Abstract

The closed-form expression for the free-space propagation of superimposed Laguerre–Gaussian beams beyond the paraxial approximation is derived, and the composite polarization singularities formed by the transverse and longitudinal electric-field components are studied in detail. It is shown that there exist composite C-points and L-lines in vector nonparaxial fields. By suitably varying a control parameter, such as the off-axis distance, relative phase, or amplitude ratio, the motion, creation, and annihilation of composite C-points may appear, and in the process the sum of topological charge remains unchanged. The shift, deformation, combination, and disappearance of composite L-lines may take place. The topological relationship holds true. The results are compared with the previous work.

© 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 (IOP, 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. M. R. Dennis, “Polarization singularities in paraxial vector fields morphology and statistics,” Opt. Commun. 213, 201-221 (2002).
    [CrossRef]
  4. 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]
  5. I. Freund, A. I. Mokhun, M. S. Soskin, O. V. Angelsky, and I. I. Mokhun, “Stokes singularity relations,” Opt. Lett. 27, 545-547 (2002).
    [CrossRef]
  6. O. V. Angelsky, A. I. Mokhun, I. I. Mokhun, and M. S. Soskin, “Interferometric methods in diagnostics of polarization singularities,” Phys. Rev. E 65, 036602 (2002).
    [CrossRef]
  7. O. V. Angelsky, A. I. Mokhun, I. I. Mokhun, and M. S. Soskin, “The relationship between topological characteristics of component vortices and polarization singularities,” Opt. Commun. 207, 57-65 (2002).
    [CrossRef]
  8. I. Freund, “Polarization singularity indices in Gaussian laser beams,” Opt. Commun. 201, 251-270 (2002).
    [CrossRef]
  9. M. S. Soskin, V. Denisenko, and I. Freund, “Optical polarization singularities and elliptic stationary points,” Opt. Lett. 28, 1475-1477 (2003).
    [CrossRef] [PubMed]
  10. 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]
  11. 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 5458, 79-85 (2004).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  16. A. A. Chernyshov, Ch. V. Felde, H. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Vector singularities of the combined beams assembled from mutually incoherent orthogonally polarized components,” J. Opt. A, Pure Appl. Opt. 11, 094010(8) (2009).
    [CrossRef]
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    [CrossRef] [PubMed]

2009 (1)

A. A. Chernyshov, Ch. V. Felde, H. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Vector singularities of the combined beams assembled from mutually incoherent orthogonally polarized components,” J. Opt. A, Pure Appl. Opt. 11, 094010(8) (2009).
[CrossRef]

2008 (2)

Ch. 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 (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]

2004 (2)

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 5458, 79-85 (2004).
[CrossRef]

K. Duan and B. Lü, “Partially coherent noparaxial beams,” Opt. Lett. 29, 800-802 (2004).
[CrossRef] [PubMed]

2003 (2)

2002 (6)

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

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]

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

O. V. Angelsky, A. I. Mokhun, I. I. Mokhun, and M. S. Soskin, “Interferometric methods in diagnostics of polarization singularities,” Phys. Rev. E 65, 036602 (2002).
[CrossRef]

O. V. Angelsky, A. I. Mokhun, I. I. Mokhun, and M. S. Soskin, “The relationship between topological characteristics of component vortices and polarization singularities,” Opt. Commun. 207, 57-65 (2002).
[CrossRef]

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

2001 (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]

1997 (1)

1994 (1)

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

1990 (1)

J. V. Hajnal, “Observations of singularities in the electric and magnetic fields of freely propagating microwaves,” Proc. R. Soc. London, Ser. A 430, 413-421 (1990).
[CrossRef]

1987 (1)

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

An, Y.

Angelsky, O. V.

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

O. V. Angelsky, A. I. Mokhun, I. I. Mokhun, and M. S. Soskin, “Interferometric methods in diagnostics of polarization singularities,” Phys. Rev. E 65, 036602 (2002).
[CrossRef]

O. V. Angelsky, A. I. Mokhun, I. I. Mokhun, and M. S. Soskin, “The relationship between topological characteristics of component vortices and polarization singularities,” Opt. Commun. 207, 57-65 (2002).
[CrossRef]

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.

Ch. 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]

Bogatyryova, H. V.

A. A. Chernyshov, Ch. V. Felde, H. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Vector singularities of the combined beams assembled from mutually incoherent orthogonally polarized components,” J. Opt. A, Pure Appl. Opt. 11, 094010(8) (2009).
[CrossRef]

Chernyshov, A. A.

A. A. Chernyshov, Ch. V. Felde, H. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Vector singularities of the combined beams assembled from mutually incoherent orthogonally polarized components,” J. Opt. A, Pure Appl. Opt. 11, 094010(8) (2009).
[CrossRef]

Ch. 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 5458, 79-85 (2004).
[CrossRef]

Dennis, M. R.

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, Ser. A 457, 141-155 (2001).
[CrossRef]

Duan, K.

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 5458, 79-85 (2004).
[CrossRef]

Felde, Ch. V.

A. A. Chernyshov, Ch. V. Felde, H. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Vector singularities of the combined beams assembled from mutually incoherent orthogonally polarized components,” J. Opt. A, Pure Appl. Opt. 11, 094010(8) (2009).
[CrossRef]

Ch. 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, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
[CrossRef] [PubMed]

Freund, I.

Hajnal, J. V.

J. V. Hajnal, “Observations of singularities in the electric and magnetic fields of freely propagating microwaves,” Proc. R. Soc. London, Ser. A 430, 413-421 (1990).
[CrossRef]

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

Hasman, E.

Kleiner, V.

Liang, C.

Lü, B.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (Univ. of California Press, 1966).

Maier, M.

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]

Maleev, I. D.

Mokhun, A. I.

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 I. I. Mokhun, “Stokes singularity relations,” Opt. Lett. 27, 545-547 (2002).
[CrossRef]

O. V. Angelsky, A. I. Mokhun, I. I. Mokhun, and M. S. Soskin, “The relationship between topological characteristics of component vortices and polarization singularities,” Opt. Commun. 207, 57-65 (2002).
[CrossRef]

O. V. Angelsky, A. I. Mokhun, I. I. Mokhun, and M. S. Soskin, “Interferometric methods in diagnostics of polarization singularities,” Phys. Rev. E 65, 036602 (2002).
[CrossRef]

Mokhun, I. I.

O. V. Angelsky, A. I. Mokhun, I. I. Mokhun, and M. S. Soskin, “Interferometric methods in diagnostics of polarization singularities,” Phys. Rev. E 65, 036602 (2002).
[CrossRef]

O. V. Angelsky, A. I. Mokhun, I. I. Mokhun, and M. S. Soskin, “The relationship between topological characteristics of component vortices and polarization singularities,” Opt. Commun. 207, 57-65 (2002).
[CrossRef]

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

Niv, A.

Nye, J. F.

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

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

Polyanskii, P. V.

A. A. Chernyshov, Ch. V. Felde, H. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Vector singularities of the combined beams assembled from mutually incoherent orthogonally polarized components,” J. Opt. A, Pure Appl. Opt. 11, 094010(8) (2009).
[CrossRef]

Ch. 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]

Schoonover, R. W.

Schwarz, U. T.

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.

A. A. Chernyshov, Ch. V. Felde, H. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Vector singularities of the combined beams assembled from mutually incoherent orthogonally polarized components,” J. Opt. A, Pure Appl. Opt. 11, 094010(8) (2009).
[CrossRef]

Ch. 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 5458, 79-85 (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 I. I. Mokhun, “Stokes singularity relations,” Opt. Lett. 27, 545-547 (2002).
[CrossRef]

O. V. Angelsky, A. I. Mokhun, I. I. Mokhun, and M. S. Soskin, “The relationship between topological characteristics of component vortices and polarization singularities,” Opt. Commun. 207, 57-65 (2002).
[CrossRef]

O. V. Angelsky, A. I. Mokhun, I. I. Mokhun, and M. S. Soskin, “Interferometric methods in diagnostics of polarization singularities,” Phys. Rev. E 65, 036602 (2002).
[CrossRef]

Swartzlander, G. A.

Visser, T. D.

Zeng, X.

Appl. Opt. (1)

J. Opt. A, Pure Appl. Opt. (1)

A. A. Chernyshov, Ch. V. Felde, H. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Vector singularities of the combined beams assembled from mutually incoherent orthogonally polarized components,” J. Opt. A, Pure Appl. Opt. 11, 094010(8) (2009).
[CrossRef]

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

JETP Lett. (1)

Ch. 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. (3)

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

O. V. Angelsky, A. I. Mokhun, I. I. Mokhun, and M. S. Soskin, “The relationship between topological characteristics of component vortices and polarization singularities,” Opt. Commun. 207, 57-65 (2002).
[CrossRef]

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

Opt. Express (2)

Opt. Lett. (4)

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. E (1)

O. V. Angelsky, A. I. Mokhun, I. I. Mokhun, and M. S. Soskin, “Interferometric methods in diagnostics of polarization singularities,” Phys. Rev. E 65, 036602 (2002).
[CrossRef]

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 (3)

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

J. V. Hajnal, “Observations of singularities in the electric and magnetic fields of freely propagating microwaves,” Proc. R. Soc. London, Ser. A 430, 413-421 (1990).
[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]

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 5458, 79-85 (2004).
[CrossRef]

Other (2)

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

R. K. Luneburg, Mathematical Theory of Optics (Univ. of California Press, 1966).

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Figures (5)

Fig. 1
Fig. 1

Contour lines of s 1 = 0 (solid curves) and s 2 = 0 (dashed curves) at the position z = 5 z R for different values of the off-axis distance (a) a = 2.50 λ , (b) a = 2.90 λ , (c) a = 3.90 λ , (d) a = 3.95 λ . “◼”—left-handed C-point, “●”—right-handed C-point. The calculation parameters are given in the text.

Fig. 2
Fig. 2

Critical off-axis distance a λ for the creation of a pair of C c and D c labeled “◻” and for the annihilation of a pair of A c and D c labeled “◯” versus the propagation distance z z R .

Fig. 3
Fig. 3

Contour lines of s 1 = 0 (solid curves) and s 2 = 0 (dashed curves) at z = 5 z R for different values of the relative phase (a) β = π 100 , (b) β = π 40 .

Fig. 4
Fig. 4

Contour lines of s 3 = 0 at z = 5 z R for different values of the relative phase (a) β = π 30 , (b) β = π 10 , (c) β = π 7 .

Fig. 5
Fig. 5

Contour lines of s 1 = 0 (solid curve), s 2 = 0 (dashed curve), and s 3 = 0 (dotted curves) at z = 5 z R , “●”—left-handed C-point, “◼”—right-handed C-point, “▲”—phase singularity of E x , “★”— s 31 vortex.

Equations (27)

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

E x ν ( ρ ν , θ ν , z ) = E 0 ν w 0 ν w ν ( 2 ρ ν w ν ) | l ν | exp ( ρ ν 2 w ν 2 ) L p ν | l ν | ( 2 ρ ν 2 w ν 2 ) × exp { i [ k z + k ρ ν 2 2 R ν ( z ) + l ν θ ν ( 2 p ν + | l ν | + 1 ) arctan ( z z R ν ) ] } ,
E y ν ( ρ ν , θ ν , z ) = 0 , ( ν = 1 , 2 ) ,
E x ( ρ , θ , 0 ) = E x 1 ( ρ 1 , θ 1 , 0 ) exp ( i β ) + E x 2 ( ρ 2 , θ 2 , 0 ) ,
E x ( x 0 , y 0 , 0 ) = E 01 2 w 01 ( x 0 a + i y 0 ) exp [ ( x 0 a ) 2 + y 0 2 w 01 2 + i β ] + E 02 2 w 02 ( x 0 + b i y 0 ) exp [ ( x 0 + b ) 2 + y 0 2 w 02 2 ] .
E x ( x , y , z ) = 1 2 π E x ( x 0 , y 0 , 0 ) G ( r , r 0 ) z d x 0 d y 0 ,
E y ( x , y , z ) = 1 2 π E y ( x 0 , y 0 , 0 ) G ( r , r 0 ) z d x 0 d y 0 ,
E z ( x , y , z ) = 1 2 π [ E x ( x 0 , y 0 , 0 ) G ( r , r 0 ) x + E y ( x 0 , y 0 , 0 ) G ( r , r 0 ) y ] d x 0 d y 0 ,
G ( r , r 0 ) = exp ( i k | r r 0 | ) | r r 0 | ,
| r r 0 | r + x 0 2 + y 0 2 2 x x 0 2 y y 0 2 r ,
E x ( x , y , z ) = i k π z 2 λ r 3 exp ( i k r ) { E 01 [ i ( a x ) + y ] w 01 g 1 2 exp ( h 1 g 1 a 2 w 01 2 + i β ) E 02 [ i ( b + x ) + y ] w 02 g 2 2 exp ( h 2 g 2 b 2 w 02 2 ) } ,
E y ( x , y , z ) = 0 ,
E z ( x , y , z ) = i π 2 2 λ r 4 exp ( i k r ) { E 01 w 01 5 g 1 3 exp ( h 1 g 1 a 2 w 01 2 + i β ) { 4 a 2 r 2 ( g 1 w 01 2 1 ) + 2 a k r w 01 2 ( 2 i x y ) + w 01 4 k 2 x ( x + i y ) 2 w 01 4 g 1 r [ r + i k x ( x + i y ) ] } E 02 g 2 3 w 02 5 exp ( h 2 g 2 b 2 w 02 2 ) { 4 b 2 r 2 ( g 2 w 02 2 1 ) 2 b k r w 02 2 ( 2 i x + y ) k 2 x w 02 4 ( x i y ) + 2 g 2 w 02 4 r [ r + k x ( i x + y ) ] } } ,
g 1 = 1 w 01 2 i k 2 r , g 2 = 1 w 02 2 i k 2 r ,
h 1 = ( a w 01 2 i k x 2 r ) 2 + ( i k y 2 r ) 2 , h 2 = ( b w 02 2 + i k x 2 r ) 2 + ( i k y 2 r ) 2 .
α 1 = | E z ( x , y , z ) | ,
δ 1 = arg [ E z ( x , y , z ) ] ,
α 2 = | E x ( , y , z ) | ,
δ 2 = arg [ E x ( x , y , z ) ] .
S 0 = α 1 2 + α 2 2 ,
S 1 = α 1 2 α 2 2 ,
S 2 = 2 α 1 α 2 cos δ ,
S 3 = 2 α 1 α 2 sin δ ,
s 12 = s 1 + i s 2 ,
s 23 = s 2 + i s 3 ,
s 31 = s 3 + i s 1 .
Re [ E z ( x , y , z ) ] Im [ E x ( x , y , z ) ] Re [ E x ( x , y , z ) ] Im [ E z ( x , y , z ) ] = 0 .
2 σ k ( k ) q i j = ( k ) σ i q j k = ( k ) σ j q i k ( i , j , k = 1 , 2 , 3 ) ,

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