Abstract

Polarization singularity lattices exhibit richer features and varieties than their scalar counterparts, namely phase vortex lattices. Lattices consisting of only C-points or only V-points of different Stokes indices are possible by phase and polarization engineering. In this article we show the generation of one generic and three non-generic lattices—all abiding the enlarged sign principle in six-beam interference. Two of them are vector fields and two of them are ellipse fields. In the vector fields, one lattice consists of V-points of the same magnitude and the other consists of V-points of different magnitude of the Poincare-Hopf index. Similarly in the two ellipse fields, the same and different C-point index lattices are there. Interestingly all the C-points are of the same handedness. In one particular case the C-point lattice is interlaced with saddle points, in which formation of C-lines is also noticed. These are saddles in both azimuth and ellipticity distributions. The governing rules for realizing these lattices are given and these lattices are experimentally demonstrated.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (2)

2017 (3)

2016 (2)

2015 (1)

2013 (1)

2012 (1)

2011 (2)

2010 (1)

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

2009 (1)

R. W. Schoonover and T. D. Visser, “Creating polarization singularities with an N-pinhole interferometer,” Phys. Rev. A 79, 1–7 (2009).
[Crossref]

2004 (1)

2002 (4)

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

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

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

1995 (1)

I. Freund, “Saddles, singularities, and extrema in random phase fields,” Phys. Rev. E 52, 2348–2360 (1995).
[Crossref]

1994 (1)

I. Freund and N. Shvartsman, “Wave-field phase singularities: The sign principle,” Phys. Rev. A 50, 5164–5172 (1994).
[Crossref] [PubMed]

1987 (1)

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

Born, M.

M. Born and E. Wolf, Principle of Optics (Cambridge University Press, 1999).
[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] [PubMed]

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

Chen, Y.

Dennis, M. R.

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

Freund, I.

I. Freund, “Möbius strips and twisted ribbons in intersecting Gauss-Laguerre beams,” Opt. Commun. 284, 3816–3845 (2011).
[Crossref]

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

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

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

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

I. Freund, “Saddles, singularities, and extrema in random phase fields,” Phys. Rev. E 52, 2348–2360 (1995).
[Crossref]

I. Freund and N. Shvartsman, “Wave-field phase singularities: The sign principle,” Phys. Rev. A 50, 5164–5172 (1994).
[Crossref] [PubMed]

Galushko, Y.

Gbur, G.

Goldstein, D.

D. Goldstein, Polarized Light (CRC Press, 2011).

Hajnal, J. V.

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

Kharitonova, Y.

Khrobatin, R.

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] [PubMed]

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

Mokhun, A. I.

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

Mokhun, I.

Pal, S. K.

Pang, X.

Peng, X.

Ram, B. Bhargava

Ruchi,

Schoonover, R. W.

R. W. Schoonover and T. D. Visser, “Creating polarization singularities with an N-pinhole interferometer,” Phys. Rev. A 79, 1–7 (2009).
[Crossref]

Senthilkumaran, P.

Shvartsman, N.

I. Freund and N. Shvartsman, “Wave-field phase singularities: The sign principle,” Phys. Rev. A 50, 5164–5172 (1994).
[Crossref] [PubMed]

Song, Q.

Soskin, M. S.

A. I. Mokhun, M. S. Soskin, and I. Freund, “Elliptic 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]

Viktorovskaya, Y.

Visser, T. D.

X. Pang, G. Gbur, and T. D. Visser, “Cycle of phase, coherence and polarization singularities in Young’s three-pinhole experiment,” Opt. Express 23, 34093–34108 (2015).
[Crossref]

R. W. Schoonover and T. D. Visser, “Creating polarization singularities with an N-pinhole interferometer,” Phys. Rev. A 79, 1–7 (2009).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principle of Optics (Cambridge University Press, 1999).
[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] [PubMed]

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

Xin, Y.

Ye, D.

Yu, R.

Zdunek, M.

Zhao, Q.

Appl. Opt. (1)

J. Opt. (1)

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

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

Opt. Commun. (5)

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

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

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

I. Freund, “Möbius strips and twisted ribbons in intersecting Gauss-Laguerre beams,” Opt. Commun. 284, 3816–3845 (2011).
[Crossref]

Opt. Express (4)

Opt. Lett. (5)

Phys. Rev. A (2)

I. Freund and N. Shvartsman, “Wave-field phase singularities: The sign principle,” Phys. Rev. A 50, 5164–5172 (1994).
[Crossref] [PubMed]

R. W. Schoonover and T. D. Visser, “Creating polarization singularities with an N-pinhole interferometer,” Phys. Rev. A 79, 1–7 (2009).
[Crossref]

Phys. Rev. E (1)

I. Freund, “Saddles, singularities, and extrema in random phase fields,” Phys. Rev. E 52, 2348–2360 (1995).
[Crossref]

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

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

Other (2)

D. Goldstein, Polarized Light (CRC Press, 2011).

M. Born and E. Wolf, Principle of Optics (Cambridge University Press, 1999).
[Crossref]

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Figures (8)

Fig. 1
Fig. 1 The wave vectors of each of the radially polarized beams are shown as black dots on a circle in the k-space. In (a–d), the initial constant phase offset given to beams numbered 1 to 6 leads to an accumulated phase as indicated at the center.
Fig. 2
Fig. 2 Simulated six-beam interference intensity patterns (a–d) for the four lattices E60, E61, E62, and E63. Insets: S12 Stokes intensity patterns.
Fig. 3
Fig. 3 Simulated polarization distributions of phase engineered vector and ellipse fields: (a) E60; (b) E61; (c) E62; and (d) E63 (Inset-Stokes phase). Blue, red, and green colours are used to indicate left, right handed SOPs and C-lines respectively.
Fig. 4
Fig. 4 Simulated index inversed polarization distributions of the ellipse fields: (a) E60; (b) E61; (c) E62; and (d) E63 (Inset-Stokes phase).
Fig. 5
Fig. 5 Zero contours of S1 (red) and S2 (black) of the field E62. The field is sufficiently perturbed to see the exact contours of S1 = 0 and S2 = 0 at several points that are numbered. The R and I denote real and imaginary part of S12 Stokes field.
Fig. 6
Fig. 6 Experimental setup: Microscope Objective (MO); Pin hole (PH); Lenses (L); Fourier filter (FF); S-wave plate (PC); Stokes camera (SC); Spatial light modulator (SLM). (a)–(d) Phase distributions displayed onto the SLM; (e)–(h) Recorded intensity distributions of the lattices.
Fig. 7
Fig. 7 Experimentally obtained polarization distributions of fields: (a) E60; (b) E61; (c) E62; and (d) E63.
Fig. 8
Fig. 8 Experimentally obtained polarization distributions of index inversed fields: (a) E60; (b) E61; (c) E62; and (d) E63.

Tables (1)

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Table 1 Synthesis of Various Polarization Singularity Lattices

Equations (1)

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E R = j = 1 6 ( E j , l ) = j = 1 6 r ^ e i ( k j r + j ϕ l / 6 ) ,