Abstract

The tight focusing of vortex carrying beams is studied using the Debye–Wolf diffraction integral in the presence of primary astigmatism. The roles of topological charge, polarization distribution of the input beam, and handedness of the beam polarization are investigated in the intensity distribution of the focal plane of a high-numerical-aperture lens. The effect of tight focusing in the presence of astigmatism on the dark core of the azimuthally polarized beam is also investigated and compared with the dark core of a circularly polarized vortex beam. The effect of an aberration has been discussed in the context of the fluorescent spot size in the focal plane of a stimulated emission depletion microscope for two different polarization setups.

© 2009 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  32. A. Wada, T. Ohtani, Y. Miyamoto, and M. Takeda, “Propagation analysis of the Laguerre-Gaussian beam with astigmatism,” J. Opt. Soc. Am. A 22, 2746-2755 (2005).
    [CrossRef]
  33. R. K. Singh, P. Senthilkumaran, and K. Singh, “Influence of astigmatism and defocusing on the focusing of a singular beam,” Opt. Commun. 270, 128-138 (2007).
    [CrossRef]
  34. R. Kant, “An analytical method of vector diffraction for focusing optical systems with Seidel aberrations II: astigmatism and coma,” J. Mod. Opt. 42, 299-320 (1995).
    [CrossRef]
  35. D. P. Biss and T. G. Brown, “Primary aberrations in focused radially polarized vortex beams,” Opt. Express 12, 384-393 (2004).
    [CrossRef] [PubMed]
  36. J. J. M. Braat, P. Dirksen, A. J. E. M. Janssen, and A. S. van de Nes, “Extended Nijboer-Zernike representation of the vector field in the focal region of an aberrated high-aperture optical system,” J. Opt. Soc. Am. A 20, 2281-2292 (2003).
    [CrossRef]
  37. R. K. Singh, P. Senthilkumaran, and K. Singh, “Effect of primary spherical aberration on high-numerical-aperture focusing of a Laguerre-Gaussian beam,” J. Opt. Soc. Am. A 25, 1307-1318 (2008).
    [CrossRef]
  38. R. K. Singh, P. Senthilkumaran, and K. Singh, “Effect of primary coma on the focusing of a Laguere-Gaussian beam by a high numerical aperture system; vectorial diffraction theory,” J. Opt. A, Pure Appl. Opt. 10, 075008 (2008).
    [CrossRef]
  39. Y. Iketaki, T. Watanabe, M. Sakai, S. Ishiuchi, M. Fujii, and T. Watanabe, “Theoretical investigation of the point-spread function given by super-resolving fluorescence microscopy using two-color fluorescence dip spectroscopy,” Opt. Eng. (Bellingham) 44, 033602 (2005).
    [CrossRef]

2008 (9)

Y. Zhao, G. Milne, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Quantitative force mapping of an optical vortex trap,” Appl. Phys. Lett. 92, 161111 (2008).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, “Effect of primary coma on the focusing of a Laguere-Gaussian beam by a high numerical aperture system; vectorial diffraction theory,” J. Opt. A, Pure Appl. Opt. 10, 075008 (2008).
[CrossRef]

P. Török, P. R. T. Munro, and Em. E. Kriezis, “High numerical aperture vectorial imaging in coherent optical microscopes,” Opt. Express 16, 507-523 (2008).
[CrossRef] [PubMed]

S. S. Sherif, M. R. Foreman, and P. Török, “Eigenfunction expansion of the electric fields in the focal region of a high numerical aperture focusing system,” Opt. Express 16, 3397-3407 (2008).
[CrossRef] [PubMed]

G. M. Lerman and U. Levy, “Effect of radial polarization and apodization on spot size under tight focusing conditions,” Opt. Express 16, 4567-4581 (2008).
[CrossRef] [PubMed]

Z. Zhang, J. Pu, and X. Wang, “Tight focusing of radially and azimuthally polarized vortex beams through a uniaxial birefringent crystal,” Appl. Opt. 47, 1963-1967 (2008).
[CrossRef] [PubMed]

R. K. Singh, P. Senthilkumaran, and K. Singh, “Effect of primary spherical aberration on high-numerical-aperture focusing of a Laguerre-Gaussian beam,” J. Opt. Soc. Am. A 25, 1307-1318 (2008).
[CrossRef]

W. Gao, “Effects of coherence and vector properties of the light on the resolution limit in stimulated emission depletion fluorescence microscopy,” J. Opt. Soc. Am. A 25, 1378-1382 (2008).
[CrossRef]

R. Martínez-Herrero and P. M. Mejías, “Propagation of light fields with radial or azimuthal polarization distribution at a transverse plane,” Opt. Express 16, 9021-9033 (2008).
[CrossRef] [PubMed]

2007 (6)

Y. Iketaki, T. Watanabe, N. Bokor, and M. Fujii, “Investigation of the center intensity of first- and second-order Laguerre-Gaussian beams with linear and circular polarization,” Opt. Lett. 32, 2357-2359 (2007).
[CrossRef] [PubMed]

Z. Bomzon and M. Gu, “Space-variant geometrical phases in focused cylindrical light beams,” Opt. Lett. 32, 3017-3019 (2007).
[CrossRef] [PubMed]

R. K. Singh, P. Senthilkumaran, and K. Singh, “Influence of astigmatism and defocusing on the focusing of a singular beam,” Opt. Commun. 270, 128-138 (2007).
[CrossRef]

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 1-20 (2007).
[CrossRef]

N. Bokor, Y. Iketaki, T. Watanabe, K. Daigoku, N. Davidson, and M. Fujii, “On polarization effects in fluorescence depletion microscopy,” Opt. Commun. 272, 263-268 (2007).
[CrossRef]

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef] [PubMed]

2006 (3)

L. E. Helseth, “Smallest focal hole,” Opt. Commun. 257, 1-8 (2006).
[CrossRef]

K. I. Willig, J. Keller, M. Bossi, and S. W. Hell, “STED microscopy resolves nanoparticle assemblies,” New J. Phys. 8, 1-8 (2006).
[CrossRef]

B. R. Boruah and M. A. Neil, “Susceptibility to and correction of azimuthal aberrations in singular light beams,” Opt. Express 14, 10377-10385 (2006).
[CrossRef] [PubMed]

2005 (2)

A. Wada, T. Ohtani, Y. Miyamoto, and M. Takeda, “Propagation analysis of the Laguerre-Gaussian beam with astigmatism,” J. Opt. Soc. Am. A 22, 2746-2755 (2005).
[CrossRef]

Y. Iketaki, T. Watanabe, M. Sakai, S. Ishiuchi, M. Fujii, and T. Watanabe, “Theoretical investigation of the point-spread function given by super-resolving fluorescence microscopy using two-color fluorescence dip spectroscopy,” Opt. Eng. (Bellingham) 44, 033602 (2005).
[CrossRef]

2004 (5)

D. P. Biss and T. G. Brown, “Primary aberrations in focused radially polarized vortex beams,” Opt. Express 12, 384-393 (2004).
[CrossRef] [PubMed]

P. Török and P. R. T. Munro, “The use of Gauss-Laguerre vector beams in STED microscopy,” Opt. Express 12, 3605-3617 (2004).
[CrossRef] [PubMed]

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

M. D. Levenson, T. Ebihara, G. Dai, Y. Morikawa, N. Hayashi, and S. M. Tan, “Optical vortex masks for via levels,” J. Microlithogr., Microfabr., Microsyst. 3, 293-304 (2004).
[CrossRef]

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Transformation of higher-order optical vortices upon focusing by an astigmatic lens,” Opt. Commun. 241, 237-247 (2004).
[CrossRef]

2003 (4)

2001 (2)

C. J. R. Sheppard, “High-aperture beams,” J. Opt. Soc. Am. A 18, 1579-1587 (2001).
[CrossRef]

T. A. Klar, E. Engel, and S. W. Hell, “Breaking Abbe's diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,” Phys. Rev. E 64, 066613 (2001).
[CrossRef]

2000 (1)

K. Bahlmann and S. W. Hell, “Electric field depolarization in high aperture focusing with emphasis on annular apertures,” J. Microsc. 200, 59-67 (2000).
[CrossRef] [PubMed]

1995 (1)

R. Kant, “An analytical method of vector diffraction for focusing optical systems with Seidel aberrations II: astigmatism and coma,” J. Mod. Opt. 42, 299-320 (1995).
[CrossRef]

1965 (1)

A. Boivin and E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. 138, B1561-B1565 (1965).
[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. London, Ser. A 253, 358-379 (1959).
[CrossRef]

1919 (1)

V. S. Ignatowsky, “Diffraction by a lens of arbitrary aperture,” Trans. Opt. Inst. 1, 1-36 (1919).

Bahlmann, K.

K. Bahlmann and S. W. Hell, “Electric field depolarization in high aperture focusing with emphasis on annular apertures,” J. Microsc. 200, 59-67 (2000).
[CrossRef] [PubMed]

Bekshaev, A. Ya.

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Transformation of higher-order optical vortices upon focusing by an astigmatic lens,” Opt. Commun. 241, 237-247 (2004).
[CrossRef]

Bernet, S.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 1-20 (2007).
[CrossRef]

Biss, D. P.

Boivin, A.

A. Boivin and E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. 138, B1561-B1565 (1965).
[CrossRef]

Bokor, N.

Y. Iketaki, T. Watanabe, N. Bokor, and M. Fujii, “Investigation of the center intensity of first- and second-order Laguerre-Gaussian beams with linear and circular polarization,” Opt. Lett. 32, 2357-2359 (2007).
[CrossRef] [PubMed]

N. Bokor, Y. Iketaki, T. Watanabe, K. Daigoku, N. Davidson, and M. Fujii, “On polarization effects in fluorescence depletion microscopy,” Opt. Commun. 272, 263-268 (2007).
[CrossRef]

Bomzon, Z.

Boruah, B. R.

Bossi, M.

K. I. Willig, J. Keller, M. Bossi, and S. W. Hell, “STED microscopy resolves nanoparticle assemblies,” New J. Phys. 8, 1-8 (2006).
[CrossRef]

Braat, J. J. M.

Brown, T. G.

Chiu, D. T.

Y. Zhao, G. Milne, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Quantitative force mapping of an optical vortex trap,” Appl. Phys. Lett. 92, 161111 (2008).
[CrossRef]

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef] [PubMed]

Dai, G.

M. D. Levenson, T. Ebihara, G. Dai, Y. Morikawa, N. Hayashi, and S. M. Tan, “Optical vortex masks for via levels,” J. Microlithogr., Microfabr., Microsyst. 3, 293-304 (2004).
[CrossRef]

Daigoku, K.

N. Bokor, Y. Iketaki, T. Watanabe, K. Daigoku, N. Davidson, and M. Fujii, “On polarization effects in fluorescence depletion microscopy,” Opt. Commun. 272, 263-268 (2007).
[CrossRef]

Davidson, N.

N. Bokor, Y. Iketaki, T. Watanabe, K. Daigoku, N. Davidson, and M. Fujii, “On polarization effects in fluorescence depletion microscopy,” Opt. Commun. 272, 263-268 (2007).
[CrossRef]

Dirksen, P.

Dorn, R.

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

Ebihara, T.

M. D. Levenson, T. Ebihara, G. Dai, Y. Morikawa, N. Hayashi, and S. M. Tan, “Optical vortex masks for via levels,” J. Microlithogr., Microfabr., Microsyst. 3, 293-304 (2004).
[CrossRef]

Edgar, J. S.

Y. Zhao, G. Milne, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Quantitative force mapping of an optical vortex trap,” Appl. Phys. Lett. 92, 161111 (2008).
[CrossRef]

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef] [PubMed]

Engel, E.

E. Engel, N. Huse, T. A. Klar, and S. W. Hell, “Creating λ/3 focal holes with a Mach-Zehnder interferometer,” Appl. Phys. B 77, 11-17 (2003).
[CrossRef]

T. A. Klar, E. Engel, and S. W. Hell, “Breaking Abbe's diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,” Phys. Rev. E 64, 066613 (2001).
[CrossRef]

Foreman, M. R.

Fujii, M.

Y. Iketaki, T. Watanabe, N. Bokor, and M. Fujii, “Investigation of the center intensity of first- and second-order Laguerre-Gaussian beams with linear and circular polarization,” Opt. Lett. 32, 2357-2359 (2007).
[CrossRef] [PubMed]

N. Bokor, Y. Iketaki, T. Watanabe, K. Daigoku, N. Davidson, and M. Fujii, “On polarization effects in fluorescence depletion microscopy,” Opt. Commun. 272, 263-268 (2007).
[CrossRef]

Y. Iketaki, T. Watanabe, M. Sakai, S. Ishiuchi, M. Fujii, and T. Watanabe, “Theoretical investigation of the point-spread function given by super-resolving fluorescence microscopy using two-color fluorescence dip spectroscopy,” Opt. Eng. (Bellingham) 44, 033602 (2005).
[CrossRef]

Fürhapter, S.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 1-20 (2007).
[CrossRef]

Gan, X.

Ganic, D.

Gao, W.

Gu, M.

Hayashi, N.

M. D. Levenson, T. Ebihara, G. Dai, Y. Morikawa, N. Hayashi, and S. M. Tan, “Optical vortex masks for via levels,” J. Microlithogr., Microfabr., Microsyst. 3, 293-304 (2004).
[CrossRef]

Hell, S. W.

K. I. Willig, J. Keller, M. Bossi, and S. W. Hell, “STED microscopy resolves nanoparticle assemblies,” New J. Phys. 8, 1-8 (2006).
[CrossRef]

E. Engel, N. Huse, T. A. Klar, and S. W. Hell, “Creating λ/3 focal holes with a Mach-Zehnder interferometer,” Appl. Phys. B 77, 11-17 (2003).
[CrossRef]

T. A. Klar, E. Engel, and S. W. Hell, “Breaking Abbe's diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,” Phys. Rev. E 64, 066613 (2001).
[CrossRef]

K. Bahlmann and S. W. Hell, “Electric field depolarization in high aperture focusing with emphasis on annular apertures,” J. Microsc. 200, 59-67 (2000).
[CrossRef] [PubMed]

Helseth, L. E.

L. E. Helseth, “Smallest focal hole,” Opt. Commun. 257, 1-8 (2006).
[CrossRef]

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

Huse, N.

E. Engel, N. Huse, T. A. Klar, and S. W. Hell, “Creating λ/3 focal holes with a Mach-Zehnder interferometer,” Appl. Phys. B 77, 11-17 (2003).
[CrossRef]

Ignatowsky, V. S.

V. S. Ignatowsky, “Diffraction by a lens of arbitrary aperture,” Trans. Opt. Inst. 1, 1-36 (1919).

Iketaki, Y.

Y. Iketaki, T. Watanabe, N. Bokor, and M. Fujii, “Investigation of the center intensity of first- and second-order Laguerre-Gaussian beams with linear and circular polarization,” Opt. Lett. 32, 2357-2359 (2007).
[CrossRef] [PubMed]

N. Bokor, Y. Iketaki, T. Watanabe, K. Daigoku, N. Davidson, and M. Fujii, “On polarization effects in fluorescence depletion microscopy,” Opt. Commun. 272, 263-268 (2007).
[CrossRef]

Y. Iketaki, T. Watanabe, M. Sakai, S. Ishiuchi, M. Fujii, and T. Watanabe, “Theoretical investigation of the point-spread function given by super-resolving fluorescence microscopy using two-color fluorescence dip spectroscopy,” Opt. Eng. (Bellingham) 44, 033602 (2005).
[CrossRef]

Ishiuchi, S.

Y. Iketaki, T. Watanabe, M. Sakai, S. Ishiuchi, M. Fujii, and T. Watanabe, “Theoretical investigation of the point-spread function given by super-resolving fluorescence microscopy using two-color fluorescence dip spectroscopy,” Opt. Eng. (Bellingham) 44, 033602 (2005).
[CrossRef]

Janssen, A. J. E. M.

Jeffries, G. D. M.

Y. Zhao, G. Milne, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Quantitative force mapping of an optical vortex trap,” Appl. Phys. Lett. 92, 161111 (2008).
[CrossRef]

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef] [PubMed]

Jesacher, A.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 1-20 (2007).
[CrossRef]

Kant, R.

R. Kant, “An analytical method of vector diffraction for focusing optical systems with Seidel aberrations II: astigmatism and coma,” J. Mod. Opt. 42, 299-320 (1995).
[CrossRef]

Keller, J.

K. I. Willig, J. Keller, M. Bossi, and S. W. Hell, “STED microscopy resolves nanoparticle assemblies,” New J. Phys. 8, 1-8 (2006).
[CrossRef]

Klar, T. A.

E. Engel, N. Huse, T. A. Klar, and S. W. Hell, “Creating λ/3 focal holes with a Mach-Zehnder interferometer,” Appl. Phys. B 77, 11-17 (2003).
[CrossRef]

T. A. Klar, E. Engel, and S. W. Hell, “Breaking Abbe's diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,” Phys. Rev. E 64, 066613 (2001).
[CrossRef]

Kriezis, Em. E.

Lerman, G. M.

Leuchs, G.

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

Levenson, M. D.

M. D. Levenson, T. Ebihara, G. Dai, Y. Morikawa, N. Hayashi, and S. M. Tan, “Optical vortex masks for via levels,” J. Microlithogr., Microfabr., Microsyst. 3, 293-304 (2004).
[CrossRef]

Levy, U.

Luneburg, R.

R. Luneburg, Mathematical Theory of Optics (U. of California Press, 1966).

Mahajan, V. N.

V. N. Mahajan, Optical Imaging and Aberrations, Part 2: Wave and Diffraction Optics (SPIE, 2001).

Martínez-Herrero, R.

Maurer, C.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 1-20 (2007).
[CrossRef]

McGloin, D.

Y. Zhao, G. Milne, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Quantitative force mapping of an optical vortex trap,” Appl. Phys. Lett. 92, 161111 (2008).
[CrossRef]

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef] [PubMed]

Mejías, P. M.

Milne, G.

Y. Zhao, G. Milne, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Quantitative force mapping of an optical vortex trap,” Appl. Phys. Lett. 92, 161111 (2008).
[CrossRef]

Miyamoto, Y.

Morikawa, Y.

M. D. Levenson, T. Ebihara, G. Dai, Y. Morikawa, N. Hayashi, and S. M. Tan, “Optical vortex masks for via levels,” J. Microlithogr., Microfabr., Microsyst. 3, 293-304 (2004).
[CrossRef]

Munro, P. R. T.

Neil, M. A.

Ohtani, T.

Pu, J.

Quabis, S.

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

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. London, Ser. A 253, 358-379 (1959).
[CrossRef]

Ritsch-Marte, M.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 1-20 (2007).
[CrossRef]

Sakai, M.

Y. Iketaki, T. Watanabe, M. Sakai, S. Ishiuchi, M. Fujii, and T. Watanabe, “Theoretical investigation of the point-spread function given by super-resolving fluorescence microscopy using two-color fluorescence dip spectroscopy,” Opt. Eng. (Bellingham) 44, 033602 (2005).
[CrossRef]

Senthilkumaran, P.

R. K. Singh, P. Senthilkumaran, and K. Singh, “Effect of primary spherical aberration on high-numerical-aperture focusing of a Laguerre-Gaussian beam,” J. Opt. Soc. Am. A 25, 1307-1318 (2008).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, “Effect of primary coma on the focusing of a Laguere-Gaussian beam by a high numerical aperture system; vectorial diffraction theory,” J. Opt. A, Pure Appl. Opt. 10, 075008 (2008).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, “Influence of astigmatism and defocusing on the focusing of a singular beam,” Opt. Commun. 270, 128-138 (2007).
[CrossRef]

Sheppard, C. J. R.

Sherif, S. S.

Singh, K.

R. K. Singh, P. Senthilkumaran, and K. Singh, “Effect of primary spherical aberration on high-numerical-aperture focusing of a Laguerre-Gaussian beam,” J. Opt. Soc. Am. A 25, 1307-1318 (2008).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, “Effect of primary coma on the focusing of a Laguere-Gaussian beam by a high numerical aperture system; vectorial diffraction theory,” J. Opt. A, Pure Appl. Opt. 10, 075008 (2008).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, “Influence of astigmatism and defocusing on the focusing of a singular beam,” Opt. Commun. 270, 128-138 (2007).
[CrossRef]

Singh, R. K.

R. K. Singh, P. Senthilkumaran, and K. Singh, “Effect of primary spherical aberration on high-numerical-aperture focusing of a Laguerre-Gaussian beam,” J. Opt. Soc. Am. A 25, 1307-1318 (2008).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, “Effect of primary coma on the focusing of a Laguere-Gaussian beam by a high numerical aperture system; vectorial diffraction theory,” J. Opt. A, Pure Appl. Opt. 10, 075008 (2008).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, “Influence of astigmatism and defocusing on the focusing of a singular beam,” Opt. Commun. 270, 128-138 (2007).
[CrossRef]

Soskin, M. S.

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Transformation of higher-order optical vortices upon focusing by an astigmatic lens,” Opt. Commun. 241, 237-247 (2004).
[CrossRef]

Takeda, M.

Tan, S. M.

M. D. Levenson, T. Ebihara, G. Dai, Y. Morikawa, N. Hayashi, and S. M. Tan, “Optical vortex masks for via levels,” J. Microlithogr., Microfabr., Microsyst. 3, 293-304 (2004).
[CrossRef]

Török, P.

van de Nes, A. S.

Vasnetsov, M. V.

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Transformation of higher-order optical vortices upon focusing by an astigmatic lens,” Opt. Commun. 241, 237-247 (2004).
[CrossRef]

Wada, A.

Wang, X.

Watanabe, T.

Y. Iketaki, T. Watanabe, N. Bokor, and M. Fujii, “Investigation of the center intensity of first- and second-order Laguerre-Gaussian beams with linear and circular polarization,” Opt. Lett. 32, 2357-2359 (2007).
[CrossRef] [PubMed]

N. Bokor, Y. Iketaki, T. Watanabe, K. Daigoku, N. Davidson, and M. Fujii, “On polarization effects in fluorescence depletion microscopy,” Opt. Commun. 272, 263-268 (2007).
[CrossRef]

Y. Iketaki, T. Watanabe, M. Sakai, S. Ishiuchi, M. Fujii, and T. Watanabe, “Theoretical investigation of the point-spread function given by super-resolving fluorescence microscopy using two-color fluorescence dip spectroscopy,” Opt. Eng. (Bellingham) 44, 033602 (2005).
[CrossRef]

Y. Iketaki, T. Watanabe, M. Sakai, S. Ishiuchi, M. Fujii, and T. Watanabe, “Theoretical investigation of the point-spread function given by super-resolving fluorescence microscopy using two-color fluorescence dip spectroscopy,” Opt. Eng. (Bellingham) 44, 033602 (2005).
[CrossRef]

Willig, K. I.

K. I. Willig, J. Keller, M. Bossi, and S. W. Hell, “STED microscopy resolves nanoparticle assemblies,” New J. Phys. 8, 1-8 (2006).
[CrossRef]

Wolf, E.

A. Boivin and E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. 138, B1561-B1565 (1965).
[CrossRef]

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

Zhang, Z.

Zhao, Y.

Y. Zhao, G. Milne, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Quantitative force mapping of an optical vortex trap,” Appl. Phys. Lett. 92, 161111 (2008).
[CrossRef]

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef] [PubMed]

Appl. Opt. (1)

Appl. Phys. B (1)

E. Engel, N. Huse, T. A. Klar, and S. W. Hell, “Creating λ/3 focal holes with a Mach-Zehnder interferometer,” Appl. Phys. B 77, 11-17 (2003).
[CrossRef]

Appl. Phys. Lett. (1)

Y. Zhao, G. Milne, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Quantitative force mapping of an optical vortex trap,” Appl. Phys. Lett. 92, 161111 (2008).
[CrossRef]

J. Microlithogr., Microfabr., Microsyst. (1)

M. D. Levenson, T. Ebihara, G. Dai, Y. Morikawa, N. Hayashi, and S. M. Tan, “Optical vortex masks for via levels,” J. Microlithogr., Microfabr., Microsyst. 3, 293-304 (2004).
[CrossRef]

J. Microsc. (1)

K. Bahlmann and S. W. Hell, “Electric field depolarization in high aperture focusing with emphasis on annular apertures,” J. Microsc. 200, 59-67 (2000).
[CrossRef] [PubMed]

J. Mod. Opt. (1)

R. Kant, “An analytical method of vector diffraction for focusing optical systems with Seidel aberrations II: astigmatism and coma,” J. Mod. Opt. 42, 299-320 (1995).
[CrossRef]

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

R. K. Singh, P. Senthilkumaran, and K. Singh, “Effect of primary coma on the focusing of a Laguere-Gaussian beam by a high numerical aperture system; vectorial diffraction theory,” J. Opt. A, Pure Appl. Opt. 10, 075008 (2008).
[CrossRef]

J. Opt. Soc. Am. A (5)

New J. Phys. (2)

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 1-20 (2007).
[CrossRef]

K. I. Willig, J. Keller, M. Bossi, and S. W. Hell, “STED microscopy resolves nanoparticle assemblies,” New J. Phys. 8, 1-8 (2006).
[CrossRef]

Opt. Commun. (5)

L. E. Helseth, “Smallest focal hole,” Opt. Commun. 257, 1-8 (2006).
[CrossRef]

N. Bokor, Y. Iketaki, T. Watanabe, K. Daigoku, N. Davidson, and M. Fujii, “On polarization effects in fluorescence depletion microscopy,” Opt. Commun. 272, 263-268 (2007).
[CrossRef]

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Transformation of higher-order optical vortices upon focusing by an astigmatic lens,” Opt. Commun. 241, 237-247 (2004).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, “Influence of astigmatism and defocusing on the focusing of a singular beam,” Opt. Commun. 270, 128-138 (2007).
[CrossRef]

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

Opt. Eng. (Bellingham) (1)

Y. Iketaki, T. Watanabe, M. Sakai, S. Ishiuchi, M. Fujii, and T. Watanabe, “Theoretical investigation of the point-spread function given by super-resolving fluorescence microscopy using two-color fluorescence dip spectroscopy,” Opt. Eng. (Bellingham) 44, 033602 (2005).
[CrossRef]

Opt. Express (8)

Opt. Lett. (2)

Phys. Rev. (1)

A. Boivin and E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. 138, B1561-B1565 (1965).
[CrossRef]

Phys. Rev. E (1)

T. A. Klar, E. Engel, and S. W. Hell, “Breaking Abbe's diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,” Phys. Rev. E 64, 066613 (2001).
[CrossRef]

Phys. Rev. Lett. (2)

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef] [PubMed]

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

Proc. R. Soc. London, Ser. A (1)

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

Trans. Opt. Inst. (1)

V. S. Ignatowsky, “Diffraction by a lens of arbitrary aperture,” Trans. Opt. Inst. 1, 1-36 (1919).

Other (3)

R. Luneburg, Mathematical Theory of Optics (U. of California Press, 1966).

L.Allen L, S.M.Barnett, and M.J.Padgett, eds., Optical Angular Momentum (Institute of Physics, 2003).
[CrossRef]

V. N. Mahajan, Optical Imaging and Aberrations, Part 2: Wave and Diffraction Optics (SPIE, 2001).

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Figures (14)

Fig. 1
Fig. 1

Schematic representation of optical geometry.

Fig. 2
Fig. 2

Intensity distribution ( E 2 ) of an LC vortex beam ( m = 1 ) at the focal plane of a lens with α = 75 ° ; A a = ( a ) 0.0, (b) 0.5, (c) 1.0, (d) 1.5; for m = 2 and A a = ( e ) 0.0, (f) 0.5, (g) 1.0, (h) 1.5.

Fig. 3
Fig. 3

Squares of polarization components of an LC vortex beam ( m = 1 ) at the focal plane of a lens with α = 75 ° ; A a = 0.0 : (a) E x 2 , (b) E y 2 , (c) E z 2 ; and A a = 0.5 : (d) E x 2 , (e) E y 2 , (f) E z 2 .

Fig. 4
Fig. 4

Same as in Fig. 3; m = 2 .

Fig. 5
Fig. 5

Phase distribution of the longitudinal polarization component at the focal plane of a lens with α = 75 ° for beam with m = 1 and A a = ( a ) 0.0, (b) 0.5; for m = 2 and A a = ( c ) 0.0, (d) 0.5.

Fig. 6
Fig. 6

Intensity distribution of an LC vortex beam with m = 1 and A a = 0.5 , focused by a lens with α = 75 ° , in the transverse plane corresponding to (a) u = 1.0 , (b) u = 1.0 , (c) u = 3.16 , (d) u = 3.16 .

Fig. 7
Fig. 7

Intensity distribution ( E 2 ) of a RC vortex beam with m = 1 at the focal plane of an air aplanatic system with α = 75 ° and A a = ( a ) 0.0, (b) 0.5, (c) 1.0, (d) 1.5; for m = 2 and A a = ( e ) 0.0, (f) 0.5, (g) 1.0, (h) 1.5.

Fig. 8
Fig. 8

Squares of polarization components of a RC vortex beam with m = 1 at the focal plane of a lens with α = 75 ° and A a = 0.0 : (a) E x 2 , (b) E y 2 , (c) E z 2 ; and A a = 0.5 : (d) E x 2 , (e) E y 2 , (f) E z 2 .

Fig. 9
Fig. 9

Same as in Fig. 8; m = 2 .

Fig. 10
Fig. 10

Intensity distribution ( E 2 ) of azimuthally polarized beam at the focal plane of a lens with α = 75 ° and A a = ( a ) 0.0, (b) 0.5, (c) 1.0, (d) 1.5.

Fig. 11
Fig. 11

Intensity distribution in the x z plane of a lens with α = 75 ° for a LC vortex beam with m = 1 ; (a) A a = 0.0 , (b) A a = 0.5 ; for an azimuthally polarized beam with m = 0 ; A a = ( c ) 0.0, (d) 0.5.

Fig. 12
Fig. 12

Encircled energy at the focal plane of a lens with α = 75 ° ; for an azimuthally polarized beam with m = 0 for A a = ( a ) 0.0, (b) 0.5; for a LC vortex beam with m = 1 and A a = ( c ) 0.0, (d) 0.5; for a RC vortex beam with m = 1 and A a = ( e ) 0.0, (f) 0.5. Encircled energy at the focus plane of a lens with α = 10 ° for A a = ( g ) 0.0, (h) 0.5.

Fig. 13
Fig. 13

Intensity profiles of the fluorescence spot in the focal plane for a circular–circular polarization setup (a) A a = 0.0 ; and A a = 0.5 along the (b) x and (c) y axes.

Fig. 14
Fig. 14

Intensity profiles for the fluorescence spot in the focal plane for the azimuthal–linear polarization setup (a) A a = 0.0 and 0.5 along the x axis; A a = ( b ) 0.0 and (c) 0.5 along the y axis.

Equations (48)

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E 0 ( θ , ϕ ) = A 1 ( θ ) exp ( i m ϕ ) ,
E ( P ) = i k 2 π Ω a ( s x , s y ) s z exp [ i k { Φ ( s x , s y ) s r ( P ) } ] d s x d s y ,
x = r ( θ , ϕ ) sin θ cos ϕ ,
y = r ( θ , ϕ ) sin θ sin ϕ ,
z = r ( θ , ϕ ) cos θ .
r = r ( θ , ϕ ) { sin θ cos ϕ i ̂ + sin θ sin ϕ j ̂ + cos θ k ̂ } ,
r ( θ , ϕ ) = f + A a ρ 2 cos 2 ϕ ,
Φ = A a ρ 2 cos 2 ϕ ,
s = ( s x i ̂ , s y j ̂ , s z k ̂ ) = a θ × a ϕ a θ × a ϕ ,
s x = 1 σ ( sin θ cos ϕ 1 r r θ cos θ cos ϕ + 1 r sin θ r ϕ sin ϕ ) ,
s y = 1 σ ( sin θ sin ϕ 1 r r θ cos θ sin ϕ 1 r sin θ r ϕ cos ϕ ) ,
s z = 1 σ ( cos θ + 1 r sin θ r θ ) ,
σ = [ 1 + 1 r 2 { ( r θ ) 2 + 1 sin 2 θ ( r ϕ ) 2 } ] 1 2 .
1 σ 1 ϴ + o ( ϴ ) ,
ϴ = 1 2 r 2 [ ( r θ ) 2 + 1 sin 2 θ ( r ϕ ) 2 ] ,
a P ( θ , ϕ ) = ( s k ) 1 2 ( s x 2 + s y 2 ) [ A ( s y 2 + s x 2 s z ) + B ( s x s y + s x s y s z ) A ( s x s y + s x s y s z ) + B ( s x 2 + s y 2 s z ) [ A ( s x ) + B ( s y ) ] ( s x 2 + s y 2 ) ] ,
P ( θ , ϕ ) = 1 ( s x 2 + s y 2 ) [ A ( s y 2 + s x 2 s z ) + B ( s x s y + s x s y s z ) A ( s x s y + s x s y s z ) + B ( s x 2 + s y 2 s z ) [ A ( s x ) + B ( s y ) ] ( s x 2 + s y 2 ) ] ,
P ( θ , ϕ ) = [ A { cos θ cos 2 ϕ + sin 2 ϕ } + B { cos θ sin ϕ cos ϕ sin ϕ cos ϕ } A { cos θ sin ϕ cos ϕ sin ϕ cos ϕ } + B { cos θ sin 2 ϕ + cos 2 ϕ } A sin θ cos ϕ B sin θ sin ϕ ] .
s = ( 1 ϴ ) n + 1 σ F ( θ , ϕ ) ,
F x ( θ , ϕ ) = ( 1 r r θ cos θ cos ϕ + 1 r sin θ r ϕ sin ϕ ) ,
F y ( θ , ϕ ) = ( 1 r r θ cos θ sin ϕ 1 r sin θ r ϕ cos ϕ ) ,
F z ( θ , ϕ ) = ( 1 r sin θ r θ ) .
E ( v , u ) = i k f 2 π 0 α 0 2 π A 2 ( θ ) exp ( i m ϕ ) P ( θ , ϕ ) exp { i k [ Φ + ( ϴ n 1 σ F ) r ( P ) ] } × exp [ ( i v sin θ sin α ) cos ( ϕ ϕ P ) ] exp ( i u cos θ sin 2 α ) J d ϕ d θ ,
I ( u , v ) = E x 2 + E y 2 + E z 2 ,
v = k r P sin θ P sin α ,
u = k r P cos θ P sin 2 α ,
r ( P ) = r ( P ) { sin θ P cos ϕ P i ̂ + sin θ P sin ϕ P j ̂ + cos θ P k ̂ } .
[ E x ( v x , v y ) E y ( v x , v y ) E z ( v x , v y ) ] = ( i f λ ) 0 α 0 2 π A 2 ( θ ) exp ( i m ϕ ) 1 ( s x 2 + s y 2 ) [ ( s y 2 + s x 2 s z ) ( s x s y + s x s y s z ) s x ( s x 2 + s y 2 ) ] × exp { i k [ Φ + ( ϴ n 1 σ F ) r ( P ) ] } × exp [ i v sin α sin θ cos ( ϕ ϕ P ) ] J d ϕ d θ ,
J = ( s x θ s y ϕ s x ϕ s y θ ) .
E 0 ( θ , ϕ ) = e i m ϕ ( E 1 + e i ϕ E 2 ) ,
[ E x ( v x , v y ) E y ( v x , v y ) E z ( v x , v y ) ] = ( i f λ ) 0 α 0 2 π A 2 ( θ ) exp ( i m ϕ ) 1 ( s x 2 + s y 2 ) [ ( s y 2 + s x 2 s z ) ± i ( s x s y + s x s y s z ) ( s x s y + s x s y s z ) ± i ( s x 2 + s y 2 s z ) [ ( s x ) ± i ( s y ) ] ( s x 2 + s y 2 ) ] × exp { i k [ Φ + ( ϴ n 1 σ F ) r ( P ) ] } × exp [ i v sin α sin θ cos ( ϕ ϕ P ) ] J d ϕ d θ .
E x LC ( v ) = 2 π i m exp ( i m ϕ P ) I m 2 π i m + 2 exp [ i ( m + 2 ) ϕ P ] I m + 2 ,
E y LC ( v ) = i 2 π i m exp ( i m ϕ P ) I m + i 2 π i m + 2 exp [ i ( m + 2 ) ϕ P ] I m + 2 ,
E z LC ( v ) = 4 π i m + 1 exp [ i ( m + 1 ) ϕ P ] I m + 1 .
E x RC ( v ) = 2 π i m exp ( i m ϕ P ) I m 2 π i m 2 exp [ i ( m 2 ) ϕ P ] I m 2 ,
E y RC ( v ) = i 2 π i m exp ( i m ϕ P ) I m i 2 π i m 2 exp [ i ( m 2 ) ϕ P ] I m 2 ,
E z RC ( v ) = 4 π i m 1 exp [ i ( m 1 ) ϕ P ] I m 1 .
I m ( v , u ) = ( i f λ ) 0 α cos 1 2 θ ( 1 + cos θ ) J m [ v sin α sin θ ] exp [ i u sin 2 α cos θ ] sin θ d θ ,
I m ± 1 ( v , u ) = ( i f λ ) 0 α cos 1 2 θ J m ± 1 [ v sin α sin θ ] exp [ i u sin 2 α cos θ ] sin 2 θ d θ ,
I m ± 2 ( v , u ) = ( i f λ ) 0 α cos 1 2 θ ( 1 cos θ ) J m ± 2 [ v sin α sin θ ] exp [ i u sin 2 α cos θ ] sin θ d θ ,
[ E x ( v x , v y ) E y ( v x , v y ) E z ( v x , v y ) ] = ( i f λ ) 0 α 0 2 π A 2 ( θ ) exp ( i m ϕ ) 1 ( s x 2 + s y 2 ) [ sin ϕ ( s y 2 + s x 2 s z ) + ( cos ϕ ) ( s x s y + s x s y s z ) sin ϕ ( s x s y + s x s y s z ) + ( cos ϕ ) ( s x 2 + s y 2 s z ) [ sin ϕ ( s x ) + ( cos ϕ ) ( s y ) ] ( s x 2 + s y 2 ) ] × exp { i k [ Φ + ( ϴ n 1 σ F ) r ( P ) ] } × exp [ i v sin α sin θ cos ( ϕ ϕ P ) ] J d ϕ d θ .
E x ( u , v ) = [ i 2 π i m + 1 e i ( m + 1 ) ϕ I m + 1 A + i 2 π i m 1 e i ( m 1 ) ϕ I m 1 A ] ,
E y ( u , v ) = [ 2 π i m + 1 e i ( m + 1 ) ϕ I m + 1 A + 2 π i m 1 e i ( m 1 ) ϕ I m 1 A ] ,
E z ( u , v ) = 0 ,
I m ± 1 A ( u , v ) = ( i f λ ) 0 α cos 1 2 θ sin θ J m ± 1 ( v sin α sin θ ) exp [ i u sin 2 α cos θ ] d θ .
W ( ν 0 , u ) = 0 2 π 0 ν 0 I ( v , ϕ ) v d v d ϕ .
D vect = 1 [ 1 + C ( n d E e ) 2 ] ,
I f ( x P , y P ) D ( x P , y P ) I P ( x P , y P ) ,

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