Abstract

Hollow beam formation of radially and azimuthally polarized vortex beams, which has arbitrary topological charge, is analytically discussed under the strong focusing condition. The expressions for the electric fields of the focused vector-vortex beams are obtained based on a vector diffraction theory. The order of the Bessel function of the first kind appearing in the expressions indicates the ability to form hollow beams. Similar discussion is applied for different vortex beams, which are expressed by linear combination of radially and azimuthally polarized beams. Calculations of intensity profiles across the focus are also presented.

© 2008 Optical Society of America

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References

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  1. K. S. Youngworth and T. G. Brown, “Focusing of high numerical apterture cylindrical-vector beams,” Opt. Express 7, 77-87 (2000).
    [CrossRef] [PubMed]
  2. G. Machavariani, Y. Lumer, I. Moshe, and S. Jackel, “Effect of the spatial phase element on the radial-polarization (0,1)* LG beam,” Opt. Commun. 271, 190-196 (2007).
    [CrossRef]
  3. H. Kawauchi, Y. Kozawa, S. Sato, T. Sato, and S. Kawakami, “Simultaneous generation of helical beams with linear and radial polarization by use of a segmented half-wave plate,” Opt. Lett. 33, 399-401 (2008).
    [CrossRef] [PubMed]
  4. L. E. Helseth, “Focusing of atoms with strongly confined light potentials,” Opt. Commun. 212, 343-352 (2002).
    [CrossRef]
  5. G. Donnert, J. Keller, R. Medda, M. A. Andrei, S. O. Rizzoli, R. Lührmann, R. Jahn, C. Eggeling, and S. W. Hell, “Macromolecular-scale resolution in biological fluorescence microscopy,” Proc. Natl. Acad. Sci. U.S.A. 31, 11440-11445 (2006).
    [CrossRef]
  6. 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]
  7. G. N. Watson, Theory of Bessel Functions (Cambridge U. Press, 1944).
  8. L. E. Helseth, “Smallest focal hole,” Opt. Commun. 257, 1-8 (2006).
    [CrossRef]

2008

2007

G. Machavariani, Y. Lumer, I. Moshe, and S. Jackel, “Effect of the spatial phase element on the radial-polarization (0,1)* LG beam,” Opt. Commun. 271, 190-196 (2007).
[CrossRef]

2006

G. Donnert, J. Keller, R. Medda, M. A. Andrei, S. O. Rizzoli, R. Lührmann, R. Jahn, C. Eggeling, and S. W. Hell, “Macromolecular-scale resolution in biological fluorescence microscopy,” Proc. Natl. Acad. Sci. U.S.A. 31, 11440-11445 (2006).
[CrossRef]

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

2002

L. E. Helseth, “Focusing of atoms with strongly confined light potentials,” Opt. Commun. 212, 343-352 (2002).
[CrossRef]

2000

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]

Andrei, M. A.

G. Donnert, J. Keller, R. Medda, M. A. Andrei, S. O. Rizzoli, R. Lührmann, R. Jahn, C. Eggeling, and S. W. Hell, “Macromolecular-scale resolution in biological fluorescence microscopy,” Proc. Natl. Acad. Sci. U.S.A. 31, 11440-11445 (2006).
[CrossRef]

Brown, T. G.

Donnert, G.

G. Donnert, J. Keller, R. Medda, M. A. Andrei, S. O. Rizzoli, R. Lührmann, R. Jahn, C. Eggeling, and S. W. Hell, “Macromolecular-scale resolution in biological fluorescence microscopy,” Proc. Natl. Acad. Sci. U.S.A. 31, 11440-11445 (2006).
[CrossRef]

Eggeling, C.

G. Donnert, J. Keller, R. Medda, M. A. Andrei, S. O. Rizzoli, R. Lührmann, R. Jahn, C. Eggeling, and S. W. Hell, “Macromolecular-scale resolution in biological fluorescence microscopy,” Proc. Natl. Acad. Sci. U.S.A. 31, 11440-11445 (2006).
[CrossRef]

Hell, S. W.

G. Donnert, J. Keller, R. Medda, M. A. Andrei, S. O. Rizzoli, R. Lührmann, R. Jahn, C. Eggeling, and S. W. Hell, “Macromolecular-scale resolution in biological fluorescence microscopy,” Proc. Natl. Acad. Sci. U.S.A. 31, 11440-11445 (2006).
[CrossRef]

Helseth, L. E.

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

L. E. Helseth, “Focusing of atoms with strongly confined light potentials,” Opt. Commun. 212, 343-352 (2002).
[CrossRef]

Jackel, S.

G. Machavariani, Y. Lumer, I. Moshe, and S. Jackel, “Effect of the spatial phase element on the radial-polarization (0,1)* LG beam,” Opt. Commun. 271, 190-196 (2007).
[CrossRef]

Jahn, R.

G. Donnert, J. Keller, R. Medda, M. A. Andrei, S. O. Rizzoli, R. Lührmann, R. Jahn, C. Eggeling, and S. W. Hell, “Macromolecular-scale resolution in biological fluorescence microscopy,” Proc. Natl. Acad. Sci. U.S.A. 31, 11440-11445 (2006).
[CrossRef]

Kawakami, S.

Kawauchi, H.

Keller, J.

G. Donnert, J. Keller, R. Medda, M. A. Andrei, S. O. Rizzoli, R. Lührmann, R. Jahn, C. Eggeling, and S. W. Hell, “Macromolecular-scale resolution in biological fluorescence microscopy,” Proc. Natl. Acad. Sci. U.S.A. 31, 11440-11445 (2006).
[CrossRef]

Kozawa, Y.

Lührmann, R.

G. Donnert, J. Keller, R. Medda, M. A. Andrei, S. O. Rizzoli, R. Lührmann, R. Jahn, C. Eggeling, and S. W. Hell, “Macromolecular-scale resolution in biological fluorescence microscopy,” Proc. Natl. Acad. Sci. U.S.A. 31, 11440-11445 (2006).
[CrossRef]

Lumer, Y.

G. Machavariani, Y. Lumer, I. Moshe, and S. Jackel, “Effect of the spatial phase element on the radial-polarization (0,1)* LG beam,” Opt. Commun. 271, 190-196 (2007).
[CrossRef]

Machavariani, G.

G. Machavariani, Y. Lumer, I. Moshe, and S. Jackel, “Effect of the spatial phase element on the radial-polarization (0,1)* LG beam,” Opt. Commun. 271, 190-196 (2007).
[CrossRef]

Medda, R.

G. Donnert, J. Keller, R. Medda, M. A. Andrei, S. O. Rizzoli, R. Lührmann, R. Jahn, C. Eggeling, and S. W. Hell, “Macromolecular-scale resolution in biological fluorescence microscopy,” Proc. Natl. Acad. Sci. U.S.A. 31, 11440-11445 (2006).
[CrossRef]

Moshe, I.

G. Machavariani, Y. Lumer, I. Moshe, and S. Jackel, “Effect of the spatial phase element on the radial-polarization (0,1)* LG beam,” Opt. Commun. 271, 190-196 (2007).
[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. London, Ser. A 253, 358-379 (1959).
[CrossRef]

Rizzoli, S. O.

G. Donnert, J. Keller, R. Medda, M. A. Andrei, S. O. Rizzoli, R. Lührmann, R. Jahn, C. Eggeling, and S. W. Hell, “Macromolecular-scale resolution in biological fluorescence microscopy,” Proc. Natl. Acad. Sci. U.S.A. 31, 11440-11445 (2006).
[CrossRef]

Sato, S.

Sato, T.

Watson, G. N.

G. N. Watson, Theory of Bessel Functions (Cambridge U. Press, 1944).

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]

Youngworth, K. S.

Opt. Commun.

L. E. Helseth, “Focusing of atoms with strongly confined light potentials,” Opt. Commun. 212, 343-352 (2002).
[CrossRef]

G. Machavariani, Y. Lumer, I. Moshe, and S. Jackel, “Effect of the spatial phase element on the radial-polarization (0,1)* LG beam,” Opt. Commun. 271, 190-196 (2007).
[CrossRef]

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

Opt. Express

Opt. Lett.

Proc. Natl. Acad. Sci. U.S.A.

G. Donnert, J. Keller, R. Medda, M. A. Andrei, S. O. Rizzoli, R. Lührmann, R. Jahn, C. Eggeling, and S. W. Hell, “Macromolecular-scale resolution in biological fluorescence microscopy,” Proc. Natl. Acad. Sci. U.S.A. 31, 11440-11445 (2006).
[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]

Other

G. N. Watson, Theory of Bessel Functions (Cambridge U. Press, 1944).

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Figures (5)

Fig. 1
Fig. 1

Calculated intensity profiles for radially polarized beams with topological charge m across the focus in the plane perpendicular to the beam axis.

Fig. 2
Fig. 2

Calculated intensity profiles for azimuthally polarized beams with topological charge m across the focus in the plane perpendicular to the beam axis.

Fig. 3
Fig. 3

Calculated intensity profiles for left-handed circularly polarized beams with topological charge m across the focus in the plane perpendicular to the beam axis.

Fig. 4
Fig. 4

Calculated intensity profiles for hybrid mode beams with topological charge m across the focus in the plane perpendicular to the beam axis.

Fig. 5
Fig. 5

Calculated intensity profiles for linearly polarized beams with topological charge m across the focus in the plane perpendicular to the beam axis.

Equations (32)

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E ρ ( rad ) ( r ) = i A π 0 α 0 2 π cos 1 2 θ sin θ cos θ cos ( ϕ ϕ S ) Y ( θ , ϕ ) exp { i k [ z S cos θ + ρ S sin θ cos ( ϕ ϕ S ) ] } d ϕ d θ ,
E ϕ ( rad ) ( r ) = i A π 0 α 0 2 π cos 1 2 θ sin θ cos θ sin ( ϕ ϕ S ) Y ( θ , ϕ ) exp { i k [ z S cos θ + ρ S sin θ cos ( ϕ ϕ S ) ] } d ϕ d θ ,
E z ( rad ) ( r ) = i A π 0 α 0 2 π cos 1 2 θ sin 2 θ Y ( θ , ϕ ) exp { i k [ z S cos θ + ρ S sin θ cos ( ϕ ϕ S ) ] } d ϕ d θ ,
e ρ ( rad ) ( r ) = 0 2 π cos ( ϕ ϕ S ) exp ( i m ϕ ) exp [ i k ρ S sin θ cos ( ϕ ϕ S ) ] d ϕ ,
e ϕ ( rad ) ( r ) = 0 2 π sin ( ϕ ϕ S ) exp ( i m ϕ ) exp [ i k ρ S sin θ cos ( ϕ ϕ S ) ] d ϕ ,
e z ( rad ) ( r ) = 0 2 π exp ( i m ϕ ) exp [ i k ρ S sin θ cos ( ϕ ϕ S ) ] d ϕ .
J m ( z ) = 1 2 π α 2 π + α exp [ i ( m ϕ z sin ϕ ) ] d ϕ = 1 2 π i m α 2 π + α exp [ i ( m ϕ + z cos ϕ ) ] d ϕ ,
e ρ ( rad ) ( r ) = π i m + 1 exp ( i m ϕ S ) [ J m + 1 ( β ) J m 1 ( β ) ] ,
e ϕ ( rad ) ( r ) = π i m exp ( i m ϕ S ) [ J m + 1 ( β ) + J m 1 ( β ) ] ,
e z ( rad ) ( r ) = 2 π i m exp ( i m ϕ S ) J m ( β ) .
E ρ ( rad ) ( r ) = A i m exp ( i m ϕ S ) 0 α cos 1 2 θ sin θ cos θ y ( θ ) exp ( i k z S cos θ ) [ J m + 1 ( β ) J m 1 ( β ) ] d θ ,
E ϕ ( rad ) ( r ) = A i m + 1 exp ( i m ϕ S ) 0 α cos 1 2 θ sin θ cos θ y ( θ ) exp ( i k z S cos θ ) [ J m + 1 ( β ) + J m 1 ( β ) ] d θ ,
E z ( rad ) ( r ) = 2 A i m + 1 exp ( i m ϕ S ) 0 α cos 1 2 θ sin 2 θ y ( θ ) exp ( i k z S cos θ ) J m ( β ) d θ .
E ρ ( azi ) ( r ) = i A π 0 α 0 2 π cos 1 2 θ sin θ sin ( ϕ ϕ S ) Y ( θ , ϕ ) exp [ i k z S cos θ + ρ S sin θ cos ( ϕ ϕ S ) ] d ϕ d θ ,
E ϕ ( azi ) ( r ) = i A π 0 α 0 2 π cos 1 2 θ sin θ cos ( ϕ ϕ S ) Y ( θ , ϕ ) exp [ i k z S cos θ + ρ S sin θ cos ( ϕ ϕ S ) ] d ϕ d θ ,
E z ( azi ) ( r ) = 0 .
E ρ ( azi ) ( r ) = A i m + 1 exp ( i m ϕ S ) 0 α cos 1 2 θ sin θ y ( θ ) exp ( i k z S cos θ ) [ J m + 1 ( β ) + J m 1 ( β ) ] d θ ,
E ϕ ( azi ) ( r ) = A i m exp ( i m ϕ S ) 0 α cos 1 2 θ sin θ y ( θ ) exp ( i k z S cos θ ) [ J m + 1 ( β ) J m 1 ( β ) ] d θ ,
E z ( azi ) ( r ) = 0 .
1 2 Y ( θ , ϕ ) ( x + i y ) = 1 2 y ( θ ) exp ( i m ϕ ) ( x + i y ) = 1 2 y ( θ ) exp [ i ( m + 1 ) ϕ ] ( ρ + i φ ) ,
1 2 Y ( θ , ϕ ) ( x i y ) = 1 2 y ( θ ) exp ( i m ϕ ) ( x i y ) = 1 2 y ( θ ) exp [ i ( m + 1 ) ϕ ] ( ρ i φ ) ,
E ρ ( R c ) ( r ) = A i m + 1 2 exp [ i ( m + 1 ) ϕ S ] 0 α cos 1 2 θ sin θ cos θ y ( θ ) exp ( i k z S cos θ ) [ J m + 2 ( β ) J m ( β ) ] d θ A i m + 1 2 exp [ i ( m + 1 ) ϕ S ] 0 α cos 1 2 θ sin θ y ( θ ) exp ( i k z S cos θ ) [ J m + 2 ( β ) + J m ( β ) ] d θ ,
E ϕ ( R c ) ( r ) = A i m 2 exp [ i ( m + 1 ) ϕ S ] 0 α cos 1 2 θ sin θ cos θ y ( θ ) exp ( i k z S cos θ ) [ J m + 2 ( β ) + J m ( β ) ] d θ A i m 2 exp [ i ( m + 1 ) ϕ S ] 0 α cos 1 2 θ sin θ y ( θ ) exp ( i k z S cos θ ) [ J m + 2 ( β ) J m ( β ) ] d θ ,
E z ( R c ) ( r ) = 2 A i m exp [ i ( m + 1 ) ϕ S ] 0 α cos 1 2 θ sin 2 θ y ( θ ) exp ( i k z S cos θ ) J m + 1 ( β ) d θ .
1 2 exp ( i m ϕ ) ρ [ ρ cos ( 2 ϕ ) φ sin ( 2 ϕ ) ] ,
1 2 exp ( i m ϕ ) ρ [ ρ sin ( 2 ϕ ) + φ cos ( 2 ϕ ) ] .
exp ( i m ϕ ) ρ x = 1 2 exp ( i m ϕ ) ρ ( ρ cos ϕ φ sin ϕ ) ,
exp ( i m ϕ ) ρ y = 1 2 exp ( i m ϕ ) ρ ( ρ sin ϕ + φ cos ϕ ) .
exp ( i m ϕ ) x x = 1 2 exp ( i m ϕ ) ρ ( 1 + cos 2 ϕ 2 ρ sin 2 ϕ 2 ϕ ) ,
exp ( i m ϕ ) y y = 1 2 exp ( i m ϕ ) ρ ( 1 cos 2 ϕ 2 ρ + sin 2 ϕ 2 ϕ ) ,
exp ( i m ϕ ) y x = 1 2 exp ( i m ϕ ) ρ ( sin 2 ϕ 2 ρ 1 cos 2 ϕ 2 ϕ ) ,
exp ( i m ϕ ) x y = 1 2 exp ( i m ϕ ) ρ ( sin 2 ϕ 2 ρ + 1 + cos 2 ϕ 2 ϕ ) .

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