Abstract

The theoretical analyses in this paper show that a highly focused double-ring radially polarized Laguerre–Gaussian beam with a topological charge of 1 (R-LG11) can generate a small three-dimensional (3D) dark spot surrounded by an almost 100% uniform light shell in all directions. The cleanness and size of the 3D dark spot, the uniformity and strength of the light shell surrounding the dark spot, and the light efficiency all depend on the truncation parameter β of the R-LG11 beam and the numerical aperture (NA) of the system. When β=1.6 and the NA is close to its utmost, an almost 100% uniform light shell surrounding the 3D dark spot can be achieved and the dark spot is very clean. If the NA is lowered but β is increased to 1.95, we can also achieve an almost 100% uniform light shell and light efficiency can reach 90%, but the disadvantage is that the center of the dark spot is not too clean. A not-too-clean 3D dark spot, if the light shell surrounding it is very uniform, is acceptable for many applications. Therefore, 3D dark spots surrounded by a high uniform light shell, generated by simply adjusting the truncation parameter of the R-LG11 beam and the NA of the system, are useful for superresolution fluorescence microscopy, dark spot microscopy, and the dark spot trap.

© 2010 Optical Society of America

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References

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2010

Y. Zhang, B. Ding, and T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81, 023831 (2010).
[CrossRef]

2009

2008

B. Harke, C. K. Ullal, J. Keller, and S. W. Hell, “Three-dimensional nanoscopy of colloidal crystals,” Nano Lett. 8, 1309–1313 (2008).
[CrossRef] [PubMed]

M. L. Terraciano, M. Bashkansky, and F. K. Fatemi, “Faraday spectroscopy of atoms confined in a dark optical trap,” Phys. Rev. A 77, 063417 (2008).
[CrossRef]

Y. Kozawa and S. Sato, “Dark-spot formation by vector beams,” Opt. Lett. 33, 2326–2328 (2008).
[CrossRef] [PubMed]

2007

S. Yan and B. Yao, “Radiation forces of a highly focused radially polarized beam on spherical particles,” Phys. Rev. A 76, 053836 (2007).
[CrossRef]

J. Lin, M. Wei, H. Liang, K. Lin, and W. Hsieh, “Generation of supercontinuum bottle beam using an axicon,” Opt. Express 15, 2940–2946 (2007).
[CrossRef] [PubMed]

N. Bokor and N. Davidson, “Tight parabolic dark spot with high numerical aperture focusing with a circular pi phase plate,” Opt. Commun. 270, 145–150 (2007).
[CrossRef]

2006

2005

2004

2003

2002

1999

H. Kim and Y. H. Lee, “Hermite–Gaussian and Laguerre–Gaussian beams beyond the paraxial approximation,” Opt. Commun. 169, 9–16 (1999).
[CrossRef]

1997

T. Kuga, Y. Torii, N. Shiokawa, and T. Hirano, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

1996

1995

1959

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]

Alonso, M. A.

Bashkansky, M.

M. L. Terraciano, M. Bashkansky, and F. K. Fatemi, “Faraday spectroscopy of atoms confined in a dark optical trap,” Phys. Rev. A 77, 063417 (2008).
[CrossRef]

Bokor, N.

N. Bokor and N. Davidson, “Tight parabolic dark spot with high numerical aperture focusing with a circular pi phase plate,” Opt. Commun. 270, 145–150 (2007).
[CrossRef]

Booker, G. R.

Borghi, R.

Bouma, B. E.

Dally, A.

Davidson, N.

N. Bokor and N. Davidson, “Tight parabolic dark spot with high numerical aperture focusing with a circular pi phase plate,” Opt. Commun. 270, 145–150 (2007).
[CrossRef]

Ding, B.

Y. Zhang, B. Ding, and T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81, 023831 (2010).
[CrossRef]

Fatemi, F. K.

M. L. Terraciano, M. Bashkansky, and F. K. Fatemi, “Faraday spectroscopy of atoms confined in a dark optical trap,” Phys. Rev. A 77, 063417 (2008).
[CrossRef]

Foo, G.

Fujii, M.

Gahagan, K. T.

Harke, B.

B. Harke, C. K. Ullal, J. Keller, and S. W. Hell, “Three-dimensional nanoscopy of colloidal crystals,” Nano Lett. 8, 1309–1313 (2008).
[CrossRef] [PubMed]

Hell, S. W.

B. Harke, C. K. Ullal, J. Keller, and S. W. Hell, “Three-dimensional nanoscopy of colloidal crystals,” Nano Lett. 8, 1309–1313 (2008).
[CrossRef] [PubMed]

Herman, R. M.

Hirano, T.

T. Kuga, Y. Torii, N. Shiokawa, and T. Hirano, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

Hsieh, W.

Iketaki, Y.

Isenhower, L.

Keller, J.

B. Harke, C. K. Ullal, J. Keller, and S. W. Hell, “Three-dimensional nanoscopy of colloidal crystals,” Nano Lett. 8, 1309–1313 (2008).
[CrossRef] [PubMed]

Kim, H.

H. Kim and Y. H. Lee, “Hermite–Gaussian and Laguerre–Gaussian beams beyond the paraxial approximation,” Opt. Commun. 169, 9–16 (1999).
[CrossRef]

Kozawa, Y.

Kuga, T.

T. Kuga, Y. Torii, N. Shiokawa, and T. Hirano, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

Laczik, Z.

Lee, Y. H.

H. Kim and Y. H. Lee, “Hermite–Gaussian and Laguerre–Gaussian beams beyond the paraxial approximation,” Opt. Commun. 169, 9–16 (1999).
[CrossRef]

Liang, H.

Lin, J.

Lin, K.

Omatsu, T.

Palacios, D. M.

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]

Saffman, M.

Sakai, M.

Santarsiero, M.

Sato, S.

Shiokawa, N.

T. Kuga, Y. Torii, N. Shiokawa, and T. Hirano, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

Suyama, T.

Y. Zhang, B. Ding, and T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81, 023831 (2010).
[CrossRef]

Swartzlander, G. A.

Tearney, G. J.

Terraciano, M. L.

M. L. Terraciano, M. Bashkansky, and F. K. Fatemi, “Faraday spectroscopy of atoms confined in a dark optical trap,” Phys. Rev. A 77, 063417 (2008).
[CrossRef]

Torii, Y.

T. Kuga, Y. Torii, N. Shiokawa, and T. Hirano, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

Török, P.

Ullal, C. K.

B. Harke, C. K. Ullal, J. Keller, and S. W. Hell, “Three-dimensional nanoscopy of colloidal crystals,” Nano Lett. 8, 1309–1313 (2008).
[CrossRef] [PubMed]

Varga, P.

Watanabe, T.

Wei, M.

Wiggins, T. A.

Williams, W.

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

Yamamoto, K.

Yan, S.

S. Yan and B. Yao, “Radiation forces of a highly focused radially polarized beam on spherical particles,” Phys. Rev. A 76, 053836 (2007).
[CrossRef]

Yao, B.

S. Yan and B. Yao, “Radiation forces of a highly focused radially polarized beam on spherical particles,” Phys. Rev. A 76, 053836 (2007).
[CrossRef]

Yelin, D.

Yonezawa, K.

Zhang, Y.

Y. Zhang, B. Ding, and T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81, 023831 (2010).
[CrossRef]

J. Opt. Soc. Am. A

Nano Lett.

B. Harke, C. K. Ullal, J. Keller, and S. W. Hell, “Three-dimensional nanoscopy of colloidal crystals,” Nano Lett. 8, 1309–1313 (2008).
[CrossRef] [PubMed]

Opt. Commun.

N. Bokor and N. Davidson, “Tight parabolic dark spot with high numerical aperture focusing with a circular pi phase plate,” Opt. Commun. 270, 145–150 (2007).
[CrossRef]

H. Kim and Y. H. Lee, “Hermite–Gaussian and Laguerre–Gaussian beams beyond the paraxial approximation,” Opt. Commun. 169, 9–16 (1999).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

Y. Zhang, B. Ding, and T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81, 023831 (2010).
[CrossRef]

S. Yan and B. Yao, “Radiation forces of a highly focused radially polarized beam on spherical particles,” Phys. Rev. A 76, 053836 (2007).
[CrossRef]

M. L. Terraciano, M. Bashkansky, and F. K. Fatemi, “Faraday spectroscopy of atoms confined in a dark optical trap,” Phys. Rev. A 77, 063417 (2008).
[CrossRef]

Phys. Rev. Lett.

T. Kuga, Y. Torii, N. Shiokawa, and T. Hirano, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

Proc. R. Soc. London Ser. A

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]

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Figures (7)

Fig. 1
Fig. 1

Sketch map of the focusing system generating (a) a 3D dark spot and (b) the phase distribution of the helical phase plate in (a): P, helical phase plate; A, circular aperture; and L, objective with high NA.

Fig. 2
Fig. 2

Light efficiency versus the truncation parameter of the R - LG 11 beam.

Fig. 3
Fig. 3

Calculated intensity distributions near the focus for a radially polarized R - LG 11 vortex beam with the truncation parameter of β = 1.58 . (a)–(c) show the intensity distributions of the x, y, and z components, respectively, while (d) is the intensity distribution of the total field. The vertical and horizontal axes are in the x and z directions in units of wavelength, respectively.

Fig. 4
Fig. 4

Calculated intensity distributions near the focus for a radially polarized R - LG 11 vortex beam with the truncation parameter of β = 1.95 . All signs are the same as in Fig. 3.

Fig. 5
Fig. 5

Dependence of U, V, C, and S describing the 3D dark spot pattern on the truncation parameter β of the R - LG 11 beam when NA = 0.95 n 1 .

Fig. 6
Fig. 6

Dependence of U, V, C, and S describing the 3D dark spot pattern on the NA of the focusing system when β = 1.60 .

Fig. 7
Fig. 7

Dependence of U, V, C, and S describing the 3D dark spot pattern on the NA of the focusing system when β = 1.95 .

Equations (9)

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E i ( θ , ϕ ) = E 0 A 11 ( θ , ϕ ) ( cos ϕ sin ϕ 0 ) ,
A 11 ( θ , ϕ ) = A 11 ( θ ) exp ( i ϕ ) ,
A 11 ( θ ) = ( 2 β sin θ sin α ) exp [ ( β sin θ sin α ) 2 ] L 1 1 [ 2 ( β sin θ sin α ) 2 ] .
E = ( E x E y E z ) = i k 1 f E 0 2 π 0 α 0 2 π cos θ ( cos θ cos ϕ cos θ sin ϕ sin θ ) A 11 ( θ ) sin θ exp ( i ϕ ) exp { i k 1 [ z cos θ + r sin θ cos ( ϕ φ ) ] } d ϕ d θ ,
E = K ( i [ exp ( i φ ) I 2 exp ( i φ ) I 0 ) ] [ exp ( i φ ) I 2 + exp ( i φ ) I 0 ) ] 2 I 1 ) ,
I 1 = 0 α A 11 ( θ ) cos θ sin 2 θ J 1 ( k 1 r sin θ ) exp ( i k 1 z cos θ ) d θ ,
I 2 = 0 α A 11 ( θ ) cos θ cos θ sin θ J 2 ( k 1 r sin θ ) exp ( i k 1 z cos θ ) d θ ,
I 0 = 0 α A 11 ( θ ) cos θ cos θ sin θ J 0 ( k 1 r sin θ ) exp ( i k 1 z cos θ ) d θ .
V = 4 π a b 2 / 3 ,

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