Abstract

We present a general theory of spin-to-orbital angular momentum (AM) conversion of light in focusing, scattering, and imaging optical systems. Our theory employs universal geometric transformations of non-paraxial optical fields in such systems and allows for direct calculation and comparison of the AM conversion efficiency in different physical settings. Observations of the AM conversions using local intensity distributions and far-field polarimetric measurements are discussed.

© 2011 OSA

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  40. P. Török, P. D. Higdon, and T. Wilson, “On the general properties of polarized light conventional and confocal microscopes,” Opt. Commun. 148(4-6), 300–315 (1998).
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  41. A. Bekshaev, K. Y. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13(5), 053001 (2011).
    [CrossRef]
  42. R. Bhandari, “Polarization of light and topological phases,” Phys. Rep. 281(1), 2–64 (1997).
    [CrossRef]
  43. M. A. Alonso and G. W. Forbes, “Uncertainty products for nonparaxial wave fields,” J. Opt. Soc. Am. A 17(12), 2391–2402 (2000).
    [CrossRef] [PubMed]
  44. M. A. Alonso, “The effect of orbital angular momentum and helicity in the uncertainty-type relations between focal spot size and angular spread,” J. Opt. 13(6), 064016 (2011).
    [CrossRef]
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  47. 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(3), 030404 (2008).
    [CrossRef] [PubMed]
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    [CrossRef]
  50. V. Rossetto and A. C. Maggs, “Writhing geometry of stiff polymers and scattered light,” Eur. Phys. J. B 29(2), 323–326 (2002).
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    [CrossRef] [PubMed]
  55. Y. A. Kravtsov, B. Bieg, and K. Y. Bliokh, “Stokes-vector evolution in a weakly anisotropic inhomogeneous medium,” J. Opt. Soc. Am. A 24(10), 3388–3396 (2007).
    [CrossRef] [PubMed]
  56. M. V. Berry, “Paraxial beams of spinning light,” Proc. SPIE 3487, 6–11 (1998).
    [CrossRef]

2011

M. R. Foreman and P. Török, “Spin-orbit coupling and conservation of angular momentum flux in non-paraxial imaging of forbidden radiation,” New J. Phys. 13(6), 063041 (2011).
[CrossRef]

L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, “Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications,” J. Opt. 13(6), 064001 (2011).
[CrossRef]

F. Manni, K. Lagoudakis, T. Paraïso, R. Cerna, Y. Léger, T. Liew, I. Shelykh, A. Kavokin, F. Morier-Genoud, and B. Deveaud-Plédran, “Spin-to-orbital angular momentum conversion in semiconductor microcavities,” Phys. Rev. B 83(24), 241307 (2011).
[CrossRef]

A. Bekshaev, K. Y. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13(5), 053001 (2011).
[CrossRef]

M. A. Alonso, “The effect of orbital angular momentum and helicity in the uncertainty-type relations between focal spot size and angular spread,” J. Opt. 13(6), 064016 (2011).
[CrossRef]

2010

O. G. Rodríguez-Herrera, D. Lara, and C. Dainty, “Far-field polarization-based sensitivity to sub-resolution displacements of a sub-resolution scatterer in tightly focused fields,” Opt. Express 18(6), 5609–5628 (2010).
[CrossRef] [PubMed]

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(6), 063825 (2010).
[CrossRef]

L. T. Vuong, A. J. L. Adam, J. M. Brok, P. C. M. Planken, and H. P. Urbach, “Electromagnetic spin-orbit interactions via scattering of subwavelength apertures,” Phys. Rev. Lett. 104(8), 083903 (2010).
[CrossRef] [PubMed]

Y. Gorodetski, S. Nechayev, V. Kleiner, and E. Hasman, “Plasmonic Aharonov-Bohm effect: Optical spin as the magnetic flux parameter,” Phys. Rev. B 82(12), 125433 (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(25), 253601 (2010).
[CrossRef] [PubMed]

2009

P. B. Monteiro, P. A. M. Neto, and H. M. Nussenzveig, “Angular momentum of focused beams: Beyond the paraxial approximation,” Phys. Rev. A 79(3), 033830 (2009).
[CrossRef]

D. Haefner, S. Sukhov, and A. Dogariu, “Spin hall effect of light in spherical geometry,” Phys. Rev. Lett. 102(12), 123903 (2009).
[CrossRef] [PubMed]

Y. Gorodetski, N. Shitrit, I. Bretner, V. Kleiner, and E. Hasman, “Observation of optical spin symmetry breaking in nanoapertures,” Nano Lett. 9(8), 3016–3019 (2009).
[CrossRef] [PubMed]

V. Garbin, G. Volpe, E. Ferrari, M. Versluis, D. Cojoc, and D. Petrov, “Mie scattering distinguishes the topological charge of an optical vortex: a homage to Gustav Mie,” New J. Phys. 11(1), 013046 (2009).
[CrossRef]

E. Brasselet, N. Murazawa, H. Misawa, and S. Juodkazis, “Optical vortices from liquid crystal droplets,” Phys. Rev. Lett. 103(10), 103903 (2009).
[CrossRef] [PubMed]

C.-F. Li, “Spin and orbital angular momentum of a class of nonparaxial light beams having a globally defined polarization,” Phys. Rev. A 80(6), 063814 (2009).
[CrossRef]

E. Brasselet, Y. Izdebskaya, V. Shvedov, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Dynamics of optical spin-orbit coupling in uniaxial crystals,” Opt. Lett. 34(7), 1021–1023 (2009).
[CrossRef] [PubMed]

Y. Zhao, D. Shapiro, D. McGloin, D. T. Chiu, and S. Marchesini, “Direct observation of the transfer of orbital angular momentum to metal particles from a focused circularly polarized Gaussian beam,” Opt. Express 17(25), 23316–23322 (2009).
[CrossRef] [PubMed]

M. V. Berry, “Optical currents,” J. Opt. A, Pure Appl. Opt. 11(9), 094001 (2009).
[CrossRef]

2008

C. Schwartz and A. Dogariu, “Backscattered polarization patterns determined by conservation of angular momentum,” J. Opt. Soc. Am. A 25(2), 431–436 (2008).
[CrossRef] [PubMed]

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(3), 030404 (2008).
[CrossRef] [PubMed]

T. A. Nieminen, A. B. Stilgoe, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Angular momentum of a strongly focused Gaussian beam,” J. Opt. A, Pure Appl. Opt. 10(11), 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(4), 043903 (2008).
[CrossRef] [PubMed]

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser & Photon. Rev. 2(4), 299–313 (2008).
[CrossRef]

2007

2006

C. Schwartz and A. Dogariu, “Backscattered polarization patterns, optical vortices, and the angular momentum of light,” Opt. Lett. 31(8), 1121–1123 (2006).
[CrossRef] [PubMed]

A. Dogariu and C. Schwartz, “Conservation of angular momentum of light in single scattering,” Opt. Express 14(18), 8425–8433 (2006).
[CrossRef] [PubMed]

Z. Bomzon, M. Gu, and J. Shamir, “Angular momentum and geometric phases in tightly-focused circularly polarized plane waves,” Appl. Phys. Lett. 89(24), 241104 (2006).
[CrossRef]

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[CrossRef] [PubMed]

2005

2004

2003

A. Ciattoni, G. Cincotti, and C. Palma, “Angular momentum dynamics of a paraxial beam in a uniaxial crystal,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(3 Pt 2), 036618 (2003).
[CrossRef] [PubMed]

2002

V. Rossetto and A. C. Maggs, “Writhing geometry of stiff polymers and scattered light,” Eur. Phys. J. B 29(2), 323–326 (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(21), 1875–1877 (2002).
[CrossRef] [PubMed]

2000

1998

M. V. Berry, “Paraxial beams of spinning light,” Proc. SPIE 3487, 6–11 (1998).
[CrossRef]

P. Török, P. D. Higdon, and T. Wilson, “On the general properties of polarized light conventional and confocal microscopes,” Opt. Commun. 148(4-6), 300–315 (1998).
[CrossRef]

1997

R. Bhandari, “Polarization of light and topological phases,” Phys. Rep. 281(1), 2–64 (1997).
[CrossRef]

1994

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,” JETP 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,” JETP Lett. 59, 766–769 (1994).

S. J. van Enk and G. Nienhuis, “Spin and orbital angular momentum of photons,” Europhys. Lett. 25(7), 497–501 (1994).
[CrossRef]

S. J. van Enk and G. Nienhuis, “Commutation rules and eigenvalues of spin and orbital angular momentum of radiation fields,” J. Mod. Opt. 41(5), 963–977 (1994).
[CrossRef]

S. M. Barnett and L. Allen, “Orbital angular-momentum and nonparaxial light-beams,” Opt. Commun. 110(5-6), 670–678 (1994).
[CrossRef]

1987

M. V. Berry, “Interpreting the anholonomy of coiled light,” Nature 326(6110), 277–278 (1987).
[CrossRef]

1977

G. Moe and W. Happer, “Conservation of angular momentum for light propagating in a transparent anisotropic medium,” J. Phys. B 10(7), 1191–1208 (1977).
[CrossRef]

1959

E. Wolf, ““Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. Roy. Soc. London,” Ser. A 253, 349–357 (1959).

B. Richards and E. Wolf, ““Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[CrossRef]

Adam, A. J. L.

L. T. Vuong, A. J. L. Adam, J. M. Brok, P. C. M. Planken, and H. P. Urbach, “Electromagnetic spin-orbit interactions via scattering of subwavelength apertures,” Phys. Rev. Lett. 104(8), 083903 (2010).
[CrossRef] [PubMed]

Aiello, A.

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(6), 063825 (2010).
[CrossRef]

Allen, L.

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser & Photon. Rev. 2(4), 299–313 (2008).
[CrossRef]

S. M. Barnett and L. Allen, “Orbital angular-momentum and nonparaxial light-beams,” Opt. Commun. 110(5-6), 670–678 (1994).
[CrossRef]

Alonso, M. A.

M. A. Alonso, “The effect of orbital angular momentum and helicity in the uncertainty-type relations between focal spot size and angular spread,” J. Opt. 13(6), 064016 (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(6), 063825 (2010).
[CrossRef]

M. A. Alonso and G. W. Forbes, “Uncertainty products for nonparaxial wave fields,” J. Opt. Soc. Am. A 17(12), 2391–2402 (2000).
[CrossRef] [PubMed]

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,” JETP Lett. 59, 232–234 (1994).

Barnett, S. M.

S. M. Barnett and L. Allen, “Orbital angular-momentum and nonparaxial light-beams,” Opt. Commun. 110(5-6), 670–678 (1994).
[CrossRef]

Bekshaev, A.

A. Bekshaev, K. Y. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13(5), 053001 (2011).
[CrossRef]

Berry, M. V.

M. V. Berry, “Optical currents,” J. Opt. A, Pure Appl. Opt. 11(9), 094001 (2009).
[CrossRef]

M. V. Berry, “Paraxial beams of spinning light,” Proc. SPIE 3487, 6–11 (1998).
[CrossRef]

M. V. Berry, “Interpreting the anholonomy of coiled light,” Nature 326(6110), 277–278 (1987).
[CrossRef]

Bhandari, R.

R. Bhandari, “Polarization of light and topological phases,” Phys. Rep. 281(1), 2–64 (1997).
[CrossRef]

Bieg, B.

Biener, G.

Bliokh, K. Y.

A. Bekshaev, K. Y. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13(5), 053001 (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(25), 253601 (2010).
[CrossRef] [PubMed]

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(6), 063825 (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(3), 030404 (2008).
[CrossRef] [PubMed]

Y. A. Kravtsov, B. Bieg, and K. Y. Bliokh, “Stokes-vector evolution in a weakly anisotropic inhomogeneous medium,” J. Opt. Soc. Am. A 24(10), 3388–3396 (2007).
[CrossRef] [PubMed]

Bokor, N.

Bomzon, Z.

Z. Bomzon and M. Gu, “Space-variant geometrical phases in focused cylindrical light beams,” Opt. Lett. 32(20), 3017–3019 (2007).
[CrossRef] [PubMed]

Z. Bomzon, M. Gu, and J. Shamir, “Angular momentum and geometric phases in tightly-focused circularly polarized plane waves,” Appl. Phys. Lett. 89(24), 241104 (2006).
[CrossRef]

Brasselet, E.

Bretner, I.

Y. Gorodetski, N. Shitrit, I. Bretner, V. Kleiner, and E. Hasman, “Observation of optical spin symmetry breaking in nanoapertures,” Nano Lett. 9(8), 3016–3019 (2009).
[CrossRef] [PubMed]

Brok, J. M.

L. T. Vuong, A. J. L. Adam, J. M. Brok, P. C. M. Planken, and H. P. Urbach, “Electromagnetic spin-orbit interactions via scattering of subwavelength apertures,” Phys. Rev. Lett. 104(8), 083903 (2010).
[CrossRef] [PubMed]

Calvo, G. F.

Cerna, R.

F. Manni, K. Lagoudakis, T. Paraïso, R. Cerna, Y. Léger, T. Liew, I. Shelykh, A. Kavokin, F. Morier-Genoud, and B. Deveaud-Plédran, “Spin-to-orbital angular momentum conversion in semiconductor microcavities,” Phys. Rev. B 83(24), 241307 (2011).
[CrossRef]

Chiu, D. T.

Y. Zhao, D. Shapiro, D. McGloin, D. T. Chiu, and S. Marchesini, “Direct observation of the transfer of orbital angular momentum to metal particles from a focused circularly polarized Gaussian beam,” Opt. Express 17(25), 23316–23322 (2009).
[CrossRef] [PubMed]

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(7), 073901 (2007).
[CrossRef] [PubMed]

Ciattoni, A.

A. Ciattoni, G. Cincotti, and C. Palma, “Angular momentum dynamics of a paraxial beam in a uniaxial crystal,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(3 Pt 2), 036618 (2003).
[CrossRef] [PubMed]

Cincotti, G.

A. Ciattoni, G. Cincotti, and C. Palma, “Angular momentum dynamics of a paraxial beam in a uniaxial crystal,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(3 Pt 2), 036618 (2003).
[CrossRef] [PubMed]

Cojoc, D.

V. Garbin, G. Volpe, E. Ferrari, M. Versluis, D. Cojoc, and D. Petrov, “Mie scattering distinguishes the topological charge of an optical vortex: a homage to Gustav Mie,” New J. Phys. 11(1), 013046 (2009).
[CrossRef]

Dainty, C.

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(25), 253601 (2010).
[CrossRef] [PubMed]

O. G. Rodríguez-Herrera, D. Lara, and C. Dainty, “Far-field polarization-based sensitivity to sub-resolution displacements of a sub-resolution scatterer in tightly focused fields,” Opt. Express 18(6), 5609–5628 (2010).
[CrossRef] [PubMed]

Darsht, M. Y.

M. Y. Darsht, B. Y. Zel’dovich, I. V. Kataevskaya, and N. D. Kundikova, “Formation of an isolated wavefront dislocation,” Zh. Eksp. Theor. Fiz. 107, 1464[JETP 80, 817 (1995)].

Desyatnikov, A. S.

Deveaud-Plédran, B.

F. Manni, K. Lagoudakis, T. Paraïso, R. Cerna, Y. Léger, T. Liew, I. Shelykh, A. Kavokin, F. Morier-Genoud, and B. Deveaud-Plédran, “Spin-to-orbital angular momentum conversion in semiconductor microcavities,” Phys. Rev. B 83(24), 241307 (2011).
[CrossRef]

Dogariu, A.

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(7), 073901 (2007).
[CrossRef] [PubMed]

Ferrari, E.

V. Garbin, G. Volpe, E. Ferrari, M. Versluis, D. Cojoc, and D. Petrov, “Mie scattering distinguishes the topological charge of an optical vortex: a homage to Gustav Mie,” New J. Phys. 11(1), 013046 (2009).
[CrossRef]

Forbes, G. W.

Foreman, M. R.

M. R. Foreman and P. Török, “Spin-orbit coupling and conservation of angular momentum flux in non-paraxial imaging of forbidden radiation,” New J. Phys. 13(6), 063041 (2011).
[CrossRef]

Franke-Arnold, S.

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser & Photon. Rev. 2(4), 299–313 (2008).
[CrossRef]

Fujii, M.

Garbin, V.

V. Garbin, G. Volpe, E. Ferrari, M. Versluis, D. Cojoc, and D. Petrov, “Mie scattering distinguishes the topological charge of an optical vortex: a homage to Gustav Mie,” New J. Phys. 11(1), 013046 (2009).
[CrossRef]

Gorodetski, Y.

Y. Gorodetski, S. Nechayev, V. Kleiner, and E. Hasman, “Plasmonic Aharonov-Bohm effect: Optical spin as the magnetic flux parameter,” Phys. Rev. B 82(12), 125433 (2010).
[CrossRef]

Y. Gorodetski, N. Shitrit, I. Bretner, V. Kleiner, and E. Hasman, “Observation of optical spin symmetry breaking in nanoapertures,” Nano Lett. 9(8), 3016–3019 (2009).
[CrossRef] [PubMed]

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(3), 030404 (2008).
[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(4), 043903 (2008).
[CrossRef] [PubMed]

Gu, M.

Z. Bomzon and M. Gu, “Space-variant geometrical phases in focused cylindrical light beams,” Opt. Lett. 32(20), 3017–3019 (2007).
[CrossRef] [PubMed]

Z. Bomzon, M. Gu, and J. Shamir, “Angular momentum and geometric phases in tightly-focused circularly polarized plane waves,” Appl. Phys. Lett. 89(24), 241104 (2006).
[CrossRef]

Haefner, D.

D. Haefner, S. Sukhov, and A. Dogariu, “Spin hall effect of light in spherical geometry,” Phys. Rev. Lett. 102(12), 123903 (2009).
[CrossRef] [PubMed]

Happer, W.

G. Moe and W. Happer, “Conservation of angular momentum for light propagating in a transparent anisotropic medium,” J. Phys. B 10(7), 1191–1208 (1977).
[CrossRef]

Hasman, E.

Y. Gorodetski, S. Nechayev, V. Kleiner, and E. Hasman, “Plasmonic Aharonov-Bohm effect: Optical spin as the magnetic flux parameter,” Phys. Rev. B 82(12), 125433 (2010).
[CrossRef]

Y. Gorodetski, N. Shitrit, I. Bretner, V. Kleiner, and E. Hasman, “Observation of optical spin symmetry breaking in nanoapertures,” Nano Lett. 9(8), 3016–3019 (2009).
[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(4), 043903 (2008).
[CrossRef] [PubMed]

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(3), 030404 (2008).
[CrossRef] [PubMed]

E. Hasman, G. Biener, A. Niv, and V. Kleiner, “Space-variant polarization manipulation,” Prog. Opt. 47, 215–289 (2005).
[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(21), 1875–1877 (2002).
[CrossRef] [PubMed]

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, Pure Appl. Opt. 10(11), 115005 (2008).
[CrossRef]

Higdon, P. D.

P. Török, P. D. Higdon, and T. Wilson, “On the general properties of polarized light conventional and confocal microscopes,” Opt. Commun. 148(4-6), 300–315 (1998).
[CrossRef]

Iketaki, Y.

Izdebskaya, Y.

Jaillon, F.

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(7), 073901 (2007).
[CrossRef] [PubMed]

Juodkazis, S.

E. Brasselet, N. Murazawa, H. Misawa, and S. Juodkazis, “Optical vortices from liquid crystal droplets,” Phys. Rev. Lett. 103(10), 103903 (2009).
[CrossRef] [PubMed]

Karimi, E.

L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, “Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications,” J. Opt. 13(6), 064001 (2011).
[CrossRef]

Kataevskaya, I. V.

M. Y. Darsht, B. Y. Zel’dovich, I. V. Kataevskaya, and N. D. Kundikova, “Formation of an isolated wavefront dislocation,” Zh. Eksp. Theor. Fiz. 107, 1464[JETP 80, 817 (1995)].

Kavokin, A.

F. Manni, K. Lagoudakis, T. Paraïso, R. Cerna, Y. Léger, T. Liew, I. Shelykh, A. Kavokin, F. Morier-Genoud, and B. Deveaud-Plédran, “Spin-to-orbital angular momentum conversion in semiconductor microcavities,” Phys. Rev. B 83(24), 241307 (2011).
[CrossRef]

Kivshar, Y. S.

Kleiner, V.

Y. Gorodetski, S. Nechayev, V. Kleiner, and E. Hasman, “Plasmonic Aharonov-Bohm effect: Optical spin as the magnetic flux parameter,” Phys. Rev. B 82(12), 125433 (2010).
[CrossRef]

Y. Gorodetski, N. Shitrit, I. Bretner, V. Kleiner, and E. Hasman, “Observation of optical spin symmetry breaking in nanoapertures,” Nano Lett. 9(8), 3016–3019 (2009).
[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(4), 043903 (2008).
[CrossRef] [PubMed]

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(3), 030404 (2008).
[CrossRef] [PubMed]

E. Hasman, G. Biener, A. Niv, and V. Kleiner, “Space-variant polarization manipulation,” Prog. Opt. 47, 215–289 (2005).
[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(21), 1875–1877 (2002).
[CrossRef] [PubMed]

Kravtsov, Y. A.

Krolikowski, W.

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,” JETP Lett. 59, 766–769 (1994).

M. Y. Darsht, B. Y. Zel’dovich, I. V. Kataevskaya, and N. D. Kundikova, “Formation of an isolated wavefront dislocation,” Zh. Eksp. Theor. Fiz. 107, 1464[JETP 80, 817 (1995)].

Lacoste, D.

Lagoudakis, K.

F. Manni, K. Lagoudakis, T. Paraïso, R. Cerna, Y. Léger, T. Liew, I. Shelykh, A. Kavokin, F. Morier-Genoud, and B. Deveaud-Plédran, “Spin-to-orbital angular momentum conversion in semiconductor microcavities,” Phys. Rev. B 83(24), 241307 (2011).
[CrossRef]

Lara, D.

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(25), 253601 (2010).
[CrossRef] [PubMed]

O. G. Rodríguez-Herrera, D. Lara, and C. Dainty, “Far-field polarization-based sensitivity to sub-resolution displacements of a sub-resolution scatterer in tightly focused fields,” Opt. Express 18(6), 5609–5628 (2010).
[CrossRef] [PubMed]

Léger, Y.

F. Manni, K. Lagoudakis, T. Paraïso, R. Cerna, Y. Léger, T. Liew, I. Shelykh, A. Kavokin, F. Morier-Genoud, and B. Deveaud-Plédran, “Spin-to-orbital angular momentum conversion in semiconductor microcavities,” Phys. Rev. B 83(24), 241307 (2011).
[CrossRef]

Li, C.-F.

C.-F. Li, “Spin and orbital angular momentum of a class of nonparaxial light beams having a globally defined polarization,” Phys. Rev. A 80(6), 063814 (2009).
[CrossRef]

Liew, T.

F. Manni, K. Lagoudakis, T. Paraïso, R. Cerna, Y. Léger, T. Liew, I. Shelykh, A. Kavokin, F. Morier-Genoud, and B. Deveaud-Plédran, “Spin-to-orbital angular momentum conversion in semiconductor microcavities,” Phys. Rev. B 83(24), 241307 (2011).
[CrossRef]

Maggs, A. C.

V. Rossetto and A. C. Maggs, “Writhing geometry of stiff polymers and scattered light,” Eur. Phys. J. B 29(2), 323–326 (2002).
[CrossRef]

Manni, F.

F. Manni, K. Lagoudakis, T. Paraïso, R. Cerna, Y. Léger, T. Liew, I. Shelykh, A. Kavokin, F. Morier-Genoud, and B. Deveaud-Plédran, “Spin-to-orbital angular momentum conversion in semiconductor microcavities,” Phys. Rev. B 83(24), 241307 (2011).
[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(16), 163905 (2006).
[CrossRef] [PubMed]

Marchesini, S.

Marrucci, L.

L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, “Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications,” J. Opt. 13(6), 064001 (2011).
[CrossRef]

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[CrossRef] [PubMed]

McGloin, D.

Y. Zhao, D. Shapiro, D. McGloin, D. T. Chiu, and S. Marchesini, “Direct observation of the transfer of orbital angular momentum to metal particles from a focused circularly polarized Gaussian beam,” Opt. Express 17(25), 23316–23322 (2009).
[CrossRef] [PubMed]

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(7), 073901 (2007).
[CrossRef] [PubMed]

Misawa, H.

E. Brasselet, N. Murazawa, H. Misawa, and S. Juodkazis, “Optical vortices from liquid crystal droplets,” Phys. Rev. Lett. 103(10), 103903 (2009).
[CrossRef] [PubMed]

Moe, G.

G. Moe and W. Happer, “Conservation of angular momentum for light propagating in a transparent anisotropic medium,” J. Phys. B 10(7), 1191–1208 (1977).
[CrossRef]

Monteiro, P. B.

P. B. Monteiro, P. A. M. Neto, and H. M. Nussenzveig, “Angular momentum of focused beams: Beyond the paraxial approximation,” Phys. Rev. A 79(3), 033830 (2009).
[CrossRef]

Morier-Genoud, F.

F. Manni, K. Lagoudakis, T. Paraïso, R. Cerna, Y. Léger, T. Liew, I. Shelykh, A. Kavokin, F. Morier-Genoud, and B. Deveaud-Plédran, “Spin-to-orbital angular momentum conversion in semiconductor microcavities,” Phys. Rev. B 83(24), 241307 (2011).
[CrossRef]

Murazawa, N.

E. Brasselet, N. Murazawa, H. Misawa, and S. Juodkazis, “Optical vortices from liquid crystal droplets,” Phys. Rev. Lett. 103(10), 103903 (2009).
[CrossRef] [PubMed]

Nagali, E.

L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, “Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications,” J. Opt. 13(6), 064001 (2011).
[CrossRef]

Nechayev, S.

Y. Gorodetski, S. Nechayev, V. Kleiner, and E. Hasman, “Plasmonic Aharonov-Bohm effect: Optical spin as the magnetic flux parameter,” Phys. Rev. B 82(12), 125433 (2010).
[CrossRef]

Neto, P. A. M.

P. B. Monteiro, P. A. M. Neto, and H. M. Nussenzveig, “Angular momentum of focused beams: Beyond the paraxial approximation,” Phys. Rev. A 79(3), 033830 (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, Pure Appl. Opt. 10(11), 115005 (2008).
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Nienhuis, G.

S. J. van Enk and G. Nienhuis, “Spin and orbital angular momentum of photons,” Europhys. Lett. 25(7), 497–501 (1994).
[CrossRef]

S. J. van Enk and G. Nienhuis, “Commutation rules and eigenvalues of spin and orbital angular momentum of radiation fields,” J. Mod. Opt. 41(5), 963–977 (1994).
[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(4), 043903 (2008).
[CrossRef] [PubMed]

E. Hasman, G. Biener, A. Niv, and V. Kleiner, “Space-variant polarization manipulation,” Prog. Opt. 47, 215–289 (2005).
[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(21), 1875–1877 (2002).
[CrossRef] [PubMed]

Nussenzveig, H. M.

P. B. Monteiro, P. A. M. Neto, and H. M. Nussenzveig, “Angular momentum of focused beams: Beyond the paraxial approximation,” Phys. Rev. A 79(3), 033830 (2009).
[CrossRef]

Ostrovskaya, E. A.

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(6), 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(25), 253601 (2010).
[CrossRef] [PubMed]

Padgett, M.

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser & Photon. Rev. 2(4), 299–313 (2008).
[CrossRef]

Palma, C.

A. Ciattoni, G. Cincotti, and C. Palma, “Angular momentum dynamics of a paraxial beam in a uniaxial crystal,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(3 Pt 2), 036618 (2003).
[CrossRef] [PubMed]

Paparo, D.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[CrossRef] [PubMed]

Paraïso, T.

F. Manni, K. Lagoudakis, T. Paraïso, R. Cerna, Y. Léger, T. Liew, I. Shelykh, A. Kavokin, F. Morier-Genoud, and B. Deveaud-Plédran, “Spin-to-orbital angular momentum conversion in semiconductor microcavities,” Phys. Rev. B 83(24), 241307 (2011).
[CrossRef]

Petrov, D.

V. Garbin, G. Volpe, E. Ferrari, M. Versluis, D. Cojoc, and D. Petrov, “Mie scattering distinguishes the topological charge of an optical vortex: a homage to Gustav Mie,” New J. Phys. 11(1), 013046 (2009).
[CrossRef]

Piccirillo, B.

L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, “Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications,” J. Opt. 13(6), 064001 (2011).
[CrossRef]

Picón, A.

Planken, P. C. M.

L. T. Vuong, A. J. L. Adam, J. M. Brok, P. C. M. Planken, and H. P. Urbach, “Electromagnetic spin-orbit interactions via scattering of subwavelength apertures,” Phys. Rev. Lett. 104(8), 083903 (2010).
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Richards, B.

B. Richards and E. Wolf, ““Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
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Rodríguez-Herrera, O. G.

O. G. Rodríguez-Herrera, D. Lara, and C. Dainty, “Far-field polarization-based sensitivity to sub-resolution displacements of a sub-resolution scatterer in tightly focused fields,” Opt. Express 18(6), 5609–5628 (2010).
[CrossRef] [PubMed]

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(25), 253601 (2010).
[CrossRef] [PubMed]

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,” JETP Lett. 59, 766–769 (1994).

Rossetto, V.

D. Lacoste, V. Rossetto, F. Jaillon, and H. Saint-Jalmes, “Geometric depolarization in patterns formed by backscattered light,” Opt. Lett. 29(17), 2040–2042 (2004).
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V. Rossetto and A. C. Maggs, “Writhing geometry of stiff polymers and scattered light,” Eur. Phys. J. B 29(2), 323–326 (2002).
[CrossRef]

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, Pure Appl. Opt. 10(11), 115005 (2008).
[CrossRef]

Saint-Jalmes, H.

Santamato, E.

L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, “Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications,” J. Opt. 13(6), 064001 (2011).
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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,” JETP Lett. 59, 232–234 (1994).

Schwartz, C.

Sciarrino, F.

L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, “Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications,” J. Opt. 13(6), 064001 (2011).
[CrossRef]

Shamir, J.

Z. Bomzon, M. Gu, and J. Shamir, “Angular momentum and geometric phases in tightly-focused circularly polarized plane waves,” Appl. Phys. Lett. 89(24), 241104 (2006).
[CrossRef]

Shapiro, D.

Shelykh, I.

F. Manni, K. Lagoudakis, T. Paraïso, R. Cerna, Y. Léger, T. Liew, I. Shelykh, A. Kavokin, F. Morier-Genoud, and B. Deveaud-Plédran, “Spin-to-orbital angular momentum conversion in semiconductor microcavities,” Phys. Rev. B 83(24), 241307 (2011).
[CrossRef]

Shitrit, N.

Y. Gorodetski, N. Shitrit, I. Bretner, V. Kleiner, and E. Hasman, “Observation of optical spin symmetry breaking in nanoapertures,” Nano Lett. 9(8), 3016–3019 (2009).
[CrossRef] [PubMed]

Shvedov, V.

Slussarenko, S.

L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, “Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications,” J. Opt. 13(6), 064001 (2011).
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A. Bekshaev, K. Y. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13(5), 053001 (2011).
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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, Pure Appl. Opt. 10(11), 115005 (2008).
[CrossRef]

Sukhov, S.

D. Haefner, S. Sukhov, and A. Dogariu, “Spin hall effect of light in spherical geometry,” Phys. Rev. Lett. 102(12), 123903 (2009).
[CrossRef] [PubMed]

Török, P.

M. R. Foreman and P. Török, “Spin-orbit coupling and conservation of angular momentum flux in non-paraxial imaging of forbidden radiation,” New J. Phys. 13(6), 063041 (2011).
[CrossRef]

P. Török, P. D. Higdon, and T. Wilson, “On the general properties of polarized light conventional and confocal microscopes,” Opt. Commun. 148(4-6), 300–315 (1998).
[CrossRef]

Urbach, H. P.

L. T. Vuong, A. J. L. Adam, J. M. Brok, P. C. M. Planken, and H. P. Urbach, “Electromagnetic spin-orbit interactions via scattering of subwavelength apertures,” Phys. Rev. Lett. 104(8), 083903 (2010).
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L. T. Vuong, A. J. L. Adam, J. M. Brok, P. C. M. Planken, and H. P. Urbach, “Electromagnetic spin-orbit interactions via scattering of subwavelength apertures,” Phys. Rev. Lett. 104(8), 083903 (2010).
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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,” JETP Lett. 59, 232–234 (1994).

Zhao, Y.

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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,” JETP 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,” JETP Lett. 59, 766–769 (1994).

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S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser & Photon. Rev. 2(4), 299–313 (2008).
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M. R. Foreman and P. Török, “Spin-orbit coupling and conservation of angular momentum flux in non-paraxial imaging of forbidden radiation,” New J. Phys. 13(6), 063041 (2011).
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[CrossRef] [PubMed]

D. Haefner, S. Sukhov, and A. Dogariu, “Spin hall effect of light in spherical geometry,” Phys. Rev. Lett. 102(12), 123903 (2009).
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L. T. Vuong, A. J. L. Adam, J. M. Brok, P. C. M. Planken, and H. P. Urbach, “Electromagnetic spin-orbit interactions via scattering of subwavelength apertures,” Phys. Rev. Lett. 104(8), 083903 (2010).
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Proc. R. Soc. Lond. A Math. Phys. Sci.

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[CrossRef]

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M. Y. Darsht, B. Y. Zel’dovich, I. V. Kataevskaya, and N. D. Kundikova, “Formation of an isolated wavefront dislocation,” Zh. Eksp. Theor. Fiz. 107, 1464[JETP 80, 817 (1995)].

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M. Born and E. Wolf, Principles of Optics, 7th edn. (Pergamon, 2005).

A. Bekshaev and S. Sviridova, “Mechanical action of inhomogeneously polarized optical fields and detection of the internal energy flows,” arXiv:1102.3514.

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Figures (8)

Fig. 1
Fig. 1

Focusing of light by a spherical high-NA lens. The incident paraxial field E 0 ( x,y ) is refracted in the meridional plane and transformed to a spectrum of plane waves (rays) E ˜ ( θ,ϕ ) with the k -vectors distributed on the sphere in k-space: θ( 0, θ c ) , ϕ( 0,2π ) . The electric field vectors are parallel-transported along each partial ray with the local helicity being conserved.

Fig. 8
Fig. 8

Left: OAM angle-resolved density l z ( θ ) converted from SAM for the cases of focusing (red) [Eq. (10)], scattering (blue) [Eq. (24)], and imaging (black) [Eq. (30)], with circularly polarized incident light, σ=+1 , =0 . Right: The integral OAM, L z , converted from SAM, vs. the aperture angle θ c for focusing [Eqs. (12) and (13)], scattering [Eq. (25)], and imaging systems [Eq. (31)]. The solid, dashed, and dotted curves for focusing represent incident paraxial beams with =0 , =1 , and =2 ( σ=+1 everywhere). For scattering, the dependence is obtained by formally replacing the upper limit of integration in Eqs. (25): π θ c ; analytical results above are recovered at θ c =π or π/2 . The semitransparent curves indicate the corresponding quantities for the SAM, i.e., s z ( θ ) and S z , satisfying the conservation law (14). For comparison, the right-hand panel also displays the OAM and SAM values for nonparaxial vector Bessel beams (yellow) [29], which achieve the total spin-to-orbital conversion at the aperture angle θ c =π/2 , see [14].

Fig. 2
Fig. 2

Focal intensity distributions I σ ( ρ,0 ) , Eq. (17), in comparison with the radii R σ , Eq. (18), (vertical lines) for =1,2,3 , σ=±1 , and different values of the aperture angles θ c . It is seen that the structure of radii (18) underlies positions of the first maxima of intensity (17), cf [29].

Fig. 3
Fig. 3

Akin to focusing, Fig. 1, dipole scattering of the incident paraxial field E 0 ( x,y ) by a small spherical particle transforms it into a spectrum of plane waves E ˜ ( θ,ϕ ) with the spherically-distributed k -vectors.

Fig. 4
Fig. 4

Scheme of the ‘lens-scatterer-lens’ imaging system. First, the incident paraxial light E 0 is focused by a high-NA lens, then a small specimen in the focus scatters the nonparaxial focused field, and finally the scattered light is collected by the second high-NA lens. The output paraxial field E( θ,ϕ ) has a space-variant polarization distribution and bears information about the spin-orbit coupling inside the system. This offers an efficient tool to retrieve fine subwavelength information about the specimen.

Fig. 5
Fig. 5

Distributions of the Stokes parameters S =( S 1 , S 2 , S 3 ) , Eqs. (32) and (33), in the exit pupil of the optical microscope (Fig. 4) in the case of the on-axis location of the specimen, r s =0 , and left-hand circular polarization of the incident light, σ=1 . The normalized coordinates x ˜ =x/( fsin θ c ) and y ˜ =y/( fsin θ c ) are used. The four-fold patterns in the S 1 and S 2 distributions are the signature of the generation of the right-hand polarized component with optical vortex e 2iϕ , i.e., the spin-to-orbital AM conversion (29). The aperture angle is θ c =3π/8 .

Fig. 6
Fig. 6

Distributions of the output field intensity I( θ,ϕ ) in the imaging system Fig. 4 for the incident right-hand circularly polarized light ( σ=1 ) at different subwavelength displacements of the scattering particle: x s =0,λ/3,λ/3 . The transverse shift of the center of gravity of the field, Y σ x s signifies the spin Hall effect of light. Parameters are the same as in Fig. 5.

Fig. 7
Fig. 7

Intensities and phases of the right- and left-hand circularly polarized components in the output field for the on-axis ( r s =0 ) and off-axis ( x s =0.28λ ) positions of the scatterer (Fig. 4). The incident field is right-hand circularly polarized ( σ=1 ), the aperture of the system corresponds to sin θ c =0.922 . The charge-2 optical vortex and nonzero intensity in the left-hand polarized component is clearly seen for the on-axis particle. Displacement of the scatterer induces splitting of the charge-2 vortex into two charge-1 vortices (cf [34].), strong deformation of the intensity of the right-hand component (responsible for the spin-Hall effect and Y 0 ), and smooth gradient of the phase in the right-hand component which yields P x 0 .

Equations (36)

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u ± = u x ±i u y 2 , E ± = E x i E y 2 ,
E L = V ^ E C , V ^ = 1 2 ( 1 1 0 i i 0 0 0 2 ).
L ^ z =i ϕ , V ^ S ^ z V ^ = σ ^ =diag( 1,1,0 ).
k x =k x f , k y =k y f , k z =k z f ,
E ˜ cosθ U ^ ( θ,ϕ ) E 0 , U ^ ( θ,ϕ )=( a b e 2iϕ 2ab e iϕ b e 2iϕ a 2ab e iϕ 2ab e iϕ 2ab e iϕ ab ),
E ˜ H = U ^ E ˜ E 0 E 0 H .
| ,σ cosθ [ a| ,σ b| +2σ,σ 2ab | +σ,0 ].
l z ( θ,ϕ )=i E ˜ * ϕ E ˜ E ˜ * E ˜ , s z ( θ,ϕ )= E ˜ * σ ^ E ˜ E ˜ * E ˜ .
E 0 σ e σ E ( θ,ϕ ), E = F | | ( θ ) e iϕ ,
l z ( θ )=+σ Φ B ( θ ), s z ( θ )=σ[ 1 Φ B ( θ ) ], Φ B ( θ )=( 1cosθ )=2b.
L z =i dΩ E ˜ * ϕ E ˜ dΩ E ˜ * E ˜ , S z = dΩ E ˜ * σ ^ E ˜ dΩ E ˜ * E ˜ .
L z =+σ Φ ¯ B , S z =σ( 1 Φ ¯ B ), Φ ¯ B =( 1cos θ ¯ ),
cos θ ¯ = c P z W = 0 θ c | F | | ( θ ) | 2 cos 2 θsinθdθ 0 θ c | F | | ( θ ) | 2 cosθsinθdθ .
l z + s z = L z + S z =+σ.
cos θ ¯ = 2(1+| |) sin ( 2+2| | ) θ c [ 2 | | | |! ( 2| |+3 )!! 1 3 cos 3 θ c F 2 1 ( 3 2 ,| |, 5 2 ; cos 2 θ c ) ].
E( r ) dΩ E ˜ ( θ,ϕ ) e iΦ( θ,ϕ,r ) ,Φ=k( xsinθcosϕ+ysinθsinϕ+zcosθ ).
I σ ( ρ,z ) a J ( ξ ) 2 + b J +2σ ( ξ ) 2 +2 ab J +σ ( ξ ) 2 .
R σ L z ksin θ ¯ = +σ( 1cos θ ¯ ) ksin θ ¯ .
E ˜ ( θ,ϕ,r ) r ×[ r × E 0 ( 0 ) ] r ,
E ˜ ( θ,ϕ ) 1 r Π ^ ( θ,ϕ ) E 0 ( 0 ), Π ^ = 1 2 ( 1+ a 1 b 1 e 2iϕ 2 a 1 b 1 e iϕ b 1 e 2iϕ 1+ a 1 2 a 1 b 1 e iϕ 2 a 1 b 1 e iϕ 2 a 1 b 1 e iϕ 2 b 1 2 ).
E ˜ H = U ^ E ˜ p ^ z U ^ E 0 p ^ U ^ E 0H
| 0,σ 1 2 [ (1+ a 1 )| 0,σ b 1 | 2σ,σ 2 a 1 b 1 | σ,0 ].
|σ H a |σ H b e 2iσϕ | σ H .
l z =σ 1 cos 2 θ 1+ cos 2 θ , s z =σ 2 cos 2 θ 1+ cos 2 θ .
L z =σ 0 π ( 1 cos 2 θ )sinθ dθ 0 π ( 1+ cos 2 θ )sinθdθ = 1 2 σ, S z =σ 2 0 π cos 2 θsinθdθ 0 π ( 1+ cos 2 θ )sinθdθ = 1 2 σ.
l z =σP( θ ), s z =σ[ 1P( θ ) ],P( θ )= 2 σ σ + σ = sin 2 θ 1+ cos 2 θ ,
Φ k r s R /f=k( x s sin θ cos ϕ + y s sin θ sin ϕ + z s cos θ ), Φk r s R/f=k( x s sinθcosϕ+ y s sinθsinϕ+ z s cosθ ).
T ^ (0) = A( θ c ) cosθ ( a b e 2iϕ b e 2iϕ a ),
| 0,σ a| 0,σ b| 2σ,σ .
l z =σ( 1 2cosθ 1+ cos 2 θ ), s z =σ 2cosθ 1+ cos 2 θ .
L z =σ 13cos θ c +3 cos 2 θ c cos 3 θ c 43cos θ c cos 3 θ c , S z =σ 3 sin 2 θ c 43cos θ c cos 3 θ c ,
S ( θ,ϕ )= E ˜ * σ ^ E ˜ E ˜ * E ˜ ,
S ( θ,ϕ )= 1 a 2 + b 2 [ 2abcos2ϕ,2absin2ϕ,σ( a 2 b 2 ) ].
T ^ (1) =ik B( θ c )sinθ 2 cosθ ( ρ s e iϕ ρ s e iϕ ρ s e iϕ ρ s e iϕ ).
R = d Ω R ( E σ* E σ ) d Ω E σ* E σ ,
X =0, Y =σfk x s 3B sin 4 θ c 2A( 43cos θ c cos 3 θ c ) .

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