Abstract

With the help of a tensor method, an analytical formula is derived for the cross-spectral density matrix of a twisted electromagnetic Gaussian Schell-model (EGSM) beam (i.e., EGSM beam with twist phase) propagating in a uniaxial crystal orthogonal to the optical axis. The twist phase-induced changes of the statistical properties, such as the spectral density, the degree of polarization and the degree of coherence, of an EGSM beam propagating in a uniaxial crystal are illustrated numerically. It is found that the distributions of the spectral density, the degree of polarization and the degree of coherence of a twisted EGSM beam in a uniaxial crystal all exhibit non-circular symmetries, which are quite different from those of a twisted EGSM beam in isotropic medium or in free space. One may use uniaxial crystal to determine whether an EGSM beam carries twist phase or not.

© 2015 Optical Society of America

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References

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    [Crossref]
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    [Crossref] [PubMed]

2015 (3)

2014 (4)

2013 (1)

2012 (4)

2011 (3)

2010 (5)

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

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

X. Du and D. Zhao, “Propagation of uniformly polarized stochastic electromagnetic beams in uniaxial crystals,” J. Electromagn. Waves Appl. 24(7), 971–981 (2010).
[Crossref]

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]

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]

2009 (4)

D. Liu and Z. Zhou, “Propagation of partially polarized, partially coherent beams in uniaxially crystals orthogonal to the optical axis,” Eur. Phys. J. D 54(1), 95–101 (2009).
[Crossref]

D. Liu and Z. Zhou, “Generalized Stokes parameters of stochastic electromagnetic beams propagating through uniaxial crystals orthogonal to the optical axis,” J. Opt. A 11(6), 065710 (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]

2008 (5)

2007 (2)

2006 (2)

2005 (5)

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, 1611–1618 (2005).
[Crossref]

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A 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 and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249(4-6), 379–385 (2005).
[Crossref]

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

2004 (1)

2003 (2)

2002 (3)

2001 (2)

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. 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 (1)

1992 (1)

1980 (1)

F. Gori, “Collet–Wolf sources and multimode lasers,” Opt. Commun. 34(3), 301–305 (1980).
[Crossref]

1979 (1)

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29(3), 256–260 (1979).
[Crossref]

1978 (1)

E. Wolf and E. Collett, “Partially coherent sources which produce the same far-field intensity distribution as a laser,” Opt. Commun. 25(3), 293–296 (1978).
[Crossref]

Agrawal, G. P.

Bastiaans, M. J.

Basu, S.

Baykal, Y.

Borghi, R.

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]

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]

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]

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]

Y. Shen, L. Liu, C. Zhao, Y. Yuan, and Y. Cai, “Second-order moments of an electromagnetic Gaussian Schell-model beam in a uniaxial crystal,” J. Opt. Soc. Am. A 31(2), 238–245 (2014).
[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]

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]

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. Cai, “Generation of various partially coherent beams and their propagation properties in turbulent atmosphere: a review,” Proc. SPIE 7924, 792402 (2011).
[Crossref]

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

L. Zhang and Y. Cai, “Statistical properties of a nonparaxial Gaussian Schell-model beam in a uniaxial crystal,” Opt. Express 19(14), 13312–13325 (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]

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]

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]

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]

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

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. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (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] [PubMed]

Chen, R.

Chen, Y.

Chu, X.

Ciattoni, A.

Cincotti, G.

Collett, E.

E. Wolf and E. Collett, “Partially coherent sources which produce the same far-field intensity distribution as a laser,” Opt. Commun. 25(3), 293–296 (1978).
[Crossref]

De Santis, P.

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29(3), 256–260 (1979).
[Crossref]

Dogariu, A.

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

Dong, Y.

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]

Du, X.

X. Du and D. Zhao, “Propagation of uniformly polarized stochastic electromagnetic beams in uniaxial crystals,” J. Electromagn. Waves Appl. 24(7), 971–981 (2010).
[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.

Friberg, A. T.

Gori, F.

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]

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]

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

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

F. Gori, “Collet–Wolf sources and multimode lasers,” Opt. Commun. 34(3), 301–305 (1980).
[Crossref]

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29(3), 256–260 (1979).
[Crossref]

Guattari, G.

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29(3), 256–260 (1979).
[Crossref]

He, S.

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006).
[Crossref]

Hyde, M. W.

James, D. F. V.

Kaivola, M.

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(1), 016615 (2002).
[Crossref] [PubMed]

Korotkova, O.

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]

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, “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]

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

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

D. Liu and Z. Zhou, “Propagation of partially polarized, partially coherent beams in uniaxially crystals orthogonal to the optical axis,” Eur. Phys. J. D 54(1), 95–101 (2009).
[Crossref]

D. Liu and Z. Zhou, “Generalized Stokes parameters of stochastic electromagnetic beams propagating through uniaxial crystals orthogonal to the optical axis,” J. Opt. A 11(6), 065710 (2009).
[Crossref]

Liu, L.

Liu, X.

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.

Palma, C.

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.

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, 1611–1618 (2005).
[Crossref]

Ramírez-Sánchez, V.

Roychowdhury, H.

H. Roychowdhury, G. P. Agrawal, and E. Wolf, “Changes in the spectrum, in the spectral degree of polarization, and in the spectral degree of coherence of a partially coherent beam propagating through a gradient-index fiber,” J. Opt. Soc. Am. A 23(4), 940–948 (2006).
[Crossref] [PubMed]

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249(4-6), 379–385 (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, 1611–1618 (2005).
[Crossref]

Salem, M.

Santarsiero, M.

Serna, J.

Setälä, T.

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(1), 016615 (2002).
[Crossref] [PubMed]

Shen, Y.

Shevchenko, A.

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(1), 016615 (2002).
[Crossref] [PubMed]

Shirai, T.

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

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.

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]

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]

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]

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]

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]

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

Wolf, E.

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

H. Roychowdhury, G. P. Agrawal, and E. Wolf, “Changes in the spectrum, in the spectral degree of polarization, and in the spectral degree of coherence of a partially coherent beam propagating through a gradient-index fiber,” J. Opt. Soc. Am. A 23(4), 940–948 (2006).
[Crossref] [PubMed]

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]

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, 1611–1618 (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]

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

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]

E. Wolf and E. Collett, “Partially coherent sources which produce the same far-field intensity distribution as a laser,” Opt. Commun. 25(3), 293–296 (1978).
[Crossref]

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.

Yao, M.

Yuan, Y.

Zhang, L.

Zhang, M.

Zhao, C.

Zhao, D.

X. Du and D. Zhao, “Propagation of uniformly polarized stochastic electromagnetic beams in uniaxial crystals,” J. Electromagn. Waves Appl. 24(7), 971–981 (2010).
[Crossref]

Zhou, G.

Zhou, Z.

D. Liu and Z. Zhou, “Generalized Stokes parameters of stochastic electromagnetic beams propagating through uniaxial crystals orthogonal to the optical axis,” J. Opt. A 11(6), 065710 (2009).
[Crossref]

D. Liu and Z. Zhou, “Propagation of partially polarized, partially coherent beams in uniaxially crystals orthogonal to the optical axis,” Eur. Phys. J. D 54(1), 95–101 (2009).
[Crossref]

Zhu, S.

Appl. Phys. B (2)

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. 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]

Appl. Phys. Lett. (1)

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006).
[Crossref]

Eur. Phys. J. D (1)

D. Liu and Z. Zhou, “Propagation of partially polarized, partially coherent beams in uniaxially crystals orthogonal to the optical axis,” Eur. Phys. J. D 54(1), 95–101 (2009).
[Crossref]

J. Electromagn. Waves Appl. (1)

X. Du and D. Zhao, “Propagation of uniformly polarized stochastic electromagnetic beams in uniaxial crystals,” J. Electromagn. Waves Appl. 24(7), 971–981 (2010).
[Crossref]

J. Mod. Opt. (1)

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, 1611–1618 (2005).
[Crossref]

J. Opt. A (2)

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

D. Liu and Z. Zhou, “Generalized Stokes parameters of stochastic electromagnetic beams propagating through uniaxial crystals orthogonal to the optical axis,” J. Opt. A 11(6), 065710 (2009).
[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. Am. A (16)

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]

D. F. V. James, “Change 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]

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

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]

E. Tervonen, A. T. Friberg, and J. Turunen, “Gaussian Schell-model beams generated with synthetic acousto-optic holograms,” J. Opt. Soc. Am. A 9(5), 796–803 (1992).
[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]

Y. Shen, L. Liu, C. Zhao, Y. Yuan, and Y. Cai, “Second-order moments of an electromagnetic Gaussian Schell-model beam in a uniaxial crystal,” J. Opt. Soc. Am. A 31(2), 238–245 (2014).
[Crossref] [PubMed]

H. Roychowdhury, G. P. Agrawal, and E. Wolf, “Changes in the spectrum, in the spectral degree of polarization, and in the spectral degree of coherence of a partially coherent beam propagating through a gradient-index fiber,” J. Opt. Soc. Am. A 23(4), 940–948 (2006).
[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]

A. Ciattoni and C. Palma, “Optical propagation in uniaxial crystals orthogonal to the optical axis: paraxial theory and beyond,” J. Opt. Soc. Am. A 20(11), 2163–2171 (2003).
[Crossref] [PubMed]

A. Ciattoni, G. Cincotti, and C. Palma, “Propagation of cylindrically symmetric fields in uniaxial crystals,” J. Opt. Soc. Am. A 19(4), 792–796 (2002).
[Crossref] [PubMed]

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. (8)

E. Wolf and E. Collett, “Partially coherent sources which produce the same far-field intensity distribution as a laser,” Opt. Commun. 25(3), 293–296 (1978).
[Crossref]

F. Gori, “Collet–Wolf sources and multimode lasers,” Opt. Commun. 34(3), 301–305 (1980).
[Crossref]

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29(3), 256–260 (1979).
[Crossref]

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

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

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

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

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

Opt. Express (8)

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]

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. 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]

L. Zhang and Y. Cai, “Statistical properties of a nonparaxial Gaussian Schell-model beam in a uniaxial crystal,” Opt. Express 19(14), 13312–13325 (2011).
[Crossref] [PubMed]

G. Zhou, R. Chen, and X. Chu, “Propagation of Airy beams in uniaxial crystals orthogonal to the optical axis,” Opt. Express 20(3), 2196–2205 (2012).
[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]

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

Opt. Lett. (11)

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

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

O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source,” Opt. Lett. 29(11), 1173–1175 (2004).
[Crossref] [PubMed]

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

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

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]

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, 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]

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. E Stat. Nonlin. Soft Matter Phys. (1)

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(1), 016615 (2002).
[Crossref] [PubMed]

Proc. SPIE (1)

Y. Cai, “Generation of various partially coherent beams and their propagation properties in turbulent atmosphere: a review,” Proc. SPIE 7924, 792402 (2011).
[Crossref]

Prog. Electromagnetics Res. (1)

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]

Other (3)

Y. Cai, F. Wang, C. Zhao, S. Zhu, G. Wu, and Y. Dong, Partially coherent vector beams: from theory to experiment, in Vectorial Optical Fields: Fundamentals and Applications, Q. Zhen, ed. (World Scientific, 2013), Chap. 7, pp. 221–273.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

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Figures (11)

Fig. 1
Fig. 1 Geometry of the propagation of a beam in a uniaxial crystal orthogonal to the optical axis.
Fig. 2
Fig. 2 Normalized spectral densities I x / I xmax , I y / I ymax , I/ I max of an EGSM beam without twist phase at several propagation distances in isotropic medium ( n e = n o ).
Fig. 3
Fig. 3 Normalized spectral densities I x / I xmax , I y / I ymax , I/ I max of a twisted EGSM beam at z=3000μm in isotropic medium ( n e = n o ) for different values of the twist factors γ xx , γ yy .
Fig. 4
Fig. 4 Normalized spectral densities I x / I xmax , I y / I ymax , I/ I max of an EGSM beam without twist phase at several propagation distances in a uniaxial crystal ( n e =1.5 n o ).
Fig. 5
Fig. 5 Normalized spectral densities I x / I xmax , I y / I ymax , I/ I max of a twisted EGSM beam at z=3000μm in a uniaxial crystal ( n e =1.5 n o ) for different values of the twist factors γ xx with γ yy =0 .
Fig. 6
Fig. 6 Normalized spectral densities I x / I xmax , I y / I ymax , I/ I max of a twisted EGSM beam at z=3000μm in a uniaxial crystal ( n e =1.5 n o ) for different values of the twist factors γ yy with γ xx =0 .
Fig. 7
Fig. 7 Normalized spectral densities I x / I xmax , I y / I ymax , I/ I max of a twisted EGSM beam at z=3000μm in a uniaxial crystal ( n e =1.5 n o ) for different values of the twist factors γ xx , γ yy .
Fig. 8
Fig. 8 Distribution of the degree of polarization of a twisted EGSM beam at z=3000μm in isotropic medium ( n e = n o ) for different values of the twist factors γ xx , γ yy .
Fig. 9
Fig. 9 Distribution of the degree of polarization of a twisted EGSM beam at z=3000μm in a uniaxial crystal ( n e =1.5 n o ) for different values of the twist factors γ xx , γ yy .
Fig. 10
Fig. 10 Distribution of the degree of coherence of a twisted EGSM beam at z=3000μm in isotropic medium ( n e = n o ) for different values of the twist factors γ xx , γ yy .
Fig. 11
Fig. 11 Distribution of the degree of coherence of a twisted EGSM beam at z=3000μm in a uniaxial crystal ( n e =1.5 n o ) for different values of the twist factors γ xx , γ yy .

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

W ( r 1 , r 2 ;ω )=( W xx ( r 1 , r 2 ;ω ) W xy ( r 1 , r 2 ;ω ) W yx ( r 1 , r 2 ;ω ) W yy ( r 1 , r 2 ;ω ) ),
W αβ ( r 1 , r 2 ;ω) = A α A β B αβ exp [ r 1 2 4 σ α 2 r 2 2 4 σ β 2 ( r 1 r 2 ) 2 2 δ αβ 2 ik 2 γ αβ ( r 1 r 2 ) T J( r 1 + r 2 ) ], (α=x,y;β=x,y),
J=( 0 1 1 0 ).
W αβ ( r ˜ ;ω)= A α A β B αβ exp( r ˜ T M ˜ 0αβ 1 r ˜ ), (α=x,y;β=x,y),
M ˜ 0αβ 1 =( 1 2 ( 1 2 σ a 2 + 1 δ αβ 2 )I 1 2 δ αβ 2 I+ ik 2 γ αβ J 1 2 δ αβ 2 I+ ik 2 γ αβ J T 1 2 ( 1 2 σ β 2 + 1 δ αβ 2 )I ).
ε=( n e 2 0 0 0 n o 2 0 0 0 n o 2 ),
W xx ( ρ x1 , ρ y1 , ρ x2 , ρ y2 ;ω)= k 2 n o 2 4 π 2 z 2 W xx ( x 1 , y 1 , x 2 , y 2 ;ω) d x 1 d y 1 d x 2 d y 2 ×exp{ k 2iz n e [ n o 2 ( ρ x1 x 1 ) 2 + n e 2 ( ρ y y 1 ) 2 ] k 2iz n e [ n o 2 ( ρ x2 x 2 ) 2 + n e 2 ( ρ y2 y 2 ) 2 ] },
W yy ( ρ x1 , ρ y1 , ρ x2 , ρ y2 ;ω)= k 2 n o 2 4 π 2 z 2 W yy ( x 1 , y 1 , x 2 , y 2 ;ω) d x 1 d y 1 d x 2 d y 2 ×exp{ k n o 2iz [ ( ρ x1 x 1 ) 2 + ( ρ y y 1 ) 2 ] k n o 2iz [ ( ρ x2 x 2 ) 2 + ( ρ y2 y 2 ) 2 ] },
W xy ( ρ x1 , ρ y1 , ρ x2 , ρ y2 ;ω)= k 2 n o 2 4 π 2 z 2 exp[ikz( n o n e )] W xy ( x 1 , y 1 , x 2 , y 2 ;ω) d x 1 d y 1 d x 2 d y 2 ×exp{ k 2iz n e [ n o 2 ( ρ x1 x 1 ) 2 + n e 2 ( ρ y y 1 ) 2 ] k n o 2iz [ ( ρ x2 x 2 ) 2 + ( ρ y2 y 2 ) 2 ] },
W yx ( ρ x1 , ρ y1 , ρ x2 , ρ y2 ;ω)= W xy * ( ρ x2 , ρ y2 , ρ x1 , ρ y1 ;ω),
W xx ( ρ ˜ ;ω)= 1 π 2 [det ( B ˜ xx ) 1/2 ] W xx ( r ˜ ;ω) exp( r ˜ T B ˜ xx 1 r ˜ ρ ˜ T B ˜ xx 1 ρ ˜ +2 r ˜ T B ˜ xx 1 ρ ˜ )d r ˜ ,
W yy ( ρ ˜ ;ω)= 1 π 2 [det ( B ˜ yy ) 1/2 ] W yy ( r ˜ ;ω) exp( r ˜ T B ˜ yy 1 r ˜ ρ ˜ T B ˜ yy 1 ρ ˜ +2 r ˜ T B ˜ yy 1 ρ ˜ )d r ˜ ,
W xy ( ρ ˜ ;ω)= A x A y B xy π 2 [det ( B ˜ xy ) 1/2 ] exp[ikz( n o n e )] × W xy ( r ˜ ;ω) exp( r ˜ T B ˜ xy 1 r ˜ ρ ˜ T B ˜ xy 1 ρ ˜ +2 r ˜ T B ˜ xy 1 ρ ˜ )d r ˜ ,
B ˜ xx =[ 2iz n e k n o 2 0 0 0 0 2iz k n e 0 0 0 0 2iz n e k n o 2 0 0 0 0 2iz k n e ], B ˜ yy =[ 2iz k n o 0 0 0 0 2iz k n o 0 0 0 0 2iz k n o 0 0 0 0 2iz k n o ], B ˜ xy =[ 2iz n e k n o 2 0 0 0 0 2iz k n e 0 0 0 0 2iz k n o 0 0 0 0 2iz k n o ].
W xx ( ρ ˜ ;ω)= A x 2 B xx [det( I ˜ + B ˜ xx M ˜ 0xx 1 )] 1/2 exp( ρ ˜ T M ˜ 1xx 1 ρ ˜ ),
W yy ( ρ ˜ ;ω)= A y 2 B yy [det( I ˜ + B ˜ yy M ˜ 0yy 1 )] 1/2 exp( ρ ˜ T M ˜ 1yy 1 ρ ˜ ),
W xy ( ρ ˜ ;ω)= A x A y B xy exp[ikz( n o n e )] [det( I ˜ + B ˜ xy M ˜ 0xy 1 )] 1/2 exp( ρ T M ˜ 1xy 1 ρ),
M ˜ 1xx 1 = ( M ˜ 0xx + B ˜ xx ) 1 , M ˜ 1yy 1 = ( M ˜ 0yy + B ˜ yy ) 1 , M ˜ 1xy 1 = ( M ˜ 0xy + B ˜ xy ) 1 .
I( ρ;ω )=Tr W ( ρ,ρ;ω )= W xx ( ρ,ρ;ω )+ W yy ( ρ,ρ;ω )= I x ( ρ;ω )+ I y ( ρ;ω ).
P(ρ;ω)= 1 4Det W (ρ,ρ;ω) [Tr W (ρ,ρ;ω)] 2 ,
μ( ρ 1 , ρ 2 ;ω )= Tr W ( ρ 1 , ρ 2 ;ω ) Tr W ( ρ 1 , ρ 1 ;ω )Tr W ( ρ 2 , ρ 2 ;ω ) .

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