Abstract

The state of polarization of strongly focused, radially polarized electromagnetic fields is examined. It is found that several types of polarization singularities exist. Their relationship is investigated, and it is demonstrated that on smoothly varying a system parameter, such as the aperture angle of the lens, different polarization singularities can annihilate each other. For example, the evolution of a lemon into a monstar and its subsequent annihilation with a star is studied. Also, the quite rare collision of a C-line and an L-line, resulting in a V-point, is observed.

© 2006 Optical Society of America

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References

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  1. J.F. Nye and M.V. Berry, "Dislocations in wave trains," Proc. R. Soc. Lond. A 336, 165-190 (1974).
    [CrossRef]
  2. J.F. Nye, Natural Focusing and Fine Structure of Light (IOP Publishing, Bristol, 1999).
  3. M.S. Soskin and M.V. Vasnetsov, in Progress in Optics, edited by E. Wolf (Elsevier, Amsterdam, 2001), Vol. 42.
    [CrossRef]
  4. G.P. Karman, M.W. Beijersbergen, A. van Duijl, and J.P. Woerdman, "Creation and annihilation of phase singularities in a focal field," Opt. Lett. 22, 1503-1505 (1997).
    [CrossRef]
  5. H.F. Schouten, T.D. Visser, D. Lenstra and H. Blok, "Light transmission through a sub-wavelength slit: waveguiding and optical vortices," Phys. Rev. E 67, 036608 (2003).
    [CrossRef]
  6. H.F. Schouten, G. Gbur, T.D. Visser, D. Lenstra and H. Blok, "Creation and annihilation of phase singularities near a sub-wavelength slit," Opt. Express 11, 371-380 (2003)
    [CrossRef] [PubMed]
  7. H.F. Schouten, T.D. Visser, G. Gbur, D. Lenstra and H. Blok, "Diffraction of light by narrow slits in plates of different materials," J. Opt. A 6, S277-S280 (2004).
    [CrossRef]
  8. D.W. Diehl and T.D. Visser, "Phase singularities of the longitudinal field components in high-aperture systems," J. Opt. Soc. Am. A 21, 2103-2108 (2004).
    [CrossRef]
  9. H.F. Schouten, G. Gbur, T.D. Visser and E. Wolf, "Phase singularities of the coherence functions in Young’s interference pattern," Opt. Lett. 28, 968-970 (2003).
    [CrossRef] [PubMed]
  10. G. Gbur and T.D. Visser, "Coherence vortices in partially coherent beams," Opt. Commun. 222, 117-125 (2003).
    [CrossRef]
  11. D.G. Fischer and T.D. Visser, "Spatial correlation properties of partially coherent focused fields," J. Opt. Soc. Am. A 21, 2097-2102 (2004).
    [CrossRef]
  12. G. Gbur and T.D. Visser, "Phase singularities and coherence vortices in linear optical systems," Opt. Commun. 259, 428-435 (2005).
    [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).
    [PubMed]
  14. M.V. Berry and M.R. Dennis, "Polarization singularities in isotropic random vector waves," Proc. R. Soc. Lond. A 457, pp. 141-155 (2001).
    [CrossRef]
  15. M.R. Dennis, "Polarization singularities in paraxial vector fields: morphology and statistics," Opt. Commun. 213, pp. 201-221 (2002).
    [CrossRef]
  16. I. Freund, A.I. Mokhun, M.S. Soskin, O.V. Angelsky, and I.I. Mokhun, "Stokes singularity relations," Opt. Lett. 27, pp. 545-547 (2002).
    [CrossRef]
  17. M.S. Soskin, V. Denisenko, and I. Freund, "Optical polarization singularities and elliptic stationary points," Opt. Lett. 28, pp. 1475-1477 (2003).
    [CrossRef] [PubMed]
  18. A.I. Mokhun, M.S. Soskin, and I. Freund, "Elliptic critical points: C -points, a -lines, and the sign rule," Opt. Lett. 27, pp. 995-997 (2002).
    [CrossRef]
  19. T.D. Visser and J.T. Foley, "On the wavefront-spacing of focused, radially polarized beams," J. Opt. Soc. Am. A 22, pp. 2527-2531 (2005).
    [CrossRef]
  20. D.W. Diehl, R.W. Schoonover and T.D. Visser, "The structure of focused, radially polarized fields," Opt. Express 14, pp. 3030-3038 (2006),
    [CrossRef] [PubMed]
  21. 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, pp. 358-379 (1959).
    [CrossRef]
  22. J.D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), Sec. 7.2.
  23. V.G. Denisenko, R.I. Egorov and M.S. Soskin, "Measurement of the morphological forms of polarization singularities and their statistical weight in optical vector fields," JETP Lett. 80, pp. 17-19 (2004).
    [CrossRef]
  24. I. Freund, "Polarization singularity indices in Gaussian laser beams," Opt. Commun. 201, pp. 251-270 (2002).
    [CrossRef]
  25. E.T. Copson, An Introduction to the Theory of Functions of a Complex Variable (Oxford University Press, London, 1935). See especially Sec. 4.51.

2006 (1)

2005 (2)

G. Gbur and T.D. Visser, "Phase singularities and coherence vortices in linear optical systems," Opt. Commun. 259, 428-435 (2005).
[CrossRef]

T.D. Visser and J.T. Foley, "On the wavefront-spacing of focused, radially polarized beams," J. Opt. Soc. Am. A 22, pp. 2527-2531 (2005).
[CrossRef]

2004 (4)

D.G. Fischer and T.D. Visser, "Spatial correlation properties of partially coherent focused fields," J. Opt. Soc. Am. A 21, 2097-2102 (2004).
[CrossRef]

H.F. Schouten, T.D. Visser, G. Gbur, D. Lenstra and H. Blok, "Diffraction of light by narrow slits in plates of different materials," J. Opt. A 6, S277-S280 (2004).
[CrossRef]

D.W. Diehl and T.D. Visser, "Phase singularities of the longitudinal field components in high-aperture systems," J. Opt. Soc. Am. A 21, 2103-2108 (2004).
[CrossRef]

V.G. Denisenko, R.I. Egorov and M.S. Soskin, "Measurement of the morphological forms of polarization singularities and their statistical weight in optical vector fields," JETP Lett. 80, pp. 17-19 (2004).
[CrossRef]

2003 (5)

2002 (4)

A.I. Mokhun, M.S. Soskin, and I. Freund, "Elliptic critical points: C -points, a -lines, and the sign rule," Opt. Lett. 27, pp. 995-997 (2002).
[CrossRef]

M.R. Dennis, "Polarization singularities in paraxial vector fields: morphology and statistics," Opt. Commun. 213, pp. 201-221 (2002).
[CrossRef]

I. Freund, A.I. Mokhun, M.S. Soskin, O.V. Angelsky, and I.I. Mokhun, "Stokes singularity relations," Opt. Lett. 27, pp. 545-547 (2002).
[CrossRef]

I. Freund, "Polarization singularity indices in Gaussian laser beams," Opt. Commun. 201, pp. 251-270 (2002).
[CrossRef]

2001 (1)

M.V. Berry and M.R. Dennis, "Polarization singularities in isotropic random vector waves," Proc. R. Soc. Lond. A 457, pp. 141-155 (2001).
[CrossRef]

1997 (1)

1974 (1)

J.F. Nye and M.V. Berry, "Dislocations in wave trains," Proc. R. Soc. Lond. A 336, 165-190 (1974).
[CrossRef]

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, pp. 358-379 (1959).
[CrossRef]

Angelsky, O.V.

Beijersbergen, M.W.

Berry, M.V.

M.V. Berry and M.R. Dennis, "Polarization singularities in isotropic random vector waves," Proc. R. Soc. Lond. A 457, pp. 141-155 (2001).
[CrossRef]

J.F. Nye and M.V. Berry, "Dislocations in wave trains," Proc. R. Soc. Lond. A 336, 165-190 (1974).
[CrossRef]

Blok, H.

H.F. Schouten, T.D. Visser, G. Gbur, D. Lenstra and H. Blok, "Diffraction of light by narrow slits in plates of different materials," J. Opt. A 6, S277-S280 (2004).
[CrossRef]

H.F. Schouten, T.D. Visser, D. Lenstra and H. Blok, "Light transmission through a sub-wavelength slit: waveguiding and optical vortices," Phys. Rev. E 67, 036608 (2003).
[CrossRef]

H.F. Schouten, G. Gbur, T.D. Visser, D. Lenstra and H. Blok, "Creation and annihilation of phase singularities near a sub-wavelength slit," Opt. Express 11, 371-380 (2003)
[CrossRef] [PubMed]

Denisenko, V.

Denisenko, V.G.

V.G. Denisenko, R.I. Egorov and M.S. Soskin, "Measurement of the morphological forms of polarization singularities and their statistical weight in optical vector fields," JETP Lett. 80, pp. 17-19 (2004).
[CrossRef]

Dennis, M.R.

M.R. Dennis, "Polarization singularities in paraxial vector fields: morphology and statistics," Opt. Commun. 213, pp. 201-221 (2002).
[CrossRef]

M.V. Berry and M.R. Dennis, "Polarization singularities in isotropic random vector waves," Proc. R. Soc. Lond. A 457, pp. 141-155 (2001).
[CrossRef]

Diehl, D.W.

Egorov, R.I.

V.G. Denisenko, R.I. Egorov and M.S. Soskin, "Measurement of the morphological forms of polarization singularities and their statistical weight in optical vector fields," JETP Lett. 80, pp. 17-19 (2004).
[CrossRef]

Fischer, D.G.

Foley, J.T.

Freund, I.

Gbur, G.

G. Gbur and T.D. Visser, "Phase singularities and coherence vortices in linear optical systems," Opt. Commun. 259, 428-435 (2005).
[CrossRef]

H.F. Schouten, T.D. Visser, G. Gbur, D. Lenstra and H. Blok, "Diffraction of light by narrow slits in plates of different materials," J. Opt. A 6, S277-S280 (2004).
[CrossRef]

G. Gbur and T.D. Visser, "Coherence vortices in partially coherent beams," Opt. Commun. 222, 117-125 (2003).
[CrossRef]

H.F. Schouten, G. Gbur, T.D. Visser, D. Lenstra and H. Blok, "Creation and annihilation of phase singularities near a sub-wavelength slit," Opt. Express 11, 371-380 (2003)
[CrossRef] [PubMed]

H.F. Schouten, G. Gbur, T.D. Visser and E. Wolf, "Phase singularities of the coherence functions in Young’s interference pattern," Opt. Lett. 28, 968-970 (2003).
[CrossRef] [PubMed]

Karman, G.P.

Lenstra, D.

H.F. Schouten, T.D. Visser, G. Gbur, D. Lenstra and H. Blok, "Diffraction of light by narrow slits in plates of different materials," J. Opt. A 6, S277-S280 (2004).
[CrossRef]

H.F. Schouten, T.D. Visser, D. Lenstra and H. Blok, "Light transmission through a sub-wavelength slit: waveguiding and optical vortices," Phys. Rev. E 67, 036608 (2003).
[CrossRef]

H.F. Schouten, G. Gbur, T.D. Visser, D. Lenstra and H. Blok, "Creation and annihilation of phase singularities near a sub-wavelength slit," Opt. Express 11, 371-380 (2003)
[CrossRef] [PubMed]

Mokhun, A.I.

Mokhun, I.I.

Nye, J.F.

J.F. Nye and M.V. Berry, "Dislocations in wave trains," Proc. R. Soc. Lond. A 336, 165-190 (1974).
[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, pp. 358-379 (1959).
[CrossRef]

Schoonover, R.W.

Schouten, H.F.

H.F. Schouten, T.D. Visser, G. Gbur, D. Lenstra and H. Blok, "Diffraction of light by narrow slits in plates of different materials," J. Opt. A 6, S277-S280 (2004).
[CrossRef]

H.F. Schouten, T.D. Visser, D. Lenstra and H. Blok, "Light transmission through a sub-wavelength slit: waveguiding and optical vortices," Phys. Rev. E 67, 036608 (2003).
[CrossRef]

H.F. Schouten, G. Gbur, T.D. Visser, D. Lenstra and H. Blok, "Creation and annihilation of phase singularities near a sub-wavelength slit," Opt. Express 11, 371-380 (2003)
[CrossRef] [PubMed]

H.F. Schouten, G. Gbur, T.D. Visser and E. Wolf, "Phase singularities of the coherence functions in Young’s interference pattern," Opt. Lett. 28, 968-970 (2003).
[CrossRef] [PubMed]

Soskin, M.S.

van Duijl, A.

Visser, T.D.

D.W. Diehl, R.W. Schoonover and T.D. Visser, "The structure of focused, radially polarized fields," Opt. Express 14, pp. 3030-3038 (2006),
[CrossRef] [PubMed]

T.D. Visser and J.T. Foley, "On the wavefront-spacing of focused, radially polarized beams," J. Opt. Soc. Am. A 22, pp. 2527-2531 (2005).
[CrossRef]

G. Gbur and T.D. Visser, "Phase singularities and coherence vortices in linear optical systems," Opt. Commun. 259, 428-435 (2005).
[CrossRef]

H.F. Schouten, T.D. Visser, G. Gbur, D. Lenstra and H. Blok, "Diffraction of light by narrow slits in plates of different materials," J. Opt. A 6, S277-S280 (2004).
[CrossRef]

D.G. Fischer and T.D. Visser, "Spatial correlation properties of partially coherent focused fields," J. Opt. Soc. Am. A 21, 2097-2102 (2004).
[CrossRef]

D.W. Diehl and T.D. Visser, "Phase singularities of the longitudinal field components in high-aperture systems," J. Opt. Soc. Am. A 21, 2103-2108 (2004).
[CrossRef]

H.F. Schouten, G. Gbur, T.D. Visser and E. Wolf, "Phase singularities of the coherence functions in Young’s interference pattern," Opt. Lett. 28, 968-970 (2003).
[CrossRef] [PubMed]

H.F. Schouten, T.D. Visser, D. Lenstra and H. Blok, "Light transmission through a sub-wavelength slit: waveguiding and optical vortices," Phys. Rev. E 67, 036608 (2003).
[CrossRef]

G. Gbur and T.D. Visser, "Coherence vortices in partially coherent beams," Opt. Commun. 222, 117-125 (2003).
[CrossRef]

H.F. Schouten, G. Gbur, T.D. Visser, D. Lenstra and H. Blok, "Creation and annihilation of phase singularities near a sub-wavelength slit," Opt. Express 11, 371-380 (2003)
[CrossRef] [PubMed]

Woerdman, J.P.

Wolf, E.

H.F. Schouten, G. Gbur, T.D. Visser and E. Wolf, "Phase singularities of the coherence functions in Young’s interference pattern," Opt. Lett. 28, 968-970 (2003).
[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, pp. 358-379 (1959).
[CrossRef]

J. Opt. A (1)

H.F. Schouten, T.D. Visser, G. Gbur, D. Lenstra and H. Blok, "Diffraction of light by narrow slits in plates of different materials," J. Opt. A 6, S277-S280 (2004).
[CrossRef]

J. Opt. Soc. Am. A (3)

JETP Lett. (1)

V.G. Denisenko, R.I. Egorov and M.S. Soskin, "Measurement of the morphological forms of polarization singularities and their statistical weight in optical vector fields," JETP Lett. 80, pp. 17-19 (2004).
[CrossRef]

Opt. Commun. (4)

I. Freund, "Polarization singularity indices in Gaussian laser beams," Opt. Commun. 201, pp. 251-270 (2002).
[CrossRef]

G. Gbur and T.D. Visser, "Phase singularities and coherence vortices in linear optical systems," Opt. Commun. 259, 428-435 (2005).
[CrossRef]

M.R. Dennis, "Polarization singularities in paraxial vector fields: morphology and statistics," Opt. Commun. 213, pp. 201-221 (2002).
[CrossRef]

G. Gbur and T.D. Visser, "Coherence vortices in partially coherent beams," Opt. Commun. 222, 117-125 (2003).
[CrossRef]

Opt. Express (2)

Opt. Lett. (5)

Phys. Rev. E (1)

H.F. Schouten, T.D. Visser, D. Lenstra and H. Blok, "Light transmission through a sub-wavelength slit: waveguiding and optical vortices," Phys. Rev. E 67, 036608 (2003).
[CrossRef]

Proc. R. Soc. Lond. A (2)

J.F. Nye and M.V. Berry, "Dislocations in wave trains," Proc. R. Soc. Lond. A 336, 165-190 (1974).
[CrossRef]

M.V. Berry and M.R. Dennis, "Polarization singularities in isotropic random vector waves," Proc. R. Soc. Lond. A 457, pp. 141-155 (2001).
[CrossRef]

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, pp. 358-379 (1959).
[CrossRef]

Other (5)

J.D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), Sec. 7.2.

J.F. Nye, Natural Focusing and Fine Structure of Light (IOP Publishing, Bristol, 1999).

M.S. Soskin and M.V. Vasnetsov, in Progress in Optics, edited by E. Wolf (Elsevier, Amsterdam, 2001), Vol. 42.
[CrossRef]

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).
[PubMed]

E.T. Copson, An Introduction to the Theory of Functions of a Complex Variable (Oxford University Press, London, 1935). See especially Sec. 4.51.

Supplementary Material (4)

» Media 1: MOV (2144 KB)     
» Media 2: MOV (181 KB)     
» Media 3: MOV (1265 KB)     
» Media 4: MOV (990 KB)     

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Figures (10)

Fig. 1.
Fig. 1.

Illustration of a high numerical aperture system focusing a radially polarized beam.

Fig. 2.
Fig. 2.

The Poincar é sphere with cartesian axes (s 1, s 2, s 3).

Fig. 3.
Fig. 3.

Illustrating the symmetry properties of the polarization ellipse.

Fig. 4.
Fig. 4.

The normalized Stokes parameters s 1 and s 3 in the focal plane for increasing values of the dimensionless radial distance v. In this example α=π/4 and β=1.5. [Media 1]

Fig. 5.
Fig. 5.

The loci of linear polarization, i.e. contours of s 3=0, in the focal region. Because of the rotational symmetry of the field, these are tori centered on the u-axis. The contours are superposed on a color-coded phase map of ez . Phase singularities of ez are located at points where all different colors converge. The open circles indicate phase singularities of the other field component, eρ . In this example α=π/4 and β=0.5.

Fig. 6.
Fig. 6.

A color-coded phase map of (a) e+ and (b) e-, both with lines of linear polarization (solid black curves) superposed. Left-handedC-points are phase singularities of e+ in panel (a), whereas right-handed C-points are singularities of e- in panel (b). In this example β=0.5 and α=π/4.

Fig. 7.
Fig. 7.

Local orientation of the major axis of the electric polarization ellipse shown for three different values of the semi-aperture angle α with the beam parameter β kept fixed at 0.5. The local straight-line orientations of the major axes are shown in red to aid the eye. In panel a (α=52°), a star (above) and a lemon (below) are seen. In panel b (α=61°), the lemon has transitioned into a monstar. In panel c (α=65°), the annihilation leaves only a straight-line orientation of the major axis.

Fig. 8.
Fig. 8.

A color-coded phase map of the left-handed component e- with L-lines, shown in black, superposed. The semi-aperture angle α is increased from 35° to 65.5°, with β kept fixed at 0.5. [Media 2]

Fig. 9.
Fig. 9.

The real and imaginary parts of ez along the optical axis normalized to Im[ez (0,0)] as α ranges from 55.0° to 58.0°, with β kept fixed at 1.5. [Media 3]

Fig. 10.
Fig. 10.

A color-coded phase map of e+ with L-lines (solid black curves) superposed as the semi-aperture angle α ranges from 50.0° to 66.0° and with β kept fixed at 1.5. [Media 4]

Tables (2)

Tables Icon

Table 1. The behavior of various quantities that characterize the state of polarization under reflection of the point of observation in the focal plane.

Tables Icon

Table 2. Comparison of the approximate location of the C-surface to its actual location. In these examples β=0.5.

Equations (35)

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

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 ,
v = k ρ P sin α ,
e z ( u , v ) = 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 ( v sin θ sin α ) d θ ,
e ρ ( u , v ) = 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 ( v sin θ sin α ) d θ ,
e ( u , v ) = e z ( u , v ) z ̂ + e ρ ( u , v ) ρ ̂ ,
a 1 = e z ( u , v ) ,
δ 1 = arg [ e z ( u , v ) ] ,
a 2 = e ρ ( u , v ) ,
δ 2 = arg [ e ρ ( u , v ) ] .
S 0 = a 1 2 + a 2 2 ,
S 1 = a 1 2 a 2 2 ,
S 2 = 2 a 1 a 2 cos δ ,
S 3 = 2 a 1 a 2 sin δ ,
δ = δ 2 δ 1 .
ψ = 1 2 arctan ( s 2 s 1 ) ,
e z ( u , v ) = e z * ( u , v ) ,
e ρ ( u , v ) = e ρ * ( u , v ) ,
s 2 ( 0 , v ) = 0 ; ψ ( 0 , v ) = 0 .
Re [ e z ( u , v ) ] Im [ e ρ ( u , v ) ] = Im [ e z ( u , v ) ] Re [ e ρ ( u , v ) ] .
Re [ e z ( u , v ) ] = Im [ e z ( u , v ) ] = 0 .
e ( u , v ) = e + ( u , v ) c ̂ + ( u , v ) + e ( u , v ) c ̂ ( u , v ) ,
e ± = ( e z i e ρ ) 2 , c ̂ ± = ( z ̂ ± i ρ ̂ ) 2 .
e z ( u , v ) = i k f 2 0 α θ 3 ( 1 β 2 θ 2 ) exp ( i u sin 2 α ) d θ ,
e ρ ( u , v ) = k f 2 0 α θ 3 ( 1 β 2 θ 2 ) exp ( i u sin 2 α ) v 2 sin α d θ .
e + ( u , v ) = i k f 2 2 exp ( i u sin 2 α ) [ 1 v 2 sin α ] 0 α θ 3 ( 1 β 2 θ 2 ) d θ .
2 L q = L q ρ + L q z ,

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