Abstract

In order to research the statistical properties of Gaussian beam propagation through an arbitrary thickness random phase screen for adaptive optics and laser communication application in the laboratory, we establish mathematic models of statistical quantities, which are based on the Rytov method and the thin phase screen model, involved in the propagation process. And the analytic results are developed for an arbitrary thickness phase screen based on the Kolmogorov power spectrum. The comparison between the arbitrary thickness phase screen and the thin phase screen shows that it is more suitable for our results to describe the generalized case, especially the scintillation index.

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  1. C. Chao, L. Hu, and Q. Mu, “Bandwidth requires of adaptive optical system for horizontal turbulence correction,” Opt. Precision Eng. 18(10), 2137–3142 (2010).
  2. B. Wang, Z.-y. Wang, J.-l. Wang, J.-Y. Zhao, Y.-H. Wu, S.-X Zhang, L. Dong, and M. Wen, “Phase-diverse speckle imaging with two cameras,” Opt. Precision Eng. 19(6), 1384–1390 (2011).
    [CrossRef]
  3. H. G. Booker, J. A. Ferguson, and H. O. Vats, “Comparison between the extended-medium and the phase-screen scintillation theories,” J. Atmos. Terr. Phys. 47(38), 1–399 (1985).
    [CrossRef]
  4. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).
  5. L. C. Andrews, R. L. Phillips, and A. R. Weeks, “Propagation of a Gaussian-beam wave through a random phase screen,” Waves Random Media 7(2), 229–244 (1997).
    [CrossRef]
  6. L. C. Andrews, W. B. Miller, and J. C. Ricklin, “Propagation through complex optical system: a phase screen analysis,” SPIE 2312, 122–129 (1994).
    [CrossRef]
  7. L. C. Andrews and W. B. Miller, “Single- and double-pass propagation through complex paraxial optical systems,” J. Opt. Soc. Am. A 12(1), 137–150 (1995).
    [CrossRef]
  8. S. V. Mantravadi, T. A. Rhoadarmer, and R. S. Glas, “Simple laboratory system for generating well-controlled atmospheric-like turbulence,” Proc. SPIE 5553, 290–300 (2004).
    [CrossRef]
  9. X. J. Gan, J. Guo, and Y. Y. Fu, “The simulating turbulence method of laser propagation in the inner field,” J. Phys. Conf. Ser. 48, 907–910 (2006).
    [CrossRef]
  10. B. D. Zhang, S. Qin, and X. S. Wang, “Accurate and fast simulation of Kolmogorov phase screen by combining spectral method with Zernike polynomials method,” Chin. Opt. Lett. 8(10), 969–971 (2010).
  11. M. X. Qian, W. Y. Zhu, and R. Rao, “Phase screen distribution for simulating laser propagation along an inhomogeneous atmospheric path,” Acta Phys. Sinica 58(9), 6633–6639 (2009).
  12. M. Gao and Z. Wu, “Experiments of effect of beam spreading of far-field on aiming deviation,” Opt. Precision Eng. 18(3), 602–608 (2010).
  13. Y.-h. Gao, Z.-y. An, N.-n. Li, W.-x Zhao, and J.-s. Wang, “Optical design of Gaussian beam shaping,” Opt. Precision Eng. 19(7), 1464–1471 (2011).
    [CrossRef]
  14. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (New York: McGraw-Hill, 1961).
  15. D. L. Fried and J. B. Seidman, “Laser-beam scintillation in the atmosphere,” J. Opt. Soc. Am. 57(2), 181–185 (1967).
    [CrossRef]

2011 (2)

B. Wang, Z.-y. Wang, J.-l. Wang, J.-Y. Zhao, Y.-H. Wu, S.-X Zhang, L. Dong, and M. Wen, “Phase-diverse speckle imaging with two cameras,” Opt. Precision Eng. 19(6), 1384–1390 (2011).
[CrossRef]

Y.-h. Gao, Z.-y. An, N.-n. Li, W.-x Zhao, and J.-s. Wang, “Optical design of Gaussian beam shaping,” Opt. Precision Eng. 19(7), 1464–1471 (2011).
[CrossRef]

2010 (3)

B. D. Zhang, S. Qin, and X. S. Wang, “Accurate and fast simulation of Kolmogorov phase screen by combining spectral method with Zernike polynomials method,” Chin. Opt. Lett. 8(10), 969–971 (2010).

M. Gao and Z. Wu, “Experiments of effect of beam spreading of far-field on aiming deviation,” Opt. Precision Eng. 18(3), 602–608 (2010).

C. Chao, L. Hu, and Q. Mu, “Bandwidth requires of adaptive optical system for horizontal turbulence correction,” Opt. Precision Eng. 18(10), 2137–3142 (2010).

2009 (1)

M. X. Qian, W. Y. Zhu, and R. Rao, “Phase screen distribution for simulating laser propagation along an inhomogeneous atmospheric path,” Acta Phys. Sinica 58(9), 6633–6639 (2009).

2006 (1)

X. J. Gan, J. Guo, and Y. Y. Fu, “The simulating turbulence method of laser propagation in the inner field,” J. Phys. Conf. Ser. 48, 907–910 (2006).
[CrossRef]

2004 (1)

S. V. Mantravadi, T. A. Rhoadarmer, and R. S. Glas, “Simple laboratory system for generating well-controlled atmospheric-like turbulence,” Proc. SPIE 5553, 290–300 (2004).
[CrossRef]

1997 (1)

L. C. Andrews, R. L. Phillips, and A. R. Weeks, “Propagation of a Gaussian-beam wave through a random phase screen,” Waves Random Media 7(2), 229–244 (1997).
[CrossRef]

1995 (1)

1994 (1)

L. C. Andrews, W. B. Miller, and J. C. Ricklin, “Propagation through complex optical system: a phase screen analysis,” SPIE 2312, 122–129 (1994).
[CrossRef]

1985 (1)

H. G. Booker, J. A. Ferguson, and H. O. Vats, “Comparison between the extended-medium and the phase-screen scintillation theories,” J. Atmos. Terr. Phys. 47(38), 1–399 (1985).
[CrossRef]

1967 (1)

An, Z.-y.

Y.-h. Gao, Z.-y. An, N.-n. Li, W.-x Zhao, and J.-s. Wang, “Optical design of Gaussian beam shaping,” Opt. Precision Eng. 19(7), 1464–1471 (2011).
[CrossRef]

Andrews, L. C.

L. C. Andrews, R. L. Phillips, and A. R. Weeks, “Propagation of a Gaussian-beam wave through a random phase screen,” Waves Random Media 7(2), 229–244 (1997).
[CrossRef]

L. C. Andrews and W. B. Miller, “Single- and double-pass propagation through complex paraxial optical systems,” J. Opt. Soc. Am. A 12(1), 137–150 (1995).
[CrossRef]

L. C. Andrews, W. B. Miller, and J. C. Ricklin, “Propagation through complex optical system: a phase screen analysis,” SPIE 2312, 122–129 (1994).
[CrossRef]

Booker, H. G.

H. G. Booker, J. A. Ferguson, and H. O. Vats, “Comparison between the extended-medium and the phase-screen scintillation theories,” J. Atmos. Terr. Phys. 47(38), 1–399 (1985).
[CrossRef]

Chao, C.

C. Chao, L. Hu, and Q. Mu, “Bandwidth requires of adaptive optical system for horizontal turbulence correction,” Opt. Precision Eng. 18(10), 2137–3142 (2010).

Dong, L.

B. Wang, Z.-y. Wang, J.-l. Wang, J.-Y. Zhao, Y.-H. Wu, S.-X Zhang, L. Dong, and M. Wen, “Phase-diverse speckle imaging with two cameras,” Opt. Precision Eng. 19(6), 1384–1390 (2011).
[CrossRef]

Ferguson, J. A.

H. G. Booker, J. A. Ferguson, and H. O. Vats, “Comparison between the extended-medium and the phase-screen scintillation theories,” J. Atmos. Terr. Phys. 47(38), 1–399 (1985).
[CrossRef]

Fried, D. L.

Fu, Y. Y.

X. J. Gan, J. Guo, and Y. Y. Fu, “The simulating turbulence method of laser propagation in the inner field,” J. Phys. Conf. Ser. 48, 907–910 (2006).
[CrossRef]

Gan, X. J.

X. J. Gan, J. Guo, and Y. Y. Fu, “The simulating turbulence method of laser propagation in the inner field,” J. Phys. Conf. Ser. 48, 907–910 (2006).
[CrossRef]

Gao, M.

M. Gao and Z. Wu, “Experiments of effect of beam spreading of far-field on aiming deviation,” Opt. Precision Eng. 18(3), 602–608 (2010).

Gao, Y.-h.

Y.-h. Gao, Z.-y. An, N.-n. Li, W.-x Zhao, and J.-s. Wang, “Optical design of Gaussian beam shaping,” Opt. Precision Eng. 19(7), 1464–1471 (2011).
[CrossRef]

Glas, R. S.

S. V. Mantravadi, T. A. Rhoadarmer, and R. S. Glas, “Simple laboratory system for generating well-controlled atmospheric-like turbulence,” Proc. SPIE 5553, 290–300 (2004).
[CrossRef]

Guo, J.

X. J. Gan, J. Guo, and Y. Y. Fu, “The simulating turbulence method of laser propagation in the inner field,” J. Phys. Conf. Ser. 48, 907–910 (2006).
[CrossRef]

Hu, L.

C. Chao, L. Hu, and Q. Mu, “Bandwidth requires of adaptive optical system for horizontal turbulence correction,” Opt. Precision Eng. 18(10), 2137–3142 (2010).

Li, N.-n.

Y.-h. Gao, Z.-y. An, N.-n. Li, W.-x Zhao, and J.-s. Wang, “Optical design of Gaussian beam shaping,” Opt. Precision Eng. 19(7), 1464–1471 (2011).
[CrossRef]

Mantravadi, S. V.

S. V. Mantravadi, T. A. Rhoadarmer, and R. S. Glas, “Simple laboratory system for generating well-controlled atmospheric-like turbulence,” Proc. SPIE 5553, 290–300 (2004).
[CrossRef]

Miller, W. B.

L. C. Andrews and W. B. Miller, “Single- and double-pass propagation through complex paraxial optical systems,” J. Opt. Soc. Am. A 12(1), 137–150 (1995).
[CrossRef]

L. C. Andrews, W. B. Miller, and J. C. Ricklin, “Propagation through complex optical system: a phase screen analysis,” SPIE 2312, 122–129 (1994).
[CrossRef]

Mu, Q.

C. Chao, L. Hu, and Q. Mu, “Bandwidth requires of adaptive optical system for horizontal turbulence correction,” Opt. Precision Eng. 18(10), 2137–3142 (2010).

Phillips, R. L.

L. C. Andrews, R. L. Phillips, and A. R. Weeks, “Propagation of a Gaussian-beam wave through a random phase screen,” Waves Random Media 7(2), 229–244 (1997).
[CrossRef]

Qian, M. X.

M. X. Qian, W. Y. Zhu, and R. Rao, “Phase screen distribution for simulating laser propagation along an inhomogeneous atmospheric path,” Acta Phys. Sinica 58(9), 6633–6639 (2009).

Qin, S.

Rao, R.

M. X. Qian, W. Y. Zhu, and R. Rao, “Phase screen distribution for simulating laser propagation along an inhomogeneous atmospheric path,” Acta Phys. Sinica 58(9), 6633–6639 (2009).

Rhoadarmer, T. A.

S. V. Mantravadi, T. A. Rhoadarmer, and R. S. Glas, “Simple laboratory system for generating well-controlled atmospheric-like turbulence,” Proc. SPIE 5553, 290–300 (2004).
[CrossRef]

Ricklin, J. C.

L. C. Andrews, W. B. Miller, and J. C. Ricklin, “Propagation through complex optical system: a phase screen analysis,” SPIE 2312, 122–129 (1994).
[CrossRef]

Seidman, J. B.

Vats, H. O.

H. G. Booker, J. A. Ferguson, and H. O. Vats, “Comparison between the extended-medium and the phase-screen scintillation theories,” J. Atmos. Terr. Phys. 47(38), 1–399 (1985).
[CrossRef]

Wang, B.

B. Wang, Z.-y. Wang, J.-l. Wang, J.-Y. Zhao, Y.-H. Wu, S.-X Zhang, L. Dong, and M. Wen, “Phase-diverse speckle imaging with two cameras,” Opt. Precision Eng. 19(6), 1384–1390 (2011).
[CrossRef]

Wang, J.-l.

B. Wang, Z.-y. Wang, J.-l. Wang, J.-Y. Zhao, Y.-H. Wu, S.-X Zhang, L. Dong, and M. Wen, “Phase-diverse speckle imaging with two cameras,” Opt. Precision Eng. 19(6), 1384–1390 (2011).
[CrossRef]

Wang, J.-s.

Y.-h. Gao, Z.-y. An, N.-n. Li, W.-x Zhao, and J.-s. Wang, “Optical design of Gaussian beam shaping,” Opt. Precision Eng. 19(7), 1464–1471 (2011).
[CrossRef]

Wang, X. S.

Wang, Z.-y.

B. Wang, Z.-y. Wang, J.-l. Wang, J.-Y. Zhao, Y.-H. Wu, S.-X Zhang, L. Dong, and M. Wen, “Phase-diverse speckle imaging with two cameras,” Opt. Precision Eng. 19(6), 1384–1390 (2011).
[CrossRef]

Weeks, A. R.

L. C. Andrews, R. L. Phillips, and A. R. Weeks, “Propagation of a Gaussian-beam wave through a random phase screen,” Waves Random Media 7(2), 229–244 (1997).
[CrossRef]

Wen,, M.

B. Wang, Z.-y. Wang, J.-l. Wang, J.-Y. Zhao, Y.-H. Wu, S.-X Zhang, L. Dong, and M. Wen, “Phase-diverse speckle imaging with two cameras,” Opt. Precision Eng. 19(6), 1384–1390 (2011).
[CrossRef]

Wu, Y.-H.

B. Wang, Z.-y. Wang, J.-l. Wang, J.-Y. Zhao, Y.-H. Wu, S.-X Zhang, L. Dong, and M. Wen, “Phase-diverse speckle imaging with two cameras,” Opt. Precision Eng. 19(6), 1384–1390 (2011).
[CrossRef]

Wu, Z.

M. Gao and Z. Wu, “Experiments of effect of beam spreading of far-field on aiming deviation,” Opt. Precision Eng. 18(3), 602–608 (2010).

Zhang, B. D.

Zhang, S.-X

B. Wang, Z.-y. Wang, J.-l. Wang, J.-Y. Zhao, Y.-H. Wu, S.-X Zhang, L. Dong, and M. Wen, “Phase-diverse speckle imaging with two cameras,” Opt. Precision Eng. 19(6), 1384–1390 (2011).
[CrossRef]

Zhao, J.-Y.

B. Wang, Z.-y. Wang, J.-l. Wang, J.-Y. Zhao, Y.-H. Wu, S.-X Zhang, L. Dong, and M. Wen, “Phase-diverse speckle imaging with two cameras,” Opt. Precision Eng. 19(6), 1384–1390 (2011).
[CrossRef]

Zhao, W.-x

Y.-h. Gao, Z.-y. An, N.-n. Li, W.-x Zhao, and J.-s. Wang, “Optical design of Gaussian beam shaping,” Opt. Precision Eng. 19(7), 1464–1471 (2011).
[CrossRef]

Zhu, W. Y.

M. X. Qian, W. Y. Zhu, and R. Rao, “Phase screen distribution for simulating laser propagation along an inhomogeneous atmospheric path,” Acta Phys. Sinica 58(9), 6633–6639 (2009).

Acta Phys. Sinica (1)

M. X. Qian, W. Y. Zhu, and R. Rao, “Phase screen distribution for simulating laser propagation along an inhomogeneous atmospheric path,” Acta Phys. Sinica 58(9), 6633–6639 (2009).

Chin. Opt. Lett. (1)

J. Atmos. Terr. Phys. (1)

H. G. Booker, J. A. Ferguson, and H. O. Vats, “Comparison between the extended-medium and the phase-screen scintillation theories,” J. Atmos. Terr. Phys. 47(38), 1–399 (1985).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

J. Phys. Conf. Ser. (1)

X. J. Gan, J. Guo, and Y. Y. Fu, “The simulating turbulence method of laser propagation in the inner field,” J. Phys. Conf. Ser. 48, 907–910 (2006).
[CrossRef]

Opt. Precision Eng. (4)

M. Gao and Z. Wu, “Experiments of effect of beam spreading of far-field on aiming deviation,” Opt. Precision Eng. 18(3), 602–608 (2010).

Y.-h. Gao, Z.-y. An, N.-n. Li, W.-x Zhao, and J.-s. Wang, “Optical design of Gaussian beam shaping,” Opt. Precision Eng. 19(7), 1464–1471 (2011).
[CrossRef]

C. Chao, L. Hu, and Q. Mu, “Bandwidth requires of adaptive optical system for horizontal turbulence correction,” Opt. Precision Eng. 18(10), 2137–3142 (2010).

B. Wang, Z.-y. Wang, J.-l. Wang, J.-Y. Zhao, Y.-H. Wu, S.-X Zhang, L. Dong, and M. Wen, “Phase-diverse speckle imaging with two cameras,” Opt. Precision Eng. 19(6), 1384–1390 (2011).
[CrossRef]

Proc. SPIE (1)

S. V. Mantravadi, T. A. Rhoadarmer, and R. S. Glas, “Simple laboratory system for generating well-controlled atmospheric-like turbulence,” Proc. SPIE 5553, 290–300 (2004).
[CrossRef]

SPIE (1)

L. C. Andrews, W. B. Miller, and J. C. Ricklin, “Propagation through complex optical system: a phase screen analysis,” SPIE 2312, 122–129 (1994).
[CrossRef]

Waves Random Media (1)

L. C. Andrews, R. L. Phillips, and A. R. Weeks, “Propagation of a Gaussian-beam wave through a random phase screen,” Waves Random Media 7(2), 229–244 (1997).
[CrossRef]

Other (2)

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (New York: McGraw-Hill, 1961).

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Figures (5)

Fig. 1
Fig. 1

The model of Gaussian beam propagation through a random phase screen

Fig. 2
Fig. 2

Comparison between mean irradiance of thin phase screen and that of arbitrary thickness (T: thin phase screen; A: arbitrary thickness phase screen; d 3 = L 3/L)

Fig. 3
Fig. 3

Comparison between normalized MCF of thin phase screen and that of arbitrary thickness (T: thin phase screen; A: arbitrary thickness phase screen; d 3 = L 3/L)

Fig. 4
Fig. 4

Comparison the DOC of thin and arbitrary thickness phase screen (T: thin phase screen; A: arbitrary thickness phase screen)

Fig. 5
Fig. 5

Comparison of scintillation index with different thickness (T: thin phase screen; A: arbitrary thickness phase screen)

Equations (57)

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z = 0 : U 0 ( r , 0 ) = a 0 exp ( r W 0 2 i k r 2 2 F 0 ) = a 0 exp ( 1 2 α 0 k r 2 ) ,
z = L : U 0 ( r , L ) = a 0 p ( L ) exp [ i k L + ( r W 2 i k r 2 2 F ) ] = a 0 p ( L ) exp [ i k L 1 2 ( α 0 k r 2 p ( L ) ) ] ,
Θ 0 = 1 L F 0 Λ 0 = 2 L k W 0 2 .
W = W 0 Θ 0 2 + Λ 0 2 F = F 0 ( Θ 0 2 + Λ 0 2 ) ( Θ 1 ) Θ 0 2 + Λ 0 2 Θ 0 .
z = L 1 Θ 1 = 1 + L 1 F 1 Λ 1 = 2 L 1 k W 0 2 ,
z = L 1 + L 2 Θ 2 = 1 + L 2 F 2 Λ 1 = 2 L 2 k W 0 2 ,
z = L 1 + L 2 + L 3 Θ 3 = 1 + L 3 F 3 Λ 1 = 2 L 3 k W 0 2 .
1 p ( L ) = Θ i Λ = i = 1 3 ( Θ i i Λ i ) .
U ( r , L ) = U 0 ( r , L ) exp [ Ψ ( r , L ) ] = U 0 ( r , L ) exp [ ψ 1 ( r , L ) + ψ 2 ( r , L ) + ] ,
E 1 ( 0 , 0 ) = ψ 2 ( r , L ) + 1 2 ψ 1 2 ( r , L ) = 2 π 2 k 2 0 L d z 0 κ Φ n ( κ , z ) d k ,
E 2 ( r 1 , r 2 ) = ψ 1 ( r 1 , L ) ψ 1 * ( r 2 , L ) = 4 π 2 k 2 0 L d z 0 d κ κ Φ n ( κ , z ) exp [ Λ L κ 2 ( 1 Z L ) 2 k ] × J 0 { κ | [ 1 Θ ¯ ( 1 Z L ) ] p 2 i Λ ( 1 Z L ) r | } ,
E 3 ( r 1 , r 2 ) = ψ 1 ( r 1 , L ) ψ 1 ( r 2 , L ) = 4 π 2 k 2 0 L d z 0 d κ κ Φ n ( κ , z ) exp { i κ 2 γ L k ( 1 Z L ) [ 1 Θ ¯ ( 1 Z L ) ] } × exp [ Λ L κ 2 ( 1 Z L ) 2 k ] ) J 0 { κ ρ [ 1 ( Θ ¯ + i Λ ) ( 1 Z L ) ] } ,
1 z L = ξ ( 0 z L , L 3 L ξ ( L 2 + L 3 ) L ) .
Φ n ( κ ) = 0.033 C n 2 κ 11 / 3 ,
Φ n ( κ ) = { 0.033 C ^ n 2 κ 11 / 3 L 1 z L 1 + L 2 0 o t h e r s ,
E ^ 1 ( 0 , 0 ) = ψ 2 ( r , L ) + 1 2 ψ 1 2 ( r , L ) = 2 π 2 k 2 L L 3 L L 2 + L 3 L d ξ 0 κ Φ n ( κ ) d k ,
E ^ 2 ( r 1 , r 2 ) = ψ 1 ( r 1 , L ) ψ 1 * ( r 2 , L ) = 4 π 2 k 2 L L 3 L L 2 + L 3 L d ξ 0 d κ κ Φ n ( κ ) exp [ Λ L κ 2 ξ 2 k ] × J 0 { κ | [ 1 Θ ¯ ξ ] p 2 i Λ ξ r | } ,
E ^ 3 ( r 1 , r 2 ) = ψ 1 ( r 1 , L ) ψ 1 * ( r 2 , L ) = 4 π 2 k 2 L L 3 L L 2 + L 3 L d ξ 0 d κ κ Φ n ( κ ) exp { i κ 2 γ L k ξ [ 1 Θ ¯ ξ ] } × exp [ Λ L κ 2 ξ 2 k ] ) J 0 { κ ρ [ 1 ( Θ ¯ + i Λ ) ξ ] } ,
σ R 2 = 8 π 2 k 2 0 L d z 0 d κ κ Φ n ( κ ) [ 1 cos ( κ 2 ( L z ) k ) ] = 1.23 C n 2 k 7 6 L 11 6 .
σ ^ R 2 = 8 π 2 k 2 L L 3 L L 2 + L 3 L d ξ 0 d κ κ Φ n ( κ ) [ 1 cos ( κ 2 L ξ k ) ] .
Φ n ( κ ) = 0.033 C n 2 exp ( κ 2 / κ m 2 ) ( κ 2 + κ 0 2 ) 11 / 6 ,
U ( a ; c ; x ) = 1 Γ ( a ) 0 exp ( x t ) t a 1 ( 1 + t ) c a 1 d t ( a > 0 , Re ( x ) > 0 ) ,
U ( a ; c ; x ) Γ ( 1 c ) Γ ( 1 + a c ) + Γ ( c 1 ) Γ ( a ) x 1 c ( | x | 1 ) ,
σ ^ R 2 4.7466 C n 2 k 7 6 L 11 6 Re { i 5 6 L 3 L L 2 + L 3 L ξ 5 6 d ξ } = 1.23 C n 2 k 7 6 L 11 6 [ ( L 2 + L 3 L ) 11 6 ( L 3 L ) 11 6 ] ,
C n 2 = C ^ n 2 [ ( L 2 + L 3 L ) 11 6 ( L 3 L ) 11 6 ] .
C n 2 L A 11 6 = C ^ n 2 L 11 6 [ ( L 2 + L 3 L ) 11 6 ( L 3 L ) 11 6 ] ,
Γ 2 ( r 1 , r 2 , L ) = U ( r 1 , L ) U * ( r 2 , L ) = U 0 ( r 1 , L ) U 0 * ( r 2 , L ) exp [ 2 E 1 ( 0 , 0 ) + E 2 ( r 1 , r 2 ) ] = Γ 2 0 ( r 1 , r 2 , L ) exp [ 2 E 1 ( 0 , 0 ) + E 2 ( r 1 , r 2 ) ] ,
Γ 2 0 ( r 1 , r 2 , L ) = U 0 ( r 1 , L ) U 0 * ( r 2 , L ) = W 0 2 W exp ( 2 r 2 W 2 ρ 2 2 W 2 i k F p r ) ,
Γ 2 ( r 1 , r 2 , L ) = Γ 2 0 ( r 1 , r 2 , L ) exp [ 2 E 1 ( 0 , 0 ) + E 2 ( r 1 , r 2 ) ] = Γ 2 0 ( r 1 , r 2 , L ) exp [ σ r 2 ( r 1 , L ) + σ r 2 ( r 1 , L ) T 1 2 Δ ( r 1 , r 2 , L ) ] .
σ ^ r 2 ( r , L ) = 1 2 [ E 2 ( r , r ) E 2 ( 0 , 0 ) ] = 2 π 2 k 2 L L 3 L L 2 + L 3 L d ξ 0 d κ κ Φ n ( κ ) exp [ Λ L κ 2 ξ 2 k ] [ I 0 ( 2 Λ r ξ κ ) 1 ] ,
T ^ = 2 E 1 ( 0 , 0 ) E 2 ( 0 , 0 ) = 4 π 2 k 2 L L 3 L L 2 + L 3 L d ξ 0 d κ κ Φ n ( κ ) [ 1 exp ( Λ L ξ 2 κ 2 k ) ] ,
Δ ^ ( r 1 , r 2 , L ) = E 2 ( r 1 , r 1 ) + E 2 ( r 2 , r 2 ) 2 E 2 ( r 1 , r 2 ) = 4 π 2 k 2 L L 3 L L 2 + L 3 L d ξ 0 d κ κ Φ n ( κ ) exp ( Λ L κ 2 ξ 2 k ) × { I 0 ( 2 Λ r 1 ξ κ ) + I 0 ( 2 Λ r 2 ξ κ ) 2 J 0 [ | ( 1 Θ ¯ ξ ) p - 2 i Λ ξ r | κ ] } ,
I ^ ( r , L ) = Γ 2 ( r , r , L ) = W 0 2 W exp ( 2 r 2 W 2 ) exp [ 2 σ ^ r 2 ( r , L ) T ^ ] .
I 0 ( x ) = n = 0 ( x / 2 ) 2 n n ! Γ ( n + 1 ) 0 e s t t x 1 d t = Γ ( x ) s x [ Re ( x ) > 0 , Re ( s ) > 0 ] ,
σ ^ r 2 ( r , L ) = 0.815 C ^ n 2 k 7 6 L 11 6 Λ 5 6 [ ( L 3 + L 2 L ) 8 3 ( L 3 L ) 8 3 ] [ 1 F 1 1 ( 5 6 ; 1 ; 2 r 2 W 2 ) ] ,
T ^ 1.63 C ^ n 2 k 7 6 L 11 6 Λ 5 6 [ ( L 3 + L 2 L ) 8 3 ( L 3 L ) 8 3 ] .
I ( r , L ) = W 0 2 W exp ( 2 r 2 W 2 ) exp [ 1.63 C n 2 k 7 6 L 11 6 Λ 5 6 [ ( L 3 + L 2 L ) 8 3 ( L 3 L ) 8 3 ] F 1 1 ( 5 6 ; 1 ; 2 r 2 W 2 ) ] .
I ( r , L ) = W 0 2 W exp ( 2 r 2 W 2 ) exp [ 1.93 σ ^ R 2 ( Λ d 3 ) 5 6 F 1 1 ( 5 6 ; 1 ; 2 r 2 W 2 ) ] ,
σ ^ R 2 = 2.25 C ^ n 2 k 7 6 L 2 L 3 5 6
Γ ^ 2 ( ρ , L ) = W 0 2 W 2 exp ( 1 4 Λ k ρ 2 L ) exp [ T ^ 1 2 d ^ ( ρ , L ) ] ,
d ^ ( ρ , L ) = 8 π 2 k 2 L L 3 L L 2 + L 3 L d ξ 0 d κ κ Φ n ( κ ) exp ( Λ L κ 2 ξ 2 k ) { 1 J 0 [ ( 1 Θ ¯ ξ ) κ ρ ] } .
J 0 ( x ) = n = 0 ( 1 ) n ( x / 2 ) 2 n n ! Γ ( n + 1 ) F 1 1 ( a ; c ; x ) = n = 0 ( a ) n x n ( c ) n n !
d ^ ( ρ , L ) = 8.70 C ^ n 2 k 7 6 L 11 6 Λ 5 6 H ,
Η = L 3 L L 2 + L 3 L ξ 5 3 { F 1 1 [ 5 6 ; 1 ; ( 1 Θ ¯ ξ ) 2 k ρ 2 4 Λ L ξ 2 ] 1 } d ξ .
Γ ^ 2 ( ρ , L ) Γ ^ 2 ( 0 , L ) = exp [ 1 4 Λ k ρ 2 L 1 2 d ^ ( ρ , L ) ] ,
Γ ^ 2 ( ρ , L ) Γ ^ 2 ( 0 , L ) = exp [ 1 4 Λ k ρ 2 L 1.179 σ ^ R 2 | 1 Θ ¯ d 3 | 5 3 ( k ρ 2 L ) 5 6 ] ,
DOC ( r 1 , r 2 , L ) = | Γ 2 ( r 1 , r 2 , L ) | Γ 2 ( r 1 , r 1 , L ) Γ 2 ( r 2 , r 2 , L ) = exp [ 1 2 D ( r 1 , r 2 , L ) ] ,
D ^ ( ρ , L ) = 8 π 2 k 2 L L 3 L L 2 + L 3 L d ξ 0 d κ κ Φ n ( κ ) exp ( Λ L κ 2 ξ 2 k ) { I 0 ( Λ ρ ξ κ ) J 0 [ ( 1 Θ ¯ ξ ) κ ρ ] } = d ^ ( ρ , L ) + 4 σ ^ r 2 ( ρ 2 , L ) .
D ^ ( ρ , L ) = 7.074 σ ^ R 2 ( Λ d 3 2 ) 5 6 { F 1 1 [ 5 6 ; 1 ; ( 1 Θ ¯ d 3 ) 2 k ρ 2 4 Λ L d 3 2 ] F 1 1 ( 5 6 ; 1 ; k Λ ρ 2 4 L ) } .
σ I 2 ( r , L ) = exp [ 4 σ χ 2 ( r , L ) ] 1 4 σ χ 2 ( r , L ) = 2 Re [ E 2 ( r , r ) + E 3 ( r , r ) ] .
σ I 2 ( r , L ) = σ I , r 2 ( r , L ) + σ I , l 2 ( L ) = 4 σ r 2 ( r , L ) + σ I , l 2 ( L ) .
σ ^ I , l 2 ( L ) = 8 π 2 k 2 L L 3 L L 2 + L 3 L d ξ 0 d κ κ Φ n ( κ ) exp [ Λ L ξ 2 k κ 2 ] { 1 cos [ L κ 2 k ξ ( 1 Θ ¯ ξ ) ] } .
0 x λ 1 ( e μ x e ν x ) d x = Γ ( x ) ( μ λ ν λ ) Re ( λ ) > 2 , Re ( μ ) > 0 , Re ( ν ) > 0 ,
0 x t μ 1 ( 1 + β t ) ν d t = x μ μ F 2 1 ( ν , μ ; 1 + μ ; β x ) Re ( μ ) > 0 ,
σ ^ I , l 2 ( L ) = 4.74 C ^ n 2 k 7 6 L 11 6 Re { i 5 6 ( L 3 + L 2 L ) 11 / 6 F 2 1 [ 5 6 , 11 6 ; 17 6 ; ( Θ ¯ + i Λ ) L 3 + L 2 L ] i 5 6 ( L 3 L ) 11 / 6 F 2 1 [ 5 6 , 11 6 ; 17 6 ; ( Θ ¯ + i Λ ) L 3 L ] 11 16 Λ 5 6 [ ( L 3 + L 2 L ) 8 / 3 ( L 3 L ) 8 / 3 ] } .
σ ^ I 2 ( r , L ) = 4 σ ^ r 2 ( r , L ) + σ ^ I , l 2 ( L ) = 3.26 C ^ n 2 k 7 6 L 11 6 Λ 5 6 [ ( L 3 + L 2 L ) 8 3 ( L 3 L ) 8 3 ] [ 1 F 1 1 ( 5 6 ; 1 ; 2 r 2 W 2 ) ] + 4.74 C ^ n 2 k 7 6 L 11 6 Re { i 5 6 ( L 3 + L 2 L ) 11 / 6 F 2 1 [ 5 6 , 11 6 ; 17 6 ; ( Θ ¯ + i Λ ) L 3 + L 2 L ] i 5 6 ( L 3 L ) 11 / 6 F 2 1 [ 5 6 , 11 6 ; 17 6 ; ( Θ ¯ + i Λ ) L 3 L ] 11 16 Λ 5 6 [ ( L 3 + L 2 L ) 8 / 3 ( L 3 L ) 8 / 3 ] } . ( r W )
σ ^ I 2 ( r , L ) = 6.45 σ ^ R 2 ( Λ d 3 ) 5 6 ( r 2 W 2 ) + 3.87 σ ^ R 2 { [ ( Λ d 3 ) 2 + ( 1 Θ ¯ d 3 ) 2 ] 5 12 cos [ 5 6 arctan ( 1 Θ ¯ d 3 Λ d 3 ) ] ( Λ d 3 ) 5 6 } . ( r W )

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