Abstract

Experimental generation of a radially polarized (RP) beam with controllable spatial coherence (i.e., partially coherent RP beam) was reported recently [Appl. Phys. Lett. 100, 051108 (2012)]. In this paper, we carry out theoretical and experimental studies of the statistical properties in Young’s two-slit interference pattern formed with a partially coherent RP beam. An approximate analytical expression for the cross-spectral density matrix of a partially coherent RP beam in the observation plane is obtained, and it is found that the statistical properties, such as the intensity, the degree of coherence and the degree of polarization, are strongly affected by the spatial coherence of the incident beam. Our experimental results are consistent with the theoretical predictions, and may be useful in some applications, where light field with special statistical properties are required.

© 2014 Optical Society of America

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2014 (4)

X. Liu, F. Wang, C. Wei, and Y. Cai, “Experimental study of turbulence-induced beam wander and deformation of a partially coherent beam,” Opt. Lett. 39(11), 3336–3339 (2014).
[Crossref] [PubMed]

J. Qi, W. Wang, X. Li, X. Wang, W. Sun, J. Liao, and Y. Nie, “Double-slit interference of radially polarized vortex beams,” Opt. Eng. 53(4), 044107 (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]

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

2013 (4)

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]

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]

X. Liu, Y. Shen, L. Liu, F. Wang, and Y. Cai, “Experimental demonstration of vortex phase-induced reduction in scintillation of a partially coherent beam,” Opt. Lett. 38(24), 5323–5326 (2013).
[Crossref] [PubMed]

J. Zhang, Z. Wang, B. Cheng, Q. Wang, B. Wu, X. Shen, L. Zheng, Y. Xu, and Q. Lin, “Atom cooling by partially spatially coherent lasers,” Phys. Rev. A 88(2), 023416 (2013).
[Crossref]

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]

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]

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]

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. Li, X. L. Wang, H. Zhao, L. J. Kong, K. Lou, B. Gu, C. Tu, and H. T. Wang, “Young’s two-slit interference of vector light fields,” Opt. Lett. 37(11), 1790–1792 (2012).
[Crossref] [PubMed]

2011 (6)

2010 (2)

Y. Cai and F. Wang, “Tensor method for treating the propagation of scalar and electromagnetic Gaussian Schell-model beams: a review,” Open Opt. J. 4(1), 1–20 (2010).
[Crossref]

T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104(17), 173902 (2010).
[Crossref] [PubMed]

2009 (4)

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

C. Zhao, Y. Cai, X. Lu, and H. T. Eyyuboğlu, “Radiation force of coherent and partially coherent flat-topped beams on a Rayleigh particle,” Opt. Express 17(3), 1753–1765 (2009).
[Crossref] [PubMed]

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]

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]

2008 (2)

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]

A. Luis, “Modulation of coherence of vectorial electromagnetic waves in the Young interferometer,” Opt. Lett. 33(13), 1497–1499 (2008).
[Crossref] [PubMed]

2007 (2)

C. H. Gan, G. Gbur, and T. D. Visser, “Surface plasmons modulate the spatial coherence of light in Young’s interference experiment,” Phys. Rev. Lett. 98(4), 043908 (2007).
[Crossref] [PubMed]

Y. Cai and U. Peschel, “Second-harmonic generation by an astigmatic partially coherent beam,” Opt. Express 15(23), 15480–15492 (2007).
[Crossref] [PubMed]

2006 (5)

2005 (3)

H. Roychowdhury and E. Wolf, “Young’s interference experiment with light of any state coherence and of polarization,” Opt. Commun. 252(4–6), 268–274 (2005).
[Crossref]

G. S. Agarwal, A. Dogariu, T. D. Visser, and E. Wolf, “Generation of complete coherence in Young’s interference experiment with random mutually uncorrelated electromagnetic beams,” Opt. Lett. 30(2), 120–122 (2005).
[Crossref] [PubMed]

Y. Cai and S. Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(5 Pt 2), 056607 (2005).
[Crossref] [PubMed]

2004 (3)

2003 (5)

2002 (2)

2001 (2)

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]

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]

2000 (2)

G. P. Agrawal and E. Wolf, “Propagation-induced polarization changes in partially coherent optical beams,” J. Opt. Soc. Am. A 17(11), 2019–2023 (2000).
[Crossref] [PubMed]

H. C. Kandpal and J. S. Vaishya, “Experimental study of coherence properties of light fields in the region of superposition in Young’s interference experiment,” Opt. Commun. 186(1–3), 15–20 (2000).
[Crossref]

1999 (2)

S. A. Ponomarenko and E. Wolf, “Coherence properties of light in Young’s interference pattern formed with partially coherent light,” Opt. Commun. 170(1–3), 1–8 (1999).
[Crossref]

D. Ding and X. Liu, “Approximate description of Bessel, Bessel–Gauss, and Gaussian beams with finite aperture,” J. Opt. Soc. Am. A 16(6), 1286–1293 (1999).
[Crossref]

1998 (1)

1994 (1)

1991 (1)

D. F. V. James and E. Wolf, “Spectral changes produced in Young’s interference experiment,” Opt. Commun. 81(3–4), 150–154 (1991).
[Crossref]

1988 (1)

J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83(5), 1752–1756 (1988).
[Crossref]

1987 (1)

M. S. Zubairy and J. K. McIver, “Second-harmonic generation by a partially coherent beam,” Phys. Rev. A 36(1), 202–206 (1987).
[Crossref] [PubMed]

1984 (1)

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53(11), 1057–1060 (1984).
[Crossref]

1983 (1)

1802 (1)

T. Young, “The bakerian lecture: on the theory of light and colours,” Philos. Trans. R. Soc. Lond. 92(0), 12–48 (1802).
[Crossref]

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]

Agarwal, G. S.

Agrawal, G. P.

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]

Arinaga, S.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53(11), 1057–1060 (1984).
[Crossref]

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.

F. Gori, M. Santarsiero, R. Borghi, and E. Wolf, “Effects of coherence on the degree of polarization in a young interference pattern,” Opt. Lett. 31(6), 688–690 (2006).
[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]

Breazeale, M. A.

J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83(5), 1752–1756 (1988).
[Crossref]

Brown, T. G.

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.

X. Liu, F. Wang, C. Wei, and Y. Cai, “Experimental study of turbulence-induced beam wander and deformation of a partially coherent beam,” Opt. Lett. 39(11), 3336–3339 (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. Cai, Y. Chen, and F. Wang, “Generation and propagation of partially coherent beams with non-conventional correlation functions: a review,” J. Opt. Soc. Am. A 31(9), 2083–2096 (2014).
[Crossref]

X. Liu, Y. Shen, L. Liu, F. Wang, and Y. Cai, “Experimental demonstration of vortex phase-induced reduction in scintillation of a partially coherent beam,” Opt. Lett. 38(24), 5323–5326 (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]

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]

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).
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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).
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J. Zhang, Z. Wang, B. Cheng, Q. Wang, B. Wu, X. Shen, L. Zheng, Y. Xu, and Q. Lin, “Atom cooling by partially spatially coherent lasers,” Phys. Rev. A 88(2), 023416 (2013).
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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).
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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).
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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).
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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).
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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).
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R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
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Feng, F.

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T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104(17), 173902 (2010).
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T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93(6), 068103 (2004).
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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).
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C. Ding, Y. Cai, O. Korotkova, Y. Zhang, and L. Pan, “Scattering-induced changes in the temporal coherence length and the pulse duration of a partially coherent plane-wave pulse,” Opt. Lett. 36(4), 517–519 (2011).
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K. P. Singh and M. Kumar, “Electron acceleration by a radially polarized laser pulse during ionization of low density gases,” Phys. Rev. ST Accel. Beams 14(3), 030401 (2011).
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Y. Li, H. Lee, and E. Wolf, “Spectra, coherence and polarization in Young’s interference pattern formed by stochastic electromagnetic beams,” Opt. Commun. 265(1), 63–72 (2006).
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R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
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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).
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J. Qi, W. Wang, X. Li, X. Wang, W. Sun, J. Liao, and Y. Nie, “Double-slit interference of radially polarized vortex beams,” Opt. Eng. 53(4), 044107 (2014).
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J. Zhang, Z. Wang, B. Cheng, Q. Wang, B. Wu, X. Shen, L. Zheng, Y. Xu, and Q. Lin, “Atom cooling by partially spatially coherent lasers,” Phys. Rev. A 88(2), 023416 (2013).
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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).
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T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93(6), 068103 (2004).
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Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53(11), 1057–1060 (1984).
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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).
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J. Qi, W. Wang, X. Li, X. Wang, W. Sun, J. Liao, and Y. Nie, “Double-slit interference of radially polarized vortex beams,” Opt. Eng. 53(4), 044107 (2014).
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Peschel, U.

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J. Qi, W. Wang, X. Li, X. Wang, W. Sun, J. Liao, and Y. Nie, “Double-slit interference of radially polarized vortex beams,” Opt. Eng. 53(4), 044107 (2014).
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R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
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H. Roychowdhury and E. Wolf, “Young’s interference experiment with light of any state coherence and of polarization,” Opt. Commun. 252(4–6), 268–274 (2005).
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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).
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Santarsiero, M.

F. Gori, M. Santarsiero, R. Borghi, and E. Wolf, “Effects of coherence on the degree of polarization in a young interference pattern,” Opt. Lett. 31(6), 688–690 (2006).
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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).
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Setälä, T.

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J. Zhang, Z. Wang, B. Cheng, Q. Wang, B. Wu, X. Shen, L. Zheng, Y. Xu, and Q. Lin, “Atom cooling by partially spatially coherent lasers,” Phys. Rev. A 88(2), 023416 (2013).
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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).
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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).
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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).
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T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93(6), 068103 (2004).
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J. Qi, W. Wang, X. Li, X. Wang, W. Sun, J. Liao, and Y. Nie, “Double-slit interference of radially polarized vortex beams,” Opt. Eng. 53(4), 044107 (2014).
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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).
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Tu, C.

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H. C. Kandpal and J. S. Vaishya, “Experimental study of coherence properties of light fields in the region of superposition in Young’s interference experiment,” Opt. Commun. 186(1–3), 15–20 (2000).
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T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104(17), 173902 (2010).
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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).
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[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).
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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).
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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).
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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).
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Wang, Q.

J. Zhang, Z. Wang, B. Cheng, Q. Wang, B. Wu, X. Shen, L. Zheng, Y. Xu, and Q. Lin, “Atom cooling by partially spatially coherent lasers,” Phys. Rev. A 88(2), 023416 (2013).
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J. Qi, W. Wang, X. Li, X. Wang, W. Sun, J. Liao, and Y. Nie, “Double-slit interference of radially polarized vortex beams,” Opt. Eng. 53(4), 044107 (2014).
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J. Zhang, Z. Wang, B. Cheng, Q. Wang, B. Wu, X. Shen, L. Zheng, Y. Xu, and Q. Lin, “Atom cooling by partially spatially coherent lasers,” Phys. Rev. A 88(2), 023416 (2013).
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T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104(17), 173902 (2010).
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J. Zhang, Z. Wang, B. Cheng, Q. Wang, B. Wu, X. Shen, L. Zheng, Y. Xu, and Q. Lin, “Atom cooling by partially spatially coherent lasers,” Phys. Rev. A 88(2), 023416 (2013).
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Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53(11), 1057–1060 (1984).
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Figures (10)

Fig. 1
Fig. 1

Illustration of the notation relating to Young’s two-slit interference experiment with a partially coherent beam.

Fig. 2
Fig. 2

Normalized intensity distributions (contour graphs) I ( u ) / I max ( u ) , I x ( u ) / I max ( u ) , I y ( u ) / I max ( u ) of the Young’s interference pattern formed with a partially coherent RP beam at z = 700mm for two different values of the transverse spatial coherence width δ 0 . (a)-(a2) δ 0 = 0.99 mm ; (b)-(b2) δ 0 = 0.26 mm .

Fig. 3
Fig. 3

Square of the degree of coherence μ 2 ( u , 0 ) (contour graph) and the corresponding cross line (v = 0) of the Young’s interference pattern formed with a partially coherent RP beam at z = 700mm for two different values of the transverse spatial coherence width δ 0 .

Fig. 4
Fig. 4

Degree of polarization (contour graph) and the corresponding cross line (v = 0) of the Young’s interference pattern formed with a partially coherent RP beam at z = 700mm for two different values of the transverse spatial coherence width δ 0 .

Fig. 5
Fig. 5

Experimental setup for generating a partially coherent RP beam and measuring the intensity and the degree of coherence in Young’s two-slit interference pattern formed with the generated beam. LP, linear polarizer; M, mirror; L1, L2, thin lenses; RGGD, rotating ground-glass disk; GAF, Gaussian amplitude filter; RPC, radial polarization converter; T-S, two-slit; BS, 50:50 nonpolarization beam splitter; BPA, beam profile analyzer; CCD, charge-coupled devices; PC, personal computers.

Fig. 6
Fig. 6

Experimental results of the intensity distribution and the corresponding cross line (y = 0, dotted curve) of the generated partially coherent RP beam in the output plane of RPC. The solid curve denotes the theoretical fit of the experimental data.

Fig. 7
Fig. 7

Experimental results of the square of the degree of coherence μ 2 ( r 1 , r 2 = 0 ) and the corresponding cross line (y1 = 0, dotted curve) of the generated partially coherent RP beam in the output plane of RPC for two different values of the spatial coherence width. The solid curve denotes the theoretical fit of the experimental data.

Fig. 8
Fig. 8

Experimental results of the intensity distributions I ( u ) , I x ( u ) , I y ( u ) of the Young’s two-slit interference pattern formed with the generated partially coherent RP beam at z = 700mm for two different values of the transverse spatial coherence width δ 0 . (a)-(a2) δ 0 = 0.99 mm ; (b)-(b2) δ 0 = 0.26 mm .

Fig. 9
Fig. 9

Experimental results of the square of the degree of coherence μ 2 ( u , 0 ) of the Young’s interference pattern formed with a partially coherent RP beam at z = 700mm for two different values of the transverse spatial coherence width δ 0 . (a) δ 0 = 0.99 mm ; (b) δ 0 = 0.26 mm .

Fig. 10
Fig. 10

Experimental results of the degree of polarization (cross line v = 0) of the Young’s interference pattern formed with the generated partially coherent RP beam at z = 700mm for two different values of the transverse spatial coherence width δ 0 . The solid lines denote theoretical results calculated by Eq. (17) with the beam parameters measured in experiment..

Equations (18)

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W ( r 1 , r 2 ) = ( W x x ( r 1 , r 2 ) W x y ( r 1 , r 2 ) W y x ( r 1 , r 2 ) W y y ( r 1 , r 2 ) ) ,
W α β ( r 1 , r 2 ) = E α * ( r 1 ) E β ( r 2 ) , ( α = x , y ; β = x , y ) ,
W α β ( r 1 , r 2 ) = α 1 β 2 σ 0 2 exp [ r 1 2 + r 2 2 σ 0 2 ( r 1 r 2 ) 2 2 δ 0 2 ] , ( α , β = x , y ) ,
H ( x , y ) = m = 1 M A m { exp [ B m ( b / 2 ) 2 ( x a b 2 ) 2 ] + exp [ B m ( b / 2 ) 2 ( x + a + b 2 ) 2 ] } ,
W 1 α β ( r 1 , r 2 ) = W α β ( r 1 , r 2 ) H ( x 1 , y 1 ) H * ( x 2 , y 2 ) = α 1 β 2 σ 0 2 exp ( r 1 2 + r 2 2 σ 0 2 ) exp ( ( r 1 r 2 ) 2 2 δ 0 2 ) × m = 1 M i = 1 M A m A i * { exp [ B m ( b / 2 ) 2 ( x 1 a b 2 ) 2 ] + exp [ B m ( b / 2 ) 2 ( x 1 + a + b 2 ) 2 ] } × { exp [ B i * ( b / 2 ) 2 ( x 2 a b 2 ) 2 ] + exp [ B i * ( b / 2 ) 2 ( x 2 + a + b 2 ) 2 ] } .
W α β ( u 1 , u 2 ) = 1 λ 2 z 2 W 1 α β ( r 1 , r 2 ) × exp [ i k 2 z ( r 1 2 r 2 2 ) + i k z ( r 1 u 1 r 2 u 2 ) i k 2 z ( u 1 2 u 2 2 ) ] d 2 r 1 d 2 r 2 ,
W x x ( u 1 , u 2 ) = k 2 16 σ 0 2 z 2 M 1 1 / 2 Π 0 1 / 2 m = 1 M i = 1 M A m A i * M 1 B 3 / 2 Ω B 3 / 2 exp [ k 2 4 z 2 Π 0 ( δ 0 2 v 1 2 M 1 v 2 ) 2 ] × { ( δ 0 2 + N 1 m Δ 11 + δ 0 2 Δ 11 2 2 Ω B ) exp [ Δ 11 2 4 Ω B + N 1 m 2 4 M 1 B ] + ( δ 0 2 + N 2 m Π 21 + δ 0 2 Π 21 2 2 Ω B ) × exp [ Π 21 2 4 Ω B + N 2 m 2 4 M 1 B ] + ( δ 0 2 + N 1 m Δ 12 + δ 0 2 Δ 12 2 2 Ω B ) exp [ Δ 12 2 4 Ω B + N 1 m 2 4 M 1 B ] + ( δ 0 2 + N 2 m Π 22 + δ 0 2 Π 22 2 2 Ω B ) × exp [ Π 22 2 4 Ω B + N 2 m 2 4 M 1 B ] } exp [ i k 2 z ( u 1 2 u 2 2 ) ] × exp [ k 2 v 1 2 4 z 2 M 1 O 2 ( b / 2 ) 2 ( B m + B i * ) ] ,
W y y ( u 1 , u 2 ) = k 2 16 σ 0 2 z 2 M 1 3 / 2 Π 0 3 / 2 m = 1 M i = 1 M A m A i * M 1 B 1 / 2 Ω B 1 / 2 exp [ k 2 4 z 2 Π 0 ( δ 0 2 v 1 2 M 1 v 2 ) 2 ] × [ δ 0 2 k 2 z 2 ( δ 0 2 v 1 2 M 1 v 2 ) v 1 k 2 δ 0 2 2 z 2 Π 0 ( δ 0 2 v 1 2 M 1 v 2 ) 2 ] exp [ i k 2 z ( u 1 2 u 2 2 ) ] × { exp [ Δ 11 2 4 Ω B + N 1 m 2 4 M 1 B ] + exp [ Π 21 2 4 Ω B + N 2 m 2 4 M 1 B ] + exp [ Δ 12 2 4 Ω B + N 1 m 2 4 M 1 B ] + exp [ Π 22 2 4 Ω B + N 2 m 2 4 M 1 B ] } exp [ k 2 v 1 2 4 z 2 M 1 O 2 ( b / 2 ) 2 ( B m + B i * ) ] ,
W x y ( u 1 , u 2 ) = i k 3 32 σ 0 2 z 3 M 1 1 / 2 M 1 B 3 / 2 Π 0 3 / 2 Ω B 3 / 2 ( δ 0 2 v 1 2 M 1 v 2 ) m = 1 M i = 1 M A m A i * exp [ i k 2 z ( u 1 2 u 2 2 ) ] × { ( 2 N 1 m Ω B + Δ 11 δ 0 2 ) exp [ Δ 11 2 4 Ω B + N 1 m 2 4 M 1 B ] + ( 2 N 2 m Ω B Π 21 δ 0 2 ) exp [ Π 21 2 4 Ω B + N 2 m 2 4 M 1 B ] + ( 2 N 1 m Ω B + Δ 12 δ 0 2 ) exp [ Δ 12 2 4 Ω B + N 1 m 2 4 M 1 B ] + ( 2 N 2 m Ω B Π 22 δ 0 2 ) exp [ Π 22 2 4 Ω B + N 2 m 2 4 M 1 B ] } × exp [ k 2 4 z 2 Π 0 ( δ 0 2 v 1 2 M 1 v 2 ) 2 k 2 4 M 1 z 2 v 1 2 O 2 ( b / 2 ) 2 ( B m + B i * ) ] ,
W y x ( u 1 , u 2 ) = W x y * ( u 2 , u 1 ) ,
O = a + b 2 , M 1 = 1 σ 0 2 + 1 2 δ 0 2 + i k 2 z , M 2 = 1 σ 0 2 + 1 2 δ 0 2 i k 2 z , M 1 B = B m ( b / 2 ) 2 + M 1 , N 1 m = 2 B m O ( b / 2 ) 2 + i k u 1 z , N 2 m = 2 B m O ( b / 2 ) 2 i k u 1 z , Ω B = M 2 + B i * ( b / 2 ) 2 δ 0 4 4 M 1 B , Π 0 = M 2 δ 0 4 4 M 1 , Δ 11 = 2 B i * O ( b / 2 ) 2 i k u 2 z + δ 0 2 N 1 m 2 M 1 B , Δ 12 = 2 B i * O ( b / 2 ) 2 i k u 2 z + δ 0 2 N 1 m 2 M 1 B , Π 21 = 2 B i * O ( b / 2 ) 2 + i k u 2 z + δ 0 2 N 2 m 2 M 1 B , Π 22 = 2 B i * O ( b / 2 ) 2 + i k u 2 z + δ 0 2 N 2 m 2 M 1 B .
I ( u ) = T r W ( u , u ) = W x x ( u , u ) + W y y ( u , u ) = I x ( u ) + I y ( u ) .
μ 2 ( u 1 , u 2 ) = T r [ W ( u 1 , u 2 ) W ( u 1 , u 2 ) ] T r [ W ( u 1 , u 1 ) ] T r [ W ( u 2 , u 2 ) ] ,
T r [ W ( u 1 , u 2 ) W ( u 1 , u 2 ) ] = α , β | W α β ( u 1 , u 2 ) | 2 , ( α , β = x , y ) .
F ( 4 ) ( u 1 , u 2 ) = T r [ W ( u 1 , u 1 ) ] T r [ W ( u 2 , u 2 ) ] + α , β | W α β ( u 1 , u 2 ) | 2 .
f ( 4 ) ( u 1 , u 2 ) = 1 + μ 2 ( u 1 , u 2 ) .
P ( u ) = 1 4 D e t [ W ( u ) ] T r 2 [ W ( u ) ] .
W ( r 1 , r 2 ) = C 0 exp [ ( r 1 2 + r 2 2 ) / σ 2 ( r 1 r 2 ) 2 / 2 δ 2 ] .

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