Abstract

Cylindrical vector beams have been proposed and demonstrated for applications ranging from microscopy to high energy physics. In this paper, we analyze the three-dimensional field distributions of radial and azimuthal beams focused near a dielectric interface. We give particular attention to the classic problem of high numerical aperture focusing from an immersion lens to a glass-air interface and find that the use of radially and azimuthally polarized illumination for this type of imaging provides an impressive lateral confinement of the fields over a wide range of interface positions.

© Optical Society of America

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References

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  1. B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems II.Structure of the image field in an aplanatic system," Proc. Roy. Soc. A 253, 358-379 (1959).
    [CrossRef]
  2. Hao Ling and Shung-Wu Lee, "Focusing of electromagnetic waves through a dielectric interface," J. Opt. Soc. Am. A 1, 965-973 (1984).
    [CrossRef]
  3. P. T�r�k, P.Varga, G. R. Booker, "Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices:structure of the electromagnetic field.I," J. Opt. Soc. Am. A 12, 2136-2144 (1995).
    [CrossRef]
  4. S. H. Wiersma, P.T�r�k, T. D. Visser, P. Varga, "Comparison of different heories for focusing through a plane interface," J. Opt. Soc. Am. A 14, 1482-1490 (1997).
    [CrossRef]
  5. Lars Egil Helseth, "Roles of polarization,phase,and amplitude in solid immersion lens systems," Opt. Commun. 191, 161-172 (2001).
    [CrossRef]
  6. T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, "Circularly symmetrical operation of a concentric-circle-grating,surface-emitting, AlGaAs/GaAs Quantum-well semiconductor-laser," Appl. Phys. Lett. 60, 1921-1923 (1992).
    [CrossRef]
  7. R. H. Jordan and D. G. Hall, "Free-space azimuthal paraxial wave equation: the azimuthal Bessel-Gauss beam solution," Opt. Lett. 19, 427-429 (1994).
    [CrossRef] [PubMed]
  8. D. G. Hall, "Vector-beam solutions of Maxwell's wave equation," Opt. Lett. 21, 9-11 (1996).
    [CrossRef] [PubMed]
  9. P. L. Greene and D. G.Hall, "Diffraction characteristics of the azimuthal Bessel-Gauss beam, " J. Opt. Soc. Am. A 13, 962-966 (1996).
    [CrossRef]
  10. P. L. Greene and D. G. Hall, "Properties and diffraction of vector Bessel-Gauss beams," J. Opt. Soc. Am. A 15, 3020-3027 (1998).
    [CrossRef]
  11. P. L. Greene and D. G .Hall, "Focal shift in vector beams," Opt. Express 4, 411-419 (1999), http://www.opticsexpress.org/tocv4n10.htm
    [CrossRef] [PubMed]
  12. C. J. R. Sheppard and S. Saghafi, "Transverse-electric and transverse-magnetic beam modes beyond the paraxial approximation," Opt. Lett. 24, 1543-1545 (1999).
    [CrossRef]
  13. S. C. Tidwell, D. H. Ford, and W. D. Kimura, "Generating radially polarized beams interferometrically," Appl. Opt. 29, 2234-2239 (1990).
    [CrossRef] [PubMed]
  14. K. S. Youngworth and T. G. Brown, "Inhomogeneous polarization in scanning optical microscopy," Proc. SPIE 3919 (2000).
    [CrossRef]
  15. M. Stalder and M. Schadt, "Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters," Opt. Lett. 21, 1948-1949 (1996).
    [CrossRef] [PubMed]
  16. R. Yamaguchi, T. Nose, and S. Sato, "Liquid-crystal polarizers with axially symmetric properties," Jpn. J. Appl. Phys. 28, 1730-1731 (1989).
    [CrossRef]
  17. E. G. Churin, J. Hossfeld, and T. Tschudi, "Polarization configurations with singular point formed by computer generated holograms," Opt.Commun. 99, 13-17 (1993).
    [CrossRef]
  18. K. S. Youngworth and T. G. Brown, "Focusing of high numerical aperture cylindrical-vector beams," Opt. Express 7, 77-87 (2000), http://www.opticsexpress.org/oearchive/source/22809.htm
    [CrossRef]
  19. S. Quabis, R. Dorn, M. Eberler, O. G. Gl�ckl, G. Leuchs, "Focusing light to a tighter spot," Opt. Commun. 179, 1-7 (2000).
    [CrossRef]
  20. B. Sick, B. Hecht, and L. Novotny, "Orientational imaging of single molecules by annular illumination," Phys. Rev. Lett. 85, 4482-4485,(2000).
    [CrossRef] [PubMed]
  21. L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, "Longitudinal field modes probed by single molecules," Phys. Rev. Lett. 86, 5251-5254 (2001).
    [CrossRef] [PubMed]
  22. J .Enderlein, "Theoretical study of detection of a dipole emitter through an objective with high numerical aperture," Opt. Lett. 25, 634-636 (2000).
    [CrossRef]
  23. T. Ha, T. A. Laurence, D. S. Chemla, and S. Weiss, "Polarization spectroscopy of single fluorescent molecules," J. Phys. Chem. B 103, 6839-6850 (1999)
    [CrossRef]

Other

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems II.Structure of the image field in an aplanatic system," Proc. Roy. Soc. A 253, 358-379 (1959).
[CrossRef]

Hao Ling and Shung-Wu Lee, "Focusing of electromagnetic waves through a dielectric interface," J. Opt. Soc. Am. A 1, 965-973 (1984).
[CrossRef]

P. T�r�k, P.Varga, G. R. Booker, "Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices:structure of the electromagnetic field.I," J. Opt. Soc. Am. A 12, 2136-2144 (1995).
[CrossRef]

S. H. Wiersma, P.T�r�k, T. D. Visser, P. Varga, "Comparison of different heories for focusing through a plane interface," J. Opt. Soc. Am. A 14, 1482-1490 (1997).
[CrossRef]

Lars Egil Helseth, "Roles of polarization,phase,and amplitude in solid immersion lens systems," Opt. Commun. 191, 161-172 (2001).
[CrossRef]

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, "Circularly symmetrical operation of a concentric-circle-grating,surface-emitting, AlGaAs/GaAs Quantum-well semiconductor-laser," Appl. Phys. Lett. 60, 1921-1923 (1992).
[CrossRef]

R. H. Jordan and D. G. Hall, "Free-space azimuthal paraxial wave equation: the azimuthal Bessel-Gauss beam solution," Opt. Lett. 19, 427-429 (1994).
[CrossRef] [PubMed]

D. G. Hall, "Vector-beam solutions of Maxwell's wave equation," Opt. Lett. 21, 9-11 (1996).
[CrossRef] [PubMed]

P. L. Greene and D. G.Hall, "Diffraction characteristics of the azimuthal Bessel-Gauss beam, " J. Opt. Soc. Am. A 13, 962-966 (1996).
[CrossRef]

P. L. Greene and D. G. Hall, "Properties and diffraction of vector Bessel-Gauss beams," J. Opt. Soc. Am. A 15, 3020-3027 (1998).
[CrossRef]

P. L. Greene and D. G .Hall, "Focal shift in vector beams," Opt. Express 4, 411-419 (1999), http://www.opticsexpress.org/tocv4n10.htm
[CrossRef] [PubMed]

C. J. R. Sheppard and S. Saghafi, "Transverse-electric and transverse-magnetic beam modes beyond the paraxial approximation," Opt. Lett. 24, 1543-1545 (1999).
[CrossRef]

S. C. Tidwell, D. H. Ford, and W. D. Kimura, "Generating radially polarized beams interferometrically," Appl. Opt. 29, 2234-2239 (1990).
[CrossRef] [PubMed]

K. S. Youngworth and T. G. Brown, "Inhomogeneous polarization in scanning optical microscopy," Proc. SPIE 3919 (2000).
[CrossRef]

M. Stalder and M. Schadt, "Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters," Opt. Lett. 21, 1948-1949 (1996).
[CrossRef] [PubMed]

R. Yamaguchi, T. Nose, and S. Sato, "Liquid-crystal polarizers with axially symmetric properties," Jpn. J. Appl. Phys. 28, 1730-1731 (1989).
[CrossRef]

E. G. Churin, J. Hossfeld, and T. Tschudi, "Polarization configurations with singular point formed by computer generated holograms," Opt.Commun. 99, 13-17 (1993).
[CrossRef]

K. S. Youngworth and T. G. Brown, "Focusing of high numerical aperture cylindrical-vector beams," Opt. Express 7, 77-87 (2000), http://www.opticsexpress.org/oearchive/source/22809.htm
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. G. Gl�ckl, G. Leuchs, "Focusing light to a tighter spot," Opt. Commun. 179, 1-7 (2000).
[CrossRef]

B. Sick, B. Hecht, and L. Novotny, "Orientational imaging of single molecules by annular illumination," Phys. Rev. Lett. 85, 4482-4485,(2000).
[CrossRef] [PubMed]

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, "Longitudinal field modes probed by single molecules," Phys. Rev. Lett. 86, 5251-5254 (2001).
[CrossRef] [PubMed]

J .Enderlein, "Theoretical study of detection of a dipole emitter through an objective with high numerical aperture," Opt. Lett. 25, 634-636 (2000).
[CrossRef]

T. Ha, T. A. Laurence, D. S. Chemla, and S. Weiss, "Polarization spectroscopy of single fluorescent molecules," J. Phys. Chem. B 103, 6839-6850 (1999)
[CrossRef]

Supplementary Material (2)

» Media 1: MOV (621 KB)     
» Media 2: MOV (1578 KB)     

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Figures (7)

Fig. 1.
Fig. 1.

Aplanatic lens focusing onto an interface

Fig. 2.
Fig. 2.

a. Focusing of radial component with incident medium (left side) n=1.518 and exiting medium (right side) n=1. The NA for the system is 1.4. The white line represents the interface of the system. The color scale is a log base 10 scale in arbitrary units. Axes are in units of wavelength. The picture on the left is the radial component. The picture on the right is the longitudinal component.

Fig. 2.
Fig. 2.

b. Focusing of radial component with incident medium (left side) n=1 and exiting medium (right side) n=3.55. The NA for the system is .85. The white line represents the interface of the system. The color scale is a log base 10 scale in arbitrary units. Axes are in the units of wavelength. The picture on the left is the radial component. The picture on the right is the longitudinal component.

Fig. 3.
Fig. 3.

a. Azimuthally polarized light focused onto an interface with NA=1.4 and entering medium n=1.518, and exiting medium n=1. The white line represents the interface of the system. The color scale is a log base 10 scale in arbitrary units. Axes are in the units of wavelengths.

Fig. 3.
Fig. 3.

b. Azimuthally polarized light focused onto an interface with NA=.85 and entering medium n=1, and exiting medium n=3.55. The white line represents the interface of the system. The color scale is a log base 10 scale in arbitrary units. Axes are in units of wavelengths.

Fig. 4.
Fig. 4.

(2.1 MB) Movie of the defocusing of a radial beam. The beam is entering through a medium of oil (n=1.518) and exiting into a medium of air (n=1). The white line represents the position of the interface. The color scale is a logarithmic base 10 scale. Axes are in units of wavelengths.[Media 2]

Fig. 5.
Fig. 5.

Full width of half maximum of the focal spot for light of linear and radial polarization (total electric field intensity) and the longitudinal component of the radially polarized beams focused through an immersion lens (NA=1.4). Blue stars correspond to the radially polarized beam (total intensity), red triangles to linearly polarized beams, and green squares represent the longitudinal component of the radial polarization. Both interface positions and the FWHM are given in wavelengths. An interface at a negative position is inside the geometrical focus.

Equations (11)

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E f ( θ , ϕ ) = o ( θ ) cos 1 2 θ [ sin ϕ cos ϕ 0 ] .
E f ( ρ , z ) = 0 θ max o ( θ ) E f cos 1 2 θ sin θ d θ ,
E f ( ρ , z ) = 0 θ max o ( θ ) E f cos 1 2 θ sin θ d θ ,
E t ( ρ , z ) = 0 θ max o ( θ ) E f cos 1 2 θ sin θ d θ ,
E f = e i k 1 z cos θ [ 0 J 1 ( ρ k 1 sin θ ) 0 ] ,
E r = r s e i k 1 z cos θ e i 2 k 1 z 0 cos θ [ 0 J 1 ( ρ k 1 sin θ ) 0 ] ,
E t = t s e i k 1 z 0 cos θ e i k 2 1 ( k 1 k 2 sin θ ) 2 ( z z 0 ) [ 0 J 1 ( ρ k 1 sin θ ) 0 ] .
E f = e i k 1 z cos θ [ i cos θ J 1 ( ρ k 1 sin θ ) 0 sin θ J 0 ( ρ k 1 sin θ ) ] ,
E r = r p e i k 1 z cos θ e i 2 k 1 z o cos θ [ i cos θ J 1 ( ρ k 1 sin θ ) 0 sin θ J 0 ( ρ k 1 sin θ ) ] ,
E t = t p e i k 1 z 0 cos θ e i k 2 1 ( k 1 k 2 sin θ ) 2 ( z z 0 ) [ i 1 ( k 1 k 2 sin θ ) 2 J 1 ( ρ k 1 sin θ ) 0 k 1 k 2 sin θ J 0 ( ρ k 1 sin θ ) ] .
o ( θ ) = E o exp ( β 2 sin 2 θ sin 2 θ max ) J 1 ( 2 β sin θ sin θ max ) .

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