Abstract

Analytical nonparaxial propagation formula for the cross-spectral density matrix of a cylindrical vector partially coherent beam in free space is derived based on the generalized Raleigh-Sommerfeld diffraction integrals. Statistical properties, such as average intensity, degree of polarization and degree of coherence, of a nonparaxial cylindrical vector partially coherent field are illustrated numerically, and compared with that of a paraxial cylindrical vector partially coherent beam. It is found that the statistical properties of a nonparaxial cylindrical vector partially coherent field are much different from that of a paraxial cylindrical vector partially coherent beam, and are closely determined by the initial beam width and correlation coefficients. Our results will be useful for modulating the properties of a nonparaxial cylindrical vector partially coherent field.

© 2012 OSA

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2012

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

X. Li and Y. Cai, “Nonparaxial propagation of a partially coherent dark hollow beam,” Appl. Phys. B102(1), 205–213 (2011).
[CrossRef]

L. Zhang and Y. Cai, “Propagation of a twisted anisotropic Gaussian-Schell model beam beyond the paraxial approximation,” Appl. Phys. B103(4), 1001–1008 (2011).
[CrossRef]

L. Zhang and Y. Cai, “Statistical properties of a nonparaxial Gaussian Schell-model beam in a uniaxial crystal,” Opt. Express19(14), 13312–13325 (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]

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. Express19(7), 5979–5992 (2011).
[CrossRef] [PubMed]

Y. Dong, Y. Cai, and C. Zhao, “Degree of polarization of a tightly focused partially coherent dark hollow beam,” Appl. Phys. B105(2), 405–414 (2011).
[CrossRef]

G. Wu and Y. Cai, “Modulation of spectral intensity, polarization and coherence of a stochastic electromagnetic beam,” Opt. Express19(9), 8700–8714 (2011).
[CrossRef] [PubMed]

2010

S. Zhu and Y. Cai, “M2-factor of a stochastic electromagnetic beam in a Gaussian cavity,” Opt. Express18(26), 27567–27581 (2010).
[CrossRef] [PubMed]

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

S. Zhu and Y. Cai, “Spectral shift of a twisted electromagnetic Gaussian Schell-model beam focused by a thin lens,” Appl. Phys. B99(1-2), 317–323 (2010).
[CrossRef]

S. Zhu, Y. Cai, and O. Korotkova, “Propagation factor of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Express18(12), 12587–12598 (2010).
[CrossRef] [PubMed]

B. J. Roxworthy and K. C. Toussaint., “Optical trapping with pi-phase cylindrical vector beams,” New J. Phys.12(7), 073012 (2010).
[CrossRef]

Y. Kozawa and S. Sato, “Optical trapping of micrometer-sized dielectric particles by cylindrical vector beams,” Opt. Express18(10), 10828–10833 (2010).
[CrossRef] [PubMed]

R. Zheng, C. Gu, A. Wang, L. Xu, and H. Ming, “An all-fiber laser generating cylindrical vector beam,” Opt. Express18(10), 10834–10838 (2010).
[CrossRef] [PubMed]

Z. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A82(3), 033836 (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. B101(1-2), 361–369 (2010).
[CrossRef]

2009

2008

2007

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys.9(3), 78 (2007).
[CrossRef]

C. F. Li, “Integral transformation solution of free-space cylindrical vector beams and prediction of modified Bessel-Gaussian vector beams,” Opt. Lett.32(24), 3543–3545 (2007).
[CrossRef] [PubMed]

2006

D. Deng, “Nonparaxial propagation of radially polarized light beams,” J. Opt. Soc. Am. B23(6), 1228–1234 (2006).
[CrossRef]

D. P. Biss, K. S. Youngworth, and T. G. Brown, “Dark-field imaging with cylindrical-vector beams,” Appl. Opt.45(3), 470–479 (2006).
[CrossRef] [PubMed]

K. J. Moh, X. C. Yuan, J. Bu, D. K. Y. Low, and R. E. Burge, “Direct noninterference cylindrical vector beam generation applied in the femtosecond regime,” Appl. Phys. Lett.89(25), 251114 (2006).
[CrossRef]

2005

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]

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]

2004

2003

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A312(5-6), 263–267 (2003).
[CrossRef]

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

Q. Zhan, “Radiation forces on a dielectric sphere produced by highly focused cylindrical vector beams,” J. Opt. A, Pure Appl. Opt.5(3), 229–232 (2003).
[CrossRef]

G. V. Bogatyryova, C. V. Fel’de, P. V. Polyanskii, S. A. Ponomarenko, M. S. Soskin, and E. Wolf, “Partially coherent vortex beams with a separable phase,” Opt. Lett.28(11), 878–880 (2003).
[CrossRef] [PubMed]

2002

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]

Q. Zhan and J. R. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express10(7), 324–331 (2002).
[PubMed]

2001

2000

1998

A. A. Tovar, “Production and propagation of cylindrically polarized Laguerre–Gaussian laser beams,” J. Opt. Soc. Am. A15(10), 2705–2711 (1998).
[CrossRef]

H. Laabs, “Propagation of Hermite–Gaussian beams beyond the paraxial approximation,” Opt. Commun.147(1-3), 1–4 (1998).
[CrossRef]

Baykal, Y.

Bernet, S.

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys.9(3), 78 (2007).
[CrossRef]

Biss, D. P.

Bogatyryova, G. V.

Borghi, R.

Brown, T. G.

Bu, J.

K. J. Moh, X. C. Yuan, J. Bu, D. K. Y. Low, and R. E. Burge, “Direct noninterference cylindrical vector beam generation applied in the femtosecond regime,” Appl. Phys. Lett.89(25), 251114 (2006).
[CrossRef]

Burge, R. E.

K. J. Moh, X. C. Yuan, J. Bu, D. K. Y. Low, and R. E. Burge, “Direct noninterference cylindrical vector beam generation applied in the femtosecond regime,” Appl. Phys. Lett.89(25), 251114 (2006).
[CrossRef]

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, Y. Cai, C. Zhao, and M. Yao, “Statistics properties of a cylindrical vector partially coherent beam,” Opt. Express19(7), 5979–5992 (2011).
[CrossRef] [PubMed]

L. Zhang and Y. Cai, “Propagation of a twisted anisotropic Gaussian-Schell model beam beyond the paraxial approximation,” Appl. Phys. B103(4), 1001–1008 (2011).
[CrossRef]

L. Zhang and Y. Cai, “Statistical properties of a nonparaxial Gaussian Schell-model beam in a uniaxial crystal,” Opt. Express19(14), 13312–13325 (2011).
[CrossRef] [PubMed]

X. Li and Y. Cai, “Nonparaxial propagation of a partially coherent dark hollow beam,” Appl. Phys. B102(1), 205–213 (2011).
[CrossRef]

G. Wu and Y. Cai, “Modulation of spectral intensity, polarization and coherence of a stochastic electromagnetic beam,” Opt. Express19(9), 8700–8714 (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]

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, and C. Zhao, “Degree of polarization of a tightly focused partially coherent dark hollow beam,” Appl. Phys. B105(2), 405–414 (2011).
[CrossRef]

S. Zhu and Y. Cai, “M2-factor of a stochastic electromagnetic beam in a Gaussian cavity,” Opt. Express18(26), 27567–27581 (2010).
[CrossRef] [PubMed]

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

S. Zhu and Y. Cai, “Spectral shift of a twisted electromagnetic Gaussian Schell-model beam focused by a thin lens,” Appl. Phys. B99(1-2), 317–323 (2010).
[CrossRef]

S. Zhu, Y. Cai, and O. Korotkova, “Propagation factor of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Express18(12), 12587–12598 (2010).
[CrossRef] [PubMed]

O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for LIDAR systems operating through turbulent atmosphere,” Appl. Phys. B94(4), 681–690 (2009).
[CrossRef]

C. Zhao, Y. Cai, and O. Korotkova, “Radiation force of scalar and electromagnetic twisted Gaussian Schell-model beams,” Opt. Express17(24), 21472–21487 (2009).
[CrossRef] [PubMed]

F. Wang, Y. Cai, and O. Korotkova, “Partially coherent standard and elegant Laguerre-Gaussian beams of all orders,” Opt. Express17(25), 22366–22379 (2009).
[CrossRef] [PubMed]

Y. Cai, O. Korotkova, H. T. Eyyuboğlu, and Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express16(20), 15834–15846 (2008).
[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, 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. Express16(11), 7665–7673 (2008).
[CrossRef] [PubMed]

Chen, J.

X. Wang, J. Ding, J. Qin, J. Chen, Y. Fan, and H. Wang, “Configurable three-dimensional optical cage generated from cylindrical vector beams,” Opt. Commun.282(17), 3421–3425 (2009).
[CrossRef]

Cheng, W.

Deng, D.

Ding, J.

X. Wang, J. Ding, J. Qin, J. Chen, Y. Fan, and H. Wang, “Configurable three-dimensional optical cage generated from cylindrical vector beams,” Opt. Commun.282(17), 3421–3425 (2009).
[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, Y. Cai, C. Zhao, and M. Yao, “Statistics properties of a cylindrical vector partially coherent beam,” Opt. Express19(7), 5979–5992 (2011).
[CrossRef] [PubMed]

Y. Dong, Y. Cai, and C. Zhao, “Degree of polarization of a tightly focused partially coherent dark hollow beam,” Appl. Phys. B105(2), 405–414 (2011).
[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]

Duan, K.

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.

Fan, Y.

X. Wang, J. Ding, J. Qin, J. Chen, Y. Fan, and H. Wang, “Configurable three-dimensional optical cage generated from cylindrical vector beams,” Opt. Commun.282(17), 3421–3425 (2009).
[CrossRef]

Fel’de, C. V.

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]

Furhapter, S.

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys.9(3), 78 (2007).
[CrossRef]

Gori, F.

F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. A25(5), 1016–1021 (2008).
[CrossRef] [PubMed]

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

Gu, C.

Halterman, K.

Haus, J. W.

Jesacher, A.

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys.9(3), 78 (2007).
[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]

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]

S. Zhu, Y. Cai, and O. Korotkova, “Propagation factor of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Express18(12), 12587–12598 (2010).
[CrossRef] [PubMed]

Z. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A82(3), 033836 (2010).
[CrossRef]

O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for LIDAR systems operating through turbulent atmosphere,” Appl. Phys. B94(4), 681–690 (2009).
[CrossRef]

C. Zhao, Y. Cai, and O. Korotkova, “Radiation force of scalar and electromagnetic twisted Gaussian Schell-model beams,” Opt. Express17(24), 21472–21487 (2009).
[CrossRef] [PubMed]

F. Wang, Y. Cai, and O. Korotkova, “Partially coherent standard and elegant Laguerre-Gaussian beams of all orders,” Opt. Express17(25), 22366–22379 (2009).
[CrossRef] [PubMed]

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

Y. Cai, O. Korotkova, H. T. Eyyuboğlu, and Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express16(20), 15834–15846 (2008).
[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]

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, 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]

O. Korotkova and E. Wolf, “Spectral degree of coherence of a random three-dimensional electromagnetic field,” J. Opt. Soc. Am. A21(12), 2382–2385 (2004).
[CrossRef] [PubMed]

Kozawa, Y.

Kurti, R. S.

Laabs, H.

H. Laabs, “Propagation of Hermite–Gaussian beams beyond the paraxial approximation,” Opt. Commun.147(1-3), 1–4 (1998).
[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, C. F.

Li, X.

X. Li and Y. Cai, “Nonparaxial propagation of a partially coherent dark hollow beam,” Appl. Phys. B102(1), 205–213 (2011).
[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. B101(1-2), 361–369 (2010).
[CrossRef]

Liu, X.

Low, D. K. Y.

K. J. Moh, X. C. Yuan, J. Bu, D. K. Y. Low, and R. E. Burge, “Direct noninterference cylindrical vector beam generation applied in the femtosecond regime,” Appl. Phys. Lett.89(25), 251114 (2006).
[CrossRef]

Lü, B.

Maurer, C.

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys.9(3), 78 (2007).
[CrossRef]

Ming, H.

Moh, K. J.

K. J. Moh, X. C. Yuan, J. Bu, D. K. Y. Low, and R. E. Burge, “Direct noninterference cylindrical vector beam generation applied in the femtosecond regime,” Appl. Phys. Lett.89(25), 251114 (2006).
[CrossRef]

Mondello, A.

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

Petrov, D.

G. Volpe and D. Petrov, “Generation of cylindrical vector beams with few-mode fibers excited by Laguerre-Gaussian beams,” Opt. Commun.237(1-3), 89–95 (2004).
[CrossRef]

Piquero, G.

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

Polyanskii, P. V.

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]

Ponomarenko, S. A.

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]

Z. Zhang, J. Pu, and X. Wang, “Focusing of partially coherent Bessel-Gaussian beams through a high-numerical-aperture objective,” Opt. Lett.33(1), 49–51 (2008).
[CrossRef] [PubMed]

Qin, J.

X. Wang, J. Ding, J. Qin, J. Chen, Y. Fan, and H. Wang, “Configurable three-dimensional optical cage generated from cylindrical vector beams,” Opt. Commun.282(17), 3421–3425 (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]

Ramírez-Sánchez, V.

Ritsch-Marte, M.

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys.9(3), 78 (2007).
[CrossRef]

Roxworthy, B. J.

B. J. Roxworthy and K. C. Toussaint., “Optical trapping with pi-phase cylindrical vector beams,” New J. Phys.12(7), 073012 (2010).
[CrossRef]

Salem, M.

Santarsiero, M.

Sato, S.

Setälä, 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]

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]

Shirai, T.

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]

Shori, R. K.

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]

Soskin, M. S.

Tong, Z.

Z. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A82(3), 033836 (2010).
[CrossRef]

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

Toussaint, K. C.

B. J. Roxworthy and K. C. Toussaint., “Optical trapping with pi-phase cylindrical vector beams,” New J. Phys.12(7), 073012 (2010).
[CrossRef]

Tovar, A. A.

Volpe, G.

G. Volpe and D. Petrov, “Generation of cylindrical vector beams with few-mode fibers excited by Laguerre-Gaussian beams,” Opt. Commun.237(1-3), 89–95 (2004).
[CrossRef]

Wang, A.

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, Y. Cai, and O. Korotkova, “Partially coherent standard and elegant Laguerre-Gaussian beams of all orders,” Opt. Express17(25), 22366–22379 (2009).
[CrossRef] [PubMed]

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. B101(1-2), 361–369 (2010).
[CrossRef]

X. Wang, J. Ding, J. Qin, J. Chen, Y. Fan, and H. Wang, “Configurable three-dimensional optical cage generated from cylindrical vector beams,” Opt. Commun.282(17), 3421–3425 (2009).
[CrossRef]

Wang, X.

X. Wang, J. Ding, J. Qin, J. Chen, Y. Fan, and H. Wang, “Configurable three-dimensional optical cage generated from cylindrical vector beams,” Opt. Commun.282(17), 3421–3425 (2009).
[CrossRef]

Z. Zhang, J. Pu, and X. Wang, “Focusing of partially coherent Bessel-Gaussian beams through a high-numerical-aperture objective,” Opt. Lett.33(1), 49–51 (2008).
[CrossRef] [PubMed]

Wardlaw, M. J.

Watson, E.

O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for LIDAR systems operating through turbulent atmosphere,” Appl. Phys. B94(4), 681–690 (2009).
[CrossRef]

Wolf, E.

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

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]

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]

O. Korotkova and E. Wolf, “Spectral degree of coherence of a random three-dimensional electromagnetic field,” J. Opt. Soc. Am. A21(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. A312(5-6), 263–267 (2003).
[CrossRef]

G. V. Bogatyryova, C. V. Fel’de, P. V. Polyanskii, S. A. Ponomarenko, M. S. Soskin, and E. Wolf, “Partially coherent vortex beams with a separable phase,” Opt. Lett.28(11), 878–880 (2003).
[CrossRef] [PubMed]

Wu, G.

Xu, L.

Yao, M.

Youngworth, K. S.

Yuan, X. C.

K. J. Moh, X. C. Yuan, J. Bu, D. K. Y. Low, and R. E. Burge, “Direct noninterference cylindrical vector beam generation applied in the femtosecond regime,” Appl. Phys. Lett.89(25), 251114 (2006).
[CrossRef]

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]

L. Zhang and Y. Cai, “Statistical properties of a nonparaxial Gaussian Schell-model beam in a uniaxial crystal,” Opt. Express19(14), 13312–13325 (2011).
[CrossRef] [PubMed]

L. Zhang and Y. Cai, “Propagation of a twisted anisotropic Gaussian-Schell model beam beyond the paraxial approximation,” Appl. Phys. B103(4), 1001–1008 (2011).
[CrossRef]

Zhang, Y.

Zhang, Z.

Zhao, C.

Zheng, R.

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. B101(1-2), 361–369 (2010).
[CrossRef]

Zhu, S.

Adv. Opt. Photon.

Appl. Opt.

Appl. Phys. B

O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for LIDAR systems operating through turbulent atmosphere,” Appl. Phys. B94(4), 681–690 (2009).
[CrossRef]

S. Zhu and Y. Cai, “Spectral shift of a twisted electromagnetic Gaussian Schell-model beam focused by a thin lens,” Appl. Phys. B99(1-2), 317–323 (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. B101(1-2), 361–369 (2010).
[CrossRef]

X. Li and Y. Cai, “Nonparaxial propagation of a partially coherent dark hollow beam,” Appl. Phys. B102(1), 205–213 (2011).
[CrossRef]

L. Zhang and Y. Cai, “Propagation of a twisted anisotropic Gaussian-Schell model beam beyond the paraxial approximation,” Appl. Phys. B103(4), 1001–1008 (2011).
[CrossRef]

Y. Dong, Y. Cai, and C. Zhao, “Degree of polarization of a tightly focused partially coherent dark hollow beam,” Appl. Phys. B105(2), 405–414 (2011).
[CrossRef]

Appl. Phys. Lett.

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]

K. J. Moh, X. C. Yuan, J. Bu, D. K. Y. Low, and R. E. Burge, “Direct noninterference cylindrical vector beam generation applied in the femtosecond regime,” Appl. Phys. Lett.89(25), 251114 (2006).
[CrossRef]

J. Mod. Opt.

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. A, Pure Appl. Opt.

Q. Zhan, “Radiation forces on a dielectric sphere produced by highly focused cylindrical vector beams,” J. Opt. A, Pure Appl. Opt.5(3), 229–232 (2003).
[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]

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]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

New J. Phys.

B. J. Roxworthy and K. C. Toussaint., “Optical trapping with pi-phase cylindrical vector beams,” New J. Phys.12(7), 073012 (2010).
[CrossRef]

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys.9(3), 78 (2007).
[CrossRef]

Opt. Commun.

X. Wang, J. Ding, J. Qin, J. Chen, Y. Fan, and H. Wang, “Configurable three-dimensional optical cage generated from cylindrical vector beams,” Opt. Commun.282(17), 3421–3425 (2009).
[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]

G. Volpe and D. Petrov, “Generation of cylindrical vector beams with few-mode fibers excited by Laguerre-Gaussian beams,” Opt. Commun.237(1-3), 89–95 (2004).
[CrossRef]

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

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

H. Laabs, “Propagation of Hermite–Gaussian beams beyond the paraxial approximation,” Opt. Commun.147(1-3), 1–4 (1998).
[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]

Opt. Express

L. Zhang and Y. Cai, “Statistical properties of a nonparaxial Gaussian Schell-model beam in a uniaxial crystal,” Opt. Express19(14), 13312–13325 (2011).
[CrossRef] [PubMed]

Y. Dong, Y. Cai, C. Zhao, and M. Yao, “Statistics properties of a cylindrical vector partially coherent beam,” Opt. Express19(7), 5979–5992 (2011).
[CrossRef] [PubMed]

F. Wang, Y. Cai, and O. Korotkova, “Partially coherent standard and elegant Laguerre-Gaussian beams of all orders,” Opt. Express17(25), 22366–22379 (2009).
[CrossRef] [PubMed]

C. Zhao, Y. Cai, and O. Korotkova, “Radiation force of scalar and electromagnetic twisted Gaussian Schell-model beams,” Opt. Express17(24), 21472–21487 (2009).
[CrossRef] [PubMed]

G. Wu and Y. Cai, “Modulation of spectral intensity, polarization and coherence of a stochastic electromagnetic beam,” Opt. Express19(9), 8700–8714 (2011).
[CrossRef] [PubMed]

S. Zhu and Y. Cai, “M2-factor of a stochastic electromagnetic beam in a Gaussian cavity,” Opt. Express18(26), 27567–27581 (2010).
[CrossRef] [PubMed]

Y. Cai, O. Korotkova, H. T. Eyyuboğlu, and Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express16(20), 15834–15846 (2008).
[CrossRef] [PubMed]

S. Zhu, Y. Cai, and O. Korotkova, “Propagation factor of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Express18(12), 12587–12598 (2010).
[CrossRef] [PubMed]

Y. Kozawa and S. Sato, “Optical trapping of micrometer-sized dielectric particles by cylindrical vector beams,” Opt. Express18(10), 10828–10833 (2010).
[CrossRef] [PubMed]

R. Zheng, C. Gu, A. Wang, L. Xu, and H. Ming, “An all-fiber laser generating cylindrical vector beam,” Opt. Express18(10), 10834–10838 (2010).
[CrossRef] [PubMed]

R. S. Kurti, K. Halterman, R. K. Shori, and M. J. Wardlaw, “Discrete cylindrical vector beam generation from an array of optical fibers,” Opt. Express17(16), 13982–13988 (2009).
[CrossRef] [PubMed]

Q. Zhan and J. R. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express10(7), 324–331 (2002).
[PubMed]

K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express7(2), 77–87 (2000).
[CrossRef] [PubMed]

D. P. Biss and T. G. Brown, “Cylindrical vector beam focusing through a dielectric interface,” Opt. Express9(10), 490–497 (2001).
[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. Express16(11), 7665–7673 (2008).
[CrossRef] [PubMed]

W. Cheng, J. W. Haus, and Q. Zhan, “Propagation of vector vortex beams through a turbulent atmosphere,” Opt. Express17(20), 17829–17836 (2009).
[CrossRef] [PubMed]

Opt. Lett.

C. F. Li, “Integral transformation solution of free-space cylindrical vector beams and prediction of modified Bessel-Gaussian vector beams,” Opt. Lett.32(24), 3543–3545 (2007).
[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]

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]

Z. Zhang, J. Pu, and X. Wang, “Focusing of partially coherent Bessel-Gaussian beams through a high-numerical-aperture objective,” Opt. Lett.33(1), 49–51 (2008).
[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]

K. Duan and B. Lü, “Partially coherent nonparaxial beams,” Opt. Lett.29(8), 800–802 (2004).
[CrossRef] [PubMed]

Y. Zhang and B. Lü, “Propagation of the Wigner distribution function for partially coherent nonparaxial beams,” Opt. Lett.29(23), 2710–2712 (2004).
[CrossRef] [PubMed]

G. V. Bogatyryova, C. V. Fel’de, P. V. Polyanskii, S. A. Ponomarenko, M. S. Soskin, and E. Wolf, “Partially coherent vortex beams with a separable phase,” Opt. Lett.28(11), 878–880 (2003).
[CrossRef] [PubMed]

Phys. Lett. A

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A312(5-6), 263–267 (2003).
[CrossRef]

Phys. Rev. A

Z. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A82(3), 033836 (2010).
[CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys.

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]

Phys. Rev. Lett.

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

Other

E. Wolf, Introduction to the theory of coherence and polarization of light (Cambridge U. Press, 2007).

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1966).

A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (U. S. Department of Commerce, 1970).

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Figures (11)

Fig. 1
Fig. 1

Normalized intensity distributions (contour graphs) I/ I max , ( I x + I y )/ I max , I z / I max and the corresponding cross lines (y = x) of a nonparaxial cylindrical vector partially coherent LG field at several propagation distances with ρ= x 2 + y 2 and w 0 =10λ . I p / I pmax denotes the normalized intensity distribution calculated by the paraxial propagation formulae.

Fig. 2
Fig. 2

Normalized intensity distributions (contour graphs) I/ I max , ( I x + I y )/ I max , I z / I max and the corresponding cross lines (y = x) of a nonparaxial cylindrical vector partially coherent LG field at several propagation distances with ρ= x 2 + y 2 and w 0 =λ . I p / I pmax denotes the normalized intensity distribution calculated by the paraxial propagation formulae.

Fig. 3
Fig. 3

Normalized intensity distributions (contour graphs) I/ I max , ( I x + I y )/ I max , I z / I max and the corresponding cross lines (y = x) of a nonparaxial cylindrical vector partially coherent LG field at several propagation distances with ρ= x 2 + y 2 and w 0 =0.5λ . I p / I pmax denotes the normalized intensity distribution calculated by the paraxial propagation formulae.

Fig. 4
Fig. 4

Normalized intensity distributions (contour graphs) I/ I max , ( I x + I y )/ I max , I z / I max and the corresponding cross lines (y = x) of a nonparaxial cylindrical vector partially coherent LG field at z=10 z r with ρ= x 2 + y 2 and w 0 =λ for different values of the correlation coefficients σ xx , σ xy , σ yy .

Fig. 5
Fig. 5

Degree of polarization and the corresponding cross line (y = 0) of a nonparaxial cylindrical vector partially coherent LG field at the source plane (z = 0).

Fig. 6
Fig. 6

Degree of polarization and the corresponding cross line (y = x) of a nonparaxial cylindrical vector partially coherent field at several propagation distances with w 0 =10λ and ρ= x 2 + y 2 .

Fig. 7
Fig. 7

Degree of polarization and the corresponding cross line (y = x) of a nonparaxial cylindrical vector partially coherent LG field at several propagation distances with w 0 =λ and ρ= x 2 + y 2 .

Fig. 8
Fig. 8

Degree of polarization and the corresponding cross line (y = x) of a nonparaxial cylindrical vector partially coherent field at several propagation distances with w 0 =0.5λ and ρ= x 2 + y 2 .

Fig. 9
Fig. 9

Degree of polarization and the corresponding cross line (y = 0) of a nonparaxial cylindrical vector partially coherent LG field at z= z r for different values of the correlation coefficients σ xx , σ xy , σ yy .

Fig. 10
Fig. 10

Modulus of the spectral degree of coherence of a nonparaxial cylindrical vector partially coherent LG field between two transverse points (x1,0) and (x2,0) at several propagation distances for different values of w 0 .

Fig. 11
Fig. 11

Modulus of the spectral degree of coherence of a nonparaxial cylindrical vector partially coherent LG field between two transverse points (x1, 0) and (x2, 0) at z= z r for different values of the correlation coefficients σ xx , σ xy , σ yy .

Equations (47)

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W ( x 1 , y 1 , x 2 , y 2 ,z )=( W xx ( x 1 , y 1 , x 2 , y 2 ,z ) W xy ( x 1 , y 1 , x 2 , y 2 ,z ) W xz ( x 1 , y 1 , x 2 , y 2 ,z ) W yx ( x 1 , y 1 , x 2 , y 2 ,z ) W yy ( x 1 , y 1 , x 2 , y 2 ,z ) W yz ( x 1 , y 1 , x 2 , y 2 ,z ) W zx ( x 1 , y 1 , x 2 , y 2 ,z ) W zy ( x 1 , y 1 , x 2 , y 2 ,z ) W zz ( x 1 , y 1 , x 2 , y 2 ,z ) ),
W αβ ( x 1 , y 1 , x 2 , y 2 ,z )= E α * ( x 1 , y 1 ,z ) E β ( x 2 , y 2 ,z ) ,( α,β=x,y,z )
E α (x,y,z)= 1 2π E α ( x 0 , y 0 ,0) z [ exp( ikR ) R ]d x 0 d y 0 ,(α=x,y)
E z ( x,y,z )= 1 2π { E x ( x 0 , y 0 ,0 ) x [ exp( ikR ) R ] + E y ( x 0 , y 0 ,0 ) y [ exp( ikR ) R ] }d x 0 d y 0 ,
α [ exp( ikR ) R ]= ikexp( ikR ) R 2 ( α α 0 ),(α=x,y)
z [ exp( ikR ) R ]= ikzexp( ikR ) R 2 .
W αβ ( x 1 , y 1 , x 2 , y 2 ,z)= k 2 z 2 4 π 2 W αβ ( x 10 , y 10 , x 20 , y 20 ,0) exp[ ik( R 1 R 2 ) ] R 1 2 R 2 2 ×d x 10 d y 10 d x 20 d y 20 ,(α,β=x,y)
W αz ( x 1 , y 1 , x 2 , y 2 ,z)= k 2 z 4 π 2 [ W αx ( x 10 , y 10 , x 20 , y 20 ,0) exp[ ik( R 1 R 2 ) ] R 1 2 R 2 2 ( x 2 x 20 ) + W αy ( x 10 , y 10 , x 20 , y 20 ,0) exp[ ik( R 1 R 2 ) ] R 1 2 R 2 2 ( y 2 y 20 ) ]d x 10 d y 10 d x 20 d y 20 ,(α=x,y)
W zz ( x 1 , y 1 , x 2 , y 2 ,z)= k 2 4 π 2 [ W xx ( x 10 , y 10 , x 20 , y 20 ,0) exp[ ik( R 1 R 2 ) ] R 1 2 R 2 2 ×( x 1 x 10 )( x 2 x 20 )+ W xy ( x 10 , y 10 , x 20 , y 20 ,0) exp[ ik( R 1 R 2 ) ] R 1 2 R 2 2 ( x 1 x 10 )( y 2 y 20 ) + W yx ( x 10 , y 10 , x 20 , y 20 ,0) exp[ ik( R 1 R 2 ) ] R 1 2 R 2 2 ( y 1 y 10 )( x 2 x 20 ) + W yy ( x 10 , y 10 , x 20 , y 20 ,0) exp[ ik( R 1 R 2 ) ] R 1 2 R 2 2 ( y 1 y 10 )( y 2 y 20 ) ]d x 10 d y 10 d x 20 d y 20 ,
W αβ ( x 10 , y 10 , x 20 , y 20 ,0 )= E α * ( x 10 , y 10 ,0 ) E β ( x 20 , y 20 ,0 ) . ( α,β=x,y )
E ( r,ϕ,0 )=exp( r 2 w 0 2 ) ( 2 r 2 w 0 2 ) ( n±1 )/2 L p n±1 ( 2 r 2 w 0 2 ){ cos( nϕ ) e ϕ sin( nϕ ) e r ±sin( nϕ ) e ϕ +cos( nϕ ) e r },
E ( x 0 , y 0 ,0 )={ E x ( x 0 , y 0 ,0 ) e x + E y ( x 0 , y 0 ,0 ) e y E y ( x 0 , y 0 ,0 ) e x + E x ( x 0 , y 0 ,0 ) e y } =exp( x 0 2 + y 0 2 w 0 2 ) (1) p 2 2p+n±1 p! { 1 2i m=0 p s=0 n±1 i s [ 1 (1) s ]( p m )( n±1 s ) H 2m+n±1s ( 2 x 0 w 0 ) H 2p2m+s ( 2 y 0 w 0 ) e x 1 2 m=0 p s=0 n±1 i s [ 1+ (1) s ]( p m )( n±1 s ) H 2m+n±1s ( 2 x 0 w 0 ) H 2p2m+s ( 2 y 0 w 0 ) e x + 1 2 m=0 p s=0 n±1 i s [ 1+ (1) s ]( p m )( n±1 s ) H 2m+n±1s ( 2 x 0 w 0 ) H 2p2m+s ( 2 y 0 w 0 ) e y + 1 2i m=0 p s=0 n±1 i s [ 1 (1) s ]( p m )( n±1 s ) H 2m+n±1s ( 2 x 0 w 0 ) H 2p2m+s ( 2 y 0 w 0 ) e y },
W ( x 10 , y 10 , x 20 , y 20 ,0 )=( W xx ( x 10 , y 10 , x 20 , y 20 ,0 ) W xy ( x 10 , y 10 , x 20 , y 20 ,0 )0 W yx ( x 10 , y 10 , x 20 , y 20 ,0 ) W yy ( x 10 , y 10 , x 20 , y 20 ,0 )0 000 ),
W αβ ( x 10 , y 10 , x 20 , y 20 ,0 )= A αβ 1 4 exp[ x 10 2 + x 20 2 + y 10 2 + y 20 2 w 0 2 ( x 10 x 20 ) 2 + ( y 10 y 20 ) 2 2 σ αβ 2 ] × 1 2 4p+2n±2 (p!) 2 m=0 p s=0 n±1 l=0 p h=0 n±1 i s (i) h [ 1+ C α (1) s ][ 1+ C β (1) h ]( p m )( p l )( n±1 s )( n±1 h ) × H 2m+n±1s ( 2 x 10 w 0 ) H 2l+n±1h ( 2 x 20 w 0 ) H 2p2m+s ( 2 y 10 w 0 ) H 2p2l+h ( 2 y 20 w 0 ),(α,β=x,y)
W yx ( x 10 , y 10 , x 20 , y 20 ,0 )= [ W xy ( x 20 , y 20 , x 10 , y 10 ,0 ) ] * ,
A αβ ={ 1α=β=x, i B xy α=x,β=y 1α=β=y, , C α ={ 1α=x, 1α=y,
W 1xx ( x 10 , y 10 , x 20 , y 20 ,0 )= W yy ( x 10 , y 10 , x 20 , y 20 ,0 ), W 1yy ( x 10 , y 10 , x 20 , y 20 ,0 )= W xx ( x 10 , y 10 , x 20 , y 20 ,0 ), W 1xy ( x 10 , y 10 , x 20 , y 20 ,0 )= W yx ( x 10 , y 10 , x 20 , y 20 ,0 ), W 1xy ( x 10 , y 10 , x 20 , y 20 ,0 )= W xy ( x 10 , y 10 , x 20 , y 20 ,0 ),
R i r i + x i0 2 + y i0 2 2 x i x i0 2 y i y i0 2 r i ,(i=1,2)
W αβ ( x 1 , y 1 , x 2 , y 2 ,z )= A αβ k 2 z 2 exp[ ik( r 1 r 2 ) ] 16 r 1 2 r 2 2 M 1αβ 1 2 5( 2p+n±1 )/2 (p!) 2 ( 1 2 M 1αβ w 0 2 ) ( n±1+2p )/2 ×exp{ k 2 ( x 1 2 + y 1 2 ) 4 M 1αβ r 1 2 k 2 4 M 2αβ [ ( x 2 r 2 x 1 2 M 1αβ r 1 σ αβ 2 ) 2 + ( y 2 r 2 y 1 2 M 1αβ r 1 σ αβ 2 ) 2 ] } × m=0 p s=0 n±1 l=0 p h=0 n±1 c 1 =0 2m+n±1s d 1 =0 c 1 /2 e 1 =0 (2l+n±1h)/2 c 2 =0 2p2m+s d 2 =0 c 2 /2 e 2 =0 (2p2l+h)/2 ( 2p2m+s c 2 )( 2m+n±1s c 1 ) ×( p m )( p l )( n±1 s )( n±1 h )[ 1+ C α (1) s ][ 1+ C β (1) h ] (1) d 1 + d 2 + e 1 + e 2 i s (i) h (2i) ( c 1 + c 2 2 d 1 2 d 2 +n±1+2p2 e 1 2 e 2 ) × ( 2 2 w 0 ) 2p+n±12 e 2 2 e 1 c 1 ! d 1 !( c 1 2 d 1 )! c 2 ! d 2 !( c 2 2 d 2 )! (2l+n±1h)! e 1 !(2l+n±1h2 e 1 )! (2p2l+h)! e 2 !(2p2l+h2 e 2 )! × 1 ( M 2αβ ) c 1 + c 2 +n±1+2p2 d 1 2 d 2 2 e 1 2 e 2 +2 [ 2 σ αβ 2 ( w 0 2 M 1αβ 2 2 M 1αβ ) 1/2 ] c 1 + c 2 2 d 1 2 d 2 × H c 1 2 d 1 +2l+n±1h2 e 1 [ k 2 M 2αβ ( x 1 2 M 1αβ r 1 σ αβ 2 x 2 r 2 ) ] H 2m+n±1s c 1 [ ik x 1 r 1 ( w 0 2 M 1αβ 2 2 M 1αβ ) 1/2 ] × H c 2 2 d 2 +2p2l+h2 e 2 [ k 2 M 2αβ ( y 1 2 M 1αβ r 1 σ αβ 2 y 2 r 2 ) ] H 2p2m+s c 2 [ ik y 1 r 1 ( w 0 2 M 1αβ 2 2 M 1αβ ) 1/2 ],(α,β=x,y)
W yx ( x 1 , y 1 , x 2 , y 2 ,z)= W xy * ( x 2 , y 2 , x 1 , y 1 ,z),
W xz ( x 1 , y 1 , x 2 , y 2 ,z)= m=0 p s=0 n±1 l=0 p h=0 n±1 c 1 =0 2m+n±1s d 1 =0 c 1 /2 e 1 =0 ( 2l+n±1h ) /2 c 2 =0 2p2m+s d 2 =0 c 2 /2 e 2 =0 (2p2l+h)/2 Q 1 ×[ 1 (1) s ]{ Q 3xx Q 4xx [ 1 (1) h ] H c 2 2 d 2 +2p2l+h2 e 2 [ k 2 M 2xx ( y 1 2 M 1xx r 1 σ xx 2 y 2 r 2 ) ] Q 2 Q 5xy [ 1+ (1) h ] H c 1 2 d 1 +2l+n±1h2 e 1 [ k 2 M 2xy ( x 1 2 M 1xy r 1 σ xy 2 x 2 r 2 ) ] },
W yz ( x 1 , y 1 , x 2 , y 2 ,z)= m=0 p s=0 n±1 l=0 p h=0 n±1 c 1 =0 2m+n±1s d 1 =0 c 1 /2 e 1 =0 ( 2l+n±1h ) /2 c 2 =0 2p2m+s d 2 =0 c 2 /2 e 2 =0 (2p2l+h)/2 Q 1 [ 1+ (1) s ]{ Q 2 Q 4xy [ 1 (1) h ] H c 2 2 d 2 +2p2l+h2 e 2 [ k 2 M 2xy ( y 1 2 M 1xy r 1 σ xy 2 y 2 r 2 ) ] + Q 3yy Q 5yy [ 1+ (1) h ] H c 1 2 d 1 +2l+n±1h2 e 1 [ k 2 M 2yy ( x 1 2 M 1yy r 1 σ yy 2 x 2 r 2 ) ] },
W zx ( x 1 , y 1 , x 2 , y 2 ,z)= W xz * ( x 2 , y 2 , x 1 , y 1 ,z),
W zy ( x 1 , y 1 , x 2 , y 2 ,z)= W yz * ( x 2 , y 2 , x 1 , y 1 ,z),
W zz ( x 1 , y 1 , x 2 , y 2 ,z)= W zzxx ( x 1 , y 1 , x 2 , y 2 ,z)+ W zzxy ( x 1 , y 1 , x 2 , y 2 ,z) + W zzyx ( x 1 , y 1 , x 2 , y 2 ,z)+ W zzyy ( x 1 , y 1 , x 2 , y 2 ,z),
W zzxx ( x 1 , y 1 , x 2 , y 2 ,z)= m=0 p s=0 n±1 l=0 p h=0 n±1 c 1 =0 ( 2m+n±1s )/2 d 1 =0 ( 2l+n±1h )/2 c 2 =0 2p2m+s d 2 =0 c 2 /2 e 2 =0 (2p2l+h)/2 Q 6xx Q 7xx × H c 2 2 d 2 +2p2l+h2 e 2 [ k 2 M 2xx ( y 1 2 M 1xx r 1 σ xx 2 y 2 r 2 ) ]{ x 1 f 1 =0 2m+n±1s2 c 1 g 1 =0 f 1 /2 Q 8xx ( 2m+n±1s2 c 1 f 1 ) × H 2m+n±1s2 c 1 f 1 ( k x 1 2 M 1xx r 1 ) [ 2i M 2xx x 2 H f 1 2 g 1 +2l+n±1h2 d 1 ( k 2 M 2xx ( x 1 2 M 1xx r 1 σ xx 2 x 2 r 2 ) ) H f 1 2 g 1 +2l+n±1h2 d 1 +1 ( k 2 M 2xx ( x 1 2 M 1xx r 1 σ xx 2 x 2 r 2 ) ) ] f 2 =0 2m+n±1s2 c 1 +1 g 2 =0 f 2 /2 Q 9xx ( 2m+n±1s2 c 1 +1 f 2 ) × H 2m+n±1s2 c 1 +1 f 2 ( k x 1 2 M 1xx r 1 ) [ 2i M 2xx x 2 H f 2 2 g 2 +2l+n±1h2 d 1 ( k 2 M 2xx ( x 1 2 M 1xx r 1 σ xx 2 x 2 r 2 ) ) H f 2 2 g 2 +2l+n±1h2 d 1 +1 ( k 2 M 2xx ( x 1 2 M 1xx r 1 σ xx 2 x 2 r 2 ) ) ] },
W zzyy ( x 1 , y 1 , x 2 , y 2 ,z)= m=0 p s=0 n±1 l=0 p h=0 n±1 c 2 =0 2m+n±1s d 2 =0 c 2 /2 e 2 =0 ( 2l+n±1h ) /2 c 1 =0 ( 2p2m+s )/2 d 1 =0 ( 2p2l+h )/2 Q 6yy ( 2m+n±1s c 2 ) × ( 2i ) ( c 2 2 d 2 +n±12 e 2 +4p2m+s2 c 1 2 d 1 +2) ( 1 2 w 0 2 M 1yy ) ( 2m+n±1s ) /2 ( 2 2 w 0 ) n±12 e 2 +4p2m+s2 c 1 2 d 1 × ( 2l+n±1h )! e 2 !(2l+n±1h2 e 2 )! ( 2p2m+s )! c 1 !( 2p2m+s2 c 1 )! ( 2p2l+h )! d 1 !( 2p2l+h2 d 1 )! 1 ( M 1yy ) 2p2m+s2 c 1 +3 × H c 2 2 d 2 +2l+n±1h2 e 2 [ k 2 M 2yy ( x 1 2 M 1yy r 1 σ yy 2 x 2 r 2 ) ] H 2m+n±1s c 2 [ ik x 1 ( w 0 2 M 1yy 2 r 1 2 2 M 1yy r 1 2 ) 1/2 ] ×{ y 1 f 1 =0 2p2m+s2 c 1 g 1 =0 f 1 /2 Q 8yy ( 2p2m+s2 c 1 f 1 ) H 2p2m+s2 c 1 f 1 ( k y 1 2 M 1yy r 1 ) [ 2i M 2yy y 2 H 2p2l+h2 d 2 + f 1 2 g 1 ( k 2 M 2yy ( y 1 2 M 1yy r 1 σ yy 2 y 2 r 2 ) ) H 2p2l+h2 d 1 + f 1 2 g 1 +1 ( k 2 M 2yy ( y 1 2 M 1yy r 1 σ yy 2 y 2 r 2 ) ) ] f 2 =0 2p2m+s2 c 1 +1 g 2 =0 f 2 /2 Q 9yy ( 2p2m+s2 c 1 +1 f 2 ) H 2p2m+s2 c 1 +1 f 2 ( k y 1 2 M 1yy r 1 ) [ 2i M 2yy y 2 H 2p2l+h2 d 1 + f 2 2 g 2 ( k 2 M 2yy ( y 1 2 M 1yy r 1 σ yy 2 y 2 r 2 ) ) H 2p2l+h2 d 1 + f 2 2 g 2 +1 ( k 2 M 2yy ( y 1 2 M 1yy r 1 σ yy 2 y 2 r 2 ) ) ] }.
W zzyx ( x 1 , y 1 , x 2 , y 2 ,z)= [ W zzxy ( x 2 , y 2 , x 1 , y 1 ,z) ] * ,
W zzxy ( x 1 , y 1 , x 2 , y 2 ,z)= m=0 p s=0 n±1 l=0 p h=0 n±1 c 1 =0 ( 2m+n±1s )/2 d 1 =0 ( 2l+n±1h )/2 c 2 =0 2p2m+s d 2 =0 c 2 /2 e 2 =0 ( 2p2l+h ) /2 Q 5xy Q 6xy Q 7xy ×{ x 1 f 1 =0 2m+n±1s2 c 1 g 1 =0 f 1 /2 Q 8xy ( 2m+n±1s2 c 1 f 1 ) H 2m+n±1s2 c 1 f 1 ( k x 1 2 M 1xy r 1 ) H f 1 2 g 1 +2l+n±1h2 d 1 [ k 2 M 2xy ( x 1 2 M 1xy r 1 σ xy 2 x 2 r 2 ) ] f 2 =0 2m+n±1s2 c 1 +1 g 2 =0 f 2 /2 Q 9xy ( 2m+n±1s2 c 1 +1 f 2 ) × H 2m+n±1s2 c 1 +1 f 2 ( k x 1 2 M 1xy r 1 ) H f 2 2 g 2 +2l+n±1h2 d 1 [ k 2 M 2xy ( x 1 2 M 1xy r 1 σ xy 2 x 2 r 2 ) ] },
M 1αβ =1/ w 0 2 +1/(2 σ αβ 2 )ik/(2 r 1 ), (α,β=x,y)
M 2αβ =1/ w 0 2 +1/(2 σ αβ 2 )+ik/(2 r 2 )1/(4 M 1αβ σ αβ 4 ), (α,β=x,y)
Q 1 = k 2 zexp[ ik( r 1 r 2 ) ] 16 r 1 2 r 2 2 2 5( 2p+n±1 ) /2 (p!) 2 (1) d 1 + d 2 + e 1 + e 2 i s (i) h ( 2i ) ( c 1 + c 2 2 d 1 2 d 2 +n±1+2p2 e 1 2 e 2 +1) × ( 2 2 w 0 ) n±1+2p2 e 1 2 e 2 ( 2m+n±1s c 1 )( 2p2m+s c 2 )( p m )( p l )( n±1 s )( n±1 h ) × c 1 ! d 1 !( c 1 2 d 1 )! c 2 ! d 2 !( c 2 2 d 2 )! ( 2l+n±1h )! e 1 !(2l+n±1h2 e 1 )! (2p2l+h)! e 2 !(2p2l+h2 e 2 )! ,
Q 2 = i B xy M 1xy 1 ( M 2xy ) c 1 + c 2 2 d 1 2 d 2 +n±1+2p2 e 1 2 e 2 +3 [ 2 σ xy 2 ( w 0 2 M 1xy 2 2 M 1xy ) 1/2 ] c 1 + c 2 2 d 1 2 d 2 × ( 1 2 M 1xy w 0 2 ) ( n±1+2p )/2 exp{ k 2 ( x 1 2 + y 1 2 ) 4 M 1xy r 1 2 k 2 4 M 2xy [ ( x 2 r 2 x 1 2 M 1xy r 1 σ xy 2 ) 2 + ( y 2 r 2 y 1 2 M 1xy r 1 σ xy 2 ) 2 ] } × H 2m+n±1s c 1 [ ik x 1 ( w 0 2 M 1xy 2 r 1 2 2 M 1xy r 1 2 ) 1/2 ] H 2p2m+s c 2 ( ik y 1 r 1 ( w 0 2 M 1xy 2 2 M 1xy ) 1/2 ),
Q 3αα = 1 M 1αα 1 ( M 2αα ) c 1 + c 2 2 d 1 2 d 2 +n±1+2p2 e 1 2 e 2 +3 [ 2 σ αα 2 ( w 0 2 M 1αα 2 2 M 1αα ) 1/2 ] c 1 + c 2 2 d 1 2 d 2 × ( w 0 2 M 1αα 2 w 0 2 M 1αα ) ( 2p+n±1 ) /2 exp{ k 2 ( x 1 2 + y 1 2 ) 4 M 1αα r 1 2 k 2 4 M 2αα [ ( x 2 r 2 x 1 2 M 1αα r 1 σ αα 2 ) 2 + ( y 2 r 2 y 1 2 M 1αα r 1 σ αα 2 ) 2 ] } × H 2m+n±1s c 1 [ ik x 1 r 1 ( w 0 2 M 1αα 2 2 M 1αα ) 1/2 ] H 2p2m+s c 2 [ ik y 1 r 1 ( w 0 2 M 1αα 2 2 M 1αα ) 1/2 ],(α=x,y)
Q 4xα =2i M 2xα x 2 H c 1 2 d 1 +2l+n±1h2 e 1 [ k 2 M 2xα ( x 1 2 M 1xα r 1 σ xα 2 x 2 r 2 ) ] H c 1 2 d 1 +2l+n±1h2 e 1 +1 [ k 2 M 2xα ( x 1 2 M 1xα r 1 σ xα 2 x 2 r 2 ) ],(α=x,y)
Q 5αy =2i M 2αy y 2 H c 2 2 d 2 +2p2l+h2 e 2 [ k 2 M 2αy ( y 1 2 M 1αy r 1 σ αy 2 y 2 r 2 ) ] H c 2 2 d 2 +2p2l+h2 e 2 +1 [ k 2 M 2αy ( y 1 2 M 1αy r 1 σ αy 2 y 2 r 2 ) ],(α=x,y)
Q 6αβ = A αβ exp{ k 2 ( x 1 2 + y 1 2 ) 4 M 1αβ r 1 2 k 2 4 M 2αβ [ ( x 2 r 2 x 1 2 M 1αβ r 1 σ αβ 2 ) 2 + ( y 2 r 2 y 1 2 M 1αβ r 1 σ αβ 2 ) 2 ] } × k 2 exp[ ik( r 1 r 2 ) ] 16 r 1 2 r 2 2 2 5( 2p+n±1 )/2 (p!) 2 ( p m )( p l )( n±1 s )( n±1 h ) i s (i) h (1) c 1 + d 1 + d 2 + e 2 [ 1+ C α (1) s ] 1+ C β (1) h 2 ( 2 c 1 +1 )/2 × 1 ( M 2αβ ) n±12 d 1 + c 2 2 d 2 +2p2 e 2 +3 [ 2 σ αβ 2 ( w 0 2 M 1αβ 2 2 M 1αβ ) 1/2 ] c 2 2 d 2 c 2 ! d 2 !( c 2 2 d 2 )! ,(α,β=x,y)
Q 7xα =( 2p2m+s c 2 ) ( 2i ) ( 2m+2( n±1 )s2 c 1 2 d 1 + c 2 2 d 2 +2p2 e 2 +2 ) ( 2 2 w 0 ) 2m+2( n±1 )s2 c 1 2 d 1 +2p2 e 2 × ( 2m+n±1s )! c 1 !( 2m+n±1s2 c 1 )! ( 2l+n±1h )! d 1 !( 2l+n±1h2 d 1 )! (2p2l+h)! e 2 !(2p2l+h2 e 2 )! ( 1 2 M 1xα w 0 2 ) ( 2p2m+s )/2 × 1 ( M 1xα ) 2m+n±1s2 c 1 +3 H 2p2m+s c 2 [ ik y 1 ( w 0 2 M 1xα 2 r 1 2 2 M 1xα r 1 2 ) 1/2 ],(α=x,y)
Q 8αβ = 2 M 1αβ (1) g 1 (2i) ( f 1 2 g 1 1) f 1 ! g 1 !( f 1 2 g 1 )! 1 ( M 2αβ ) f 1 2 g 1 ( 2 i M 1αβ σ αβ 2 ) f 1 2 g 1 ,(α,β=x,y)
Q 9αβ = (1) g 2 (2i) ( f 2 2 g 2 ) f 2 ! g 2 !( f 2 2 g 2 )! 1 ( M 2αβ ) f 2 2 g 2 ( 2 i M 1αβ σ αβ 2 ) f 2 2 g 2 .(α,β=x,y)
exp[ ( xy ) 2 ] H n ( ax )dx= π ( 1 a 2 ) n/2 H n ( ay ( 1 a 2 ) 1/2 ),
x n exp[ ( xβ ) 2 ] dx= ( 2i ) n π H n ( iβ ),
H n ( x+y )= 1 2 n/2 k=0 n ( n k ) H k ( 2 x ) H nk ( 2 y ),
H n ( x )= k=0 n/2 ( 1 ) k n! k!( n2k )! ( 2x ) n2k .
I(x,y,z)= I x (x,y,z)+ I y (x,y,z)+ I z (x,y,z) = W xx (x,y,x,y,z)+ W yy (x,y,x,y,z)+ W zz (x,y,x,y,z),
P(x,y,z)= p 1 (x,y,z) p 2 (x,y,z) p 1 (x,y,z)+ p 2 (x,y,z)+ p 3 (x,y,z) ,
μ( x 1 x 2 , y 1 y 2 ,z)= Tr W ( x 1 , y 1 , x 2 , y 2 ,z) Tr W ( x 1 , y 1 , x 1 , y 1 ,z) Tr W ( x 2 , y 2 , x 2 , y 2 ,z) ,

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