Abstract

We describe the evolution of a paraxial electromagnetic wave characterizing by a non-uniform polarization distribution with singularities and propagating in a weakly anisotropic medium. Our approach is based on the Stokes vector evolution equation applied to a non-uniform initial polarization field. In the case of a homogeneous medium, this equation is integrated analytically. This yields a 3-dimensional distribution of the polarization parameters containing singularities, i.e. C-lines of circular polarization and L-surfaces of linear polarization. The general theory is applied to specific examples of the unfolding of a vectorial vortex in birefringent and dichroic media.

© 2008 Optical Society of America

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    [CrossRef]
  5. J. F. Nye, "Lines of circular polarization in electromagnetic wave fields," Proc. R. Soc. London A 389, 279-290 (1983).
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  6. J. F. Nye and J. V. Hajnal, "The wave structure of monochromatic electromagnetic radiation," Proc. R. Soc. London A 409, 21-36 (1987).
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  7. M. V. Berry and M. R. Dennis, "Polarization singularities in isotropic random vector waves," Proc. R. Soc. London A 457, 141−155 (2001).
    [CrossRef]
  8. I.  Freund, "Polarization singularity indices in Gaussian laser beams," Opt. Commun. 201, 251-270 (2002).
    [CrossRef]
  9. M. R. Dennis, "Polarization singularities in paraxial vector fields: morphology and statistics," Opt. Commun. 213, 201-221 (2002).
    [CrossRef]
  10. J.  Masajada and B. Dubik, "Optical vortex generation by three plane wave interference," Opt. Commun. 198, 21-27 (2001).
    [CrossRef]
  11. I.  Freund, N. Shvartsman, and V. Freilikher, "Optical dislocation networks in highly random media," Opt. Commun. 101, 247-264 (1993).
    [CrossRef]
  12. E.  Hasman, G.  Biener, A.  Niv, and V.  Kleiner, "Space-variant polarization manipulation," Prog. Opt. 47, 215-289 (2005).
    [CrossRef]
  13. A.  Niv, G.  Biener, V.  Kleiner, and E. Hasman, "Manipulation of the Pancharatnam phase in vectorial vortices," Opt. Express 14, 4208-4220 (2006).
    [CrossRef] [PubMed]
  14. K. Yu. Bliokh, "Geometrical optics of beams with vortices: Berry phase and orbital angular momentum Hall effect," Phys. Rev. Lett. 97, 043901 (2006).
    [CrossRef] [PubMed]
  15. 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]
  16. 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).
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  21. C.  Brosseau, "Evolution of the Stokes parameters in optically anisotropic media," Opt. Lett. 20, 1221-1223 (1995).
    [CrossRef] [PubMed]
  22. H.  Kuratsuji and S.  Kakigi, "Maxwell-Schrödinger equation for polarized light and evolution of the Stokes parameters," Phys. Rev. Lett. 80, 1888-1891 (1998).
    [CrossRef]
  23. S. E.  Segre, "New formalism for the analysis of polarization evolution for radiation in a weakly nonuniform, fully anisotropic medium: a magnetized plasma," J. Opt. Soc. Am. A 18, 2601-2606 (2001).
    [CrossRef]
  24. R.  Botet, H.  Kuratsuji, and R.  Seto, "Novel aspects of evolution of the Stokes parameters for an electromagnetic wave in anisotropic media," Prog. Theor. Phys. 116, 285-294 (2006).
    [CrossRef]
  25. K. Y.  Bliokh, D. Y.  Frolov, and Y. A.  Kravtsov, "Non-Abelian evolution of electromagnetic waves in a weakly anisotropic inhomogeneous medium," Phys. Rev. A 75, 053821 (2007).
    [CrossRef]
  26. Y. A.  Kravtsov, B.  Bieg, and K. Y.  Bliokh, "Stokes-vector evolution in a weakly anisotropic inhomogeneous medium," J. Opt. Soc. Am. A 24, 3388-3396 (2007).
    [CrossRef]
  27. R.  Barakat, "Bilinear constraints between elements of the 4x4 Mueller-Jones transfer matrix of polarization theory," Opt. Commun. 38, 159-161 (1981).
    [CrossRef]
  28. R.  Simon, "The connection between Mueller and Jones matrices of polarization optics," Opt. Commun. 42, 293-297 (1982).
    [CrossRef]
  29. S. R.  Cloude, "Group theory and polarization algebra," Optik 75, 26-32 (1986).
  30. T.  Opartny and J.  Perina, "Non-image-forming polarization optical devices and Lorentz transformation - an analogy," Phys. Lett. A 181, 199-202 (1993).
    [CrossRef]
  31. D.  Han, Y. S.  Kim, and M. E.  Noz, "Polarization optics and bilinear representation of the Lorentz group," Phys. Lett. A. 219, 26-32 (1996).
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    [CrossRef]

2007 (3)

K. Y.  Bliokh, D. Y.  Frolov, and Y. A.  Kravtsov, "Non-Abelian evolution of electromagnetic waves in a weakly anisotropic inhomogeneous medium," Phys. Rev. A 75, 053821 (2007).
[CrossRef]

Y. A.  Kravtsov, B.  Bieg, and K. Y.  Bliokh, "Stokes-vector evolution in a weakly anisotropic inhomogeneous medium," J. Opt. Soc. Am. A 24, 3388-3396 (2007).
[CrossRef]

A. D.  Kiselev, "Singularities in polarization resolved angular patterns: transmittance of nematic liquid crystal cells," J. Phys.: Condens. Matter 19, 246102 (2007).
[CrossRef]

2006 (4)

R.  Botet, H.  Kuratsuji, and R.  Seto, "Novel aspects of evolution of the Stokes parameters for an electromagnetic wave in anisotropic media," Prog. Theor. Phys. 116, 285-294 (2006).
[CrossRef]

A.  Niv, G.  Biener, V.  Kleiner, and E. Hasman, "Manipulation of the Pancharatnam phase in vectorial vortices," Opt. Express 14, 4208-4220 (2006).
[CrossRef] [PubMed]

K. Yu. Bliokh, "Geometrical optics of beams with vortices: Berry phase and orbital angular momentum Hall effect," Phys. Rev. Lett. 97, 043901 (2006).
[CrossRef] [PubMed]

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]

2005 (2)

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]

E.  Hasman, G.  Biener, A.  Niv, and V.  Kleiner, "Space-variant polarization manipulation," Prog. Opt. 47, 215-289 (2005).
[CrossRef]

2002 (2)

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

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

2001 (4)

J.  Masajada and B. Dubik, "Optical vortex generation by three plane wave interference," Opt. Commun. 198, 21-27 (2001).
[CrossRef]

M. S. Soskin and M. V. Vasnetsov, "Singular optics," Prog. Opt. 42, 219-276 (2001).
[CrossRef]

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

S. E.  Segre, "New formalism for the analysis of polarization evolution for radiation in a weakly nonuniform, fully anisotropic medium: a magnetized plasma," J. Opt. Soc. Am. A 18, 2601-2606 (2001).
[CrossRef]

1999 (1)

L. Allen, M. J. Padgett, and M. Babiker, "The orbital angular momentum of light," Prog. Opt. 39, 291-372 (1999).
[CrossRef]

1998 (1)

H.  Kuratsuji and S.  Kakigi, "Maxwell-Schrödinger equation for polarized light and evolution of the Stokes parameters," Phys. Rev. Lett. 80, 1888-1891 (1998).
[CrossRef]

1996 (1)

D.  Han, Y. S.  Kim, and M. E.  Noz, "Polarization optics and bilinear representation of the Lorentz group," Phys. Lett. A. 219, 26-32 (1996).
[CrossRef]

1995 (2)

C. S.  Brown and A. E.  Bak, "Unified formalism for polarization optics with application to polarimetry on a twisted optical fiber," Opt. Eng. 34, 1625-1635 (1995).
[CrossRef]

C.  Brosseau, "Evolution of the Stokes parameters in optically anisotropic media," Opt. Lett. 20, 1221-1223 (1995).
[CrossRef] [PubMed]

1993 (2)

I.  Freund, N. Shvartsman, and V. Freilikher, "Optical dislocation networks in highly random media," Opt. Commun. 101, 247-264 (1993).
[CrossRef]

T.  Opartny and J.  Perina, "Non-image-forming polarization optical devices and Lorentz transformation - an analogy," Phys. Lett. A 181, 199-202 (1993).
[CrossRef]

1987 (1)

J. F. Nye and J. V. Hajnal, "The wave structure of monochromatic electromagnetic radiation," Proc. R. Soc. London A 409, 21-36 (1987).
[CrossRef]

1986 (1)

S. R.  Cloude, "Group theory and polarization algebra," Optik 75, 26-32 (1986).

1983 (1)

J. F. Nye, "Lines of circular polarization in electromagnetic wave fields," Proc. R. Soc. London A 389, 279-290 (1983).
[CrossRef]

1982 (1)

R.  Simon, "The connection between Mueller and Jones matrices of polarization optics," Opt. Commun. 42, 293-297 (1982).
[CrossRef]

1981 (1)

R.  Barakat, "Bilinear constraints between elements of the 4x4 Mueller-Jones transfer matrix of polarization theory," Opt. Commun. 38, 159-161 (1981).
[CrossRef]

1978 (1)

Allen, L.

L. Allen, M. J. Padgett, and M. Babiker, "The orbital angular momentum of light," Prog. Opt. 39, 291-372 (1999).
[CrossRef]

Azzam, R. M. A.

Babiker, M.

L. Allen, M. J. Padgett, and M. Babiker, "The orbital angular momentum of light," Prog. Opt. 39, 291-372 (1999).
[CrossRef]

Bak, A. E.

C. S.  Brown and A. E.  Bak, "Unified formalism for polarization optics with application to polarimetry on a twisted optical fiber," Opt. Eng. 34, 1625-1635 (1995).
[CrossRef]

Barakat, R.

R.  Barakat, "Bilinear constraints between elements of the 4x4 Mueller-Jones transfer matrix of polarization theory," Opt. Commun. 38, 159-161 (1981).
[CrossRef]

Berry, M. V.

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

Bieg, B.

Biener, G.

A.  Niv, G.  Biener, V.  Kleiner, and E. Hasman, "Manipulation of the Pancharatnam phase in vectorial vortices," Opt. Express 14, 4208-4220 (2006).
[CrossRef] [PubMed]

E.  Hasman, G.  Biener, A.  Niv, and V.  Kleiner, "Space-variant polarization manipulation," Prog. Opt. 47, 215-289 (2005).
[CrossRef]

Bliokh, K. Y.

Y. A.  Kravtsov, B.  Bieg, and K. Y.  Bliokh, "Stokes-vector evolution in a weakly anisotropic inhomogeneous medium," J. Opt. Soc. Am. A 24, 3388-3396 (2007).
[CrossRef]

K. Y.  Bliokh, D. Y.  Frolov, and Y. A.  Kravtsov, "Non-Abelian evolution of electromagnetic waves in a weakly anisotropic inhomogeneous medium," Phys. Rev. A 75, 053821 (2007).
[CrossRef]

Bliokh, K. Yu.

K. Yu. Bliokh, "Geometrical optics of beams with vortices: Berry phase and orbital angular momentum Hall effect," Phys. Rev. Lett. 97, 043901 (2006).
[CrossRef] [PubMed]

Botet, R.

R.  Botet, H.  Kuratsuji, and R.  Seto, "Novel aspects of evolution of the Stokes parameters for an electromagnetic wave in anisotropic media," Prog. Theor. Phys. 116, 285-294 (2006).
[CrossRef]

Brosseau, C.

Brown, C. S.

C. S.  Brown and A. E.  Bak, "Unified formalism for polarization optics with application to polarimetry on a twisted optical fiber," Opt. Eng. 34, 1625-1635 (1995).
[CrossRef]

Cloude, S. R.

S. R.  Cloude, "Group theory and polarization algebra," Optik 75, 26-32 (1986).

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

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

Dubik, B.

J.  Masajada and B. Dubik, "Optical vortex generation by three plane wave interference," Opt. Commun. 198, 21-27 (2001).
[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]

Freilikher, V.

I.  Freund, N. Shvartsman, and V. Freilikher, "Optical dislocation networks in highly random media," Opt. Commun. 101, 247-264 (1993).
[CrossRef]

Freund, I.

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

I.  Freund, N. Shvartsman, and V. Freilikher, "Optical dislocation networks in highly random media," Opt. Commun. 101, 247-264 (1993).
[CrossRef]

Frolov, D. Y.

K. Y.  Bliokh, D. Y.  Frolov, and Y. A.  Kravtsov, "Non-Abelian evolution of electromagnetic waves in a weakly anisotropic inhomogeneous medium," Phys. Rev. A 75, 053821 (2007).
[CrossRef]

Hajnal, J. V.

J. F. Nye and J. V. Hajnal, "The wave structure of monochromatic electromagnetic radiation," Proc. R. Soc. London A 409, 21-36 (1987).
[CrossRef]

Han, D.

D.  Han, Y. S.  Kim, and M. E.  Noz, "Polarization optics and bilinear representation of the Lorentz group," Phys. Lett. A. 219, 26-32 (1996).
[CrossRef]

Hasman, E.

A.  Niv, G.  Biener, V.  Kleiner, and E. Hasman, "Manipulation of the Pancharatnam phase in vectorial vortices," Opt. Express 14, 4208-4220 (2006).
[CrossRef] [PubMed]

E.  Hasman, G.  Biener, A.  Niv, and V.  Kleiner, "Space-variant polarization manipulation," Prog. Opt. 47, 215-289 (2005).
[CrossRef]

Kakigi, S.

H.  Kuratsuji and S.  Kakigi, "Maxwell-Schrödinger equation for polarized light and evolution of the Stokes parameters," Phys. Rev. Lett. 80, 1888-1891 (1998).
[CrossRef]

Kim, Y. S.

D.  Han, Y. S.  Kim, and M. E.  Noz, "Polarization optics and bilinear representation of the Lorentz group," Phys. Lett. A. 219, 26-32 (1996).
[CrossRef]

Kiselev, A. D.

A. D.  Kiselev, "Singularities in polarization resolved angular patterns: transmittance of nematic liquid crystal cells," J. Phys.: Condens. Matter 19, 246102 (2007).
[CrossRef]

Kleiner, V.

A.  Niv, G.  Biener, V.  Kleiner, and E. Hasman, "Manipulation of the Pancharatnam phase in vectorial vortices," Opt. Express 14, 4208-4220 (2006).
[CrossRef] [PubMed]

E.  Hasman, G.  Biener, A.  Niv, and V.  Kleiner, "Space-variant polarization manipulation," Prog. Opt. 47, 215-289 (2005).
[CrossRef]

Kravtsov, Y. A.

K. Y.  Bliokh, D. Y.  Frolov, and Y. A.  Kravtsov, "Non-Abelian evolution of electromagnetic waves in a weakly anisotropic inhomogeneous medium," Phys. Rev. A 75, 053821 (2007).
[CrossRef]

Y. A.  Kravtsov, B.  Bieg, and K. Y.  Bliokh, "Stokes-vector evolution in a weakly anisotropic inhomogeneous medium," J. Opt. Soc. Am. A 24, 3388-3396 (2007).
[CrossRef]

Kuratsuji, H.

R.  Botet, H.  Kuratsuji, and R.  Seto, "Novel aspects of evolution of the Stokes parameters for an electromagnetic wave in anisotropic media," Prog. Theor. Phys. 116, 285-294 (2006).
[CrossRef]

H.  Kuratsuji and S.  Kakigi, "Maxwell-Schrödinger equation for polarized light and evolution of the Stokes parameters," Phys. Rev. Lett. 80, 1888-1891 (1998).
[CrossRef]

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]

Masajada, J.

J.  Masajada and B. Dubik, "Optical vortex generation by three plane wave interference," Opt. Commun. 198, 21-27 (2001).
[CrossRef]

Niv, A.

A.  Niv, G.  Biener, V.  Kleiner, and E. Hasman, "Manipulation of the Pancharatnam phase in vectorial vortices," Opt. Express 14, 4208-4220 (2006).
[CrossRef] [PubMed]

E.  Hasman, G.  Biener, A.  Niv, and V.  Kleiner, "Space-variant polarization manipulation," Prog. Opt. 47, 215-289 (2005).
[CrossRef]

Noz, M. E.

D.  Han, Y. S.  Kim, and M. E.  Noz, "Polarization optics and bilinear representation of the Lorentz group," Phys. Lett. A. 219, 26-32 (1996).
[CrossRef]

Nye, J. F.

J. F. Nye and J. V. Hajnal, "The wave structure of monochromatic electromagnetic radiation," Proc. R. Soc. London A 409, 21-36 (1987).
[CrossRef]

J. F. Nye, "Lines of circular polarization in electromagnetic wave fields," Proc. R. Soc. London A 389, 279-290 (1983).
[CrossRef]

Opartny, T.

T.  Opartny and J.  Perina, "Non-image-forming polarization optical devices and Lorentz transformation - an analogy," Phys. Lett. A 181, 199-202 (1993).
[CrossRef]

Padgett, M. J.

L. Allen, M. J. Padgett, and M. Babiker, "The orbital angular momentum of light," Prog. Opt. 39, 291-372 (1999).
[CrossRef]

Perina, J.

T.  Opartny and J.  Perina, "Non-image-forming polarization optical devices and Lorentz transformation - an analogy," Phys. Lett. A 181, 199-202 (1993).
[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]

Segre, S.E.

Seto, R.

R.  Botet, H.  Kuratsuji, and R.  Seto, "Novel aspects of evolution of the Stokes parameters for an electromagnetic wave in anisotropic media," Prog. Theor. Phys. 116, 285-294 (2006).
[CrossRef]

Shvartsman, N.

I.  Freund, N. Shvartsman, and V. Freilikher, "Optical dislocation networks in highly random media," Opt. Commun. 101, 247-264 (1993).
[CrossRef]

Simon, R.

R.  Simon, "The connection between Mueller and Jones matrices of polarization optics," Opt. Commun. 42, 293-297 (1982).
[CrossRef]

Soskin, M. S.

M. S. Soskin and M. V. Vasnetsov, "Singular optics," Prog. Opt. 42, 219-276 (2001).
[CrossRef]

Vasnetsov, M. V.

M. S. Soskin and M. V. Vasnetsov, "Singular optics," Prog. Opt. 42, 219-276 (2001).
[CrossRef]

J. Opt. Soc. Am. (1)

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

J. Phys.: Condens. Matter (1)

A. D.  Kiselev, "Singularities in polarization resolved angular patterns: transmittance of nematic liquid crystal cells," J. Phys.: Condens. Matter 19, 246102 (2007).
[CrossRef]

Opt. Commun. (6)

R.  Barakat, "Bilinear constraints between elements of the 4x4 Mueller-Jones transfer matrix of polarization theory," Opt. Commun. 38, 159-161 (1981).
[CrossRef]

R.  Simon, "The connection between Mueller and Jones matrices of polarization optics," Opt. Commun. 42, 293-297 (1982).
[CrossRef]

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

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

J.  Masajada and B. Dubik, "Optical vortex generation by three plane wave interference," Opt. Commun. 198, 21-27 (2001).
[CrossRef]

I.  Freund, N. Shvartsman, and V. Freilikher, "Optical dislocation networks in highly random media," Opt. Commun. 101, 247-264 (1993).
[CrossRef]

Opt. Eng. (1)

C. S.  Brown and A. E.  Bak, "Unified formalism for polarization optics with application to polarimetry on a twisted optical fiber," Opt. Eng. 34, 1625-1635 (1995).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Optik (1)

S. R.  Cloude, "Group theory and polarization algebra," Optik 75, 26-32 (1986).

Phys. Lett. A (1)

T.  Opartny and J.  Perina, "Non-image-forming polarization optical devices and Lorentz transformation - an analogy," Phys. Lett. A 181, 199-202 (1993).
[CrossRef]

Phys. Lett. A. (1)

D.  Han, Y. S.  Kim, and M. E.  Noz, "Polarization optics and bilinear representation of the Lorentz group," Phys. Lett. A. 219, 26-32 (1996).
[CrossRef]

Phys. Rev. A (1)

K. Y.  Bliokh, D. Y.  Frolov, and Y. A.  Kravtsov, "Non-Abelian evolution of electromagnetic waves in a weakly anisotropic inhomogeneous medium," Phys. Rev. A 75, 053821 (2007).
[CrossRef]

Phys. Rev. Lett. (3)

H.  Kuratsuji and S.  Kakigi, "Maxwell-Schrödinger equation for polarized light and evolution of the Stokes parameters," Phys. Rev. Lett. 80, 1888-1891 (1998).
[CrossRef]

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Figures (5)

Fig. 1.
Fig. 1.

Distributions of Stokes vectors (upper panel) and of polarization ellipses (lower panel) for vectorial vortices Eqs. (25) with a C-point in the center, Eq. (26), at different values of azimuthal index m. Hereafter, δ=0 and directions of Stokes vectors are naturally depicted in the real space with the s 1, s 2, and s 3 components pointing along the x, y, and z axis, respectively.

Fig. 2.
Fig. 2.

C-lines and L-surfaces under propagation of the vectorial vortex, Eqs. (25) and (26), with m=3 and χ=1 in a linearly-birefringent medium with ω=(1, 0,0). Dimensionless coordinates ξ=x/ρ*, η=y/ρ*, and ζz/π are used, whereas red and blue colors indicate C-lines with χ=1 and χ=-1, respectively.

Fig. 3.
Fig. 3.

Distributions of Stokes vectors, Eq. (12), (upper panel) and of polarization ellipses (lower panel) for vectorial vortex, Eqs. (25) and (26), with m=3 and χ=1, propagating in a linearly-birefringent medium with ω=(1, 0,0), Fig. 2. Distributions are shown at different propagation distances ζz/π within half-period. Red and blue colors indicate areas with right-hand (s 3>0) and left-hand (s 3<0) polarizations. C-points are marked by dots (upper panel) and crosses (lower panel).

Fig. 4.
Fig. 4.

C-lines and L-surfaces under propagation of the vectorial vortex, Eqs. (25) and (26), with m=3 and χ=1 in a dichroic medium. Pictures (a) and (b) correspond to circularly- and linearly-dichroic media with σ=(0,0,-1) and σ=(1,0,0), respectively. Dimensionless coordinates ξ=x/ρ*, η=y/ρ*, and ζz are used.

Fig. 5.
Fig. 5.

Distributions of Stokes vectors, Eq. (18), (upper panel) and of polarization ellipses (lower panel) for vectorial vortex, Eqs. (25) and (26), with m=3 and χ=1, propagating in a linearly-dichroic medium with σ=(1,0,0), Fig. 4b. Distributions are shown at different propagation distances ζz. C-points are marked by dots (upper panel) and crosses (lower panel).

Equations (41)

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s ( x , y , 0 ) = s 0 ( x , y ) ,
s z = m ̂ s ,
s 1 = 0 , s 2 = 0 ,
s 3 = 0 .
z P z z R , z D .
S z = M ̂ S ,
ε ̂ = ε 0 I ̂ 2 + ν ̂ ( ε 0 + ν x x ν x y ν y x ε 0 + ν y y ) .
M ̂ = ( Im G 0 Im G 1 Im G 2 Im G 3 Im G 1 Im G 0 Re G 3 Re G 2 Im G 2 Re G 3 Im G 0 Re G 1 Im G 3 Re G 2 Re G 1 Im G 0 ) .
G = k 0 2 ε 0 [ ( ν x x + ν y y ) , ( ν x x ν y y ) , ( ν x y + ν y x ) , i ( ν x y ν y x ) ] T ,
M ̂ = Im G 0 I ̂ 4 + ( 0 Im G 1 Im G 2 Im G 3 Im G 1 0 0 0 Im G 2 0 0 0 Im G 3 0 0 0 ) + ( 0 0 0 0 0 0 Re G 3 Re G 2 0 Re G 3 0 Re G 1 0 Re G 2 Re G 1 0 ) ,
s z = ( Ω + s × ) × s ,
s z = Ω × s .
s = s 0 cos ( Ω z ) + ( ω × s 0 ) sin ( Ω z ) + ( ω s 0 ) ω [ 1 cos ( Ω z ) ] .
s 01 = 0 , s 02 = 0 ,
s 03 = 0 .
s 01 = 0 , tan ( Ω z ) = s 02 s 03 ,
tan ( Ω z ) = s 03 s 02 .
s z = ( s × ) × s .
s = 2 ( 1 + A 0 ) e z + ( 1 A 0 ) e z s 0 + ( 1 + A 0 ) e z 2 A 0 ( 1 A 0 ) e z ( 1 + A 0 ) e z + ( 1 A 0 ) e z σ ,
s 01 = 0 , s 02 = 0 .
( 1 + A 0 ) e z ( 1 A 0 ) e z ( 1 + A 0 ) e z + ( 1 A 0 ) e z = 0 ,
tan h ( z ) = s 03 σ 3 .
s 03 = 0 .
( 1 + A 0 ) e z ( 1 A 0 ) e z ( 1 + A 0 ) e z + ( 1 A 0 ) e z = 0 , s 02 = 0 ,
tanh ( z ) = s 01 , s 02 = 0 .
s 01 = 1 f 2 ( ρ ) cos [ m ( φ δ ) ] ,
s 02 = 1 f 2 ( ρ ) sin [ m ( φ δ ) ] ,
s 03 = f ( ρ ) .
f = χ 1 + ( ρ ρ * ) 2 ,
1 f ( ρ ) 2 cos [ m ( φ δ ) ] = 0 , tan ( Ω z ) = 1 f 2 ( ρ ) sin [ m ( φ δ ) ] f ( ρ ) ,
tan ( Ω z ) = f ( ρ ) 1 f 2 ( ρ ) sin [ m ( φ δ ) ] .
φ n = δ + π ( 2 n + 1 ) 2 m , tan ( Ω z ) = ± 1 f 2 ( ρ ) f ( ρ ) = ± χ ρ ρ * .
Γ = a m ( a ) χ ( a ) = const .
tanh ( z ) = f ( ρ ) σ 3 .
tanh ( z ) = 1 f 2 ( ρ ) cos [ m ( φ δ ) ] , 1 f 2 ( ρ ) sin [ m ( φ δ ) ] = 0 .
φ n = δ + π n m , tanh ( z ) = 1 f 2 ( ρ ) ,
s = A σ ( B × σ ) .
A z = ( 1 A 2 ) ,
B z = A B .
A = ( 1 + A 0 ) e z ( 1 A 0 ) e z ( 1 + A 0 ) e z + ( 1 A 0 ) e z ,
B = 2 ( 1 + A 0 ) e z + ( 1 A 0 ) e z B 0 ,

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