Abstract

Based on the vectorial Rayleigh–Sommerfeld integrals, the analytical expressions for azimuthal-variant vector fields diffracted by an annular aperture are presented. This helps us to investigate the propagation behaviors and the focusing properties of apertured azimuthal-variant vector fields under nonparaxial and paraxial approximations. The diffraction by a circular aperture, a circular disk, or propagation in free space can be treated as special cases of this general result. Simulation results show that the transverse intensity, longitudinal intensity, and far-field divergence angle of nonparaxially apertured azimuthal-variant vector fields depend strongly on the azimuthal index, the outer truncation parameter and the inner truncation parameter of the annular aperture, as well as the ratio of the waist width to the wavelength. Moreover, the multiple-ring-structured intensity pattern of the focused azimuthal-variant vector field, which originates from the diffraction effect caused by an annular aperture, is experimentally demonstrated.

© 2014 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing local field structure of focused ultrafast pulses,” Phys. Rev. Lett. 106, 123901 (2011).
    [CrossRef]
  2. C. Min, Z. Shen, J. Shen, Y. Zhang, H. Fang, G. Yuan, L. Du, S. Zhu, T. Lei, and X. Yuan, “Focused plasmonic trapping of metallic particles,” Nat. Commun. 4, 3891 (2013).
    [CrossRef]
  3. G. Bautista, M. J. Huttunen, J. Mäkitalo, J. M. Kontio, J. Simonen, and M. Kauranen, “Second-harmonic generation imaging of metal nano-objects with cylindrical vector beams,” Nano Lett. 12, 3207–3212 (2012).
    [CrossRef]
  4. Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Radially and azimuthally polarized beams generated by space-variant dielectric subwavelength gratings,” Opt. Lett. 27, 285–287 (2002).
    [CrossRef]
  5. X. L. Wang, J. P. Ding, W. J. Ni, C. S. Guo, and H. T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. 32, 3549–3551 (2007).
    [CrossRef]
  6. Z. Y. Rong, Y. J. Han, S. Z. Wang, and C. S. Guo, “Generation of arbitrary vector beams with cascaded liquid crystal spatial light modulators,” Opt. Express 22, 1636–1644 (2014).
    [CrossRef]
  7. M. Khajavikhan, A. Simic, M. Katz, J. H. Lee, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Thresholdless nanoscale coaxial lasers,” Nature 482, 204–207 (2012).
    [CrossRef]
  8. K. Ding, L. Yin, M. T. Hill, Z. Liu, P. J. van Veldhoven, and C. Z. Ning, “An electrical injection metallic cavity nanolaser with azimuthal polarization,” Appl. Phys. Lett. 102, 041110 (2013).
    [CrossRef]
  9. K. Yu, A. Lakhani, and M. C. Wu, “Subwavelength metal-optic semiconductor nanopatch lasers,” Opt. Express 18, 8790–8799 (2010).
    [CrossRef]
  10. L. Pan and D. B. Bogy, “Data storage: heat-assisted magnetic recording,” Nat. Photonics 3, 189–190 (2009).
    [CrossRef]
  11. Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato, R. J. Saykally, J. Liphardt, and P. D. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447, 1098–1101 (2007).
    [CrossRef]
  12. J. Li, Y. Chen, and Q. Cao, “Analytical vectorial structure of Bessel-Gauss beam in the near field,” Opt. Laser Technol. 45, 734–747 (2013).
    [CrossRef]
  13. B. Lü and K. Duan, “Nonparaxial propagation of vectorial Gaussian beams diffracted at a circular aperture,” Opt. Lett. 28, 2440–2442 (2003).
    [CrossRef]
  14. X. Jia, Y. Wang, and B. Li, “Nonparaxial analyses of radially polarized beams diffracted at a circular aperture,” Opt. Express 18, 7064–7075 (2010).
    [CrossRef]
  15. X. Jia and Y. Wang, “Vectorial structure of far field of cylindrically polarized beams diffracted at a circular aperture,” Opt. Lett. 36, 295–297 (2011).
    [CrossRef]
  16. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
    [CrossRef]
  17. Y. Yang, X. Li, and K. Duan, “Nonparaxial propagation of vectorial hollow Gaussian beams diffracted at an annular aperture,” Opt. Eng. 50, 078001 (2011).
    [CrossRef]
  18. J. L. Chen, “Nonparaxial propagation of a radially polarized beam diffracted by an annular aperture,” Chin. Phys. Lett. 28, 124202 (2011).
    [CrossRef]
  19. R. K. Luneburg, Mathematical Theory of Optics (University of California, Berkeley, 1966).
  20. V. V. Kotlyar and A. A. Kovalev, “Nonparaxial propagation of a Gaussian optical vortex with initial radial polarization,” J. Opt. Soc. Am. A 27, 372–380 (2010).
    [CrossRef]
  21. B. Gu and Y. Cui, “Nonparaxial and paraxial focusing of azimuthal-variant vector beams,” Opt. Express 20, 17684–17694 (2012).
    [CrossRef]
  22. K. Duan and B. Lü, “Vectorial nonparaxial propagation equation of elliptical Gaussian beams in the presence of a rectangular aperture,” J. Opt. Soc. Am. A 21, 1613–1620 (2004).
    [CrossRef]
  23. P. Liu, B. Lü, and K. Duan, “Propagation of vectorial nonparaxial Gaussian beams through an annular aperture,” Opt. Laser Technol. 38, 133–137 (2006).
    [CrossRef]

2014 (1)

2013 (3)

J. Li, Y. Chen, and Q. Cao, “Analytical vectorial structure of Bessel-Gauss beam in the near field,” Opt. Laser Technol. 45, 734–747 (2013).
[CrossRef]

C. Min, Z. Shen, J. Shen, Y. Zhang, H. Fang, G. Yuan, L. Du, S. Zhu, T. Lei, and X. Yuan, “Focused plasmonic trapping of metallic particles,” Nat. Commun. 4, 3891 (2013).
[CrossRef]

K. Ding, L. Yin, M. T. Hill, Z. Liu, P. J. van Veldhoven, and C. Z. Ning, “An electrical injection metallic cavity nanolaser with azimuthal polarization,” Appl. Phys. Lett. 102, 041110 (2013).
[CrossRef]

2012 (3)

G. Bautista, M. J. Huttunen, J. Mäkitalo, J. M. Kontio, J. Simonen, and M. Kauranen, “Second-harmonic generation imaging of metal nano-objects with cylindrical vector beams,” Nano Lett. 12, 3207–3212 (2012).
[CrossRef]

M. Khajavikhan, A. Simic, M. Katz, J. H. Lee, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Thresholdless nanoscale coaxial lasers,” Nature 482, 204–207 (2012).
[CrossRef]

B. Gu and Y. Cui, “Nonparaxial and paraxial focusing of azimuthal-variant vector beams,” Opt. Express 20, 17684–17694 (2012).
[CrossRef]

2011 (4)

X. Jia and Y. Wang, “Vectorial structure of far field of cylindrically polarized beams diffracted at a circular aperture,” Opt. Lett. 36, 295–297 (2011).
[CrossRef]

Y. Yang, X. Li, and K. Duan, “Nonparaxial propagation of vectorial hollow Gaussian beams diffracted at an annular aperture,” Opt. Eng. 50, 078001 (2011).
[CrossRef]

J. L. Chen, “Nonparaxial propagation of a radially polarized beam diffracted by an annular aperture,” Chin. Phys. Lett. 28, 124202 (2011).
[CrossRef]

C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing local field structure of focused ultrafast pulses,” Phys. Rev. Lett. 106, 123901 (2011).
[CrossRef]

2010 (3)

2009 (1)

L. Pan and D. B. Bogy, “Data storage: heat-assisted magnetic recording,” Nat. Photonics 3, 189–190 (2009).
[CrossRef]

2007 (2)

Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato, R. J. Saykally, J. Liphardt, and P. D. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447, 1098–1101 (2007).
[CrossRef]

X. L. Wang, J. P. Ding, W. J. Ni, C. S. Guo, and H. T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. 32, 3549–3551 (2007).
[CrossRef]

2006 (1)

P. Liu, B. Lü, and K. Duan, “Propagation of vectorial nonparaxial Gaussian beams through an annular aperture,” Opt. Laser Technol. 38, 133–137 (2006).
[CrossRef]

2004 (1)

2003 (2)

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

B. Lü and K. Duan, “Nonparaxial propagation of vectorial Gaussian beams diffracted at a circular aperture,” Opt. Lett. 28, 2440–2442 (2003).
[CrossRef]

2002 (1)

Bautista, G.

G. Bautista, M. J. Huttunen, J. Mäkitalo, J. M. Kontio, J. Simonen, and M. Kauranen, “Second-harmonic generation imaging of metal nano-objects with cylindrical vector beams,” Nano Lett. 12, 3207–3212 (2012).
[CrossRef]

Biener, G.

Bogy, D. B.

L. Pan and D. B. Bogy, “Data storage: heat-assisted magnetic recording,” Nat. Photonics 3, 189–190 (2009).
[CrossRef]

Bomzon, Z.

Cao, Q.

J. Li, Y. Chen, and Q. Cao, “Analytical vectorial structure of Bessel-Gauss beam in the near field,” Opt. Laser Technol. 45, 734–747 (2013).
[CrossRef]

Chen, J. L.

J. L. Chen, “Nonparaxial propagation of a radially polarized beam diffracted by an annular aperture,” Chin. Phys. Lett. 28, 124202 (2011).
[CrossRef]

Chen, Y.

J. Li, Y. Chen, and Q. Cao, “Analytical vectorial structure of Bessel-Gauss beam in the near field,” Opt. Laser Technol. 45, 734–747 (2013).
[CrossRef]

Cui, Y.

Ding, J. P.

Ding, K.

K. Ding, L. Yin, M. T. Hill, Z. Liu, P. J. van Veldhoven, and C. Z. Ning, “An electrical injection metallic cavity nanolaser with azimuthal polarization,” Appl. Phys. Lett. 102, 041110 (2013).
[CrossRef]

Dorn, R.

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

Du, L.

C. Min, Z. Shen, J. Shen, Y. Zhang, H. Fang, G. Yuan, L. Du, S. Zhu, T. Lei, and X. Yuan, “Focused plasmonic trapping of metallic particles,” Nat. Commun. 4, 3891 (2013).
[CrossRef]

Duan, K.

Y. Yang, X. Li, and K. Duan, “Nonparaxial propagation of vectorial hollow Gaussian beams diffracted at an annular aperture,” Opt. Eng. 50, 078001 (2011).
[CrossRef]

P. Liu, B. Lü, and K. Duan, “Propagation of vectorial nonparaxial Gaussian beams through an annular aperture,” Opt. Laser Technol. 38, 133–137 (2006).
[CrossRef]

K. Duan and B. Lü, “Vectorial nonparaxial propagation equation of elliptical Gaussian beams in the presence of a rectangular aperture,” J. Opt. Soc. Am. A 21, 1613–1620 (2004).
[CrossRef]

B. Lü and K. Duan, “Nonparaxial propagation of vectorial Gaussian beams diffracted at a circular aperture,” Opt. Lett. 28, 2440–2442 (2003).
[CrossRef]

Fainman, Y.

M. Khajavikhan, A. Simic, M. Katz, J. H. Lee, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Thresholdless nanoscale coaxial lasers,” Nature 482, 204–207 (2012).
[CrossRef]

Fang, H.

C. Min, Z. Shen, J. Shen, Y. Zhang, H. Fang, G. Yuan, L. Du, S. Zhu, T. Lei, and X. Yuan, “Focused plasmonic trapping of metallic particles,” Nat. Commun. 4, 3891 (2013).
[CrossRef]

Gu, B.

Guo, C. S.

Han, Y. J.

Hasman, E.

Hill, M. T.

K. Ding, L. Yin, M. T. Hill, Z. Liu, P. J. van Veldhoven, and C. Z. Ning, “An electrical injection metallic cavity nanolaser with azimuthal polarization,” Appl. Phys. Lett. 102, 041110 (2013).
[CrossRef]

Hnatovsky, C.

C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing local field structure of focused ultrafast pulses,” Phys. Rev. Lett. 106, 123901 (2011).
[CrossRef]

Huttunen, M. J.

G. Bautista, M. J. Huttunen, J. Mäkitalo, J. M. Kontio, J. Simonen, and M. Kauranen, “Second-harmonic generation imaging of metal nano-objects with cylindrical vector beams,” Nano Lett. 12, 3207–3212 (2012).
[CrossRef]

Jia, X.

Katz, M.

M. Khajavikhan, A. Simic, M. Katz, J. H. Lee, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Thresholdless nanoscale coaxial lasers,” Nature 482, 204–207 (2012).
[CrossRef]

Kauranen, M.

G. Bautista, M. J. Huttunen, J. Mäkitalo, J. M. Kontio, J. Simonen, and M. Kauranen, “Second-harmonic generation imaging of metal nano-objects with cylindrical vector beams,” Nano Lett. 12, 3207–3212 (2012).
[CrossRef]

Khajavikhan, M.

M. Khajavikhan, A. Simic, M. Katz, J. H. Lee, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Thresholdless nanoscale coaxial lasers,” Nature 482, 204–207 (2012).
[CrossRef]

Kleiner, V.

Kontio, J. M.

G. Bautista, M. J. Huttunen, J. Mäkitalo, J. M. Kontio, J. Simonen, and M. Kauranen, “Second-harmonic generation imaging of metal nano-objects with cylindrical vector beams,” Nano Lett. 12, 3207–3212 (2012).
[CrossRef]

Kotlyar, V. V.

Kovalev, A. A.

Krolikowski, W.

C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing local field structure of focused ultrafast pulses,” Phys. Rev. Lett. 106, 123901 (2011).
[CrossRef]

Lakhani, A.

Lee, J. H.

M. Khajavikhan, A. Simic, M. Katz, J. H. Lee, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Thresholdless nanoscale coaxial lasers,” Nature 482, 204–207 (2012).
[CrossRef]

Lei, T.

C. Min, Z. Shen, J. Shen, Y. Zhang, H. Fang, G. Yuan, L. Du, S. Zhu, T. Lei, and X. Yuan, “Focused plasmonic trapping of metallic particles,” Nat. Commun. 4, 3891 (2013).
[CrossRef]

Leuchs, G.

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

Li, B.

Li, J.

J. Li, Y. Chen, and Q. Cao, “Analytical vectorial structure of Bessel-Gauss beam in the near field,” Opt. Laser Technol. 45, 734–747 (2013).
[CrossRef]

Li, X.

Y. Yang, X. Li, and K. Duan, “Nonparaxial propagation of vectorial hollow Gaussian beams diffracted at an annular aperture,” Opt. Eng. 50, 078001 (2011).
[CrossRef]

Liphardt, J.

Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato, R. J. Saykally, J. Liphardt, and P. D. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447, 1098–1101 (2007).
[CrossRef]

Liu, P.

P. Liu, B. Lü, and K. Duan, “Propagation of vectorial nonparaxial Gaussian beams through an annular aperture,” Opt. Laser Technol. 38, 133–137 (2006).
[CrossRef]

Liu, Z.

K. Ding, L. Yin, M. T. Hill, Z. Liu, P. J. van Veldhoven, and C. Z. Ning, “An electrical injection metallic cavity nanolaser with azimuthal polarization,” Appl. Phys. Lett. 102, 041110 (2013).
[CrossRef]

Lomakin, V.

M. Khajavikhan, A. Simic, M. Katz, J. H. Lee, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Thresholdless nanoscale coaxial lasers,” Nature 482, 204–207 (2012).
[CrossRef]

Lü, B.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California, Berkeley, 1966).

Mäkitalo, J.

G. Bautista, M. J. Huttunen, J. Mäkitalo, J. M. Kontio, J. Simonen, and M. Kauranen, “Second-harmonic generation imaging of metal nano-objects with cylindrical vector beams,” Nano Lett. 12, 3207–3212 (2012).
[CrossRef]

Min, C.

C. Min, Z. Shen, J. Shen, Y. Zhang, H. Fang, G. Yuan, L. Du, S. Zhu, T. Lei, and X. Yuan, “Focused plasmonic trapping of metallic particles,” Nat. Commun. 4, 3891 (2013).
[CrossRef]

Mizrahi, A.

M. Khajavikhan, A. Simic, M. Katz, J. H. Lee, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Thresholdless nanoscale coaxial lasers,” Nature 482, 204–207 (2012).
[CrossRef]

Nakayama, Y.

Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato, R. J. Saykally, J. Liphardt, and P. D. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447, 1098–1101 (2007).
[CrossRef]

Ni, W. J.

Ning, C. Z.

K. Ding, L. Yin, M. T. Hill, Z. Liu, P. J. van Veldhoven, and C. Z. Ning, “An electrical injection metallic cavity nanolaser with azimuthal polarization,” Appl. Phys. Lett. 102, 041110 (2013).
[CrossRef]

Onorato, R. M.

Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato, R. J. Saykally, J. Liphardt, and P. D. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447, 1098–1101 (2007).
[CrossRef]

Pan, L.

L. Pan and D. B. Bogy, “Data storage: heat-assisted magnetic recording,” Nat. Photonics 3, 189–190 (2009).
[CrossRef]

Pauzauskie, P. J.

Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato, R. J. Saykally, J. Liphardt, and P. D. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447, 1098–1101 (2007).
[CrossRef]

Quabis, S.

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

Radenovic, A.

Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato, R. J. Saykally, J. Liphardt, and P. D. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447, 1098–1101 (2007).
[CrossRef]

Rode, A.

C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing local field structure of focused ultrafast pulses,” Phys. Rev. Lett. 106, 123901 (2011).
[CrossRef]

Rong, Z. Y.

Saykally, R. J.

Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato, R. J. Saykally, J. Liphardt, and P. D. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447, 1098–1101 (2007).
[CrossRef]

Shen, J.

C. Min, Z. Shen, J. Shen, Y. Zhang, H. Fang, G. Yuan, L. Du, S. Zhu, T. Lei, and X. Yuan, “Focused plasmonic trapping of metallic particles,” Nat. Commun. 4, 3891 (2013).
[CrossRef]

Shen, Z.

C. Min, Z. Shen, J. Shen, Y. Zhang, H. Fang, G. Yuan, L. Du, S. Zhu, T. Lei, and X. Yuan, “Focused plasmonic trapping of metallic particles,” Nat. Commun. 4, 3891 (2013).
[CrossRef]

Shvedov, V.

C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing local field structure of focused ultrafast pulses,” Phys. Rev. Lett. 106, 123901 (2011).
[CrossRef]

Simic, A.

M. Khajavikhan, A. Simic, M. Katz, J. H. Lee, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Thresholdless nanoscale coaxial lasers,” Nature 482, 204–207 (2012).
[CrossRef]

Simonen, J.

G. Bautista, M. J. Huttunen, J. Mäkitalo, J. M. Kontio, J. Simonen, and M. Kauranen, “Second-harmonic generation imaging of metal nano-objects with cylindrical vector beams,” Nano Lett. 12, 3207–3212 (2012).
[CrossRef]

Slutsky, B.

M. Khajavikhan, A. Simic, M. Katz, J. H. Lee, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Thresholdless nanoscale coaxial lasers,” Nature 482, 204–207 (2012).
[CrossRef]

van Veldhoven, P. J.

K. Ding, L. Yin, M. T. Hill, Z. Liu, P. J. van Veldhoven, and C. Z. Ning, “An electrical injection metallic cavity nanolaser with azimuthal polarization,” Appl. Phys. Lett. 102, 041110 (2013).
[CrossRef]

Wang, H. T.

Wang, S. Z.

Wang, X. L.

Wang, Y.

Wu, M. C.

Yang, P. D.

Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato, R. J. Saykally, J. Liphardt, and P. D. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447, 1098–1101 (2007).
[CrossRef]

Yang, Y.

Y. Yang, X. Li, and K. Duan, “Nonparaxial propagation of vectorial hollow Gaussian beams diffracted at an annular aperture,” Opt. Eng. 50, 078001 (2011).
[CrossRef]

Yin, L.

K. Ding, L. Yin, M. T. Hill, Z. Liu, P. J. van Veldhoven, and C. Z. Ning, “An electrical injection metallic cavity nanolaser with azimuthal polarization,” Appl. Phys. Lett. 102, 041110 (2013).
[CrossRef]

Yu, K.

Yuan, G.

C. Min, Z. Shen, J. Shen, Y. Zhang, H. Fang, G. Yuan, L. Du, S. Zhu, T. Lei, and X. Yuan, “Focused plasmonic trapping of metallic particles,” Nat. Commun. 4, 3891 (2013).
[CrossRef]

Yuan, X.

C. Min, Z. Shen, J. Shen, Y. Zhang, H. Fang, G. Yuan, L. Du, S. Zhu, T. Lei, and X. Yuan, “Focused plasmonic trapping of metallic particles,” Nat. Commun. 4, 3891 (2013).
[CrossRef]

Zhang, Y.

C. Min, Z. Shen, J. Shen, Y. Zhang, H. Fang, G. Yuan, L. Du, S. Zhu, T. Lei, and X. Yuan, “Focused plasmonic trapping of metallic particles,” Nat. Commun. 4, 3891 (2013).
[CrossRef]

Zhu, S.

C. Min, Z. Shen, J. Shen, Y. Zhang, H. Fang, G. Yuan, L. Du, S. Zhu, T. Lei, and X. Yuan, “Focused plasmonic trapping of metallic particles,” Nat. Commun. 4, 3891 (2013).
[CrossRef]

Appl. Phys. Lett. (1)

K. Ding, L. Yin, M. T. Hill, Z. Liu, P. J. van Veldhoven, and C. Z. Ning, “An electrical injection metallic cavity nanolaser with azimuthal polarization,” Appl. Phys. Lett. 102, 041110 (2013).
[CrossRef]

Chin. Phys. Lett. (1)

J. L. Chen, “Nonparaxial propagation of a radially polarized beam diffracted by an annular aperture,” Chin. Phys. Lett. 28, 124202 (2011).
[CrossRef]

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

Nano Lett. (1)

G. Bautista, M. J. Huttunen, J. Mäkitalo, J. M. Kontio, J. Simonen, and M. Kauranen, “Second-harmonic generation imaging of metal nano-objects with cylindrical vector beams,” Nano Lett. 12, 3207–3212 (2012).
[CrossRef]

Nat. Commun. (1)

C. Min, Z. Shen, J. Shen, Y. Zhang, H. Fang, G. Yuan, L. Du, S. Zhu, T. Lei, and X. Yuan, “Focused plasmonic trapping of metallic particles,” Nat. Commun. 4, 3891 (2013).
[CrossRef]

Nat. Photonics (1)

L. Pan and D. B. Bogy, “Data storage: heat-assisted magnetic recording,” Nat. Photonics 3, 189–190 (2009).
[CrossRef]

Nature (2)

Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato, R. J. Saykally, J. Liphardt, and P. D. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447, 1098–1101 (2007).
[CrossRef]

M. Khajavikhan, A. Simic, M. Katz, J. H. Lee, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Thresholdless nanoscale coaxial lasers,” Nature 482, 204–207 (2012).
[CrossRef]

Opt. Eng. (1)

Y. Yang, X. Li, and K. Duan, “Nonparaxial propagation of vectorial hollow Gaussian beams diffracted at an annular aperture,” Opt. Eng. 50, 078001 (2011).
[CrossRef]

Opt. Express (4)

Opt. Laser Technol. (2)

J. Li, Y. Chen, and Q. Cao, “Analytical vectorial structure of Bessel-Gauss beam in the near field,” Opt. Laser Technol. 45, 734–747 (2013).
[CrossRef]

P. Liu, B. Lü, and K. Duan, “Propagation of vectorial nonparaxial Gaussian beams through an annular aperture,” Opt. Laser Technol. 38, 133–137 (2006).
[CrossRef]

Opt. Lett. (4)

Phys. Rev. Lett. (2)

C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing local field structure of focused ultrafast pulses,” Phys. Rev. Lett. 106, 123901 (2011).
[CrossRef]

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

Other (1)

R. K. Luneburg, Mathematical Theory of Optics (University of California, Berkeley, 1966).

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 (11)

Fig. 1.
Fig. 1.

Schematic of the propagation of a lowest-order Laguerre–Gaussian beam diffracted by an annular aperture. The cross section of the incident beam is plotted in the left part of the figure, in which the blue arrows indicate the direction of the polarization.

Fig. 2.
Fig. 2.

Nonparaxial intensity patterns in the transverse plane for z=10zR (top row) and in the longitudinal plane for y=0 (bottom row) of an azimuthal-variant vector field diffracted by an annular aperture when m=2, φ0=0, ω0=2λ, σa=1.0, and σb=0.5. The intensity patterns are normalized by IGMax(x,y,z).

Fig. 3.
Fig. 3.

Nonparaxial intensity patterns in the transverse plane for z=10zR (top row) and in the longitudinal plane for y=0 (bottom row) of azimuthal-variant vector fields with different topological charges m diffracted by an annular aperture when φ0=0, ω0=2λ, σa=1.0, and σb=0.5. The intensity patterns are normalized by IGMax(x,y,z).

Fig. 4.
Fig. 4.

Nonparaxial intensity patterns in the transverse plane for z=10zR (top row) and in the longitudinal plane for y=0 (bottom row) of azimuthal-variant vector fields diffracted by an annular aperture when m=3, φ0=0, ω0=2λ, and σa=1.0 and different values of σb. The intensity patterns are normalized by IGMax(x,y,z).

Fig. 5.
Fig. 5.

Nonparaxial intensity patterns in the transverse plane for z=10zR (top row) and in the longitudinal plane for y=0 (bottom row) of azimuthal-variant vector fields diffracted by an annular aperture when m=3, φ0=0, ω0=2λ, and σb=0.5 and different values of σa. The intensity patterns are normalized by IGMax(x,y,z).

Fig. 6.
Fig. 6.

Nonparaxial intensity patterns in the transverse plane for z=10zR (top row) and in the longitudinal plane for y=0 (bottom row) of azimuthal-variant vector fields diffracted by an annular aperture when m=3, φ0=0, σa=1.0, and σb=0.5 and different values of ω0. The intensity patterns are normalized by IGMax(x,y,z).

Fig. 7.
Fig. 7.

Ratio η=IZMax(x,y,10zR)/ITMax(x,y,10zR) of nonparaxially apertured azimuthal-variant vector fields with φ0=0 and different m versus (a) ω0, (b) σa, and (c) σb.

Fig. 8.
Fig. 8.

Far-field divergence angle Ω of nonparaxially apertured azimuthal-variant vector fields for φ0=0, θ=0, and different m versus (a) ω0, (b) σa, and (c) σb.

Fig. 9.
Fig. 9.

Experimentally measured intensity patterns of an apertured azimuthal-variant vector field without and with a linear polarizer when m=4, φ0=0, ω0=2.88mm, a=2.37mm, b=2.13mm, and λ=532nm. Red arrows in (a) illustrate the schematics of SoP.

Fig. 10.
Fig. 10.

Nonparaxial intensity patterns of an azimuthal-variant vector field with and without annular aperture at the focal plane when m=4, φ0=0, ω0=2.88mm, a=2.37mm, b=2.13mm, λ=532nm, and f=10mm. The intensity patterns are normalized by IGMax(x,y,f).

Fig. 11.
Fig. 11.

Theoretically predicted and experimentally measured intensity patterns of an apertured azimuthal-variant vector field at the focal plane when m=4, φ0=0, ω0=2.88mm, a=2.37mm, b=2.13mm, λ=532nm, and f=150mm. Red arrows in (a) illustrate the schematics of SoP. The dimension for all of these images is 300μm×300μm.

Equations (19)

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

E⃗(r,ϕ,0)=Ex(r,ϕ,0)e^x+Ey(r,ϕ,0)e^y,
Ex(r,ϕ,0)=2E0rω0exp(μr2)cos(mϕ+φ0)T(r),
Ey(r,ϕ,0)=2E0rω0exp(μr2)sin(mϕ+φ0)T(r),
T(r)={1,bra0,otherwise.
Ex(ρ,θ,z)=(i)m+12E0kzω0ξ2eikξcos(mθ+φ0)×baexp(δr2)Jm(βr)r2dr,
Ey(ρ,θ,z)=(i)m+12E0kzω0ξ2eikξsin(mθ+φ0)×baexp(δr2)Jm(βr)r2dr,
Ez(ρ,θ,z)=(i)m+12E0kω0ξ2eikξcos(mθθ+φ0)×ba[ρJm(βr)irJm1(βr)]exp(δr2)r2dr,
Ex(ρ,θ,z)=(i)m+12E0kzω0ξ2eikξcos(mθ+φ0)×l=0(1)lβm+2l2m+2l+1l!(m+l)!δj[Γ(j,b2δ)Γ(j,a2δ)],
Ey(ρ,θ,z)=(i)m+12E0kzω0ξ2eikξsin(mθ+φ0)×l=0(1)lβm+2l2m+2l+1l!(m+l)!δj[Γ(j,b2δ)Γ(j,a2δ)],
Ez(ρ,θ,z)=(i)m+12E0kω0ξ2eikξcos(mθθ+φ0)×{ρl=0(1)lβm+2l2m+2l+1l!(m+l)!δj[Γ(j,b2δ)Γ(j,a2δ)]il=0(1)lβm+2l+12m+2l1l!(m+l1)!δj[Γ(j,b2δ)Γ(j,a2δ)]},
Ex(ρ,θ,z)=(i)m+12E0kzω0ξ2eikξcos(mθ+φ0)×l=0(1)lβm+2l2m+2l+1l!(m+l)!δj[Γ(j)Γ(j,a2δ)],
Ey(ρ,θ,z)=(i)m+12E0kzω0ξ2eikξsin(mθ+φ0)×l=0(1)lβm+2l2m+2l+1l!(m+l)!δj[Γ(j)Γ(j,a2δ)],
Ez(ρ,θ,z)=(i)m+12E0kω0ξ2eikξcos(mθθ+φ0)×l=0(1)lβm+2l[ρ2i(m+l)/β]2m+2l+1l!(m+l)!δj[Γ(j)Γ(j,a2δ)],
Ex(ρ,θ,z)=(i)m+12E0kzω0ξ2eikξcos(mθ+φ0)×l=0(1)lβm+2l2m+2l+1l!(m+l)!δjΓ(j,b2δ),
Ey(ρ,θ,z)=(i)m+12E0kzω0ξ2eikξsin(mθ+φ0)×l=0(1)lβm+2l2m+2l+1l!(m+l)!δjΓ(j,b2δ),
Ez(ρ,θ,z)=(i)m+12E0kω0ξ2eikξcos(mθθ+φ0)×l=0(1)lβm+2l[ρ2i(m+l)/β]2m+2l+1l!(m+l)!δjΓ(j,b2δ).
Ex(ρ,θ,z)=(i)m+12E0kzω0ξ2eikξcos(mθ+φ0)×l=0(1)lβm+2l2m+2l+1l!(m+l)!δjΓ(j),
Ey(ρ,θ,z)=(i)m+12E0kzω0ξ2eikξsin(mθ+φ0)×l=0(1)lβm+2l2m+2l+1l!(m+l)!δjΓ(j),
Ez(ρ,θ,z)=(i)m+12E0kω0ξ2eikξcos(mθθ+φ0)×l=0(1)lβm+2l[ρ2i(m+l)/β]2m+2l+1l!(m+l)!δjΓ(j).

Metrics