Abstract

In this paper we present experimental studies on diffraction of V-point singularities through equilateral and isosceles right triangular apertures. When V-point index, also called Poincare-Hopf index (η), of the optical field is +1, the diffraction disintegrates it into two monstars/lemons. When V-point index η is −1, diffraction produces two stars. The diffraction pattern, unlike phase singularity, is insensitive to polarity of the polarization singularity and the intensity pattern remains invariant. Higher order V-point singularities are generated using Sagnac interferometer and it is observed that the diffraction disintegrates them into lower order C-points.

© 2017 Optical Society of America

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References

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    [Crossref]
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  7. B. S. B. Ram, P. Senthilkumaran, and A. Sharma, “Polarization based spatial filtering for directional and non-directional edge enhancement using S-waveplate,” Appl. Opt. 56, 3171–3178 (2017).
    [Crossref]
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    [Crossref]
  9. M. Verma, S.K. Pal, A. Jejusaria, and P. Senthilkumaran, “Separation of spin and orbital angular momentum states from cylindrical vector beams,” Optik 132, 121–126 (2017).
    [Crossref]
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    [Crossref]
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2017 (3)

B. S. B. Ram, P. Senthilkumaran, and A. Sharma, “Polarization based spatial filtering for directional and non-directional edge enhancement using S-waveplate,” Appl. Opt. 56, 3171–3178 (2017).
[Crossref]

M. Verma, S.K. Pal, A. Jejusaria, and P. Senthilkumaran, “Separation of spin and orbital angular momentum states from cylindrical vector beams,” Optik 132, 121–126 (2017).
[Crossref]

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]

2016 (4)

B. Zhang, Z. Chen, H. Sun, J. Xia, and J. Ding, “Vectorial optical vortex filtering for edge enhancement,” J. Opt. 18, 035703 (2016).
[Crossref]

S. K. Pal and P. Senthilkumaran, “Cultivation of lemon fields,” Opt. Express. 24, 28008–28013 (2016).
[Crossref] [PubMed]

A. Aadhi, P. Vaity, P. Chithrabhanu, S. G. Reddy, S. Prabhakar, and R. P. Singh, “Non-coaxial superposition of vector vortex beams,” Appl. Opt. 55, 1107–1111 (2016).
[Crossref] [PubMed]

E. Otte, C. Alpmann, and C. Denz, “Higher-order polarization singularities in tailored vector beams,” J. Opt. 18, 074017 (2016).
[Crossref]

2015 (2)

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

M. Bahl and P. Senthilkumaran, “Energy circulations in singular beams diffracted through an isosceles right triangular aperture,” Phy. Rev. A. 92, 013831 (2015).
[Crossref]

2014 (1)

2013 (3)

Y. Liu, S. Sun, J. Pu, and B. Lu, “Propagation of an optical vortex beam through a diamond-shaped aperture,” Opt. Laser. Tech. 45, 473–479 (2013).
[Crossref]

J. Qi, X. Li, W. Wang, X. Wang, W. Sun, and J. Liao, “Generation and double slit interference of higher order vector beams,” Appl. Opt. 52, 8369–8375 (2013).
[Crossref]

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

2012 (2)

2011 (3)

T. G. Brown, “Unconventional Polarization States: Beam Propagation, Focusing, and Imaging,” Prog. Opt. 56, 81–129 (2011)
[Crossref]

L. E. E. de Araujo and M. E. Anderson, “Measuring vortex charge with a triangular aperture,” Opt. Lett. 36, 787–789 (2011).
[Crossref] [PubMed]

A. Mourka, J. Baumgartl, C. Shanor, K. Dholakia, and E. M. Wright, “Visualization of the birth of an optical vortex using diffraction from a triangular aperture,” Opt. Express. 19, 5760–5771 (2011).
[Crossref] [PubMed]

2010 (1)

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chavez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using lights orbital angular momentum,” Phy. Rev. Lett. 105, 053904 (2010).
[Crossref]

2009 (1)

D. P. Ghai, P. Senthilkumaran, and R. S. Sirohi, “Single-slit diffraction of an optical beam with phase singularity,” Opt. Lasers. Eng. 47, 123–126 (2009).
[Crossref]

2007 (1)

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. R. Marte, “Tailoring of arbitrary optical vector beams,” New J. Phy. 9, 78 (2007).
[Crossref]

2002 (2)

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

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

2001 (2)

I. Freund, “Polarization flowers,” Opt. Commun. 199, 47–63 (2001).
[Crossref]

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[Crossref]

Aadhi, A.

Alpmann, C.

E. Otte, C. Alpmann, and C. Denz, “Higher-order polarization singularities in tailored vector beams,” J. Opt. 18, 074017 (2016).
[Crossref]

Ambuj, A.

Anderson, M. E.

Andrews, D. L.

D. L. Andrews and M. Babiker, “The angular momentum of light,” (Cambridge. Univ. Press, 2012).
[Crossref]

Babiker, M.

D. L. Andrews and M. Babiker, “The angular momentum of light,” (Cambridge. Univ. Press, 2012).
[Crossref]

Bahl, M.

M. Bahl and P. Senthilkumaran, “Energy circulations in singular beams diffracted through an isosceles right triangular aperture,” Phy. Rev. A. 92, 013831 (2015).
[Crossref]

Baumgartl, J.

A. Mourka, J. Baumgartl, C. Shanor, K. Dholakia, and E. M. Wright, “Visualization of the birth of an optical vortex using diffraction from a triangular aperture,” Opt. Express. 19, 5760–5771 (2011).
[Crossref] [PubMed]

Bernet, S.

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. R. Marte, “Tailoring of arbitrary optical vector beams,” New J. Phy. 9, 78 (2007).
[Crossref]

Bonaccorso, F.

Brown, T. G.

T. G. Brown, “Unconventional Polarization States: Beam Propagation, Focusing, and Imaging,” Prog. Opt. 56, 81–129 (2011)
[Crossref]

Chavez-Cerda, S.

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chavez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using lights orbital angular momentum,” Phy. Rev. Lett. 105, 053904 (2010).
[Crossref]

Chen, Z.

B. Zhang, Z. Chen, H. Sun, J. Xia, and J. Ding, “Vectorial optical vortex filtering for edge enhancement,” J. Opt. 18, 035703 (2016).
[Crossref]

Chithrabhanu, P.

de Araujo, L. E. E.

Dennis, M. R.

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, 074017 (2016).
[Crossref]

Dholakia, K.

A. Mourka, J. Baumgartl, C. Shanor, K. Dholakia, and E. M. Wright, “Visualization of the birth of an optical vortex using diffraction from a triangular aperture,” Opt. Express. 19, 5760–5771 (2011).
[Crossref] [PubMed]

Ding, J.

B. Zhang, Z. Chen, H. Sun, J. Xia, and J. Ding, “Vectorial optical vortex filtering for edge enhancement,” J. Opt. 18, 035703 (2016).
[Crossref]

Donato, M. G.

Ferrari, A. C.

Fonseca, E. J. S.

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chavez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using lights orbital angular momentum,” Phy. Rev. Lett. 105, 053904 (2010).
[Crossref]

Freund, I.

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

I. Freund, “Polarization flowers,” Opt. Commun. 199, 47–63 (2001).
[Crossref]

Furhapter, S.

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. R. Marte, “Tailoring of arbitrary optical vector beams,” New J. Phy. 9, 78 (2007).
[Crossref]

Ghai, D. P.

D. P. Ghai, P. Senthilkumaran, and R. S. Sirohi, “Single-slit diffraction of an optical beam with phase singularity,” Opt. Lasers. Eng. 47, 123–126 (2009).
[Crossref]

Gu, B.

Gucciardi, P. G.

Hickmann, J. M.

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chavez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using lights orbital angular momentum,” Phy. Rev. Lett. 105, 053904 (2010).
[Crossref]

Jejusaria, A.

M. Verma, S.K. Pal, A. Jejusaria, and P. Senthilkumaran, “Separation of spin and orbital angular momentum states from cylindrical vector beams,” Optik 132, 121–126 (2017).
[Crossref]

Jesacher, A.

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. R. Marte, “Tailoring of arbitrary optical vector beams,” New J. Phy. 9, 78 (2007).
[Crossref]

Jones, P. H.

Joseph, J.

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

Joshi, S.

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

Kandpal, H.

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

Kong, L. J.

Kozawa, Y.

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

Li, X.

Li, Y.

Liao, J.

Liu, Y.

Y. Liu, S. Sun, J. Pu, and B. Lu, “Propagation of an optical vortex beam through a diamond-shaped aperture,” Opt. Laser. Tech. 45, 473–479 (2013).
[Crossref]

Lou, K.

Lu, B.

Y. Liu, S. Sun, J. Pu, and B. Lu, “Propagation of an optical vortex beam through a diamond-shaped aperture,” Opt. Laser. Tech. 45, 473–479 (2013).
[Crossref]

Marago, O. M.

Marte, M. R.

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. R. Marte, “Tailoring of arbitrary optical vector beams,” New J. Phy. 9, 78 (2007).
[Crossref]

Maurer, C.

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. R. Marte, “Tailoring of arbitrary optical vector beams,” New J. Phy. 9, 78 (2007).
[Crossref]

Mourka, A.

A. Mourka, J. Baumgartl, C. Shanor, K. Dholakia, and E. M. Wright, “Visualization of the birth of an optical vortex using diffraction from a triangular aperture,” Opt. Express. 19, 5760–5771 (2011).
[Crossref] [PubMed]

Otte, E.

E. Otte, C. Alpmann, and C. Denz, “Higher-order polarization singularities in tailored vector beams,” J. Opt. 18, 074017 (2016).
[Crossref]

Pal, S. K.

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]

S. K. Pal and P. Senthilkumaran, “Cultivation of lemon fields,” Opt. Express. 24, 28008–28013 (2016).
[Crossref] [PubMed]

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

Pal, S.K.

M. Verma, S.K. Pal, A. Jejusaria, and P. Senthilkumaran, “Separation of spin and orbital angular momentum states from cylindrical vector beams,” Optik 132, 121–126 (2017).
[Crossref]

Prabhakar, S.

Pu, J.

Y. Liu, S. Sun, J. Pu, and B. Lu, “Propagation of an optical vortex beam through a diamond-shaped aperture,” Opt. Laser. Tech. 45, 473–479 (2013).
[Crossref]

Qi, J.

Ram, B. S. B.

Reddy, S. G.

Ruchi,

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]

Sato, S.

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

Sayed, R.

Senthilkumaran, P.

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]

M. Verma, S.K. Pal, A. Jejusaria, and P. Senthilkumaran, “Separation of spin and orbital angular momentum states from cylindrical vector beams,” Optik 132, 121–126 (2017).
[Crossref]

B. S. B. Ram, P. Senthilkumaran, and A. Sharma, “Polarization based spatial filtering for directional and non-directional edge enhancement using S-waveplate,” Appl. Opt. 56, 3171–3178 (2017).
[Crossref]

S. K. Pal and P. Senthilkumaran, “Cultivation of lemon fields,” Opt. Express. 24, 28008–28013 (2016).
[Crossref] [PubMed]

M. Bahl and P. Senthilkumaran, “Energy circulations in singular beams diffracted through an isosceles right triangular aperture,” Phy. Rev. A. 92, 013831 (2015).
[Crossref]

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

D. P. Ghai, P. Senthilkumaran, and R. S. Sirohi, “Single-slit diffraction of an optical beam with phase singularity,” Opt. Lasers. Eng. 47, 123–126 (2009).
[Crossref]

Shanor, C.

A. Mourka, J. Baumgartl, C. Shanor, K. Dholakia, and E. M. Wright, “Visualization of the birth of an optical vortex using diffraction from a triangular aperture,” Opt. Express. 19, 5760–5771 (2011).
[Crossref] [PubMed]

Sharma, A.

Singh, R. P.

Singh, S.

Sirohi, R. S.

D. P. Ghai, P. Senthilkumaran, and R. S. Sirohi, “Single-slit diffraction of an optical beam with phase singularity,” Opt. Lasers. Eng. 47, 123–126 (2009).
[Crossref]

Soares, W. C.

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chavez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using lights orbital angular momentum,” Phy. Rev. Lett. 105, 053904 (2010).
[Crossref]

Soskin, M. S.

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[Crossref]

Sun, H.

B. Zhang, Z. Chen, H. Sun, J. Xia, and J. Ding, “Vectorial optical vortex filtering for edge enhancement,” J. Opt. 18, 035703 (2016).
[Crossref]

Sun, S.

Y. Liu, S. Sun, J. Pu, and B. Lu, “Propagation of an optical vortex beam through a diamond-shaped aperture,” Opt. Laser. Tech. 45, 473–479 (2013).
[Crossref]

Sun, W.

Tu, C.

Vaity, P.

Vasi, S.

Vasnetsov, M. V.

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[Crossref]

Verma, M.

M. Verma, S.K. Pal, A. Jejusaria, and P. Senthilkumaran, “Separation of spin and orbital angular momentum states from cylindrical vector beams,” Optik 132, 121–126 (2017).
[Crossref]

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

Vyas, R.

Vyas, S.

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

Wang, H. T.

Wang, W.

Wang, X.

Wang, X. L.

Wright, E. M.

A. Mourka, J. Baumgartl, C. Shanor, K. Dholakia, and E. M. Wright, “Visualization of the birth of an optical vortex using diffraction from a triangular aperture,” Opt. Express. 19, 5760–5771 (2011).
[Crossref] [PubMed]

Xia, J.

B. Zhang, Z. Chen, H. Sun, J. Xia, and J. Ding, “Vectorial optical vortex filtering for edge enhancement,” J. Opt. 18, 035703 (2016).
[Crossref]

Zhang, B.

B. Zhang, Z. Chen, H. Sun, J. Xia, and J. Ding, “Vectorial optical vortex filtering for edge enhancement,” J. Opt. 18, 035703 (2016).
[Crossref]

Zhao, H.

Appl. Opt. (3)

J. Mod. Opt. (1)

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

J. Opt. (2)

B. Zhang, Z. Chen, H. Sun, J. Xia, and J. Ding, “Vectorial optical vortex filtering for edge enhancement,” J. Opt. 18, 035703 (2016).
[Crossref]

E. Otte, C. Alpmann, and C. Denz, “Higher-order polarization singularities in tailored vector beams,” J. Opt. 18, 074017 (2016).
[Crossref]

New J. Phy. (1)

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. R. Marte, “Tailoring of arbitrary optical vector beams,” New J. Phy. 9, 78 (2007).
[Crossref]

Opt. Commun. (4)

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

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

I. Freund, “Polarization flowers,” Opt. Commun. 199, 47–63 (2001).
[Crossref]

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]

Opt. Express. (3)

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

S. K. Pal and P. Senthilkumaran, “Cultivation of lemon fields,” Opt. Express. 24, 28008–28013 (2016).
[Crossref] [PubMed]

A. Mourka, J. Baumgartl, C. Shanor, K. Dholakia, and E. M. Wright, “Visualization of the birth of an optical vortex using diffraction from a triangular aperture,” Opt. Express. 19, 5760–5771 (2011).
[Crossref] [PubMed]

Opt. Laser. Tech. (1)

Y. Liu, S. Sun, J. Pu, and B. Lu, “Propagation of an optical vortex beam through a diamond-shaped aperture,” Opt. Laser. Tech. 45, 473–479 (2013).
[Crossref]

Opt. Lasers. Eng. (1)

D. P. Ghai, P. Senthilkumaran, and R. S. Sirohi, “Single-slit diffraction of an optical beam with phase singularity,” Opt. Lasers. Eng. 47, 123–126 (2009).
[Crossref]

Opt. Lett. (4)

Optik (1)

M. Verma, S.K. Pal, A. Jejusaria, and P. Senthilkumaran, “Separation of spin and orbital angular momentum states from cylindrical vector beams,” Optik 132, 121–126 (2017).
[Crossref]

Phy. Rev. A. (1)

M. Bahl and P. Senthilkumaran, “Energy circulations in singular beams diffracted through an isosceles right triangular aperture,” Phy. Rev. A. 92, 013831 (2015).
[Crossref]

Phy. Rev. Lett. (1)

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chavez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using lights orbital angular momentum,” Phy. Rev. Lett. 105, 053904 (2010).
[Crossref]

Prog. Opt. (2)

T. G. Brown, “Unconventional Polarization States: Beam Propagation, Focusing, and Imaging,” Prog. Opt. 56, 81–129 (2011)
[Crossref]

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[Crossref]

Other (1)

D. L. Andrews and M. Babiker, “The angular momentum of light,” (Cambridge. Univ. Press, 2012).
[Crossref]

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Figures (5)

Fig. 1
Fig. 1 Experimental setup: (a) Sagnac Interferometer, (b) Triangular apertures, (c) 2f experimental configuration. L:lens, M:mirror, H/QP1,2:half/quarter waveplate, PBS:polarizing beam splitter, SPP:spiral phase plate, T:aperture, SC:Stokes camera, PC:computer
Fig. 2
Fig. 2 Simulated(left) and experimentally generated(right) V-point singularities of Poincare-Hopf index |η|=1 to 3. For every |η| the polarization distributions are classified into four types. The polarization distribution is superimposed on the Stokes phase ϕ12.
Fig. 3
Fig. 3 Simulated far field pattern: Top row shows the intensity distribution for diffraction through equilateral (left) and isosceles right triangular (right) apertures. Two triangles connecting the diffraction spots in the top three frames are drawn to indicate the superposition of diffraction due to charge +m and −m phase singularities (shown separately in the insets) of V-point singularities. Stokes phase shown in rows 2 to 5 correspond to the diffraction of beam types I–IV. The V-point index for each column is given in the top frame. Note, the incident V-point singularity has disintegrated into C-points of same polarity.
Fig. 4
Fig. 4 Experimentally observed far field patterns: Top row shows the intensity distribution for diffraction through equilateral (left) and isosceles right triangle (right) apertures. Stokes phase shown in rows 2 to 5 correspond to the diffraction of beam types I–IV. The V-point index for each column is given in the top frame. Note the incident V-point singularity has disintegrated into C-points of same polarity.
Fig. 5
Fig. 5 Stokes phase variation inside the equilateral triangular aperture due to V-point singularities (a) |η|=1, (b) |η|=2 and (c) |η|=3. Note the lack of symmetry in (b)

Equations (2)

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E = t ( x , y ) ( E x x ^ + E y y ^ )
T ( u , v ) j ^ = C i λ f t ( x , y ) E j j ^ e i 2 π λ f ( u x + v y ) d x d y

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