Abstract

A new kind of partially coherent vector beam named vector Hermite-Gaussian correlated Schell-model (HGCSM) beam is introduced as a natural extension of recently introduced scalar HGCSM beam. The realizability and beam conditions for a vector HGCSM beam with uniform state of polarization (SOP) or non-uniform SOP are derived, respectively. Furthermore, analytical formulae for a vector HGCSM beam propagating in free space are derived, and the propagation properties of a vector HGCSM beam with uniform SOP or non-uniform SOP in free space are studied and analyzed in detail. We find that the behaviors of a vector HGCSM beam on propagation are quite different from those of a conventional vector partially coherent beam with uniform SOP or non-uniform SOP, and modulating the structures of the correlation functions cannot only modulate the intensity distribution, but also the state of polarization, the degree of polarization and the polarization singularities of a partially coherent vector beam on propagation. Furthermore, we report experimental generation of a radially polarized HGCSM beam for the first time. Our results provide a novel way for polarization modulation.

© 2016 Optical Society of America

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References

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    [Crossref]
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    [Crossref] [PubMed]
  42. Y. Chen and Y. Cai, “Generation of a controllable optical cage by focusing a Laguerre-Gaussian correlated Schell-model beam,” Opt. Lett. 39(9), 2549–2552 (2014).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  48. J. Yu, Y. Chen, L. Liu, X. Liu, and Y. Cai, “Splitting and combining properties of an elegant Hermite-Gaussian correlated Schell-model beam in Kolmogorov and non-Kolmogorov turbulence,” Opt. Express 23(10), 13467–13481 (2015).
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    [Crossref] [PubMed]

2015 (10)

A. T. Friberg and T. D. Visser, “Scintillation of electromagnetic beams generated by quasi-homogeneous sources,” Opt. Commun. 335, 82–85 (2015).
[Crossref]

M. W. Hyde, S. Basu, D. G. Voelz, and X. Xiao, “Experimental generating any desired partially coherent Schell-model sources using phase-only control,” J. Appl. Phys. 118(9), 093102 (2015).
[Crossref]

Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91(1), 013823 (2015).
[Crossref]

X. Liu and D. Zhao, “Trapping two types of particles with a focused generalized Multi-Gaussian Schell model beam,” Opt. Commun. 354, 250–255 (2015).
[Crossref]

D. Voelz, X. Xiao, and O. Korotkova, “Numerical modeling of Schell-model beams with arbitrary far-field patterns,” Opt. Lett. 40(3), 352–355 (2015).
[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]

J. Lehtolahti, M. Kuittinen, J. Turunen, and J. Tervo, “Coherence modulation by deterministic rotating diffusers,” Opt. Express 23(8), 10453–10466 (2015).
[Crossref] [PubMed]

X. Liu, F. Wang, M. Zhang, and Y. Cai, “Experimental demonstration of ghost imaging with an electromagnetic Gaussian Schell-model beam,” J. Opt. Soc. Am. A 32(5), 910–920 (2015).
[Crossref] [PubMed]

R. Chriki, M. Nixon, V. Pal, C. Tradonsky, G. Barach, A. A. Friesem, and N. Davidson, “Manipulating the spatial coherence of a laser source,” Opt. Express 23(10), 12989–12997 (2015).
[Crossref] [PubMed]

J. Yu, Y. Chen, L. Liu, X. Liu, and Y. Cai, “Splitting and combining properties of an elegant Hermite-Gaussian correlated Schell-model beam in Kolmogorov and non-Kolmogorov turbulence,” Opt. Express 23(10), 13467–13481 (2015).
[Crossref] [PubMed]

2014 (10)

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]

O. Korotkova, S. Avramov-Zamurovic, C. Nelson, R. Malek-Madani, Y. Gu, and G. Gbur, “Scintillation reduction in multi-Gaussian Schell-model beams propagating in atmospheric turbulence,” Proc. SPIE 9224, 92240M (2014).
[Crossref]

C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
[Crossref] [PubMed]

Y. Chen, F. Wang, C. Zhao, and Y. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22(5), 5826–5838 (2014).
[Crossref] [PubMed]

R. V. Vinu, M. K. Sharma, R. K. Singh, and P. Senthilkumaran, “Generation of spatial coherence comb using Dammann grating,” Opt. Lett. 39(8), 2407–2410 (2014).
[Crossref] [PubMed]

Y. Chen and Y. Cai, “Generation of a controllable optical cage by focusing a Laguerre-Gaussian correlated Schell-model beam,” Opt. Lett. 39(9), 2549–2552 (2014).
[Crossref] [PubMed]

Y. Chen, L. Liu, F. Wang, C. Zhao, and Y. Cai, “Elliptical Laguerre-Gaussian correlated Schell-model beam,” Opt. Express 22(11), 13975–13987 (2014).
[Crossref] [PubMed]

Y. Cai, Y. Chen, and F. Wang, “Generation and propagation of partially coherent beams with nonconventional correlation functions: a review [invited],” J. Opt. Soc. Am. A 31(9), 2083–2096 (2014).
[Crossref] [PubMed]

F. Wang, C. Liang, Y. Yuan, and Y. Cai, “Generalized multi-Gaussian correlated Schell-model beam: from theory to experiment,” Opt. Express 22(19), 23456–23464 (2014).
[Crossref] [PubMed]

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

2013 (4)

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[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. Gu and G. Gbur, “Scintillation of nonuniformly correlated beams in atmospheric turbulence,” Opt. Lett. 38(9), 1395–1397 (2013).
[Crossref] [PubMed]

F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[Crossref] [PubMed]

2012 (5)

S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
[Crossref] [PubMed]

Z. Tong and O. Korotkova, “Nonuniformly correlated light beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012).
[Crossref] [PubMed]

G. Wu, F. Wang, and Y. Cai, “Coherence and polarization properties of a radially polarized beam with variable spatial coherence,” Opt. Express 20(27), 28301–28318 (2012).
[Crossref] [PubMed]

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]

2011 (4)

2010 (1)

Z. Tong, Y. Cai, and O. Korotkova, “Ghost imaging with electromagnetic stochastic beams,” Opt. Commun. 283(20), 3838–3845 (2010).
[Crossref]

2009 (2)

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

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

2008 (4)

2007 (1)

2006 (1)

2005 (2)

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249(4–6), 379–385 (2005).
[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]

2004 (1)

2003 (1)

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

2001 (1)

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]

1998 (1)

1994 (1)

Avramov-Zamurovic, S.

O. Korotkova, S. Avramov-Zamurovic, C. Nelson, R. Malek-Madani, Y. Gu, and G. Gbur, “Scintillation reduction in multi-Gaussian Schell-model beams propagating in atmospheric turbulence,” Proc. SPIE 9224, 92240M (2014).
[Crossref]

Barach, G.

Basu, S.

M. W. Hyde, S. Basu, D. G. Voelz, and X. Xiao, “Experimental generating any desired partially coherent Schell-model sources using phase-only control,” J. Appl. Phys. 118(9), 093102 (2015).
[Crossref]

Borghi, R.

F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 25(5), 1016–1021 (2008).
[Crossref] [PubMed]

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]

Brown, D. P.

Brown, T. G.

Cai, Y.

Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91(1), 013823 (2015).
[Crossref]

J. Yu, Y. Chen, L. Liu, X. Liu, and Y. Cai, “Splitting and combining properties of an elegant Hermite-Gaussian correlated Schell-model beam in Kolmogorov and non-Kolmogorov turbulence,” Opt. Express 23(10), 13467–13481 (2015).
[Crossref] [PubMed]

X. Liu, F. Wang, M. Zhang, and Y. Cai, “Experimental demonstration of ghost imaging with an electromagnetic Gaussian Schell-model beam,” J. Opt. Soc. Am. A 32(5), 910–920 (2015).
[Crossref] [PubMed]

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]

C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
[Crossref] [PubMed]

Y. Chen, F. Wang, C. Zhao, and Y. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22(5), 5826–5838 (2014).
[Crossref] [PubMed]

Y. Chen and Y. Cai, “Generation of a controllable optical cage by focusing a Laguerre-Gaussian correlated Schell-model beam,” Opt. Lett. 39(9), 2549–2552 (2014).
[Crossref] [PubMed]

Y. Chen, L. Liu, F. Wang, C. Zhao, and Y. Cai, “Elliptical Laguerre-Gaussian correlated Schell-model beam,” Opt. Express 22(11), 13975–13987 (2014).
[Crossref] [PubMed]

Y. Cai, Y. Chen, and F. Wang, “Generation and propagation of partially coherent beams with nonconventional correlation functions: a review [invited],” J. Opt. Soc. Am. A 31(9), 2083–2096 (2014).
[Crossref] [PubMed]

F. Wang, C. Liang, Y. Yuan, and Y. Cai, “Generalized multi-Gaussian correlated Schell-model beam: from theory to experiment,” Opt. Express 22(19), 23456–23464 (2014).
[Crossref] [PubMed]

F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[Crossref] [PubMed]

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[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]

G. Wu, F. Wang, and Y. Cai, “Coherence and polarization properties of a radially polarized beam with variable spatial coherence,” Opt. Express 20(27), 28301–28318 (2012).
[Crossref] [PubMed]

F. Wang, G. Wu, X. Liu, S. Zhu, and Y. Cai, “Experimental measurement of the beam parameters of an electromagnetic Gaussian Schell-model source,” Opt. Lett. 36(14), 2722–2724 (2011).
[Crossref] [PubMed]

Z. Tong, Y. Cai, and O. Korotkova, “Ghost imaging with electromagnetic stochastic beams,” Opt. Commun. 283(20), 3838–3845 (2010).
[Crossref]

Chen, Y.

Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91(1), 013823 (2015).
[Crossref]

J. Yu, Y. Chen, L. Liu, X. Liu, and Y. Cai, “Splitting and combining properties of an elegant Hermite-Gaussian correlated Schell-model beam in Kolmogorov and non-Kolmogorov turbulence,” Opt. Express 23(10), 13467–13481 (2015).
[Crossref] [PubMed]

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. Chen, F. Wang, C. Zhao, and Y. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22(5), 5826–5838 (2014).
[Crossref] [PubMed]

Y. Chen, L. Liu, F. Wang, C. Zhao, and Y. Cai, “Elliptical Laguerre-Gaussian correlated Schell-model beam,” Opt. Express 22(11), 13975–13987 (2014).
[Crossref] [PubMed]

Y. Cai, Y. Chen, and F. Wang, “Generation and propagation of partially coherent beams with nonconventional correlation functions: a review [invited],” J. Opt. Soc. Am. A 31(9), 2083–2096 (2014).
[Crossref] [PubMed]

Y. Chen and Y. Cai, “Generation of a controllable optical cage by focusing a Laguerre-Gaussian correlated Schell-model beam,” Opt. Lett. 39(9), 2549–2552 (2014).
[Crossref] [PubMed]

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[Crossref]

Chriki, R.

Davidson, N.

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]

Eyyuboglu, H. T.

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[Crossref]

Fonseca, E. J. S.

I. Vidal, E. J. S. Fonseca, and J. M. Hickmann, “Light polarization control during free-space propagation using coherence,” Phys. Rev. A 84(3), 033836 (2011).
[Crossref]

Friberg, A. T.

A. T. Friberg and T. D. Visser, “Scintillation of electromagnetic beams generated by quasi-homogeneous sources,” Opt. Commun. 335, 82–85 (2015).
[Crossref]

T. Shirai, H. Kellock, T. Setälä, and A. T. Friberg, “Visibility in ghost imaging with classical partially polarized electromagnetic beams,” Opt. Lett. 36(15), 2880–2882 (2011).
[Crossref] [PubMed]

Friesem, A. A.

Gbur, G.

O. Korotkova, S. Avramov-Zamurovic, C. Nelson, R. Malek-Madani, Y. Gu, and G. Gbur, “Scintillation reduction in multi-Gaussian Schell-model beams propagating in atmospheric turbulence,” Proc. SPIE 9224, 92240M (2014).
[Crossref]

Y. Gu and G. Gbur, “Scintillation of nonuniformly correlated beams in atmospheric turbulence,” Opt. Lett. 38(9), 1395–1397 (2013).
[Crossref] [PubMed]

Gori, F.

F. Gori, V. R. Sanchez, 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, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 25(5), 1016–1021 (2008).
[Crossref] [PubMed]

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
[Crossref] [PubMed]

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]

Gu, J.

Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91(1), 013823 (2015).
[Crossref]

Gu, Y.

O. Korotkova, S. Avramov-Zamurovic, C. Nelson, R. Malek-Madani, Y. Gu, and G. Gbur, “Scintillation reduction in multi-Gaussian Schell-model beams propagating in atmospheric turbulence,” Proc. SPIE 9224, 92240M (2014).
[Crossref]

Y. Gu and G. Gbur, “Scintillation of nonuniformly correlated beams in atmospheric turbulence,” Opt. Lett. 38(9), 1395–1397 (2013).
[Crossref] [PubMed]

Hickmann, J. M.

I. Vidal, E. J. S. Fonseca, and J. M. Hickmann, “Light polarization control during free-space propagation using coherence,” Phys. Rev. A 84(3), 033836 (2011).
[Crossref]

Hyde, M. W.

M. W. Hyde, S. Basu, D. G. Voelz, and X. Xiao, “Experimental generating any desired partially coherent Schell-model sources using phase-only control,” J. Appl. Phys. 118(9), 093102 (2015).
[Crossref]

James, D.

Kellock, H.

Korotkova, O.

D. Voelz, X. Xiao, and O. Korotkova, “Numerical modeling of Schell-model beams with arbitrary far-field patterns,” Opt. Lett. 40(3), 352–355 (2015).
[Crossref] [PubMed]

C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
[Crossref] [PubMed]

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]

O. Korotkova, S. Avramov-Zamurovic, C. Nelson, R. Malek-Madani, Y. Gu, and G. Gbur, “Scintillation reduction in multi-Gaussian Schell-model beams propagating in atmospheric turbulence,” Proc. SPIE 9224, 92240M (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 and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
[Crossref] [PubMed]

Z. Tong and O. Korotkova, “Nonuniformly correlated light beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012).
[Crossref] [PubMed]

Z. Tong, Y. Cai, and O. Korotkova, “Ghost imaging with electromagnetic stochastic beams,” Opt. Commun. 283(20), 3838–3845 (2010).
[Crossref]

O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281(9), 2342–2348 (2008).
[Crossref]

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249(4–6), 379–385 (2005).
[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]

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]

Kuittinen, M.

Lajunen, H.

Lehtolahti, J.

Liang, C.

Liu, L.

J. Yu, Y. Chen, L. Liu, X. Liu, and Y. Cai, “Splitting and combining properties of an elegant Hermite-Gaussian correlated Schell-model beam in Kolmogorov and non-Kolmogorov turbulence,” Opt. Express 23(10), 13467–13481 (2015).
[Crossref] [PubMed]

Y. Chen, L. Liu, F. Wang, C. Zhao, and Y. Cai, “Elliptical Laguerre-Gaussian correlated Schell-model beam,” Opt. Express 22(11), 13975–13987 (2014).
[Crossref] [PubMed]

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.

X. Liu and D. Zhao, “Trapping two types of particles with a focused generalized Multi-Gaussian Schell model beam,” Opt. Commun. 354, 250–255 (2015).
[Crossref]

J. Yu, Y. Chen, L. Liu, X. Liu, and Y. Cai, “Splitting and combining properties of an elegant Hermite-Gaussian correlated Schell-model beam in Kolmogorov and non-Kolmogorov turbulence,” Opt. Express 23(10), 13467–13481 (2015).
[Crossref] [PubMed]

X. Liu, F. Wang, M. Zhang, and Y. Cai, “Experimental demonstration of ghost imaging with an electromagnetic Gaussian Schell-model beam,” J. Opt. Soc. Am. A 32(5), 910–920 (2015).
[Crossref] [PubMed]

C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
[Crossref] [PubMed]

F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[Crossref] [PubMed]

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[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, G. Wu, X. Liu, S. Zhu, and Y. Cai, “Experimental measurement of the beam parameters of an electromagnetic Gaussian Schell-model source,” Opt. Lett. 36(14), 2722–2724 (2011).
[Crossref] [PubMed]

Ma, L.

Malek-Madani, R.

O. Korotkova, S. Avramov-Zamurovic, C. Nelson, R. Malek-Madani, Y. Gu, and G. Gbur, “Scintillation reduction in multi-Gaussian Schell-model beams propagating in atmospheric turbulence,” Proc. SPIE 9224, 92240M (2014).
[Crossref]

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]

Nelson, C.

O. Korotkova, S. Avramov-Zamurovic, C. Nelson, R. Malek-Madani, Y. Gu, and G. Gbur, “Scintillation reduction in multi-Gaussian Schell-model beams propagating in atmospheric turbulence,” Proc. SPIE 9224, 92240M (2014).
[Crossref]

Nixon, M.

Pal, V.

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.

Qu, J.

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[Crossref]

Ramírez-Sánchez, V.

Roychowdhury, H.

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249(4–6), 379–385 (2005).
[Crossref]

Saastamoinen, T.

Sahin, S.

Salem, M.

Sanchez, V. R.

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

Santarsiero, M.

F. Gori, V. R. Sanchez, 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, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 25(5), 1016–1021 (2008).
[Crossref] [PubMed]

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
[Crossref] [PubMed]

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]

Schoonover, R. W.

Senthilkumaran, P.

Setälä, T.

Sharma, M. K.

Shirai, T.

T. Shirai, H. Kellock, T. Setälä, and A. T. Friberg, “Visibility in ghost imaging with classical partially polarized electromagnetic beams,” Opt. Lett. 36(15), 2880–2882 (2011).
[Crossref] [PubMed]

F. Gori, V. R. Sanchez, 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]

Singh, R. K.

Tervo, J.

Tong, Z.

Z. Tong and O. Korotkova, “Nonuniformly correlated light beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012).
[Crossref] [PubMed]

Z. Tong, Y. Cai, and O. Korotkova, “Ghost imaging with electromagnetic stochastic beams,” Opt. Commun. 283(20), 3838–3845 (2010).
[Crossref]

Tradonsky, C.

Turunen, J.

Vidal, I.

I. Vidal, E. J. S. Fonseca, and J. M. Hickmann, “Light polarization control during free-space propagation using coherence,” Phys. Rev. A 84(3), 033836 (2011).
[Crossref]

Vinu, R. V.

Visser, T. D.

A. T. Friberg and T. D. Visser, “Scintillation of electromagnetic beams generated by quasi-homogeneous sources,” Opt. Commun. 335, 82–85 (2015).
[Crossref]

R. W. Schoonover and T. D. Visser, “Polarization singularities of focused, radially polarized fields,” Opt. Express 14(12), 5733–5745 (2006).
[Crossref] [PubMed]

Voelz, D.

Voelz, D. G.

M. W. Hyde, S. Basu, D. G. Voelz, and X. Xiao, “Experimental generating any desired partially coherent Schell-model sources using phase-only control,” J. Appl. Phys. 118(9), 093102 (2015).
[Crossref]

Wang, F.

Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91(1), 013823 (2015).
[Crossref]

X. Liu, F. Wang, M. Zhang, and Y. Cai, “Experimental demonstration of ghost imaging with an electromagnetic Gaussian Schell-model beam,” J. Opt. Soc. Am. A 32(5), 910–920 (2015).
[Crossref] [PubMed]

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. Cai, Y. Chen, and F. Wang, “Generation and propagation of partially coherent beams with nonconventional correlation functions: a review [invited],” J. Opt. Soc. Am. A 31(9), 2083–2096 (2014).
[Crossref] [PubMed]

F. Wang, C. Liang, Y. Yuan, and Y. Cai, “Generalized multi-Gaussian correlated Schell-model beam: from theory to experiment,” Opt. Express 22(19), 23456–23464 (2014).
[Crossref] [PubMed]

Y. Chen, L. Liu, F. Wang, C. Zhao, and Y. Cai, “Elliptical Laguerre-Gaussian correlated Schell-model beam,” Opt. Express 22(11), 13975–13987 (2014).
[Crossref] [PubMed]

C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
[Crossref] [PubMed]

Y. Chen, F. Wang, C. Zhao, and Y. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22(5), 5826–5838 (2014).
[Crossref] [PubMed]

F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[Crossref] [PubMed]

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[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]

G. Wu, F. Wang, and Y. Cai, “Coherence and polarization properties of a radially polarized beam with variable spatial coherence,” Opt. Express 20(27), 28301–28318 (2012).
[Crossref] [PubMed]

F. Wang, G. Wu, X. Liu, S. Zhu, and Y. Cai, “Experimental measurement of the beam parameters of an electromagnetic Gaussian Schell-model source,” Opt. Lett. 36(14), 2722–2724 (2011).
[Crossref] [PubMed]

Wolf, E.

M. Salem and E. Wolf, “Coherence-induced polarization changes in light beams,” Opt. Lett. 33(11), 1180–1182 (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, 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]

Wu, G.

Xiao, X.

D. Voelz, X. Xiao, and O. Korotkova, “Numerical modeling of Schell-model beams with arbitrary far-field patterns,” Opt. Lett. 40(3), 352–355 (2015).
[Crossref] [PubMed]

M. W. Hyde, S. Basu, D. G. Voelz, and X. Xiao, “Experimental generating any desired partially coherent Schell-model sources using phase-only control,” J. Appl. Phys. 118(9), 093102 (2015).
[Crossref]

Yu, J.

Yuan, Y.

F. Wang, C. Liang, Y. Yuan, and Y. Cai, “Generalized multi-Gaussian correlated Schell-model beam: from theory to experiment,” Opt. Express 22(19), 23456–23464 (2014).
[Crossref] [PubMed]

F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[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]

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[Crossref]

Zhan, Q.

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

Zhang, M.

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. Chen, F. Wang, C. Zhao, and Y. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22(5), 5826–5838 (2014).
[Crossref] [PubMed]

Y. Chen, L. Liu, F. Wang, C. Zhao, and Y. Cai, “Elliptical Laguerre-Gaussian correlated Schell-model beam,” Opt. Express 22(11), 13975–13987 (2014).
[Crossref] [PubMed]

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]

Zhao, D.

X. Liu and D. Zhao, “Trapping two types of particles with a focused generalized Multi-Gaussian Schell model beam,” Opt. Commun. 354, 250–255 (2015).
[Crossref]

Zhu, S.

Adv. Opt. Photonics (1)

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

Appl. Phys. Lett. (2)

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. Appl. Phys. (1)

M. W. Hyde, S. Basu, D. G. Voelz, and X. Xiao, “Experimental generating any desired partially coherent Schell-model sources using phase-only control,” J. Appl. Phys. 118(9), 093102 (2015).
[Crossref]

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

F. Gori, V. R. Sanchez, 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]

J. Opt. Soc. Am. A (4)

Opt. Commun. (7)

X. Liu and D. Zhao, “Trapping two types of particles with a focused generalized Multi-Gaussian Schell model beam,” Opt. Commun. 354, 250–255 (2015).
[Crossref]

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[Crossref]

O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281(9), 2342–2348 (2008).
[Crossref]

A. T. Friberg and T. D. Visser, “Scintillation of electromagnetic beams generated by quasi-homogeneous sources,” Opt. Commun. 335, 82–85 (2015).
[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]

Z. Tong, Y. Cai, and O. Korotkova, “Ghost imaging with electromagnetic stochastic beams,” Opt. Commun. 283(20), 3838–3845 (2010).
[Crossref]

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249(4–6), 379–385 (2005).
[Crossref]

Opt. Express (10)

G. Wu, F. Wang, and Y. Cai, “Coherence and polarization properties of a radially polarized beam with variable spatial coherence,” Opt. Express 20(27), 28301–28318 (2012).
[Crossref] [PubMed]

Y. Chen, L. Liu, F. Wang, C. Zhao, and Y. Cai, “Elliptical Laguerre-Gaussian correlated Schell-model beam,” Opt. Express 22(11), 13975–13987 (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]

R. Chriki, M. Nixon, V. Pal, C. Tradonsky, G. Barach, A. A. Friesem, and N. Davidson, “Manipulating the spatial coherence of a laser source,” Opt. Express 23(10), 12989–12997 (2015).
[Crossref] [PubMed]

J. Lehtolahti, M. Kuittinen, J. Turunen, and J. Tervo, “Coherence modulation by deterministic rotating diffusers,” Opt. Express 23(8), 10453–10466 (2015).
[Crossref] [PubMed]

Y. Chen, F. Wang, C. Zhao, and Y. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22(5), 5826–5838 (2014).
[Crossref] [PubMed]

F. Wang, C. Liang, Y. Yuan, and Y. Cai, “Generalized multi-Gaussian correlated Schell-model beam: from theory to experiment,” Opt. Express 22(19), 23456–23464 (2014).
[Crossref] [PubMed]

J. Yu, Y. Chen, L. Liu, X. Liu, and Y. Cai, “Splitting and combining properties of an elegant Hermite-Gaussian correlated Schell-model beam in Kolmogorov and non-Kolmogorov turbulence,” Opt. Express 23(10), 13467–13481 (2015).
[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]

R. W. Schoonover and T. D. Visser, “Polarization singularities of focused, radially polarized fields,” Opt. Express 14(12), 5733–5745 (2006).
[Crossref] [PubMed]

Opt. Lett. (16)

F. Wang, G. Wu, X. Liu, S. Zhu, and Y. Cai, “Experimental measurement of the beam parameters of an electromagnetic Gaussian Schell-model source,” Opt. Lett. 36(14), 2722–2724 (2011).
[Crossref] [PubMed]

D. Voelz, X. Xiao, and O. Korotkova, “Numerical modeling of Schell-model beams with arbitrary far-field patterns,” Opt. Lett. 40(3), 352–355 (2015).
[Crossref] [PubMed]

Y. Chen and Y. Cai, “Generation of a controllable optical cage by focusing a Laguerre-Gaussian correlated Schell-model beam,” Opt. Lett. 39(9), 2549–2552 (2014).
[Crossref] [PubMed]

Y. Gu and G. Gbur, “Scintillation of nonuniformly correlated beams in atmospheric turbulence,” Opt. Lett. 38(9), 1395–1397 (2013).
[Crossref] [PubMed]

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Phys. Lett. A (1)

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Proc. SPIE (1)

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Other (2)

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

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Figures (13)

Fig. 1
Fig. 1 Density plots of the modulus of the correlation functions | μ x x | , | μ y y | and | μ x y | for different values of the mode orders m α β and n α β with δ 0 x x = δ 0 y y = δ 0 x y = 0.2 mm .
Fig. 2
Fig. 2 Density plot of the normalized intensity distribution of an electromagnetic HGCSM beam with mxx = myy = nxx = nyy = 5 at several propagation distances in free space.
Fig. 3
Fig. 3 Density plot of the normalized intensity distribution of an electromagnetic HGCSM beam at z = 5m for different values of mxx, myy, nxx, nyy.
Fig. 4
Fig. 4 Density plot of the normalized intensity distribution and the distribution of the SOP of an electromagnetic HGCSM beam with mxx = nxx = 1, myy = nyy = 5, mxy = nxy = 3 at several propagation distances in free space.
Fig. 5
Fig. 5 Density plot of the normalized intensity distribution and the distribution of the SOP of an electromagnetic HGCSM beam at z = 5m in free space for different values of mxx, myy, mxy, nxx, nyy, nxy.
Fig. 6
Fig. 6 Density plot of the normalized intensity distribution and the distribution of the SOP of an electromagnetic HGCSM beam with mxx = myy = mxy = nxx = nyy = nxy = 5 at several propagation distances in free space.
Fig. 7
Fig. 7 Density plot of the DOP of an electromagnetic HGCSM beam at several propagation distances in free space for different values of mxx, myy, mxy, nxx, nyy, nxy.
Fig. 8
Fig. 8 Density plot of the normalized intensity distribution and the distribution of the SOP of a radially polarized HGCSM beam at several propagation distances in free space for different values of m and n.
Fig. 9
Fig. 9 Distribution of the polarization singularities of a radially polarized HGCSM beam at several propagation distances in free space for different values of m and n.
Fig. 10
Fig. 10 Distribution of the DOP of a radially polarized HGCSM beam at several propagation distances in free space for different values of m and n.
Fig. 11
Fig. 11 Experimental setup for generating a radially polarized HGCSM beam and measuring its focused intensity. BE, beam expander; RM, reflecting mirror; SLM, spatial light modulator; CA, circular aperture; RGGD, rotating ground-disk; L1 and L2, thin lenses; GAF, Gaussian amplitude filter; RPC, radial polarization converter; LP1 and LP2, linear polarizers; CCD, charge-coupled device; PC1 and PC2, personal computers.
Fig. 12
Fig. 12 Experimental results of intensity distribution of the generated radially polarized HGCSM beam with m = n = 5 focused by thin lens L2 with focal length f2 = 400 mm and its corresponding components I x x ( ρ ) and I y y ( ρ ) at different propagation distance z.
Fig. 13
Fig. 13 Experimental results of intensity distribution of the generated radially polarized HGCSM beam with m = 5, n = 0 focused by thin lens L2 with focal length f2 = 400 mm and its corresponding components I x x ( ρ ) and I y y ( ρ ) at different propagation distance z.

Equations (36)

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Γ ^ ( r 1 , r 2 ) = ( Γ x x ( r 1 , r 2 ) Γ x y ( r 1 , r 2 ) Γ y x ( r 1 , r 2 ) Γ y y ( r 1 , r 2 ) ) ,
Γ α β ( r 1 , r 2 ) = E α ( r 1 ) E β ( r 2 ) , ( α , β = x , y )
Γ α β ( r 1 , r 2 ) = p α β ( v ) α ( r 1 , v ) β ( r 2 , v ) d 2 v , ( α , β = x , y ) ,
p ^ ( v ) = ( p x x ( v ) p x y ( v ) p y x ( v ) p y y ( v ) ) ,
p x x ( v ) 0 , p y y ( v ) 0 , p x x ( v ) p y y ( v ) | p x y ( v ) | 2 0.
p α β ( v ) = 2 m α β + n α β + 5 δ 0 α β 2 m α β + 2 n α β + 2 H 2 m α β ( 0 ) H 2 n α β ( 0 ) π ( 1 ) m α β + n α β B α β × v x 2 m α β v y 2 n α β exp ( δ 0 α β 2 2 v x 2 ) exp ( δ 0 α β 2 2 v y 2 ) ,
α ( r , v ) = F α ( r ) exp ( 2 π i v r ) .
Γ α β ( r 1 , r 2 ) = F α ( r 1 ) F β ( r 2 ) μ α β ( r 1 , r 2 ) ,
μ α β ( r 1 , r 2 ) = B α β H 2 m α β ( 0 ) H 2 n α β ( 0 ) exp ( ( x 2 x 1 ) 2 2 δ 0 α β 2 ) H 2 m α β ( ( x 2 x 1 ) 2 δ 0 α β ) × exp ( ( y 2 y 1 ) 2 2 δ 0 α β 2 ) H 2 n α β ( ( y 2 y 1 ) 2 δ 0 α β ) , ( α , β = x , y ) .
| B α β | = 1 , ϕ α β =0, ( α = β ) , | B α β | 1 , ( α β ) , | B x y | = | B y x | , ϕ x y = ϕ y x , δ 0 x y = δ 0 y x .
1 H 2 m x x ( 0 ) H 2 m y y ( 0 ) 1 H 2 n x x ( 0 ) H 2 n y y ( 0 ) 2 m x x + m y y + n x x + n y y + 10 δ 0 x x 2 m x x + 2 n x x + 2 δ 0 y y 2 m y y + 2 n y y + 2 ( 1 ) m x x + m y y + n x x + n y y v x 2 m x x + 2 m y y v y 2 n x x + 2 n y y exp ( δ 0 x x 2 + δ 0 y y 2 2 v x 2 ) exp ( δ 0 x x 2 + δ 0 y y 2 2 v y 2 ) 1 H 2 m x y ( 0 ) H 2 m x y ( 0 ) 1 H 2 n x y ( 0 ) H 2 n x y ( 0 ) 2 2 m x y + 2 n x y + 10 | B x y | 2 δ 0 x y 2 m x y + 2 n x y + 2 δ 0 x y 2 m x y + 2 n x y + 2 ( 1 ) 2 m x y + 2 n x y . v x 4 m x y v y 4 n x y exp ( δ 0 x y 2 v x 2 ) exp ( δ 0 x y 2 v y 2 )
m x x + m y y = 2 m x y , n x x + n y y = 2 n x y .
δ 0 x x 2 m x x + 2 n x x + 2 H 2 m x x ( 0 ) H 2 m y y ( 0 ) δ 0 y y 2 m y y + 2 n y y + 2 H 2 n x x ( 0 ) H 2 n y y ( 0 ) exp ( δ 0 x x 2 + δ 0 y y 2 2 v x 2 δ 0 x x 2 + δ 0 y y 2 2 v y 2 ) 1 [ H 2 m x y ( 0 ) ] 2 1 [ H 2 n x y ( 0 ) ] 2 | B x y | 2 δ 0 x y 4 m x y + 4 n x y + 4 exp ( δ 0 x y 2 v x 2 δ 0 x y 2 v y 2 )
δ 0 x x 2 + δ 0 y y 2 2 δ 0 x y [ H 2 m x y ( 0 ) ] 2 δ 0 x x 2 m x x + 2 n x x + 2 H 2 m x x ( 0 ) H 2 m y y ( 0 ) [ H 2 n x y ( 0 ) ] 2 δ 0 y y 2 m y y + 2 n y y + 2 H 2 n x x ( 0 ) H 2 n y y ( 0 ) 1 | B x y | 2 4 m x y + 4 n x y + 4 .
[ H 2 m x y ( 0 ) ] 2 H 2 m x x ( 0 ) H 2 m y y ( 0 ) [ H 2 n x y ( 0 ) ] 2 H 2 n x x ( 0 ) H 2 n y y ( 0 ) 1 | B x y | 2 1.
I ( ρ ) = ( 2 π k / ρ ) 2 cos 2 θ [ Γ ˜ x x ( k s , k s ) + Γ ˜ y y ( k s , k s ) ] ,
Γ ˜ α α ( f 1 , f 2 ) = 1 ( 2 π ) 4 Γ α α ( r 1 , r 2 ) exp [ i ( f 1 r 1 + f 2 r 2 ) ] d 2 r 1 d 2 r 2 .
1 4 σ 0 2 + 1 δ 0 x x 2 2 π 2 λ 2 , 1 4 σ 0 2 + 1 δ 0 y y 2 2 π 2 λ 2 .
θ ( x , y ) = arc tan ( y / x ) .
θ ( r ) = 1 2 arc tan [ 2 Re [ Γ x y ( r , r ) ] Γ x x ( r , r ) Γ y y ( r , r ) ] ,
ε ( r ) = A 2 ( r ) A 1 ( r ) ,
A 1 ( r ) = 1 2 [ [ Γ x x ( r , r ) Γ y y ( r , r ) ] 2 + 4 | Γ x y ( r , r ) | 2 + [ Γ x x ( r , r ) Γ y y ( r , r ) ] 2 + 4 [ Re Γ x y ( r , r ) ] 2 ] 1 / 2 ,
A 2 ( r ) = 1 2 [ [ Γ x x ( r , r ) Γ y y ( r , r ) ] 2 + 4 | Γ x y ( r , r ) | 2 [ Γ x x ( r , r ) Γ y y ( r , r ) ] 2 + 4 [ Re Γ x y ( r , r ) ] 2 ] 1 / 2 .
B x y = B y x = 1 , δ 0 x x = δ 0 y y = δ 0 x y = δ 0 , m x x = m y y = m x y = m , n x x = n y y = n x y = n .
Γ α β ( ρ 1 , ρ 2 ) = 1 ( λ z ) 2 exp [ i k 2 z ( ρ 1 2 ρ 2 2 ) ] × Γ α β ( r 1 , r 2 ) exp [ i k 2 z ( r 1 2 r 2 2 ) ] exp [ i k z ( r 1 ρ 1 r 2 ρ 2 ) ] d 2 r 1 d 2 r 2 ,
I ( ρ ) = Γ x x ( ρ , ρ ) + Γ y y ( ρ , ρ ) = I x x ( ρ ) + I y y ( ρ ) ,
P ( ρ ) = 1 4 Det Γ ^ ( ρ , ρ ) [ Tr Γ ^ ( ρ , ρ ) ] 2 .
Γ α β ( ρ , ρ ) = A α A β B α β k 2 σ 0 2 2 z 2 1 H 2 m α β ( 0 ) H 2 n α β ( 0 ) 1 a α β exp [ ( k z ) 2 ( ρ x 2 + ρ y 2 ) 4 a α β ] × ( 1 1 2 a α β δ 0 α β 2 ) m α β H 2 m α β [ i k z ρ x 2 a α β ( 2 a α β δ 0 α β 2 1 ) 1 / 2 ] × ( 1 1 2 a α β δ 0 α β 2 ) n α β H 2 n α β [ i k z ρ y 2 a α β ( 2 a α β δ 0 α β 2 1 ) 1 / 2 ] , ( α , β = x , y ) ,
a α β = 1 8 σ 0 2 + 1 2 δ 0 α β 2 + σ 0 2 2 ( k z ) 2 .
Γ x x ( ρ , ρ ) = 1 4 π 2 ( λ z ) 2 1 H 2 m x x ( 0 ) H 2 n x x ( 0 ) 1 a × exp [ ( k z ) 2 ( ρ x 2 + ρ y 2 ) 4 a ] H 2 n x x [ i k z ρ y 2 a ( 2 a δ 0 2 1 ) 1 / 2 ] ( 1 1 2 a δ 0 2 ) n x x × { 2 σ 0 2 { ( 1 1 2 a δ 0 2 ) m x x H 2 m x x [ i k z ρ x 2 a ( 2 a δ 0 2 1 ) 1 / 2 ] σ 0 2 δ 0 2 ( k z ) 2 Ω x x } 1 2 δ 0 2 Ω x x } ,
Γ y y ( ρ , ρ ) = 1 4 π 2 ( λ z ) 2 1 H 2 m y y ( 0 ) H 2 n y y ( 0 ) 1 a × exp [ ( k z ) 2 ( ρ x 2 + ρ y 2 ) 4 a ] H 2 m y y [ i k z ρ x 2 a ( 2 a δ 0 2 1 ) 1 / 2 ] ( 1 1 2 a δ 0 2 ) m y y × { 2 σ 0 2 { ( 1 1 2 a δ 0 2 ) n y y H 2 n y y [ i k z ρ y 2 a ( 2 a δ 0 2 1 ) 1 / 2 ] σ 0 2 δ 0 2 ( k z ) 2 Ω y y } 1 2 δ 0 2 Ω y y } ,
Γ x y ( ρ , ρ ) = 1 4 [ σ 0 4 ( k z ) 2 + 1 4 ] δ 0 2 π 2 ( λ z ) 2 1 H 2 m x y ( 0 ) H 2 n x y ( 0 ) 1 a exp [ ( k z ) 2 ( ρ x 2 + ρ y 2 ) 4 a ] × p = 0 min ( 2 m x y , 1 ) 2 p p ! ( 2 m x y p ) ( 1 p ) ( 1 1 2 a δ 0 2 ) m x y + 0.5 p H 2 m x y + 1 2 p [ i k z ρ x 2 a ( 2 a δ 0 2 1 ) 1 / 2 ] × q = 0 min ( 2 n x y , 1 ) 2 q q ! ( 2 n x y q ) ( 1 q ) ( 1 1 2 a δ 0 2 ) n x y + 0.5 q H 2 n x y + 1 2 q [ i k z ρ y 2 a ( 2 a δ 0 2 1 ) 1 / 2 ] ,
Γ y x ( ρ , ρ ) = Γ x y ( ρ , ρ ) ,
a = 1 8 σ 0 2 + 1 2 δ 0 2 + σ 0 2 2 ( k z ) 2 ,
Ω x x = ( 1 1 2 a δ 0 2 ) m x x H 2 m x x [ i k z ρ x 2 a ( 2 a δ 0 2 1 ) 1 / 2 ] + 1 2 p = 0 min ( 2 m x x , 2 ) 2 p p ! ( 2 m x x p ) × ( 2 p ) ( 1 1 2 a δ 0 2 ) m x x + 1 p H 2 m x x + 2 2 p [ i k z ρ x 2 a ( 2 a δ 0 2 1 ) 1 / 2 ] ,
Ω y y = ( 1 1 2 a δ 0 2 ) n y y H 2 n y y [ i k z ρ y 2 a [ 2 a δ 0 2 1 ] 1 / 2 ] + 1 2 q = 0 min ( 2 n y y , 2 ) 2 q q ! ( 2 n y y q ) × ( 2 q ) ( 1 1 2 a δ 0 2 ) n y y + 1 q H 2 n y y + 2 2 q [ i k z ρ y 2 a ( 2 a δ 0 2 1 ) 1 / 2 ] .

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