Abstract

Cylindrical-vector beams are of increasing recent interest for their role in novel laser resonators and their applications to electron acceleration and scanning microscopy. In this paper, we calculate cylindrical-vector fields, near the focal region of an aplanatic lens, and briefly discuss some applications. We show that, in the particular case of a tightly focused, radially polarized beam, the polarization shows large inhomogeneities in the focal region, while the azimuthally polarized beam is purely transverse even at very high numerical apertures.

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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. A. Boivin and E. Wolf, "Electromagnetic field in the neighborhood of the focus of a coherent beam," Phys. Rev. B 138, 1561-1565 (1965).
    [CrossRef]
  3. A. Yoshida and T. Asakura, "Electromagnetic field near the focus of Gaussian beams," Optik 41, 281-292 (1974).
  4. A. Yoshida and T. Asakura, "Electromagnetic field in the focal plane of a coherent beam from a wide-angular annular-aperture system," Optik 40, 322-331 (1974).
  5. T. Wilson, R. Juskaitis, and P. Higdon, "The imaging of dielectric point scatterers in conventional and confocal polarisation microscopes," Opt. Commun. 141, 298-313 (1997).
    [CrossRef]
  6. P. Tšršk, P.D. Higdon, and T. Wilson, "On the general properties of polarised light conventional and confocal microscopes," 148, 300-315 (1998).
  7. 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]
  8. 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]
  9. D. G. Hall, "Vector-beam solutions of Maxwell's wave equation," Opt. Lett. 21, 9-11 (1996).
    [CrossRef] [PubMed]
  10. 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]
  11. 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]
  12. 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]
  13. 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]
  14. 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]
  15. R. Yamaguchi, T. Nose, and S. Sato, "Liquid-crystal polarizers with axially symmetrical properties", Jpn. J. Appl. Phys. 28, 1730-1731 (1989).
    [CrossRef]
  16. D. Pohl, "Operation of a ruby laser in the purely transverse electric mode TE 01," Appl. Phys. Lett. 20, 266- 267, 1972.
    [CrossRef]
  17. J. J. Wynne, "Generation of the rotationally symmetric TE 01 and TM01 modes from a wavelength-tunable laser," IEEE J. Quant. Elec. QE-10, 125-127 (1974).
    [CrossRef]
  18. M. E. Marhic and E. Garmire, "Low-order TE 0q operation of a CO2 laser for transmission through circular metallic waveguides," Appl. Phys. Lett. 38, 743-745 (1981).
    [CrossRef]
  19. S. C. Tidwell, G. H. Kim, and W. D. Kimura, "Efficient radially polarized laser-beam generation with a double interferometer," Appl. Opt. 32, 5222-5229 (1993).
    [CrossRef] [PubMed]
  20. S. C. Tidwell, D. H. Ford, and W. D. Kimura, "Generating radially polarized beams interferometrically," Appl. Opt. 29, 2234-2239 (1990).
    [CrossRef] [PubMed]
  21. 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]
  22. K. S. Youngworth and T. G. Brown, "Inhomogeneous polarization in scanning optical microscopy," Proc. SPIE 3919 (2000)
    [CrossRef]
  23. B. Hecht, B. Sick, U. P. Wild, and L. Novotny, "Orientational imaging of single molecules by annular illumination," submitted to Phys. Rev. Lett.
  24. J. Enderlein, "Theoretical study of detection of a dipole emitter through an objective with high numerical aperture," Opt. Lett. 25, 634-636 (2000).
    [CrossRef]
  25. 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 (25)

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]

A. Boivin and E. Wolf, "Electromagnetic field in the neighborhood of the focus of a coherent beam," Phys. Rev. B 138, 1561-1565 (1965).
[CrossRef]

A. Yoshida and T. Asakura, "Electromagnetic field near the focus of Gaussian beams," Optik 41, 281-292 (1974).

A. Yoshida and T. Asakura, "Electromagnetic field in the focal plane of a coherent beam from a wide-angular annular-aperture system," Optik 40, 322-331 (1974).

T. Wilson, R. Juskaitis, and P. Higdon, "The imaging of dielectric point scatterers in conventional and confocal polarisation microscopes," Opt. Commun. 141, 298-313 (1997).
[CrossRef]

P. Tšršk, P.D. Higdon, and T. Wilson, "On the general properties of polarised light conventional and confocal microscopes," 148, 300-315 (1998).

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]

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 symmetrical properties", Jpn. J. Appl. Phys. 28, 1730-1731 (1989).
[CrossRef]

D. Pohl, "Operation of a ruby laser in the purely transverse electric mode TE 01," Appl. Phys. Lett. 20, 266- 267, 1972.
[CrossRef]

J. J. Wynne, "Generation of the rotationally symmetric TE 01 and TM01 modes from a wavelength-tunable laser," IEEE J. Quant. Elec. QE-10, 125-127 (1974).
[CrossRef]

M. E. Marhic and E. Garmire, "Low-order TE 0q operation of a CO2 laser for transmission through circular metallic waveguides," Appl. Phys. Lett. 38, 743-745 (1981).
[CrossRef]

S. C. Tidwell, G. H. Kim, and W. D. Kimura, "Efficient radially polarized laser-beam generation with a double interferometer," Appl. Opt. 32, 5222-5229 (1993).
[CrossRef] [PubMed]

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

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, "Inhomogeneous polarization in scanning optical microscopy," Proc. SPIE 3919 (2000)
[CrossRef]

B. Hecht, B. Sick, U. P. Wild, and L. Novotny, "Orientational imaging of single molecules by annular illumination," submitted to Phys. Rev. Lett.

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 (1)

» Media 1: MOV (2183 KB)     

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Figures (10)

Fig. 1.
Fig. 1.

Geometry of the problem. The input apodization is defined in plane 0, conjugate to focal sphere 1.

Fig. 2.
Fig. 2.

Normalized intensity of the transverse (radial) component of a high-NA (1.32) radially polarized beam at focus and through focus. Intensities of 0 and 1 correspond to black and white, respectively. The units of x, y, ρ, and z are in wavelengths.

Fig. 3.
Fig. 3.

Normalized intensity of the longitudinal (z-) component of a high-NA (1.32) radially polarized beam at focus and through focus. Intensities of 0 and 1 correspond to black and white, respectively. The units of x, y, ρ, and z are in wavelengths.

Fig. 4.
Fig. 4.

The ratio of the maximum intensities of the longitudinal (max I z) and radial transverse (max Iρ) fields versus the focusing angle (α). The dotted line shows where max Iz becomes larger than max Iρ. For the example, we used an objective with NA=1.32 in oil, or α≈1.05 radians; max Iz is 1.4 times larger than max Iρ.

Fig. 5.
Fig. 5.

Normalized intensity of the transverse (azimuthal) component of a high-NA (1.32) azimuthally polarized beam at focus and through focus. Intensities of 0 and 1 correspond to black and white, respectively. The units of x, y, ρ, and z are in wavelengths.

Fig. 6.
Fig. 6.

Movie (2.1 MB) of an azimuthally polarized beam that was created by using a modified Mach-Zehnder interferometer to convert linear polarization to azimuthal polarization.

Fig. 7.
Fig. 7.

A small scatterer on a metalized region of a silicon integrated circuit taken in dark-field confocal (IPC) mode with azimuthal polarization with a 40x (NA=0.65) objective.

Fig. 8.
Fig. 8.

Normalized intensity of the transverse (y-) component of a high-NA (1.32) linearly polarized beam. Intensities of 0 and 1 correspond to black and white, respectively. The units of x and y are in wavelengths.

Fig. 9.
Fig. 9.

Normalized intensity of the longitudinal (z-) component of a high-NA (1.32) linearly polarized beam. Note that the intensity is normalized from 0 to 0.1 instead of 0 to 1 as in previous figures. The units of x and y are in wavelengths.

Fig. 10.
Fig. 10.

Normalized intensity of the transverse (x-) component of a high-NA (1.32) linearly polarized beam. Note that the intensity is normalized from 0 to 0.01. The units of x and y are in wavelengths.

Equations (14)

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e 0 = l 0 [ e r ( 0 ) g 0 + e ϕ ( 0 ) ( g 0 × k ̂ ) ] ,
e ( s ) = ik 2 π Ω a 1 ( θ , ϕ ) e ik ( s ̂ 1 r ) d Ω
a 1 = f 1 cos 1 2 ( θ ) l 0 ( θ ) [ e r ( 0 ) g 1 + e ϕ ( 0 ) ( g 1 × s 1 ) ] .
g 1 = cos θ ( cos ϕ i ̂ + sin ϕ j ̂ ) + sin θ k ̂ .
e ( s ) = [ e x ( s ) e y ( s ) e z ( s ) ] = iA π 0 α 0 2 π sin θ cos 1 2 θ l 0 ( θ ) e ik ( z S cos θ + ρ S sin θ cos ( ϕ ϕ S ) ) [ cos θ cos ϕ cos θ sin ϕ sin θ ] d ϕ d θ
e ϕ ( s ) = e y ( s ) cos ϕ S e x ( s ) sin ϕ S
e ρ ( s ) = e x ( s ) cos ϕ S e y ( s ) sin ϕ S
e ρ ( s ) = iA π 0 α 0 2 π cos 1 2 θ sin θ cos θ cos ( ϕ ϕ S ) l 0 ( θ ) e ik ( z S cos θ + ρ S sin θ cos ( ϕ ϕ S ) ) d ϕ d θ
e z ( s ) = iA π 0 α 0 2 π cos 1 2 θ sin 2 θ l 0 ( θ ) e ik ( z S cos θ + ρ S sin θ cos ( ϕ ϕ S ) ) d ϕ d θ
e ρ ( s ) ( ρ s , z s ) = A 0 α cos 1 2 θ sin ( 2 θ ) l 0 ( θ ) J 1 ( k ρ s sin θ ) e ikz s cos θ d θ
e z ( s ) ( ρ s , z s ) = 2 i A 0 α cos 1 2 θ sin 2 θ l 0 ( θ ) J 0 ( k ρ s sin θ ) e ikz s cos θ d θ
e ( s ) = [ e x ( s ) e y ( s ) e z ( s ) ] = iA π 0 α 0 2 π sin θ cos 1 2 θ l 0 ( θ ) e ik ( z S cos θ + ρ S sin θ cos ( ϕ ϕ S ) ) [ sin ϕ cos ϕ 0 ] d ϕ d θ
e ϕ ( s ) ( ρ S , z S ) = 2 A 0 α cos 1 2 θ sin θ l 0 ( θ ) J 1 ( k ρ S sin θ ) e ikz S cos θ d θ
l 0 ( θ ) = exp [ β 0 2 ( sin θ sin α ) 2 ] J 1 ( 2 β 0 sin θ sin α )

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