Abstract

The dynamic behavior of spectral Stokes singularities (vortices) of stochastic electromagnetic beams propagating through an astigmatic lens is studied based on the spectral Stokes parameters and complex spectral Stokes fields. It is found that there exist S12, S23, and S31 spectral Stokes vortices. By suitably varying propagation distance, spatial correlation length, off-axis distance or astigmatic coefficient, the motion, creation, annihilation, and changes in the degree of polarization of S12, S23, and S31 vortices, in particular, the inversion of handedness of S12 vortices may take place. A comparison with the previous work is also made.

© 2010 Optical Society of America

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References

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  1. J. F. Nye, Natural Focusing and Fine Structure of Light (Institute of Physics Publishing, 1999).
  2. M. V. Berry and M. R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. London, Ser. A 457, 141-155 (2001).
    [CrossRef]
  3. C. V. Felde, A. A. Chernyshov, G. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Polarization singularities in partially coherent combined beams,” JETP Lett. 88, 418-422 (2008).
    [CrossRef]
  4. F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
    [CrossRef] [PubMed]
  5. F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Stokes parameters in the unfolding of an optical vortex through a birefringent crystal,” Opt. Express 14, 11402-11411 (2006).
    [CrossRef] [PubMed]
  6. K. Y. Bliokh, A. Niv, V. Kleiner, and E. Hasman, “Singular polarimetry: Evolution of polarization singularities in electromagnetic waves propagating in a weakly anisotropic medium,” Opt. Express 16, 695-709 (2008).
    [CrossRef] [PubMed]
  7. M. S. Soskin, V. G. Denisenko, and R. I. Egorov, “Singular Stokes-polarimetry as new technique for metrology and inspection of polarized speckle fields,” Proc. SPIE , 545879-85 (2004).
    [CrossRef]
  8. I. Freund, “Polarization singularity indices in Gaussian laser beams,” Opt. Commun. 201, 251-270 (2002).
    [CrossRef]
  9. I. Freund, “Poincare vortices,” Opt. Lett. 26, 1996-1998 (2001).
    [CrossRef]
  10. I. Freund, A. I. Mokhun, M. S. Soskin, O. V. Angelsky, and Mokhun II, “Stokes singularity relations,” Opt. Lett. 27, 545-547 (2002).
    [CrossRef]
  11. A. I. Mokhun, M. S. Soskin, and I. Freund, “Elliptic critical points: C-points, a-lines, and the sign rule,” Opt. Lett. 27, 995-997 (2002).
    [CrossRef]
  12. M. S. Soskin, V. Denisenko, and I. Freund, “Optical polarization singularities and elliptic stationary points,” Opt. Lett. 28, 1475-1477 (2003).
    [CrossRef] [PubMed]
  13. H. W. Yan and B. D. Lu, “Spectral Stokes singularities of stochastic electromagnetic beams,” Opt. Lett. 34, 1933-1935 (2009).
    [CrossRef] [PubMed]
  14. O. Korotkova and E. Wolf, “Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett. 30, 198-200 (2005).
    [CrossRef] [PubMed]
  15. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge Univ. Press, 2007).
  16. A. T. Friberg and J. Turunen, “Imaging of Gaussian Schell-model sources,” J. Opt. Soc. Am. A 5, 713-720 (1988).
    [CrossRef]
  17. H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249, 379-385 (2005).
    [CrossRef]
  18. I. Freund and N. Shvartsman, “Wave-field phase singularities--the sign principle,” Phys. Rev. A 50, 5164-5172 (1994).
    [CrossRef] [PubMed]

2009 (1)

2008 (2)

C. V. Felde, A. A. Chernyshov, G. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Polarization singularities in partially coherent combined beams,” JETP Lett. 88, 418-422 (2008).
[CrossRef]

K. Y. Bliokh, A. Niv, V. Kleiner, and E. Hasman, “Singular polarimetry: Evolution of polarization singularities in electromagnetic waves propagating in a weakly anisotropic medium,” Opt. Express 16, 695-709 (2008).
[CrossRef] [PubMed]

2006 (1)

2005 (3)

F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
[CrossRef] [PubMed]

O. Korotkova and E. Wolf, “Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett. 30, 198-200 (2005).
[CrossRef] [PubMed]

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249, 379-385 (2005).
[CrossRef]

2004 (1)

M. S. Soskin, V. G. Denisenko, and R. I. Egorov, “Singular Stokes-polarimetry as new technique for metrology and inspection of polarized speckle fields,” Proc. SPIE , 545879-85 (2004).
[CrossRef]

2003 (1)

2002 (3)

2001 (2)

I. Freund, “Poincare vortices,” Opt. Lett. 26, 1996-1998 (2001).
[CrossRef]

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

1994 (1)

I. Freund and N. Shvartsman, “Wave-field phase singularities--the sign principle,” Phys. Rev. A 50, 5164-5172 (1994).
[CrossRef] [PubMed]

1988 (1)

Angelsky, O. V.

Berry, M. V.

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

Bliokh, K. Y.

Bogatyryova, G. V.

C. V. Felde, A. A. Chernyshov, G. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Polarization singularities in partially coherent combined beams,” JETP Lett. 88, 418-422 (2008).
[CrossRef]

Chernyshov, A. A.

C. V. Felde, A. A. Chernyshov, G. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Polarization singularities in partially coherent combined beams,” JETP Lett. 88, 418-422 (2008).
[CrossRef]

Denisenko, V.

Denisenko, V. G.

M. S. Soskin, V. G. Denisenko, and R. I. Egorov, “Singular Stokes-polarimetry as new technique for metrology and inspection of polarized speckle fields,” Proc. SPIE , 545879-85 (2004).
[CrossRef]

Dennis, M. R.

F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Stokes parameters in the unfolding of an optical vortex through a birefringent crystal,” Opt. Express 14, 11402-11411 (2006).
[CrossRef] [PubMed]

F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
[CrossRef] [PubMed]

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

Egorov, R. I.

M. S. Soskin, V. G. Denisenko, and R. I. Egorov, “Singular Stokes-polarimetry as new technique for metrology and inspection of polarized speckle fields,” Proc. SPIE , 545879-85 (2004).
[CrossRef]

Felde, C. V.

C. V. Felde, A. A. Chernyshov, G. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Polarization singularities in partially coherent combined beams,” JETP Lett. 88, 418-422 (2008).
[CrossRef]

Flossmann, F.

F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Stokes parameters in the unfolding of an optical vortex through a birefringent crystal,” Opt. Express 14, 11402-11411 (2006).
[CrossRef] [PubMed]

F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
[CrossRef] [PubMed]

Freund, I.

Friberg, A. T.

Hasman, E.

Kleiner, V.

Korotkova, O.

O. Korotkova and E. Wolf, “Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett. 30, 198-200 (2005).
[CrossRef] [PubMed]

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249, 379-385 (2005).
[CrossRef]

Lu, B. D.

Maier, M.

F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Stokes parameters in the unfolding of an optical vortex through a birefringent crystal,” Opt. Express 14, 11402-11411 (2006).
[CrossRef] [PubMed]

F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
[CrossRef] [PubMed]

Mokhun,

Mokhun, A. I.

Niv, A.

Nye, J. F.

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

Polyanskii, P. V.

C. V. Felde, A. A. Chernyshov, G. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Polarization singularities in partially coherent combined beams,” JETP Lett. 88, 418-422 (2008).
[CrossRef]

Roychowdhury, H.

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249, 379-385 (2005).
[CrossRef]

Schwarz, U. T.

F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Stokes parameters in the unfolding of an optical vortex through a birefringent crystal,” Opt. Express 14, 11402-11411 (2006).
[CrossRef] [PubMed]

F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
[CrossRef] [PubMed]

Shvartsman, N.

I. Freund and N. Shvartsman, “Wave-field phase singularities--the sign principle,” Phys. Rev. A 50, 5164-5172 (1994).
[CrossRef] [PubMed]

Soskin, M. S.

C. V. Felde, A. A. Chernyshov, G. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Polarization singularities in partially coherent combined beams,” JETP Lett. 88, 418-422 (2008).
[CrossRef]

M. S. Soskin, V. G. Denisenko, and R. I. Egorov, “Singular Stokes-polarimetry as new technique for metrology and inspection of polarized speckle fields,” Proc. SPIE , 545879-85 (2004).
[CrossRef]

M. S. Soskin, V. Denisenko, and I. Freund, “Optical polarization singularities and elliptic stationary points,” Opt. Lett. 28, 1475-1477 (2003).
[CrossRef] [PubMed]

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

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

Turunen, J.

Wolf, E.

O. Korotkova and E. Wolf, “Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett. 30, 198-200 (2005).
[CrossRef] [PubMed]

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge Univ. Press, 2007).

Yan, H. W.

J. Opt. Soc. Am. A (1)

JETP Lett. (1)

C. V. Felde, A. A. Chernyshov, G. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Polarization singularities in partially coherent combined beams,” JETP Lett. 88, 418-422 (2008).
[CrossRef]

Opt. Commun. (2)

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249, 379-385 (2005).
[CrossRef]

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

Opt. Express (2)

Opt. Lett. (6)

Phys. Rev. A (1)

I. Freund and N. Shvartsman, “Wave-field phase singularities--the sign principle,” Phys. Rev. A 50, 5164-5172 (1994).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
[CrossRef] [PubMed]

Proc. R. Soc. London, Ser. A (1)

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

Proc. SPIE (1)

M. S. Soskin, V. G. Denisenko, and R. I. Egorov, “Singular Stokes-polarimetry as new technique for metrology and inspection of polarized speckle fields,” Proc. SPIE , 545879-85 (2004).
[CrossRef]

Other (2)

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

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge Univ. Press, 2007).

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Figures (5)

Fig. 1
Fig. 1

Spectral Stokes vortices of an electromagnetic GSM vortex beam at the source plane z = 0 . The calculation parameters are seen in the text.

Fig. 2
Fig. 2

Evolution of spectral Stokes vortices of an electromagnetic GSM vortex beam in the focused field. (a) z = 0.6 f , (b) z = 0.73 f , (c) z = 0.7465 f , (d) z = 0.75 f , (e) z = 0.81 f , (f) z = f .

Fig. 3
Fig. 3

Variation of spectral Stokes vortices at z = f with the spatial correlation length δ y y . (a) δ y y = 0.23 mm , (b) δ y y = 0.25 mm .

Fig. 4
Fig. 4

Variation of spectral Stokes vortices at z = f with the off-axis distance s y . (a) s y = 0.1 mm , (b) s y = 1 mm .

Fig. 5
Fig. 5

Variation of spectral Stokes vortices at z = f with the astigmatism coefficient C 6 . (a) C 6 = 0.134 × 10 3 mm 1 , (b) C 6 = 0.130 × 10 3 mm 1 , (c) C 6 = 0 .

Equations (20)

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W u v ( x 01 , y 01 , x 02 , y 02 , 0 ) = A u A v B u v w 0 2 ( x 01 s u i y 01 ) ( x 02 s v + i y 02 ) exp [ ( x 01 s u ) 2 + y 01 2 w 0 2 ] × exp [ ( x 02 s v ) 2 + y 02 2 w 0 2 ] exp [ ( x 01 x 02 s u + s v ) 2 + ( y 01 y 02 ) 2 2 δ u v 2 ] ,
( u , v = x , y , unless otherwise stated )
W u v ( x 1 , y 1 , x 2 , y 2 , z ) = ( k 2 π B ) 2 exp [ i k D 2 B ( x 1 2 + y 1 2 x 2 2 y 2 2 ) ] W u v ( x 01 , y 01 , x 02 , y 02 , 0 ) exp [ i k C 6 ( x 01 2 x 02 2 y 01 2 + y 02 2 ) ] × exp [ i k A 2 B ( x 01 2 + y 01 2 x 02 2 y 02 2 ) ] × exp [ i k B ( x 1 x 01 + y 1 y 01 x 2 x 02 y 2 y 02 ) ] d x 01 d y 01 d x 02 d y 02 ,
( A B C D ) = ( 1 z f z 1 f 1 ) ,
W u v ( x 1 , y 1 , x 2 , y 2 , z ) = Q u v A u A v B u v w 0 2 { β x q u v ( x ) α u v ( x ) p u v ( x ) + β y q u v ( y ) α u v ( y ) p u v ( y ) + p u v ( x ) + 2 q u v ( x ) 2 4 α u v ( x ) δ u v 2 p u v ( x ) 2 + p u v ( y ) + 2 q u v ( y ) 2 4 α u v ( y ) δ u v 2 p u v ( y ) 2 + i α u v ( x ) [ β x + q u v ( x ) 2 δ u v 2 p u v ( x ) ] q u v ( y ) p u v ( y ) i α u v ( y ) [ β y + q u v ( y ) 2 δ u v 2 p u v ( y ) ] q u v ( x ) p u v ( x ) } ,
Q u v = ( k 2 B ) 2 exp [ i k 2 π B ( x 1 2 + y 1 2 x 2 2 y 2 2 ) ] 1 α u v ( x ) p u v ( x ) α u v ( y ) p u v ( y ) exp [ ( i k C 6 i k A 2 B ) ( s u s v ) + i k B ( x 1 s u x 2 s v ) + β x 2 α u v ( x ) + β y 2 α u v ( y ) + q u v ( x ) 2 p u v ( x ) + q u v ( y ) 2 p u v ( y ) ] ,
β x = i k ( x 1 A s u + 2 B C 6 s u ) 2 B , β y = i k y 1 2 B ,
α u v ( x ) = 1 w 0 2 + i k A 2 B i k C 6 + 1 2 δ u v 2 ,
p u v ( x ) = 1 w 0 2 i k A 2 B + i k C 6 + 1 2 δ u v 2 1 4 δ u v 4 α u v ( x ) ,
q u v ( x ) = i k 2 B ( x 2 A s v + 2 B C 6 s v x 1 A s u + 2 B C 6 s u 2 δ u v 2 α u v ( x ) ) ,
α u v ( y ) = 1 w 0 2 + i k A 2 B + i k C 6 + 1 2 δ u v 2 ,
p u v ( y ) = 1 w 0 2 i k A 2 B i k C 6 + 1 2 δ u v 2 1 4 δ u v 4 α u v ( y ) ,
q u v ( y ) = i k 2 B ( y 2 y 1 2 δ u v 2 α u v ( y ) ) .
S 1 ( x , y , x , y , z ) = s 0 1 [ W x x ( x , y , x , y , z ) W y y ( x , y , x , y , z ) ] ,
S 2 ( x , y , x , y , z ) = s 0 1 [ W x y ( x , y , x , y , z ) + W y x ( x , y , x , y , z ) ] ,
S 3 ( x , y , x , y , z ) = i s 0 1 [ W y x ( x , y , x , y , z ) W x y ( x , y , x , y , z ) ] ,
s 0 ( x , y , x , y , z ) = W x x ( x , y , x , y , z ) + W y y ( x , y , x , y , z ) .
S 12 = S 1 + i S 2 ,
S 23 = S 2 + i S 3 ,
S 31 = S 3 + i S 1 .

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