Abstract

We develop a method for generating focused vector beams with circular polarization at any transverse plane. Based on the Richards-Wolf vector model, we derive analytical expressions to describe the propagation of these set of beams near the focal area. Since the polarization and the amplitude of the input beam are not uniform, an interferometric system capable of generating spatially-variant polarized beams has to be used. In particular, this wavefront is manipulated by means of spatial light modulators displaying computer generated holograms and subsequently focused using a high numerical aperture objective lens. Experimental results using a NA = 0.85 system are provided: irradiance and Stokes images of the focused field at different planes near the focal plane are presented and compared with those obtained by numerical simulation.

© 2014 Optical Society of America

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References

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  1. R. Dorn, S. Quabis, G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev, Lett. 91, 233901 (2003).
    [CrossRef]
  2. N. Davidson, N. Bokor, “High-numerical-aperture focusing of radially polarized doughnut beams with a parabolic mirror and a flat diffractive lens,” Opt. Lett. 29, 1318–1320 (2004).
    [CrossRef] [PubMed]
  3. M. Leutenegger, R. Rao, R. A. Leitgeb, T. Lasser, “Fast focus field calculations,” Opt. Express 14, 11277–11291 (2006).
    [CrossRef] [PubMed]
  4. Y. Kozawa, S. Sato, “Sharper focal spot formed by higher-order radially polarized laser beams,” J. Opt. Soc. Am. A 24, 1793–1798 (2007).
    [CrossRef]
  5. H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nature Photon. 2, 501–505 (2008).
    [CrossRef]
  6. G. Lerman, U. Levy, “Effect of radial polarization and apodization on spot size under tight focusing conditions,” Opt. Express 16, 4567–4581 (2008).
    [CrossRef] [PubMed]
  7. X. Hao, C. Kuang, T. Wang, X. Liu, “Phase encoding for sharper focus of the azimuthally polarized beam,” Opt. Lett. 35, 3928–3930 (2010).
    [CrossRef] [PubMed]
  8. S. N. Khonina, S. G. Volotovsky, “Controlling the contribution of the electric field components to the focus of a high-aperture lens using binary phase structures,” J. Opt. Soc. Am. A 27, 2188–2197 (2010).
    [CrossRef]
  9. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1, 1–57 (2009).
    [CrossRef]
  10. R. Martinez-Herrero, I. Juvells, A. Carnicer, “On the physical realizability of highly focused electromagnetic field distributions,” Opt. Lett. 38, 2065–2067 (2013).
    [CrossRef] [PubMed]
  11. M. R. Foreman, S. S. Sherif, P. R. T. Munro, P. Török, “Inversion of the Debye-Wolf diffraction integral using an eigenfunction representation of the electric fields in the focal region,” Opt. Express 16, 4901–4917 (2008).
    [CrossRef] [PubMed]
  12. K. Jahn, N. Bokor, “Solving the inverse problem of high numerical aperture focusing using vector Slepian harmonics and vector Slepian multipole fields”, Opt. Commun. 288, 13–16 (2013).
    [CrossRef]
  13. C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
    [CrossRef]
  14. H.-T. Wang, X.-L. Wang, Y. Li, J. Chen, C.-S. Guo, J. Ding, “A new type of vector fields with hybrid states of polarization,” Opt. Express 18, 10786–10795 (2010).
    [CrossRef] [PubMed]
  15. I. Moreno, C. Iemmi, J. Campos, M. Yzuel, “Jones matrix treatment for optical Fourier processors with structured polarization,” Opt. Express 19, 4583–4594 (2011).
    [CrossRef] [PubMed]
  16. F. Kenny, D. Lara, O. G. Rodríguez-Herrera, C. Dainty, “Complete polarization and phase control for focus-shaping in high-na microscopy,” Opt. Express 20, 14015–14029 (2012).
    [CrossRef] [PubMed]
  17. D. Maluenda, I. Juvells, R. Martínez-Herrero, A. Carnicer, “Reconfigurable beams with arbitrary polarization and shape distributions at a given plane,” Opt. Express 21, 5432–5439 (2013).
    [CrossRef] [PubMed]
  18. W. Han, Y. Yang, W. Cheng, Q. Zhan, “Vectorial optical field generator for the creation of arbitrarily complex fields,” Opt. Express 21, 20692–20706 (2013).
    [CrossRef] [PubMed]
  19. E. H. Waller, G. von Freymann, “Independent spatial intensity, phase and polarization distributions,” Opt. Express 21, 28167–28174 (2013).
    [CrossRef]
  20. Z.-Y. Rong, Y.-J. Han, S.-Z. Wang, C.-S Guo, “Generation of arbitrary vector beams with cascaded liquid crystal spatial light modulators,” Opt. Express 22, 1636–1644 (2014).
    [CrossRef] [PubMed]
  21. C.-S. Guo, Z.-Y. Rong, S.-Z. Wang, “Double-channel vector spatial light modulator for generation of arbitrary complex vector beams,” Opt. Lett. 39, 386–389 (2014).
    [CrossRef] [PubMed]
  22. G. Brakenhoff, P. Blom, P. Barends, “Confocal scanning light microscopy with high aperture immersion lenses,” J. Microsc. 117, 219–232 (1979).
    [CrossRef]
  23. C. Sheppard, A. Choudhury, “Annular pupils, radial polarization, and superresolution,” Appl. Opt. 43, 4322–4327 (2004).
    [CrossRef] [PubMed]
  24. K. Kitamura, K. Sakai, S. Noda, “Sub-wavelength focal spot with long depth of focus generated by radially polarized, narrow-width annular beam,” Opt. Express 18, 4518–4525 (2010).
    [CrossRef] [PubMed]
  25. D. Biss, T. Brown, “Polarization-vortex-driven second-harmonic generation,” Opt. Lett. 28, 923–925 (2003).
    [CrossRef] [PubMed]
  26. D. Oron, E. Tal, Y. Silberberg, “Depth-resolved multiphoton polarization microscopy by third-harmonic generation,” Opt. Lett. 28, 2315–2317 (2003).
    [CrossRef] [PubMed]
  27. O. Masihzadeh, P. Schlup, R. A. Bartels, “Enhanced spatial resolution in third-harmonic microscopy through polarization switching,” Opt. Lett. 34, 1240–1242 (2009).
    [CrossRef] [PubMed]
  28. Y. Gorodetski, A. Niv, V. Kleiner, E. Hasman, “Observation of the spin-based plasmonic effect in nanoscale structures,” Phys. Rev. Lett. 101, 043903 (2008).
    [CrossRef] [PubMed]
  29. L. Vuong, A. Adam, J. Brok, P. Planken, H. Urbach, “Electromagnetic spin-orbit interactions via scattering of subwavelength apertures,” Phys. Rev. Lett. 104, 083903 (2010).
    [CrossRef] [PubMed]
  30. L. D. Barron, Molecular Light Scattering and Optical Activity (Cambridge University, 2004).
    [CrossRef]
  31. Y. Inoue, V. Ramamurthy, Chiral Photochemistry (CRC, 2004).
  32. A. Turpin, Y. V. Loiko, T. K. Kalkandjiev, J. Mompart, “Multiple rings formation in cascaded conical refraction,” Opt. Lett. 38, 1455–1457 (2013).
    [CrossRef] [PubMed]
  33. B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” P. Roy. Soc. London A Mat. 253, 358–379 (1959).
    [CrossRef]
  34. V. Arrizón, L. González, R. Ponce, A. Serrano-Heredia, “Computer-generated holograms with optimum bandwidths obtained with twisted-nematic liquid-crystal displays,” Appl. Opt. 44, 1625–1634 (2005).
    [CrossRef] [PubMed]
  35. V. Arrizón, “Complex modulation with a twisted-nematic liquid-crystal spatial light modulator: double-pixel approach,” Opt. Lett. 28, 1359–1361 (2003).
    [CrossRef] [PubMed]
  36. M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).
    [CrossRef]

2014 (2)

2013 (6)

2012 (1)

2011 (1)

2010 (5)

2009 (2)

2008 (4)

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nature Photon. 2, 501–505 (2008).
[CrossRef]

G. Lerman, U. Levy, “Effect of radial polarization and apodization on spot size under tight focusing conditions,” Opt. Express 16, 4567–4581 (2008).
[CrossRef] [PubMed]

M. R. Foreman, S. S. Sherif, P. R. T. Munro, P. Török, “Inversion of the Debye-Wolf diffraction integral using an eigenfunction representation of the electric fields in the focal region,” Opt. Express 16, 4901–4917 (2008).
[CrossRef] [PubMed]

Y. Gorodetski, A. Niv, V. Kleiner, E. Hasman, “Observation of the spin-based plasmonic effect in nanoscale structures,” Phys. Rev. Lett. 101, 043903 (2008).
[CrossRef] [PubMed]

2007 (2)

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[CrossRef]

Y. Kozawa, S. Sato, “Sharper focal spot formed by higher-order radially polarized laser beams,” J. Opt. Soc. Am. A 24, 1793–1798 (2007).
[CrossRef]

2006 (1)

2005 (1)

2004 (2)

2003 (4)

1979 (1)

G. Brakenhoff, P. Blom, P. Barends, “Confocal scanning light microscopy with high aperture immersion lenses,” J. Microsc. 117, 219–232 (1979).
[CrossRef]

1959 (1)

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” P. Roy. Soc. London A Mat. 253, 358–379 (1959).
[CrossRef]

Adam, A.

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

Arrizón, V.

Barends, P.

G. Brakenhoff, P. Blom, P. Barends, “Confocal scanning light microscopy with high aperture immersion lenses,” J. Microsc. 117, 219–232 (1979).
[CrossRef]

Barron, L. D.

L. D. Barron, Molecular Light Scattering and Optical Activity (Cambridge University, 2004).
[CrossRef]

Bartels, R. A.

Bernet, S.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[CrossRef]

Biss, D.

Blom, P.

G. Brakenhoff, P. Blom, P. Barends, “Confocal scanning light microscopy with high aperture immersion lenses,” J. Microsc. 117, 219–232 (1979).
[CrossRef]

Bokor, N.

K. Jahn, N. Bokor, “Solving the inverse problem of high numerical aperture focusing using vector Slepian harmonics and vector Slepian multipole fields”, Opt. Commun. 288, 13–16 (2013).
[CrossRef]

N. Davidson, N. Bokor, “High-numerical-aperture focusing of radially polarized doughnut beams with a parabolic mirror and a flat diffractive lens,” Opt. Lett. 29, 1318–1320 (2004).
[CrossRef] [PubMed]

Born, M.

M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).
[CrossRef]

Brakenhoff, G.

G. Brakenhoff, P. Blom, P. Barends, “Confocal scanning light microscopy with high aperture immersion lenses,” J. Microsc. 117, 219–232 (1979).
[CrossRef]

Brok, J.

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

Brown, T.

Campos, J.

Carnicer, A.

Chen, J.

Cheng, W.

Chong, C. T.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nature Photon. 2, 501–505 (2008).
[CrossRef]

Choudhury, A.

Dainty, C.

Davidson, N.

Ding, J.

Dorn, R.

R. Dorn, S. Quabis, G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev, Lett. 91, 233901 (2003).
[CrossRef]

Foreman, M. R.

Fürhapter, S.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[CrossRef]

González, L.

Gorodetski, Y.

Y. Gorodetski, A. Niv, V. Kleiner, E. Hasman, “Observation of the spin-based plasmonic effect in nanoscale structures,” Phys. Rev. Lett. 101, 043903 (2008).
[CrossRef] [PubMed]

Guo, C.-S

Guo, C.-S.

Han, W.

Han, Y.-J.

Hao, X.

Hasman, E.

Y. Gorodetski, A. Niv, V. Kleiner, E. Hasman, “Observation of the spin-based plasmonic effect in nanoscale structures,” Phys. Rev. Lett. 101, 043903 (2008).
[CrossRef] [PubMed]

Iemmi, C.

Inoue, Y.

Y. Inoue, V. Ramamurthy, Chiral Photochemistry (CRC, 2004).

Jahn, K.

K. Jahn, N. Bokor, “Solving the inverse problem of high numerical aperture focusing using vector Slepian harmonics and vector Slepian multipole fields”, Opt. Commun. 288, 13–16 (2013).
[CrossRef]

Jesacher, A.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[CrossRef]

Juvells, I.

Kalkandjiev, T. K.

Kenny, F.

Khonina, S. N.

Kitamura, K.

Kleiner, V.

Y. Gorodetski, A. Niv, V. Kleiner, E. Hasman, “Observation of the spin-based plasmonic effect in nanoscale structures,” Phys. Rev. Lett. 101, 043903 (2008).
[CrossRef] [PubMed]

Kozawa, Y.

Kuang, C.

Lara, D.

Lasser, T.

Leitgeb, R. A.

Lerman, G.

Leuchs, G.

R. Dorn, S. Quabis, G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev, Lett. 91, 233901 (2003).
[CrossRef]

Leutenegger, M.

Levy, U.

Li, Y.

Liu, X.

Loiko, Y. V.

Lukyanchuk, B.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nature Photon. 2, 501–505 (2008).
[CrossRef]

Maluenda, D.

Martinez-Herrero, R.

Martínez-Herrero, R.

Masihzadeh, O.

Maurer, C.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[CrossRef]

Mompart, J.

Moreno, I.

Munro, P. R. T.

Niv, A.

Y. Gorodetski, A. Niv, V. Kleiner, E. Hasman, “Observation of the spin-based plasmonic effect in nanoscale structures,” Phys. Rev. Lett. 101, 043903 (2008).
[CrossRef] [PubMed]

Noda, S.

Oron, D.

Planken, P.

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

Ponce, R.

Quabis, S.

R. Dorn, S. Quabis, G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev, Lett. 91, 233901 (2003).
[CrossRef]

Ramamurthy, V.

Y. Inoue, V. Ramamurthy, Chiral Photochemistry (CRC, 2004).

Rao, R.

Richards, B.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” P. Roy. Soc. London A Mat. 253, 358–379 (1959).
[CrossRef]

Ritsch-Marte, M.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[CrossRef]

Rodríguez-Herrera, O. G.

Rong, Z.-Y.

Sakai, K.

Sato, S.

Schlup, P.

Serrano-Heredia, A.

Sheppard, C.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nature Photon. 2, 501–505 (2008).
[CrossRef]

C. Sheppard, A. Choudhury, “Annular pupils, radial polarization, and superresolution,” Appl. Opt. 43, 4322–4327 (2004).
[CrossRef] [PubMed]

Sherif, S. S.

Shi, L.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nature Photon. 2, 501–505 (2008).
[CrossRef]

Silberberg, Y.

Tal, E.

Török, P.

Turpin, A.

Urbach, H.

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

Volotovsky, S. G.

von Freymann, G.

Vuong, L.

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

Waller, E. H.

Wang, H.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nature Photon. 2, 501–505 (2008).
[CrossRef]

Wang, H.-T.

Wang, S.-Z.

Wang, T.

Wang, X.-L.

Wolf, E.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” P. Roy. Soc. London A Mat. 253, 358–379 (1959).
[CrossRef]

M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).
[CrossRef]

Yang, Y.

Yzuel, M.

Zhan, Q.

Adv. Opt. Photon. (1)

Appl. Opt. (2)

J. Microsc. (1)

G. Brakenhoff, P. Blom, P. Barends, “Confocal scanning light microscopy with high aperture immersion lenses,” J. Microsc. 117, 219–232 (1979).
[CrossRef]

J. Opt. Soc. Am. A (2)

Nature Photon. (1)

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nature Photon. 2, 501–505 (2008).
[CrossRef]

New J. Phys. (1)

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[CrossRef]

Opt. Commun. (1)

K. Jahn, N. Bokor, “Solving the inverse problem of high numerical aperture focusing using vector Slepian harmonics and vector Slepian multipole fields”, Opt. Commun. 288, 13–16 (2013).
[CrossRef]

Opt. Express (11)

M. R. Foreman, S. S. Sherif, P. R. T. Munro, P. Török, “Inversion of the Debye-Wolf diffraction integral using an eigenfunction representation of the electric fields in the focal region,” Opt. Express 16, 4901–4917 (2008).
[CrossRef] [PubMed]

K. Kitamura, K. Sakai, S. Noda, “Sub-wavelength focal spot with long depth of focus generated by radially polarized, narrow-width annular beam,” Opt. Express 18, 4518–4525 (2010).
[CrossRef] [PubMed]

H.-T. Wang, X.-L. Wang, Y. Li, J. Chen, C.-S. Guo, J. Ding, “A new type of vector fields with hybrid states of polarization,” Opt. Express 18, 10786–10795 (2010).
[CrossRef] [PubMed]

I. Moreno, C. Iemmi, J. Campos, M. Yzuel, “Jones matrix treatment for optical Fourier processors with structured polarization,” Opt. Express 19, 4583–4594 (2011).
[CrossRef] [PubMed]

F. Kenny, D. Lara, O. G. Rodríguez-Herrera, C. Dainty, “Complete polarization and phase control for focus-shaping in high-na microscopy,” Opt. Express 20, 14015–14029 (2012).
[CrossRef] [PubMed]

D. Maluenda, I. Juvells, R. Martínez-Herrero, A. Carnicer, “Reconfigurable beams with arbitrary polarization and shape distributions at a given plane,” Opt. Express 21, 5432–5439 (2013).
[CrossRef] [PubMed]

W. Han, Y. Yang, W. Cheng, Q. Zhan, “Vectorial optical field generator for the creation of arbitrarily complex fields,” Opt. Express 21, 20692–20706 (2013).
[CrossRef] [PubMed]

E. H. Waller, G. von Freymann, “Independent spatial intensity, phase and polarization distributions,” Opt. Express 21, 28167–28174 (2013).
[CrossRef]

Z.-Y. Rong, Y.-J. Han, S.-Z. Wang, C.-S Guo, “Generation of arbitrary vector beams with cascaded liquid crystal spatial light modulators,” Opt. Express 22, 1636–1644 (2014).
[CrossRef] [PubMed]

G. Lerman, U. Levy, “Effect of radial polarization and apodization on spot size under tight focusing conditions,” Opt. Express 16, 4567–4581 (2008).
[CrossRef] [PubMed]

M. Leutenegger, R. Rao, R. A. Leitgeb, T. Lasser, “Fast focus field calculations,” Opt. Express 14, 11277–11291 (2006).
[CrossRef] [PubMed]

Opt. Lett. (9)

N. Davidson, N. Bokor, “High-numerical-aperture focusing of radially polarized doughnut beams with a parabolic mirror and a flat diffractive lens,” Opt. Lett. 29, 1318–1320 (2004).
[CrossRef] [PubMed]

X. Hao, C. Kuang, T. Wang, X. Liu, “Phase encoding for sharper focus of the azimuthally polarized beam,” Opt. Lett. 35, 3928–3930 (2010).
[CrossRef] [PubMed]

C.-S. Guo, Z.-Y. Rong, S.-Z. Wang, “Double-channel vector spatial light modulator for generation of arbitrary complex vector beams,” Opt. Lett. 39, 386–389 (2014).
[CrossRef] [PubMed]

D. Biss, T. Brown, “Polarization-vortex-driven second-harmonic generation,” Opt. Lett. 28, 923–925 (2003).
[CrossRef] [PubMed]

D. Oron, E. Tal, Y. Silberberg, “Depth-resolved multiphoton polarization microscopy by third-harmonic generation,” Opt. Lett. 28, 2315–2317 (2003).
[CrossRef] [PubMed]

O. Masihzadeh, P. Schlup, R. A. Bartels, “Enhanced spatial resolution in third-harmonic microscopy through polarization switching,” Opt. Lett. 34, 1240–1242 (2009).
[CrossRef] [PubMed]

R. Martinez-Herrero, I. Juvells, A. Carnicer, “On the physical realizability of highly focused electromagnetic field distributions,” Opt. Lett. 38, 2065–2067 (2013).
[CrossRef] [PubMed]

V. Arrizón, “Complex modulation with a twisted-nematic liquid-crystal spatial light modulator: double-pixel approach,” Opt. Lett. 28, 1359–1361 (2003).
[CrossRef] [PubMed]

A. Turpin, Y. V. Loiko, T. K. Kalkandjiev, J. Mompart, “Multiple rings formation in cascaded conical refraction,” Opt. Lett. 38, 1455–1457 (2013).
[CrossRef] [PubMed]

P. Roy. Soc. London A Mat. (1)

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” P. Roy. Soc. London A Mat. 253, 358–379 (1959).
[CrossRef]

Phys. Rev, Lett. (1)

R. Dorn, S. Quabis, G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev, Lett. 91, 233901 (2003).
[CrossRef]

Phys. Rev. Lett. (2)

Y. Gorodetski, A. Niv, V. Kleiner, E. Hasman, “Observation of the spin-based plasmonic effect in nanoscale structures,” Phys. Rev. Lett. 101, 043903 (2008).
[CrossRef] [PubMed]

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

Other (3)

L. D. Barron, Molecular Light Scattering and Optical Activity (Cambridge University, 2004).
[CrossRef]

Y. Inoue, V. Ramamurthy, Chiral Photochemistry (CRC, 2004).

M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Notation and sketch of a highly focused optical system.

Fig. 2
Fig. 2

Irradiance maps for a circularly-polarized highly-focused beam (NA = 0.85): (a) |E+|2, (b) |Ez|2 and (c) I. (d) Profiles of I at z = 0 (red) z = −3λ (blue), z = −5λ (magenta) and z = −7λ (black).

Fig. 3
Fig. 3

Sketch of the optical setup. Light source: HeNe laser λ= 633nm; P1, P2 and P3: linear polarizers; PBS1 and PBS2; polarizing beam splitters; M1 and M2: mirrors; HWP: half-wave plate: QWP: quarter-wave plate; SLM1 and SLM2: spatial light modulators; L1, L2 and L3: lenses; BS: beam splitter; MO: microscope objective.

Fig. 4
Fig. 4

A certain complex value C is generated as a combination of phasors ML and MR (that belong to the modulation response curve), and EL and ER that are diffracted off-axis and removed. The inset shows the subset of C values used to generate the holograms.

Fig. 5
Fig. 5

Experimental results at the observation plane: the first row corresponds to the image captured by camera 2. These images are normalized to its corresponding maximum. The second row shows the profile of the experimental images (black dots) and the numeric evaluation of I(red solid line) for z = −3.5λ, −5λ and −7λ.

Fig. 6
Fig. 6

Stokes images of the focused field at z = −3.5λ.

Tables (1)

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Table 1 Si values for the three transverse planes z analyzed.

Equations (24)

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E ( r , ϕ , z ) = A 0 θ 0 0 2 π cos θ [ f 1 ( θ , φ ) e 1 ( φ ) + f 2 ( θ , φ ) e 2 o ( θ , φ ) ] × e i k r sin θ cos ( ϕ φ ) e i k z cos θ sin θ d θ d φ ,
f 1 ( θ , φ ) = E S ( θ , φ ) e 1 ( φ )
f 2 ( θ , φ ) = E S ( θ , φ ) e 2 i ( φ ) ,
e 1 ( φ ) = ( sin φ , cos φ , 0 )
e 2 i ( φ ) = ( cos φ , sin φ , 0 )
e 2 0 ( θ , φ ) = ( cos θ cos φ , cos θ sin φ , sin θ ) .
u ± = 1 2 ( 1 , ± i , 0 ) u z = ( 0 , 0 , 1 ) ,
E ± ( r , ϕ , z ) = A 2 0 θ 0 0 2 π cos θ [ i f 1 ( θ , φ ) + cos θ f 2 ( θ , φ ) ] × × e i k r sin θ cos ( ϕ φ ) e i k z cos θ e i φ sin θ d θ d φ
E ± ( r , ϕ , z ) = A 0 θ 0 0 2 π cos θ sin θ f 2 ( θ , φ ) e i k r sin θ cos ( ϕ φ ) e i k z cos θ sin θ d θ d φ .
f 1 ( θ , φ ) = ± i cos θ g ( θ , φ )
f 2 ( θ , φ ) = g ( θ , φ )
E + ( r , ϕ , z ) = 2 A 2 0 θ 0 0 2 π cos θ cos θ g ( θ ) e i φ e i k sin θ r cos ( ϕ φ ) e i k z cos θ sin θ d θ d φ
E ( r , ϕ , z ) = 0
E z ( r , ϕ , z ) = A 0 θ 0 0 2 π cos θ sin θ g ( θ ) e i k sin θ r cos ( ϕ φ ) e i k z cos θ sin θ d θ d φ .
E + ( r , ϕ , z ) = i 4 π A 2 e i ϕ 0 θ 0 cos θ cos θ g ( θ ) J 1 ( k r sin θ ) e i k z cos θ sin θ d θ
E z ( r , z ) = 2 π A 0 θ 0 cos θ sin θ g ( θ ) J 0 ( k r sin θ ) e i k z cos θ sin θ d θ ,
E S = ( cos φ i cos θ sin φ ) g ( θ ) e x + ( sin φ + i cos θ cos φ ) g ( θ ) e y
C x ( ρ , φ ) = cos φ i 1 ρ 2 sin φ
C y ( ρ , φ ) = sin φ + i 1 ρ 2 sin φ .
S 0 = I ( 0 ° , 0 ) + I ( 90 ° , 0 )
S 1 = I ( 0 ° , 0 ) I ( 90 ° , 0 )
S 2 = I ( 45 ° , 0 ) I ( 135 ° , 0 )
S 3 = I ( 45 ° , π / 2 ) I ( 135 ° , π / 2 ) ,
S i 2 = S i 2 ( k , l ) S 0 2 ( k , l ) i = 1 , 2 , 3

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