Abstract

A new class of twisted Schell-model array correlated sources are introduced based on Mercer’s expansion. It turns out that such sources can be expressed as superposition of fully coherent Laguerre-Gaussian modes, and the twistable condition is established. Furthermore, on the basis of a stretched coordinate system and a quadratic approximation, analytical expressions for the mutual coherence function of an anisotropic non-Kolmogorov turbulence and the cross-spectral density of a twisted Gaussian Schell-model array beam are rigorously derived. Due to the presence of the twist phase, the beam spot and the degree of coherence rotate as they propagate, but their rotation centers are different. It is shown that the anisotropy of turbulence causes an anisotropic beam spreading in the horizontal and vertical directions. However, impressing a twist phase on source beams can significantly inhibit this effect. For an anticipated atmospheric channel condition, a comprehensive selection of initial optical signal parameters, receiver aperture size and receiver capability, etc., is necessary. Our work is helpful for exploring new forms of twistable sources, and promotes guidance on optimization of partial coherent beam applications.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (4)

2017 (5)

2016 (7)

S. Zhu, J. Wang, X. Liu, Y. Cai, and Z. Li, “Generation of arbitrary radially polarized array beams by manipulating correlation structure,” Appl. Phys. Lett. 109(16), 161904 (2016).
[Crossref]

M. W. Hyde, S. Bose-Pillai, D. G. Voelz, and X. Xiao, “Generation of vector partially coherent optical sources using phase-only spatial light modulators,” Phys. Rev. Appl. 6(6), 064030 (2016).
[Crossref]

G. R. Salla, C. Perumangattu, S. Prabhakar, A. Anwar, and R. P. Singh, “Recovering the vorticity of a light beam after scattering,” Appl. Phys. Lett. 107(2), 021104 (2016).
[Crossref]

J. Wang, S. Zhu, and Z. Li, “Vector properties of a tunable random electromagnetic beam in non-Kolmogrov turbulence,” Chin. Opt. Lett. 14(8), 80101–80105 (2016).
[Crossref]

J. Wang, S. Zhu, H. Wang, Y. Cai, and Z. Li, “Second-order statistics of a radially polarized cosine-Gaussian correlated Schell-model beam in anisotropic turbulence,” Opt. Express 24(11), 11626–11639 (2016).
[Crossref] [PubMed]

X. Liu, J. Yu, Y. Cai, and S. A. Ponomarenko, “Propagation of optical coherence lattices in the turbulent atmosphere,” Opt. Lett. 41(18), 4182–4185 (2016).
[Crossref] [PubMed]

Y. Baykal, Y. Luo, and X. Ji, “Scintillations of higher order laser beams in anisotropic atmospheric turbulence,” Appl. Opt. 55(33), 9422–9426 (2016).
[Crossref] [PubMed]

2015 (5)

2014 (6)

2013 (2)

2012 (1)

2010 (3)

2009 (3)

2008 (1)

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008).
[Crossref]

2007 (1)

2006 (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]

2002 (2)

2001 (1)

2000 (1)

1996 (1)

E. Wolf and D. F. James, “Correlation-induced spectral changes,” Rep. Prog. Phys. 59(6), 771–818 (1996).
[Crossref]

1994 (2)

A. T. Friberg, B. 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]

A. I. Kon, “Qualitative theory of amplitude and phase fluctuations in a medium with anisotropic turbulent irregularity,” Waves Random Complex Media 4(3), 297–306 (1994).
[Crossref]

1993 (2)

1991 (1)

1972 (1)

Agarwal, G. S.

Andrews, L. C.

L. C. Andrews, R. L. Phillips, and R. Crabbs, “Propagation of a Gaussian-beam wave in general anisotropic turbulence,” Proc. SPIE 9224, 922402 (2014).
[Crossref]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008).
[Crossref]

Anwar, A.

G. R. Salla, C. Perumangattu, S. Prabhakar, A. Anwar, and R. P. Singh, “Recovering the vorticity of a light beam after scattering,” Appl. Phys. Lett. 107(2), 021104 (2016).
[Crossref]

Baykal, Y.

Borghi, R.

Bose-Pillai, S.

M. W. Hyde, S. Bose-Pillai, D. G. Voelz, and X. Xiao, “Generation of vector partially coherent optical sources using phase-only spatial light modulators,” Phys. Rev. Appl. 6(6), 064030 (2016).
[Crossref]

Cai, Y.

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017).
[Crossref]

S. Zhu, J. Wang, X. Liu, Y. Cai, and Z. Li, “Generation of arbitrary radially polarized array beams by manipulating correlation structure,” Appl. Phys. Lett. 109(16), 161904 (2016).
[Crossref]

X. Liu, J. Yu, Y. Cai, and S. A. Ponomarenko, “Propagation of optical coherence lattices in the turbulent atmosphere,” Opt. Lett. 41(18), 4182–4185 (2016).
[Crossref] [PubMed]

J. Wang, S. Zhu, H. Wang, Y. Cai, and Z. Li, “Second-order statistics of a radially polarized cosine-Gaussian correlated Schell-model beam in anisotropic turbulence,” Opt. Express 24(11), 11626–11639 (2016).
[Crossref] [PubMed]

S. Zhu, Y. Chen, J. Wang, H. Wang, Z. Li, and Y. Cai, “Generation and propagation of a vector cosine-Gaussian correlated beam with radial polarization,” Opt. Express 23(26), 33099–33115 (2015).
[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]

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

Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91, 013832 (2014).

F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[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]

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]

Chen, Y.

Crabbs, R.

L. C. Andrews, R. L. Phillips, and R. Crabbs, “Propagation of a Gaussian-beam wave in general anisotropic turbulence,” Proc. SPIE 9224, 922402 (2014).
[Crossref]

de Sande, J. C. G.

Ding, C.

Ferrero, V.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008).
[Crossref]

Friberg, A. T.

Gbur, G.

G. Gbur and T. D. Visser, “The structure of partially coherent fields,” Prog. Opt. 55, 285–341 (2010).
[Crossref]

G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19(8), 1592–1598 (2002).
[Crossref] [PubMed]

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, 013832 (2014).

Guattari, G.

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.

M. W. Hyde, S. Bose-Pillai, D. G. Voelz, and X. Xiao, “Generation of vector partially coherent optical sources using phase-only spatial light modulators,” Phys. Rev. Appl. 6(6), 064030 (2016).
[Crossref]

James, D. F.

E. Wolf and D. F. James, “Correlation-induced spectral changes,” Rep. Prog. Phys. 59(6), 771–818 (1996).
[Crossref]

Ji, X.

Koivurova, M.

Kon, A. I.

A. I. Kon, “Qualitative theory of amplitude and phase fluctuations in a medium with anisotropic turbulent irregularity,” Waves Random Complex Media 4(3), 297–306 (1994).
[Crossref]

Korotkova, O.

Kumar, A.

Li, J.

Li, Z.

Lindfors, K.

Liu, L.

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017).
[Crossref]

Liu, X.

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017).
[Crossref]

S. Zhu, J. Wang, X. Liu, Y. Cai, and Z. Li, “Generation of arbitrary radially polarized array beams by manipulating correlation structure,” Appl. Phys. Lett. 109(16), 161904 (2016).
[Crossref]

X. Liu, J. Yu, Y. Cai, and S. A. Ponomarenko, “Propagation of optical coherence lattices in the turbulent atmosphere,” Opt. Lett. 41(18), 4182–4185 (2016).
[Crossref] [PubMed]

F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[Crossref] [PubMed]

Luo, Y.

Ma, L.

Maluenda, D.

Mao, Y.

Martínez-Herrero, R.

Mei, Z.

Mejías, P. M.

Mukunda, N.

Pan, L.

Perumangattu, C.

G. R. Salla, C. Perumangattu, S. Prabhakar, A. Anwar, and R. P. Singh, “Recovering the vorticity of a light beam after scattering,” Appl. Phys. Lett. 107(2), 021104 (2016).
[Crossref]

Phillips, R. L.

L. C. Andrews, R. L. Phillips, and R. Crabbs, “Propagation of a Gaussian-beam wave in general anisotropic turbulence,” Proc. SPIE 9224, 922402 (2014).
[Crossref]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008).
[Crossref]

Piquero, G.

Ponomarenko, S. A.

Prabhakar, S.

G. R. Salla, C. Perumangattu, S. Prabhakar, A. Anwar, and R. P. Singh, “Recovering the vorticity of a light beam after scattering,” Appl. Phys. Lett. 107(2), 021104 (2016).
[Crossref]

S. G. Reddy, A. Kumar, S. Prabhakar, and R. P. Singh, “Experimental generation of ring-shaped beams with random sources,” Opt. Lett. 38(21), 4441–4444 (2013).
[Crossref] [PubMed]

Ramírez-Sánchez, V.

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross–spectral density matrices,” J. Opt. A 11(8), 085706 (2009).
[Crossref]

Reddy, S. G.

Sahin, S.

Salla, G. R.

G. R. Salla, C. Perumangattu, S. Prabhakar, A. Anwar, and R. P. Singh, “Recovering the vorticity of a light beam after scattering,” Appl. Phys. Lett. 107(2), 021104 (2016).
[Crossref]

Santarsiero, M.

Setälä, T.

Shirai, T.

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross–spectral density matrices,” J. Opt. A 11(8), 085706 (2009).
[Crossref]

Simon, R.

Singh, R. P.

G. R. Salla, C. Perumangattu, S. Prabhakar, A. Anwar, and R. P. Singh, “Recovering the vorticity of a light beam after scattering,” Appl. Phys. Lett. 107(2), 021104 (2016).
[Crossref]

S. G. Reddy, A. Kumar, S. Prabhakar, and R. P. Singh, “Experimental generation of ring-shaped beams with random sources,” Opt. Lett. 38(21), 4441–4444 (2013).
[Crossref] [PubMed]

Sundar, K.

Tang, M.

Tervonen, B.

Toselli, I.

Turunen, J.

Vasara, A.

Visser, T. D.

G. Gbur and T. D. Visser, “The structure of partially coherent fields,” Prog. Opt. 55, 285–341 (2010).
[Crossref]

Voelz, D. G.

M. W. Hyde, S. Bose-Pillai, D. G. Voelz, and X. Xiao, “Generation of vector partially coherent optical sources using phase-only spatial light modulators,” Phys. Rev. Appl. 6(6), 064030 (2016).
[Crossref]

I. Toselli, O. Korotkova, X. Xiao, and D. G. Voelz, “SLM-based laboratory simulations of Kolmogorov and non-Kolmogorov anisotropic turbulence,” Appl. Opt. 54(15), 4740–4744 (2015).
[Crossref] [PubMed]

Wan, L.

Wang, F.

Wang, H.

Wang, J.

Wolf, E.

Xiao, X.

M. W. Hyde, S. Bose-Pillai, D. G. Voelz, and X. Xiao, “Generation of vector partially coherent optical sources using phase-only spatial light modulators,” Phys. Rev. Appl. 6(6), 064030 (2016).
[Crossref]

I. Toselli, O. Korotkova, X. Xiao, and D. G. Voelz, “SLM-based laboratory simulations of Kolmogorov and non-Kolmogorov anisotropic turbulence,” Appl. Opt. 54(15), 4740–4744 (2015).
[Crossref] [PubMed]

Yu, J.

Yuan, Y.

Yura, H. T.

Zhao, D.

Zhu, S.

Appl. Opt. (3)

Appl. Phys. Lett. (3)

G. R. Salla, C. Perumangattu, S. Prabhakar, A. Anwar, and R. P. Singh, “Recovering the vorticity of a light beam after scattering,” Appl. Phys. Lett. 107(2), 021104 (2016).
[Crossref]

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[Crossref]

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Figures (7)

Fig. 1
Fig. 1 Propagation of a TGSMA beam in anisotropic turbulence.
Fig. 2
Fig. 2 Contour plots of normalized spectral density distributions and the corresponding cross line of a TGSMA beam at several different propagation distances. (a1)-(a4) ε y = 1 , μ = 0 ; (b1)-(b4) ε y = 3 , μ = 0 ; (c1)-(c4) ε y = 3 ; (d1)-(d4) ε y = 3 , a = 2 δ 0 = 12 mm .
Fig. 3
Fig. 3 Contour plots of the normalized spectral density distribution and the corresponding cross line of a TGSMA beam at z = 5km. (a1)-(a4) μ = 0 ; (b1)-(b4) μ = 0.5 km 1 ; (c1)-(c4) μ = 1 km 1 .
Fig. 4
Fig. 4 On-axis spectral density as a function of ξ and ε y at z = 5km, (a1)-(a4) δ 0 = 2 cm ; (b1)-(b4) δ 0 = 1 cm .
Fig. 5
Fig. 5 The modulus of the DOC η ( u , v , 0 , 0 ) and the corresponding cross line of a TGSMA beam at several different propagation distances. (a1)-(a4) ε y = 1 , μ = 0 ; (b1)-(b4) ε y = 3 , μ = 0 ; (c1)-(c4) ε y = 3 ; (d1)-(d4) ε y = 3 , a = 2 δ 0 = 12 mm .
Fig. 6
Fig. 6 The modulus of the DOC η ( u , v , 0 , 0 ) and the corresponding cross line of a TGSMA beam at z = 5km for different turbulence statistics and initial coherence. (a1)-(a4) μ = 0 ; (b1)-(b4) μ = 0.5 km 1 ; (c1)-(c4) μ = 1 km 1 .
Fig. 7
Fig. 7 The modulus of the DOC η ( u , 0 , 0 , 0 ) for different values of power law ξ and anisotropic factor ε y at z = 5km. (a1)-(a3) ξ = 11 / 3 ; (b1)-(b3) ε y = 3 .

Equations (25)

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W ( r 1 , r 2 ) = S ( r 1 ) S ( r 2 ) g ( r 1 r 2 ) χ ( r 1 , r 2 ) ,
g ( r 1 r 2 ) = 1 C 0 n x = N N n y = M M exp [ ( r 1 r 2 ) 2 2 δ 0 2 ] × exp [ i π n x ( x 1 x 2 ) / a ] exp [ i π n y ( y 1 y 2 ) / a ] ,
χ ( r 1 , r 2 ) = exp ( i μ k r 1 × r 2 ) ,
W ¯ ( r 1 , r 2 ) = j = 0 , 1 / 2 , 1 , ... m = j j α j , m ϕ j , m * ( r 1 ) ϕ j , m ( r 2 ) ,
ϕ j , m ( r ) = k u π [ ( j | m | ) ! ( j + | m | ) ! ] 1 2 ( r k μ ) 2 | m | L j | m | 2 | m | ( k μ r 2 ) exp ( k μ r 2 2 ) exp ( i 2 m θ ) ,
α j , m = 2 π δ 0 2 1 + k u δ 0 2 ( 1 k u δ 0 2 1 + k u δ 0 2 ) j + m .
α n = W ¯ ( r ) L n ( k μ r 2 ) exp ( k μ r 2 / 2 ) d 2 r , ( n = 0 , 1 , 2 , 3... ) ,
α n = 1 C 0 n x = N N n y = M M 2 π 0 J 0 ( β r ) L n ( k μ r 2 ) exp [ 1 2 ( δ 0 2 + k μ ) r 2 ] r d r ,
0 2 π exp ( i a r cos θ + i b r sin θ ) d θ = 2 π J 0 ( r a 2 + b 2 ) .
α n = 1 C 0 n x = N N n y = M M 2 π δ 0 2 1 + k u δ 0 2 ( 1 k u δ 0 2 1 + k u δ 0 2 ) n exp [ β 2 δ 0 2 2 ( 1 + k u δ 0 2 ) ] L n ( β 2 k u δ 0 4 k 2 u 2 δ 0 4 1 ) .
W ( r 1 , r 2 ) = 1 C 0 n x = N N n y = M M exp ( x 1 2 + x 2 2 4 w x 2 y 1 2 + y 2 2 4 w y 2 ) exp ( ( r 1 r 2 ) 2 2 δ 0 2 ) × exp [ i π n x a ( x 1 x 2 ) ] exp [ i π n y a ( y 1 y 2 ) ] exp ( i k u r 1 × r 2 ) .
W ( ρ 1 , ρ 2 ) = ( 1 λ z ) 2 W ( r 1 , r 2 ) exp [ i k 2 z ( r 1 2 2 ρ 1 r 1 + 2 ρ 2 r 2 r 2 2 ) ] × exp [ i k 2 z ( ρ 1 2 ρ 2 2 ) ] exp [ Ψ ( r 1 , ρ 1 ) + Ψ * ( r 2 , ρ 2 ) ] d 2 r 1 d 2 r 2 ,
exp [ Ψ ( r 1 , ρ 1 ) + Ψ * ( r 2 , ρ 2 ) ] = exp ( 2 π k 2 z 0 1 d t 0 Φ n ( κ ) d 3 κ { 1 exp [ t ρ d κ + ( 1 t ) r d κ ] } ) ,
Φ n ( κ x , κ y , κ z ) = ε x ε y C ˜ n 2 A ( ξ ) ( ε x 2 κ x 2 + ε y 2 κ y 2 + κ z 2 ) ξ / 2 ,
A ( ξ ) = Γ ( ξ 1 ) cos ( π ξ / 2 ) 4 π 2 , 3 < ξ < 4 ,
κ x = q x ε x = q cos φ ε x , κ y = q y ε y = q sin φ ε y , d κ x d κ y = d q x d q y ε x ε y = q d q d φ ε x ε y , q = q x 2 + q y 2 , ρ d ( u d ε x , v d ε y ) , r d ( x d ε x , y d ε y ) .
exp [ Ψ ( r 1 , ρ 1 ) + Ψ * ( r 2 , ρ 2 ) ] = exp ( 4 π 2 A ( ξ ) C ˜ n 2 k 2 z 0 1 d t 0 q 1 ξ { 1 J 0 [ q | t ρ d + ( 1 t ) r d | ] } d q ) ,
0 x 1 p [ 1 J 0 ( x | Q | ) ] d x = 2 p p | Q | p 2 Γ ( p / 2 ) Γ ( p / 2 ) , ( 2< Re( p ) < 4 ) ,
exp [ Ψ ( r 1 , ρ 1 ) + Ψ * ( r 2 , ρ 2 ) ] = exp [ ξ Γ ( ξ 1 ) Γ ( ξ / 2 ) 2 ξ ( ξ 1 ) Γ ( ξ / 2 ) cos ( π ξ 2 ) C ˜ n 2 k 2 z ( ρ d ξ 1 r d ξ 1 | ρ d r d | ) ] .
exp [ Ψ ( r 1 , ρ 1 ) + Ψ * ( r 2 , ρ 2 ) ] exp [ 1 ρ 0 2 ( z ) ( u d 2 + u d x d + x d 2 ε x 2 + v d 2 + v d y d + y d 2 ε y 2 ) ] ,
ρ 0 ( z ) = [ ξ Γ ( ξ 1 ) Γ ( ξ / 2 ) cos ( π ξ / 2 ) 2 ξ ( ξ 1 ) Γ ( ξ / 2 ) C ˜ n 2 k 2 z ] 1 2 ξ ,
W ( ρ 1 , ρ 2 ) = n x = N N n y = M M V ( ρ 1 , ρ 2 ) exp ( γ u 1 2 4 Χ ( + ) + γ v 1 2 4 ϒ ( + ) + Ω u 2 4 Χ ( ) + Ω v 2 4 ϒ ( ) ) ,
V ( ρ 1 , ρ 2 ) = k 2 4 C 0 z 2 Χ ( + ) ϒ ( + ) Χ ( ) ϒ ( ) exp [ i k 2 z ( ρ 1 2 ρ 2 2 ) ρ d 2 ρ 0 2 ( z ) ] γ u 1 = i k u 1 z + i π n x a u d ε x 2 ρ 0 2 ( z ) , γ u 2 = i k u 2 z + i π n x a u d ε x 2 ρ 0 2 ( z ) , Δ x = 1 2 δ 0 2 + 1 ε x 2 ρ 0 2 ( z ) , γ v 1 = i k v 1 z + i π n y a v d ε y 2 ρ 0 2 ( z ) , γ v 2 = i k v 2 z + i π n y a v d ε y 2 ρ 0 2 ( z ) , Δ y = 1 2 δ 0 2 + 1 ε y 2 ρ 0 2 ( z ) , Χ ( + ) = Δ x + 1 4 w x 2 + i k 2 z , Χ ( ) = Δ x + 1 4 w x 2 i k 2 z Δ x 2 Χ ( + ) + k 2 μ 2 4 ϒ ( + ) , Ω u = Δ x γ u 1 Χ ( + ) γ u 2 + i k μ γ v 1 2 ϒ ( + ) , ϒ ( + ) = Δ y + 1 4 w y 2 + i k 2 z , ϒ ( ) = Δ y + 1 4 w y 2 i k 2 z Δ y 2 ϒ ( + ) + k 2 μ 2 4 Χ ( + ) + k 2 μ 2 4 Χ ( ) ( Δ y ϒ ( + ) Δ x Χ ( + ) ) 2 , Ω v = Δ y γ v 1 ϒ ( + ) γ v 2 i k μ γ u 1 2 Χ ( + ) + i k μ Ω u 2 Χ ( ) ( Δ y ϒ ( + ) Δ x Χ ( + ) ) .
S ( ρ ) = W ( ρ , ρ ) ,
η ( ρ 1 , ρ 2 ) = W ( ρ 1 , ρ 2 ) W ( ρ 1 , ρ 1 ) W ( ρ 2 , ρ 2 ) .

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