Abstract

We show that, by uniformly modulating the amplitude or polarization of one half of the input beam, a tunable three-dimensional (3D) polarization field near the focus of a 4Pi focusing system can be generated. If the input field is radially polarized and modulated by an amplitude-phase modulator, the longitudinal component of the focused field will partially convert to the transversal one according to the modulation factor and a 3D linear polarization state is created. If the input field is circularly polarized in one half and elliptically polarized in another half, the focal field will have elliptical polarization with the normal to the polarization ellipse being 3D controllable, corresponding to a 3D controllable spin angular momentum.

© 2013 OSA

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    [CrossRef] [PubMed]
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  7. B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett.85(21), 4482–4485 (2000).
    [CrossRef] [PubMed]
  8. X. Li, T.-H. Lan, C.-H. Tien, and M. Gu, “Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam,” Nat Commun3, 998 (2012).
    [CrossRef] [PubMed]
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    [CrossRef]
  10. A. Picón, A. Benseny, J. Mompart, J. R. Vázquez de Aldana, L. Plaja, G. F. Calvo, and L. Roso, “Transferring orbital and spin angular momenta of light to atoms,” New J. Phys.12(8), 083053 (2010).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  15. W. Chen and Q. Zhan, “Three dimensional polarization control in 4Pi microscopy,” Opt. Commun.284(1), 52–56 (2011).
    [CrossRef]
  16. C.-F. Li, “Spin and orbital angular momentum of a class of nonparaxial light beams having a globally defined polarization,” Phys. Rev. A80(6), 063814 (2009).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  23. A. Aiello, N. Lindlein, C. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin Hall effect of light,” Phys. Rev. Lett.103(10), 100401 (2009).
    [CrossRef] [PubMed]
  24. H. Shpaisman, D. B. Ruffner, and D. G. Grier, “Light-driven three-dimensional rotational motion of dandelion-shaped microparticles,” Appl. Phys. Lett.102(7), 071103 (2013).
    [CrossRef]
  25. P. Galajda and P. Ormos, “Complex micromachines produced and driven by light,” Appl. Phys. Lett.78(2), 249–251 (2001).
    [CrossRef]
  26. A. Bekshaev, K. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt.13(5), 053001 (2011).
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    [CrossRef]

2013 (1)

H. Shpaisman, D. B. Ruffner, and D. G. Grier, “Light-driven three-dimensional rotational motion of dandelion-shaped microparticles,” Appl. Phys. Lett.102(7), 071103 (2013).
[CrossRef]

2012 (4)

J. Wang, W. Chen, and Q. Zhan, “Creation of uniform three-dimensional optical chain through tight focusing of space-variant polarized beams,” J. Opt.14(5), 055004 (2012).
[CrossRef]

X. Li, T.-H. Lan, C.-H. Tien, and M. Gu, “Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam,” Nat Commun3, 998 (2012).
[CrossRef] [PubMed]

Z. Chen and D. Zhao, “4Pi focusing of spatially modulated radially polarized vortex beams,” Opt. Lett.37(8), 1286–1288 (2012).
[CrossRef] [PubMed]

W. Zhu and W. She, “Electrically controlling spin and orbital angular momentum of a focused light beam in a uniaxial crystal,” Opt. Express20(23), 25876–25883 (2012).
[CrossRef] [PubMed]

2011 (4)

W. Chen and Q. Zhan, “Three dimensional polarization control in 4Pi microscopy,” Opt. Commun.284(1), 52–56 (2011).
[CrossRef]

S. N. Khonina and I. Golub, “Optimization of focusing of linearly polarized light,” Opt. Lett.36(3), 352–354 (2011).
[CrossRef] [PubMed]

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

M. Mansuripur, “Spin and orbital angular momenta of electromagnetic waves in free space,” Phys. Rev. A84(3), 033838 (2011).
[CrossRef]

2010 (2)

A. Picón, A. Benseny, J. Mompart, J. R. Vázquez de Aldana, L. Plaja, G. F. Calvo, and L. Roso, “Transferring orbital and spin angular momenta of light to atoms,” New J. Phys.12(8), 083053 (2010).
[CrossRef]

W. Chen and Q. Zhan, “Diffraction limited focusing with controllable arbitrary three-dimensional polarization,” J. Opt.12(4), 045707 (2010).
[CrossRef]

2009 (2)

C.-F. Li, “Spin and orbital angular momentum of a class of nonparaxial light beams having a globally defined polarization,” Phys. Rev. A80(6), 063814 (2009).
[CrossRef]

A. Aiello, N. Lindlein, C. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin Hall effect of light,” Phys. Rev. Lett.103(10), 100401 (2009).
[CrossRef] [PubMed]

2008 (1)

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics2(8), 501–505 (2008).
[CrossRef]

2007 (1)

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]

2006 (1)

A. F. Abouraddy and K. C. Toussaint., “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett.96(15), 153901 (2006).
[CrossRef] [PubMed]

2005 (1)

R. Zambrini and S. M. Barnett, “Local transfer of angular momentum to matter,” J. Mod. Opt.52, 1045–1052 (2005).
[CrossRef]

2004 (1)

2001 (1)

P. Galajda and P. Ormos, “Complex micromachines produced and driven by light,” Appl. Phys. Lett.78(2), 249–251 (2001).
[CrossRef]

2000 (3)

J. M. Bueno, “Polarimetry using liquid-crystal variable retarders: theory and calibration,” J. Opt. A, Pure Appl. Opt.2(3), 216–222 (2000).
[CrossRef]

B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett.85(21), 4482–4485 (2000).
[CrossRef] [PubMed]

K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express7(2), 77–87 (2000).
[CrossRef] [PubMed]

1999 (1)

E. J. Sánchez, L. Novotny, and X. S. Xie, “Near-field fluorescence microscopy based on two-photon excitation with metal tips,” Phys. Rev. Lett.82(20), 4014–4017 (1999).
[CrossRef]

1998 (1)

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser trapped microscopic particles,” Nature394(6691), 348–350 (1998).
[CrossRef]

1992 (1)

1959 (1)

B. Richards and E. Wolf, “Electomagnetic diffraction in optical systems: Structure of the image field in an aplanatic system,” Proc. Roy. Soc. London Series A253(1274), 358–379 (1959).
[CrossRef]

Abouraddy, A. F.

A. F. Abouraddy and K. C. Toussaint., “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett.96(15), 153901 (2006).
[CrossRef] [PubMed]

Aiello, A.

A. Aiello, N. Lindlein, C. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin Hall effect of light,” Phys. Rev. Lett.103(10), 100401 (2009).
[CrossRef] [PubMed]

Barnett, S. M.

R. Zambrini and S. M. Barnett, “Local transfer of angular momentum to matter,” J. Mod. Opt.52, 1045–1052 (2005).
[CrossRef]

Bekshaev, A.

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

Benseny, A.

A. Picón, A. Benseny, J. Mompart, J. R. Vázquez de Aldana, L. Plaja, G. F. Calvo, and L. Roso, “Transferring orbital and spin angular momenta of light to atoms,” New J. Phys.12(8), 083053 (2010).
[CrossRef]

Bliokh, K.

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

Bokor, N.

Brown, T.

Bueno, J. M.

J. M. Bueno, “Polarimetry using liquid-crystal variable retarders: theory and calibration,” J. Opt. A, Pure Appl. Opt.2(3), 216–222 (2000).
[CrossRef]

Calvo, G. F.

A. Picón, A. Benseny, J. Mompart, J. R. Vázquez de Aldana, L. Plaja, G. F. Calvo, and L. Roso, “Transferring orbital and spin angular momenta of light to atoms,” New J. Phys.12(8), 083053 (2010).
[CrossRef]

Chen, W.

J. Wang, W. Chen, and Q. Zhan, “Creation of uniform three-dimensional optical chain through tight focusing of space-variant polarized beams,” J. Opt.14(5), 055004 (2012).
[CrossRef]

W. Chen and Q. Zhan, “Three dimensional polarization control in 4Pi microscopy,” Opt. Commun.284(1), 52–56 (2011).
[CrossRef]

W. Chen and Q. Zhan, “Diffraction limited focusing with controllable arbitrary three-dimensional polarization,” J. Opt.12(4), 045707 (2010).
[CrossRef]

Chen, Z.

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

Chong, C. T.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics2(8), 501–505 (2008).
[CrossRef]

Davidson, N.

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]

Friese, M. E. J.

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser trapped microscopic particles,” Nature394(6691), 348–350 (1998).
[CrossRef]

Galajda, P.

P. Galajda and P. Ormos, “Complex micromachines produced and driven by light,” Appl. Phys. Lett.78(2), 249–251 (2001).
[CrossRef]

Golub, I.

Grier, D. G.

H. Shpaisman, D. B. Ruffner, and D. G. Grier, “Light-driven three-dimensional rotational motion of dandelion-shaped microparticles,” Appl. Phys. Lett.102(7), 071103 (2013).
[CrossRef]

Gu, M.

X. Li, T.-H. Lan, C.-H. Tien, and M. Gu, “Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam,” Nat Commun3, 998 (2012).
[CrossRef] [PubMed]

Hecht, B.

B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett.85(21), 4482–4485 (2000).
[CrossRef] [PubMed]

Heckenberg, N. R.

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser trapped microscopic particles,” Nature394(6691), 348–350 (1998).
[CrossRef]

Hell, S.

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]

Khonina, S. N.

Lan, T.-H.

X. Li, T.-H. Lan, C.-H. Tien, and M. Gu, “Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam,” Nat Commun3, 998 (2012).
[CrossRef] [PubMed]

Leuchs, G.

A. Aiello, N. Lindlein, C. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin Hall effect of light,” Phys. Rev. Lett.103(10), 100401 (2009).
[CrossRef] [PubMed]

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. A80(6), 063814 (2009).
[CrossRef]

Li, X.

X. Li, T.-H. Lan, C.-H. Tien, and M. Gu, “Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam,” Nat Commun3, 998 (2012).
[CrossRef] [PubMed]

Lindlein, N.

A. Aiello, N. Lindlein, C. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin Hall effect of light,” Phys. Rev. Lett.103(10), 100401 (2009).
[CrossRef] [PubMed]

Lukyanchuk, B.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics2(8), 501–505 (2008).
[CrossRef]

Mansuripur, M.

M. Mansuripur, “Spin and orbital angular momenta of electromagnetic waves in free space,” Phys. Rev. A84(3), 033838 (2011).
[CrossRef]

Marquardt, C.

A. Aiello, N. Lindlein, C. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin Hall effect of light,” Phys. Rev. Lett.103(10), 100401 (2009).
[CrossRef] [PubMed]

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

Mompart, J.

A. Picón, A. Benseny, J. Mompart, J. R. Vázquez de Aldana, L. Plaja, G. F. Calvo, and L. Roso, “Transferring orbital and spin angular momenta of light to atoms,” New J. Phys.12(8), 083053 (2010).
[CrossRef]

Nieminen, T. A.

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser trapped microscopic particles,” Nature394(6691), 348–350 (1998).
[CrossRef]

Novotny, L.

B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett.85(21), 4482–4485 (2000).
[CrossRef] [PubMed]

E. J. Sánchez, L. Novotny, and X. S. Xie, “Near-field fluorescence microscopy based on two-photon excitation with metal tips,” Phys. Rev. Lett.82(20), 4014–4017 (1999).
[CrossRef]

Ormos, P.

P. Galajda and P. Ormos, “Complex micromachines produced and driven by light,” Appl. Phys. Lett.78(2), 249–251 (2001).
[CrossRef]

Picón, A.

A. Picón, A. Benseny, J. Mompart, J. R. Vázquez de Aldana, L. Plaja, G. F. Calvo, and L. Roso, “Transferring orbital and spin angular momenta of light to atoms,” New J. Phys.12(8), 083053 (2010).
[CrossRef]

Plaja, L.

A. Picón, A. Benseny, J. Mompart, J. R. Vázquez de Aldana, L. Plaja, G. F. Calvo, and L. Roso, “Transferring orbital and spin angular momenta of light to atoms,” New J. Phys.12(8), 083053 (2010).
[CrossRef]

Richards, B.

B. Richards and E. Wolf, “Electomagnetic diffraction in optical systems: Structure of the image field in an aplanatic system,” Proc. Roy. Soc. London Series A253(1274), 358–379 (1959).
[CrossRef]

Roso, L.

A. Picón, A. Benseny, J. Mompart, J. R. Vázquez de Aldana, L. Plaja, G. F. Calvo, and L. Roso, “Transferring orbital and spin angular momenta of light to atoms,” New J. Phys.12(8), 083053 (2010).
[CrossRef]

Rubinsztein-Dunlop, H.

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser trapped microscopic particles,” Nature394(6691), 348–350 (1998).
[CrossRef]

Ruffner, D. B.

H. Shpaisman, D. B. Ruffner, and D. G. Grier, “Light-driven three-dimensional rotational motion of dandelion-shaped microparticles,” Appl. Phys. Lett.102(7), 071103 (2013).
[CrossRef]

Sánchez, E. J.

E. J. Sánchez, L. Novotny, and X. S. Xie, “Near-field fluorescence microscopy based on two-photon excitation with metal tips,” Phys. Rev. Lett.82(20), 4014–4017 (1999).
[CrossRef]

She, W.

Sheppard, C.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics2(8), 501–505 (2008).
[CrossRef]

Shi, L.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics2(8), 501–505 (2008).
[CrossRef]

Shpaisman, H.

H. Shpaisman, D. B. Ruffner, and D. G. Grier, “Light-driven three-dimensional rotational motion of dandelion-shaped microparticles,” Appl. Phys. Lett.102(7), 071103 (2013).
[CrossRef]

Sick, B.

B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett.85(21), 4482–4485 (2000).
[CrossRef] [PubMed]

Soskin, M.

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

Stelzer, E. H. K.

Tien, C.-H.

X. Li, T.-H. Lan, C.-H. Tien, and M. Gu, “Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam,” Nat Commun3, 998 (2012).
[CrossRef] [PubMed]

Toussaint, K. C.

A. F. Abouraddy and K. C. Toussaint., “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett.96(15), 153901 (2006).
[CrossRef] [PubMed]

Vázquez de Aldana, J. R.

A. Picón, A. Benseny, J. Mompart, J. R. Vázquez de Aldana, L. Plaja, G. F. Calvo, and L. Roso, “Transferring orbital and spin angular momenta of light to atoms,” New J. Phys.12(8), 083053 (2010).
[CrossRef]

Wang, H.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics2(8), 501–505 (2008).
[CrossRef]

Wang, J.

J. Wang, W. Chen, and Q. Zhan, “Creation of uniform three-dimensional optical chain through tight focusing of space-variant polarized beams,” J. Opt.14(5), 055004 (2012).
[CrossRef]

Wolf, E.

B. Richards and E. Wolf, “Electomagnetic diffraction in optical systems: Structure of the image field in an aplanatic system,” Proc. Roy. Soc. London Series A253(1274), 358–379 (1959).
[CrossRef]

Xie, X. S.

E. J. Sánchez, L. Novotny, and X. S. Xie, “Near-field fluorescence microscopy based on two-photon excitation with metal tips,” Phys. Rev. Lett.82(20), 4014–4017 (1999).
[CrossRef]

Youngworth, K.

Zambrini, R.

R. Zambrini and S. M. Barnett, “Local transfer of angular momentum to matter,” J. Mod. Opt.52, 1045–1052 (2005).
[CrossRef]

Zhan, Q.

J. Wang, W. Chen, and Q. Zhan, “Creation of uniform three-dimensional optical chain through tight focusing of space-variant polarized beams,” J. Opt.14(5), 055004 (2012).
[CrossRef]

W. Chen and Q. Zhan, “Three dimensional polarization control in 4Pi microscopy,” Opt. Commun.284(1), 52–56 (2011).
[CrossRef]

W. Chen and Q. Zhan, “Diffraction limited focusing with controllable arbitrary three-dimensional polarization,” J. Opt.12(4), 045707 (2010).
[CrossRef]

Zhao, D.

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

Zhu, W.

Appl. Phys. Lett. (2)

H. Shpaisman, D. B. Ruffner, and D. G. Grier, “Light-driven three-dimensional rotational motion of dandelion-shaped microparticles,” Appl. Phys. Lett.102(7), 071103 (2013).
[CrossRef]

P. Galajda and P. Ormos, “Complex micromachines produced and driven by light,” Appl. Phys. Lett.78(2), 249–251 (2001).
[CrossRef]

J. Mod. Opt. (1)

R. Zambrini and S. M. Barnett, “Local transfer of angular momentum to matter,” J. Mod. Opt.52, 1045–1052 (2005).
[CrossRef]

J. Opt. (3)

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

W. Chen and Q. Zhan, “Diffraction limited focusing with controllable arbitrary three-dimensional polarization,” J. Opt.12(4), 045707 (2010).
[CrossRef]

J. Wang, W. Chen, and Q. Zhan, “Creation of uniform three-dimensional optical chain through tight focusing of space-variant polarized beams,” J. Opt.14(5), 055004 (2012).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

J. M. Bueno, “Polarimetry using liquid-crystal variable retarders: theory and calibration,” J. Opt. A, Pure Appl. Opt.2(3), 216–222 (2000).
[CrossRef]

J. Opt. Soc. Am. A (1)

Nat Commun (1)

X. Li, T.-H. Lan, C.-H. Tien, and M. Gu, “Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam,” Nat Commun3, 998 (2012).
[CrossRef] [PubMed]

Nat. Photonics (1)

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics2(8), 501–505 (2008).
[CrossRef]

Nature (1)

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser trapped microscopic particles,” Nature394(6691), 348–350 (1998).
[CrossRef]

New J. Phys. (1)

A. Picón, A. Benseny, J. Mompart, J. R. Vázquez de Aldana, L. Plaja, G. F. Calvo, and L. Roso, “Transferring orbital and spin angular momenta of light to atoms,” New J. Phys.12(8), 083053 (2010).
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Figures (5)

Fig. 1
Fig. 1

Illustration of the scheme for generating 3D linear polarization. The blue arrows represent the polarization vectors of light field propagating towards the focus, while the light blue arrow represents the resultant polarization vector at geometrical focus. The color rectangle shows the intensity distribution in the xz plane near the focus. And the intensity and polarization distributions within the central focal spot are shown by projections on three orthogonal planes. In calculation, the configure factors (a,ϕ0) = (0.5,-π/2), NA = 0.95.

Fig. 2
Fig. 2

The normalized intensity and polarization distributions in the focal plane (a~f) with polarization orientation (a) (χll) = (0,π), (b) (π/6,π), (c) (π/3,π), (d) (π/2,π), (e) (π/2,-π/2), (f) (π/2,-π/4), which correspond to configure factors (a,ϕ0) = (1,π/2), (0.23,π/2), (−0.3,π/2), (−1,π/2), (−1,π), and (−1,5π/4), respectively. (g) The dependence of χl on a. (h) Projection of intensity and polarization distributions on three orthogonal planes within the FWHM region when a = −1, and ϕ0 = 5π/4. The FWHMs along x, y, and z axes are 0.56λ, 0.82λ, and 0.3λ, respectively. In calculation, NA is chosen to be 0.95.

Fig. 3
Fig. 3

The dependence of intensity at geometrical focus on the modulator factor a. The intensity at the case of a = 1 is set to be 1. In calculation, NA is 0.95.

Fig. 4
Fig. 4

Illustration of the scheme for generating 3D elliptical polarization. The blue arrows represent the SAM vectors of local light fields before and after objectives, while the light blue arrow represents the resultant SAM vector at focus.

Fig. 5
Fig. 5

(a~c) Projection of intensity and polarization distributions on three orthogonal planes within the central spot with configure factors (a) (δ1,δ2,ϕ0) = (π/2,π/2,/2), (b) (π/2,-π/5,0), and (c) (π/2,-π/2,π/4), respectively. (d) The dependences of χσ (blue) and χs (red) on δ2 for δ1 = ± π/2. (e) The dependences of |σ| (blue) and |s| (red) on δ2 for δ1 = ± π/2. In (d) and (e), the dotted and solid lines correspond to δ1 = ± π/2, respectively. In calculation, NA is chosen to be 0.95.

Equations (14)

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[ E x (r,φ,z) E y (r,φ,z) E z (r,φ,z) ]= iC π 0 α 0 2π X(θ) l 0 (θ)T(ϕ)sinθexp{ik[zcosθ+rsinθcos(ϕφ)]}[ cosθcosϕ cosθsinϕ sinθ ]dϕdθ,
l 0 (θ)=exp[ ( βsinθ sinα ) 2 ] J 1 ( 2βsinθ sinα ),
T L (ϕ)={ 1 ϕ 0 ϕ< ϕ 0 +π a ϕ 0 +πϕ< ϕ 0 +2π ,
E(r,φ,z)=[ E L x (r,φ,z)+ E R x (r,φ,z)] e x +[ E L y (r,φ,z)+ E R y (r,φ,z)] e y +[ E L z (r,φ,z) E R z (r,φ,z)] e z .
E LCVR (ϕ)=A(ϕ) e + +B(ϕ) e ={ e i ϕ 0 2 (cos δ 1 2 +sin δ 1 2 ) e + + e i ϕ 0 2 (cos δ 1 2 sin δ 1 2 ) e ϕ 0 ϕ< ϕ 0 +π e i ϕ 0 2 (cos δ 2 2 +sin δ 2 2 ) e + + e i ϕ 0 2 (cos δ 2 2 sin δ 2 2 ) e ϕ 0 +πϕ< ϕ 0 +2π .
[ E L/R x (r,φ,z) E L/R y (r,φ,z) E L/R z (r,φ,z) ]= iC 2 π 0 α 0 2π X(θ) l 0 (θ)sinθexp{ik[zcosθ+rsinθcos(ϕφ)]+iϕ} ×[ (cosθcosϕisinϕ) A L/R (ϕ)+(cosθcosϕ+isinϕ) e i2ϕ B L/R (ϕ) (cosθsinϕ+icosϕ) A L/R (ϕ)+(cosθsinϕicosϕ) e i2ϕ B L/R (ϕ) sinθ[ A L/R (ϕ)+ e i2ϕ B L/R (ϕ)] ]dϕdθ.
s= ε 0 /w Im[( E y * E z ) e x +( E z * E x ) e y +( E x * E y ) e z ].
σ= Ω s dV / Ω E * ·E dV ,
s= ε 0 4w r×[ ×Im( E * ×E) ] = ε 0 2w Im[ 2 E y * E z y (y E y * E z ) z (z E y * E z )+y x ( E z * E x )+z x ( E x * E y ) 2 E z * E x x (x E z * E z ) z (z E z * E x )+z y ( E x * E y )+x y ( E y * E z ) 2 E x * E y x (x E x * E y ) y (y E x * E y )+x z ( E y * E z )+y z ( E z * E x ) ].
E= E x e x + E y e y + E z e z =a e i δ x e x +b e i δ y e y +c e i δ z e z .
E=A[ a A e x + bcosΔ δ 3 A e y + ccosΔ δ 2 A e z ]+iB[ bsinΔ δ 3 B e y + csinΔ δ 2 B e z ] =A e 1 +iB e 2 ,
e 3 = a B 2 e x +bcsinΔ δ 1 (csinΔ δ 2 e y +bsinΔ δ 3 e z ) B a 2 B 2 + b 2 c 2 sin 2 Δ δ 1 ,
E= 1 2B [ ( b 2 sin2Δ δ 3 + c 2 sin2Δ δ 2 +i2B) e 2 2 a 2 B 2 + (bcsinΔ δ 1 ) 2 e 3 ] = E 2 e 2 + E 3 e 3 .
s 3 = 2Im( E 2 * E 3 ) | E 2 | 2 +| E 3 | 2 = 2 (absinΔ δ 3 ) 2 + (acsinΔ δ 2 ) 2 + (bcsinΔ δ 1 ) 2 a 2 + b 2 + c 2 = 2 [Im( E x * E y )] 2 + [Im( E y * E z )] 2 + [Im( E z * E x )] 2 | E x | 2 +| E y | 2 +| E z | 2 .

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