Abstract

We demonstrate the possibility of creating multiple spherical spots in a 4π focusing system with a radially polarized beam. Using spherical waves to expand the plane wave factor in the Richards–Wolf integral, it is found that a proper spatial modulation in the amplitude of the input field with radial polarization can form multiple spherical spots with a focusing system satisfying the Herschel condition. These spots are distributed symmetrically about the focus on the optical axis with variable positions and intensities. Although we consider only the case of three spherical spots in this paper, generalization to the multiple-spots case will present no difficulty.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. Y. I. Salamin, “Electron acceleration from rest in vacuum by an axicon Gaussian laser beam,” Phys. Rev. A 73, 043402 (2006).
    [Crossref]
  2. P.-L. Fortin, M. Piché, and C. Varin, “Direct-field electron acceleration with ultrafast radially polarized laser beams: scaling laws and optimization,” J. Phys. B 43, 025401 (2010).
    [Crossref]
  3. N. Hayazawa, Y. Saito, and S. Kawata, “Detection and characterization of longitudinal field for tip-enhanced Raman spectroscopy,” Appl. Phys. Lett. 85, 6239–6241 (2004).
    [Crossref]
  4. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12, 3377–3382 (2004).
    [Crossref] [PubMed]
  5. S. Yan and B. Yao, “Radiation forces of a highly focused radially polarized beam on spherical particles,” Phys. Rev. A 76, 053836 (2007).
    [Crossref]
  6. H. Kawauchi, K. Yonezawa, Y. Kozawa, and S. Sato, “Calculation of optical trapping forces on a dielectric sphere in the ray optics regime produced by a radially polarized laser beam,” Opt. Lett. 32, 1839–1841 (2007).
    [Crossref] [PubMed]
  7. T. A. Nieminen, N. R. Heckenberg, and H. R. Dunlop, “Forces in optical tweezers with radially and azimuthally polarized trapping beams,” Opt. Lett. 33, 122–124 (2008).
    [Crossref] [PubMed]
  8. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
    [Crossref] [PubMed]
  9. H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nature Photon. 2, 501–505 (2008).
    [Crossref]
  10. N. Bokor and N. Davidson, “Toward a spherical spot distribution with 4π focusing of radially polarized light,” Opt. Lett. 29, 1968–1970 (2004).
    [Crossref] [PubMed]
  11. W. Chen and Q. Zhan, “Creating a spherical focal spot with spatially modulated radial polarization in 4Pi microscopy,” Opt. Lett. 34, 2444–2446 (2009).
    [Crossref] [PubMed]
  12. 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]
  13. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical–vector beams,” Opt. Express 7, 77–87 (2000).
    [Crossref] [PubMed]
  14. J. A. Stratton, Electromagnetic Theory (McGraw-Hill Book, 1941).

2010 (1)

P.-L. Fortin, M. Piché, and C. Varin, “Direct-field electron acceleration with ultrafast radially polarized laser beams: scaling laws and optimization,” J. Phys. B 43, 025401 (2010).
[Crossref]

2009 (1)

2008 (2)

T. A. Nieminen, N. R. Heckenberg, and H. R. Dunlop, “Forces in optical tweezers with radially and azimuthally polarized trapping beams,” Opt. Lett. 33, 122–124 (2008).
[Crossref] [PubMed]

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nature Photon. 2, 501–505 (2008).
[Crossref]

2007 (2)

2006 (1)

Y. I. Salamin, “Electron acceleration from rest in vacuum by an axicon Gaussian laser beam,” Phys. Rev. A 73, 043402 (2006).
[Crossref]

2004 (3)

2003 (1)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref] [PubMed]

2000 (1)

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

Bokor, N.

Brown, T. G.

Chen, W.

Chong, C. T.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nature Photon. 2, 501–505 (2008).
[Crossref]

Davidson, N.

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref] [PubMed]

Dunlop, H. R.

Fortin, P. -L.

P.-L. Fortin, M. Piché, and C. Varin, “Direct-field electron acceleration with ultrafast radially polarized laser beams: scaling laws and optimization,” J. Phys. B 43, 025401 (2010).
[Crossref]

Hayazawa, N.

N. Hayazawa, Y. Saito, and S. Kawata, “Detection and characterization of longitudinal field for tip-enhanced Raman spectroscopy,” Appl. Phys. Lett. 85, 6239–6241 (2004).
[Crossref]

Heckenberg, N. R.

Kawata, S.

N. Hayazawa, Y. Saito, and S. Kawata, “Detection and characterization of longitudinal field for tip-enhanced Raman spectroscopy,” Appl. Phys. Lett. 85, 6239–6241 (2004).
[Crossref]

Kawauchi, H.

Kozawa, Y.

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref] [PubMed]

Lukyanchuk, B.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nature Photon. 2, 501–505 (2008).
[Crossref]

Nieminen, T. A.

Piché, M.

P.-L. Fortin, M. Piché, and C. Varin, “Direct-field electron acceleration with ultrafast radially polarized laser beams: scaling laws and optimization,” J. Phys. B 43, 025401 (2010).
[Crossref]

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref] [PubMed]

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]

Saito, Y.

N. Hayazawa, Y. Saito, and S. Kawata, “Detection and characterization of longitudinal field for tip-enhanced Raman spectroscopy,” Appl. Phys. Lett. 85, 6239–6241 (2004).
[Crossref]

Salamin, Y. I.

Y. I. Salamin, “Electron acceleration from rest in vacuum by an axicon Gaussian laser beam,” Phys. Rev. A 73, 043402 (2006).
[Crossref]

Sato, S.

Sheppard, C.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nature Photon. 2, 501–505 (2008).
[Crossref]

Shi, L.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nature Photon. 2, 501–505 (2008).
[Crossref]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill Book, 1941).

Varin, C.

P.-L. Fortin, M. Piché, and C. Varin, “Direct-field electron acceleration with ultrafast radially polarized laser beams: scaling laws and optimization,” J. Phys. B 43, 025401 (2010).
[Crossref]

Wang, H.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nature Photon. 2, 501–505 (2008).
[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. London, Ser. A 253, 358–379 (1959).
[Crossref]

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]

Yonezawa, K.

Youngworth, K. S.

Zhan, Q.

Appl. Phys. Lett. (1)

N. Hayazawa, Y. Saito, and S. Kawata, “Detection and characterization of longitudinal field for tip-enhanced Raman spectroscopy,” Appl. Phys. Lett. 85, 6239–6241 (2004).
[Crossref]

J. Phys. B (1)

P.-L. Fortin, M. Piché, and C. Varin, “Direct-field electron acceleration with ultrafast radially polarized laser beams: scaling laws and optimization,” J. Phys. B 43, 025401 (2010).
[Crossref]

Nature Photon. (1)

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nature Photon. 2, 501–505 (2008).
[Crossref]

Opt. Express (2)

Opt. Lett. (4)

Phys. Rev. A (2)

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

Y. I. Salamin, “Electron acceleration from rest in vacuum by an axicon Gaussian laser beam,” Phys. Rev. A 73, 043402 (2006).
[Crossref]

Phys. Rev. Lett. (1)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref] [PubMed]

Proc. R. Soc. London, Ser. 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. London, Ser. A 253, 358–379 (1959).
[Crossref]

Other (1)

J. A. Stratton, Electromagnetic Theory (McGraw-Hill Book, 1941).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

Geometry of the 4 π focusing system consisting of two confocal high-NA objective lenses, illuminated by two counter-propagating radially polarized doughnut beams with a relative π phase shift.

Fig. 2
Fig. 2

Input field amplitude as a function of θ: l 0 ( θ ) corresponds to the fundamental radial polarization mode (solid line), l 3 ( θ ) corresponds to generation of three identical spherical spots (dotted line), and l 2 ( θ ) corresponds to generation of two identical spherical spots (dashed line).

Fig. 3
Fig. 3

Normalized intensity distributions of electric fields of (a), (b) three and (c) two spherical spots with all intensities normalized to the maximum of the total intensity. (a) The total intensity distribution in X Z plane. (b) Linescans of corresponding transversal and axial intensity distributions of (a); I 0 ( x ) , I ( x ) , and I + ( x ) are transversal linescans for the three spherical spots overlaid on the axial linescan I ( z ) . (c) Intensity linescans of electric fields of two identical spherical spots with I ( x ) and I + ( x ) denoting the transversal intensity distribution.

Equations (15)

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

E ρ ( ρ , z ) = 0 θ max l ( θ ) X ( θ ) sin   2 θ J 1 ( k ρ   sin   θ ) e i h z d θ ,
E z ( ρ , z ) = i 2 0 θ max l ( θ ) X ( θ ) ( sin   θ ) 2 J 0 ( k ρ   sin   θ ) e i h z d θ .
E ρ ( ρ , z ) = 0 π l ( θ ) sin   2 θ J 1 ( k ρ   sin   θ ) e i h z d θ ,
E z ( ρ , z ) = i 2 0 π l ( θ ) ( sin   θ ) 2 J 0 ( k ρ   sin   θ ) e i h z d θ .
J 0 ( k ρ   sin   θ ) e i h z = n = 0 i n ( 2 n + 1 ) P n ( cos   θ ) P n ( cos   α ) j n ( k R ) ,
E z = i 2 n = 0 A n j n ( k R ) P n ( cos   α ) ,
A n = i n ( 2 n + 1 ) 0 π l ( θ ) P n ( cos   θ ) ( sin   θ ) 2 d θ .
F = i 2 C 0 j 0 ( k R ) + i 2 C [ j 0 ( k | R e z z 0 | ) + j 0 ( k | R + e z z 0 | ) ] ,
j 0 ( k | R ± e z z 0 | ) = n = 0 ( 2 n + 1 ) P n ( cos   α ) j n ( ± k z 0 ) j n ( k R ) .
F = i 2 n = 0 ( 2 n + 1 ) [ δ 0 n C 0 + 2 C j n ( k z 0 ) ] j n ( k R ) P n ( cos   α ) .
A n = δ 0 n C 0 + 2 C ( 2 n + 1 ) j n ( k z 0 ) ,     n = even ,
A n = 0 ,     n = odd ,
i n 2 A n = 2 n + 1 2 1 1 ( 1 x 2 ) 1 / 2 l ( x ) P n ( x ) d x .
l ( x ) = ( 1 x 2 ) 1 / 2 k = 0 ( 1 ) k A 2 k 2 P 2 k ( x ) .
l ( x ) = l 0 ( x ) k = 0 ( 1 ) k A 2 k 2 P 2 k ( x ) = l 0 ( x ) A ( x ) .

Metrics