Abstract

The high-NA focusing properties of vector vortex beams are studied theoretically and experimentally. The vector vortex beams are generated by space-variant segmented subwavelength metallic gratings first. Then the mathematical expressions for the focused fields are derived based on the vector diffraction theory, and some numerical simulations are presented that show that the focused fields are not dark at the center and the focusing spot size of vector vortex beams with high topological charges approaches the diffraction limitation at high NA. Finally, to verify the theoretical analysis, the tightly focused fields are measured based on a confocal microscopy system when the NA of the objective lens is 0.90. The research results confirm the potential of vector vortex beams in some applications, such as optical trapping, laser printing, lithography, and material processing.

© 2011 Optical Society of America

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  1. M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
    [CrossRef]
  2. Z. Bomzon, V. Kleiner, and E. Hasman, “Pancharatnam–Berry phase in space-variant polarization-state manipulations with subwavelength gratings,” Opt. Lett. 26, 1424–1426 (2001).
    [CrossRef]
  3. G. Biener, A. Niv, V. Kleiner, and E. Hasman, “Formation of helical beams by use of Pancharatnam–Berry phase optical elements,” Opt. Lett. 27, 1875–1877 (2002).
    [CrossRef]
  4. A. Niv, G. Biener, V. Kleiner, and E. Hasman, “Spiral phase elements obtained by use of discrete space-variant subwavelength gratings,” Opt. Commun. 251, 306–314 (2005).
    [CrossRef]
  5. A. Niv, G. Biener, V. Kleiner, and E. Hasman, “Manipulation of the Pancharatnam phase in vectorial vortices,” Opt. Express 14, 4208–4220 (2006).
    [CrossRef] [PubMed]
  6. Z. Bomzon, M. Gu, and J. Shamir, “Angular momentum and geometrical phases in tight-focused circularly polarized plane waves,” Appl. Phys. Lett. 89, 241104 (2006).
    [CrossRef]
  7. Q. Zhan and J. R. Leger, “Interferometric measurement of Berry’s phase in space-variant polarization manipulations,” Opt. Commun. 213, 241–245 (2002).
    [CrossRef]
  8. Q. Zhan, “Properties of circularly polarized vortex beams,” Opt. Lett. 31, 867–869 (2006).
    [CrossRef] [PubMed]
  9. L. Rao, J. Pu, Z. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical-aperture lens,” Opt. Laser Technol. 41, 241–246 (2009).
    [CrossRef]
  10. J. Pu and Z. Zhang, “Tight focusing of spirally polarized vortex beams,” Opt. Laser Technol. 42, 186–191 (2010).
    [CrossRef]
  11. D. Zhang, P. Wang, X. Jiao, C. Ming, G. Yuan, Y. Deng, H. Ming, L. Zhang, and W. Liu, “Polarization properties of subwavelength metallic gratings in visible light band,” Appl. Phys. B 85, 139–146 (2006).
    [CrossRef]
  12. G. Machavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Efficient extracavity generation of radially and azimuthally polarized beams,” Opt. Lett. 32, 1468–1470 (2007).
    [CrossRef] [PubMed]
  13. M. Born and E. Wolf, Principles of Optics (Pergamon, 1999), pp. 20–65.
  14. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358–379 (1959).
    [CrossRef]
  15. C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
    [CrossRef]
  16. M. Rashid, O. Maragò, and P. Jones, “Focusing of high order cylindrical vector Beams,” J. Opt. A: Pure Appl. Opt. 11, 065204 (2009).
    [CrossRef]
  17. Z. Zhou, Q. Tan, and G. Jin, “Cylindrically polarized vortex beams generated by subwavelength concentric Al metallic gratings,” J. Opt. 13, 075004 (2011).
    [CrossRef]

2011 (1)

Z. Zhou, Q. Tan, and G. Jin, “Cylindrically polarized vortex beams generated by subwavelength concentric Al metallic gratings,” J. Opt. 13, 075004 (2011).
[CrossRef]

2010 (1)

J. Pu and Z. Zhang, “Tight focusing of spirally polarized vortex beams,” Opt. Laser Technol. 42, 186–191 (2010).
[CrossRef]

2009 (2)

L. Rao, J. Pu, Z. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical-aperture lens,” Opt. Laser Technol. 41, 241–246 (2009).
[CrossRef]

M. Rashid, O. Maragò, and P. Jones, “Focusing of high order cylindrical vector Beams,” J. Opt. A: Pure Appl. Opt. 11, 065204 (2009).
[CrossRef]

2007 (2)

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[CrossRef]

G. Machavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Efficient extracavity generation of radially and azimuthally polarized beams,” Opt. Lett. 32, 1468–1470 (2007).
[CrossRef] [PubMed]

2006 (4)

Q. Zhan, “Properties of circularly polarized vortex beams,” Opt. Lett. 31, 867–869 (2006).
[CrossRef] [PubMed]

A. Niv, G. Biener, V. Kleiner, and E. Hasman, “Manipulation of the Pancharatnam phase in vectorial vortices,” Opt. Express 14, 4208–4220 (2006).
[CrossRef] [PubMed]

D. Zhang, P. Wang, X. Jiao, C. Ming, G. Yuan, Y. Deng, H. Ming, L. Zhang, and W. Liu, “Polarization properties of subwavelength metallic gratings in visible light band,” Appl. Phys. B 85, 139–146 (2006).
[CrossRef]

Z. Bomzon, M. Gu, and J. Shamir, “Angular momentum and geometrical phases in tight-focused circularly polarized plane waves,” Appl. Phys. Lett. 89, 241104 (2006).
[CrossRef]

2005 (1)

A. Niv, G. Biener, V. Kleiner, and E. Hasman, “Spiral phase elements obtained by use of discrete space-variant subwavelength gratings,” Opt. Commun. 251, 306–314 (2005).
[CrossRef]

2002 (2)

Q. Zhan and J. R. Leger, “Interferometric measurement of Berry’s phase in space-variant polarization manipulations,” Opt. Commun. 213, 241–245 (2002).
[CrossRef]

G. Biener, A. Niv, V. Kleiner, and E. Hasman, “Formation of helical beams by use of Pancharatnam–Berry phase optical elements,” Opt. Lett. 27, 1875–1877 (2002).
[CrossRef]

2001 (2)

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. A 253, 358–379 (1959).
[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, 78 (2007).
[CrossRef]

Biener, G.

Bomzon, Z.

Z. Bomzon, M. Gu, and J. Shamir, “Angular momentum and geometrical phases in tight-focused circularly polarized plane waves,” Appl. Phys. Lett. 89, 241104 (2006).
[CrossRef]

Z. Bomzon, V. Kleiner, and E. Hasman, “Pancharatnam–Berry phase in space-variant polarization-state manipulations with subwavelength gratings,” Opt. Lett. 26, 1424–1426 (2001).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1999), pp. 20–65.

Chen, Z.

L. Rao, J. Pu, Z. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical-aperture lens,” Opt. Laser Technol. 41, 241–246 (2009).
[CrossRef]

Deng, Y.

D. Zhang, P. Wang, X. Jiao, C. Ming, G. Yuan, Y. Deng, H. Ming, L. Zhang, and W. Liu, “Polarization properties of subwavelength metallic gratings in visible light band,” Appl. Phys. B 85, 139–146 (2006).
[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, 78 (2007).
[CrossRef]

Gu, M.

Z. Bomzon, M. Gu, and J. Shamir, “Angular momentum and geometrical phases in tight-focused circularly polarized plane waves,” Appl. Phys. Lett. 89, 241104 (2006).
[CrossRef]

Hasman, E.

Jackel, S.

Jesacher, A.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[CrossRef]

Jiao, X.

D. Zhang, P. Wang, X. Jiao, C. Ming, G. Yuan, Y. Deng, H. Ming, L. Zhang, and W. Liu, “Polarization properties of subwavelength metallic gratings in visible light band,” Appl. Phys. B 85, 139–146 (2006).
[CrossRef]

Jin, G.

Z. Zhou, Q. Tan, and G. Jin, “Cylindrically polarized vortex beams generated by subwavelength concentric Al metallic gratings,” J. Opt. 13, 075004 (2011).
[CrossRef]

Jones, P.

M. Rashid, O. Maragò, and P. Jones, “Focusing of high order cylindrical vector Beams,” J. Opt. A: Pure Appl. Opt. 11, 065204 (2009).
[CrossRef]

Kleiner, V.

Leger, J. R.

Q. Zhan and J. R. Leger, “Interferometric measurement of Berry’s phase in space-variant polarization manipulations,” Opt. Commun. 213, 241–245 (2002).
[CrossRef]

Liu, W.

D. Zhang, P. Wang, X. Jiao, C. Ming, G. Yuan, Y. Deng, H. Ming, L. Zhang, and W. Liu, “Polarization properties of subwavelength metallic gratings in visible light band,” Appl. Phys. B 85, 139–146 (2006).
[CrossRef]

Lumer, Y.

Machavariani, G.

Maragò, O.

M. Rashid, O. Maragò, and P. Jones, “Focusing of high order cylindrical vector Beams,” J. Opt. A: Pure Appl. Opt. 11, 065204 (2009).
[CrossRef]

Maurer, C.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[CrossRef]

Meir, A.

Ming, C.

D. Zhang, P. Wang, X. Jiao, C. Ming, G. Yuan, Y. Deng, H. Ming, L. Zhang, and W. Liu, “Polarization properties of subwavelength metallic gratings in visible light band,” Appl. Phys. B 85, 139–146 (2006).
[CrossRef]

Ming, H.

D. Zhang, P. Wang, X. Jiao, C. Ming, G. Yuan, Y. Deng, H. Ming, L. Zhang, and W. Liu, “Polarization properties of subwavelength metallic gratings in visible light band,” Appl. Phys. B 85, 139–146 (2006).
[CrossRef]

Moshe, I.

Niv, A.

Pu, J.

J. Pu and Z. Zhang, “Tight focusing of spirally polarized vortex beams,” Opt. Laser Technol. 42, 186–191 (2010).
[CrossRef]

L. Rao, J. Pu, Z. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical-aperture lens,” Opt. Laser Technol. 41, 241–246 (2009).
[CrossRef]

Rao, L.

L. Rao, J. Pu, Z. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical-aperture lens,” Opt. Laser Technol. 41, 241–246 (2009).
[CrossRef]

Rashid, M.

M. Rashid, O. Maragò, and P. Jones, “Focusing of high order cylindrical vector Beams,” J. Opt. A: Pure Appl. Opt. 11, 065204 (2009).
[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. 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, 78 (2007).
[CrossRef]

Shamir, J.

Z. Bomzon, M. Gu, and J. Shamir, “Angular momentum and geometrical phases in tight-focused circularly polarized plane waves,” Appl. Phys. Lett. 89, 241104 (2006).
[CrossRef]

Soskin, M. S.

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[CrossRef]

Tan, Q.

Z. Zhou, Q. Tan, and G. Jin, “Cylindrically polarized vortex beams generated by subwavelength concentric Al metallic gratings,” J. Opt. 13, 075004 (2011).
[CrossRef]

Vasnetsov, M. V.

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[CrossRef]

Wang, P.

D. Zhang, P. Wang, X. Jiao, C. Ming, G. Yuan, Y. Deng, H. Ming, L. Zhang, and W. Liu, “Polarization properties of subwavelength metallic gratings in visible light band,” Appl. Phys. B 85, 139–146 (2006).
[CrossRef]

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. A 253, 358–379 (1959).
[CrossRef]

M. Born and E. Wolf, Principles of Optics (Pergamon, 1999), pp. 20–65.

Yei, P.

L. Rao, J. Pu, Z. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical-aperture lens,” Opt. Laser Technol. 41, 241–246 (2009).
[CrossRef]

Yuan, G.

D. Zhang, P. Wang, X. Jiao, C. Ming, G. Yuan, Y. Deng, H. Ming, L. Zhang, and W. Liu, “Polarization properties of subwavelength metallic gratings in visible light band,” Appl. Phys. B 85, 139–146 (2006).
[CrossRef]

Zhan, Q.

Q. Zhan, “Properties of circularly polarized vortex beams,” Opt. Lett. 31, 867–869 (2006).
[CrossRef] [PubMed]

Q. Zhan and J. R. Leger, “Interferometric measurement of Berry’s phase in space-variant polarization manipulations,” Opt. Commun. 213, 241–245 (2002).
[CrossRef]

Zhang, D.

D. Zhang, P. Wang, X. Jiao, C. Ming, G. Yuan, Y. Deng, H. Ming, L. Zhang, and W. Liu, “Polarization properties of subwavelength metallic gratings in visible light band,” Appl. Phys. B 85, 139–146 (2006).
[CrossRef]

Zhang, L.

D. Zhang, P. Wang, X. Jiao, C. Ming, G. Yuan, Y. Deng, H. Ming, L. Zhang, and W. Liu, “Polarization properties of subwavelength metallic gratings in visible light band,” Appl. Phys. B 85, 139–146 (2006).
[CrossRef]

Zhang, Z.

J. Pu and Z. Zhang, “Tight focusing of spirally polarized vortex beams,” Opt. Laser Technol. 42, 186–191 (2010).
[CrossRef]

Zhou, Z.

Z. Zhou, Q. Tan, and G. Jin, “Cylindrically polarized vortex beams generated by subwavelength concentric Al metallic gratings,” J. Opt. 13, 075004 (2011).
[CrossRef]

Appl. Phys. B (1)

D. Zhang, P. Wang, X. Jiao, C. Ming, G. Yuan, Y. Deng, H. Ming, L. Zhang, and W. Liu, “Polarization properties of subwavelength metallic gratings in visible light band,” Appl. Phys. B 85, 139–146 (2006).
[CrossRef]

Appl. Phys. Lett. (1)

Z. Bomzon, M. Gu, and J. Shamir, “Angular momentum and geometrical phases in tight-focused circularly polarized plane waves,” Appl. Phys. Lett. 89, 241104 (2006).
[CrossRef]

J. Opt. (1)

Z. Zhou, Q. Tan, and G. Jin, “Cylindrically polarized vortex beams generated by subwavelength concentric Al metallic gratings,” J. Opt. 13, 075004 (2011).
[CrossRef]

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

M. Rashid, O. Maragò, and P. Jones, “Focusing of high order cylindrical vector Beams,” J. Opt. A: Pure Appl. Opt. 11, 065204 (2009).
[CrossRef]

New J. Phys. (1)

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[CrossRef]

Opt. Commun. (2)

A. Niv, G. Biener, V. Kleiner, and E. Hasman, “Spiral phase elements obtained by use of discrete space-variant subwavelength gratings,” Opt. Commun. 251, 306–314 (2005).
[CrossRef]

Q. Zhan and J. R. Leger, “Interferometric measurement of Berry’s phase in space-variant polarization manipulations,” Opt. Commun. 213, 241–245 (2002).
[CrossRef]

Opt. Express (1)

Opt. Laser Technol. (2)

L. Rao, J. Pu, Z. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical-aperture lens,” Opt. Laser Technol. 41, 241–246 (2009).
[CrossRef]

J. Pu and Z. Zhang, “Tight focusing of spirally polarized vortex beams,” Opt. Laser Technol. 42, 186–191 (2010).
[CrossRef]

Opt. Lett. (4)

Proc. R. Soc. 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. A 253, 358–379 (1959).
[CrossRef]

Prog. Opt. (1)

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[CrossRef]

Other (1)

M. Born and E. Wolf, Principles of Optics (Pergamon, 1999), pp. 20–65.

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Figures (4)

Fig. 1
Fig. 1

Local structure of a segmented SMG.

Fig. 2
Fig. 2

(a) SEM image of the grating stripe structure near the center of the fabricated device and (b) photo of manufactured devices.

Fig. 3
Fig. 3

Intensity distributions at focus for (a)  l = 3 , (b)  l = 4 , and (c)  l = 5 , respectively, under the condition that NA = 0.90 , and along the x axis for (d)  l = 4 , while NA = 0.90 and NA = 1.00 , respectively.

Fig. 4
Fig. 4

Calculated and measured intensities distribution of the vector vortex beams for l = 3 (left column) and l = 4 (right column) at focus. (a), (b) Calculated results, (c), (d) measured results, and (e), (f) intensity distributions along x axis for calculated and measured results, respectively.

Tables (1)

Tables Icon

Table 1 Measured Intensity Distribution Sampled by CCD through Four Measurements at 488 nm

Equations (10)

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E in = e i ϕ ( e r + i e ϕ ) ,
E out = e i ϕ cos [ ( l 1 ) ϕ ] e r + i e i ϕ sin [ ( l 1 ) ϕ ] e ϕ = e i l ϕ e r = e i l ϕ { cos [ ( l 1 ) ϕ ] e r + sin [ ( l 1 ) ϕ ] e ϕ } .
E in = A ( r ) exp ( i l ϕ ) { cos [ ( l 1 ) ϕ ] e r + sin [ ( l 1 ) ϕ ] e ϕ } ,
E ( r S , ϕ S , z S ) = [ E r ( S ) E ϕ ( S ) E z ( S ) ] = i a 0 π 0 α d θ 0 2 π l 0 ( θ ) cos θ sin θ exp ( i l ϕ ) exp { i k [ z S cos θ r S sin θ cos ( ϕ ϕ S ) ] } × [ cos [ ( l 1 ) ϕ ] cos θ cos ( ϕ ϕ s ) sin [ ( l 1 ) ϕ ] sin ( ϕ ϕ s ) cos [ ( l 1 ) ϕ ] cos θ sin ( ϕ ϕ s ) + sin [ ( l 1 ) ϕ ] cos ( ϕ ϕ s ) cos [ ( l 1 ) ϕ ] sin θ ] d ϕ ,
exp [ i ( l 1 ) ϕ ] = cos [ ( l 1 ) ϕ ] + i sin [ ( l 1 ) ϕ ] ,
0 2 π exp [ i m ϕ + i t cos ( ϕ γ ) ] d ϕ = 2 π i m exp ( i m γ ) J m ( t ) ,
E r ( S ) ( r s , ϕ s , z s ) = i a 0 2 0 α l 0 ( θ ) sin θ cos θ exp ( i k z s cos θ ) × { i 2 l exp [ i ( 2 l 1 ) ϕ s ] { cos θ [ J 2 l ( k r s sin θ ) J 2 l 2 ( k r s sin θ ) ] + J 2 l ( k r s sin θ ) + J 2 l 2 ( k r s sin θ ) } + exp ( i ϕ s ) { cos θ [ J 0 ( k r s sin θ ) J 2 ( k r s sin θ ) ] + [ J 0 ( k r s sin θ ) + J 2 ( k r s sin θ ) ] } } d θ ,
E ϕ ( S ) ( r s , ϕ s , z s ) = i a 0 2 0 α l 0 ( θ ) sin θ cos θ exp ( i k z s cos θ ) × { i 2 l exp [ i ( 2 l 1 ) ϕ s ] { cos θ [ J 2 l ( k r s sin θ ) + J 2 l 2 ( k r s sin θ ) ] + J 2 l ( k r s sin θ ) J 2 l 2 ( k r s sin θ ) } exp ( i ϕ s ) { cos θ [ J 2 ( k r s sin θ ) + J 0 ( k r s sin θ ) ] + J 2 ( k r s sin θ ) J 0 ( k r s sin θ ) } } d θ ,
E z ( S ) ( r s , ϕ s , z s ) = a 0 0 α l 0 ( θ ) sin 2 θ cos θ exp ( i k z s cos θ ) × { i 2 l exp [ i ( 2 l 1 ) ϕ s ] J 2 l 1 ( k r s sin θ ) exp [ i ϕ s ] J 1 ( k r s sin θ ) } d θ .
l 0 ( θ ) = exp [ β 2 ( sin θ sin α ) 2 ] ( 2 β sin θ sin α ) l L p l ( 2 β 2 sin 2 θ sin 2 α ) ,

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