Abstract

We demonstrate the possibility of creating curved optical tubes in a 4Pi focusing system. The focal fields of such optical tubes have interesting properties: the energy is concentered in the neighborhood of a prescribed three-dimensional (3D) curve while the cross section is of hollow shape. The creation of these optical tubes is based on the annular focal spot of a vortex beam, which is employed as a building block. An optical tube is thus obtained by covering the central-axis curve of the tube by various such building blocks. Each building block has a certain orientation and position, realized by a rotation plus a certain translation. The spatial spectrum (the input field as well) of the optical tube is obtained by linearly superposing the spectrum of each transformed building block. The curve is rather arbitrary. Three examples of optical tubes: a torus, a solenoid and a trefoil knot are given, showing a good agreement with the expected results.

© 2015 Optical Society of America

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References

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    [Crossref]
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2015 (2)

2013 (4)

2012 (2)

Z. Li, S. Yan, B. Yao, M. Lei, B. Ma, P. Gao, D. Dan, and R. Rupp, “Theoretical prediction of three-dimensional shifting of a spherical focal spot in a 4Pi focusing system,” J. Opt. 14(5), 055706 (2012).
[Crossref]

Z. Chen and D. Zhao, “4Pi focusing of spatially modulated radially polarized vortex beams,” Opt. Lett. 37(8), 1286–1288 (2012).
[Crossref] [PubMed]

2011 (2)

2010 (2)

2009 (1)

2008 (1)

2006 (1)

2004 (1)

2001 (1)

C. M. Blanca, J. Bewersdorf, and S. Hell, “Single sharp spot in fluorescence microscopy of two opposing lenses,” Appl. Phys. Lett. 79(15), 2321–2323 (2001).
[Crossref]

2000 (1)

1997 (1)

1992 (2)

S. Hell and E. H. K. Stelzer, “Properties of a 4Pi confocal fluorescence microscope,” J. Opt. Soc. Am. A 9(12), 2159–2166 (1992).
[Crossref]

S. Hell and E. H. K. Stelzer, “Fundamental improvement of resolution with a 4Pi-confocal fluorescence microscope using two-photon excitation,” Opt. Commun. 93(5-6), 277–282 (1992).
[Crossref]

1970 (1)

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24(4), 156–159 (1970).
[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]

Abramochkin, E.

Alieva, T.

Ashkin, A.

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24(4), 156–159 (1970).
[Crossref]

Berns, M. W.

Bewersdorf, J.

C. M. Blanca, J. Bewersdorf, and S. Hell, “Single sharp spot in fluorescence microscopy of two opposing lenses,” Appl. Phys. Lett. 79(15), 2321–2323 (2001).
[Crossref]

Blanca, C. M.

C. M. Blanca, J. Bewersdorf, and S. Hell, “Single sharp spot in fluorescence microscopy of two opposing lenses,” Appl. Phys. Lett. 79(15), 2321–2323 (2001).
[Crossref]

Bokor, N.

Brown, T.

Castro, I.

Chen, G. Y.

Chen, W.

Chen, Z.

Z. Chen, J. Pu, and D. Zhao, “Generating and shifting a spherical focal spot in a 4Pi focusing system illuminated by azimuthally polarized beams,” Phys. Lett. A 377(34-36), 2231–2234 (2013).
[Crossref]

Z. Chen and D. Zhao, “4Pi focusing of spatially modulated radially polarized vortex beams,” Opt. Lett. 37(8), 1286–1288 (2012).
[Crossref] [PubMed]

Chiou, A. E.

Dan, D.

Z. Li, S. Yan, B. Yao, M. Lei, B. Ma, P. Gao, D. Dan, and R. Rupp, “Theoretical prediction of three-dimensional shifting of a spherical focal spot in a 4Pi focusing system,” J. Opt. 14(5), 055706 (2012).
[Crossref]

Davidson, N.

Divitt, S.

Dogterom, M.

Gao, P.

Z. Li, S. Yan, B. Yao, M. Lei, B. Ma, P. Gao, D. Dan, and R. Rupp, “Theoretical prediction of three-dimensional shifting of a spherical focal spot in a 4Pi focusing system,” J. Opt. 14(5), 055706 (2012).
[Crossref]

Grier, D. G.

Hell, S.

C. M. Blanca, J. Bewersdorf, and S. Hell, “Single sharp spot in fluorescence microscopy of two opposing lenses,” Appl. Phys. Lett. 79(15), 2321–2323 (2001).
[Crossref]

S. Hell and E. H. K. Stelzer, “Fundamental improvement of resolution with a 4Pi-confocal fluorescence microscope using two-photon excitation,” Opt. Commun. 93(5-6), 277–282 (1992).
[Crossref]

S. Hell and E. H. K. Stelzer, “Properties of a 4Pi confocal fluorescence microscope,” J. Opt. Soc. Am. A 9(12), 2159–2166 (1992).
[Crossref]

Lee, S. H.

Lei, M.

Z. Li, S. Yan, B. Yao, M. Lei, B. Ma, P. Gao, D. Dan, and R. Rupp, “Theoretical prediction of three-dimensional shifting of a spherical focal spot in a 4Pi focusing system,” J. Opt. 14(5), 055706 (2012).
[Crossref]

S. Yan, B. Yao, W. Zhao, and M. Lei, “Generation of multiple spherical spots with a radially polarized beam in a 4π focusing system,” J. Opt. Soc. Am. A 27(9), 2033–2037 (2010).
[Crossref] [PubMed]

Li, Z.

Z. Li, S. Yan, B. Yao, M. Lei, B. Ma, P. Gao, D. Dan, and R. Rupp, “Theoretical prediction of three-dimensional shifting of a spherical focal spot in a 4Pi focusing system,” J. Opt. 14(5), 055706 (2012).
[Crossref]

Ma, B.

Z. Li, S. Yan, B. Yao, M. Lei, B. Ma, P. Gao, D. Dan, and R. Rupp, “Theoretical prediction of three-dimensional shifting of a spherical focal spot in a 4Pi focusing system,” J. Opt. 14(5), 055706 (2012).
[Crossref]

Moroz, A.

Novotny, L.

Pu, J.

Z. Chen, J. Pu, and D. Zhao, “Generating and shifting a spherical focal spot in a 4Pi focusing system illuminated by azimuthally polarized beams,” Phys. Lett. A 377(34-36), 2231–2234 (2013).
[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]

Rodrigo, J. A.

Roichman, Y.

Rondin, L.

Rupp, R.

Z. Li, S. Yan, B. Yao, M. Lei, B. Ma, P. Gao, D. Dan, and R. Rupp, “Theoretical prediction of three-dimensional shifting of a spherical focal spot in a 4Pi focusing system,” J. Opt. 14(5), 055706 (2012).
[Crossref]

S. Yan, B. Yao, and R. Rupp, “Shifting the spherical focus of a 4Pi focusing system,” Opt. Express 19(2), 673–678 (2011).
[Crossref] [PubMed]

Shanblatt, E. R.

She, W.

Sonek, G. J.

Song, F.

Stelzer, E. H. K.

S. Hell and E. H. K. Stelzer, “Properties of a 4Pi confocal fluorescence microscope,” J. Opt. Soc. Am. A 9(12), 2159–2166 (1992).
[Crossref]

S. Hell and E. H. K. Stelzer, “Fundamental improvement of resolution with a 4Pi-confocal fluorescence microscope using two-photon excitation,” Opt. Commun. 93(5-6), 277–282 (1992).
[Crossref]

van Blaaderen, A.

van der Horst, A.

van Oostrum, P. D. J.

Wang, H. T.

Wang, 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. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[Crossref]

Yan, S.

Yao, B.

Youngworth, K.

Yu, Y.

Zhan, Q.

Zhao, D.

Z. Chen, J. Pu, and D. Zhao, “Generating and shifting a spherical focal spot in a 4Pi focusing system illuminated by azimuthally polarized beams,” Phys. Lett. A 377(34-36), 2231–2234 (2013).
[Crossref]

Z. Chen and D. Zhao, “4Pi focusing of spatially modulated radially polarized vortex beams,” Opt. Lett. 37(8), 1286–1288 (2012).
[Crossref] [PubMed]

Zhao, W.

Zhu, W.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

C. M. Blanca, J. Bewersdorf, and S. Hell, “Single sharp spot in fluorescence microscopy of two opposing lenses,” Appl. Phys. Lett. 79(15), 2321–2323 (2001).
[Crossref]

J. Opt. (1)

Z. Li, S. Yan, B. Yao, M. Lei, B. Ma, P. Gao, D. Dan, and R. Rupp, “Theoretical prediction of three-dimensional shifting of a spherical focal spot in a 4Pi focusing system,” J. Opt. 14(5), 055706 (2012).
[Crossref]

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (1)

Opt. Commun. (1)

S. Hell and E. H. K. Stelzer, “Fundamental improvement of resolution with a 4Pi-confocal fluorescence microscope using two-photon excitation,” Opt. Commun. 93(5-6), 277–282 (1992).
[Crossref]

Opt. Express (7)

Opt. Lett. (6)

Phys. Lett. A (1)

Z. Chen, J. Pu, and D. Zhao, “Generating and shifting a spherical focal spot in a 4Pi focusing system illuminated by azimuthally polarized beams,” Phys. Lett. A 377(34-36), 2231–2234 (2013).
[Crossref]

Phys. Rev. Lett. (1)

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24(4), 156–159 (1970).
[Crossref]

Proc. R. Soc. Lond. A Math. Phys. Sci. (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]

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Figures (4)

Fig. 1
Fig. 1 Geometry of a 4Pi focusing system.
Fig. 2
Fig. 2 Generation of an optical torus and the corresponding incident field A (θ, ϕ) in the 4Pi focusing system. (a),(b): Magnitudes of A (θ, ϕ) on the left (k z > 0) and right (kz < 0) lenses. (c),(d): Phases of A (θ, ϕ) on the left (k z > 0) and right (kz < 0) lenses. (e),(f): Distributions of the total intensity |(E)|2 in the xy and xz planes. (g): 3D view of the total intensity.
Fig. 3
Fig. 3 Generation of an optical hollow solenoid and the corresponding incident field A (θ, ϕ) in the 4Pi focusing system. (a),(b): Magnitudes of A (θ, ϕ) on the left (k z > 0) and right (kz < 0) lenses. (c),(d): Phases of A (θ, ϕ) on the left (k z > 0) and right (kz < 0) lenses. (e),(f): Distributions of the total intensity |(E)|2 in the xy and yz planes. (g): 3D view of the total intensity.
Fig. 4
Fig. 4 Generation of an optical hollow trefoil knot and the corresponding incident field A (θ, ϕ) in the 4pi focusing system.. (a),(b): Magnitudes of A (θ, ϕ) on the left (k z > 0) and right (kz < 0) lenses. (c),(d): Phases of A (θ, ϕ) on the left (k z > 0) and right (kz < 0) lenses. (e),(f): Distributions of the total intensity |(E)|2 in the xy and xz planes. (g): 3D view of the total intensity.

Equations (6)

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A ( x ) = A ( θ , ϕ ) exp ( i k x ) sin θ d θ d φ
T ( x 0 ) = d x 0 ( s ) / d s d x 0 ( s ) / d s ,
n ( x 0 ) = d v ( s ) / d s T 0 ( s ) d v / d s d v ( s ) / d s T 0 ( s ) d v / d s .
( R i j ) ( x 0 ) = [ n 1 B 1 T 1 n 2 B 2 T 2 n 3 B 3 T 3 ] ( x 0 ) ,
( sin θ cos ϕ sin θ sin ϕ cos θ ) = ( R i j ) ( x 0 ) ( sin θ cos ϕ sin θ sin ϕ cos θ ) .
A ( θ , ϕ ) = δ A ( x 0 ) = TR δ A 0 exp [ i β l ( s ) ] exp [ i k x 0 ( s ) + i m ϕ ' ] d l ( s ) .

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