Abstract

We report here the controlled generation of polarization singularities (PS) in the output beam from a two-mode optical fiber, obtained via coherent superposition of the fundamental (HE11) mode with the vortex (CV±1±/IV±1±) modes by selectively coupling circularly polarized (σ=±1) Gaussian (TEM00) input beam at two different launch angles. The PS in the output beam are characterized using the complex Stokes fields and interferometry measurements, based on the two-mode optical fiber which are an effective means to generate, manipulate, study, and use isolated PS, depending strongly on the input launch conditions.

© 2011 Optical Society of America

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References

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  1. M. V. Berry, “Singularities in wave and rays,” in Les Houches Session XXV-Physics of Defects, R.Balian, M.Kleman, and J.P.Poirier, eds. (North-Holland, 1981).
  2. J. F. Nye, Natural Focusing and the Fine Structure of Light (Institute of Physics, 1999).
  3. M. S. Soskin and M. V. Vasnetsov, “Singular optics,” in Progress in Optics, E.Wolf, ed. (Elsevier, 2001), Vol.  42, Chap. 4.
  4. I. Freund, M. S. Soskin, and A. I. Mokhun, “Elliptic critical points in paraxial optical fields,” Opt. Commun. 208, 223–253(2002).
    [CrossRef]
  5. M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213, 201–221 (2002).
    [CrossRef]
  6. M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” in Progress in Optics, E.Wolf, ed. (Elsevier, 2009), Vol.  53, pp. 293–363.
    [CrossRef]
  7. J. V. Hajnal, “Singularities in the transverse fields of electromagnetic waves. II. Observations on the eectric field,” Proc. R. Soc. A 414, 447–468 (1987).
    [CrossRef]
  8. J. V. Hajnal, “Compound modulated scatterer measuring system,” IEE Proc. H 134, 350–356 (1987).
    [CrossRef]
  9. J. V. Hajnal, “Observations of singularities in the electric and magnetic fields of freely propagating microwaves,” Proc. R. Soc. A 430, 413–421 (1990).
    [CrossRef]
  10. O. V. Angelsky, Optical Correlation: Techniques and Applications (SPIE, 2007).
    [CrossRef]
  11. N. I. Petrov, “Evolution of polarization in an inhomogeneous isotropic medium,” J. Exp. Theor. Phys. 85, 1085–1093 (1997).
    [CrossRef]
  12. A. Snyder and J. D. Love, Optical Waveguide Theory(Chapman & Hall, 1983).
  13. M. S. Soskin, V. Denisenko, and I. Freund, “Optical polarization singularities and elliptic stationary points,” Opt. Lett. 28, 1475–1477 (2003).
    [CrossRef] [PubMed]
  14. I. Freund, “Polarization singularities in three dimensional optical fields: the next frontier,” Ukr. J. Phys. 49, 370–376(2004).
  15. I. Mokhun and R. Khrobatin, “Shift of application point of angular momentum in the area of elementary polarization singularity,” J. Opt. A 10, 064015 (2008).
    [CrossRef]
  16. N. K. Viswanathan and V. V. G. Krishna Inavalli, “Generation of optical vector beams using a two-mode fiber,” Opt. Lett. 34, 1189–1191 (2009).
    [CrossRef] [PubMed]
  17. V. V. G. Krishna Inavalli and N. K. Viswanathan, “Switchable vector vortex beam generation using an optical fiber,” Opt. Commun. 283, 861–864 (2010).
    [CrossRef]
  18. V. V. G. Krishna Inavalli and N. K. Viswanathan, “Rotational Doppler-effect due to selective excitation of vector-vortex field in optical fiber,” Opt. Express 19, 448–457 (2011).
    [CrossRef]
  19. O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, “Interferometric methods in diagnostics of polarization singularities,” Phys. Rev. E 65, 036602 (2002).
    [CrossRef]
  20. A. I. Mokhun, M. S. Soskin, and I. Freund, “Elliptical critical points: C-points, a-lines, and the sign rule,” Opt. Lett. 27, 995–997 (2002).
    [CrossRef]
  21. M. S. Soskin, V. Denisenko, and I. Freund, “Optical polarization singularities and elliptic stationary points,” Opt. Lett. 28, 1475–1477 (2003).
    [CrossRef] [PubMed]
  22. A. V. Volyar and T. A. Fadeeva, “Optics of singularities of a low mode fiber: optical vortices,” Opt. Spectros. 85, 272–280(1998).
  23. G. Volpe and D. Petrov, “Generation of cylindrical vector beams with few-mode fibers excited by Laguerre-Gaussian beams,” Opt. Commun. 237, 89–95 (2004).
    [CrossRef]
  24. A. V. Volyar, V. Z. Zhilaitis, and T. A. Fadeeva, “Optical vortices in low-mode fibers: III. Dislocation reactions, phase transitions, and topological birefringence,” Opt. Spectrosc. 88, 397–405 (2000).
    [CrossRef]
  25. M. Born and E. W. Wolf, Principles of Optics, 6th ed.(Cambridge University, 1980).
  26. A. V. Dooghin, N. D. Kundikova, V. S. Liberman, and B. Y. Zel’dovich, “Optical Magnus effect,” Phys. Rev. A 45, 8204–8208 (1992).
    [CrossRef] [PubMed]
  27. O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science 319, 787–790(2008).
    [CrossRef] [PubMed]

2011 (1)

2010 (1)

V. V. G. Krishna Inavalli and N. K. Viswanathan, “Switchable vector vortex beam generation using an optical fiber,” Opt. Commun. 283, 861–864 (2010).
[CrossRef]

2009 (1)

2008 (2)

I. Mokhun and R. Khrobatin, “Shift of application point of angular momentum in the area of elementary polarization singularity,” J. Opt. A 10, 064015 (2008).
[CrossRef]

O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science 319, 787–790(2008).
[CrossRef] [PubMed]

2004 (2)

G. Volpe and D. Petrov, “Generation of cylindrical vector beams with few-mode fibers excited by Laguerre-Gaussian beams,” Opt. Commun. 237, 89–95 (2004).
[CrossRef]

I. Freund, “Polarization singularities in three dimensional optical fields: the next frontier,” Ukr. J. Phys. 49, 370–376(2004).

2003 (2)

2002 (4)

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

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

I. Freund, M. S. Soskin, and A. I. Mokhun, “Elliptic critical points in paraxial optical fields,” Opt. Commun. 208, 223–253(2002).
[CrossRef]

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

2000 (1)

A. V. Volyar, V. Z. Zhilaitis, and T. A. Fadeeva, “Optical vortices in low-mode fibers: III. Dislocation reactions, phase transitions, and topological birefringence,” Opt. Spectrosc. 88, 397–405 (2000).
[CrossRef]

1998 (1)

A. V. Volyar and T. A. Fadeeva, “Optics of singularities of a low mode fiber: optical vortices,” Opt. Spectros. 85, 272–280(1998).

1997 (1)

N. I. Petrov, “Evolution of polarization in an inhomogeneous isotropic medium,” J. Exp. Theor. Phys. 85, 1085–1093 (1997).
[CrossRef]

1992 (1)

A. V. Dooghin, N. D. Kundikova, V. S. Liberman, and B. Y. Zel’dovich, “Optical Magnus effect,” Phys. Rev. A 45, 8204–8208 (1992).
[CrossRef] [PubMed]

1990 (1)

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

1987 (2)

J. V. Hajnal, “Singularities in the transverse fields of electromagnetic waves. II. Observations on the eectric field,” Proc. R. Soc. A 414, 447–468 (1987).
[CrossRef]

J. V. Hajnal, “Compound modulated scatterer measuring system,” IEE Proc. H 134, 350–356 (1987).
[CrossRef]

Angelsky, O. V.

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

O. V. Angelsky, Optical Correlation: Techniques and Applications (SPIE, 2007).
[CrossRef]

Berry, M. V.

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

Born, M.

M. Born and E. W. Wolf, Principles of Optics, 6th ed.(Cambridge University, 1980).

Denisenko, V.

Dennis, M. R.

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

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” in Progress in Optics, E.Wolf, ed. (Elsevier, 2009), Vol.  53, pp. 293–363.
[CrossRef]

Dooghin, A. V.

A. V. Dooghin, N. D. Kundikova, V. S. Liberman, and B. Y. Zel’dovich, “Optical Magnus effect,” Phys. Rev. A 45, 8204–8208 (1992).
[CrossRef] [PubMed]

Fadeeva, T. A.

A. V. Volyar, V. Z. Zhilaitis, and T. A. Fadeeva, “Optical vortices in low-mode fibers: III. Dislocation reactions, phase transitions, and topological birefringence,” Opt. Spectrosc. 88, 397–405 (2000).
[CrossRef]

A. V. Volyar and T. A. Fadeeva, “Optics of singularities of a low mode fiber: optical vortices,” Opt. Spectros. 85, 272–280(1998).

Freund, I.

Hajnal, J. V.

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

J. V. Hajnal, “Singularities in the transverse fields of electromagnetic waves. II. Observations on the eectric field,” Proc. R. Soc. A 414, 447–468 (1987).
[CrossRef]

J. V. Hajnal, “Compound modulated scatterer measuring system,” IEE Proc. H 134, 350–356 (1987).
[CrossRef]

Hosten, O.

O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science 319, 787–790(2008).
[CrossRef] [PubMed]

Khrobatin, R.

I. Mokhun and R. Khrobatin, “Shift of application point of angular momentum in the area of elementary polarization singularity,” J. Opt. A 10, 064015 (2008).
[CrossRef]

Krishna Inavalli, V. V. G.

Kundikova, N. D.

A. V. Dooghin, N. D. Kundikova, V. S. Liberman, and B. Y. Zel’dovich, “Optical Magnus effect,” Phys. Rev. A 45, 8204–8208 (1992).
[CrossRef] [PubMed]

Kwiat, P.

O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science 319, 787–790(2008).
[CrossRef] [PubMed]

Liberman, V. S.

A. V. Dooghin, N. D. Kundikova, V. S. Liberman, and B. Y. Zel’dovich, “Optical Magnus effect,” Phys. Rev. A 45, 8204–8208 (1992).
[CrossRef] [PubMed]

Love, J. D.

A. Snyder and J. D. Love, Optical Waveguide Theory(Chapman & Hall, 1983).

Mokhun, A. I.

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

I. Freund, M. S. Soskin, and A. I. Mokhun, “Elliptic critical points in paraxial optical fields,” Opt. Commun. 208, 223–253(2002).
[CrossRef]

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

Mokhun, I.

I. Mokhun and R. Khrobatin, “Shift of application point of angular momentum in the area of elementary polarization singularity,” J. Opt. A 10, 064015 (2008).
[CrossRef]

Mokhun, I. I.

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

Nye, J. F.

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

O’Holleran, K.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” in Progress in Optics, E.Wolf, ed. (Elsevier, 2009), Vol.  53, pp. 293–363.
[CrossRef]

Padgett, M. J.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” in Progress in Optics, E.Wolf, ed. (Elsevier, 2009), Vol.  53, pp. 293–363.
[CrossRef]

Petrov, D.

G. Volpe and D. Petrov, “Generation of cylindrical vector beams with few-mode fibers excited by Laguerre-Gaussian beams,” Opt. Commun. 237, 89–95 (2004).
[CrossRef]

Petrov, N. I.

N. I. Petrov, “Evolution of polarization in an inhomogeneous isotropic medium,” J. Exp. Theor. Phys. 85, 1085–1093 (1997).
[CrossRef]

Snyder, A.

A. Snyder and J. D. Love, Optical Waveguide Theory(Chapman & Hall, 1983).

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]

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, “Elliptical critical points: C-points, a-lines, and the sign rule,” Opt. Lett. 27, 995–997 (2002).
[CrossRef]

I. Freund, M. S. Soskin, and A. I. Mokhun, “Elliptic critical points in paraxial optical fields,” Opt. Commun. 208, 223–253(2002).
[CrossRef]

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

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” in Progress in Optics, E.Wolf, ed. (Elsevier, 2001), Vol.  42, Chap. 4.

Vasnetsov, M. V.

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” in Progress in Optics, E.Wolf, ed. (Elsevier, 2001), Vol.  42, Chap. 4.

Viswanathan, N. K.

Volpe, G.

G. Volpe and D. Petrov, “Generation of cylindrical vector beams with few-mode fibers excited by Laguerre-Gaussian beams,” Opt. Commun. 237, 89–95 (2004).
[CrossRef]

Volyar, A. V.

A. V. Volyar, V. Z. Zhilaitis, and T. A. Fadeeva, “Optical vortices in low-mode fibers: III. Dislocation reactions, phase transitions, and topological birefringence,” Opt. Spectrosc. 88, 397–405 (2000).
[CrossRef]

A. V. Volyar and T. A. Fadeeva, “Optics of singularities of a low mode fiber: optical vortices,” Opt. Spectros. 85, 272–280(1998).

Wolf, E. W.

M. Born and E. W. Wolf, Principles of Optics, 6th ed.(Cambridge University, 1980).

Zel’dovich, B. Y.

A. V. Dooghin, N. D. Kundikova, V. S. Liberman, and B. Y. Zel’dovich, “Optical Magnus effect,” Phys. Rev. A 45, 8204–8208 (1992).
[CrossRef] [PubMed]

Zhilaitis, V. Z.

A. V. Volyar, V. Z. Zhilaitis, and T. A. Fadeeva, “Optical vortices in low-mode fibers: III. Dislocation reactions, phase transitions, and topological birefringence,” Opt. Spectrosc. 88, 397–405 (2000).
[CrossRef]

IEE Proc. H (1)

J. V. Hajnal, “Compound modulated scatterer measuring system,” IEE Proc. H 134, 350–356 (1987).
[CrossRef]

J. Exp. Theor. Phys. (1)

N. I. Petrov, “Evolution of polarization in an inhomogeneous isotropic medium,” J. Exp. Theor. Phys. 85, 1085–1093 (1997).
[CrossRef]

J. Opt. A (1)

I. Mokhun and R. Khrobatin, “Shift of application point of angular momentum in the area of elementary polarization singularity,” J. Opt. A 10, 064015 (2008).
[CrossRef]

Opt. Commun. (4)

V. V. G. Krishna Inavalli and N. K. Viswanathan, “Switchable vector vortex beam generation using an optical fiber,” Opt. Commun. 283, 861–864 (2010).
[CrossRef]

I. Freund, M. S. Soskin, and A. I. Mokhun, “Elliptic critical points in paraxial optical fields,” Opt. Commun. 208, 223–253(2002).
[CrossRef]

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

G. Volpe and D. Petrov, “Generation of cylindrical vector beams with few-mode fibers excited by Laguerre-Gaussian beams,” Opt. Commun. 237, 89–95 (2004).
[CrossRef]

Opt. Express (1)

Opt. Lett. (4)

Opt. Spectros. (1)

A. V. Volyar and T. A. Fadeeva, “Optics of singularities of a low mode fiber: optical vortices,” Opt. Spectros. 85, 272–280(1998).

Opt. Spectrosc. (1)

A. V. Volyar, V. Z. Zhilaitis, and T. A. Fadeeva, “Optical vortices in low-mode fibers: III. Dislocation reactions, phase transitions, and topological birefringence,” Opt. Spectrosc. 88, 397–405 (2000).
[CrossRef]

Phys. Rev. A (1)

A. V. Dooghin, N. D. Kundikova, V. S. Liberman, and B. Y. Zel’dovich, “Optical Magnus effect,” Phys. Rev. A 45, 8204–8208 (1992).
[CrossRef] [PubMed]

Phys. Rev. E (1)

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

Proc. R. Soc. A (2)

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

J. V. Hajnal, “Singularities in the transverse fields of electromagnetic waves. II. Observations on the eectric field,” Proc. R. Soc. A 414, 447–468 (1987).
[CrossRef]

Science (1)

O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science 319, 787–790(2008).
[CrossRef] [PubMed]

Ukr. J. Phys. (1)

I. Freund, “Polarization singularities in three dimensional optical fields: the next frontier,” Ukr. J. Phys. 49, 370–376(2004).

Other (7)

O. V. Angelsky, Optical Correlation: Techniques and Applications (SPIE, 2007).
[CrossRef]

A. Snyder and J. D. Love, Optical Waveguide Theory(Chapman & Hall, 1983).

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” in Progress in Optics, E.Wolf, ed. (Elsevier, 2009), Vol.  53, pp. 293–363.
[CrossRef]

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

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

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” in Progress in Optics, E.Wolf, ed. (Elsevier, 2001), Vol.  42, Chap. 4.

M. Born and E. W. Wolf, Principles of Optics, 6th ed.(Cambridge University, 1980).

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Figures (6)

Fig. 1
Fig. 1

(a) Polarization ellipse showing the orientation angle α, the major and minor axes A and B, and the parameter ω. (b) Poincare sphere, showing the position vector corresponding to a polarization state ( S 0 , S 1 , S 2 , S 3 ) , where p is the degree of polarization.

Fig. 2
Fig. 2

(a) Schematic of the experimental setup to generate and characterize PS. BS, beam splitter; P, polarizer; QWP, quarter wave plate; L 1 and L 2 , objective lenses; TMF, two-mode fiber; M 1 and M 2 , mirrors; A, analyzer; CCD, detector. (b) Skew rays coupled into the TMF for two different launch angles.

Fig. 3
Fig. 3

Output from the two-mode fiber for four different input launch conditions. (a)–(d). Corresponding interference patterns at an analyzer angle of 40 ° are shown. (e)–(h). Regions containing the forklet in each figure are marked with a white circle.

Fig. 4
Fig. 4

Intensity plots of | s 3 | showing the shape of the s-line, which is a closed curve. Points with an intensity value more than a small positive value δ are saturated and shown as white, to enhance contrast and clearly show the region where | s 3 | 0 . Blue dots indicate a LCP C-point, while the red dots indicate RCP C-points. The inset in each image shows the correspondence with the shape of the s-line obtained by interferometry, where each fork location is represented by a square. Plots correspond to the (a) LCP input I 1 , (b) RCP input I 1 , (c) LCP input I 2 , and (d) RCP input I 2 .

Fig. 5
Fig. 5

Orientation of the major axis of the polarization ellipse ( α = arg ( S 12 ) ) in the region around the C-point is shown in a hue plot. Plots correspond to the C-point in the output of the fiber for (a) the LCP input I 1 ( σ = + 1 / 2 ), (b) the RCP input I 1 ( σ = 1 / 2 ), (c) the LCP input I 2 ( σ = 1 / 2 ), and (d) the RCP input I 2 ( σ = + 1 / 2 ). The index of the C-point is determined by finding the value of the integral in Eq. (5), the contour being any loop enclosing the C-point described as counterclockwise. The color coding is according to the bar given to the extreme right.

Fig. 6
Fig. 6

Demonstration of the polarization and phase transforming properties of a DP. The input beam with well-defined PS and ellipse orientation is modified significantly upon passing through a DP oriented at 0 ° . Green dots are C-point and red dots are phase point in the beam.

Equations (5)

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e t ( CP 11 e + ) e t ( LP 11 e x ) + i e t ( LP 11 e y ) , e t ( CP 11 o + ) e t ( LP 11 o x ) i e t ( LP 11 o y ) , e t ( CP 11 e ) e t ( LP 11 e x ) i e t ( LP 11 e y ) , e t ( CP 11 o ) e t ( LP 11 o x ) + i e t ( LP 11 o y ) .
CV + 1 + CP 11 e + + i CP 11 o + , CV 1 CP 11 e i CP 11 o , IV 1 + CP 11 e + i CP 11 o + , IV + 1 CP 11 e + i CP 11 o .
S 0 = | E x | 2 + | E y | 2 = I ( 0 ° , 0 ° ) + I ( 90 ° , 0 ° ) , S 1 = | E x | 2 | E y | 2 = I ( 0 ° , 0 ° ) I ( 90 ° , 0 ° ) , S 2 = E x E y * E x * E y = I ( 45 ° , 0 ° ) I ( 135 ° , 0 ° ) , S 3 = i ( E x * E y E x E y * ) = I ( 45 ° , 90 ° ) I ( 135 ° , 90 ° ) .
s i = S i S 1 2 + S 2 2 + S 3 2 = S i p S 0 ; i = 1 , 2 , 3.
I = 1 2 π d α ,

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