Abstract

Based on the extended Huygens-Fresnel principle, we have derived the analytical expression of the average intensity of optical coherence lattices (OCLs) in oceanic turbulence with anisotropy, and then the beam quality parameters including the Strehl ratio (SR) and the power-in-the-bucket (PIB) are obtained. One can find that the OCLs will eventually evolve into Gaussian shape with the periodicity reciprocity gradually breaking down when propagating through the anisotropic ocean water, and that the trend of evolving into Gaussian can be accelerated for increasing the ratio of temperature and salinity contributions to the refractive index spectrum ω, the lattice constant a and the rate of dissipation of mean square temperature χT or decreasing the anisotropic factor ξ and the rate of dissipation of turbulent kinetic energy per unit mass of fluid ε. Further, the SR and PIB in the target plane under the effects of oceanic parameters are discussed in detail, and the SR and PIB can be increased for the larger ξ and ε or the smaller χT and ω, namely, the beam quality becomes better. Our results can find potential application in the future optical communication system in an oceanic environment.

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

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References

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2017 (3)

2016 (8)

J. Gao, Y. Zhu, D. L. Wang, Y. X. Zhang, Z. D. Hu, and M. J. Cheng, “Bessel-Gauss photon beams with fractional order vortex propagation in weak non-Kolmogorov turbulence,” Photon. Res. 4(2), 30–34 (2016).
[Crossref]

Y. Li, Y. Zhang, Y. Zhu, and M. Chen, “Effects of anisotropic turbulence on average polarizability of Gaussian Schell-model quantized beams through ocean link,” Appl. Opt. 55(19), 5234–5239 (2016).
[Crossref] [PubMed]

Y. Wu, Y. Zhang, and Y. Zhu, “Average intensity and directionality of partially coherent model beams propagating in turbulent ocean,” J. Opt. Soc. Am. A 33(8), 1451–1458 (2016).
[Crossref]

F. D. Kashani and M. Yousefi, “Analyzing the propagation behavior of coherence and polarization degrees of a phase-locked partially coherent radial flat-topped array laser beam in underwater turbulence,” Appl. Opt. 55(23), 6311–6320 (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. Chen, S. A. Ponomarenko, and Y. Cai, “Experimental generation of optical coherence lattices,” Appl. Phys. Lett. 109(6), 061107 (2016).
[Crossref]

Y. Wu, Y. Zhang, Y. Li, and Z. Hu, “Beam wander of Gaussian-Schell model beams propagating through oceanic turbulence,” Opt. Commun. 371, 59–66 (2016).
[Crossref]

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 (7)

Y. Huang, P. Huang, F. Wang, G. Zhao, and A. Zeng, “The influence of oceanic turbulence on the beam quality parameters of partially coherent Hermite-Gaussian linear array beams,” Opt. Commun. 336(1), 146–152 (2015).
[Crossref]

D. Zhi, Y. Chen, R. Tao, Y. Ma, P. Zhou, and L. Si, “Average spreading and beam quality evolution of Gaussian array beams propagating through oceanic turbulence,” Laser Phys. Lett. 12(11), 116001 (2015).
[Crossref]

C. Ding, L. Liao, H. Wang, Y. Zhang, and L. Pan, “Effect of oceanic turbulence on the propagation of cosine-Gaussian correlated Schell-model beams,” J. Opt. 17(3), 035615 (2015).
[Crossref]

L. Ma and S. A. Ponomarenko, “Free-space propagation of optical coherence lattices and periodicity reciprocity,” Opt. Express 23(2), 1848–1856 (2015).
[Crossref] [PubMed]

L. Lu, P. Zhang, C. Fan, and C. Qiao, “Influence of oceanic turbulence on propagation of a radial Gaussian beam array,” Opt. Express 23(3), 2827–28362015.
[Crossref] [PubMed]

M. Tang and D. Zhao, “Regions of spreading of Gaussian array beams propagating through oceanic turbulence,” Appl. Opt. 54(11), 3407–3411 (2015).
[Crossref] [PubMed]

J. Yu, Y. Chen, L. Liu, X. Liu, and Y. Cai, “Splitting and combining properties of an elegant Hermite-Gaussian correlated Schell-model beam in Kolmogorov and non-Kolmogorov turbulence,” Opt. Express 23(10), 13467–13481 (2015).
[Crossref] [PubMed]

2014 (4)

2012 (1)

N. Farwell and O. Korotkova, “Intensity and coherence properties of light in oceanic turbulence,” Opt. Commun. 285(6), 872–875 (2012).
[Crossref]

2008 (1)

X. Xiao, X. Ji, and B. Lü, “The influence of turbulence on propagation properties of partially coherent sinh-Gaussian beams and their beam quality in the far field,” Opt. Laser Technol. 40(1), 129–136 (2008).
[Crossref]

2006 (1)

W. Lu, L. Liu, and J. Sun, “Influence of temperature and salinity fluctuations on propagation behaviour of partially coherent beams in oceanic turbulence,” J. Opt. 8(12), 1052–1058 (2006).

2002 (1)

1978 (1)

R. J. Hill, “Optical propagation in turbulent water,” J. Opt. Soc. Am. A 68(8), 1067–1072 (1978).
[Crossref]

Ata, Y.

Bai, Y.

X. Huang, Y. Bai, and X. Fu, “Propagation factors of partially coherent model beams in oceanic turbulence,” IEEE Photonics J. 9(5) 1–11 (2017).

Baykal, Y.

Cai, Y.

Chen, M.

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.

Y. Chen, S. A. Ponomarenko, and Y. Cai, “Experimental generation of optical coherence lattices,” Appl. Phys. Lett. 109(6), 061107 (2016).
[Crossref]

J. Yu, Y. Chen, L. Liu, X. Liu, and Y. Cai, “Splitting and combining properties of an elegant Hermite-Gaussian correlated Schell-model beam in Kolmogorov and non-Kolmogorov turbulence,” Opt. Express 23(10), 13467–13481 (2015).
[Crossref] [PubMed]

D. Zhi, Y. Chen, R. Tao, Y. Ma, P. Zhou, and L. Si, “Average spreading and beam quality evolution of Gaussian array beams propagating through oceanic turbulence,” Laser Phys. Lett. 12(11), 116001 (2015).
[Crossref]

Cheng, M. J.

Ding, C.

C. Ding, L. Liao, H. Wang, Y. Zhang, and L. Pan, “Effect of oceanic turbulence on the propagation of cosine-Gaussian correlated Schell-model beams,” J. Opt. 17(3), 035615 (2015).
[Crossref]

Duan, Z.

Fan, C.

Farwell, N.

N. Farwell and O. Korotkova, “Intensity and coherence properties of light in oceanic turbulence,” Opt. Commun. 285(6), 872–875 (2012).
[Crossref]

Fu, X.

X. Huang, Y. Bai, and X. Fu, “Propagation factors of partially coherent model beams in oceanic turbulence,” IEEE Photonics J. 9(5) 1–11 (2017).

Gao, J.

Gao, Z.

Gbur, G.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 2007).

Guo, M.

Hill, R. J.

R. J. Hill, “Optical propagation in turbulent water,” J. Opt. Soc. Am. A 68(8), 1067–1072 (1978).
[Crossref]

Hu, Z.

Y. Wu, Y. Zhang, Y. Li, and Z. Hu, “Beam wander of Gaussian-Schell model beams propagating through oceanic turbulence,” Opt. Commun. 371, 59–66 (2016).
[Crossref]

Hu, Z. D.

Huang, P.

Y. Huang, P. Huang, F. Wang, G. Zhao, and A. Zeng, “The influence of oceanic turbulence on the beam quality parameters of partially coherent Hermite-Gaussian linear array beams,” Opt. Commun. 336(1), 146–152 (2015).
[Crossref]

Huang, X.

X. Huang, Y. Bai, and X. Fu, “Propagation factors of partially coherent model beams in oceanic turbulence,” IEEE Photonics J. 9(5) 1–11 (2017).

Huang, Y.

Y. Huang, P. Huang, F. Wang, G. Zhao, and A. Zeng, “The influence of oceanic turbulence on the beam quality parameters of partially coherent Hermite-Gaussian linear array beams,” Opt. Commun. 336(1), 146–152 (2015).
[Crossref]

Y. Huang, B. Zhang, Z. Gao, G. Zhao, and Z. Duan, “Evolution behavior of Gaussian Schell-model vortex beams propagating through oceanic turbulence,” Opt. Express 22(15), 17723–17734 (2014).
[Crossref] [PubMed]

Ji, X.

X. Xiao, X. Ji, and B. Lü, “The influence of turbulence on propagation properties of partially coherent sinh-Gaussian beams and their beam quality in the far field,” Opt. Laser Technol. 40(1), 129–136 (2008).
[Crossref]

Ji, X. L.

Kashani, F. D.

Korotkova, O.

N. Farwell and O. Korotkova, “Intensity and coherence properties of light in oceanic turbulence,” Opt. Commun. 285(6), 872–875 (2012).
[Crossref]

Li, Y.

Y. Wu, Y. Zhang, Y. Li, and Z. Hu, “Beam wander of Gaussian-Schell model beams propagating through oceanic turbulence,” Opt. Commun. 371, 59–66 (2016).
[Crossref]

Y. Li, Y. Zhang, Y. Zhu, and M. Chen, “Effects of anisotropic turbulence on average polarizability of Gaussian Schell-model quantized beams through ocean link,” Appl. Opt. 55(19), 5234–5239 (2016).
[Crossref] [PubMed]

Liao, L.

C. Ding, L. Liao, H. Wang, Y. Zhang, and L. Pan, “Effect of oceanic turbulence on the propagation of cosine-Gaussian correlated Schell-model beams,” J. Opt. 17(3), 035615 (2015).
[Crossref]

Liu, L.

Liu, X.

Lu, L.

Lu, W.

W. Lu, L. Liu, and J. Sun, “Influence of temperature and salinity fluctuations on propagation behaviour of partially coherent beams in oceanic turbulence,” J. Opt. 8(12), 1052–1058 (2006).

Lü, B.

X. Xiao, X. Ji, and B. Lü, “The influence of turbulence on propagation properties of partially coherent sinh-Gaussian beams and their beam quality in the far field,” Opt. Laser Technol. 40(1), 129–136 (2008).
[Crossref]

Ma, L.

Ma, Y.

D. Zhi, Y. Chen, R. Tao, Y. Ma, P. Zhou, and L. Si, “Average spreading and beam quality evolution of Gaussian array beams propagating through oceanic turbulence,” Laser Phys. Lett. 12(11), 116001 (2015).
[Crossref]

Pan, L.

C. Ding, L. Liao, H. Wang, Y. Zhang, and L. Pan, “Effect of oceanic turbulence on the propagation of cosine-Gaussian correlated Schell-model beams,” J. Opt. 17(3), 035615 (2015).
[Crossref]

Peng, X.

Ponomarenko, S. A.

Qiao, C.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 2007).

Si, L.

D. Zhi, Y. Chen, R. Tao, Y. Ma, P. Zhou, and L. Si, “Average spreading and beam quality evolution of Gaussian array beams propagating through oceanic turbulence,” Laser Phys. Lett. 12(11), 116001 (2015).
[Crossref]

Sun, J.

W. Lu, L. Liu, and J. Sun, “Influence of temperature and salinity fluctuations on propagation behaviour of partially coherent beams in oceanic turbulence,” J. Opt. 8(12), 1052–1058 (2006).

Tang, M.

Tao, R.

D. Zhi, Y. Chen, R. Tao, Y. Ma, P. Zhou, and L. Si, “Average spreading and beam quality evolution of Gaussian array beams propagating through oceanic turbulence,” Laser Phys. Lett. 12(11), 116001 (2015).
[Crossref]

Wang, D. L.

Wang, F.

Y. Huang, P. Huang, F. Wang, G. Zhao, and A. Zeng, “The influence of oceanic turbulence on the beam quality parameters of partially coherent Hermite-Gaussian linear array beams,” Opt. Commun. 336(1), 146–152 (2015).
[Crossref]

Wang, H.

C. Ding, L. Liao, H. Wang, Y. Zhang, and L. Pan, “Effect of oceanic turbulence on the propagation of cosine-Gaussian correlated Schell-model beams,” J. Opt. 17(3), 035615 (2015).
[Crossref]

Wolf, E.

Wu, Y.

Y. Wu, Y. Zhang, Y. Li, and Z. Hu, “Beam wander of Gaussian-Schell model beams propagating through oceanic turbulence,” Opt. Commun. 371, 59–66 (2016).
[Crossref]

Y. Wu, Y. Zhang, and Y. Zhu, “Average intensity and directionality of partially coherent model beams propagating in turbulent ocean,” J. Opt. Soc. Am. A 33(8), 1451–1458 (2016).
[Crossref]

Xiao, X.

X. Xiao, X. Ji, and B. Lü, “The influence of turbulence on propagation properties of partially coherent sinh-Gaussian beams and their beam quality in the far field,” Opt. Laser Technol. 40(1), 129–136 (2008).
[Crossref]

Yousefi, M.

Yu, J.

Zeng, A.

Y. Huang, P. Huang, F. Wang, G. Zhao, and A. Zeng, “The influence of oceanic turbulence on the beam quality parameters of partially coherent Hermite-Gaussian linear array beams,” Opt. Commun. 336(1), 146–152 (2015).
[Crossref]

Zhang, B.

Zhang, P.

Zhang, Y.

Y. Wu, Y. Zhang, Y. Li, and Z. Hu, “Beam wander of Gaussian-Schell model beams propagating through oceanic turbulence,” Opt. Commun. 371, 59–66 (2016).
[Crossref]

Y. Wu, Y. Zhang, and Y. Zhu, “Average intensity and directionality of partially coherent model beams propagating in turbulent ocean,” J. Opt. Soc. Am. A 33(8), 1451–1458 (2016).
[Crossref]

Y. Li, Y. Zhang, Y. Zhu, and M. Chen, “Effects of anisotropic turbulence on average polarizability of Gaussian Schell-model quantized beams through ocean link,” Appl. Opt. 55(19), 5234–5239 (2016).
[Crossref] [PubMed]

C. Ding, L. Liao, H. Wang, Y. Zhang, and L. Pan, “Effect of oceanic turbulence on the propagation of cosine-Gaussian correlated Schell-model beams,” J. Opt. 17(3), 035615 (2015).
[Crossref]

Zhang, Y. X.

Zhao, D.

Zhao, G.

Y. Huang, P. Huang, F. Wang, G. Zhao, and A. Zeng, “The influence of oceanic turbulence on the beam quality parameters of partially coherent Hermite-Gaussian linear array beams,” Opt. Commun. 336(1), 146–152 (2015).
[Crossref]

Y. Huang, B. Zhang, Z. Gao, G. Zhao, and Z. Duan, “Evolution behavior of Gaussian Schell-model vortex beams propagating through oceanic turbulence,” Opt. Express 22(15), 17723–17734 (2014).
[Crossref] [PubMed]

Zhi, D.

D. Zhi, Y. Chen, R. Tao, Y. Ma, P. Zhou, and L. Si, “Average spreading and beam quality evolution of Gaussian array beams propagating through oceanic turbulence,” Laser Phys. Lett. 12(11), 116001 (2015).
[Crossref]

Zhou, P.

D. Zhi, Y. Chen, R. Tao, Y. Ma, P. Zhou, and L. Si, “Average spreading and beam quality evolution of Gaussian array beams propagating through oceanic turbulence,” Laser Phys. Lett. 12(11), 116001 (2015).
[Crossref]

Zhu, Y.

Appl. Opt. (3)

Appl. Phys. Lett. (1)

Y. Chen, S. A. Ponomarenko, and Y. Cai, “Experimental generation of optical coherence lattices,” Appl. Phys. Lett. 109(6), 061107 (2016).
[Crossref]

IEEE Photonics J. (1)

X. Huang, Y. Bai, and X. Fu, “Propagation factors of partially coherent model beams in oceanic turbulence,” IEEE Photonics J. 9(5) 1–11 (2017).

J. Opt. (2)

C. Ding, L. Liao, H. Wang, Y. Zhang, and L. Pan, “Effect of oceanic turbulence on the propagation of cosine-Gaussian correlated Schell-model beams,” J. Opt. 17(3), 035615 (2015).
[Crossref]

W. Lu, L. Liu, and J. Sun, “Influence of temperature and salinity fluctuations on propagation behaviour of partially coherent beams in oceanic turbulence,” J. Opt. 8(12), 1052–1058 (2006).

J. Opt. Soc. Am. A (5)

Laser Phys. Lett. (1)

D. Zhi, Y. Chen, R. Tao, Y. Ma, P. Zhou, and L. Si, “Average spreading and beam quality evolution of Gaussian array beams propagating through oceanic turbulence,” Laser Phys. Lett. 12(11), 116001 (2015).
[Crossref]

Opt. Commun. (4)

Y. Huang, P. Huang, F. Wang, G. Zhao, and A. Zeng, “The influence of oceanic turbulence on the beam quality parameters of partially coherent Hermite-Gaussian linear array beams,” Opt. Commun. 336(1), 146–152 (2015).
[Crossref]

Y. Wu, Y. Zhang, Y. Li, and Z. Hu, “Beam wander of Gaussian-Schell model beams propagating through oceanic turbulence,” Opt. Commun. 371, 59–66 (2016).
[Crossref]

N. Farwell and O. Korotkova, “Intensity and coherence properties of light in oceanic turbulence,” Opt. Commun. 285(6), 872–875 (2012).
[Crossref]

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]

Opt. Express (6)

Opt. Laser Technol. (1)

X. Xiao, X. Ji, and B. Lü, “The influence of turbulence on propagation properties of partially coherent sinh-Gaussian beams and their beam quality in the far field,” Opt. Laser Technol. 40(1), 129–136 (2008).
[Crossref]

Opt. Lett. (2)

Photon. Res. (1)

Other (1)

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 2007).

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Figures (5)

Fig. 1
Fig. 1 The average intensity of OCLs versus propagation distance in free space for (a) a = 1, N = 10 and σ0 = 0.01m, and in anisotropic ocean for (b) a = 0.5, N = 10, σ0 = 0.01m, ξ = 1, ω = −2.5, ε = 10−5m2/s3 and χT = 10−9K2/s and (c) a = 1, N = 10, σ0 = 0.01m, ξ = 1, ω = −2.5, ε = 10−5m2/s3 and χT = 10−9K2/s.
Fig. 2
Fig. 2 The average intensity of OCLs in anisotropic ocean at Z = 2 under different turbulence parameters with a = 1, N = 10, σ0 = 0.01m for (a) ω = −2.5, ε = 10−5m2/s3 and χT = 10−9K2/s, (b) ξ = 1, ε = 10−5m2/s3 and χT = 10−9K2/s, (c) ξ = 1, ω = −2.5 and χT = 10−9K2/s and (d) ξ = 1, ω = −2.5 and ε = 10−5m2/s3.
Fig. 3
Fig. 3 The Strehl ratio versus propagation distance in ocean turbulence under the the effects of the number of lattice lobes N with a = 1, ξ = 1, σ0 = 0.01m, ω = −2.5, ε = 10−5m2/s3.
Fig. 4
Fig. 4 The Strehl ratio under different oceanic turbulence parameters with Z = 2, a = 1, N = 5 and σ0 = 0.01m for (a) ε = 10−5m2/s3, χT = 10−9K2/s, (b) ξ = 1, χT = 10−9K2/s and (c) ξ = 1, ε = 10−5m2/s3.
Fig. 5
Fig. 5 The power-in-the-bucket (PIB) of OCLs in anisotropic ocean versus different parameters of w, ξ, ε, ω and χT with σ0 = 0.01m, (a) ω = −2.5, ε = 10−5m2/s3, a = 1, ξ = 1 and χT = 10−9K2/s, (b) Z = 3, ε = 10−5m2/s3, a = 1, ξ = 1 and χT = 10−9K2/s, (c) w = 5, Z = 3, ε = 10−5m2/s3, a = 1 and χT = 10−9K2/s, (d) w = 5, Z = 3, ξ = 1, a = 1 and χT = 10−9K2/s and (e) w = 5, Z = 3, ξ = 1, a = 1 and ε = 10−5m2/s3.

Equations (21)

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W ( x 1 , y 1 , x 2 , y 2 , 0 ) = s = x y n s = 0 N v n s π exp [ s 1 2 + s 2 2 2 σ 0 2 2 i π n s ( s 1 s 2 ) a σ 0 ] ,
W ( ρ 1 , ρ 2 , z ) = ( k 2 π z ) 2 d 2 r 1 d 2 r 1 W ( r 1 , r 2 , 0 ) exp { i k 2 z [ ( ρ 1 r 1 ) 2 ( ρ 2 r 2 ) 2 ] } × exp [ Ψ * ( ρ 1 , r 1 ) + Ψ ( ρ 2 , r 2 ) ] .
exp [ Ψ * ( ρ 1 , r 1 ) + Ψ ( ρ 2 , r 2 ) ] = exp [ ( r 1 r 2 ) 2 + ( r 1 r 2 ) ( ρ 1 ρ 2 ) + ( ρ 1 ρ 2 ) 2 ρ o c ξ 2 ] ,
ρ o c ξ 2 = π 2 k 2 z ξ 4 3 0 κ 3 ψ ˜ a n ( κ ) d κ ,
ψ ˜ a n ( κ ) = 0.388 × 10 8 ε 1 / 3 χ T ξ 2 ( κ ) 11 / 3 [ 1 + 2.35 ( κ η ) 2 / 3 ] × [ exp ( A T δ ) + ω 2 exp ( A S δ ) 2 ω 1 exp ( A T S δ ) ] ,
ρ o c ξ = ξ | ω | [ 1.802 × 10 7 k 2 z ( ε η ) 1 / 3 χ T ( 0.483 ω 2 0.835 ω + 3.380 ) ] 1 / 2 ,
W ( ρ 1 , ρ 2 , z ) = ( k 2 π z ) 2 d 2 r 1 d 2 r 1 W ( r 1 , r 2 , 0 ) exp { i k 2 z [ ( ρ 1 r 1 ) 2 ( ρ 2 r 2 ) 2 ] } × exp [ ( r 1 r 2 ) 2 + ( r 1 r 2 ) ( ρ 1 ρ 2 ) + ( ρ 1 ρ 2 ) 2 ρ o c ξ 2 ] ,
W ( ρ 1 , ρ 2 , z ) = ( k 2 π z ) 2 exp [ i k ( ρ 1 2 ρ 2 2 ) 2 z ( ρ 1 ρ 2 ) 2 ρ o c ξ 2 ] d 2 R d 2 T × exp [ R 2 σ 0 2 ( 1 4 σ 0 2 + 1 ρ o c ξ ) T 2 i k RT z ] s = x , y n s = 0 N v n s π exp ( 2 π i n s T s a σ 0 ) × exp [ i k 2 z T ( ρ 1 + ρ 2 ) T ( ρ 1 ρ 2 ) ρ o c ξ 2 ] exp [ i k ( ρ 1 ρ 2 ) R z ] ,
exp ( p x 2 + 2 q x ) d x = π p exp ( q 2 p ) ,
x exp ( p x 2 + 2 q x ) d x = q p π p exp ( q 2 p ) ,
W ( ρ 1 , ρ 2 , z ) = s = x , y n s = 0 N v n s A 1 π exp [ i k ( s 1 2 s 2 2 ) 2 z ( s 1 s 2 ) 2 2 ρ o c ξ 2 i π n s ( s 1 s 2 ) a σ 0 A 1 2 ] × exp [ i π n s z A 2 ρ o c ξ 2 a σ 0 ( s 1 s 2 ) + i 2 A 2 ( k 2 σ 0 2 z 2 1 ρ o c ξ 2 ) ( s 1 2 s 2 2 ) ] × exp [ ( 2 A 1 2 ρ o c ξ 2 + z k ρ o c ξ 4 A 2 ) ( s 1 s 2 ) 2 ] × exp [ ( s 1 π n s z k σ 0 a ) 2 2 σ 0 2 A 1 2 ( s 2 π n s z k σ 0 a ) 2 2 σ 0 2 A 1 2 ] .
I ( ρ , z ) = W ( ρ , ρ , z ) = s = x , y n s = 0 N v n s A 1 π exp [ ( s 1 π n s z k σ 0 a ) 2 σ 0 2 A 1 2 ] ,
I ( ρ , Z ) = s = x , y n s = 0 N v n s A 1 π exp [ ( s 1 π n s Z a ) 2 A 1 2 ] .
I 0 ( ρ , Z ) = s = x , y n s = 0 N v n s A 0 π exp [ ( S 1 π n s Z a ) 2 A 0 2 ] .
S R = I m a x I 0 m a x ,
S R = ( A 0 2 s = x , y n s = 0 N exp [ ( S 1 π n s Z a ) 2 A 1 2 ] ) m a x ( A 1 2 s = x , y n s = 0 N exp [ ( S 1 π n s Z a ) 2 A 0 2 ] ) m a x .
PIB ( w , z ) = w w w w I ( x , y , z ) d x d y I ( x , y , z ) d x d y ,
PIB = w w w w s = x , y n s = 0 N v n s A 1 π exp [ ( S 1 π n s Z a ) 2 A 1 2 ] d x 1 d y 1 s = x , y n s = 0 N v n s A 1 π exp [ ( S 1 π n s Z a ) 2 A 1 2 ] d x 1 d y 1 ,
PIB = 1 144 [ A + A + + 2 erf ( w A 1 ) ] 2 ,
A = erf ( a w 5 π Z a A 1 ) + erf ( a w 4 π Z a A 1 ) + erf ( a w 3 π Z a A 1 ) + erf ( a w 2 π Z a A 1 ) + erf ( a w π Z a A 1 ) ,
A + = erf ( a w + 5 π Z a A 1 ) + erf ( a w + 4 π Z a A 1 ) + erf ( a w + 3 π Z a A 1 ) + erf ( a w + 2 π Z a A 1 ) + erf ( a w + π Z a A 1 ) .

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