Abstract

We propose a method that is used to derive the moment radius of intensity distribution in a turbulent atmosphere. From this study, we have found that the second moment radius is affected only by the first-order expansion coefficient of the wave structure function. If our attention is directed to a higher moment radius, a higher order approximation of the expansion needs to be used. As an example, the propagation of a Gaussian–Schell beam in a slant path has been studied based on the turbulent atmosphere of a three-layer model. The variation of some beam properties, such as the relative waist width, angular spread, and kurtosis parameter with the initial waist width, wavelength, and zenith angle, has been analyzed and discussed in detail. The study shows that there is little difference between the three-layer model and the Kolmogorov model in studying uplink propagation, and the difference is large for downlink propagation. The intensity profile of the Gaussian beam in turbulence does not keep a Gaussian shape unless the beam spreading due to turbulence is very large or very small.

© 2011 Optical Society of America

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2010 (4)

2009 (1)

2008 (6)

2007 (2)

2006 (4)

E. Golbraikh, H. Branover, N. S. Kopeika, and A. Zilberman, “Non-Kolmogorov atmospheric turbulence and optical signal propagation,” Nonlin. Proc. Geoph. 13, 297–301 (2006).
[CrossRef]

Y. Cai and S. He, “Propagation of various dark hollow beams in turbulent atmosphere,” Opt. Express 14, 1353–1367 (2006).
[CrossRef] [PubMed]

H. T. Eyyuboğlu, C. Arpali, and Y. Baykal, “Flat topped beams and their characteristics in turbulent media,” Opt. Express 14, 4196–4207 (2006).
[CrossRef] [PubMed]

M. S. Belen’kii, E. Cuellar, K. A. Hughes, and V. A. Rye, “Experimental study of spatial structure of turbulence at Maui Space Surveillance Site (MSSS),” Proc. SPIE 6304, 6304OU (2006).
[CrossRef]

2004 (2)

2003 (1)

M. Salem, T. Shirai, A. Dogariu, and E. Wolf, “Long-distance propagation of partially coherent beams through atmospheric turbulence,” Opt. Commun. 216, 261–265 (2003).
[CrossRef]

2002 (2)

E. Golbraikh and S. S. Moiseev, “Different spectra formation in the presence of helical transfer,” Phys. Lett. A 305, 173–175(2002).
[CrossRef]

B. Lü and S. Luo, “Analytical expression for the kurtosis parameter of flattened Gaussian beams propagating through ABCD optical systems,” J. Mod. Opt. 49, 1731–1738(2002).
[CrossRef]

2001 (1)

2000 (1)

C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47, 1111–1126 (2000).
[CrossRef]

1999 (1)

M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown II, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[CrossRef]

1996 (2)

1995 (3)

1993 (1)

G. Piquero, P. M. Mejias, and R. Martinez-Herrero, “On the propagation of the kurtosis parameter of general beams,” in Proceedings of the Workshop on Laser Beam Characterization, P.M.Mejias, H.Weber, R. Martinez-Herrero, and A.Gonzalez-Urena, eds. (SEDO, 1993), pp. 141–154.

1992 (1)

1987 (1)

1981 (1)

1980 (1)

R. L. Fante, “Electromagnetic beam propagation in turbulent media: an update,” Proc. IEEE 68, 1424–1443 (1980).
[CrossRef]

1979 (1)

1972 (1)

1971 (1)

Andrews, L. C.

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, 026003(2008).
[CrossRef]

L. C. Andrews, C. Y. Young, and W. B. Miller, “Coherence properties of a reflected optical wave in atmospheric turbulence,” J. Opt. Soc. Am. A 13, 851–861 (1996).
[CrossRef]

Arpali, C.

Barchers, J. D.

M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown II, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[CrossRef]

Baykal, Y.

Belen’kii, M. S.

M. S. Belen’kii, E. Cuellar, K. A. Hughes, and V. A. Rye, “Experimental study of spatial structure of turbulence at Maui Space Surveillance Site (MSSS),” Proc. SPIE 6304, 6304OU (2006).
[CrossRef]

M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown II, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[CrossRef]

A. S. Gurvich and M. S. Belen’kii, “Influence of stratospheric turbulence on infrared imaging,” J. Opt. Soc. Am. A 12, 2517–2522 (1995).
[CrossRef]

M. S. Belen’kii, “Effect of the stratospheric turbulence on star image motion,” Opt. Lett. 20, 1359–1361 (1995).
[CrossRef] [PubMed]

Boreman, G. D.

Branover, H.

E. Golbraikh, H. Branover, N. S. Kopeika, and A. Zilberman, “Non-Kolmogorov atmospheric turbulence and optical signal propagation,” Nonlin. Proc. Geoph. 13, 297–301 (2006).
[CrossRef]

Brown, J. M.

M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown II, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[CrossRef]

Cai, Y.

Chu, X.

Cuellar, E.

M. S. Belen’kii, E. Cuellar, K. A. Hughes, and V. A. Rye, “Experimental study of spatial structure of turbulence at Maui Space Surveillance Site (MSSS),” Proc. SPIE 6304, 6304OU (2006).
[CrossRef]

Dainty, C.

Dan, Y.

Dayton, D.

Dogariu, A.

M. Salem, T. Shirai, A. Dogariu, and E. Wolf, “Long-distance propagation of partially coherent beams through atmospheric turbulence,” Opt. Commun. 216, 261–265 (2003).
[CrossRef]

Du, X.

Eyyuboglu, H. T.

Fante, R. L.

R. L. Fante, “Electromagnetic beam propagation in turbulent media: an update,” Proc. IEEE 68, 1424–1443 (1980).
[CrossRef]

Feng, X.

X. Chu, C. Qiao, and X. Feng, “The effect of non-Kolmogorov turbulence on the propagation of cosh-Gaussian beam,” Opt. Commun. 283, 3398–3403 (2010).
[CrossRef]

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, 026003(2008).
[CrossRef]

Fugate, R. Q.

M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown II, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[CrossRef]

Gbur, G.

Golbraikh, E.

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Propagation of electromagnetic waves in Kolmogorov and non-Kolmogorov atmospheric turbulence: three-layer altitude model,” Appl. Opt. 47, 6385–6391 (2008).
[CrossRef] [PubMed]

E. Golbraikh, H. Branover, N. S. Kopeika, and A. Zilberman, “Non-Kolmogorov atmospheric turbulence and optical signal propagation,” Nonlin. Proc. Geoph. 13, 297–301 (2006).
[CrossRef]

E. Golbraikh and N. S. Kopeika, “Behavior of structure function of refraction coefficients in different turbulent fields,” Appl. Opt. 43, 6151–6156 (2004).
[CrossRef] [PubMed]

E. Golbraikh and S. S. Moiseev, “Different spectra formation in the presence of helical transfer,” Phys. Lett. A 305, 173–175(2002).
[CrossRef]

Gonglewski, J.

Guo, H.

Gurvich, A. S.

Hanson, S. G.

He, S.

Hughes, K. A.

M. S. Belen’kii, E. Cuellar, K. A. Hughes, and V. A. Rye, “Experimental study of spatial structure of turbulence at Maui Space Surveillance Site (MSSS),” Proc. SPIE 6304, 6304OU (2006).
[CrossRef]

Jiang, W.

C. Rao, W. Jiang, and N. Ling, “Adaptive-optics compensation by distributed beacons for non-Kolmogorov turbulence,” Appl. Opt. 40, 3441–3449 (2001).
[CrossRef]

C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47, 1111–1126 (2000).
[CrossRef]

Karis, S. J.

M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown II, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[CrossRef]

Kopeika, N. S.

Korotkova, O.

Ling, N.

C. Rao, W. Jiang, and N. Ling, “Adaptive-optics compensation by distributed beacons for non-Kolmogorov turbulence,” Appl. Opt. 40, 3441–3449 (2001).
[CrossRef]

C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47, 1111–1126 (2000).
[CrossRef]

Liu, Z.

Lü, B.

B. Lü and S. Luo, “Analytical expression for the kurtosis parameter of flattened Gaussian beams propagating through ABCD optical systems,” J. Mod. Opt. 49, 1731–1738(2002).
[CrossRef]

Luo, B.

Luo, S.

B. Lü and S. Luo, “Analytical expression for the kurtosis parameter of flattened Gaussian beams propagating through ABCD optical systems,” J. Mod. Opt. 49, 1731–1738(2002).
[CrossRef]

Lutomirski, R. F.

Ma, Y.

Martinez-Herrero, R.

G. Piquero, P. M. Mejias, and R. Martinez-Herrero, “On the propagation of the kurtosis parameter of general beams,” in Proceedings of the Workshop on Laser Beam Characterization, P.M.Mejias, H.Weber, R. Martinez-Herrero, and A.Gonzalez-Urena, eds. (SEDO, 1993), pp. 141–154.

G. Piquero, P. M. Mejias, and R. Martinez-Herrero, “On the propagation of the kurtosis parameter of general beams,” in Proceedings of the Workshop on Laser Beam Characterization, P.M.Mejias, H.Weber, R. Martinez-Herrero, and A.Gonzalez-Urena, eds. (SEDO, 1993), pp. 141–154.

Mejias, P. M.

G. Piquero, P. M. Mejias, and R. Martinez-Herrero, “On the propagation of the kurtosis parameter of general beams,” in Proceedings of the Workshop on Laser Beam Characterization, P.M.Mejias, H.Weber, R. Martinez-Herrero, and A.Gonzalez-Urena, eds. (SEDO, 1993), pp. 141–154.

Miller, W. B.

Moiseev, S. S.

E. Golbraikh and S. S. Moiseev, “Different spectra formation in the presence of helical transfer,” Phys. Lett. A 305, 173–175(2002).
[CrossRef]

Osmon, C. L.

M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown II, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[CrossRef]

Pérez, D. G.

Phillips, R. L.

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, 026003(2008).
[CrossRef]

Pierson, B.

Piquero, G.

G. Piquero, P. M. Mejias, and R. Martinez-Herrero, “On the propagation of the kurtosis parameter of general beams,” in Proceedings of the Workshop on Laser Beam Characterization, P.M.Mejias, H.Weber, R. Martinez-Herrero, and A.Gonzalez-Urena, eds. (SEDO, 1993), pp. 141–154.

Plonus, M. A.

Qiao, C.

X. Chu, C. Qiao, and X. Feng, “The effect of non-Kolmogorov turbulence on the propagation of cosh-Gaussian beam,” Opt. Commun. 283, 3398–3403 (2010).
[CrossRef]

Rao, C.

C. Rao, W. Jiang, and N. Ling, “Adaptive-optics compensation by distributed beacons for non-Kolmogorov turbulence,” Appl. Opt. 40, 3441–3449 (2001).
[CrossRef]

C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47, 1111–1126 (2000).
[CrossRef]

Roggemann, M. C.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[CrossRef]

Rye, V. A.

M. S. Belen’kii, E. Cuellar, K. A. Hughes, and V. A. Rye, “Experimental study of spatial structure of turbulence at Maui Space Surveillance Site (MSSS),” Proc. SPIE 6304, 6304OU (2006).
[CrossRef]

Salem, M.

M. Salem, T. Shirai, A. Dogariu, and E. Wolf, “Long-distance propagation of partially coherent beams through atmospheric turbulence,” Opt. Commun. 216, 261–265 (2003).
[CrossRef]

Shirai, T.

M. Salem, T. Shirai, A. Dogariu, and E. Wolf, “Long-distance propagation of partially coherent beams through atmospheric turbulence,” Opt. Commun. 216, 261–265 (2003).
[CrossRef]

Spielbusch, B.

Stribling, B. E.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[CrossRef]

Toselli, I.

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, 026003(2008).
[CrossRef]

Wandzura, S. M.

Wang, S. C. H.

Wang, X.

Welsh, B. M.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[CrossRef]

Wolf, E.

M. Salem, T. Shirai, A. Dogariu, and E. Wolf, “Long-distance propagation of partially coherent beams through atmospheric turbulence,” Opt. Commun. 216, 261–265 (2003).
[CrossRef]

Wu, G.

Wu, Y.

Young, C. Y.

Yu, S.

Yura, H. T.

Zhang, B.

Zhao, D.

Zhao, H.

Zhou, P.

Zhu, Y.

Zilberman, A.

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Propagation of electromagnetic waves in Kolmogorov and non-Kolmogorov atmospheric turbulence: three-layer altitude model,” Appl. Opt. 47, 6385–6391 (2008).
[CrossRef] [PubMed]

E. Golbraikh, H. Branover, N. S. Kopeika, and A. Zilberman, “Non-Kolmogorov atmospheric turbulence and optical signal propagation,” Nonlin. Proc. Geoph. 13, 297–301 (2006).
[CrossRef]

Zunino, L.

Appl. Opt. (6)

J. Mod. Opt. (2)

B. Lü and S. Luo, “Analytical expression for the kurtosis parameter of flattened Gaussian beams propagating through ABCD optical systems,” J. Mod. Opt. 49, 1731–1738(2002).
[CrossRef]

C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47, 1111–1126 (2000).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (6)

Nonlin. Proc. Geoph. (1)

E. Golbraikh, H. Branover, N. S. Kopeika, and A. Zilberman, “Non-Kolmogorov atmospheric turbulence and optical signal propagation,” Nonlin. Proc. Geoph. 13, 297–301 (2006).
[CrossRef]

Opt. Commun. (2)

M. Salem, T. Shirai, A. Dogariu, and E. Wolf, “Long-distance propagation of partially coherent beams through atmospheric turbulence,” Opt. Commun. 216, 261–265 (2003).
[CrossRef]

X. Chu, C. Qiao, and X. Feng, “The effect of non-Kolmogorov turbulence on the propagation of cosh-Gaussian beam,” Opt. Commun. 283, 3398–3403 (2010).
[CrossRef]

Opt. Eng. (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, 026003(2008).
[CrossRef]

Opt. Express (6)

Opt. Lett. (6)

Phys. Lett. A (1)

E. Golbraikh and S. S. Moiseev, “Different spectra formation in the presence of helical transfer,” Phys. Lett. A 305, 173–175(2002).
[CrossRef]

Proc. IEEE (1)

R. L. Fante, “Electromagnetic beam propagation in turbulent media: an update,” Proc. IEEE 68, 1424–1443 (1980).
[CrossRef]

Proc. SPIE (3)

M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown II, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[CrossRef]

M. S. Belen’kii, E. Cuellar, K. A. Hughes, and V. A. Rye, “Experimental study of spatial structure of turbulence at Maui Space Surveillance Site (MSSS),” Proc. SPIE 6304, 6304OU (2006).
[CrossRef]

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[CrossRef]

Other (2)

G. Piquero, P. M. Mejias, and R. Martinez-Herrero, “On the propagation of the kurtosis parameter of general beams,” in Proceedings of the Workshop on Laser Beam Characterization, P.M.Mejias, H.Weber, R. Martinez-Herrero, and A.Gonzalez-Urena, eds. (SEDO, 1993), pp. 141–154.

ITU-R. Document 3J/31-E, “On propagation data and prediction methods required for the design of space-to-earth and earth-to-space optical communication systems,” Radiocommunication Study Group Meeting, Budapest, July 2001.

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Figures (3)

Fig. 1
Fig. 1

Variation of angular spread with w 0 , σ 0 , λ, and zenith angle for collimated beam: (a) with initial waist width, where λ = 3.8 × 10 6 m , σ 0 and ξ = 0 , (b) with σ 0 , where w 0 = 0.3 m , λ = 3.8 × 10 6 m and ξ = 0 , (c) with wavelength, where w 0 = 0.3 m , σ 0 and ξ = 0 , and (d) with zenith angle, where λ = 3.8 × 10 6 m , σ 0 , and w 0 = 0.3 m .

Fig. 2
Fig. 2

Comparison of the relative waist width between the three-layer model and the Kolmogorov model for completed coherence and collimated beam: (a) with initial waist width, where λ = 1.06 × 10 6 m , ξ = 0 , (b) with wavelength, where w 0 = 0.2 m , ξ = 0 , and (c) with zenith angle, where w 0 = 0.2 m , λ = 1.06 × 10 6 m .

Fig. 3
Fig. 3

Variation of the kurtosis parameter with w 0 , λ, and zenith angle for collimated beam: (a) with initial waist width, where λ = 1.06 × 10 6 m and ξ = 0 , (b) with wavelength, where w 0 = 0.05 m and ξ = 0 , and (c) with zenith angle, where λ = 5 × 10 7 m and w 0 = 0.3 m .

Equations (28)

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W 0 ( r 01 , r 02 , 0 ) = exp ( r 01 2 + r 02 2 w 0 2 ) exp [ ( r 01 r 02 ) 2 2 σ 0 2 ] exp [ i k ( r 01 2 r 02 2 ) 2 R 0 ] ,
I ( r , L ) = ( k 2 π L ) 2 H ( q ) M ( q ) exp ( i k L r · q ) d 2 q ,
H ( q ) = W 0 ( p + q 2 , p q 2 , 0 ) exp ( i k L p · q ) d 2 p ,
M ( q ) = exp [ 0.5 D w ( q ) ] .
D w = 8 π 2 k 2 0 L 0 Φ n K { 1 J 0 [ K q ( 1 z L ) ] } d K d z ,
H ( q ) = 1 2 π w 0 2 exp ( a 0 q 2 ) ,
a 0 = 1 2 w 0 2 + k 2 w 0 2 8 ( 1 L 1 R 0 ) 2 + 1 2 σ 0 2 .
Φ n = A β ( K 2 + K 0 2 ) α / 2 exp ( K 2 K m 2 ) ,
A = sin [ ( α 3 ) π / 2 ] Γ ( α 1 ) 4 π 2
α = { α 1 0 < h H 1 α 2 H 1 < h H 2 α 3 h > H 2 ,
α = α 1 u ( h ) + ( α 2 α 1 ) u ( h H 1 ) + ( α 3 α 2 ) u ( h H 2 ) ,
1 J 0 [ K q ( 1 z L ) ] = j = 1 ( 1 ) j ( j ! ) 2 ( K q 2 ) 2 j ( 1 z L ) 2 j .
D w = 2 j = 1 a j q 2 j ,
a j = ( 1 ) j + 1 π 2 k 2 ( j ! ) 2 2 2 j 2 0 L 0 ( 1 z L ) 2 j Φ n K 2 j + 1 d K d z .
a j = ( 1 ) j + 1 π 2 k 2 ( j ! ) 2 2 2 j 1 sec ( ξ ) × 0 H ( 1 h H ) 2 j A β K m 2 + 2 j α [ Γ ( 1 + j α 2 ) + ( K 0 K m ) 2 + 2 j α Γ ( 1 j + α / 2 ) Γ ( α / 2 ) j ! ] d h .
I ( r , L ) = 1 2 π ( k w 0 2 L ) 2 exp [ ( a 0 q 2 + j = 1 a j q 2 j ) ] exp ( i k L r · q ) d 2 q .
δ ( n ) ( s ) = 1 2 π ( i x ) n exp ( i s x ) d x , n = 0 , 1 , 2 ,
δ ( n ) ( a x ) = a n 1 δ ( n ) ( x ) n = 0 , 1 , 2 ,
f ( x ) δ ( n ) ( x ) d x = ( 1 ) n f ( n ) ( 0 ) n = 0 , 1 , 2 ,
x 2 = ( 1 L R 0 ) 2 w 0 2 4 + L 2 k 2 w 0 2 + L 2 k 2 σ 0 2 + 2 L 2 a 1 k 2 ,
x 4 = 3 x 2 2 24 L 4 a 2 k 4 .
I ( r , L ) = ( w 0 w ) 2 exp ( 2 r 2 w 2 ) ,
w = ( 1 L R 0 ) 2 w 0 2 + 4 L 2 k 2 w 0 2 + 4 L 2 k 2 σ 0 2 + 8 L 2 a 1 k 2 .
C n 2 = 8.148 × 10 56 V 2 h 10 exp ( h / 1000 ) + 2.7 × 10 16 exp ( h / 1500 ) + C 0 exp ( h / 100 ) ,
β = 0.033 C n 2 A ( k L ) 1 2 ( α 11 3 ) .
θ = 2 k 1 w 0 2 + 1 σ 0 2 + 2 a 1 .
w w 0 = ( 1 L R 0 ) 2 + 4 L 2 k 2 w 0 2 ( 1 w 0 2 + 1 σ 0 2 + 2 a 1 ) .
K = 3 384 a 2 L 4 k 4 w 4 .

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