Abstract

On the basis of the vector Rayleigh–Sommerfeld diffraction integrals, the analytical expression for Gaussian vortex beams diffracted at a half-plane screen beyond the paraxial approximation is derived and used to study the polarization singularities formed by the transverse and longitudinal electric field components in the diffracted field. It is shown that there exist C-points and L-lines that depend on the off-axis parameters in the x and y directions, waist width, wavelength, and topological charge of diffracted Gaussian vortex beams, as well as on the position parameter. By a suitable variation of the off-axis and position parameters, the creation, motion, and annihilation of polarization singularities may take place, and the topological relationship holds true. Therefore, the nonparaxial beam diffraction and propagation provide a method for generating polarization singularities.

© 2009 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 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. 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. 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]
  9. I. Freund, “Polarization singularity indices in Gaussian laser beams,” Opt. Commun. 201, 251-270 (2002).
    [CrossRef]
  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. Denisenko, and I. Freund, “Optical polarization singularities and elliptic stationary points,” Opt. Lett. 28, 1475-1477 (2003).
    [CrossRef] [PubMed]
  12. M. R. Dennis, “Polarization singularity anisotropy: determining monstardom,” Opt. Lett. 33, 2572-2574 (2008).
    [CrossRef] [PubMed]
  13. R. W. Schoonover and T. D. Visser, “Polarization singularities of focused, radially polarized fields,” Opt. Express 14, 5733-5745 (2006).
    [CrossRef] [PubMed]
  14. R. K. Luneburg, Mathematical Theory of Optics (Univ. of California Press, 1966).
  15. K. Duan and B. Lü, “Partially coherent noparaxial beams,” Opt. Lett. 29, 800-802 (2004).
    [CrossRef] [PubMed]
  16. I. Freund and N. Shvartsman, “Wave-field phase singularities: The sign principle,” Phys. Rev. A 50, 5164-5172 (1994).
    [CrossRef] [PubMed]
  17. J. Masajada, “Half-plane diffraction in the case of Gaussian beams containing an optical vortex,” Opt. Commun. 175, 289-294 (2000).
    [CrossRef]
  18. P. Liu, K. Cheng, and B. Lü, “Propagation dynamics of off-axis phase singularities,” Acta Phys. Sin. 57, 1683-1688 (2008).

2008 (3)

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 (1)

2003 (1)

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]

2000 (1)

J. Masajada, “Half-plane diffraction in the case of Gaussian beams containing an optical vortex,” Opt. Commun. 175, 289-294 (2000).
[CrossRef]

1994 (1)

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

Angelsky, O. V.

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]

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.

Cheng, K.

P. Liu, K. Cheng, and B. Lü, “Propagation dynamics of off-axis phase singularities,” Acta Phys. Sin. 57, 1683-1688 (2008).

Denisenko, V.

Dennis, M. R.

M. R. Dennis, “Polarization singularity anisotropy: determining monstardom,” Opt. Lett. 33, 2572-2574 (2008).
[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 Ser. A 457, 141-155 (2001).
[CrossRef]

Duan, K.

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.

Hasman, E.

Kleiner, V.

Liu, P.

P. Liu, K. Cheng, and B. Lü, “Propagation dynamics of off-axis phase singularities,” Acta Phys. Sin. 57, 1683-1688 (2008).

Lü, B.

P. Liu, K. Cheng, and B. Lü, “Propagation dynamics of off-axis phase singularities,” Acta Phys. Sin. 57, 1683-1688 (2008).

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

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]

Masajada, J.

J. Masajada, “Half-plane diffraction in the case of Gaussian beams containing an optical vortex,” Opt. Commun. 175, 289-294 (2000).
[CrossRef]

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, Natural Focusing and Fine Structure of Light (IOP Publishing, 1999).

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.

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]

Visser, T. D.

Acta Phys. Sin. (1)

P. Liu, K. Cheng, and B. Lü, “Propagation dynamics of off-axis phase singularities,” Acta Phys. Sin. 57, 1683-1688 (2008).

Opt. Commun. (4)

J. Masajada, “Half-plane diffraction in the case of Gaussian beams containing an optical vortex,” Opt. Commun. 175, 289-294 (2000).
[CrossRef]

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

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

Opt. Express (2)

Opt. Lett. (5)

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 (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]

Other (2)

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

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

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Figures (6)

Fig. 1
Fig. 1

Schematic illustration of a half-plane screen.

Fig. 2
Fig. 2

Contour lines of s 1 = 0 (solid curves) and s 2 = 0 (dashed curves) at the position y = 3 L R for different values of the off-axis parameter b : (a) b = 0.60 w 0 , (b) b = 0.66 w 0 , (c) b = 0.69 w 0 , (d) b = 0.75 w 0 .

Fig. 3
Fig. 3

Contour lines of s 1 = 0 (solid curves) and s 2 = 0 (dashed curves) at b = 0.75 w 0 for different values of y : (a) y = 3.2 L R , (b) y = 3.4 L R .

Fig. 4
Fig. 4

Distance △ between a pair of the C-points D c and E c versus y L R .

Fig. 5
Fig. 5

Contour lines of s 3 = 0 at the position y = 2 L R for different values of the off-axis parameter b : (a) b = 0.9 w 0 , (b) b = 1.4 w 0 , (c) b = 1.5 w 0 .

Fig. 6
Fig. 6

Contour lines of s 1 = 0 (solid curve), s 2 = 0 (dashed curve), and s 3 = 0 (dotted curve) at the position y = 2 L R ; “●”represents left-handed C-point, “◼”, right-handed C-point, “▲”, phase singularity of E x or E z , “★”, s 31 vortex.

Equations (23)

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E x ( x 0 , y 0 , 0 ) = E 0 [ ( x 0 a ) + i sgn ( m ) ( y 0 b ) ] | m | exp ( x 0 2 + y 0 2 w 0 2 ) ,
E y ( x 0 , y 0 , 0 ) = 0 ,
T ( x 0 , y 0 ) = { 1 , y 0 0 0 y 0 < 0 }
E x ( x 0 , y 0 , 0 ) = T ( x 0 , y 0 ) E 0 [ ( x 0 a ) + i sgn ( m ) ( y 0 b ) ] | m | exp ( x 0 2 + y 0 2 w 0 2 ) ,
E y ( x 0 , y 0 , 0 ) = 0.
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 ) = π w 0 2 z 2 λ r [ 2 ( L R + i r ) ] 5 2 exp { i k [ r L R ( x 2 + y 2 ) 2 r ( i r + L R ) ] } { 2 ( 1 + i ) 2 π ( r i L R ) [ a ( r i L R ) + b ( i r + L R ) + i x L R ] erf i [ y L R ( 1 + i ) w 0 2 r ( i r + L R ) ] + 4 π ( i r + L R ) [ L R ( x + i y ) + b ( r i L R ) ] 4 w 0 r ( i r + L R ) ( r i L R ) exp [ i k y 2 L R 4 r ( i r + L R ) ] 4 a π ( i r + L R ) 3 + 8 π y 2 L R 2 2 ( i r + L R ) erf i [ i k y 2 L R 2 r ( i r + L R ) ] } ,
E y ( x , y , z ) = 0 ,
E z ( x , y , z ) = π w 0 2 i 2 λ [ 2 ( L R + i r ) ] 7 2 exp { k [ 2 r 2 ( L R + i r ) L R ( x 2 + y 2 ) ] 2 r ( r i L R ) } { 4 ( 1 + i ) 2 π ( r i L R ) [ ( r i L R ) ( w 0 2 + 2 x a + 2 i x b ) + 2 i x 2 L R ] erf i [ y L R ( 1 + i ) w 0 2 r ( i r + L R ) ] 4 π ( i r + L R ) [ 2 x L R ( x + i y ) + 2 x b ( r i L R ) + ( L R + i r ) ( w 0 2 2 x a ) ] 8 x w 0 r ( i r + L R ) ( r i L R ) exp [ i k y 2 L R 4 r ( i r + L R ) ] + 8 x y π L R ( i r + L R ) erf i [ i k y 2 L R 2 r ( i r + L R ) ] } ,
E = E x i + E z k ,
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 ,
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 ,

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