Abstract

In this paper, we show that arrayed polarization singularities (PSs) can be generated by interference of three polarized waves. An experimental setup was proposed and used to produce one of the available types of array structures. We then analytically identified factors that control the structures of these arrays and classified all the structures into three types. Simulation results were used to assist this analysis.

© 2013 Optical Society of America

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References

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  1. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London 336, 165–190 (1974).
    [CrossRef]
  2. J. Masajada, A. Popiolek-Masajada, and D. M. Wieliczka, “The interferometric system using optical vortices as phase markers,” Opt. Commun. 207, 85–93 (2002).
    [CrossRef]
  3. J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. 90, 133901 (2003).
    [CrossRef]
  4. J. F. Nye, Natural Focusing and Fine Structure of Light: Caustics and Wave Dislocations (CRC Press, 1999).
  5. M. R. Dennis, “Polarization singularity anisotropy: determining monstardom,” Opt. Lett. 33, 2572–2574 (2008).
    [CrossRef]
  6. J. F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. R. Soc. London 389, 279–290 (1983).
    [CrossRef]
  7. M. R. Dennis, “Fermionic out-of-plane structure of polarization singularities,” Opt. Lett. 36, 3765–3767 (2011).
    [CrossRef]
  8. I. Freund and D. A. Kessler, “Singularities in speckled speckle,” Opt. Lett. 33, 479–481 (2008).
    [CrossRef]
  9. M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213, 201–221 (2002).
    [CrossRef]
  10. U. T. Schwarz, F. Flossmann, and M. R. Dennis, “Topology of generic polarization singularities in birefringent crystals,” Topologica 2, 006 (2009).
    [CrossRef]
  11. F. Flossmann, K. O’Holleran, and M. R. Dennis, “Polarization singularities in 2D and 3D speckle fields,” Phys. Rev. Lett. 100, 203902 (2008).
    [CrossRef]
  12. F. Flossmann, U. T. Schwarz, and M. Maier, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
    [CrossRef]
  13. L. Pouget, J. Fade, C. Hamel, and M. Alouini, “Polarimetric imaging beyond the speckle grain scale,” Appl. Opt. 51, 7345–7356 (2012).
    [CrossRef]
  14. P. Kurzynowski, W. A. Woźniak, and M. Borwińska, “Regular lattices of polarization singularities: their generation and properties,” J. Opt. 12, 035406 (2010).
    [CrossRef]
  15. P. Kurzynowski, W. A. Woźniak, M. Zdunek, and M. Borwińska, “Singularities of interference of three waves with different polarization states,” Opt. Express 20, 26755–26765 (2012).
    [CrossRef]
  16. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987).
  17. J. F. Wheeldon and H. Schriemer, “Wyckoff positions and the expression of polarization singularities in photonic crystals,” Opt. Express 17, 2111–2121 (2009).
    [CrossRef]
  18. J. L. Stay and T. K. Gaylord, “Three-beam-interference lithography: contrast and crystallography,” Appl. Opt. 47, 3221–3230 (2008).
    [CrossRef]
  19. I. Freund, “Polarization singularities in optical lattices,” Opt. Lett. 29, 875–877 (2004).
    [CrossRef]

2012

2011

2010

P. Kurzynowski, W. A. Woźniak, and M. Borwińska, “Regular lattices of polarization singularities: their generation and properties,” J. Opt. 12, 035406 (2010).
[CrossRef]

2009

J. F. Wheeldon and H. Schriemer, “Wyckoff positions and the expression of polarization singularities in photonic crystals,” Opt. Express 17, 2111–2121 (2009).
[CrossRef]

U. T. Schwarz, F. Flossmann, and M. R. Dennis, “Topology of generic polarization singularities in birefringent crystals,” Topologica 2, 006 (2009).
[CrossRef]

2008

2005

F. Flossmann, U. T. Schwarz, and M. Maier, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
[CrossRef]

2004

2003

J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. 90, 133901 (2003).
[CrossRef]

2002

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

J. Masajada, A. Popiolek-Masajada, and D. M. Wieliczka, “The interferometric system using optical vortices as phase markers,” Opt. Commun. 207, 85–93 (2002).
[CrossRef]

1983

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

1974

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London 336, 165–190 (1974).
[CrossRef]

Alouini, M.

Azzam, R. M. A.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987).

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987).

Berry, M. V.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London 336, 165–190 (1974).
[CrossRef]

Borwinska, M.

P. Kurzynowski, W. A. Woźniak, M. Zdunek, and M. Borwińska, “Singularities of interference of three waves with different polarization states,” Opt. Express 20, 26755–26765 (2012).
[CrossRef]

P. Kurzynowski, W. A. Woźniak, and M. Borwińska, “Regular lattices of polarization singularities: their generation and properties,” J. Opt. 12, 035406 (2010).
[CrossRef]

Curtis, J. E.

J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. 90, 133901 (2003).
[CrossRef]

Dennis, M. R.

M. R. Dennis, “Fermionic out-of-plane structure of polarization singularities,” Opt. Lett. 36, 3765–3767 (2011).
[CrossRef]

U. T. Schwarz, F. Flossmann, and M. R. Dennis, “Topology of generic polarization singularities in birefringent crystals,” Topologica 2, 006 (2009).
[CrossRef]

M. R. Dennis, “Polarization singularity anisotropy: determining monstardom,” Opt. Lett. 33, 2572–2574 (2008).
[CrossRef]

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

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

Fade, J.

Flossmann, F.

U. T. Schwarz, F. Flossmann, and M. R. Dennis, “Topology of generic polarization singularities in birefringent crystals,” Topologica 2, 006 (2009).
[CrossRef]

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

F. Flossmann, U. T. Schwarz, and M. Maier, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
[CrossRef]

Freund, I.

Gaylord, T. K.

Grier, D. G.

J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. 90, 133901 (2003).
[CrossRef]

Hamel, C.

Kessler, D. A.

Kurzynowski, P.

P. Kurzynowski, W. A. Woźniak, M. Zdunek, and M. Borwińska, “Singularities of interference of three waves with different polarization states,” Opt. Express 20, 26755–26765 (2012).
[CrossRef]

P. Kurzynowski, W. A. Woźniak, and M. Borwińska, “Regular lattices of polarization singularities: their generation and properties,” J. Opt. 12, 035406 (2010).
[CrossRef]

Maier, M.

F. Flossmann, U. T. Schwarz, and M. Maier, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
[CrossRef]

Masajada, J.

J. Masajada, A. Popiolek-Masajada, and D. M. Wieliczka, “The interferometric system using optical vortices as phase markers,” Opt. Commun. 207, 85–93 (2002).
[CrossRef]

Nye, J. F.

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

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London 336, 165–190 (1974).
[CrossRef]

J. F. Nye, Natural Focusing and Fine Structure of Light: Caustics and Wave Dislocations (CRC Press, 1999).

O’Holleran, K.

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

Popiolek-Masajada, A.

J. Masajada, A. Popiolek-Masajada, and D. M. Wieliczka, “The interferometric system using optical vortices as phase markers,” Opt. Commun. 207, 85–93 (2002).
[CrossRef]

Pouget, L.

Schriemer, H.

Schwarz, U. T.

U. T. Schwarz, F. Flossmann, and M. R. Dennis, “Topology of generic polarization singularities in birefringent crystals,” Topologica 2, 006 (2009).
[CrossRef]

F. Flossmann, U. T. Schwarz, and M. Maier, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
[CrossRef]

Stay, J. L.

Wheeldon, J. F.

Wieliczka, D. M.

J. Masajada, A. Popiolek-Masajada, and D. M. Wieliczka, “The interferometric system using optical vortices as phase markers,” Opt. Commun. 207, 85–93 (2002).
[CrossRef]

Wozniak, W. A.

P. Kurzynowski, W. A. Woźniak, M. Zdunek, and M. Borwińska, “Singularities of interference of three waves with different polarization states,” Opt. Express 20, 26755–26765 (2012).
[CrossRef]

P. Kurzynowski, W. A. Woźniak, and M. Borwińska, “Regular lattices of polarization singularities: their generation and properties,” J. Opt. 12, 035406 (2010).
[CrossRef]

Zdunek, M.

Appl. Opt.

J. Opt.

P. Kurzynowski, W. A. Woźniak, and M. Borwińska, “Regular lattices of polarization singularities: their generation and properties,” J. Opt. 12, 035406 (2010).
[CrossRef]

Opt. Commun.

J. Masajada, A. Popiolek-Masajada, and D. M. Wieliczka, “The interferometric system using optical vortices as phase markers,” Opt. Commun. 207, 85–93 (2002).
[CrossRef]

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

Opt. Express

Opt. Lett.

Phys. Rev. Lett.

J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. 90, 133901 (2003).
[CrossRef]

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

F. Flossmann, U. T. Schwarz, and M. Maier, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
[CrossRef]

Proc. R. Soc. London

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

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London 336, 165–190 (1974).
[CrossRef]

Topologica

U. T. Schwarz, F. Flossmann, and M. R. Dennis, “Topology of generic polarization singularities in birefringent crystals,” Topologica 2, 006 (2009).
[CrossRef]

Other

J. F. Nye, Natural Focusing and Fine Structure of Light: Caustics and Wave Dislocations (CRC Press, 1999).

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987).

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Figures (5)

Fig. 1.
Fig. 1.

Schematic illustration of the system realizing the array of PS.

Fig. 2.
Fig. 2.

Classification of the arrayed PSs. The background represents the light intensity. The green ellipses show the distribution of the polarization states, and the yellow lines indicate where linear polarization occurs. The blue circles and red squares represent the circular polarization of star and mon-star type, respectively. (a) The polarization distribution obtained when A1A2A3=111. (b) The polarization distribution obtained when A1A2A3=557. (c) The arrayed PSs obtained when A1A2A3=553.

Fig. 3.
Fig. 3.

Regional relation of the three types (Ai>0,i=1,2,3). The red line indicates the region where A1=A2=A3, and the PS array is Lattice I type. The blue surfaces are the case of Lattice II type. The remaining areas are the case of Band type.

Fig. 4.
Fig. 4.

Light intensity obtained through different angles of the analyzer when the pixels are 900×900. (a) The light intensity I0° when the angle of the analyzer is 0°. (b) The light intensity I45° when the angle of the analyzer is 45°. (c) The light intensity I90° when the angle of the analyzer is 90°. (d) The light intensity I135° when the angle of the analyzer is 135°. (e) The light intensity I90°45° when the angle of the analyzer is 45° and the fast axis of the quarter-wave plate has an angle of 90° with the X axis.

Fig. 5.
Fig. 5.

Experimental results compared with simulation. (a) Simulation result with A1A2A3=5810. (b) The experimental result with the ratio of A1A2A3 is 5810. Because of the impact of the noise of optical elements, the L-lines are not smooth.

Equations (11)

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

E1=A1[10]exp(iKa1x)exp(iKb1y),
E2=A2[1/23/2]exp(iKa2x)exp(iKb2y),
E3=A3[1/23/2]exp(iKa3x)exp(iKb3y),
E=[ExEy]=E1+E2+E3=[A1+A2·exp(iKx)2+A3·exp(iKy)2A2·3exp(iKx)2A3·3exp(iKy)2].
Eα=G·E=[cos2α·Ex+12sin2α·Ey12sin2α·Ex+sin2α·Ey],
G=[cos2α12sin2α12sin2αsin2α],
S=[S0S1S2S3]=[Ex2+Ey2Ex2Ey22|Ex||Ey|cos(Δ)2|Ex||Ey|sin(Δ)],
θ=12arctan(S2S1),
ε=12arcsin(S3S12+S22+S32).
θ=12arctan2|Ex||Ey|cos(Δ)Ex2Ey2,
ε=12arcsin2|Ex||Ey|sin(Δ)Ex2+Ey2.

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