Abstract

Analytical formula for the cross-spectral density matrix of a stochastic electromagnetic Gaussian Schell-model (EGSM) beam truncated by a circular phase aperture propagating in free space is derived with the help of a tensor method, which provides a reliable and fast way for studying the propagation and transformation of a truncated EGSM beam. Statistics properties, such as the spectral intensity, the degree of coherence, the degree of polarization and the polarization ellipse of a truncated EGSM beam in free space are studied numerically. The propagation factor of a truncated EGSM beam is also analyzed. Our numerical results show that we can modulate the spectral intensity, the polarization, the coherence and the propagation factor of an EGSM beam by a circular phase aperture. It is found that the phase aperture can be used to shape the beam profile of an EGSM beam and generate electromagnetic partially coherent dark hollow or flat-topped beam, which is useful in some applications, such as optical trapping, material processing, free-space optical communications.

© 2011 OSA

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2011 (5)

X. Li, F. Wang, and Y. Cai, “An alternative model for a partially coherent elliptical dark hollow beam,” Opt. Laser Technol. 43(3), 577–585 (2011).
[CrossRef]

F. Cheng and Y. Cai, “Propagation factor of a truncated partially coherent flat-topped beam in turbulent atmosphere,” Opt. Commun. 284(1), 30–37 (2011).
[CrossRef]

F. Wang, Y. Cai, and X. Ma, “Circular partially coherent flattened Gaussian beam,” Opt. Lasers Eng. 49(4), 481–489 (2011).
[CrossRef]

G. Zhou, “Far-field structure property of a Gaussian beam diffracted by a phase aperture,” Opt. Commun. 284(1), 8–14 (2011).
[CrossRef]

Y. Dong, Y. Cai, C. Zhao, and M. Yao, “Statistics properties of a cylindrical vector partially coherent beam,” Opt. Express 19(7), 5979–5992 (2011).
[CrossRef] [PubMed]

2010 (7)

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

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

S. Zhu and Y. Cai, “M2-factor of a stochastic electromagnetic beam in a Gaussian cavity,” Opt. Express 18(26), 27567–27581 (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(8), 662–669 (2010).
[CrossRef]

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

M. Yao, Y. Cai, and O. Korotkova, “Spectral shift of a stochastic electromagnetic Gaussian Schell-model beam in a Gaussian cavity,” Opt. Commun. 283(22), 4505–4511 (2010).
[CrossRef]

C. Ding, L. Pan, and B. Lu, “Changes in the state of polarization of apertured stochastic electromagnetic modified Bessel-Gauss beams in free-space propagation,” Appl. Phys. B 99(1-2), 307–315 (2010).
[CrossRef]

2009 (8)

L. Pan, Z. Zhao, C. Ding, and B. Lu, “Effect of polarization on spectral switches in the diffraction of stochastic electromagnetic beams,” Appl. Phys. Lett. 95(18), 181112 (2009).
[CrossRef]

Y. Cai and O. Korotkova, “Twist phase-induced polarization changes in electromagnetic Gaussian Schell-model beams,” Appl. Phys. B 96(2-3), 499–507 (2009).
[CrossRef]

O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for LIDAR systems operating through turbulent atmosphere,” Appl. Phys. B 94(4), 681–690 (2009).
[CrossRef]

X. Chu, Z. Li, and Y. Wu, “Relay propagation of partially coherent flattened Gaussian beam in turbulent atmosphere,” Opt. Commun. 282(23), 4486–4489 (2009).
[CrossRef]

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

L. Pan, M. Sun, C. Ding, Z. Zhao, and B. Lü, “Effects of astigmatism on spectra, coherence and polarization of stochastic electromagnetic beams passing through an astigmatic optical system,” Opt. Express 17(9), 7310–7321 (2009).
[CrossRef] [PubMed]

Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “M2-factor of coherent and partially coherent dark hollow beams propagating in turbulent atmosphere,” Opt. Express 17(20), 17344–17356 (2009).
[CrossRef] [PubMed]

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

2008 (11)

F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schellmodel sources,” J. Opt. Soc. Am. A 25(5), 1016–1021 (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(12), 1389–1391 (2008).
[CrossRef] [PubMed]

F. Wang, Y. Cai, and Q. Lin, “Experimental observation of truncated fractional Fourier transform for a partially coherent Gaussian Schell-model beam,” J. Opt. Soc. Am. A 25(8), 2001–2010 (2008).
[CrossRef]

F. Wang and Y. Cai, “Experimental generation of a partially coherent flat-topped beam,” Opt. Lett. 33(16), 1795–1797 (2008).
[CrossRef] [PubMed]

Y. Dan and B. Zhang, “Beam propagation factor of partially coherent flat-topped beams in a turbulent atmosphere,” Opt. Express 16(20), 15563–15575 (2008).
[CrossRef] [PubMed]

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

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]

O. Korotkova, M. Yao, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “State of polarization of a stochastic electromagnetic beam in an optical resonator,” J. Opt. Soc. Am. A 25(11), 2710–2720 (2008).
[CrossRef]

B. Kanseri and H. C. Kandpal, “Experimental determination of electric cross-spectral density matrix and generalized Stokes parameters for a laser beam,” Opt. Lett. 33(20), 2410–2412 (2008).
[CrossRef] [PubMed]

M. Alavinejad and B. Ghafary, “Turbulence-induced degradation properties of partially coherent flat-topped beams,” Opt. Lasers Eng. 46(5), 357–362 (2008).
[CrossRef]

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

2007 (6)

X. Ji, E. Zhang, and B. Lu, “Changes in the spectrum and polarization of polychromatic partially coherent electromagnetic beams in the turbulent atmosphere,” Opt. Commun. 275(2), 292–300 (2007).
[CrossRef]

Z. Chen and J. Pu, “Propagation characteristics of aberrant stochastic electromagnetic beams in a turbulent atmosphere,” J. Opt. A, Pure Appl. Opt. 9(12), 1123–1130 (2007).
[CrossRef]

H. Wang, X. Wang, A. Zeng, and W. Lu, “Changes in the coherence of quasi-monochromatic electromagnetic Gaussian Schell-model beams propagating through turbulent atmosphere,” Opt. Commun. 276(2), 218–221 (2007).
[CrossRef]

X. Lu and Y. Cai, “Partially coherent circular and elliptical dark hollow beams and their paraxial propagations,” Phys. Lett. A 369(1-2), 157–166 (2007).
[CrossRef]

H. Wang, X. Wang, A. Zeng, and K. Yang, “Effects of coherence on anisotropic electromagnetic Gaussian-Schell model beams on propagation,” Opt. Lett. 32(15), 2215–2217 (2007).
[CrossRef] [PubMed]

X. Ji, X. Chen, S. Chen, X. Li, and B. Lü, “Influence of atmospheric turbulence on the spatial correlation properties of partially coherent flat-topped beams,” J. Opt. Soc. Am. A 24(11), 3554–3563 (2007).
[CrossRef]

2006 (4)

2005 (5)

T. Saastamoinen, J. Turunen, J. Tervo, T. Setälä, and A. T. Friberg, “Electromagnetic coherence theory of laser resonator modes,” J. Opt. Soc. Am. A 22(1), 103–108 (2005).
[CrossRef]

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

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

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. 52(11), 1611–1618 (2005).
[CrossRef]

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere,” Waves Random Complex Media 15(3), 353–364 (2005).
[CrossRef]

2004 (2)

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233(4-6), 225–230 (2004).
[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]

2003 (1)

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

2002 (2)

Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27(4), 216–218 (2002).
[CrossRef]

K. Ait-Ameur, “Effects of a phase aperture on the fundamental mode of a hard-apertured cavity,” J. Mod. Opt. 49(7), 1157–1168 (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, Pure Appl. Opt. 3(1), 1–9 (2001).
[CrossRef]

2000 (1)

N. Nishi, T. Jitsuno, K. Tsubakimoto, S. Matsuoka, N. Miyanaga, and M. Nakatsuka, “Two-dimensional multi-lens array with circular aperture spherical lens for flat-top irradiation of inertial confinement fusion target,” Opt. Rev. 7(3), 216–220 (2000).
[CrossRef]

1999 (1)

1997 (1)

1988 (1)

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

1984 (1)

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

1981 (1)

V. A. Zenkin, V. R. Kushnir, and L. V. Tarasov, “Phase aperture as means for selecting the lowest transverse mode,” Sov. J. Quantum Electron. 11(2), 243–245 (1981).
[CrossRef]

Ait-Ameur, K.

K. Ait-Ameur, “Effects of a phase aperture on the fundamental mode of a hard-apertured cavity,” J. Mod. Opt. 49(7), 1157–1168 (2002).
[CrossRef]

Alavinejad, M.

M. Alavinejad and B. Ghafary, “Turbulence-induced degradation properties of partially coherent flat-topped beams,” Opt. Lasers Eng. 46(5), 357–362 (2008).
[CrossRef]

Arinaga, S.

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

Baykal, Y.

Borghi, R.

Breazeale, M. A.

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

Cai, Y.

F. Wang, Y. Cai, and X. Ma, “Circular partially coherent flattened Gaussian beam,” Opt. Lasers Eng. 49(4), 481–489 (2011).
[CrossRef]

X. Li, F. Wang, and Y. Cai, “An alternative model for a partially coherent elliptical dark hollow beam,” Opt. Laser Technol. 43(3), 577–585 (2011).
[CrossRef]

F. Cheng and Y. Cai, “Propagation factor of a truncated partially coherent flat-topped beam in turbulent atmosphere,” Opt. Commun. 284(1), 30–37 (2011).
[CrossRef]

Y. Dong, Y. Cai, C. Zhao, and M. Yao, “Statistics properties of a cylindrical vector partially coherent beam,” Opt. Express 19(7), 5979–5992 (2011).
[CrossRef] [PubMed]

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, “M2-factor of a stochastic electromagnetic beam in a Gaussian cavity,” Opt. Express 18(26), 27567–27581 (2010).
[CrossRef]

M. Yao, Y. Cai, and O. Korotkova, “Spectral shift of a stochastic electromagnetic Gaussian Schell-model beam in a Gaussian cavity,” Opt. Commun. 283(22), 4505–4511 (2010).
[CrossRef]

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

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

O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for LIDAR systems operating through turbulent atmosphere,” Appl. Phys. B 94(4), 681–690 (2009).
[CrossRef]

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

Y. Cai and O. Korotkova, “Twist phase-induced polarization changes in electromagnetic Gaussian Schell-model beams,” Appl. Phys. B 96(2-3), 499–507 (2009).
[CrossRef]

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

Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “M2-factor of coherent and partially coherent dark hollow beams propagating in turbulent atmosphere,” Opt. Express 17(20), 17344–17356 (2009).
[CrossRef] [PubMed]

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(12), 1389–1391 (2008).
[CrossRef] [PubMed]

F. Wang and Y. Cai, “Experimental generation of a partially coherent flat-topped beam,” Opt. Lett. 33(16), 1795–1797 (2008).
[CrossRef] [PubMed]

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

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]

F. Wang, Y. Cai, and Q. Lin, “Experimental observation of truncated fractional Fourier transform for a partially coherent Gaussian Schell-model beam,” J. Opt. Soc. Am. A 25(8), 2001–2010 (2008).
[CrossRef]

O. Korotkova, M. Yao, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “State of polarization of a stochastic electromagnetic beam in an optical resonator,” J. Opt. Soc. Am. A 25(11), 2710–2720 (2008).
[CrossRef]

X. Lu and Y. Cai, “Partially coherent circular and elliptical dark hollow beams and their paraxial propagations,” Phys. Lett. A 369(1-2), 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(7), 1398–1407 (2006).
[CrossRef]

Y. Cai and L. Hu, “Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams through an apertured astigmatic optical system,” Opt. Lett. 31(6), 685–687 (2006).
[CrossRef] [PubMed]

Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14(4), 1353–1367 (2006).
[CrossRef] [PubMed]

Y. Cai and S. He, “Partially coherent flattened Gaussian beam and its paraxial propagation properties,” J. Opt. Soc. Am. A 23(10), 2623–2628 (2006).
[CrossRef]

Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27(4), 216–218 (2002).
[CrossRef]

Chen, S.

Chen, X.

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(8), 662–669 (2010).
[CrossRef]

Z. Chen and J. Pu, “Propagation characteristics of aberrant stochastic electromagnetic beams in a turbulent atmosphere,” J. Opt. A, Pure Appl. Opt. 9(12), 1123–1130 (2007).
[CrossRef]

Cheng, F.

F. Cheng and Y. Cai, “Propagation factor of a truncated partially coherent flat-topped beam in turbulent atmosphere,” Opt. Commun. 284(1), 30–37 (2011).
[CrossRef]

Chu, X.

X. Chu, Z. Li, and Y. Wu, “Relay propagation of partially coherent flattened Gaussian beam in turbulent atmosphere,” Opt. Commun. 282(23), 4486–4489 (2009).
[CrossRef]

Cincotti, G.

Dan, Y.

Ding, C.

C. Ding, L. Pan, and B. Lu, “Changes in the state of polarization of apertured stochastic electromagnetic modified Bessel-Gauss beams in free-space propagation,” Appl. Phys. B 99(1-2), 307–315 (2010).
[CrossRef]

L. Pan, Z. Zhao, C. Ding, and B. Lu, “Effect of polarization on spectral switches in the diffraction of stochastic electromagnetic beams,” Appl. Phys. Lett. 95(18), 181112 (2009).
[CrossRef]

L. Pan, M. Sun, C. Ding, Z. Zhao, and B. Lü, “Effects of astigmatism on spectra, coherence and polarization of stochastic electromagnetic beams passing through an astigmatic optical system,” Opt. Express 17(9), 7310–7321 (2009).
[CrossRef] [PubMed]

Dogariu, A.

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere,” Waves Random Complex Media 15(3), 353–364 (2005).
[CrossRef]

Dong, Y.

Eyyuboglu, H. T.

Friberg, A. T.

Ghafary, B.

M. Alavinejad and B. Ghafary, “Turbulence-induced degradation properties of partially coherent flat-topped beams,” Opt. Lasers Eng. 46(5), 357–362 (2008).
[CrossRef]

Gori, F.

He, S.

Hu, L.

Ji, X.

X. Ji, E. Zhang, and B. Lu, “Changes in the spectrum and polarization of polychromatic partially coherent electromagnetic beams in the turbulent atmosphere,” Opt. Commun. 275(2), 292–300 (2007).
[CrossRef]

X. Ji, X. Chen, S. Chen, X. Li, and B. Lü, “Influence of atmospheric turbulence on the spatial correlation properties of partially coherent flat-topped beams,” J. Opt. Soc. Am. A 24(11), 3554–3563 (2007).
[CrossRef]

Jiang, Z.

Jitsuno, T.

N. Nishi, T. Jitsuno, K. Tsubakimoto, S. Matsuoka, N. Miyanaga, and M. Nakatsuka, “Two-dimensional multi-lens array with circular aperture spherical lens for flat-top irradiation of inertial confinement fusion target,” Opt. Rev. 7(3), 216–220 (2000).
[CrossRef]

Kandpal, H. C.

Kanseri, B.

Kato, Y.

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

Kitagawa, Y.

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

Korotkova, O.

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

M. Yao, Y. Cai, and O. Korotkova, “Spectral shift of a stochastic electromagnetic Gaussian Schell-model beam in a Gaussian cavity,” Opt. Commun. 283(22), 4505–4511 (2010).
[CrossRef]

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

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

O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for LIDAR systems operating through turbulent atmosphere,” Appl. Phys. B 94(4), 681–690 (2009).
[CrossRef]

Y. Cai and O. Korotkova, “Twist phase-induced polarization changes in electromagnetic Gaussian Schell-model beams,” Appl. Phys. B 96(2-3), 499–507 (2009).
[CrossRef]

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

Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “M2-factor of coherent and partially coherent dark hollow beams propagating in turbulent atmosphere,” Opt. Express 17(20), 17344–17356 (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. Express 16(20), 15834–15846 (2008).
[CrossRef] [PubMed]

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

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]

O. Korotkova, M. Yao, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “State of polarization of a stochastic electromagnetic beam in an optical resonator,” J. Opt. Soc. Am. A 25(11), 2710–2720 (2008).
[CrossRef]

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

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere,” Waves Random Complex Media 15(3), 353–364 (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, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233(4-6), 225–230 (2004).
[CrossRef]

Kushnir, V. R.

V. A. Zenkin, V. R. Kushnir, and L. V. Tarasov, “Phase aperture as means for selecting the lowest transverse mode,” Sov. J. Quantum Electron. 11(2), 243–245 (1981).
[CrossRef]

Li, X.

X. Li, F. Wang, and Y. Cai, “An alternative model for a partially coherent elliptical dark hollow beam,” Opt. Laser Technol. 43(3), 577–585 (2011).
[CrossRef]

X. Ji, X. Chen, S. Chen, X. Li, and B. Lü, “Influence of atmospheric turbulence on the spatial correlation properties of partially coherent flat-topped beams,” J. Opt. Soc. Am. A 24(11), 3554–3563 (2007).
[CrossRef]

Li, Z.

X. Chu, Z. Li, and Y. Wu, “Relay propagation of partially coherent flattened Gaussian beam in turbulent atmosphere,” Opt. Commun. 282(23), 4486–4489 (2009).
[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(8), 662–669 (2010).
[CrossRef]

Liu, Z.

Lu, B.

C. Ding, L. Pan, and B. Lu, “Changes in the state of polarization of apertured stochastic electromagnetic modified Bessel-Gauss beams in free-space propagation,” Appl. Phys. B 99(1-2), 307–315 (2010).
[CrossRef]

L. Pan, Z. Zhao, C. Ding, and B. Lu, “Effect of polarization on spectral switches in the diffraction of stochastic electromagnetic beams,” Appl. Phys. Lett. 95(18), 181112 (2009).
[CrossRef]

X. Ji, E. Zhang, and B. Lu, “Changes in the spectrum and polarization of polychromatic partially coherent electromagnetic beams in the turbulent atmosphere,” Opt. Commun. 275(2), 292–300 (2007).
[CrossRef]

Lu, Q.

Lu, W.

H. Wang, X. Wang, A. Zeng, and W. Lu, “Changes in the coherence of quasi-monochromatic electromagnetic Gaussian Schell-model beams propagating through turbulent atmosphere,” Opt. Commun. 276(2), 218–221 (2007).
[CrossRef]

Lu, X.

Lü, B.

Ma, X.

F. Wang, Y. Cai, and X. Ma, “Circular partially coherent flattened Gaussian beam,” Opt. Lasers Eng. 49(4), 481–489 (2011).
[CrossRef]

Matsuoka, S.

N. Nishi, T. Jitsuno, K. Tsubakimoto, S. Matsuoka, N. Miyanaga, and M. Nakatsuka, “Two-dimensional multi-lens array with circular aperture spherical lens for flat-top irradiation of inertial confinement fusion target,” Opt. Rev. 7(3), 216–220 (2000).
[CrossRef]

Mima, K.

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

Miyanaga, N.

N. Nishi, T. Jitsuno, K. Tsubakimoto, S. Matsuoka, N. Miyanaga, and M. Nakatsuka, “Two-dimensional multi-lens array with circular aperture spherical lens for flat-top irradiation of inertial confinement fusion target,” Opt. Rev. 7(3), 216–220 (2000).
[CrossRef]

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

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]

Nakatsuka, M.

N. Nishi, T. Jitsuno, K. Tsubakimoto, S. Matsuoka, N. Miyanaga, and M. Nakatsuka, “Two-dimensional multi-lens array with circular aperture spherical lens for flat-top irradiation of inertial confinement fusion target,” Opt. Rev. 7(3), 216–220 (2000).
[CrossRef]

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

Nishi, N.

N. Nishi, T. Jitsuno, K. Tsubakimoto, S. Matsuoka, N. Miyanaga, and M. Nakatsuka, “Two-dimensional multi-lens array with circular aperture spherical lens for flat-top irradiation of inertial confinement fusion target,” Opt. Rev. 7(3), 216–220 (2000).
[CrossRef]

Pan, L.

C. Ding, L. Pan, and B. Lu, “Changes in the state of polarization of apertured stochastic electromagnetic modified Bessel-Gauss beams in free-space propagation,” Appl. Phys. B 99(1-2), 307–315 (2010).
[CrossRef]

L. Pan, Z. Zhao, C. Ding, and B. Lu, “Effect of polarization on spectral switches in the diffraction of stochastic electromagnetic beams,” Appl. Phys. Lett. 95(18), 181112 (2009).
[CrossRef]

L. Pan, M. Sun, C. Ding, Z. Zhao, and B. Lü, “Effects of astigmatism on spectra, coherence and polarization of stochastic electromagnetic beams passing through an astigmatic optical system,” Opt. Express 17(9), 7310–7321 (2009).
[CrossRef] [PubMed]

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]

Ponomarenko, S. A.

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. 52(11), 1611–1618 (2005).
[CrossRef]

Pu, J.

Z. Chen and J. Pu, “Propagation characteristics of aberrant stochastic electromagnetic beams in a turbulent atmosphere,” J. Opt. A, Pure Appl. Opt. 9(12), 1123–1130 (2007).
[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(8), 662–669 (2010).
[CrossRef]

Qu, J.

Ramírez-Sánchez, V.

Roychowdhury, H.

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. 52(11), 1611–1618 (2005).
[CrossRef]

Saastamoinen, T.

Salem, M.

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere,” Waves Random Complex Media 15(3), 353–364 (2005).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233(4-6), 225–230 (2004).
[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]

Santarsiero, M.

Setälä, T.

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]

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]

Sun, M.

Tarasov, L. V.

V. A. Zenkin, V. R. Kushnir, and L. V. Tarasov, “Phase aperture as means for selecting the lowest transverse mode,” Sov. J. Quantum Electron. 11(2), 243–245 (1981).
[CrossRef]

Tervo, J.

Tong, Z.

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

Tsubakimoto, K.

N. Nishi, T. Jitsuno, K. Tsubakimoto, S. Matsuoka, N. Miyanaga, and M. Nakatsuka, “Two-dimensional multi-lens array with circular aperture spherical lens for flat-top irradiation of inertial confinement fusion target,” Opt. Rev. 7(3), 216–220 (2000).
[CrossRef]

Turunen, J.

Vahimaa, P.

Wang, F.

Wang, H.

H. Wang, X. Wang, A. Zeng, and K. Yang, “Effects of coherence on anisotropic electromagnetic Gaussian-Schell model beams on propagation,” Opt. Lett. 32(15), 2215–2217 (2007).
[CrossRef] [PubMed]

H. Wang, X. Wang, A. Zeng, and W. Lu, “Changes in the coherence of quasi-monochromatic electromagnetic Gaussian Schell-model beams propagating through turbulent atmosphere,” Opt. Commun. 276(2), 218–221 (2007).
[CrossRef]

Wang, X.

H. Wang, X. Wang, A. Zeng, and W. Lu, “Changes in the coherence of quasi-monochromatic electromagnetic Gaussian Schell-model beams propagating through turbulent atmosphere,” Opt. Commun. 276(2), 218–221 (2007).
[CrossRef]

H. Wang, X. Wang, A. Zeng, and K. Yang, “Effects of coherence on anisotropic electromagnetic Gaussian-Schell model beams on propagation,” Opt. Lett. 32(15), 2215–2217 (2007).
[CrossRef] [PubMed]

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(4), 681–690 (2009).
[CrossRef]

Wen, J. J.

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

Wolf, E.

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]

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. 52(11), 1611–1618 (2005).
[CrossRef]

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere,” Waves Random Complex Media 15(3), 353–364 (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, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233(4-6), 225–230 (2004).
[CrossRef]

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

Wu, Y.

X. Chu, Z. Li, and Y. Wu, “Relay propagation of partially coherent flattened Gaussian beam in turbulent atmosphere,” Opt. Commun. 282(23), 4486–4489 (2009).
[CrossRef]

Yamanaka, C.

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

Yang, K.

Yao, M.

Yuan, Y.

Zeng, A.

H. Wang, X. Wang, A. Zeng, and K. Yang, “Effects of coherence on anisotropic electromagnetic Gaussian-Schell model beams on propagation,” Opt. Lett. 32(15), 2215–2217 (2007).
[CrossRef] [PubMed]

H. Wang, X. Wang, A. Zeng, and W. Lu, “Changes in the coherence of quasi-monochromatic electromagnetic Gaussian Schell-model beams propagating through turbulent atmosphere,” Opt. Commun. 276(2), 218–221 (2007).
[CrossRef]

Zenkin, V. A.

V. A. Zenkin, V. R. Kushnir, and L. V. Tarasov, “Phase aperture as means for selecting the lowest transverse mode,” Sov. J. Quantum Electron. 11(2), 243–245 (1981).
[CrossRef]

Zhang, B.

Zhang, E.

X. Ji, E. Zhang, and B. Lu, “Changes in the spectrum and polarization of polychromatic partially coherent electromagnetic beams in the turbulent atmosphere,” Opt. Commun. 275(2), 292–300 (2007).
[CrossRef]

Zhang, L.

Zhao, C.

Zhao, Z.

L. Pan, Z. Zhao, C. Ding, and B. Lu, “Effect of polarization on spectral switches in the diffraction of stochastic electromagnetic beams,” Appl. Phys. Lett. 95(18), 181112 (2009).
[CrossRef]

L. Pan, M. Sun, C. Ding, Z. Zhao, and B. Lü, “Effects of astigmatism on spectra, coherence and polarization of stochastic electromagnetic beams passing through an astigmatic optical system,” Opt. Express 17(9), 7310–7321 (2009).
[CrossRef] [PubMed]

Zhou, G.

G. Zhou, “Far-field structure property of a Gaussian beam diffracted by a phase aperture,” Opt. Commun. 284(1), 8–14 (2011).
[CrossRef]

Zhu, S.

Appl. Opt. (1)

Appl. Phys. B (3)

Y. Cai and O. Korotkova, “Twist phase-induced polarization changes in electromagnetic Gaussian Schell-model beams,” Appl. Phys. B 96(2-3), 499–507 (2009).
[CrossRef]

O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for LIDAR systems operating through turbulent atmosphere,” Appl. Phys. B 94(4), 681–690 (2009).
[CrossRef]

C. Ding, L. Pan, and B. Lu, “Changes in the state of polarization of apertured stochastic electromagnetic modified Bessel-Gauss beams in free-space propagation,” Appl. Phys. B 99(1-2), 307–315 (2010).
[CrossRef]

Appl. Phys. Lett. (1)

L. Pan, Z. Zhao, C. Ding, and B. Lu, “Effect of polarization on spectral switches in the diffraction of stochastic electromagnetic beams,” Appl. Phys. Lett. 95(18), 181112 (2009).
[CrossRef]

J. Acoust. Soc. Am. (1)

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

J. Mod. Opt. (3)

K. Ait-Ameur, “Effects of a phase aperture on the fundamental mode of a hard-apertured cavity,” J. Mod. Opt. 49(7), 1157–1168 (2002).
[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(8), 662–669 (2010).
[CrossRef]

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. 52(11), 1611–1618 (2005).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (3)

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]

Z. Chen and J. Pu, “Propagation characteristics of aberrant stochastic electromagnetic beams in a turbulent atmosphere,” J. Opt. A, Pure Appl. Opt. 9(12), 1123–1130 (2007).
[CrossRef]

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

J. Opt. Soc. Am. A (7)

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

Opt. Commun. (10)

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

H. Wang, X. Wang, A. Zeng, and W. Lu, “Changes in the coherence of quasi-monochromatic electromagnetic Gaussian Schell-model beams propagating through turbulent atmosphere,” Opt. Commun. 276(2), 218–221 (2007).
[CrossRef]

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

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Figures (10)

Fig. 1
Fig. 1

Schematic diagram of an EGSM beam truncated by a phase aperture propagating in free space.

Fig. 2
Fig. 2

Normalized spectral intensity (contour graph) and the corresponding cross line (y = 0) of an EGSM beam truncated by a circular phase aperture at several propagation distances in free space.

Fig. 3
Fig. 3

Normalized spectral intensity (cross line y = 0) of a truncated EGSM beam at z = 6m for different values of the phase delay ϕ and correlation coefficients δ x x and δ y y .

Fig. 4
Fig. 4

Modulus of the degree of coherence between two transverse positions ρ 1 = ( 1 ,   0 ) m m ,   ρ 2 = ( 1 ,   0 ) m m of a truncated EGSM beam versus the propagation distance for different values of the phase delay ϕ.

Fig. 5
Fig. 5

Modulus of the degree of coherence between two transverse positions ρ 1 = ( 1 ,   0 ) m m ,   ρ 2 = ( 1 ,   0 ) m m of a truncated EGSM beam versus the propagation distance for different values of the correlation coefficients δ x x ,   δ y y .

Fig. 6
Fig. 6

Degree of polarization (cross line y = 0) of a truncated EGSM beam at several propagation distances for different values of the phase delay ϕ.

Fig. 7
Fig. 7

On-axis polarization ellipse of the truncated EGSM beam at several propagation distances in free space.

Fig. 8
Fig. 8

Polarization ellipse of the truncated EGSM beam at z = 6m for different values of the phase delay ϕ in free space.

Fig. 9
Fig. 9

Dependence of the M 2 -factor of a truncated EGSM beam on the phase delay ϕ.

Fig. 10
Fig. 10

Dependence of the M 2 -factor of a truncated EGSM beam on the radius a.

Equations (39)

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W ( ρ x 1 , ρ y 1 , ρ x 2 , ρ y 2 , z ) = ( 1 λ z ) 2 T ( x 1 , y 1 ) T * ( x 2 , y 2 ) W ( x 1 , y 1 , x 2 , y 2 , 0 )                                        × exp [ i k 2 z ( x 1 2 x 2 2 2 x 1 ρ x 1 + 2 x 2 ρ x 2 + ρ x 1 2 ρ x 2 2 ) ]                                        × exp [ i k 2 z ( y 1 2 y 2 2 2 y 1 ρ y 1 + 2 y 2 ρ x 2 + ρ y 1 2 ρ y 2 2 ) ] d x 1 d x 2 d y 1 d y 2 ,
T ( x 1 , y 1 ) = 1 + [ exp ( i ϕ ) 1 ] H ( x 1 , y 1 ) ,
T ( x 1 , y 1 ) T * ( x 2 , y 2 ) = 1 + [ exp ( i ϕ ) 1 ] H ( x 1 , y 1 ) + [ exp ( i ϕ ) 1 ] H * ( x 2 , y 2 )                                   + [ exp ( i ϕ ) 1 ] [ exp ( i ϕ ) 1 ] H ( x 1 , y 1 ) H * ( x 2 , y 2 ) ,
H ( x 1 , y 1 ) = m = 1 M A m exp [ B m a 2 ( x 1 2 + y 1 2 ) ] ,
W ( ρ ˜ , z ) = 1 λ 2 ( det B ˜ ) 1 / 2 T ( r 1 ) T * ( r 2 ) W ( r ˜ , 0 ) exp [ i k 2 ( r ˜ T B ˜ 1 r ˜ 2 r ˜ T B ˜ 1 ρ ˜ + ρ ˜ T B ˜ 1 ρ ˜ ) ] d r ˜ ,
B ˜ = ( z I 0 I 0 I z I ) .
T ( r 1 ) T * ( r 2 ) = 1 + [ exp ( i ϕ ) 1 ] m = 1 M A m exp ( i k 2 r ˜ T B ˜ m r ˜ ) + [ exp ( i ϕ ) 1 ] p = 1 M A p * exp [ i k 2 r ˜ T B ˜ p r ˜ ]                       + [ exp ( i ϕ ) 1 ] [ exp ( i ϕ ) 1 ] m = 1 M p = 1 M A m A p * exp [ i k 2 r ˜ T B ˜ m p r ˜ ] ,
B ˜ m = ( 2 i B m k a 2 I 0 I 0 I 0 I ) ,   B ˜ p = ( 0 I 0 I 0 I 2 i B p * k a 2 I ) ,      B ˜ m p = ( 2 i B m k a 2 I 0 I 0 I 2 i B p * k a 2 I ) .  
W α β ( r ˜ , 0 ) = A α A β B α β exp [ i k 2 r ˜ T M 0 α β 1 r ˜ ] ,    ( α = x , y ; β = x , y ) ,
M 0 α β 1 = ( 1 i k ( 1 2 σ a 2 + 1 δ α β 2 ) I i k δ α β 2 I i k δ α β 2 I 1 i k ( 1 2 σ β 2 + 1 δ α β 2 ) I ) ,
W α β ( ρ ˜ , z ) = A α A β B α β { det [ I ˜ + B ˜ M 0 α β 1 ] 1 / 2 exp [ i k 2 ρ ˜ T M 1 α β 1 ρ ˜ ]                 + [ exp ( i ϕ ) 1 ] m = 1 M A m det [ I ˜ + B ˜ M 0 α β 1 + B ˜ B ˜ m ] 1 / 2 × exp [ i k 2 ρ ˜ T M 2 α β 1 ρ ˜ ]                 + [ exp ( i ϕ ) 1 ] p = 1 M A p * det [ I ˜ + B ˜ M 0 α β 1 + B ˜ B ˜ p ] 1 / 2 exp [ i k 2 ρ ˜ T M 3 α β 1 ρ ˜ ] +                  [ exp ( i ϕ ) 1 ] [ exp ( i ϕ ) 1 ] m = 1 M p = 1 M A m A p * det [ I ˜ + B ˜ M 0 α β 1 + B ˜ B ˜ m p ] 1 / 2 exp [ i k 2 ρ ˜ T M 4 α β 1 ρ ˜ ] } ,
M 1 α β 1 = ( M 0 α β + B ˜ ) 1 ,    M 2 α β 1 = [ ( M 0 α β 1 + B ˜ m ) 1 + B ˜ ] 1 M 3 α β 1 = [ ( M 0 ε β 1 + B ˜ p ) 1 + B ˜ ] 1 ,   M 4 α β 1 = [ ( M 0 α β 1 + B ˜ m p ) 1 + B ˜ ] 1 .
I ( ρ , z ) = Tr W ( ρ , ρ , z ) ,
P ( ρ , l ) = 1 4 Det W ( ρ , ρ , z ) [ Tr W ( ρ , ρ , z ) ] 2 .
μ ( ρ 1 , ρ 2 , z ) = Tr W ( ρ 1 , ρ 2 , z ) Tr W ( ρ 1 , ρ 1 , z ) Tr W ( ρ 2 , ρ 2 , z )
W ( ρ , ρ , z ) = W ( u ) ( ρ , ρ , z ) + W ( p ) ( ρ , ρ , z ) ,
W ( u ) ( ρ , ρ , z ) = ( A ( ρ , ρ , z ) 0 0 A ( ρ , ρ , z ) ) ,     W ( p ) ( ρ , ρ , z ) = ( B ( ρ , ρ , z ) D ( ρ , ρ , z ) D * ( ρ , ρ , z ) C ( ρ , ρ , z ) ) ,
A ( ρ , ρ , z ) = 1 2 [ W x x ( ρ , ρ , z ) + W y y ( ρ , ρ , z ) + [ W x x ( ρ , ρ , z ) W y y ( ρ , ρ , z ) ] 2 + 4 | W x y ( ρ , ρ , z ) | 2 ] , B ( ρ , ρ , z ) = 1 2 [ W x x ( ρ , ρ , z ) W y y ( ρ , ρ , z ) + [ W x x ( ρ , ρ , z ) W y y ( ρ , ρ , z ) ] 2 + 4 | W x y ( ρ , ρ , z ) | 2 ] , C ( ρ , ρ , z ) = 1 2 [ W y y ( ρ , ρ , z ) W x x ( ρ , ρ , z ) + [ W x x ( ρ , ρ , z ) W y y ( ρ , ρ , z ) ] 2 + 4 | W x y ( ρ , ρ , z ) | 2 ] , D ( ρ , ρ , z ) = W x y .
C ( ρ , ρ , z ) ε x ( r ) 2 ( ρ , ρ , z ) 2 Re D ( ρ , ρ , z ) ε x ( r ) ( ρ , ρ , z ) ε y ( r ) ( ρ , ρ , z ) + B ( ρ , ρ , z ) ε y ( r ) 2 ( ρ , ρ , z ) = [ Im D ( ρ , ρ , z ) ] 2
A 1 , 2 ( ρ , ρ , z ) = 1 2 [ [ W x x ( ρ , ρ , z ) W y y ( ρ , ρ , z ) ] 2 + 4 | W x y ( ρ , ρ , z ) | 2                      ± [ W x x ( ρ , ρ , z ) W y y ( ρ , ρ , z ) ] 2 + 4 [ Re W x y ( ρ , ρ , z ) ] 2 ] 1 / 2 ,
ε ( ρ , ρ , z ) = A 2 ( ρ , ρ , z ) A 1 ( ρ , ρ , z ) ,
θ ( ρ , ρ , z ) = 1 2 arctan [ 2 Re W x y ( ρ , ρ , z ) W x x ( ρ , ρ , z ) W y y ( ρ , ρ , z ) ] .
M x 2 = 4 π Δ x Δ p x ,       M y 2 = 4 π Δ y Δ p y ,
Δ x = 1 J ( x x ¯ ) 2 W ( x , y , x , y ) d x d y ,
Δ y = 1 J ( y y ¯ ) 2 W ( x , y , x , y ) d x d y ,
Δ p x = 1 J ( p x p ¯ x ) 2 W ˜ ( p x , p y , p x , p y ) d p x d p y ,
Δ p y = 1 J ( p y p ¯ y ) 2 W ˜ ( p x , p y , p x , p y ) d p x d p y ,
J = W ( x , y , x , y ) d x d y = W ˜ ( p x , p y , p x , p y ) d p x d p y ,
x ¯ = 1 J x W ( x , y , x , y ) d x d y ,
p ¯ x = 1 J p x W ˜ ( p x , p y , p x , p y ) d p x d p y .
W T α β ( r 1 , r 2 , 0 ) = T ( r 1 ) T * ( r 2 ) W α β ( r 1 , r 2 , 0 ) , ( α = x , y ; β = x , y )
W T r ( r 1 , r 2 , 0 ) = T ( r 1 ) T * ( r 2 ) W x x ( r 1 , r 2 , 0 ) + T ( r 1 ) T * ( r 2 ) W y y ( r 1 , r 2 , 0 ) .
M 2 = M x 2 = M y 2 = 2 J ( G x x + G y y ) ( Q x x + Q y y ) ,
G x x = A x 2 { 2 π σ 4 + [ exp ( i ϕ ) 1 ] m = 1 M A m π 2 α x x 1 2 + [ exp ( i ϕ ) 1 ] p = 1 M A p * π 2 α x x 2 2           + [ exp ( i ϕ ) 1 ] [ exp ( i ϕ ) 1 ] m = 1 M p = 1 M A m A p * π 2 α x x 3 2 } ,
G y y = A y 2 { 2 π σ 4 + [ exp ( i ϕ ) 1 ] m = 1 M A m π 2 α y y 1 2 + [ exp ( i ϕ ) 1 ] p = 1 M A p * π 2 α y y 2 2           + [ exp ( i ϕ ) 1 ] [ exp ( i ϕ ) 1 ] m = 1 M p = 1 M A m A p * π 2 α y y 3 2 } ,
Q x x = A x 2 { π ( 2 β x x 0 δ x x 2 + 1 ) 2 ( 2 β x x 0 δ x x 2 1 ) + ( exp ( i ϕ ) 1 ) m = 1 M A m π ( 4 β x x 0 β x x 1 δ x x 4 1 ) 2 ( β x x 0 δ x x 2 + β x x 1 δ x x 2 1 ) 2 + ( exp ( i ϕ ) 1 ) × p = 1 M A p * π ( 4 β x x 0 β x x 2 δ x x 4 1 ) 2 ( β x x 0 δ x x 2 + β x x 2 δ x x 2 1 ) 2 + [ exp ( i ϕ ) 1 ] [ exp ( i ϕ ) 1 ] m = 1 M p = 1 M A m A p * π ( 4 β x x 1 β x x 2 δ x x 4 1 ) 2 ( β x x 1 δ x x 2 + β x x 2 δ x x 2 1 ) 2 ,
Q y y = A y 2 { π ( 2 β y y 0 δ y y 2 + 1 ) 2 ( 2 β y y 0 δ y y 2 1 ) + ( exp ( i ϕ ) 1 ) m = 1 M A m π ( 4 β y y 0 β y y 1 δ y y 4 1 ) 2 ( β y y 0 δ y y 2 + β y y 1 δ y y 2 1 ) 2 + ( exp ( i ϕ ) 1 ) × p = 1 M A p * π ( 4 β y y 0 β y y 2 δ y y 4 1 ) 2 ( β y y 0 δ y y 2 + β y y 2 δ y y 2 1 ) 2 + [ exp ( i ϕ ) 1 ] [ exp ( i ϕ ) 1 ] m = 1 M p = 1 M A m A p * π ( 4 β y y 1 β y y 2 δ y y 4 1 ) 2 ( β y y 1 δ y y 2 + β y y 2 δ y y 2 1 ) 2 ,
J = A x 2 { 2 π σ 2 + [ exp ( i ϕ ) 1 ] m = 1 M A m π α x x 1 + [ exp ( i ϕ ) 1 ] p = 1 M A p * π α x x 2        + [ exp ( i ϕ ) 1 ] [ exp ( i ϕ ) 1 ] m = 1 M p = 1 M A m A p * π α x x 3 } + A y 2 { 2 π σ 2 + [ exp ( i ϕ ) 1 ] m = 1 M A m π α y y 1        + [ exp ( i ϕ ) 1 ] p = 1 M A p * π α y y 2 + [ exp ( i ϕ ) 1 ] [ exp ( i ϕ ) 1 ] m = 1 M p = 1 M A m A p * π α y y 3 } ,
α α β 1 = 1 2 σ 2 + B m a 2 , α α β 2 = 1 2 σ 2 + B p * a 2 , α α β 3 = 1 2 σ 2 + B m a 2 + B p * a 2 , β α β 0 = 1 4 σ 2 + 1 2 δ α β 2 , β α β 1 = 1 4 σ 2 + 1 2 δ α β 2 + B m a 2 , β α β 2 = 1 4 σ 2 + 1 2 δ α β 2 + B p * a 2 .

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