Abstract

Focusing properties of transverse circular polarization modes that bring light to a small focal spot are investigated. Two particular illumination polarization distributions are discussed. Rotating electric dipole polarization results in a central lobe diameter 8% smaller than for the circularly polarized aplanatic case at a NA of 0.95 in air and is also smaller than for radial polarization at NAs less than 0.90. Azimuthal polarization with a phase singularity of charge unity results in a small central lobe width that is smaller than that produced by focusing radially polarized light, having a width that is 17% smaller than for circularly polarized illumination at a NA of 0.95.

© 2011 Optical Society of America

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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. R. Soc. A 253, 358–379 (1959).
    [CrossRef]
  2. C. J. R. Sheppard and P. Török, “Electromagnetic field in the focal region of an electric dipole wave,” Optik 104, 175–177(1997).
  3. 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]
  4. S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
    [CrossRef]
  5. K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7, 77–87(2000).
    [CrossRef] [PubMed]
  6. S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “The focus of light—theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).
    [CrossRef]
  7. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901(2003).
    [CrossRef] [PubMed]
  8. J. J. Stamnes and V. Dhayalan, “Focusing of electric dipole waves,” Pure Appl. Opt. 5, 195–226 (1996).
    [CrossRef]
  9. C. J. R. Sheppard, N. K. Balla, and S. Rehman, “Performance parameters for highly-focused electromagnetic waves,” Opt. Commun. 282, 727–734 (2009).
    [CrossRef]
  10. C. J. R. Sheppard, S. Rehman, N. K. Balla, E. Y. S. Yew, and T. W. Teng, “Bessel beams: effects of polarization,” Opt. Commun. 282, 4647–4656 (2009).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  14. C. J. R. Sheppard and E. Y. S. Yew, “Performance parameters for focussing of radial polarization,” Opt. Lett. 33, 497–499(2008).
    [CrossRef] [PubMed]
  15. M. W. Kowarz, “Energy constraints in optimum apodization problems,” Opt. Commun. 110, 274–278 (1994).
    [CrossRef]
  16. J. J. Stamnes, “Focussing of perfect wave and Airy pattern formula,” Opt. Commun. 37, 311–314 (1981).
    [CrossRef]
  17. C. J. R. Sheppard and M. Gu, “Imaging by a high aperture optical system,” J. Mod. Opt. 40, 1631–1651 (1993).
    [CrossRef]
  18. N. Davidson and N. Bokor, “High-numerical aperture focusing of radially polarized doughnut beams with a parabolic mirror and a flat diffractive lens,” Opt. Lett. 29, 1318–1321 (2004).
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2009

C. J. R. Sheppard, N. K. Balla, and S. Rehman, “Performance parameters for highly-focused electromagnetic waves,” Opt. Commun. 282, 727–734 (2009).
[CrossRef]

C. J. R. Sheppard, S. Rehman, N. K. Balla, E. Y. S. Yew, and T. W. Teng, “Bessel beams: effects of polarization,” Opt. Commun. 282, 4647–4656 (2009).
[CrossRef]

2008

2004

2003

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

2001

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “The focus of light—theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).
[CrossRef]

2000

K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7, 77–87(2000).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

1999

1997

C. J. R. Sheppard and P. Török, “Electromagnetic field in the focal region of an electric dipole wave,” Optik 104, 175–177(1997).

1996

J. J. Stamnes and V. Dhayalan, “Focusing of electric dipole waves,” Pure Appl. Opt. 5, 195–226 (1996).
[CrossRef]

1995

C. Ye, “Construction of an optical rotator using quarter-wave plates and an optical retarder,” Opt. Eng. 34, 3031–3035(1995).
[CrossRef]

1994

M. W. Kowarz, “Energy constraints in optimum apodization problems,” Opt. Commun. 110, 274–278 (1994).
[CrossRef]

1993

C. J. R. Sheppard and M. Gu, “Imaging by a high aperture optical system,” J. Mod. Opt. 40, 1631–1651 (1993).
[CrossRef]

1990

1981

J. J. Stamnes, “Focussing of perfect wave and Airy pattern formula,” Opt. Commun. 37, 311–314 (1981).
[CrossRef]

1959

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]

Balla, N. K.

C. J. R. Sheppard, N. K. Balla, and S. Rehman, “Performance parameters for highly-focused electromagnetic waves,” Opt. Commun. 282, 727–734 (2009).
[CrossRef]

C. J. R. Sheppard, S. Rehman, N. K. Balla, E. Y. S. Yew, and T. W. Teng, “Bessel beams: effects of polarization,” Opt. Commun. 282, 4647–4656 (2009).
[CrossRef]

Bokor, N.

Brown, T.

Davidson, N.

Dhayalan, V.

J. J. Stamnes and V. Dhayalan, “Focusing of electric dipole waves,” Pure Appl. Opt. 5, 195–226 (1996).
[CrossRef]

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]

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “The focus of light—theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “The focus of light—theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

Ford, D. H.

Glockl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “The focus of light—theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

Gu, M.

C. J. R. Sheppard and M. Gu, “Imaging by a high aperture optical system,” J. Mod. Opt. 40, 1631–1651 (1993).
[CrossRef]

Kimura, W. D.

Kowarz, M. W.

M. W. Kowarz, “Energy constraints in optimum apodization problems,” Opt. Commun. 110, 274–278 (1994).
[CrossRef]

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]

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “The focus of light—theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

Martinez-Corral, M.

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]

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “The focus of light—theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

Rehman, S.

C. J. R. Sheppard, N. K. Balla, and S. Rehman, “Performance parameters for highly-focused electromagnetic waves,” Opt. Commun. 282, 727–734 (2009).
[CrossRef]

C. J. R. Sheppard, S. Rehman, N. K. Balla, E. Y. S. Yew, and T. W. Teng, “Bessel beams: effects of polarization,” Opt. Commun. 282, 4647–4656 (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]

Saghafi, S.

Sheppard, C. J. R.

C. J. R. Sheppard, S. Rehman, N. K. Balla, E. Y. S. Yew, and T. W. Teng, “Bessel beams: effects of polarization,” Opt. Commun. 282, 4647–4656 (2009).
[CrossRef]

C. J. R. Sheppard, N. K. Balla, and S. Rehman, “Performance parameters for highly-focused electromagnetic waves,” Opt. Commun. 282, 727–734 (2009).
[CrossRef]

C. J. R. Sheppard and E. Y. S. Yew, “Performance parameters for focussing of radial polarization,” Opt. Lett. 33, 497–499(2008).
[CrossRef] [PubMed]

C. J. R. Sheppard and M. Martinez-Corral, “Filter performance parameters for vectorial high-aperture wave-fields,” Opt. Lett. 33, 476–478 (2008).
[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]

C. J. R. Sheppard and P. Török, “Electromagnetic field in the focal region of an electric dipole wave,” Optik 104, 175–177(1997).

C. J. R. Sheppard and M. Gu, “Imaging by a high aperture optical system,” J. Mod. Opt. 40, 1631–1651 (1993).
[CrossRef]

Stamnes, J. J.

J. J. Stamnes and V. Dhayalan, “Focusing of electric dipole waves,” Pure Appl. Opt. 5, 195–226 (1996).
[CrossRef]

J. J. Stamnes, “Focussing of perfect wave and Airy pattern formula,” Opt. Commun. 37, 311–314 (1981).
[CrossRef]

Teng, T. W.

C. J. R. Sheppard, S. Rehman, N. K. Balla, E. Y. S. Yew, and T. W. Teng, “Bessel beams: effects of polarization,” Opt. Commun. 282, 4647–4656 (2009).
[CrossRef]

Tidwell, S. C.

Török, P.

C. J. R. Sheppard and P. Török, “Electromagnetic field in the focal region of an electric dipole wave,” Optik 104, 175–177(1997).

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]

Ye, C.

C. Ye, “Construction of an optical rotator using quarter-wave plates and an optical retarder,” Opt. Eng. 34, 3031–3035(1995).
[CrossRef]

Yew, E. Y. S.

C. J. R. Sheppard, S. Rehman, N. K. Balla, E. Y. S. Yew, and T. W. Teng, “Bessel beams: effects of polarization,” Opt. Commun. 282, 4647–4656 (2009).
[CrossRef]

C. J. R. Sheppard and E. Y. S. Yew, “Performance parameters for focussing of radial polarization,” Opt. Lett. 33, 497–499(2008).
[CrossRef] [PubMed]

Youngworth, K.

Appl. Opt.

Appl. Phys. B

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “The focus of light—theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).
[CrossRef]

J. Mod. Opt.

C. J. R. Sheppard and M. Gu, “Imaging by a high aperture optical system,” J. Mod. Opt. 40, 1631–1651 (1993).
[CrossRef]

Opt. Commun.

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

M. W. Kowarz, “Energy constraints in optimum apodization problems,” Opt. Commun. 110, 274–278 (1994).
[CrossRef]

J. J. Stamnes, “Focussing of perfect wave and Airy pattern formula,” Opt. Commun. 37, 311–314 (1981).
[CrossRef]

C. J. R. Sheppard, N. K. Balla, and S. Rehman, “Performance parameters for highly-focused electromagnetic waves,” Opt. Commun. 282, 727–734 (2009).
[CrossRef]

C. J. R. Sheppard, S. Rehman, N. K. Balla, E. Y. S. Yew, and T. W. Teng, “Bessel beams: effects of polarization,” Opt. Commun. 282, 4647–4656 (2009).
[CrossRef]

Opt. Eng.

C. Ye, “Construction of an optical rotator using quarter-wave plates and an optical retarder,” Opt. Eng. 34, 3031–3035(1995).
[CrossRef]

Opt. Express

Opt. Lett.

Optik

C. J. R. Sheppard and P. Török, “Electromagnetic field in the focal region of an electric dipole wave,” Optik 104, 175–177(1997).

Phys. Rev. Lett.

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. A

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]

Pure Appl. Opt.

J. J. Stamnes and V. Dhayalan, “Focusing of electric dipole waves,” Pure Appl. Opt. 5, 195–226 (1996).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Illustrating the electric field in the front focal plane of the lens, for (left) rotating ED polarization, and (right) azimuthal polarization with a phase vortex (TE1). The outer circle corresponds to a NA of unity. The dashed lines represent the field with a phase of 90 ° .

Fig. 2
Fig. 2

Normalized intensity at the focal point for a given input power. ED: rotating electric dipole field. UTE1: uniform azimuthal polarization with a phase vortex (TE1). R: radially polarized input (TM0). AP: aplanatic circularly polarized. ED is greatest for NA < 1 .

Fig. 3
Fig. 3

Normalized intensity at the focal point for a given integrated intensity in the focal plane. ED: rotating electric dipole field. PED: perfect rotating electric dipole field. UTE1: uniform azimuthal polarization with a phase vortex (TE1). R: radially polarized input (TM0). PR: perfect radially polarized input (TM0). AP: aplanatic circularly polarized. PED is greatest.

Fig. 4
Fig. 4

Normalized width of the parabolic central lobe. The values are normalized to that of a Bessel beam (annular pupil) for TE1 or R at NA = 1 . UTE1 is narrowest for NA > 0.98 . R is large for a small NA because of the radially polarized component.

Fig. 5
Fig. 5

Electric energy density at the focus of an optical system (NA of 0.95 in air) for radial, aplanatic, TE1: azimuthal and ED: rotating electric dipole polarizations in the transverse direction.

Fig. 6
Fig. 6

Comparison of electric energy density at the focus of an optical system (NA of 0.95 in air) with polarization modes such as radial, aplanatic, TE1: azimuthal and ED: rotating electric dipole, plotted with dimensionless optical units u and v.

Equations (20)

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E = Q TE [ ( i + i j ) ( i i j ) e 2 i ϕ ] = 2 i e i ϕ Q TE a ϕ ,
E = 2 cos θ Q TE i e i ϕ ( sin ϕ i cos ϕ j ) = 2 cos θ Q TE i e i ϕ a ϕ .
E = Q p ( cos θ e i ϕ a θ i e i ϕ a ϕ ) ,
E = 1 cos θ Q p i e i ϕ [ ( sin ϕ + i cos θ cos ϕ ) i ( cos ϕ i cos θ sin ϕ ) j ] = 1 cos θ Q p i e i ϕ ( a ϕ i cos θ a ρ ) ,
E = i k f [ I 0 ( i + i j ) + I 2 e 2 i ϕ ( i i j ) 2 i I 1 e i ϕ k ] ,
I 0 = 1 1 Q p ( c ) ( 1 + c 2 ) J 0 ( k ρ 1 c 2 ) exp ( i k z c ) d c , I 1 = 1 1 Q p ( c ) c 1 c 2 J 1 ( k ρ 1 c 2 ) exp ( i k z c ) d c , I 2 = 1 1 Q p ( c ) ( 1 c 2 ) J 2 ( k ρ 1 c 2 ) exp ( i k z c ) d c ,
I 0 = 1 1 Q TE ( c ) J 0 ( k ρ 1 c 2 ) exp ( i k z c ) d c , I 1 = 0 , I 2 = 1 1 Q TE ( c ) J 2 ( k ρ 1 c 2 ) exp ( i k z c ) d c .
q p ( n ) = 1 1 Q p ( c ) c n d c , q TE ( n ) = 1 1 Q TE ( c ) c n d c ,
I 0 = ( q p ( 0 ) + q p ( 2 ) ) ( k ρ ) 2 4 ( q p ( 0 ) q p ( 4 ) ) + i k z ( q p ( 1 ) + q p ( 3 ) ) 1 2 ( k z ) 2 ( q p ( 2 ) + q p ( 4 ) ) , I 1 = ( k ρ ) 2 2 ( q p ( 1 ) q p ( 3 ) ) + i ( k z ) ( k ρ ) 2 ( q p ( 2 ) q p ( 4 ) ) , I 2 = ( k ρ ) 2 8 ( q p ( 0 ) 2 q p ( 2 ) + q p ( 4 ) ) ,
I 0 = q TE ( 0 ) ( k ρ ) 2 4 ( q TE ( 0 ) q TE ( 2 ) ) + i k z q TE ( 1 ) 1 2 ( k z ) 2 q TE ( 2 ) , I 1 = 0 , I 2 = ( k ρ ) 2 8 ( q TE ( 0 ) q TE ( 2 ) ) .
W 0 = ( q p ( 0 ) + q p ( 2 ) ) 2 = ( q TE ( 0 ) ) 2 .
E = 1 2 1 1 Q p 2 ( c ) ( 1 + c 2 ) d c , = 1 2 1 1 Q TE 2 ( c ) d c .
F = 3 W 0 16 E ,
W Int = 1 2 1 1 Q p 2 ( c ) ( 1 + c 2 ) c d c , = 1 2 1 1 Q TE 2 ( c ) 1 c d c .
F I = 3 W 0 16 W Int ,
W = W 0 [ 1 G T ( k ρ ) 2 3 G A ( k z ) 2 3 ] ,
G P = ( 2 G T + G A ) / 3.
G T = 3 2 ( q p ( 0 ) + q p ( 2 ) ) ( q p ( 0 ) q p ( 4 ) ) ( q p ( 1 ) q p ( 3 ) ) 2 ( q p ( 0 ) + q p ( 2 ) ) 2 , G A = 3 ( q p ( 0 ) + q p ( 2 ) ) ( q p ( 2 ) + q p ( 4 ) ) ( q p ( 1 ) + q p ( 3 ) ) 2 ( q p ( 0 ) + q p ( 2 ) ) 2 , G P = ( q p ( 0 ) + q p ( 2 ) ) 2 + 2 ( q p ( 1 ) ) 2 2 ( q p ( 3 ) ) 2 ( q p ( 0 ) + q p ( 2 ) + 2 q m ( 1 ) ) 2 ,
G T = 3 4 ( q TE ( 0 ) q TE ( 2 ) ) q TE ( 0 ) , G A = 3 q TE ( 0 ) q TE ( 2 ) ( q TE ( 1 ) ) 2 ( q TE ( 0 ) ) 2 , G P = ( q TE ( 0 ) ) 2 ( q TE ( 1 ) ) 2 ( q TE ( 0 ) ) 2 ,
FWHM = ( 3 λ 2 2 π G T ) 1 / 2 ,

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