Abstract

We present a systematic study of the superposition of two vector Laguerre-Gaussian (LG) beams. Propagation depended field distribution obtained from the superposition of two vector LG beams has many interesting features of intensity and polarization. Characteristic inhomogeneous polarization distribution of the vector LG beam appears in the form of azimuthally modulated intensity and polarization distributions in the superposition of the beams. We found that the array of polarization singular points, whose number depends upon the azimuthal indices of the two beams, evolves during propagation of the field. The position and number of C-points generated in the field were analyzed using Stokes singularity relations. Novel intensity and polarization patterns obtained from the superposition of two vector LG beams may find applications in the field of molecular imaging, optical manipulation, atom optics, and optical lattices.

© 2013 OSA

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2012

X. Li, T. H. Lan, C. H. Tien, and M. Gu, “Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam,” Nat. Commun3, 998 (2012).
[CrossRef] [PubMed]

P. Senthilkumaran, J. Masajada, and S. Sato, “Interferometry with vortices,” Int. J. Opt.2012, 517591 (2012).

G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam-Berry phase and the angular momentum of light,” Phys. Rev. Lett.108(19), 190401 (2012).
[CrossRef] [PubMed]

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]

Y. Kozawa and S. Sato, “Focusing of higher-order radially polarized Laguerre-Gaussian beam,” J. Opt. Soc. Am. A29(11), 2439–2443 (2012).
[CrossRef] [PubMed]

P. Kurzynowski, W. A. Woźniak, M. Zdunek, and M. Borwińska, “Singularities of interference of three waves with different polarization states,” Opt. Express20(24), 26755–26765 (2012).
[CrossRef] [PubMed]

2011

I. Freund, “Möbius strip and twisted ribbon in intersecting Gauss-Laguerre beams,” Opt. Commun.284(16-17), 3816–3845 (2011).
[CrossRef]

D. Yang, J. Zhao, T. Zhao, and L. Kong, “Generation of rotating intensity blades by superposing optical vortex beams,” Opt. Commun.284(14), 3597–3600 (2011).
[CrossRef]

2010

2009

2008

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics2(8), 501–505 (2008).
[CrossRef]

K. Yu. Bliokh, A. Niv, V. Kleiner, and E. Hasman, “Singular polarimetry: Evolution of polarization singularities in electromagnetic waves propagating in a weakly anisotropic medium,” Opt. Express16(2), 695–709 (2008).
[CrossRef] [PubMed]

2007

S. Franke-Arnold, J. Leach, M. J. Padgett, V. E. Lembessis, D. Ellinas, A. J. Wright, J. M. Girkin, P. Ohberg, and A. S. Arnold, “Optical ferris wheel for ultracold atoms,” Opt. Express15(14), 8619–8625 (2007).
[CrossRef] [PubMed]

T. H. Lu, Y. F. Chen, and K. F. Huang, “Generalized hyperboloid structures of polarization singularities in Laguerre-Gaussian vector fields,” Phys. Rev. A76(6), 063809 (2007).
[CrossRef]

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys.9(3), 78 (2007).
[CrossRef]

2006

2005

2004

2003

2002

I. Freund, A. I. Mokhun, M. S. Soskin, O. V. Angelsky, and I. I. Mokhun, “Stokes singularity relations,” Opt. Lett.27(7), 545–547 (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]

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

I. Freund, “Polarization singularity indices in Gaussian laser beams,” Opt. Commun.201(4-6), 251–270 (2002).
[CrossRef]

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

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures,” Science296(5570), 1101–1103 (2002).
[CrossRef] [PubMed]

2001

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science292(5518), 912–914 (2001).
[CrossRef] [PubMed]

I. Freund, “Polarization flowers,” Opt. Commun.199(1-4), 47–63 (2001).
[CrossRef]

I. Freund, “Poincaré vortices,” Opt. Lett.26(24), 1996–1998 (2001).
[CrossRef] [PubMed]

1998

1987

J. V. Hajnal, “Singularities in the transverse field of electromagnetic waves. I Theory,” Proc. R. Soc. Lond. A Math. Phys. Sci.414(1847), 433–446 (1987).
[CrossRef]

J. V. Hajnal, “Singularities in the transverse fields of electromagnetic waves. II. Observation on the electric field,” Proc. R. Soc. Lond. A Math. Phys. Sci.414(1847), 447–468 (1987).
[CrossRef]

J. F. Nye and J. V. Hajnal, “The wave structure of monochromatic electromagnetic radiation,” Proc. R. Soc. Lond. A Math. Phys. Sci.409(1836), 21–36 (1987).
[CrossRef]

Alfano, R. R.

G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam-Berry phase and the angular momentum of light,” Phys. Rev. Lett.108(19), 190401 (2012).
[CrossRef] [PubMed]

Alonso, M. A.

Ando, T.

Angelsky, O. V.

O. V. Angelsky, A. G. Ushenko, Y. G. Ushenko, and Y. Y. Tomka, “Polarization singularities of biological tissues images,” J. Biomed. Opt.11(5), 054030 (2006).
[CrossRef] [PubMed]

I. Freund, A. I. Mokhun, M. S. Soskin, O. V. Angelsky, and I. I. Mokhun, “Stokes singularity relations,” Opt. Lett.27(7), 545–547 (2002).
[CrossRef] [PubMed]

Arlt, J.

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures,” Science296(5570), 1101–1103 (2002).
[CrossRef] [PubMed]

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science292(5518), 912–914 (2001).
[CrossRef] [PubMed]

Arnold, A. S.

Baba, T.

M. Burresi, R. J. P. Engelen, A. Opheij, D. van Oosten, D. Mori, T. Baba, and L. Kuipers, “Observation of polarization singularities at the nanoscale,” Phys. Rev. Lett.102(3), 033902 (2009).
[CrossRef] [PubMed]

Baumann, S. M.

Beckley, A. M.

Bernet, S.

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys.9(3), 78 (2007).
[CrossRef]

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Size selective trapping with optical “cogwheel” tweezers,” Opt. Express12(17), 4129–4135 (2004).
[CrossRef] [PubMed]

Bliokh, K. Yu.

Borwinska, M.

Brasselet, E.

E. Brasselet and S. Juodkazis, “Intangible pointlike tracers for liquid-crystal-based microsensors,” Phys. Rev. A82(6), 063832 (2010).
[CrossRef]

Brown, T. G.

Bryant, P. E.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science292(5518), 912–914 (2001).
[CrossRef] [PubMed]

Burge, R. E.

Burresi, M.

M. Burresi, R. J. P. Engelen, A. Opheij, D. van Oosten, D. Mori, T. Baba, and L. Kuipers, “Observation of polarization singularities at the nanoscale,” Phys. Rev. Lett.102(3), 033902 (2009).
[CrossRef] [PubMed]

Chen, Y. F.

T. H. Lu, Y. F. Chen, and K. F. Huang, “Generalized hyperboloid structures of polarization singularities in Laguerre-Gaussian vector fields,” Phys. Rev. A76(6), 063809 (2007).
[CrossRef]

Y. F. Chen, T. H. Lu, and K. F. Huang, “Hyperboloid structures formed by polarization singularities in coherent vector fields with longitudinal-transverse coupling,” Phys. Rev. Lett.97(23), 233903 (2006).
[CrossRef] [PubMed]

Chong, C. T.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics2(8), 501–505 (2008).
[CrossRef]

Courtial, J.

J. Courtial, “Self-imaging beams and the Gouy effect,” Opt. Commun.151(1-3), 1–4 (1998).
[CrossRef]

Dennis, M. R.

Dholakia, K.

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures,” Science296(5570), 1101–1103 (2002).
[CrossRef] [PubMed]

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science292(5518), 912–914 (2001).
[CrossRef] [PubMed]

Ellinas, D.

Engelen, R. J. P.

M. Burresi, R. J. P. Engelen, A. Opheij, D. van Oosten, D. Mori, T. Baba, and L. Kuipers, “Observation of polarization singularities at the nanoscale,” Phys. Rev. Lett.102(3), 033902 (2009).
[CrossRef] [PubMed]

Evans, S.

G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam-Berry phase and the angular momentum of light,” Phys. Rev. Lett.108(19), 190401 (2012).
[CrossRef] [PubMed]

Flossmann, F.

Franke-Arnold, S.

Freund, I.

I. Freund, “Möbius strip and twisted ribbon in intersecting Gauss-Laguerre beams,” Opt. Commun.284(16-17), 3816–3845 (2011).
[CrossRef]

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, “Polarization singularity indices in Gaussian laser beams,” Opt. Commun.201(4-6), 251–270 (2002).
[CrossRef]

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

I. Freund, A. I. Mokhun, M. S. Soskin, O. V. Angelsky, and I. I. Mokhun, “Stokes singularity relations,” Opt. Lett.27(7), 545–547 (2002).
[CrossRef] [PubMed]

I. Freund, “Polarization flowers,” Opt. Commun.199(1-4), 47–63 (2001).
[CrossRef]

I. Freund, “Poincaré vortices,” Opt. Lett.26(24), 1996–1998 (2001).
[CrossRef] [PubMed]

Furhapter, S.

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys.9(3), 78 (2007).
[CrossRef]

Fürhapter, S.

Galvez, E. J.

Girkin, J. M.

Gu, M.

X. Li, T. H. Lan, C. H. Tien, and M. Gu, “Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam,” Nat. Commun3, 998 (2012).
[CrossRef] [PubMed]

Hajnal, J. V.

J. V. Hajnal, “Singularities in the transverse field of electromagnetic waves. I Theory,” Proc. R. Soc. Lond. A Math. Phys. Sci.414(1847), 433–446 (1987).
[CrossRef]

J. V. Hajnal, “Singularities in the transverse fields of electromagnetic waves. II. Observation on the electric field,” Proc. R. Soc. Lond. A Math. Phys. Sci.414(1847), 447–468 (1987).
[CrossRef]

J. F. Nye and J. V. Hajnal, “The wave structure of monochromatic electromagnetic radiation,” Proc. R. Soc. Lond. A Math. Phys. Sci.409(1836), 21–36 (1987).
[CrossRef]

Hasman, E.

Huang, K. F.

T. H. Lu, Y. F. Chen, and K. F. Huang, “Generalized hyperboloid structures of polarization singularities in Laguerre-Gaussian vector fields,” Phys. Rev. A76(6), 063809 (2007).
[CrossRef]

Y. F. Chen, T. H. Lu, and K. F. Huang, “Hyperboloid structures formed by polarization singularities in coherent vector fields with longitudinal-transverse coupling,” Phys. Rev. Lett.97(23), 233903 (2006).
[CrossRef] [PubMed]

Inoue, T.

Jesacher, A.

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys.9(3), 78 (2007).
[CrossRef]

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Size selective trapping with optical “cogwheel” tweezers,” Opt. Express12(17), 4129–4135 (2004).
[CrossRef] [PubMed]

Juodkazis, S.

E. Brasselet and S. Juodkazis, “Intangible pointlike tracers for liquid-crystal-based microsensors,” Phys. Rev. A82(6), 063832 (2010).
[CrossRef]

Kalb, D. M.

Khadka, S.

Kleiner, V.

Kong, L.

D. Yang, J. Zhao, T. Zhao, and L. Kong, “Generation of rotating intensity blades by superposing optical vortex beams,” Opt. Commun.284(14), 3597–3600 (2011).
[CrossRef]

Kozawa, Y.

Kuipers, L.

M. Burresi, R. J. P. Engelen, A. Opheij, D. van Oosten, D. Mori, T. Baba, and L. Kuipers, “Observation of polarization singularities at the nanoscale,” Phys. Rev. Lett.102(3), 033902 (2009).
[CrossRef] [PubMed]

Kurzynowski, P.

Lan, T. H.

X. Li, T. H. Lan, C. H. Tien, and M. Gu, “Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam,” Nat. Commun3, 998 (2012).
[CrossRef] [PubMed]

Leach, J.

Lembessis, V. E.

Lerman, G. M.

Levy, U.

Li, X.

X. Li, T. H. Lan, C. H. Tien, and M. Gu, “Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam,” Nat. Commun3, 998 (2012).
[CrossRef] [PubMed]

Li, Y. P.

Lin, J.

Lu, T. H.

T. H. Lu, Y. F. Chen, and K. F. Huang, “Generalized hyperboloid structures of polarization singularities in Laguerre-Gaussian vector fields,” Phys. Rev. A76(6), 063809 (2007).
[CrossRef]

Y. F. Chen, T. H. Lu, and K. F. Huang, “Hyperboloid structures formed by polarization singularities in coherent vector fields with longitudinal-transverse coupling,” Phys. Rev. Lett.97(23), 233903 (2006).
[CrossRef] [PubMed]

Lü, B.

Lukyanchuk, B.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics2(8), 501–505 (2008).
[CrossRef]

Luo, Y.

MacDonald, M. P.

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures,” Science296(5570), 1101–1103 (2002).
[CrossRef] [PubMed]

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science292(5518), 912–914 (2001).
[CrossRef] [PubMed]

MacMillan, L. H.

Maier, M.

Maleev, I. D.

Masajada, J.

P. Senthilkumaran, J. Masajada, and S. Sato, “Interferometry with vortices,” Int. J. Opt.2012, 517591 (2012).

Matsumoto, N.

Maurer, C.

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M. Burresi, R. J. P. Engelen, A. Opheij, D. van Oosten, D. Mori, T. Baba, and L. Kuipers, “Observation of polarization singularities at the nanoscale,” Phys. Rev. Lett.102(3), 033902 (2009).
[CrossRef] [PubMed]

Niv, A.

Nolan, D. A.

G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam-Berry phase and the angular momentum of light,” Phys. Rev. Lett.108(19), 190401 (2012).
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M. Burresi, R. J. P. Engelen, A. Opheij, D. van Oosten, D. Mori, T. Baba, and L. Kuipers, “Observation of polarization singularities at the nanoscale,” Phys. Rev. Lett.102(3), 033902 (2009).
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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics2(8), 501–505 (2008).
[CrossRef]

Shi, L.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics2(8), 501–505 (2008).
[CrossRef]

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M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures,” Science296(5570), 1101–1103 (2002).
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[CrossRef] [PubMed]

Soskin, M. S.

Swartzlander, G. A.

Takiguchi, Yu.

Tao, S. H.

Tien, C. H.

X. Li, T. H. Lan, C. H. Tien, and M. Gu, “Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam,” Nat. Commun3, 998 (2012).
[CrossRef] [PubMed]

Tomka, Y. Y.

O. V. Angelsky, A. G. Ushenko, Y. G. Ushenko, and Y. Y. Tomka, “Polarization singularities of biological tissues images,” J. Biomed. Opt.11(5), 054030 (2006).
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Tovar, A. A.

Ushenko, A. G.

O. V. Angelsky, A. G. Ushenko, Y. G. Ushenko, and Y. Y. Tomka, “Polarization singularities of biological tissues images,” J. Biomed. Opt.11(5), 054030 (2006).
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Ushenko, Y. G.

O. V. Angelsky, A. G. Ushenko, Y. G. Ushenko, and Y. Y. Tomka, “Polarization singularities of biological tissues images,” J. Biomed. Opt.11(5), 054030 (2006).
[CrossRef] [PubMed]

van Oosten, D.

M. Burresi, R. J. P. Engelen, A. Opheij, D. van Oosten, D. Mori, T. Baba, and L. Kuipers, “Observation of polarization singularities at the nanoscale,” Phys. Rev. Lett.102(3), 033902 (2009).
[CrossRef] [PubMed]

Visser, T. D.

Volke-Sepulveda, K.

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures,” Science296(5570), 1101–1103 (2002).
[CrossRef] [PubMed]

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H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics2(8), 501–505 (2008).
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Zdunek, M.

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D. Yang, J. Zhao, T. Zhao, and L. Kong, “Generation of rotating intensity blades by superposing optical vortex beams,” Opt. Commun.284(14), 3597–3600 (2011).
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D. Yang, J. Zhao, T. Zhao, and L. Kong, “Generation of rotating intensity blades by superposing optical vortex beams,” Opt. Commun.284(14), 3597–3600 (2011).
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[CrossRef] [PubMed]

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures,” Science296(5570), 1101–1103 (2002).
[CrossRef] [PubMed]

Supplementary Material (2)

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» Media 2: AVI (1849 KB)     

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Figures (7)

Fig. 1
Fig. 1

Numerically calculated intensity distributions together with the schematic polarization vector for the vector LG beams with the radial index p = 0, (a) different polarization distributions of the vector LG beam with the azimuthal index m = 1, (b) different polarization distributions of the vector LG beams with the azimuthal index m = 2.

Fig. 2
Fig. 2

Calculated intensity distributions for the superposition of two vector LG beams (a) the same azimuthal indices (m1 = 2, m2 = 2) and type-I polarization distribution, (b) the same azimuthal indices (m1 = 2, m2 = 2) with type-I and type-II polarization distributions respectively.

Fig. 3
Fig. 3

Calculated intensity distributions for the superposition of two vector LG beams having different azimuthal indices (a) beam 1: (m1 = 1, polarization type-I), beam 2: (m2 = 2, polarization type-I), (b) beam 1: (m1 = 1 polarization type-I), beam 2: (m2 = 2, polarization type-II).

Fig. 4
Fig. 4

Propagated intensity distributions for the superposition of two vector LG beams at different z-planes (a) beam 1: (m1 = 2, type-I), beam 2: (m2 = 2, type-II), (b) beam 1: (m1 = 1 type-I), beam 2 (m2 = 5, type-I), (c) beam 1 (m1 = 1, type I), beam 2: (m2 = 5, type-II).

Fig. 5
Fig. 5

Phase φ12 of complex Stokes field S12 (Stokes vortex), (a) for vector LG beams with (m = 1) at the z = 0 plane, (b) for vector LG beams with (m = 2) at the z = 0 plane. Positive sign of the Stokes index (σ12 = + 2m) corresponds to the increase of Stokes phase in the counterclockwise directions and negative sign of Stokes index (σ12 = −2m) corresponds to the clockwise direction.

Fig. 6
Fig. 6

Calculated field distributions for the superposition of two vector LG beams (m1 = 1 type-I, and m2 = 2, type-I). (a) Intensity distributions at different z-planes, (b) polarization distributions at different z-planes. Red and blue color implies right-handed and left handed polarization for elliptical and circular polarization respectively, whereas black color represents linear polarization. (c) phase distribution φ12 obtained from complex Stokes field. C-points are marked with the squares. Positive sign of σ12 of Stokes vortices corresponds to the increase of Stokes phase in the counter-clockwise directions and negative sign of σ12 of Stokes vortices corresponds to the increase of Stokes phase in the clockwise direction.

Fig. 7
Fig. 7

Calculated field distributions for the superposition of two vector LG beams (m1 = 1 type-I, and m2 = 2, type-II). (a) Intensity distributions at different z-planes, (b) polarization distributions at different z-planes. Red and blue color implies right-handed and left handed polarization for elliptical and circular polarization respectively, whereas black color represents linear polarization. (c) phase distributions φ12 obtained from complex Stokes field. C-points are marked with the squares. Positive sign of σ12 of Stokes vortices corresponds to the increase of Stokes phase in the counter-clockwise directions and negative sign of σ12 of Stokes vortices corresponds to the increase of Stokes phase in the clockwise direction.

Tables (1)

Tables Icon

Table 1 Angular dependence of the polarization distribution for vector LG modes

Equations (12)

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E p,m ( r,ϕ,z )={ Y p,m ( r,z )[ cos( m1 )ϕ i ϕ sin( m1 )ϕ i r ]( TypeI ) Y p,m ( r,z )[ cos( m+1 )ϕ i ϕ +sin( m+1 )ϕ i r ]( TypeII ) Y p,m ( r,z )[ sin( m1 )ϕ i ϕ +cos( m1 )ϕ i r ]( TypeIII ) Y p,m ( r,z )[ sin( m+1 )ϕ i ϕ +cos( m+1 )ϕ i r ]( TypeIV ) ,
Y p,m ( r,z )= 2p! π( p+m )! 1 w(z) [ 2 r w( z ) ] | m | L p | m | [ 2 r 2 w 2 ( z ) ]exp[ r 2 w 2 ( z ) ]exp{ ik r 2 2R( z ) ikz+i( 2p+| m |+1 )ψ( z ) },
I= | E | 2 = I 01 | E p 1 , m 1 | 2 + I 02 | E p 2 , m 2 | 2 +2 I I 01 02 | E p 1 , m 1 || E p 2 , m 2 |Q( ϕ )S( z ),
Q( ϕ )={ cos( m 1 m 2 )ϕfor I 1 + I 2 ,I I 1 +I I 2 ,II I 1 +II I 2 ,I V 1 +I V 2 sin( m 1 m 2 )ϕfor I 1 +II I 2 sin( m 1 m 2 )ϕforI I 1 +I V 2 cos( m 1 + m 2 )ϕfor I 1 +I I 2 ,II I 1 +I V 2 sin( m 1 + m 2 )ϕforI I 1 +II I 2 sin( m 1 + m 2 )ϕforI V 1 + I 2
r d = w(z) 2 ( m 2 ! m 1 ! α 2 ) 1 2( m 2 m 1 ) ,
z d = z R tan( qπ | m 1 m 2 | ),
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 Im( E x * E y )
E p,m ( r,ϕ,z )={ Y p,m ( r,z )[ sin( mϕ ) i x +cos( mϕ ) i y ]( TypeI ) Y p,m ( r,z )[ sin( mϕ ) i x +cos( mϕ ) i y ]( TypeII ) Y p,m ( r,z )[ cos( mϕ ) i x +sin( mϕ ) i y ]( TypeIII ). Y p,m ( r,z )[ cos( mϕ ) i x sin( mϕ ) i y ]( TypeIV )
I 01 | E p 1 , m 1 | 2 + I 02 | E p 2 , m 2 | 2 +2 I I 01 02 | E p 1 , m 1 || E p 2 , m 2 | C ± (ϕ,z)=0,
C ± ( ϕ,z )={ cos[ ( m 1 m 2 )ϕ±( m 1 m 2 )ψ(z) ] for I 1 + I 2 ,II I 1 +II I 2 cos[ ( m 1 m 2 )ϕ( m 1 m 2 )ψ(z) ] for I I 1 +I I 2 ,I V 1 +I V 2 sin[ ( m 1 m 2 )ϕ±( m 1 m 2 )ψ(z) ] for I 1 +II I 2 sin[ ( m 1 m 2 )ϕ( m 1 m 2 )ψ(z) ] for I I 1 +I V 2 cos[ ( m 1 + m 2 )ϕ±( m 1 m 2 )ψ(z) ] for I 1 +I I 2 ,II I 1 +I V 2 sin[ ( m 1 + m 2 )ϕ( m 1 m 2 )ψ(z) ] for I I 1 +II I 2 sin[ ( m 1 + m 2 )ϕ( m 1 m 2 )ψ(z) ] for I V 1 + I 2
r c = w(z) 2 ( m 2 ! m 1 ! α 2 ) 1 2( m 2 m 1 ) .
ϕ c± = nπ( m 1 m 2 )ψ(z) m 1 m 2 for I 1 + I 2 andII I 1 +II I 2

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