Abstract

A theoretical treatment is presented for the focusing of polarized vortex beams, including the generation of Bessel beams. A combination of a phase vortex with arbitrary topological charge, and a polarization vortex of arbitrary order is considered. Results are given for both paraxial and high NA systems. Conditions for the presence of non-zero on-axis intensity are given. An interesting observation is that half-order phase vortices can exist, without the existence of any phase discontinuity. The behavior of Bessel beams with half-order phase vortices is investigated.

© 2014 Optical Society of America

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2012 (1)

2011 (1)

2010 (2)

X. Hao, C. F. Kuang, T. T. Wang, and X. Liu, “Phase encoding for sharper focus of the azimuthally polarized beam,” Opt. Lett. 35(23), 3928–3930 (2010).
[CrossRef] [PubMed]

X. Hao, C. F. Kuang, T. T. Wang, and X. Liu, “Effects of polarization on the de-excitation dark focal spot in STED microscopy,” J. Opt. 12(11), 115707 (2010).
[CrossRef]

2009 (2)

W. T. Tang, E. Y. Yew, and C. J. Sheppard, “Polarization conversion in confocal microscopy with radially polarized illumination,” Opt. Lett. 34(14), 2147–2149 (2009).
[CrossRef] [PubMed]

C. J. R. Sheppard, N. K. Balla, S. Rehman, E. Y. S. Yew, and W. T. Teng, “Bessel beams with the tightest focus,” Opt. Commun. 282, 4647–4656 (2009).
[CrossRef]

2008 (1)

2007 (1)

2005 (1)

2004 (2)

I. V. Basistiy, V. A. Pas’ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. A, Pure Appl. Opt. 6(5), S166–S169 (2004).
[CrossRef]

L. E. Helseth, “Optical vortices in focal regions,” Opt. Commun. 229(1–6), 85–91 (2004).
[CrossRef]

2003 (1)

J. Lekner, “Polarization of tightly focused laser beams,” J. Opt. A, Pure Appl. Opt. 5(1), 6–14 (2003).
[CrossRef]

2001 (6)

2000 (3)

1999 (1)

1998 (2)

P. L. Greene and D. G. Hall, “Properties and diffraction of the azimuthal Bessel-Gauss beam,” J. Opt. Soc. Am. A 15(12), 3020–3027 (1998).
[CrossRef]

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394(6691), 348–350 (1998).
[CrossRef]

1997 (1)

M. Totzeck and H. J. Tiziani, “Phase-singularities in 2D diffraction fields and interference microscopy,” Opt. Commun. 138(4–6), 365–382 (1997).
[CrossRef]

1996 (2)

1995 (3)

Z. Bouchal and M. Olivík, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42(8), 1555–1566 (1995).
[CrossRef]

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995).
[CrossRef] [PubMed]

P. Török, P. Varga, Z. Laczik, and G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A 12(2), 325–332 (1995).
[CrossRef]

1994 (3)

X. S. Xie and R. C. Dunn, “Probing single molecule dynamics,” Science 265(5170), 361–364 (1994).
[CrossRef] [PubMed]

W. P. Ambrose, P. M. Goodwin, R. A. Keller, and J. C. Martin, “Alterations of single molecule fluorescence lifetimes in near-field optical microscopy,” Science 265(5170), 364–367 (1994).
[CrossRef] [PubMed]

R. H. Jordan and D. G. Hall, “Free-space azimuthal paraxial wave equation: the azimuthal Bessel-Gauss beam solution,” Opt. Lett. 19(7), 427–429 (1994).
[CrossRef] [PubMed]

1993 (2)

G. Indebetouw, “Optical vortices and their applications,” J. Mod. Opt. 40(1), 73–87 (1993).
[CrossRef]

I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101(3–4), 247–264 (1993).
[CrossRef]

1992 (3)

V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wave-fronts,” J. Mod. Opt. 39, 985–990 (1992).

N. Heckenberg, R. McDuff, C. P. Smith, and H. Rubinstein Dunlop, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, S951–S962 (1992).

V. V. Kotlyar and V. A. Soifer, “Rotor spatial filter for analysis and synthesis of coherent fields,” Opt. Commun. 89(2–4), 159–163 (1992).

1987 (4)

J. F. Nye and J. V. Hajnal, “The wave structure of monochromatic electromagnetic waves,” Proc. R. Soc. Lond. A Math. Phys. Sci. 409(1836), 21–36 (1987).
[CrossRef]

J. V. Hajnal, “Singularities in the transverse fields of electromagnetic waves, 1. Theory,” Proc. R. Soc. Lond. A Math. Phys. Sci. 414(1847), 433–446 (1987).
[CrossRef]

J. V. Hajnal, “Singularities in the transverse fields of electromagnetic waves, 2. Observations on the electric field,” Proc. R. Soc. Lond. A Math. Phys. Sci. 414(1847), 447–468 (1987).
[CrossRef]

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[CrossRef] [PubMed]

1986 (1)

1985 (1)

W. J. Condell, “Fraunhoffer diffraction from a circular aperture with helical phase factor,” J. Opt. Soc. Am. 2(2), 206 (1985).
[CrossRef]

1983 (3)

G. P. Agrawal and M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27(3), 1693–1695 (1983).
[CrossRef]

J. F. Nye, “Polarization effects in the diffraction of electromagnetic waves: the role of disclinations,” Proc. R. Soc. Lond. A Math. Phys. Sci. 387(1792), 105–132 (1983).
[CrossRef]

J. F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. R. Soc. Lond. A Math. Phys. Sci. 389(1797), 279–290 (1983).
[CrossRef]

1981 (1)

1979 (1)

1978 (2)

C. J. R. Sheppard, “Electromagnetic field in the focal region of wide-angular annular lens and mirror systems,” IEE J. Microwaves Opt. Acoust. 2, 163–166 (1978).

C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J. Microwaves Opt. Acoust. 2, 105–112 (1978).

1974 (1)

J. F. Nye and M. Berry, “Dislocations of wave-fronts,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336(1605), 165–190 (1974).
[CrossRef]

1959 (1)

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

Agrawal, G. P.

G. P. Agrawal and M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27(3), 1693–1695 (1983).
[CrossRef]

Almazov, A. A.

Ambrose, W. P.

W. P. Ambrose, P. M. Goodwin, R. A. Keller, and J. C. Martin, “Alterations of single molecule fluorescence lifetimes in near-field optical microscopy,” Science 265(5170), 364–367 (1994).
[CrossRef] [PubMed]

Balla, N. K.

C. J. R. Sheppard, N. K. Balla, S. Rehman, E. Y. S. Yew, and W. T. Teng, “Bessel beams with the tightest focus,” Opt. Commun. 282, 4647–4656 (2009).
[CrossRef]

Basistiy, I. V.

I. V. Basistiy, V. A. Pas’ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. A, Pure Appl. Opt. 6(5), S166–S169 (2004).
[CrossRef]

Bazhenov, V. Y.

V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wave-fronts,” J. Mod. Opt. 39, 985–990 (1992).

Benfenati, F.

Berry, M.

J. F. Nye and M. Berry, “Dislocations of wave-fronts,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336(1605), 165–190 (1974).
[CrossRef]

Bianchini, P.

Bone, D. J.

Booker, G. R.

Bouchal, Z.

Z. Bouchal and M. Olivík, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42(8), 1555–1566 (1995).
[CrossRef]

Brown, T. G.

Burge, J. H.

Condell, W. J.

W. J. Condell, “Fraunhoffer diffraction from a circular aperture with helical phase factor,” J. Opt. Soc. Am. 2(2), 206 (1985).
[CrossRef]

Davis, L. W.

Debus, C.

Diaspro, A.

Dorn, R.

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “The focus of light - theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72(1), 109–113 (2001).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179(1–6), 1–7 (2000).
[CrossRef]

Drechsler, A.

Dunn, R. C.

X. S. Xie and R. C. Dunn, “Probing single molecule dynamics,” Science 265(5170), 361–364 (1994).
[CrossRef] [PubMed]

Durnin, J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[CrossRef] [PubMed]

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “The focus of light - theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72(1), 109–113 (2001).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179(1–6), 1–7 (2000).
[CrossRef]

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[CrossRef] [PubMed]

Elfstrom, H.

Fink, D.

Freilikher, V.

I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101(3–4), 247–264 (1993).
[CrossRef]

Freund, I.

I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101(3–4), 247–264 (1993).
[CrossRef]

Friese, M. E. J.

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394(6691), 348–350 (1998).
[CrossRef]

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995).
[CrossRef] [PubMed]

Galiani, S.

Glockl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “The focus of light - theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72(1), 109–113 (2001).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179(1–6), 1–7 (2000).
[CrossRef]

Goodwin, P. M.

W. P. Ambrose, P. M. Goodwin, R. A. Keller, and J. C. Martin, “Alterations of single molecule fluorescence lifetimes in near-field optical microscopy,” Science 265(5170), 364–367 (1994).
[CrossRef] [PubMed]

Greene, P. L.

Hajnal, J. V.

J. F. Nye and J. V. Hajnal, “The wave structure of monochromatic electromagnetic waves,” Proc. R. Soc. Lond. A Math. Phys. Sci. 409(1836), 21–36 (1987).
[CrossRef]

J. V. Hajnal, “Singularities in the transverse fields of electromagnetic waves, 1. Theory,” Proc. R. Soc. Lond. A Math. Phys. Sci. 414(1847), 433–446 (1987).
[CrossRef]

J. V. Hajnal, “Singularities in the transverse fields of electromagnetic waves, 2. Observations on the electric field,” Proc. R. Soc. Lond. A Math. Phys. Sci. 414(1847), 447–468 (1987).
[CrossRef]

Hall, D. G.

Hao, X.

X. Hao, C. F. Kuang, T. T. Wang, and X. Liu, “Phase encoding for sharper focus of the azimuthally polarized beam,” Opt. Lett. 35(23), 3928–3930 (2010).
[CrossRef] [PubMed]

X. Hao, C. F. Kuang, T. T. Wang, and X. Liu, “Effects of polarization on the de-excitation dark focal spot in STED microscopy,” J. Opt. 12(11), 115707 (2010).
[CrossRef]

Harke, B.

He, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995).
[CrossRef] [PubMed]

Heckenberg, N.

N. Heckenberg, R. McDuff, C. P. Smith, and H. Rubinstein Dunlop, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, S951–S962 (1992).

Heckenberg, N. R.

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394(6691), 348–350 (1998).
[CrossRef]

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995).
[CrossRef] [PubMed]

Hell, S. W.

N. Huse, A. Schönle, and S. W. Hell, “Z-polarized confocal microscopy,” J. Biomed. Opt. 6(3), 273–276 (2001).
[CrossRef] [PubMed]

Helseth, L. E.

L. E. Helseth, “Optical vortices in focal regions,” Opt. Commun. 229(1–6), 85–91 (2004).
[CrossRef]

Huse, N.

N. Huse, A. Schönle, and S. W. Hell, “Z-polarized confocal microscopy,” J. Biomed. Opt. 6(3), 273–276 (2001).
[CrossRef] [PubMed]

Indebetouw, G.

G. Indebetouw, “Optical vortices and their applications,” J. Mod. Opt. 40(1), 73–87 (1993).
[CrossRef]

Jordan, R. H.

Keller, R. A.

W. P. Ambrose, P. M. Goodwin, R. A. Keller, and J. C. Martin, “Alterations of single molecule fluorescence lifetimes in near-field optical microscopy,” Science 265(5170), 364–367 (1994).
[CrossRef] [PubMed]

Khonina, S. N.

Kotlyar, V. V.

Kuang, C. F.

X. Hao, C. F. Kuang, T. T. Wang, and X. Liu, “Phase encoding for sharper focus of the azimuthally polarized beam,” Opt. Lett. 35(23), 3928–3930 (2010).
[CrossRef] [PubMed]

X. Hao, C. F. Kuang, T. T. Wang, and X. Liu, “Effects of polarization on the de-excitation dark focal spot in STED microscopy,” J. Opt. 12(11), 115707 (2010).
[CrossRef]

Laczik, Z.

Larkin, K. G.

Lax, M.

G. P. Agrawal and M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27(3), 1693–1695 (1983).
[CrossRef]

Lekner, J.

J. Lekner, “Polarization of tightly focused laser beams,” J. Opt. A, Pure Appl. Opt. 5(1), 6–14 (2003).
[CrossRef]

Leuchs, G.

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “The focus of light - theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72(1), 109–113 (2001).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179(1–6), 1–7 (2000).
[CrossRef]

Lieb, M. A.

Lignani, G.

Liu, X.

X. Hao, C. F. Kuang, T. T. Wang, and X. Liu, “Effects of polarization on the de-excitation dark focal spot in STED microscopy,” J. Opt. 12(11), 115707 (2010).
[CrossRef]

X. Hao, C. F. Kuang, T. T. Wang, and X. Liu, “Phase encoding for sharper focus of the azimuthally polarized beam,” Opt. Lett. 35(23), 3928–3930 (2010).
[CrossRef] [PubMed]

Martin, J. C.

W. P. Ambrose, P. M. Goodwin, R. A. Keller, and J. C. Martin, “Alterations of single molecule fluorescence lifetimes in near-field optical microscopy,” Science 265(5170), 364–367 (1994).
[CrossRef] [PubMed]

McDuff, R.

N. Heckenberg, R. McDuff, C. P. Smith, and H. Rubinstein Dunlop, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, S951–S962 (1992).

Meixner, A. J.

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[CrossRef] [PubMed]

Mukunda, N.

Nieminen, T. A.

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394(6691), 348–350 (1998).
[CrossRef]

Nye, J. F.

J. F. Nye and J. V. Hajnal, “The wave structure of monochromatic electromagnetic waves,” Proc. R. Soc. Lond. A Math. Phys. Sci. 409(1836), 21–36 (1987).
[CrossRef]

J. F. Nye, “Polarization effects in the diffraction of electromagnetic waves: the role of disclinations,” Proc. R. Soc. Lond. A Math. Phys. Sci. 387(1792), 105–132 (1983).
[CrossRef]

J. F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. R. Soc. Lond. A Math. Phys. Sci. 389(1797), 279–290 (1983).
[CrossRef]

J. F. Nye and M. Berry, “Dislocations of wave-fronts,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336(1605), 165–190 (1974).
[CrossRef]

Oldfield, M. A.

Olivík, M.

Z. Bouchal and M. Olivík, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42(8), 1555–1566 (1995).
[CrossRef]

Pas’ko, V. A.

I. V. Basistiy, V. A. Pas’ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. A, Pure Appl. Opt. 6(5), S166–S169 (2004).
[CrossRef]

Patsakos, G.

Quabis, S.

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “The focus of light - theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72(1), 109–113 (2001).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179(1–6), 1–7 (2000).
[CrossRef]

Rehman, S.

C. J. R. Sheppard and S. Rehman, “Highly convergent focusing of light based on rotating dipole polarization,” Appl. Opt. 50(22), 4463–4467 (2011).
[CrossRef] [PubMed]

C. J. R. Sheppard, N. K. Balla, S. Rehman, E. Y. S. Yew, and W. T. Teng, “Bessel beams with the tightest focus,” Opt. Commun. 282, 4647–4656 (2009).
[CrossRef]

Richards, B.

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

Rubinstein Dunlop, H.

N. Heckenberg, R. McDuff, C. P. Smith, and H. Rubinstein Dunlop, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, S951–S962 (1992).

Rubinsztein-Dunlop, H.

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394(6691), 348–350 (1998).
[CrossRef]

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995).
[CrossRef] [PubMed]

Saghafi, S.

Schönle, A.

N. Huse, A. Schönle, and S. W. Hell, “Z-polarized confocal microscopy,” J. Biomed. Opt. 6(3), 273–276 (2001).
[CrossRef] [PubMed]

Sheppard, C. J.

Sheppard, C. J. R.

C. J. R. Sheppard and S. Rehman, “Highly convergent focusing of light based on rotating dipole polarization,” Appl. Opt. 50(22), 4463–4467 (2011).
[CrossRef] [PubMed]

C. J. R. Sheppard, N. K. Balla, S. Rehman, E. Y. S. Yew, and W. T. Teng, “Bessel beams with the tightest focus,” Opt. Commun. 282, 4647–4656 (2009).
[CrossRef]

C. J. R. Sheppard, “Polarization of almost-plane waves,” J. Opt. Soc. Am. A 17(2), 335–341 (2000).
[CrossRef] [PubMed]

C. J. R. Sheppard and S. Saghafi, “Transverse-electric and transverse-magnetic beam modes beyond the paraxial approximation,” Opt. Lett. 24(22), 1543–1545 (1999).
[CrossRef] [PubMed]

C. J. R. Sheppard, “Electromagnetic field in the focal region of wide-angular annular lens and mirror systems,” IEE J. Microwaves Opt. Acoust. 2, 163–166 (1978).

C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J. Microwaves Opt. Acoust. 2, 105–112 (1978).

C. J. R. Sheppard, “Polarization of beams and highly focused waves,” in ICO Topical Meeting on Polarization Optics, Polvijärvi, Finland (2003).

Shvartsman, N.

I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101(3–4), 247–264 (1993).
[CrossRef]

Simon, R.

Slyusar, V. V.

I. V. Basistiy, V. A. Pas’ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. A, Pure Appl. Opt. 6(5), S166–S169 (2004).
[CrossRef]

Smith, C. P.

N. Heckenberg, R. McDuff, C. P. Smith, and H. Rubinstein Dunlop, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, S951–S962 (1992).

Soifer, V. A.

Soskin, M. S.

I. V. Basistiy, V. A. Pas’ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. A, Pure Appl. Opt. 6(5), S166–S169 (2004).
[CrossRef]

V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wave-fronts,” J. Mod. Opt. 39, 985–990 (1992).

Sudarshan, E. C. G.

Tang, W. T.

Tarrach, G.

Teng, W. T.

C. J. R. Sheppard, N. K. Balla, S. Rehman, E. Y. S. Yew, and W. T. Teng, “Bessel beams with the tightest focus,” Opt. Commun. 282, 4647–4656 (2009).
[CrossRef]

Tiziani, H. J.

M. Totzeck and H. J. Tiziani, “Phase-singularities in 2D diffraction fields and interference microscopy,” Opt. Commun. 138(4–6), 365–382 (1997).
[CrossRef]

Török, P.

Totzeck, M.

M. Totzeck and H. J. Tiziani, “Phase-singularities in 2D diffraction fields and interference microscopy,” Opt. Commun. 138(4–6), 365–382 (1997).
[CrossRef]

Turunen, J.

Varga, P.

Vasnetsov, M. V.

I. V. Basistiy, V. A. Pas’ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. A, Pure Appl. Opt. 6(5), S166–S169 (2004).
[CrossRef]

V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wave-fronts,” J. Mod. Opt. 39, 985–990 (1992).

Vicidomini, G.

Wang, T. T.

X. Hao, C. F. Kuang, T. T. Wang, and X. Liu, “Effects of polarization on the de-excitation dark focal spot in STED microscopy,” J. Opt. 12(11), 115707 (2010).
[CrossRef]

X. Hao, C. F. Kuang, T. T. Wang, and X. Liu, “Phase encoding for sharper focus of the azimuthally polarized beam,” Opt. Lett. 35(23), 3928–3930 (2010).
[CrossRef] [PubMed]

Wilson, T.

C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J. Microwaves Opt. Acoust. 2, 105–112 (1978).

Wolf, E.

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

Xie, X. S.

X. S. Xie and R. C. Dunn, “Probing single molecule dynamics,” Science 265(5170), 361–364 (1994).
[CrossRef] [PubMed]

Yew, E. Y.

Yew, E. Y. S.

C. J. R. Sheppard, N. K. Balla, S. Rehman, E. Y. S. Yew, and W. T. Teng, “Bessel beams with the tightest focus,” Opt. Commun. 282, 4647–4656 (2009).
[CrossRef]

Youngworth, K. S.

Zhao, C.

Appl. Opt. (2)

Appl. Phys. B (1)

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “The focus of light - theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72(1), 109–113 (2001).
[CrossRef]

IEE J. Microwaves Opt. Acoust. (2)

C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J. Microwaves Opt. Acoust. 2, 105–112 (1978).

C. J. R. Sheppard, “Electromagnetic field in the focal region of wide-angular annular lens and mirror systems,” IEE J. Microwaves Opt. Acoust. 2, 163–166 (1978).

J. Biomed. Opt. (1)

N. Huse, A. Schönle, and S. W. Hell, “Z-polarized confocal microscopy,” J. Biomed. Opt. 6(3), 273–276 (2001).
[CrossRef] [PubMed]

J. Mod. Opt. (3)

V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wave-fronts,” J. Mod. Opt. 39, 985–990 (1992).

G. Indebetouw, “Optical vortices and their applications,” J. Mod. Opt. 40(1), 73–87 (1993).
[CrossRef]

Z. Bouchal and M. Olivík, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42(8), 1555–1566 (1995).
[CrossRef]

J. Opt. (1)

X. Hao, C. F. Kuang, T. T. Wang, and X. Liu, “Effects of polarization on the de-excitation dark focal spot in STED microscopy,” J. Opt. 12(11), 115707 (2010).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (2)

I. V. Basistiy, V. A. Pas’ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. A, Pure Appl. Opt. 6(5), S166–S169 (2004).
[CrossRef]

J. Lekner, “Polarization of tightly focused laser beams,” J. Opt. A, Pure Appl. Opt. 5(1), 6–14 (2003).
[CrossRef]

J. Opt. Soc. Am. (1)

W. J. Condell, “Fraunhoffer diffraction from a circular aperture with helical phase factor,” J. Opt. Soc. Am. 2(2), 206 (1985).
[CrossRef]

J. Opt. Soc. Am. A (8)

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian-Maxwell beams,” J. Opt. Soc. Am. A 3(4), 536–540 (1986).
[CrossRef]

P. L. Greene and D. G. Hall, “Diffraction characteristics of the azimuthal Bessel-gauss beam,” J. Opt. Soc. Am. A 13(5), 962–966 (1996).
[CrossRef]

P. L. Greene and D. G. Hall, “Properties and diffraction of the azimuthal Bessel-Gauss beam,” J. Opt. Soc. Am. A 15(12), 3020–3027 (1998).
[CrossRef]

C. J. R. Sheppard, “Polarization of almost-plane waves,” J. Opt. Soc. Am. A 17(2), 335–341 (2000).
[CrossRef] [PubMed]

V. V. Kotlyar, A. A. Almazov, S. N. Khonina, V. A. Soifer, H. Elfstrom, and J. Turunen, “Generation of phase singularity through diffracting a plane or Gaussian beam by a spiral phase plate,” J. Opt. Soc. Am. A 22(5), 849–861 (2005).
[CrossRef] [PubMed]

K. G. Larkin, D. J. Bone, and M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns. I. General background of the spiral phase quadrature transform,” J. Opt. Soc. Am. A 18(8), 1862–1870 (2001).
[CrossRef] [PubMed]

K. G. Larkin, “Natural demodulation of two-dimensional fringe patterns. II. Stationary phase analysis of the spiral phase quadrature transform,” J. Opt. Soc. Am. A 18(8), 1871–1881 (2001).
[CrossRef] [PubMed]

P. Török, P. Varga, Z. Laczik, and G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A 12(2), 325–332 (1995).
[CrossRef]

Nature (1)

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394(6691), 348–350 (1998).
[CrossRef]

Opt. Commun. (6)

V. V. Kotlyar and V. A. Soifer, “Rotor spatial filter for analysis and synthesis of coherent fields,” Opt. Commun. 89(2–4), 159–163 (1992).

L. E. Helseth, “Optical vortices in focal regions,” Opt. Commun. 229(1–6), 85–91 (2004).
[CrossRef]

C. J. R. Sheppard, N. K. Balla, S. Rehman, E. Y. S. Yew, and W. T. Teng, “Bessel beams with the tightest focus,” Opt. Commun. 282, 4647–4656 (2009).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179(1–6), 1–7 (2000).
[CrossRef]

I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101(3–4), 247–264 (1993).
[CrossRef]

M. Totzeck and H. J. Tiziani, “Phase-singularities in 2D diffraction fields and interference microscopy,” Opt. Commun. 138(4–6), 365–382 (1997).
[CrossRef]

Opt. Express (6)

Opt. Lett. (6)

Opt. Quantum Electron. (1)

N. Heckenberg, R. McDuff, C. P. Smith, and H. Rubinstein Dunlop, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, S951–S962 (1992).

Phys. Rev. A (1)

G. P. Agrawal and M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27(3), 1693–1695 (1983).
[CrossRef]

Phys. Rev. Lett. (2)

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995).
[CrossRef] [PubMed]

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[CrossRef] [PubMed]

Proc. R. Soc. Lond. A Math. Phys. Sci. (7)

J. F. Nye and M. Berry, “Dislocations of wave-fronts,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336(1605), 165–190 (1974).
[CrossRef]

J. F. Nye, “Polarization effects in the diffraction of electromagnetic waves: the role of disclinations,” Proc. R. Soc. Lond. A Math. Phys. Sci. 387(1792), 105–132 (1983).
[CrossRef]

J. F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. R. Soc. Lond. A Math. Phys. Sci. 389(1797), 279–290 (1983).
[CrossRef]

J. F. Nye and J. V. Hajnal, “The wave structure of monochromatic electromagnetic waves,” Proc. R. Soc. Lond. A Math. Phys. Sci. 409(1836), 21–36 (1987).
[CrossRef]

J. V. Hajnal, “Singularities in the transverse fields of electromagnetic waves, 1. Theory,” Proc. R. Soc. Lond. A Math. Phys. Sci. 414(1847), 433–446 (1987).
[CrossRef]

J. V. Hajnal, “Singularities in the transverse fields of electromagnetic waves, 2. Observations on the electric field,” Proc. R. Soc. Lond. A Math. Phys. Sci. 414(1847), 447–468 (1987).
[CrossRef]

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

Science (2)

X. S. Xie and R. C. Dunn, “Probing single molecule dynamics,” Science 265(5170), 361–364 (1994).
[CrossRef] [PubMed]

W. P. Ambrose, P. M. Goodwin, R. A. Keller, and J. C. Martin, “Alterations of single molecule fluorescence lifetimes in near-field optical microscopy,” Science 265(5170), 364–367 (1994).
[CrossRef] [PubMed]

Other (2)

J. F. Nye, Natural Focusing and Fine Structure of Light (Institute of Physics Publishing, 1999).

C. J. R. Sheppard, “Polarization of beams and highly focused waves,” in ICO Topical Meeting on Polarization Optics, Polvijärvi, Finland (2003).

Supplementary Material (4)

» Media 1: AVI (12153 KB)     
» Media 2: AVI (12153 KB)     
» Media 3: AVI (12153 KB)     
» Media 4: AVI (12153 KB)     

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Figures (7)

Fig. 1
Fig. 1

The input field polarizations (inhomogeneous, locally linear polarization) for different values of m , for n=0 , for the first (upper) case of Eq. (1). The arrow length is proportional to the field strength.

Fig. 2
Fig. 2

Input vector field at time t=0 , with the time variation shown in the Media, for (a) m=1/2,n=1/2 (Media 1), (b) m=1/2,n=1/2 (Media 2).

Fig. 3
Fig. 3

Parameters for the case m=1/2,n=1/2 . The top row shows the Stokes parameters. The bottom row show ψ,χ,α,δ, δ .

Fig. 4
Fig. 4

A cross section through the beam at y=0 for m=1/2,n=1/2 . S 0 is shown in blue, S 2 in red, and S 3 in green. The Airy disc is shown as a chained line for comparison.

Fig. 5
Fig. 5

Parameters for the case m=1/2,n=1/2 . The top row shows the Stokes parameters. The bottom row show ψ,χ,α,δ, δ .

Fig. 6
Fig. 6

The vector electric field of Bessel beams at time t=0 , with the time variation shown in the media, for (a) m=1/2,n=1/2 (Media 3), and (b) m=1/2,n=1/2 (Media 4).

Fig. 7
Fig. 7

The intensity of a non-paraxial Bessel beam of semi-angle of convergence θ=0, 60 , 75 , for (a) m=1/2,n=1/2 , and (b) m=1/2,n=1/2 .

Equations (19)

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

E 1 = ρ | m |+| n | ( cosmϕi+sinmϕj )exp( inϕ ), E 2 = ρ | m |+| n | ( sinmϕi+cosmϕj )exp( inϕ ),
E 1 = ρ | m |+| n | { cos[(m1)ϕ] a ρ +sin[(m1)ϕ] a ϕ }exp( inϕ ), E 2 = ρ | m |+| n | { sin[(m1)ϕ] a ρ +cos[(m1)ϕ] a ϕ }exp( inϕ ),
E 1 = 1 2 ρ g { ( iij )exp[ i( n+m )ϕ ]+( i+ij )exp[ i( nm )ϕ ] }, E 2 = i 2 ρ g { ( iij )exp[ i( n+m )ϕ ]( i+ij )exp[ i( nm )ϕ ] },
E r = E 1 +i E 2 = ρ g ( i+ij )exp[ i( nm )ϕ ], E l = E 1 i E 2 = ρ g ( iij )exp[ i( n+m )ϕ ],
e x = 1 2 { E x [ ( 1+cosθ )cos2ϕ( 1cosθ ) ] E y sin2ϕ( 1cosθ ) }, e y = 1 2 { E x sin2ϕ( 1cosθ ) E y [ ( 1+cosθ )+cos2ϕ( 1cosθ ) ] }, e z =( E x cosϕ+ E y sinϕ )sinθ.
e x = cos 2 θ 2 [ E x ( 1cos2ϕ tan 2 θ 2 ) E y sin2ϕ tan 2 θ 2 ], e y = cos 2 θ 2 [ E x sin2ϕ tan 2 θ 2 E y ( 1+cos2ϕ tan 2 θ 2 ) ], e z = cos 2 θ 2 [ ( E x cosϕ+ E y sinϕ )2tan θ 2 ],
E= iA π 0 α 0 2π e cos 1/2 θexp( ik r P cosε )dϕsinθdθ
cosε=cosθcos θ P +sinθsin θ P cos( ϕ ϕ P )
e x1 = sin g θ 2 ( { exp[ i( n+m )ϕ ]+exp[ i( nm )ϕ ] } cos 2 θ 2 { exp[ i( n+m2 )ϕ ]+exp[ i( nm+2 )ϕ ] } sin 2 θ 2 ), e y1 = i sin g θ 2 ( { exp[ i( n+m )ϕ ]exp[ i( nm )ϕ ] } cos 2 θ 2 +{ exp[ i( n+m2 )ϕ ]exp[ i( nm+2 )ϕ ] } sin 2 θ 2 ), e z1 = sin g θ{ exp[ i( n+m1 )ϕ ]+exp[ i( nm+1 )ϕ ]sin θ 2 cos θ 2 }.
0 2π exp( inϕ )exp[ iρcos( ϕγ ) ] dϕ=2π i n J n ( ρ )exp( inγ ),
E x1 = 1 2 ( { J n+m ( k ρ P sinθ )exp[i(n+m) ϕ P ] + (1) m J nm ( k ρ P sinθ )exp[i(nm) ϕ P ] } + tan 2 θ 2 { J n+m2 ( k ρ P sinθ )exp[i(n+m2) ϕ P ] + (1) m J nm+2 ( k ρ P sinθ )exp[i(nm+2) ϕ P ] } ) ×exp( ik z P cosθ ), E y1 = i 2 ( { J n+m ( k ρ P sinθ )exp[i(n+m) ϕ P ] (1) m J nm ( k ρ P sinθ )exp[i(nm) ϕ P ] } tan 2 θ 2 { J n+m2 ( k ρ P sinθ )exp[i(n+m2) ϕ P ] (1) m J nm+2 ( k ρ P sinθ )exp[i(nm+2) ϕ P ] } ) ×exp( ik z P cosθ ), E z1 =itan θ 2 { J n+m1 ( k ρ P sinθ )exp[i(n+m1) ϕ P ] (1) m J nm+1 ( k ρ P sinθ )exp[i(nm+1) ϕ P ] } ×exp( ik z P cosθ ).
e x2 = i sin g θ 2 ( { exp[ i(n+m)ϕ ]exp[ i(nm)ϕ ] } cos 2 θ 2 { exp[ i(n+m2)ϕ ]exp[ i(nm+2)ϕ ] } sin 2 θ 2 ), e y2 = sin g θ 2 ( { exp[ i(n+m)ϕ ]+exp[ i(nm)ϕ ] } cos 2 θ 2 +{ exp[ i(n+m2)ϕ ]+exp[ i(nm+2)ϕ ] } sin 2 θ 2 ), e z2 =i sin g θ{ exp[ i(n+m1)ϕ ]+exp[ i(nm+1)ϕ ]sin θ 2 cos θ 2 },
E x2 = i 2 ( { J n+m ( k ρ P sinθ )exp[i(n+m) ϕ P ] (1) m J nm ( k ρ P sinθ )exp[i(nm) ϕ P ] } + tan 2 θ 2 { J n+m2 ( k ρ P sinθ )exp[i(n+m2) ϕ P ] (1) m J nm+2 ( k ρ P sinθ )exp[i(nm+2) ϕ P ] } ) ×exp( ik z P cosθ ), E y2 = 1 2 ( { J n+m ( k ρ P sinθ )[i(n+m) ϕ P ] + (1) m J nm ( k ρ P sinθ )exp[i(nm) ϕ P ] } tan 2 θ 2 { J n+m2 ( k ρ P sinθ )[i(n+m2) ϕ P ] + (1) m J nm+2 ( k ρ P sinθ )[i(nm+2) ϕ P ] } ) ×exp( ik z P cosθ ), E z2 =tan θ 2 { J n+m1 ( k ρ P sinθ )exp[i(n+m1) ϕ P ] + (1) m J nm+1 ( k ρ P sinθ )exp[i(nm+1) ϕ P ] } ×exp( ik z P cosθ ).
E x1 = 1 2 { J n+m ( k ρ P sinθ )exp[i(n+m) ϕ P ] + (1) m J nm ( k ρ P sinθ )exp[i(nm) ϕ P ] }exp( ik z P cosθ ), E y1 = i 2 { J n+m ( k ρ P sinθ )exp[i(n+m) ϕ P ] (1) m J nm ( k ρ P sinθ )exp[i(nm) ϕ P ] }exp( ik z P cosθ ).
E x2 = i 2 { J n+m ( k ρ P sinθ )exp[i(n+m) ϕ P ] (1) m J nm ( k ρ P sinθ )exp[i(nm) ϕ P ] }exp( ik z P cosθ ), E y2 = 1 2 { J n+m ( k ρ P sinθ )exp[i(n+m) ϕ P ] + (1) m J nm ( k ρ P sinθ )exp[i(nm) ϕ P ] }exp( ik z P cosθ ).
E r = E 1 +i E 2 = ρ g ( i+ij )exp(ipϕ), E l = E 1 i E 2 = ρ g ( iij )exp(ipϕ).
e x = sin g θ cos 2 θ 2 exp(ipϕ){ 1exp(±i2ϕ) tan 2 θ 2 }, e y =±i sin g θ cos 2 θ 2 exp(ipϕ){ 1+exp(±i2ϕ) tan 2 θ 2 }, e z =2 sin g θ cos 2 θ 2 exp(ipϕ)exp(±iϕ)tan θ 2 .
E x ={ J p ( k ρ P sinθ )exp(ip ϕ P ) + tan 2 θ 2 J p±2 ( k ρ P sinθ )exp[i(p±2) ϕ P ] }exp( ik z P cosθ ), E y =±i{ J p ( k ρ P sinθ )exp(ip ϕ P ) tan 2 θ 2 { J p±2 ( k ρ P sinθ )exp[i(p±2) ϕ P ] }exp( ik z P cosθ ), E z =±i2tan θ 2 J p±1 ( k ρ P sinθ )exp[i(p±1) ϕ P ]exp( ik z P cosθ ).
E x = J p ( k ρ P sinθ )exp(ip ϕ P )exp( ik z P cosθ ), E y =±i J p ( k ρ P sinθ )exp(ip ϕ P )exp( ik z P cosθ ),

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