Abstract

We report a method of generating exotic optical polarization Möbius strips through tightly focusing an arbitrary vector beam. A heart-shaped Möbius strip, an “8”-shaped twin Möbius strip, and a circular Möbius strip with varying polarization twisting rate are demonstrated. The ability of tailoring three-dimensional optical polarization topologies may spur novel studies of optics and physics and find their applications in sensing, light coupling to nanostructures, light-matter interaction, and metamaterial fabrication.

© 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. C. Pickover, The Möbius Strip: Dr. August Möbius's Marvelous Band in Mathematics, Games, Literature, Art, Technology, and Cosmology (Thunder's Mouth, 2005).
  2. E. L. Starostin and G. H. van der Heijden, “The shape of a Möbius strip,” Nat. Mater. 6(8), 563–567 (2007).
    [Crossref] [PubMed]
  3. T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, “Optics. Observation of optical polarization Möbius strips,” Science 347(6225), 964–966 (2015).
    [Crossref] [PubMed]
  4. I. Freund, “Cones, spirals, and Möbius strips, in elliptically polarized light,” Opt. Commun. 249(1–3), 7–22 (2005).
    [Crossref]
  5. I. Freund, “Multitwist optical Möbius strips,” Opt. Lett. 35(2), 148–150 (2010).
    [Crossref] [PubMed]
  6. E. J. Galvez, I. Dutta, K. Beach, J. J. Zeosky, J. A. Jones, and B. Khajavi, “Multitwist Möbius strips and twisted ribbons in the polarization of paraxial light beams,” Sci. Rep. 7(1), 13653 (2017).
    [Crossref] [PubMed]
  7. A. Garcia-Etxarri, “Optical Polarization Möbius Strips on All-Dielectric Optical Scatterers,” ACS Photonics 4(5), 1159–1164 (2017).
    [Crossref]
  8. T. Bauer, M. Neugebauer, G. Leuchs, and P. Banzer, “Optical polarization Möbius strips and points of purely transverse spin density,” Phys. Rev. Lett. 117(1), 013601 (2016).
    [Crossref] [PubMed]
  9. W. Han, Y. Yang, W. Cheng, and Q. Zhan, “Vectorial optical field generator for the creation of arbitrarily complex fields,” Opt. Express 21(18), 20692–20706 (2013).
    [Crossref] [PubMed]
  10. J. Wang, W. Chen, and Q. Zhan, “Engineering of high purity ultra-long optical needle field through reversing the electric dipole array radiation,” Opt. Express 18(21), 21965–21972 (2010).
    [Crossref] [PubMed]
  11. J. Wang, W. Chen, and Q. Zhan, “Three-dimensional focus engineering using dipole array radiation pattern,” Opt. Commun. 284(12), 2668–2671 (2011).
    [Crossref]
  12. 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]
  13. Y. Yu and Q. Zhan, “Generation of uniform three-dimensional optical chain with controllable characteristics,” J. Opt. 17(10), 105606 (2015).
    [Crossref]
  14. E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 349–357 (1959).
  15. B. Richards and E. Wolf, “Electromagnetic diffraction in optical system II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
  16. M. Gu, Advanced Optical Imaging Theory (Springer, 1999).
  17. M. Leutenegger, R. Rao, R. A. Leitgeb, and T. Lasser, “Fast focus field calculations,” Opt. Express 14(23), 11277–11291 (2006).
    [Crossref] [PubMed]
  18. M. Berry, “Index formulae for singular lines of polarization,” J. Opt. A, Pure Appl. Opt. 6(7), 675–678 (2004).
    [Crossref]
  19. R. Gerchberg, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35, 237–246 (1972).
  20. H. Chen, Z. Zheng, B. F. Zhang, J. Ding, and H. T. Wang, “Polarization structuring of focused field through polarization-only modulation of incident beam,” Opt. Lett. 35(16), 2825–2827 (2010).
    [Crossref] [PubMed]
  21. M. Berry and J. Hannay, “Umbilic points on Gaussian random surfaces,” J. Phys. Math. Gen. 10(11), 1809–1821 (1977).
    [Crossref]

2017 (2)

E. J. Galvez, I. Dutta, K. Beach, J. J. Zeosky, J. A. Jones, and B. Khajavi, “Multitwist Möbius strips and twisted ribbons in the polarization of paraxial light beams,” Sci. Rep. 7(1), 13653 (2017).
[Crossref] [PubMed]

A. Garcia-Etxarri, “Optical Polarization Möbius Strips on All-Dielectric Optical Scatterers,” ACS Photonics 4(5), 1159–1164 (2017).
[Crossref]

2016 (1)

T. Bauer, M. Neugebauer, G. Leuchs, and P. Banzer, “Optical polarization Möbius strips and points of purely transverse spin density,” Phys. Rev. Lett. 117(1), 013601 (2016).
[Crossref] [PubMed]

2015 (2)

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, “Optics. Observation of optical polarization Möbius strips,” Science 347(6225), 964–966 (2015).
[Crossref] [PubMed]

Y. Yu and Q. Zhan, “Generation of uniform three-dimensional optical chain with controllable characteristics,” J. Opt. 17(10), 105606 (2015).
[Crossref]

2013 (1)

2012 (1)

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]

2011 (1)

J. Wang, W. Chen, and Q. Zhan, “Three-dimensional focus engineering using dipole array radiation pattern,” Opt. Commun. 284(12), 2668–2671 (2011).
[Crossref]

2010 (3)

2007 (1)

E. L. Starostin and G. H. van der Heijden, “The shape of a Möbius strip,” Nat. Mater. 6(8), 563–567 (2007).
[Crossref] [PubMed]

2006 (1)

2005 (1)

I. Freund, “Cones, spirals, and Möbius strips, in elliptically polarized light,” Opt. Commun. 249(1–3), 7–22 (2005).
[Crossref]

2004 (1)

M. Berry, “Index formulae for singular lines of polarization,” J. Opt. A, Pure Appl. Opt. 6(7), 675–678 (2004).
[Crossref]

1977 (1)

M. Berry and J. Hannay, “Umbilic points on Gaussian random surfaces,” J. Phys. Math. Gen. 10(11), 1809–1821 (1977).
[Crossref]

1972 (1)

R. Gerchberg, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35, 237–246 (1972).

1959 (2)

E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 349–357 (1959).

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

Banzer, P.

T. Bauer, M. Neugebauer, G. Leuchs, and P. Banzer, “Optical polarization Möbius strips and points of purely transverse spin density,” Phys. Rev. Lett. 117(1), 013601 (2016).
[Crossref] [PubMed]

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, “Optics. Observation of optical polarization Möbius strips,” Science 347(6225), 964–966 (2015).
[Crossref] [PubMed]

Bauer, T.

T. Bauer, M. Neugebauer, G. Leuchs, and P. Banzer, “Optical polarization Möbius strips and points of purely transverse spin density,” Phys. Rev. Lett. 117(1), 013601 (2016).
[Crossref] [PubMed]

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, “Optics. Observation of optical polarization Möbius strips,” Science 347(6225), 964–966 (2015).
[Crossref] [PubMed]

Beach, K.

E. J. Galvez, I. Dutta, K. Beach, J. J. Zeosky, J. A. Jones, and B. Khajavi, “Multitwist Möbius strips and twisted ribbons in the polarization of paraxial light beams,” Sci. Rep. 7(1), 13653 (2017).
[Crossref] [PubMed]

Berry, M.

M. Berry, “Index formulae for singular lines of polarization,” J. Opt. A, Pure Appl. Opt. 6(7), 675–678 (2004).
[Crossref]

M. Berry and J. Hannay, “Umbilic points on Gaussian random surfaces,” J. Phys. Math. Gen. 10(11), 1809–1821 (1977).
[Crossref]

Boyd, R. W.

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, “Optics. Observation of optical polarization Möbius strips,” Science 347(6225), 964–966 (2015).
[Crossref] [PubMed]

Chen, H.

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]

J. Wang, W. Chen, and Q. Zhan, “Three-dimensional focus engineering using dipole array radiation pattern,” Opt. Commun. 284(12), 2668–2671 (2011).
[Crossref]

J. Wang, W. Chen, and Q. Zhan, “Engineering of high purity ultra-long optical needle field through reversing the electric dipole array radiation,” Opt. Express 18(21), 21965–21972 (2010).
[Crossref] [PubMed]

Cheng, W.

Ding, J.

Dutta, I.

E. J. Galvez, I. Dutta, K. Beach, J. J. Zeosky, J. A. Jones, and B. Khajavi, “Multitwist Möbius strips and twisted ribbons in the polarization of paraxial light beams,” Sci. Rep. 7(1), 13653 (2017).
[Crossref] [PubMed]

Freund, I.

I. Freund, “Multitwist optical Möbius strips,” Opt. Lett. 35(2), 148–150 (2010).
[Crossref] [PubMed]

I. Freund, “Cones, spirals, and Möbius strips, in elliptically polarized light,” Opt. Commun. 249(1–3), 7–22 (2005).
[Crossref]

Galvez, E. J.

E. J. Galvez, I. Dutta, K. Beach, J. J. Zeosky, J. A. Jones, and B. Khajavi, “Multitwist Möbius strips and twisted ribbons in the polarization of paraxial light beams,” Sci. Rep. 7(1), 13653 (2017).
[Crossref] [PubMed]

Garcia-Etxarri, A.

A. Garcia-Etxarri, “Optical Polarization Möbius Strips on All-Dielectric Optical Scatterers,” ACS Photonics 4(5), 1159–1164 (2017).
[Crossref]

Gerchberg, R.

R. Gerchberg, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35, 237–246 (1972).

Han, W.

Hannay, J.

M. Berry and J. Hannay, “Umbilic points on Gaussian random surfaces,” J. Phys. Math. Gen. 10(11), 1809–1821 (1977).
[Crossref]

Jones, J. A.

E. J. Galvez, I. Dutta, K. Beach, J. J. Zeosky, J. A. Jones, and B. Khajavi, “Multitwist Möbius strips and twisted ribbons in the polarization of paraxial light beams,” Sci. Rep. 7(1), 13653 (2017).
[Crossref] [PubMed]

Karimi, E.

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, “Optics. Observation of optical polarization Möbius strips,” Science 347(6225), 964–966 (2015).
[Crossref] [PubMed]

Khajavi, B.

E. J. Galvez, I. Dutta, K. Beach, J. J. Zeosky, J. A. Jones, and B. Khajavi, “Multitwist Möbius strips and twisted ribbons in the polarization of paraxial light beams,” Sci. Rep. 7(1), 13653 (2017).
[Crossref] [PubMed]

Lasser, T.

Leitgeb, R. A.

Leuchs, G.

T. Bauer, M. Neugebauer, G. Leuchs, and P. Banzer, “Optical polarization Möbius strips and points of purely transverse spin density,” Phys. Rev. Lett. 117(1), 013601 (2016).
[Crossref] [PubMed]

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, “Optics. Observation of optical polarization Möbius strips,” Science 347(6225), 964–966 (2015).
[Crossref] [PubMed]

Leutenegger, M.

Marrucci, L.

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, “Optics. Observation of optical polarization Möbius strips,” Science 347(6225), 964–966 (2015).
[Crossref] [PubMed]

Neugebauer, M.

T. Bauer, M. Neugebauer, G. Leuchs, and P. Banzer, “Optical polarization Möbius strips and points of purely transverse spin density,” Phys. Rev. Lett. 117(1), 013601 (2016).
[Crossref] [PubMed]

Orlov, S.

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, “Optics. Observation of optical polarization Möbius strips,” Science 347(6225), 964–966 (2015).
[Crossref] [PubMed]

Rao, R.

Richards, B.

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

Rubano, A.

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, “Optics. Observation of optical polarization Möbius strips,” Science 347(6225), 964–966 (2015).
[Crossref] [PubMed]

Santamato, E.

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, “Optics. Observation of optical polarization Möbius strips,” Science 347(6225), 964–966 (2015).
[Crossref] [PubMed]

Starostin, E. L.

E. L. Starostin and G. H. van der Heijden, “The shape of a Möbius strip,” Nat. Mater. 6(8), 563–567 (2007).
[Crossref] [PubMed]

van der Heijden, G. H.

E. L. Starostin and G. H. van der Heijden, “The shape of a Möbius strip,” Nat. Mater. 6(8), 563–567 (2007).
[Crossref] [PubMed]

Wang, H. T.

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]

J. Wang, W. Chen, and Q. Zhan, “Three-dimensional focus engineering using dipole array radiation pattern,” Opt. Commun. 284(12), 2668–2671 (2011).
[Crossref]

J. Wang, W. Chen, and Q. Zhan, “Engineering of high purity ultra-long optical needle field through reversing the electric dipole array radiation,” Opt. Express 18(21), 21965–21972 (2010).
[Crossref] [PubMed]

Wolf, E.

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

E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 349–357 (1959).

Yang, Y.

Yu, Y.

Y. Yu and Q. Zhan, “Generation of uniform three-dimensional optical chain with controllable characteristics,” J. Opt. 17(10), 105606 (2015).
[Crossref]

Zeosky, J. J.

E. J. Galvez, I. Dutta, K. Beach, J. J. Zeosky, J. A. Jones, and B. Khajavi, “Multitwist Möbius strips and twisted ribbons in the polarization of paraxial light beams,” Sci. Rep. 7(1), 13653 (2017).
[Crossref] [PubMed]

Zhan, Q.

Y. Yu and Q. Zhan, “Generation of uniform three-dimensional optical chain with controllable characteristics,” J. Opt. 17(10), 105606 (2015).
[Crossref]

W. Han, Y. Yang, W. Cheng, and Q. Zhan, “Vectorial optical field generator for the creation of arbitrarily complex fields,” Opt. Express 21(18), 20692–20706 (2013).
[Crossref] [PubMed]

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. Wang, W. Chen, and Q. Zhan, “Three-dimensional focus engineering using dipole array radiation pattern,” Opt. Commun. 284(12), 2668–2671 (2011).
[Crossref]

J. Wang, W. Chen, and Q. Zhan, “Engineering of high purity ultra-long optical needle field through reversing the electric dipole array radiation,” Opt. Express 18(21), 21965–21972 (2010).
[Crossref] [PubMed]

Zhang, B. F.

Zheng, Z.

ACS Photonics (1)

A. Garcia-Etxarri, “Optical Polarization Möbius Strips on All-Dielectric Optical Scatterers,” ACS Photonics 4(5), 1159–1164 (2017).
[Crossref]

J. Opt. (2)

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]

Y. Yu and Q. Zhan, “Generation of uniform three-dimensional optical chain with controllable characteristics,” J. Opt. 17(10), 105606 (2015).
[Crossref]

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

M. Berry, “Index formulae for singular lines of polarization,” J. Opt. A, Pure Appl. Opt. 6(7), 675–678 (2004).
[Crossref]

J. Phys. Math. Gen. (1)

M. Berry and J. Hannay, “Umbilic points on Gaussian random surfaces,” J. Phys. Math. Gen. 10(11), 1809–1821 (1977).
[Crossref]

Nat. Mater. (1)

E. L. Starostin and G. H. van der Heijden, “The shape of a Möbius strip,” Nat. Mater. 6(8), 563–567 (2007).
[Crossref] [PubMed]

Opt. Commun. (2)

I. Freund, “Cones, spirals, and Möbius strips, in elliptically polarized light,” Opt. Commun. 249(1–3), 7–22 (2005).
[Crossref]

J. Wang, W. Chen, and Q. Zhan, “Three-dimensional focus engineering using dipole array radiation pattern,” Opt. Commun. 284(12), 2668–2671 (2011).
[Crossref]

Opt. Express (3)

Opt. Lett. (2)

Optik (Stuttg.) (1)

R. Gerchberg, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35, 237–246 (1972).

Phys. Rev. Lett. (1)

T. Bauer, M. Neugebauer, G. Leuchs, and P. Banzer, “Optical polarization Möbius strips and points of purely transverse spin density,” Phys. Rev. Lett. 117(1), 013601 (2016).
[Crossref] [PubMed]

Proc. R. Soc. Lond. A Math. Phys. Sci. (2)

E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 349–357 (1959).

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

Sci. Rep. (1)

E. J. Galvez, I. Dutta, K. Beach, J. J. Zeosky, J. A. Jones, and B. Khajavi, “Multitwist Möbius strips and twisted ribbons in the polarization of paraxial light beams,” Sci. Rep. 7(1), 13653 (2017).
[Crossref] [PubMed]

Science (1)

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, “Optics. Observation of optical polarization Möbius strips,” Science 347(6225), 964–966 (2015).
[Crossref] [PubMed]

Other (2)

M. Gu, Advanced Optical Imaging Theory (Springer, 1999).

C. Pickover, The Möbius Strip: Dr. August Möbius's Marvelous Band in Mathematics, Games, Literature, Art, Technology, and Cosmology (Thunder's Mouth, 2005).

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Figures (8)

Fig. 1
Fig. 1 Schematic of the optical configuration: An arbitrary vector beam is focused by a microscope objective lens to the focal region. The radius of the aperture stop is R, and the focal length is f. The wave vector k Ω is denoted in red arrow.
Fig. 2
Fig. 2 The flowchart of the iterative algorithm to design the polarization topology in the focal region.
Fig. 3
Fig. 3 Generation of a heart-shaped Möbius strip in the focal region. (a) The intensity and polarization distributions of the required optical fields in the pupil plane. The left-handed polarization and right-handed polarization are denoted in red and white, respectively. (b)-(c) The intensity distributions of Ex and Ey components in the pupil plane, respectively. (d)-(e) The phase distributions of Ex and Ey components in the pupil plane, respectively.
Fig. 4
Fig. 4 (a)-(c) The intensity distributions of Ex, Ey and Ez components in the z = 0 plane, respectively. (d)-(f) The phase distributions of Ex, Ey and Ez components in the z = 0 plane, respectively. (g) The total intensity distributions in the z = 0 plane. The prescribed heart-shaped path is denoted in white. (h) The polarization topology along the red heart-shaped path. The major axes of the polarization ellipses are denoted in blue and green to indicate the orientation.
Fig. 5
Fig. 5 Generation of an “8”-shaped twin Möbius strip in the focal region. (a) The intensity and polarization distributions of the required optical fields in the pupil plane. The left-handed polarization and right-handed polarization are denoted in red and white, respectively. (b)-(c) The intensity distributions of Ex and Ey components in the pupil plane, respectively. (d)-(e) The phase distributions of Ex and Ey components in the pupil plane, respectively.
Fig. 6
Fig. 6 (a)-(c) The intensity distributions of Ex, Ey and Ez components in the z = 0 plane, respectively. (d)-(f) The phase distributions of Ex, Ey and Ez components in the z = 0 plane, respectively. (g) The total intensity distributions in the z = 0 plane. The prescribed “8”-shaped path is denoted in white. (h) The polarization topology along the red “8”-shaped path. The major axes of the polarization ellipses are denoted in blue and green to indicate the orientation..
Fig. 7
Fig. 7 Generation of a circular Möbius strip with varying twisting speed in the focal region. (a) The intensity and polarization distributions of the required optical fields in the pupil plane. The left-handed polarization and right-handed polarization are denoted in red and white, respectively. (b)-(c) The intensity distributions of Ex and Ey components in the pupil plane, respectively. (d)-(e) The phase distributions of Ex and Ey components in the pupil plane, respectively.
Fig. 8
Fig. 8 (a)-(c) The intensity distributions of Ex, Ey and Ez components in the z = 0 plane, respectively. (d)-(f) The phase distributions of Ex, Ey and Ez components in the z = 0 plane, respectively. (g) The total intensity distributions in the z = 0 plane. The prescribed circular path is divided to two halves with one half denoted in solid white and the other half in dashed white. (h) The two halves are plotted in red and black, respectively. The major axes of the polarization ellipses are denoted in blue and green to indicate the orientation. The black half twists faster (2x) than the red half.

Equations (4)

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k Ω =( k x k y k z )= k 0 ( cosϕsinθ sinϕsinθ cosθ ),
{ E x_f ( r p ,Ψ, z p )= i λ 0 θ max 0 2π E x_Ω ( θ,ϕ )exp( jk r p sinθcos( ϕΨ )+jk z p cosθ )sinθdθdϕ E y_f ( r p ,Ψ, z p )= i λ 0 θ max 0 2π E y_Ω ( θ,ϕ )exp( jk r p sinθcos( ϕΨ )+jk z p cosθ )sinθdθdϕ E z_f ( r p ,Ψ, z p )= i λ 0 θ max 0 2π E z_Ω ( θ,ϕ )exp( jk r p sinθcos( ϕΨ )+jk z p cosθ )sinθdθdϕ .
{ E x_f ( x,y, z 0 )= i λ k 0 2 k 0 NA k 0 NA k 0 NA k 0 NA E x_Ω ( θ,ϕ ) exp( j k z z 0 ) cosθ exp( j( k x x+ k y y ) )d k x d k y E y_f ( x,y, z 0 )= i λ k 0 2 k 0 NA k 0 NA k 0 NA k 0 NA E y_Ω ( θ,ϕ ) exp( j k z z 0 ) cosθ exp( j( k x x+ k y y ) )d k x d k y E z_f ( x,y, z 0 )= i λ k 0 2 k 0 NA k 0 NA k 0 NA k 0 NA E z_Ω ( θ,ϕ ) exp( j k z z 0 ) cosθ exp( j( k x x+ k y y ) )d k x d k y ,
α = 1 | E E | Re( E * E E ),

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