Abstract

We present a detailed analysis of the structure of strongly focused, radially polarized electromagnetic fields. The existence of phase singularities of the two components of the electric field is demonstrated. Two different mechanisms to obtain creation or annihilation of these phase singularities are discussed. These are changing the aperture angle of the lens and the width of the beam. Also, it is shown that in the focal plane the handedness of the electric polarization ellipse is an alternating function of the radial distance. Finally, the different contributions to the electric energy density are examined.

© 2006 Optical Society of America

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References

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  1. K.S. Youngworth and T.G. Brown, "Inhomogeneous polarization in scanning optical microscopy," Proceedings of SPIE 3919, 75-85 (2000).
    [CrossRef]
  2. 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/abstract.cfm?URI=OPEX-7-2-77
    [CrossRef] [PubMed]
  3. 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]
  4. C.J.R. Sheppard and A. Choudhury, "Annular pupils, radial polarization, and superresolution," Appl. Opt. 43, 4322-4327 (2004).
    [CrossRef] [PubMed]
  5. Q. Zhan, "Trapping metallic Rayleigh particles with radial polarization," Opt. Express 12, 3377-3382 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-15-3377
    [CrossRef] [PubMed]
  6. S. Quabis, R. Dorn, M. Eberler, O. Glö ckl and G. Leuchs, "Focusing light to a tighter spot," Opt. Commun. 179, 1-7 (2000).
    [CrossRef]
  7. S. Quabis, R. Dorn, M. Eberler, O. Glö ckl and G. Leuchs, "The focus of light - theoretical calculation and experimental tomographic reconstruction," Appl. Phys. B 72, 109-113 (2001).
    [CrossRef]
  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. J.T. Foley and E. Wolf, "Wave-front spacing in the focal region of high-numerical-aperture systems," Opt. Lett. 30, 1312-1314 (2005).
    [CrossRef] [PubMed]
  10. T.D. Visser and J.T. Foley, "On the wavefront spacing of focused, radially polarized beams," J. Opt. Soc. Am. A 22, 2527-2531 (2005).
    [CrossRef]
  11. D.W. Diehl and T.D. Visser, "Phase singularities of the longitudinal field components in the focal region of a high-aperture optical system," J. Opt. Soc. Am. A 21, 2103-2108 (2004).
    [CrossRef]
  12. B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system," Proc. Royal Soc. A 253,358-379 (1959).
    [CrossRef]
  13. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, seventh (expanded) ed. (Cambridge University Press, Cambridge, 1999). See especially Sec. 4.5.1.
  14. J.F. Nye, Natural Focusing and Fine Structure of Light (Institue of Phyisics Publishing, Bristol, 1999). See especially Chapter 12.
  15. G.P. Karman, A. van Duijl and J.P. Woerdman, "Unfolding of an unstable singularity point into a ring," Opt. Lett. 23, 403-405 (1998).
    [CrossRef]

2005 (2)

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]

2001 (2)

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]

S. Quabis, R. Dorn, M. Eberler, O. Glö ckl and G. Leuchs, "The focus of light - theoretical calculation and experimental tomographic reconstruction," Appl. Phys. B 72, 109-113 (2001).
[CrossRef]

2000 (3)

K.S. Youngworth and T.G. Brown, "Inhomogeneous polarization in scanning optical microscopy," Proceedings of SPIE 3919, 75-85 (2000).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glö ckl and G. Leuchs, "Focusing light to a tighter spot," Opt. Commun. 179, 1-7 (2000).
[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/abstract.cfm?URI=OPEX-7-2-77
[CrossRef] [PubMed]

1998 (1)

1959 (1)

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

Beversluis, M.R.

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]

Brown, T.G.

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]

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/abstract.cfm?URI=OPEX-7-2-77
[CrossRef] [PubMed]

K.S. Youngworth and T.G. Brown, "Inhomogeneous polarization in scanning optical microscopy," Proceedings of SPIE 3919, 75-85 (2000).
[CrossRef]

Choudhury, A.

Diehl, D.W.

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. Glö ckl 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. Glö ckl 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. Glö ckl 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. Glö ckl and G. Leuchs, "Focusing light to a tighter spot," Opt. Commun. 179, 1-7 (2000).
[CrossRef]

Foley, J.T.

Karman, G.P.

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]

Novotny, L.

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]

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. Glö ckl 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. Glö ckl and G. Leuchs, "Focusing light to a tighter spot," Opt. Commun. 179, 1-7 (2000).
[CrossRef]

Richards, B.

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

Sheppard, C.J.R.

van Duijl, A.

Visser, T.D.

Woerdman, J.P.

Wolf, E.

J.T. Foley and E. Wolf, "Wave-front spacing in the focal region of high-numerical-aperture systems," Opt. Lett. 30, 1312-1314 (2005).
[CrossRef] [PubMed]

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

Youngworth, K.S.

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]

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/abstract.cfm?URI=OPEX-7-2-77
[CrossRef] [PubMed]

K.S. Youngworth and T.G. Brown, "Inhomogeneous polarization in scanning optical microscopy," Proceedings of SPIE 3919, 75-85 (2000).
[CrossRef]

Zhan, Q.

Appl. Opt. (1)

Appl. Phys. B (1)

S. Quabis, R. Dorn, M. Eberler, O. Glö ckl and G. Leuchs, "The focus of light - theoretical calculation and experimental tomographic reconstruction," Appl. Phys. B 72, 109-113 (2001).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

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

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. Lett. (2)

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]

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

Proc. Royal Soc. A (1)

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

Proceedings of SPIE (1)

K.S. Youngworth and T.G. Brown, "Inhomogeneous polarization in scanning optical microscopy," Proceedings of SPIE 3919, 75-85 (2000).
[CrossRef]

Other (2)

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, seventh (expanded) ed. (Cambridge University Press, Cambridge, 1999). See especially Sec. 4.5.1.

J.F. Nye, Natural Focusing and Fine Structure of Light (Institue of Phyisics Publishing, Bristol, 1999). See especially Chapter 12.

Supplementary Material (3)

» Media 1: MOV (2153 KB)     
» Media 2: MOV (476 KB)     
» Media 3: MOV (1480 KB)     

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Figures (8)

Fig. 1.
Fig. 1.

Illustration of a high numerical aperture focusing system. The radially polarized incident beam propagates along the z-axis.

Fig. 2.
Fig. 2.

Contours of |eρ (u,v)|2 (a) and |ez (u,v)|2 (b). In this example the semi-aperture angle α= π/6, and the beam parameter β = 0.6.

Fig. 3.
Fig. 3.

Contours of |eρ (u,ν)|2 (a) and |ez (u,ν)|2 (b). In this example the semi-aperture angle α= π/3, and the beam parameter β = 0.6.

Fig. 4.
Fig. 4.

(a) Plot of the normalized electric field components eρ (0,ν)/Im[ez (0,0)] (dashed curve) and Im[ez (0,ν)]/Im[ez (0,0)] (solid curve). (b) Plot of R(ν), the ratio of the lengths of the two conjugate semi-axes of the electric polarization ellipse. The different intersections with the horizontal dashed line (at which R(ν) = 1) are points at which the polarization is circular. In this example the semi-aperture angle α=π/4, and the beam parameter β= 0.6.

Fig. 5.
Fig. 5.

The electric polarization ellipse in the focal plane for selected values of the radial distance (ν = 0.00,0.71,1.42,3.05,4.00,5.00). The arrow indicates the direction in which the ellipse is being traversed. In this example the semi-aperture angle α = π/4, and the beam parameter β = 0.6.

Fig. 6.
Fig. 6.

Color-coded plot of the phase of the longitudinal electric field component ez for different values of the semi-aperture angle α. In this example β = 0.6. [Media 1]

Fig. 7.
Fig. 7.

Color-coded plot of the phase of the longitudinal electric field component ez for different values of the beam-size parameter β. In this example α= π/3. [Media 2]

Fig. 8.
Fig. 8.

Color-coded plot of the phase of the radial electric field component eρ for different values of the beam-size parameter β. When β is decreased, an Airy ring-like singularity is created. In this example α= π/4. [Media 3]

Equations (18)

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E ( r , t ) = Re [ e ( r ) exp ( i ω t ) ] ,
H ( r , t ) = Re [ h ( r ) exp ( i ω t ) ] ,
e z ( ρ P , z P ) = i k f 0 α l ( θ ) sin 2 θ cos 1 / 2 θ
× exp ( i k z p cos θ ) J 0 ( k ρ p sin θ ) d θ ,
e ρ ( ρ P , z P ) = k f 0 α l ( θ ) sin θ cos 3 / 2 θ
× exp ( i k z P cos θ ) J 1 ( k ρ P sin θ ) d θ ,
l ( θ ) = f sin θ exp [ f 2 sin 2 θ / w 0 2 ] ,
u = k z P sin 2 α ,
ν = k ρ P sin α ,
e z ( u , ν ) = i k f 2 0 α sin 3 θ cos 1 / 2 θ exp ( β 2 sin 2 θ )
× exp ( i u cos θ / sin 2 α ) J 0 ( ν sin θ sin α ) ,
e ρ ( u , ν ) = k f 2 0 α sin 2 θ cos 3 / 2 θ exp ( β 2 sin 2 θ )
× exp ( i u cos θ / sin 2 α ) J 1 ( ν sin θ sin α ) d θ ,
e z ( u , ν ) = e z * ( u , ν ) ,
e ρ ( u , ν ) = e ρ * ( u , ν ) ,
w e = ε 0 2 E 2 ( u , ν ) = ε 0 4 [ e ρ ( u , ν ) 2 + e z ( u , ν ) 2 ] ,
e ( 0 , ν ) = e ρ ( 0 , ν ) ρ ̂ + i Im [ e z ( 0 , ν ) ] z ̂ ,
R ( ν ) = Im [ e z ( 0 , ν ) ] e ρ ( 0 , ν ) .

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