Abstract

Scalar cosine-Gaussian-correlated Schell-model (CGCSM) beams of circular or rectangular symmetry were introduced just recently. In this paper, a new kind of partially coherent vector beam named vector CGCSM beam with radial polarization (i.e., radially polarized CGCSM beam) is introduced. The realizability conditions for a radially polarized CGCSM source and the beam condition for radiation generated by such source are derived. The statistical properties, such as the average intensity, the degree of coherence, the degree of polarization and the state of polarization, of a radially polarized CGCSM beam focused by a thin lens are analyzed in detail. It is found that the statistical properties of a radially polarized CGCSM beam are quite different from those of a conventional radial polarized partially coherent beam with Gaussian correlated Schell-model function. Furthermore, we first report experimental generation of a radially polarized CGCSM beam and measure its focusing properties. Our experimental results are consistent with the theoretical predictions.

© 2015 Optical Society of America

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References

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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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2015 (6)

F. Wang, X. Liu, and Y. Cai, “Propagation of partially coherent beam in turbulent atmosphere: a review (Invited review),” Prog. Electromagnetics Res. 150, 123–143 (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]

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]

R. Martínez-Herrero and F. Prado, “Polarization evolution of radially polarized partially coherent vortex fields: role of Gouy phase of Laguerre-Gauss beams,” Opt. Express 23(4), 5043–5051 (2015).
[Crossref] [PubMed]

O. Korotkova, “Can a sphere scatter light producing rectangular intensity patterns?” Opt. Lett. 40(8), 1709–1712 (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 (9)

R. Chen, L. Liu, S. Zhu, G. Wu, F. Wang, and Y. Cai, “Statistical properties of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 22(2), 1871–1883 (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 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]

L. Pan, C. Ding, and H. Wang, “Diffraction of cosine-Gaussian-correlated Schell-model beams,” Opt. Express 22(10), 11670–11679 (2014).
[Crossref] [PubMed]

Z. Chen, S. Cui, L. Zhang, C. Sun, M. Xiong, and J. Pu, “Measuring the intensity fluctuation of partially coherent radially polarized beams in atmospheric turbulence,” Opt. Express 22(15), 18278–18283 (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]

S. Zhu, F. Wang, Y. Chen, Z. Li, and Y. Cai, “Statistical properties in Young’s interference pattern formed with a radially polarized beam with controllable spatial coherence,” Opt. Express 22(23), 28697–28710 (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]

2013 (7)

2012 (6)

J. Li, Y. Salamin, B. J. Galow, and C. Keitel, “Acceleration of proton bunches by petawatt chirped radially polarized laser pulses,” Phys. Rev. A 85(6), 063832 (2012).
[Crossref]

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. Commun. 3, 998 (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. Feng, Y. Chen, C. Zhao, and Y. Cai, “Statistical properties of a nonparaxial cylindrical vector partially coherent field in free space,” Opt. Express 20(14), 15908–15927 (2012).
[Crossref] [PubMed]

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

2011 (3)

2010 (1)

2009 (4)

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]

P. Wróbel, J. Pniewski, T. J. Antosiewicz, and T. Szoplik, “Focusing radially polarized light by a concentrically corrugated silver film without a hole,” Phys. Rev. Lett. 102(18), 183902 (2009).
[Crossref] [PubMed]

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

W. Chen, D. C. Abeysinghe, R. L. Nelson, and Q. Zhan, “Plasmonic lens made of multiple concentric metallic rings under radially polarized illumination,” Nano Lett. 9(12), 4320–4325 (2009).
[Crossref] [PubMed]

2008 (4)

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]

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]

H. Wang, L. Shi, B. Lukyanchuk, C. J. R. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
[Crossref]

H. F. Schouten and T. D. Visser, “The role of correlation functions in the theory of optical wave fields,” Am. J. Phys. 76(9), 867–871 (2008).
[Crossref]

2007 (1)

2005 (1)

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

2003 (2)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

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

2002 (2)

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]

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(1), 016615 (2002).
[Crossref] [PubMed]

2001 (2)

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001).
[Crossref] [PubMed]

S. A. Ponomarenko, “A class of partially coherent beams carrying optical vortices,” J. Opt. Soc. Am. A 18(1), 150–156 (2001).
[Crossref] [PubMed]

1991 (1)

1987 (1)

F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64(4), 311–316 (1987).
[Crossref]

1983 (1)

F. Gori, “Mode propagation of the field generated by Collett-Wolf Schell-model sources,” Opt. Commun. 46(3), 149–154 (1983).
[Crossref]

1982 (1)

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 (1982).
[Crossref]

1978 (1)

E. Wolf and E. Collett, “Partially coherent sources which produce the same far-field intensity distribution as a laser,” Opt. Commun. 25(3), 293–296 (1978).
[Crossref]

1970 (1)

Abeysinghe, D. C.

W. Chen, D. C. Abeysinghe, R. L. Nelson, and Q. Zhan, “Plasmonic lens made of multiple concentric metallic rings under radially polarized illumination,” Nano Lett. 9(12), 4320–4325 (2009).
[Crossref] [PubMed]

Antosiewicz, T. J.

P. Wróbel, J. Pniewski, T. J. Antosiewicz, and T. Szoplik, “Focusing radially polarized light by a concentrically corrugated silver film without a hole,” Phys. Rev. Lett. 102(18), 183902 (2009).
[Crossref] [PubMed]

Beversluis, M. R.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001).
[Crossref] [PubMed]

Borghi, R.

Brown, D. P.

Brown, T. G.

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]

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001).
[Crossref] [PubMed]

Cai, Y.

F. Wang, X. Liu, and Y. Cai, “Propagation of partially coherent beam in turbulent atmosphere: a review (Invited review),” Prog. Electromagnetics Res. 150, 123–143 (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]

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]

S. Zhu, F. Wang, Y. Chen, Z. Li, and Y. Cai, “Statistical properties in Young’s interference pattern formed with a radially polarized beam with controllable spatial coherence,” Opt. Express 22(23), 28697–28710 (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]

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 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. 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, 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]

R. Chen, L. Liu, S. Zhu, G. Wu, F. Wang, and Y. Cai, “Statistical properties of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 22(2), 1871–1883 (2014).
[Crossref] [PubMed]

S. Zhu, X. Zhu, L. Liu, F. Wang, and Y. Cai, “Theoretical and experimental studies of the spectral changes of a polychromatic partially coherent radially polarized beam,” Opt. Express 21(23), 27682–27696 (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]

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(3), 57–65 (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. Feng, Y. Chen, C. Zhao, and Y. Cai, “Statistical properties of a nonparaxial cylindrical vector partially coherent field in free space,” Opt. Express 20(14), 15908–15927 (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]

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, “Generation of various partially coherent beams and their propagation properties in turbulent atmosphere: a review,” Proc. SPIE 7924, 792402 (2011).
[Crossref]

S. Zhu, Y. Cai, and O. Korotkova, “Propagation factor of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Express 18(12), 12587–12598 (2010).
[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, R.

Chen, W.

W. Chen, D. C. Abeysinghe, R. L. Nelson, and Q. Zhan, “Plasmonic lens made of multiple concentric metallic rings under radially polarized illumination,” Nano Lett. 9(12), 4320–4325 (2009).
[Crossref] [PubMed]

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]

S. Zhu, F. Wang, Y. Chen, Z. Li, and Y. Cai, “Statistical properties in Young’s interference pattern formed with a radially polarized beam with controllable spatial coherence,” Opt. Express 22(23), 28697–28710 (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. 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. 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(3), 57–65 (2013).
[Crossref]

Y. Dong, F. Feng, Y. Chen, C. Zhao, and Y. Cai, “Statistical properties of a nonparaxial cylindrical vector partially coherent field in free space,” Opt. Express 20(14), 15908–15927 (2012).
[Crossref] [PubMed]

Chen, Z.

Chong, C. T.

H. Wang, L. Shi, B. Lukyanchuk, C. J. R. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
[Crossref]

Collett, E.

E. Wolf and E. Collett, “Partially coherent sources which produce the same far-field intensity distribution as a laser,” Opt. Commun. 25(3), 293–296 (1978).
[Crossref]

Collins, S. A.

Cui, S.

Ding, C.

Dong, Y.

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

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(3), 57–65 (2013).
[Crossref]

Feng, F.

Friberg, A.

Friberg, A. T.

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(1), 016615 (2002).
[Crossref] [PubMed]

J. Turunen, A. Vasara, and A. T. Friberg, “Propagation invariance and self-imaging in variable-coherence optics,” J. Opt. Soc. Am. A 8(2), 282–289 (1991).
[Crossref]

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 (1982).
[Crossref]

Galow, B. J.

J. Li, Y. Salamin, B. J. Galow, and C. Keitel, “Acceleration of proton bunches by petawatt chirped radially polarized laser pulses,” Phys. Rev. A 85(6), 063832 (2012).
[Crossref]

Gbur, G.

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, 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, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64(4), 311–316 (1987).
[Crossref]

F. Gori, “Mode propagation of the field generated by Collett-Wolf Schell-model sources,” Opt. Commun. 46(3), 149–154 (1983).
[Crossref]

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, 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. Commun. 3, 998 (2012).
[Crossref] [PubMed]

Gu, Y.

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64(4), 311–316 (1987).
[Crossref]

Kaivola, M.

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(1), 016615 (2002).
[Crossref] [PubMed]

Keitel, C.

J. Li, Y. Salamin, B. J. Galow, and C. Keitel, “Acceleration of proton bunches by petawatt chirped radially polarized laser pulses,” Phys. Rev. A 85(6), 063832 (2012).
[Crossref]

Korotkova, O.

O. Korotkova, “Can a sphere scatter light producing rectangular intensity patterns?” Opt. Lett. 40(8), 1709–1712 (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]

Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
[Crossref] [PubMed]

Z. Mei and O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013).
[Crossref] [PubMed]

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

S. Zhu, Y. Cai, and O. Korotkova, “Propagation factor of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Express 18(12), 12587–12598 (2010).
[Crossref] [PubMed]

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

Lajunen, H.

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. Commun. 3, 998 (2012).
[Crossref] [PubMed]

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

Li, J.

J. Li, Y. Salamin, B. J. Galow, and C. Keitel, “Acceleration of proton bunches by petawatt chirped radially polarized laser pulses,” Phys. Rev. A 85(6), 063832 (2012).
[Crossref]

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. Commun. 3, 998 (2012).
[Crossref] [PubMed]

Li, Z.

Liang, C.

Lin, Q.

Liu, L.

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]

F. Wang, X. Liu, and Y. Cai, “Propagation of partially coherent beam in turbulent atmosphere: a review (Invited review),” Prog. Electromagnetics Res. 150, 123–143 (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]

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. 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(3), 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, 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]

Lukyanchuk, B.

H. Wang, L. Shi, B. Lukyanchuk, C. J. R. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
[Crossref]

Martínez-Herrero, R.

Mei, Z.

Nelson, R. L.

W. Chen, D. C. Abeysinghe, R. L. Nelson, and Q. Zhan, “Plasmonic lens made of multiple concentric metallic rings under radially polarized illumination,” Nano Lett. 9(12), 4320–4325 (2009).
[Crossref] [PubMed]

Novotny, L.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001).
[Crossref] [PubMed]

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64(4), 311–316 (1987).
[Crossref]

Pan, L.

Pniewski, J.

P. Wróbel, J. Pniewski, T. J. Antosiewicz, and T. Szoplik, “Focusing radially polarized light by a concentrically corrugated silver film without a hole,” Phys. Rev. Lett. 102(18), 183902 (2009).
[Crossref] [PubMed]

Ponomarenko, S. A.

Prado, F.

Pu, J.

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(3), 57–65 (2013).
[Crossref]

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

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]

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]

Saastamoinen, T.

Sahin, S.

Salamin, Y.

J. Li, Y. Salamin, B. J. Galow, and C. Keitel, “Acceleration of proton bunches by petawatt chirped radially polarized laser pulses,” Phys. Rev. A 85(6), 063832 (2012).
[Crossref]

Santarsiero, M.

Schouten, H. F.

H. F. Schouten and T. D. Visser, “The role of correlation functions in the theory of optical wave fields,” Am. J. Phys. 76(9), 867–871 (2008).
[Crossref]

Setälä, T.

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

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(1), 016615 (2002).
[Crossref] [PubMed]

Sheppard, C. J. R.

H. Wang, L. Shi, B. Lukyanchuk, C. J. R. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
[Crossref]

Shevchenko, A.

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(1), 016615 (2002).
[Crossref] [PubMed]

Shi, L.

H. Wang, L. Shi, B. Lukyanchuk, C. J. R. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
[Crossref]

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]

Sudol, R. J.

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 (1982).
[Crossref]

Sun, C.

Szoplik, T.

P. Wróbel, J. Pniewski, T. J. Antosiewicz, and T. Szoplik, “Focusing radially polarized light by a concentrically corrugated silver film without a hole,” Phys. Rev. Lett. 102(18), 183902 (2009).
[Crossref] [PubMed]

Tervo, J.

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. Commun. 3, 998 (2012).
[Crossref] [PubMed]

Turunen, J.

Vasara, A.

Visser, T. D.

H. F. Schouten and T. D. Visser, “The role of correlation functions in the theory of optical wave fields,” Am. J. Phys. 76(9), 867–871 (2008).
[Crossref]

Wang, F.

F. Wang, X. Liu, and Y. Cai, “Propagation of partially coherent beam in turbulent atmosphere: a review (Invited review),” Prog. Electromagnetics Res. 150, 123–143 (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]

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]

R. Chen, L. Liu, S. Zhu, G. Wu, F. Wang, and Y. Cai, “Statistical properties of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 22(2), 1871–1883 (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]

S. Zhu, F. Wang, Y. Chen, Z. Li, and Y. Cai, “Statistical properties in Young’s interference pattern formed with a radially polarized beam with controllable spatial coherence,” Opt. Express 22(23), 28697–28710 (2014).
[Crossref] [PubMed]

S. Zhu, X. Zhu, L. Liu, F. Wang, and Y. Cai, “Theoretical and experimental studies of the spectral changes of a polychromatic partially coherent radially polarized beam,” Opt. Express 21(23), 27682–27696 (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]

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(3), 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]

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]

Wang, H.

L. Pan, C. Ding, and H. Wang, “Diffraction of cosine-Gaussian-correlated Schell-model beams,” Opt. Express 22(10), 11670–11679 (2014).
[Crossref] [PubMed]

H. Wang, L. Shi, B. Lukyanchuk, C. J. R. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
[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), 35–43 (2005).
[Crossref]

E. Wolf and E. Collett, “Partially coherent sources which produce the same far-field intensity distribution as a laser,” Opt. Commun. 25(3), 293–296 (1978).
[Crossref]

Wróbel, P.

P. Wróbel, J. Pniewski, T. J. Antosiewicz, and T. Szoplik, “Focusing radially polarized light by a concentrically corrugated silver film without a hole,” Phys. Rev. Lett. 102(18), 183902 (2009).
[Crossref] [PubMed]

Wu, G.

Xiong, M.

Yao, M.

Youngworth, K. S.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001).
[Crossref] [PubMed]

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]

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(3), 57–65 (2013).
[Crossref]

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]

Zhan, Q.

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

W. Chen, D. C. Abeysinghe, R. L. Nelson, and Q. Zhan, “Plasmonic lens made of multiple concentric metallic rings under radially polarized illumination,” Nano Lett. 9(12), 4320–4325 (2009).
[Crossref] [PubMed]

Zhang, L.

Zhao, C.

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.

Zhu, X.

Adv. Opt. Photonics (1)

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

Am. J. Phys. (1)

H. F. Schouten and T. D. Visser, “The role of correlation functions in the theory of optical wave fields,” Am. J. Phys. 76(9), 867–871 (2008).
[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. Opt. A, Pure Appl. Opt. (1)

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]

J. Opt. Soc. Am. (1)

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

Nano Lett. (1)

W. Chen, D. C. Abeysinghe, R. L. Nelson, and Q. Zhan, “Plasmonic lens made of multiple concentric metallic rings under radially polarized illumination,” Nano Lett. 9(12), 4320–4325 (2009).
[Crossref] [PubMed]

Nat. Commun. (1)

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. Commun. 3, 998 (2012).
[Crossref] [PubMed]

Nat. Photonics (1)

H. Wang, L. Shi, B. Lukyanchuk, C. J. R. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
[Crossref]

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]

F. Gori, “Mode propagation of the field generated by Collett-Wolf Schell-model sources,” Opt. Commun. 46(3), 149–154 (1983).
[Crossref]

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 (1982).
[Crossref]

E. Wolf and E. Collett, “Partially coherent sources which produce the same far-field intensity distribution as a laser,” Opt. Commun. 25(3), 293–296 (1978).
[Crossref]

F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64(4), 311–316 (1987).
[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(3), 57–65 (2013).
[Crossref]

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

Opt. Express (14)

L. Pan, C. Ding, and H. Wang, “Diffraction of cosine-Gaussian-correlated Schell-model beams,” Opt. Express 22(10), 11670–11679 (2014).
[Crossref] [PubMed]

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S. Zhu, Y. Cai, and O. Korotkova, “Propagation factor of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Express 18(12), 12587–12598 (2010).
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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).
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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).
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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).
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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. Zhen, ed. (World Scientific, 2013), pp. 221–273.

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Figures (13)

Fig. 1
Fig. 1 Density plots of the square of the DOC q 2 ( x , y , 1 m m , 1 m m ) , q x x 2 ( x , y , 1 m m , 1 m m ) q x y 2 ( x , y , 1 m m , 1 m m ) , q y y 2 ( x , y , 1 m m , 1 m m ) of a radially polarized CGCSM beam in the source plane for different values of n .
Fig. 2
Fig. 2 Spectral intensity distribution and the corresponding cross line of a radially polarized CGCSM beam in the source plane for different values of n .
Fig. 3
Fig. 3 Spectral intensity distribution I ( ρ ) and the corresponding cross line of a radially polarized CGCSM beam focused by a thin lens at several propagation distances for different values of n .
Fig. 4
Fig. 4 Spectral intensity distribution I ( ρ ) , its composition components W x x ( ρ , ρ ) , W y y ( ρ , ρ ) , and the corresponding cross lines of a radially polarized CGCSM beam focused by a thin lens at several propagation distances with n = 1.
Fig. 5
Fig. 5 Density plot of the square of the DOC q 2 ( ρ 1 , ρ 2 = 0 ) and the corresponding cross line of a radially polarized CGCSM beam focused by a thin lens at different propagation distances for different values of n.
Fig. 6
Fig. 6 Density plot of the DOP and the corresponding cross line of a radially polarized CGCSM beam focused by a thin lens at different propagation distances for different values of n.
Fig. 7
Fig. 7 SOP of a radially polarized CGCSM beam in the source plane and in the focal plane for different values of n.
Fig. 8
Fig. 8 Experimental setup for generating a radially polarized CGCSM beam, measuring the square of the modulus of its DOC and its focused intensity. Laser, He-Ne laser; BE, beam expander; M, mirror; SLM, spatial light modulator; CA, circular aperture; LP, linear polarizer; RGGD, rotating ground-disk; L1, L2, L3, thin lenses; GAF, Gaussian amplitude filter; RPC, radial polarization converter; BS, beam splitter; CCD, charge-coupled device; BPA, beam profile analyzer; PC1, PC2; personal computers.
Fig. 9
Fig. 9 Experimental results of the intensity distribution and the corresponding cross line of the generated radially polarized CGCSM beam with n = 1 in the output plane of RPC, as well as the composition components W x x ( ρ , ρ ) and W y y ( ρ , ρ ) . The solid curve denotes the theoretical fit of the experimental data.
Fig. 10
Fig. 10 Experimental results of the square of the DOC and the corresponding cross line of the generated radially polarized CGCSM beam in the output plane of RPC with n = 1. (a) q 2 ( x , 0 , 0.2 mm , 0.2 mm ) , (b) q 2 ( x , y , 0.2 mm , 0.2 m m ) , (c) q x x 2 ( x , y , 0.2 mm , 0.2 m m ) , (d) q y y 2 ( x , y , 0.2 mm , 0.2 m m ) . The solid curve denotes the theoretical fit of the experimental data.
Fig. 11
Fig. 11 Experimental results of intensity distribution of the generated radially polarized CGCSM beam with n = 1 focused by a thin lens with f 3 = 150mm and its corresponding composition components W x x ( ρ , ρ ) and W y y ( ρ , ρ ) at different propagation distances with n = 1 .
Fig. 12
Fig. 12 Experimental results of the square of the DOC q 2 and the corresponding cross line of the generated radially polarized CGCSM beam with n = 1 focused by a thin lens with f 3 = 150mm at different propagation distances. The solid curve denotes the theoretical result.
Fig. 13
Fig. 13 Experimental results of the DOP (cross line v = 0) of the generated radially polarized CGCSM beam with n = 1 focused by a thin lens with f 3 = 150mm at different propagation distances. The solid curve denotes the theoretical result.

Equations (35)

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W α β ( r 1 , r 2 ) = A α A β α 1 β 2 exp [ r 1 2 + r 2 2 4 w 0 2 ] q α β ( r 1 r 2 ) , ( α , β = x , y ) ,
q α β ( r 1 r 2 ) = B α β cos [ 2 π n ( x 1 x 2 ) δ α β ] cos [ 2 π n ( y 1 y 2 ) δ α β ] exp [ ( r 1 r 2 ) 2 2 δ α β 2 ] ,
W α β ( r 1 , r 2 ) = W α α ( r 1 , r 1 ) W β β ( r 2 , r 2 ) μ α β ( r 1 r 2 ) , ( α , β = x , y ) ,
= α = x , y β = x , y q α β ( r 1 , r 2 ) g α ( r 1 ) g β ( r 2 ) d 2 r 1 d 2 r 2 ,
q α β ( r 1 r 2 ) = q ˜ α β ( ξ ) exp [ 2 π i ξ ( r 1 r 2 ) ] d 2 ξ .
q x y ( r 1 r 2 ) = q y x ( r 2 r 1 ) , q ˜ x y ( ξ ) = q ˜ y x ( ξ ) .
= { | g ˜ x ( ξ ) | 2 q ˜ x x ( ξ ) + | g ˜ y ( ξ ) | 2 q ˜ y y ( ξ ) + 2 Re [ g ˜ x ( ξ ) g ˜ y ( ξ ) q ˜ x y ( ξ ) ] } d 2 ξ ,
| q ˜ x y ( ξ ) | q ˜ x x ( ξ ) q ˜ y y ( ξ ) ,
| B x y | δ x y 2 cos h 2 ( 2 2 π 3 / 2 n δ x y ξ ) exp ( 4 π 2 δ x y 2 ξ 2 ) δ x x 2 δ y y 2 exp [ 4 π 2 ( δ x x 2 + δ y y 2 ) ξ 2 ] × cos h 2 ( 2 2 π 3 / 2 n δ x x ξ ) cos h 2 ( 2 2 π 3 / 2 n δ y y ξ ) .
( δ x x 2 + δ y y 2 ) / 2 δ x y δ x x δ y y / | B x y | , | B x y | 2 δ x x δ y y / ( δ x x 2 + δ y y 2 ) 1.
θ ( x , y ) = arc tan ( y / x ) .
θ ( r ) = 1 2 arc tan [ 2 Re [ W x y ( r , r ) ] W x x ( r , r ) W y y ( r , r ) ] ,
ε ( r , r ) = A ( r , r ) / A + ( r , r ) , 0 ε 1.
A ± ( r , r ) = 1 2 [ [ W x x ( r , r ) W y y ( r , r ) ] 2 + 4 | W x y ( r , r ) | 2 ± [ W x x ( r , r ) W y y ( r , r ) ] 2 + 4 ( Re [ W x y ( r , r ) ] ) 2 ] 1 / 2 .
B x y = B y x = 1 ,
δ x x = δ y y = δ x y = δ y x = δ 0 .
Λ x x ( u ) 0 , Λ y y ( u ) 0 , Λ x x ( u ) Λ y y ( u ) | Λ x y ( u ) | 2 0 ,
Λ α β ( u ) = cos h ( 2 n 2 π u x σ α β ) cos h ( 2 n 2 π u y σ α β ) exp [ 2 ( u x 2 + u y 2 ) σ α β 2 ] , ( α , β = x , y ) .
S ( ρ ) = 4 π 2 k 2 ρ 2 cos 2 φ [ W ˜ x x ( k s , k s ) + W ˜ y y ( k s , k s ) ] ,
W ˜ α α ( f 1 , f 2 ) = 1 ( 2 π ) 4 W α α ( r 1 , r 2 ) exp [ i ( f 1 r 1 + f 2 r 2 ) ] d 2 r 1 d 2 r 2 ,
W ˜ x x ( k s , k s ) = k 2 w 0 2 cos 2 φ 1 6 ρ 2 M 1 2 M 2 { e x p [ ( k s y a ) 2 2 M 2 ] + e x p [ ( k s y + a ) 2 2 M 2 ] } × { [ δ 0 2 + ( k s x + a ) 2 4 w 0 2 M 2 ] exp ( ( k s x + a ) 2 2 M 2 ) + [ δ 0 2 + ( k s x a ) 2 4 w 0 2 M 2 ] exp ( ( k s x a ) 2 2 M 2 ) } ,
W ˜ y y ( k s , k s ) = k 2 w 0 2 cos 2 φ 1 6 ρ 2 M 1 2 M 2 { e x p [ ( k s x a ) 2 2 M 2 ] + e x p [ ( k s x + a ) 2 2 M 2 ] } × { [ δ 0 2 + ( k s y + a ) 2 4 w 0 2 M 2 ] exp ( ( k s y + a ) 2 2 M 2 ) + [ δ 0 2 + ( k s y a ) 2 4 w 0 2 M 2 ] exp ( ( k s y a ) 2 2 M 2 ) } ,
a = n 2 π / δ 0 , M 1 = 1 / 4 w 0 2 + 1 / 2 δ 0 2 , M 2 = 1 / 4 w 0 2 + 1 / δ 0 2 .
e x p ( k 2 s x 2 / 2 M 2 ) 0 , e x p ( k 2 s y 2 / 2 M 2 ) 0.
1 / 4 w 0 2 + δ 0 2 2 π 2 / λ 2 .
W α β ( ρ 1 , ρ 2 ) = k 2 4 π 2 B 2 W α β ( r 1 , r 2 ) exp [ i k 2 B ( A r 2 2 2 r 2 ρ 2 + D ρ 2 2 ) ] × exp [ i k 2 B ( A r 1 2 2 r 1 ρ 1 + D ρ 1 2 ) ] d 2 r 1 d 2 r 2 ,
W x x ( ρ 1 , ρ 2 ) = k 2 Γ ( ρ 1 , ρ 2 ) 64 B 2 N 1 2 Π 2 [ e x p ( ζ v 12 2 4 N 1 Ω v 22 2 4 Π ) + e x p ( ζ v 11 2 4 N 1 Ω v 21 2 4 Π ) ] × { ( δ 0 2 ζ u 11 Ω u 21 Ω u 21 2 2 δ 0 2 Π ) exp ( ζ u 11 2 4 N 1 Ω u 21 2 4 Π ) + ( δ 0 2 ζ u 12 Ω u 22 Ω u 22 2 2 δ 0 2 Π ) exp ( ζ u 12 2 4 N 1 Ω u 22 2 4 Π ) } ,
W y y ( ρ 1 , ρ 2 ) = k 2 Γ ( ρ 1 , ρ 2 ) 64 B 2 N 1 2 Π 2 [ e x p ( ζ u 12 2 4 N 1 Ω u 22 2 4 Π ) + e x p ( ζ u 11 2 4 N 1 Ω u 21 2 4 Π ) ] × { ( δ 0 2 ζ v 11 Ω v 21 Ω v 21 2 2 δ 0 2 Π ) exp ( ζ v 11 2 4 N 1 Ω v 21 2 4 Π ) + ( δ 0 2 ζ v 12 Ω v 22 Ω v 22 2 2 δ 0 2 Π ) exp ( ζ v 12 2 4 N 1 Ω v 22 2 4 Π ) } ,
W x y ( ρ 1 , ρ 2 ) = k 2 Γ ( ρ 1 , ρ 2 ) 64 B 2 N 1 2 Π 2 { Ω v 22 e x p [ ζ v 12 2 4 N 1 Ω v 22 2 4 Π ] + Ω v 21 e x p [ ζ v 11 2 4 N 1 Ω v 21 2 4 Π ] } × { ( ζ u 11 Ω u 21 2 δ 0 2 Π ) exp [ ζ u 11 2 4 N 1 Ω u 21 2 4 Π ] + ( ζ u 12 Ω u 22 2 δ 0 2 Π ) exp [ ζ u 12 2 4 N 1 Ω u 22 2 4 Π ] } ,
W y x ( ρ 1 , ρ 2 ) = W x y * ( ρ 1 , ρ 2 ) ,
ζ u 11 = k u 1 / B + a , ζ u 12 = k u 1 / B a , ζ u 21 = k u 2 / B + a , ζ u 22 = k u 2 / B a , ζ v 11 = k v 1 / B + a , ζ v 12 = k v 1 / B a , ζ v 21 = k v 2 / B + a , ζ v 22 = k v 2 / B a , Ω u 22 = ζ u 12 / 2 δ 0 2 N 1 ζ u 22 , Ω u 21 = ζ u 11 / 2 δ 0 2 N 1 ζ u 21 , Ω v 22 = ζ v 12 / 2 δ 0 2 N 1 ζ v 22 , Ω v 21 = ζ v 11 / 2 δ 0 2 N 1 ζ v 21 , N 1 = M 1 + i k A / 2 B , Π = M 1 i k A / 2 B 1 / 4 δ 0 4 N 1 , Γ ( ρ 1 , ρ 2 ) = exp [ i k D ( ρ 1 2 ρ 2 2 ) / 2 B ] .
q 2 ( ρ 1 , ρ 2 ) = T r [ W ( ρ 1 , ρ 2 ) W ( ρ 1 , ρ 2 ) ] T r [ W ( ρ 1 , ρ 1 ) ] T r [ W ( ρ 2 , ρ 2 ) ] ,
I ( ρ ) = W x x ( ρ , ρ ) + W y y ( ρ , ρ ) ,
P ( ρ ) = 1 4 Det [ W ( ρ , ρ ) ] [ Tr W ( ρ , ρ ) ] 2 .
( A B C D ) = ( 1 z 0 1 ) ( 1 0 1 / f 1 ) ( 1 f 0 1 ) = ( 1 z / f f 1 / f 0 ) .

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