Abstract

In a recent publication [Appl. Phys. Lett, 100, 051108 (2012)], a radially polarized (RP) beam with variable spatial coherence (i.e., partially coherent RP beam) was generated experimentally. In this paper, we derive the realizability conditions for a partially coherent RP beam, and we carry out theoretical and experimental study of the coherence and polarization properties of a partially coherent RP beam. It is found that after passing through a thin lens, both the degree of coherence and the degree of polarization of a partially coherent RP beam varies on propagation, while the state of polarization of the completely polarized part of such beam remains invariant. The variations of the degree of coherence and the degree of polarization depend closely on the initial spatial coherence. Our experimental results agree well with the theoretical predictions.

© 2012 OSA

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    [CrossRef]
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    [CrossRef]
  58. 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]
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    [CrossRef]

2012 (3)

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]

2011 (6)

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]

G. Wu and Y. Cai, “Modulation of spectral intensity, polarization and coherence of a stochastic electromagnetic beam,” Opt. Express 19(9), 8700–8714 (2011).
[CrossRef] [PubMed]

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]

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]

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).
[CrossRef]

L. Zhang, F. Wang, Y. Cai, and O. Korotkova, “Degree of paraxiality of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Commun. 284(5), 1111–1117 (2011).
[CrossRef]

2010 (7)

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

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

Z. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A 82(3), 033836 (2010).
[CrossRef]

M. Yao, Y. Cai, O. Korotkova, Q. Lin, and Z. Wang, “Spatio-temporal coupling of random electromagnetic pulses interacting with reflecting gratings,” Opt. Express 18(21), 22503–22514 (2010).
[CrossRef] [PubMed]

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

Y. Luo and B. Lu, “Spectral stokes singularities of partially coherent radially polarized beams focused by a high numerical aperture objective,” J. Opt. 12(11), 115703 (2010).
[CrossRef]

H. Wang, D. Liu, and Z. Zhou, “The propagation of radially polarized partially coherent beam through an optical system in turbulent atmosphere,” Appl. Phys. B 101(1-2), 361–369 (2010).
[CrossRef]

2009 (9)

H. Lin and J. Pu, “Propagation properties of partially coherent radially polarized beam in a turbulent atmosphere,” J. Mod. Opt. 56(11), 1296–1303 (2009).
[CrossRef]

C. Zhao, Y. Cai, and O. Korotkova, “Radiation force of scalar and electromagnetic twisted Gaussian Schell-model beams,” Opt. Express 17(24), 21472–21487 (2009).
[CrossRef] [PubMed]

M. Salem and G. P. Agrawal, “Coupling of stochastic electromagnetic beams into optical fibers,” Opt. Lett. 34(18), 2829–2831 (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]

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

M. Santarsiero, R. Borghi, and V. Ramirez-Sanchez, “Synthesis of electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 26(6), 1437–1443 (2009).
[CrossRef]

B. Kanseri, S. Rath, and H. C. Kandpal, “Determination of the beam coherence-polarization matrix of a random electromagnetic beam,” IEEE J. Quantum Electron. 45(9), 1163–1167 (2009).
[CrossRef]

W. Cheng, J. W. Haus, and Q. Zhan, “Propagation of vector vortex beams through a turbulent atmosphere,” Opt. Express 17(20), 17829–17836 (2009).
[CrossRef] [PubMed]

2008 (7)

2007 (4)

F. Wang and Y. Cai, “Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics,” J. Opt. Soc. Am. A 24(7), 1937–1944 (2007).
[CrossRef] [PubMed]

M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys. B 86, 329–334 (2007).

D. Deng and Q. Guo, “Analytical vectorial structure of radially polarized light beams,” Opt. Lett. 32(18), 2711–2713 (2007).
[CrossRef] [PubMed]

T. Shirai and E. Wolf, “Correlation between intensity fluctuations in stochastic electromagnetic beams of any state of coherence and polarization,” Opt. Commun. 272(2), 289–292 (2007).
[CrossRef]

2006 (2)

2005 (4)

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248(4-6), 333–337 (2005).
[CrossRef]

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 7(5), 232–237 (2005).
[CrossRef]

O. Korotkova and E. Wolf, “Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett. 30(2), 198–200 (2005).
[CrossRef] [PubMed]

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

2004 (3)

2003 (3)

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. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11(10), 1137–1143 (2003).
[CrossRef] [PubMed]

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

2002 (3)

2001 (3)

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

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]

2000 (1)

1998 (2)

1979 (1)

P. De Santis, F. Gori, G. Guattari, and C. Palma, “an example of a collectt-wolf source,” Opt. Commun. 29(3), 256–260 (1979).
[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]

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]

Baykal, Y.

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]

Biss, D. P.

Borghi, R.

M. Santarsiero, R. Borghi, and V. Ramirez-Sanchez, “Synthesis of electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 26(6), 1437–1443 (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]

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1-3), 9–16 (2002).
[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]

Brown, T. G.

Cai, Y.

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 and Y. Cai, “Modulation of spectral intensity, polarization and coherence of a stochastic electromagnetic beam,” Opt. Express 19(9), 8700–8714 (2011).
[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]

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]

L. Zhang, F. Wang, Y. Cai, and O. Korotkova, “Degree of paraxiality of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Commun. 284(5), 1111–1117 (2011).
[CrossRef]

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

M. Yao, Y. Cai, O. Korotkova, Q. Lin, and Z. Wang, “Spatio-temporal coupling of random electromagnetic pulses interacting with reflecting gratings,” Opt. Express 18(21), 22503–22514 (2010).
[CrossRef] [PubMed]

C. Zhao, Y. Cai, and O. Korotkova, “Radiation force of scalar and electromagnetic twisted Gaussian Schell-model beams,” Opt. Express 17(24), 21472–21487 (2009).
[CrossRef] [PubMed]

M. Yao, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “Evolution of the degree of polarization of an electromagnetic Gaussian Schell-model beam in a Gaussian cavity,” Opt. Lett. 33(19), 2266–2268 (2008).
[CrossRef] [PubMed]

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

Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Opt. Express 16(11), 7665–7673 (2008).
[CrossRef] [PubMed]

F. Wang and Y. Cai, “Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics,” J. Opt. Soc. Am. A 24(7), 1937–1944 (2007).
[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]

C. Zhao, Y. Dong, G. Wu, F. Wang, Y. Cai, and O. Korotkova, “Experimental demonstration of coupling of an electromagnetic Gaussian Schell-model beam into a single-mode optical fiber,” Appl. Phys. B, doi:.
[CrossRef]

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.

Cheng, W.

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]

De Santis, P.

P. De Santis, F. Gori, G. Guattari, and C. Palma, “an example of a collectt-wolf source,” Opt. Commun. 29(3), 256–260 (1979).
[CrossRef]

Deng, D.

Ding, B.

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

Dogariu, A.

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248(4-6), 333–337 (2005).
[CrossRef]

Dong, Y.

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]

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]

C. Zhao, Y. Dong, G. Wu, F. Wang, Y. Cai, and O. Korotkova, “Experimental demonstration of coupling of an electromagnetic Gaussian Schell-model beam into a single-mode optical fiber,” Appl. Phys. B, doi:.
[CrossRef]

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]

Ellis, J.

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248(4-6), 333–337 (2005).
[CrossRef]

Eyyuboglu, H. T.

Feng, F.

Feurer, T.

M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys. B 86, 329–334 (2007).

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.

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]

Gori, F.

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]

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1-3), 9–16 (2002).
[CrossRef]

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

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

P. De Santis, F. Gori, G. Guattari, and C. Palma, “an example of a collectt-wolf source,” Opt. Commun. 29(3), 256–260 (1979).
[CrossRef]

Guattari, G.

P. De Santis, F. Gori, G. Guattari, and C. Palma, “an example of a collectt-wolf source,” Opt. Commun. 29(3), 256–260 (1979).
[CrossRef]

Guo, Q.

Haus, J. W.

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]

Kandpal, H. C.

B. Kanseri, S. Rath, and H. C. Kandpal, “Determination of the beam coherence-polarization matrix of a random electromagnetic beam,” IEEE J. Quantum Electron. 45(9), 1163–1167 (2009).
[CrossRef]

Kanseri, B.

B. Kanseri, S. Rath, and H. C. Kandpal, “Determination of the beam coherence-polarization matrix of a random electromagnetic beam,” IEEE J. Quantum Electron. 45(9), 1163–1167 (2009).
[CrossRef]

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.

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]

L. Zhang, F. Wang, Y. Cai, and O. Korotkova, “Degree of paraxiality of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Commun. 284(5), 1111–1117 (2011).
[CrossRef]

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

Z. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A 82(3), 033836 (2010).
[CrossRef]

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

M. Yao, Y. Cai, O. Korotkova, Q. Lin, and Z. Wang, “Spatio-temporal coupling of random electromagnetic pulses interacting with reflecting gratings,” Opt. Express 18(21), 22503–22514 (2010).
[CrossRef] [PubMed]

C. Zhao, Y. Cai, and O. Korotkova, “Radiation force of scalar and electromagnetic twisted Gaussian Schell-model beams,” Opt. Express 17(24), 21472–21487 (2009).
[CrossRef] [PubMed]

M. Yao, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “Evolution of the degree of polarization of an electromagnetic Gaussian Schell-model beam in a Gaussian cavity,” Opt. Lett. 33(19), 2266–2268 (2008).
[CrossRef] [PubMed]

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

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

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 7(5), 232–237 (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 and E. Wolf, “Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett. 30(2), 198–200 (2005).
[CrossRef] [PubMed]

O. Korotkova and E. Wolf, “Spectral degree of coherence of a random three-dimensional electromagnetic field,” J. Opt. Soc. Am. A 21(12), 2382–2385 (2004).
[CrossRef] [PubMed]

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]

C. Zhao, Y. Dong, G. Wu, F. Wang, Y. Cai, and O. Korotkova, “Experimental demonstration of coupling of an electromagnetic Gaussian Schell-model beam into a single-mode optical fiber,” Appl. Phys. B, doi:.
[CrossRef]

Kumar, M.

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).
[CrossRef]

Leger, J. R.

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]

Lin, H.

H. Lin and J. Pu, “Propagation properties of partially coherent radially polarized beam in a turbulent atmosphere,” J. Mod. Opt. 56(11), 1296–1303 (2009).
[CrossRef]

Lin, Q.

Liu, D.

H. Wang, D. Liu, and Z. Zhou, “The propagation of radially polarized partially coherent beam through an optical system in turbulent atmosphere,” Appl. Phys. B 101(1-2), 361–369 (2010).
[CrossRef]

Liu, X.

Lu, B.

Y. Luo and B. Lu, “Spectral stokes singularities of partially coherent radially polarized beams focused by a high numerical aperture objective,” J. Opt. 12(11), 115703 (2010).
[CrossRef]

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]

Luo, Y.

Y. Luo and B. Lu, “Spectral stokes singularities of partially coherent radially polarized beams focused by a high numerical aperture objective,” J. Opt. 12(11), 115703 (2010).
[CrossRef]

Meier, M.

M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys. B 86, 329–334 (2007).

Mondello, A.

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1-3), 9–16 (2002).
[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]

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]

Palma, C.

P. De Santis, F. Gori, G. Guattari, and C. Palma, “an example of a collectt-wolf source,” Opt. Commun. 29(3), 256–260 (1979).
[CrossRef]

Piquero, G.

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1-3), 9–16 (2002).
[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]

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.

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248(4-6), 333–337 (2005).
[CrossRef]

Pu, J.

H. Lin and J. Pu, “Propagation properties of partially coherent radially polarized beam in a turbulent atmosphere,” J. Mod. Opt. 56(11), 1296–1303 (2009).
[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]

Ramirez-Sanchez, V.

Ramírez-Sánchez, V.

Rath, S.

B. Kanseri, S. Rath, and H. C. Kandpal, “Determination of the beam coherence-polarization matrix of a random electromagnetic beam,” IEEE J. Quantum Electron. 45(9), 1163–1167 (2009).
[CrossRef]

Romanini, P.

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1-3), 9–16 (2002).
[CrossRef]

Romano, V.

M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys. B 86, 329–334 (2007).

Sahin, S.

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

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]

Salem, M.

Santarsiero, M.

M. Santarsiero, R. Borghi, and V. Ramirez-Sanchez, “Synthesis of electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 26(6), 1437–1443 (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]

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1-3), 9–16 (2002).
[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]

Setälä, T.

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]

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.

T. Shirai and E. Wolf, “Correlation between intensity fluctuations in stochastic electromagnetic beams of any state of coherence and polarization,” Opt. Commun. 272(2), 289–292 (2007).
[CrossRef]

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 7(5), 232–237 (2005).
[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, K. P.

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).
[CrossRef]

Suyama, T.

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

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.

Tong, Z.

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

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

Z. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A 82(3), 033836 (2010).
[CrossRef]

Tovar, A. A.

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]

Wang, F.

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]

L. Zhang, F. Wang, Y. Cai, and O. Korotkova, “Degree of paraxiality of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Commun. 284(5), 1111–1117 (2011).
[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]

F. Wang and Y. Cai, “Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics,” J. Opt. Soc. Am. A 24(7), 1937–1944 (2007).
[CrossRef] [PubMed]

C. Zhao, Y. Dong, G. Wu, F. Wang, Y. Cai, and O. Korotkova, “Experimental demonstration of coupling of an electromagnetic Gaussian Schell-model beam into a single-mode optical fiber,” Appl. Phys. B, doi:.
[CrossRef]

Wang, H.

H. Wang, D. Liu, and Z. Zhou, “The propagation of radially polarized partially coherent beam through an optical system in turbulent atmosphere,” Appl. Phys. B 101(1-2), 361–369 (2010).
[CrossRef]

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]

Wang, Z.

Wolf, E.

M. Salem and E. Wolf, “Coherence-induced polarization changes in light beams,” Opt. Lett. 33(11), 1180–1182 (2008).
[CrossRef] [PubMed]

T. Shirai and E. Wolf, “Correlation between intensity fluctuations in stochastic electromagnetic beams of any state of coherence and polarization,” Opt. Commun. 272(2), 289–292 (2007).
[CrossRef]

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248(4-6), 333–337 (2005).
[CrossRef]

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 7(5), 232–237 (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 and E. Wolf, “Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett. 30(2), 198–200 (2005).
[CrossRef] [PubMed]

O. Korotkova and E. Wolf, “Spectral degree of coherence of a random three-dimensional electromagnetic field,” J. Opt. Soc. Am. A 21(12), 2382–2385 (2004).
[CrossRef] [PubMed]

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]

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.

Yao, M.

Youngworth, K. S.

Zhan, Q.

Zhang, L.

L. Zhang, F. Wang, Y. Cai, and O. Korotkova, “Degree of paraxiality of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Commun. 284(5), 1111–1117 (2011).
[CrossRef]

Zhang, Y.

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

Zhao, C.

Zhou, Z.

H. Wang, D. Liu, and Z. Zhou, “The propagation of radially polarized partially coherent beam through an optical system in turbulent atmosphere,” Appl. Phys. B 101(1-2), 361–369 (2010).
[CrossRef]

Zhu, S.

Adv. Opt. Photon. (1)

Appl. Opt. (1)

Appl. Phys. B (3)

M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys. B 86, 329–334 (2007).

C. Zhao, Y. Dong, G. Wu, F. Wang, Y. Cai, and O. Korotkova, “Experimental demonstration of coupling of an electromagnetic Gaussian Schell-model beam into a single-mode optical fiber,” Appl. Phys. B, doi:.
[CrossRef]

H. Wang, D. Liu, and Z. Zhou, “The propagation of radially polarized partially coherent beam through an optical system in turbulent atmosphere,” Appl. Phys. B 101(1-2), 361–369 (2010).
[CrossRef]

Appl. Phys. Lett. (1)

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]

IEEE J. Quantum Electron. (1)

B. Kanseri, S. Rath, and H. C. Kandpal, “Determination of the beam coherence-polarization matrix of a random electromagnetic beam,” IEEE J. Quantum Electron. 45(9), 1163–1167 (2009).
[CrossRef]

J. Mod. Opt. (1)

H. Lin and J. Pu, “Propagation properties of partially coherent radially polarized beam in a turbulent atmosphere,” J. Mod. Opt. 56(11), 1296–1303 (2009).
[CrossRef]

J. Opt. (1)

Y. Luo and B. Lu, “Spectral stokes singularities of partially coherent radially polarized beams focused by a high numerical aperture objective,” J. Opt. 12(11), 115703 (2010).
[CrossRef]

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

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 7(5), 232–237 (2005).
[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 (5)

J. Opt. Soc. Am. B (1)

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. 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. (9)

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]

T. Shirai and E. Wolf, “Correlation between intensity fluctuations in stochastic electromagnetic beams of any state of coherence and polarization,” Opt. Commun. 272(2), 289–292 (2007).
[CrossRef]

L. Zhang, F. Wang, Y. Cai, and O. Korotkova, “Degree of paraxiality of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Commun. 284(5), 1111–1117 (2011).
[CrossRef]

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248(4-6), 333–337 (2005).
[CrossRef]

P. De Santis, F. Gori, G. Guattari, and C. Palma, “an example of a collectt-wolf source,” Opt. Commun. 29(3), 256–260 (1979).
[CrossRef]

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1-3), 9–16 (2002).
[CrossRef]

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

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

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

Opt. Express (13)

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

C. Zhao, Y. Cai, and O. Korotkova, “Radiation force of scalar and electromagnetic twisted Gaussian Schell-model beams,” Opt. Express 17(24), 21472–21487 (2009).
[CrossRef] [PubMed]

M. Yao, Y. Cai, O. Korotkova, Q. Lin, and Z. Wang, “Spatio-temporal coupling of random electromagnetic pulses interacting with reflecting gratings,” Opt. Express 18(21), 22503–22514 (2010).
[CrossRef] [PubMed]

G. Wu and Y. Cai, “Modulation of spectral intensity, polarization and coherence of a stochastic electromagnetic beam,” Opt. Express 19(9), 8700–8714 (2011).
[CrossRef] [PubMed]

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Figures (12)

Fig. 1
Fig. 1

Degree of polarization of a focused partially coherent RP beam at point ( 0.05mm 0.05mm ) versus the propagation distance z for different values of the correlation radii δ xx , δ yy , δ xy .

Fig. 2
Fig. 2

Intensity distributions of a focused partially coherent RP beam, its completely polarized part and its completely unpolarized part with δ xx = δ yy = δ xy =1mm.

Fig. 3
Fig. 3

Intensity distributions of a focused partially coherent RP beam, its completely polarized part and its completely unpolarized part with . δ xx = δ yy = δ xy =0.20mm

Fig. 4
Fig. 4

Variation of the normalized powers of the completely polarized part and the completely unpolarized part of a focused partially coherent RP beam (a) versus the propagation distance with δ xx = δ yy = δ xy =0.20mm , (b) versus the correlation coefficient δ xx with δ xx = δ yy = δ xy at the focal plane z = 400mm.

Fig. 5
Fig. 5

Normalized intensity (cross line, v = 0) of a focused partially coherent RP beam under the condition of η ( u ) = η ( p ) .

Fig. 6
Fig. 6

Square of the degree of coherence of a focused partially coherent RP beam at several propagation distances with δ xx = δ yy = δ xy =0.20mm .

Fig. 7
Fig. 7

Square of the degree of coherence of a focused partially coherent RP beam at the focal plane z = 400mm for different values of the correlation radii δ xx , δ yy , δ xy .

Fig. 8
Fig. 8

Experimental setup for generating a partially coherent RP beam and measuring its coherence properties. L1, L2, L3, thin lenses; RM, reflecting mirror; RGGP, rotating ground-glass plate; GAF, Gaussian amplitude filter; RPC, radial polarization converter; beam splitter; D1, D2, single photon detectors.

Fig. 9
Fig. 9

Experimental results (dotted curves) of the modulus of the square of the degree of coherence of the generated GSM beam and the corresponding Gaussian fit (solid curves) for two different focused beamspot sizes on the RGGP.

Fig. 10
Fig. 10

Experimental results (dotted curves) of the modulus of the square of the degree of coherence of a focused partially coherent RP beam at the focal plane for two different values of the coherence width δ. The solid curves are calculated by theoretical formulas.

Fig. 11
Fig. 11

Experimental setup for measuring the degree of polarization of a focused partially coherent RP beam at the focal plane. LP, linear polarizer; QWP, quarter-wave plate; CCD, charge-coupled device.

Fig. 12
Fig. 12

Experimental results of the degree of polarization of a focused partially coherent RP beam at the focal plane versus the transverse coordinate u (v = 0) for two different values of the coherence width δ. The solid curves are calculated by theoretical formulas.

Equations (46)

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E ( x,y )= E x ( x,y ) e x + E y ( x,y ) e y = x ω 0 exp( x 2 + y 2 ω 0 2 ) e x + y ω 0 exp( x 2 + y 2 ω 0 2 ) e y ,
W ( x 1 , y 1 , x 2 , y 2 ,0 )=( W xx ( x 1 , y 1 , x 2 , y 2 ,0 ) W xy ( x 1 , y 1 , x 2 , y 2 ,0 ) W yx ( x 1 , y 1 , x 2 , y 2 ,0 ) W yy ( x 1 , y 1 , x 2 , y 2 ,0 ) ),
W αβ ( x 1 , y 1 , x 2 , y 2 ,0 )= E α ( x 1 , y 1 ,0) E β * ( x 2 , y 2 ,0) . (α=x,y;β=x,y)
W xx ( x 1 , y 1 , x 2 , y 2 ,0 )= x 1 x 2 ω 0 2 exp[ x 1 2 + y 1 2 + x 2 2 + y 2 2 ω 0 2 ( x 1 - x 2 ) 2 + ( y 1 - y 2 ) 2 2 δ xx 2 ],
W xy ( x 1 , y 1 , x 2 , y 2 ,0 )= B xy x 1 y 2 ω 0 2 exp[ x 1 2 + y 1 2 + x 2 2 + y 2 2 ω 0 2 ( x 1 - x 2 ) 2 + ( y 1 - y 2 ) 2 2 δ xy 2 ],
W yx ( x 1 , y 1 , x 2 , y 2 ,0 )= W xy * ( x 2 , y 2 , x 1 , y 1 ,0 ),
W yy ( x 1 , y 1 , x 2 , y 2 ,0 )= y 1 y 2 ω 0 2 exp[ x 1 2 + y 1 2 + x 2 2 + y 2 2 ω 0 2 ( x 1 - x 2 ) 2 + ( y 1 - y 2 ) 2 2 δ yy 2 ],
W αβ ( u 1 , v 1 , u 2 , v 2 ,z)= ( 1 λ| B | ) 2 W αβ ( x 1 , y 1 , x 2 , y 2 ,0) ×exp[ ik 2B (A x 1 2 2 x 1 u 1 +D u 1 2 ) ik 2B (A y 1 2 2 y 1 v 1 +D v 1 2 ) ] ×exp[ ik 2 B * ( A * x 2 2 2 x 2 u 2 + D * u 2 2 )+ ik 2 B * ( A * y 2 2 2 y 2 v 2 + D * y 2 2 ) ]d x 1 d x 2 d y 1 d y 2 ,
W xx ( u 1 , v 1 , u 2 , v 2 ,z)== π 2 8 δ xx 2 M 1xx 2 M 2xx 3 ω 0 2 ( 1 λ| B | ) 2 exp[ ikD 2B ( u 1 2 + v 1 2 ) ik D * 2 B * ( u 2 2 + v 2 2 ) ] ×exp[ k 2 ( u 1 2 + v 1 2 ) 4 M 1xx B 2 4 k 2 M 1xx 2 B 2 δ xx 4 ( u 2 2 + v 2 2 )+ k 2 ( B * ) 2 ( u 1 2 + v 1 2 )4 k 2 M 1xx | B | 2 δ xx 2 ( u 1 u 2 + v 1 v 2 ) 16 M 1xx 2 M 2xx | B | 4 δ xx 4 ] ×[ k 2 ( 2 M 1xx B δ xx 2 u 2 B * u 1 2 M 1xx | B | 2 δ xx 2 ) 2 + 2 M 2xx k 2 δ xx 2 u 1 B ( 2 M 1xx B δ xx 2 u 2 B * u 1 2 M 1xx | B | 2 δ xx 2 )+2 M 2xx ],
W xy ( u 1 , v 1 , u 2 , v 2 ,z )= B xy k 2 π 2 8 M 1xy 2 M 2xy 3 δ xy 2 w 0 2 ( 1 λ| B | ) 2 exp[ ikD 2B ( u 1 2 + v 1 2 ) ik D * 2 B * ( u 2 2 + v 2 2 ) ] ×exp[ k 2 ( u 2 2 + v 2 2 ) 4 M 1xy B * 2 4 k 2 M 1xy 2 B * 2 δ xy 4 ( u 1 2 + v 1 2 )+ k 2 B 2 ( u 2 2 + v 2 2 )4 k 2 M 1xy | B | 2 δ xy 2 ( u 1 u 2 + v 1 v 2 ) 16 M 1xy 2 M 2xy | B | 4 δ xy 4 ] ×( u 2 2 M 1xy B * δ xy 2 u 1 B )( v 1 B 1+4 M 1xy M 2xy δ xy 4 2 M 1xy B * δ xy 2 v 2 ),
W yx ( u 1 , v 1 , u 2 , v 2 ,z )= W xy * ( u 2 , v 2 , u 1 , v 1 ,z ),
W yy ( u 1 , v 1 , u 2 , v 2 ,z )= π 2 8 δ yy 2 M 1yy 2 M 2yy 3 ω 0 2 ( 1 λ| B | ) 2 exp[ ikD 2B ( u 1 2 + v 1 2 ) ik D * 2 B * ( u 2 2 + v 2 2 ) ] ×exp[ k 2 ( u 1 2 + v 1 2 ) 4 M 1yy B 2 4 k 2 M 1yy 2 B 2 δ yy 4 ( u 2 2 + v 2 2 )+ k 2 ( B * ) 2 ( u 1 2 + v 1 2 )4 k 2 M 1yy | B | 2 δ yy 2 ( u 1 u 2 + v 1 v 2 ) 16 M 1yy 2 M 2yy | B | 4 δ yy 4 ] ×[ k 2 ( 2 M 1yy B δ yy 2 v 2 B * v 1 2 M 1yy | B | 2 δ yy 2 ) 2 + 2 M 2yy k 2 δ yy 2 v 1 B ( 2 M 1yy B δ yy 2 v 2 B * v 1 2 M 1yy | B | 2 δ yy 2 )+2 M 2yy ],
M 1xx =1/ ω 0 2 +1/ ( 2 δ xx 2 ) ikA / ( 2B ) , M 2xx =1/ ω 0 2 +1/ ( 2 δ xx 2 ) + ik A * / ( 2 B * ) 1/ ( 4 M 1xx δ xx 4 ) ,
M 1xy =1/ ω 0 2 +1/ ( 2 δ xy 2 ) + ik A * / ( 2 B * ) , M 2xy =1/ ω 0 2 +1/ ( 2 δ xy 2 ) ikA / ( 2B ) 1/ ( 4 M 1xy δ xy 4 ) ,
M 1yy =1/ ω 0 2 +1/ ( 2 δ yy 2 ) ikA / ( 2B ) , M 2yy =1/ ω 0 2 +1/ ( 2 δ yy 2 ) + ik A * / ( 2 B * ) 1/ ( 4 M 1yy δ yy 4 ) .
I( u,v,z ) =Tr W ( u 1 , v 1 , u 2 , v 2 ,z )= W xx ( u,v,u,v,z )+ W yy ( u,v,u,v,z ),
P(u,v,z)= 1 4Det W (u,v,u,v,z) [Tr W (u,v,u,v,z)] 2 ,
W (u,v,u,v,z)= W (u) (u,v,u,v,z)+ W (p) (u,v,u,v,z),
W (u) (u,v,u,v,z)=( A(u,v,u,v,z) 0 0 A(u,v,u,v,z) ),
W (p) (u,v,u,v,z)=( B(u,v,u,v,z) D(u,v,u,v,z) D * (u,v,u,v,z) C(u,v,u,v,z) ),
A(u,v,u,v,z)= 1 2 [ W xx (u,v,u,v,z)+ W yy (u,v,u,v,z) [ W xx (u,v,u,v,z) W yy (u,v,u,v,z) ] 2 +4 | W xy (u,v,u,v,z) | 2 ],
B(u,v,u,v,z)= 1 2 [ W xx (u,v,u,v,z) W yy (u,v,u,v,z) + [ W xx (u,v,u,v,z) W yy (u,v,u,v,z) ] 2 +4 | W xy (u,v,u,v,z) | 2 ,
C(u,v,u,v,z)= 1 2 [ W yy (u,v,u,v,z) W xx (u,v,u,v,z) + [ W xx (u,v,u,v,z) W yy (u,v,u,v,z) ] 2 +4 | W xy (u,v,u,v,z) | 2 ],
D(u,v,u,v,z)= W xy (u,v,u,v,z).
A 1,2 (u,v,z)= 1 2 [ [ W xx (u,v,u,v,z) W yy (u,v,u,v,z) ] 2 +4 | W xy (u,v,u,v,z) | 2 ± [ W xx (u,v,u,v,z) W yy (u,v,u,v,z) ] 2 +4 [Re W xy (u,v,u,v,z)] 2 ] 1/2 ,
ε(u,v,z)= A 2 (u,v,u,v,z) A 1 (u,v,u,v,z) ,
θ(u,v,z)= 1 2 arctan[ 2Re W xy (u,v,u,v,z) W xx (u,v,u,v,z) W yy (u,v,u,v,z) ].
I (u) ( u,v,z ) =Tr W (u) (u,v,u,v,z), I (p) ( u,v,z ) =Tr W (p) (u,v,u,v,z).
μ 2 ( u 1 , v 1 , u 2 , v 2 ,z )= Tr[ W ( u 1 , v 1 , u 2 , v 2 ,z ) W ( u 2 , v 2 , u 1 , v 1 ,z ) ] I( u 1 , v 1 ,z ) I( u 2 , v 2 ,z ) .
Tr[ W ( u 1 , v 1 , u 2 , v 2 ,z ) W ( u 2 , v 2 , u 1 , v 1 ,z ) ]= α,β | W αβ ( u 1 , v 1 , u 2 , v 2 ,z ) | 2 . (α,β=x,y)
G (2) ( u 1 , v 1 , u 2 , v 2 ,z)= I( u 1 , v 1 ,z ) I( u 2 , v 2 ,z ) + α,β | W αβ ( u 1 , v 1 , u 2 , v 2 ,z ) | 2 .
g (2) ( u 1 , v 1 , u 2 , v 2 ,z)=1+ μ 2 ( u 1 , v 1 , u 2 , v 2 ,z ).
( A B C D )=( 1 z 0 1 )( 1 0 1/f 1 )=( 1z/f z 1/f 1 ).
A 2 (u,v,u,v,z)=0, ε(u,v,u,v,z)=0, θ(u,v,u,v,z)=arctan( v/u ).
η ( l ) = I (l) ( u,v,z ) dudv I( u,v,z ) dudv , ( l=p,u )
I θ (u,v,f) = cos 2 θ W xx (u,v,u,v,f)+ sin 2 θ W yy (u,v,u,v,f) +[ W xy (u,v,u,v,f)+ W yx (u,v,u,v,f)]sinθcosθ.
W xx (u,v,u,v,f)= I 0 (u,v,f) .
W yy (u,v,u,v,f)= I π/2 (u,v,f) .
I θ,ϕ (u,v,f) = sin 2 θ W yy (u,v,u,v,f)+ cos 2 θ W xx (u,v,u,v,f) +exp(iϕ)sinθ[ W xy (u,v,u,v,f)+exp(2iϕ) W yx (u,v,u,v,f)]cosθ,
W xy (u,v,u,v,f)= 1 2 [ I π/4,0 (u,v,f) I 3π/4,0 (u,v,f) ] +i 1 2 [ I π/4,π/2 (u,v,f) I 3π/4,π/2 (u,v,f) ],
W yx (u,v,u,v,f)= W xy * (u,v,u,v,f).
δ xx 2 + δ yy 2 2 δ xy δ xx δ yy | B xy | , | B xy | 2 δ xx / δ yy + δ yy / δ xx .
A 1,2 (x,y,0)= 1 2 [ [ W xx (x,y,x,y,0) W yy (x,y,x,y,0) ] 2 +4 | W xy (x,y,x,y,0) | 2 ± [ W xx (x,y,x,y,0) W yy (x,y,x,y,0) ] 2 +4 [Re W xy (x,y,x,y,0)] 2 ] 1/2 ,
ε(x,y,0)= A 2 (x,y,x,y,0) A 1 (x,y,x,y,0) ,
θ(x,y,0)= 1 2 arctan[ 2Re W xy (x,y,x,y,0) W xx (x,y,x,y,0) W yy (x,y,x,y,0) ].
Im[ B xy ]=0, Re[ B xy ]=1,

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