Abstract

The linear birefringence of uniaxial crystal plates has been known since the 17th century, and it is widely used in numerous optical setups and devices. Here we demonstrate, both theoretically and experimentally, the fine lateral circular birefringence of such crystal plates. We show that this effect is a novel example of the spin-Hall effect of light, i.e., a transverse spin-dependent shift of the paraxial light beam transmitted through the plate. The well-known linear birefringence and the new circular birefringence form an interesting analogy with the Goos–Hänchen and Imbert–Fedorov beam shifts that appear in the light reflection at a dielectric interface. We report experimental observation of the effect in a remarkably simple system of a tilted half-wave plate and polarizers using polarimetric and quantum-weak-measurement techniques for beam-shift measurements. In view of much recent interest in spin–orbit interaction phenomena, our results could find applications in modern polarization optics and nanophotonics.

© 2016 Optical Society of America

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  43. K. Y. Bliokh, A. Y. Bekshaev, A. G. Kofman, and F. Nori, “Photon trajectories, anomalous velocities and weak measurements: a classical interpretation,” New J. Phys. 15, 073022 (2013).
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    [Crossref]

2015 (3)

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]

F. Cardano and L. Marrucci, “Spin-orbit photonics,” Nat. Photonics 9, 776–778 (2015).
[Crossref]

K. Y. Bliokh and F. Nori, “Transverse and longitudinal angular momenta of light,” Phys. Rep. 592, 1–38 (2015).
[Crossref]

2014 (3)

2013 (4)

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

K. Y. Bliokh, A. Y. Bekshaev, A. G. Kofman, and F. Nori, “Photon trajectories, anomalous velocities and weak measurements: a classical interpretation,” New J. Phys. 15, 073022 (2013).
[Crossref]

V. Kumar and N. K. Viswanathan, “Topological structures in Poynting vector field: an experimental realization,” Opt. Lett. 38, 3886–3889 (2013).
[Crossref]

F. Töppel, M. Ornigotti, and A. Aiello, “Goos-Hänchen and Imbert-Fedorov shifts from a quantum-mechanical perspective,” New J. Phys. 15, 113059 (2013).
[Crossref]

2012 (5)

A. G. Kofman, S. Ashhab, and F. Nori, “Nonperturbative theory of weak pre- and post-selected measurements,” Phys. Rep. 520, 43–133 (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]

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]

M. R. Dennis and J. B. Götte, “The analogy between optical beam shifts and quantum weak measurements,” New J. Phys. 14, 073013 (2012).
[Crossref]

J. B. Götte and M. R. Dennis, “Generalized shifts and weak values for polarization components of reflected light beams,” New J. Phys. 14, 073016 (2012).
[Crossref]

2011 (2)

K. Y. Bliokh, E. A. Ostrovskaya, M. A. Alonso, O. G. Rodríguez-Herrera, D. Lara, and C. Dainty, “Spin-to-orbital angular momentum conversion in focusing, scattering, and imaging systems,” Opt. Express 19, 26132–26149 (2011).
[Crossref]

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).
[Crossref]

2010 (2)

K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82, 063825 (2010).
[Crossref]

O. G. Rodríguez-Herrera, D. Lara, K. Y. Bliokh, E. A. Ostrovskaya, and C. Dainty, “Optical nanoprobing via spin-orbit interaction of light,” Phys. Rev. Lett. 104, 253601 (2010).
[Crossref]

2009 (4)

D. Haefner, S. Sukhov, and A. Dogariu, “Spin Hall effect of light in spherical geometry,” Phys. Rev. Lett. 102, 123903 (2009).
[Crossref]

E. Brasselet, N. Murazawa, H. Misawa, and S. Juodkazis, “Optical vortices from liquid crystal droplets,” Phys. Rev. Lett. 103, 103903 (2009).
[Crossref]

E. Brasselet, Y. Izdebskaya, V. Shvedov, A. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Dynamics of optical spin-orbit coupling in uniaxial crystals,” Opt. Lett. 34, 1021–1023 (2009).
[Crossref]

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

2008 (6)

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

A. Aiello and J. P. Woerdman, “Role of beam propagation in Goos-Hänchen and Imbert-Fedorov shifts,” Opt. Lett. 33, 1437–1439 (2008).
[Crossref]

K. Y. Bliokh, Y. Gorodetski, V. Kleiner, and E. Hasman, “Coriolis effect in optics: unified geometric phase and spin-Hall effect,” Phys. Rev. Lett. 101, 030404 (2008).
[Crossref]

K. Y. Bliokh, A. Niv, V. Kleiner, and E. Hasman, “Geometrodynamics of spinning light,” Nat. Photonics 2, 748–753 (2008).
[Crossref]

T. A. Nieminen, A. B. Stilgoe, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Angular momentum of a strongly focused Gaussian beam,” J. Opt. A 10, 115005 (2008).
[Crossref]

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]

2007 (3)

Z. Bomzon and M. Gu, “Space-variant geometrical phases in focused cylindrical light beams,” Opt. Lett. 32, 3017–3019 (2007).
[Crossref]

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[Crossref]

K. Y. Bliokh and Y. P. Bliokh, “Polarization, transverse shifts, and angular momentum conservation laws in partial reflection and refraction of an electromagnetic wave packet,” Phys. Rev. E 75, 066609 (2007).

2006 (3)

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[Crossref]

C. Schwartz and A. Dogariu, “Conservation of angular momentum of light in single scattering,” Opt. Express 14, 8425–8433 (2006).
[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]

2003 (1)

A. Ciattoni, G. Cincotti, and C. Palma, “Angular momentum dynamics of a paraxial beam in a uniaxial crystal,” Phys. Rev. E 67, 036618 (2003).
[Crossref]

2002 (2)

1998 (1)

A. D. Parks, D. W. Cullin, and D. C. Stoudt, “Observation and measurement of an optical Aharonov-Albert-Vaidman effect,” Proc. R. Soc. London A 454, 2997–3008 (1998).
[Crossref]

1994 (2)

N. B. Baranova, A. Y. Savchenko, and B. Y. Zel’dovich, “Transverse shift of a focal spot due to switching of the sign of circular-polarization,” J. Exp. Thoer. Phys. Lett. 59, 232–234 (1994).

B. Y. Zel’dovich, N. D. Kundikova, and L. F. Rogacheva, “Observed transverse shift of a focal spot upon a change in the sign of circular polarization,” J. Exp. Theor. Phys. Lett. 59, 766–769 (1994).

1991 (1)

N. W. M. Ritchie, J. G. Story, and R. G. Hulet, “Realization of a measurement of a weak value”,” Phys. Rev. Lett. 66, 1107–1110 (1991).
[Crossref]

1989 (1)

I. M. Duck, P. M. Stevenson, and E. C. G. Sudarshan, “The sense in which a ‘weak measurement’ of a spin- particle’s spin component yields a value 100,” Phys. Rev. D 40, 2112–2117 (1989).
[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]

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]

Aiello, A.

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

F. Töppel, M. Ornigotti, and A. Aiello, “Goos-Hänchen and Imbert-Fedorov shifts from a quantum-mechanical perspective,” New J. Phys. 15, 113059 (2013).
[Crossref]

K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82, 063825 (2010).
[Crossref]

A. Aiello and J. P. Woerdman, “Role of beam propagation in Goos-Hänchen and Imbert-Fedorov shifts,” Opt. Lett. 33, 1437–1439 (2008).
[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]

Alonso, M. A.

K. Y. Bliokh, E. A. Ostrovskaya, M. A. Alonso, O. G. Rodríguez-Herrera, D. Lara, and C. Dainty, “Spin-to-orbital angular momentum conversion in focusing, scattering, and imaging systems,” Opt. Express 19, 26132–26149 (2011).
[Crossref]

K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82, 063825 (2010).
[Crossref]

Ashhab, S.

A. G. Kofman, S. Ashhab, and F. Nori, “Nonperturbative theory of weak pre- and post-selected measurements,” Phys. Rep. 520, 43–133 (2012).
[Crossref]

Baranova, N. B.

N. B. Baranova, A. Y. Savchenko, and B. Y. Zel’dovich, “Transverse shift of a focal spot due to switching of the sign of circular-polarization,” J. Exp. Thoer. Phys. Lett. 59, 232–234 (1994).

Bekshaev, A. Y.

K. Y. Bliokh, A. Y. Bekshaev, A. G. Kofman, and F. Nori, “Photon trajectories, anomalous velocities and weak measurements: a classical interpretation,” New J. Phys. 15, 073022 (2013).
[Crossref]

Biener, G.

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 F. Nori, “Transverse and longitudinal angular momenta of light,” Phys. Rep. 592, 1–38 (2015).
[Crossref]

K. Y. Bliokh, A. Y. Bekshaev, A. G. Kofman, and F. Nori, “Photon trajectories, anomalous velocities and weak measurements: a classical interpretation,” New J. Phys. 15, 073022 (2013).
[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]

K. Y. Bliokh, E. A. Ostrovskaya, M. A. Alonso, O. G. Rodríguez-Herrera, D. Lara, and C. Dainty, “Spin-to-orbital angular momentum conversion in focusing, scattering, and imaging systems,” Opt. Express 19, 26132–26149 (2011).
[Crossref]

K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82, 063825 (2010).
[Crossref]

O. G. Rodríguez-Herrera, D. Lara, K. Y. Bliokh, E. A. Ostrovskaya, and C. Dainty, “Optical nanoprobing via spin-orbit interaction of light,” Phys. Rev. Lett. 104, 253601 (2010).
[Crossref]

K. Y. Bliokh, Y. Gorodetski, V. Kleiner, and E. Hasman, “Coriolis effect in optics: unified geometric phase and spin-Hall effect,” Phys. Rev. Lett. 101, 030404 (2008).
[Crossref]

K. Y. Bliokh, A. Niv, V. Kleiner, and E. Hasman, “Geometrodynamics of spinning light,” Nat. Photonics 2, 748–753 (2008).
[Crossref]

K. Y. Bliokh and Y. P. Bliokh, “Polarization, transverse shifts, and angular momentum conservation laws in partial reflection and refraction of an electromagnetic wave packet,” Phys. Rev. E 75, 066609 (2007).

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, “Polarization, transverse shifts, and angular momentum conservation laws in partial reflection and refraction of an electromagnetic wave packet,” Phys. Rev. E 75, 066609 (2007).

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]

Bomzon, Z.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Boyd, R. W.

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

Brasselet, E.

E. Brasselet, Y. Izdebskaya, V. Shvedov, A. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Dynamics of optical spin-orbit coupling in uniaxial crystals,” Opt. Lett. 34, 1021–1023 (2009).
[Crossref]

E. Brasselet, N. Murazawa, H. Misawa, and S. Juodkazis, “Optical vortices from liquid crystal droplets,” Phys. Rev. Lett. 103, 103903 (2009).
[Crossref]

Braverman, B.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).
[Crossref]

Cardano, F.

F. Cardano and L. Marrucci, “Spin-orbit photonics,” Nat. Photonics 9, 776–778 (2015).
[Crossref]

Chiu, D. T.

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[Crossref]

Ciattoni, A.

A. Ciattoni, G. Cincotti, and C. Palma, “Angular momentum dynamics of a paraxial beam in a uniaxial crystal,” Phys. Rev. E 67, 036618 (2003).
[Crossref]

Cincotti, G.

A. Ciattoni, G. Cincotti, and C. Palma, “Angular momentum dynamics of a paraxial beam in a uniaxial crystal,” Phys. Rev. E 67, 036618 (2003).
[Crossref]

Cullin, D. W.

A. D. Parks, D. W. Cullin, and D. C. Stoudt, “Observation and measurement of an optical Aharonov-Albert-Vaidman effect,” Proc. R. Soc. London A 454, 2997–3008 (1998).
[Crossref]

Dainty, C.

K. Y. Bliokh, E. A. Ostrovskaya, M. A. Alonso, O. G. Rodríguez-Herrera, D. Lara, and C. Dainty, “Spin-to-orbital angular momentum conversion in focusing, scattering, and imaging systems,” Opt. Express 19, 26132–26149 (2011).
[Crossref]

O. G. Rodríguez-Herrera, D. Lara, K. Y. Bliokh, E. A. Ostrovskaya, and C. Dainty, “Optical nanoprobing via spin-orbit interaction of light,” Phys. Rev. Lett. 104, 253601 (2010).
[Crossref]

Dennis, M. R.

M. R. Dennis and J. B. Götte, “The analogy between optical beam shifts and quantum weak measurements,” New J. Phys. 14, 073013 (2012).
[Crossref]

J. B. Götte and M. R. Dennis, “Generalized shifts and weak values for polarization components of reflected light beams,” New J. Phys. 14, 073016 (2012).
[Crossref]

Desyatnikov, A.

Dogariu, A.

D. Haefner, S. Sukhov, and A. Dogariu, “Spin Hall effect of light in spherical geometry,” Phys. Rev. Lett. 102, 123903 (2009).
[Crossref]

C. Schwartz and A. Dogariu, “Conservation of angular momentum of light in single scattering,” Opt. Express 14, 8425–8433 (2006).
[Crossref]

Dressel, J.

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

Duck, I. M.

I. M. Duck, P. M. Stevenson, and E. C. G. Sudarshan, “The sense in which a ‘weak measurement’ of a spin- particle’s spin component yields a value 100,” Phys. Rev. D 40, 2112–2117 (1989).
[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]

Edgar, J. S.

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[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]

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]

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]

K. Y. Bliokh, Y. Gorodetski, V. Kleiner, and E. Hasman, “Coriolis effect in optics: unified geometric phase and spin-Hall effect,” Phys. Rev. Lett. 101, 030404 (2008).
[Crossref]

Götte, J. B.

M. R. Dennis and J. B. Götte, “The analogy between optical beam shifts and quantum weak measurements,” New J. Phys. 14, 073013 (2012).
[Crossref]

J. B. Götte and M. R. Dennis, “Generalized shifts and weak values for polarization components of reflected light beams,” New J. Phys. 14, 073016 (2012).
[Crossref]

Gu, M.

Haefner, D.

D. Haefner, S. Sukhov, and A. Dogariu, “Spin Hall effect of light in spherical geometry,” Phys. Rev. Lett. 102, 123903 (2009).
[Crossref]

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]

K. Y. Bliokh, A. Niv, V. Kleiner, and E. Hasman, “Geometrodynamics of spinning light,” Nat. Photonics 2, 748–753 (2008).
[Crossref]

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]

K. Y. Bliokh, Y. Gorodetski, V. Kleiner, and E. Hasman, “Coriolis effect in optics: unified geometric phase and spin-Hall effect,” Phys. Rev. Lett. 101, 030404 (2008).
[Crossref]

Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Space-variant Pancharatnam-Berry phase optical elements with computer-generated subwavelength gratings,” Opt. Lett. 27, 1141–1143 (2002).
[Crossref]

G. Biener, A. Niv, V. Kleiner, and E. Hasman, “Formation of helical beams by use of Pancharatnam-Berry phase optical elements,” Opt. Lett. 27, 1875–1877 (2002).
[Crossref]

He, H.

Heckenberg, N. R.

T. A. Nieminen, A. B. Stilgoe, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Angular momentum of a strongly focused Gaussian beam,” J. Opt. A 10, 115005 (2008).
[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]

Hulet, R. G.

N. W. M. Ritchie, J. G. Story, and R. G. Hulet, “Realization of a measurement of a weak value”,” Phys. Rev. Lett. 66, 1107–1110 (1991).
[Crossref]

Izdebskaya, Y.

Jayaswal, G.

Jeffries, G. D. M.

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[Crossref]

Jordan, A. N.

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

Juodkazis, S.

E. Brasselet, N. Murazawa, H. Misawa, and S. Juodkazis, “Optical vortices from liquid crystal droplets,” Phys. Rev. Lett. 103, 103903 (2009).
[Crossref]

Kivshar, Y. S.

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]

K. Y. Bliokh, Y. Gorodetski, V. Kleiner, and E. Hasman, “Coriolis effect in optics: unified geometric phase and spin-Hall effect,” Phys. Rev. Lett. 101, 030404 (2008).
[Crossref]

K. Y. Bliokh, A. Niv, V. Kleiner, and E. Hasman, “Geometrodynamics of spinning light,” Nat. Photonics 2, 748–753 (2008).
[Crossref]

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]

G. Biener, A. Niv, V. Kleiner, and E. Hasman, “Formation of helical beams by use of Pancharatnam-Berry phase optical elements,” Opt. Lett. 27, 1875–1877 (2002).
[Crossref]

Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Space-variant Pancharatnam-Berry phase optical elements with computer-generated subwavelength gratings,” Opt. Lett. 27, 1141–1143 (2002).
[Crossref]

Kocsis, S.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).
[Crossref]

Kofman, A. G.

K. Y. Bliokh, A. Y. Bekshaev, A. G. Kofman, and F. Nori, “Photon trajectories, anomalous velocities and weak measurements: a classical interpretation,” New J. Phys. 15, 073022 (2013).
[Crossref]

A. G. Kofman, S. Ashhab, and F. Nori, “Nonperturbative theory of weak pre- and post-selected measurements,” Phys. Rep. 520, 43–133 (2012).
[Crossref]

Krolikowski, W.

Kumar, V.

Kundikova, N. D.

B. Y. Zel’dovich, N. D. Kundikova, and L. F. Rogacheva, “Observed transverse shift of a focal spot upon a change in the sign of circular polarization,” J. Exp. Theor. Phys. Lett. 59, 766–769 (1994).

Kwiat, P.

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

Lara, D.

K. Y. Bliokh, E. A. Ostrovskaya, M. A. Alonso, O. G. Rodríguez-Herrera, D. Lara, and C. Dainty, “Spin-to-orbital angular momentum conversion in focusing, scattering, and imaging systems,” Opt. Express 19, 26132–26149 (2011).
[Crossref]

O. G. Rodríguez-Herrera, D. Lara, K. Y. Bliokh, E. A. Ostrovskaya, and C. Dainty, “Optical nanoprobing via spin-orbit interaction of light,” Phys. Rev. Lett. 104, 253601 (2010).
[Crossref]

Li, Y.

Luo, H.

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]

Malik, M.

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

Manzo, C.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[Crossref]

Marrucci, L.

F. Cardano and L. Marrucci, “Spin-orbit photonics,” Nat. Photonics 9, 776–778 (2015).
[Crossref]

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[Crossref]

McGloin, D.

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[Crossref]

Merano, M.

Miatto, F. M.

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

Mirin, R. P.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).
[Crossref]

Misawa, H.

E. Brasselet, N. Murazawa, H. Misawa, and S. Juodkazis, “Optical vortices from liquid crystal droplets,” Phys. Rev. Lett. 103, 103903 (2009).
[Crossref]

Mistura, G.

Murazawa, N.

E. Brasselet, N. Murazawa, H. Misawa, and S. Juodkazis, “Optical vortices from liquid crystal droplets,” Phys. Rev. Lett. 103, 103903 (2009).
[Crossref]

Nieminen, T. A.

T. A. Nieminen, A. B. Stilgoe, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Angular momentum of a strongly focused Gaussian beam,” J. Opt. A 10, 115005 (2008).
[Crossref]

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]

K. Y. Bliokh, A. Niv, V. Kleiner, and E. Hasman, “Geometrodynamics of spinning light,” Nat. Photonics 2, 748–753 (2008).
[Crossref]

G. Biener, A. Niv, V. Kleiner, and E. Hasman, “Formation of helical beams by use of Pancharatnam-Berry phase optical elements,” Opt. Lett. 27, 1875–1877 (2002).
[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]

K. Y. Bliokh and F. Nori, “Transverse and longitudinal angular momenta of light,” Phys. Rep. 592, 1–38 (2015).
[Crossref]

K. Y. Bliokh, A. Y. Bekshaev, A. G. Kofman, and F. Nori, “Photon trajectories, anomalous velocities and weak measurements: a classical interpretation,” New J. Phys. 15, 073022 (2013).
[Crossref]

A. G. Kofman, S. Ashhab, and F. Nori, “Nonperturbative theory of weak pre- and post-selected measurements,” Phys. Rep. 520, 43–133 (2012).
[Crossref]

Ornigotti, M.

F. Töppel, M. Ornigotti, and A. Aiello, “Goos-Hänchen and Imbert-Fedorov shifts from a quantum-mechanical perspective,” New J. Phys. 15, 113059 (2013).
[Crossref]

Ostrovskaya, E. A.

K. Y. Bliokh, E. A. Ostrovskaya, M. A. Alonso, O. G. Rodríguez-Herrera, D. Lara, and C. Dainty, “Spin-to-orbital angular momentum conversion in focusing, scattering, and imaging systems,” Opt. Express 19, 26132–26149 (2011).
[Crossref]

K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82, 063825 (2010).
[Crossref]

O. G. Rodríguez-Herrera, D. Lara, K. Y. Bliokh, E. A. Ostrovskaya, and C. Dainty, “Optical nanoprobing via spin-orbit interaction of light,” Phys. Rev. Lett. 104, 253601 (2010).
[Crossref]

Palma, C.

A. Ciattoni, G. Cincotti, and C. Palma, “Angular momentum dynamics of a paraxial beam in a uniaxial crystal,” Phys. Rev. E 67, 036618 (2003).
[Crossref]

Paparo, D.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[Crossref]

Parks, A. D.

A. D. Parks, D. W. Cullin, and D. C. Stoudt, “Observation and measurement of an optical Aharonov-Albert-Vaidman effect,” Proc. R. Soc. London A 454, 2997–3008 (1998).
[Crossref]

Pidishety, S.

Prajapati, C.

Qin, Y.

Ravets, S.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).
[Crossref]

Ritchie, N. W. M.

N. W. M. Ritchie, J. G. Story, and R. G. Hulet, “Realization of a measurement of a weak value”,” Phys. Rev. Lett. 66, 1107–1110 (1991).
[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]

Rodríguez-Herrera, O. G.

K. Y. Bliokh, E. A. Ostrovskaya, M. A. Alonso, O. G. Rodríguez-Herrera, D. Lara, and C. Dainty, “Spin-to-orbital angular momentum conversion in focusing, scattering, and imaging systems,” Opt. Express 19, 26132–26149 (2011).
[Crossref]

O. G. Rodríguez-Herrera, D. Lara, K. Y. Bliokh, E. A. Ostrovskaya, and C. Dainty, “Optical nanoprobing via spin-orbit interaction of light,” Phys. Rev. Lett. 104, 253601 (2010).
[Crossref]

Rogacheva, L. F.

B. Y. Zel’dovich, N. D. Kundikova, and L. F. Rogacheva, “Observed transverse shift of a focal spot upon a change in the sign of circular polarization,” J. Exp. Theor. Phys. Lett. 59, 766–769 (1994).

Rubinsztein-Dunlop, H.

T. A. Nieminen, A. B. Stilgoe, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Angular momentum of a strongly focused Gaussian beam,” J. Opt. A 10, 115005 (2008).
[Crossref]

Savchenko, A. Y.

N. B. Baranova, A. Y. Savchenko, and B. Y. Zel’dovich, “Transverse shift of a focal spot due to switching of the sign of circular-polarization,” J. Exp. Thoer. Phys. Lett. 59, 232–234 (1994).

Schwartz, C.

Shalm, L. K.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).
[Crossref]

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]

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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]

Steinberg, A. M.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).
[Crossref]

Stevens, M. J.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).
[Crossref]

Stevenson, P. M.

I. M. Duck, P. M. Stevenson, and E. C. G. Sudarshan, “The sense in which a ‘weak measurement’ of a spin- particle’s spin component yields a value 100,” Phys. Rev. D 40, 2112–2117 (1989).
[Crossref]

Stilgoe, A. B.

T. A. Nieminen, A. B. Stilgoe, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Angular momentum of a strongly focused Gaussian beam,” J. Opt. A 10, 115005 (2008).
[Crossref]

Story, J. G.

N. W. M. Ritchie, J. G. Story, and R. G. Hulet, “Realization of a measurement of a weak value”,” Phys. Rev. Lett. 66, 1107–1110 (1991).
[Crossref]

Stoudt, D. C.

A. D. Parks, D. W. Cullin, and D. C. Stoudt, “Observation and measurement of an optical Aharonov-Albert-Vaidman effect,” Proc. R. Soc. London A 454, 2997–3008 (1998).
[Crossref]

Sudarshan, E. C. G.

I. M. Duck, P. M. Stevenson, and E. C. G. Sudarshan, “The sense in which a ‘weak measurement’ of a spin- particle’s spin component yields a value 100,” Phys. Rev. D 40, 2112–2117 (1989).
[Crossref]

Sukhov, S.

D. Haefner, S. Sukhov, and A. Dogariu, “Spin Hall effect of light in spherical geometry,” Phys. Rev. Lett. 102, 123903 (2009).
[Crossref]

Töppel, F.

F. Töppel, M. Ornigotti, and A. Aiello, “Goos-Hänchen and Imbert-Fedorov shifts from a quantum-mechanical perspective,” New J. Phys. 15, 113059 (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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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).
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Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

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]

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]

Zel’dovich, B. Y.

N. B. Baranova, A. Y. Savchenko, and B. Y. Zel’dovich, “Transverse shift of a focal spot due to switching of the sign of circular-polarization,” J. Exp. Thoer. Phys. Lett. 59, 232–234 (1994).

B. Y. Zel’dovich, N. D. Kundikova, and L. F. Rogacheva, “Observed transverse shift of a focal spot upon a change in the sign of circular polarization,” J. Exp. Theor. Phys. Lett. 59, 766–769 (1994).

Zhao, Y.

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[Crossref]

Zhou, X.

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]

J. Exp. Theor. Phys. Lett. (1)

B. Y. Zel’dovich, N. D. Kundikova, and L. F. Rogacheva, “Observed transverse shift of a focal spot upon a change in the sign of circular polarization,” J. Exp. Theor. Phys. Lett. 59, 766–769 (1994).

J. Exp. Thoer. Phys. Lett. (1)

N. B. Baranova, A. Y. Savchenko, and B. Y. Zel’dovich, “Transverse shift of a focal spot due to switching of the sign of circular-polarization,” J. Exp. Thoer. Phys. Lett. 59, 232–234 (1994).

J. Opt. (1)

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

J. Opt. A (1)

T. A. Nieminen, A. B. Stilgoe, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Angular momentum of a strongly focused Gaussian beam,” J. Opt. A 10, 115005 (2008).
[Crossref]

Nat. Photonics (3)

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]

F. Cardano and L. Marrucci, “Spin-orbit photonics,” Nat. Photonics 9, 776–778 (2015).
[Crossref]

K. Y. Bliokh, A. Niv, V. Kleiner, and E. Hasman, “Geometrodynamics of spinning light,” Nat. Photonics 2, 748–753 (2008).
[Crossref]

New J. Phys. (4)

M. R. Dennis and J. B. Götte, “The analogy between optical beam shifts and quantum weak measurements,” New J. Phys. 14, 073013 (2012).
[Crossref]

J. B. Götte and M. R. Dennis, “Generalized shifts and weak values for polarization components of reflected light beams,” New J. Phys. 14, 073016 (2012).
[Crossref]

F. Töppel, M. Ornigotti, and A. Aiello, “Goos-Hänchen and Imbert-Fedorov shifts from a quantum-mechanical perspective,” New J. Phys. 15, 113059 (2013).
[Crossref]

K. Y. Bliokh, A. Y. Bekshaev, A. G. Kofman, and F. Nori, “Photon trajectories, anomalous velocities and weak measurements: a classical interpretation,” New J. Phys. 15, 073022 (2013).
[Crossref]

Opt. Express (2)

Opt. Lett. (9)

V. Kumar and N. K. Viswanathan, “Topological structures in Poynting vector field: an experimental realization,” Opt. Lett. 38, 3886–3889 (2013).
[Crossref]

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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Figures (8)

Fig. 1.
Fig. 1. Schematics of the transmission of a paraxial beam through a tilted uniaxial-crystal plate. (a) General 3D geometry of the problem with the angle ϑ between the anisotropy axis of the plate and the beam axis z . The small angles Θ = ( Θ x , Θ y ) determine the directions of the wave vectors k in the incident beam. (b) The in-plane Θ x deflections of the wave vectors change the angle between k and the anisotropy axis and result in the well-known birefringence shift X , analogous to the GH shift [24]. (c) The view along the anisotropy axis of the crystal is shown. The transverse Θ y deflections of the wave vectors rotate the corresponding planes of the wave propagation with respect to the anisotropy axis by the angle φ Θ y / sin ϑ . This causes a new helicity-dependent transverse shift Y , i.e., a circular birefringence or spin-Hall effect similar to the IF shift [2024].
Fig. 2.
Fig. 2. Distribution of the polarization in the (a) extraordinary and (b) ordinary Gaussian beams transmitted through a uniaxial crystal plate. Right-hand- and left-hand- polarization ellipses are shown in magenta and cyan, respectively. The background gray-scale distributions show the Gaussian intensities of the beams, x -shifted due to ordinary linear birefringence. Transverse y -dependent separations of opposite helicities and tilts of the polarization ellipses indicate the σ - and χ -dependent transverse birefringence, described by the Y e , o and Y terms in Eqs. (5)–(13). The parameters used here are k w 0 = 10 , ϑ = π / 4 , Φ 0 = 2 π / 3 , and d Φ 0 / d ϑ = 5 .
Fig. 3.
Fig. 3. Longitudinal (in-plane) and transverse (out-of-plane) shifts of the beam transmitted through a tilted uniaxial crystal plate [Eqs. (11) and (12)]. These plots correspond to a 1 mm thick quartz plate and wavelength λ = 2 π / k = 632.8    nm . The polarizations are (a)  τ = 1 (extraordinary wave) and (b)  σ = 1 (left-hand circular). For the opposite polarizations, τ = 1 and σ = 1 , the beam shifts have opposite sign, which signifies the usual in-plane linear birefringence between the ordinary and extraordinary waves, as well as the transverse circular birefringence, i.e., the spin-Hall effect of light.
Fig. 4.
Fig. 4. Schematics of the experimental setups used for the (a) polarimetric measurements and (b) quantum weak measurements of the beam shifts.
Fig. 5.
Fig. 5. Polarimetrically measured phase difference Φ 0 ( ϑ ) [Eq. (20)] between the ordinary and extraordinary modes for the beam transmitted through the tilted half-wave plate. The inset shows the actual measured phase in the range ( π , π ) , whereas the main plot shows the unwrapped phase.
Fig. 6.
Fig. 6. Experimentally measured distributions of the local Stokes parameter s 3 ( R ) [Eq. (21)] in the (a) extraordinary and (b) ordinary beams transmitted through the tilted half-wave plate with ϑ 35 ° and Φ 0 ( ϑ ) π (see Fig. 5). The y splitting of opposite spin states with s 3 > 0 and s 3 < 0 corresponds to the splitting of opposite polarization ellipticities in Fig. 2 and signals the spin-Hall effect of light.
Fig. 7.
Fig. 7. Transverse intensity distributions I ( R ) in the o -polarized beam transmitted through the tilted half-wave plate and post-selected in the almost e -polarized state with ϵ = 1.4 × 10 2 , 0 , 1.4 × 10 2 (see explanations in the text). The two-hump Hermite–Gaussian distribution at ϵ = 0 corresponds to Eq. (8), while the opposite shifts Y z weak at ϵ = ± 1.4 · 10 2 are the spin-Hall shifts amplified via quantum weak measurements [Eq. (16)]. Like in Fig. 6, the tilt angle ϑ 35 ° corresponds to Φ 0 ( ϑ ) π , i.e., maximizes the spin-Hall effect ( 1 cos Φ 0 ) .
Fig. 8.
Fig. 8. Transverse beam shifts Y z weak determined via quantum weak measurements (Fig. 7) and theoretically calculated using Eq. (16) with the phase Φ 0 ( ϑ ) taken from polarimetric measurements (Fig. 5). The measurements are done for the e -polarized (a) and o -polarized (b) input beams, and the post-selection with ϵ = 1.4 · 10 2 (see explanations in the text).

Tables (1)

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Table 1. Basic Spin–Orbit Interaction Effects, Spin-to-Orbital Angular Momentum Conversion, and Spin-Hall Effect, in Inhomogeneous Isotropic and Anisotropic Homogeneous Systemsa

Equations (22)

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k = k z z ¯ + k x x ¯ + k y y ¯ k ( 1 Θ 2 2 ) z ¯ + k Θ x x ¯ + k Θ y y ¯ ,
E ˜ ( Θ ) ( α β ) exp [ ( k w 0 ) 2 Θ 2 4 ] ,
E ( R ) E ˜ ( Θ ) e i k · r d 2 Θ ( α β ) exp [ R 2 w 0 2 ] .
E ˜ ( 0 ) = M ^ 0 E ˜ ( 0 ) , M ^ 0 = ( e i Φ 0 / 2 0 0 e i Φ 0 / 2 ) ,
E ˜ ( Θ ) = M ˜ ^ ( Θ ) E ˜ ( Θ ) , M ˜ ^ ( e i Φ 0 / 2 ( 1 + Θ x X e ) e i Φ 0 / 2 Θ y Y e e i Φ 0 / 2 Θ y Y o e i Φ 0 / 2 ( 1 + Θ x X o ) ) ,
X e , o = i 2 d Φ 0 d ϑ , Y e , o = [ 1 exp ( ± i Φ 0 ) ] cot ϑ .
E ( R ) = M ^ ( R ) E ( R ) , M ^ ( e i Φ 0 / 2 ( 1 + i x z R X e ) i e i Φ 0 / 2 y z R Y e i e i Φ 0 / 2 y z R Y o e i Φ 0 / 2 ( 1 + i x z R X o ) ) .
E ( R ) ( 1 + i x z R X e i e i Φ 0 y z R Y o ) exp [ R 2 w 0 2 ] .
E y ( R ) y z R Y o exp [ R 2 w 0 2 ] .
X ^ = i k 1 ( X e 0 0 X o ) , Y ^ = i k 1 ( 0 e i Φ 0 Y e e i Φ 0 Y o 0 ) .
X = ψ | X ^ | ψ = τ 1 2 k d Φ 0 d ϑ ,
Y = ψ | Y ^ | ψ = cot ϑ k [ σ ( 1 cos Φ 0 ) + χ sin Φ 0 ] .
τ = | α | 2 | β | 2 ,       χ = 2 Re ( α * β ) ,       σ = 2 Im ( α * β ) .
R weak = Re ϕ | R ^ | ψ ϕ | ψ ,       Θ weak = 1 z R Im ϕ | R ^ | ψ ϕ | ψ .
R z weak = R weak + z Θ weak .
Y z weak = 1 ϵ k sin Φ 0 cot ϑ + z z R 1 ϵ k ( 1 cos Φ 0 ) cot ϑ .
Φ 0 ( ϑ ) = k [ n o d o ( ϑ ) n ˜ e ( ϑ ) d e ( ϑ ) ] .
d e ( ϑ ) = n ˜ e ( ϑ ) d n ˜ e 2 ( ϑ ) cos 2 ϑ ,       d o ( ϑ ) = n o d n o 2 cos 2 ϑ .
S ¯ 0 = I ¯ ( 0 ° , 0 ° ) + I ¯ ( 0 ° , 90 ° ) , S ¯ 1 = I ¯ ( 0 ° , 0 ° ) I ¯ ( 0 ° , 90 ° ) , S ¯ 2 = I ¯ ( 0 ° , 45 ° ) I ¯ ( 0 ° , 135 ° ) , S ¯ 3 = I ¯ ( 90 ° , 45 ° ) I ¯ ( 90 ° , 135 ° ) .
Φ 0 = tan 1 ( S ¯ 3 S ¯ 2 ) .
s 3 ( R ) = S 3 ( R ) S 0 ( R ) ,
A = 1 | ϵ | z z R 929 .

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