Abstract

We introduce partially coherent vector sources with periodic spatial coherence properties, which we term vector optical coherence lattices (VOCLs), as an extension of recently introduced scalar OCLs. We derive the realizability conditions and propagation formulae for radially polarized VOCLs (i.e., a typical kind of VOCLs). We show that radially polarized VOCLs display nontrivial propagation properties and generate controllable intensity lattices in the far zone of the source (or in the focal plane of a lens). By adjusting source coherence, one can obtain intensity lattices with bright or dark nodes. The latter can be employed to simultaneously trap multiple particles or atoms as well as in free-space optical communications. We also report the experimental generation of radially polarized VOCLs and we characterize VOCLs propagation properties.

© 2017 Optical Society of America

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References

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2017 (1)

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017).
[Crossref]

2016 (2)

Y. Chen, S. A. Ponomarenko, and Y. Cai, “Experimental generation of optical coherence lattices,” Appl. Phys. Lett. 109(6), 061107 (2016).
[Crossref]

X. Liu, J. Yu, Y. Cai, and S. A. Ponomarenko, “Propagation of optical coherence lattices in the turbulent atmosphere,” Opt. Lett. 41(18), 4182–4185 (2016).
[Crossref] [PubMed]

2015 (1)

2014 (5)

L. Zhu, J. Yu, D. Zhang, M. Sun, and J. Chen, “Multifocal spot array generated by fractional Talbot effect phase-only modulation,” Opt. Express 22(8), 9798–9808 (2014).
[Crossref] [PubMed]

L. Ma and S. A. Ponomarenko, “Optical coherence gratings and lattices,” Opt. Lett. 39(23), 6656–6659 (2014).
[Crossref] [PubMed]

G. Wu, F. Wang, and Y. Cai, “Generation and self-healing of a radially polarized Bessel-Gaussian beam,” Phys. Rev. A 89(4), 043807 (2014).
[Crossref]

M. Kumar and J. Joseph, “Optical generation of a spatially variant two-dimensional lattice structure by using a phase only spatial light modulator,” Appl. Phys. Lett. 105(5), 051102 (2014).
[Crossref]

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[Crossref]

2013 (2)

F. Wang, X. Liu, L. Liu, Y. Yuan, and Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 103(9), 091102 (2013).
[Crossref]

M. Sakamoto, K. Oka, R. Morita, and N. Murakami, “Stable and flexible ring-shaped optical-lattice generation by use of axially symmetric polarization elements,” Opt. Lett. 38(18), 3661–3664 (2013).
[Crossref] [PubMed]

2012 (2)

Y. Dong, F. Wang, C. Zhao, and Y. Cai, “Effect of spatial coherence on propagation, tight focusing and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86(1), 013840 (2012).
[Crossref]

F. Wang, Y. Cai, Y. Dong, and O. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 100(5), 051108 (2012).
[Crossref]

2011 (1)

2010 (3)

Y. Zhang, B. Ding, and T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81(2), 023831 (2010).
[Crossref]

S. Sahin, Z. Tong, and O. Korotkova, “Sensing of semi-rough targets embedded in atmospheric turbulence by means of stochastic electromagnetic beams,” Opt. Commun. 283(22), 4512–4518 (2010).
[Crossref]

P. Kurzynowski, W. A. Wozniak, and M. Borwinska, “Regular lattices of polarization singularities: their generation and properties,” J. Opt. 12(3), 035406 (2010).
[Crossref]

2009 (3)

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
[Crossref]

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

D. P. Brown and T. G. Brown, “Coherence measurements applied to critical and Köhler vortex illumination,” Proc. SPIE 7184, 71840B (2009).
[Crossref]

2008 (2)

2007 (1)

2005 (6)

E. Betzig, “Excitation strategies for optical lattice microscopy,” Opt. Express 13(8), 3021–3036 (2005).
[Crossref] [PubMed]

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246(1–3), 35–43 (2005).
[Crossref]

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Changes in polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. 52(11), 1611–1618 (2005).
[Crossref]

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere,” Waves Random Complex Media 15(3), 353–364 (2005).
[Crossref]

I. Bloch, “Ultracold quantum gases in optical lattices,” Nat. Phys. 1(1), 23–30 (2005).
[Crossref]

M. Takamoto, F. L. Hong, R. Higashi, and H. Katori, “An optical lattice clock,” Nature 435(7040), 321–324 (2005).
[Crossref] [PubMed]

2004 (2)

2003 (3)

J. Tervo, T. Setälä, and A. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11(10), 1137–1143 (2003).
[Crossref] [PubMed]

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426(6965), 421–424 (2003).
[Crossref] [PubMed]

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5–6), 263–267 (2003).
[Crossref]

2002 (1)

2001 (3)

2000 (1)

1998 (2)

F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23(4), 241–243 (1998).
[Crossref] [PubMed]

D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold bosonic atoms in optical lattices,” Phys. Rev. Lett. 81(15), 3108–3111 (1998).
[Crossref]

1994 (1)

P. Senthilkumaran and R. S. Sirohi, “Michelson interferometers in tandem for array generation,” Opt. Commun. 105(3–4), 158–160 (1994).
[Crossref]

Baykal, Y.

Betzig, E.

Biss, D.

Bloch, I.

I. Bloch, “Ultracold quantum gases in optical lattices,” Nat. Phys. 1(1), 23–30 (2005).
[Crossref]

Borghi, R.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 3(1), 1–9 (2001).
[Crossref]

Borwinska, M.

P. Kurzynowski, W. A. Wozniak, and M. Borwinska, “Regular lattices of polarization singularities: their generation and properties,” J. Opt. 12(3), 035406 (2010).
[Crossref]

Brown, D. P.

Brown, T.

Brown, T. G.

Bruder, C.

D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold bosonic atoms in optical lattices,” Phys. Rev. Lett. 81(15), 3108–3111 (1998).
[Crossref]

Cai, Y.

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017).
[Crossref]

Y. Chen, S. A. Ponomarenko, and Y. Cai, “Experimental generation of optical coherence lattices,” Appl. Phys. Lett. 109(6), 061107 (2016).
[Crossref]

X. Liu, J. Yu, Y. Cai, and S. A. Ponomarenko, “Propagation of optical coherence lattices in the turbulent atmosphere,” Opt. Lett. 41(18), 4182–4185 (2016).
[Crossref] [PubMed]

G. Wu, F. Wang, and Y. Cai, “Generation and self-healing of a radially polarized Bessel-Gaussian beam,” Phys. Rev. A 89(4), 043807 (2014).
[Crossref]

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[Crossref]

F. Wang, X. Liu, L. Liu, Y. Yuan, and Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 103(9), 091102 (2013).
[Crossref]

F. Wang, Y. Cai, Y. Dong, and O. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 100(5), 051108 (2012).
[Crossref]

Y. Dong, F. Wang, C. Zhao, and Y. Cai, “Effect of spatial coherence on propagation, tight focusing and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86(1), 013840 (2012).
[Crossref]

Y. Dong, Y. Cai, C. Zhao, and M. Yao, “Statistics properties of a cylindrical vector partially coherent beam,” Opt. Express 19(7), 5979–5992 (2011).
[Crossref] [PubMed]

Y. Cai, O. Korotkova, H. T. Eyyuboğlu, and Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express 16(20), 15834–15846 (2008).
[Crossref] [PubMed]

Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27(4), 216–218 (2002).
[Crossref] [PubMed]

Chen, J.

Chen, Y.

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017).
[Crossref]

Y. Chen, S. A. Ponomarenko, and Y. Cai, “Experimental generation of optical coherence lattices,” Appl. Phys. Lett. 109(6), 061107 (2016).
[Crossref]

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[Crossref]

Cirac, J. I.

D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold bosonic atoms in optical lattices,” Phys. Rev. Lett. 81(15), 3108–3111 (1998).
[Crossref]

Dholakia, K.

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426(6965), 421–424 (2003).
[Crossref] [PubMed]

Ding, B.

Y. Zhang, B. Ding, and T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81(2), 023831 (2010).
[Crossref]

Dogariu, A.

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere,” Waves Random Complex Media 15(3), 353–364 (2005).
[Crossref]

Dong, Y.

Y. Dong, F. Wang, C. Zhao, and Y. Cai, “Effect of spatial coherence on propagation, tight focusing and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86(1), 013840 (2012).
[Crossref]

F. Wang, Y. Cai, Y. Dong, and O. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 100(5), 051108 (2012).
[Crossref]

Y. Dong, Y. Cai, C. Zhao, and M. Yao, “Statistics properties of a cylindrical vector partially coherent beam,” Opt. Express 19(7), 5979–5992 (2011).
[Crossref] [PubMed]

Eyyuboglu, H. T.

Friberg, A.

Gardiner, C. W.

D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold bosonic atoms in optical lattices,” Phys. Rev. Lett. 81(15), 3108–3111 (1998).
[Crossref]

Gori, F.

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 3(1), 1–9 (2001).
[Crossref]

F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23(4), 241–243 (1998).
[Crossref] [PubMed]

Guérineau, N.

Harchaoui, B.

Heggarty, K.

Higashi, R.

M. Takamoto, F. L. Hong, R. Higashi, and H. Katori, “An optical lattice clock,” Nature 435(7040), 321–324 (2005).
[Crossref] [PubMed]

Hong, F. L.

M. Takamoto, F. L. Hong, R. Higashi, and H. Katori, “An optical lattice clock,” Nature 435(7040), 321–324 (2005).
[Crossref] [PubMed]

Jaksch, D.

D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold bosonic atoms in optical lattices,” Phys. Rev. Lett. 81(15), 3108–3111 (1998).
[Crossref]

Joseph, J.

M. Kumar and J. Joseph, “Optical generation of a spatially variant two-dimensional lattice structure by using a phase only spatial light modulator,” Appl. Phys. Lett. 105(5), 051102 (2014).
[Crossref]

Katori, H.

M. Takamoto, F. L. Hong, R. Higashi, and H. Katori, “An optical lattice clock,” Nature 435(7040), 321–324 (2005).
[Crossref] [PubMed]

Kivshar, Y.

Korotkova, O.

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[Crossref]

F. Wang, Y. Cai, Y. Dong, and O. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 100(5), 051108 (2012).
[Crossref]

S. Sahin, Z. Tong, and O. Korotkova, “Sensing of semi-rough targets embedded in atmospheric turbulence by means of stochastic electromagnetic beams,” Opt. Commun. 283(22), 4512–4518 (2010).
[Crossref]

Y. Cai, O. Korotkova, H. T. Eyyuboğlu, and Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express 16(20), 15834–15846 (2008).
[Crossref] [PubMed]

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246(1–3), 35–43 (2005).
[Crossref]

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere,” Waves Random Complex Media 15(3), 353–364 (2005).
[Crossref]

O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source,” Opt. Lett. 29(11), 1173–1175 (2004).
[Crossref] [PubMed]

Kumar, M.

M. Kumar and J. Joseph, “Optical generation of a spatially variant two-dimensional lattice structure by using a phase only spatial light modulator,” Appl. Phys. Lett. 105(5), 051102 (2014).
[Crossref]

Kurzynowski, P.

P. Kurzynowski, W. A. Wozniak, and M. Borwinska, “Regular lattices of polarization singularities: their generation and properties,” J. Opt. 12(3), 035406 (2010).
[Crossref]

Lin, Q.

Liu, L.

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017).
[Crossref]

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[Crossref]

F. Wang, X. Liu, L. Liu, Y. Yuan, and Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 103(9), 091102 (2013).
[Crossref]

Liu, X.

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017).
[Crossref]

X. Liu, J. Yu, Y. Cai, and S. A. Ponomarenko, “Propagation of optical coherence lattices in the turbulent atmosphere,” Opt. Lett. 41(18), 4182–4185 (2016).
[Crossref] [PubMed]

F. Wang, X. Liu, L. Liu, Y. Yuan, and Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 103(9), 091102 (2013).
[Crossref]

Ma, L.

MacDonald, M. P.

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426(6965), 421–424 (2003).
[Crossref] [PubMed]

Mondello, A.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 3(1), 1–9 (2001).
[Crossref]

Morita, R.

Murakami, N.

Oka, K.

Ostrovskaya, E.

Piquero, G.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 3(1), 1–9 (2001).
[Crossref]

Ponomarenko, S. A.

Y. Chen, S. A. Ponomarenko, and Y. Cai, “Experimental generation of optical coherence lattices,” Appl. Phys. Lett. 109(6), 061107 (2016).
[Crossref]

X. Liu, J. Yu, Y. Cai, and S. A. Ponomarenko, “Propagation of optical coherence lattices in the turbulent atmosphere,” Opt. Lett. 41(18), 4182–4185 (2016).
[Crossref] [PubMed]

L. Ma and S. A. Ponomarenko, “Free-space propagation of optical coherence lattices and periodicity reciprocity,” Opt. Express 23(2), 1848–1856 (2015).
[Crossref] [PubMed]

L. Ma and S. A. Ponomarenko, “Optical coherence gratings and lattices,” Opt. Lett. 39(23), 6656–6659 (2014).
[Crossref] [PubMed]

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Changes in polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. 52(11), 1611–1618 (2005).
[Crossref]

Primot, J.

Ramírez-Sánchez, V.

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

Roychowdhury, H.

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Changes in polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. 52(11), 1611–1618 (2005).
[Crossref]

Sahin, S.

S. Sahin, Z. Tong, and O. Korotkova, “Sensing of semi-rough targets embedded in atmospheric turbulence by means of stochastic electromagnetic beams,” Opt. Commun. 283(22), 4512–4518 (2010).
[Crossref]

Sakamoto, M.

Salem, M.

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere,” Waves Random Complex Media 15(3), 353–364 (2005).
[Crossref]

O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source,” Opt. Lett. 29(11), 1173–1175 (2004).
[Crossref] [PubMed]

Santarsiero, M.

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 3(1), 1–9 (2001).
[Crossref]

Senthilkumaran, P.

S. Vyas and P. Senthilkumaran, “Interferometric optical vortex array generator,” Appl. Opt. 46(15), 2893–2898 (2007).
[Crossref] [PubMed]

P. Senthilkumaran and R. S. Sirohi, “Michelson interferometers in tandem for array generation,” Opt. Commun. 105(3–4), 158–160 (1994).
[Crossref]

Setälä, T.

Shirai, T.

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

Simon, R.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 3(1), 1–9 (2001).
[Crossref]

Sirohi, R. S.

P. Senthilkumaran and R. S. Sirohi, “Michelson interferometers in tandem for array generation,” Opt. Commun. 105(3–4), 158–160 (1994).
[Crossref]

Spalding, G. C.

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426(6965), 421–424 (2003).
[Crossref] [PubMed]

Sun, M.

Suyama, T.

Y. Zhang, B. Ding, and T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81(2), 023831 (2010).
[Crossref]

Takamoto, M.

M. Takamoto, F. L. Hong, R. Higashi, and H. Katori, “An optical lattice clock,” Nature 435(7040), 321–324 (2005).
[Crossref] [PubMed]

Tervo, J.

Tong, Z.

S. Sahin, Z. Tong, and O. Korotkova, “Sensing of semi-rough targets embedded in atmospheric turbulence by means of stochastic electromagnetic beams,” Opt. Commun. 283(22), 4512–4518 (2010).
[Crossref]

Vyas, S.

Wang, F.

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[Crossref]

G. Wu, F. Wang, and Y. Cai, “Generation and self-healing of a radially polarized Bessel-Gaussian beam,” Phys. Rev. A 89(4), 043807 (2014).
[Crossref]

F. Wang, X. Liu, L. Liu, Y. Yuan, and Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 103(9), 091102 (2013).
[Crossref]

Y. Dong, F. Wang, C. Zhao, and Y. Cai, “Effect of spatial coherence on propagation, tight focusing and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86(1), 013840 (2012).
[Crossref]

F. Wang, Y. Cai, Y. Dong, and O. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 100(5), 051108 (2012).
[Crossref]

Wolf, E.

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246(1–3), 35–43 (2005).
[Crossref]

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Changes in polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. 52(11), 1611–1618 (2005).
[Crossref]

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere,” Waves Random Complex Media 15(3), 353–364 (2005).
[Crossref]

O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source,” Opt. Lett. 29(11), 1173–1175 (2004).
[Crossref] [PubMed]

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5–6), 263–267 (2003).
[Crossref]

Wozniak, W. A.

P. Kurzynowski, W. A. Wozniak, and M. Borwinska, “Regular lattices of polarization singularities: their generation and properties,” J. Opt. 12(3), 035406 (2010).
[Crossref]

Wu, G.

G. Wu, F. Wang, and Y. Cai, “Generation and self-healing of a radially polarized Bessel-Gaussian beam,” Phys. Rev. A 89(4), 043807 (2014).
[Crossref]

Yao, M.

Youngworth, K.

Yu, J.

Yuan, Y.

F. Wang, X. Liu, L. Liu, Y. Yuan, and Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 103(9), 091102 (2013).
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Zhan, Q.

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
[Crossref]

Zhang, D.

Zhang, Y.

Y. Zhang, B. Ding, and T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81(2), 023831 (2010).
[Crossref]

Zhao, C.

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[Crossref]

Y. Dong, F. Wang, C. Zhao, and Y. Cai, “Effect of spatial coherence on propagation, tight focusing and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86(1), 013840 (2012).
[Crossref]

Y. Dong, Y. Cai, C. Zhao, and M. Yao, “Statistics properties of a cylindrical vector partially coherent beam,” Opt. Express 19(7), 5979–5992 (2011).
[Crossref] [PubMed]

Zhu, L.

Zoller, P.

D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold bosonic atoms in optical lattices,” Phys. Rev. Lett. 81(15), 3108–3111 (1998).
[Crossref]

Adv. Opt. Photonics (1)

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (4)

M. Kumar and J. Joseph, “Optical generation of a spatially variant two-dimensional lattice structure by using a phase only spatial light modulator,” Appl. Phys. Lett. 105(5), 051102 (2014).
[Crossref]

Y. Chen, S. A. Ponomarenko, and Y. Cai, “Experimental generation of optical coherence lattices,” Appl. Phys. Lett. 109(6), 061107 (2016).
[Crossref]

F. Wang, Y. Cai, Y. Dong, and O. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 100(5), 051108 (2012).
[Crossref]

F. Wang, X. Liu, L. Liu, Y. Yuan, and Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 103(9), 091102 (2013).
[Crossref]

J. Mod. Opt. (1)

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Changes in polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. 52(11), 1611–1618 (2005).
[Crossref]

J. Opt. (1)

P. Kurzynowski, W. A. Wozniak, and M. Borwinska, “Regular lattices of polarization singularities: their generation and properties,” J. Opt. 12(3), 035406 (2010).
[Crossref]

J. Opt. A, Pure Appl. Opt. (2)

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 3(1), 1–9 (2001).
[Crossref]

Nat. Phys. (1)

I. Bloch, “Ultracold quantum gases in optical lattices,” Nat. Phys. 1(1), 23–30 (2005).
[Crossref]

Nature (2)

M. Takamoto, F. L. Hong, R. Higashi, and H. Katori, “An optical lattice clock,” Nature 435(7040), 321–324 (2005).
[Crossref] [PubMed]

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426(6965), 421–424 (2003).
[Crossref] [PubMed]

Opt. Commun. (3)

P. Senthilkumaran and R. S. Sirohi, “Michelson interferometers in tandem for array generation,” Opt. Commun. 105(3–4), 158–160 (1994).
[Crossref]

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246(1–3), 35–43 (2005).
[Crossref]

S. Sahin, Z. Tong, and O. Korotkova, “Sensing of semi-rough targets embedded in atmospheric turbulence by means of stochastic electromagnetic beams,” Opt. Commun. 283(22), 4512–4518 (2010).
[Crossref]

Opt. Express (10)

Y. Cai, O. Korotkova, H. T. Eyyuboğlu, and Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express 16(20), 15834–15846 (2008).
[Crossref] [PubMed]

J. Tervo, T. Setälä, and A. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11(10), 1137–1143 (2003).
[Crossref] [PubMed]

Y. Dong, Y. Cai, C. Zhao, and M. Yao, “Statistics properties of a cylindrical vector partially coherent beam,” Opt. Express 19(7), 5979–5992 (2011).
[Crossref] [PubMed]

D. P. Brown and T. G. Brown, “Partially correlated azimuthal vortex illumination: coherence and correlation measurements and effects in imaging,” Opt. Express 16(25), 20418–20426 (2008).
[Crossref] [PubMed]

E. Betzig, “Excitation strategies for optical lattice microscopy,” Opt. Express 13(8), 3021–3036 (2005).
[Crossref] [PubMed]

E. Ostrovskaya and Y. Kivshar, “Photonic crystals for matter waves: Bose-Einstein condensates in optical lattices,” Opt. Express 12(1), 19–29 (2004).
[Crossref] [PubMed]

K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000).
[Crossref] [PubMed]

D. Biss and T. Brown, “Cylindrical vector beam focusing through a dielectric interface,” Opt. Express 9(10), 490–497 (2001).
[Crossref] [PubMed]

L. Zhu, J. Yu, D. Zhang, M. Sun, and J. Chen, “Multifocal spot array generated by fractional Talbot effect phase-only modulation,” Opt. Express 22(8), 9798–9808 (2014).
[Crossref] [PubMed]

L. Ma and S. A. Ponomarenko, “Free-space propagation of optical coherence lattices and periodicity reciprocity,” Opt. Express 23(2), 1848–1856 (2015).
[Crossref] [PubMed]

Opt. Lett. (7)

Phys. Lett. A (1)

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5–6), 263–267 (2003).
[Crossref]

Phys. Rev. A (4)

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[Crossref]

Y. Dong, F. Wang, C. Zhao, and Y. Cai, “Effect of spatial coherence on propagation, tight focusing and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86(1), 013840 (2012).
[Crossref]

G. Wu, F. Wang, and Y. Cai, “Generation and self-healing of a radially polarized Bessel-Gaussian beam,” Phys. Rev. A 89(4), 043807 (2014).
[Crossref]

Y. Zhang, B. Ding, and T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81(2), 023831 (2010).
[Crossref]

Phys. Rev. Lett. (1)

D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold bosonic atoms in optical lattices,” Phys. Rev. Lett. 81(15), 3108–3111 (1998).
[Crossref]

Proc. SPIE (1)

D. P. Brown and T. G. Brown, “Coherence measurements applied to critical and Köhler vortex illumination,” Proc. SPIE 7184, 71840B (2009).
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Prog. Opt. (1)

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017).
[Crossref]

Waves Random Complex Media (1)

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere,” Waves Random Complex Media 15(3), 353–364 (2005).
[Crossref]

Other (2)

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

Y. Cai, F. Wang, C. Zhao, S. Zhu, G. Wu, and Y. Dong, “Partially coherent vector beams: from theory to experiment,” in Vectorial Optical Fields: Fundamentals and Applications, Q. Zhan, ed. (World Scientific, 2013), Chap. 7, pp. 221–273.

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Figures (9)

Fig. 1
Fig. 1

Density plot of the square of the the degree of coherence μ 2 ( x 1 , y 1 ,1mm,1mm ) of radially polarized VOCLs for different values of M and N with and in the source plane.

Fig. 2
Fig. 2

Density plot of the normalized intensity distribution I( ρ )/ I max ( ρ ) , the corresponding components I x ( ρ )/ I ymax ( ρ ) , I y ( ρ )/ I ymax ( ρ ) , and the distribution of the SOP of radially polarized VOCLs at several propagation distances in free space with M = N = 3 and.

Fig. 3
Fig. 3

Density plot of the normalized intensity distribution I( ρ )/ I max ( ρ ) , the corresponding components I x ( ρ )/ I ymax ( ρ ) , I y ( ρ )/ I ymax ( ρ ) , and the distribution of the SOP of radially polarized VOCLs at several propagation distances in free space with M = N = 3 and δ 0 =0.4mm .

Fig. 4
Fig. 4

Density plot of the normalized intensity distribution I( ρ )/ I max ( ρ ) , the corresponding components I x ( ρ )/ I ymax ( ρ ) , I y ( ρ )/ I ymax ( ρ ) , and the distribution of the SOP of radially polarized VOCLs at several propagation distances in free space with M = N = 3 and δ 0 =0.32mm .

Fig. 5
Fig. 5

Density plot of the normalized intensity distribution I( ρ )/ I max ( ρ ) of radially polarized VOCLs at z=10km in free space for different values of M and N with δ 0 =3mm .

Fig. 6
Fig. 6

Experimental setup for generating radially polarized VOCLs, measuring the degree of coherence and the focused intensity. Laser, Nd: YAG laser; LP, linear polarizer; BE, beam expander; AM, amplitude mask; L1, L2 and L3, thin lenses; RGGD, rotating ground-glass disk; MC, motion controller; GAF, Gaussian amplitude filter; RPC, radial polarization converter; BS, beam splitter; CCD, charge-coupled device; BPA, beam profile analyzer; PC, personal computer.

Fig. 7
Fig. 7

Experimental results of the squared modulus of the degree of coherence | μ( x 1 , y 1 ,1mm,1mm ) | 2 of the generated radially polarized VOCLs just behind the RPC with δ 0 =0.37mm andfor different values of M and N, (a) M=N=2 , (b) M=N=3 , (c) M=N=5 .

Fig. 8
Fig. 8

Experimental results of the intensity distribution of the generated radially polarized VOCLs with M=N=3, δ 0 =0.37mm and d=1mm focused by the thin lens L3 with focal length f 3 =15cm and its corresponding components I x and I y at several propagation distances.

Fig. 9
Fig. 9

Experimental results of the intensity distribution of the generated radially polarized VOCLs with M=N=3 and d=1mm focused by the thin lens L3 with focal length f 3 =15cm in the focal plane for different values of the coherence parameter, (a) δ 0 =0.6mm , (b) δ 0 =0.4mm , (c) δ 0 =0.37mm .

Equations (26)

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μ( r 1 , r 2 )= 2 M m=1 M J 1 ( | r 1 r 2 |/ 2 δ ) | r 1 r 2 |/ 2 δ exp[ i V 0m ( r 1 r 2 ) ] .
Γ αβ ( r 1 , r 2 )= E α * ( r 1 ) E β ( r 2 ) , ( α,β=x,y ).
Γ αβ ( r 1 , r 2 )= P αβ ( v ) H α ( r 1 ,v ) H β * ( r 2 ,v ) d 2 v,
P αα ( v )0, P xx ( v ) P yy ( v ) | P xy ( v ) | 2 0.
H α ( r,v )= i λf T α exp[ iπ λf ( v 2 2rv ) ],
P αβ ( v )= B αβ MN m= ( M1 ) /2 ( M1 ) /2 n= ( N1 ) /2 ( N1 ) /2 circ( v v mn a αβ ) .
Γ αβ ( r 1 , r 2 )= C 0 T α * T β γ αβ ( r 1 , r 2 ),
γ αβ ( r 1 , r 2 )= 2 B αβ MN m= ( M1 ) /2 ( M1 ) /2 n= ( N1 ) /2 ( N1 ) /2 J 1 ( | r 1 r 2 | / 2 δ 0αβ ) | r 1 r 2 | / 2 δ 0αβ exp[ i2π λf v mn ( r 1 r 2 ) ].
| B αβ |=1, ϕ αβ =0, ( α=β ), | B αβ |1,( αβ ), | B xy |=| B yx |, ϕ xy = ϕ yx , δ 0xy = δ 0yx .
m= ( M1 ) /2 ( M1 ) /2 n= ( N1 ) /2 ( N1 ) /2 circ( v v mn a xx ) m= ( M1 ) /2 ( M1 ) /2 n= ( N1 ) /2 ( N1 ) /2 circ( v v mn a yy ) | B xy | 2 | m= ( M1 ) /2 ( M1 ) /2 n= ( N1 ) /2 ( N1 ) /2 circ( v v mn a xy ) | 2 .
a xy min( a xx , a yy ).
δ 0xy max( δ 0xx , δ 0yy ).
A 1,2 ( r )= 1 2 [ ( Γ xx ( r,r ) Γ yy ( r,r ) ) 2 +4 | Γ xy ( r,r ) | 2 ± ( Γ xx ( r,r ) Γ yy ( r,r ) ) 2 +4Re | Γ xy ( r,r ) | 2 ] 1/2 ,
ε( r )= A 2 ( r ) / A 1 ( r ) ,
θ( r )= 1 2 arctan[ 2Re[ Γ xy ( r,r ) ] Γ xx ( r,r ) Γ yy ( r,r ) ].
B xy = B yx =1, δ 0xx = δ 0yy = δ 0xy = δ 0yx = δ 0 .
γ xx ( r 1 , r 2 )= γ yy ( r 1 , r 2 )= γ xy ( r 1 , r 2 )= γ yx ( r 1 , r 2 ).
μ 2 ( r 1 , r 2 )= α,β | Γ αβ ( r 1 , r 2 ) | 2 α,β Γ αα ( r 1 , r 1 ) Γ ββ ( r 2 , r 2 ) .
μ 2 ( r 1 , r 2 )= γ 2 αβ ( r 1 , r 2 ) ( α,β=x,y ).
Γ αβ ( ρ 1 , ρ 2 ) = 1 ( λB ) 2 exp[ ikD 2B ( ρ 1 2 ρ 2 2 ) ] Γ αβ ( r 1 , r 2 ) exp[ ikA 2B ( r 1 2 r 2 2 ) ]exp[ ik B ( r 1 ρ 1 r 2 ρ 2 ) ] d 2 r 1 d 2 r 2 ,
Γ xx ( ρ 1 , ρ 2 )= C 0 π 2 λ 2 B 2 Δ 2 MN h=0 s=0 h m= ( M1 ) /2 ( M1 ) /2 n= ( N1 ) /2 ( N1 ) /2 ( 1 ) h 2 3h s!( hs )!( h+1 )! δ 0 2h Q 2s n ( ρ sy , ρ dy ) ×[ ( ρ dx 2 B 2 k 2 w s 2 ) Q 2( hs ) m ( ρ sx , ρ dx )+2A ρ dx Q 2( hs )+1 m ( ρ sx , ρ dx ) Δ 2 Q 2( hs )+2 m ( ρ sx , ρ dx ) ],
Γ yy ( ρ 1 , ρ 2 )= C 0 π 2 λ 2 B 2 Δ 2 MN h=0 s=0 h m= ( M1 ) /2 ( M1 ) /2 n= ( N1 ) /2 ( N1 ) /2 ( 1 ) h 2 3h s!( hs )!( h+1 )! δ 0 2h Q 2( hs ) m ( ρ sx , ρ dx ) ×[ ( ρ dy 2 B 2 k 2 w s 2 ) Q 2s n ( ρ sy , ρ dy )+2A ρ dx Q 2s+1 n ( ρ sy , ρ dy ) Δ 2 Q 2s+2 n ( ρ sy , ρ dy ) ],
Γ xy ( ρ 1 , ρ 2 )= C 0 π 2 λ 2 B 2 Δ 2 MN h=0 s=0 h m= ( M1 ) /2 ( M1 ) /2 n= ( N1 ) /2 ( N1 ) /2 ( 1 ) h 2 3h s!( hs )!( h+1 )! δ 0 2h ×[ i ρ dx Q 2( hs ) m ( ρ sx , ρ dx )( iA B 2 w s 2 k ) Q 2( hs )+1 m ( ρ sx , ρ dx ) ][ i ρ dy Q 2s n ( ρ sy , ρ dy )( iA+ B 2 w s 2 k ) Q 2s+1 n ( ρ sy , ρ dy ) ],
Γ yx ( ρ 1 , ρ 2 )= Γ xy * ( ρ 2 , ρ 1 ),
Q t p ( ρ sα , ρ dα )= ( iB 2 k w s Δ ) t exp( ikD B ρ sα ρ dα ) H t ( 1 2 w s Δ ( B f pd ρ sα +i k w s 2 A B ρ dα ) ). ×exp( k 2 w s 2 2 B 2 ρ dα 2 )exp( 1 2 w s 2 Δ 2 ( B f pd ρ sα +i k w s 2 A B ρ dα ) 2 )
I( ρ )= Γ xx ( ρ,ρ )+ Γ yy ( ρ,ρ )= I x ( ρ )+ I y ( ρ ).

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