Abstract

The generation and dynamics of polarization singularities have been underresearched for years, while the focusing property of the topological configuration has not been explored much. In this paper, we simulated the generation of low-order polarization singularities with a circular Airy beam and explored the focusing property of the synthetic light field during propagation due to the autofocusing of the component. Our work researched the focusing properties of the polarization singularity configuration, which may help to develop its application prospect.

© 2017 Optical Society of America

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References

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    [Crossref]
  2. M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
    [Crossref]
  3. J. F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. R. Soc. London A 389, 279–290 (1983).
    [Crossref]
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    [Crossref]
  5. M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213, 201–221 (2002).
    [Crossref]
  6. F. Flossmann, K. O’Holleran, M. R. Dennis, and M. J. Padgett, “Polarization singularities in 2D and 3D speckle fields,” Phys. Rev. Lett. 100, 203902 (2008).
    [Crossref]
  7. I. Freund and D. A. Kessler, “Singularities in speckled speckle,” Opt. Lett. 33, 479–481 (2008).
    [Crossref]
  8. V. Kumar and N. K. Viswanthan, “Polarization singularities and fiber modal decomposition,” Proc. SPIE 8637, 86371A (2013).
    [Crossref]
  9. E. J. Galvez, B. L. Rojec, and K. R. McCullough, “Imaging optical singularities: understanding the duality of C-points and optical vortices,” Proc. SPIE 8637, 863706 (2013).
    [Crossref]
  10. E. J. Galvez, B. L. Rojec, and K. Beach, “Mapping of all polarization-singularity C-point morphologies,” Proc. SPIE 8999, 899901 (2014).
    [Crossref]
  11. R. Yu, D. Ye, Y. Xin, Y. Chen, and Q. Zhao, “Distributions of amplitude and phase around C-points: lemon, mon-star and star,” J. Opt. Soc. Korea 20, 192–198 (2016).
    [Crossref]
  12. M. V. Vasnetsov, M. S. Soskin, and V. A. Pas’ko, “Topological configurations of cross-coupled polarization singularities in a space-variant vector field,” Opt. Commun. 363, 181–187 (2016).
    [Crossref]
  13. D. Ye, X. Peng, Q. Zhao, and Y. Chen, “Numerical generation of a polarization singularity array with modulated amplitude and phase,” J. Opt. Soc. Am. A 33, 1705–1709 (2016).
    [Crossref]
  14. E. J. Galvez and B. Khajavi, “Monstar disclinations in the polarization of singular optical beams,” J. Opt. Soc. Am. A 34, 568–575 (2017).
    [Crossref]
  15. S. K. Pal, R. Ruchi, and P. Senthilkumaran, “C-point and V-point singularity lattice formation and index sign conversion methods,” Opt. Commun. 393, 156–168 (2017).
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    [Crossref]
  28. F. Wang, C. Zhao, Y. Dong, Y. Dong, and Y. Cai, “Generation and tight-focusing properties of cylindrical vector circular Airy beams,” Appl. Phys. B 117, 905–913 (2014).
    [Crossref]
  29. F. Cardano, E. Karimi, L. Marrucci, C. Lisio, and E. Santamato, “Generation and dynamics of optical beams with polarization singularities,” Opt. Express 21, 8815–8820 (2013).
    [Crossref]
  30. B. Khajavi and E. J. Galvez, “High-order disclinations in space-variant polarization,” J. Opt. 18, 084003 (2016).
    [Crossref]
  31. E. Otte, C. Alpmann, and C. Denz, “Higher-order polarization singularities in tailored vector beams,” J. Opt. 18, 074012 (2016).
    [Crossref]
  32. E. J. Galvez and B. Khajavi, “High-order disclinations in the polarization of light,” Proc. SPIE 9764, 97640R (2016).
    [Crossref]
  33. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (W. H. Freeman, 2004).
  34. A. I. Konukhov and L. A. Melnikov, “Optical vortices in a vector field: the general definition based on the analogy with topological solitons in a 2D ferromagnet, and examples from the polarization transverse patterns in a laser,” J. Opt. B 3, S139–S144 (2001).
    [Crossref]
  35. I. Freund, “Poincaré vortices,” Opt. Lett. 26, 1996–1998 (2001).
    [Crossref]

2017 (2)

S. K. Pal, R. Ruchi, and P. Senthilkumaran, “C-point and V-point singularity lattice formation and index sign conversion methods,” Opt. Commun. 393, 156–168 (2017).
[Crossref]

E. J. Galvez and B. Khajavi, “Monstar disclinations in the polarization of singular optical beams,” J. Opt. Soc. Am. A 34, 568–575 (2017).
[Crossref]

2016 (7)

R. Yu, D. Ye, Y. Xin, Y. Chen, and Q. Zhao, “Distributions of amplitude and phase around C-points: lemon, mon-star and star,” J. Opt. Soc. Korea 20, 192–198 (2016).
[Crossref]

D. Ye, X. Peng, Q. Zhao, and Y. Chen, “Numerical generation of a polarization singularity array with modulated amplitude and phase,” J. Opt. Soc. Am. A 33, 1705–1709 (2016).
[Crossref]

H. Garcia-Gracia and J. C. Gutiérrez-Vega, “Polarization singularities in nondiffracting Mathieu-Poincaré beams,” J. Opt. 18, 014006 (2016).
[Crossref]

M. V. Vasnetsov, M. S. Soskin, and V. A. Pas’ko, “Topological configurations of cross-coupled polarization singularities in a space-variant vector field,” Opt. Commun. 363, 181–187 (2016).
[Crossref]

B. Khajavi and E. J. Galvez, “High-order disclinations in space-variant polarization,” J. Opt. 18, 084003 (2016).
[Crossref]

E. Otte, C. Alpmann, and C. Denz, “Higher-order polarization singularities in tailored vector beams,” J. Opt. 18, 074012 (2016).
[Crossref]

E. J. Galvez and B. Khajavi, “High-order disclinations in the polarization of light,” Proc. SPIE 9764, 97640R (2016).
[Crossref]

2015 (1)

2014 (2)

F. Wang, C. Zhao, Y. Dong, Y. Dong, and Y. Cai, “Generation and tight-focusing properties of cylindrical vector circular Airy beams,” Appl. Phys. B 117, 905–913 (2014).
[Crossref]

E. J. Galvez, B. L. Rojec, and K. Beach, “Mapping of all polarization-singularity C-point morphologies,” Proc. SPIE 8999, 899901 (2014).
[Crossref]

2013 (4)

V. Kumar and N. K. Viswanthan, “Polarization singularities and fiber modal decomposition,” Proc. SPIE 8637, 86371A (2013).
[Crossref]

E. J. Galvez, B. L. Rojec, and K. R. McCullough, “Imaging optical singularities: understanding the duality of C-points and optical vortices,” Proc. SPIE 8637, 863706 (2013).
[Crossref]

F. Cardano, E. Karimi, L. Marrucci, C. Lisio, and E. Santamato, “Generation and dynamics of optical beams with polarization singularities,” Opt. Express 21, 8815–8820 (2013).
[Crossref]

S. Liu, M. Wang, P. Li, P. Zhang, and J. Zhao, “Abrupt polarization transition of vector autofocusing Airy beams,” Opt. Lett. 38, 2416–2418 (2013).
[Crossref]

2012 (1)

2011 (2)

2010 (2)

2009 (1)

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[Crossref]

2008 (4)

2007 (2)

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32, 979–981 (2007).
[Crossref]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
[Crossref]

2004 (1)

M. V. Berry, M. R. Dennis, and R. L. Lee, “Polarization singularities in the clear sky,” New J. Phys. 6, 162 (2004).
[Crossref]

2002 (1)

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

2001 (2)

A. I. Konukhov and L. A. Melnikov, “Optical vortices in a vector field: the general definition based on the analogy with topological solitons in a 2D ferromagnet, and examples from the polarization transverse patterns in a laser,” J. Opt. B 3, S139–S144 (2001).
[Crossref]

I. Freund, “Poincaré vortices,” Opt. Lett. 26, 1996–1998 (2001).
[Crossref]

1987 (1)

J. V. Hajnal, “Singularities in the transverse fields of electromagnetic waves. I. Theory,” Proc. R. Soc. London A 414, 433–446 (1987).
[Crossref]

1983 (1)

J. F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. R. Soc. London A 389, 279–290 (1983).
[Crossref]

Alonso, M. A.

Alpmann, C.

E. Otte, C. Alpmann, and C. Denz, “Higher-order polarization singularities in tailored vector beams,” J. Opt. 18, 074012 (2016).
[Crossref]

Beach, K.

E. J. Galvez, B. L. Rojec, and K. Beach, “Mapping of all polarization-singularity C-point morphologies,” Proc. SPIE 8999, 899901 (2014).
[Crossref]

Beckley, A. M.

Berry, M. V.

M. V. Berry, M. R. Dennis, and R. L. Lee, “Polarization singularities in the clear sky,” New J. Phys. 6, 162 (2004).
[Crossref]

Broky, J.

Brown, T. G.

Cai, Y.

F. Wang, C. Zhao, Y. Dong, Y. Dong, and Y. Cai, “Generation and tight-focusing properties of cylindrical vector circular Airy beams,” Appl. Phys. B 117, 905–913 (2014).
[Crossref]

Cardano, F.

Chen, X.

Chen, Y.

Chen, Z.

Christodoulides, D. N.

Dennis, M. R.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[Crossref]

F. Flossmann, K. O’Holleran, M. R. Dennis, and M. J. Padgett, “Polarization singularities in 2D and 3D speckle fields,” Phys. Rev. Lett. 100, 203902 (2008).
[Crossref]

M. V. Berry, M. R. Dennis, and R. L. Lee, “Polarization singularities in the clear sky,” New J. Phys. 6, 162 (2004).
[Crossref]

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

Denz, C.

E. Otte, C. Alpmann, and C. Denz, “Higher-order polarization singularities in tailored vector beams,” J. Opt. 18, 074012 (2016).
[Crossref]

Dogariu, A.

Dong, Y.

F. Wang, C. Zhao, Y. Dong, Y. Dong, and Y. Cai, “Generation and tight-focusing properties of cylindrical vector circular Airy beams,” Appl. Phys. B 117, 905–913 (2014).
[Crossref]

F. Wang, C. Zhao, Y. Dong, Y. Dong, and Y. Cai, “Generation and tight-focusing properties of cylindrical vector circular Airy beams,” Appl. Phys. B 117, 905–913 (2014).
[Crossref]

Efremidis, N. K.

Flossmann, F.

F. Flossmann, K. O’Holleran, M. R. Dennis, and M. J. Padgett, “Polarization singularities in 2D and 3D speckle fields,” Phys. Rev. Lett. 100, 203902 (2008).
[Crossref]

Freund, I.

Galvez, E. J.

E. J. Galvez and B. Khajavi, “Monstar disclinations in the polarization of singular optical beams,” J. Opt. Soc. Am. A 34, 568–575 (2017).
[Crossref]

E. J. Galvez and B. Khajavi, “High-order disclinations in the polarization of light,” Proc. SPIE 9764, 97640R (2016).
[Crossref]

B. Khajavi and E. J. Galvez, “High-order disclinations in space-variant polarization,” J. Opt. 18, 084003 (2016).
[Crossref]

E. J. Galvez, B. L. Rojec, and K. Beach, “Mapping of all polarization-singularity C-point morphologies,” Proc. SPIE 8999, 899901 (2014).
[Crossref]

E. J. Galvez, B. L. Rojec, and K. R. McCullough, “Imaging optical singularities: understanding the duality of C-points and optical vortices,” Proc. SPIE 8637, 863706 (2013).
[Crossref]

Garcia-Gracia, H.

H. Garcia-Gracia and J. C. Gutiérrez-Vega, “Polarization singularities in nondiffracting Mathieu-Poincaré beams,” J. Opt. 18, 014006 (2016).
[Crossref]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (W. H. Freeman, 2004).

Gutiérrez-Vega, J. C.

H. Garcia-Gracia and J. C. Gutiérrez-Vega, “Polarization singularities in nondiffracting Mathieu-Poincaré beams,” J. Opt. 18, 014006 (2016).
[Crossref]

Hajnal, J. V.

J. V. Hajnal, “Singularities in the transverse fields of electromagnetic waves. I. Theory,” Proc. R. Soc. London A 414, 433–446 (1987).
[Crossref]

Hnatovsky, C.

Huang, K.

Jiang, Y.

Karimi, E.

Karpinski, P.

Kessler, D. A.

Khajavi, B.

E. J. Galvez and B. Khajavi, “Monstar disclinations in the polarization of singular optical beams,” J. Opt. Soc. Am. A 34, 568–575 (2017).
[Crossref]

E. J. Galvez and B. Khajavi, “High-order disclinations in the polarization of light,” Proc. SPIE 9764, 97640R (2016).
[Crossref]

B. Khajavi and E. J. Galvez, “High-order disclinations in space-variant polarization,” J. Opt. 18, 084003 (2016).
[Crossref]

Konukhov, A. I.

A. I. Konukhov and L. A. Melnikov, “Optical vortices in a vector field: the general definition based on the analogy with topological solitons in a 2D ferromagnet, and examples from the polarization transverse patterns in a laser,” J. Opt. B 3, S139–S144 (2001).
[Crossref]

Krolikowski, W.

Kumar, V.

V. Kumar and N. K. Viswanthan, “Polarization singularities and fiber modal decomposition,” Proc. SPIE 8637, 86371A (2013).
[Crossref]

Lee, R. L.

M. V. Berry, M. R. Dennis, and R. L. Lee, “Polarization singularities in the clear sky,” New J. Phys. 6, 162 (2004).
[Crossref]

Li, P.

Lisio, C.

Liu, S.

Lu, X.

Marrucci, L.

McCullough, K. R.

E. J. Galvez, B. L. Rojec, and K. R. McCullough, “Imaging optical singularities: understanding the duality of C-points and optical vortices,” Proc. SPIE 8637, 863706 (2013).
[Crossref]

Melnikov, L. A.

A. I. Konukhov and L. A. Melnikov, “Optical vortices in a vector field: the general definition based on the analogy with topological solitons in a 2D ferromagnet, and examples from the polarization transverse patterns in a laser,” J. Opt. B 3, S139–S144 (2001).
[Crossref]

Mills, M. S.

Nye, J. F.

J. F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. R. Soc. London A 389, 279–290 (1983).
[Crossref]

O’Holleran, K.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[Crossref]

F. Flossmann, K. O’Holleran, M. R. Dennis, and M. J. Padgett, “Polarization singularities in 2D and 3D speckle fields,” Phys. Rev. Lett. 100, 203902 (2008).
[Crossref]

Orakash, J.

Otte, E.

E. Otte, C. Alpmann, and C. Denz, “Higher-order polarization singularities in tailored vector beams,” J. Opt. 18, 074012 (2016).
[Crossref]

Padgett, M. J.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[Crossref]

F. Flossmann, K. O’Holleran, M. R. Dennis, and M. J. Padgett, “Polarization singularities in 2D and 3D speckle fields,” Phys. Rev. Lett. 100, 203902 (2008).
[Crossref]

Pal, S. K.

S. K. Pal, R. Ruchi, and P. Senthilkumaran, “C-point and V-point singularity lattice formation and index sign conversion methods,” Opt. Commun. 393, 156–168 (2017).
[Crossref]

Papazoglou, D. G.

Pas’ko, V. A.

M. V. Vasnetsov, M. S. Soskin, and V. A. Pas’ko, “Topological configurations of cross-coupled polarization singularities in a space-variant vector field,” Opt. Commun. 363, 181–187 (2016).
[Crossref]

Peng, X.

Rojec, B. L.

E. J. Galvez, B. L. Rojec, and K. Beach, “Mapping of all polarization-singularity C-point morphologies,” Proc. SPIE 8999, 899901 (2014).
[Crossref]

E. J. Galvez, B. L. Rojec, and K. R. McCullough, “Imaging optical singularities: understanding the duality of C-points and optical vortices,” Proc. SPIE 8637, 863706 (2013).
[Crossref]

Ruchi, R.

S. K. Pal, R. Ruchi, and P. Senthilkumaran, “C-point and V-point singularity lattice formation and index sign conversion methods,” Opt. Commun. 393, 156–168 (2017).
[Crossref]

Santamato, E.

Senthilkumaran, P.

S. K. Pal, R. Ruchi, and P. Senthilkumaran, “C-point and V-point singularity lattice formation and index sign conversion methods,” Opt. Commun. 393, 156–168 (2017).
[Crossref]

Sheng, Y.

Shvedov, V.

Siviloglou, G. A.

Soskin, M. S.

M. V. Vasnetsov, M. S. Soskin, and V. A. Pas’ko, “Topological configurations of cross-coupled polarization singularities in a space-variant vector field,” Opt. Commun. 363, 181–187 (2016).
[Crossref]

Tzortzakis, S.

Vasnetsov, M. V.

M. V. Vasnetsov, M. S. Soskin, and V. A. Pas’ko, “Topological configurations of cross-coupled polarization singularities in a space-variant vector field,” Opt. Commun. 363, 181–187 (2016).
[Crossref]

Viswanthan, N. K.

V. Kumar and N. K. Viswanthan, “Polarization singularities and fiber modal decomposition,” Proc. SPIE 8637, 86371A (2013).
[Crossref]

Wang, F.

F. Wang, C. Zhao, Y. Dong, Y. Dong, and Y. Cai, “Generation and tight-focusing properties of cylindrical vector circular Airy beams,” Appl. Phys. B 117, 905–913 (2014).
[Crossref]

Wang, M.

Xin, Y.

Ye, D.

Yu, R.

Zhang, P.

Zhang, Z.

Zhao, C.

F. Wang, C. Zhao, Y. Dong, Y. Dong, and Y. Cai, “Generation and tight-focusing properties of cylindrical vector circular Airy beams,” Appl. Phys. B 117, 905–913 (2014).
[Crossref]

Zhao, J.

Zhao, Q.

Zhu, W.

Appl. Phys. B (1)

F. Wang, C. Zhao, Y. Dong, Y. Dong, and Y. Cai, “Generation and tight-focusing properties of cylindrical vector circular Airy beams,” Appl. Phys. B 117, 905–913 (2014).
[Crossref]

J. Opt. (3)

H. Garcia-Gracia and J. C. Gutiérrez-Vega, “Polarization singularities in nondiffracting Mathieu-Poincaré beams,” J. Opt. 18, 014006 (2016).
[Crossref]

B. Khajavi and E. J. Galvez, “High-order disclinations in space-variant polarization,” J. Opt. 18, 084003 (2016).
[Crossref]

E. Otte, C. Alpmann, and C. Denz, “Higher-order polarization singularities in tailored vector beams,” J. Opt. 18, 074012 (2016).
[Crossref]

J. Opt. B (1)

A. I. Konukhov and L. A. Melnikov, “Optical vortices in a vector field: the general definition based on the analogy with topological solitons in a 2D ferromagnet, and examples from the polarization transverse patterns in a laser,” J. Opt. B 3, S139–S144 (2001).
[Crossref]

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

J. Opt. Soc. Korea (1)

New J. Phys. (1)

M. V. Berry, M. R. Dennis, and R. L. Lee, “Polarization singularities in the clear sky,” New J. Phys. 6, 162 (2004).
[Crossref]

Opt. Commun. (3)

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

M. V. Vasnetsov, M. S. Soskin, and V. A. Pas’ko, “Topological configurations of cross-coupled polarization singularities in a space-variant vector field,” Opt. Commun. 363, 181–187 (2016).
[Crossref]

S. K. Pal, R. Ruchi, and P. Senthilkumaran, “C-point and V-point singularity lattice formation and index sign conversion methods,” Opt. Commun. 393, 156–168 (2017).
[Crossref]

Opt. Express (5)

Opt. Lett. (8)

Phys. Rev. Lett. (2)

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
[Crossref]

F. Flossmann, K. O’Holleran, M. R. Dennis, and M. J. Padgett, “Polarization singularities in 2D and 3D speckle fields,” Phys. Rev. Lett. 100, 203902 (2008).
[Crossref]

Proc. R. Soc. London A (2)

J. F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. R. Soc. London A 389, 279–290 (1983).
[Crossref]

J. V. Hajnal, “Singularities in the transverse fields of electromagnetic waves. I. Theory,” Proc. R. Soc. London A 414, 433–446 (1987).
[Crossref]

Proc. SPIE (4)

V. Kumar and N. K. Viswanthan, “Polarization singularities and fiber modal decomposition,” Proc. SPIE 8637, 86371A (2013).
[Crossref]

E. J. Galvez, B. L. Rojec, and K. R. McCullough, “Imaging optical singularities: understanding the duality of C-points and optical vortices,” Proc. SPIE 8637, 863706 (2013).
[Crossref]

E. J. Galvez, B. L. Rojec, and K. Beach, “Mapping of all polarization-singularity C-point morphologies,” Proc. SPIE 8999, 899901 (2014).
[Crossref]

E. J. Galvez and B. Khajavi, “High-order disclinations in the polarization of light,” Proc. SPIE 9764, 97640R (2016).
[Crossref]

Prog. Opt. (1)

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[Crossref]

Other (1)

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (W. H. Freeman, 2004).

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Figures (5)

Fig. 1.
Fig. 1.

Intensity distribution of (a) circular Airy beam with optical vortex and (b) fundamental Gauss beam with parameter w 0 = 0.2    mm , r 0 = 1    mm and a = 0.1 .

Fig. 2.
Fig. 2.

Polarization states of the synthetic field in the initial plane. (a)  e ^ 1 is left-handed, e ^ 2 is right-handed, and the polarization configuration is a lemon. (b)  e ^ 1 is right-handed, e ^ 2 is left-handed, and the polarization configuration is a star.

Fig. 3.
Fig. 3.

Intensity distribution of (a) circular Airy beam and (b) circular Airy beam with optical vortex at z = 464    mm .

Fig. 4.
Fig. 4.

Polarization states of the synthetic light field at (a)  z = 100    mm , (b)  z = 300    mm , (c)  z = 464    mm , and (d)  z = 500    mm .

Fig. 5.
Fig. 5.

Stokes phases of the synthetic field at the distance of (a)  z = 0 , (b)  z = 100    mm , (c)  z = 300    mm , (d)  z = 400    mm , (e)  z = 420    mm , (f)  z = 450    mm , (g)  z = 464    mm , and (h)  z = 500    mm . The color bars to the right of the figures in each row denote the values of the phases.

Equations (2)

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E P ( r ; γ ) = cos γ e ^ 1 U 00 ( r ) + sin γ e ^ 2 U 01 ( r ) ,
U ( r , φ , z = 0 ) = A 0 · A i ( r 0 r w 0 ) exp ( a r 0 r w 0 ) · ( r e i φ ) l ,

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