Abstract

Following our earlier work (F. Flossmann et al., Phys. Rev. Lett. 95 253901 (2005)), we describe the fine polarization structure of a beam containing optical vortices propagating through a birefringent crystal, both experimentally and theoretically.We emphasize here the zero surfaces of the Stokes parameters in three-dimensional space, two transverse dimensions and the third corresponding to optical path length in the crystal. We find that the complicated network of polarization singularities reported earlier – lines of circular polarization (C lines) and surfaces of linear polarization (L surfaces) – can be understood naturally in terms of the zeros of the Stokes parameters.

© 2006 Optical Society of America

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References

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  1. J. F. Nye, Natural focusing and fine structure of light: caustics and wave dislocations, (IoP Publishing, 1999).
  2. M. S. Soskin and M. Vasnetsov, "Singular optics," Prog. Opt. 42219-276 (2001).
    [CrossRef]
  3. J.F. Nye, "Lines of circular polarization in electromagnetic wave fields," Proc. R. Soc. Lond. A 389279-290 (1983).
    [CrossRef]
  4. M. R. Dennis, "Polarization singularities in paraxial vector fields: morphology and statistics," Opt. Commun. 213201-221 (2002).
    [CrossRef]
  5. M. V. Berry, "Singularities in waves and rays," in Les Houches, Session XXV - Physics of Defects, R. Balian, M. Kléman, and J.-P. Poirier, eds., (North-Holland, 1981).
  6. K. O’Holleran, M. J. Padgett, and M. R. Dennis, "Topology of optical vortex lines formed by the interference of three, four and five plane waves," Opt. Express 143039-3044 (2006).
    [CrossRef] [PubMed]
  7. M. V. Berry and M. R. Dennis, "Phase singularities in isotropic random waves," Proc. R. Soc. Lond. A 4562059-2079 (2000).
    [CrossRef]
  8. L. Allen, M. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes, Phys. Rev. A 458185-8189 (1992).
    [CrossRef] [PubMed]
  9. F. Flossmann, U. T. Schwarz, and M. Maier "Propagation dynamics of optical vortices in Laguerre-Gaussian beams," Opt. Commun. 250218-230 (2005).
    [CrossRef]
  10. R. Dorn, S. Quabis, and G. Leuchs, "Sharper focus for a radially polarized light beam," Phys. Rev. Lett. 91233901 (2003).
    [CrossRef] [PubMed]
  11. J. V. Hajnal, "Observation on singularities in the electric and magnetic fields of freely propagating microwaves," Proc. R. Soc. Lond. A 430413-421 (1990).
    [CrossRef]
  12. A. Niv, G. Biener, V. Kleiner, and E. Hasman, "Manipulation of the Pancharatnam phase in vectorial vortices," Opt. Express 144208-4220 (2006).
    [CrossRef] [PubMed]
  13. M. V. Berry and M. R. Dennis, "The optical singularities of birefringent dichroic chiral crystals," Proc. R. Soc. Lond. A 4591261-1292 (2003).
    [CrossRef]
  14. M. V. Berry and M. R. Jeffrey, "Conical diffraction: Hamilton’s diabolic point at the heart of crystal optics," Prog. Opt.in press.
  15. Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, "Fine structure of singular beams in crystals: colours and polarization," J. Opt. A: Pure Appl. Opt. 6S217-S228 (2004).
    [CrossRef]
  16. 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. 95253901 (2005).
    [CrossRef] [PubMed]
  17. B. Riemann, in Bernhard Riemann’s gesammelte mathematische Werke und wissenschaftlicher Nachlass, edited by H. Weber (Teubner, Leipzig, 1892) p. 301.
  18. R. M. A. Azzam and N. M. Bashara, Ellipsometry and polarized light (North-Holland, 1977).
  19. D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized light in optics and spectroscopy (Academic Press, San Diego, 1990).
  20. I. Freund, A. I. Mokhun, M. S. Soskin, O. V. Angelsky, and I. I. Mokhun, "Stokes singularity relations," Opt. Lett. 27545-547 (2002).
    [CrossRef]
  21. D. J. Struik, Lectures on Classical Differential Geometry (Dover, 1988).
  22. A. Volyar, V. Shvedov, T. Fadeyeva, A. S. Desyatnikov, D. N. Neshev, W. Krolikowski, and Y. S. Kivshar, "Generation of single-charge optical vortices with an uniaxial crystal" Opt. Express 143724-3729 (2006).
    [CrossRef]
  23. M. R. Dennis, "Braided nodal lines in wave superpositions," New J. Phys. 5134 (2003).
    [CrossRef]
  24. F. Flossmann, Singularit¨aten von Phase und Polarisation des Lichts (PhD Thesis, University of Regensburg, 2006).

2006 (3)

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. 95253901 (2005).
[CrossRef] [PubMed]

F. Flossmann, U. T. Schwarz, and M. Maier "Propagation dynamics of optical vortices in Laguerre-Gaussian beams," Opt. Commun. 250218-230 (2005).
[CrossRef]

2004 (1)

Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, "Fine structure of singular beams in crystals: colours and polarization," J. Opt. A: Pure Appl. Opt. 6S217-S228 (2004).
[CrossRef]

2003 (3)

M. R. Dennis, "Braided nodal lines in wave superpositions," New J. Phys. 5134 (2003).
[CrossRef]

R. Dorn, S. Quabis, and G. Leuchs, "Sharper focus for a radially polarized light beam," Phys. Rev. Lett. 91233901 (2003).
[CrossRef] [PubMed]

M. V. Berry and M. R. Dennis, "The optical singularities of birefringent dichroic chiral crystals," Proc. R. Soc. Lond. A 4591261-1292 (2003).
[CrossRef]

2002 (2)

M. R. Dennis, "Polarization singularities in paraxial vector fields: morphology and statistics," Opt. Commun. 213201-221 (2002).
[CrossRef]

I. Freund, A. I. Mokhun, M. S. Soskin, O. V. Angelsky, and I. I. Mokhun, "Stokes singularity relations," Opt. Lett. 27545-547 (2002).
[CrossRef]

2001 (1)

M. S. Soskin and M. Vasnetsov, "Singular optics," Prog. Opt. 42219-276 (2001).
[CrossRef]

2000 (1)

M. V. Berry and M. R. Dennis, "Phase singularities in isotropic random waves," Proc. R. Soc. Lond. A 4562059-2079 (2000).
[CrossRef]

1992 (1)

L. Allen, M. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes, Phys. Rev. A 458185-8189 (1992).
[CrossRef] [PubMed]

1990 (1)

J. V. Hajnal, "Observation on singularities in the electric and magnetic fields of freely propagating microwaves," Proc. R. Soc. Lond. A 430413-421 (1990).
[CrossRef]

1983 (1)

J.F. Nye, "Lines of circular polarization in electromagnetic wave fields," Proc. R. Soc. Lond. A 389279-290 (1983).
[CrossRef]

Allen, L.

L. Allen, M. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes, Phys. Rev. A 458185-8189 (1992).
[CrossRef] [PubMed]

Angelsky, O. V.

Beijersbergen, M.

L. Allen, M. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes, Phys. Rev. A 458185-8189 (1992).
[CrossRef] [PubMed]

Berry, M. V.

M. V. Berry and M. R. Dennis, "The optical singularities of birefringent dichroic chiral crystals," Proc. R. Soc. Lond. A 4591261-1292 (2003).
[CrossRef]

M. V. Berry and M. R. Dennis, "Phase singularities in isotropic random waves," Proc. R. Soc. Lond. A 4562059-2079 (2000).
[CrossRef]

M. V. Berry and M. R. Jeffrey, "Conical diffraction: Hamilton’s diabolic point at the heart of crystal optics," Prog. Opt.in press.

Biener, G.

Dennis, M. R.

K. O’Holleran, M. J. Padgett, and M. R. Dennis, "Topology of optical vortex lines formed by the interference of three, four and five plane waves," Opt. Express 143039-3044 (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. 95253901 (2005).
[CrossRef] [PubMed]

M. R. Dennis, "Braided nodal lines in wave superpositions," New J. Phys. 5134 (2003).
[CrossRef]

M. V. Berry and M. R. Dennis, "The optical singularities of birefringent dichroic chiral crystals," Proc. R. Soc. Lond. A 4591261-1292 (2003).
[CrossRef]

M. R. Dennis, "Polarization singularities in paraxial vector fields: morphology and statistics," Opt. Commun. 213201-221 (2002).
[CrossRef]

M. V. Berry and M. R. Dennis, "Phase singularities in isotropic random waves," Proc. R. Soc. Lond. A 4562059-2079 (2000).
[CrossRef]

Desyatnikov, A. S.

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, "Sharper focus for a radially polarized light beam," Phys. Rev. Lett. 91233901 (2003).
[CrossRef] [PubMed]

Egorov, Y. A.

Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, "Fine structure of singular beams in crystals: colours and polarization," J. Opt. A: Pure Appl. Opt. 6S217-S228 (2004).
[CrossRef]

Fadeyeva, T.

Fadeyeva, T. A.

Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, "Fine structure of singular beams in crystals: colours and polarization," J. Opt. A: Pure Appl. Opt. 6S217-S228 (2004).
[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. 95253901 (2005).
[CrossRef] [PubMed]

F. Flossmann, U. T. Schwarz, and M. Maier "Propagation dynamics of optical vortices in Laguerre-Gaussian beams," Opt. Commun. 250218-230 (2005).
[CrossRef]

Freund, I.

Hajnal, J. V.

J. V. Hajnal, "Observation on singularities in the electric and magnetic fields of freely propagating microwaves," Proc. R. Soc. Lond. A 430413-421 (1990).
[CrossRef]

Hasman, E.

Jeffrey, M. R.

M. V. Berry and M. R. Jeffrey, "Conical diffraction: Hamilton’s diabolic point at the heart of crystal optics," Prog. Opt.in press.

Kivshar, Y. S.

Kleiner, V.

Krolikowski, W.

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, "Sharper focus for a radially polarized light beam," Phys. Rev. Lett. 91233901 (2003).
[CrossRef] [PubMed]

Maier, M.

F. Flossmann, U. T. Schwarz, and M. Maier "Propagation dynamics of optical vortices in Laguerre-Gaussian beams," Opt. Commun. 250218-230 (2005).
[CrossRef]

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. 95253901 (2005).
[CrossRef] [PubMed]

Mokhun, A. I.

Mokhun, I. I.

Neshev, D. N.

Niv, A.

Nye, J.F.

J.F. Nye, "Lines of circular polarization in electromagnetic wave fields," Proc. R. Soc. Lond. A 389279-290 (1983).
[CrossRef]

O’Holleran, K.

Padgett, M. J.

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, "Sharper focus for a radially polarized light beam," Phys. Rev. Lett. 91233901 (2003).
[CrossRef] [PubMed]

Schwarz, U. T.

F. Flossmann, U. T. Schwarz, and M. Maier "Propagation dynamics of optical vortices in Laguerre-Gaussian beams," Opt. Commun. 250218-230 (2005).
[CrossRef]

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. 95253901 (2005).
[CrossRef] [PubMed]

Shvedov, V.

Soskin, M. S.

Spreeuw, R. J. C.

L. Allen, M. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes, Phys. Rev. A 458185-8189 (1992).
[CrossRef] [PubMed]

Vasnetsov, M.

M. S. Soskin and M. Vasnetsov, "Singular optics," Prog. Opt. 42219-276 (2001).
[CrossRef]

Volyar, A.

Volyar, A. V.

Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, "Fine structure of singular beams in crystals: colours and polarization," J. Opt. A: Pure Appl. Opt. 6S217-S228 (2004).
[CrossRef]

Woerdman, J. P.

L. Allen, M. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes, Phys. Rev. A 458185-8189 (1992).
[CrossRef] [PubMed]

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

Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, "Fine structure of singular beams in crystals: colours and polarization," J. Opt. A: Pure Appl. Opt. 6S217-S228 (2004).
[CrossRef]

New J. Phys. (1)

M. R. Dennis, "Braided nodal lines in wave superpositions," New J. Phys. 5134 (2003).
[CrossRef]

Opt. Commun. (2)

F. Flossmann, U. T. Schwarz, and M. Maier "Propagation dynamics of optical vortices in Laguerre-Gaussian beams," Opt. Commun. 250218-230 (2005).
[CrossRef]

M. R. Dennis, "Polarization singularities in paraxial vector fields: morphology and statistics," Opt. Commun. 213201-221 (2002).
[CrossRef]

Opt. Express (3)

Opt. Lett. (1)

Phys. Rev. A (1)

L. Allen, M. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes, Phys. Rev. A 458185-8189 (1992).
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

R. Dorn, S. Quabis, and G. Leuchs, "Sharper focus for a radially polarized light beam," Phys. Rev. Lett. 91233901 (2003).
[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. 95253901 (2005).
[CrossRef] [PubMed]

Proc. R. Soc. Lond. A (4)

J. V. Hajnal, "Observation on singularities in the electric and magnetic fields of freely propagating microwaves," Proc. R. Soc. Lond. A 430413-421 (1990).
[CrossRef]

M. V. Berry and M. R. Dennis, "The optical singularities of birefringent dichroic chiral crystals," Proc. R. Soc. Lond. A 4591261-1292 (2003).
[CrossRef]

J.F. Nye, "Lines of circular polarization in electromagnetic wave fields," Proc. R. Soc. Lond. A 389279-290 (1983).
[CrossRef]

M. V. Berry and M. R. Dennis, "Phase singularities in isotropic random waves," Proc. R. Soc. Lond. A 4562059-2079 (2000).
[CrossRef]

Prog. Opt. (2)

M. S. Soskin and M. Vasnetsov, "Singular optics," Prog. Opt. 42219-276 (2001).
[CrossRef]

M. V. Berry and M. R. Jeffrey, "Conical diffraction: Hamilton’s diabolic point at the heart of crystal optics," Prog. Opt.in press.

Other (7)

B. Riemann, in Bernhard Riemann’s gesammelte mathematische Werke und wissenschaftlicher Nachlass, edited by H. Weber (Teubner, Leipzig, 1892) p. 301.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and polarized light (North-Holland, 1977).

D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized light in optics and spectroscopy (Academic Press, San Diego, 1990).

J. F. Nye, Natural focusing and fine structure of light: caustics and wave dislocations, (IoP Publishing, 1999).

M. V. Berry, "Singularities in waves and rays," in Les Houches, Session XXV - Physics of Defects, R. Balian, M. Kléman, and J.-P. Poirier, eds., (North-Holland, 1981).

D. J. Struik, Lectures on Classical Differential Geometry (Dover, 1988).

F. Flossmann, Singularit¨aten von Phase und Polarisation des Lichts (PhD Thesis, University of Regensburg, 2006).

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Figures (7)

Fig. 1.
Fig. 1.

(a) Representation of the Poincaré sphere. In abstract three-dimensional space, the Stokes parameters are Cartesian axes for a sphere, whose points represent states of elliptic polarization. 2α and 2ω are the azimuth and colatitude angles respectively for spherical coordinates with S 3 as the axis. Not shown are 2δ and 2β, which are the azimuth and latitude angles for spherical coordinates with S 1 as the axis. (b) Representation of the ellipse parameters. α is the angle of the major axis, ω the arctangent of the ratio of minor and major axes (signed by handedness).

Fig. 2.
Fig. 2.

(a) Experimental setup scheme. A beam with uniform polarization d in and varying complex amplitude ψ(x, y) (incorporating a Gaussian factor with waist width w 0=0.7mm), is synthesized from a He–Ne laser using the polarizer Pol2 and spatial light modulator SLM. Higher diffraction orders are removed by a telescope (lenses L3 and L4) and a pinhole P. The KDP birefringent crystal is located in the waist plane. The Stokes parameters for each outgoing x, y are measured by a CCD camera using different combinations of the polarizer Pol3 and λ/4-plate. A second laser propagating through the analyser gives a fixed reference point. (b) Orientation of crystal, and ordinary and extraordinary rays (angles are exaggerated). The normal to the crystal surface is initially in the z-direction, and the optic axis is the xz-plane. Λ is varied by rotating the crystal about the y-axis, which does not change the polarizations d 1,2. Typically two Λ-periods are measured, corresponding to Δγ=0.112°. (c) and (d) Intensity profiles of input and outgoing beams.

Fig. 3.
Fig. 3.

The polarization field for a constant phase shift Λ, for an input beam ψ=(x+iy)exp(-(x 2+y 2)/w02). Background intensity is the output intensity S 0, the small ellipses represent the polarization state (as in Fig. 1(b)), with C points of circular polarization represented by colored circles, the L line of linear polarization by a blue curve. (a) Input linear polarization with α in=β in=45°,δ in=0° and Λ=π+0.1. (b) Input circular polarization with β in=δ in=45°, and Λ=π/2+0.1. (c) Theoretical plot for the situation represented in (a) and (b), based on Eq. (3).

Fig. 4.
Fig. 4.

Representations of the experimentally-determined polarization field for fixed Λ=0, with Stokes parameter zero contours (light brown S 1=0, dark orange S 2=0, light blue S 3=0). The S 1,S 2=0 lines intersect at C points, represented by colored circles, and the S 2,S 3=0 lines intersect at the positions of the refracted vortices xs, y=0. (a) Intensity and ellipse field. (b) Polarization streamlines. The curve tangent at each point represents the ellipse axis orientations α. (c) Orientation of ellipse axis α as hue plot. The singular nature of the C points, with index ±1/2, is exhibited clearly in (b) and (c), where α is not defined.

Fig. 5.
Fig. 5.

Experimentally-determined zero surfaces of the Stokes parameters in x, y,Λ-space, and polarization singularities, for an input field with a strength 1 vortex. (a) Surface S 1=0, with C lines plotted on the surface. (b) Surface S 2=0, with C lines on surface. (c) Surfaces S 1=S 2=0, with C lines along the intersection. (d) L surface (S 3=0), with C lines off surface. The S 1=0 surface is invariant with respect to Λ, and the S 2=0 and S 3=0 surfaces are the same up to a quarter Λ-period, and are topologically Riemann’s minimal surface.

Fig. 6.
Fig. 6.

Zero surfaces of the measured Stokes parameters in x, y,Λ-space, and polarization singularities, for an input field with an input Gaussian (no vortices). (a) Surfaces S 1=0 (consisting of a single sheet with constant x), and surfaces S 2=0, corresponding (approximately) to planes of constant Λ. The two sets of surfaces intersect along C lines. (b) L surfaces (S 3=0), corresponding (approximately) to planes of constant Λ.

Fig. 7.
Fig. 7.

C lines (intersection of S 1,S 2=0 surfaces) and L surface (S 3=0) in x, y,Λ-space, for an input field with a strength 2 vortex. (a) Experimental plot. (b) Theoretical plot. The theoretical L surface corresponds to two intersecting Riemann minimal surfaces. The reconstruction of the self-intersections in the experiment is very sensitive to experimental imperfections.

Equations (14)

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E = I ( cos ω cos α i sin ω sin α , cos ω sin α + i sin ω cos α )
= I ( cos β exp ( i δ ) , sin β exp ( i δ ) ) ,
S 0 S 1 S 2 S 3 = E x 2 + E y 2 , = E x 2 E y 2 = E x * E y + E x E y * = i ( E x * E y E x E y * ) = I , = I 0 ° I 90 ° = I 45 ° I 135 ° = I rhcp I lhcp = I cos 2 α cos 2 ω = I sin 2 α cos 2 ω = I sin 2 ω = I cos 2 β , = I sin 2 β cos 2 δ , = I sin 2 β sin 2 δ . }
E ( x , y , Λ ) = j = 1,2 ψ ( x + ( 1 ) j s , y ) ( d in · d j ) d j exp ( i ( 1 ) j Λ 2 )
= ( ψ ( x s , y ) cos β in exp ( i [ Λ 2 + δ in ] ) , ψ ( x + s , y ) sin β in exp ( i [ Λ 2 + δ in ] ) ) .
S 0 = ψ 2 cos 2 β in + ψ + 2 sin 2 β in
S 1 = ψ 2 cos 2 β in ψ + 2 sin 2 β in
S 2 = sin 2 β in ( Re { ψ * ψ + } cos ( Λ + 2 δ in ) + Im { ψ * ψ + } sin ( Λ + 2 δ in ) )
S 3 = sin 2 β in ( Re { ψ * ψ + } sin ( Λ + 2 δ in ) + Im { ψ * ψ + } cos ( Λ + 2 δ in ) )
S 1 : ( x s ) 2 + y 2 ( x + s ) 2 + y 2 exp ( 8 sx w 0 2 ) = tan 2 β in S 2 : x 2 + ( y + s tan Λ ) 2 = s 2 sec 2 Λ S 3 : x 2 + ( y s cot Λ ) 2 = s 2 cos ec 2 Λ } charge 1 vortex input .
S 1 : x = [ w 0 2 log tan 2 β in ] 8 s S 2 : cos Λ = 0 S 3 : sin Λ = 0 } Gaussian beam input .
S 1 : ( x s ) 2 + y 2 ( x + s ) 2 + y 2 exp ( 8 sx n w 0 2 ) = tan 2 n β in S 2 : x 2 + ( y s cot ( [ Λ + 2 π ( m + 1 4 ) ] n ) ) 2 = s 2 cosec 2 ( [ Λ + 2 π ( m + 1 4 ) ] n ) S 3 : x 2 + ( y s cot ( [ Λ + 2 π m ] n ) ) 2 = s 2 cos ec 2 ( [ Λ + 2 π m ] n ) }
m = 0 , 1 , . . . , n 1 , charge n vortex input .
ρ = E y E x = S 2 + i S 3 S 0 + S 1 = exp ( i Λ ) d in , y d in , y ψ ( x + s , y ) ψ ( x s , y ) = exp ( i [ Λ + 2 δ in ] ) tan 2 β in ψ ( x + s , y ) ψ ( x s , y ) .

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