Abstract

We derive the analytical formula for the orbital angular momentum (OAM) flux of a stochastic electromagnetic beam carrying twist phase [i.e., twisted electromagnetic Gaussian Schell-model (TEGSM) beam] in the source plane with the help of the Wigner distribution function. Furthermore, we derive the general expression of the OAM flux of a TEGSM beam on propagation with the help of a tensor method. As numerical examples, we explore the evolution properties of the OAM flux of a TEGSM beam propagating through a cylindrical thin lens or a uniaxial crystal. It is found that the OAM flux of a TEGSM beam closely depends on its twist factors and degree of polarization in the source plane, and one can modulate the OAM flux of a TEGSM beam by a cylindrical thin lens or a uniaxial crystal. Our results may be useful in some applications, such as particle manipulation and free-space optical communications, where light beam with OAM is preferred.

© 2015 Optical Society of America

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References

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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
  52. S. Zhu, L. Liu, Y. Chen, and Y. Cai, “State of polarization and propagation factor of a stochastic electromagnetic beam in a gradient-index fiber,” J. Opt. Soc. Am. A 30(11), 2306–2313 (2013).
    [Crossref] [PubMed]
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    [Crossref]
  54. X. Liu, F. Wang, M. Zhang, and Y. Cai, “Experimental demonstration of ghost imaging with an electromagnetic Gaussian Schell-model beam,” J. Opt. Soc. Am. A 32(5), 910–920 (2015).
    [Crossref] [PubMed]
  55. X. Liu, F. Wang, L. Liu, C. Zhao, and Y. Cai, “Generation and propagation of an electromagnetic Gaussian Schell-model vortex beam,” J. Opt. Soc. Am. A 32(11), 2058–2065 (2015).
    [Crossref]
  56. 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]
  57. S. A. Ponomarenko and G. P. Agrawal, “Asymmetric incoherent vector solitons,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(3), 036604 (2004).
    [Crossref] [PubMed]
  58. S. Zhu and Y. Cai, “Degree of polarization of a twisted electromagnetic Gaussian Schell-model beam in a Gaussian cavity filled with gain media,” Prog. Electromagn. Res. B 21, 171–187 (2010).
  59. L. Liu, Y. Chen, L. Guo, and Y. Cai, “Twist phase-induced changes of the statistical properties of a stochastic electromagnetic beam propagating in a uniaxial crystal,” Opt. Express 23(9), 12454–12467 (2015).
    [Crossref] [PubMed]
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2015 (6)

2014 (3)

2013 (1)

2012 (5)

C. Zhao, Y. Dong, G. Wu, F. Wang, Y. Cai, and O. Korotkova, “Experimental demonstration of coupling of an electromagnetic Gaussian Schell-model beam into a single-mode optical fiber,” Appl. Phys. B 108(4), 891–895 (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(1), 013840 (2012).
[Crossref]

C. Zhao, F. Wang, Y. Dong, Y. Han, and Y. Cai, “Effect of spatial coherence on determining the topological charge of a vortex beam,” Appl. Phys. Lett. 101(26), 261104 (2012).
[Crossref]

F. Wang, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Twist phase-induced reduction in scintillation of a partially coherent beam in turbulent atmosphere,” Opt. Lett. 37(2), 184–186 (2012).
[Crossref] [PubMed]

Z. Tong and O. Korotkova, “Beyond the classical Rayleigh limit with twisted light,” Opt. Lett. 37(13), 2595–2597 (2012).
[Crossref] [PubMed]

2011 (2)

2010 (5)

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]

S. Zhu and Y. Cai, “Degree of polarization of a twisted electromagnetic Gaussian Schell-model beam in a Gaussian cavity filled with gain media,” Prog. Electromagn. Res. B 21, 171–187 (2010).

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

F. Wang and Y. Cai, “Second-order statistics of a twisted gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 18(24), 24661–24672 (2010).
[Crossref] [PubMed]

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

2009 (2)

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. 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]

2008 (5)

2007 (1)

2005 (5)

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

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. Soc. A 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, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246(1–3), 35–43 (2005).
[Crossref]

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249(4–6), 379–385 (2005).
[Crossref]

2004 (3)

S. A. Ponomarenko and G. P. Agrawal, “Asymmetric incoherent vector solitons,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(3), 036604 (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]

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93(6), 068103 (2004).
[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)

2001 (4)

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]

S. A. Ponomarenko, “A class of partially coherent beams carrying optical vortices,” J. Opt. Soc. Am. A 18(1), 150–156 (2001).
[Crossref] [PubMed]

S. A. Ponomarenko, “Twisted Gaussian Schell-model solitons,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(3), 036618 (2001).
[Crossref] [PubMed]

J. Serna and J. M. Movilla, “Orbital angular momentum of partially coherent beams,” Opt. Lett. 26(7), 405–407 (2001).
[Crossref] [PubMed]

2000 (2)

1998 (2)

1994 (2)

1993 (2)

R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10(1), 95–109 (1993).
[Crossref]

A. Beléndez, L. Carretero, and A. Fimia, “The use of partially coherent light to reduce the efficiency of silver halide noise gratings,” Opt. Commun. 98(4–6), 236–240 (1993).
[Crossref]

1985 (1)

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31(4), 2419–2434 (1985).
[Crossref] [PubMed]

1984 (2)

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]

A. E. Attard, “Matrix optical analysis of skew rays in mixed systems of spherical and orthogonal cylindrical lenses,” Appl. Opt. 23(16), 2706–2709 (1984).
[Crossref] [PubMed]

1975 (1)

Agrawal, G. P.

S. A. Ponomarenko and G. P. Agrawal, “Asymmetric incoherent vector solitons,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(3), 036604 (2004).
[Crossref] [PubMed]

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

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]

Attard, A. E.

Bastiaans, M. J.

Basu, S.

Baykal, Y.

Beléndez, A.

A. Beléndez, L. Carretero, and A. Fimia, “The use of partially coherent light to reduce the efficiency of silver halide noise gratings,” Opt. Commun. 98(4–6), 236–240 (1993).
[Crossref]

Borghi, R.

Cai, Y.

F. Wang, X. Liu, and Y. Cai, “Propagation of partially coherent beam in turbulent atmosphere: a review (Invited review),” Prog. Electromagnetics Res. 150, 123–143 (2015).
[Crossref]

X. Liu, F. Wang, M. Zhang, and Y. Cai, “Experimental demonstration of ghost imaging with an electromagnetic Gaussian Schell-model beam,” J. Opt. Soc. Am. A 32(5), 910–920 (2015).
[Crossref] [PubMed]

L. Liu, Y. Chen, L. Guo, and Y. Cai, “Twist phase-induced changes of the statistical properties of a stochastic electromagnetic beam propagating in a uniaxial crystal,” Opt. Express 23(9), 12454–12467 (2015).
[Crossref] [PubMed]

X. Liu, F. Wang, L. Liu, C. Zhao, and Y. Cai, “Generation and propagation of an electromagnetic Gaussian Schell-model vortex beam,” J. Opt. Soc. Am. A 32(11), 2058–2065 (2015).
[Crossref]

Y. Cai and S. Zhu, “Orbital angular moment of a partially coherent beam propagating through an astigmatic ABCD optical system with loss or gain,” Opt. Lett. 39(7), 1968–1971 (2014).
[Crossref] [PubMed]

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

S. Zhu, L. Liu, Y. Chen, and Y. Cai, “State of polarization and propagation factor of a stochastic electromagnetic beam in a gradient-index fiber,” J. Opt. Soc. Am. A 30(11), 2306–2313 (2013).
[Crossref] [PubMed]

F. Wang, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Twist phase-induced reduction in scintillation of a partially coherent beam in turbulent atmosphere,” Opt. Lett. 37(2), 184–186 (2012).
[Crossref] [PubMed]

Y. Dong, F. Wang, C. Zhao, and Y. Cai, “Effect of spatial coherence on propagation, tight focusing and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86(1), 013840 (2012).
[Crossref]

C. Zhao, F. Wang, Y. Dong, Y. Han, and Y. Cai, “Effect of spatial coherence on determining the topological charge of a vortex beam,” Appl. Phys. Lett. 101(26), 261104 (2012).
[Crossref]

C. Zhao, Y. Dong, G. Wu, F. Wang, Y. Cai, and O. Korotkova, “Experimental demonstration of coupling of an electromagnetic Gaussian Schell-model beam into a single-mode optical fiber,” Appl. Phys. B 108(4), 891–895 (2012).
[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]

C. Zhao and Y. Cai, “Trapping two types of particles using a focused partially coherent elegant Laguerre-Gaussian beam,” Opt. Lett. 36(12), 2251–2253 (2011).
[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]

F. Wang and Y. Cai, “Second-order statistics of a twisted gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 18(24), 24661–24672 (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, “Degree of polarization of a twisted electromagnetic Gaussian Schell-model beam in a Gaussian cavity filled with gain media,” Prog. Electromagn. Res. B 21, 171–187 (2010).

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

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]

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

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

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

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

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

Carretero, L.

A. Beléndez, L. Carretero, and A. Fimia, “The use of partially coherent light to reduce the efficiency of silver halide noise gratings,” Opt. Commun. 98(4–6), 236–240 (1993).
[Crossref]

Chen, Y.

Davidson, F. M.

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.

C. Zhao, Y. Dong, G. Wu, F. Wang, Y. Cai, and O. Korotkova, “Experimental demonstration of coupling of an electromagnetic Gaussian Schell-model beam into a single-mode optical fiber,” Appl. Phys. B 108(4), 891–895 (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(1), 013840 (2012).
[Crossref]

C. Zhao, F. Wang, Y. Dong, Y. Han, and Y. Cai, “Effect of spatial coherence on determining the topological charge of a vortex beam,” Appl. Phys. Lett. 101(26), 261104 (2012).
[Crossref]

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.

Fimia, A.

A. Beléndez, L. Carretero, and A. Fimia, “The use of partially coherent light to reduce the efficiency of silver halide noise gratings,” Opt. Commun. 98(4–6), 236–240 (1993).
[Crossref]

Friberg, A. T.

Gori, F.

Guattari, G.

Guo, L.

Gureyev, T. E.

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93(6), 068103 (2004).
[Crossref] [PubMed]

Han, Y.

C. Zhao, F. Wang, Y. Dong, Y. Han, and Y. Cai, “Effect of spatial coherence on determining the topological charge of a vortex beam,” Appl. Phys. Lett. 101(26), 261104 (2012).
[Crossref]

Hyde, M. W.

James, D. F. V.

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]

Kermisch, D.

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.

S. Basu, M. W. Hyde, X. Xiao, D. G. Voelz, and O. Korotkova, “Computational approaches for generating electromagnetic Gaussian Schell-model sources,” Opt. Express 22(26), 31691–31707 (2014).
[Crossref] [PubMed]

C. Zhao, Y. Dong, G. Wu, F. Wang, Y. Cai, and O. Korotkova, “Experimental demonstration of coupling of an electromagnetic Gaussian Schell-model beam into a single-mode optical fiber,” Appl. Phys. B 108(4), 891–895 (2012).
[Crossref]

Z. Tong and O. Korotkova, “Beyond the classical Rayleigh limit with twisted light,” Opt. Lett. 37(13), 2595–2597 (2012).
[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]

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

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

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

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

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 and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249(4–6), 379–385 (2005).
[Crossref]

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

Lin, Q.

Liu, L.

Liu, X.

Mayo, S. C.

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93(6), 068103 (2004).
[Crossref] [PubMed]

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.

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]

Movilla, J. M.

Mukunda, N.

R. Simon and N. Mukunda, “Twist phase in Gaussian-beam optics,” J. Opt. Soc. Am. A 15(9), 2373–2382 (1998).
[Crossref]

R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10(1), 95–109 (1993).
[Crossref]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31(4), 2419–2434 (1985).
[Crossref] [PubMed]

Nakatsuka, M.

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]

Paganin, D. M.

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93(6), 068103 (2004).
[Crossref] [PubMed]

Peschel, U.

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.

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.

S. A. Ponomarenko and G. P. Agrawal, “Asymmetric incoherent vector solitons,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(3), 036604 (2004).
[Crossref] [PubMed]

S. A. Ponomarenko, “Twisted Gaussian Schell-model solitons,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(3), 036618 (2001).
[Crossref] [PubMed]

S. A. Ponomarenko, “A class of partially coherent beams carrying optical vortices,” J. Opt. Soc. Am. A 18(1), 150–156 (2001).
[Crossref] [PubMed]

Ramírez-Sánchez, V.

Ricklin, J. C.

Roychowdhury, H.

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249(4–6), 379–385 (2005).
[Crossref]

Salem, M.

Santarsiero, M.

Serna, J.

Shirai, T.

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. Soc. A 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]

R. Simon and N. Mukunda, “Twist phase in Gaussian-beam optics,” J. Opt. Soc. Am. A 15(9), 2373–2382 (1998).
[Crossref]

R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10(1), 95–109 (1993).
[Crossref]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31(4), 2419–2434 (1985).
[Crossref] [PubMed]

Stevenson, A. W.

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93(6), 068103 (2004).
[Crossref] [PubMed]

Sudarshan, E. C. G.

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31(4), 2419–2434 (1985).
[Crossref] [PubMed]

Tervonen, E.

Tong, Z.

Z. Tong and O. Korotkova, “Beyond the classical Rayleigh limit with twisted light,” Opt. Lett. 37(13), 2595–2597 (2012).
[Crossref] [PubMed]

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

Turunen, J.

Voelz, D. G.

Wang, F.

F. Wang, X. Liu, and Y. Cai, “Propagation of partially coherent beam in turbulent atmosphere: a review (Invited review),” Prog. Electromagnetics Res. 150, 123–143 (2015).
[Crossref]

X. Liu, F. Wang, L. Liu, C. Zhao, and Y. Cai, “Generation and propagation of an electromagnetic Gaussian Schell-model vortex beam,” J. Opt. Soc. Am. A 32(11), 2058–2065 (2015).
[Crossref]

X. Liu, F. Wang, M. Zhang, and Y. Cai, “Experimental demonstration of ghost imaging with an electromagnetic Gaussian Schell-model beam,” J. Opt. Soc. Am. A 32(5), 910–920 (2015).
[Crossref] [PubMed]

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

F. Wang, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Twist phase-induced reduction in scintillation of a partially coherent beam in turbulent atmosphere,” Opt. Lett. 37(2), 184–186 (2012).
[Crossref] [PubMed]

Y. Dong, F. Wang, C. Zhao, and Y. Cai, “Effect of spatial coherence on propagation, tight focusing and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86(1), 013840 (2012).
[Crossref]

C. Zhao, Y. Dong, G. Wu, F. Wang, Y. Cai, and O. Korotkova, “Experimental demonstration of coupling of an electromagnetic Gaussian Schell-model beam into a single-mode optical fiber,” Appl. Phys. B 108(4), 891–895 (2012).
[Crossref]

C. Zhao, F. Wang, Y. Dong, Y. Han, and Y. Cai, “Effect of spatial coherence on determining the topological charge of a vortex beam,” Appl. Phys. Lett. 101(26), 261104 (2012).
[Crossref]

F. Wang, G. Wu, X. Liu, S. Zhu, and Y. Cai, “Experimental measurement of the beam parameters of an electromagnetic Gaussian Schell-model source,” Opt. Lett. 36(14), 2722–2724 (2011).
[Crossref] [PubMed]

F. Wang and Y. Cai, “Second-order statistics of a twisted gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 18(24), 24661–24672 (2010).
[Crossref] [PubMed]

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

Wilkins, S. W.

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93(6), 068103 (2004).
[Crossref] [PubMed]

Wolf, E.

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

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. Soc. A 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, “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]

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

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

Wu, G.

C. Zhao, Y. Dong, G. Wu, F. Wang, Y. Cai, and O. Korotkova, “Experimental demonstration of coupling of an electromagnetic Gaussian Schell-model beam into a single-mode optical fiber,” Appl. Phys. B 108(4), 891–895 (2012).
[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]

Xiao, X.

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]

Yao, M.

Zhang, M.

Zhao, C.

X. Liu, F. Wang, L. Liu, C. Zhao, and Y. Cai, “Generation and propagation of an electromagnetic Gaussian Schell-model vortex beam,” J. Opt. Soc. Am. A 32(11), 2058–2065 (2015).
[Crossref]

C. Zhao, F. Wang, Y. Dong, Y. Han, and Y. Cai, “Effect of spatial coherence on determining the topological charge of a vortex beam,” Appl. Phys. Lett. 101(26), 261104 (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(1), 013840 (2012).
[Crossref]

C. Zhao, Y. Dong, G. Wu, F. Wang, Y. Cai, and O. Korotkova, “Experimental demonstration of coupling of an electromagnetic Gaussian Schell-model beam into a single-mode optical fiber,” Appl. Phys. B 108(4), 891–895 (2012).
[Crossref]

C. Zhao and Y. Cai, “Trapping two types of particles using a focused partially coherent elegant Laguerre-Gaussian beam,” Opt. Lett. 36(12), 2251–2253 (2011).
[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]

Zhu, S.

Zhu, S. Y.

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

Appl. Opt. (1)

Appl. Phys. B (2)

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. Dong, G. Wu, F. Wang, Y. Cai, and O. Korotkova, “Experimental demonstration of coupling of an electromagnetic Gaussian Schell-model beam into a single-mode optical fiber,” Appl. Phys. B 108(4), 891–895 (2012).
[Crossref]

Appl. Phys. Lett. (1)

C. Zhao, F. Wang, Y. Dong, Y. Han, and Y. Cai, “Effect of spatial coherence on determining the topological charge of a vortex beam,” Appl. Phys. Lett. 101(26), 261104 (2012).
[Crossref]

J. Opt. A, Pure Appl. Opt. (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]

J. Opt. Soc. A (1)

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

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (13)

J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. A 19(9), 1794–1802 (2002).
[Crossref] [PubMed]

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

R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10(1), 95–109 (1993).
[Crossref]

A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11(6), 1818–1826 (1994).
[Crossref]

R. Simon and N. Mukunda, “Twist phase in Gaussian-beam optics,” J. Opt. Soc. Am. A 15(9), 2373–2382 (1998).
[Crossref]

M. J. Bastiaans, “Wigner distribution function applied to twisted Gaussian light propagating in first-order optical systems,” J. Opt. Soc. Am. A 17(12), 2475–2480 (2000).
[Crossref] [PubMed]

S. A. Ponomarenko, “A class of partially coherent beams carrying optical vortices,” J. Opt. Soc. Am. A 18(1), 150–156 (2001).
[Crossref] [PubMed]

D. F. V. James, “Changes of polarization of light beams on propagation in free space,” J. Opt. Soc. Am. A 11(5), 1641–1643 (1994).
[Crossref]

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

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

S. Zhu, L. Liu, Y. Chen, and Y. Cai, “State of polarization and propagation factor of a stochastic electromagnetic beam in a gradient-index fiber,” J. Opt. Soc. Am. A 30(11), 2306–2313 (2013).
[Crossref] [PubMed]

X. Liu, F. Wang, M. Zhang, and Y. Cai, “Experimental demonstration of ghost imaging with an electromagnetic Gaussian Schell-model beam,” J. Opt. Soc. Am. A 32(5), 910–920 (2015).
[Crossref] [PubMed]

X. Liu, F. Wang, L. Liu, C. Zhao, and Y. Cai, “Generation and propagation of an electromagnetic Gaussian Schell-model vortex beam,” J. Opt. Soc. Am. A 32(11), 2058–2065 (2015).
[Crossref]

Open Opt. J. (1)

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

Opt. Commun. (6)

A. Beléndez, L. Carretero, and A. Fimia, “The use of partially coherent light to reduce the efficiency of silver halide noise gratings,” Opt. Commun. 98(4–6), 236–240 (1993).
[Crossref]

Z. Tong, Y. Cai, and O. Korotkova, “Ghost imaging with electromagnetic stochastic beams,” Opt. Commun. 283(20), 3838–3845 (2010).
[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, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246(1–3), 35–43 (2005).
[Crossref]

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

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249(4–6), 379–385 (2005).
[Crossref]

Opt. Express (7)

Opt. Lett. (13)

Z. Tong and O. Korotkova, “Beyond the classical Rayleigh limit with twisted light,” Opt. Lett. 37(13), 2595–2597 (2012).
[Crossref] [PubMed]

J. Serna and J. M. Movilla, “Orbital angular momentum of partially coherent beams,” Opt. Lett. 26(7), 405–407 (2001).
[Crossref] [PubMed]

Y. Cai and S. Zhu, “Orbital angular moment of a partially coherent beam propagating through an astigmatic ABCD optical system with loss or gain,” Opt. Lett. 39(7), 1968–1971 (2014).
[Crossref] [PubMed]

F. Wang, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Twist phase-induced reduction in scintillation of a partially coherent beam in turbulent atmosphere,” Opt. Lett. 37(2), 184–186 (2012).
[Crossref] [PubMed]

R. Borghi, F. Gori, G. Guattari, and M. Santarsiero, “Twisted Schell-model beams with axial symmetry,” Opt. Lett. 40(19), 4504–4507 (2015).
[Crossref] [PubMed]

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

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

C. Zhao and Y. Cai, “Trapping two types of particles using a focused partially coherent elegant Laguerre-Gaussian beam,” Opt. Lett. 36(12), 2251–2253 (2011).
[Crossref] [PubMed]

F. Gori and M. Santarsiero, “Twisted Gaussian Schell-model beams as series of partially coherent modified Bessel-Gauss beams,” Opt. Lett. 40(7), 1587–1590 (2015).
[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, 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]

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

Phys. Lett. A (1)

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

Phys. Rev. A (2)

Y. Dong, F. Wang, C. Zhao, and Y. Cai, “Effect of spatial coherence on propagation, tight focusing and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86(1), 013840 (2012).
[Crossref]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31(4), 2419–2434 (1985).
[Crossref] [PubMed]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (3)

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Figures (8)

Fig. 1
Fig. 1 Dependence of the normalized OAM flux of an anisotropic TEGSM beam on its DOP in the source plane for two different cases (a) A y < A x , (b) A y > A x .
Fig. 2
Fig. 2 Dependence of the normalized OAM flux of an anisotropic TEGSM beam on its twist factors in the source plane.
Fig. 3
Fig. 3 Schematic of a cylindrical thin lens, which is located in the source plane.
Fig. 4
Fig. 4 Normalized OAM flux J z / J z ( φ = 0 ) and its components J z x / J z ( φ = 0 ) , J z y / J z ( φ = 0 ) of an anisotropic TEGSM beam propagating through a cylindrical thin lens versus the orientation angle φ
Fig. 5
Fig. 5 Geometry of the propagation of a beam in a uniaxial crystal orthogonal to the optical axis.
Fig. 6
Fig. 6 Normalized OAM flux J z ( z ) / J z ( 0 ) and its components J z x ( z ) / J z ( 0 ) , J z y ( z ) / J z ( 0 ) of an anisotropic TEGSM beam propagating in a uniaxial crystal for different values of n e / n o .
Fig. 7
Fig. 7 Normalized OAM flux J z ( z ) / J z ( 0 ) and its components J z x ( z ) / J z ( 0 ) , J z y ( z ) / J z ( 0 ) of an anisotropic TEGSM beam propagating in a uniaxial crystal for different values of the twist factor μ 0 x x .
Fig. 8
Fig. 8 Normalized OAM flux J z ( z ) / J z ( 0 ) and its components J z x ( z ) / J z ( 0 ) , J z y ( z ) / J z ( 0 ) of an anisotropic TEGSM beam propagating in a uniaxial crystal for different values of the twist factor μ 0 y y .

Equations (41)

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W ( r 1 , r 2 ) = ( W x x ( r 1 , r 2 ) W x y ( r 1 , r 2 ) W y x ( r 1 , r 2 ) W y y ( r 1 , r 2 ) ) ,
W α β ( r 1 , r 2 ) = A α A β B α β exp [ 1 4 r 1 T ( σ 0 α 2 ) 1 r 1 1 4 r 2 T ( σ 0 β 2 ) 1 r 2 1 2 ( r 1 r 2 ) T ( δ 0 α β 2 ) 1 ( r 1 r 2 ) ] × exp [ i k 2 ( r 1 r 2 ) T ( R 0 α β 1 + μ 0 α β J ) ( r 1 + r 2 ) ] , ( α = x , y ; β = x , y ) ,
σ 0 α 2 = ( σ 0 α 11 2 σ 0 α 12 2 σ 0 α 12 2 σ 0 α 22 2 ) , σ 0 β 2 = ( σ 0 β 11 2 σ 0 β 12 2 σ 0 β 12 2 σ 0 β 22 2 ) , R 0 α β = ( R 0 α β 11 R 0 α β 12 R 0 α β 12 R 0 α β 22 ) , δ 0 α β 2 = ( δ 0 α β 11 2 δ 0 α β 12 2 δ 0 α β 12 2 δ 0 α β 22 2 ) .
J = ( 0 1 1 0 ) .
W T r ( r 1 , r 2 ) = Tr W ( r 1 , r 2 ) = W x x ( r 1 , r 2 ) + W y y ( r 1 , r 2 ) .
W ( r , p ) = ( k 2 π ) 2 { W x x ( r + r 2 , r r 2 ) + W y y ( r + r 2 , r r 2 ) } exp { i k p T r } d r = ( k 2 π ) 2 W x x ( r + r 2 , r r 2 ) exp { i k p T r } d r + ( k 2 π ) 2 W y y ( r + r 2 , r r 2 ) exp { i k p T r } d r = W x x ( r , p ) + W y y ( r , p ) .
r = r 1 + r 2 2 , r ' = r 1 r 2 .
W x x ( r + r 2 , r r 2 ) = A x 2 exp { 1 2 r T L 0 x x r 1 2 ( r ) T ( 1 4 L 0 x x + N 0 x x ) ( r ) i k ( r ) T K 0 x x r } ,
W y y ( r + r 2 , r r 2 ) = A y 2 exp { 1 2 r T L 0 y y r 1 2 ( r ) T ( 1 4 L 0 y y + N 0 y y ) ( r ) i k ( r ) T K 0 y y r } .
L 0 x x = ( σ 0 x 2 ) 1 , N 0 x x = ( δ 0 x x 2 ) 1 , K 0 x x = ( R 0 x x + μ 0 x x J ) ,
L 0 y y = ( σ 0 y 2 ) 1 , N 0 y y = ( δ 0 y y 2 ) 1 , K 0 y y = ( R 0 y y + μ 0 y y J ) .
W x x ( r , p ) = ( k 2 π ) 2 W x x ( r + r 2 , r r 2 ) exp { i k p T r } d r = 2 π Det ( σ 0 x ) A x 2 1 ( 2 π ) 2 ( Det v 0 x ) 1 2 exp { 1 2 q T v 0 x q } ,
W y y ( r , p ) = ( k 2 π ) 2 W y y ( r + r 2 , r r 2 ) exp { i k p T r } d r = 2 π Det ( σ 0 y ) A y 2 1 ( 2 π ) 2 ( Det v 0 y ) 1 2 exp { 1 2 q T v 0 y q } .
v 0 x = [ L 0 x x + k 2 K 0 x x T ( 1 4 L 0 x x + N 0 x x ) 1 K 0 x x k 2 K 0 x x T ( 1 4 L 0 x x + N 0 x x ) 1 k 2 ( 1 4 L 0 x x + N 0 x x ) 1 K 0 x x k 2 ( 1 4 L 0 x x + N 0 x x ) 1 ] ,
v 0 y = [ L 0 y y + k 2 K 0 y y T ( 1 4 L 0 y y + N 0 y y ) 1 K 0 y y k 2 K 0 y y T ( 1 4 L 0 y y + N 0 y y ) 1 k 2 ( 1 4 L 0 y y + N 0 y y ) 1 K 0 y y k 2 ( 1 4 L 0 y y + N 0 y y ) 1 ] .
[ x 2 x y x u x v x y y 2 y u y v x u y u u 2 u v x v y v u v v 2 ] = 1 I 0 q T q { W x x ( r , p ) + W y y ( r , p ) } d r d p = 1 I 0 ( 2 π Det ( σ 0 x ) A x 2 v 0 x 1 + 2 π Det ( σ 0 y ) A y 2 v 0 y 1 ) ,
v 0 x 1 = [ L 0 x x 1 L 0 x x 1 K 0 x x T K 0 x x L 0 x x 1 K 0 x x L 0 x x 1 K 0 x x T + ( 1 4 L 0 x x + N 0 x x ) ] ,
v 0 y 1 = [ L 0 y y 1 L 0 y y 1 K 0 y y T K 0 y y L 0 y y 1 K 0 y y L 0 y y 1 K 0 y y T + ( 1 4 L 0 y y + N 0 y y ) ] .
J z = I 0 c ( x v y u ) .
J z ( 0 ) = J z x ( 0 ) + J z y ( 0 ) ,
J z x ( 0 ) = π A x 2 Det ( σ 0 x ) c { T r [ ( R 0 x x 1 ( σ 0 x 2 ) ( σ 0 x 2 ) R 0 x x 1 ) J ] 2 μ 0 x x T r [ ( σ 0 x 2 ) ] } ,
J z y ( 0 ) = π A y 2 Det ( σ 0 y ) c { T r [ ( R 0 y y 1 ( σ 0 y 2 ) ( σ 0 y 2 ) R 0 y y 1 ) J ] 2 μ 0 y y T r [ ( σ 0 y 2 ) ] } .
W ( r 1 , r 2 ) = ( W x x ( r 1 , r 2 ) 0 0 W y y ( r 1 , r 2 ) ) .
P 0 = 1 4 Det W ( r 1 , r 2 ) [ Tr W ( r 1 , r 2 ) ] 2 .
W α β ( r ˜ ) = 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 = ( R 0 α β 1 i 2 k ( σ 0 α 2 ) 1 i k ( δ 0 α β 2 ) 1 i k ( δ 0 α β 2 ) 1 + μ 0 α β J i k ( δ 0 α β ) 1 + μ 0 α β J T R 0 α β 1 i 2 k ( σ 0 β 2 ) 1 i k ( δ 0 α β 2 ) 1 ) .
W α β ( ρ ˜ ) = A α A β B α β H α β exp ( i k 2 ρ ˜ T M 1 α β 1 ρ ˜ ) , ( α = x , y ; β = x , y ) ,
M 1 α β 1 = ( m 11 α β m 12 α β m 13 α β m 14 α β m 21 α β m 22 α β m 23 α β m 24 α β m 31 α β m 32 α β m 33 α β m 34 α β m 41 α β m 42 α β m 43 α β m 44 α β ) = ( R 1 α β 1 i 2 k ( σ 1 α 2 ) 1 i k ( δ 1 α β 2 ) 1 i k ( δ 1 α β 2 ) 1 + μ 1 α β J i k ( δ 1 α β ) 1 + μ 1 α β J T R 1 α β 1 i 2 k ( σ 1 β 2 ) 1 i k ( δ 1 α β 2 ) 1 ) .
( σ 1 α 2 ) 1 = ( i k ( m 11 α β + m 33 α β + 2 m 13 α β ) i k ( m 12 α β + m 34 α β + m 14 α β + m 32 α β ) i k ( m 12 α β + m 34 α β + m 14 α β + m 32 α β ) i k ( m 22 α β + m 44 α β + 2 m 24 α β ) ) , ( δ 1 α β 2 ) 1 = ( i k m 13 α β i k ( m 14 α β + m 32 α β ) / 2 i k ( m 14 α β + m 32 α β ) / 2 i k m 24 α β ) , R 1 α β 1 = 1 2 ( m 11 α β m 33 α β m 12 α β m 34 α β m 12 α β m 34 α β m 22 α β m 44 α β ) , μ 1 α β = m 14 α β m 32 α β 2 .
J z ( z ) = J z x ( z ) + J z y ( z ) ,
J z x ( z ) = π A x 2 H x x Det ( σ 1 x ) c { T r [ ( R 1 x x 1 ( σ 1 x 2 ) ( σ 1 x 2 ) R 1 x x 1 ) J ] 2 μ 1 x x T r [ ( σ 1 x 2 ) ] } ,
J z y ( z ) = π A y 2 H y y Det ( σ 1 y ) c { T r [ ( R 1 y y 1 ( σ 1 y 2 ) ( σ 1 y 2 ) R 1 y y 1 ) J ] 2 μ 1 y y T r [ ( σ 1 y 2 ) ] } .
W α β ( ρ ˜ ) = A α A β B α β [ Det ( A ¯ + B ¯ M 0 α β 1 ) ] 1 / 2 exp ( i k 2 ρ ˜ T M 1 α β 1 ρ ˜ ) ,
M 1 α β 1 = ( C ¯ + D ¯ M 0 α β 1 ) ( A ¯ + B ¯ M 0 α β 1 ) 1 .
A ¯ = ( A 0 I 0 I A ) , B ¯ = ( B 0 I 0 I B ) , C ¯ = ( C 0 I 0 I C ) , D ¯ = ( D 0 I 0 I D ) ,
A = D = I , B = 0 I , C = ( cos 2 φ / f sin 2 φ / 2 f sin 2 φ / 2 f sin 2 φ / f ) ,
ε = ( n e 2 0 0 0 n o 2 0 0 0 n o 2 ) ,
W x x ( ρ x 1 , ρ y 1 , ρ x 2 , ρ y 2 ) = A x 2 [ Det ( I ˜ + B x x M 0 x x 1 ) ] 1 / 2 exp ( i k 2 ρ ˜ T M 1 x x 1 ρ ˜ ) ,
W y y ( ρ x 1 , ρ y 1 , ρ x 2 , ρ y 2 ) = A y 2 [ Det ( I ˜ + B y y M 0 y y 1 ) ] 1 / 2 exp ( i k 2 ρ ˜ T M 1 y y 1 ρ ˜ ) ,
B x x = [ z n e n o 2 0 0 0 0 z n e 0 0 0 0 z n e n o 2 0 0 0 0 z n e ] , B y y = [ z n o 0 0 0 0 z n o 0 0 0 0 z n o 0 0 0 0 z n o ] ,
M 1 x x 1 = ( M 0 x x + B x x ) 1 , M 1 y y 1 = ( M 0 y y + B y y ) 1 .

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