Abstract

Based on the generalized Raleigh–Sommerfeld diffraction integrals, analytical nonparaxial propagation formulas for the elements of the cross-spectral density matrix of a vector partially coherent dark hollow beam (DHB) in free space are derived. The effect of spatial coherence and beam waist sizes on the statistical properties of a nonparaxial vector DHB is studied numerically. It is found that one can modulate the statistical properties of a nonparaxial vector DHB by varying its initial spatial coherence, which will be useful in some applications where nonparaxial beams are commonly encountered.

© 2013 Optical Society of America

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2013 (1)

Y. Yuan, Y. Chen, C. Liang, Y. Cai, and Y. Baykal, “Effect of spatial coherence on the scintillation properties of a dark hollow beam in turbulent atmosphere,” Appl. Phys. B 110, 519–529 (2013).
[CrossRef]

2012 (6)

G. Taherabadi, M. Alavynejad, F. D. Kashani, B. Ghafary, and M. Yousefi, “Changes in the spectral degree of polarization of a partially coherent dark hollow beam in the turbulent atmosphere for on-axis and off-axis propagation point,” Opt. Commun. 285, 2017–2021 (2012).
[CrossRef]

G. Zhou, Y. Cai, and X. Chu, “Propagation of a partially coherent hollow vortex Gaussian beam in turbulent atmosphere,” Opt. Express 20, 9897–9910 (2012).
[CrossRef]

Q. Sun, K. Zhou, G. Fang, G. Zhang, Z. Liu, and S. Liu, “Hollow sinh-Gaussian beams and their paraxial properties,” Opt. Express 20, 9682–9691 (2012).
[CrossRef]

F. Wang, Y. Cai, Y. Dong, and O. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 100, 051108 (2012).
[CrossRef]

Y. Dong, F. Wang, C. Zhao, and Y. Cai, “Effect of spatial coherence on propagation, tight focusing and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86, 013840 (2012).
[CrossRef]

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

2011 (10)

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, 2722–2724 (2011).
[CrossRef]

Y. Dong, Y. Cai, C. Zhao, and M. Yao, “Statistics properties of a cylindrical vector partially coherent beam,” Opt. Express 19, 5979–5992 (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, 1111–1117 (2011).
[CrossRef]

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

Y. Yang, X. Li, and K. Duan, “Nonparaxial propagation of vectorial hollow Gaussian beams diffracted at an annular aperture,” Opt. Eng. 50, 078001 (2011).
[CrossRef]

X. Li and Y. Cai, “Nonparaxial propagation of a partially coherent dark hollow beam,” Appl. Phys. B 102, 205–213 (2011).
[CrossRef]

Y. Nie, H. Ma, X. Li, W. Hu, and J. Yang, “Generation of dark hollow femtosecond pulsed beam by phase-only liquid crystal spatial light modulator,” Appl. Opt. 50, 4174–4179 (2011).
[CrossRef]

B. K. Yadav and H. C. Kandpal, “Spectral anomalies of polychromatic DHGB and its applications in FSO,” J. Lightwave Technol. 29, 960–966 (2011).
[CrossRef]

X. Li, F. Wang, and Y. Cai, “An alternative model for a partially coherent elliptical dark hollow beam,” Opt. Laser Technol. 43, 577–585 (2011).
[CrossRef]

Y. Dong, Y. Cai, and C. Zhao, “Degree of polarization of a tightly focused partially coherent dark hollow beam,” Appl. Phys. B 105, 405–414 (2011).
[CrossRef]

2010 (8)

H. Ma, P. Zhou, X. Wang, Y. Ma, F. Xi, X. Xu, and Z. Liu, “Near-diffraction-limited annular flattop beam shaping with dual phase only liquid crystal spatial light modulators,” Opt. Express 18, 8251–8260 (2010).
[CrossRef]

Y. Qiu, Z. Chen, and L. Liu, “Partially coherent dark hollow beams propagating through real ABCD optical systems in a turbulent atmosphere,” J. Mod. Opt. 57, 662–669 (2010).
[CrossRef]

H. Wang and X. Li, “Propagation of partially coherent controllable dark hollow beams with various symmetries in turbulent atmosphere,” Opt. Lasers Eng. 48, 48–57 (2010).
[CrossRef]

G. Zhou, “Non-paraxial investigation in the far field properties of controllable dark-hollow beams diffracted by a circular aperture,” J. Opt. Soc. Am. A 27, 890–894 (2010).
[CrossRef]

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

S. Zhu and Y. Cai, “M2-factor of a stochastic electromagnetic beam in a Gaussian cavity,” Opt. Express 18, 27567–27581 (2010).
[CrossRef]

Z. Tong, Y. Cai, and O. Korotkova, “Ghost imaging with electromagnetic stochastic beams,” Opt. Commun. 283, 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. B 99, 317–323 (2010).
[CrossRef]

2009 (7)

2008 (13)

H. T. Eyyuboğlu, “Propagation and coherence properties of higher order partially coherent dark hollow beams in turbulence,” Opt. Laser Technol. 40, 156–166 (2008).
[CrossRef]

C. Zhao, Y. Cai, F. Wang, X. Lu, and Y. Wang, “Generation of a high-quality partially coherent dark hollow beam with a multimode fiber,” Opt. Lett. 33, 1389–1391 (2008).
[CrossRef]

Y. Chen, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation properties of dark hollow beams in a weak turbulent atmosphere,” Appl. Phys. B 90, 87–92 (2008).
[CrossRef]

Z. Liu, J. Dai, X. Sun, and S. Liu, “Generation of hollow Gaussian beam by phase-only filtering,” Opt. Express 16, 19926–19933 (2008).
[CrossRef]

Y. Zhang, “Generation of thin and hollow beams by the axicon with a large open angle,” Opt. Commun. 281, 508–514 (2008).
[CrossRef]

Y. Cai, Z. Wang, and Q. Lin, “An alternative theoretical model for an anomalous hollow beam,” Opt. Express 16, 15254–15267 (2008).
[CrossRef]

O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281, 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. Express 16, 15834–15846 (2008).
[CrossRef]

D. Deng, H. Yu, S. Xu, G. Tian, and Z. Fan, “Nonparaxial propagation of vectorial hollow Gaussian beams,” J. Opt. Soc. Am. B 25, 83–87 (2008).
[CrossRef]

G. Wu, Q. Lou, and J. Zhou, “Analytical vectorial structure of hollow Gaussian beams in the far field,” Opt. Express 16, 6417–6424 (2008).
[CrossRef]

Y. Cai and F. Wang, “Partially coherent anomalous hollow beam and its paraxial propagation,” Phys. Lett. A 372, 4654–4660 (2008).
[CrossRef]

Z. Mei and D. Zhao, “Non-paraxial propagation of controllable dark-hollow beams,” J. Opt. Soc. Am. A 25, 537–542 (2008).
[CrossRef]

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, 2266–2268 (2008).
[CrossRef]

2007 (2)

Y. Cai, “Model for an anomalous hollow beam and its paraxial propagation,” Opt. Lett. 32, 3179–3181 (2007).
[CrossRef]

X. Lu and Y. Cai, “Partially coherent circular and elliptical dark hollow beams and their paraxial propagations,” Phys. Lett. A 369, 157–166 (2007).
[CrossRef]

2006 (5)

2005 (4)

Z. Mei and D. Zhao, “Controllable dark-hollow beams and their propagation characteristics,” J. Opt. Soc. Am. A 22, 1898–1902 (2005).
[CrossRef]

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

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A 7, 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, 35–43 (2005).
[CrossRef]

2004 (4)

2003 (2)

Y. Cai, X. Lu, and Q. Lin, “Hollow Gaussian beams and their propagation properties,” Opt. Lett. 28, 1084–1086 (2003).
[CrossRef]

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

2002 (1)

T. Setala, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

2001 (1)

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

2000 (1)

1997 (1)

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

1994 (2)

H. S. Lee, B. W. Stewart, K. Choi, and H. Fenichel, “Holographic nondiverging hollow beam,” Phys. Rev. A 49, 4922–4927 (1994).
[CrossRef]

D. F. V. James, “Changes of polarization of light beams on propagation in free space,” J. Opt. Soc. Am. A 11, 1641–1643 (1994).
[CrossRef]

1987 (1)

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Alavynejad, M.

G. Taherabadi, M. Alavynejad, F. D. Kashani, B. Ghafary, and M. Yousefi, “Changes in the spectral degree of polarization of a partially coherent dark hollow beam in the turbulent atmosphere for on-axis and off-axis propagation point,” Opt. Commun. 285, 2017–2021 (2012).
[CrossRef]

Baykal, Y.

Borghi, R.

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

Cai, Y.

Y. Yuan, Y. Chen, C. Liang, Y. Cai, and Y. Baykal, “Effect of spatial coherence on the scintillation properties of a dark hollow beam in turbulent atmosphere,” Appl. Phys. B 110, 519–529 (2013).
[CrossRef]

G. Zhou, Y. Cai, and X. Chu, “Propagation of a partially coherent hollow vortex Gaussian beam in turbulent atmosphere,” Opt. Express 20, 9897–9910 (2012).
[CrossRef]

F. Wang, Y. Cai, Y. Dong, and O. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 100, 051108 (2012).
[CrossRef]

Y. Dong, F. Wang, C. Zhao, and Y. Cai, “Effect of spatial coherence on propagation, tight focusing and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86, 013840 (2012).
[CrossRef]

G. Wu, F. Wang, and Y. Cai, “Coherence and polarization properties of a radially polarized beam with variable spatial coherence,” Opt. Express 20, 28301–28318 (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, 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, 2722–2724 (2011).
[CrossRef]

Y. Dong, Y. Cai, C. Zhao, and M. Yao, “Statistics properties of a cylindrical vector partially coherent beam,” Opt. Express 19, 5979–5992 (2011).
[CrossRef]

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

Y. Dong, Y. Cai, and C. Zhao, “Degree of polarization of a tightly focused partially coherent dark hollow beam,” Appl. Phys. B 105, 405–414 (2011).
[CrossRef]

X. Li, F. Wang, and Y. Cai, “An alternative model for a partially coherent elliptical dark hollow beam,” Opt. Laser Technol. 43, 577–585 (2011).
[CrossRef]

X. Li and Y. Cai, “Nonparaxial propagation of a partially coherent dark hollow beam,” Appl. Phys. B 102, 205–213 (2011).
[CrossRef]

S. Zhu and Y. Cai, “Spectral shift of a twisted electromagnetic Gaussian Schell-model beam focused by a thin lens,” Appl. Phys. B 99, 317–323 (2010).
[CrossRef]

S. Zhu and Y. Cai, “M2-factor of a stochastic electromagnetic beam in a Gaussian cavity,” Opt. Express 18, 27567–27581 (2010).
[CrossRef]

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

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

O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for LIDAR systems operating through turbulent atmosphere,” Appl. Phys. B 94, 681–690 (2009).
[CrossRef]

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

Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “M2-factor coherent and partially coherent dark hollow beams propagating in turbulent atmosphere,” Opt. Express 17, 17344–17356 (2009).
[CrossRef]

C. Zhao, Y. Cai, F. Wang, X. Lu, and Y. Wang, “Generation of a high-quality partially coherent dark hollow beam with a multimode fiber,” Opt. Lett. 33, 1389–1391 (2008).
[CrossRef]

Y. Chen, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation properties of dark hollow beams in a weak turbulent atmosphere,” Appl. Phys. B 90, 87–92 (2008).
[CrossRef]

Y. Cai and F. Wang, “Partially coherent anomalous hollow beam and its paraxial propagation,” Phys. Lett. A 372, 4654–4660 (2008).
[CrossRef]

Y. Cai, Z. Wang, and Q. Lin, “An alternative theoretical model for an anomalous hollow beam,” Opt. Express 16, 15254–15267 (2008).
[CrossRef]

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, 2266–2268 (2008).
[CrossRef]

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

Y. Cai, “Model for an anomalous hollow beam and its paraxial propagation,” Opt. Lett. 32, 3179–3181 (2007).
[CrossRef]

X. Lu and Y. Cai, “Partially coherent circular and elliptical dark hollow beams and their paraxial propagations,” Phys. Lett. A 369, 157–166 (2007).
[CrossRef]

Y. Cai and L. Zhang, “Coherent and partially coherent dark hollow beams with rectangular symmetry and paraxial propagation,” J. Opt. Soc. Am. B 23, 1398–1407 (2006).
[CrossRef]

Y. Cai and S. He, “Propagation of hollow Gaussian beams through apertured paraxial optical systems,” J. Opt. Soc. Am. A 23, 1410–1418 (2006).
[CrossRef]

Y. Cai and D. Ge, “Propagation of various dark hollow beams through an apertured paraxial ABCD optical system,” Phys. Lett. A 357, 72–80 (2006).
[CrossRef]

Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14, 1353–1367 (2006).
[CrossRef]

Y. Cai and Q. Lin, “Hollow elliptical Gaussian beam and its propagation through aligned and misaligned paraxial optical systems,” J. Opt. Soc. Am. A 21, 1058–1065 (2004).
[CrossRef]

Y. Cai, X. Lu, and Q. Lin, “Hollow Gaussian beams and their propagation properties,” Opt. Lett. 28, 1084–1086 (2003).
[CrossRef]

Chávez-Cerda, S.

Chen, Y.

Y. Yuan, Y. Chen, C. Liang, Y. Cai, and Y. Baykal, “Effect of spatial coherence on the scintillation properties of a dark hollow beam in turbulent atmosphere,” Appl. Phys. B 110, 519–529 (2013).
[CrossRef]

Y. Chen, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation properties of dark hollow beams in a weak turbulent atmosphere,” Appl. Phys. B 90, 87–92 (2008).
[CrossRef]

Chen, Z.

Y. Qiu, Z. Chen, and L. Liu, “Partially coherent dark hollow beams propagating through real ABCD optical systems in a turbulent atmosphere,” J. Mod. Opt. 57, 662–669 (2010).
[CrossRef]

Choi, K.

H. S. Lee, B. W. Stewart, K. Choi, and H. Fenichel, “Holographic nondiverging hollow beam,” Phys. Rev. A 49, 4922–4927 (1994).
[CrossRef]

Chu, X.

Dai, J.

Deng, D.

D. Deng and Q. Guo, “Exact nonparaxial propagation of a hollow Gaussian beam,” J. Opt. Soc. Am. A 26, 2044–2049 (2009).
[CrossRef]

D. Deng, H. Yu, S. Xu, G. Tian, and Z. Fan, “Nonparaxial propagation of vectorial hollow Gaussian beams,” J. Opt. Soc. Am. B 25, 83–87 (2008).
[CrossRef]

Dogariu, A.

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248, 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, 051108 (2012).
[CrossRef]

Y. Dong, F. Wang, C. Zhao, and Y. Cai, “Effect of spatial coherence on propagation, tight focusing and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86, 013840 (2012).
[CrossRef]

Y. Dong, Y. Cai, C. Zhao, and M. Yao, “Statistics properties of a cylindrical vector partially coherent beam,” Opt. Express 19, 5979–5992 (2011).
[CrossRef]

Y. Dong, Y. Cai, and C. Zhao, “Degree of polarization of a tightly focused partially coherent dark hollow beam,” Appl. Phys. B 105, 405–414 (2011).
[CrossRef]

Duan, K.

Y. Yang, X. Li, and K. Duan, “Nonparaxial propagation of vectorial hollow Gaussian beams diffracted at an annular aperture,” Opt. Eng. 50, 078001 (2011).
[CrossRef]

K. Duan and B. Lü, “Partially coherent vectorial nonparaxial beams,” J. Opt. Soc. Am. A 21, 1924–1932 (2004).
[CrossRef]

Ellis, J.

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

Erdelyi, A.

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

Eyyuboglu, H. T.

Fan, Z.

Fang, G.

Fenichel, H.

H. S. Lee, B. W. Stewart, K. Choi, and H. Fenichel, “Holographic nondiverging hollow beam,” Phys. Rev. A 49, 4922–4927 (1994).
[CrossRef]

Friberg, A. T.

T. Setala, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

Gao, W.

J. Yin, W. Gao, and Y. Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics, E. Wolf, ed., Vol. 44 (North-Holland, 2003), pp. 119–204.

Gao, Z.

Z. Gao and B. Lu, “Nonparaxial dark-hollow Gaussian beam,” Chin. Phys. Lett. 23, 106–109 (2006).
[CrossRef]

Ge, D.

Y. Cai and D. Ge, “Propagation of various dark hollow beams through an apertured paraxial ABCD optical system,” Phys. Lett. A 357, 72–80 (2006).
[CrossRef]

Ghafary, B.

G. Taherabadi, M. Alavynejad, F. D. Kashani, B. Ghafary, and M. Yousefi, “Changes in the spectral degree of polarization of a partially coherent dark hollow beam in the turbulent atmosphere for on-axis and off-axis propagation point,” Opt. Commun. 285, 2017–2021 (2012).
[CrossRef]

Gori, F.

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

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Guo, Q.

D. Deng and Q. Guo, “Exact nonparaxial propagation of a hollow Gaussian beam,” J. Opt. Soc. Am. A 26, 2044–2049 (2009).
[CrossRef]

Gutiérrez-Vega, J. C.

He, S.

Hirano, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

Hu, W.

Iturbe-Castillo, M. D.

James, D. F. V.

Kaivola, M.

T. Setala, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

Kandpal, H. C.

Kashani, F. D.

G. Taherabadi, M. Alavynejad, F. D. Kashani, B. Ghafary, and M. Yousefi, “Changes in the spectral degree of polarization of a partially coherent dark hollow beam in the turbulent atmosphere for on-axis and off-axis propagation point,” Opt. Commun. 285, 2017–2021 (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, 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, 1111–1117 (2011).
[CrossRef]

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

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

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

O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for LIDAR systems operating through turbulent atmosphere,” Appl. Phys. B 94, 681–690 (2009).
[CrossRef]

Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “M2-factor coherent and partially coherent dark hollow beams propagating in turbulent atmosphere,” Opt. Express 17, 17344–17356 (2009).
[CrossRef]

O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281, 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. Express 16, 15834–15846 (2008).
[CrossRef]

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, 2266–2268 (2008).
[CrossRef]

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A 7, 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, 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, 1173–1175 (2004).
[CrossRef]

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

Kuga, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

Lee, H. S.

H. S. Lee, B. W. Stewart, K. Choi, and H. Fenichel, “Holographic nondiverging hollow beam,” Phys. Rev. A 49, 4922–4927 (1994).
[CrossRef]

Li, X.

Y. Nie, H. Ma, X. Li, W. Hu, and J. Yang, “Generation of dark hollow femtosecond pulsed beam by phase-only liquid crystal spatial light modulator,” Appl. Opt. 50, 4174–4179 (2011).
[CrossRef]

X. Li, F. Wang, and Y. Cai, “An alternative model for a partially coherent elliptical dark hollow beam,” Opt. Laser Technol. 43, 577–585 (2011).
[CrossRef]

X. Li and Y. Cai, “Nonparaxial propagation of a partially coherent dark hollow beam,” Appl. Phys. B 102, 205–213 (2011).
[CrossRef]

Y. Yang, X. Li, and K. Duan, “Nonparaxial propagation of vectorial hollow Gaussian beams diffracted at an annular aperture,” Opt. Eng. 50, 078001 (2011).
[CrossRef]

H. Wang and X. Li, “Propagation of partially coherent controllable dark hollow beams with various symmetries in turbulent atmosphere,” Opt. Lasers Eng. 48, 48–57 (2010).
[CrossRef]

Liang, C.

Y. Yuan, Y. Chen, C. Liang, Y. Cai, and Y. Baykal, “Effect of spatial coherence on the scintillation properties of a dark hollow beam in turbulent atmosphere,” Appl. Phys. B 110, 519–529 (2013).
[CrossRef]

Lin, Q.

Liu, L.

Y. Qiu, Z. Chen, and L. Liu, “Partially coherent dark hollow beams propagating through real ABCD optical systems in a turbulent atmosphere,” J. Mod. Opt. 57, 662–669 (2010).
[CrossRef]

Liu, S.

Liu, X.

Liu, Z.

Lou, Q.

Lu, B.

Z. Gao and B. Lu, “Nonparaxial dark-hollow Gaussian beam,” Chin. Phys. Lett. 23, 106–109 (2006).
[CrossRef]

Lu, X.

Lü, B.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California, 1966).

Ma, H.

Ma, Y.

Magnus, W.

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

Mei, Z.

Mondello, A.

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

Nie, Y.

Oberhettinger, F.

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

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[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 3, 1–9 (2001).
[CrossRef]

Ponomarenko, S.

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

Qiu, Y.

Y. Qiu, Z. Chen, and L. Liu, “Partially coherent dark hollow beams propagating through real ABCD optical systems in a turbulent atmosphere,” J. Mod. Opt. 57, 662–669 (2010).
[CrossRef]

Qu, J.

Salem, M.

Santarsiero, M.

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

Sasada, H.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

Setala, T.

T. Setala, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

Shevchenko, A.

T. Setala, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

Shimizu, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

Shiokawa, N.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

Shirai, T.

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A 7, 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 3, 1–9 (2001).
[CrossRef]

Stewart, B. W.

H. S. Lee, B. W. Stewart, K. Choi, and H. Fenichel, “Holographic nondiverging hollow beam,” Phys. Rev. A 49, 4922–4927 (1994).
[CrossRef]

Sun, Q.

Sun, X.

Taherabadi, G.

G. Taherabadi, M. Alavynejad, F. D. Kashani, B. Ghafary, and M. Yousefi, “Changes in the spectral degree of polarization of a partially coherent dark hollow beam in the turbulent atmosphere for on-axis and off-axis propagation point,” Opt. Commun. 285, 2017–2021 (2012).
[CrossRef]

Tian, G.

Tong, Z.

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

Torii, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

Wang, F.

Y. Dong, F. Wang, C. Zhao, and Y. Cai, “Effect of spatial coherence on propagation, tight focusing and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86, 013840 (2012).
[CrossRef]

F. Wang, Y. Cai, Y. Dong, and O. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 100, 051108 (2012).
[CrossRef]

G. Wu, F. Wang, and Y. Cai, “Coherence and polarization properties of a radially polarized beam with variable spatial coherence,” Opt. Express 20, 28301–28318 (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, 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, 2722–2724 (2011).
[CrossRef]

X. Li, F. Wang, and Y. Cai, “An alternative model for a partially coherent elliptical dark hollow beam,” Opt. Laser Technol. 43, 577–585 (2011).
[CrossRef]

C. Zhao, Y. Cai, F. Wang, X. Lu, and Y. Wang, “Generation of a high-quality partially coherent dark hollow beam with a multimode fiber,” Opt. Lett. 33, 1389–1391 (2008).
[CrossRef]

Y. Cai and F. Wang, “Partially coherent anomalous hollow beam and its paraxial propagation,” Phys. Lett. A 372, 4654–4660 (2008).
[CrossRef]

Wang, H.

H. Wang and X. Li, “Propagation of partially coherent controllable dark hollow beams with various symmetries in turbulent atmosphere,” Opt. Lasers Eng. 48, 48–57 (2010).
[CrossRef]

Wang, X.

Wang, Y.

Wang, Z.

Watson, E.

O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for LIDAR systems operating through turbulent atmosphere,” Appl. Phys. B 94, 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, 35–43 (2005).
[CrossRef]

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

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

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

O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source,” Opt. Lett. 29, 1173–1175 (2004).
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Xi, F.

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Yang, J.

Yang, Y.

Y. Yang, X. Li, and K. Duan, “Nonparaxial propagation of vectorial hollow Gaussian beams diffracted at an annular aperture,” Opt. Eng. 50, 078001 (2011).
[CrossRef]

Yao, M.

Yin, J.

J. Yin, W. Gao, and Y. Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics, E. Wolf, ed., Vol. 44 (North-Holland, 2003), pp. 119–204.

Yousefi, M.

G. Taherabadi, M. Alavynejad, F. D. Kashani, B. Ghafary, and M. Yousefi, “Changes in the spectral degree of polarization of a partially coherent dark hollow beam in the turbulent atmosphere for on-axis and off-axis propagation point,” Opt. Commun. 285, 2017–2021 (2012).
[CrossRef]

Yu, H.

Yuan, Y.

Y. Yuan, Y. Chen, C. Liang, Y. Cai, and Y. Baykal, “Effect of spatial coherence on the scintillation properties of a dark hollow beam in turbulent atmosphere,” Appl. Phys. B 110, 519–529 (2013).
[CrossRef]

Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “M2-factor coherent and partially coherent dark hollow beams propagating in turbulent atmosphere,” Opt. Express 17, 17344–17356 (2009).
[CrossRef]

Zhan, Q.

Zhang, G.

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, 1111–1117 (2011).
[CrossRef]

Y. Cai and L. Zhang, “Coherent and partially coherent dark hollow beams with rectangular symmetry and paraxial propagation,” J. Opt. Soc. Am. B 23, 1398–1407 (2006).
[CrossRef]

Zhang, Y.

Y. Zhang, “Generation of thin and hollow beams by the axicon with a large open angle,” Opt. Commun. 281, 508–514 (2008).
[CrossRef]

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Zhao, D.

Zhou, G.

Zhou, J.

Zhou, K.

Zhou, P.

Zhu, S.

Zhu, Y.

J. Yin, W. Gao, and Y. Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics, E. Wolf, ed., Vol. 44 (North-Holland, 2003), pp. 119–204.

Adv. Opt. Photon. (1)

Appl. Opt. (1)

Appl. Phys. B (6)

Y. Chen, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation properties of dark hollow beams in a weak turbulent atmosphere,” Appl. Phys. B 90, 87–92 (2008).
[CrossRef]

Y. Dong, Y. Cai, and C. Zhao, “Degree of polarization of a tightly focused partially coherent dark hollow beam,” Appl. Phys. B 105, 405–414 (2011).
[CrossRef]

Y. Yuan, Y. Chen, C. Liang, Y. Cai, and Y. Baykal, “Effect of spatial coherence on the scintillation properties of a dark hollow beam in turbulent atmosphere,” Appl. Phys. B 110, 519–529 (2013).
[CrossRef]

X. Li and Y. Cai, “Nonparaxial propagation of a partially coherent dark hollow beam,” Appl. Phys. B 102, 205–213 (2011).
[CrossRef]

O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for LIDAR systems operating through turbulent atmosphere,” Appl. Phys. B 94, 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. B 99, 317–323 (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, 051108 (2012).
[CrossRef]

Chin. Phys. Lett. (1)

Z. Gao and B. Lu, “Nonparaxial dark-hollow Gaussian beam,” Chin. Phys. Lett. 23, 106–109 (2006).
[CrossRef]

J. Lightwave Technol. (1)

J. Mod. Opt. (1)

Y. Qiu, Z. Chen, and L. Liu, “Partially coherent dark hollow beams propagating through real ABCD optical systems in a turbulent atmosphere,” J. Mod. Opt. 57, 662–669 (2010).
[CrossRef]

J. Opt. A (2)

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

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

J. Opt. Soc. Am. A (11)

D. F. V. James, “Changes of polarization of light beams on propagation in free space,” J. Opt. Soc. Am. A 11, 1641–1643 (1994).
[CrossRef]

G. Zhou, “Analytical vectorial structure of controllable dark-hollow beams in the far field,” J. Opt. Soc. Am. A 26, 1654–1660 (2009).
[CrossRef]

G. Zhou, “Analytical vectorial structure of controllable dark-hollow beams close to the source,” J. Opt. Soc. Am. A 26, 2386–2395 (2009).
[CrossRef]

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Figures (10)

Fig. 1.
Fig. 1.

Normalized intensity distributions (contour graphs) I/Imax, (Ix+Iy)/Imax, and Iz/Imax, and the corresponding cross lines (y=0) of a nonparaxial vector partially coherent DHB at several propagation distances with w0x=w0y=10λ. Ip/Ipmax represents the normalized intensity distribution calculated by the paraxial propagation formulas.

Fig. 2.
Fig. 2.

Normalized intensity distributions (contour graphs) I/Imax, (Ix+Iy)/Imax, and Iz/Imax, and the corresponding cross lines (y=0) of a nonparaxial vector partially coherent DHB at several propagation distances with w0x=w0y=0.5λ. Ip/Ipmax represents the normalized intensity distribution calculated by the paraxial propagation formulas.

Fig. 3.
Fig. 3.

Normalized intensity distributions (contour graphs) I/Imax, (Ix+Iy)/Imax, and Iz/Imax, and the corresponding cross lines (y=0) of a nonparaxial vector partially coherent DHB at several propagation distances with w0x=w0y=0.1λ. Ip/Ipmax represents the normalized intensity distribution calculated by the paraxial propagation formulas.

Fig. 4.
Fig. 4.

Normalized intensity distributions (contour graphs) I/Imax, (Ix+Iy)/Imax, and Iz/Imax, and the corresponding cross lines (y=0) of a nonparaxial vector partially coherent DHB at z=zr with w0x=w0y=10λ for different values of the coherence widths (σgx and σgy).

Fig. 5.
Fig. 5.

Degree of polarization and the corresponding cross line (y=x) of a nonparaxial vector partially coherent DHB at several propagation distances with w0x=w0y=10λ.

Fig. 6.
Fig. 6.

Degree of polarization and the corresponding cross line (y=x) of a nonparaxial vector partially coherent DHB at several propagation distances with w0x=w0y=3λ.

Fig. 7.
Fig. 7.

Degree of polarization and the corresponding cross line (y=x) of a nonparaxial vector partially coherent DHB at several propagation distances with w0x=w0y=1λ.

Fig. 8.
Fig. 8.

Degree of polarization and the corresponding cross line (y=0) of a nonparaxial vector partially coherent DHB at z=zr with w0x=w0y=10λ for different values of the coherence widths (σgx and σgy).

Fig. 9.
Fig. 9.

Modulus of the degree of coherence of a nonparaxial vector partially coherent DHB between two transverse points (x1, 0) and (x2, 0) at several propagation distances for different values of the beam waist sizes w0x and w0y.

Fig. 10.
Fig. 10.

Modulus of the spectral degree of coherence of a nonparaxial vector partially coherent DHB between two transverse points (x1, 0) and (x2, 0) at z=zr for different values of the coherence widths (σgx and σgy).

Equations (46)

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Eα(x,y,z)=12πEα(x0,y0,0)z[exp(ikR)R]dx0dy0,(α=x,y),
Ez(x,y,z)=12π{Ex(x0,y0,0)x[exp(ikR)R]+Ey(x0,y0,0)y[exp(ikR)R]}dx0dy0,
α[exp(ikR)R]=ikexp(ikR)R2(αα0),(α=x,y),
z[exp(ikR)R]=ikzexp(ikR)R2.
W⃗(x10,y10,x20,y20,0)=(Wxx(x10,y10,x20,y20,0)Wxy(x10,y10,x20,y20,0)0Wyx(x10,y10,x20,y20,0)Wyy(x10,y10,x20,y20,0)0000),
Wαβ(x10,y10,x20,y20,0)=Eα*(x10,y10,0)Eβ(x20,y20,0),(α,β=x,y).
W⃗(x1,y1,x2,y2,z)=(Wxx(x1,y1,x2,y2,z)Wxy(x1,y1,x2,y2,z)Wxz(x1,y1,x2,y2,z)Wyx(x1,y1,x2,y2,z)Wyy(x1,y1,x2,y2,z)Wyz(x1,y1,x2,y2,z)Wzx(x1,y1,x2,y2,z)Wzy(x1,y1,x2,y2,z)Wzz(x1,y1,x2,y2,z)),
Wαβ(x1,y1,x2,y2,z)=Eα*(x1,y1,z)Eβ(x2,y2,z),(α,β=x,y,z).
Wαβ(x1,y1,x2,y2,z)=k2z24π2Wαβ(x10,y10,x20,y20,0)exp[ik(R1R2)]R12R22×dx10dy10dx20dy20,(α,β=x,y),
Wαz(x1,y1,x2,y2,z)=k2z4π2[Wαx(x10,y10,x20,y20,0)exp[ik(R1R2)]R12R22(x2x20)+Wαy(x10,y10,x20,y20,0)exp[ik(R1R2)]R12R22(y2y20)]dx10dy10dx20dy20,(α=x,y),
Wzz(x1,y1,x2,y2,z)=k24π2[Wxx(x10,y10,x20,y20,0)exp[ik(R1R2)]R12R22(x1x10)(x2x20)+Wxy(x10,y10,x20,y20,0)exp[ik(R1R2)]R12R22(x1x10)(y2y20)+Wyx(x10,y10,x20,y20,0)exp[ik(R1R2)]R12R22(y1y10)(x2x20)+Wyy(x10,y10,x20,y20,0)exp[ik(R1R2)]R12R22(y1y10)(y2y20)]dx10dy10dx20dy20.
W⃗(x10,y10,x20,y20,0)=(Wxx(x10,y10,x20,y20,0)000Wyy(x10,y10,x20,y20,0)0000),
Wαβ0(x10,y10,x20,y20,0)=m=1Mn=1N(1)m+nMN(Mm)(Nn)[exp(nx102+mx202w0α2ny102+my202w0α2)exp(nx102w0α2mx202pαw0α2ny102w0α2my202pαw0α2)exp(nx102pαw0α2mx202w0α2ny102pαw0α2my202w0α2)+exp(nx102pαw0α2mx202pαw0α2ny102pαw0α2my202pαw0α2)]exp[(x10x20)22σgα2(y10y20)22σgα2],(α=β=x,y).
Riri+xi02+yi022xixi02yiyi02ri,(i=1,2),
exp[p2x2+qx]dx=exp(q24p2)πp,
xexp[p2x2+qx]dx=exp(q24p2)πpq2p2,
x2exp[p2x2+qx]dx=exp(q24p2)πp(12p2+q24p4),
Wxx(x1,y1,x2,y2,z)=m=1Mn=1N(1)m+nMN(Mm)(Nn)z2k2exp[ik(r2r1)]r22r12{1Axexp[k2(x22+y22)4s2xr22]exp[k2s2xAx(x1r1k2fσx2x22s2xr2)2k2s2xAx(y1r1k2fσx2y22s2xr2)2]1Bxexp[k2(x22+y22)4s4xr22]exp[k2s4xBx(x1r1k2fσx2x22s4xr2)2k2s4xBx(y1r1k2fσx2y22s4xr2)2]1Cxexp[k2(x22+y22)4s2xr22]exp[k2s2xCx(x1r1k2fσx2x22s2xr2)2k2s4xBx(y1r1k2fσx2y22s2xr2)2]+1Dxexp[k2(x22+y22)4s4xr22]exp[k2s4xDx(x1r1k2fσx2x22s4xr2)2k2s4xDx(y1r1k2fσx2y22s4xr2)2]},
Wyy(x1,y1,x2,y2,z)=(zλ)2m=1Mn=1N(1)m+nMN(Mm)(Nn)exp[ik(r2r1)]r22r12{4π2Ayexp[k2(x22+y22)4s2yr22]exp[k2s2yAy(x1r1k2fσy2x22s2yr2)2k2s2yAy(y1r1k2fσy2y22s2yr2)2]4π2Byexp[k2(x22+y22)4s4yr22]exp[k2s4yBy(x1r1k2fσy2x22s4yr2)2k2s4yBy(y1r1k2fσy2y22s4yr2)2]4π2Cyexp[k2(x22+y22)4s2yr22]exp[k2s2yCy(x1r1k2fσy2x22s2yr2)2k2s2yCy(y1r1k2fσy2y22s2yr2)2]+4π2Dyexp[k2(x22+y22)4s4yr22]exp[k2s4yDy(x1r1k2fσy2x22s4yr2)2k2s4yDy(y1r1k2fσy2y22s4yr2)2]},
Wxy(x1,y1,x2,y2,z)=Wyx(x1,y1,x2,y2,z)=0,
Wzz(x1,y1,x2,y2,z)=m=1Mn=1N(1)m+nMN(Nm)(Nn)k2exp[ik(r2r1)]r22r12×[Wzz1xWzz2xWzz3x+Wzz4x+Wzz1yWzz2yWzz3y+Wzz4y],
Wzz1x=1Axexp[k24s2xr22(x22+y22)]exp{k2s2xAx[(x1r1k2fσx2x22s2xr2)2+(y1r1k2fσx2y22s2xr2)2]}×{x1x22iks2xx2Ax(x1r1k2fσx2x22s2xr2)x1[ik3fσx2Ax(x1r1k2fσx2x22s2xr2)ikx22s2xr2]+[k2fσx2Ax2s2xk4fσx2Ax2(x1r1k2fσx2x22s2xr2)2]+k2x2Axr2(x1r1k2fσx2x22s2xr2)},
Wzz2x=1Bxexp[k24s4xr22(x22+y22)]exp{k2s4xBx[(x1r1k2fσx2x22s4xr2)2+(y1r1k2fσx2y22s4xr2)2]}×{x1x22iks4xx2Bx(x1r1k2fσx2x22s4xr2)x1[ik3fσx2Bx(x1r1k2fσx2x22s4xr2)ikx22s4xr2]+[k2fσx2Bx2s4xk4fσx2Bx2(x1r1k2fσx2x22s4xr2)2]+k2x2Bxr2(x1r1k2fσx2x22s4xr2)},
Wzz3x=1Cxexp[k24s2xr22(x22+y22)]exp{k2s2xCx[(x1r1k2fσx2x22s2xr2)2+(y1r1k2fσx2y22s2xr2)2]}×{x1x22iks2xx2Cx(x1r1k2fσx2x22s2xr2)x1[ik3fσx2Cx(x1r1k2fσx2x22s2xr2)ikx22s2xr2]+[k2fσx2Cx2s2xk4fσx2Cx2(x1r1k2fσx2x22s2xr2)2]+k2x2Cxr2(x1r1k2fσx2x22s2xr2)},
Wzz4x=1Dxexp[k24s4xr22(x22+y22)]exp{k2s4xDx[(x1r1k2fσx2x22s4xr2)2+(y1r1k2fσx2y22s4xr2)2]}×{x1x22iks4xx2Dx(x1r1k2fσx2x22s4xr2)x1[ik3fσx2Dx(x1r1k2fσx2x22s4xr2)ikx22s4xr2]+[k2fσx2Dx2s4xk4fσx2Dx2(x1r1k2fσx2x22s4xr2)2]+k2x2Dxr2(x1r1k2fσx2x22s4xr2)},
Wzz1y=1Ayexp[k24s2yr22(x22+y22)]exp{k2s2yAy[(x1r1k2fσy2x22s2yr2)2+(y1r1k2fσy2y22s2yr2)2]}×{y1y22iks2yy2Ay(y1r1k2fσy2y22s2yr2)y1[ik3fσy2Ay(y1r1k2fσy2y22s2yr2)iky22s2yr2]+[k2fσy2Ay2s2yk4fσy2Ay2(y1r1k2fσy2y22s2yr2)2]+k2y2Ayr2(y1r1k2fσy2y22s2yr2)},
Wzz2y(ρ1,ρ2,z)=1Byexp[k24s3yr22(x22+y22)]exp{k2s3yBy[(x1r1k2fσy2x22s3yr2)2+(y1r1k2fσy2y22s3yr2)2]}×{y1y22iks3yy2By(y1r1k2fσy2y22s3yr2)y1[ik3fσy2By(y1r1k2fσy2y22s3xr2)iky22s3yr2]+[k2fσy2By2s3yk4fσy2By2(y1r1k2fσy2y22s3yr2)2]+k2y2Byr2(y1r1k2fσy2y22s3yr2)},
Wzz3y=1Cyexp[k24s2yr22(x22+y22)]exp{k2s2yCy[(x1r1k2fσy2x22s2yr2)2+(y1r1k2fσy2y22s2yr2)2]}×{y1y22iks2yy2Cy(y1r1k2fσy2y22s2yr2)y1[ik3fσy2Cy(y1r1k2fσy2y22s2yr2)iky22s2yr2]+[k2fσy2Cy2s2yk4fσy2Cy2(y1r1k2fσy2y22s2yr2)2]+k2y2Cyr2(y1r1k2fσy2y22s2yr2)},
Wzz4y=1Dyexp[k24s3yr22(x22+y22)]exp{k2s3yDy[(x1r1k2fσy2x22s3yr2)2+(y1r1k2fσy2y22s3yr2)2]}}×{y1y22iks3yy2Dy(y1r1k2fσy2y22s3yr2)y1[ik3fσy2Dy(y1r1k2fσy2y22s3yr2)iky22s3yr2]+[k2fσy2Dx2s3yk4fσy2Dy2(y1r1k2fσy2y22s3yr2)2]+k2y2Dyr2(y1r1k2fσy2y22s3yr2)},
Wxz(x1,y1,x2,y2,z)=m=1Mn=1N(1)m+nMN(Nm)(Nn)zk2exp[ik(r2r1)]r22r12{1Ax[x2ik3fσx2Ax(x1r1k2fσx2x22s2xr2)+ikx22s2xr2]×exp[k24s2xr22(x22+y22)k2s2xAx(x1r1k2fσx2x22s2xr2)2k2s2xAx(y1r1k2fσx2y22s2xr2)2]1Bx[x2ik3fσx2Bx(x1r1k2fσx2x22s3xr2)+ikx22s3xr2]exp[k24s3xr22(x22+y22)k2s3xBx(x1r1k2fσx2x22s3xr2)2]exp[k2s3xBx(y1r1k2fσx2y22s3xr2)2]1Cx[x2ik3fσx2Cx(x1r1k2fσx2x22s2xr2)+ikx22s2xr2]exp[k24s2xr22(x22+y22)]exp[k2s2xCx(x1r1k2fσx2x22s2xr2)2k2s2xCx(y1r1k2fσx2y22s2xr2)2]+1Dx[x2ik3fσx2Dx(x1r1k2fσx2x22s3xr2)+ikx22s3xr2]exp[k24s3xr22(x22+y22)]exp[k2s3xDx(x1r1k2fσx2x22s3xr2)2k2s3xDx(y1r1k2fσx2y22s3xr2)2]},
Wyz(x1,y1,x2,y2,z)=m=1Mn=1N(1)m+nMN(Nm)(Nn)zk2exp[ik(r2r1)]r22r12{1Ay[y2ik3fσy2Ay(y1r1k2fσy2y22s2yr2)+iky22s2yr2]×exp[k24s2yr22(x22+y22)k2s2yAy(x1r1k2fσy2x22s2yr2)2k2s2yAy(y1r1k2fσy2y22s2yr2)2]1By[y2ik3fσy2By(y1r1k2fσy2y22s3yr2)+iky22s3yr2]exp[k24s3yr22(x22+y22)k2s3yBy(x1r1k2fσy2x22s3yr2)2]×exp[k2s3yBy(y1r1k2fσy2y22s3yr2)2]1Cy[y2ik3fσy2Cy(y1r1k2fσy2y22s2yr2)+iky22s2yr2]exp[k24s2yr22(x22+y22)]×exp[k2s2yCy(x1r1k2fσx2x22s2yr2)2k2s2yCy(y1r1k2fσy2y22s2yr2)2]+1Dy[y2ik3fσy2Dy(y1r1k2fσy2y22s3yr2)+iky22s3yr2]exp[k24s3yr22(x22+y22)]exp[k2s3yDy(x1r1k2fσy2x22s3yr2)2k2s3yDy(y1r1k2fσy2y22s3yr2)2],
Wzx(x1,y1,x2,y2,z)=Wxz*(x2,y2,x1,y1,z),
Wzy(x1,y1,x2,y2,z)=Wyz*(x2,y2,x1,y1,z),
fx=1kw0x,fσx=1kσgx,fy=1kw0y,fσy=1kσgy,s1x=nw0x2+12σgx2+ik2r1,s2x=mw0x2+12σgx2ik2r2,s3x=npxw0x2+12σgx2+ik2r1,s4x=mpxw0x2+12σgx2ik2r2,s1y=nw0y2+12σgy2+ik2r1,s2y=mw0y2+12σgy2ik2r2,s3y=npyw0y2+12σgy2+ik2r1,s4y=mpyw0y2+12σgy2ik2r2,Ax=4s1xs2xk4fσx4,Bx=4s1xs4xk4fσx4,Ay=4s1ys2yk4fσy4,By=4s1ys4yk4fσy4,Cx=4s3xs2xk4fσx4,Dx=4s3xs4xk4fσx4,Cy=4s3ys2yk4fσy4,Dy=4s3ys4yk4fσy4.
I(x,y,z)=Ix(x,y,z)+Iy(x,y,z)+Iz(x,y,z)=Wxx(x,y,x,y,z)+Wyy(x,y,x,y,z)+Wzz(x,y,x,y,z),
P(x,y,z)=p1(x,y,z)p2(x,y,z)p1(x,y,z)+p2(x,y,z)+p3(x,y,z),
μ(x1x2,y1y2,z)=TrW⃗(x1,y1,x2,y2,z)TrW⃗(x1,y1,x1,y1,z)TrW⃗(x2,y2,x2,y2,z),
r1z+((x12+y12)/2z),r2z+((x22+y22)/2z),
Wpxx(x1,y1,x2,y2,z)=m=1Mn=1N(1)m+nMN(Nm)(Nn)k2z2exp[ik(x22+y222zx12+y122z)]{1Axexp[k24s2xz2(x22+y22)]exp[k2s2xAxz2(x1k2fσx2x22s2x)2k2s2xAxz2(y1k2fσx2y22s2x)2]1Bxexp[k24s4xz2(x22+y22)]exp[k2s4xBxz2(x1k2fσx2x22s4x)2k2s4xBxz2(y1k2fσx2y22s4x)2]1Cxexp[k24s2xz2(x22+y22)]exp[k2s2xCxz2(x1k2fσx2x22s2x)2k2s2xCxz2(y1k2fσx2y22s2x)2]+1Dxexp[k24s4xz2(x22+y22)]exp[k2s4xDxz2(x1k2fσx2x22s4x)2k2s4xDxz2(y1k2fσx2y22s4x)2]},
Wpyy(x1,y1,x2,y2,z)=m=1Mn=1N(1)m+nMN(Nm)(Nn)k2z2exp[ik(x22+y222zx12+y122z)]×{1Ayexp[k24s2yz2(x22+y22)]exp[k2s2yAyz2(x1k2fσy2x22s2y)2k2s2yAyz2(y1k2fσx2y22s2x)2]1Byexp[k24s4yz2(x22+y22)]exp[k2s4yByz2(x1k2fσy2x22s4y)2k2s4yByz2(y1k2fσy2y22s4y)2]1Cyexp[k24s2yz2(x22+y22)]exp[k2s2yCyz2(x1k2fσy2x22s2y)2k2s2yCyz2(y1k2fσy2y22s2y)2]+1Dyexp[k24s4yz2(x22+y22)]exp[k2s4yDyz2(x1k2fσy2x22s4y)2k2s4yDyz2(y1k2fσy2y22s4y)2]},
Wpxy(x1,y1,x2,y2,z)=Wpyx(x1,y1,x2,y2,z)=0,
Wpxz(x1,y1,x2,y2,z)=Wpzx(x1,y1,x2,y2,z)=0,
Wpyz(x1,y1,x2,y2,z)=Wpzy(x1,y1,x2,y2,z)=0,
Wpzz(x1,y1,x2,y2,z)=0,
fx=1kw0x,fσx=1kσgx,fy=1kw0y,fσy=1kσgy,s1x=nw0x2+12σgx2+ik2z,s2x=mw0x2+12σgx2ik2z,s3x=npxw0x2+12σgx2+ik2z,s4x=mpxw0x2+12σgx2ik2z,s1y=nw0y2+12σgy2+ik2z,s2y=mw0y2+12σgy2ik2z,s3y=npyw0y2+12σgy2+ik2z,s4y=mpyw0y2+12σgy2ik2z,Ax=4s1xs2xk4fσx4,Bx=4s1xs3xk4fσx4Cx=4s4xs2xk4fσx4,Dx=4s4xs3xk4fσx4,Ay=4s1ys2yk4fσy4,By=4s1ys4yk4fσy4,Cy=4s3ys2yk4fσy4,Dy=4s3ys4yk4fσy4.
I(x,y,z)=Ix(x,y,z)+Iy(x,y,z)=Wxx(x,y,x,y,z)+Wyy(x,y,x,y,z),

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