Abstract

Dark focal spot shaping is investigated by vector diffraction theory in the focal region of a hyperbolic-cosine beam that contains one on-axis spiral optical vortex. Results show that a dark focal shape can be altered considerably by the decentered parameters in cosh parts of the beam and topological charge of the vortex. Many novel and interesting dark focal shapes may appear, including rhombic, quadrangular, cross-shaped, and foursquare dark foci. Some dark focal spot chains can also occur. In addition, the numerical aperture of the focusing system can also affect dark focal shapes remarkably, which may lead to dark focal spots that disappear or change focal shape due to the depolarization effect for a high numerical aperture. All of the above dark focal shapes can be used in an optical manipulation system to construct alterable optical traps.

© 2010 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
  3. D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810-816 (2003).
    [CrossRef] [PubMed]
  4. M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426, 421-424 (2003).
    [CrossRef] [PubMed]
  5. V. Garces-Chaves, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145-147 (2002).
    [CrossRef]
  6. L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optical trapped microscopic particles,” Science 292, 912-914 (2001).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  20. Z. Hricha and A. Belafhal, “Focusing properties and focal shift in hyperbolic-cosine-Gaussian beams,” Opt. Commun. 253, 242-249 (2005).
    [CrossRef]
  21. X. Gao, “Focusing properties of the hyperbolic-cosine-Gaussian beam induced by phase plate,” Phys. Lett. A 360, 330-335 (2006).
    [CrossRef]
  22. X. Gao and J. Li, “Focal shift of apodized truncated hyperbolic-cosine-Gaussian beam,” Opt. Commun. 273, 21-27 (2007).
    [CrossRef]
  23. B. Lü, H. Ma, and B. Zhang, “Propagation properties of cosh-Gaussian beams,” Opt. Commun. 164, 165-170 (1999).
    [CrossRef]
  24. X. Gao, J. Wang, H. Gu, and S. Hu, “Focusing of hyperbolic-cosine-Gaussian beam with a non-spiral vortex,” Optik (Stuttgart) 120, 201-206 (2009).
    [CrossRef]
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    [CrossRef] [PubMed]

2009

X. Gao, J. Wang, H. Gu, and S. Hu, “Focusing of hyperbolic-cosine-Gaussian beam with a non-spiral vortex,” Optik (Stuttgart) 120, 201-206 (2009).
[CrossRef]

2007

X. Gao and J. Li, “Focal shift of apodized truncated hyperbolic-cosine-Gaussian beam,” Opt. Commun. 273, 21-27 (2007).
[CrossRef]

X. Gao and J. Wang, “Focal evolution induced by combination of nonspiral and spiral phase plates,” Chin. Opt. Lett. 5(5), 257-259 (2007).

2006

X. Du and D. Zhao, “Elliptical cosh-Gaussian beams,” Opt. Commun. 265, 418-424 (2006).
[CrossRef]

X. Gao, “Focusing properties of the hyperbolic-cosine-Gaussian beam induced by phase plate,” Phys. Lett. A 360, 330-335 (2006).
[CrossRef]

2005

Z. Hricha and A. Belafhal, “Focusing properties and focal shift in hyperbolic-cosine-Gaussian beams,” Opt. Commun. 253, 242-249 (2005).
[CrossRef]

D. Cojoc, V. Garbin, E. Ferrari, L. Businaro, F. Romamato, and E. Di. Fabrizio, “Laser trapping and micro-manipulation using optical vortices,” Microelectron. Eng. 77, 125-131 (2005).
[CrossRef]

2004

2003

2002

D. Ganic, X. Gan, M. Gu, M. Hain, S. Somalingam, S. Stankovic, and T. Tschudi, “Generation of doughnut laser beams by use of a liquid-crystal cell with a conversion efficiency near 100%,” Opt. Lett. 27, 1351-1353 (2002).
[CrossRef]

V. Garces-Chaves, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145-147 (2002).
[CrossRef]

2001

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optical trapped microscopic particles,” Science 292, 912-914 (2001).
[CrossRef] [PubMed]

2000

1999

B. Lü, H. Ma, and B. Zhang, “Propagation properties of cosh-Gaussian beams,” Opt. Commun. 164, 165-170 (1999).
[CrossRef]

1997

D. Rozas, Z. S. Sacks, and G. A. Swartzlander, Jr., “Experimental observation of fluidlike motion of optical vortices,” Phys. Rev. Lett. 79, 3399-3402 (1997).
[CrossRef]

L. W. Casperson, D. G. Hall, and A. A. Tovar, “Sinusoidal-Gaussian beams in complex optical systems,” J. Opt. Soc. Am. A 14, 3341-3348 (1997).
[CrossRef]

1992

K. Visscher and G. J. Brakenhoff, “Theoretical study of optically induced forces on spherical particles in a single beam trap I: Rayleigh scatterers,” Optik (Stuttgart) 89, 174-180 (1992).

1989

Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330, 769-771 (1989).
[CrossRef]

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403-408 (1989).
[CrossRef]

1986

Arlt, J.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optical trapped microscopic particles,” Science 292, 912-914 (2001).
[CrossRef] [PubMed]

J. Arlt and M. J. Padgett, “Generation of a beam with a dark focus surrounded by regions of higher intensity: the optical bottle beam,” Opt. Lett. 25, 191-193 (2000).
[CrossRef]

Ashkin,

Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330, 769-771 (1989).
[CrossRef]

Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11, 288-290 (1986).
[CrossRef] [PubMed]

Belafhal, A.

Z. Hricha and A. Belafhal, “Focusing properties and focal shift in hyperbolic-cosine-Gaussian beams,” Opt. Commun. 253, 242-249 (2005).
[CrossRef]

Bjorkholm, J. E.

Brakenhoff, G. J.

K. Visscher and G. J. Brakenhoff, “Theoretical study of optically induced forces on spherical particles in a single beam trap I: Rayleigh scatterers,” Optik (Stuttgart) 89, 174-180 (1992).

Bryant, P. E.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optical trapped microscopic particles,” Science 292, 912-914 (2001).
[CrossRef] [PubMed]

Businaro, L.

D. Cojoc, V. Garbin, E. Ferrari, L. Businaro, F. Romamato, and E. Di. Fabrizio, “Laser trapping and micro-manipulation using optical vortices,” Microelectron. Eng. 77, 125-131 (2005).
[CrossRef]

Casperson, L. W.

Cheong, W. C.

Chu, S.

Cojoc, D.

D. Cojoc, V. Garbin, E. Ferrari, L. Businaro, F. Romamato, and E. Di. Fabrizio, “Laser trapping and micro-manipulation using optical vortices,” Microelectron. Eng. 77, 125-131 (2005).
[CrossRef]

Coullet, P.

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403-408 (1989).
[CrossRef]

Dholakia, K.

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426, 421-424 (2003).
[CrossRef] [PubMed]

V. Garces-Chaves, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145-147 (2002).
[CrossRef]

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optical trapped microscopic particles,” Science 292, 912-914 (2001).
[CrossRef] [PubMed]

Du, X.

X. Du and D. Zhao, “Elliptical cosh-Gaussian beams,” Opt. Commun. 265, 418-424 (2006).
[CrossRef]

Dziedzic, J. M.

Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330, 769-771 (1989).
[CrossRef]

Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11, 288-290 (1986).
[CrossRef] [PubMed]

Fabrizio, E. Di.

D. Cojoc, V. Garbin, E. Ferrari, L. Businaro, F. Romamato, and E. Di. Fabrizio, “Laser trapping and micro-manipulation using optical vortices,” Microelectron. Eng. 77, 125-131 (2005).
[CrossRef]

Fei, Z.

X. Gao, Z. Fei, W. Xu, and F. I. Gan, “Focus splitting induced by a pure phase-shifting apodizer,” Opt. Commun. 239, 55-59 (2004).
[CrossRef]

Ferrari, E.

D. Cojoc, V. Garbin, E. Ferrari, L. Businaro, F. Romamato, and E. Di. Fabrizio, “Laser trapping and micro-manipulation using optical vortices,” Microelectron. Eng. 77, 125-131 (2005).
[CrossRef]

Gan, F. I.

X. Gao, Z. Fei, W. Xu, and F. I. Gan, “Focus splitting induced by a pure phase-shifting apodizer,” Opt. Commun. 239, 55-59 (2004).
[CrossRef]

Gan, X.

Ganic, D.

Gao, X.

X. Gao, J. Wang, H. Gu, and S. Hu, “Focusing of hyperbolic-cosine-Gaussian beam with a non-spiral vortex,” Optik (Stuttgart) 120, 201-206 (2009).
[CrossRef]

X. Gao and J. Li, “Focal shift of apodized truncated hyperbolic-cosine-Gaussian beam,” Opt. Commun. 273, 21-27 (2007).
[CrossRef]

X. Gao and J. Wang, “Focal evolution induced by combination of nonspiral and spiral phase plates,” Chin. Opt. Lett. 5(5), 257-259 (2007).

X. Gao, “Focusing properties of the hyperbolic-cosine-Gaussian beam induced by phase plate,” Phys. Lett. A 360, 330-335 (2006).
[CrossRef]

X. Gao, Z. Fei, W. Xu, and F. I. Gan, “Focus splitting induced by a pure phase-shifting apodizer,” Opt. Commun. 239, 55-59 (2004).
[CrossRef]

Garbin, V.

D. Cojoc, V. Garbin, E. Ferrari, L. Businaro, F. Romamato, and E. Di. Fabrizio, “Laser trapping and micro-manipulation using optical vortices,” Microelectron. Eng. 77, 125-131 (2005).
[CrossRef]

Garces-Chaves, V.

V. Garces-Chaves, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145-147 (2002).
[CrossRef]

Gbur, G.

Gil, L.

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403-408 (1989).
[CrossRef]

Grier, D. G.

Gu, H.

X. Gao, J. Wang, H. Gu, and S. Hu, “Focusing of hyperbolic-cosine-Gaussian beam with a non-spiral vortex,” Optik (Stuttgart) 120, 201-206 (2009).
[CrossRef]

Gu, M.

Hain, M.

Hall, D. G.

Hricha, Z.

Z. Hricha and A. Belafhal, “Focusing properties and focal shift in hyperbolic-cosine-Gaussian beams,” Opt. Commun. 253, 242-249 (2005).
[CrossRef]

Hu, S.

X. Gao, J. Wang, H. Gu, and S. Hu, “Focusing of hyperbolic-cosine-Gaussian beam with a non-spiral vortex,” Optik (Stuttgart) 120, 201-206 (2009).
[CrossRef]

Ladavac, K.

Lee, W. M.

Li, J.

X. Gao and J. Li, “Focal shift of apodized truncated hyperbolic-cosine-Gaussian beam,” Opt. Commun. 273, 21-27 (2007).
[CrossRef]

Lü, B.

B. Lü, H. Ma, and B. Zhang, “Propagation properties of cosh-Gaussian beams,” Opt. Commun. 164, 165-170 (1999).
[CrossRef]

Ma, H.

B. Lü, H. Ma, and B. Zhang, “Propagation properties of cosh-Gaussian beams,” Opt. Commun. 164, 165-170 (1999).
[CrossRef]

MacDonald, M. P.

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426, 421-424 (2003).
[CrossRef] [PubMed]

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optical trapped microscopic particles,” Science 292, 912-914 (2001).
[CrossRef] [PubMed]

McGloin, D.

V. Garces-Chaves, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145-147 (2002).
[CrossRef]

Melville, H.

V. Garces-Chaves, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145-147 (2002).
[CrossRef]

Padgett, M. J.

Paterson, L.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optical trapped microscopic particles,” Science 292, 912-914 (2001).
[CrossRef] [PubMed]

Rocca, F.

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403-408 (1989).
[CrossRef]

Romamato, F.

D. Cojoc, V. Garbin, E. Ferrari, L. Businaro, F. Romamato, and E. Di. Fabrizio, “Laser trapping and micro-manipulation using optical vortices,” Microelectron. Eng. 77, 125-131 (2005).
[CrossRef]

Rozas, D.

D. Rozas, Z. S. Sacks, and G. A. Swartzlander, Jr., “Experimental observation of fluidlike motion of optical vortices,” Phys. Rev. Lett. 79, 3399-3402 (1997).
[CrossRef]

Sacks, Z. S.

D. Rozas, Z. S. Sacks, and G. A. Swartzlander, Jr., “Experimental observation of fluidlike motion of optical vortices,” Phys. Rev. Lett. 79, 3399-3402 (1997).
[CrossRef]

Sibbett, W.

V. Garces-Chaves, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145-147 (2002).
[CrossRef]

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optical trapped microscopic particles,” Science 292, 912-914 (2001).
[CrossRef] [PubMed]

Somalingam, S.

Spalding, G. C.

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426, 421-424 (2003).
[CrossRef] [PubMed]

Stankovic, S.

Swartzlander, G. A.

D. Rozas, Z. S. Sacks, and G. A. Swartzlander, Jr., “Experimental observation of fluidlike motion of optical vortices,” Phys. Rev. Lett. 79, 3399-3402 (1997).
[CrossRef]

Tovar, A. A.

Tschudi, T.

Visscher, K.

K. Visscher and G. J. Brakenhoff, “Theoretical study of optically induced forces on spherical particles in a single beam trap I: Rayleigh scatterers,” Optik (Stuttgart) 89, 174-180 (1992).

Visser, T. D.

Wang, J.

X. Gao, J. Wang, H. Gu, and S. Hu, “Focusing of hyperbolic-cosine-Gaussian beam with a non-spiral vortex,” Optik (Stuttgart) 120, 201-206 (2009).
[CrossRef]

X. Gao and J. Wang, “Focal evolution induced by combination of nonspiral and spiral phase plates,” Chin. Opt. Lett. 5(5), 257-259 (2007).

Xu, W.

X. Gao, Z. Fei, W. Xu, and F. I. Gan, “Focus splitting induced by a pure phase-shifting apodizer,” Opt. Commun. 239, 55-59 (2004).
[CrossRef]

Yamane, T.

Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330, 769-771 (1989).
[CrossRef]

Yuan, X. C.

Zhang, B.

B. Lü, H. Ma, and B. Zhang, “Propagation properties of cosh-Gaussian beams,” Opt. Commun. 164, 165-170 (1999).
[CrossRef]

Zhao, D.

X. Du and D. Zhao, “Elliptical cosh-Gaussian beams,” Opt. Commun. 265, 418-424 (2006).
[CrossRef]

Chin. Opt. Lett.

J. Opt. Soc. Am. A

Microelectron. Eng.

D. Cojoc, V. Garbin, E. Ferrari, L. Businaro, F. Romamato, and E. Di. Fabrizio, “Laser trapping and micro-manipulation using optical vortices,” Microelectron. Eng. 77, 125-131 (2005).
[CrossRef]

Nature

Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330, 769-771 (1989).
[CrossRef]

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810-816 (2003).
[CrossRef] [PubMed]

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426, 421-424 (2003).
[CrossRef] [PubMed]

V. Garces-Chaves, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145-147 (2002).
[CrossRef]

Opt. Commun.

X. Gao, Z. Fei, W. Xu, and F. I. Gan, “Focus splitting induced by a pure phase-shifting apodizer,” Opt. Commun. 239, 55-59 (2004).
[CrossRef]

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403-408 (1989).
[CrossRef]

X. Du and D. Zhao, “Elliptical cosh-Gaussian beams,” Opt. Commun. 265, 418-424 (2006).
[CrossRef]

Z. Hricha and A. Belafhal, “Focusing properties and focal shift in hyperbolic-cosine-Gaussian beams,” Opt. Commun. 253, 242-249 (2005).
[CrossRef]

X. Gao and J. Li, “Focal shift of apodized truncated hyperbolic-cosine-Gaussian beam,” Opt. Commun. 273, 21-27 (2007).
[CrossRef]

B. Lü, H. Ma, and B. Zhang, “Propagation properties of cosh-Gaussian beams,” Opt. Commun. 164, 165-170 (1999).
[CrossRef]

Opt. Express

Opt. Lett.

Optik (Stuttgart)

K. Visscher and G. J. Brakenhoff, “Theoretical study of optically induced forces on spherical particles in a single beam trap I: Rayleigh scatterers,” Optik (Stuttgart) 89, 174-180 (1992).

X. Gao, J. Wang, H. Gu, and S. Hu, “Focusing of hyperbolic-cosine-Gaussian beam with a non-spiral vortex,” Optik (Stuttgart) 120, 201-206 (2009).
[CrossRef]

Phys. Lett. A

X. Gao, “Focusing properties of the hyperbolic-cosine-Gaussian beam induced by phase plate,” Phys. Lett. A 360, 330-335 (2006).
[CrossRef]

Phys. Rev. Lett.

D. Rozas, Z. S. Sacks, and G. A. Swartzlander, Jr., “Experimental observation of fluidlike motion of optical vortices,” Phys. Rev. Lett. 79, 3399-3402 (1997).
[CrossRef]

Science

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optical trapped microscopic particles,” Science 292, 912-914 (2001).
[CrossRef] [PubMed]

Other

M. Gu, Advanced Optical Imaging Theory (Springer, 2000).

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Figures (12)

Fig. 1
Fig. 1

Intensity distributions for θ 1 = 0 , β x = β y = 2 , and m = 1 . (a) NA = 0.3 and (b) NA = 0.95 .

Fig. 2
Fig. 2

Intensity distributions for θ 1 = 0 , β x = β y = 5 , and m = 1 . (a) NA = 0.3 , (b) NA = 0.95 , and (c) corresponding intensity curve along the y axis.

Fig. 3
Fig. 3

Intensity distributions for θ 1 = 0 , β x = β y = 5 , and m = 2 . (a) NA = 0.3 and (b) NA = 0.95 .

Fig. 4
Fig. 4

Intensity distributions for θ 1 = 0 , β x = β y = 5 , and m = 3 . (a) NA = 0.3 , (b) NA = 0.95 , and (c) corresponding intensity curve along the y axis.

Fig. 5
Fig. 5

Intensity distributions for θ 1 = 0 , β x = β y = 5 , and m = 4 . (a) NA = 0.3 , (b) NA = 0.95 , and (c) corresponding intensity curve along the y axis.

Fig. 6
Fig. 6

Intensity distributions for θ 1 = 0 , β x = β y = 8 , and m = 1 . (a) NA = 0.3 and (b) NA = 0.95 .

Fig. 7
Fig. 7

Intensity distributions for θ 1 = 0 , β x = β y = 8 , and m = 2 . (a) NA = 0.3 and (b) NA = 0.95 .

Fig. 8
Fig. 8

Intensity distributions for θ 1 = 0 , β x = β y = 8 , and m = 3 . (a) NA = 0.3 and (b) NA = 0.95 .

Fig. 9
Fig. 9

Intensity distributions for θ 1 = 0 , β x = β y = 8 , and m = 4 . (a) NA = 0.3 and (b) NA = 0.95 .

Fig. 10
Fig. 10

Intensity distributions for θ 1 = 0.1 θ 2 , m = 3 , and β x = β y = 8 . (a) NA = 0.4 and (b) NA = 0.95 .

Fig. 11
Fig. 11

Intensity distributions of electric field components in (a) the z axis direction and (b) the x axis direction for θ 1 = 0.1 θ 2 , NA = 0.95 , m = 3 , and β x = β y = 8 .

Fig. 12
Fig. 12

Dependence of the ratio of optical intensity of the electric field component in the z axis direction to that in the x axis direction on the NA for θ 1 = 0.1 θ 2 , m = 3 , and β x = β y = 8 .

Equations (3)

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

E ( θ , φ ) = A 0 cosh [ NA 1 β x sin ( θ ) cos ( φ ) ] cosh [ NA 1 β y sin ( θ ) sin ( φ ) ] exp [ sin 2 ( θ ) NA 2 w 2 ] exp ( i m φ ) ,
E ( ρ , ψ , z ) = 1 λ θ 1 θ 2 0 2 π E ( θ , φ ) { [ cos θ + sin 2 φ ( 1 cos θ ) ] x + cos φ sin φ ( cos θ 1 ) y + cos φ sin θ z } exp [ i k ρ sin θ cos ( φ ψ ) ] exp ( i k z cos θ ) sin θ d θ d φ ,
E ( ρ , ψ , z ) = A 0 λ θ 1 θ 2 0 2 π { cosh [ NA 1 β x sin ( θ ) cos ( φ ) ] cosh [ NA 1 β x sin ( θ ) cos ( φ ) ] cosh [ NA 1 β y sin ( θ ) sin ( φ ) ] exp [ sin 2 ( θ ) NA 2 w 2 ] exp ( i m φ ) } { [ cos θ + sin 2 φ ( 1 cos θ ) ] x + cos φ sin φ ( cos θ 1 ) y + cos φ sin θ z } exp [ i k ρ sin θ cos ( φ ψ ) ] exp ( i k z cos θ ) sin θ d θ d φ .

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