Abstract

The evolution properties of the normalized intensity distribution, the spectral degree of coherence (SDOC), and the spectral degree of polarization (SDOP) of the radially polarized multi-Gaussian Schell-model (MGSM) beam in uniaxial crystals are illustrated. Numerical results show that the intensity distribution of the radially polarized MGSM beam gradually evolves from a doughnut shape into an elliptical symmetric flattop shape and retains its elliptical flattop shape on further propagation in anisotropic crystals. The evolution behavior of the SDOC and SDOP for the radially polarized MGSM beam is quite different from that of the linearly polarized one. In addition, the influences of the spatial coherence length δ0, beam index M, and the ratio of the extraordinary refractive index to the ordinary refractive index ne/no of the uniaxial crystals on the evolution properties of the normalized intensity distribution, the SDOC, and the SDOP of the radially polarized MGSM beam are discussed in detail.

© 2018 Optical Society of America

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J. Opt. Soc. Am. A 34(9) 1703-1710 (2017)

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2018 (3)

2017 (4)

H. Xu, L. Huang, Z. Sheng, X. Tang, and J. Qu, “Propagation properties of an orthogonal cosine-Gaussian Schell-model beam in uniaxial crystals orthogonal to the optical axis,” J. Opt. 19, 085601 (2017).
[Crossref]

Y. Mao and Z. Mei, “Propagation properties of the rectangular multi-Gaussian Schell-model beams in uniaxial crystals orthogonal to the optical axis,” IEEE Photon. J. 9, 6100410 (2017).
[Crossref]

Y. Zhang, L. Pan, and Y. Cai, “Propagation of correlation singularities of a partially coherent Laguerre–Gaussian electromagnetic beam in a uniaxial crystal,” IEEE Photon. J. 9, 6101213 (2017).
[Crossref]

C. Ping, C. Liang, F. Wang, and Y. Cai, “Radially polarized multi-Gaussian Schell-model beam and its tight focusing properties,” Opt. Express 25, 32475–32490 (2017).
[Crossref]

2016 (1)

X. Chen and D. Zhao, “Propagation properties of electromagnetic rectangular multi-Gaussian Schell-model beams in oceanic turbulence,” Opt. Commun. 372, 137–143 (2016).
[Crossref]

2015 (5)

2014 (7)

2013 (4)

Z. Mei, O. Korotkova, and E. Shchepakina, “Electromagnetic multi-Gaussian Schell-model beams,” J. Opt. 15, 025705 (2013).
[Crossref]

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[Crossref]

F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38, 1814–1816 (2013).
[Crossref]

Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38, 91–93 (2013).
[Crossref]

2012 (4)

2011 (1)

2009 (4)

2008 (1)

2007 (1)

2006 (2)

2005 (1)

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

2003 (2)

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, 2163–2171 (2003).
[Crossref]

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

2002 (1)

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Brown, D. P.

Brown, T. G.

Cai, Y.

S. Sahin, M. Zhang, Y. Chen, and Y. Cai, “Transmission of a polychromatic electromagnetic multi-Gaussian Schell-model beam in an inhomogeneous gradient-index fiber,” J. Opt. Soc. Am. A 35, 1604–1611 (2018).
[Crossref]

C. Ping, C. Liang, F. Wang, and Y. Cai, “Radially polarized multi-Gaussian Schell-model beam and its tight focusing properties,” Opt. Express 25, 32475–32490 (2017).
[Crossref]

Y. Zhang, L. Pan, and Y. Cai, “Propagation of correlation singularities of a partially coherent Laguerre–Gaussian electromagnetic beam in a uniaxial crystal,” IEEE Photon. J. 9, 6101213 (2017).
[Crossref]

Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91, 013823 (2015).
[Crossref]

Z. Zhu, L. Liu, F. Wang, and Y. Cai, “Evolution properties of a Laguerre–Gaussian correlated Schell-model beam propagating in uniaxial crystals orthogonal to the optical axis,” J. Opt. Soc. Am. A 32, 374–380 (2015).
[Crossref]

F. Wang, C. Liang, Y. Yuan, and Y. Cai, “Generalized multi-Gaussian correlated Schell-model beam: from theory to experiment,” Opt. Express 22, 23456–23464 (2014).
[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, 2083–2096 (2014).
[Crossref]

Y. Zhang, L. Liu, C. Zhao, and Y. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378, 750–754 (2014).
[Crossref]

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[Crossref]

F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38, 1814–1816 (2013).
[Crossref]

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

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

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

Chan, V.

Chen, R.

Chen, X.

X. Chen and D. Zhao, “Propagation properties of electromagnetic rectangular multi-Gaussian Schell-model beams in oceanic turbulence,” Opt. Commun. 372, 137–143 (2016).
[Crossref]

Chen, Y.

Chu, X.

Ciattoni, A.

Davidson, F.

Deng, D.

Dong, Y.

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

Eyyuboglu, H.

Eyyuboglu, H. T.

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[Crossref]

Firberg, A.

Gbur, G.

Gori, F.

Gu, J.

Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91, 013823 (2015).
[Crossref]

Huang, L.

H. Xu, L. Huang, Z. Sheng, X. Tang, and J. Qu, “Propagation properties of an orthogonal cosine-Gaussian Schell-model beam in uniaxial crystals orthogonal to the optical axis,” J. Opt. 19, 085601 (2017).
[Crossref]

Korotkova, O.

Lajunen, H.

Liang, C.

Liu, D.

Liu, L.

Liu, X.

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[Crossref]

F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38, 1814–1816 (2013).
[Crossref]

Lu, X.

Ma, L.

Mao, Y.

Y. Mao and Z. Mei, “Propagation properties of the rectangular multi-Gaussian Schell-model beams in uniaxial crystals orthogonal to the optical axis,” IEEE Photon. J. 9, 6100410 (2017).
[Crossref]

Martínez-Herrero, R.

Mei, Z.

Y. Mao and Z. Mei, “Propagation properties of the rectangular multi-Gaussian Schell-model beams in uniaxial crystals orthogonal to the optical axis,” IEEE Photon. J. 9, 6100410 (2017).
[Crossref]

Z. Mei, “Two types of sinc Schell-model beams and their propagation characteristics,” Opt. Lett. 39, 4188–4191 (2014).
[Crossref]

Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38, 91–93 (2013).
[Crossref]

Z. Mei, O. Korotkova, and E. Shchepakina, “Electromagnetic multi-Gaussian Schell-model beams,” J. Opt. 15, 025705 (2013).
[Crossref]

Mejías, P.

Norrman, A.

Palma, C.

Pan, L.

Y. Zhang, L. Pan, and Y. Cai, “Propagation of correlation singularities of a partially coherent Laguerre–Gaussian electromagnetic beam in a uniaxial crystal,” IEEE Photon. J. 9, 6101213 (2017).
[Crossref]

Ping, C.

Ponomarenko, S.

Qu, J.

H. Xu, L. Huang, Z. Sheng, X. Tang, and J. Qu, “Propagation properties of an orthogonal cosine-Gaussian Schell-model beam in uniaxial crystals orthogonal to the optical axis,” J. Opt. 19, 085601 (2017).
[Crossref]

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[Crossref]

Ricklin, J.

Saastamoinen, T.

Sahin, S.

Sanchez, V. R.

F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. 11, 085706 (2009).
[Crossref]

Santarsiero, M.

F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. 11, 085706 (2009).
[Crossref]

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32, 3531–3533 (2007).
[Crossref]

Schoonover, R. W.

Shchepakina, E.

O. Korotkova and E. Shchepakina, “Rectangular multi-Gaussian Schell-model beams in atmospheric turbulence,” J. Opt. 16, 045704 (2014).
[Crossref]

Z. Mei, O. Korotkova, and E. Shchepakina, “Electromagnetic multi-Gaussian Schell-model beams,” J. Opt. 15, 025705 (2013).
[Crossref]

O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29, 2159–2164 (2012).
[Crossref]

Sheng, Z.

H. Xu, L. Huang, Z. Sheng, X. Tang, and J. Qu, “Propagation properties of an orthogonal cosine-Gaussian Schell-model beam in uniaxial crystals orthogonal to the optical axis,” J. Opt. 19, 085601 (2017).
[Crossref]

Shirai, T.

F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. 11, 085706 (2009).
[Crossref]

Tang, M.

Tang, X.

H. Xu, L. Huang, Z. Sheng, X. Tang, and J. Qu, “Propagation properties of an orthogonal cosine-Gaussian Schell-model beam in uniaxial crystals orthogonal to the optical axis,” J. Opt. 19, 085601 (2017).
[Crossref]

Visser, T. D.

Wang, F.

C. Ping, C. Liang, F. Wang, and Y. Cai, “Radially polarized multi-Gaussian Schell-model beam and its tight focusing properties,” Opt. Express 25, 32475–32490 (2017).
[Crossref]

Z. Zhu, L. Liu, F. Wang, and Y. Cai, “Evolution properties of a Laguerre–Gaussian correlated Schell-model beam propagating in uniaxial crystals orthogonal to the optical axis,” J. Opt. Soc. Am. A 32, 374–380 (2015).
[Crossref]

Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91, 013823 (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, 2083–2096 (2014).
[Crossref]

F. Wang, C. Liang, Y. Yuan, and Y. Cai, “Generalized multi-Gaussian correlated Schell-model beam: from theory to experiment,” Opt. Express 22, 23456–23464 (2014).
[Crossref]

F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38, 1814–1816 (2013).
[Crossref]

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[Crossref]

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

Wang, Y.

Wolf, E.

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

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

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Xie, J.

Xu, H.

H. Xu, L. Huang, Z. Sheng, X. Tang, and J. Qu, “Propagation properties of an orthogonal cosine-Gaussian Schell-model beam in uniaxial crystals orthogonal to the optical axis,” J. Opt. 19, 085601 (2017).
[Crossref]

Ye, J.

Yin, H.

Yuan, Y.

Zhang, J.

Zhang, M.

Zhang, Y.

Y. Zhang, L. Pan, and Y. Cai, “Propagation of correlation singularities of a partially coherent Laguerre–Gaussian electromagnetic beam in a uniaxial crystal,” IEEE Photon. J. 9, 6101213 (2017).
[Crossref]

Y. Zhang, L. Liu, C. Zhao, and Y. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378, 750–754 (2014).
[Crossref]

Zhao, C.

Y. Zhang, L. Liu, C. Zhao, and Y. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378, 750–754 (2014).
[Crossref]

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

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

Zhao, D.

X. Chen and D. Zhao, “Propagation properties of electromagnetic rectangular multi-Gaussian Schell-model beams in oceanic turbulence,” Opt. Commun. 372, 137–143 (2016).
[Crossref]

M. Tang and D. Zhao, “Propagation of multi-Gaussian Schell-model vortex beams in isotropic random media,” Opt. Express 23, 32766–32776 (2015).
[Crossref]

Zheng, X.

Zhou, G.

Zhou, Z.

Zhu, S.

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

Zhu, Z.

IEEE Photon. J. (2)

Y. Zhang, L. Pan, and Y. Cai, “Propagation of correlation singularities of a partially coherent Laguerre–Gaussian electromagnetic beam in a uniaxial crystal,” IEEE Photon. J. 9, 6101213 (2017).
[Crossref]

Y. Mao and Z. Mei, “Propagation properties of the rectangular multi-Gaussian Schell-model beams in uniaxial crystals orthogonal to the optical axis,” IEEE Photon. J. 9, 6100410 (2017).
[Crossref]

J. Lightwave Technol. (1)

J. Opt. (4)

H. Xu, L. Huang, Z. Sheng, X. Tang, and J. Qu, “Propagation properties of an orthogonal cosine-Gaussian Schell-model beam in uniaxial crystals orthogonal to the optical axis,” J. Opt. 19, 085601 (2017).
[Crossref]

F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. 11, 085706 (2009).
[Crossref]

O. Korotkova and E. Shchepakina, “Rectangular multi-Gaussian Schell-model beams in atmospheric turbulence,” J. Opt. 16, 045704 (2014).
[Crossref]

Z. Mei, O. Korotkova, and E. Shchepakina, “Electromagnetic multi-Gaussian Schell-model beams,” J. Opt. 15, 025705 (2013).
[Crossref]

J. Opt. Soc. Am. A (9)

J. Ricklin and F. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. A 19, 1794–1802 (2002).
[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, 2163–2171 (2003).
[Crossref]

D. Liu and Z. Zhou, “Propagation of partially coherent flat-topped beams in uniaxial crystals orthogonal to the optical axis,” J. Opt. Soc. Am. A 26, 924–930 (2009).
[Crossref]

O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29, 2159–2164 (2012).
[Crossref]

G. Gbur, “Partially coherent beam propagation in atmospheric turbulence [invited],” J. Opt. Soc. Am. A 31, 2038–2045 (2014).
[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, 2083–2096 (2014).
[Crossref]

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[Crossref]

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[Crossref]

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[Crossref]

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[Crossref]

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Figures (10)

Fig. 1.
Fig. 1. Geometry of the propagation of a beam in uniaxial crystals orthogonal to the optical axis.
Fig. 2.
Fig. 2. Contour graphs of the normalized intensity distribution of the radially polarized MGSM beams propagating in uniaxial crystals at several different propagation distances for different values of the spatial coherence length δ 0 with M = 10 and n e / n o = 1.2 .
Fig. 3.
Fig. 3. Contour graphs of the normalized intensity distribution of the radially polarized MGSM beams propagating in uniaxial crystals at several different propagation distances for different values of the beam index M with spatial coherence length δ 0 = 0.1 w 0 and n e / n o = 1.2 .
Fig. 4.
Fig. 4. Contour graphs of the normalized intensity distribution of the radially polarized MGSM beam propagating in uniaxial crystals at several different propagation distances for different values of the ratio n e / n o with beam index M = 10 and spatial coherence length δ 0 = 0.1 w 0 .
Fig. 5.
Fig. 5. Modulus of the SDOC of the radially polarized MGSM beams propagating in uniaxial crystals at several different propagation distances for different values of the spatial coherence length δ 0 with M = 10 and n e / n o = 1.2 .
Fig. 6.
Fig. 6. Modulus of the SDOC of the radially polarized MGSM beams propagating in uniaxial crystals at several different propagation distances for different values of the beam index M with spatial coherence length δ 0 = 0.5 w 0 and n e / n o = 1.2 .
Fig. 7.
Fig. 7. Modulus of the SDOC of the radially polarized MGSM beam propagating in uniaxial crystals at several different propagation distances for different values of the ratio n e / n o with beam index M = 10 and spatial coherence length δ 0 = 0.5 w 0 .
Fig. 8.
Fig. 8. Distribution of the SDOP of the radially polarized MGSM beam propagating in uniaxial crystals at several different propagation distances for different values of the spatial coherence length δ 0 with M = 10 and n e / n o = 1.2 .
Fig. 9.
Fig. 9. Distribution of the SDOP of the radially polarized MGSM beam propagating in uniaxial crystals at several different propagation distances for different values of the beam index M with spatial coherence length δ 0 = 0.5 w 0 and n e / n o = 1.2 .
Fig. 10.
Fig. 10. Distribution of the SDOP of the radially polarized MGSM beam propagating in uniaxial crystals at several different propagation distances for different values of the ratio n e / n o with beam index M = 10 and spatial coherence length δ 0 = 0.5 w 0 .

Equations (14)

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E ( r ; ω ) = E x ( r ; ω ) e x + E y ( r ; ω ) e y = x w 0 exp ( r 2 4 w 0 2 ) e x + y w 0 exp ( r 2 4 w 0 2 ) e y ,
W α β ( r 1 , r 2 , ω ) = E α * ( r 1 ; ω ) E β ( r 2 ; ω ) , ( α , β = x , y ) ,
W α β ( 0 ) ( r 1 , r 2 , 0 ) = α 1 β 2 w 0 2 exp ( r 1 2 + r 2 2 4 w 0 2 ) μ α β ( r 1 r 2 ) ,
μ α β ( r 1 r 2 ) = 1 C 0 m = 1 M ( M m ) ( 1 ) m 1 m exp [ ( r 1 r 2 ) 2 2 m δ 0 2 ] ,
ϵ = ( n e 2 0 0 0 n o 2 0 0 0 n o 2 ) ,
W α β ( ρ 1 , ρ 2 , z ) = k 0 2 n o 2 4 π 2 z 2 W α β ( 0 ) ( r 1 , r 2 , 0 ) × exp { i k 0 2 z n e [ n o 2 ( x 1 ρ x 1 ) 2 + n e 2 ( y 1 ρ y 1 ) 2 ] } × exp { i k 0 2 z n e [ n o 2 ( x 2 ρ x 2 ) 2 + n e 2 ( y 2 ρ y 2 ) 2 ] } d r 1 d r 2 ,
W x x ( ρ 1 , ρ 2 , z ) = k 0 2 n o 2 4 w 0 2 z 2 1 C 0 m = 1 M ( M m ) ( 1 ) m 1 m 1 ( a 2 d 2 ) 1 / 2 ( a 1 d 1 ) 3 / 2 × ( c 0 d 1 Δ γ x 2 + b 1 ρ x 2 Δ γ x + c 0 2 ) exp ( Π x 2 + Π y 2 ) ,
W x y ( ρ 1 , ρ 2 , z ) = k 0 2 n o 2 4 w 0 2 z 2 1 C 0 m = 1 M ( M m ) ( 1 ) m 1 m 1 ( a 1 d 2 ) 1 / 2 ( d 1 a 2 ) 3 / 2 × Δ γ x ( c 0 d 2 Δ γ y + b 2 ρ y 2 ) exp ( Π x 2 + Π y 2 ) ,
W y x ( ρ 1 , ρ 2 , z ) = k 0 2 n o 2 4 w 0 2 z 2 1 C 0 m = 1 M ( M m ) ( 1 ) m 1 m 1 ( a 2 d 1 ) 1 / 2 ( a 1 d 2 ) 3 / 2 × Δ γ y ( c 0 d 1 Δ γ x + b 1 ρ x 2 ) exp ( Π x 2 + Π y 2 ) ,
W y y ( ρ 1 , ρ 2 , z ) = k 0 2 n o 2 4 w 0 2 z 2 1 C 0 m = 1 M ( M m ) ( 1 ) m 1 m 1 ( a 1 d 1 ) 1 / 2 ( a 2 d 2 ) 3 / 2 × ( c 0 d 2 Δ γ y 2 + b 2 ρ y 2 Δ γ y + c 0 2 ) exp ( Π x 2 + Π y 2 ) ,
a 1 = 1 4 w 0 2 + 1 2 m δ 0 2 + i k 0 n o 2 2 z n e , b 1 = i k 0 n o 2 2 z n e , c 0 = 1 2 m δ 0 2 , d 1 = a 1 * c 0 2 a 1 , Δ γ x = b 1 ( c 0 a 1 ρ x 2 ρ x 1 ) , a 2 = 1 4 w 0 2 + 1 2 m δ 0 2 + i k 0 n e 2 z , b 2 = i k 0 n e 2 z , d 2 = a 2 * c 0 2 a 2 , Δ γ y = b 2 ( c 0 a 2 ρ y 2 ρ y 1 ) , Π x 2 = b 1 ( ρ x 1 2 ρ x 2 2 ) + b 1 2 a 1 ρ x 2 2 + Δ γ x 2 d 1 , Π y 2 = b 2 ( ρ y 1 2 ρ y 2 2 ) + b 2 2 a 2 ρ y 2 2 + Δ γ y 2 d 2 .
I ( ρ , z ) = W x x ( ρ , ρ , z ) + W y y ( ρ , ρ , z ) ,
P ( ρ , z ) = 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 ) ] .

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