Abstract

In this paper, the general formula for tightly focusing radially polarized beams (RPB) superposed with off-axis vortex arrays is derived based on Richard-Wolf vector diffraction theory. The off-axis vortex breaks the rotational symmetry of the energy flow along the axial direction and leads to the spatial redistribution of intensity within the focal plane. The dependence of the consequent focal intensity redistribution on the off-axis distance of vortices as well as the numerical aperture of the lens is theoretically studied. Based on this intriguing feature, generation of equilateral-polygon-like flat-top focus (EPFF) with a flat-top area on the level of sub-λ2 is realized. The demonstrated method provides new opportunities for focus shaping and holds great potentials in optical manipulation and laser fabrication.

© 2017 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Focus shaping of the radially polarized Laguerre-Gaussian-correlated Schell-model vortex beams

Hua-Feng Xu, Yuan Zhou, Hong-Wei Wu, Hua-Jun Chen, Zong-Qiang Sheng, and Jun Qu
Opt. Express 26(16) 20076-20088 (2018)

Focus shaping of partially coherent radially polarized vortex beam with tunable topological charge

Hua-Feng Xu, Rui Zhang, Zong-Qiang Sheng, and Jun Qu
Opt. Express 27(17) 23959-23969 (2019)

Spherical and sub-wavelength longitudinal magnetization generated by 4π tightly focusing radially polarized vortex beams

Zhongquan Nie, Weiqiang Ding, Dongyu Li, Xueru Zhang, Yuxiao Wang, and Yinglin Song
Opt. Express 23(2) 690-701 (2015)

References

  • View by:
  • |
  • |
  • |

  1. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
  2. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
    [PubMed]
  3. 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. Commun. 3(8), 998 (2012).
    [PubMed]
  4. S. Segawa, Y. Kozawa, and S. Sato, “Resolution enhancement of confocal microscopy by subtraction method with vector beams,” Opt. Lett. 39(11), 3118–3121 (2014).
    [PubMed]
  5. L. Huang, H. Guo, J. Li, L. Ling, B. Feng, and Z. Y. Li, “Optical trapping of gold nanoparticles by cylindrical vector beam,” Opt. Lett. 37(10), 1694–1696 (2012).
    [PubMed]
  6. Y. Kozawa and S. Sato, “Optical trapping of micrometer-sized dielectric particles by cylindrical vector beams,” Opt. Express 18(10), 10828–10833 (2010).
    [PubMed]
  7. J. Xu, Z. J. Yang, J. X. Li, and W. P. Zang, “Electron acceleration by a tightly focused cylindrical vector Gaussian beam,” Laser Phys. Lett. 14(2), 025301 (2017).
  8. V. Parigi, V. D’Ambrosio, C. Arnold, L. Marrucci, F. Sciarrino, and J. Laurat, “Storage and retrieval of vector beams of light in a multiple-degree-of-freedom quantum memory,” Nat. Commun. 6, 7706 (2015).
    [PubMed]
  9. T. G. Jabbour and S. M. Kuebler, “Vectorial beam shaping,” Opt. Express 16(10), 7203–7213 (2008).
    [PubMed]
  10. H. Wang, L. Shi, and B. Lukyanchuk, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(6), 501–505 (2008).
  11. Y. Zhao, Q. Zhan, Y. Zhang, and Y. P. Li, “Creation of a three-dimensional optical chain for controllable particle delivery,” Opt. Lett. 30(8), 848–850 (2005).
    [PubMed]
  12. X. Wang, L. Gong, Z. Zhu, B. Gu, and Q. Zhan, “Creation of identical multiple focal spots with three-dimensional arbitrary shifting,” Opt. Express 25(15), 17737–17745 (2017).
    [PubMed]
  13. Y. Yu and Q. Zhan, “Creation of identical multiple focal spots with prescribed axial distribution,” Sci. Rep. 5, 14673 (2015).
    [PubMed]
  14. L. Z. Rao, J. X. Pu, Z. Y. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical aperture lens,” Opt. Laser Technol. 41(3), 241–246 (2009).
  15. L. Wei and H. P. Urbach, “Shaping the focal field of radially/azimuthally polarized phase vortex with Zernike polynomials,” J. Opt. 18(6), 25–28 (2016).
  16. F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a Subwavelength Needle with Ultra-long Focal Length by Focusing Azimuthally Polarized Light,” Sci. Rep. 5, 9977 (2015).
    [PubMed]
  17. G. Indebetouw, “Optical Vortices and Their Propagation,” J. Mod. Opt. 40(1), 73–87 (1993).
  18. B. Richards and E. Wolf, “Electromagnetic Diffraction in Optical Systems. II. Structure of the Image Field in an Aplanatic,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
  19. J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009).
    [PubMed]

2017 (2)

J. Xu, Z. J. Yang, J. X. Li, and W. P. Zang, “Electron acceleration by a tightly focused cylindrical vector Gaussian beam,” Laser Phys. Lett. 14(2), 025301 (2017).

X. Wang, L. Gong, Z. Zhu, B. Gu, and Q. Zhan, “Creation of identical multiple focal spots with three-dimensional arbitrary shifting,” Opt. Express 25(15), 17737–17745 (2017).
[PubMed]

2016 (1)

L. Wei and H. P. Urbach, “Shaping the focal field of radially/azimuthally polarized phase vortex with Zernike polynomials,” J. Opt. 18(6), 25–28 (2016).

2015 (3)

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a Subwavelength Needle with Ultra-long Focal Length by Focusing Azimuthally Polarized Light,” Sci. Rep. 5, 9977 (2015).
[PubMed]

V. Parigi, V. D’Ambrosio, C. Arnold, L. Marrucci, F. Sciarrino, and J. Laurat, “Storage and retrieval of vector beams of light in a multiple-degree-of-freedom quantum memory,” Nat. Commun. 6, 7706 (2015).
[PubMed]

Y. Yu and Q. Zhan, “Creation of identical multiple focal spots with prescribed axial distribution,” Sci. Rep. 5, 14673 (2015).
[PubMed]

2014 (1)

2012 (2)

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. Commun. 3(8), 998 (2012).
[PubMed]

L. Huang, H. Guo, J. Li, L. Ling, B. Feng, and Z. Y. Li, “Optical trapping of gold nanoparticles by cylindrical vector beam,” Opt. Lett. 37(10), 1694–1696 (2012).
[PubMed]

2010 (1)

2009 (3)

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

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009).
[PubMed]

L. Z. Rao, J. X. Pu, Z. Y. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical aperture lens,” Opt. Laser Technol. 41(3), 241–246 (2009).

2008 (2)

H. Wang, L. Shi, and B. Lukyanchuk, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(6), 501–505 (2008).

T. G. Jabbour and S. M. Kuebler, “Vectorial beam shaping,” Opt. Express 16(10), 7203–7213 (2008).
[PubMed]

2005 (1)

2003 (1)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[PubMed]

1993 (1)

G. Indebetouw, “Optical Vortices and Their Propagation,” J. Mod. Opt. 40(1), 73–87 (1993).

1959 (1)

B. Richards and E. Wolf, “Electromagnetic Diffraction in Optical Systems. II. Structure of the Image Field in an Aplanatic,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).

Arnold, C.

V. Parigi, V. D’Ambrosio, C. Arnold, L. Marrucci, F. Sciarrino, and J. Laurat, “Storage and retrieval of vector beams of light in a multiple-degree-of-freedom quantum memory,” Nat. Commun. 6, 7706 (2015).
[PubMed]

Bade, K.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009).
[PubMed]

Chen, Z. Y.

L. Z. Rao, J. X. Pu, Z. Y. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical aperture lens,” Opt. Laser Technol. 41(3), 241–246 (2009).

D’Ambrosio, V.

V. Parigi, V. D’Ambrosio, C. Arnold, L. Marrucci, F. Sciarrino, and J. Laurat, “Storage and retrieval of vector beams of light in a multiple-degree-of-freedom quantum memory,” Nat. Commun. 6, 7706 (2015).
[PubMed]

Decker, M.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009).
[PubMed]

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[PubMed]

Feng, B.

Gansel, J. K.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009).
[PubMed]

Gong, L.

Gu, B.

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. Commun. 3(8), 998 (2012).
[PubMed]

Guo, H.

Hong, M.

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a Subwavelength Needle with Ultra-long Focal Length by Focusing Azimuthally Polarized Light,” Sci. Rep. 5, 9977 (2015).
[PubMed]

Huang, K.

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a Subwavelength Needle with Ultra-long Focal Length by Focusing Azimuthally Polarized Light,” Sci. Rep. 5, 9977 (2015).
[PubMed]

Huang, L.

Indebetouw, G.

G. Indebetouw, “Optical Vortices and Their Propagation,” J. Mod. Opt. 40(1), 73–87 (1993).

Jabbour, T. G.

Jiao, J.

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a Subwavelength Needle with Ultra-long Focal Length by Focusing Azimuthally Polarized Light,” Sci. Rep. 5, 9977 (2015).
[PubMed]

Kozawa, Y.

Kuebler, S. M.

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. Commun. 3(8), 998 (2012).
[PubMed]

Laurat, J.

V. Parigi, V. D’Ambrosio, C. Arnold, L. Marrucci, F. Sciarrino, and J. Laurat, “Storage and retrieval of vector beams of light in a multiple-degree-of-freedom quantum memory,” Nat. Commun. 6, 7706 (2015).
[PubMed]

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[PubMed]

Li, J.

Li, J. X.

J. Xu, Z. J. Yang, J. X. Li, and W. P. Zang, “Electron acceleration by a tightly focused cylindrical vector Gaussian beam,” Laser Phys. Lett. 14(2), 025301 (2017).

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. Commun. 3(8), 998 (2012).
[PubMed]

Li, Y. P.

Li, Z. Y.

Linden, S.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009).
[PubMed]

Ling, L.

Lukyanchuk, B.

H. Wang, L. Shi, and B. Lukyanchuk, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(6), 501–505 (2008).

Luo, X.

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a Subwavelength Needle with Ultra-long Focal Length by Focusing Azimuthally Polarized Light,” Sci. Rep. 5, 9977 (2015).
[PubMed]

Marrucci, L.

V. Parigi, V. D’Ambrosio, C. Arnold, L. Marrucci, F. Sciarrino, and J. Laurat, “Storage and retrieval of vector beams of light in a multiple-degree-of-freedom quantum memory,” Nat. Commun. 6, 7706 (2015).
[PubMed]

Parigi, V.

V. Parigi, V. D’Ambrosio, C. Arnold, L. Marrucci, F. Sciarrino, and J. Laurat, “Storage and retrieval of vector beams of light in a multiple-degree-of-freedom quantum memory,” Nat. Commun. 6, 7706 (2015).
[PubMed]

Pu, J. X.

L. Z. Rao, J. X. Pu, Z. Y. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical aperture lens,” Opt. Laser Technol. 41(3), 241–246 (2009).

Qin, F.

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a Subwavelength Needle with Ultra-long Focal Length by Focusing Azimuthally Polarized Light,” Sci. Rep. 5, 9977 (2015).
[PubMed]

Qiu, C.

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a Subwavelength Needle with Ultra-long Focal Length by Focusing Azimuthally Polarized Light,” Sci. Rep. 5, 9977 (2015).
[PubMed]

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[PubMed]

Rao, L. Z.

L. Z. Rao, J. X. Pu, Z. Y. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical aperture lens,” Opt. Laser Technol. 41(3), 241–246 (2009).

Richards, B.

B. Richards and E. Wolf, “Electromagnetic Diffraction in Optical Systems. II. Structure of the Image Field in an Aplanatic,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).

Rill, M. S.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009).
[PubMed]

Saile, V.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009).
[PubMed]

Sato, S.

Sciarrino, F.

V. Parigi, V. D’Ambrosio, C. Arnold, L. Marrucci, F. Sciarrino, and J. Laurat, “Storage and retrieval of vector beams of light in a multiple-degree-of-freedom quantum memory,” Nat. Commun. 6, 7706 (2015).
[PubMed]

Segawa, S.

Shi, L.

H. Wang, L. Shi, and B. Lukyanchuk, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(6), 501–505 (2008).

Thiel, M.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009).
[PubMed]

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. Commun. 3(8), 998 (2012).
[PubMed]

Urbach, H. P.

L. Wei and H. P. Urbach, “Shaping the focal field of radially/azimuthally polarized phase vortex with Zernike polynomials,” J. Opt. 18(6), 25–28 (2016).

von Freymann, G.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009).
[PubMed]

Wang, H.

H. Wang, L. Shi, and B. Lukyanchuk, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(6), 501–505 (2008).

Wang, X.

Wegener, M.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009).
[PubMed]

Wei, L.

L. Wei and H. P. Urbach, “Shaping the focal field of radially/azimuthally polarized phase vortex with Zernike polynomials,” J. Opt. 18(6), 25–28 (2016).

Wolf, E.

B. Richards and E. Wolf, “Electromagnetic Diffraction in Optical Systems. II. Structure of the Image Field in an Aplanatic,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).

Wu, J.

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a Subwavelength Needle with Ultra-long Focal Length by Focusing Azimuthally Polarized Light,” Sci. Rep. 5, 9977 (2015).
[PubMed]

Xu, J.

J. Xu, Z. J. Yang, J. X. Li, and W. P. Zang, “Electron acceleration by a tightly focused cylindrical vector Gaussian beam,” Laser Phys. Lett. 14(2), 025301 (2017).

Yang, Z. J.

J. Xu, Z. J. Yang, J. X. Li, and W. P. Zang, “Electron acceleration by a tightly focused cylindrical vector Gaussian beam,” Laser Phys. Lett. 14(2), 025301 (2017).

Yei, P.

L. Z. Rao, J. X. Pu, Z. Y. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical aperture lens,” Opt. Laser Technol. 41(3), 241–246 (2009).

Yu, Y.

Y. Yu and Q. Zhan, “Creation of identical multiple focal spots with prescribed axial distribution,” Sci. Rep. 5, 14673 (2015).
[PubMed]

Zang, W. P.

J. Xu, Z. J. Yang, J. X. Li, and W. P. Zang, “Electron acceleration by a tightly focused cylindrical vector Gaussian beam,” Laser Phys. Lett. 14(2), 025301 (2017).

Zhan, Q.

X. Wang, L. Gong, Z. Zhu, B. Gu, and Q. Zhan, “Creation of identical multiple focal spots with three-dimensional arbitrary shifting,” Opt. Express 25(15), 17737–17745 (2017).
[PubMed]

Y. Yu and Q. Zhan, “Creation of identical multiple focal spots with prescribed axial distribution,” Sci. Rep. 5, 14673 (2015).
[PubMed]

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

Y. Zhao, Q. Zhan, Y. Zhang, and Y. P. Li, “Creation of a three-dimensional optical chain for controllable particle delivery,” Opt. Lett. 30(8), 848–850 (2005).
[PubMed]

Zhang, Y.

Zhao, Y.

Zhu, Z.

Adv. Opt. Photonics (1)

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

J. Mod. Opt. (1)

G. Indebetouw, “Optical Vortices and Their Propagation,” J. Mod. Opt. 40(1), 73–87 (1993).

J. Opt. (1)

L. Wei and H. P. Urbach, “Shaping the focal field of radially/azimuthally polarized phase vortex with Zernike polynomials,” J. Opt. 18(6), 25–28 (2016).

Laser Phys. Lett. (1)

J. Xu, Z. J. Yang, J. X. Li, and W. P. Zang, “Electron acceleration by a tightly focused cylindrical vector Gaussian beam,” Laser Phys. Lett. 14(2), 025301 (2017).

Nat. Commun. (2)

V. Parigi, V. D’Ambrosio, C. Arnold, L. Marrucci, F. Sciarrino, and J. Laurat, “Storage and retrieval of vector beams of light in a multiple-degree-of-freedom quantum memory,” Nat. Commun. 6, 7706 (2015).
[PubMed]

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. Commun. 3(8), 998 (2012).
[PubMed]

Nat. Photonics (1)

H. Wang, L. Shi, and B. Lukyanchuk, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(6), 501–505 (2008).

Opt. Express (3)

Opt. Laser Technol. (1)

L. Z. Rao, J. X. Pu, Z. Y. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical aperture lens,” Opt. Laser Technol. 41(3), 241–246 (2009).

Opt. Lett. (3)

Phys. Rev. Lett. (1)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[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,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).

Sci. Rep. (2)

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a Subwavelength Needle with Ultra-long Focal Length by Focusing Azimuthally Polarized Light,” Sci. Rep. 5, 9977 (2015).
[PubMed]

Y. Yu and Q. Zhan, “Creation of identical multiple focal spots with prescribed axial distribution,” Sci. Rep. 5, 14673 (2015).
[PubMed]

Science (1)

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009).
[PubMed]

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1 The sketch for the generation of EPFF. Without loss of generality, triangle-like flat-top focus by using three off-axis vortices is taken as an example. (a) Arrangement of three vortices (red circular) in the pupil plane. Each vortex has equal distance from the optical axis. (b) Illustration of the phase distribution of the incident beam with three off-axis vortices, off-axis vortex induced asymmetric energy flow in the axial direction, and intensity distribution of triangle-like flat-top focus at the focal plane. (c) The optical system for generating EPFF where P (ρ, ϕ) and Q (r, φ) denote points in the object space and the image space, respectively.
Fig. 2
Fig. 2 (a) Variation tendency of the optical intensity along the y axis in the focal plane as the increase of off-axis distance r0 of vortices at different NA value. The data is normalized to the maximum value of the optical intensity along the y axis in the focal plane for each r0, respectively. (b) Transverse intensities and their constituent radial and azimuthal components obtained at variant r0 focused by an objective lens with NA = 0.2 in Fig. 2(a).
Fig. 3
Fig. 3 (a) Dependence of the intensity distribution along the y axis in the focal plane on the NA of the objective lens. The data are normalized to the maximum value of the intensity along the y axis in the focal plane for each NA respectively. (b) Transverse intensity patterns and their constituent radial and azimuthal components obtained by different NA when the off-axis distance r0 is fixed at 0.8. The scale in Fig. 3(b) is different for each NA.
Fig. 4
Fig. 4 Normalized Poynting vector field (color density plots) and the energy flow (white lines) along the axial direction obtained by focusing a RPB superposed with three off-axis vortices under the condition of NA = 0.6 and (a) r0 = 0.1w (b) r0 = 0.5w and (c) r0 = 0.9w, respectively. The red wire frame indicates the propagation section of the phase singularity in the axial direction.
Fig. 5
Fig. 5 Focal intensity distribution shaped by tightly focusing a RPB superposed with three off-axis vortices. (a), (b), (c) and (d) are the azimuthal, radial, longitudinal components and total intensity pattern in the x-y plane, respectively. The white line indicates the intensity profile along the x-axis and the y-axis. (e) and (f) are the total intensity pattern in the x-z plane and the y-z plane, respectively.
Fig. 6
Fig. 6 Arrangement of off-axis vortices in the incident beam and corresponding EPFF. The first column shows the location and arrangement of off-axis vortex arrays. The second to the fifth column corresponds to the radial, azimuthal, longitudinal components and the total intensity pattern in the x-y plane, respectively. From top to bottom, the number N of vortices used are in turn 2, 4, 5 and 6 corresponding to bar-type-like, square-like, pentagon-like and hexagon-like flat-top focus, respectively. The flat-top areas, from top to bottom, are 0.14λ2, 0.33λ2, 0.31λ2 and 0.35λ2, respectively.

Equations (14)

Equations on this page are rendered with MathJax. Learn more.

E in0 (ρ,ϕ)= E ρ e ^ ρ = E 0  exp( ρ 2 w 2 ) e ^ ρ
E in (ρ,ϕ)= E in0 (ρ,ϕ) (ρ e jϕ ρ 1 e j ϕ 1 ) m 1
E in (ρ,ϕ)= E in0 (ρ,ϕ) (ρ e jϕ ρ 1 e j ϕ 1 ) m 1 (ρ e jϕ ρ 2 e j ϕ 2 ) m 2 ...× (ρ e jϕ ρ N e j ϕ N ) m N               = E in0 (ρ,ϕ) i=1 N (ρ e jϕ ρ i e j ϕ i ) m i
E in (ρ,ϕ)= E in0 (ρ,ϕ) i=0 M A n ρ Mi e j(Mi)ϕ
M= n=1 N m n
E (r,φ,z)= jkf 2π 0 α dθ 0 2π P(θ) ( E in cosθcosϕφ) E in cosθsin(ϕφ)            E in sinθ )  ×exp[jk(zcosθ+rsinθcos(φϕ))]sinθdϕ
E (r,φ,z)= ( E r E φ   E z )= jkf E 0   2π n=0 M 0 α dθ 0 2π A n ρ Mn e j(Mn)ϕ cosθ sinθ ( cosθcos(ϕφ) cosθsin(ϕφ)        sinθ ) ×exp( ρ 2 w 2 )exp[jk(zcosθ+rsinθcos(φϕ))]dϕ
E r (r,φ,z)=  k E 0   2 l=0 M A Ml j l+2 f l+1 exp(jlφ) 0 α (sinθ) l+1 (cosθ) 3 2 exp( f 2 (sinθ) 2 w 2 )                     ×exp[jkzcosθ] [  J l+1 (krsinθ)  J l1 (krsinθ) ] dθ
E φ (r,φ,z)=  k E 0   2 l=0 M A Ml j l+1 f l+1 exp(jlφ) 0 α (sinθ) l+1 (cosθ) 3 2 exp( f 2 (sinθ) 2 w 2 )                     ×exp[jkzcosθ] [  J l+1 (krsinθ)+  J l1 (krsinθ) ]dθ
E z (r,φ,z)=k E 0   l=0 M A Ml j l+1   f l+1 exp(jlφ) 0 α (sinθ) l+2 (cosθ) 1 2                        ×exp( f 2 (sinθ) 2 w 2 )exp[jkzcosθ]  J l (krsinθ)dθ
H r (r,φ,z)= i k ( 1 r E z φ E φ z )
H φ (r,φ,z)= i k ( E r z E z r )
H z (r,φ,z)= i k 1 r ( (ρ E φ ) r E r φ )
S = c 4π Re( E × H )

Metrics