum Hall regime,” Phys. Rev. A 96, 043814 (2017).
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  15. W. J. M. Kort-Kamp, “Topological phase transitions in the photonic spin Hall effect,” Phys. Rev. Lett. 119, 147401 (2017).
    [Crossref]
  16. X. Zhou, X. Li, H. Luo, and S. Wen, “Identifying graphene layers via spin Hall effect of light,” Appl. Phys. Lett. 101, 251602 (2012).
    [Crossref]
  17. S. Chen, C. Mi, L. Cai, M. Liu, H. Luo, and S. Wen, “Observation of the Goos–Hänchen shift in graphene via weak measurements,” Appl. Phys. Lett. 110, 031105 (2017).
    [Crossref]
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    [Crossref]
  22. A. Nemilentsau, T. Low, and G. Hanson, “Anisotropic 2D materials for tunable hyperbolic plasmonics,” Phys. Rev. Lett. 116, 066804 (2016).
    [Crossref]
  23. X. Ling, X. Zhou, X. Yi, W. Shu, Y. Liu, S. Chen, H. Luo, S. Wen, and D. Fan, “Giant photonic spin Hall effect in momentum space in a structured metamaterial with spatially varying birefringence,” Light Sci. Appl. 4, e290 (2015).
    [Crossref]
  24. C. Mi, S. Chen, X. Zhou, K. Tian, H. Luo, and S. Wen, “Observation of tiny polarization rotation rate in total internal reflection via weak measurements,” Photon. Res. 5, 92–96 (2017).
    [Crossref]
  25. M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. London Ser. A 392, 45–57 (1984).
    [Crossref]
  26. 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]
  27. C. Mi, S. Chen, W. Wu, W. Zhang, X. Zhou, X. Ling, W. Shu, H. Luo, and S. Wen, “Precise identification of graphene layers at the air-prism interface via a pseudo-Brewster angle,” Opt. Lett. 42, 4135–4138 (2017).
    [Crossref]
  28. 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]
  29. V. Tran, R. Soklaski, Y. Liang, and L. Yang, “Layer-controlled band gap and anisotropic excitons in few-layer black phosphorus,” Phys. Rev. B 89, 235319 (2014).
    [Crossref]
  30. L. Fei and L. Yang, “Strain-engineering the anisotropic electrical conductance of few-layer black phosphorus,” Nano Lett. 14, 2884–2889 (2014).
    [Crossref]
  31. 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]
  32. Y. Xiang, X. Jiang, Q. You, J. Guo, and X. Dai, “Enhanced spin Hall effect of reflected light with guided-wave surface plasmon resonance,” Photon. Res. 5, 467–472 (2017).
    [Crossref]

2017 (10)

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]

L. Cai, M. Liu, S. Chen, Y. Liu, W. Shu, H. Luo, and S. Wen, “Quantized photonic spin Hall effect in graphene,” Phys. Rev. A 95, 013809 (2017).
[Crossref]

S. Chen, C. Mi, L. Cai, M. Liu, H. Luo, and S. Wen, “Observation of the Goos–Hänchen shift in graphene via weak measurements,” Appl. Phys. Lett. 110, 031105 (2017).
[Crossref]

M. Liu, L. Cai, S. Chen, Y. Liu, H. Luo, and S. Wen, “Strong spin-orbit interaction of light on the surface of atomically thin crystals,” Phys. Rev. A 95, 063827 (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. Wu, S. Chen, C. Mi, W. Zhang, H. Luo, and S. Wen, “Giant quantized Goos–Hänchen effect on the surface of graphene in the quantum Hall regime,” Phys. Rev. A 96, 043814 (2017).
[Crossref]

W. J. M. Kort-Kamp, “Topological phase transitions in the photonic spin Hall effect,” Phys. Rev. Lett. 119, 147401 (2017).
[Crossref]

C. Mi, S. Chen, X. Zhou, K. Tian, H. Luo, and S. Wen, “Observation of tiny polarization rotation rate in total internal reflection via weak measurements,” Photon. Res. 5, 92–96 (2017).
[Crossref]

C. Mi, S. Chen, W. Wu, W. Zhang, X. Zhou, X. Ling, W. Shu, H. Luo, and S. Wen, “Precise identification of graphene layers at the air-prism interface via a pseudo-Brewster angle,” Opt. Lett. 42, 4135–4138 (2017).
[Crossref]

Y. Xiang, X. Jiang, Q. You, J. Guo, and X. Dai, “Enhanced spin Hall effect of reflected light with guided-wave surface plasmon resonance,” Photon. Res. 5, 467–472 (2017).
[Crossref]

2016 (5)

A. Nemilentsau, T. Low, and G. Hanson, “Anisotropic 2D materials for tunable hyperbolic plasmonics,” Phys. Rev. Lett. 116, 066804 (2016).
[Crossref]

W. J. M. Kort-Kamp, N. A. Sinitsyn, and D. A. R. Dalvit, “Quantized beam shifts in graphene,” Phys. Rev. B 93, 081410 (2016).
[Crossref]

M. Merano, “Fresnel coefficients of a two-dimensional atomic crystal,” Phys. Rev. A 93, 013832 (2016).
[Crossref]

M. Merano, “Transverse electric surface mode in atomically thin boron-nitride,” Opt. Lett. 41, 2668–2671 (2016).
[Crossref]

M. Merano, “Optical beam shifts in graphene and single-layer boron-nitride,” Opt. Lett. 41, 5780–5783 (2016).
[Crossref]

2015 (2)

K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9, 796–808 (2015).
[Crossref]

X. Ling, X. Zhou, X. Yi, W. Shu, Y. Liu, S. Chen, H. Luo, S. Wen, and D. Fan, “Giant photonic spin Hall effect in momentum space in a structured metamaterial with spatially varying birefringence,” Light Sci. Appl. 4, e290 (2015).
[Crossref]

2014 (5)

V. Tran, R. Soklaski, Y. Liang, and L. Yang, “Layer-controlled band gap and anisotropic excitons in few-layer black phosphorus,” Phys. Rev. B 89, 235319 (2014).
[Crossref]

L. Fei and L. Yang, “Strain-engineering the anisotropic electrical conductance of few-layer black phosphorus,” Nano Lett. 14, 2884–2889 (2014).
[Crossref]

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]

F. Xia, H. Wang, and Y. Jia, “Rediscovering black phosphorus as an anisotropic layered material for optoelectronics and electronics,” Nat. Commun. 5, 4458 (2014).
[Crossref]

T. Low, R. Roldán, H. Wang, F. Xia, P. Avouris, L. M. Moreno, and F. Guinea, “Plasmons and screening in monolayer and multilayer black phosphorus,” Phys. Rev. Lett. 113, 106802 (2014).
[Crossref]

2012 (2)

X. Zhou, X. Li, 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]

2011 (1)

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]

2010 (1)

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4, 611–622 (2010).
[Crossref]

2008 (1)

O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science 319, 787–790 (2008).
[Crossref]

2007 (1)

A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6, 183–191 (2007).
[Crossref]

2006 (1)

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]

2004 (1)

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

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]

1984 (1)

M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. London Ser. A 392, 45–57 (1984).
[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]

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]

Avouris, P.

T. Low, R. Roldán, H. Wang, F. Xia, P. Avouris, L. M. Moreno, and F. Guinea, “Plasmons and screening in monolayer and multilayer black phosphorus,” Phys. Rev. Lett. 113, 106802 (2014).
[Crossref]

Berry, M. V.

M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. London Ser. A 392, 45–57 (1984).
[Crossref]

Bliokh, K. Y.

K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9, 796–808 (2015).
[Crossref]

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]

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]

Bonaccorso, F.

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4, 611–622 (2010).
[Crossref]

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]

Cai, L.

M. Liu, L. Cai, S. Chen, Y. Liu, H. Luo, and S. Wen, “Strong spin-orbit interaction of light on the surface of atomically thin crystals,” Phys. Rev. A 95, 063827 (2017).
[Crossref]

S. Chen, C. Mi, L. Cai, M. Liu, H. Luo, and S. Wen, “Observation of the Goos–Hänchen shift in graphene via weak measurements,” Appl. Phys. Lett. 110, 031105 (2017).
[Crossref]

L. Cai, M. Liu, S. Chen, Y. Liu, W. Shu, H. Luo, and S. Wen, “Quantized photonic spin Hall effect in graphene,” Phys. Rev. A 95, 013809 (2017).
[Crossref]

Chen, S.

L. Cai, M. Liu, S. Chen, Y. Liu, W. Shu, H. Luo, and S. Wen, “Quantized photonic spin Hall effect in graphene,” Phys. Rev. A 95, 013809 (2017).
[Crossref]

W. Wu, S. Chen, C. Mi, W. Zhang, H. Luo, and S. Wen, “Giant quantized Goos–Hänchen effect on the surface of graphene in the quantum Hall regime,” Phys. Rev. A 96, 043814 (2017).
[Crossref]

S. Chen, C. Mi, L. Cai, M. Liu, H. Luo, and S. Wen, “Observation of the Goos–Hänchen shift in graphene via weak measurements,” Appl. Phys. Lett. 110, 031105 (2017).
[Crossref]

M. Liu, L. Cai, S. Chen, Y. Liu, H. Luo, and S. Wen, “Strong spin-orbit interaction of light on the surface of atomically thin crystals,” Phys. Rev. A 95, 063827 (2017).
[Crossref]

C. Mi, S. Chen, X. Zhou, K. Tian, H. Luo, and S. Wen, “Observation of tiny polarization rotation rate in total internal reflection via weak measurements,” Photon. Res. 5, 92–96 (2017).
[Crossref]

C. Mi, S. Chen, W. Wu, W. Zhang, X. Zhou, X. Ling, W. Shu, H. Luo, and S. Wen, “Precise identification of graphene layers at the air-prism interface via a pseudo-Brewster angle,” Opt. Lett. 42, 4135–4138 (2017).
[Crossref]

X. Ling, X. Zhou, X. Yi, W. Shu, Y. Liu, S. Chen, H. Luo, S. Wen, and D. Fan, “Giant photonic spin Hall effect in momentum space in a structured metamaterial with spatially varying birefringence,” Light Sci. Appl. 4, e290 (2015).
[Crossref]

Chen, Z.

Dai, X.

Dalvit, D. A. R.

W. J. M. Kort-Kamp, N. A. Sinitsyn, and D. A. R. Dalvit, “Quantized beam shifts in graphene,” Phys. Rev. B 93, 081410 (2016).
[Crossref]

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]

Fan, D.

X. Ling, X. Zhou, X. Yi, W. Shu, Y. Liu, S. Chen, H. Luo, S. Wen, and D. Fan, “Giant photonic spin Hall effect in momentum space in a structured metamaterial with spatially varying birefringence,” Light Sci. Appl. 4, e290 (2015).
[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]

Fei, L.

L. Fei and L. Yang, “Strain-engineering the anisotropic electrical conductance of few-layer black phosphorus,” Nano Lett. 14, 2884–2889 (2014).
[Crossref]

Ferrari, A. C.

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4, 611–622 (2010).
[Crossref]

Geim, A. K.

A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6, 183–191 (2007).
[Crossref]

Guan, H.

Guinea, F.

T. Low, R. Roldán, H. Wang, F. Xia, P. Avouris, L. M. Moreno, and F. Guinea, “Plasmons and screening in monolayer and multilayer black phosphorus,” Phys. Rev. Lett. 113, 106802 (2014).
[Crossref]

Guo, J.

Hanson, G.

A. Nemilentsau, T. Low, and G. Hanson, “Anisotropic 2D materials for tunable hyperbolic plasmonics,” Phys. Rev. Lett. 116, 066804 (2016).
[Crossref]

Hasan, T.

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4, 611–622 (2010).
[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]

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]

Jia, Y.

F. Xia, H. Wang, and Y. Jia, “Rediscovering black phosphorus as an anisotropic layered material for optoelectronics and electronics,” Nat. Commun. 5, 4458 (2014).
[Crossref]

Jiang, M.

Jiang, X.

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]

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]

W. J. M. Kort-Kamp, N. A. Sinitsyn, and D. A. R. Dalvit, “Quantized beam shifts in graphene,” Phys. Rev. B 93, 081410 (2016).
[Crossref]

Kwiat, P.

O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science 319, 787–790 (2008).
[Crossref]

Li, X.

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

Liang, Y.

V. Tran, R. Soklaski, Y. Liang, and L. Yang, “Layer-controlled band gap and anisotropic excitons in few-layer black phosphorus,” Phys. Rev. B 89, 235319 (2014).
[Crossref]

Ling, X.

C. Mi, S. Chen, W. Wu, W. Zhang, X. Zhou, X. Ling, W. Shu, H. Luo, and S. Wen, “Precise identification of graphene layers at the air-prism interface via a pseudo-Brewster angle,” Opt. Lett. 42, 4135–4138 (2017).
[Crossref]

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]

X. Ling, X. Zhou, X. Yi, W. Shu, Y. Liu, S. Chen, H. Luo, S. Wen, and D. Fan, “Giant photonic spin Hall effect in momentum space in a structured metamaterial with spatially varying birefringence,” Light Sci. Appl. 4, e290 (2015).
[Crossref]

Liu, M.

M. Liu, L. Cai, S. Chen, Y. Liu, H. Luo, and S. Wen, “Strong spin-orbit interaction of light on the surface of atomically thin crystals,” Phys. Rev. A 95, 063827 (2017).
[Crossref]

S. Chen, C. Mi, L. Cai, M. Liu, H. Luo, and S. Wen, “Observation of the Goos–Hänchen shift in graphene via weak measurements,” Appl. Phys. Lett. 110, 031105 (2017).
[Crossref]

L. Cai, M. Liu, S. Chen, Y. Liu, W. Shu, H. Luo, and S. Wen, “Quantized photonic spin Hall effect in graphene,” Phys. Rev. A 95, 013809 (2017).
[Crossref]

Liu, Y.

L. Cai, M. Liu, S. Chen, Y. Liu, W. Shu, H. Luo, and S. Wen, “Quantized photonic spin Hall effect in graphene,” Phys. Rev. A 95, 013809 (2017).
[Crossref]

M. Liu, L. Cai, S. Chen, Y. Liu, H. Luo, and S. Wen, “Strong spin-orbit interaction of light on the surface of atomically thin crystals,” Phys. Rev. A 95, 063827 (2017).
[Crossref]

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]

X. Ling, X. Zhou, X. Yi, W. Shu, Y. Liu, S. Chen, H. Luo, S. Wen, and D. Fan, “Giant photonic spin Hall effect in momentum space in a structured metamaterial with spatially varying birefringence,” Light Sci. Appl. 4, e290 (2015).
[Crossref]

Low, T.

A. Nemilentsau, T. Low, and G. Hanson, “Anisotropic 2D materials for tunable hyperbolic plasmonics,” Phys. Rev. Lett. 116, 066804 (2016).
[Crossref]

T. Low, R. Roldán, H. Wang, F. Xia, P. Avouris, L. M. Moreno, and F. Guinea, “Plasmons and screening in monolayer and multilayer black phosphorus,” Phys. Rev. Lett. 113, 106802 (2014).
[Crossref]

Lu, H.

Luo, H.

C. Mi, S. Chen, W. Wu, W. Zhang, X. Zhou, X. Ling, W. Shu, H. Luo, and S. Wen, “Precise identification of graphene layers at the air-prism interface via a pseudo-Brewster angle,” Opt. Lett. 42, 4135–4138 (2017).
[Crossref]

C. Mi, S. Chen, X. Zhou, K. Tian, H. Luo, and S. Wen, “Observation of tiny polarization rotation rate in total internal reflection via weak measurements,” Photon. Res. 5, 92–96 (2017).
[Crossref]

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]

M. Liu, L. Cai, S. Chen, Y. Liu, H. Luo, and S. Wen, “Strong spin-orbit interaction of light on the surface of atomically thin crystals,” Phys. Rev. A 95, 063827 (2017).
[Crossref]

S. Chen, C. Mi, L. Cai, M. Liu, H. Luo, and S. Wen, “Observation of the Goos–Hänchen shift in graphene via weak measurements,” Appl. Phys. Lett. 110, 031105 (2017).
[Crossref]

L. Cai, M. Liu, S. Chen, Y. Liu, W. Shu, H. Luo, and S. Wen, “Quantized photonic spin Hall effect in graphene,” Phys. Rev. A 95, 013809 (2017).
[Crossref]

W. Wu, S. Chen, C. Mi, W. Zhang, H. Luo, and S. Wen, “Giant quantized Goos–Hänchen effect on the surface of graphene in the quantum Hall regime,” Phys. Rev. A 96, 043814 (2017).
[Crossref]

X. Ling, X. Zhou, X. Yi, W. Shu, Y. Liu, S. Chen, H. Luo, S. Wen, and D. Fan, “Giant photonic spin Hall effect in momentum space in a structured metamaterial with spatially varying birefringence,” Light Sci. Appl. 4, e290 (2015).
[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. Li, 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]

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]

Merano, M.

Mi, C.

S. Chen, C. Mi, L. Cai, M. Liu, H. Luo, and S. Wen, “Observation of the Goos–Hänchen shift in graphene via weak measurements,” Appl. Phys. Lett. 110, 031105 (2017).
[Crossref]

W. Wu, S. Chen, C. Mi, W. Zhang, H. Luo, and S. Wen, “Giant quantized Goos–Hänchen effect on the surface of graphene in the quantum Hall regime,” Phys. Rev. A 96, 043814 (2017).
[Crossref]

C. Mi, S. Chen, X. Zhou, K. Tian, H. Luo, and S. Wen, “Observation of tiny polarization rotation rate in total internal reflection via weak measurements,” Photon. Res. 5, 92–96 (2017).
[Crossref]

C. Mi, S. Chen, W. Wu, W. Zhang, X. Zhou, X. Ling, W. Shu, H. Luo, and S. Wen, “Precise identification of graphene layers at the air-prism interface via a pseudo-Brewster angle,” Opt. Lett. 42, 4135–4138 (2017).
[Crossref]

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]

Moreno, L. M.

T. Low, R. Roldán, H. Wang, F. Xia, P. Avouris, L. M. Moreno, and F. Guinea, “Plasmons and screening in monolayer and multilayer black phosphorus,” Phys. Rev. Lett. 113, 106802 (2014).
[Crossref]

Murakami, S.

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

Nagaosa, N.

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

Nemilentsau, A.

A. Nemilentsau, T. Low, and G. Hanson, “Anisotropic 2D materials for tunable hyperbolic plasmonics,” Phys. Rev. Lett. 116, 066804 (2016).
[Crossref]

Nori, F.

K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9, 796–808 (2015).
[Crossref]

Novoselov, K. S.

A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6, 183–191 (2007).
[Crossref]

Onoda, M.

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

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]

Rodríguez-Fortuño, F. J.

K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9, 796–808 (2015).
[Crossref]

Roldán, R.

T. Low, R. Roldán, H. Wang, F. Xia, P. Avouris, L. M. Moreno, and F. Guinea, “Plasmons and screening in monolayer and multilayer black phosphorus,” Phys. Rev. Lett. 113, 106802 (2014).
[Crossref]

Shu, W.

L. Cai, M. Liu, S. Chen, Y. Liu, W. Shu, H. Luo, and S. Wen, “Quantized photonic spin Hall effect in graphene,” Phys. Rev. A 95, 013809 (2017).
[Crossref]

C. Mi, S. Chen, W. Wu, W. Zhang, X. Zhou, X. Ling, W. Shu, H. Luo, and S. Wen, “Precise identification of graphene layers at the air-prism interface via a pseudo-Brewster angle,” Opt. Lett. 42, 4135–4138 (2017).
[Crossref]

X. Ling, X. Zhou, X. Yi, W. Shu, Y. Liu, S. Chen, H. Luo, S. Wen, and D. Fan, “Giant photonic spin Hall effect in momentum space in a structured metamaterial with spatially varying birefringence,” Light Sci. Appl. 4, e290 (2015).
[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]

Sinitsyn, N. A.

W. J. M. Kort-Kamp, N. A. Sinitsyn, and D. A. R. Dalvit, “Quantized beam shifts in graphene,” Phys. Rev. B 93, 081410 (2016).
[Crossref]

Soklaski, R.

V. Tran, R. Soklaski, Y. Liang, and L. Yang, “Layer-controlled band gap and anisotropic excitons in few-layer black phosphorus,” Phys. Rev. B 89, 235319 (2014).
[Crossref]

Sun, Z.

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4, 611–622 (2010).
[Crossref]

Tian, K.

Tran, V.

V. Tran, R. Soklaski, Y. Liang, and L. Yang, “Layer-controlled band gap and anisotropic excitons in few-layer black phosphorus,” Phys. Rev. B 89, 235319 (2014).
[Crossref]

Vaidman, L.

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]

Wang, H.

T. Low, R. Roldán, H. Wang, F. Xia, P. Avouris, L. M. Moreno, and F. Guinea, “Plasmons and screening in monolayer and multilayer black phosphorus,” Phys. Rev. Lett. 113, 106802 (2014).
[Crossref]

F. Xia, H. Wang, and Y. Jia, “Rediscovering black phosphorus as an anisotropic layered material for optoelectronics and electronics,” Nat. Commun. 5, 4458 (2014).
[Crossref]

Wen, S.

C. Mi, S. Chen, X. Zhou, K. Tian, H. Luo, and S. Wen, “Observation of tiny polarization rotation rate in total internal reflection via weak measurements,” Photon. Res. 5, 92–96 (2017).
[Crossref]

C. Mi, S. Chen, W. Wu, W. Zhang, X. Zhou, X. Ling, W. Shu, H. Luo, and S. Wen, “Precise identification of graphene layers at the air-prism interface via a pseudo-Brewster angle,” Opt. Lett. 42, 4135–4138 (2017).
[Crossref]

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]

M. Liu, L. Cai, S. Chen, Y. Liu, H. Luo, and S. Wen, “Strong spin-orbit interaction of light on the surface of atomically thin crystals,” Phys. Rev. A 95, 063827 (2017).
[Crossref]

S. Chen, C. Mi, L. Cai, M. Liu, H. Luo, and S. Wen, “Observation of the Goos–Hänchen shift in graphene via weak measurements,” Appl. Phys. Lett. 110, 031105 (2017).
[Crossref]

L. Cai, M. Liu, S. Chen, Y. Liu, W. Shu, H. Luo, and S. Wen, “Quantized photonic spin Hall effect in graphene,” Phys. Rev. A 95, 013809 (2017).
[Crossref]

W. Wu, S. Chen, C. Mi, W. Zhang, H. Luo, and S. Wen, “Giant quantized Goos–Hänchen effect on the surface of graphene in the quantum Hall regime,” Phys. Rev. A 96, 043814 (2017).
[Crossref]

X. Ling, X. Zhou, X. Yi, W. Shu, Y. Liu, S. Chen, H. Luo, S. Wen, and D. Fan, “Giant photonic spin Hall effect in momentum space in a structured metamaterial with spatially varying birefringence,” Light Sci. Appl. 4, e290 (2015).
[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. Li, 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, W.

C. Mi, S. Chen, W. Wu, W. Zhang, X. Zhou, X. Ling, W. Shu, H. Luo, and S. Wen, “Precise identification of graphene layers at the air-prism interface via a pseudo-Brewster angle,” Opt. Lett. 42, 4135–4138 (2017).
[Crossref]

W. Wu, S. Chen, C. Mi, W. Zhang, H. Luo, and S. Wen, “Giant quantized Goos–Hänchen effect on the surface of graphene in the quantum Hall regime,” Phys. Rev. A 96, 043814 (2017).
[Crossref]

Xia, F.

F. Xia, H. Wang, and Y. Jia, “Rediscovering black phosphorus as an anisotropic layered material for optoelectronics and electronics,” Nat. Commun. 5, 4458 (2014).
[Crossref]

T. Low, R. Roldán, H. Wang, F. Xia, P. Avouris, L. M. Moreno, and F. Guinea, “Plasmons and screening in monolayer and multilayer black phosphorus,” Phys. Rev. Lett. 113, 106802 (2014).
[Crossref]

Xiang, Y.

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]

Yang, L.

V. Tran, R. Soklaski, Y. Liang, and L. Yang, “Layer-controlled band gap and anisotropic excitons in few-layer black phosphorus,” Phys. Rev. B 89, 235319 (2014).
[Crossref]

L. Fei and L. Yang, “Strain-engineering the anisotropic electrical conductance of few-layer black phosphorus,” Nano Lett. 14, 2884–2889 (2014).
[Crossref]

Yi, X.

X. Ling, X. Zhou, X. Yi, W. Shu, Y. Liu, S. Chen, H. Luo, S. Wen, and D. Fan, “Giant photonic spin Hall effect in momentum space in a structured metamaterial with spatially varying birefringence,” Light Sci. Appl. 4, e290 (2015).
[Crossref]

You, Q.

Yu, J.

Zayats, A. V.

K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9, 796–808 (2015).
[Crossref]

Zhang, J.

Zhang, W.

C. Mi, S. Chen, W. Wu, W. Zhang, X. Zhou, X. Ling, W. Shu, H. Luo, and S. Wen, “Precise identification of graphene layers at the air-prism interface via a pseudo-Brewster angle,” Opt. Lett. 42, 4135–4138 (2017).
[Crossref]

W. Wu, S. Chen, C. Mi, W. Zhang, H. Luo, and S. Wen, “Giant quantized Goos–Hänchen effect on the surface of graphene in the quantum Hall regime,” Phys. Rev. A 96, 043814 (2017).
[Crossref]

Zhou, X.

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]

C. Mi, S. Chen, W. Wu, W. Zhang, X. Zhou, X. Ling, W. Shu, H. Luo, and S. Wen, “Precise identification of graphene layers at the air-prism interface via a pseudo-Brewster angle,” Opt. Lett. 42, 4135–4138 (2017).
[Crossref]

C. Mi, S. Chen, X. Zhou, K. Tian, H. Luo, and S. Wen, “Observation of tiny polarization rotation rate in total internal reflection via weak measurements,” Photon. Res. 5, 92–96 (2017).
[Crossref]

X. Ling, X. Zhou, X. Yi, W. Shu, Y. Liu, S. Chen, H. Luo, S. Wen, and D. Fan, “Giant photonic spin Hall effect in momentum space in a structured metamaterial with spatially varying birefringence,” Light Sci. Appl. 4, e290 (2015).
[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. Li, 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.

Appl. Phys. Lett. (2)

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

S. Chen, C. Mi, L. Cai, M. Liu, H. Luo, and S. Wen, “Observation of the Goos–Hänchen shift in graphene via weak measurements,” Appl. Phys. Lett. 110, 031105 (2017).
[Crossref]

Light Sci. Appl. (1)

X. Ling, X. Zhou, X. Yi, W. Shu, Y. Liu, S. Chen, H. Luo, S. Wen, and D. Fan, “Giant photonic spin Hall effect in momentum space in a structured metamaterial with spatially varying birefringence,” Light Sci. Appl. 4, e290 (2015).
[Crossref]

Nano Lett. (1)

L. Fei and L. Yang, “Strain-engineering the anisotropic electrical conductance of few-layer black phosphorus,” Nano Lett. 14, 2884–2889 (2014).
[Crossref]

Nat. Commun. (1)

F. Xia, H. Wang, and Y. Jia, “Rediscovering black phosphorus as an anisotropic layered material for optoelectronics and electronics,” Nat. Commun. 5, 4458 (2014).
[Crossref]

Nat. Mater. (1)

A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6, 183–191 (2007).
[Crossref]

Nat. Photonics (2)

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4, 611–622 (2010).
[Crossref]

K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9, 796–808 (2015).
[Crossref]

Opt. Lett. (3)

Photon. Res. (3)

Phys. Rev. A (6)

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]

M. Merano, “Fresnel coefficients of a two-dimensional atomic crystal,” Phys. Rev. A 93, 013832 (2016).
[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]

L. Cai, M. Liu, S. Chen, Y. Liu, W. Shu, H. Luo, and S. Wen, “Quantized photonic spin Hall effect in graphene,” Phys. Rev. A 95, 013809 (2017).
[Crossref]

M. Liu, L. Cai, S. Chen, Y. Liu, H. Luo, and S. Wen, “Strong spin-orbit interaction of light on the surface of atomically thin crystals,” Phys. Rev. A 95, 063827 (2017).
[Crossref]

W. Wu, S. Chen, C. Mi, W. Zhang, H. Luo, and S. Wen, “Giant quantized Goos–Hänchen effect on the surface of graphene in the quantum Hall regime,” Phys. Rev. A 96, 043814 (2017).
[Crossref]

Phys. Rev. B (2)

W. J. M. Kort-Kamp, N. A. Sinitsyn, and D. A. R. Dalvit, “Quantized beam shifts in graphene,” Phys. Rev. B 93, 081410 (2016).
[Crossref]

V. Tran, R. Soklaski, Y. Liang, and L. Yang, “Layer-controlled band gap and anisotropic excitons in few-layer black phosphorus,” Phys. Rev. B 89, 235319 (2014).
[Crossref]

Phys. Rev. Lett. (6)

T. Low, R. Roldán, H. Wang, F. Xia, P. Avouris, L. M. Moreno, and F. Guinea, “Plasmons and screening in monolayer and multilayer black phosphorus,” Phys. Rev. Lett. 113, 106802 (2014).
[Crossref]

A. Nemilentsau, T. Low, and G. Hanson, “Anisotropic 2D materials for tunable hyperbolic plasmonics,” Phys. Rev. Lett. 116, 066804 (2016).
[Crossref]

W. J. M. Kort-Kamp, “Topological phase transitions in the photonic spin Hall effect,” Phys. Rev. Lett. 119, 147401 (2017).
[Crossref]

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]

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

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]

Proc. R. Soc. London Ser. A (1)

M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. London Ser. A 392, 45–57 (1984).
[Crossref]

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]

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]

Science (1)

O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science 319, 787–790 (2008).
[Crossref]

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Figures (5)

Fig. 1.
Fig. 1. Schematic illustration of the wave reflection at a surface of black phosphorus in a Cartesian coordinate system. A black phosphorus sheet is placed on the top of a homogeneous and isotropic substrate. The photonic SHE occurs on the reflecting surface and exhibits in-plane and transverse spin Hall shifts.
Fig. 2.
Fig. 2. Real and imaginary parts of the conductivity of the 2D atomic crystal as a function of frequency. Parameters are set as η=0.01  eV, ωx=1  eV, and ωy=0.35  eV. The frequency of interband electron transitions is present at ωy=0.35  eV. (a), (b) The optical axis is chosen as ϕ=0°. (c), (d) The optical axis is chosen as ϕ=30°. The doping concentration of the 2D atomic crystal is n=5×1013  cm2.
Fig. 3.
Fig. 3. (a) Real and (b) imaginary parts of the conductivity of the 2D atomic crystal as a function of optical axis angles. The parameters of the 2D atomic crystal are n=5×1013  cm2 and ω=0.1  eV. (c) Real and (d) imaginary parts of the conductivity as a function of doping concentration. The optical axis is chosen as ϕ=30°. Other parameters are the same as in Fig. 2.
Fig. 4.
Fig. 4. (a) In-plane and (b) transverse spin-dependent shifts on the surface of anisotropic 2D atomic crystal as a function of optical axis angle and frequency. The incident light impinges on the substrate at θi=60°, the refractive index of the substrate is assumed as 2, and the doping concentration is n=10×1013  cm2.
Fig. 5.
Fig. 5. (a) In-plane and (b) transverse spin Hall shifts on the anisotropic 2D atomic crystal as a function of the optical axis angle and doping concentration. The frequency ω=0.1  eV. Other parameters are the same as in Fig. 4.

Equations (34)

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

σii=Ie2ω+Iηnmi+sie24[Θ(ωωi)+Iπln|ωωiω+ωi|],
M0=(σxx00σyy),
M=MlM0Mr.
Ml=(cosϕsin(ϕ)sinϕcosϕ),
Mr=(cosϕsinϕsin(ϕ)cosϕ),
σpp=σxxcos2ϕ+σyysin2ϕ,
σps=σsp=(σxxσyy)cosϕsinϕ,
σss=σyycos2ϕ+σxxsin2ϕ.
Eis+Ers=Ets,
cosθi(EipErp)=cosθrEtp,
cosθiZ0(EisErs)=(σss+cosθrZ)Ets+σspcosθrEtp,
1Z0(Eip+Erp)=(σppcosθr+1Z)EtpσpsEts.
rpp=α+TαL+βα+Tα+L+β,
rss=αTα+L+βα+Tα+L+β,
rps=rsp=γα+Tα+L+β.
|H(ki,r)=|P(ki,r)kiyki,rcotθi,r|S(ki,r),
|V(ki,r)=|S(ki,r)+kiyki,rcotθi,r|P(ki,r),
mR=(rpprpsrsprss).
|H(ki)[rppkrycotθ(rpsrsp)k0]|H(kr)+[rspkrycotθ(rpp+rss)k0]|V(kr),
|V(ki)[rps+krycotθ(rpp+rss)k0]|H(kr)+(rss2krycotθrpsk0)|V(kr),
|H=12(|++|),
|V=12i(||+),
|ϕ=w02πexp[w02(kix2+kiy2)4],
|ϕrHrpp+irps2exp(+ikrxδxH+ikryδyH)|+|ϕ+rppirsp2exp(ikrxδxHikryδyH)||ϕ,
|ϕrVrpsirss2exp(+ikrxδxV+ikryδyV)|+|ϕ+rps+irss2exp(ikrxδxVikryδyV)||ϕ.
δxH=rppk0(rpp2+rsp2)rpsθirspk0(rpp2+rsp2)rppθi,
δxV=rpsk0(rss2+rps2)rssθirssk0(rss2+rps2)rpsθi.
δyH=(rpp+rss)rppk0(rpp2+rsp2)cotθi(rpsrsp)rspk0(rpp2+rsp2)cotθi,
δyV=(rpp+rss)rssk0(rss2+rps2)cotθi+(rpsrsp)rpsk0(rss2+rps2)cotθi.
xr±H,V=ϕrH,V|krx|ϕrH,VϕrH,V|ϕrH,V.
xr±H=1k0Re[rpprpp2+rsp2rpsθirsprpp2+rsp2rppθi],
xr±V=1k0Re[rpsrss2+rps2rssθirssrss2+rps2rpsθi].
yr±H,V=ϕrH,V|kry|ϕrH,VϕrH,V|ϕrH,V.
yr±H=1k0Re[(rpp+rss)rpprpp2+rsp2cotθi(rpsrsp)rsprpp2+rsp2cotθi],yr±V=1k0Re[(rpp+rss)rssk0(rss2+rps2)cotθi+(rpsrsp)rpsk0(rss2+rps2)cotθi].

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