Abstract

The sign rule requires that adjacent singularities on a contour have opposite signs and hence cultivation of lemon only fields poses problem as all lemons have positive index. In this paper we show that the interference of three linearly polarized plane waves can create regions of ellipse and vector fields in 2-dimensions (2D) in which a lemon lattice is interlaced in a V-point lattice. The lemons appear at intensity maxima of the lattice structure while the V-points take care of index conservation by sitting at intensity minima. In the Stokes field S12, lemons and disclinations (V-points) appear as phase vortices of topological charge + 1 and −2 respectively. In the polarization distribution the constant azimuth lines (a-lines) are seen running through lemon and disclination alternatively obeying the sign rule [Opt. Lett. 27, 995 (2002) [CrossRef]  ]. We envisage that such polarization lattice structure may lead to novel concept of structured polarization illumination methods in super resolution microscopy.

© 2016 Optical Society of America

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References

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  1. J. F. Nye and M. V. Berry, “Dislocations in wavetrains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336(1605), 165–190 (1974).
    [Crossref]
  2. V. Y. Bazhenov, M. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wave-fronts,” J. Mod. Opt. 39(5), 985–990 (1992).
    [Crossref]
  3. M. Padgett and L. Allen, “Light with a twist in its tail,” Contemp. Phys. 41(5), 275–285 (2000).
    [Crossref]
  4. S. Vyas and P. Senthilkumaran, “Interferometric optical vortex array generator,” Appl. Opt. 46(15), 2893–2898 (2007).
    [Crossref] [PubMed]
  5. J. Xavier, S. Vyas, P. Senthilkumaran, and J. Joseph, “Tailored complex 3D vortex lattice structures by perturbed multiples of three-plane waves,” Appl. Opt. 51(12), 1872–1878 (2012).
    [Crossref] [PubMed]
  6. J. F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. Roy. Soc. A 389(1797), 279–290 (1983).
  7. J. V. Hajnal, “Singularities in the transverse fields of electromagnetic waves. I. theory,” Proc. Roy. Soc. A 414(1847), 433–446 (1987).
  8. J. V. Hajnal, “Singularities in the transverse fields of electromagnetic waves. II. Observations on the electric field,” Proc. Roy. Soc. A 414(1847), 447–468 (1987).
  9. M. V. Berry, “The electric and magnetic polarization singularities of paraxial waves,” J. Opt. A, Pure Appl. Opt. 6(5), 475–481 (2004).
    [Crossref]
  10. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1 (2009).
    [Crossref]
  11. Q. Zhan and J. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10(7), 324–331 (2002).
    [Crossref] [PubMed]
  12. K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000).
    [Crossref] [PubMed]
  13. M. Verma, S. K. Pal, S. Joshi, P. Senthilkumaran, J. Joseph, and H. C. Kandpal, “Singularities in cylindrical vector beams,” J. Mod. Opt. 62(13), 1068–1075 (2015).
    [Crossref]
  14. M. V. Berry and M. R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. Lond. A 457(2005), 141–155 (2001).
  15. E. J. Galvez, S. Khadka, W. H. Schubert, and S. Nomoto, “Poincaré-beam patterns produced by nonseparable superpositions of Laguerre-Gauss and polarization modes of light,” Appl. Opt. 51(15), 2925–2934 (2012).
    [Crossref] [PubMed]
  16. M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213(4–6), 201–221 (2002).
    [Crossref]
  17. I. Freund, “Polarization singularities in optical lattices,” Opt. Lett. 29(8), 875–877 (2004).
    [Crossref] [PubMed]
  18. S. Vyas, Y. Kozawa, and S. Sato, “Polarization singularities in superposition of vector beams,” Opt. Express 21(7), 8972–8986 (2013).
    [Crossref] [PubMed]
  19. I. Freund, M. Soskin, and A. I. Mokhun, “Elliptic critical points in paraxial optical fields,” Opt. Commun. 208(4–6), 223–253 (2002).
    [Crossref]
  20. I. Freund, “Polarization singularity indices in Gaussian laser beams,” Opt. Commun. 201(4), 251–270 (2002).
    [Crossref]
  21. F. Cardano, E. Karimi, S. Slussarenko, L. Marrucci, C. de Lisio, and E. Santamato, “Polarization pattern of vector vortex beams generated by q-plates with different topological charges,” Appl. Opt. 51(10), C1–C6 (2012).
    [Crossref] [PubMed]
  22. V. Kumar, G. M. Philip, and N. K. Viswanathan, “Formation and morphological transformation of polarization singularities: hunting the monstar,” J. Opt. 15(4), 1–4 (2013).
    [Crossref]
  23. M. V. Vasnetsov, M. S. Soskin, V. A. Pas’ko, and V. I. Vasil’ev, “A monstar portrait in the interior,” J. Opt. 18(034003), 1–8 (2016).
  24. A. I. Mokhun, M. S. Soskin, and I. Freund, “Elliptic critical points: C-points, a-lines, and the sign rule,” Opt. Lett. 27(12), 995–997 (2002).
    [Crossref] [PubMed]
  25. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).
  26. D. Goldstein, Polarized Light, 3rd ed. (CRC, 2011).

2016 (1)

M. V. Vasnetsov, M. S. Soskin, V. A. Pas’ko, and V. I. Vasil’ev, “A monstar portrait in the interior,” J. Opt. 18(034003), 1–8 (2016).

2015 (1)

M. Verma, S. K. Pal, S. Joshi, P. Senthilkumaran, J. Joseph, and H. C. Kandpal, “Singularities in cylindrical vector beams,” J. Mod. Opt. 62(13), 1068–1075 (2015).
[Crossref]

2013 (2)

V. Kumar, G. M. Philip, and N. K. Viswanathan, “Formation and morphological transformation of polarization singularities: hunting the monstar,” J. Opt. 15(4), 1–4 (2013).
[Crossref]

S. Vyas, Y. Kozawa, and S. Sato, “Polarization singularities in superposition of vector beams,” Opt. Express 21(7), 8972–8986 (2013).
[Crossref] [PubMed]

2012 (3)

2009 (1)

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1 (2009).
[Crossref]

2007 (1)

2004 (2)

I. Freund, “Polarization singularities in optical lattices,” Opt. Lett. 29(8), 875–877 (2004).
[Crossref] [PubMed]

M. V. Berry, “The electric and magnetic polarization singularities of paraxial waves,” J. Opt. A, Pure Appl. Opt. 6(5), 475–481 (2004).
[Crossref]

2002 (5)

M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213(4–6), 201–221 (2002).
[Crossref]

I. Freund, M. Soskin, and A. I. Mokhun, “Elliptic critical points in paraxial optical fields,” Opt. Commun. 208(4–6), 223–253 (2002).
[Crossref]

I. Freund, “Polarization singularity indices in Gaussian laser beams,” Opt. Commun. 201(4), 251–270 (2002).
[Crossref]

Q. Zhan and J. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10(7), 324–331 (2002).
[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(12), 995–997 (2002).
[Crossref] [PubMed]

2001 (1)

M. V. Berry and M. R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. Lond. A 457(2005), 141–155 (2001).

2000 (2)

1992 (1)

V. Y. Bazhenov, M. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wave-fronts,” J. Mod. Opt. 39(5), 985–990 (1992).
[Crossref]

1987 (2)

J. V. Hajnal, “Singularities in the transverse fields of electromagnetic waves. I. theory,” Proc. Roy. Soc. A 414(1847), 433–446 (1987).

J. V. Hajnal, “Singularities in the transverse fields of electromagnetic waves. II. Observations on the electric field,” Proc. Roy. Soc. A 414(1847), 447–468 (1987).

1983 (1)

J. F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. Roy. Soc. A 389(1797), 279–290 (1983).

1974 (1)

J. F. Nye and M. V. Berry, “Dislocations in wavetrains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336(1605), 165–190 (1974).
[Crossref]

Allen, L.

M. Padgett and L. Allen, “Light with a twist in its tail,” Contemp. Phys. 41(5), 275–285 (2000).
[Crossref]

Bazhenov, V. Y.

V. Y. Bazhenov, M. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wave-fronts,” J. Mod. Opt. 39(5), 985–990 (1992).
[Crossref]

Berry, M. V.

M. V. Berry, “The electric and magnetic polarization singularities of paraxial waves,” J. Opt. A, Pure Appl. Opt. 6(5), 475–481 (2004).
[Crossref]

M. V. Berry and M. R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. Lond. A 457(2005), 141–155 (2001).

J. F. Nye and M. V. Berry, “Dislocations in wavetrains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336(1605), 165–190 (1974).
[Crossref]

Brown, T.

Cardano, F.

de Lisio, C.

Dennis, M. R.

M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213(4–6), 201–221 (2002).
[Crossref]

M. V. Berry and M. R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. Lond. A 457(2005), 141–155 (2001).

Freund, I.

I. Freund, “Polarization singularities in optical lattices,” Opt. Lett. 29(8), 875–877 (2004).
[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(12), 995–997 (2002).
[Crossref] [PubMed]

I. Freund, M. Soskin, and A. I. Mokhun, “Elliptic critical points in paraxial optical fields,” Opt. Commun. 208(4–6), 223–253 (2002).
[Crossref]

I. Freund, “Polarization singularity indices in Gaussian laser beams,” Opt. Commun. 201(4), 251–270 (2002).
[Crossref]

Galvez, E. J.

Hajnal, J. V.

J. V. Hajnal, “Singularities in the transverse fields of electromagnetic waves. I. theory,” Proc. Roy. Soc. A 414(1847), 433–446 (1987).

J. V. Hajnal, “Singularities in the transverse fields of electromagnetic waves. II. Observations on the electric field,” Proc. Roy. Soc. A 414(1847), 447–468 (1987).

Joseph, J.

M. Verma, S. K. Pal, S. Joshi, P. Senthilkumaran, J. Joseph, and H. C. Kandpal, “Singularities in cylindrical vector beams,” J. Mod. Opt. 62(13), 1068–1075 (2015).
[Crossref]

J. Xavier, S. Vyas, P. Senthilkumaran, and J. Joseph, “Tailored complex 3D vortex lattice structures by perturbed multiples of three-plane waves,” Appl. Opt. 51(12), 1872–1878 (2012).
[Crossref] [PubMed]

Joshi, S.

M. Verma, S. K. Pal, S. Joshi, P. Senthilkumaran, J. Joseph, and H. C. Kandpal, “Singularities in cylindrical vector beams,” J. Mod. Opt. 62(13), 1068–1075 (2015).
[Crossref]

Kandpal, H. C.

M. Verma, S. K. Pal, S. Joshi, P. Senthilkumaran, J. Joseph, and H. C. Kandpal, “Singularities in cylindrical vector beams,” J. Mod. Opt. 62(13), 1068–1075 (2015).
[Crossref]

Karimi, E.

Khadka, S.

Kozawa, Y.

Kumar, V.

V. Kumar, G. M. Philip, and N. K. Viswanathan, “Formation and morphological transformation of polarization singularities: hunting the monstar,” J. Opt. 15(4), 1–4 (2013).
[Crossref]

Leger, J.

Marrucci, L.

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(12), 995–997 (2002).
[Crossref] [PubMed]

I. Freund, M. Soskin, and A. I. Mokhun, “Elliptic critical points in paraxial optical fields,” Opt. Commun. 208(4–6), 223–253 (2002).
[Crossref]

Nomoto, S.

Nye, J. F.

J. F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. Roy. Soc. A 389(1797), 279–290 (1983).

J. F. Nye and M. V. Berry, “Dislocations in wavetrains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336(1605), 165–190 (1974).
[Crossref]

Padgett, M.

M. Padgett and L. Allen, “Light with a twist in its tail,” Contemp. Phys. 41(5), 275–285 (2000).
[Crossref]

Pal, S. K.

M. Verma, S. K. Pal, S. Joshi, P. Senthilkumaran, J. Joseph, and H. C. Kandpal, “Singularities in cylindrical vector beams,” J. Mod. Opt. 62(13), 1068–1075 (2015).
[Crossref]

Pas’ko, V. A.

M. V. Vasnetsov, M. S. Soskin, V. A. Pas’ko, and V. I. Vasil’ev, “A monstar portrait in the interior,” J. Opt. 18(034003), 1–8 (2016).

Philip, G. M.

V. Kumar, G. M. Philip, and N. K. Viswanathan, “Formation and morphological transformation of polarization singularities: hunting the monstar,” J. Opt. 15(4), 1–4 (2013).
[Crossref]

Santamato, E.

Sato, S.

Schubert, W. H.

Senthilkumaran, P.

Slussarenko, S.

Soskin, M.

I. Freund, M. Soskin, and A. I. Mokhun, “Elliptic critical points in paraxial optical fields,” Opt. Commun. 208(4–6), 223–253 (2002).
[Crossref]

V. Y. Bazhenov, M. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wave-fronts,” J. Mod. Opt. 39(5), 985–990 (1992).
[Crossref]

Soskin, M. S.

M. V. Vasnetsov, M. S. Soskin, V. A. Pas’ko, and V. I. Vasil’ev, “A monstar portrait in the interior,” J. Opt. 18(034003), 1–8 (2016).

A. I. Mokhun, M. S. Soskin, and I. Freund, “Elliptic critical points: C-points, a-lines, and the sign rule,” Opt. Lett. 27(12), 995–997 (2002).
[Crossref] [PubMed]

Vasil’ev, V. I.

M. V. Vasnetsov, M. S. Soskin, V. A. Pas’ko, and V. I. Vasil’ev, “A monstar portrait in the interior,” J. Opt. 18(034003), 1–8 (2016).

Vasnetsov, M. V.

M. V. Vasnetsov, M. S. Soskin, V. A. Pas’ko, and V. I. Vasil’ev, “A monstar portrait in the interior,” J. Opt. 18(034003), 1–8 (2016).

V. Y. Bazhenov, M. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wave-fronts,” J. Mod. Opt. 39(5), 985–990 (1992).
[Crossref]

Verma, M.

M. Verma, S. K. Pal, S. Joshi, P. Senthilkumaran, J. Joseph, and H. C. Kandpal, “Singularities in cylindrical vector beams,” J. Mod. Opt. 62(13), 1068–1075 (2015).
[Crossref]

Viswanathan, N. K.

V. Kumar, G. M. Philip, and N. K. Viswanathan, “Formation and morphological transformation of polarization singularities: hunting the monstar,” J. Opt. 15(4), 1–4 (2013).
[Crossref]

Vyas, S.

Xavier, J.

Youngworth, K.

Zhan, Q.

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1 (2009).
[Crossref]

Q. Zhan and J. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10(7), 324–331 (2002).
[Crossref] [PubMed]

Adv. Opt. Photonics (1)

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1 (2009).
[Crossref]

Appl. Opt. (4)

Contemp. Phys. (1)

M. Padgett and L. Allen, “Light with a twist in its tail,” Contemp. Phys. 41(5), 275–285 (2000).
[Crossref]

J. Mod. Opt. (2)

M. Verma, S. K. Pal, S. Joshi, P. Senthilkumaran, J. Joseph, and H. C. Kandpal, “Singularities in cylindrical vector beams,” J. Mod. Opt. 62(13), 1068–1075 (2015).
[Crossref]

V. Y. Bazhenov, M. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wave-fronts,” J. Mod. Opt. 39(5), 985–990 (1992).
[Crossref]

J. Opt. (2)

V. Kumar, G. M. Philip, and N. K. Viswanathan, “Formation and morphological transformation of polarization singularities: hunting the monstar,” J. Opt. 15(4), 1–4 (2013).
[Crossref]

M. V. Vasnetsov, M. S. Soskin, V. A. Pas’ko, and V. I. Vasil’ev, “A monstar portrait in the interior,” J. Opt. 18(034003), 1–8 (2016).

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

M. V. Berry, “The electric and magnetic polarization singularities of paraxial waves,” J. Opt. A, Pure Appl. Opt. 6(5), 475–481 (2004).
[Crossref]

Opt. Commun. (3)

M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213(4–6), 201–221 (2002).
[Crossref]

I. Freund, M. Soskin, and A. I. Mokhun, “Elliptic critical points in paraxial optical fields,” Opt. Commun. 208(4–6), 223–253 (2002).
[Crossref]

I. Freund, “Polarization singularity indices in Gaussian laser beams,” Opt. Commun. 201(4), 251–270 (2002).
[Crossref]

Opt. Express (3)

Opt. Lett. (2)

Proc. R. Soc. Lond. A (1)

M. V. Berry and M. R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. Lond. A 457(2005), 141–155 (2001).

Proc. R. Soc. Lond. A Math. Phys. Sci. (1)

J. F. Nye and M. V. Berry, “Dislocations in wavetrains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336(1605), 165–190 (1974).
[Crossref]

Proc. Roy. Soc. A (3)

J. F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. Roy. Soc. A 389(1797), 279–290 (1983).

J. V. Hajnal, “Singularities in the transverse fields of electromagnetic waves. I. theory,” Proc. Roy. Soc. A 414(1847), 433–446 (1987).

J. V. Hajnal, “Singularities in the transverse fields of electromagnetic waves. II. Observations on the electric field,” Proc. Roy. Soc. A 414(1847), 447–468 (1987).

Other (2)

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

D. Goldstein, Polarized Light, 3rd ed. (CRC, 2011).

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Figures (5)

Fig. 1
Fig. 1 Schematic representation of the wave vectors of three interfering plane waves.
Fig. 2
Fig. 2 Simulated intensity pattern of (a) x-component of resultant field, (b) y-component of resultant field, (c) transverse intensity distribution of resultant field.
Fig. 3
Fig. 3 Simulated patterns of (a) Stokes field intensity embedded with zero contour lines of real and imaginary part of the Stokes field S 12 , (b) phase variation of the Stokes field S 12 , (c) phase contour lines, (d) polarization ellipse field: lemons are shown by red points, V-points are shown by black points, (e) magnified version of (d) to show arrangement of lemons around a V-point.
Fig. 4
Fig. 4 (a) Experimental setup. SF, spatial filter assembly (MO and PH); L, lenses; BS, beam splitter; SLM, spatial light modulator; FF, Fourier filter; PC, polarization Converter; HWP, half wave plate; SC, Stokes camera, (b) simulated three beam interference phase pattern.
Fig. 5
Fig. 5 Experimental results (a) intensity pattern, (b) and (c) Polarization distribution.

Equations (4)

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k m = k 0 { s i n θ m cos ξ m , s i n θ m s i n ξ m , c o s θ m }
E m = ( x ^ E m x + y ^ E m y + z ^ E m z ) exp ( i k m r )
S 0 = | E x | 2 + | E y | 2 S 1 = S 0 1 ( | E x | 2 | E y | 2 ) S 2 = 2 S 0 1 Re ( E x * E y ) S 3 = 2 S 0 1 I m ( E x * E y )
2 I c = 1 π r γ 0 d R = 1 2 π r φ 12 d R = l

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