Abstract

We propose an approach to the generation of nondiffracting quasi-circularly polarized beams by a highly focusing azimuthally polarized beam using an amplitude modulated spiral phase hologram. Numerical verifications are implemented in the calculation of the electromagnetic fields and Poynting vector field near the focus based on the vector diffraction theory, and the polarization of the wavefront near the focal plane is analyzed in detail by calculating the Stokes polarization parameters. It is found that the electric field, magnetic field, and Poynting vector field can simultaneously be uniform and nondiverging over a relatively long axial range of 7.23λ. In the transverse plane, the ellipticity and azimuthal angle of the local polarization ellipse varies from point to point. No polarization singularity and phase singularity are found at the beam center, which makes the bright spot possible.

© 2011 Optical Society of America

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References

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    [CrossRef]
  2. H. F. Wang, L. P. Shi, G. Q. Yuan, X. S. Miao, W. L. Tan, and C. T. Chong, “Subwavelength and super-resolution nondiffraction beam,” Appl. Phys. Lett. 89, 171102 (2006).
    [CrossRef]
  3. C. C. Sun and C. K. Liu, “Ultrasmall focusing spot with a long depth of focus based on polarization and phase modulation,” Opt. Lett. 28, 99–101 (2003).
    [CrossRef] [PubMed]
  4. M. Erdelyi, Z. L. Horvath, G. Szabo, Z. Bor, F. K. Tittel, J. R. Cavallaro, and M. C. Smayling, “Generation of diffraction-free beams for applications in optical microlithography,” J. Vac. Sci. Technol. B 15, 287–292 (1997).
    [CrossRef]
  5. L. B. Liu, C. Liu, W. C. Howe, C. J. R. Sheppard, and N. G. Chen, “Binary-phase spatial filter for real-time swept-source optical coherence microscopy,” Opt. Lett. 32, 2375–2377 (2007).
    [CrossRef] [PubMed]
  6. R. A. Leitgeb, M. Villiger, A. H. Bachmann, L. Steinmann, and T. Lasser, “Extended focus depth for Fourier domain optical coherence microscopy,” Opt. Lett. 31, 2450–2452 (2006).
    [CrossRef] [PubMed]
  7. Q. Q. Zhang, J. G. Wang, M. W. Wang, J. Bu, S. W. Zhu, R. Wang, B. Z. Gao, and X.-C. Yuan, “A modified fractal zone plate with extended depth of focus in spectral domain optical coherence tomorgraphy,” J. Opt. 13, 055301 (2011).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  12. J. Lin, K. Yin, Y. D. Li, and J. B. Tan, “Achievement of longitudinally polarized focusing with long focal depth by amplitude modulation,” Opt. Lett. 36, 1185–1187 (2011).
    [CrossRef] [PubMed]
  13. E. J. Botcherby, R. Juskaitis, and T. Wilson, “Scanning two photon fluorescence microscopy with extended depth of field,” Opt. Commun. 268, 253–260 (2006).
    [CrossRef]
  14. H. F. Wang and F. X. Gan, “Phase-shifting apodizers for increasing focal depth,” Appl. Opt. 41, 5263–5266 (2002).
    [CrossRef] [PubMed]
  15. B. Tian and P. X. Pu, “Tight focusing of a double-ring-shaped azimuthally polarized beam,” Opt. Lett. 36, 2014–2016(2011).
    [CrossRef] [PubMed]
  16. X. Hao, C. F. Kuang, T. T. Wang, and X. Liu, “Phase encoding for sharper focus of the azimuthally polarized beam,” Opt. Lett. 35, 3928–3930 (2010).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  19. K. J. Moh, X.-C. Yuan, J. Bu, R. E. Burge, and Bruce Z. Gao, “Generating radial or azimuthal polarization by axial sampling of circularly polarized vortex beams,” Appl. Opt. 46, 7544–7551(2007).
    [CrossRef] [PubMed]
  20. B. Richard and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplannatic system,” Proc. R. Soc. A 253, 358–379 (1959).
    [CrossRef]
  21. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7, 77–87(2000).
    [CrossRef] [PubMed]
  22. R. H. Jordan and D. G. Hall, “Free-space azimuthal paraxial wave equation: the azimuthal Bessel-Gauss beam solution,” Opt. Lett. 19, 427–429 (1994).
    [CrossRef] [PubMed]
  23. P. L. Greene and D. G. Hall, “Diffraction characteristics of the azimuthal Bessel-Gauss beam,” J. Opt. Soc. Am. A 13, 962–966(1996).
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  25. Z. Bomzon, M. Gu, and J. Shamir, “Angular momentum and geometrical phases in tight-focused circularly polarized plane waves,” Appl. Phys. Lett. 89, 241104 (2006).
    [CrossRef]
  26. M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34, 1401–1407 (1987).
    [CrossRef]
  27. L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
    [CrossRef] [PubMed]

2011

2010

2009

2008

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

2007

2006

R. A. Leitgeb, M. Villiger, A. H. Bachmann, L. Steinmann, and T. Lasser, “Extended focus depth for Fourier domain optical coherence microscopy,” Opt. Lett. 31, 2450–2452 (2006).
[CrossRef] [PubMed]

E. J. Botcherby, R. Juskaitis, and T. Wilson, “Scanning two photon fluorescence microscopy with extended depth of field,” Opt. Commun. 268, 253–260 (2006).
[CrossRef]

H. F. Wang, L. P. Shi, G. Q. Yuan, X. S. Miao, W. L. Tan, and C. T. Chong, “Subwavelength and super-resolution nondiffraction beam,” Appl. Phys. Lett. 89, 171102 (2006).
[CrossRef]

M. A. Golub, V. Shurman, and I. Grossinger, “Extended focus diffractive optical element for Gaussian laser beams,” Appl. Opt. 45, 144–150 (2006).
[CrossRef] [PubMed]

Z. Bomzon, M. Gu, and J. Shamir, “Angular momentum and geometrical phases in tight-focused circularly polarized plane waves,” Appl. Phys. Lett. 89, 241104 (2006).
[CrossRef]

2004

2003

2002

2001

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef] [PubMed]

2000

1999

1997

M. Erdelyi, Z. L. Horvath, G. Szabo, Z. Bor, F. K. Tittel, J. R. Cavallaro, and M. C. Smayling, “Generation of diffraction-free beams for applications in optical microlithography,” J. Vac. Sci. Technol. B 15, 287–292 (1997).
[CrossRef]

1996

1994

1987

M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34, 1401–1407 (1987).
[CrossRef]

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[CrossRef]

1959

B. Richard and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplannatic system,” Proc. R. Soc. A 253, 358–379 (1959).
[CrossRef]

Bachmann, A. H.

Berry, M. V.

M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34, 1401–1407 (1987).
[CrossRef]

Beversluis, M. R.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef] [PubMed]

Bomzon, Z.

Z. Bomzon, M. Gu, and J. Shamir, “Angular momentum and geometrical phases in tight-focused circularly polarized plane waves,” Appl. Phys. Lett. 89, 241104 (2006).
[CrossRef]

Bor, Z.

M. Erdelyi, Z. L. Horvath, G. Szabo, Z. Bor, F. K. Tittel, J. R. Cavallaro, and M. C. Smayling, “Generation of diffraction-free beams for applications in optical microlithography,” J. Vac. Sci. Technol. B 15, 287–292 (1997).
[CrossRef]

Botcherby, E. J.

E. J. Botcherby, R. Juskaitis, and T. Wilson, “Scanning two photon fluorescence microscopy with extended depth of field,” Opt. Commun. 268, 253–260 (2006).
[CrossRef]

Brown, T. G.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef] [PubMed]

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

Bu, J.

Q. Q. Zhang, J. G. Wang, M. W. Wang, J. Bu, S. W. Zhu, R. Wang, B. Z. Gao, and X.-C. Yuan, “A modified fractal zone plate with extended depth of focus in spectral domain optical coherence tomorgraphy,” J. Opt. 13, 055301 (2011).
[CrossRef]

K. J. Moh, X.-C. Yuan, J. Bu, R. E. Burge, and Bruce Z. Gao, “Generating radial or azimuthal polarization by axial sampling of circularly polarized vortex beams,” Appl. Opt. 46, 7544–7551(2007).
[CrossRef] [PubMed]

Burge, R. E.

Campos, J.

Cavallaro, J. R.

M. Erdelyi, Z. L. Horvath, G. Szabo, Z. Bor, F. K. Tittel, J. R. Cavallaro, and M. C. Smayling, “Generation of diffraction-free beams for applications in optical microlithography,” J. Vac. Sci. Technol. B 15, 287–292 (1997).
[CrossRef]

Chen, J.

Chen, N. G.

Chong, C. T.

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

H. F. Wang, L. P. Shi, G. Q. Yuan, X. S. Miao, W. L. Tan, and C. T. Chong, “Subwavelength and super-resolution nondiffraction beam,” Appl. Phys. Lett. 89, 171102 (2006).
[CrossRef]

Collet, E.

E. Collet, Polarized Light (Dekker, 1993), Chap 4, pp. 33–66.

Cottrell, D. M.

Davis, J. A.

Durnin, J.

Erdelyi, M.

M. Erdelyi, Z. L. Horvath, G. Szabo, Z. Bor, F. K. Tittel, J. R. Cavallaro, and M. C. Smayling, “Generation of diffraction-free beams for applications in optical microlithography,” J. Vac. Sci. Technol. B 15, 287–292 (1997).
[CrossRef]

Fang, Z. L.

Flores, A.

Gan, F. X.

Gao, B. Z.

Q. Q. Zhang, J. G. Wang, M. W. Wang, J. Bu, S. W. Zhu, R. Wang, B. Z. Gao, and X.-C. Yuan, “A modified fractal zone plate with extended depth of focus in spectral domain optical coherence tomorgraphy,” J. Opt. 13, 055301 (2011).
[CrossRef]

Gao, Bruce Z.

Golub, M. A.

Greene, P. L.

Grossinger, I.

Gu, M.

Z. Bomzon, M. Gu, and J. Shamir, “Angular momentum and geometrical phases in tight-focused circularly polarized plane waves,” Appl. Phys. Lett. 89, 241104 (2006).
[CrossRef]

Hall, D. G.

Hao, X.

Horvath, Z. L.

M. Erdelyi, Z. L. Horvath, G. Szabo, Z. Bor, F. K. Tittel, J. R. Cavallaro, and M. C. Smayling, “Generation of diffraction-free beams for applications in optical microlithography,” J. Vac. Sci. Technol. B 15, 287–292 (1997).
[CrossRef]

Howe, W. C.

Huang, K.

Jordan, R. H.

Juskaitis, R.

E. J. Botcherby, R. Juskaitis, and T. Wilson, “Scanning two photon fluorescence microscopy with extended depth of field,” Opt. Commun. 268, 253–260 (2006).
[CrossRef]

Kang, X. L.

Kuang, C. F.

Lasser, T.

Leitgeb, R. A.

Li, Y. D.

Li, Y. P.

Lin, J.

Liu, C.

Liu, C. K.

Liu, L. B.

Liu, X.

Lukyanchuk, B.

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

Miao, X. S.

H. F. Wang, L. P. Shi, G. Q. Yuan, X. S. Miao, W. L. Tan, and C. T. Chong, “Subwavelength and super-resolution nondiffraction beam,” Appl. Phys. Lett. 89, 171102 (2006).
[CrossRef]

Moh, K. J.

Moreno, I.

Novotny, L.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef] [PubMed]

Pu, P. X.

Richard, B.

B. Richard and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplannatic system,” Proc. R. Soc. A 253, 358–379 (1959).
[CrossRef]

Shamir, J.

Z. Bomzon, M. Gu, and J. Shamir, “Angular momentum and geometrical phases in tight-focused circularly polarized plane waves,” Appl. Phys. Lett. 89, 241104 (2006).
[CrossRef]

Sheppard, C.

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

Sheppard, C. J. R.

Shi, L. P.

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

H. F. Wang, L. P. Shi, G. Q. Yuan, X. S. Miao, W. L. Tan, and C. T. Chong, “Subwavelength and super-resolution nondiffraction beam,” Appl. Phys. Lett. 89, 171102 (2006).
[CrossRef]

Shi, P.

Shurman, V.

Smayling, M. C.

M. Erdelyi, Z. L. Horvath, G. Szabo, Z. Bor, F. K. Tittel, J. R. Cavallaro, and M. C. Smayling, “Generation of diffraction-free beams for applications in optical microlithography,” J. Vac. Sci. Technol. B 15, 287–292 (1997).
[CrossRef]

Steinmann, L.

Sun, C. C.

Szabo, G.

M. Erdelyi, Z. L. Horvath, G. Szabo, Z. Bor, F. K. Tittel, J. R. Cavallaro, and M. C. Smayling, “Generation of diffraction-free beams for applications in optical microlithography,” J. Vac. Sci. Technol. B 15, 287–292 (1997).
[CrossRef]

Tan, J. B.

Tan, W. L.

H. F. Wang, L. P. Shi, G. Q. Yuan, X. S. Miao, W. L. Tan, and C. T. Chong, “Subwavelength and super-resolution nondiffraction beam,” Appl. Phys. Lett. 89, 171102 (2006).
[CrossRef]

Tian, B.

Tittel, F. K.

M. Erdelyi, Z. L. Horvath, G. Szabo, Z. Bor, F. K. Tittel, J. R. Cavallaro, and M. C. Smayling, “Generation of diffraction-free beams for applications in optical microlithography,” J. Vac. Sci. Technol. B 15, 287–292 (1997).
[CrossRef]

Villiger, M.

Wang, H. F.

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

H. F. Wang, L. P. Shi, G. Q. Yuan, X. S. Miao, W. L. Tan, and C. T. Chong, “Subwavelength and super-resolution nondiffraction beam,” Appl. Phys. Lett. 89, 171102 (2006).
[CrossRef]

H. F. Wang and F. X. Gan, “Phase-shifting apodizers for increasing focal depth,” Appl. Opt. 41, 5263–5266 (2002).
[CrossRef] [PubMed]

Wang, J. G.

Q. Q. Zhang, J. G. Wang, M. W. Wang, J. Bu, S. W. Zhu, R. Wang, B. Z. Gao, and X.-C. Yuan, “A modified fractal zone plate with extended depth of focus in spectral domain optical coherence tomorgraphy,” J. Opt. 13, 055301 (2011).
[CrossRef]

Wang, M. R.

Wang, M. W.

Q. Q. Zhang, J. G. Wang, M. W. Wang, J. Bu, S. W. Zhu, R. Wang, B. Z. Gao, and X.-C. Yuan, “A modified fractal zone plate with extended depth of focus in spectral domain optical coherence tomorgraphy,” J. Opt. 13, 055301 (2011).
[CrossRef]

Wang, R.

Q. Q. Zhang, J. G. Wang, M. W. Wang, J. Bu, S. W. Zhu, R. Wang, B. Z. Gao, and X.-C. Yuan, “A modified fractal zone plate with extended depth of focus in spectral domain optical coherence tomorgraphy,” J. Opt. 13, 055301 (2011).
[CrossRef]

Wang, T. T.

Wilson, T.

E. J. Botcherby, R. Juskaitis, and T. Wilson, “Scanning two photon fluorescence microscopy with extended depth of field,” Opt. Commun. 268, 253–260 (2006).
[CrossRef]

Wolf, E.

B. Richard and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplannatic system,” Proc. R. Soc. A 253, 358–379 (1959).
[CrossRef]

Yang, J. J.

Yin, K.

Youngworth, K. S.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef] [PubMed]

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

Yuan, G. Q.

H. F. Wang, L. P. Shi, G. Q. Yuan, X. S. Miao, W. L. Tan, and C. T. Chong, “Subwavelength and super-resolution nondiffraction beam,” Appl. Phys. Lett. 89, 171102 (2006).
[CrossRef]

Yuan, X.-C.

Yzuel, M. J.

Zhang, Q. Q.

Q. Q. Zhang, J. G. Wang, M. W. Wang, J. Bu, S. W. Zhu, R. Wang, B. Z. Gao, and X.-C. Yuan, “A modified fractal zone plate with extended depth of focus in spectral domain optical coherence tomorgraphy,” J. Opt. 13, 055301 (2011).
[CrossRef]

Zhang, X. B.

Zhao, X.

Zhu, S. W.

Q. Q. Zhang, J. G. Wang, M. W. Wang, J. Bu, S. W. Zhu, R. Wang, B. Z. Gao, and X.-C. Yuan, “A modified fractal zone plate with extended depth of focus in spectral domain optical coherence tomorgraphy,” J. Opt. 13, 055301 (2011).
[CrossRef]

J. Chen, X.-C. Yuan, X. Zhao, Z. L. Fang, and S. W. Zhu, “Generalized approach to modifying optical vortices with suppressed sidelobes using Bessel-like functions,” Opt. Lett. 34, 3289–3291(2009).
[CrossRef] [PubMed]

Appl. Opt.

Appl. Phys. Lett.

H. F. Wang, L. P. Shi, G. Q. Yuan, X. S. Miao, W. L. Tan, and C. T. Chong, “Subwavelength and super-resolution nondiffraction beam,” Appl. Phys. Lett. 89, 171102 (2006).
[CrossRef]

Z. Bomzon, M. Gu, and J. Shamir, “Angular momentum and geometrical phases in tight-focused circularly polarized plane waves,” Appl. Phys. Lett. 89, 241104 (2006).
[CrossRef]

J. Mod. Opt.

M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34, 1401–1407 (1987).
[CrossRef]

J. Opt.

Q. Q. Zhang, J. G. Wang, M. W. Wang, J. Bu, S. W. Zhu, R. Wang, B. Z. Gao, and X.-C. Yuan, “A modified fractal zone plate with extended depth of focus in spectral domain optical coherence tomorgraphy,” J. Opt. 13, 055301 (2011).
[CrossRef]

J. Opt. Soc. Am. A

J. Vac. Sci. Technol. B

M. Erdelyi, Z. L. Horvath, G. Szabo, Z. Bor, F. K. Tittel, J. R. Cavallaro, and M. C. Smayling, “Generation of diffraction-free beams for applications in optical microlithography,” J. Vac. Sci. Technol. B 15, 287–292 (1997).
[CrossRef]

Nat. Photon.

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

Opt. Commun.

E. J. Botcherby, R. Juskaitis, and T. Wilson, “Scanning two photon fluorescence microscopy with extended depth of field,” Opt. Commun. 268, 253–260 (2006).
[CrossRef]

Opt. Express

Opt. Lett.

R. H. Jordan and D. G. Hall, “Free-space azimuthal paraxial wave equation: the azimuthal Bessel-Gauss beam solution,” Opt. Lett. 19, 427–429 (1994).
[CrossRef] [PubMed]

B. Tian and P. X. Pu, “Tight focusing of a double-ring-shaped azimuthally polarized beam,” Opt. Lett. 36, 2014–2016(2011).
[CrossRef] [PubMed]

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

J. Chen, X.-C. Yuan, X. Zhao, Z. L. Fang, and S. W. Zhu, “Generalized approach to modifying optical vortices with suppressed sidelobes using Bessel-like functions,” Opt. Lett. 34, 3289–3291(2009).
[CrossRef] [PubMed]

L. B. Liu, C. Liu, W. C. Howe, C. J. R. Sheppard, and N. G. Chen, “Binary-phase spatial filter for real-time swept-source optical coherence microscopy,” Opt. Lett. 32, 2375–2377 (2007).
[CrossRef] [PubMed]

R. A. Leitgeb, M. Villiger, A. H. Bachmann, L. Steinmann, and T. Lasser, “Extended focus depth for Fourier domain optical coherence microscopy,” Opt. Lett. 31, 2450–2452 (2006).
[CrossRef] [PubMed]

C. C. Sun and C. K. Liu, “Ultrasmall focusing spot with a long depth of focus based on polarization and phase modulation,” Opt. Lett. 28, 99–101 (2003).
[CrossRef] [PubMed]

K. Huang, P. Shi, X. L. Kang, X. B. Zhang, and Y. P. Li, “Design of DOE for generating a needle of a strong longitudinally polarized field,” Opt. Lett. 35, 965–967 (2010).
[CrossRef] [PubMed]

J. Lin, K. Yin, Y. D. Li, and J. B. Tan, “Achievement of longitudinally polarized focusing with long focal depth by amplitude modulation,” Opt. Lett. 36, 1185–1187 (2011).
[CrossRef] [PubMed]

Phys. Rev. Lett.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef] [PubMed]

Proc. R. Soc. A

B. Richard and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplannatic system,” Proc. R. Soc. A 253, 358–379 (1959).
[CrossRef]

Other

E. Collet, Polarized Light (Dekker, 1993), Chap 4, pp. 33–66.

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Figures (5)

Fig. 1
Fig. 1

Schematic of the setup: A linearly polarized beam illuminates on the amplitude modulated phase hologram and is converted to the desired azimuthal polarization vortex beam by the combination of a quarter waveplate and azimuthal-type polarization analyzer. The polarization state after each optical element is briefly given in the brackets. The topological charge of the spiral phase hologram is 2.

Fig. 2
Fig. 2

Electric field intensity distributions in the y z plane for (a)  N = 1 , (b)  N = 2 , (c)  N = 3 after phase modulation, and (d) without amplitude and phase modulation. Normalized axial electric field intensity profiles (e) and corresponding transmission functions of the amplitude mask (f) for different N. M = 0.638 .

Fig. 3
Fig. 3

(a) Cross section of the ellipticity of the local polarization ellipse and (b) the azimuthal angle distribution in the focal plane. The orientation of the long axis of the local polarization ellipse is indicated by the line.

Fig. 4
Fig. 4

Phase distributions in the focal plane for N = 3 . No phase singularity is found at the beam center.

Fig. 5
Fig. 5

Normalized intensity distributions of the magnetic field (a) and the time-averaged Poynting vector field (color plot) and the energy flow (blue arrows) in the (b)  y z and (c)  x y cross sections. N = 3 and M = 0.638 .

Tables (1)

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Table 1 Comparison of Key Parameters Without and With Amplitude Modulation

Equations (10)

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T N ( θ ) = p = 1 N C p cos [ 2 M π ( 2 p 1 ) cos θ ] ,
E ( r , ϕ , z ) = [ E r E ϕ E z ] = i A π 0 α 0 2 π cos θ sin θ t ( θ , φ ) l 0 ( θ ) [ sin ( ϕ φ ) cos ( ϕ φ ) 0 ] e i k 0 [ z cos θ + r sin θ cos ( φ ϕ ) ] d φ d θ
l 0 ( θ ) = exp [ ( β sin θ sin α ) 2 ] J 1 ( 2 γ sin θ sin α ) ,
E ( r , ϕ , z ) = [ E r , E ϕ , E z ] = [ A e i ϕ ( I 0 + I 2 ) , i A e i ϕ ( I 0 I 2 ) , 0 ] ,
I n ( r ) = 0 α cos θ sin θ T ( θ ) l 0 ( θ ) e i k 0 z cos θ J n ( k 0 r sin θ ) d θ = 0 α cos θ sin θ l 0 ( θ ) p = 1 N { C p 2 [ e i k 0 ( z + M λ ( 2 p 1 ) ) cos θ + e i k 0 ( z M λ ( 2 p 1 ) ) cos θ ] } J n ( k 0 r sin θ ) d θ .
( S 0 S 1 S 2 S 3 ) = ( E x E x * + E y E y * E x E x * E y E y * E x E y * + E y E x * i ( E x E y * E y E x * ) ) ,
tan 2 ψ = S 2 / S 1 = tan [ 2 ϕ + arg ( I 0 * I 2 ) ] ,
sin 2 χ = S 3 / S 0 = ( | I 0 | 2 | I 2 | 2 ) / ( | I 0 | 2 + | I 2 | 2 ) .
[ H r , H ϕ , H z ] = 1 i ω μ 0 [ E ϕ z , E ρ z , ( E ϕ ρ 1 ρ E ρ ϕ ) ] ,
[ S r , S ϕ , S z ] = [ E ϕ H z * , E ρ H z * , E ρ H ϕ * E ϕ H ρ * ] ,

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