Abstract

Based on the Richards-Wolf formalism, we obtain for the first time a set of explicit analytical expressions that completely describe a light field with a double higher-order singularity (phase and polarization), as well as distributions of its intensity and energy flux near the focus. A light field with the double singularity is an optical vortex with a topological charge m and with nth-order cylindrical polarization (azimuthal or radial). From the theory developed, rather general predictions follow. 1) For any singularity orders m and n, the intensity distribution near the focus has a symmetry of order 2(n – 1), while the longitudinal component of the Poynting vector has always an axially symmetric distribution. 2) If n = m + 2, there is a reverse energy flux on the optical axis near the focus, which is comparable in magnitude with the forward flux. 3) If m0, forward and reverse energy fluxes rotate along a spiral around the optical axis, whereas at m = 0 the energy flux is irrotational. 4) For any values of m and n, there is a toroidal energy flux in the focal area near the dark rings in the distribution of the longitudinal component of the Poynting vector.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. 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, 358–379 (1959).
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
  5. B. Chen and J. Pu, “Tight focusing of elliptically polarized vortex beams,” Appl. Opt. 48(7), 1288–1294 (2009).
    [Crossref] [PubMed]
  6. M. Rashid, O. M. Marago, and P. H. Jones, “Focusing of high order cylindrical vector beams,” J. Opt. A, Pure Appl. Opt. 11(6), 065204 (2009).
    [Crossref]
  7. G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107(5), 053601 (2011).
    [Crossref] [PubMed]
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    [Crossref]
  11. L. Gong, Y. Ren, W. Liu, M. Wang, M. Zhong, Z. Wang, and Y. Li, “Generation of cylindrical polarized vector vortex beams with digital micromirror device,” J. Appl. Phys. 116(18), 183105 (2014).
    [Crossref]
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    [Crossref]
  13. Y. Han, L. Chen, Y. G. Liu, Z. Wang, H. Zhang, K. Yang, and K. C. Chou, “Orbital angular momentum transition of light using a cylindrical vector beam,” Opt. Lett. 43(9), 2146–2149 (2018).
    [Crossref] [PubMed]
  14. Y. Li, Z. Zhu, X. Wang, L. Gong, M. Wang, and S. Nie, “Propagation evolution of an off-axis high-order cylindrical vector beam,” J. Opt. Soc. Am. A 31(11), 2356–2361 (2014).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  17. V. V. Kotlyar, A. G. Nalimov, and A. A. Kovalev, “Helical reverse flux of light of a focused optical vortex,” J. Opt. 20(9), 095603 (2018).
    [Crossref]
  18. V. V. Kotlyar, A. G. Nalimov, and S. S. Stafeev, “Energy backflow in the focus of an optical vortex,” Laser Phys. 28(12), 126203 (2018).
    [Crossref]
  19. S. S. Stafeev, A. G. Nalimov, and V. V. Kotlyar, “Energy backflow in the focal spot of a cylindrical vector beam,” Comput. Opt. 42(5), 744–750 (2018).
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
  22. M. V. Vasnetsov, V. N. Gorshkov, I. G. Marienko, and M. S. Soskin, “Wave front motion in the vicinity of a phase dislocation: optical vortex,” Opt. Spectrosc. 88(2), 260–265 (2000).
    [Crossref]
  23. MV Berry, Optical currents, J. Opt. A Pure Appl. 11(1), 1–12 (2009).
  24. Y. Roichman, B. Sun, A. Stolarski, and D. G. Grier, “Influence of nonconservative optical forces on the dynamics of optically trapped colloidal spheres: the fountain of probability,” Phys. Rev. Lett. 101(12), 128301 (2008).
    [Crossref] [PubMed]
  25. V. V. Kotlyar, S. S. Stafeev, and A. G. Nalimov, “Energy backflow in the focus of a light beam with phase or polarization singularity,” Phys. Rev. A. 99(3), 033840 (2019).
    [Crossref]
  26. Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124(5–6), 529–541 (1996).
    [Crossref]

2019 (1)

V. V. Kotlyar, S. S. Stafeev, and A. G. Nalimov, “Energy backflow in the focus of a light beam with phase or polarization singularity,” Phys. Rev. A. 99(3), 033840 (2019).
[Crossref]

2018 (7)

X. Zhang, R. Chen, and A. Wang, “Focusing properties of cylindrical vector vortex beams,” Opt. Commun. 414, 10–15 (2018).
[Crossref]

Y. Han, L. Chen, Y. G. Liu, Z. Wang, H. Zhang, K. Yang, and K. C. Chou, “Orbital angular momentum transition of light using a cylindrical vector beam,” Opt. Lett. 43(9), 2146–2149 (2018).
[Crossref] [PubMed]

S. Matsusaka, Y. Kozawa, and S. Sato, “Micro-hole drilling by tightly focused vector beams,” Opt. Lett. 43(7), 1542–1545 (2018).
[Crossref] [PubMed]

V. V. Kotlyar, A. A. Kovalev, and A. G. Nalimov, “Energy density and energy flux in the focus of an optical vortex: reverse flux of light energy,” Opt. Lett. 43(12), 2921–2924 (2018).
[Crossref] [PubMed]

V. V. Kotlyar, A. G. Nalimov, and A. A. Kovalev, “Helical reverse flux of light of a focused optical vortex,” J. Opt. 20(9), 095603 (2018).
[Crossref]

V. V. Kotlyar, A. G. Nalimov, and S. S. Stafeev, “Energy backflow in the focus of an optical vortex,” Laser Phys. 28(12), 126203 (2018).
[Crossref]

S. S. Stafeev, A. G. Nalimov, and V. V. Kotlyar, “Energy backflow in the focal spot of a cylindrical vector beam,” Comput. Opt. 42(5), 744–750 (2018).
[Crossref]

2014 (3)

2013 (1)

T. Wang, C. Kuang, X. Hao, and X. Liu, “Focusing properties of cylindrical vector vortex beams with high numerical aperture objective,” Optik (Stuttg.) 124(21), 4762–4765 (2013).
[Crossref]

2012 (1)

2011 (2)

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107(5), 053601 (2011).
[Crossref] [PubMed]

A. Holleczek, A. Aiello, C. Gabriel, C. Marquardt, and G. Leuchs, “Classical and quantum properties of cylindrically polarized states of light,” Opt. Express 19(10), 9714–9736 (2011).
[Crossref] [PubMed]

2009 (4)

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
[Crossref]

B. Chen and J. Pu, “Tight focusing of elliptically polarized vortex beams,” Appl. Opt. 48(7), 1288–1294 (2009).
[Crossref] [PubMed]

M. Rashid, O. M. Marago, and P. H. Jones, “Focusing of high order cylindrical vector beams,” J. Opt. A, Pure Appl. Opt. 11(6), 065204 (2009).
[Crossref]

MV Berry, Optical currents, J. Opt. A Pure Appl. 11(1), 1–12 (2009).

2008 (1)

Y. Roichman, B. Sun, A. Stolarski, and D. G. Grier, “Influence of nonconservative optical forces on the dynamics of optically trapped colloidal spheres: the fountain of probability,” Phys. Rev. Lett. 101(12), 128301 (2008).
[Crossref] [PubMed]

2002 (1)

2000 (2)

K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000).
[Crossref] [PubMed]

M. V. Vasnetsov, V. N. Gorshkov, I. G. Marienko, and M. S. Soskin, “Wave front motion in the vicinity of a phase dislocation: optical vortex,” Opt. Spectrosc. 88(2), 260–265 (2000).
[Crossref]

1998 (1)

M. V. Berry, “Wave dislocation reactions in non-paraxial Gaussian beams,” J. Mod. Opt. 45(9), 1845–1858 (1998).
[Crossref]

1996 (1)

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124(5–6), 529–541 (1996).
[Crossref]

1959 (1)

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, 358–379 (1959).

Aiello, A.

Alfano, R. R.

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107(5), 053601 (2011).
[Crossref] [PubMed]

Asakura, T.

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124(5–6), 529–541 (1996).
[Crossref]

Berry, M. V.

M. V. Berry, “Wave dislocation reactions in non-paraxial Gaussian beams,” J. Mod. Opt. 45(9), 1845–1858 (1998).
[Crossref]

Berry, MV

MV Berry, Optical currents, J. Opt. A Pure Appl. 11(1), 1–12 (2009).

Brown, T.

Chen, B.

Chen, L.

Chen, R.

X. Zhang, R. Chen, and A. Wang, “Focusing properties of cylindrical vector vortex beams,” Opt. Commun. 414, 10–15 (2018).
[Crossref]

Chen, S.

Chou, K. C.

Gabriel, C.

Gong, L.

Y. Li, Z. Zhu, X. Wang, L. Gong, M. Wang, and S. Nie, “Propagation evolution of an off-axis high-order cylindrical vector beam,” J. Opt. Soc. Am. A 31(11), 2356–2361 (2014).
[Crossref] [PubMed]

L. Gong, Y. Ren, W. Liu, M. Wang, M. Zhong, Z. Wang, and Y. Li, “Generation of cylindrical polarized vector vortex beams with digital micromirror device,” J. Appl. Phys. 116(18), 183105 (2014).
[Crossref]

Gorshkov, V. N.

M. V. Vasnetsov, V. N. Gorshkov, I. G. Marienko, and M. S. Soskin, “Wave front motion in the vicinity of a phase dislocation: optical vortex,” Opt. Spectrosc. 88(2), 260–265 (2000).
[Crossref]

Grier, D. G.

Y. Roichman, B. Sun, A. Stolarski, and D. G. Grier, “Influence of nonconservative optical forces on the dynamics of optically trapped colloidal spheres: the fountain of probability,” Phys. Rev. Lett. 101(12), 128301 (2008).
[Crossref] [PubMed]

Han, Y.

Hao, X.

T. Wang, C. Kuang, X. Hao, and X. Liu, “Focusing properties of cylindrical vector vortex beams with high numerical aperture objective,” Optik (Stuttg.) 124(21), 4762–4765 (2013).
[Crossref]

Harada, Y.

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124(5–6), 529–541 (1996).
[Crossref]

Holleczek, A.

Jones, P. H.

M. Rashid, O. M. Marago, and P. H. Jones, “Focusing of high order cylindrical vector beams,” J. Opt. A, Pure Appl. Opt. 11(6), 065204 (2009).
[Crossref]

Kotlyar, V. V.

V. V. Kotlyar, S. S. Stafeev, and A. G. Nalimov, “Energy backflow in the focus of a light beam with phase or polarization singularity,” Phys. Rev. A. 99(3), 033840 (2019).
[Crossref]

S. S. Stafeev, A. G. Nalimov, and V. V. Kotlyar, “Energy backflow in the focal spot of a cylindrical vector beam,” Comput. Opt. 42(5), 744–750 (2018).
[Crossref]

V. V. Kotlyar, A. G. Nalimov, and A. A. Kovalev, “Helical reverse flux of light of a focused optical vortex,” J. Opt. 20(9), 095603 (2018).
[Crossref]

V. V. Kotlyar, A. G. Nalimov, and S. S. Stafeev, “Energy backflow in the focus of an optical vortex,” Laser Phys. 28(12), 126203 (2018).
[Crossref]

V. V. Kotlyar, A. A. Kovalev, and A. G. Nalimov, “Energy density and energy flux in the focus of an optical vortex: reverse flux of light energy,” Opt. Lett. 43(12), 2921–2924 (2018).
[Crossref] [PubMed]

Kovalev, A. A.

Kozawa, Y.

Kuang, C.

T. Wang, C. Kuang, X. Hao, and X. Liu, “Focusing properties of cylindrical vector vortex beams with high numerical aperture objective,” Optik (Stuttg.) 124(21), 4762–4765 (2013).
[Crossref]

Leger, J.

Leuchs, G.

Li, Y.

Y. Li, Z. Zhu, X. Wang, L. Gong, M. Wang, and S. Nie, “Propagation evolution of an off-axis high-order cylindrical vector beam,” J. Opt. Soc. Am. A 31(11), 2356–2361 (2014).
[Crossref] [PubMed]

L. Gong, Y. Ren, W. Liu, M. Wang, M. Zhong, Z. Wang, and Y. Li, “Generation of cylindrical polarized vector vortex beams with digital micromirror device,” J. Appl. Phys. 116(18), 183105 (2014).
[Crossref]

Ling, X.

Liu, W.

L. Gong, Y. Ren, W. Liu, M. Wang, M. Zhong, Z. Wang, and Y. Li, “Generation of cylindrical polarized vector vortex beams with digital micromirror device,” J. Appl. Phys. 116(18), 183105 (2014).
[Crossref]

Liu, X.

T. Wang, C. Kuang, X. Hao, and X. Liu, “Focusing properties of cylindrical vector vortex beams with high numerical aperture objective,” Optik (Stuttg.) 124(21), 4762–4765 (2013).
[Crossref]

Liu, Y.

Liu, Y. G.

Luo, H.

Marago, O. M.

M. Rashid, O. M. Marago, and P. H. Jones, “Focusing of high order cylindrical vector beams,” J. Opt. A, Pure Appl. Opt. 11(6), 065204 (2009).
[Crossref]

Marienko, I. G.

M. V. Vasnetsov, V. N. Gorshkov, I. G. Marienko, and M. S. Soskin, “Wave front motion in the vicinity of a phase dislocation: optical vortex,” Opt. Spectrosc. 88(2), 260–265 (2000).
[Crossref]

Marquardt, C.

Matsusaka, S.

Milione, G.

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107(5), 053601 (2011).
[Crossref] [PubMed]

Nalimov, A. G.

V. V. Kotlyar, S. S. Stafeev, and A. G. Nalimov, “Energy backflow in the focus of a light beam with phase or polarization singularity,” Phys. Rev. A. 99(3), 033840 (2019).
[Crossref]

S. S. Stafeev, A. G. Nalimov, and V. V. Kotlyar, “Energy backflow in the focal spot of a cylindrical vector beam,” Comput. Opt. 42(5), 744–750 (2018).
[Crossref]

V. V. Kotlyar, A. G. Nalimov, and S. S. Stafeev, “Energy backflow in the focus of an optical vortex,” Laser Phys. 28(12), 126203 (2018).
[Crossref]

V. V. Kotlyar, A. G. Nalimov, and A. A. Kovalev, “Helical reverse flux of light of a focused optical vortex,” J. Opt. 20(9), 095603 (2018).
[Crossref]

V. V. Kotlyar, A. A. Kovalev, and A. G. Nalimov, “Energy density and energy flux in the focus of an optical vortex: reverse flux of light energy,” Opt. Lett. 43(12), 2921–2924 (2018).
[Crossref] [PubMed]

Nie, S.

Nolan, D. A.

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107(5), 053601 (2011).
[Crossref] [PubMed]

Pu, J.

Rashid, M.

M. Rashid, O. M. Marago, and P. H. Jones, “Focusing of high order cylindrical vector beams,” J. Opt. A, Pure Appl. Opt. 11(6), 065204 (2009).
[Crossref]

Ren, Y.

L. Gong, Y. Ren, W. Liu, M. Wang, M. Zhong, Z. Wang, and Y. Li, “Generation of cylindrical polarized vector vortex beams with digital micromirror device,” J. Appl. Phys. 116(18), 183105 (2014).
[Crossref]

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, 358–379 (1959).

Roichman, Y.

Y. Roichman, B. Sun, A. Stolarski, and D. G. Grier, “Influence of nonconservative optical forces on the dynamics of optically trapped colloidal spheres: the fountain of probability,” Phys. Rev. Lett. 101(12), 128301 (2008).
[Crossref] [PubMed]

Sato, S.

Soskin, M. S.

M. V. Vasnetsov, V. N. Gorshkov, I. G. Marienko, and M. S. Soskin, “Wave front motion in the vicinity of a phase dislocation: optical vortex,” Opt. Spectrosc. 88(2), 260–265 (2000).
[Crossref]

Stafeev, S. S.

V. V. Kotlyar, S. S. Stafeev, and A. G. Nalimov, “Energy backflow in the focus of a light beam with phase or polarization singularity,” Phys. Rev. A. 99(3), 033840 (2019).
[Crossref]

V. V. Kotlyar, A. G. Nalimov, and S. S. Stafeev, “Energy backflow in the focus of an optical vortex,” Laser Phys. 28(12), 126203 (2018).
[Crossref]

S. S. Stafeev, A. G. Nalimov, and V. V. Kotlyar, “Energy backflow in the focal spot of a cylindrical vector beam,” Comput. Opt. 42(5), 744–750 (2018).
[Crossref]

Stolarski, A.

Y. Roichman, B. Sun, A. Stolarski, and D. G. Grier, “Influence of nonconservative optical forces on the dynamics of optically trapped colloidal spheres: the fountain of probability,” Phys. Rev. Lett. 101(12), 128301 (2008).
[Crossref] [PubMed]

Sun, B.

Y. Roichman, B. Sun, A. Stolarski, and D. G. Grier, “Influence of nonconservative optical forces on the dynamics of optically trapped colloidal spheres: the fountain of probability,” Phys. Rev. Lett. 101(12), 128301 (2008).
[Crossref] [PubMed]

Sztul, H. I.

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107(5), 053601 (2011).
[Crossref] [PubMed]

Vasnetsov, M. V.

M. V. Vasnetsov, V. N. Gorshkov, I. G. Marienko, and M. S. Soskin, “Wave front motion in the vicinity of a phase dislocation: optical vortex,” Opt. Spectrosc. 88(2), 260–265 (2000).
[Crossref]

Wang, A.

X. Zhang, R. Chen, and A. Wang, “Focusing properties of cylindrical vector vortex beams,” Opt. Commun. 414, 10–15 (2018).
[Crossref]

Wang, M.

L. Gong, Y. Ren, W. Liu, M. Wang, M. Zhong, Z. Wang, and Y. Li, “Generation of cylindrical polarized vector vortex beams with digital micromirror device,” J. Appl. Phys. 116(18), 183105 (2014).
[Crossref]

Y. Li, Z. Zhu, X. Wang, L. Gong, M. Wang, and S. Nie, “Propagation evolution of an off-axis high-order cylindrical vector beam,” J. Opt. Soc. Am. A 31(11), 2356–2361 (2014).
[Crossref] [PubMed]

Wang, T.

T. Wang, C. Kuang, X. Hao, and X. Liu, “Focusing properties of cylindrical vector vortex beams with high numerical aperture objective,” Optik (Stuttg.) 124(21), 4762–4765 (2013).
[Crossref]

Wang, X.

Wang, Z.

Y. Han, L. Chen, Y. G. Liu, Z. Wang, H. Zhang, K. Yang, and K. C. Chou, “Orbital angular momentum transition of light using a cylindrical vector beam,” Opt. Lett. 43(9), 2146–2149 (2018).
[Crossref] [PubMed]

L. Gong, Y. Ren, W. Liu, M. Wang, M. Zhong, Z. Wang, and Y. Li, “Generation of cylindrical polarized vector vortex beams with digital micromirror device,” J. Appl. Phys. 116(18), 183105 (2014).
[Crossref]

Wen, S.

Wolf, E.

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, 358–379 (1959).

Yang, K.

Youngworth, K.

Zhan, Q.

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
[Crossref]

Q. Zhan and J. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10(7), 324–331 (2002).
[Crossref] [PubMed]

Zhang, H.

Zhang, X.

X. Zhang, R. Chen, and A. Wang, “Focusing properties of cylindrical vector vortex beams,” Opt. Commun. 414, 10–15 (2018).
[Crossref]

Zhong, M.

L. Gong, Y. Ren, W. Liu, M. Wang, M. Zhong, Z. Wang, and Y. Li, “Generation of cylindrical polarized vector vortex beams with digital micromirror device,” J. Appl. Phys. 116(18), 183105 (2014).
[Crossref]

Zhou, X.

Zhu, Z.

Adv. Opt. Photonics (1)

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
[Crossref]

Appl. Opt. (1)

Comput. Opt. (1)

S. S. Stafeev, A. G. Nalimov, and V. V. Kotlyar, “Energy backflow in the focal spot of a cylindrical vector beam,” Comput. Opt. 42(5), 744–750 (2018).
[Crossref]

J. Appl. Phys. (1)

L. Gong, Y. Ren, W. Liu, M. Wang, M. Zhong, Z. Wang, and Y. Li, “Generation of cylindrical polarized vector vortex beams with digital micromirror device,” J. Appl. Phys. 116(18), 183105 (2014).
[Crossref]

J. Mod. Opt. (1)

M. V. Berry, “Wave dislocation reactions in non-paraxial Gaussian beams,” J. Mod. Opt. 45(9), 1845–1858 (1998).
[Crossref]

J. Opt. (1)

V. V. Kotlyar, A. G. Nalimov, and A. A. Kovalev, “Helical reverse flux of light of a focused optical vortex,” J. Opt. 20(9), 095603 (2018).
[Crossref]

J. Opt. A Pure Appl. (1)

MV Berry, Optical currents, J. Opt. A Pure Appl. 11(1), 1–12 (2009).

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

M. Rashid, O. M. Marago, and P. H. Jones, “Focusing of high order cylindrical vector beams,” J. Opt. A, Pure Appl. Opt. 11(6), 065204 (2009).
[Crossref]

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

Laser Phys. (1)

V. V. Kotlyar, A. G. Nalimov, and S. S. Stafeev, “Energy backflow in the focus of an optical vortex,” Laser Phys. 28(12), 126203 (2018).
[Crossref]

Opt. Commun. (2)

X. Zhang, R. Chen, and A. Wang, “Focusing properties of cylindrical vector vortex beams,” Opt. Commun. 414, 10–15 (2018).
[Crossref]

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124(5–6), 529–541 (1996).
[Crossref]

Opt. Express (3)

Opt. Lett. (4)

Opt. Spectrosc. (1)

M. V. Vasnetsov, V. N. Gorshkov, I. G. Marienko, and M. S. Soskin, “Wave front motion in the vicinity of a phase dislocation: optical vortex,” Opt. Spectrosc. 88(2), 260–265 (2000).
[Crossref]

Optik (Stuttg.) (1)

T. Wang, C. Kuang, X. Hao, and X. Liu, “Focusing properties of cylindrical vector vortex beams with high numerical aperture objective,” Optik (Stuttg.) 124(21), 4762–4765 (2013).
[Crossref]

Phys. Rev. A. (1)

V. V. Kotlyar, S. S. Stafeev, and A. G. Nalimov, “Energy backflow in the focus of a light beam with phase or polarization singularity,” Phys. Rev. A. 99(3), 033840 (2019).
[Crossref]

Phys. Rev. Lett. (2)

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107(5), 053601 (2011).
[Crossref] [PubMed]

Y. Roichman, B. Sun, A. Stolarski, and D. G. Grier, “Influence of nonconservative optical forces on the dynamics of optically trapped colloidal spheres: the fountain of probability,” Phys. Rev. Lett. 101(12), 128301 (2008).
[Crossref] [PubMed]

Proc. R. Soc. Lond. A Math. Phys. Sci. (1)

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, 358–379 (1959).

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Figures (8)

Fig. 1
Fig. 1 Direction of polarization disregarding the phase (a,d), phase distribution (b,e) and polarization direction with the phase taken into account (c,f) in the beam of Eq. (1) with the numbers m = 1, n = 3 (a-c) and m = 2, n = 4 (d-f).
Fig. 2
Fig. 2 Polarization direction in a cylindrical second-order vectorial beam (m = 0, n = 2).
Fig. 3
Fig. 3 Distributions of intensity I = |Ex|2 + |Ey|2 + |Ez|2 (a) and of the Poynting vector components Sz (b), Sx (c), Sy (d) in focus (m = 1, n = 3).
Fig. 4
Fig. 4 Distributions of intensity I = |Ex|2 + |Ey|2 + |Ez|2 (a) and of the Poynting vector components Sz (b), Sx (c), Sy (d) in focus (m = 2, n = 4).
Fig. 5
Fig. 5 Intensity distribution and directions of the Poynting vector (white arrows) in the longitudinal plane ZY when focusing an optical vortex with the topological charge m = 2 and with linear polarization (n = 0).
Fig. 6
Fig. 6 Intensity distribution and directions of the Poynting vector (white arrows) in the longitudinal plane ZY when focusing a second-order cylindrical vector beam (m = 0, n = 2).
Fig. 7
Fig. 7 Intensity distribution (a) and directions of the Poynting vector (31) (white arrows) (b) of a superposition of two Bessel modes in the plane xz near the focus. Calculation parameters: wavelength λ = 532 nm, focal length f = 100λ, beam order (m, n) = (0, 2) (i.e. a vortex-free beam with second-order cylindrical polarization), cone angles of Bessel modes θ1 = 25 deg, θ2 = 85 deg, superposition coefficients C1 = C2 = 1, calculation area –2.5λ ≤ x ≤ 2.5λ, –λ ≤ z ≤ λ. Yellow dots (a) and red dots (b) show the centers of the toroidal vorices, while the yellow arrows (a) show the directions of Poynting vector around these centers.
Fig. 8
Fig. 8 Dependence of the longitudinal component of the scattering (blue dotted line), gradient (green dashed line), and total force (black solid line) on the axial coordinate z for a 10 nm radius particle with a refractive index of 1.5 placed on the optical axis (r = 0) near the focus of a polarization vortex (m = 0, n = 2).

Equations (36)

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E=A(θ) e imφ ( sinnφ cosnφ ),
E 0 =A(θ) e imφ .
E=A(θ) e imφ ( sinnφ cosnφ )= i 2 A(θ) e i(m+n)φ ( 1 i ) i 2 A(θ) e i(mn)φ ( 1 i ).
E x = 1 2 i m+n e i(m+n)φ ( I 0,m+n + e i2φ I 2,m+n2 ) 1 2 i mn e i(mn)φ ( I 0,mn + e i2φ I 2,mn+2 ), E y = 1 2 i m+n+1 e i(m+n)φ ( I 0,m+n e i2φ I 2,m+n2 ) 1 2 i mn+1 e i(mn)φ ( I 0,mn e i2φ I 2,mn+2 ), E z = i m+n1 e i(m+n1)φ I 1,m+n1 + i mn+1 e i(mn+1)φ I 1,mn+1 .
I 0,ν =( πf λ ) 0 α sinθ cos 1/2 θ(1+cosθ)A(θ) e ikzcosθ J ν (x)dθ , I 1,ν =( πf λ ) 0 α sin 2 θ cos 1/2 θA(θ) e ikzcosθ J ν (x)dθ , I 2,ν =( πf λ ) 0 α sinθ cos 1/2 θ(1cosθ)A(θ) e ikzcosθ J ν (x)dθ ,
A(θ)= J 1 ( 2β sinθ sinα )exp[ β 2 ( sinθ sinα ) 2 ],
E x = 1 2 i m ( e i( m2 )φ I 2,m2 e i( m+2 )φ I 2,m+2 ), E y = 1 2 i m+1 ( 2 e imφ I 0,m e i( m2 )φ I 2,m2 e i( m+2 )φ I 2,m+2 ), E z = i m+1 ( e i(m1)φ I 1,m1 + e i(m+1)φ I 1,m+1 ).
E x = i n+1 [ I 0,n sinnφ+ I 2,n2 sin( n2 )φ ], E y = i n+1 [ I 0,n cosnφ+ I 2,n2 cos( n2 )φ ], E z =2 i n I 1,n1 sin( n1 )φ.
E x =( I 0,1 + I 2,1 )sinφ, E y =( I 0,1 + I 2,1 )cosφ, E z =0.
H x = 1 2 i m+n+1 e i(m+n)φ ( I 0,m+n + e i2φ I 2,m+n2 )+ + 1 2 i mn+1 e i(mn)φ ( I 0,mn + e i2φ I 2,mn+2 ), H y = 1 2 i m+n e i(m+n)φ ( I 0,m+n e i2φ I 2,m+n2 ) 1 2 i mn e i(mn)φ ( I 0,mn e i2φ I 2,mn+2 ), H z = i m+n e i(m+n1)φ I 1,m+n1 + i mn e i(mn+1)φ I 1,mn+1 .
S z = 1 2 ( | I 0,m+n | 2 + | I 0,mn | 2 | I 2,m+n2 | 2 | I 2,mn+2 | 2 ).
S z,n=m+2 (z=r=0)= 1 2 I 2,0 2 = ( πf 2 λ 0 α sinθ cos 1/2 θ (1cosθ)A(θ)dθ ) 2 .
I= 1 2 ( | I 0,m+n | 2 + | I 0,mn | 2 + | I 2,m+n2 | 2 + | I 2,mn+2 | 2 )+ | I 1,m+n1 | 2 + | I 1,mn+1 | 2 + + ( 1 ) n+1 Re{ e 2i( n1 )φ ( I 0,mn * I 2,m+n2 + I 2,mn+2 * I 0,m+n 2 I 1,mn+1 * I 1,m+n1 ) }.
I= 1 2 ( I 0,m+n 2 + I 0,mn 2 + I 2,m+n2 2 + I 2,mn+2 2 )+ I 1,m+n1 2 + I 1,mn+1 2 + + ( 1 ) n+1 ( I 0,m+n I 2,mn+2 + I 0,mn I 2,m+n2 2 I 1,m+n1 I 1,mn+1 )cos2( n1 )φ.
S r =Im{ I 0,m+n I * 1,m+n1 I 0,mn I * 1,mn+1 I 2,m+n2 I * 1,m+n1 + I 2,mn+2 I * 1,mn+1 },
S φ =Re{ I 0,m+n I * 1,m+n1 + I 0,mn I * 1,mn+1 + I 2,m+n2 I * 1,m+n1 + I 2,mn+2 I * 1,mn+1 }
S x =Q(r)sinφ, S y =Q(r)cosφ,
Q(r)= I 1,m+n1 ( I 0,m+n + I 2,m+n2 )+ I 1,mn+1 ( I 0,mn + I 2,mn+2 ).
S z = I 0,n 2 I 2,n2 2 .
A(θ)=δ(θ θ 0 ),
S z = A 2 J n 2 (krsin θ 0 ) B 2 J n2 2 (krsin θ 0 ), A=( πλ f )sin θ 0 cos 1/2 θ 0 (1+cos θ 0 ), B=( πλ f )sin θ 0 cos 1/2 θ 0 (1cos θ 0 ).
| A J n (krsin θ 0 ) |=| B J n2 (krsin θ 0 ) |.
r p = γ n,p ksin θ 0 , J n ( γ n,p )=0,p=1,2,3....
I L = 1 2 ( 2 I 0,m 2 +2 I 1,m1 2 +2 I 1,m+1 2 + I 2,m2 2 + I 2,m+2 2 ) cos2φ( I 0,m I 2,m+2 + I 0,m I 2,m2 2 I 1,m1 I 1,m+1 ).
S L,z = 1 2 ( 2 I 0,m 2 I 2,m2 2 I 2,m+2 2 )
E= A(θ) e imφ 2 ( 1 iσ ),H= A(θ) e imφ 2 ( iσ 1 ),
E 1x = i m1 2 2 e imφ ( I 0,m + γ + e i2φ I 2,m+2 + γ e i2φ I 2,m2 ), E 1y = i m 2 2 e imφ ( σ I 0,m γ + e i2φ I 2,m+2 + γ e i2φ I 2,m2 ), E 1z = i m 2 e imφ ( γ + e iφ I 1,m+1 γ e iφ I 1,m1 ),
H 1x = i m 2 2 e imφ ( σ I 0,m + γ + e i2φ I 2,m+2 γ e i2φ I 2,m2 ), H 1y = i m1 2 2 e imφ ( σ I 0,m γ + e i2φ I 2,m+2 γ e i2φ I 2,m2 ), H 1z = i m+1 2 e imφ ( γ + e iφ I 1,m+1 + γ e iφ I 1,m1 ),
I 1 =( 1+ σ 2 8 ) I 0,m 2 + γ + 2 4 ( 2 I 1,m+1 2 + I 2,m+2 2 ) + γ 2 4 ( 2 I 1,m1 2 + I 2,m2 2 )+cos2φ( 1 σ 2 8 ) ×( I 0,m I 2,m+2 + I 0,m I 2,m2 2 I 1,m1 I 1,m+1 ).
S 1z = 1 8 [ ( 1+ σ 2 ) I 0,m 2 γ + 2 I 2,m+2 2 γ 2 I 2,m2 2 ].
S 1z+ = 1 8 ( 2 I 0,m 2 I 2,m+2 2 ), S 1z = 1 8 ( 2 I 0,m 2 I 2,m2 2 ).
S z = | I 0,n | 2 | I 2,n2 | 2 = A 1 2 J n 2 ( x 1 ) B 1 2 J n2 2 ( x 1 )+ A 2 2 J n 2 ( x 2 ) B 2 2 J n2 2 ( x 2 ) +2[ A 1 A 2 J n ( x 1 ) J n ( x 2 ) B 1 B 2 J n2 ( x 1 ) J n2 ( x 2 ) ] xcos( kzcos θ 1 kzcos θ 2 +arg C 1 arg C 2 ),
A p =( πλ f )| C p |sin θ p cos 1/2 θ p ( 1+cos θ p ), B p =( πλ f )| C p |sin θ p cos 1/2 θ p ( 1cos θ p ), x p =krsin θ p ,
F s = e z ( 8π n 2 3c ) k 4 a 6 ( n 1 2 n 2 2 n 1 2 +2 n 2 2 ) 2 S z ,
F g =( 2π n 2 c ) a 3 ( n 1 2 n 2 2 n 1 2 +2 n 2 2 ) | E | 2 ,
I= | E | 2 = I 0,2 2 + I 2,0 2 +2 I 1,1 2 2( I 0,2 I 2,0 + I 1,1 2 )cos2φ.

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